Wetting and Spreading Dynamics: 138 (Surfactant Science)

Author: Victor M. Starov | Manuel G. Velarde | Clayton J. Radke

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WETTING AND SPREADING DYNAMICS

© 2007 by Taylor & Francis Group, LLC

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 Chemical and Biomolecular Engineering Department 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

© 2007 by Taylor & Francis Group, LLC

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 HansFriedrich 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

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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. Cationic Surfactants: Organic Chemistry, edited by James M. Richmond 35. Alkylene Oxides and Their Polymers, F. E. Bailey, Jr., and Joseph V. Koleske 36. Interfacial Phenomena in Petroleum Recovery, edited by Norman R. Morrow 37. Cationic Surfactants: Physical Chemistry, edited by Donn N. Rubingh and Paul M. Holland 38. Kinetics and Catalysis in Microheterogeneous Systems, edited by M. Grätzel and K. Kalyanasundaram 39. Interfacial Phenomena in Biological Systems, edited by Max Bender 40. Analysis of Surfactants, Thomas M. Schmitt (see Volume 96) 41. Light Scattering by Liquid Surfaces and Complementary Techniques, edited by Dominique Langevin 42. Polymeric Surfactants, Irja Piirma 43. Anionic Surfactants: Biochemistry, Toxicology, Dermatology. Second Edition, Revised and Expanded, edited by Christian Gloxhuber and Klaus Künstler 44. Organized Solutions: Surfactants in Science and Technology, edited by Stig E. Friberg and Björn Lindman 45. Defoaming: Theory and Industrial Applications, edited by P. R. Garrett 46. Mixed Surfactant Systems, edited by Keizo Ogino and Masahiko Abe 47. Coagulation and Flocculation: Theory and Applications, edited by Bohuslav Dobiás

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48. Biosurfactants: Production Properties Applications, edited by Naim Kosaric 49. Wettability, edited by John C. Berg 50. Fluorinated Surfactants: Synthesis Properties Applications, Erik Kissa 51. Surface and Colloid Chemistry in Advanced Ceramics Processing, edited by Robert J. Pugh and Lennart Bergström 52. Technological Applications of Dispersions, edited by Robert B. McKay 53. Cationic Surfactants: Analytical and Biological Evaluation, edited by John Cross and Edward J. Singer 54. Surfactants in Agrochemicals, Tharwat F. Tadros 55. Solubilization in Surfactant Aggregates, edited by Sherril D. Christian and John F. Scamehorn 56. Anionic Surfactants: Organic Chemistry, edited by Helmut W. Stache 57. Foams: Theory, Measurements, and Applications, edited by Robert K. Prud’homme and Saad A. Khan 58. The Preparation of Dispersions in Liquids, H. N. Stein 59. Amphoteric Surfactants: Second Edition, edited by Eric G. Lomax 60. Nonionic Surfactants: Polyoxyalkylene Block Copolymers, edited by Vaughn M. Nace 61. Emulsions and Emulsion Stability, edited by Johan Sjöblom 62. Vesicles, edited by Morton Rosoff 63. Applied Surface Thermodynamics, edited by A. W. Neumann and Jan K. Spelt 64. Surfactants in Solution, edited by Arun K. Chattopadhyay and K. L. Mittal 65. Detergents in the Environment, edited by Milan Johann Schwuger 66. Industrial Applications of Microemulsions, edited by Conxita Solans and Hironobu Kunieda 67. Liquid Detergents, edited by Kuo-Yann Lai 68. Surfactants in Cosmetics: Second Edition, Revised and Expanded, edited by Martin M. Rieger and Linda D. Rhein 69. Enzymes in Detergency, edited by Jan H. van Ee, Onno Misset, and Erik J. Baas 70. Structure-Performance Relationships in Surfactants, edited by Kunio Esumi and Minoru Ueno 71. Powdered Detergents, edited by Michael S. Showell 72. Nonionic Surfactants: Organic Chemistry, edited by Nico M. van Os 73. Anionic Surfactants: Analytical Chemistry, Second Edition, Revised and Expanded, edited by John Cross

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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

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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

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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, Victor M. Starov, Manuel G. Velarde, and Clayton J. Radke

© 2007 by Taylor & Francis Group, LLC

WETTING AND SPREADING DYNAMICS Victor M. Starov Loughborough University Loughborough, U.K.

Manuel G. Velarde Instituto Pluridisciplinar Madrid, Spain

Clayton J. Radke University of California at Berkeley Berkeley, California, U.S.A.

Boca Raton London New York

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

© 2007 by Taylor & Francis Group, LLC

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2007 by Taylor & 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-13: 978-1-57444-540-4 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable eﬀorts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. 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 Starov, V. M. Wetting and spreading dynamics / Victor Starov, Manuel Velarde, and Clayton Radke. p. cm. -- (Surfactant science ; 138) Includes bibliographical references and index. ISBN-13: 978-1-57444-540-4 (alk. paper) 1. Wetting. 2. Surface (Chemistry) I. Velarde, Manuel G. (Manuel García) II. Radke, Clayton. III. Title. IV. Series. QD506.S7835 2007 541’.33--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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Contents Preface .............................................................................................................. xvii Acknowledgments ..............................................................................................xxi Chapter 1

Surface Forces and the Equilibrium of Liquids on Solid Substrates.........................................................................................1

Introduction ...........................................................................................................1 1.1 Wetting and Young’s Equation ....................................................................2 1.2 Surface Forces and Disjoining Pressure ...................................................11 Components of the Disjoining Pressure ...................................................13 Molecular or Dispersion Component............................................13 The Electrostatic Component of the Disjoining Pressure ............19 Structural Component of the Disjoining Pressure ........................21 1.3 Static Hysteresis of Contact Angle ...........................................................23 Static Hysteresis of Contact Angles from Microscopic Point of View: Surface Forces ................................................................28 References ...........................................................................................................30 Chapter 2

Equilibrium Wetting Phenomena ..................................................31

Introduction .........................................................................................................31 2.1 Thin Liquid Films on Flat Solid Substrates .............................................31 Equilibrium Droplets on the Solid Substrate under Oversaturation (Pe < 0) ..........................................................................................36 Flat Films at the Equilibrium with Menisci (Pe > 0) ...............................38 S-Shaped Isotherms of Disjoining Pressure in the Special Case S– < S+.... 40 2.2 Nonflat Equilibrium Liquid Shapes on Flat Surfaces...............................41 General Consideration ...............................................................................42 Microdrops: The Case Pe > 0....................................................................47 Microscopic Equilibrium Periodic Films..................................................49 Microscopic Equilibrium Depressions on β-Films...................................54 2.3 Equilibrium Contact Angle of Menisci and Drops: Liquid Shape in the Transition Zone from the Bulk Liquid to the Flat Films in Front.....56 Equilibrium of Liquid in a Flat Capillary: Partial Wetting Case .............57 Meniscus in a Flat Capillary .....................................................................60 Meniscus in a Flat Capillary: Profile of the Transition Zone ..................63 Partial Wetting: Macroscopic Liquid Drops .............................................65 Profile of the Transition Zone in the Case of Droplets............................71 Axisymmetric Drops .................................................................................71 © 2007 by Taylor & Francis Group, LLC

Meniscus in a Cylindrical Capillary .........................................................72 Appendix 1 ................................................................................................73 2.4 Profile of the Transition Zone between a Wetting Film and the Meniscus of the Bulk Liquid in the Case of Complete Wetting..............74 2.5 Thickness of Wetting Films on Rough Solid Substrates..........................81 2.6 Wetting Films on Locally Heterogeneous Surfaces: Hydrophilic Surface with Hydrophobic Inclusions.......................................................90 2.7 Thickness and Stability of Liquid Films on Nonplanar Surfaces ..........100 2.8 Pressure on Wetting Perimeter and Deformation of Soft Solids............106 2.9 Deformation of Fluid Particles in the Contact Zone ..............................113 Two Identical Cylindrical Drops or Bubbles..........................................115 Interaction of Cylindrical Droplets of Different Radii...........................119 Shape of a Liquid Interlayer between Interacting Droplets: Critical Radius ..........................................................................................123 2.10 Line Tension ............................................................................................130 Comparison with Experimental Data and Discussion ............................142 2.11 Capillary Interaction between Solid Bodies ...........................................144 Appendix 2 ..............................................................................................152 Equilibrium Liquid Shape Close to a Vertical Plate...................152 2.12 Liquid Profiles on Curved Interfaces, Effective Disjoining Pressure. Equilibrium Contact Angles of Droplets on Outer/Inner Cylindrical Surfaces and Menisci inside Cylindrical Capillary ................................154 Liquid Profiles on Curved Surface: Derivation of Governing Equations .....................................................................................154 Equilibrium Contact Angle of a Droplet on an Outer Surface of Cylindrical Capillaries.................................................................159 Equilibrium Contact Angle of a Meniscus inside Cylindrical Capillaries ....................................................................................161 References .........................................................................................................163 Chapter 3

Kinetics of Wetting......................................................................165

Introduction .......................................................................................................165 3.1 Spreading of Droplets of Nonvolatile Liquids over Flat Solid Substrates: Qualitative Consideration .....................................................174 Capillary Regime of Spreading...............................................................179 Similarity Solution of Equation 3.18 and Equation 3.19 .......................181 Gravitational Spreading...........................................................................186 Similarity Solution ..................................................................................187 Spreading of Very Thin Droplets ............................................................190 3.2 The Spreading of Liquid Drops over Dry Surfaces: Influence of Surface Forces.....................................................................197 Case n = 2 ...............................................................................................205 Case n = 3 ...............................................................................................205

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Comparison with Experiments ................................................................209 Conclusions..............................................................................................211 Appendix 1 ..............................................................................................211 Appendix 2 ..............................................................................................213 Appendix 3 ..............................................................................................214 Appendix 4 ..............................................................................................216 3.3 Spreading of Drops over a Surface Covered with a Thin Layer of the Same Liquid ............................................................................................217 3.4 Quasi-Steady-State Approach to the Kinetics of Spreading...................225 3.5 Dynamic Advancing Contact Angle and the Form of the Moving Meniscus in Flat Capillaries in the Case of Complete Wetting .............235 Appendix 5 ..............................................................................................242 3.6 Motion of Long Drops in Thin Capillaries in the Case of Complete Wetting.....................................................................................................245 Appendix 6 ..............................................................................................255 3.7 Coating of a Liquid Film on a Moving Thin Cylindrical Fiber.............259 Statement of the Problem........................................................................260 Derivation of the Equation for the Liquid–Liquid Interface Profile ......262 Immobile Meniscus .................................................................................264 Matching of Asymptotic Solutions in Zones I and II (Figure 3.17)......265 Equilibrium Case (Ca = 0)......................................................................267 Numerical Results ...................................................................................269 3.8 Blow-Off Method for Investigation of Boundary Viscosity of Volatile Liquids .....................................................................................................270 Boundary Viscosity..................................................................................270 Theory of the Method .............................................................................271 Experimental Part ........................................................................284 Conclusions..............................................................................................287 3.9 Combined Heat and Mass Transfer in Tapered Capillaries with Bubbles under the Action of a Temperature Gradient............................287 Cylindrical Capillaries.............................................................................292 Tapered Capillaries ..................................................................................293 3.10 Static Hysteresis of Contact Angle .........................................................296 Equilibrium Contact Angles ....................................................................297 Static Hysteresis of the Contact Angle of Menisci ................................301 Static Hysteresis Contact Angles of Drops.............................................308 Conclusions..............................................................................................312 References .........................................................................................................312 Chapter 4

Spreading over Porous Substrates...............................................315

Introduction .......................................................................................................315 4.1 Spreading of Liquid Drops over Saturated Porous Layers ....................315 Theory......................................................................................................316

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Liquid inside the Drop (0 < z < h(t,r)) .......................................316 Inside the Porous Layer beneath the Drop (–D < z < 0, 0 < r < L) ...............................................................318 Materials and Methods ................................................................325 Results and Discussion. Experimental Determination of Effective Lubrication Coefficient ω........................................327 4.2 Spreading of Liquid Drops over Dry Porous Layers: Complete Wetting Case............................................................................................331 Theory......................................................................................................332 Inside the Porous Layer outside the Drop (–D < z < 0, L < r < l) ................................................................338 Experimental Part ........................................................................343 Independent Determination of Kp pc ............................................344 Results and Discussion................................................................345 Appendix 1 ..............................................................................................351 4.3 Spreading of Liquid Drops over Thick Porous Substrates: Complete Wetting Case............................................................................................354 Theory......................................................................................................355 Inside the Porous Substrate.........................................................358 Experimental Part ........................................................................358 Results and Discussion................................................................360 Spreading of Silicone Oil Drops of Different Viscosity over Identical Glass Filters..................................................................363 Spreading of Silicone Oil Drops over Filters with Similar Properties but Made of Different Materials................................364 Spreading of Silicone Oil Drops with the Same Viscosity (η = 5P) over Glass Filters with Different Porosity and Average Pore Size .......................................................................366 Conclusions..................................................................................368 4.4 Spreading of Liquid Drops from a Liquid Source .................................369 Theory......................................................................................................370 Experimental Set-Up and Results ...........................................................374 Materials and Methods ................................................................374 Results and Discussion................................................................376 Conclusions..................................................................................379 Appendix 2 ..............................................................................................379 Capillary Regime, Complete Wetting .........................................380 Gravitational Regime, Complete Wetting ...................................384 Partial Wetting .............................................................................387 References .........................................................................................................388 Chapter 5

Dynamics of Wetting or Spreading in the Presence of Surfactants ...................................................................................389

Introduction .......................................................................................................390 © 2007 by Taylor & Francis Group, LLC

5.1

5.2

5.3

5.4

5.5

5.6

Spreading of Aqueous Surfactant Solutions over Porous Layers...........390 Experimental Methods and Materials [1] ...............................................391 Spreading on Porous Substrates (Figure 4.4) .............................391 Measurement of Static Advancing and Receding Contact Angles on Nonporous Substrates ................................................391 Results and Discussion................................................................393 Advancing and Hydrodynamic Receding Contact Angles on Porous Nitrocellulose Membranes .........................................398 Static Hysteresis of the Contact Angle of SDS Solution Drops on Smooth Nonporous Nitrocellulose Substrate.........................400 Conclusions..................................................................................403 Spontaneous Capillary Imbibition of Surfactant Solutions into Hydrophobic Capillaries..........................................................................403 Theory......................................................................................................406 Concentration below CMC..........................................................410 Concentration above CMC..........................................................413 Spontaneous Capillary Rise in Hydrophobic Capillaries ...........417 Appendix 1 ..............................................................................................419 Capillary Imbibition of Surfactant Solutions in Porous Media and Thin Capillaries: Partial Wetting Case....................................................421 Theory......................................................................................................422 Concentration below CMC..........................................................424 Concentration above CMC..........................................................432 Experimental Part ........................................................................434 Results and Discussions ..............................................................435 Spreading of Surfactant Solutions over Hydrophobic Substrates ..........436 Theory......................................................................................................437 Experiment: Materials .................................................................442 Monitoring Method .....................................................................442 Results and Discussion................................................................443 Spreading of Non-Newtonian Liquids over Solid Substrates ................445 Governing Equation for the Evolution of the Profile of the Spreading Drop .............................................................................................446 Gravitational Regime of Spreading.........................................................452 Capillary Regime of Spreading...............................................................455 Discussion................................................................................................459 Spreading of an Insoluble Surfactant over Thin Viscose Liquid Layers ......................................................................................................460 Theory and Relation to Experiment........................................................462 The First Spreading Stage...........................................................465 The Second Spreading Stage ......................................................470 Experimental Results...................................................................473 Appendix 2 ..............................................................................................475 Derivation of Governing Equations for Time Evolution of Both Film Thickness and Surfactant Surface Concentration .....475

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Appendix 3 ..............................................................................................476 Influence of Capillary Forces during Initial Stage of Spreading .....................................................................................476 Appendix 4 ..............................................................................................478 Derivation of Boundary Condition at the Moving Shock Front.............................................................................................478 Appendix 5 ..............................................................................................479 Matching of Asymptotic Solutions at the Moving Shock Front.............................................................................................479 Appendix 6 ..............................................................................................480 Solution of the Governing Equations for the Second Stage of Spreading.................................................................................480 5.7 Spreading of Aqueous Droplets Induced by Overturning of Amphiphilic Molecules or Their Fragments in the Surface Layer of an Initially Hydrophobic Substrate.....................................................481 Theory and Derivation of Basic Equations.............................................482 Boundary Conditions...............................................................................487 Solution of the Problem ..............................................................493 Comparison between Theory and Experimental Data............................497 References .........................................................................................................499 Conclusions .......................................................................................................501 Frequently Used Equations...............................................................................502 Navier–Stokes Equations.........................................................................502 Navier-Stokes Equations in the Case of Two-Dimensional Flow ..........504 Capillary Pressure....................................................................................505 List of Main Symbols Used..............................................................................505 Greek........................................................................................................505 Latin.........................................................................................................506 Subscripts.................................................................................................506

© 2007 by Taylor & Francis Group, LLC

Preface This book is for anyone who has recently started to be interested in, or is already involved in, research or applications of wetting and spreading, i.e., for newcomers and practitioners alike. Its contents are not a comprehensive and critical review of the existing research literature. Needless to say, it rather reflects the authors’ recent scientific interests and understanding. The authors presume that the reader using this book has some knowledge in thermodynamics, fluid mechanics, and transport phenomena. Yet the book has been written in an almost self-contained manner, and it should be possible for a graduate student, scientist, or engineer with a reasonable background in differential equations to follow it. Although in various parts we have used the phrase “it can be shown …” or the like, the authors have tried to go as deep into the details of derivation of results as required to make the book useful. The term wetting commonly refers to the displacement of air from a solid surface. Throughout this book we shall be discussing wetting and spreading features of liquids, which partially (the most important example being water and aqueous solutions) or completely (oils) wet the solids or other liquids. Wetting water films occur everywhere, even in the driest deserts or in the sauna and bathtub, although you might not see them with the naked eye because they are too thin or because they seem to disappear too quickly. Water is essential for life. It may very well be that without water, life would have not have started on Earth. In fact no life seems possible without fluids! Life, as we know it started in a little “pond,” the “primordial soup” leading to the first replicating bio-related amino acids. In the processes of wetting or spreading, three phases — air, liquid, and solids — meet along a line, which is referred to as a three-phase contact line. Recall the spreading drop and the drop edge, which is the three-phase contact line. In the vicinity of a three-phase contact line, the thickness of the droplet becomes very thin and, even more, virtually tends to zero. In a thin water layer, new very special surface forces come into play. These forces are well known in colloid science: forces in thin layers between interfaces of neighbor particles, droplets, and bubbles in suspensions and emulsions. Understanding of the importance of surface forces in colloid science has resulted in substantial progress in this area. In fact, it is the reason why colloid science is referred to nowadays as colloid and interface science. Surface forces of the same nature act in thin liquid layers in the vicinity of the three-phase contact lines in the course of wetting and spreading. Surprisingly, the importance of surface forces has been much less recognized in wetting and spreading than it deserves. In Chapter 1 through Chapter 3 we will try to convince © 2007 by Taylor & Francis Group, LLC

the reader that virtually all wetting and spreading phenomena are determined by the surface forces acting in a tiny vicinity of the three-phase contact line. Water is, indeed, a strange liquid. For example, if you place a glass bottle full of pure water (H2O) in the deep freezer, the bottle will break as water increases in volume while solidifying as ice, an anomalous property relative to other liquids. Life (fish) in frozen lakes would not be possible without the anomalous behavior of water around 4˚C. We shall see that a property of water relative to “surface” forces is key to understanding its wetting and spreading features. We will also find that surface forces (frequently also referred to as disjoining pressure) have a very peculiar shape, in the case of water and aqueous solutions. This fact is critical for the existence of our life in a way which is yet to be understood. Wetting and spreading are dramatically affected by SURFace ACTtive AgeNTS (in short, surfactants). Their molecules have a hydrophilic head (ionic or nonionic) with affinity for water and a hydrophobic tail (a hydrocarbon group), which is repelled by an aqueous phase. Fatty acids, alcohols, and some proteins (natural polymers), and washing liquids, powders, and detergents all act as surfactants. It is the reason why the kinetics of wetting and spreading of surfactant solutions is under investigation in this book. On the other hand a number of solid substrates — printing materials, textiles, hairs — when in contact with liquids are porous in different degrees. In spite of much experimental and practical experience in the area, only a limited number of publications are available in the literature that deals with fundamental aspects of the phenomenon. We show in this book that spreading kinetics over porous substrates differs substantially as compared with spreading over nonporous substrates. Aiming at a logical progression in the problems treated with discussion at each level, building albeit not rigidly, upon the material that came earlier, the book can be divided into two parts: Chapter 1 to Chapter 3 form one part, and Chapter 4 and Chapter 5 constitute the other. Chapter 1 is key to the former in that its reading is a must for the understanding of Chapter 2 and Chapter 3. To a large extent Chapter 4 and Chapter 5 can be read independently from the preceding chapters, yet they are tied to each other and to the previous three. Chapter 1 introduces surface forces and a detailed critical analysis of the current understanding of Young’s equation, the building block in most wetting and spreading research and in a number of publications. The surface forces are also frequently referred to in the literature as colloidal forces and disjoining pressure. All these terms are used as equivalents in this book, following appropriate clarification of concepts, terminology, and origins. Colloidal forces act in thin liquid films and layers when thickness goes down to about 10–5 cm = 0.1 μm = 102 nm. Below this thickness the surface forces or disjoining pressure become so increasingly powerful that they dominate all other forces (for example, capillary forces and gravity). Accordingly, surface forces determine the wetting properties of liquids in contact with solid substrates. One purpose of Chapter 1 through Chapter 3 is to show that progress in the area of equilibrium and dynamics of wetting demands due consideration of surface forces action in the vicinity of © 2007 by Taylor & Francis Group, LLC

the three-phase contact line. Chapter 2 and Chapter 3 look sequentially at the equilibrium and kinetics or dynamics of wetting, showing that the action of surface forces determines all equilibrium and kinetics features of liquids in contact with solids. Note that Chapter 3 cannot be read and understood without reading the introduction to the chapter. Colloidal forces or disjoining pressure are well known and widely used in colloid science to account for equilibrium and dynamics of colloidal suspensions and emulsions. The current theory behind colloidal forces between colloidal particles, drops, and bubbles is the DLVO theory, an acronym made after the names of Derjaguin (B.V.), Landau (L.D.), Verwey (E.J.W.) and Overbeek (J.Th.G.). The same forces act in the vicinity of the three-phase contact line, and their action is as important in this case as it is in the case of colloids. Unfortunately, most authors currently ignore the action of colloidal forces when discussing the equilibrium and dynamics of wetting. It is our belief that this has hampered progress in the area of wetting phenomena for decades. Chapter 4 and Chapter 5 are devoted to a detailed discussion of some recent, albeit still fragmentary, developments regarding the kinetics of spreading over porous solid substrates, including the case of hydrophobic substrates in the presence of surfactants. Noteworthy are some new and universal spreading laws in the case of spreading over thin porous layers discussed in Chapter 4. Some arguments and theory in Chapter 5 are experiment-discussion oriented and heuristic or semiempirical in approach (Section 5.4 and Section 5.5) and should be judged accordingly. To our understanding, little is well established about spreading over hydrophobic substrates in the presence of surfactants. Our treatment of the spontaneous adsorption of surfactant molecules on a bare hydrophobic substrate ahead of the moving liquid front, making an initially hydrophobic substrate partially hydrophilic, allows a good description of a number of phenomena. Yet an understanding of the actual mechanism of transfer of surfactant molecules in a vicinity of the three-phase contact line will require considerable theoretical and experimental efforts. We close the book with a few comments and warnings in a chapter of conclusions. Victor M. Starov Loughborough University, Leicestershire, United Kingdom Manuel G. Velarde Instituto Pluridisciplinar, Universidad Complutense, Madrid, Spain Clayton J. Radke University of California at Berkeley

© 2007 by Taylor & Francis Group, LLC

Acknowledgments In 1974 Victor M. Starov met Prof. Nikolay V. Churaev, the beginning of a collaboration that has continued for more than 30 years and for which author Starov would like to express very special thanks. Churaev involved Starov in the investigation of wetting and spreading phenomena in the former Surface Forces Department, Moscow Institute of Physical Chemistry (MIPCh), Russian Academy of Sciences. This collaboration soon included a number of other colleagues from MIPCh; appreciation is extended to these, especially professors Boris V. Derjaguin, Georgy A. Martynov, Vladimir D. Sobolev, and Zinoviy M. Zorin. In 1981 Starov took the position of head of the Department of Applied Mathematics, Moscow University of Food Industry. He organized a weekly seminar there, where virtually all problems presented in this book were either solved, initiated, or at least discussed. These seminars were carried on until the Soviet Union collapsed. Author Starov would like to thank all members of the seminar but especially professors Anatoly N. Filippov and Vasily V. Kalinin, and Drs. Yury E. Solomentsev, Vladimir I. Ivanov, Sergey I. Vasin, and Vjacheslav G. Zhdanov. In 1987, the University of Sofia celebrated its centennial. This book’s first two authors, Victor M. Starov and Manuel G. Velarde, were honored by being chosen by Prof. Ivan B. Ivanov to be centennial lecturers at his university. Beyond being an honor, this was a lucky event in their lives. Both knew of Ivanov for quite some time but had not met him earlier nor had they worked together in the same field, although both had common interests in the interfacial phenomena. While in Sofia, hearing each other lecturing and discussing science “and beyond,” they felt that it would be interesting to work together one day, particularly in exploring the consequences of surface tension and surface tension gradients, the latter of which, e.g., creates flow or alters an existing one (the Marangoni effect). In 1991 Starov was able to visit with Manuel G. Velarde at the Instituto Pluridisciplinar of the Universidad Complutense, Madrid, Spain. Both were fortunate once more in being visited by Dr. Alain de Ryck, a young French scientist and brilliant experimentalist. He produced experiments where both Starov and Velarde were able to observe the striking role of the Marangoni effect in the spreading of a surfactant droplet over the thin aqueous layer. Later, the scientific relationship between the first two authors of this book was strengthened by the visit of Prof. Vladimir D. Sobolev, MIPCh, an outstanding scientist who went beyond being a highly skilled experimentalist. His work cemented the earlier mentioned scientific relationship and collaboration between Starov and Velarde. It was further enhanced when the former moved from Moscow to the Chemical Engineering Department, Loughborough University, United Kingdom, in 1999. There, Sobolev also worked with both Starov and Velarde, and this was the © 2007 by Taylor & Francis Group, LLC

beginning of numerous Loughborough–Madrid exchanges involving also several younger colleagues: Drs. Serguei R. Kosvintsev, Serguei A. Zhdanov, and Andre L. Zuev. Then in 2001, the first two authors of this book jointly organized a summer school on wetting and spreading dynamics and related phenomena at El Escorial, Madrid, under the sponsorship of the Universidad Complutense Summer Programme. Economic support also came from the European Union (under the ICOPAC Network), the European Space Agency (ESA), Fuchs Iberica, L’Oreal, Inescop, and Unilever, Spain. Among the prestigious speakers from Bulgaria, France, Germany, Israel, the United States, and Spain was one of the invited lecturers, the third author of this book, Clayton J. Radke. We decided not to produce proceedings of that school, but soon after, the three future coauthors of this book started thinking of writing a joint monograph. Indeed, the present book is the result of our concern about the lack of systematized knowledge on wetting and spreading dynamics, i.e., the lack of a monograph for the use of basic and applied scientists, applied mathematicians, chemists, and engineers. Two other schools are also worth mentioning. One on complex fluids, wetting, and spreading-related topics, coordinated by Velarde, took place in 1999 at La Rabida, Huelva, Spain. The other course, much more focused on spreading problems, coordinated by Starov, was scheduled in 2003 at CISM (International Center for Mechanical Sciences) in Udine, Italy. There are proceedings of the latter (“Fluid mechanics of surfactant and polymer solutions,” edited by Starov and Ivanov; Springer Verlag, 2004)) but not of the former. In the past few years several other workshops, discussion meetings, and international conferences took place in Madrid and Loughborough on the subject. The authors would like to express their gratitude to Nadezda V. Starova. Without her energy, endless patience, kindness, and expertise, this book most surely would have never been finished. We are also happy to thank Maria-Jesus Martin (Madrid) for her help in the preparation of the manuscript. We wish to express our gratitude to the coauthors of our joint publications: Nikolay N. Churaev, Boris V. Derjaguin (deceased), Ivan B. Ivanov, Vladimir I. Ivanov, Vasiliy V. Kalinin, Olga A. Kiseleva, Serguei R. Kosvintsev, Georgy A. Martynov, David Quere, Alain de Ryck, Ramon G. Rubio, Victor M. Rudoy, Vladimir D. Sobolev, Serguei A. Zhdanov, Pavel P. Zolotarev, and Zinoviy M. Zorin. We also would like to recognize the following colleagues, fruitful discussions with whom stimulated our research: Anne-Marie Cazabat, Pierre-Gilles de Gennes, Benoit Goyeau, George (Bud) Homsy, Dominique Langevin, Francisco Monroy, Alex T. Nikolov, Francisco Ortega, Len Pismen, Yves Pomeau, Uwe Thiele, and Darsh T. Wasan. Preparation of the manuscript was supported by a grant from the Royal Society, United Kingdom, which we would like to acknowledge. We wish to particularly acknowledge the support by Prof. John Enderby. The final revision of the manuscript was done while Manuel G. Velarde was Del Amo Foundation Visiting Professor with the Department of Mechanical Engineering and Environ© 2007 by Taylor & Francis Group, LLC

mental Sciences of the University of California at Santa Barbara. This was possible thanks to the hospitality of Prof. George M. Homsy. Last but not least, we acknowledge the support for the research leading to this book which came from the Engineering and Physical Sciences Research Council, United Kingdom (Grants EP/D077869 and EP/C528557), and from the Ministerio de Educacion y Ciencia, Spain (Grants MAT2003-01517, BQU200301556, and VEVES).

© 2007 by Taylor & Francis Group, LLC

1

Surface Forces and the Equilibrium of Liquids on Solid Substrates

INTRODUCTION In this chapter, we shall give a brief account of the theory and experimental evidence of the action of surface forces, i.e., forces needed to account for phenomena occurring near surfaces, very thin layers, corners, borders, contact lines, etc. All forces do originate at the microscopic level, but we shall look at the phenomenological, macroscopic manifestations of those forces. In particular, we shall emphasize the role of the so-called disjoining pressure. Such terminology is a bit misleading because, in a number of cases, action would be just the opposite: conjoining pressure (attraction). However, current use or historical reasons lead us to maintain the term disjoining pressure, whatever the particular situation might be. The disjoining pressure acts in the vicinity of the three-phase contact line, and its action becomes dominant, e.g., as a liquid proﬁle approaches a solid substrate, or with colloidal particles or drops. In the study of wetting and spreading processes, its importance seems less common than in colloid science, in spite of the same nature of the forces and the same level of necessity. The relationship between the disjoining pressure and the thickness of a liquid ﬁlm is frequently referred to as disjoining pressure isotherm because it is generally measured at a given temperature. It is noteworthy that for water and aqueous solutions, the disjoining pressure isotherm has an S-shape, hence alerting us to instability, metastability, and bistability (in the spirit of van der Waals and Maxwell description of thermodynamic equilibrium phases). Our life is very much tuned to the properties of water (as carbon is also linked to life). To what extent does the S-shape of the disjoining pressure isotherm of water affect life? This is an interesting question to be answered. However, we do not address this problem in this book. We shall start with a discussion about a well-known and much used Young’s equation in spreading and wetting dynamics. We advise reading the original paper by Young, as it is a masterpiece of scientiﬁc literature. Yet we hope to convince the reader of the ill-founded thermodynamic support of the (historical) standard form of such relationship. We shall argue and prove that the thermodynamically

1 © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

sound Young’s equation, which is frequently referred to as the Derjaguin–Frumkin equation, is only possible if due account is given to the disjoining pressure. We shall see that the disjoining pressure action either in the case of complete or partial wetting always leads to the formation of a thin liquid layer in the vicinity of the three-phase contact line. The latter results in a microscopic ﬂow that is determined by both the disjoining pressure action and the topography of the surface (roughness, heterogeneity, chemical, or otherwise). As a result, never is there a real three-phase contact line but only an apparent macroscopic contact line. Then, we shall describe and comment upon the three most used components of the disjoining pressure. Finally, we shall consider at the heuristic level the static contact angle hysteresis when, for example, a drop spreads on a smooth and homogeneous solid substrate. We shall show that microscopic ﬂow in the vicinity of the apparent three-phase contact line is unavoidable. The complication introduced by such microscopic ﬂow seems responsible for the present lack of a sound theory of the kinetics of spreading in the case of partial wetting, in contrast to the case of complete wetting, where the theory is well developed and leads to a quite good agreement with most, if not all, experimental observations.

1.1 WETTING AND YOUNG’S EQUATION Why do droplets of different liquids deposited on identical solid substrates behave so differently? Why do identical droplets — for example, aqueous droplets deposited on different substrates — behave so differently? When we attempt to make a uniform layer of mercury on a glass surface, we ﬁnd it impossible. Each time we try, the mercury layer will immediately form a droplet, which is a spherical cap with the contact angle bigger than π/2 (Figure 1.1). Note, the contact angle is always measured inside the liquid phase (Figure 1.1 to Figure 1.3). However, it is easy to make an oil layer (hexane or decane) on the same glass surface; for this purpose an oil droplet can be deposited on the same glass substrate, and it will spread out completely (Figure1.3). In this case, the contact angle decreases with time down to a zero value. Now let us try the same procedure with an ordinary tap water droplet. An aqueous droplet deposited on the same glass substrate spreads out only partially down to some contact angle, θ, which is in between 0 and π/2 (Figure 1.2). That is, an aqueous droplet on a glass surface behaves in a way that is intermediate between the behavior of the mercury and oil. These three cases (Figure 1.1, Figure 1.2, and Figure 1.3) are referred to as: nonwetting, partial wetting, and complete wetting, respectively.

θ

FIGURE 1.1 Nonwetting case: contact angle is bigger than π/2. © 2007 by Taylor & Francis Group, LLC

Surface Forces and the Equilibrium of Liquids on Solid Substrates

3

θ

FIGURE 1.2 Partial wetting case: the contact angle is in between 0 and π/2.

θ(t)

FIGURE 1.3 Complete wetting case: the droplet spreads out completely, and only the dynamic contact angle can be measured, which tends to become zero over time.

γ γsv

θ γsl

R

ℜ

FIGURE 1.4 Interfacial tensions at the three-phase contact line. R is the radius of the droplet base, ℜ is the radius of the droplet. The droplet is small enough, and the gravity action can be neglected.

Now let us try to make a water layer on a Teﬂon surface. We will be unable to do this, exactly in the same way as we were unable to in the case of mercury on a glass surface. That is, the same aqueous droplet can spread out partially on a glass substrate and does not spread at all on a Teﬂon substrate. The inability to spread on the Teﬂon surface indicates that the wetting or nonwetting is not a property of the liquid but rather a property of the liquid–solid pair. In broader terms, complete wetting, partial wetting, and nonwetting behavior are determined by the nature of both the liquid and the solid substrate. Let us consider a picture presented in Figure 1.4. Let us say that the threephase contact line is the line where three phases: liquid, solid, and vapor meet. Consideration of forces in the tangential direction at the three-phase contact line results in the well-known Young’s equation, which connects three interfacial tensions, γsl , γsv , and γ with the value of the equilibrium contact angle, θNY (Figure 1.4), where γsl , γsv , and γ are solid–liquid, solid–vapor, and liquid–vapor interfacial tensions, respectively: cos θNY = (γsv – γsl)/γ © 2007 by Taylor & Francis Group, LLC

(1.1)

4

Wetting and Spreading Dynamics

Note, we marked the equilibrium contact angle in Equation 1.1 as θNY , and we see in the following section that there is a good reason for that. According to Figure 1.4, the complete wetting case corresponds to the case when all forces cannot be compensated in the tangential direction at any contact angle, that is, if γsv > γsl + γ. Partial wetting case, according to Equation 1.1, corresponds to 0 < cos θNY < 1, and, last, the nonwetting case corresponds to 1 < cos θNY < 0. That is, Equation 1.1 reduces complete wettability, partial wettability, and nonwettability cases to the determination of three interfacial tensions, γsl , γsv , and γ. It looks like everything is very easy and straightforward. However, as we see in the following section, the situation is far more complex than that. Let us try to deduce Equation 1.1 using a rigorous theoretical procedure based on the consideration of the excess free energy of the system presented in Figure 1.4. Let us assume that the excess free energy of the small droplet (neglecting the gravity action) is as follows: 2 Φ = γS + PV e + πR ( γ sl − γ sv ),

(1.2)

where S is the area of the liquid–air interface, Pe = Pa – Pl is the excess pressure inside the liquid, Pa is the pressure in the ambient air, Pl is the pressure inside the liquid, and R is the radius of the drop base. The last term on the right-hand side of Equation 1.2 gives the difference between the energy of the part of the bare surface covered by the liquid drop as compared with the energy of the same solid surface without the droplet. Note that the excess pressure, Pe, is negative in the case of liquid droplets (concave liquid–air interface) and positive in the case of meniscus in partially or completely wetted capillaries (convex liquid–air interface). Let h(r) be the unknown proﬁle of the liquid droplet; then the excess free energy, as given by Equation 1.2, can be rewritten as R

∫

Φ = 2π r 0

( 1 + h′ + P h + γ 2

e

sl

)

− γ sv dr

(1.3)

Now we use one of the most fundamental principles: any proﬁle, h(r), in the latter expression, should give the minimum value of the excess free energy as in Equation 1.2. Details of the procedure are given in the next chapter (see Section 2.2). Under equilibrium conditions, the excess free energy should reach its minimum value. The mathematical expressions for this requirement are the following conditions: (1) the ﬁrst variation of the free energy, δΦ, should be zero, (2) the second variation, δ 2 Φ, should be positive, and (3) the transversality condition at the drop perimeter at the three-phase contact line — that is, at r = R — should be satisﬁed. In Section 2.2, these conditions are discussed in more detail, and it © 2007 by Taylor & Francis Group, LLC

Surface Forces and the Equilibrium of Liquids on Solid Substrates

5

is shown that actually one extra condition should be fulﬁlled. However, at the moment we will ignore this extra condition, because it is easy to check that this condition is always satisﬁed in the case of the excess free energy given by Equation 1.3. Conditions 1 and 2 are actually identical to those for a minimum of regular functions. Condition 3 is usually forgotten and deduced using a different consideration. Condition 1 results in the Euler equation, which gives an equation for the drop proﬁle: ∂f d ∂f − = 0, ∂h dr ∂h ′ where f = r  γ 1 + h′ 2 + Pe h + γ sl − γ sv  ,   or   γ d  r r dr   

  h′  =P e 1  2 1 + h′  2    

(1.4)

( )

Solution of the latter equation is a part of the sphere of radius 2γ/Pe (Figure 1.4). The second condition gives: ∂2 f > 0, ∂h ′ 2 or

(

γ

1 + h ′2

)

3/ 2

> 0,

which is always satisﬁed. The latter means that Equation 1.4 really gives a minimum value to the excess free energy in Equation 1.3. Now the third, transversality condition is as follows:  ∂f   f − h′ ∂h′  = 0  r = R © 2007 by Taylor & Francis Group, LLC

6

Wetting and Spreading Dynamics

or   r γh′ 2  r  γ 1 + h′ + Pe h + γ sl − γ sv  − h′   1 + h′ 2  Taking into account that h

r=R

  = 0. r = R

= 0, we conclude from the previous equation

  γ + γ sl − γ sv  = 0  2 r = R  1 + h′

(1.5)

Figure 1.4 shows that h′

r=R

= − tan θ NY .

Substitution of the latter expression in Equation 1.5 results in Equation 1.1. To summarize: application of the rigorous mathematical procedure to excess free energy given by Equation 1.3 results in: 1. A spherical proﬁle of the droplet with a radius of the curvature R=−

2γ , Pe

(1.6)

2. The Young’s equation (Equation 1.1) for the equilibrium contact angle θNY . The equation for the equilibrium contact angle shows that the derivation of Young’s equation (Equation 1.1) is based on a ﬁrm theoretical basis if we adopt the expression for the free energy, Equation 1.3. The free energy equation consideration means that Young’s equation (Equation 1.1) is valid only when the adopted expression for the excess free energy (Equation 1.3) is valid. Consideration of thin ﬁlms on curved surfaces was undertaken by I. Ivanov and P. Kralchevsky in [9]. Let us ask ourselves a question: How many equilibrium states can a thermodynamic system have? The answer is well-known: either one or, in some special cases, two or even more states that are separated from each other by potential barriers. According to condition 1 and condition 2, we get an inﬁnite and continuous set of equilibrium states, which are not separated from each other by potential barriers. Young’s equation does not specify the equilibrium volume of the droplet, V, or the excess pressure inside the drop, Pe, which can be any negative value. Both volume of the droplet and the excess pressure can be arbitrary. The latter means that the volume of the droplet is not speciﬁed; a droplet of any volume can be at the equilibrium. © 2007 by Taylor & Francis Group, LLC

Surface Forces and the Equilibrium of Liquids on Solid Substrates

7

That means Young’s equation (Equation 1.1) is in drastic contradiction with thermodynamics. Why is it so? Where is there a mistake? Deﬁnitely not in the derivation. That means we should go back to basics. Is the expression for the excess free energy (Equation 1.3) correct? At equilibrium, the following three equilibrium considerations should hold: 1. Liquid in the droplet must be in equilibrium with its own vapor. 2. Liquid in the droplet must be in equilibrium with the solid. 3. Vapor must be in equilibrium with the solid substrate. Step by step, in the following section, we show that none of these three equilibriums are taken into account by the expression for the excess free energy (Equation 1.3). The ﬁrst requirement in the list above results in the equality of chemical potentials of the liquid molecules in vapor and inside the droplet. This results in the following expression of the excess pressure, Pe: Pe =

RT p ln s , vm p

(1.7)

where νm is the molar volume of liquid, ps is the pressure of the saturated vapor at the temperature T, R is the gas constant (do not confuse with the radius of the drop base), p is the vapor pressure that is in equilibrium with the liquid droplet. Equation 1.7 determines the unique equilibrium excess pressure Pe and, hence, according to Equation 1.6, the unique radius of the droplet, ℜ. We remind the reader now that the excess pressure inside the drop, Pe, should be negative (pressure inside the droplet is bigger than the pressure in the ambient air). That means that the right-hand side in Equation 1.7 should also be negative, but negativity is possible only if p > ps, that is, the droplets can be at equilibrium only with oversaturated vapor. It is really troublesome because the equilibration process goes on for hours, and it is necessary to keep oversaturated vapor over a solid substrate under investigation for hours. To the best of our knowledge, nobody can do that, which would mean that it is difﬁcult to experimentally investigate equilibrium droplets on the solid substrate. There is a plethora of investigations published on the literature of the equilibrium contact angles of droplets on solid substrate. The previous consideration shows that the contact angles measured are mostly not in equilibrium. It is a different story when contact angles were really measured. Only in Chapter 3 will we be ready to clarify the subject completely (see Section 3.10). Unfortunately, this is not the end of the troubles encountered with Young’s equation (Equation 1.1) because now we should consider requirements of the equilibriums 2 and 3. Let us assume that we can create, at least theoretically, an oversaturated vapor over the solid substrate and wait long enough until the equilibrium is reached. Now the liquid molecules in the vapor are at equilibrium © 2007 by Taylor & Francis Group, LLC

8

Wetting and Spreading Dynamics

with the liquid molecules in the droplet. Note that the solid–liquid interfacial tension, γsl , differs from the solid–vapor interfacial tension, γsv . If they are not different, then according to Equation 1.1, the contact angle is equal to 90o (an intermediate case between partial wetting and nonwetting). In the case of partial wetting or complete wetting, γsl < γsv . The latter expression means that the presence of liquid on the solid substrate results in lower surface tension as compared with the surface tension of the bare solid surface, γsv . Now back to our theoretical case of the liquid droplet on the solid substrate in equilibrium with the oversaturated vapor. We should now take into account the equilibrium between the liquid vapor and the solid surface; it is unavoidable as the liquid molecule adsorption on the solid substrate and the presence of liquid molecules on the surface changes the initial surface tension. This means that the liquid molecules from the vapor must adsorb on the solid substrate outside the liquid droplet under consideration. The latter consideration results in the formation of an adsorption liquid ﬁlm on the surface and a new interfacial tension, γhv , where h is the thickness of the adsorbed layer. It may be said that it does not make sense to talk about a monolayer, or in the best case, several layers of the adsorbed liquid molecules on the solid substrate, as the inﬂuence on the macroscopic droplet will be negligible. Let us consider a simple but important example of the presence of a single monolayer drastically changing the wetting property. Take a microscope glass cover and put an aqueous droplet on this surface. The droplet will form a contact angle, which considerably depends on the type of the glass, and in some special case (which we consider now), it will be as small as 10˚. Now let us place a monolayer of oil on the glass surface (reminder: a monolayer is a layer with thickness of 1 molecule). Now again, let us place a water droplet on a new glass surface covered by a monolayer of oil. The droplet will form a contact angle that is higher than 90˚. That is, the presence of only one tiny monolayer changed partial wetting to nonwetting. Now, back to the droplets on the solid surface at equilibrium with the oversaturated vapor. As we now understand, the adsorption of vapor on the solid substrate is very important, and instead of the interfacial tension of the bare solid surface, γsv , we should use γhv . The latter interfacial tension is to be investigated in Chapter 2 (Section 2.1). The previous consideration shows that to investigate equilibrium liquid droplets, the following procedure should be followed: •

•

The solid substrate under investigation should be kept in the atmosphere of the oversaturated vapor until equilibrium adsorption of vapor on the solid substrate is reached, and a new interfacial tension γhv should be measured. Then, the droplet, which has a size that should be in equilibrium with the oversaturated vapor, should be deposited and kept until the equilibrium is reached.

© 2007 by Taylor & Francis Group, LLC

Surface Forces and the Equilibrium of Liquids on Solid Substrates

•

9

Young’s equation (Equation 1.1) should now be rewritten as cos θe =

γ hv − γ sl . γ

(1.8)

The characteristic time scale of the latter processes depends on the liquid volatibility and viscosity and is, in general, of hours of magnitude. No such kind of experiment has ever been attempted in the atmosphere of an oversaturated vapor to the best of the author’s knowledge. This would mean that equilibrium liquid droplets of volatile liquids probably have never been observed experimentally. It is obvious from the same reasons as given before that the thickness of the adsorbed layer, h, depends on the vapor pressure in the ambient air; that is, γhv is a function of the pressure in the ambient air and, hence, according to Equation 1.8, the contact angle changes with vapor pressure. Is this dependency strong or weak? The answer will be given in Chapter 2 (Section 2.1 and Section 2.3). Is this the end of the problems with Young’s equation (Equation 1.1)? Unfortunately not, because we still did not consider the last, but not the least, requirement of the equilibrium (3). In Figure 1.5, an equilibrium liquid droplet is presented in contact with an equilibrium-adsorbed liquid ﬁlm on the solid surface. What happens in the vicinity of the line where they meet? Is the situation presented in Figure 1.5 possible? The answer is obvious: such sharp transition from the liquid droplet to the liquid ﬁlm is impossible. On the line shown by the arrow, the capillary pressure will be inﬁnite. Hence, it should be a smooth transition from the ﬂat equilibrium liquid ﬁlm on the solid surfaces to the spherical droplet, as shown in Figure 1.6, where this smooth transition is shown. Let us call this region, where transition from a ﬂat ﬁlm to the droplet takes place, a transition zone. The presence of the transition zone leads us into much bigger problems than before, because pure capillary forces cannot keep the liquid in this zone in equilibrium; the liquid proﬁle is concave (hence, the capillary pressure under the liquid surface is higher than in the ambient air) to the right from the arrow in Figure 1.6, and the liquid proﬁle is convex (hence, the capillary

θ

FIGURE 1.5 Cross section of an equilibrium liquid droplet (at oversaturation) in contact with an equilibrium-adsorbed liquid ﬁlm on the solid substrate. What happens on the line (shown by an arrow) where they meet?

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FIGURE 1.6 Transition zone from the ﬂat equilibrium liquid ﬁlm on a solid surface to the liquid droplet. The arrow shows the point to the right where the liquid proﬁle is concave and to the left where the proﬁle is convex.

pressure under the liquid surface is lower than in the ambient air) to the left from the arrow in Figure 1.6. We come back to this paradox a bit later. Now we only remark that the consideration of this paradox was one of the motivations to replace the name colloid science with a new name, colloid and interface science. However, for a moment let us forget about the transition zone. The preceding consideration shows that Young’s equation can probably be used only in the case of nonvolatile liquids, as we have too many problems with volatile liquids. Can a liquid really be a nonvolatile one? Usually, low volatibility means liquids with big molecules that have high viscosity and a corresponding higher characteristic time scale of equilibration process with the oversaturated vapor. In spite of that, let us assume that the liquid is nonvolatile. In the case of partial wetting, as we have already seen, liquid droplets cannot be in equilibrium with a bare solid surface. There should always be at equilibrium an adsorption layer of the liquid molecules on the solid substrate in front of the droplet on the bare solid surface. If the liquid is volatile, then this layer is created by means of evaporation and adsorption. However, if the liquid is nonvolatile, the same layer should be created by means of ﬂow from the droplet edge onto the solid substrate. As a result, at equilibrium the solid substrate is covered by an equilibrium liquid layer of thickness, h. The thickness of the equilibrium liquid ﬁlm, h, is determined (as we see in the following section), by the potential of action of surface forces. The characteristic time scale of this process is hours, because it is determined by the ﬂow in the thinnest part in the vicinity of the apparent three-phase contact line, where the viscose resistance is very high. During these hours, evaporation of the liquid from the droplet cannot be ignored, and we go back to the problem of volatibility. Let us assume, however, that the equilibrium ﬁlm, after all, forms in front of the liquid droplet, and we have waited enough for the equilibrium. However, now we have again the following three interfacial tensions: γ, γsl , and γvh , which are liquid–vapor interfacial tension, solid–liquid interfacial tension and solid substrate, covered with the liquid ﬁlm of thickness h–vapor interfacial tensions. We can refer back to the same problem as in the case of volatile liquid. We can neither measure the interfacial tension, γvh, nor use it in Equation 1.8. However, there is an answer, and the answer will be given in Section 2.1. © 2007 by Taylor & Francis Group, LLC

Surface Forces and the Equilibrium of Liquids on Solid Substrates

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This would mean that even in the case of nonvolatile liquids, the applicability of Young’s equation (Equation 1.1) still remains questionable. In view of the preceding features, from now on in this book we shall be using apparent three-phase contact line because there is no such line at the microscopic scale.

1.2 SURFACE FORCES AND DISJOINING PRESSURE The presence of adsorbed liquid layers on a solid substrate is a result of the action of some special forces, referred to as surface forces. Let us go back to Figure 1.6 and consider the transition zone between the droplet and the ﬂat liquid ﬁlms in front of it. It looks like the proﬁle presented in Figure 1.6 cannot be in equilibrium because capillary pressure should change the sign inside the transition zone, and it is in contradiction with the requirement of the constancy of the capillary pressure everywhere inside the droplet. Some additional forces are missing. The mentioned problem was under consideration by a number of scientists for more than a century. Their efforts resulted in considerable reconsideration of the nature of wetting phenomena. A new class of phenomena has been introduced [1]: surface phenomena, which are determined by the special forces acting in thin liquid ﬁlms or layers in the vicinity of the apparent three-phase contact line. Surface forces are well-known and are widely used in colloid and interface science. They determine the stability and behavior of colloidal suspensions and emulsions. In the case of emulsions/suspensions, their properties and behavior (stability, instability, rheology, interactions, and so on) are completely determined by surface forces acting between colloidal particles or droplets. This theory is widely referred to as the DLVO theory [1] after the names of four scientists who developed the theory: Derjaguin, Landau, Vervey, and Overbeek. No doubt that all colloidal particles have a rough surface and, in a number of cases, even chemically inhomogeneous surfaces (living cells, for example). Roughness and inhomogeneity of colloidal particles can modify substantial surface forces: their nature, magnitude, and range of action. However, the roughness and inhomogeneity of the surface of the colloidal particles does not inﬂuence the main phenomenon; all their interactions and properties are determined by the action of the surface forces [2]. There is something unconventional about wetting studies as compared with analogous studies in colloid and interface science. It is widely (and erroneously) accepted that roughness and nonuniformity of the solid substrate in contact with liquids can in itself explain wetting features, without consideration of the surface forces acting in a vicinity of the apparent three-phase contact line. As a result, the inﬂuence of surface forces on the kinetics of wetting and spreading is much less recognized than in the study of colloidal suspensions and emulsions, in spite of the same nature of surface forces. It has been established that the range of action of surface forces is usually of the order of 0.1 µm [1]. Note that in the vicinity of the apparent three-phase © 2007 by Taylor & Francis Group, LLC

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1 h

3 2

FIGURE 1.7 Measurement of interaction between two thick plates 1 and 2, possibly made of different materials, with a thin layer 3 in between.

contact line, r = R (Figure 1.4), the liquid proﬁle, h(r), tends to be of zero thickness. This thickness means that close to the three-phase contact line, surface forces come into play and their inﬂuence cannot be ignored. A manifestation of the action of surface forces is the disjoining pressure. To explain the nature of the disjoining pressure, let us consider the interaction of two thick, plain, and parallel surfaces divided by a thin liquid layer of thickness h (aqueous electrolyte solution, for example). The surfaces are not necessarily of the same nature as two important examples show: (1) one is air, one is a liquid ﬁlm, and one is solid support, and (2) both surfaces are air, and one is a liquid ﬁlm. Example 1 is referred to as a liquid film on a solid support and models the liquid layer in the vicinity of the three-phase contact line, Example 2 is referred to as a free liquid film. There is a range of experimental methods to measure the interaction forces between these two surfaces as a function of the thickness, h (gravity action is already taken into account) (Figure 1.7) [1,3,4]. If h is bigger than ≅10–5 cm = 0.1 µm, then the interaction force is equal to zero. However, if h < 10–5 cm, then an interaction force appears. This force can depend on the thickness, h, in a very peculiar way. The interaction forces divided by the surface area of the plate has a dimension of pressure and is referred to as the disjoining pressure [1]. Note that this term is a bit misleading, because the mentioned force can be both disjoining (repulsion between surfaces) and conjoining (attraction between surfaces). Now we discuss the physical phenomena behind the existence of surface forces. Let us consider a liquid–air interface. It is obvious that the physical properties of the very ﬁrst layer on the interface are substantially different from the properties of the liquid (in bulk) far from the interface. What can we say about the properties of the second, third, and other layers? It is understandable that the physical properties do not change by jumping from the very ﬁrst layer on the interface to the subsequent layers, but the change proceeds in a continuous way. This continuous change results in the formation of a special layer, which we refer to as the boundary layer, where all properties differ from corresponding bulk properties. Do not confuse the introduced boundary layer with a boundary layer in hydrodynamics; they have nothing do to with each other. Such boundary layers exist in proximity to any interface: solid–liquid, liquid–liquid, or liquid–air. In the vicinity of the apparent three-phase contact

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Surface Forces and the Equilibrium of Liquids on Solid Substrates

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2 1 2

3

4

FIGURE 1.8 The liquid proﬁle in the vicinity of the apparent three-phase contact line: (1) bulk liquid, where boundary layers do not overlap, (2) boundary layer in the vicinity liquid–air and liquid–solid interfaces, (3) a region where boundary layers overlap, and (4) ﬂat thin equilibrium ﬁlm. The latter two are the regions where disjoining pressure acts.

line (Figure 1.8), these boundary layers overlap. The overlapping of boundary layers is the physical phenomenon that results in existence of surface forces. The surface force per unit area has a dimension of pressure and is referred to as disjoining pressure, as we have already mentioned in the preceding section. Let the thickness of the boundary layers be δ. In the vicinity of the three-phase contact line, the thickness of a droplet, h, is small enough, that is, h ~ δ, and hence boundary layers overlap (Figure 1.8), which results in the creation of disjoining pressure. The above mentioned characteristic scale, δ ~ 10–5 cm, determines the characteristic thickness where disjoining pressure acts. This thickness is referred to as the range of disjoining (or surface forces) action, ts. The main conclusion: the pressure in thin layers close to the three-phase contact line is different from the pressure in the bulk liquid, and it depends on the thickness of the layer, h, and varies with the thickness, h. In the following, we brieﬂy review the physical phenomena that result in the formation of the above mentioned surface forces and disjoining pressure.

COMPONENTS

OF THE

DISJOINING PRESSURE

Several physical phenomena have been identiﬁed for the appearance of the disjoining pressure. Here, we consider only three of them. Molecular or Dispersion Component Let us start with the most investigated molecular or dispersion component of surface forces. Note that in a number of cases, this component is the weakest among all the other components considered in the following section. Surprisingly, this component is used more frequently than others. It is well known that at relatively large distances (but still in the range of angstroms, that is, 108 cm) all neutral molecules interact with each other, and the energy of this interaction is proportional to const/r 6, where r is the distance between molecules. This is apparent by examining two surfaces made of different

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materials placed inside an aqueous electrolyte solution at a distance, h, from each other (Figure 1.7). Calculation of the molecular contribution to disjoining pressure, Πm, has been approached in two ways: from the approximation of interactions as a pairwise additive, and from a ﬁeld theory of many-body interactions in condensed matter. The simpler and, historically, earlier approach followed a theory based on summing individual London–van der Waals interactions between molecules pair-bypair, undertaken by Hamaker [1]. The more sophisticated, modern theory of Πm was developed (see review [1]) based on the consideration of a ﬂuctuating electromagnetic ﬁeld. In the following, we give an expression for the molecular component of the disjoining pressure, Πm, for a ﬁlm of uniform thickness, h, between two semiinﬁnite phases in vacuum (for simplicity). The expression is [1]:

kT Πm = 3 πc

∞

∞

∑∫ N =0 1

−1   2 pξ N h  s1 + p s2 + p p ξ  exp( ) − 1 c    s1 − p s2 − p 

( (

2 3 N

)( )(

( (

) )

)( )(

) )

 s1 + pε 1 s2 + pε 2  2 pξ N h   + exp  − 1  c    s3 − pε 3 s2 − pε 2 

−1

   dp  

where c is the speed of light, s1 ≡ (ε1 – 1 + p2)1/2, s2 ≡ (ε2 – 1 + p2)1/2, and the dielectric constants ε1, ε2 are functions of imaginary frequency ω ≡ iξ, given by:

ε (iξ) = 1 +

2 π

∞

ω ε ′′(ω ) dω , 2 + ξ2

∫ω 0

where ε ′′ (ω ) is the imaginary component of the dielectric constant. In the limiting case of ﬁlm thickness h, small in comparison with the characteristic wavelength, λ, of the adsorption spectra of the bodies, the molecular component of disjoining pressure is inversely proportional to the cube of ﬁlm thickness [1]: Πm = 8 π 2h3

∞

∫ 0

 ε1 (iξ) − 1  ε 2 (iξ) − 1 A d ξ = H3 . h  ε1 (iξ) + 1  ε 2 (iξ) + 1

(1.9)

In the limiting case of h, large in comparison to λ on the other hand, disjoining pressure turns out to be inversely proportional to the fourth power of ﬁlm thickness [1]:

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Surface Forces and the Equilibrium of Liquids on Solid Substrates

Πm =

c 32 π 2h 4

∞ ∞

∫∫ 0 1

x3 p2

( (

)( )(

15

) )

 s10 + p s20 + p e x − 1] −1 [ − s p − s p 20  10 +[

(s (s

)( )(s

10

+ pε 10 s20 + pε 20

10

− pε 10

20

− pε 20

)e )

x

 B − 1] −1  dpdx = 4 , h 

where s10 ≡ (ε10 – 1 + p2)1/2, s20 ≡ (ε20 – 1 + p2)1/2, and ε10, ε20 are the electrostatic values of the dielectric constants, i.e., the values of the dielectric constant at ξ = 0. There are corresponding expressions for the molecular component of the disjoining pressure of ﬁlms of nonpolar liquids. Those expressions are presented in Reference 1. However, the functional dependency  AH  3 , h<λ h Π m (h) =  B  h 4 , h > λ remains valid. In the following section, we use only the expression derived from Equation 1.9 for the molecular component because the contribution of the disjoining pressure at “big” ﬁlm or layer thickness at h > λ is relatively small as compared with the ﬁrst part at h < λ. For a sufﬁciently long time it was believed that the Lifshitz theory of van der Waals forces was but an elegant formalism, as the necessary dielectric constants across the entire frequency range could not readily be determined. Then Parsegian and Ninham discovered a technique for calculating those properties to an adequate approximation from dielectric data (see the review [1]). Precise measurements of Πm both in-thin and thin-liquid ﬁlms are in good agreement with the theory predictions [1]. However, the latter theory does not apply to ﬁlms so thin as to have dielectric properties that vary with thickness. Using the ﬁrst historically approximate direct summation of all molecular interaction in the system, we obtain the following expression for the molecular or dispersion components of the disjoining pressure:

Πm = −

AH , AH = A33 + A12 − A13 − A23 , 6 πh 3

(1.10)

where AH is referred to as the Hamaker constant, after the scientist who carried out these calculations around a half-century ago [1]. The Hamaker constant, AH ,

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depends on the properties of the phases 1, 2, and 3 through the Hamaker constants, Aij , of phases i and j. Equation 1.10 shows that the Hamaker constant can be either positive (attraction) or negative (repulsion). Note that the functional dependency of the molecular component of the disjoining pressure, according to Equation 1.10, coincides with the exact Equation 1.9. However, the precise value of the Hamaker constant, according to direct summation in Equation 1.10, can be completely wrong. This is the reason why a number of approximations have been developed to precisely calculate the Hamaker constant [1]. In the case of oil droplets on the glass surface, when the dispersion component is the only component of the disjoining pressure acting in thin ﬁlms, the dispersion interaction is repulsive, i.e., the Hamaker constant is negative. In the following, we mostly consider the latter situation (thin liquid ﬁlms on solid substrates) where the Hamaker constant is negative. For this purpose, we rewrite Equation 1.10 as

Πm =

A A , A=− H 6π h3

(1.11)

and just that constant, A, is referred to as the Hamaker constant. Note that the positive Hamaker constant, A, now indicates a repulsion, and the negative constant indicates an attraction. The characteristic value of the Hamaker constant is A ~ 10–14 erg (oil ﬁlms on glass, quartz, or mica surfaces). This value of the Hamaker constant shows that when the liquid layer is at a thickness of h ~ 10–7 cm, the dispersion component of the disjoining pressure is Πm ~ 10–14/10–21 = 107 dyn/cm2. Let us consider a small oil droplet of a radius ℜ ~ 0.1 cm on a solid substrate (Figure 1.4); the surface tension of oils is about γ ≅ 30 dyn/cm. The capillary pressure inside the spherical part of the droplet is 2 γ 2 ⋅ 30 ~ = 6 ⋅ 10 2 dyn /cm 2 . ℜ 0.1 This value shows that in the vicinity of the three-phase contact line, the capillary pressure is much smaller than the disjoining pressure. Let us assume for a moment that the droplet shape remains spherical until the contact with the solid substrate. However, as we have already seen in the preceding section, the capillary pressure is much smaller than the disjoining pressure and cannot counterbalance the disjoining pressure. This means that the disjoining pressure action substantially distorts the spherical shape of droplets in the vicinity of the three-phase contact line. Droplets cannot retain their spherical shape up to the contact line. See further consideration of the proﬁle of liquid droplets in Section 2.3. Before further discussing the next electrostatic component of disjoining pressure, a few words should be said about the electrical double layer.

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Electrical Double Layers Neutral molecules of many salts, acids, and alkalis dissociate into ions (cations and anions) in water, forming aqueous electrolyte solutions. For example, NaCl dissociates with the formation of a cation Na+ and an anion Cl–. Even more, if we assume water completely pure without any salts, acids, and so on, then the water molecule itself, H2O, also dissociates according to the following dissociation reaction: H2O ↔ H+ + OH–. That is, even in pure water, both cations, H+, and anions, OH–, are always present. Note that the two ions H+ and OH– play the most important role in kinetics of wetting and spreading of aqueous solutions. The total charge of cations is completely counterbalanced by the total charge of anions in the bulk of the liquid. These ions are called free ions. All free ions can be transferred both by means of convection (by the ﬂow of water) and by diffusion, if a gradient of concentration of any ion is imposed. Ions also can be transferred under the action of the gradient of electric potential (electromigration), either imposed or spontaneous. In the aqueous electrolyte solutions, the majority of solid surfaces acquire a charge. Before mentioning the mechanism of formation of this charge, let us emphasize that these charges are mostly ﬁxed rigidly on the solid surface and can usually be moved only with the solids. There are two main mechanisms of formation of the charge of the solid surface in aqueous electrolyte solutions: the dissociation of surface groups (brieﬂy discussed in the following section) and the unequal adsorption of different types of ions. A considerable number of solid surfaces have the following type of surface groups on the solid–liquid interface R-OH, where R- is the group that is rigidly connected to the solid. The –OH groups can dissociate in aqueous solutions, which results in the formation of negatively charged groups, R-O– returning the H+ ion into the solution. According to this mechanism or a similar one, many of the solid surfaces (actually majority) in aqueous solutions acquire a negative surface charge. It is obvious that this charge strongly depends on the pH of solution, i.e., depending on the concentration of H+ ions in the volume of solution; pH = log cH+, where cH+ is the concentration of H+ in mol/l. Note that pH = 7 corresponds to the neutral solution, pH < 7 is an acidic solution, and pH > 7 corresponds to an alkaline solution. In all processes, the free and bound ions behave in different ways: free ions can freely be moved, but the bound ions only move with the solid surface. Let us consider the distribution of ions in the close vicinity of a negatively charged surface in contact with aqueous electrolyte solution, for example, NaCl. NaCl dissociates as NaCl → Na+ + Cl–. The electroneutrality condition requires equal concentrations of cations and anions in the bulk solution, far from the charged surface. However, close to the charged surface, according to Coulomb’s law, the free cations Na+ are attracted by the negatively charged solid surface, and the negatively charged ions Cl– are repulsed from the same surface. As a result, the concentration of cations is higher near the surface, and the concentration of anions © 2007 by Taylor & Francis Group, LLC

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is lower than the corresponding concentration in the bulk solution. We recall the process of diffusion. The basic task of diffusion is to destroy all nonuniformities in the distribution of ions. In this case, the diffusion will attempt to make an exact opposite, in comparison with Coulomb’s interaction, to decrease the concentration of cations near the surface and to increase the concentration of anions. As a result of these two opposed trends near the negatively charged surface, a layer of ﬁnite thickness is created in which the concentration of cations reaches its maximum near the surface and monotonically decreases into the depths of the solution to its bulk value, whereas the concentration of anions monotonically grows from its minimum value near the surface to its bulk value in the depths of the solution. This layer, where the concentration of cations and anions differ from their bulk values, is referred to as a diffusive part of the electrical double layer. The characteristic thickness of the diffusive part of electrical double layer is the Debye length, Rd. The characteristic value of the Debye length is Rd =

3 ⋅ 10 −8 C

cm,

where the electrolyte concentration, C, should be expressed in mol/l. This expression shows that the higher the electrolyte concentration, the thinner is the electrical double layer. For example, at C = 10–4 mol/l, Rd = 3·10–6 cm (which is considered as a large thickness), whereas at C = 10–2 mol/l, Rd = 3·10–7 cm (which is considered as a very small thickness). The electrical double layer is formed from two parts; the ﬁrst part is the charged surface (usually negatively charged) with immobile ions, whereas the second part is the diffusive part. The electrical potential of the charged solid surface is referred to as the zeta potential (ζ). A characteristic value of the ζ potential is equal to RT/F = 25 mV, where R is the universal gas constant, T is the absolute temperature in °K, and F is the Faraday constant. The difference in mobility of free mobile ions in the diffusive part of electrical double layer and on the charged surface determines the electrokinetic phenomena, which are totally determined by properties of electrical double layer. Electrokinetic Phenomena Currently, a number of electrokinetic phenomena have been discovered and investigated. Only one of them is brieﬂy discussed as follows: the streaming potential. Let us consider the ﬂow of an electrolyte solution in a capillary with negatively charged walls (for example, a glass or quartz capillary). In the initial state, the feed solution and the receiving solution have equal concentrations of electrolyte. The electrolyte solution starts to ﬂow after a pressure difference is applied to both sides of the capillary. This ﬂow involves mobile cations in a electrical double layer near the solid negatively charged walls of the capillary into a convective motion, which is an electric current. As a result of the convective electric current, the concentration of cations increases in the receiving solution and the excess positive charges accumulate there. These excess charges cause the appearance of © 2007 by Taylor & Francis Group, LLC

Surface Forces and the Equilibrium of Liquids on Solid Substrates

19

an electric potential difference between the entrance and the end of the capillary, which generates the electric current in the direction opposite to the direction of the ﬂow. This electric current destroys the emerging surplus of cations in the outﬂowing solution. Electric potential difference appearing between the ends of capillary, in this case, is called a streaming potential. Let us note that the total electric current in the system is equal to zero, i.e., there is no electric current in the system, in spite of an electric potential difference between the ends of the capillary. The Electrostatic Component of the Disjoining Pressure Now we shall continue the examination of the next component of the disjoining pressure, the electrostatic component. Let us return to the examination of two charged surfaces (not necessarily of the same nature) in aqueous electrolyte solutions (Figure 1.9a and Figure 1.9b). The surfaces are assumed to have equal charges or opposite charges, i.e., there are electrical double layers near each of them. The sign of the charge of the diffusive part of each electrical double layer is opposite to the sign of the charge of the corresponding surface. If the width of clearance between surfaces is h >> Rd , the electrical double layers of surfaces do not overlap (Figure 1.9a), and there is no electrostatic interaction of surfaces. However, if the thickness of the clearance, h, is comparable with the thickness of the electrical double layer, Rd , then electrical double layers overlap, and it results in an interaction between –

Rd

+

– +

– +

– +

– – + +

– +

– – – + +

ζ1

h

(a)

–

+ (b) –

– +

+

+ –

– + + + –

+ –

+ –

+ –

– +

+ + –

+ + – –

–

–

– +

+ +

+ –

–

+ + – – –

+ –

– + + –

–

+

–

+ + – –

ζ2 ζ1 h ζ2

FIGURE 1.9 (a) ζ1 and ζ2 are negative. Distance between two negatively charged surfaces, h, is bigger than the thickness of the Debye layers, Rd. Electrical double layers do not overlap, and there is no electrostatic interaction between these surfaces; ζ1 and ζ2 are electrical potentials of charged surfaces. (b) ζ1 and ζ2 are negative. Distance between two negatively charged surfaces, h, is smaller or comparable with the thickness of the electrical double layer, Rd. Electrical double layers of both surfaces overlap, which results in an interaction that is repulsion, in the case under consideration. © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

Rd

+

–

+

+ –

–

+

+ –

–

+

+

–

–

+

+ –

–

+

h

(a)

–

+ (b)

ζ1

–

+

–

–

+ + –

+

+ – + + – – –

+ –

+ – + –

+ + – –

+ –

+

+ +

– –

+

+ + – – –

+ –

+

–

+

+

– + + – –

– + – –

ζ2 ζ1 h ζ2

FIGURE 1.10 (a) ζ1 > 0 and ζ2 < 0. Distance between two surfaces, baring the opposite charges, h, is bigger than the thickness of the Debye layers, Rd . Electrical double layers do not overlap, and there is no electrostatic interaction between these surfaces. ζ1 and ζ2 are electrical potentials of charged surfaces. (a) ζ1 > 0 and ζ2 < 0. Distance between two surfaces with opposite charges, h, is smaller or comparable with the thickness of the electrical double layer, Rd. Electrical double layers overlap, which results in an interaction that is attraction, in the case under consideration.

the surfaces. If the surfaces are equally charged, their diffusive layers are equally charged as well, i.e., the repulsion appears as a result of their overlapping (the electrostatic component of the disjoining pressure is positive in this case). If the surfaces have opposite charges, an attraction would ensue as a result of the overlapping of opposite charges. The electrostatic component of the disjoining pressure is negative in this case (Figure 1.10a and Figure 1.10b). There are a number of approximate expressions for the electrostatic component of the disjoining pressure [1]. For example, in the case of low ζ potentials of both surfaces, the following relation is valid [1]:

Πe ( h ) =

(

)

2 2 εκ 2 2ζ1ζ2 cosh κh − ζ1 + ζ2 , 8π sinh 2 κh

(1.12)

where ε is the dielectric constant of water and 1/κ = Rd , respectively. ζ potential is considered to be low if the corresponding dimensionless potential Fζ < 1. RT

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Note that in the case of oppositely charged surfaces and at relatively small distances, the following expression for the electrostatic component of the disjoining pressure is valid [1]:

(

ε ς1 − ς 2 Πe ( h ) = − 8π h2

)

2

,

(1.13)

which is always attraction. It is necessary to be very careful with the latter expression because in this case, the attraction can change to repulsion at a point beyond certain small critical distances [1]. Equation 1.12 and Equation 1.13 show that the disjoining pressure does not vanish even in cases when only one of the two surfaces is charged (for example, ζ1 = 0). The physical reason for this phenomenon is the deformation of the electrical double layer, if the distance between the surfaces is smaller than the Debye radius. The theory for the calculation of the disjoining pressure based on the two indicated components, i.e., dispersion, Πm(h), and electrostatic, Πe(h), is referred to as the DLVO theory. According to the DLVO theory, the total disjoining pressure is a sum of the two components, i.e., Π(h) = Πm(h) + Πe(h). The DLVO theory made possible the explanation of a range of experimental data on the stability of colloidal suspensions/emulsions as well as the static and the kinetics of wetting. However, it has been understood later that only these two components are insufﬁcient for explaining the phenomena in thin liquid ﬁlms, layers and in colloidal dispersions. There is a requirement of a third important component of disjoining pressure, which becomes equally important in aqueous electrolyte solutions. Structural Component of the Disjoining Pressure This component of disjoining pressure is caused by the orientation of water molecules in a vicinity of aqueous solution–solid interface or aqueous solution–air interface. Keep in mind that all water molecules can be modeled as an electric dipole. In the vicinity of a negatively charged interface, a positive part of water dipoles is attracted to the surface. That is, the negative part of dipoles are directed oppositely and the next set of water dipoles is facing a negatively charged part of dipoles, which in its turn, results in the orientation of the next layer of dipoles and so on. However, thermal ﬂuctuations try to destroy this orientation (Figure 1.11). As a result of these two opposite trends, there is a formation of a ﬁnite layer, where the structure of water dipoles differs from the completely random bulk structure. This layer is frequently referred to as the hydration layer. If we now have two interfaces with hydration layers close to each of them (or even one of them), then at a close separation, comparable with the thickness of the hydration

© 2007 by Taylor & Francis Group, LLC

22

Wetting and Spreading Dynamics

Hydration layer

FIGURE 1.11 Formation of a hydration layer of water dipoles in the vicinity of a negatively charged interface. The darker part of water dipoles is positively charged, whereas the lighter part is negatively charged.

layer, these surfaces “feel each other,” that is, hydration layers overlap. This overlapping results either in attraction or repulsion of these two surfaces. Unfortunately, until now, there is no ﬁrm theoretical background on the structural component of the disjoining pressure, and we are unable to deduce theoretically those cases in which the structure formation results in an attraction and those in which it results in a repulsion. As a consequence, only a semiempirical equation exists, which gives a dependence of the structural component of disjoining pressure on the thickness of the liquid ﬁlm [1]:

()

Π S h = Ke −λh ,

(1.14)

where K and λ are constants. There is a clear physical meaning of the parameter 1/λ, which is the correlation length of water molecules in aqueous solutions. This parameter further gives 1/λ ~ 10–15 Å, which is the characteristic thickness of the hydration layer. However, we are still far from a complete understanding of the preexponential factor K, which can be extracted on the current stage only from experimental measurements of the disjoining pressure. Currently, it is assumed [1] that the disjoining pressure of thin aqueous ﬁlms is equal to the sum of the three components

()

()

()

()

Π h = Πm h + Πe h + Πs h .

(1.15)

In Figure 1.12, the dependences of the disjoining pressure on the thickness of a ﬂat liquid ﬁlm are presented for the cases of the complete wetting (curve 1 that corresponds to a dispersion or molecular component of disjoining pressure, Πm(h)) and partial wetting (curve 2 that corresponds to a sum of all three components of the disjoining pressure, according to Equation 1.15). Disjoining pressure presented by curve 1 in Figure 1.12 corresponds to a case of complete wetting, for © 2007 by Taylor & Francis Group, LLC

Surface Forces and the Equilibrium of Liquids on Solid Substrates

23

Π 1

h

2

3

FIGURE 1.12 Types of isotherms of disjoining pressure, which are under consideration below: (1) complete wetting, observed for oil ﬁlms on quartz, glass, metal surfaces [1]; (2) partial wetting, observed for aqueous ﬁlms on quartz, glass, metal surfaces [1]; (3) nonwetting case.

example, oil droplets on glass substrate, whereas curve 2 corresponds to the case of partial wetting, for example, aqueous electrolyte solutions on glass substrates. In Reference 1 to 4, a number of experimental data on measurement of disjoining pressure are presented. The dependency (Equation 1.11) has been ﬁrmly conﬁrmed in the case of oil thin ﬁlms on glass, quartz, and metal surfaces, which corresponds to the case of complete wetting. In Figure 1.13, experimental data and calculations according to Equation 1.15 for aqueous thin ﬁlms are presented. In Reference 1, all necessary details concerning experimental data presented in Figure 1.13 are given.

1.3 STATIC HYSTERESIS OF CONTACT ANGLE The previous consideration shows that the situation with Young’s equation (Equation 1.1) is far more difﬁcult than it is usually assumed. This equation is supposed to describe the equilibrium contact angle. We explained in Section 1.1 that the latter equation does not comply with any of the three requirements of the equilibrium: liquid–vapor equilibrium, liquid–solid equilibrium, and vapor–solid equilibrium. However, there is a phenomenon that is far more important than the previous ones from a practical point of view. It is called the static hysteresis of contact angle. The derivation of Equation 1.1 and further considerations show that the given equation (or its modiﬁcations) determines only one unique equilibrium contact © 2007 by Taylor & Francis Group, LLC

24

Wetting and Spreading Dynamics

4

Π-10–3, dyn/cm2

2

4

α

β

0

1000

2000

h, Å

–2 1

2

3

(a)

2

Π . 10–5, dyn/cm2

2 1

0

200

4

400

t0

1

h, Å

–1

–2

3 (b)

FIGURE 1.13 Calculated and experimentally-measured isotherms of disjoining pressure, Π(h), of the ﬁlms of water on a quartz surface at concentration of KCl: C = 10–5 mol/l, pH = 7, and dimensionless ς potential of the quartz surface equals to 6 [1]. (a) Within the region of large thicknesses: dimensionless ς potential of the ﬁlm–air interface equals to 2.2 (curve 1), 1 (curve 2), and 0 (curve 3); (b) within the region of small thicknesses: dimensionless ς potential of the ﬁlm–air interface equals to 2.2 (curve 1). The structural component, ΠS(h), of the disjoining pressure isotherm and electrostatic component, Πe(h), are indicated by curves 2 and 3, respectively. Curves 4 in both part (a) and (b) are calculated according to Equation 1.13.

© 2007 by Taylor & Francis Group, LLC

Surface Forces and the Equilibrium of Liquids on Solid Substrates

θ

1

25

R

2

3

FIGURE 1.14 Schematic presentation of a liquid droplet on a horizontal solid substrate, which is slowly pumped through the liquid source in the drop center. R is the radius of the drop base; θ is the contact angle; (1) liquid drop, (2) solid substrate with a small oriﬁce in the center, (3) liquid source (syringe).

angle. Static hysteresis of contact angle results in an inﬁnite number of equilibrium contact angles of the drop on the solid surface, not the unique contact angle, θe, but the whole range of contact angles, θr < θe < θa, where θr and θa are the corresponding static receding and advancing contact angles. The meaning of static advancing and receding contact angles is apparent when we consider a liquid droplet on a horizontal substrate that is slowly being pumped through an oriﬁce in the solid substrate (Figure 1.14). Let us assume that, in some way, an initial contact angle of the droplet was equal to the equilibrium one. When we carefully and slowly pump the liquid through an oriﬁce in the center, the contact angle will grow. However, the radius of the drop base will not change until a critical value of the contact angle, θa, is reached. Further pumping will result in spreading of the drop. If we start from the same equilibrium contact angle and then pump out the liquid through the same oriﬁce, the contact angle will decrease further, but the droplet will not shrink until the critical contact angle, θr , is reached. After that, the droplet will start to recede. For example, in the case of water droplets on a smooth homogeneous glass surface that is specially treated for purity, θr ~ 0°–5°, whereas θa is in the range of 40°–60°. It is usually believed that the static hysteresis of contact angle is determined by the surface roughness and/or heterogeneity (Figure 1.15). Figure 1.15b presents the magniﬁed vicinity of the three-phase contact line of the same droplet as in Figure 1.15a. This picture gives a qualitative explanation of the phenomenon of the static hysteresis of contact angle, which is widely adopted in the literature. The static hysteresis of contact angle is connected with multiple equilibrium positions on the drop edge on a rough surface. No doubt

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26

Wetting and Spreading Dynamics

θef (a)

θi

θef

Solid (b) θ θef

Solid (c)

FIGURE 1.15 (a) Droplet on a solid substrate with a small roughness, which is invisible to a naked eye, (b) magniﬁcation of the apparent three-phase contact line, (c) magniﬁcation on the apparent three-phase contact line with the rough surface covered by the liquid ﬁlm that ﬂows out from the droplet. An arrow shows the zone where a microscopic motion occurs.

that a roughness and/or a chemical heterogeneity of the solid substrate contribute substantially to the contact angle hysteresis. As already mentioned, the static hysteresis of the contact angle is usually related to the heterogeneity of the surface, either geometric (roughness) [6,7] or chemical [8]. In this case, it is assumed that at each point of the surface the equilibrium value of the contact angle of that point is established, depending only on the local properties of the substrate. As a result, a whole series of local thermodynamic equilibrium states can be realized, corresponding to a certain interval of values of the angle. The maximum value corresponds to the value of the advancing contact angle, θa, and the minimum value corresponds to the receding contact angle, θr . According to such a model, the dependency of contact angle on velocity of motion should be as presented in Figure 1.16. There is no doubt that heterogeneity affects the wetting process. However, heterogeneity of the surface is apparently not the sole reason for hysteresis of the contact angle. This follows from the fact that not all predictions made on the © 2007 by Taylor & Francis Group, LLC

Surface Forces and the Equilibrium of Liquids on Solid Substrates

27

θa

θr v

FIGURE 1.16 Dependency of the contact angle on the velocity of advancing (v > 0) or receding (v < 0) meniscus.

basis of this theory have turned out to be true [10,11]. Besides, hysteresis has been observed in cases of quite smooth and uniform surfaces [12–17]. Further, the static hysteresis of contact angle is present even on surfaces that are deﬁnitely free liquid ﬁlms [17–18]. Now we recall that in the vicinity of the apparent three-phase contact line, surface forces (disjoining pressure) disturb the liquid proﬁle substantially, and the picture presented in Figure 1.15b is impossible. Immediately after the droplet is deposited, the disjoining pressure comes into play. This pressure results in a coverage of the substrate in the vicinity of the apparent three-phase contact line by a thin liquid ﬁlm. This would mean that the liquid edge is always in contact with an already wetted, rough solid substrate. A more realistic picture is depicted in Figure 1.15c, which describes the situation more adequately in the vicinity of the apparent three-phase contact line. Equilibrium and hysteresis contact angles on rough surfaces have never been considered from this point of view before and are the subject of future investigations. These considerations would suggest that the picture presented in Figure 1.16 cannot be realized either on smooth or rough substrate. This is the reason why we consider the static hysteresis on completely smooth substrates. In earlier studies [5], a completely new concept of hysteresis of contact angle on smooth homogeneous substrates has been suggested. This mechanism will be discussed in Section 3.10. In the following, we give a qualitative description of this phenomenon. The picture presented in Figure 1.16 is in contradiction with the thermodynamics, which requires a unique equilibrium contact angle, θe , on smooth homogeneous substrates. The latter means that at any contact angle, θ, in the range θr < θ < θa and different from the equilibrium one, the liquid droplet cannot be at the equilibrium but in the state of a very slow “microscopic” motion. More detailed observations and theoretical considerations show (see Section 3.10) that at any contact angle different from the equilibrium one, θe , the liquid droplet is in a state of slow microscopic motion, which is located in the tiny vicinity of the apparent three-phase contact line. The microscopic motion abruptly becomes “macroscopic” after the critical contact angles θa or θr are reached. This observation shows that the dependency presented in Figure 1.16 should be replaced by a more complicated but realistic dependency as shown in Figure 1.17. © 2007 by Taylor & Francis Group, LLC

28

Wetting and Spreading Dynamics

θa θe

θr v

FIGURE 1.17 At any deviation from the equilibrium contact angle θe, the liquid drop is in the state of a slow microscopic motion, which abruptly transforms into a state of macroscopic motion after critical contact angles θa or θr are reached.

The presence of the contact angle hysteresis indicates that the actual equilibrium contact angle is very difﬁcult to obtain experimentally even if we neglect the equilibrium with vapor and solid substrate. Static Hysteresis of Contact Angles from Microscopic Point of View: Surface Forces At this stage, we are capable of explaining the nature of the hysteresis of contact angles via the S-shape of the isotherm of disjoining pressure (curve 2 in Figure 1.12) in the case of partial wetting. More details are given in Section 3.10. First of all, we recall what the hysteresis of contact angle in capillaries means. Let us consider a meniscus in the case of partial wetting in a capillary (Figure 1.18a and Figure 1.18b). Note that the capillary is in contact with a reservoir where the pressure, Pa – Pe, in the reservoir is lower than the atmospheric pressure, Pa . If we increase the pressure under the meniscus, then the meniscus does not move but changes its curvature to compensate for the excess pressure and, as a consequence, the contact angle increases accordingly. The meniscus does not move until a critical pressure and critical contact angle, θa, are reached. After further increase in pressure, the meniscus starts to advance. A similar phenomenon takes place if we decrease the pressure under the meniscus; it does not recede until a critical pressure and corresponding critical contact angle, θr , are reached. This indicates that in the whole range of contact angles, θr < θ < θa, the meniscus does not move macroscopically. It is obvious that on the smooth homogeneous solid substrate only one contact angle corresponds to the equilibrium position, and all the rest do not. Based on that idea, in Figure 1.17 we present a dependency of the contact angle on the velocity of motion, which shows that all contact angles, θ, in the range, θr < θ < θa, correspond to a slow microscopic advancing or receding of the meniscus. This microscopic motion abruptly changes to macroscopic as soon as θr or θa are reached. Explanation of the dependence presented in Figure 1.18 is based on the S-shaped isotherm of disjoining pressure in the case of partial wetting. This shape © 2007 by Taylor & Francis Group, LLC

Surface Forces and the Equilibrium of Liquids on Solid Substrates

1

ρa

1

θa

ρr

2

2

4 3 (a)

29

θr

4 3

(b)

FIGURE 1.18 Hysteresis of contact angle in capillaries in the case of partial wetting (Sshaped isotherm of disjoining pressure). (a) Advancing contact angle. (1) a spherical meniscus of radius ρa, (2) transition zone with a point dangerous marked (see explanation in the text), (3) zone of ﬂow, (4) ﬂat ﬁlms. Close to the marked point, a dashed line shows the proﬁle of the transition zone just after the contact angle reaches the critical value θa, which indicates a beginning of the caterpillar motion. (b) Receding contact angle. (1) a spherical meniscus of radius ρr < ρa, (2) transition zone with a point dangerous marked (see explanation in the text), (3) zone of ﬂow, (4) ﬂat ﬁlms. Close to the marked point, a dashed line shows the proﬁle of the transition zone just after the contact angle reaches the critical value θa.

determines a very special shape of the transition zone in the case of the equilibrium meniscus (see Section 2.3). In the case of pressure increases behind the meniscus (Figure 1.18a), a detailed consideration (Section 3.10) of the transition zone indicates that close to a “dangerous” point marked in Figure 1.18a, the slope of the proﬁle becomes steeper with increasing pressure. In the range of very thin ﬁlms (region 3 in Figure 1.18a), there is a zone of ﬂow. Viscous resistance in this region is very high, hence the very slow advancement of the meniscus. After a certain critical pressure behind the meniscus is reached, the slope at the dangerous point reaches π/2, and the ﬂow proceeds stepwise, occupying the region of thick ﬁlms. Thus, the fast “caterpillar” motion begins, as shown in Figure1.18a. In the case of pressure decreases behind the meniscus, the event proceeds according to Figure 1.18b. Again, up to a certain critical pressure, the slope in the transition zone close to the point marked dangerous becomes more and more ﬂat. In the range of very thin ﬁlms (region 3 in Figure 1.18b), there is a zone of ﬂow. Viscous resistance in this region again is very high. This is why the receding of the meniscus proceeds in a very slow manner. After the attainment of the critical pressure behind the meniscus, the proﬁle in the vicinity of the dangerous point shows a discontinuous behavior, which is obviously impossible. That means the meniscus will start to slide along a thick β-ﬁlm, moving relatively fast and leaving behind the thick β-ﬁlm. The latter phenomenon (the presence of a thick β-ﬁlm behind the receding meniscus of aqueous solutions in quartz capillaries) has been discovered experimentally (see discussion in Section 3.10). This discovery supports our arguments explaining static contact angle hysteresis on smooth homogeneous substrates. © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

REFERENCES 1. Deryaguin, B.V., Churaev, N.V., and Muller, V.M., Surface Forces, Consultants Bureau, Plenum Press, New York, 1987. 2. Russel, W.B., Saville, D.A., and Schowalter, W.R., Colloidal Dispersions, Cambridge University Press, Cambridge, U.K., 1999. 3. Exerowa, D. and Kruglyakov, P., Foam and Foam Films: Theory, Experiment, Application, Elsevier, New York, 1998. 4. Israelashvili, J.N., Intermolecular and Surface Forces, Academic Press, London, 1991. 5. Starov, V.M., Adv. Colloid Interface Sci., 39, 147, 1992. 6. Wenzel, R., Ind. Eng. Chem., 28, 988, 1936. 7. Deryagin, B.V., Dokl. Akad. Nauk SSSR [in Russian], 51, 357, 1946. 8. Johnson, R.E. and Dettre, R.H., Surface and Colloid Science, Vol. 2, Wiley, New York, 1969, p. 85. 9. Ivanov, I.B. and Kralchevsky, P.A., In “Thin liquid ﬁlms. Fundmentals and Applications.” Ivanov, I.B. (ed.). Surfactant Science Series, Marcel Dekker Inc., New York and Basel, v. 29 (1988). 10. Schwartz, A.M., Racier, C.A., and Huey, E., Adv. Chem. Ser., 43, 250, 1964. 11. Neumann, A.W., Renzow, D., Renmuth, H., and Richter, I.E., Fortsch. Ber. Kolloide Polym., 55, 49, 1971. 12. Holland, L., The Properties of Glass Surfaces, London, 1964, p. 364. 13. Zorin, Z.M., Sobolev, V.D., and Churaev, N.V., Surface Forces in Thin Films and Disperse Systems [in Russian], Nauka, Moscow, 1972, p. 214. 14. Romanov, E.A., Kokorev, D.T., and Churaev, N.V., Int. J. Heat Mass Transfer, 16, 549, 1973. 15. Neumann, A.W., Z. Phys. Chem. (Frankfurt), 41, 339, 1964. 16. Zheleznyi, B.V., Dokl. Akad. Nauk SSSR [in Russian], 207, 647, 1972. 17. Platikanov, D., Nedyalkov, M., and Petkova, V. Advances in Colloid and Interface Science, Vol. 100–102, 2003, pp. 185–203. 18. Petkova, V., Platikanov, D., and Nedyalkov, M., Adv. Colloid and Interface Sci., 104, 37, 2003.

© 2007 by Taylor & Francis Group, LLC

2

Equilibrium Wetting Phenomena

INTRODUCTION In this chapter, we shall discuss equilibrium liquid shapes on solid substrates, which demands equilibrium of liquid–vapor, liquid–solid, and vapor–solid. Not always do authors take into account all the three equilibria. The vapor–solid equilibrium determines the presence of adsorbed liquid layers on solid surfaces for both complete and partial wetting. Even at this moment, we and everything around are covered by a thin water ﬁlm. The thickness of aqueous ﬁlms depends on the humidity in the room, and the adsorption is exactly at equilibrium with the surrounding humidity, no matter how low or high. The presence of liquid layers on solid substrates is determined by the action of surface forces (the disjoining pressure), which was discussed in Chapter 1. The disjoining pressure isotherm is normally dealt with because it is usually measured at a constant temperature. In the case of water and aqueous solutions, the disjoining pressure is S-shaped. Water and aqueous solutions are crucial for life. Does the peculiar shape of the disjoining pressure isotherm of water and aqueous solutions in some way determine our existence? It is well known that all properties of water and aqueous solutions are vitally important for life. This means that the peculiar shape of disjoining pressure isotherms of water and aqueous solutions, in some unknown way, determines the existence of our life. At the moment we do not know how the process works. Further in this chapter, we investigate the inﬂuence of the combined action of disjoining pressure and capillary forces on the equilibrium shapes of liquids on solid substrates.

2.1 THIN LIQUID FILMS ON FLAT SOLID SUBSTRATES In this section, we shall consider the properties and stability of liquid ﬁlms on solid substrates under partial or complete wetting conditions. We shall account for the disjoining pressure action alternatively with the action of surface forces. As discussed in Chapter 1, the adsorption of liquid on solid substrates is a manifestation of the action of surface forces. But before we start, let us recall that partially or completely wetted solid surfaces, at equilibrium, are always covered by a liquid ﬁlm that is at equilibrium with the vapor pressure, p, of the surrounding air. The free energy of such a solid covered substrate is lower than the free energy of the corresponding bare solid substrate. Hence, in all cases here 31 © 2007 by Taylor & Francis Group, LLC

32

Wetting and Spreading Dynamics

to be considered, there is no real three-phase contact line at equilibrium because the whole solid surface is covered by a ﬂat equilibrium liquid ﬁlm (on occasion we shall mention the apparent contact line). In this section, we consider properties and stability of liquid ﬁlms in the case of both partial and complete wetting. The two terms that are equally used are: disjoining pressure action and the action of surface forces. As we already discussed in Chapter 1, the adsorption of liquid on solid substrates is a manifestation the action of surface forces. This means that the latter forces must be taken into account if we are to consider equilibrium states of liquid ﬁlms on solid substrates. We also noted that, in all cases under consideration, there is no real three-phase contact line at the equilibrium because the whole solid surface is covered by ﬂat equilibrium liquid ﬁlm. The excess free energy per unit area of a ﬂat equilibrium liquid ﬁlm of thickness he on a solid substrate at equilibrium with the vapor in the surrounding air is: Φ/S = γ + Pe he + fD (he ) + γ sl − γ sv ,

(2.1)

where S is the surface covered by the liquid ﬁlm, and fD (he) is the potential of surface forces; γ, γsl, and γsv are liquid–air, solid–liquid, and liquid–vapor interfacial tensions, respectively; the excess pressure Pe = Pa – Pl , where Pl is the pressure inside the liquid ﬁlm, and Pa is the pressure in the ambient air. Note that, according to the spontaneous adsorption of liquid molecules in partial or complete wetting cases, the latter excess free energy should be negative; otherwise the liquid molecules would not adsorb at all. Owing to the equilibrium of the liquid ﬁlm with the vapor, the excess pressure, Pe, cannot be left as an arbitrary constant; it is determined by the equality of chemical potentials of liquid molecules in the ﬁlm and in the vapor. This requirement results in the well-known Kelvin’s equation: Pe =

RT p ln s , vm p

(2.2)

where R, T, and vm are the universal gas constants, the absolute temperature, and the liquid molar volume, respectively; ps and p correspond to the pressures of the saturated vapor and the vapor at which the liquid ﬁlm is at equilibrium. The latter expression shows that the excess pressure, Pe, cannot be ﬁxed arbitrarily but is determined by the vapor pressure in the ambient air, p. It must be noted that Equation 2.2 expresses the equality of chemical potentials of water molecules in vapor and liquid phases. The excess free energy, according to Equation 2.1, is a function of the variable, he, which is the thickness of the equilibrium ﬁlm. Hence, the usual conditions of thermodynamic equilibrium should hold, which give a minimum value to the excess free energy. Those conditions are: © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

33

dΦ = 0, dhe

d2Φ > 0. dhe2

The ﬁrst requirement results in Pe = Π( he ) ,

(2.3)

d Π( he ) < 0, dhe

(2.4)

and the second requirement yields

where Π( h ) = −

df D ( h ) dh

is referred to as the disjoining pressure [1]. The disjoining pressure, Π(h), is the physical property that can be experimentally measured. That is, the consideration that follows is based on the consideration of the disjoining pressure. Using the previous deﬁnition, we can rewrite the excess free energy fD (h) as: ∞

f D (h) =

∫ Π(h)dh . h

Equation 2.3 determines the thickness of the equilibrium liquid ﬁlm, he, via disjoining pressure isotherm. Equation 2.4 gives the well-known stability condition of ﬂat equilibrium liquid ﬁlms [1]. According to the stability condition (2.4), all ﬂat equilibrium ﬁlms are stable in the case of complete wetting (curve 1, Figure 2.1), and only ﬁlms in the range of thickness 0 to tmin (these ﬁlms are referred to in the following section as α-films, which are absolutely stable) and at h > tmax (the latter ﬁlms are referred to as β-films, and it is shown that they are metastable) in the case of partial wetting (curve 2 in Figure 2.1) are stable. Hence, only those α- and β-ﬁlms can exist as ﬂat ﬁlms. Note again that the S-shaped disjoining pressure isotherms (curve 2 in Figure 2.1) are characteristic shapes in the case of water and aqueous solutions. All properties of water and aqueous solutions are vitally important for life. The latter means that the peculiar shape of disjoining pressure of water and aqueous solutions presented in Figure 2.1 in some way determines the existence of life. At the moment we do not know how, but the peculiar shape of curve 2 in Figure 2.1 does tell us something about the process. © 2007 by Taylor & Francis Group, LLC

34

Wetting and Spreading Dynamics Π 1

Πmax

tmin t0

S+ tmax

S–

h

2 –Πmin

FIGURE 2.1 Two types of isotherms of disjoining pressure, which are under consideration below: 1 — complete wetting, 2 — partial wetting. Isotherms for partial wetting are observed for water ﬁlms on almost all surfaces, for example, on quartz, glass, and metal surfaces [2,3]. Isotherms of type 1 are observed in a number of cases of complete wetting, for example, at oil ﬁlms on quartz, glass, and metal surfaces [4].

We can rewrite the expression for the excess free energy of the ﬁlm Equation 2.1 using the disjoining pressure in the following way: ∞

∫

Φ /S = γ + Pe he + Π(h)dh + γ sl − γ sv .

(2.5)

he

The latter expression gives the excess free energy via a measurable physical dependency, Π(h), which is the disjoining pressure isotherm. Now we can rewrite expression (2.5), of the excess free energy of thin liquid ﬁlms as Φ/S = γ svhe − γ sv ,

(2.6)

where ∞

γ svhe = γ + Pe he +

∫ Π(h)dh + γ

sl

(2.7)

he

is the “interfacial tension” (actually the excess free energy) of the solid substrate covered with the liquid ﬁlm of thickness he. The preceding expression determines the unknown value of γsvhe in Young’s equation 1.8 in Chapter 1, Section 1.1: © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

35

cos θe =

γ svh − γ sl . γ

(2.8)

Combination of Equation 2.7 and Equation 2.8 results in ∞

γ + Pe he + cos θe =

∫ Π(h)dh

he

γ

.

(2.9)

The latter equation is the well-known Derjaguin–Frumkin equation for the equilibrium contact angle, which has been deduced using a different thermodynamic consideration [1]. Equation 2.9 is a very important equation and is deduced in Section 2.3 in a different way. Because –1 < cos θe < 1, using Equation 2.9 we conclude that the integral on the right-hand side should be negative. This requirement is satisﬁed in the case of partial wetting (see curve 2 in Figure 2.1). ∞

∫ Π(h)dh < 0 .

(2.10)

he

The latter inequality is satisﬁed if S− > S+ , (see Figure 2.1).

(2.11)

In the case of complete wetting, the right hand side in Equation 2.9 is always positive, that is, equilibrium droplets cannot exist on the solid substrate either under oversaturation or undersaturation; they spread out completely and evaporate. However, equilibrium menisci (at undersaturation) can exist in capillaries. That is, the behavior of droplets and menisci in the case of complete wetting is completely different. Using Equation 2.9, we can rewrite the expression for the excess free energy of a ﬂat liquid ﬁlm in Equation 2.1 as

(

)

Φ/S = γ cos θe + γ sl − γ sv = γ cos θe − cos θ NY .

(2.12)

According to spontaneous adsorption of liquid molecules on solid substrates, in the case of partial and complete wetting, the right-hand side value corresponding to the excess free energy (Equation 2.12) is negative, and hence,

© 2007 by Taylor & Francis Group, LLC

36

Wetting and Spreading Dynamics

cos θe − cos θ NY < 0, θe > θ NY .

(2.13)

That is, even if γsl and γsv are measured, the contact angle according to the original Young’s equation, cos θ NY =

γ sv − γ sl , γ

(2.14)

is smaller than the real equilibrium contact angle. Frequently, a contact angle determined according to Equation 2.14 is identiﬁed with a static advancing contact angle, θa. It is obvious that the static advancing contact angle, θa, is bigger than the equilibrium contact angle, θa > θe. If we now compare this inequality with Equation 2.13, we can conclude that there is no justiﬁcation for the identiﬁcation of θNY and θa because θNY < θe < θa.

EQUILIBRIUM DROPLETS ON THE SOLID SUBSTRATE OVERSATURATION (Pe < 0)

UNDER

As we already noticed, the excess pressure, Pe, is negative at oversaturation according to Equation 2.2. The equilibrium ﬁlm or ﬁlms are determined according to Equation 2.3 at both undersaturation and oversaturation. Figure 2.2 shows that, in the case of complete wetting, there are no ﬂat equilibrium ﬁlms on solid substrates under oversaturation because the line, Pe < 0, does not intersect (curve 1 in Figure 2.2). Hence, there are also no equilibrium droplets on completely wettable solids at oversaturation; they are in the surrounding air. Π

t0

he

tmin

hu tmax

h

Pe

–Πmin

FIGURE 2.2 Two equilibrium ﬂat ﬁlms on solid substrates under oversaturation: stable ﬁlm of thickness he and unstable ﬁlm of thickness hu. © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

37

However, in the case of partial wetting, Equation 2.3 has two solutions (Figure 2.2). According to the stability condition of ﬂat ﬁlms in Equation 2.4, one of them corresponds to the stable equilibrium ﬁlm of thickness he, and the second one corresponds to the unstable ﬁlm of thickness hu (Figure 2.2). This would suggest that equilibrium droplets in the case of partial wetting are “sitting” on the stable equilibrium ﬁlm of thickness he. However, even in the case of partial wetting, equilibrium droplets can exist on the solid substrate only in a limited interval of oversaturation, which is determined by 0 < Pe < –Πmin (Figure 2.2) or using Equation 2.2 in the following range of oversaturated pressure, p, over the solid substrate

1<

v Π  p < exp  m min  . ps  RT 

(2.15)

If Πmin is in the range 106–107 dyn/cm2, then the latter inequality takes the following form: 1<

p v Π < 1 + m min ≈ 1.001 − 1.01; ps RT

that is, the equilibrium droplets in the case of partial wetting exist only in a very limited interval of oversaturation on the solid substrates. Beyond this interval, at higher oversaturation, neither equilibrium liquid ﬁlms nor droplets exist on the solid substrate as in the case of complete wetting. Probably, the critical oversaturation pcr v Π  pcr = exp  m min  , ps  RT  determined from Equation 2.15, corresponds to the beginning of homogeneous nucleation, and at higher oversaturations, homogeneous nucleation is more favorable. Let ℜ be the radius of the equilibrium droplet. According to the deﬁnition of the capillary pressure, Pe = −

2γ . ℜ

Hence, the radius of equilibrium drops is ℜ= © 2007 by Taylor & Francis Group, LLC

2γ . − Pe

38

Wetting and Spreading Dynamics

In the above-mentioned narrow interval of oversaturation, the radius of the equilibrium drops changes from inﬁnity at p → ps to ℜcr =

2γ Πmin

at p = pcr . If Πmin ≈ 106 dyn/cm2 and γ ≈ 72 dyn, then ℜcr ≈

144 = 1.44 µm. 10 6

that is, the critical size is out of the range of the action of surface forces and the droplet size is sufﬁciently big. However, if Πmin ≈ 107 dyn/cm2, then ℜcr ≈

144 = 0.144 µm = 1440 A 7 10

and the whole droplet is in the range of the action of surface forces. In this case, the drop is so small that it does not have anywhere (even on the very top) a spherical part that is undisturbed by surface forces.

FLAT FILMS

AT THE

EQUILIBRIUM

WITH

MENISCI (Pe > 0)

Equation 2.3 and Figure 2.3 show that, in the case of complete wetting, there is only one equilibrium ﬂat ﬁlm, hc, which is stable according to the stability condition (2.4). Π

1 Πmax Pe

he

tmin

hu

hβ

hc

h

2

FIGURE 2.3 Disjoining pressure isotherm in the case of complete wetting (1), and partial wetting (2). In thick capillaries (H > γ/Πmax), there are three solutions of Equation 2.3. © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

39

In the case of partial wetting, (Figure 2.3) and Equation 2.3 show different solutions in the case of Pe > Πmax and Pe < Πmax. If Pe > Πmax, Equation 2.3 has only one solution that is stable (according to the stability condition 2.4) and is referred to as α-film. In the second case, Pe < Πmax (Figure 2.3) and Equation 2.3 show three solutions, one of which corresponds to the stable equilibrium α-ﬁlm with thickness, he. The second solution of Equation 2.3, hu , is unstable according to the stability condition in Equation 2.4, and the third solution, hβ , is stable again according to the same stability condition in Equation 2.4. The latter ﬁlms are referred to as β-films. Note that the thickness of an equilibrium ﬁlm in the case of complete wetting, hc, is bigger than the thickness of β- ﬁlm, hβ, in the case of partial wetting. Let us compare the excess free energy of ﬂat α- and β-ﬁlms, he and hβ. According to the deﬁnition, this difference is equal to

(

hβ

) ∫

∆ αβ =  Φ(hβ ) − Φ(hα )  S = Pe hβ − he − Π(h)dh .

(2.16)

he

The difference (hβ – he) is always positive (Figure 2.3) in the case of partial wetting, S > S– , according to Equation 2.11. Hence, the integral on the right-hand side of Equation 2.16 is negative. Hence, the excess free energy of β-ﬁlms is higher than the excess free energy of α-ﬁlms. This means that β-ﬁlms are less stable than α-ﬁlms, and that is why β-ﬁlms are referred to as metastable ﬁlms, and α-ﬁlms as absolutely stable films. It is necessary to make additional comments on α-ﬁlms and β-ﬁlms in the case of partial wetting. If we increase the vapor pressure over partially wettable surfaces from p = 0 to the saturation pressure, ps, then we can observe the formation of only α-ﬁlms on the solid substrate. The thickness of these ﬁlms changes correspondingly (according to Equation 2.3 and Figure 2. 3) from zero at p = 0 to t0 ≈ 70 Å [1]. However, β-ﬁlms cannot be obtained in the course of the adsorption process; they can be obtained only by decreasing the thickness of very thick ﬁlms down to the equilibrium thickness of the β-ﬁlm. This is why α-ﬁlms are referred to as adsorption films (because they can be obtained in the course of adsorption), and β-ﬁlms are referred to as wetting films. Let ρ be the radius of the curvature of a meniscus in a ﬂat capillary (a meniscus between two parallel plates). According to the deﬁnition of the capillary pressure, Pe =

γ . ρ

Let us introduce ρmax = © 2007 by Taylor & Francis Group, LLC

γ Πmax

40

Wetting and Spreading Dynamics

(Figure 2.3), and consider Pe > Πmax (Figure 2.3). We deﬁne a capillary as a “thin” capillary if ρ < ρmax. In such capillaries, only thin α-ﬁlms can be at equilibrium with the meniscus, and equilibrium β-ﬁlms do not exist in such thin capillaries. However, if the capillary is “thick,” that is, ρ > ρmax, then in such capillaries, both α- and β-ﬁlms can be at equilibrium with the meniscus. However, β-ﬁlms are metastable. If we adopt γ ~ 70 dyn/cm and Πmax ~ 104 dyn/cm2 for estimations, then ρmax ~ 7⋅10–3 cm.

S-SHAPED ISOTHERMS CASE S– < S+

OF

DISJOINING PRESSURE

IN THE

SPECIAL

Let us consider the case when the disjoining pressure isotherm is S-shaped as in Figure 2.3, curve 2. However, let us assume that ∞

∫ Π(h)dh > 0 ,

he

that is, S– < S+ (Figure 2. 3). In this case, from Equation 2.16 we conclude: ∆ αβ

Pe = 0

= S− − S+ < 0.

The latter means that at low Pe (or high humidity), β-ﬁlms are more stable than α-ﬁlms. It is easy to check using Equation 2.16 that ∆ αβ is an increasing function of Pe because d ∆ αβ ( Pe ) > 0. dPe Hence, ∆ αβ can become positive at some value of Pe, and after that, thick β-ﬁlms become less stable than thin α-ﬁlms. This instability occurs if ∆ αβ ( Pe )

Pe = Πmax

> 0.

In this case, if Pe increases from zero (where thick β-ﬁlms are more stable than thin α-ﬁlms), it reaches a critical value Pcr , such as ∆ αβ < 0 at 0 < Pe < Pcr , and ∆ αβ > 0 at Pcr < Pe < Πmax. This would indicate that in the range 0 < Pe < Πcr , thick β-ﬁlms are more stable than thin α-ﬁlms; however, at Pcr < Pe < Πmax, α-ﬁlms become more stable than β-ﬁlms. This consideration shows that a cycle presented in Figure 2.4 with a spontaneous and reversible transition from α-ﬁlms © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

41

Π

Πmax C Pcr

B

A

D tmin

tmax

h

−Πmin

FIGURE 2.4 S-shaped disjoining pressure isotherm with S– < S+. Reversible transition from α- to β-ﬁlms along DA at Pe < Pcr and from β- to α-ﬁlms along BC at Pe > Pcr .

to β-ﬁlms (along DA) should take place with a decrease in Pe (along CD). Also, a spontaneous reversible transition from β-ﬁlms to α-ﬁlms (along BC) should take place with an increase in Pe (along AB). Such spontaneous reversible transitions have been discovered by Exerowa et al. [5–8]. In their experiments, the disjoining pressure isotherm was S-shaped, but the minimum value of the disjoining pressure isotherm was positive, which means that the condition ∞

∫ Π(h)dh > 0 ,

he

that is, S– < S+, was satisﬁed.

2.2 NONFLAT EQUILIBRIUM LIQUID SHAPES ON FLAT SURFACES In thin ﬂat liquid ﬁlms (oil and aqueous thin ﬁlms, thin ﬁlms of aqueous electrolyte and surfactant solutions, and both free ﬁlms and ﬁlms on solid substrates), the disjoining pressure acts alone and determines their thickness. However, if the ﬁlm surface is curved or uneven, both the disjoining and the capillary pressures act together. In the case of partial wetting, their simultaneous action is expected to yield nonﬂat equilibrium shapes. For instance, due to the S-shaped disjoining pressure isotherm, microdrops, microdepressions, and equilibrium periodic ﬁlms could exist on ﬂat solid substrates. We shall establish a criteria for both existence and stability of such nonﬂat equilibrium liquid shapes. On the other hand, we © 2007 by Taylor & Francis Group, LLC

42

Wetting and Spreading Dynamics

shall see that a transition from thick ﬁlms to thinner ﬁlms can proceed transitorily via nonﬂat states with microdepressions and wavy shapes, both of which can be more stable than ﬂat ﬁlms in some range of hydrostatic pressures [30]. All mentioned nonﬂat equilibrium shapes on ﬂat solid substrates are to be discovered experimentally. The equilibrium contact angle of either an equilibrium drop or a meniscus in the capillary can be expressed via disjoining pressure isotherm (see Section 2.3, Figure 2.3):

cos θe =

1 1+ γ

∞

∫ Π(h)dh

he

1−

he H

1 ≈ 1+ γ

∞

∫ Π(h)dh ≈ 1 −

he

S− − S+ , γ

(2.17)

where Pe is the excess pressure (positive in the case of the meniscus and negative in the case of drops), he is the equilibrium of an absolutely stable α-ﬁlm (Figure 2.3), H is the radius of the capillary in the case of meniscus and the maximum height in the case of drops. Equation 2.17 shows that the partial wetting case corresponds to S– > S+, that is, S-shaped isotherm 2 in Figure 2.3. Equation 2.17 also shows that the equilibrium contact angle is completely determined by the shape of the disjoining pressure in the case of molecular smooth substrates. No doubt that the surface roughness inﬂuences the apparent value of the contact angle. However, it is obvious that the roughness cannot result in a transition from the nonwetting to the partial wetting case or from the partial wetting to the complete wetting case. That is why, in this section, only molecularly smooth solid substrates are under consideration. The inﬂuence of roughness and chemical heterogeneity is considered in Section 2.4 and Section 2.5. The main idea of this section is to show that the simultaneous action of the capillary pressure and S-shaped disjoining pressure isotherm results in the formation of nonﬂat equilibrium liquid shapes even on smooth homogeneous solid substrates. Again, we should emphasize that the shape is speciﬁc for water and aqueous solutions and hence is vitally important for life. However, we are still completely unaware of the way in which it is important. Consideration of the equilibrium nonﬂat liquid layers allows the suggestion of a new scenario of rupture of thick metastable β-ﬁlms and their transition to absolutely stable α-ﬁlms (see the following section).

GENERAL CONSIDERATION The excess free energy, Φ, of a liquid layer, drop, or meniscus on a solid substrate can be expressed in the following way: Φ = γS + PV e + Φ D − Φref , © 2007 by Taylor & Francis Group, LLC

(2.18)

Equilibrium Wetting Phenomena

43

where S, V, and ΦD are excesses of the vapor–liquid interfacial area, the excess volume, and the excess energy associated with the action of surface forces, respectively; γ is the liquid–vapor interfacial tension; Pe is the excess pressure (see Section 2.1); Φref is the excess free energy of a reference state (see the following section). The gravity action is neglected in Equation 2.18. The excess pressure, Pe, is introduced as in Section 1.2 as Pe = Pa − Pl ,

(2.19)

where Pa is the pressure in the ambient air, and Pl is the pressure inside the liquid. The latter pressure is referred to as the hydrostatic pressure inside the liquid. This means that Pe > 0 in the case of a meniscus, and Pe < 0 in the case of a drop. Pe is uniquely determined by the ambient vapor pressure p, according to Equation 2.2 (Section 2.1). Equation 2.2 shows that Pe > 0 corresponds to an undersaturation, whereas Pe < 0 corresponds to an oversaturation. That is, menisci can be at equilibrium at undersaturation, and drops can be at equilibrium at oversaturation. In the following text, only two-dimensional equilibrium systems are under consideration for the sake of simplicity. In this case, the excess free energy in Equation 2.18 can be rewritten as:

Φ=

∫

  γ  

(

∞ ∞   1 + h′ 2 − 1 + Pe h − he + Π(h)dh − Π(h)  dx.  h he 

)

(

)

∫

∫

(2.20)

Integration, in the preceding equation, is taking over the whole space occupied by the system. Note that the excess free energy in Equation 2.20 is selected as an excess over the energy of a reference state, which is the state of the same ﬂat surface covered by a stable equilibrium α-ﬁlm. A selection of any reference state results in an additive constant in the expression 2.20. However, the reference state is important during the consideration of the liquid proﬁles in the vicinity of the apparent three-phase contact line. Any liquid proﬁle, h(x), which gives the minimum value to the excess free energy, Φ, according to Equation 2.20, describes an equilibrium liquid conﬁguration. For the existence of the minimum of the excess free energy 2.20, the following four conditions should be satisﬁed: 1. δΦ = 0 2.

∂2 f >0 ∂h ′ 2 where f = γ

(

)

(

)

∞

∫

∞

∫

1 + h′ 2 − 1 + Pe h − he + Π(h)dh − Π(h)

© 2007 by Taylor & Francis Group, LLC

h

he

44

Wetting and Spreading Dynamics

3. Solution of Jacoby’s equation, u(x), should not vanish at any position, x, inside the region under consideration except for boundaries of the region of integration in Equation 2.20. 4. The transversality condition at the apparent three-phase contact line should be satisﬁed. It provides the condition of a smooth transition from a nonﬂat liquid proﬁle to a ﬂat equilibrium ﬁlm in front. The transversality condition reads  ∂f   f − h′ ∂h′  = 0, B  where B is the position of the three-phase contact line. The preceding condition can be rewritten using the aforementioned deﬁnition of f as:  γ  

(

  =0 1 + h′ − 1 + Pe h − he + Π(h)dh − Π(h) − 2  1 + h ′ h he  B 2

)

(

)

∞

∫

∞

∫

γh′ 2

This condition shows that the three-phase contact line should be determined at the intersection of the liquid proﬁle with the equilibrium liquid ﬁlm of thickness he, and not at the intersection with the solid substrate as usually assumed. This further results in

(

)

 h′ 2  2  1 + h′ − 1 −  =0  1 + h′ 2  B or   1   = 1,  1 + h′ 2  B which is obviously satisﬁed only at

( h′ )

B

= 0 or h ′( he ) = 0

(2.21)

This transversality condition is discussed in Section 2.3. There it is shown that the condition actually means © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

45

h ′ → 0, x → ∞ ,

(2.22)

and the meaning of x → ∞ is clariﬁed, which is, “tends to the end of the transition zone.” The ﬁrst condition (1) results in the well-known equation γh ′′ + Π( h ) = Pe , (1 + h ′ 2 )3/ 2

(2.23)

which should be referred to as Derjaguin’s equation because Derjaguin was the ﬁrst to introduce it [1]. The ﬁrst term on the left-hand side of Equation 2.23 corresponds to the capillary pressure, and the second term represents the disjoining pressure action. If the thickness of the liquid is out of the range of the disjoining pressure action, then Equation 2.23 describes either a ﬂat (Pe = 0) liquid surface, a spherical drop proﬁle (at Pe < 0), or a spherical meniscus proﬁle (at Pe > 0). The second condition (2) is always satisﬁed because γ ∂2 f = > 0. 2 (1 + h ′ 2 )3/ 2 ∂h ′ The third condition (3) results in Jacoby’s equation: d γ u′ d Π( h ) + u=0 2 3/ 2 dx (1 + h ′ ) dh

(2.24)

Direct differentiation of Equation 2.23 results in d γh ′′ d Π( h ) dh + = 0. dx (1 + h ′ 2 )3/ 2 dh dx Comparison of Equation 2.23 and Equation 2.24 shows that the solution of Jacoby’s equation is: u = const ⋅ h ′ .

(2.25)

Hence, if h′(x) does not vanish anywhere inside the system under consideration, then the system is stable; however, if h′(x0) = 0 and x0 is different from the ends of the system under consideration (that is, inside the range of integration in Equation 2.20), then the system is unstable. © 2007 by Taylor & Francis Group, LLC

46

Wetting and Spreading Dynamics

The second-order differential Equation 2.23 can be integrated once, which gives: ∞

∫

C − Pe h − Π(h)dh 1 1 + h′

2

=

h

(2.26)

,

γ

where C is an integration constant to be determined. The important observation is that the right-hand side of the preceding equation should always be positive. In the case of a meniscus in a ﬂat capillary, the integration constant, C, is determined from the following condition: at the capillary center, h′(H) = –∞, which gives C = Pe H, where H is the half-width of the capillary (see Section 2.3). In the case of equilibrium droplets, the constant should be selected using a different condition at the droplet apex, h = H: h′(H) = 0 (see Section 2.3), which results in C = γ + Pe H. An alternate way of selection of the integration constant, C, is by using the transversality condition (2.21). The integration constant, C, in this section is selected individually according to boundary conditions in each case under consideration. In the case of equilibrium liquid drops and menisci (see Section 2.3), they are supposed to be always at equilibrium with ﬂat ﬁlms with which they are in contact with in the front. Only the capillary pressure acts inside the spherical parts of drops or menisci, and only the disjoining pressure acts inside thin ﬂat ﬁlms. However, there is a transition zone between the bulk liquid (drops or menisci) and the thin ﬂat ﬁlm in front of them. In this transition zone, both the capillary pressure and the disjoining pressure act simultaneously (see Section 2.3 for more details). A proﬁle of the transition zone between a meniscus in a ﬂat capillary and a thin α-ﬁlm in front of it, in the case of partial wetting, is presented in Figure 2.5. It shows that the liquid proﬁle is not always concave but changes its curvature inside the transition zone. Just this peculiar liquid shape in the transition zone determines the static hysteresis of contact angle (see Chapter 3) h

h a

hβ

b

tmax hu tmin

θe

he Pe

Π

xβ xmax xu xmin

x

FIGURE 2.5 Partial wetting. Magniﬁcation of the liquid proﬁle inside the transition zone in “thick capillaries.” S-shaped disjoining pressure isotherm (left side, a) and the liquid proﬁle in the transition zone (right side, b). © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

47

and a number of other equilibrium and nonequilibrium macroscopic liquid properties on solid substrates (Chapter 2 and Chapter 3). In the transition zone (Figure 2.5), all thicknesses are presented from very thick (outside the range of the disjoining pressure action) to thin α-ﬁlms. This means that the stability condition of ﬂat ﬁlms (Equation 2.4, Section 2.1) cannot be used any more because this condition is valid only in the case of ﬂat ﬁlms. The more sophisticated Jacoby’s condition (3) should be used instead, which shows that the transition zone is stable if h′(x) does not vanish anywhere inside the transition zone. A peculiar shape of the transition zone, where both the capillary pressure and the disjoining pressure are equally important, provides an idea to consider solutions of Equation 2.26 in the case of the S-shaped isotherm and to see if this equation has other stable solutions different from ﬂat liquid ﬁlms of a constant thickness. Each of these solutions (if any) corresponds to a nonﬂat liquid layer, whose stability should be checked using Jacoby’s condition (3). In the following text, we show that such nonﬂat equilibrium liquid shapes can exist.

MICRODROPS: THE CASE Pe > 0 In the following text, we consider the possibility of existence of microdrops, that is, drops with an apex in the range of inﬂuence of the disjoining pressure. In this case, the drop does not have a spherical part even at the drop apex because its shape is distorted everywhere by the disjoining pressure action. The liquid proﬁle, h(x), that is to be determined, is obtained by the integration from Equation 2.23 with an integration constant, C, as described by Equation 2.26. The transversality condition (2.21) at h = he gives h′(he) = 0, which means the drop edge approaches the equilibrium ﬁlm of thickness he on the solid surface at zero microscopic contact angle. This condition allows the determination of the integration constant in Equation 2.26 as ∞

C = γ + Pe he +

∫ Π(h)dh.

he

Hence, the drop proﬁle is described by the following equation:

h′ = −

γ2

( γ − L (h) )

2

− 1,

where h

∫

L (h) = Pe (h − he ) − Π(h)dh. he

© 2007 by Taylor & Francis Group, LLC

(2.27)

48

Wetting and Spreading Dynamics Π

Πmax Pe 0

h

–Πmin L

0

h+ he

hu

h hβ

FIGURE 2.6 Determination of the microdrop apex. Upper part: S-shaped disjoining pressure isotherm, lower part: L(h). L(h+) = 0 determines the drop apex, h+.

The expression under the square root in Equation 2.27 should be positive, that is, the following condition should be satisﬁed: 0 ≤ L (h) ≤ γ ,

(2.28)

where the ﬁrst equality corresponds to the zero derivative, and the second one corresponds to the inﬁnite derivative. Let h+ be the apex of the microdrop. The upper part of Figure 2.6 shows the S-shaped dependence of the disjoining pressure isotherm, Π(h), whereas the lower part of Figure 2.6 shows the curve L(h) that has a value of maximum or minimum thickness, which are solutions of Pe = Π(h). At the apex of the drop, when h = h+, the ﬁrst derivative should be zero, that is, h′(h+) = 0, or from Equation 2.27, L(h+) = 0 .

(2.29)

The origin is placed at the center of the drop. In the following text, we consider only the situation that corresponds to the formation of microdrops at undersaturation, that is, at Pe > 0. Equilibrium macrodrops at oversaturation, that is, at Pe < 0, are considered in the Section 2.3. At 0 < Pe < Πmax, the equation Pe = Π(h) has three roots (Figure 2.6), the smallest of which corresponds to the equilibrium ﬂat α-ﬁlm of thickness, he. For © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

49

h+ he x

FIGURE 2.7 Proﬁle of an equilibrium microdrop. Note that the apex of the microdrop is in the range of the disjoining pressure action, that is, the drop does not have any spherical part (even at the drop apex).

the existence of microdrops, the following conditions should be satisﬁed: h″ < 0 at h = h+, and h″ > 0 as h → he (Figure 2.7); hence, the following inequality should be satisﬁed: hu < h+ < hβ. At the drop apex h′(h+) = 0, and hence, according to Equation 2.29, h+

Pe (h+ − he ) =

∫ Π(h)dh,

(2.30)

he

and the solution of this equation, h+, should be located in the following range: hu < h+ < hβ (see the lower part in Figure 2.6). The left-hand side in Equation 2.30 is positive, and so should be the righthand side. Hence, a sufﬁcient condition for the existence of equilibrium microdrops is as follows: S– < S+, that is, the disjoining pressure isotherm should be S-shaped but the equilibrium contact angle should be equal to zero. This means that the equilibrium microdrops do not exist either in the case of partial wetting, when S– > S+, or in the case of a “regular” complete wetting, when the disjoining pressure decreases in a monotonous way, as for example, Π( h ) =

A . h3

However, conditions for the existence of equilibrium microdrops are satisﬁed in the experiments mentioned already by Exerowa et al. [5–7].

MICROSCOPIC EQUILIBRIUM PERIODIC FILMS In this section, we consider the possibility of existence of equilibrium periodic liquid ﬁlms that are situated completely in the range of the disjoining pressure action (partial wetting, S-shaped disjoining pressure isotherm). Undersaturation is under consideration, that is, Pe > 0. © 2007 by Taylor & Francis Group, LLC

50

Wetting and Spreading Dynamics h

h+

h– x 0

x–

FIGURE 2.8 Equilibrium periodic ﬁlm. h+ denotes maximum thickness, and h– denotes minimum thickness; x– is the length of the half-period of the ﬁlm.

Let h+ and h be the maximum and minimum heights of an equilibrium periodic ﬁlm (Figure 2.8). Derivatives should be zero at h = h– and h = h+ or h′(h–) = h′(h+) = 0. Using the second of these two conditions in Equation 2.26, we can determine the integration constant C as ∞

C = γ + Pe h+ +

∫ Π(h)dh .

h+

Hence, the proﬁle of the equilibrium periodic ﬁlm is described by the following equation: h′ = −

γ2

( γ − L (h))

2

−1,

(2.31)

+

where h+

∫

L + (h) = − Pe (h+ − h) + Π(h)dh.

(2.32)

h

The following condition 0 ≤ L+ ( h ) ≤ γ should be satisﬁed to have the positive expression under the square root in Equation 2.31. The origin is placed at the position of the maximum height (Figure 2.8) and Equation 2.31 is written for a half period of the periodic ﬁlm from x = 0, which corresponds to the position of the maximum, to x = x–, which corresponds to the position of the minimum height (Figure 2.8). Notice that h–, h+, and x– are to be determined. As we must have h′(h–) = 0 at the position of the minimum height of the ﬁlm, it follows from Equation 2.31 that h+

Pe (h+ − h− ) =

∫ Π(h)dh.

h−

© 2007 by Taylor & Francis Group, LLC

(2.33)

Equilibrium Wetting Phenomena

51

The latter condition relates the two unknown thicknesses h– and h+: h– = h– (h+). Equation 2.23 can be rewritten as

(1 + h ′ ) h ′′ =

3/ 2

( P − Π(h)).

γ

e

The latter equation and Figure 2.8 show that near the minimum height, h = h–, the liquid proﬁle is convex, h″ > 0, that is, Pe > Π(h+); similarly, near the maximum height, h = h+, h″ < 0 if Pe < Π(h+). At Pe > 0, for every pressure Pe there exists either no solution at all of Equation 2.33, or there exists an interval of values h+ min < h+ < h+ max, where h+ min, h+ max are determined by the following conditions h–(h+ min) = he, h–(h+ max) = hu. In the following text, we give a method for determining the unique value of h+, that is, the value that is actually realized at the equilibrium. The excess free energy of a half period, x– (Figure 2.8), of the periodic ﬁlm is given by the same relation (2.20) where we, however, omit the additive constant determined by the reference state. The latter, as we see in the following text, is unimportant. From Equation 2.31, we can express dh

dx = −

(

.

γ2 γ − L+ ( h )

)

2

−1

After substitution of the latter expression into Equation 2.20, we arrive at ∞

h+

Φ=

∫

h−

∫

γ 1 + h′ 2 + Π(h)dh + Pe h h

dh.

γ2  γ − L+ (h) 

2

(2.34)

−1

The latter expression includes only one undetermined parameter, h+, because h– is expressed via h+ according to Equation 2.33. Only shapes with the minimum value of the excess free energy, Equation 2.34, can be realized, i.e., the unknown h+ should be determined using the following conditions: ∂Φ = 0, ∂h+ © 2007 by Taylor & Francis Group, LLC

∂2 Φ > 0. ∂h+2

(2.35)

52

Wetting and Spreading Dynamics Π

Πmax Pe

tmin h−

0 t0

h t2

tmax h+ hβ

ts

−Πmin

FIGURE 2.9 Isotherm of disjoining pressure used for calculations of the excess free energy of periodic ﬁlms.

As the volume of the half period per unit length of the periodic ﬁlm is x+

V=

∫ hdx = V (h ), +

x−

conditions (2.35) are identical to the usual thermodynamic conditions of equilibrium: ∂Φ = 0, ∂V

∂2 Φ > 0. ∂V 2

Conditions 2.35 completely determine the equilibrium shape of the periodic ﬁlm. The procedure suggested in Equation 2.35 (minimization of the excess free energy) is consistent with Euler’s Equation (Equation 2.23), which minimizes the same excess free energy. Computer calculations shown in the following text indicate that there is a unique h+ value satisfying conditions (2.35). These conditions prove the thermodynamic stability of periodic ﬁlms. For calculations of the dependence of the excess free energy on h+, according to Equation 2.34, a disjoining pressure isotherm should be selected. It is selected for the calculation as follows (Figure 2.9): t0 = 10 −6 cm, tmin = 1.5 ⋅ 10 −6 cm, t2 = 2 ⋅ 10 −6 cm, tmax = 3 ⋅ 10 −6 cm, ts = 5 ⋅ 10 −6 cm, Πmax = 10 6 dyn/cm 2 , Πmin = 10 7 dyn/cm 2 , γ = 72 dyn/cm This choice approximately corresponds to aqueous ﬁlms on quartz surface [1] and, according to Equation 2.17, gives: © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

53

Φ . 104, erg

50 2

40

1

30 20 10 2

3

4 h+ . 106, cm

5

FIGURE 2.10 The excess free energy of periodic ﬁlms, Φ(h+), calculated according to Equation 2.34, using the isotherm of disjoining pressure presented in Figure 2.9. (1) Pe = 0.7⋅106 dyn/cm2, (2) Pe = 0.2⋅106 dyn/cm2.

cos θe ≈ 1 −

S− − S+ ≈ 0.94. γ

Using the adopted disjoining pressure isotherm (Figure 2.9), the dependence of the excess free energy on h+ is calculated according to Equation 2.34. The excess pressure, Pe, varied from 0 to Πmax = 10 6 dyn/cm 2 . Two calculated dependences are shown in Figure 2.10. Each of these plots has a sharp minimum value. The minimum value determines the unique h+ value, which is realized at the equilibrium. According to Equation 2.34 and using the isotherm of disjoining pressure presented in Figure 2.9, the calculated dependences are: 1.

Pe = 0.7⋅106 dyn/cm2.

2. Pe = 0.2⋅106 dyn/cm2.

Let us compare the excess free energy of the corresponding β-ﬁlm, Φβ, of the same length as a half period of the periodic ﬁlm, ∞    Φβ = γ + Π(h)dh + Pe hβ  x − ,   hβ  

∫

with the energy of a half period of the periodic ﬁlm. The comparison is presented in Table 2.1, which shows that at Pe < 0.6 · 10–6 dyn/cm2, the excess free energy © 2007 by Taylor & Francis Group, LLC

54

Wetting and Spreading Dynamics

TABLE 2.1 Calculated Excess Free Energy of the Periodic Film and β-Film Pe⋅106 dyn/cm2

F⋅⋅106 dyn (the periodic film)

Fβ⋅106 dyn β-film) (β

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1130 1212 1303 1299 1446 1633 1727 2227 2250

1115 1200 1290 1290 1438 1635 1730 2230 2265

of (β-ﬁlms is lower than the corresponding energy of the periodic ﬁlms (that is, (β-ﬁlms are more stable); however, at Pe > 0.6 · 10–6 dyn/cm2, and the free energy of the periodic ﬁlm becomes lower. This would mean that close to the maximum value of the disjoining pressure isotherm, Πmax, periodic ﬁlms are more stable than β-ﬁlms, that is, the periodic ﬁlms are a transitional state before rupture of β-ﬁlms. It was previously mentioned that periodic ﬁlms exist only in the case of partial wetting, that is, if S– > S+. Periodic ﬁlms are to be experimentally discovered. The case p/ps > 1, when Pe < 0, can be treated similarly. It is possible to show that in this case, the maximum thickness of periodic ﬁlms can be outside the range of the disjoining pressure action, that is, periodic ﬁlms in this case are actually a periodic array of drops.

MICROSCOPIC EQUILIBRIUM DEPRESSIONS

ON

β-FILMS

In this section, an existence of equilibrium depressions on the surface of thick β-ﬁlms is considered (Figure 2.11). A minimum thickness of a depression is marked as h– (Figure 2.11). The derivative should be zero at the top of the depression, that is, h′(hβ) = 0. Using the preceding condition, an integration constant in Equation 2.26 can be determined as follows: ∞

C = γ + Pe hβ +

∫ Π(h)dh .

hβe

After which Equation 2.26 can be rewritten as © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

55

hβ

h–

FIGURE 2.11 Schematic presentation of an equilibrium depression on the β-ﬁlm with thickness hβ. h denotes the minimum thickness of the depression.

1 1 + h′

2

=

γ − L− ( h ) , γ

(2.36)

where hβ

∫

L− (h) = − Pe (hβ − h) + Π(h)dh. h

The right-hand side in Equation 2.36 should be positive, and it gives the following restrictions: 0 ≤ L– (h) ≤ γ. The derivative should be zero at the bottom of the depression, that is, h′(h–) = 0. This condition gives an equation for the determination of h: hβ

L− (h− ) = − Pe (hβ − h− ) +

∫ Π(h)dh = 0.

(2.37)

h−

The procedure of determination of h– is shown in Figure 2.12. Π Πmax Pe 0 −Πmin

h

L–

0

he h–

hu

hβ

h

FIGURE 2.12 Determining the minimum thickness of the depression, h–, S-shaped disjoining pressure isotherm (the upper part), and function L–(h) (the lower part).

© 2007 by Taylor & Francis Group, LLC

56

Wetting and Spreading Dynamics

It is possible to show that 1. h– always exists in the case of partial wetting, that is, if S– > S+. 2. Equilibrium depressions have lower excess free energy than the corresponding ﬂat β-ﬁlms above certain critical value of Pe. Now we can suggest a new scenario of the transition from thick β-ﬁlms to thin α-ﬁlms in the case of partial wetting. At low values of Pe, β-ﬁlms are more stable than the equilibrium depressions or periodic ﬁlms. However, above some critical value of Pe, β-ﬁlms have higher energy as compared to the equilibrium depressions. That means isolated depressions develop on the β-ﬁlm. At further increases of Pe , their excess free energy exceeds the corresponding value of a periodic ﬁlm, and a transition from isolated depressions to a periodic ﬁlm takes place. As we mentioned previously, there is a critical value of Pe above which periodic ﬁlms cannot exist any more. This results in a transition from the periodic ﬁlm to the α-ﬁlm, with the microdrops sitting on it. This transition can be described as “a rupture.” Residual microdrops cannot be at equilibrium with the α-ﬁlm in the case of partial wetting and gradually disappear by evaporation and/or hydrodynamic ﬂow. To summarize: we have shown that in the case of S-shaped disjoining pressure isotherm microdrops, microdepressions and equilibrium periodic ﬁlms are possible on ﬂat solid substrates. Criteria have been provided for both existence and stability of these nonﬂat equilibrium liquid shapes. It has been suggested that transition from thick ﬁlms to thinner ﬁlms goes via intermediate nonﬂat states like microdepressions or periodic ﬁlms, which are more stable than ﬂat ﬁlms in some hydrostatic pressure ranges. Flat liquid ﬁlms are unstable in the region between α- and β-ﬁlms and hence cannot be experimentally observed. However, the predicted nonﬂat stable liquid layers (microdepressions and periodic ﬁlms) are located in this unstable region. Accordingly, experimental measurements of proﬁles of these nonﬂat layers open the possibility of determining the disjoining pressure isotherm in the unstable region.

2.3 EQUILIBRIUM CONTACT ANGLE OF MENISCI AND DROPS: LIQUID SHAPE IN THE TRANSITION ZONE FROM THE BULK LIQUID TO THE FLAT FILMS IN FRONT In this section, we shall show that the disjoining pressure action determines the peculiar shape of liquid inside the transition zone from the bulk to the ﬂat liquid ﬁlms ahead for both menisci and drops. Thus, the equilibrium contact angle is determined using the disjoining pressure isotherm. In Figure 1.12, three types of disjoining pressure isotherms are presented, which correspond to three different situations presented in Figure 1.1. Disjoining

© 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

57

pressure isotherms can be directly measured, not in the whole range of ﬁlm thickness, as we already discussed in Section 2.1 but only in the regions where ﬂat ﬁlms are stable (see the stability condition of ﬂat ﬁlms in Equation 2.4). This would indicate that the experimental measurements of the disjoining pressure can be undertaken only in the case of ﬂat and absolutely stable α-ﬁlms and metastable β-ﬁlms. In this section, only partial wetting cases are under consideration. Complete wetting cases are under consideration in Section 2.4. There is no doubt that the surface roughness inﬂuences the apparent value of the contact angle. However, it is obvious that the roughness cannot result in a transition from nonwetting to partial wetting or from partial wetting to complete wetting. That is why only molecularly smooth solid substrates are under consideration in the following section. Consideration of equilibrium liquid states on rough substrates, when both capillary forces and surface forces are taken into account, is a challenging subject to be developed in the future.

EQUILIBRIUM OF LIQUID PARTIAL WETTING CASE

IN A

FLAT CAPILLARY:

The excess free energy, Φ, of a liquid layer, drop, or meniscus on a solid substrate can be expressed by Equation 2.20 (see Section 2.2). Equation 2.2 and Equation 2.19 show that the case Pe > 0 corresponds to the case of menisci or other nonﬂat liquid shapes (see Section 2.2) at equilibrium with undersaturated vapor; and the case Pe < 0 corresponds to the case of drops or other nonﬂat liquid shapes (see Section 2.2) at equilibrium with oversaturated vapor. The difference between a volatile and a nonvolatile liquid determines only the path and the rate of a transition to the equilibrium state but not the equilibrium state itself. In the following section, only the equilibrium states are under consideration, and hence it is not speciﬁed in this chapter whether the liquid is volatile or nonvolatile. As already mentioned in Chapter 1, all solid surfaces in contact with a volatile or nonvolatile liquid at equilibrium are covered by a thin liquid ﬁlm. The thickness of this equilibrium ﬁlm is determined by the action of surface forces (disjoining pressure isotherm). That is, the choice of the reference state is uniquely determined in order to consider the vicinity of the three-phase contact line at the equilibrium state of a bulk liquid in contact with a solid substrate; the reference state is the state of solid substrate covered with the equilibrium liquid ﬁlm. That is why a reference state that has a plane parallel ﬁlm with the lowest possible equilibrium thickness (that is, α-ﬁlms introduced in Section 2.1), which corresponds to the vapor pressure p in the ambient air, is selected. In this section, twodimensional equilibrium menisci in a ﬂat chamber with a half-width H or twodimensional equilibrium liquid drops are considered for simplicity. Extension of the derivation, in the following text, to axial symmetry is brieﬂy discussed at the end of this section.

© 2007 by Taylor & Francis Group, LLC

58

Wetting and Spreading Dynamics

According to this selection, Equation 2.20 can be rewritten as F=

∫ {γ 

(

)

}

1 + h′ 2 − 1 + Pe h − he + fD (h) − fD (he ) dx , 

(2.38)

where he is the thickness of the equilibrium plane parallel α-ﬁlm, and fD(h) is the density of the energy of surface forces. Two substantial simpliﬁcations are adopted in the expressions for the free energy in Equation 2.38: 1. The density of the energy of surface forces, fD(h), depends only on the ﬁlm thickness, h, and is independent of the derivatives of the ﬁlm thickness. 2. The interfacial tension retains its bulk value, γ. The ﬁrst assumption means that only proﬁles with low slope can be described using such approximation. The only attempt to take into account a dependency of the surface forces, fD(h), on the ﬁrst derivative of the liquid proﬁle of dispersion forces has been undertaken in Reference 8. However, the calculations in Reference 8 are based on a direct summation of molecular forces. These forces are well known to be of nonadditive nature [1]. Probably, this was the reason why a controversial nonzero equilibrium contact angle has been predicted in the case of complete wetting [8]. That is why consideration of surface forces in the case of nonﬂat proﬁles remains a challenge, and we use this assumption (1) even in cases where it is not rigorously valid. Actually, the two assumptions, (1) and (2), are strongly interconnected. If the density of energy of surface forces, fD(h, h′), depends on the derivative of the ﬁlm proﬁle, h′, then the tangential stress on the surface of the liquid is unbalanced. However, if we adopt both assumptions (1) and (2), at least from this point of view, we do not have any contradictions; constant interfacial tension results in zero tangential stress under equilibrium conditions. Let us brieﬂy discuss what happens if the density of energy of surface forces, fD(h, h′), depends on the derivative of the ﬁlm proﬁle, h′. In this case, Equation 2.38 takes the following form: Φ=

∫ {γ 

(

)

}

1 + h′ 2 − 1 + Pe h − he + fD (h, h′) − fD (he ) dx. 

At equilibrium, the ﬁrst condition (1) (Section 2.2) must be satisﬁed, which results in γh ′′ ∂f D d 2 f D d 2 fD − + + h ′ = Pe . h ′′ dhdh ′ (1 + h ′ 2 )3/ 2 ∂h dh ′ 2 © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

59

Let us introduce the following functions:

a ( h, h ′) =

(

d 2 fD 1 + h ′2 dh ′ 2

)

3/ 2

, Π( h, h ′) = −

∂f D d 2 f D + h′ ; ∂h dhdh ′

then the latter equation takes the following form:

( γ + a (h, h ′)) h ′′ + Π(h, h ′) = P . e

(1 + h ′ 2 )3/ 2

This means that (1) The effective surface tension, γ + a ( h, h ′) , depends on both thickness, h, and the slope, h′. (2) The effective disjoining pressure now depends on both the thickness and slope values as the effective surface tension. The consequences of such dependences, as well as the physical meaning of these effective values are to be understood. This is the reason why we use the approximation adopted in Equation 2.38, that is, the density of the energy of surface forces, fD (h), depends only on the ﬁlm thickness, h, and is independent of the derivatives of the ﬁlm thickness. Integration, as seen in Equation 2.38, is taking over the whole space occupied by the ﬂat meniscus. Any liquid proﬁle, h(x), which gives the minimum value to the excess free energy, Φ, according to Equation 2.23, describes an equilibrium liquid conﬁguration on the planar surface. For the existence of the minimum value, the four conditions introduced in Section 2.2 should be satisﬁed (see conditions (1)–(4) and the discussion there). The ﬁrst requirement (1) shows that the liquid proﬁle gives minimum or maximum to the excess free energy, Φ, whereas two other requirements, (2) and (3), prove that the proﬁle provides a minimum value to the excess free energy Φ. It is necessary to note that both requirements (2) and (3) must be satisﬁed; only in this case the excess free energy (Equation 2.38) has a minimum value. The requirement (1) results in the Euler’s equation (Equation 2.23), which for the ﬁrst time has been suggested by Derjaguin [1] and should be referred to as Derjaguin’s equation, where disjoining pressure is introduced as Π( h ) = −

df D ( h ) . dh

If the requirement (3) is not satisﬁed, then the solution of Equation 2.23 does not provide a stable solution. Condition (3) shows that Equation 2.23 can be integrated once, which results in Equation 2.26. Note that the right-hand side of Equation 2.26 should always be positive. © 2007 by Taylor & Francis Group, LLC

60

MENISCUS

Wetting and Spreading Dynamics IN A

FLAT CAPILLARY

In the case of a meniscus in a ﬂat capillary, the integration constant, C, is determined from the condition at the capillary center: h′( H ) = −∞,

(2.39)

which gives ∞

∫

C = Pe H + Π(h)dh, H

where H is the half-width of the capillary. Using this constant, Equation 2.26 can be rewritten as

(

H

)

Pe H − h − 1 1 + h ′2

=

∫ Π(h)dh h

γ

.

(2.40)

This equation describes an equilibrium proﬁle of the meniscus in ﬂat capillaries. Let us consider the solution of Equation 2.23 in more detail. This equation determines the liquid proﬁle in three different regions: (1) A spherical meniscus, which is not disturbed by the action of surface forces. That is, the disjoining pressure action can be neglected, and we arrive at a regular Laplace equation: γh ′′ = Pe . (1 + h ′ 2 )3/ 2

(2.41)

(2) In the case of a ﬂat liquid ﬁlm in front of the meniscus, Π( h ) = Pe .

(2.42)

(3) At a transition zone in between, both the capillary force and the disjoining pressure are equally important. Note that Equation 2.42 coincides with Equation 2.3, but we keep this equation for convenience. In the following text, we consider only “macroscopic capillaries.” In these capillaries, the radius, H, is much bigger than the range of action of surface forces. Let the radius of the action of surface forces be ts ≈ 10–5 cm = 1000 Å = 0.1 µ = 100 nm, that is, at h > ts: Π(h) = 0. In this case, Equation 2.23 can be rewritten at h > ts as Equation 2.41 with boundary conditions © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

61

h (0 ) = H

(2.43)

and Equation 2.39. Solution of Equation 2.41 with boundary conditions according to Equation 2.39 and Equation 2.43 gives a spherical proﬁle 2

2

 γ   γ  ( H − h )2 +  − x  =   ,  Pe   Pe 

(2.44)

that is, a spherical meniscus of radius ρe =

γ . Pe

The preceding equation describes an idealized proﬁle of a spherical meniscus. Intersection of this proﬁle with the thin equilibrium ﬁlm of thickness he deﬁnes the apparent three-phase contact line and the macroscopic equilibrium contact angle θe (Figure 2.13). A simple geometrical consideration shows that Pe =

γ cos θe . H

(2.45)

At h = he: h′2 = 0, and we conclude from Equation 2.45 and Equation 2.40: γ cos θe H − he − H

(

1=

)

H

∫ Π(h)dh

he

,

γ

θe

(2.46)

ρe 2H

1 2

3

he x

FIGURE 2.13 Proﬁle of a meniscus in a ﬂat capillary. 1 — a spherical part of the meniscus of curvature ρe, 2 — transition zone between the spherical meniscus and ﬂat ﬁlms in front, 3 — ﬂat equilibrium liquid ﬁlm of thickness he. Further in the text, the liquid proﬁle inside the transition zone will be considered in more detail.

© 2007 by Taylor & Francis Group, LLC

62

Wetting and Spreading Dynamics

which allows the determination of the contact angle via disjoining pressure isotherm as

cos θe =

1 1+ γ

H

∫ Π(h)dh

he

1−

he H

1 ≈ 1+ γ

∞

∫ Π(h)dh,

at he < ts << H .

(2.47)

he

The preceding equation, for the ﬁrst time, has been deduced by Derjaguin and Frumkin [1] and named after these two scientists as the Derjagun–Frumkin equation. Note that this equation was deduced using completely different arguments from those previously mentioned and for many years was considered as independent of Equation 2.23. The derivation given in the preceding section shows that Derjaguin–Frumkin Equation 2.47 is a direct consequence of Equation 2.23. Also note that the same equation for the contact angle has been deduced in Section 2.1 in a different way. Equation 2.47 can be approximately rewritten (Figure 2.1) as: cos θe ≈ 1 −

S− − S+ . γ

(2.48)

The preceding equation shows that cos θe < 1 only if S– > S+ (Figure 2.1). Now, at last, we can precisely deﬁne the term partial wetting: (1) S-shaped disjoining pressure isotherm (curve 2 on Figure 2.1) and (2) S– > S+. Let us consider the case when Pe < Πmax, γ cos θ < Π max, H or H > Hcr , where H cr ~

γ . Π max

We refer to such capillaries as thick capillaries. In the case of aqueous solutions, γ ~ 70 dyn/cm, Π max ~ 10 5 erg, and hence, H cr ~ 7 ⋅ 10 −4 cm ~ 10 −3 cm . Otherwise, the capillary is referred to as a thin capillary, that is, the capillary is thin if its thickness H is in the range ts << H < Hcr , where ts is the radius of the disjoining pressure action. According to the deﬁnition of thin capillaries, these capillaries are still big enough when compared with the radius of the action of surface forces, ts. If the capillary radius is compared with the radius of action of surface forces, ts, © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

63

then such a capillary should be referred to as a microscopic capillary. Only thin, macroscopic capillaries are under consideration in this book. In the case of thick macroscopic capillaries, Equation 2.42 has three solutions, one of which corresponds to the stable equilibrium α-ﬁlm with thickness he. The excess free energy of α-ﬁlms is equal to zero, according to our choice in Equation 2.38. The second solution of Equation 2.42 in this case, hu, is unstable according to the stability condition (2.4, Section 2.1), and the third solution, hβ, is β-ﬁlm, which is also stable according to the same stability condition (2.4, Section 2.1). It has been shown in Section 2.1 that β-ﬁlms have higher excess free energy as compared with α-ﬁlms, that is, β-ﬁlms are less stable and eventually rupture to thinner and absolutely stable α-ﬁlms. However, in thin capillaries, Equation 2.42 has only one solution (not shown in Figure 2.3), which is an absolutely stable α-ﬁlm.

MENISCUS IN A FLAT CAPILLARY: PROFILE OF THE TRANSITION ZONE Let us estimate the length of the transition zone, L. Inside the transition zone, the capillary pressure and disjoining pressure are of the same order of magnitude of

Pe ~

γ . H

According to Equation 2.23, inside the transition zone, the capillary pressure can be estimated as γh ′′ 1 + h′

2

~

γhe γh γ or 2e ~ . 2 H L L

From the latter estimation, we conclude that L ~ he H .

(2.49)

In the case of he ~ 10–6 cm, H ~ 0.01 cm, and the latter estimation gives L ~ 1 µm. Note that the same estimation of the length of the transition zone is also valid in the case of droplets. Now let us rewrite Equation 2.23 in the following form: h ′′ =

© 2007 by Taylor & Francis Group, LLC

1 (1 + h ′ 2 )3/ 2 [ Pe − Π( h )] . γ

(2.50)

64

Wetting and Spreading Dynamics

The preceding equation shows that the sign of the second derivative is determined by the difference Pe – Π(h). In the case of thick capillaries, that is, H > Hcr , Figure 2.5 shows that h″ > 0, in the following range of thickness: hβ < h < H; the proﬁle is concave. h″ <, at hu < h < hβ; the proﬁle is convex. h″ > 0, at he < h < hu; the proﬁle is concave again. This would mean that the proﬁle of the liquid inside the transition zone does not remain concave all the way through the transition zone, but it changes its curvature in two inﬂection points: h( xβ ) = hβ , h( xu ) = hu (Figure 2.5). The magniﬁcation of the liquid proﬁle inside the transition zone is schematically shown in Figure 2.5. Now an important question arises: ﬂat thin ﬁlms in the range of thickness from tmin to tmax are unstable according to the stability condition (2.4, Section 2.1). We would like to emphasize that the aforementioned condition is the stability condition of ﬂat ﬁlms. As already discussed in Section 2.2 and in this section, the stability condition (3) of Section 2.2 of nonﬂat liquid layers is completely different, and according to Equation 2.25, the condition is satisﬁed inside the transition zone (see Figure 2.5 where h′ is positive everywhere). Nobody should expect any convergence of the two stability conditions (2.4, Section 2.1) of ﬂat ﬁlms and (condition 3, Section 2.2) of nonﬂat ﬁlms; they are completely different. A qualitative physical explanation of the stability of the transition zone inside the “dangerous” range of thickness from tmin to tmax is as follows. The extent of the dangerous region from xmax to xmin (Figure 2.5) is small enough, that is, any ﬂuctuation inside this dangerous region is dampened by the neighboring stable regions from both sides (Figure 2.5). The liquid proﬁle inside the transition zone in the case of thin capillaries, that is, H < Hcr, is much simpler (Figure 2.14), and it does not have any inﬂection points as in the case of thick capillaries. Here, according to Equation 2.50, the liquid proﬁle is always concave. The stability of the liquid shape inside the dangerous region of thickness from tmin to tmax is proven in precisely the same way as in the case of thick capillaries. Note that in all cases under consideration, there is no real three-phase contact line at the equilibrium because the whole solid surface is covered by a ﬂat equilibrium liquid ﬁlm. This is the reason why we refer to it as an apparent contact line. The transversality condition (4 of Section 2.2) at the apparent threephase contact line results in Equation 2.21 (Section 2.2): h ′( he ) = 0.

(2.51)

Let us consider the latter condition in more detail in Appendix 1. This consideration shows that in general cases (except for a very special model isotherm of disjoining pressure), the transition from nonﬂat transition zone to ﬂat © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena a

h

h

65 b

hβ tmax hu tmin θe

he Pe

x

Π

FIGURE 2.14 Magniﬁcation of the transition zone in the case of partial wetting in “thin capillaries.”

equilibrium ﬁlms goes very smoothly and asymptotically as x → ∞ (the sense of the latter statement is clariﬁed in Appendix 1). That is, in a general case, there is no ﬁnal point where the transition zone ends, but it approaches the ﬂat ﬁlm asymptotically.

PARTIAL WETTING: MACROSCOPIC LIQUID DROPS We have to remind ourselves that Pe < 0 in this part because the liquid drops can be at equilibrium with oversaturated vapor only. A macroscopic drop means that the drop apex, H, is outside the range of surface forces (or disjoining pressure) action. Microscopic drops, that is, drops with the apex in the range of the disjoining pressure action, are considered in Section 2.2. The equilibrium ﬁlms are determined according to Equation 2.42. Note that in the case of complete wetting, there are no equilibrium ﬂat ﬁlms on solid substrates because the line Pe < 0 does not intersect (curve 1 in Figure 2.1). Hence, there are no equilibrium droplets on completely wettable solids under oversaturation. However, in the case of partial wetting, Equation 2.42 has two solutions (Figure 2.17, left-hand side). According to the stability condition of ﬂat ﬁlms (condition 4, Section 2.2), one of them corresponds to the stable equilibrium ﬁlm of thickness he, and the second one corresponds to the unstable ﬁlm of thickness hu (Figure 2.17, left-hand side). The latter means that equilibrium droplets in the case of partial wetting are sitting on the stable equilibrium ﬁlm of thickness he. However, even in the case of partial wetting, equilibrium droplets can exist on the solid substrate only in a limited interval of oversaturation, which is determined by the following inequality: 0 > Pe > −Πmin , or using Equation 2.2 in the following range of oversaturated pressure over the solid substrate, 1< © 2007 by Taylor & Francis Group, LLC

v Π  p < exp  m min  . ps  RT 

(2.52)

66

Wetting and Spreading Dynamics

If Πmin is in the range 106–107 dyn/cm2, then the latter inequality takes the following form 1<

p v Π < 1 + m min ≈ 1.001 − 1.01 , ps RT

that is, the equilibrium droplets exist only in a very limited interval of oversaturation. Beyond this interval, at higher oversaturation, neither equilibrium liquid ﬁlms nor droplets exist on the solid substrate as in the case of complete wetting. Probably, the critical oversaturation pcr : v Π  pcr = exp  m min  ps  RT  determined using Equation 2.52 corresponds to the beginning of homogeneous nucleation, and values below this critical limit would indicate that a heterogeneous nucleation is more favorable. The radius of curvature of an equilibrium drop is ℜe =

2γ . − Pe

In the aforementioned narrow interval of oversaturation, the radii of the equilibrium drops change from inﬁnity at p → ps to ℜcr =

2γ . Πmin

If Πmin ≈ 106 dyn/cm2 and γ ≈ 72 dyn, then ℜcr ≈

144 = 1.44 µm, 10 6

that is, the critical size is out of the range of the action of surface forces. However, if Πmin ≈ 107 dyn/cm2, then ℜcr ≈

144 = 0.144 µm = 1440 Å, 10 7

and the whole droplet is in the range of the action of surface forces. That is, in the latter case, the drop is so small that it does not have anywhere (even at the very apex) a spherical part undisturbed by the action of the disjoining pressure.

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Equilibrium Wetting Phenomena

67

In the following text, we consider only two-dimensional drops for simplicity. Three-dimensional axisymmetric drops and menisci in cylindrical capillaries are brieﬂy considered towards the end of this section. On the drop apex, H, the derivative vanishes: h ′( H ) = 0 . Using the latter condition and Equation 2.26, we arrive at the following integration constant ∞

C = γ + Pe H +

∫ Π(h)dh . H

In this case, Equation 2.26 transforms as follows: H

γ + Pe ( H − h ) − 1 1 + h′

2

=

∫ Π(h)dh h

γ

.

(2.53)

The preceding equation describes the proﬁle of equilibrium liquid drop on ﬂat solid substrate at Pe < 0. As in the case of a meniscus, the whole proﬁle of a droplet can be subdivided into three parts: 1. A spherical part of the drop 2. A transition zone where both capillary pressure and disjoining pressure are equally important 3. A region of ﬂat equilibrium liquid ﬁlm in front of the drop Outside the range of the disjoining pressure action, we can neglect the action of surface forces in Equation 2.53, and these equations describe the proﬁle of a spherical drop: 1 1 + h ′2

=

γ + Pe ( H − h ) . γ

(2.54)

The preceding equation describes an idealized proﬁle of a spherical droplet. Intersection of this proﬁle with the thin equilibrium ﬁlm of thickness he deﬁnes the apparent three-phase contact line and the macroscopic equilibrium contact angle h′(he) = –tgθe. Substitution of this expression into Equation 2.54 results in Pe = −

© 2007 by Taylor & Francis Group, LLC

γ (1 − cos θe ) . H

68

Wetting and Spreading Dynamics

Casting this expression into Equation 2.53 at h = he results in the following deﬁnition of the contact angle in the case of drops on a ﬂat substrate:

cos θe = 1 +

1 γ

H

∫ Π(h)dh

he

1−

he H

1 ≈ 1+ γ

∞

∫ Π(h)dh,

at ts << H ,

(2.55)

he

which is similar to Equation 2.47 in the case of the meniscus. At he << H, expressions for the equilibrium contact angles of the menisci (2.47) and droplets (2.55) coincide. However, it is necessary to note that expressions for the equilibrium contact angle in the case of menisci (Equation 2.47) and drops (Equation 2.55) are still different; integration in these expressions, even in the case of thick capillaries and big drops, ts << H, starts from different values of he. In the case of drops, this value is always bigger than that of the menisci. This results in different values of equilibrium contact angles in the two cases. Let us rewrite Equation 2.53 for the drop proﬁle in the identical form, using the transversality condition at h = he (h′(he) = 0). This results in

h′ = −

γ2

( γ − L (h) )

2

− 1,

(2.56)

where h

∫

L (h) = Pe (h − he ) − Π(h)dh.

(2.57)

he

The expression under the square root in Equation 2.56 should be positive, that is the following condition should be satisﬁed: 0 ≤ L (h) ≤ γ ,

(2.58)

where the ﬁrst equality corresponds to the zero derivative, and the second equality corresponds to the inﬁnite derivative. Note that beyond the radius of the action of surface forces, ts, that is, at h > ts: L(h) = Pe (h – he) becomes a straight line (lower part in Figure 2.15). Now, it is important to mention that the droplet is in equilibrium with the vapor in the surrounding air. This means that the droplet volume cannot be ﬁxed,

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Equilibrium Wetting Phenomena

69

Π

he

hu

ts h

Pe –Πmin

L γ Lmax

he

hu

ts

H h

FIGURE 2.15 Determining the droplet height, H. Upper part: disjoining pressure isotherm; lower part: shape of function L(h) (see deﬁnition given by Equation 2.58).

and hence the droplet height, H, cannot be ﬁxed. That is, the drop height, H, is to be determined in the following section. The upper part of Figure 2.15 shows the S-shaped dependence of the disjoining pressure isotherm, Π(h), whereas the lower part of Figure 2.15 shows the dependency of L(h), which has maximum or minimum thickness which are solutions of Pe = Π(h). At the apex of the drop, when h = H, the ﬁrst derivative should be zero, that is, h′(H) = 0 or from Equation 2.58: L(H) = 0.

(2.59)

In this section, macrodrops, that is, drops that have their apex outside the range of the disjoining pressure action (partial wetting), are considered. In this case, max L(h) = L(hu) > 0 and L(h) → –∞ as h → +∞ (see Figure 2.15, lower part). Therefore, Equation 2.59 always has a solution hu < H < ∞ (the lower part of Figure 2.15). If the value of H lies outside the range of the inﬂuence of the disjoining pressure, then it becomes possible to determine the equilibrium macroscopic contact angle of the drop. Depending on the value of the radius of the curvature in the central part of the drop, ℜ, three different possibilities can occur (Figure 2.16a–Figure 2.16c):

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70

Wetting and Spreading Dynamics H he θe x

ᑬ (a) H

ᑬ

he

θe x

(b) H

ᑬ he x (c)

FIGURE 2.16 Determining the equilibrium contact angle of droplets. (a) partial wetting, (b) nonwetting, and (c) complete nonwetting.

1. ℜ > H he, which corresponds to the contact angle 0 < θe < π/2, the partial wetting case. 2. ℜ < H he but 2ℜ > H – he, which corresponds to the macroscopic contact angle π/2 < θe < π, the nonwetting case. 3. If 2ℜ < H – he, despite the apex of the drop being outside the range of inﬂuence of the disjoining pressure, it is impossible to determine the macroscopic contact angle, as there is no intersection of the circle of radius ℜ with the solid surface (Figure 2.16c). The last case can be referred to as the complete nonwetting case and can be referred to as θe > π, similar to the case of complete wetting, when cos θe > 1 (see Section 2.4). It is interesting to note that probably cases 2 and 3 (Figure 2.16) have never been observed experimentally. It means that either such disjoining pressure isotherms do not exist in nature or such cases are yet to be discovered. It is possible to check (using Equation 2.55) whether in partial wetting θe(Pe), dependence increases with decreasing Pe, i.e., the drop elevates itself above the solid surface as Pe decreases, and at Pe = –Πmin, the drop separates itself from the © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

71

solid surface and goes into the surrounding air. It corresponds to a transition from a heterogeneous to a homogeneous nucleation.

PROFILE

OF THE

TRANSITION ZONE

IN THE

CASE

OF

DROPLETS

Equation 2.50 and Figure 2.17 (left-hand side) show that, in the case of droplets, there is only one inﬂection point on the drop proﬁle at h(xu) = hu. Hence, the drop proﬁle inside the transition zone is shown in Figure 2.17 (right-hand side). Deviations from the spherical proﬁle start immediately as the surface forces step into action at h < ts. Note that those deviations are in the opposite direction as compared to the drop proﬁle (Figure 2.5 and Figure 2.14).

AXISYMMETRIC DROPS In this case, expression for the excess free energy takes the following form:

∫

{

(

}

)

Φ = 2π r γ  1 + h′ 2 − 1 + Pe h − he + fD (h) − fD (he ) dr,   which gives the following equation for the liquid proﬁle of an axisymmetric drop: γ d  rh ′   + Π( h ) = Pe ,  r dr  1 + h ′ 2 

(2.60)

where Pe = −

2γ , ℜ

h t

hu

θe

he –Πmin

Pe

Π

xs

xu

FIGURE 2.17 The drop proﬁle inside the transition zone. © 2007 by Taylor & Francis Group, LLC

x

72

Wetting and Spreading Dynamics

which results in Pe = −

2 γ (1 − cos θe ) H

for drops. Note the multiplier 2 in the latter expressions. Unfortunately, Equation 2.60 cannot be integrated as is done in the case of two-dimensional menisci and drops. However, the latter equation can be rewritten as γh ′′

(

1 + h ′2

+

)

3/ 2

γ r

h′ 1 + h ′2

+ Π( h ) = Pe .

(2.61)

The ﬁrst term on the left-hand side of the preceding equation is due to the ﬁrst curvature (similar to the case of the two-dimensional menisci or drops in Equation 2.23), and the second term is due to the second curvature, which is shown in the following text to be small as compared to the ﬁrst term. The characteristic length of the transition region, L, is given by Equation 2.49: L ~ he H . The latter expression shows that L << H. Let us estimate the ratio of the second term to the ﬁrst term on the left-hand side of Equation 2.61: γ r

γh′′

h′ 1 + h′

(1 + h′ )

2

2

3/ 2

~

h′ h /L L h ~ = ~ e << 1. 2 rh′′ Hh /L H H

The latter estimation shows that the second term on the left-hand side of Equation 2.61 is small as compared to the ﬁrst term and can be neglected in the transition region. After that, Equation 2.61 can be integrated, and in a similar manner as Equation 2.23, Equation 2.26 can be recovered. Outside the region of the action surface forces, Equation 2.60 can be easily solved. This solution is the “outer solution,” whereas the solutions obtained in the previously mentioned method all are “inner solutions.” The matching of these two asymptotic solutions gives the real proﬁle in the case of axial symmetry (see this procedure in the case of complete wetting in Section 2.4).

MENISCUS

IN A

CYLINDRICAL CAPILLARY

In this case, the expression for the excess free energy is as follows: Φ=

∫ {2πγ ( H − h) (

+ πPe  H − he  © 2007 by Taylor & Francis Group, LLC

(

)

1 + h′ 2 − H − he  

) − ( H − h )  + 2πH  f 2

2

D

}

(h) − fD (he )  dx

Equilibrium Wetting Phenomena

73

where H is the radius of the cylindrical capillary. The aforementioned procedure also results in γh ′′

(1 + h ′ ) 2

3/ 2

+

γ 1 H Π( h ) = Pe . + H − h 1 + h ′2 H − h

Note that the disjoining pressure in this case is H Π( h ) , H−h which is different from the disjoining pressure of ﬂat ﬁlms Π(h). We shall discuss this further in Section 2.7.

APPENDIX 1 Let us assume that the transition zone proﬁle does not tend asymptotically to the equilibrium thickness he but meets the ﬁlm at the ﬁnal point x = x0. In this case, in the vicinity of this point, we approximate the disjoining pressure isotherm by a linear dependency Π( h ) ≈ Π( he ) − a ( h − he ) , where a = −Π′( he ) is a positive value; he is a stable ﬂat liquid ﬁlm, and the derivative of the disjoining pressure should be negative and Π( he ) = Pe . The liquid proﬁle in this region has a low slope, which means Equation 2.23 can be rewritten as γh ′′ + a ( h − he ) = 0 . Solution of the latter equation is h ( x ) = he + C 1 exp(αx ) + C 2 exp(−αx ) ,

(A1.1)

where α=

a , γ

and C1 and C2 are two integration constants. At x = x0, according to Equation 2.21, the following two boundary conditions should be satisﬁed: h ( x0 ) = he . h ′ ( x 0 ) = 0. © 2007 by Taylor & Francis Group, LLC

74

Wetting and Spreading Dynamics

Equation A1.1 and the two boundary conditions result in the following system of algebraic equations for the determination of the integration constants C1 and C2: C1 exp(αx0 ) + C 2 exp(−αx0 ) = 0 . C1 exp(αx0 ) − C 2 exp(−αx0 ) = 0 . The only solution of the aforementioned system is C1 = C2 = 0, which is obviously a contradiction. Hence, the only possibility is that C1 = 0, and the liquid proﬁle has the following form if h → he: h ( x ) = he + C 2 exp(−αx ). That is, the liquid proﬁle in the transition zone tends asymptotically to the equilibrium thickness he and does not meet the equilibrium ﬂat ﬁlm in any ﬁnal point x0. Note that in the case when Π( h ) = ∞ , at h < t0, our assumption on linearization of the disjoining pressure isotherm is not valid anymore, and the only special case here is when the transition zone proﬁle meets the equilibrium ﬂat liquid ﬁlm at the ﬁnal point x0. In this case, the upper limit of the integration in Equation 2.1 should be replaced by x0. Note, that

α=

a γ

gives a new scale of the transition zone, which is 1/α. It is possible to check whether the new scales and the previous one given by Equation 2.49 are of the same order of magnitude. Indeed, 1 = α

γ γ ~ = Π′(he ) Π(he ) /he

γhe ~ Pe

γhe = he H = L. γ /H

2.4 PROFILE OF THE TRANSITION ZONE BETWEEN A WETTING FILM AND THE MENISCUS OF THE BULK LIQUID IN THE CASE OF COMPLETE WETTING The proﬁle of a liquid in the transition zone between a capillary meniscus and a wetting ﬁlm has been calculated for two types of disjoining pressure isotherms

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Equilibrium Wetting Phenomena

75

(both in the case of complete wetting). As discussed in Section 2.3, wetting ﬁlms ahead of the meniscus are separated from the capillary meniscus by a transition zone where the surface forces and the capillary forces act simultaneously. As the measurements of equilibrium contact angles and surface curvature of bulk liquids should be carried out outside the transition zone, its size and proﬁle are of interest. Moreover, the shape of the liquid proﬁle in the transition zone supplies information concerning the disjoining pressure isotherm of liquid ﬁlms on a given solid substrate. In the following text, we consider a transition zone under equilibrium conditions between a capillary meniscus between two parallel plates and wetting ﬁlms in front (Figure 2.18). The width of the capillary, 2H, is assumed to be much larger than the thickness of the equilibrium ﬂat ﬁlm, he. In the case under consideration, the thickness of the liquid layer, h(x), is a function of a single coordinate, x, directed along the capillary surface. It was already shown in Section 2.2 that the meniscus proﬁle comes to the ﬂat liquid ﬁlm at zero contact angle, according to the transversality condition (2.21), and the condition for that follows from Equation 2.26 as: ∞

γ = Pe ( H − he ) −

∫ Π(h)dh ,

he

where the equilibrium thickness, he, is determined as Π( he ) = Pe . h

2H Pe

1 0

2

h∗

3

he x

FIGURE 2.18 Complete wetting case. Schematic representation of a circular capillary meniscus (1), transition zone (2), wetting ﬁlms (3) in a ﬂat capillary. Continuation of a spherical meniscus (broken line) does not intersect either the solid walls of the capillary or the thin liquid ﬁlm of thickness he in front of the meniscus. The radius of the curvature of the meniscus, Pe, is smaller than the half-width H.

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76

Wetting and Spreading Dynamics

In the case of complete wetting and disjoining pressure isotherms of the type Π(h) = A/hn, Equation 2.21 and Equation 2.26 result in γ = Pe ( H − he ) −

A , ( n − 1) hen −1

and A = Pe , hen where n is the exponent in the expression for the isotherm, γ is the surface tension of the bulk liquid, and H is the half-width of the capillary. That is, we have two equations with two unknown values, Pe and he. The preceding two equations determine the equilibrium pressure, Pe, via the thickness of the equilibrium ﬂat ﬁlm, he . This results in Pe =

γ γ n he > he . = , h* = n n −1 H − h∗ H− he n −1

(2.62)

Note that the equilibrium pressure is equal to Pe =

γ γ , = Pe H − h∗

according to Equation 2.62. This means that the radius of the curvature of the meniscus (in the case of the complete wetting) Pe = H − h* < H , and the continuation of the spherical meniscus does not intersect either the capillary walls or the ﬂat liquid ﬁlms in front of the meniscus (Figure 2.18). According to Section 2.2 and Section 2.3, the proﬁle of the meniscus, the transition zone, and the ﬂat wetting ﬁlms in front are described by Equation 2.23, which in the case under consideration becomes γh′′

(1 + h′ ) 2

3/ 2

+

A = Pe , hn

(2.63)

where h(x) is the local liquid proﬁle, A/hn is the local disjoining pressure isotherm. Inside the transition zone (Figure 2.18), the liquid proﬁle has a very low slope,

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Equilibrium Wetting Phenomena

77

that is, (dh/dx)2 << 1 is satisﬁed, and the disjoining pressure isotherm, A/hn, of ﬂat ﬁlms can be safely used at each point of the liquid layer of varying thickness. As previously mentioned, we assume that the surface tension of the ﬁlm is the same as that of the bulk liquid. The following consideration is carried out for the case of complete wetting. As the liquid proﬁle inside the transition zone satisﬁes the condition (dh/dx)2 << 1, the low slope approximation can be used. Introducing the dimensionless variables, ξ = h/he and y = [x – (H – h*)]/l, where l is a length scale to be determined along x. It is shown in the following text that y is the local variable inside the transition zone. Using these notations in Equation 2.63, we arrive at ξ′′ +

1 = 1, ξn

(2.64)

where ξ = ξ(y), and the length scale is selected as

(

)

l = he H − h* .

(2.65)

Note that the latter selection is in excellent agreement with our previous estimation of the length of the transition zone in Section 2.3 (Equation 2.49). The thickness of the equilibrium ﬂat ﬁlm, he, is determined as before from the condition A/h ne = Pe . According to Jacoby’s condition, the dependency ξ(y) is a monotonic one. Therefore, as the independent variable, y, does not appear explicitly in Equation 2.64, we can introduce a new unknown function, ξ′ = function(ξ). Taking into account that ξ′(1) = 0, Equation 2.64 can be rewritten as

(

)

ξ′ = − 2 ξ − 1 +

 2  1 − 1 . n −1  (n − 1)  ξ 

(2.66)

The preceding equation is solved for the most important cases: n = 3 and n = 2. Films that obey the Π = A/h3 law (that is, n = 3) correspond to the case where the disjoining pressure of the ﬁlm is determined by dispersion forces [1]. For nonpolar liquids on solid dielectrics, we can adopt A = 10–14 erg [1] and γ = 30 dyn/cm. In particular, for n = 3, it follows from Equation 2.66 that ξ′ = −

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ξ −1 2ξ + 1. ξ

78

Wetting and Spreading Dynamics

Upon integration, the solution of the latter equation is

2ξ + 1 +

1 3

2ξ + 1 − 3

ln

2ξ + 1 + 3

(

)

= − y+C ,

(2.67)

where C is the integration constant. If ξ( y ) >> 1 in Equation 2.67, that is, in the region where the transition zone and the meniscus meet each other, we conclude that ( y + C )2 . 2

ξ( y ) ≈

(2.68)

In order to determine C, we consider the meniscus shape corresponding to large h values for which it can be assumed in Equation 2.63 that the disjoining pressure can be neglected, which results in γh′′

( )

1 + h′    2

3/ 2

= Pe =

γ . H − h*

Integration of this equation with the boundary condition, h(0) = H, results in the following solution for the spherical meniscus:

( H − h ) − ( x − ( H − h ))

2

2

h( x ) = H −

*

*

.

Using the same local variables previously mentioned for the transition zone, ξ(y) and y << 1, in the latter equation, we conclude that

ξ( y ) ≈

n y2 + . n −1 2

(2.69)

Comparison of Equations 2.68 and Equation 2.69 results in C = 0. That is, Equation 2.67 can now be rewritten as

2ξ + 1 +

© 2007 by Taylor & Francis Group, LLC

1 3

ln

2ξ + 1 − 3 2ξ + 1 + 3

= − y.

(2.70)

Equilibrium Wetting Phenomena

79

An “ideal proﬁle,” ξ i ( y ) , is as shown in the following function:  n y2 + , y < 0.  ξi (y ) =  n − 1 2  1, y > 0

(2.71)

The “real proﬁle,” calculated according to Equation 2.70, and the ideal proﬁle, according to Equation 2.71 at n = 3, are presented in Figure 2.19. It shows that the extent of the transition zone can be roughly estimated from y = 1.2 to y = 1.7. That is, the total length of the transition zone, L, in dimensional units is

(

)

L = 2.9 l = 4.3 he H − h* . Figure 2.19 shows that the maximum deviation of the real proﬁle from ideal proﬁle is at y = 0, that is, at the position corresponding to minimum value of continuation of the spherical meniscus, and the maximum deviation is around 2.5 he. Figure 2.19 also shows that the deviation of the real proﬁle from the ideal proﬁle is roughly symmetrical from both sides, pertaining to the position of the maximum deviation at y = 0. ξ

3.0

2.5

2.0 2 1.5

1.0 1 0.5 y –2

–1

0

1

2

3

FIGURE 2.19 Real proﬁle (1) inside the transition zone calculated according to Equation 2.70 and the ideal proﬁle (2,3) according to Equation 2.71 at n = 3.

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80

Wetting and Spreading Dynamics

The isotherm Π = A/h2 corresponds to thick β ﬁlms of water, which are stabilized by the ionic–electrostatic component of the disjoining pressure [1]. In this case, integration of Equation 2.66 at n = 2 results in

ξ+

ξ −1 1 y , =− ln 2 ξ +1 2

(2.72)

where the integration constant is equal to zero, which is concluded in precisely the same way as in the case, n = 3. Table 2.2 shows that although the absolute thickness of the transition zone, L, decreases with a decrease in the capillary radius, H, the relative transition zone thickness, L/H, increases with a decrease in H. The L values vary within a range from 37 ηm for thick ﬁlms (he ~ 500 Å) to 0.2 ηm for thin ﬁlms (he ~ 30 Å). In the case n = 2, calculations made using Equation 2.72 and adopting A = 2⋅10–7 dyn, γ = 72 dyn/cm (Table 2.3), and the ideal proﬁle (according to Equation 2.71 at n = 2) are presented in Figure 2.20. The ﬁgure shows that the extent of the transition zone can be roughly estimated from y = –2.4 to y = 1.9. That is, the total length of the transition zone, L, in dimensional units is

(

)

L = 4.3 l = 4.3 he H − h* . It also shows that the maximum deviation of the real proﬁle from the ideal proﬁle is at y = 0, that is, at the position that corresponds to the minimum value of the

TABLE 2.2 Characteristics of the Transition Zone for the Case of the Π = A/h3 Isotherm H, cm he, Å L, cm

0.3 445 3.7·10–3

0.2 405 2.88·10–3

0.1 322 1.81·10–3

5.10–2 256 1.14·10–3

10–2 150 3.92·10–4

10–3 70 0.85·10–4

10–4 32 1.81·10–5

10–3 166 2.04·10–4

10–4 53 3.64·10–5

TABLE 2.3 Characteristics of the Transition Zone for the Case of the Π = A/h2 Isotherm H, cm he, Å L, cm

1 5250 3.62·10–2

0.5 3730 2.1·10–2

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0.2 2640 115·10–2

0.1 1660 6.44·10–3

10–2 525 1.15·10–3

Equilibrium Wetting Phenomena

81 ξ

7 6 5 4

2

3 2 1 1 y –3

–2

–1

0

1

2

3

FIGURE 2.20 Real proﬁle (1) inside the transition zone calculated according to Equation 2.72 and the ideal proﬁle (2,3) according to Equation 2.71 at n = 2.

continuation of the spherical meniscus, and the maximum deviation is around 2.5he. However, now the deviation is bigger from the meniscus side than from the ﬂat ﬁlm side. Figure 2.20 also shows that the deviation of the real proﬁle from the ideal proﬁle is not symmetrical from both sides, from the position of the maximum deviation at y = 0, but decreases more rapidly from the ﬂat liquid ﬁlm side and is extended more into the depth from the meniscus side. This behavior is different from the case n = 3. The thickness of the L regions is larger at n = 2 than at n = 3 at equal H values; the maximum L values are about 362 µm for thick ﬁlm in wide slots (H = 1 cm, he ~ 0.525 µm) and 0.36 µm for thin ﬁlms (H = 1 µm, he = 53 Å). Thus, the transition zone is very much extended for low capillary meniscus pressures and thick liquid ﬁlms. The radius of the meniscus curvature must be studied outside the transition zone, i.e., at a distance bigger than L, from the apparent three-phase contact line.

2.5 THICKNESS OF WETTING FILMS ON ROUGH SOLID SUBSTRATES Let us now consider the case of thin equilibrium liquid ﬁlms on rough solid substrates when there is complete wetting. In such a case, it is possible to measure – – only the mean thickness of the ﬁlm, h. It appears that the mean thickness, h, of wetted ﬁlms on rough solid surfaces is bigger than the corresponding thickness – of a ﬂat ﬁlm, he, on a smooth substrate. It also appears that the mean thickness, h, approaches the value, he, at high and low ﬁlm thicknesses when the latter are © 2007 by Taylor & Francis Group, LLC

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smaller or bigger –in relation to the characteristic scale of roughness of the solid substrate, α. For h >> α, the effect of the roughness is made negligible, and the surface– of the ﬁlm at the interface with the gas becomes practically smooth. When h << α, the ﬁlm copies the surface of the substrate, maintaining a constant value of the ﬁlm per unit area of surface. Hence, the maximum deviation of the thickness of liquid ﬁlms on a rough substrate– from the corresponding thickness on a ﬂat substrate should be expected when h ≈ α. When a wetting ﬁlm of uniform thickness covers a curved surface, its equilibrium with the vapor of the same liquid is determined by Equation 2.23 in Section 2.2, which can be rewritten as γK + Π( h ) =

RT p ln s = Pe , vm p

(2.73)

where K is the capillary pressure due to the local curvature of the surface of the ﬁlm; γ is the surface tension; Π(h) is the disjoining pressure, which is a function of the local thickness of the ﬁlm h; R is the gas constant; T is the temperature; vm is the molar volume of the liquid; p is the equilibrium pressure of the vapor above the ﬁlm; and ps is the pressure of the saturated vapor of the liquid. Let us make a further examination for nonpolar one-component liquids (complete wetting), where the disjoining pressure isotherm, Π(h), is determined only by the dispersion forces. The isotherm of the disjoining pressure of a ﬂat ﬁlm in this case has the same form as in the previous Section 2.4. Π(h) = A/h n ,

(2.74)

where n = 3 for small and n = 4 for large thicknesses of the ﬁlm. The Hamaker constant A is characterized on the basis of the spectral characteristics of the ﬁlm and the substrate [9]. In the general case, the disjoining pressure in thin ﬁlms on a curved substrate, Πr ( h ) , is different from the corresponding disjoining pressure in ﬁlms on a ﬂat substrate, Π(h). Thus, for example, for ﬁlms at the internal surface of a capillary of radius, r, the disjoining pressure, Πr , (in the approximation h << r) is given by [10]    h Πr = Π( h ) 1 +  2r  

∞

∫ 0

 ∆ 32 ⋅ ∆ 21 ∆ 32 + ∆ 21 d ξ   , ∞   ∆ 32 ⋅ ∆ 21dξ  0

(

)

(2.75)

∫

where Π(h) is the disjoining pressure of a ﬂat ﬁlm of the same thickness on a ﬂat substrate; © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

83

∆ ik =

εi − ε k , εi + ε k

ε(iξ) is the dielectric permeability, which is a function of the angular frequency ξ, taken at the imaginary axis [11]. The subscripts 1, 2, and 3 relate to the gas, the ﬁlm, and the solid substrate, respectively. At h/r → 0, Πr → Π. Quantitative evaluations for ﬁlms of decane on the surface of quartz show that the contribution of the second term on the right-hand side of Equation 2.75 is relatively small. Thus, with h/r ≤ 0.2, the difference between Πr and Π does not exceed 2.5%. Thus, under the condition of very small curvature, h/r, the isotherm (Equation 2.74) of the disjoining pressure of ﬂat ﬁlms can be used with a sufﬁcient precision. Real surfaces, as a rule, have a roughness. In this case, the local thickness of the ﬁlm is a function of the coordinate, and a mean value of the thickness of the – – ﬁlm, h, should be used. The problem is how signiﬁcantly the mean thickness, h, differs from the thickness of a ﬂat ﬁlm on an ideally smooth surface of the same nature, and how these differences affect the roughness (or topology) of the surface. Let us consider a simpliﬁed model of one-dimensional roughness (Figure 2.21), where the proﬁle of the surface is a function of one coordinate x. Let Hs (x) be the equation of the surface of the substrate, and H(x) be the equation of the surface of the ﬁlm, forming a boundary with the gas. The local thickness of the ﬁlm is determined as h( x ) = H ( x ) − Hs ( x ) .

(2.76)

Using the latter notations, Equation 2.73 takes the following form γ H ′′ 1 + ( H ′)2   

3/ 2

+

A = Pe , hn

(2.77)

H

H(x) h(x)

0

Hs(x)

x

FIGURE 2.21 Calculations of the thickness of wetting ﬁlms on a rough cylindrical surface. Hs(x) is the proﬁle of the solid substrate, H(x) is the liquid proﬁle, and h(x) = H(x) – Hs(x) is the ﬁlm thickness. © 2007 by Taylor & Francis Group, LLC

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where H′ and H″ are the ﬁrst and the second derivatives of H(x), and  RT   ps  ln Pe =   vm   p  is determined by the vapor pressure in the ambient air. On a plane substrate, the liquid curvature is zero and Equation 2.77 gives A = Pe . hen

(2.78)

Substitution of Equation 2.78 into Equation 2.77 results in  A A d γ H ′′ = − n − n  . 1/ 2 dx 1 + ( H ′)2  he  h  

(2.79)

Let us introduce the average values of any function, ϕ, of a random variable as follows: ϕ = lim

X →∞

1 X

X

∫

ϕ( x ) dx ≈

0

1 X

X

∫ ϕ( x)dx,

X >> λ ,

0

where λ is a characteristic scale of surface roughness in the direction x; the overbar means averaged over random substrate. Let us integrate both sides of Equation 2.79 from 0 to X. We assume that the surface is statistically homogeneous, that is, there is no preferable positive or negative curvature of the liquid ﬁlm. We can subdivide the whole interval of integration form 0 to X into a big number, N, of subintervals of a small length, λ = X/N. After that, 1 X

X

∫ 0

1 Kdx = X =

N

xi +1

∑∫ i=0

1 1  λ  N

Kdx =

xi

1 X

N

∑ (sin θ(x

N

∑ i=0

i +1

) − sin θ( xi )

i=0

sin θ( xi +1 ) −

1 N

N

∑ i=0

)

 1  _____ _____  sin θ( xi ) =  sin θ− sin θ = 0,  λ  

xi = ∆ ⋅ i, i = 0, 1, 2, … N , x N = X , where θ is the local slope of the liquid proﬁle. Hence, in the case of random and statistically homogeneous roughness, the average value of the left-hand side of Equation 2.79 vanishes, and the average of the right-hand side results in © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

1 X

X

85

 A

∫  h

n

−

0

A dx = 0. hen 

(2.80)

Equation 2.80 can be rewritten in the following form: 1 X

X

∫ h dx = P , A

n

e

0

that is, the measured average disjoining pressure on a rough substrate coincides with the disjoining pressure on a corresponding ﬂat substrate. Hence the latter expression can be rewritten as A = Pe . hn

(2.81)

Let us recall a well-known theorem from the probability theory. Let us consider a concave function, ϕ, of random variable, h. Then ______

ϕ(h) > ϕ(h ),

that is, the average of the concave function is bigger than the function of the average. The disjoining pressure isotherm of the type under consideration is a concave function of h because the second derivative is positive:  A  ′′ n(n − 1) A  h n  = h n− 2 > 0. Application of the aforementioned theorem results in A A > n. n h h Comparison of this inequality and Equation 2.81 results in A A > n, n he h

© 2007 by Taylor & Francis Group, LLC

86

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and hence, h > he .

(2.82)

The preceding inequality shows that, on a rough solid substrate, the measured average thickness of the liquid ﬁlm is bigger than the equilibrium thickness of the ﬁlm on a corresponding ﬂat substrate. In the following text, we investigate how the thickness of the ﬁlm on a rough substrate inﬂuences the disjoining pressure measurements. For that purpose, we consider a model rough surface of the following kind: Hs (x) = α cos kx

(2.83)

that is, a periodic roughness with an amplitude α. This allows us to restrict our consideration to x in the following range: −

π π <x< . k k

In the case of the model roughness (Equation 2.83), we consider only half of the period. Hence, the following boundary conditions are satisﬁed: H ′ (0 ) = 0 ,

(2.84)

H ′(X) = 0,

(2.85)

and

where X = π/k is the half period. To simplify the calculation shown in the following text, we assume that δ = α · k << 1. Then, the model roughness has a low slope, and the solution of Equation 2.77 can be expanded in a series in terms of the small parameter H = he + H1 + H 2 +, …,

(2.86)

where Hi (x) ~ δi, i = 1, 2, …. We limit ourselves to the ﬁrst two terms of the expansion, as the nonlinearity of the curvature is of the third order of smallness. Using these notations, we can write A A nA n(n + 1) A = − H1 + H 2 +…− H s + H1 + H 2 +…− H s h n hen hen+1 2hen+ 2

(

© 2007 by Taylor & Francis Group, LLC

)

(

)

2

+….

Equilibrium Wetting Phenomena

87

Substitution of the preceding expression into integral (Equation 2.80) and collecting only the terms of the ﬁrst and second order pertaining to the small parameter δ, we conclude that, in the ﬁrst order, X

∫ ( H − H )dx = 0 , 1

(2.87)

s

0

and collecting the two terms of the second order, nA − n +1 he

X

∫ 0

n ( n + 1) A H 2 dx + hen +2

X

∫ ( H − H ) dx = 0 . 2

1

s

0

The latter equation can be rewritten as X

∫ 0

( n + 1) H 2 dx = he

X

∫ ( H − H ) dx . 2

1

(2.88)

s

0

Let us now determine the average thickness as 1 h = he + X

X

∫( o

)

1 H1 + H 2 +…− H s dx ≈ he + X

X

∫( o

)

1 H1 − H s dx + X

X

∫ H dx. 2

o

According to Equation 2.87, the ﬁrst term on the right-hand side of the preceding equation is equal to zero, and the second term is given by Equation 2.88. Hence, the latter equation can now be rewritten as ( n + 1) h = he + Xhe

X

∫ ( H − H ) dx . 2

1

s

(2.89)

0

The integral on the right-hand side of the preceding equation is always positive. Hence, we come to the same conclusion as in the general case (see the inequality Equation 2.82 previously mentioned): the average ﬁlm thickness on a rough substrate is always bigger than the corresponding ﬁlm thickness on a ﬂat substrate. Equation 2.89 shows that we have to determine only the ﬁrst function, H1(x), – to calculate the second correction to the average thickness, h.

© 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

Substituting the expansion (2.86) into Equation 2.77 and collecting the terms of an identical order, we obtain A = Pe , H 0n

(2.90)

which shows that  A H0 =    Pe 

1/ n

is a constant that is equal to the thickness of the equilibrium ﬁlm on a ﬂat substrate, he. The next equation is γ H1′′−

(

)

nA ⋅ H1 − H s = 0 hen+1

(2.91)

with the periodic boundary conditions  π H1′ 0 = H1′   = 0.  k

()

(2.92)

Introducing the notation a2 =

nA , γhen +1

instead of Equation 2.91 we arrive at H1′′− a 2 H1 + αa 2 ⋅ cos⋅ kx = 0.

(2.93)

Solution of Equation 2.93 with the boundary conditions given by Equation 2.92 is H1 =

αa 2 ⋅ cos⋅ kx . a2 + k 2

For a crest of the sinusoid, from Equation 2.94 we get  π αk 2 h   = he − 2 ,  k a + k2 © 2007 by Taylor & Francis Group, LLC

(2.94)

Equilibrium Wetting Phenomena

89

and for a trough,

()

h 0 = he +

αk 2 , a + k2 2

i.e., the ﬁlm is thicker in a depression than on a convexity. Using Equation 2.94, we conclude:

H1 − H s = −

αk 2 ⋅ cos⋅ kx . a2 + k2

Substitution of this expression into Equation 2.89 results in the following second approximation of the mean thickness of the ﬁlm:

h = he +

(

)

α 2 k 4 γPe1/ n n + 1 1/ n

2A

  nPe1+1/ n + k2   1/ n   γA

2

.

(2.95)

The preceding equation shows that ∆ = h − he tends to zero, at both   p Pe → ∞  or → 0 ps   and  Pe → 0  or 

 p → 1 . ps 

The difference in the thickness of the ﬁlms ∆ = h − he on a rough and smooth substrate goes via a maximum deviation at some value of Pe max, which can be determined by direct differentiation of the second term on the right-hand side of Equation 2.95. This results in

Pe max –

 γk 2  =   n (2 n + 1) 

( )

n / n +1

1/ n +1 ⋅A ( ).

(2.96)

At h >> α, the effect of the roughness is damped, and the surface of the ﬁlm – at the interface with the gas becomes practically smooth. At h << α, the ﬁlm copies the surface of the substrate. Hence, the maximum deviation of the thickness © 2007 by Taylor & Francis Group, LLC

90

Wetting and Spreading Dynamics lg h

1 lg hmax

2

lg Pe max

lg Pe

FIGURE 2.22 Disjoining pressure isotherm Pe(he) = A/hn. Sketch of deviations of the – measured average liquid ﬁlm thickness, h, from the predicted ﬁlm thickness on a ﬂat surface, he . (1) thickness of the ﬁlm on a ﬂat substrate; (2) average ﬁlm thickness on a rough substrate.

of liquid ﬁlms on a rough substrate from the corresponding thickness on a ﬂat – substrate should be expected at h ≈ α . – The approximate form of the function h (Pe) is shown in Figure 2.22 (curve 2). The straight line (1) illustrates the disjoining pressure isotherm of ﬂat ﬁlms according to Equation 2.74. The maximum deviations of the isotherm (curve 2) from the isotherm (line 1) correspond to the values of Pe max. Because polished surfaces, as a rule, have grooves left by the solid grains of the polishing pastes, these surfaces are expected to have a qualitative picture presented in Figure 2.22. This was experimentally observed in Reference 12 for wetting ﬁlms of tetradecane on the polished surfaces of steel, where the qualitative picture presented in Figure 2.22 was experimentally observed in the case of disjoining pressure isotherm for complete wetting, n = 3 (Figure 2.23).

2.6 WETTING FILMS ON LOCALLY HETEROGENEOUS SURFACES: HYDROPHILIC SURFACE WITH HYDROPHOBIC INCLUSIONS Liquid ﬁlms on heterogeneous solids, that is, solids where solid hydrophilic surfaces include hydrophobic spots, are considered in this section [31,32]. Changes in the proﬁle of a wetting ﬁlm over the spots and its eventual breakdown are expected. The combined action of the disjoining pressure and capillary forces should allow the prediction of the critical width of the hydrophobic spot on such heterogeneous substrates before wetting ﬁlm breakdown. Needless to say, this critical size is supposed to depend on the parameters of the disjoining pressure isotherms of the hydrophilic and hydrophobic parts and the relative vapor pressure in the surrounding medium. Let us take two different thin liquid ﬁlm disjoining pressure isotherms for the hydrophilic and hydrophobic parts of the substrate. © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

91

2.6

lg h

2.5

2.4

2.3 2.2 2.7

2.9

3.1

3.3

3.5

lg Π

FIGURE 2.23 Experimental data on average ﬁlm thickness of tetradecane on steel, obtained by continuous thinning of the ﬁlm (open symbols) and subsequent thickening (closed symbols). Solid line according to the disjoining pressure isotherm Π = A/h3 [12].

Complete wetting is assumed for the hydrophilic part and partial wetting for the hydrophobic or less hydrophilic spot. The equilibrium proﬁles should be calculated according to the spot size and the value θe of the equilibrium contact angle of the hydrophobic spot. It is shown in the following text that the critical width of a hydrophobic spot decreases by an order of magnitude with an increase in the contact angle of the more hydrophobic spot from 10˚ to 90˚. Let us consider a ﬂat hydrophilic surface covered by a sufﬁciently thick wetting ﬁlm in the presence of a hydrophobic strip of width 2L (Figure 2.24a). The origin of the x-axis corresponds to the middle of the strip. As a reference system, a state was chosen in which each part of the surface is covered by a corresponding equilibrium ﬁlm and an interaction between the ﬁlms is absent. The equilibrium thickness of each of the ﬁlms, He and he, is determined by the corresponding disjoining pressure isotherms, Π(H) and π(h), respectively:

( )

( )

Π H e = π he = Pe =

RT p ⋅ ln s , vm p

(2.97)

where Pe is the excess pressure of the ﬁlm as compared with a bulk liquid at the same temperature T, R is the gas constant, vm is the molar volume of the liquid, and p and ps are the equilibrium and saturated vapor pressure, respectively. Between two idealized states of the ﬁlms as shown in Figure 2.24a, a transition zone is formed, the possible shape of which is shown in Figure 2.24b, and Figure 2.24c. Consideration of the variations in the free energy of the system (similar to Section 2.1) results in the following set of differential equations © 2007 by Taylor & Francis Group, LLC

92

Wetting and Spreading Dynamics H h He a he

H b

x L

0 h

He H(x)

h(x) he

x

L

0 H h

He c x0

he

x L

0

FIGURE 2.24 Schematic representation of a wetting ﬁlm on a solid surface containing a hydrophobic spot at –L < x < L. (a) Reference system: stepwise ﬁlm proﬁle, equilibrium ﬁlm thickness He and he, on the corresponding hydrophilic and hydrophobic parts, if each part is unbounded. (b) Depression formation over a hydrophobic spot when the width L of a hydrophobic part is smaller than critical value Lc . (c) Rupture of a wetting ﬁlm if the width of the hydrophobic part L > Lc , and formation of a thin ﬁlm on the hydrophobic part.

enabling calculation of the proﬁle of the transition zone for hydrophobic and hydrophilic regions of the surface, respectively:

()

γh′′ + π h = Pe ,

( )

γH ′′ + Π H = Pe ,

at 0 < x < L ,

at

x > L,

(2.98)

where γ is the surface tension of the liquid; h′, H′ and h″, H″ are the ﬁrst and the second derivatives of the ﬁlm thickness over x within the zones 0 < x < L and x > L, respectively. © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

93

The conditions according to Equation 2.98 are applicable in the case of low slope proﬁles, when (h′) << 1 and (H′) << 1. The ﬁrst terms on the left-hand side of Equation 2.98 determine the local capillary pressure, and the second terms determine the local disjoining pressure. The pressure Pe sets the chemical potential of the system and the relative vapor pressure according to Equation 2.97. In the case of ﬂat ﬁlms, the capillary terms are equal to zero. Let us now formulate the boundary conditions that are used for solving Equation 2.98. The ﬁrst condition characterizes the equilibrium state of the thick hydrophilic ﬁlm, far from the hydrophobic spot:

()

H x → H e at x → ∞.

(2.99)

The second condition reﬂects the symmetry of the system

()

h′ 0 = 0.

(2.100)

To match the proﬁles at x = L, the following two conditions of continuity of the liquid proﬁle should be used:

( )

( )

(2.101)

( )

( )

(2.102)

h L =H L ; and h′ L = H ′ L .

As already discussed in Section 2.2, not all solutions of Equation 2.98 with boundary conditions (2.99 through Equation 2.102) describe the stable equilibrium proﬁles, but only those that satisfy Jacoby’s condition (3) (Section 2.2). This condition in application to the previously discussed problem has the following form:

( )

()

γ U ′ ′ + π′ h U = 0, at 0 < x < L;

( )

( )

γ V ′ ′ + Π′ H V = 0, at x > L,

(2.103)

with the following boundary condition:

()

U 0 = 0, and continuity at x = L, where U(x) and V(x) are the Jacoby functions. The requirement of the stability of the solution is as follows: the solution of Equation © 2007 by Taylor & Francis Group, LLC

94

Wetting and Spreading Dynamics

2.103 does not vanish anywhere except at the point x = 0 and x = ∞. The solution is given as follows: After differentiating the conditions given by Equation 2.98 over x, we obtain:

( γh′′ )′ + π′ ( h ) h′ = 0, at 0 < x < L; ( γH ′′ )′ + Π′ ( H ) H ′ = 0, at x > L .

(2.104)

Comparing the given equations with Equation 2.103, we conclude that U(x) = const·h′(x) and V(x) = const·H′(x). That means, the proﬁles h(x) and H(x) must behave in a monotonous way inside the corresponding zones, as shown in Figure 2.24b and Figure 2.24c. Nonmonotonous behavior results in the loss of stability. Further calculations are made using simpliﬁed expressions for isotherms of disjoining pressure (Figure 2.25) consisting of linear parts: ∞, h < he ,   π h =  a ( h − t ), he < h < ts ,  0, h > ts

(2.105)

 A(t − H ), 0 < h < ts , Π h = 0, h > ts .

(2.106)

()

()

+Π P2 1 Pe ts He

he

H h

2 –Pmin –Π

FIGURE 2.25 Simpliﬁed forms of disjoining pressure isotherms of the ﬁlms formed on the hydrophilic surface, Π(h) (curve I) and on its hydrophobic part, π(h) (curve 2). © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

95

The thickness, ts, characterizes the range of action of surface forces and is selected identically for both the disjoining pressure isotherms. The equilibrium thickness, he, is independent of the vapor pressure in the surrounding air, p, according to adopted isotherm, Equation 2.105. The ﬁlm thickness, He, according to Equation 2.106, depends on the vapor pressure in the surrounding medium. The parameters A=

P2 P and a = min ts ts − he

determine the slope of the isotherms. For the ﬁlms on hydrophilic surface (curve 1, Figure 2.25), the isotherm ranges from Π = 0 (when

p = 1 and He = ts) to Π = P2, ps

when He = 0. At some deﬁnite value of the pressure, Pe (between 0 and P2), the ﬁlm thickness equals He (Figure 2.25). Such a form of the isotherm corresponds to the case only when the repulsion forces (Π > 0) act in the ﬁlm and complete wetting takes place. Equilibrium ﬁlms on a hydrophobic surface (curve 2 in Figure 2.25) have smaller thickness he, which is adopted to be independent of relative vapor pressure in the range of Pe higher than –Pmin. At h > he, attractive forces act in the ﬁlms (Π < 0), which makes the ﬁlms unstable in this region of thickness. Stable ﬁlms in the system are considered only at undersaturation, that is, at p <1 ps and Pe > 0. The less hydrophilic (hydrophobic) spot may be characterized by the value of the contact angle, θe, which is a droplet of the liquid form on the hydrophobic substrate. The contact angle, θe, is calculated on the basis of the equation deduced in Section 2.1 (Equation 2.9) using the disjoining pressure isotherm, π(h), of the ﬁlms on the hydrophobic substrate: ts

∫ ()

ts

∫ ()

1 1 1 cos θe = 1 + Pe he + ⋅ π h dh ≈ 1 + ⋅ π h dh . γ γ γ he

(2.107)

he

Substituting the model disjoining pressure isotherm, Equation 2.105, into Equation 2.107, we obtain: © 2007 by Taylor & Francis Group, LLC

96

Wetting and Spreading Dynamics

cos θe = 1 −

(

a ts − he 2γ

)

2

.

(2.108)

In the framework of the adopted model, we can characterize the state of the hydrophobic surface by the contact angle, θe, that is calculated using Equation 2.108. This equation includes parameters a, ts, and he of the isotherm, as well as the surface tension γ of the liquid. We would like to analyze two possible situations that are schematically shown in Figure 2.24b and Figure 2.24c. In the ﬁrst case, the transition zones between hydrophilic and hydrophobic parts overlap, and the ﬁlm thickness in the middle (at x = 0) is higher than the equilibrium ﬁlm thickness of the hydrophobic spot, he. In the second case, in the middle of a wider hydrophobic spot, the ﬁlm thickness is equal to the equilibrium value, he, and deviation from this thickness starts only at x > x0. According to the transversality requirement discussed in Section 2.2 (condition 4), the condition h′(x0) = 0 holds at x = x0. Let us consider the ﬁrst case, Figure 2.24b. Equation 2.98, which determines the ﬁlm proﬁle, takes the following form using the disjoining pressure isotherms given by Equation 2.105 and Equation 2.106:

(

)

γh′′ + a h − ts = Pe ,

(

)

γH ′′ + A ts − H = Pe .

(2.109)

(2.110)

The solution of Equation 2.110 that satisﬁes the boundary condition (2.99) is: 1/ 2   A  H = H e + C 2 exp  − x − L    , γ    

(

)

(2.111)

where C 2 is an integration constant. Solution of Equation 2.109 is different for the ﬁrst and the second cases (Figure 2.24b and Figure 2.24c, respectively). In the ﬁrst case (Figure 2.24b), the solution that satisﬁes the symmetry condition (2.100) has the form

h=t+

 a Pe + C1 ⋅ cos⋅  x . a  γ ⋅

(2.112)

The integration constants, C1 and C2, should be determined using boundary conditions (Equation 2.101 and Equation 2.102) at x = L, at the border between © 2007 by Taylor & Francis Group, LLC

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the hydrophilic and hydrophobic zones. It follows from Jacoby’s conditions that the proﬁle within the hydrophobic zone (between x = –L and x = L) is stable when the following restriction is satisﬁed:

( )

L a/γ

1/ 2

<π

In this case, the solutions of Equation 2.110 vanish only at the point x = 0. This restriction gives an estimation of the critical width of the hydrophobic spot when the stability condition is violated. Therefore, this condition gives the critical size of the hydrophobic spot:

( )

L < π ⋅ γ /a

1/ 2

.

(2.113)

From the two boundary conditions (Equation 2.101 and Equation 2.102), and Equation 2.111 and Equation 2.112, two algebraic equations are obtained, which are used for the determination of the unknown constants C1 and C2: C2 −

Pe Pe = + C1 cos⋅ β , A a

( )

C2 A/a

1/ 2

(2.114)

= C1 sin β,

(2.115)

( )

1/ 2

where β = L a/γ . Taking into account Equation 2.113, the constant β can range between 0 and π …, and hence, sin β > 0. Therefore, parameters C1 and C2 have the identical sign. Substitution of the expression for C2 from Equation 2.114 into Equation 2.115 results in

( ) ( )

( )

C1 = − Pe⋅  1/A + 1/a   cos β − a /A 

1/ 2

sin β  . 

(2.116)

The proﬁle of the liquid within the hydrophobic zone must have monotonously increasing thickness at the point x between 0 and L (Figure 2.24b), which means that the value of the parameter C1 must be negative. However, the latter is possible only when cos β >

a sin β . A

This leads to a stronger restriction of the critical width of the hydrophobic zone as compared with Equation 2.113: L < (γ/a)1/2 arctan (A/a)1/2. Owing to © 2007 by Taylor & Francis Group, LLC

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( )

1/ 2

arctan A/a

<

π , 2

a more accurate deﬁnition of the critical width of the hydrophobic zone when the wetting ﬁlm ruptures is:

( )

1/ 2

L = γ /a

π . 2

(2.117)

Further analysis shows that a stronger limitation of the critical L values exists, which follows from the condition that ﬁlm thickness at x = 0 cannot be lower than he, where he, is the equilibrium ﬁlm thickness at the center of the hydrophobic spot at any given pressure, Pe, and corresponding value of the vapor pressure, p/ps, in the surrounding media. From Equation 2.112 for the ﬁlm proﬁle and Equation 2.116 for the parameter C1 at x = 0, the expression for the thickness of the ﬁlm in the center, h0 is: h0 = t −

( ) ( )

( )

Pe − Pe  1/A + 1/a   cos β − a /A  a

1/ 2

sin β  . 

(2.118)

Hence, the preceding equation and the condition h0 = he ﬁnally determine the critical length of the hydrophobic spot, Lc:

( )

cos βc − a /A

1/ 2

 P P  P sin βc = t − e − he   e + e  , a   A a 

(2.119)

where βc = Lc (γ/a)1/2. As distinct from the previous approximations (Equation 2.113 and Equation 2.117), the critical width of the hydrophobic zone, Lc, according to Equation 2.119 depends on the parameters of both isotherms, a and A, as well as on the relative vapor pressure in the surrounding media that is characterized by the pressure Pe. The proﬁles of a transition zone beyond the hydrophobic spot, at x > L, is calculated using Equation 2.111 and Equation 2.115. At x = L, the two proﬁles are matched according to boundary conditions (Equation 2.101 and Equation 2.102). Let us ﬁnd the critical width of the hydrophobic zone using a simpliﬁed deﬁnition given by Equation 2.117. In this case, the value of Lc may be calculated dependent on the degree of surface hydrophobicity that is characterized by the contact angle. Substituting the expression for 1 − cos θe from Equation 2.108 into Equation 2.117, we conclude

(

(

Lc = π ts − he © 2007 by Taylor & Francis Group, LLC

)

(

)

1/ 2

2  2 1 − cos θe 

)

(

≈ 1.1 ⋅ ts 1 − cos θe

)

1/ 2

.2

(2.120)

Equilibrium Wetting Phenomena

99

Lc/ts

10

5

0

π/3

π

2π/3 θ

FIGURE 2.26 Calculated according to Equation 2.24, dependence of the critical width of a hydrophobic spot, Lc , at which wetting ﬁlm ruptures, on the value of contact angle θ that characterizes the hydrophobic spot on the surface.

The results of calculations of the dependence of Lc/ts on the contact angle θe are shown in Figure 2.26. These results show that the values of the critical width, Lc, decrease with an increasing contact angle and fall sharply at θe > 30˚. Suppose the range of action of surface forces, ts, is of the order of 10–6 cm; we may then conclude that the critical width of a hydrophobic spot decreases from Lc ≈ 10–5 cm at θe = 10˚ to Lc ≈ 10–6 cm at θe = 180˚. Note that the Equation 2.119 shows that at p → 1, ps wetting ﬁlm thickness, He, approaches its highest value ts, and the critical width tends to decrease. The prediction of the theory is in line with experimental investigations of wetting ﬁlm stability on heterogeneous methylated glass surfaces [13]. Film rupturing is sensitive to the contact angle values at θe < 45˚. At larger values of contact angles, the effect is practically not dependent on the degree of hydrophobicity of the hydrophobic spot. In a similar way, the length x0 (the extension of a part of the hydrophobic spot covered with an equilibrium ﬁlm with the thickness he) may be calculated (Figure 2.24c):

( )( )

x 0 = β − ε Pe  γ /a © 2007 by Taylor & Francis Group, LLC

1/ 2

,

(2.121)

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Wetting and Spreading Dynamics

( )

where the function ε Pe is determined from the following equation:

( )

cos ε − a /A

1/ 2

(

) (

)

(

)

sin ε =  Pe /A + Pe /a  t + Pe /a − he  .

(2.122)

Therefore, the solutions obtained open the possibility of studying the critical size of a hydrophobic spot, Lc, dependent on the parameters of disjoining pressure isotherms. When the width, L, of a hydrophobic spot is smaller than Lc , a depression cavity is formed over the spot (Figure 2.24b), and this is reﬂected on the equilibrium liquid proﬁle, h(x), which describes a proﬁle of a relatively thick ﬁlm even over the hydrophobic spot. However, at L ≥ Lc , the thick wetting ﬁlm ruptures, and a part of the hydrophobic surface becomes almost “dry.” Further increase in the dimension of the hydrophobic spot leads to the expansion of a dry part of the surface (Figure 2.24c). In view of the preceding features, we can say that the presence of more hydrophobic spots on smooth hydrophilic substrates results in the formation of depressions where the ﬁlm thickness is lower than the thickness on the rest of the substrate. Accordingly, the presence of more hydrophobic spots results in a lower mean thickness of the ﬁlm relative to the thickness on a uniform hydrophilic substrate. Thus the surface roughness and the presence of hydrophobic spots on the surface inﬂuence the mean thickness of the equilibrium ﬁlm in opposite ways: the presence of roughness results in an increase in the mean ﬁlm thickness (Section 2.5), whereas the presence of hydrophobic inclusions leads to the contrary.

2.7 THICKNESS AND STABILITY OF LIQUID FILMS ON NONPLANAR SURFACES In this section, conditions are determined about the stability of liquid ﬁlms on cylindrical and spherical solid substrates. We shall see that stability is determined by an effective disjoining pressure isotherm, Πeff (h), which differs from the corresponding disjoining pressure isotherm of (ﬂat) liquid ﬁlms on ﬂat (planar) solid substrates. The effective disjoining pressure on curved surfaces is considered in more detail in Section 2.12. An analysis is given of the different types of isotherms Πeff (h) relating the ﬁlm thickness h to the total change in pressure in the ﬁlm relative to the bulk phase of the same liquid. When a liquid ﬁlm covers a planar solid surface, its equilibrium with the vapor is determined by Equation 2.23 (see Section 2.2), which takes into account the simultaneous action of both the capillary pressure and the disjoining pressure in these liquid layers. In the following text, we consider thin liquid layers on curved surfaces, for example, the inner and outer surfaces of cylinders. Let a be the inner or outer radius of the cylinder. If the ﬁlm thickness, h, is much smaller than the cylinder radius, (h << a), we can use, as a ﬁrst approximation, the isotherm of the disjoining pressure, Π(h), of a planar layer. However, in the general case, the © 2007 by Taylor & Francis Group, LLC

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disjoining pressure in thin liquid ﬁlms on a curved substrate, Πa, is different from the corresponding disjoining pressure on a ﬂat surface, Π. In Section 2.5, we already gave an estimation of the distortion of dispersion forces in this case. It has been shown that at h/a ≤ 0.2, the difference between Πa and Π does not exceed 2.5%. Capillary effects, as we see in the following text, have a considerably more pronounced inﬂuence on the thickness and stability of ﬁlms on nonplanar surfaces. Hence, in subsequent calculations, we use the disjoining pressure isotherm, Π(h), of ﬂat liquid ﬁlms, the same that is used in Equation 2.23 (Section 2.2). Let us ﬁrst examine equilibrium ﬁlms on the convex surface of a cylinder with a radius, a. The conditions of the equilibrium have the form given by Equation 2.249 and Equation 2.251 (see Section 2.12 for the derivation), where the effective isotherm of disjoining pressure is given by Equation 2.245, which is rewritten as

Πeff ( h ) =

a a+h

 γ Π( h ) − a  ,  

(2.123)

where γ is the surface tension, Πeff (h) is the effective disjoining pressure isotherm of the curved ﬁlm, and the excess pressure Pe is given by Equation 2.2 in Section 2.1. According to Equation 2.123, the Πeff (h) dependency intersects the axis of thickness at Π(h) = γ/a. Consequently, when p/ps = 1, the ﬁlm on a curved surface has a ﬁnite thickness h0 that is determined by the speciﬁc form of the disjoining pressure isotherm, Π(h). Note that according to Equation 2.123, the effective disjoining pressure isotherm depends on the radius of the cylinder, a. For planar ﬁlms in Section 2.1, we deduced the condition of stability of ﬂat ﬁlms on ﬂat solid substrate: dΠ = Π′( h ) < 0 . dh Applying the similar method to a curved ﬁlm and using the effective disjoining isotherm Πeff (h), we conclude (see Section 2.12) that the stability condition of liquid ﬁlms of uniform thickness on the outer cylindrical surface reads Πeff ′ ( he ) < 0 .

(2.124)

In comparison with the stability condition Π′( h ) < 0 , the condition (2.124) depends on the vapor pressure in the surrounding air; in the case of undersaturation (Pe > 0), the latter condition is less restrictive than on a planar substrate, whereas at oversaturation, the condition is more restrictive, as it makes ﬁlms on a convex surface less stable than on a ﬂat surface. © 2007 by Taylor & Francis Group, LLC

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It is interesting to note that in the case of complete wetting,

Π( h ) =

A , h3

the equilibrium adsorption does not take place under oversaturation on a ﬂat surface. However, according to effective disjoining pressure given by Equation 2.123, adsorption is possible on an outer cylindrical surface at oversaturations, in the range from  γv  p/ps = 1 to p /ps = exp  m  > 1.  aRT  However, adsorption ﬁlms are stable only in the region of thicknesses where the stability condition (2.124) is satisﬁed (see Section 2.12). We now calculate the critical thickness h* for the isotherm corresponding to complete wetting A/h3, using the stability condition (2.124). We conclude that  3a 2 A  h* ≈    γ 

1/ 4

.

(2.125)

For ﬁlms of decane on a cylindrical quartz surface, assuming that A = 1.6⋅10–13 erg [14] and γ = 23 dyn/cm, we ﬁnd, using Equation 2.125, that for a = 10–3, 10–4, and 10–5 cm, h* = 1190, 376, and 119 Å, respectively. However, the loss of stability occurs only at Πeff < 0 (and hence when p/ps > 1), i.e., in the region of oversaturation. When p/ps < 1, the ﬁlms remain stable, but their thickness (in contrast to planar ﬁlms) do not tend to inﬁnity as p/ps → 1 but rather toward a limiting value  Aa  h0 =   γ 

1/ 3

.

Let us now consider the more complex case of partial wetting, when the isotherm for planar ﬁlms, Π(h), intersects the axis of thickness (see Chapter 1). The form of such an S-shaped isotherm is shown schematically in Figure 2.27 (curve 1). Curves 2–4 are possible variants of the effective disjoining pressure, Πeff (h), in accordance with Equation 2.123. The values Πeff (h) > 0 correspond

© 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

103

Πeﬀ (h) Π(h)

1 tmin

0

tmax t1

t2 h

2 3

FIGURE 2.27 Effective disjoining pressure, Πeff (h), for ﬁlms on convex surface. Upper part: 1 — disjoining pressure isotherm of ﬂat liquid ﬁlms on ﬂat solid substrate, Π(h); 2–3 effective disjoining pressure isotherms, Πeff (h), at different radiis of the outer cylindrical surface, a: a2 > a3.

here to the region of undersaturation (p/ps ≤ 1), and values Π(h) < 0 correspond to the region of oversaturation (p/ps > 1). As in Section 2.2, stable ﬁlms with thicknesses h < tmin are referred to as α-ﬁlms, and metastable ﬁlms with thicknesses h > tmax as β-ﬁlms (Figure 2.27). According to Figure 2.27 (curves 2,3), as the radius of the cylinder, a, decreases, then both the region of metastable ﬁlms on curved substrates and the region of stable α-ﬁlms shrink. A new phenomenon appears: a ﬁnite extent of the region of stable β-ﬁlms (Figure 2.27, curve 2). For the Πeff (h) isotherm shown by curve 2 (Figure 2.27), the region of thicknesses, where β-ﬁlms are stable, is bounded by thicknesses from t1 to t2. In the case of even smaller radius of the cylinder, the region of existence of stable β-ﬁlms at p/ps < 1 disappears completely (curve 3 in Figure 2.27). Figure 2.27 shows that a decrease in radius of cylinders, a, leads to a decrease in adsorption (at undersaturation, p/ps ≤ 1). The latter phenomenon has nothing to do with the nature of the cylinders and depends only on the geometric features, i.e., the curvature of the surface. Let us now analyze the effective disjoining pressure isotherms, Πeff (h), on a concave surface, for example, on the inner surface of a cylindrical capillary, that is, inside the cylindrical capillary of radius a. The equation for the equilibrium of liquid and vapor in this case has the following form (see Section 2.12): a  γ Π( h ) +  = Πeff h = Pe . a − h  a

()

© 2007 by Taylor & Francis Group, LLC

(2.126)

104

Wetting and Spreading Dynamics Πeﬀ (h)

3

Π(h) 2

1 0

t1

t2

h

FIGURE 2.28 Effective disjoining pressure, Πeff (h), for ﬁlms on the inner surface of a cylinder of radius a (concave surfaces). 1 — disjoining pressure isotherm of ﬂat liquid ﬁlms on ﬂat solid substrate, Π(h); 2–3 effective disjoining pressure isotherms, Πeff (h), at different curvature of cylindrical capillaries, a: a2 > a3.

The corresponding stability condition becomes Πeff ′ ( he ) < 0.

(2.127)

Here, in contrast to convex surfaces, the ﬁlm thickness, h, is evidently limited by the value of a. However, long before h approaches the inner radius, a, of the capillary, it becomes necessary to account for the inﬂuence of overlapping ﬁelds of surface forces of all sections of the capillary surface. For slit pores, the corresponding evaluations have been made [15]. We limit ourselves in this section, as previously, to an analysis of the solution for rather large values of a, when the condition h << a is fulﬁlled. Figure 2.28 schematically shows the isotherms of a planar ﬁlm Π(h) (curve 1) and the effective disjoining pressure isotherms, Πeff (h), (curves 2 and 3) corresponding to the case of an inner surface of a cylindrical capillary. With decreasing capillary radius, a, the region of stable state of α-ﬁlms is narrowed, but their thickness is bigger than the corresponding thickness of ﬁlms on a planar surface. The appearance of a lower limit of ﬁlm stability, t1, and an upper limit, t2, also contributes to the existence of a narrowed region of β-ﬁlms stability. For the disjoining pressure isotherm shown as curve 2 (Figure 2.28), βﬁlms can exist only in the interval of ﬁlm thicknesses from t1 to t2, corresponding to a certain interval of p/ps in the region of undersaturation. In narrower cylindrical pores (curve 3, Figure 2.28), only thin α-ﬁlms are stable, and β-ﬁlms disappear completely. These conclusions are supported by published experimental data. The existence of an upper limit of stability for β-ﬁlms in cylindrical capillaries was experimentally discovered [16]. It has also been experimentally observed [17] that in glass cylindrical capillaries with radius a > 0.4 µm, thin α-ﬁlms are formed with thickness h ≈ 50–60 Å. However, in thinner capillaries with a = 0.2–0.3 µm (with a corresponding reduction of p/ps), thicker β-ﬁlms appear with thickness © 2007 by Taylor & Francis Group, LLC

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h ≈ 300–400 Å. Both the experimental observations correspond to the isotherm Πeff (h) shown by curve 2 in Figure 2.28. To conclude this section, we examine the case of complete wetting, when the isotherm Π(h) of a planar ﬁlm can be represented by A/h3. According to Equation 2.127, the critical thickness h* can be calculated when the ﬁlm of a uniform thickness loses its stability. Based on Equation 2.126 and Equation 2.127, we can conclude that h* = (3Aa2/γ)1/4. At h ≥ h*, the ﬁlms on the inner capillary surface lose stability, and the liquid changes into a more stable state, forming a capillary condensate. For a ~ 10–4 cm, A ~ 10–14 erg, and γ ~ 30 dyn/cm, we obtain a thickness h* of the order of 10–6 cm. Thus, the condition h << a is fulﬁlled over the entire interval of ﬁlm thicknesses that is physically realizable in a capillary. Even with a ~ 10–6 cm, the values of h* are no greater than 10–7 cm. As an example, we show in Figure 2.29 curves plotted on the basis of Equation 2.126 for the thickness of a decane ﬁlm, h, on the inner surface of quartz capillaries, as a function of the relative vapor pressure, p/ps. For this purpose, in Equation 2.126 we use a disjoining pressure isotherm as given by A/h3, and Pe is replaced by its values from Equation 2.2, which results in a  A γ  RT p ln s . + = a − h  h 3 a  vm p

(2.128)

2

h, Å

40

20 3 1

0 0.6

0.8 p/ps

1

FIGURE 2.29 Adsorption isotherms, h (p/ps), for ﬁlms of decane in quartz capillaries with radius a = ∝ (1), 10–5 cm (2), and 10–6 cm (3). © 2007 by Taylor & Francis Group, LLC

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In the calculations shown, it is assumed (the same as previously) that A = 8.5⋅10–15 erg, γ = 23 dyn/cm, vm = 195 cm3/mole, and T = 293˚K. Curve 1 in Figure 2.29 is the isotherm h(p/ps) at a = ∞, i.e., for a planar surface. Curves 2 and 3 were plotted for capillaries with radii a = 10–5 and 10–6 cm, respectively. Whereas on the planar surface, h → ∞ when p/ps → 1 (curve 1), we ﬁnd that, in the capillaries, the isotherm breaks off at p/ps = 0.98 (curve 2) and p/ps = 0.79 (curve 3). Breakoff of the isotherms corresponds to loss of ﬁlm stability, in accordance with Equation 2.127. Note that in the entire region of physically realizable ﬁlm thicknesses, h ≤ h*, the condition h << a is fulﬁlled. That is, capillary condensation starts much earlier than when the capillary is ﬁlled with the liquid. Even though the thickness of adsorbed ﬁlms with equal values of p/ps is bigger in a capillary with a smaller capillary radius, the region of their existence is curtailed quite substantially. When h > h*, the ﬁlms lose stability, and the capillary is ﬁlled with condensate. The properties of ﬁlms on curved surfaces that have been examined in this section, i.e., ﬁlm stability and thickness, should be taken into account to investigate processes of polymolecular adsorption in ﬁne porous solids, the surface of which have convex and concave portions. These properties should also be taken into account to study adsorbate transfer processes occurring in these solids. To conclude, analogous calculations can be carried out for any form of the disjoining pressure isotherms, Π(h) (or isotherms of adsorption on a planar surface), including isotherms obtained experimentally. We have selected the isotherm A/h3, which corresponds to adsorbate or adsorbent interaction solely due to dispersion forces, only in order to illustrate the solutions obtained.

2.8 PRESSURE ON WETTING PERIMETER AND DEFORMATION OF SOFT SOLIDS We shall continue the consideration of the simultaneous action of capillary and disjoining stresses applied on thin liquid layers. On that basis, the distribution of normal pressure on a solid substrate in the vicinity of the apparent three-phase contact line is going to be determined. In the general case, the substrate is subjected to both tensile and compressive stresses. The extent of the zone in which normal pressure acts corresponds to the extent of the transition zone between the bulk part of the liquid under the drop or meniscus and the equilibrium ﬂat ﬁlms in front [33]. We shall be concerned with the inﬂuence of the vertical components of capillary and surface forces acting on a solid support in the vicinity of the apparent three-phase contact line. For a rigid inelastic substrate, the vertical component of those forces can be ignored, as its action does not lead to shape change. For an elastic solid substrate, the change in equilibrium conditions under the inﬂuence of surface deformation is known. Within the framework of the model of an absolutely rigid substrate, the vertical component of those forces can be ignored, as its action does not lead to © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

107 h

h3

h ts

1

h2

F

h1

he x

t1 2 t0

3

θe xNY

x0

Π

0

F+

0

0 −Π

f(x)

4 x_ x

0

x+ F–

FIGURE 2.30 Proﬁle of transition zone h(x) between bulk liquid and ﬂat wetting ﬁlm (1), disjoining pressure isotherms Π(h) (2,3), and proﬁle of normal forces acting on substrate (Equation 2.131) in the case when a model disjoining pressure isotherm (2) is adopted; xNY is the position where the vertical force is exerted.

any change in the shape of the substrate. For an elastic solid substrate, the change in equilibrium conditions under the inﬂuence of surface deformation was investigated [18,19]. It has been assumed that the force, F, acts in a certain zone within a thickness δ, corresponding to the thickness of the surface layer of the liquid where the pressure is anisotropic. The thickness of the zone, δ, within the framework of this theory, remained undetermined. Through an examination of the transition zone between the bulk liquid and the ﬁlm covering the substrate surface (Figure 2.30), we can relate the force on the substrate to the disjoining pressure, Π(h), and to the action of capillary forces. This approach is based on an analysis of the distribution of the disjoining pressure within the limits of the transition zone; the shape and extent of this zone have been investigated in Section 2.3 and Section 2.4. Our subsequent analysis of the problem of pressure on a wetting perimeter is performed within the framework of the theory, which as previously mentioned, includes a simultaneous action of the capillary and disjoining pressure in the liquid layers presented in the previous sections of this chapter. It was shown in Section 2.3 that the equilibrium contact angle, θe, can be expressed via the disjoining pressure isotherm, Π(h), by Equation 2.47. The vertical component of the forces acting on the wetting line, F = γ sinθe, does not vanish, when θe > 0, i.e., in the case of partial wetting. This corresponds, in accordance with Section 2.1 through Section 2.3, to an S-shaped disjoining pressure isotherm, Π(h), entering the region of negative values in a certain interval of ﬁlm thicknesses (see Figure 2.1). © 2007 by Taylor & Francis Group, LLC

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In the following text, examine the transition zone between a ﬂat meniscus and a ﬁlm in front at θe > 0 (curve 1, Figure 2.30), that is, in the case of partial wetting. The bulk liquid here forms a wedge, changing over, as x → ∞, to a ﬂat ﬁlm with a thickness he. With this model, we can limit ourselves to a onedimensional solution of the problem without any loss of generality of the analysis. The coordinate origin x = 0 is taken as a point of the proﬁle lying beyond the limits of action of surface forces (Figure 2.30). The proﬁle of the transition zone, h(x), can be obtained by solving Equation 2.23 deduced in Section 2.2, which includes the combined action of capillary and disjoining pressure: γh′′

( )

1 + h′ 2   

3/ 2

()

+ Π h = Pe ,

(2.129)

where h′ = dh/dx; h″ = dh 2/dx2; and Pe is the excess pressure of the drop or meniscus. In the region of the ﬂat equilibrium ﬁlm, h″ = 0 and Π(he) = Pe. In the bulk part of the liquid, beyond the limits of action of surface forces, Π = 0 and Pe = ±γ/ℜ, where ℜ is the radius of surface curvature of the drop or concave meniscus. For a planar wedge (Figure 2.30), ℜ = ∞ and Pe = 0. The resultant forces in the vertical direction caused by the pressure on the solid substrate, ∞

F=

∫ Π ( h)· dx,

(2.130)

0

is calculated in the following text based on Equation 2.129. Replacing the disjoining pressure, Π(h), in Equation 2.130 by its expression from Equation 2.129, and keeping in mind that Pe = 0, we obtain

∞

F = −γ

∫ 0

    γh′ 0 γ ⋅ tan θe ee d  h′ = γ ⋅ sin θe . (2.131) dx = =  1/ 2  2 2 dx   + tan θ 1  e   1 +  h′ 0  1 + h′    

( )

() ()

In the derivation of the latter expression, we have used the boundary conditions h′(∞) = 0 and h′ (0) = tanθe . Thus, integration of the local disjoining pressure leads to the same value of the total vertical force caused by the pressure on the substrate as Young’s equation does. However, in contrast to Young’s equation, the mentioned forces are not exerted on a particular point but distributed over the whole region where the disjoining pressure acts, that is, over the transition zone. Note that the actual © 2007 by Taylor & Francis Group, LLC

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equilibrium contact angle, θe, is different from that calculated according to Young’s equation, θNY (see the discussion in Section 2.1). Now let us consider how the pressure on the substrate is distributed inside the transition zone, i.e., in the vicinity of the apparent three-phase contact line. Figure 2.30 presents the transition region that is subject to the action of pressure on the solid substrate. This region is determined by the difference between the coordinates of the points at which Π(h) = 0 (large h) and Π(he) = 0. The length scale of the transition zone has been estimated in Section 2.3 and calculated in the case of complete wetting in Section 2.4. We now perform quantitative evaluations of the extent of this region in the case of partial wetting, using a model disjoining pressure isotherm made up of linear sections (curve 2, Figure 2.30). Such a simpliﬁcation, while retaining the basic properties of the real isotherm (curve 3, Figure 2.30), enables us to obtain an analytical solution of the problem of the pressure distribution inside the transition zone. The equation of the model isotherm used has the following form:

( (

) )

 a t0 − h ,   Π h =  b t2 − h ,   0, 

()

0 < h < t1 t1 ≤ h ≤ ts ,

(2.132)

h ≥ ts

where ts corresponds to the ﬁnite radius of the action of the surface forces; t0 = he is the thickness of the equilibrium ﬁlm with Π(t0) = 0; and the parameters a and b characterize the slopes of the linear sections of the disjoining pressure isotherm in the region of thicknesses from 0 to t1 and from t1 to ts, respectively. The coordinate origin x = 0 is selected at h = ts. The proﬁle of the transition zone corresponding to the model disjoining pressure isotherm (Equation 2.132) is depicted by curve 1 in Figure 2.30. We subdivide the whole proﬁle of the liquid into three sections: h1(x) for the region of thicknesses from he = t0 to h = t1, where the liquid surface is concave; h2(x) for the region of thicknesses from t1 to ts, where the surface is convex; and h3(x) for the region of the wedge, where h > ts and the surface is ﬂat. For contact angles that are not too large, we can assume that (h′)2 << 1. Then, for each of these three zones, Equation 2.129 can be rewritten in the following form:

( γ ⋅ h′′+ b ( t

) − h ) = 0,

γ ⋅ h1′′+ a t0 − h1 = 0, 2

γh3′′= 0.

© 2007 by Taylor & Francis Group, LLC

s

2

(2.133)

110

Wetting and Spreading Dynamics

The proﬁle of the liquid in these three regions should be linked at the points (0, t2) and (x0, t1) by the conditions of equality of thicknesses and of the derivatives, h′: h1(x0) = h2(x0) = t1 and h1′ (x0) = h2′ (x0); h2(0) = h3(0) = ts, and h2′ (0) = h3′ (0). The following boundary condition should be used for a smooth transition from the bulk liquid to the ﬂat equilibrium ﬁlm: h1(x) → t0 as x → ∞ (Section 2.3, Appendix 1). The solution of the Equation 2.133, with the preceding boundary conditions, has the following form: γ a

()

h1 x = t0 + C1

 C1 + C2   C − C  1 2

1/ 2 a / b

 a exp  − x , γ 

1/ 2  b γ 2 2  h2 x = ts −  C1 − C2  sinh  x , γ b  

()

(

)

()

(

)

(2.134)

1/ 2

h3 x = ts + x  C12 − C22  ,   where C1 =

(

)

a t1 − t0 , C2 = γ

(

)

b ts − t1 . γ

Using the solutions obtained from Equation 2.134, we can ﬁnd the distribution of force on the substrate: f(x) = Π(x) = Π [h(x)]. The proﬁle f(x) corresponding to the model isotherm in Equation 2.132 is shown in Figure 2.30 as curve 4. At the point x = x0, the force f changes its sign abruptly in the same way as the disjoining pressure Π (curve 2). In the case of real isotherms (curve 3), the transition from compressive to tensile stresses is not that sharp; however, the qualitative form of the proﬁle of the local force f(x) is retained. It is important to emphasize that the solid substrate is not subjected to a tensile force only (i.e., force directed upward) as had been assumed previously. Young’s equation gives only the net value of the force F; but, as can be seen from Figure 2.30, both tensile forces (F+ > 0) and compressive forces (F < 0) contribute to the resultant force. The point of application of the resultant force F can be found from the equation ∞

( )∫ ( )

x NY = 1/F · xΠ x · dx , 0

where the value of F is determined using Equation 2.130. After substituting the values of Π[h(x)] from Equation 2.132 and Equation 2.134, and further integration, we obtain © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

111

(

x NY = ts − t0

)

tan θe .

According to the preceding expression, the value of xNY is close but does not coincide with the predicted position based on Young’s equation, that is, with the position of the line of contact between the wedge and ﬁlm in the absence of a transition zone (see Figure 2.30). Also note that the position of the application of the force, xNY , (Figure 2.30) is located at the intersection of the continuation of the bulk liquid proﬁle with continuation of the equilibrium ﬁlm of thickness he and not with the solid substrate. The positions of the maximum and minimum forces, f+(x) and f(x), are shifted from the position xNY into the depth of the bulk liquid. We now ﬁnd the values of the resultants separately for the tensile (F+) and the compressive (F–) forces ∞

F+ =

∫ f ( x ) ⋅ dx = − γ ⋅ C , +

1

0

(2.135)

∞

F− =



∫ f ( x ) ⋅ dx = γ ⋅ C  1 − −

1

0

2 1 − C2 /C1  

(

)

Summation of the latter expressions for F+ and F– gives the total resultant force value, F = γ.sinθe. This follows from Equation 2.47 after substituting the equation of the disjoining pressure isotherm (Equation 2.132), which gives

( )(

)

cos θe = 1 − 1 / 2 · C12 − C22 .

(2.136)

The latter expression as given by Equation 2.133 is used for relatively small contact angles. That is, the following relation is used: θe ~ tanθe. For the points of application of the resultant forces, F+ and F–, we obtain

x+ =

x− =

respectively.

© 2007 by Taylor & Francis Group, LLC

{

(

)( (

)

}

  γ ln  C1 + C2 C1 − C2  − 1 ⋅ < x0 , 2 b  2 1 − 1 − C2 /C1   

)

 1 a  C1 + C2   γ ⋅ ln 1 +    > x0 , a  2 b  C1 − C2  

(2.137)

112

Wetting and Spreading Dynamics h

P

a he

H 1

θe

Pe

he +Π Pe 0

–Π h

b ᑬ

he

2

he

Pe

θe

0 Pe

+Π

he

h

h

P

c

–Π

H 3

4

Pe

he +Π Pe 0

5 Pe 0

FIGURE 2.31 Proﬁles of transition zone, isotherms Π(h), and proﬁles of normal forces in the case of partial wetting for (a) a concave meniscus (curve 1) and (b) a drop (curve 2). (c) For a meniscus, in the case of complete wetting (curve 3). Deformation of the substrate is shown to be proportional to the local force applied.

The approach that we use in the preceding expression can also be utilized in the case of a concave meniscus in a capillary, or in the case of drops in partial wetting, that is, forming a ﬁnite contact angle, θe, with the equilibrium ﬁlms on a solid substrate. In Figure 2.31, we qualitatively show the form of distribution of the excess forces f(x), normal to the substrate for a meniscus in a ﬂat capillary with a width of 2H (Figure 2.31a, curve 1, Pe > 0) and for a drop (Figure 2.31b, curve 2, Pe < 0). In the two latter cases, we took into account the fact that, in the bulk part of the meniscus or drop, the substrate is subjected to an excess pressure Pe = const (excess in comparison with the pressure in the surrounding gas phase). An equal hydrodynamic pressure acts at the equilibrium state in the transition zone and in the ﬂat ﬁlm as well. Therefore, the values of f(x) must be determined as the difference f(x) = Π(h) –Pe .

(2.138)

In the bulk part of the meniscus or drop, Π(h) = 0, and hence, f(x) = Pe . In a ﬂat ﬁlm, Π(he) = Pe, and hence, f(x) = 0. From Equation 2.129 and the deﬁnition given by Equation 2.138, we note that the value of the local force f(x) = Π(h) – Pe is determined by the local curvature of the liquid surface: f(x) = γ · K(x), where K(x) is the local curvature © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

113

of the liquid surface. The local curvature, K(x), depends on the shape of the liquid in the transition zone, h(x), which in turn is determined by the combined action of capillary forces and disjoining pressure. In conclusion, we examine the case of complete wetting where the meniscus does not intersect the plane of the substrate and does not form a contact angle with the substrate. As was shown previously in Section 2.4, as in the case of partial wetting, a transition zone is formed between the spherical part of the meniscus and the ﬂat equilibrium ﬁlms in front. The distribution of excess force f(x) is shown for this case by the curve 3 in Figure 2.31c. This plot shows that for Π(h) isotherms that decrease monotonically and lie entirely in the positive region, Π > 0 (curve 4), the pressure on the substrate f(x) also decreases monotonically from the value Pe under the bulk meniscus to 0 as h → he. The extent of the zone of action f(x) coincides with the extent of the transition zone, L, which has already been evaluated in Section 2.3 as L ~ h e H , where H is the half-width of the ﬂat capillary, and he is the equilibrium thickness of the ﬁlm in front. Thus, even in the case of complete wetting, when Young’s equation gives F = 0, a certain pressure f(x) acts on the substrate beyond the limits of the bulk part of the spherical meniscus. This force may also be of a sign-alternating character if the isotherm (still corresponding to the complete wetting) intersects the vertical line P = Pe (curve 5 in Figure 2.31c). We would like to emphasize that the consideration based on the combined action of the disjoining pressure and capillary pressure inside the transition zone is not only more precise than Young’s equation but also gives a far broader picture of the events in the vicinity of the apparent three-phase contact line. The same approach can be used for the calculation of normal forces in the case of nonﬂat liquid layers that were discussed in Section 2.2. All those nonﬂat layers are located inside the range of action of the surface forces (that is, h(x) < ts); there is no bulk part of the nonﬂat layer. The pressure on the substrate is determined by the values of f(x) = γK(x), where the curvature K varies with the radial coordinate x in all parts of the nonﬂat layer. The next step should be the determination of the proﬁle of a substrate deformed under the inﬂuence of the distributed force f(x). In Figure 2.31, we adopted for simplicity that the deformation of the solid substrate is proportional to the local force. This assumption is deﬁnitely an oversimpliﬁcation and should be replaced by a more realistic hypothesis. Note that we have taken into account only the normal component of force acting on a solid substrate. Adding the effect of the tangential component of the force inside the transition zone represents a challenging problem.

2.9 DEFORMATION OF FLUID PARTICLES IN THE CONTACT ZONE The hydrostatic pressure in thin liquid ﬁlms intervening between two drops or bubbles differs from the pressure inside the drops or bubbles. This difference is © 2007 by Taylor & Francis Group, LLC

114

Wetting and Spreading Dynamics

caused by the action of both capillary and surface forces. The manifestation of the action of surface forces is the disjoining pressure, which has a special S-shaped form in the case of partial wetting (aqueous thin ﬁlms and thin ﬁlms of aqueous electrolyte and surfactant solutions). Disjoining pressure solely acts in thin ﬂat liquid ﬁlms and determines their thickness. If the ﬁlm surface is curved, then both the disjoining and the capillary pressures act simultaneously. A theory is developed in the following text enabling one to calculate the shape of the liquid interlayer between emulsion droplets or between gas bubbles of different radii under equilibrium conditions, taking into account both the local disjoining pressure of the interlayer and the local curvature of its surfaces [33–36]. The model of solid nondeforming particles is frequently used when carrying out an analysis of the forces acting between colloidal particles. However, real droplets or bubbles and even soft solid particles within the contact zone can deform, which can change the conditions of their equilibrium. In this case, interaction is not limited only to the zone of a ﬂat contact but is extended onto the surrounding parts within the range of action of surface forces. For elastic solid particles, such a problem was discussed [20–21]. Here, the case of droplets or bubbles is considered (e.g., emulsions, gas bubbles in a liquid), where shape changes very easily under the inﬂuence of surface forces (Figure 2.32). We now take into account both the ﬁnite thickness of a liquid interlayer between droplets or bubbles and the variation in thickness in the transition zone between the interlayer and the equilibrium bulk liquid phase. It was already mentioned that there is a problem in using the approach of thickness-dependent interfacial tension: if we try to use the Navier–Stokes equation for description of ﬂow or equilibrium in thin liquid ﬁlms, a thickness-dependent surface tension results in an unbalanced tangential stress on the surface of thin ﬁlms. This is the reason why this particular approach is not used in this book. In the following text, we use the same approach as in the previous sections of this chapter, which takes into account the interlayer thickness and the effect of the transition zone between the thin interlayer and the bulk liquid. This effect is equivalent to the line tension that is considered in Section 2.10. A low slope and constant surface tension approximations are used. Then, as was shown earlier in Section 2.1 through Section 2.3, it is possible to use the equation taking into account both the disjoining pressure and the capillary pressure in the interlayer. 1 Pl

Pd

h(x) x

2

2 1

FIGURE 2.32 Two identical drops or bubbles (1) at equilibrium in a surrounding liquid, (2); h(x) is a half thickness of the liquid ﬁlm between two drops or bubbles. © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

115

TWO IDENTICAL CYLINDRICAL DROPS

OR

BUBBLES

First, let us consider a case of two identical cylindrical drops or bubbles (Figure 2.33). It is assumed for simplicity that the radius of action of surface forces is limited to a certain distance, ts. Beyond this distance, the surface of droplets retains a constant curvature radius, R, and is not disturbed by surface forces. The interacting droplets are considered as being surrounded by a liquid with constant pressure, Pl . The thickness of the liquid interlayer, 2h(x), varies from 2h0 on the axis of symmetry at x = 0 to 2h = ts at x = x0 (Figure 2.33). For the drop proﬁles not disturbed by the surface forces, we use the designation hs at y < B, and Hs for y > B (Figure 2.33). The liquid in the droplets is assumed to be incompressible, which leads to the condition of the constancy of the volume of droplets per unit length, R0

V =4

∫ (H

0 s

)

− hs0 ⋅ dx = π ( R 0 )2 = const ,

0

where the superscript 0 marks an undisturbed isolated droplet prior to contact. Note that, in general, the system of two droplets is thermodynamically unstable with regard to coalescence. Therefore, the following calculations give the conditions of the metastable equilibrium of droplets separated by a thin interlayer of the surrounding liquid. The excess free energy of the two cylindrical drops (per unit length of cylindrical droplets), Φ, is equal to Φ = γS + ΦD + PeV, where S is the total

hs(x) h(x)

2B

Hs (x)

R

Pl

ts

ϕ

2h0

ϕ θe

x

2x0 R

FIGURE 2.33 The equilibrium proﬁle of interlayer, h(x), between two cylindrical droplets of the same radius, R. © 2007 by Taylor & Francis Group, LLC

116

Wetting and Spreading Dynamics

interfacial area per unit length, γ is the interfacial tension, ΦD is the excess free energy per unit length determined by the surface forces action, Pe is the excess pressure, and V is the volume per unit length. In the case of cylindrical drops or bubbles S, FD , and V are as follows: R

S=4

x0

R

∫

1 + H s′ dx + 4 2

∫

1 + hs′ dx + 4 2

x0

0

∫

1 + h′ 2 dx

0

,

∞  Φ D = 2  Π(h)dh  dx , and   0  2h  x0

∫∫

x0

R

V =4

∫ ( H − h ) ⋅ dx + 4 ∫ ( H − h) dx = π R s

s

2 0

s

x0

= const ,

(2.139)

0

where h(x), hs (x), Hs (x), x0, and radius, R are determined in Figure 2.33; ts is the radius of the action of surface forces. Substitution of the latter expression into the excess free energy results in   Φ = 4γ  

R

∫

x0

R

1 + H s′ dx + 4 2

∫

1 + hs′ dx + 4 2

x0

0

∫ 0

x0 ∞     1 + h′ dx + 2  Π(h)dh  dx    0  2h   2

∫∫

x0   R  + Pe 4 H s − hs ⋅ dx + 4 H s − h dx  .   0   x0

∫(

)

∫(

)

A variation of the excess free energy with respect to h(x), hs (x), Hs (x), and the two values x0 and R results in the following equations: γ · h′′ 1 + h′ 2 

−3/ 2

γ · hS′′ 1 + hS′ 2 

−3/ 2

γ · H S′′ 1 + H S′ 2 

( )

+ Π 2h = − Pe = − Pe

−3/ 2

(2.140)

= Pe

where the last two equations give the equation of a circle of radius R. The latter immediately determines the unknown excess pressure as © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

117

Pe = −

γ . R

After that, the ﬁrst part of Equation 2.140 takes the following form: γ ⋅ h′′ 1 + h′ 2 

−3/ 2

( )

+ Π 2h =

γ , R

(2.141)

where the ﬁrst term on the left-hand side is due to the capillary pressure, and the second term is determined by the local value of the disjoining pressure, Π(2h). The boundary conditions for Equation 2.141 are as follows:

( )

( )

( )

( )

( )

h x 0 = hs x 0 , h′ x 0 = hs′ x 0 , h x 0 = ts / 2,

( )

( )

( )

()

()

( )

hs R = H s R ; hS′ R = − H S′ R = ∞, h′ 0 = H S′ 0 = 0.

(2.142) (2.143) (2.144)

We remind ourselves that hs describes the proﬁle of the droplet at 2hs > ts. If the droplets are located at a distance, 2h0 > ts, then Π(2h) = 0; and in this case, the solution of Equation 2.141 gives the proﬁle hs(x), which corresponds to the circular form of the cross section of nondeformed droplets that do not interact with one another. However, at 2h0 < ts , the interlayer is under the effect of both surface forces, whose contribution is determined by the term Π(2h) and the capillary forces, depending on the local curvature of the interlayer surfaces. Equation 2.141 with boundary conditions (2.142 through 2.144) and with the condition of constancy of volume (2.139) provide a solution to the problem; it enables us to determine the proﬁle of an interlayer, h(x), between the droplets at the known isotherm of disjoining pressure, Π(2h). An example of the proﬁle of the droplets in the transition zone, in the case of the S-shaped disjoining pressure isotherm, is shown in Figure 2.34. Note that, in general, the thickness in the central part between drops is different from the equilibrium thickness he, and this is the reason why it is referred to in Figure 2.33 as h0. The latter two thicknesses coincide if the central part between the two drops is ﬂat. The solution thus obtained may be veriﬁed in the following way: at the equilibrium state, the total force of interaction of droplets per unit length, F, should be equal to zero. According to Section 2.8, this force x0

∫ ( )

F = 2 Π 2h · dx. 0

© 2007 by Taylor & Francis Group, LLC

(2.145)

118

Wetting and Spreading Dynamics h 2 ts 2 tmax 1

2 tmin

2 θe

2 he

3 Pe

3 x0

Π

FIGURE 2.34 Partial wetting. S-shaped disjoining pressure isotherm (left side) and the liquid proﬁle in the transition zone (right side). Magniﬁcation of the upper part of the transition zone between the drop or bubble and the thin liquid interlayer. 1 — real deformed proﬁle of the drop or bubble, 2 — ideal spherical proﬁle, when the inﬂuence of disjoining pressure has been ignored, 3 — thin liquid interlayer of thickness 2he. x0 should be replaced by r0 in the case of a spherical drop/bubble.

Substituting the expression for Π(2h) from Equation 2.140 and Equation 2.141 into Equation 2.145 and then carrying out integration, we obtain:   h′ x 0   F = 2γ  x0 − . 2  1 +  h′ x 0    

( ) ( )

(2.146)

In view of the boundary condition (2.142), at the point x = x0, both values of h(x) = hs(x), and their derivatives, h′ = h′s , are equal. This enables one to express h′(x0) through the central angle ϕ (Figure 2.33) and the values x0 and R:

( )

(

h′ x 0 = tan ϕ = x 0 R 2 − x 02

)

−1/ 2

.

(2.147)

Substituting the latter expression for h′(x0) into Equation 2.146, we obtain F = 0, as should be at the equilibrium. Thus, it should be expected that the conditions of equilibrium given by Equation 2.141, and by the boundary conditions (2.142 through 2.144), correspond to zero interaction forces between the droplets. It should be noted that the angle ϕ has a value that is very close to that of the contact angle θe, to be determined at the point of intersection of the continuation of the undisturbed proﬁle of a droplet with axis x (Figure 2.33: ho > he or with the continuation of the equilibrium ﬁlm (Figure 2.34: h0 = he). The values of θe and ϕ practically coincide when the interlayer thickness is small as compared with R and when x0 is not too small. This enables us to use Equation 2.147 for the calculation of the contact angles. Thus, derived values of x0, B, θe, and the droplet proﬁle, h(x), give the full solution of the problem, where the distance between the centers of droplets, B, (Figure 2.33) is: © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

119

(

B = tsf + 2 R 2 − x 02

INTERACTION

OF

)

1/ 2

CYLINDRICAL DROPLETS

(

)

= ts + 2 x 0 / tan ϕ . OF

(2.148)

DIFFERENT RADII

Let us now consider a more complicated case of interaction of cylindrical droplets of different radii, R2 > R1 (Figure 2.35a). Applying the same method of minimization of the excess free energy as in the preceding section, we obtain the following equations:

( )

2 γ · h1′′ 1 + h1′   

 2 γ ⋅ h2′′ 1 + h2′   

( )

−3/ 2

−3/ 2

()

(2.149)

()

(2.150)

+ Π t = γ /R1 ,

− Π t = − γ /R2 ,

where h1(x) and h2(x) are measured from an arbitrary plane that is perpendicular to the axis of symmetry, and t(x) = h1(x) – h2(x), is the thickness of the interlayer. Equation 2.149 and Equation 2.150 enable the determination of two proﬁles, h1(x) and h2(x). Equation 2.149 and Equation 2.150 along with the boundary conditions

()

()

( )

h1′ 0 = h2′ 0 = 0; t x 0 = ts ;

y

ϕ

h1s(x)

R1

h1s(x)

B h (x) 2s

r

ϕ1 t(x)

h1(x)

h2(x)

(2.151)

ϕ1

tsf t

2x0

ϕ2

R1

2x0 ϕ2

R2

H2s(x)

(a)

R2

x

(b)

FIGURE 2.35 The equilibrium proﬁle of interlayer t(x) = h1.(x) – h2 (x) between two droplets of different radii, R1 and R2, in the general case (a); and in the simpliﬁed case (b), when the transition region is neglected. © 2007 by Taylor & Francis Group, LLC

120

Wetting and Spreading Dynamics

( )

(

)

( )

(

)

h1′ x 0 = tan ϕ1 = x 0 R12 − x02

h2′ x 0 = tan ϕ 2 = x 0 R22 − x02

−1/ 2

;

−1/ 2

,

(2.152)

and with the conditions of constancy of volumes, give a solution to the problem. After the addition of Equation 2.149 and Equation 2.150, we obtain

( γ /R ) − ( γ /R ) = ∆P , 1

2

(2.153)

k

which means that the curvature of the whole interlayer equalizes the capillary pressure drop between the droplets. In a similar way, we may solve the same problem for two droplets of different composition with different interfacial tensions, γ1 and γ2, of the ﬁrst and the second droplet, respectively. In this case, even for the droplets of the same radius, the interlayer on the whole proves to be curved owing to the appearance of a capillary pressure drop, ∆Pk = (γ1 – γ2)/R. In the case of a not strongly curved interlayer between droplets, the term (h′)2 may be neglected as compared with 1. From Equation 2.149, Equation 2.150, and Equation 2.153, taking into account that t(x) = h1(x) – h2(x) and t″(x) = h″1(x) –h″2 (x), we obtain:  1 1 γ ⋅ t ′′ + 2 ⋅ Π t = γ  +  .  R1 R2 

()

(2.154)

This equation should be subjected to the following boundary conditions:

()

( )

(

( )

 t ′ 0 = 0; t x 0 = ts ; t ′ x 0 = x 0  R12 − x 02 

)

−1/ 2

(

+ R22 − x 02

)

−1/ 2

 

(2.155)

and coupled with the conditions of constancy of volumes, which determine the unknown radii R1 and R2. The latter conditions and Equation 2.154 and Equation 2.155 enable the calculation of t(x), determining the variable thickness of the interlayer. Thereafter, on substituting the known dependence Π[t(x)] = Π(x) into Equation 2.150, it is possible to obtain from

()

γ · h2′′ − Π x = − γ /R2

(2.156)

the proﬁle h2(x) of the lower surface of the interlayer. The boundary conditions for Equation 2.156 are as follows: © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

()

121

( )

( )

h2′ 0 = 0; h2 x 0 = R cos ϕ 2 ; h2′ x 0 = tan ϕ 2 .

(2.157)

It has been taken into account in the preceding expression that angle ϕ 2 is determined from Equation 2.152. The sum, h2(x) + t(x), gives the proﬁle of the upper surface of the interlayer. Calculation of the interaction of cylindrical droplets of different radii (R2 > R1) can be simpliﬁed if we assume that the interlayer is of a constant thickness. This means that the effect of a transition zone is neglected, which is justiﬁed only at x 0 >> ts . In this case, the curvature of each surface of the interlayer is constant (Figure 2.35b), and Equation 2.149 and Equation 2.150 may be rewritten in the following way:

( γ /r ) + Π (t ) = γ /R ,

(2.158)

(

(2.159)

1

)

()

 γ / r + t  − Π t = − γ /R2 .  

This system of equations enables one to determine two unknown values: t, the interlayer thickness; and r, the radius of curvature of its surface on the side of the smaller droplet. By summing up and subtracting the terms in Equation 2.158 and Equation 2.159, we obtain (at r >> t):

(

)

r ~ 2 R1R2 R2 − R1 ,

()

(

(2.160)

)

Π t ~ γ /R1 − γ /r ~ γ · R2 + R1 2 R1R2 .

(2.161)

If the shape of disjoining pressure isotherm, Π(t), is known, then Equation 2.161 determines the equilibrium thickness t = const of the curved interlayer. At R2 >> R1, Equation 2.160 and Equation 2.161 result in: r ~ 2 R1 ,

()

Π t ~ γ /2 R1.

(2.160’) (2.161’)

It should be noted that Equation 2.160 and Equation 2.161 may be derived by another method using the concept of disjoining pressure: Π = Pd − Pl , © 2007 by Taylor & Francis Group, LLC

(2.162)

122

Wetting and Spreading Dynamics

where Pd is the pressure under the interlayer surface, and Pl is the pressure in the bulk phase in which the interlayer is at equilibrium. On the side of a smaller droplet, we have Pd = Pl + (γ/R1) – (γ/r); on the side of a larger droplet, Pd = Pl + (γ/R2) + γ(r + (t). As the disjoining pressure does not depend on the side of the interlayer from which it is determined, then from Equation 2.162 it follows that:

(

) ( ) (

)

(

)

Π = γ R1 − γ r = γ R2 +  γ r + t  ,

(2.163)

which coincides with Equation 2.158 through Equation 2.161. However, determination of r and t does not completely solve the problem because the position of the center of the interlayer curvature remains unknown. Its position can be determined by minimizing the value of the free energy of the system ∞   Φ = 2 γ ⋅ R2 π − ϕ 2 + 2 γ ⋅ R1 π − ϕ1 + 2r ϕ  2γ + Π ξ d ξ    t  

(

)

(

)

∫ ()

(2.164)

and by taking into account the condition of the constancy of the volume of droplets:     1 1 R12  π − ϕ1 + sin 2ϕ1  + r 2  ϕ − sin 2ϕ  = π · R102 = const , 2 2         1 1 2 = const , R  π − ϕ 2 + sin 2ϕ 2  − r 2  ϕ − sin 2ϕ  = π ⋅ R20 2 2    

(2.165)

2 2

where R10 and R20 are the radii of the undisturbed droplets (at h0 > tsf ). It is possible to express all the values via x0: x0 = R1 sinϕ1; x0 = R2 sinϕ2; and x0 = r sinϕ. Then the condition ∂Φ/∂x0 = 0 is used to determine the value of x0, corresponding to the equilibrium position of the droplets. In carrying out the aforementioned procedure, the values of t and r can be expressed through R1, R2, and γ, in accordance with Equation 2.160 and Equation 2.161. In conclusion, let us consider the equilibrium conditions for the most common case of two spherical droplets of different radii and compositions. Using the previously mentioned method of minimization of the excess free energy, we obtain:     2γ h1′′ h1′ +Π t = 1 , γ1  + 3/ 2 1/ 2  2 R1  1 + h′ 2  r 1 + h1′   1       

( )

© 2007 by Taylor & Francis Group, LLC

( )

()

(2.166)

Equilibrium Wetting Phenomena

123

    2γ h2′′ h2′ −Π t = − 2 , γ2  + 3/ 2 1/ 2  2 2 R2  1 + h′  r 1 + h2′   2          

( )

()

( )

(2.167)

where r is now the radial coordinate. The boundary conditions for Equation 2.166 and Equation 2.167 are given by Equation 2.151 through Equation 2.152, where x0 should be replaced by r0. In this case, conditions of the constancy of the volume of droplets can be written as: R1

V1 = 2π

∫ (H

1s

r0

)

− h1s ⋅ rdr + 2π

r0

∫ (H

1s

)

− h rdr =

0

R2

V2 = 2π

∫ (H

2s

)

− h2 s ⋅ rdr + 2π

r0

r0

∫ (H

1s

0

)

4 3 πR10 = const 3

− h rdr =

4 3 πR20 = const 3

At (h′)2 << 1, Equation 2.166 and Equation 2.167 can be simpliﬁed. Summing up these equations would give:

t ′′ +

 1 t′  1 1 1 +  +  ⋅Π t = 2 +  , r  γ1 γ 2   R1 R2 

()

(2.168)

where t(r) = h1(r) – h2(r). Solution of Equation 2.168 describes the variations in the thickness, t(r), of the interlayer between the droplets. The distance between the centers of the droplets can be obtained using the same methods as are used in the case of cylindrical droplets. Thus, the use of the isotherms of disjoining pressure of thin interlayers enables us to solve the problem of the equilibrium of droplets in contact with one another and to calculate the shape of the deformed droplets and of the interlayer.

SHAPE OF A LIQUID INTERLAYER BETWEEN INTERACTING DROPLETS: CRITICAL RADIUS As seen in the previous section, the shape of a liquid interlayer between interacting droplets depends on their size, interfacial tensions, and the shape of the disjoining pressure. The calculations presented in the following text describe the shape of an interlayer between droplets under equilibrium conditions until its possible rupture. © 2007 by Taylor & Francis Group, LLC

124

Wetting and Spreading Dynamics Π

Π Π1

ts

0

t0

h

−Πmin

−Π

h

0

−Π

(a)

(b)

FIGURE 2.36 The disjoining pressure isotherm, Π(h), as used in calculations.

In carrying out further calculations, we assume that the interlayer thickness, h(r), varies within the contact region but not very sharply, that is, the approximation (h′)2 << 1 can be used. This enables the usage of the isotherm of disjoining pressure of ﬂat interlayers, Π(h), in equations of equilibrium, as well as to simplify the expression for the local curvature of the interlayer surfaces. Let us consider the interactions of two identical spherical droplets (Figure 2.37). In this case, the equation of the interlayer proﬁle, h(r), can be derived by solving Equation 2.168: h′  γ h′′ +  + Π h = 2 γ R = Pe ,  2 4

()

(2.169)

where h′ = dh/dr; h″ = d 2h/dr2; γ is the interfacial tension; and R is the radius of droplets. For carrying out quantitative calculations, we use the model isotherm of disjoining pressure, Π (h), in the following form (Figure 2.36a): 0, h ≥ ts Π h = . a (t0 − h), 0 ≤ h ≤ ts

()

(2.170)

Such an isotherm (Figure 2.36a) provides a possibility for obtaining an analytical solution of the problem, and at the same time, it possesses the main properties of real isotherms (Figure 2.36b) corresponding to the attraction of droplets at large separations, i.e., t0 < h < ts, and to their repulsion at small distances, at h < t0. Here, the thickness ts corresponds to the radius of action of surface forces. Beyond its limits, at h > ts, the interaction forces vanish, that is, Π = 0. Parameter a = (Π1 + Π min)/ts determines the slope of the isotherm. Solution of Equation 2.169 together with the isotherm as given by (Equation 2.170) has the following form: © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

125

()

(

)

()

h r = t0 − 2γ aR + A · I 0 z ,

(2.171)

where z = r(2a/γ)1/2, and I0 is Bessel’s function of an imaginary variable. The constant A is determined from the boundary condition, h(r0) = ts, where r0 is the radius of the zone of deformation (Figure 2.37), and z0 = r0(2a/γ)1/2

(

)

( )

A = ts − t0 + 2γ aR  I 0 z0 .

(2.172)

Substituting Equation 2.172 into Equation 2.171, we obtain an equation determining the proﬁle of the interlayer between droplets:

()

h r = t0 −

() ( )

2γ  2γ  I 0 z . + ts − t0 + ⋅ aR  aR  I 0 z0

(2.173)

Accordingly, the minimum distance between the surfaces of droplets is equal to

()

h0 = h 0 = t0 −

2γ  2γ  1 ⋅ . +  ts − t0 +  aR  aR  I 0 z0

( )

(2.174)

In Equation 2.173 and Equation 2.174, the value of r0 still remains to be determined. For this purpose, let us use the condition (2.145): r0

F=

∫ Π ( h) ⋅ 2πr ⋅ dr = 0.

(2.175)

0

Substituting into Equation 2.175 the equation of isotherm (Equation 2.170) and replacing h(r) by its expression from Equation 2.143, we obtain the following expression: 4 aR  2γ  r = tsf − t0 +  aR  γ  2 0

()

r0

I0 z

∫ r I ( z ) · dr. 0

0

(2.176)

0

In the case when z0 ≥ 5 (sufﬁciently large drops), the integral in Equation 2.176 is equal to r0 ( γ / 2a)1/ 2 . In this case, Equation 2.176 results in the following expression for the radius of the contact zone:

(

r0 = 2aγ

© 2007 by Taylor & Francis Group, LLC

)

1/ 2

(

)

 R ts − t0 1 + .  2 a  

(2.177)

126

Wetting and Spreading Dynamics

Using Equation 2.177, the relative extension of the contact zone, where the effect of the surface forces is pronounced, can be expressed as:

(

r0 = 2a ⋅ γ R

 ts − t0 1   2 γ + aR  .

)

1/ 2

(2.178)

At 1/aR << (ts − t0 )/ 2 γ or R >> 2γ/a(ts – t0) the ratio r0/R tends to the value

(

 a  ts − t0 ·   2γ 

)

12

which is independent of the radius of the drop, R. This would indicate the geometric similarity of all large ﬂuid droplets that are deformed by the contact interaction. For the droplets of small radius, R, one may use another approximation for I0(z), which is valid at z < 1: I0(z) = 1 + (z2/4). In this case, integration in Equation 2.176 yields: r02 =

(

(

2 R ts − t0

)(

)

1 − aR / 2 γ ⋅ ts − t0

)

.

(2.179)

Let us compare the preceding expression with the solution for solid spheres. Their equilibrium state (Figure 2.37b) can also be characterized by the interaction region of radius r0, within which the surface forces exert their effect. The proﬁle of the solid sphere can be represented as 2  h = h0 + 2 R ⋅ 1 − 1 − r / R  .  

(

)

R

R

R

h(x) 2B 2h0

θe 2r0 R

ts

2h0

R

R

(a)

θe 2r0

2r0

(b)

(c)

FIGURE 2.37 The schematic representation of the contact interaction of the ﬂuid (a,c) and the solid (b) particles. In the case (c), the thickness in the center coincides with the equilibrium thickness he. © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

127

At r0 << R, its approximate form may be used:

(

)

h ≅ h0 + r 2 R .

(2.179′)

The minimum distance, h0, between the solid particles can be determined from the same condition (2.145) and Derjaguin’s known approximation [1]: h0

( )

∫ ()

F h0 = π· R Π h · dh.

(2.180)

∞

Replacing the lower integration limit by ts and using the model disjoining pressure isotherm of Equation 2.170 (after integration in Equation 2.180), we obtain the following expression: h0 = 2t0 – t1. On substituting this expression into Equation 2.179 and taking into account that h(r0 ) = ts, we get:

(

)

r02 = 2 R ts − t0 .

(2.181)

The same value of r0 is also obtained by solving Equation 2.179 for small ﬂuid droplets. This means that very small emulsion droplets and gas bubbles practically behave as solid particles; they do not deform in the contact zone. Let us call the critical size of ﬂuid droplets R*, such that at R << R*; their interaction does not differ from that of solid spheres. Comparing Equation 2.179 and Equation 2.181, we note that these coincide under the condition that the second term in the denominator of Equation 2.179 is much smaller than unity. This condition allows the determination of R* as R* =

(

2γ . ts − t0 · a

)

(2.182)

The distance between the centers of the droplets can be determined using Equation 2.148 and simple geometrical considerations:

(

B = ts + 2 R 2 − r02

)

1/ 2

.

(2.183)

Accordingly, the contact angle, θe, can also be determined at the point of intersection between the nondeformed surface of a droplet and the r-axis (Figure 2.37a):

(

)

(

)

1/ 2

2 cos θe = B / 2 R = ts / 2 R + 1 − r0 /R  .   © 2007 by Taylor & Francis Group, LLC

(2.184)

128

Wetting and Spreading Dynamics

This expression holds only for relatively large droplets. As has already been pointed out, small droplets (R < R*) are practically nondeformable, and their undisturbed, circular proﬁle does not intersect the r-axis. In the case of small droplets, similar to that of complete wetting (see Section 2.4), the contact angle is absent. Let us now numerically calculate the proﬁles of ﬂuid droplets in the contact zone using Equation 2.173, and compare them with two other known models: the model of solid nondeformable particles (Figure 2.37b), and the model of a ﬂat interlayer between similar droplets (Figure 2.37c). In the latter model, the effect of a transition zone between the surrounding bulk medium and the ﬂat portion of the interlayer is not taken into account. In the case of ﬂat interlayers, let us use the equations of equilibrium of thin ﬂat ﬁlms (Equation 2.47 in Section 2.3) which is modiﬁed for the case of two droplets or bubbles:

(

∞

)

2 ⋅ γ cos θe − 1 =

∫ Π ( h ) ⋅ dh + 2 P ⋅ h , e

e

(2.185)

2 he

where θe is the equilibrium contact angle determined at the point of intersection of the continuation of the nondeformed part of a sphere and the surface of the equilibrium ﬂat ﬁlm. In that case, the thickness of the ﬂat interlayer, 2he, is determined using the disjoining pressure isotherm, Π(h), as Π = Pe, where Pe = 2γ /R is the capillary pressure drop at the spherical interface. The contact area of droplets at he << R is equal to πr02 = πR 2 ⋅ sin 2 θe , where r0 is the radius of the contact zone. Substituting Equation 2.170 of the disjoining pressure isotherm into Equation 2.185, we obtain the following expression:

(

) (

cos θe = 1 − a / 4 γ ·  ts − t0 

) − ( 2γ /aR )  + (1/R )· t − ( 2γ /aR ) . (2.186) 2

2

0

The calculations were carried out for the model disjoining pressure isotherm, Π(h), (Equation 2.170) preset by the following parameters: γ = 30 dyn/cm; ts = 3 · 10–6 cm; t0 = 2 · 10–6 cm; Π3 = Π(0) = 3 · 106 dyn/cm2, and Πmin = –Π (ts) = –1.5 · 106 dyn/cm2, which gives a = 1.5 · 1012 dyn/cm3. According to Equation 2.185, the adopted values correspond to the contact angle θe = 9˚. The left-hand part of the plots in Figure 2.38 relates to the spherical droplets having radius R = 10–3 cm, and the right-hand part to R = 10–4 cm. As appears from the left-hand part, the large-sized droplets deform considerably, thus forming a practically ﬂat contact zone. Proﬁles 1 and 2 are close to each other, and the transition zone occupies rather a small region immediately near the contact perimeter. Now, deviations from the proﬁle of solid particles are very large (curve 3). Thus, in the case of R >> R*, the conditions of equilibrium of the droplets may be described with in the framework of the theory of ﬂat interlayers, i.e., on the basis of Equation 2.185. © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

129

0.2 r R r0

1

2

r0

2

3

1 R = 10–3

R = 10–4

0.1

3

20

10

0

10

20

h, nm

FIGURE 2.38 The proﬁles, h(r/R), of the contact zone of spherical droplets, R = 10–3 cm (to the left) and the R = 10–4 cm (to the right), calculated while taking into account the transition zone (1), in the approximation of a ﬂat interlayer (2), and solid particles (3).

A decrease in the size of droplets (the right-hand side of the graph in Figure 2.38) makes the proﬁles of solid particles (curve 3) and droplets (curve 1) approach one another. In the central part of the contact region, the ﬂat interlayer region reduces, whereas the differences from the proﬁle of the ﬂat interlayers (curve 2) increases. A further decrease in the droplet radius causes a still greater approach of the proﬁles of the ﬂuid and the solid particles to one another and a disappearance of the ﬂat portion of the interlayer. As already shown, at R << R*, the known solutions for the interaction of solid spheres may be used. In this connection, it is important to evaluate the values of R*. This allows the determination of the regions of applicability of different solutions — namely, ﬂat interlayers for R >> R*; solid particles for R << R*; and ﬁnally, the approach developed for the intermediate values of R. As appears from Equation 2.182, the critical radius R* depends on the parameters of the disjoining pressure isotherm (a, ts, and t0) and the interface tension. Under otherwise equal conditions, a decrease in the interface tension reduces the values of R*, thus limiting the region of the applicability of the theory of solid spheres. Let us evaluate R*, assuming γ = 50 dyn/ cm, and (ts –t0) = 10–6 cm. The values of parameter a will be varied, which, in accordance with Equation 2.186, is equivalent to a change in the contact angle θe (omitting small terms):

(

)(

)

2

cos θe = 1 − a / 4 γ · ts − t0 . © 2007 by Taylor & Francis Group, LLC

(2.187)

130

Wetting and Spreading Dynamics

The calculations show that the values of R* decrease as θe increases. Thus, at small values of θe close to zero, R* = 10–3 cm; at θe = 5–6˚, R* = 10–4 cm; at θe = 20–30˚, R* = 5 × 10–6 cm; and at θe = 90˚, R* = 5 × 10–7 cm. In this manner, an increase in the contact angle, corresponding to the enhancement of the droplet interaction, causes a decrease in the critical radius R*. Now, this means that in the case of a strong interparticle interaction, the approach of solid spheres becomes less applicable; yet, in the case of weakly interacting droplets, the particular approach may be used even for relatively large-sized droplets. The solutions obtained allow the evaluation of R* and help to choose the corresponding equations for the calculation of the equilibrium shape of the droplets within the contact zone.

2.10 LINE TENSION The presence of the transition zone between a drop or a bubble and thin liquid interlayers can be described in terms of line tension, τ, a concept ﬁrst introduced by Gibbs (see for example [22]). In the case of surface tension, the transition zone between the liquid and vapor is replaced by a plane of tension with excess surface energy, γ. By analogy, the transition zone between a drop or a bubble and the thin liquid interlayer may be replaced by a three-phase contact line with an excess linear energy, τ. In contrast to surface tension deﬁned always as positive, the value of the line tension may be positive and negative. When positive, it contracts the wetting perimeter, whereas the perimeter expands if the line tension is negative [33–36]. A number of attempts have been made to improve Young’s equation and to make it more theoretically justiﬁed. The most important of them is the introduction of line tension, τ. In Section 2.3, it has been shown that the drop proﬁle cannot keep its spherical shape up to the three-phase contact line. It has been shown in the same section that the action of surface forces results in a substantial deviation of the drop shape, in the vicinity of the three-phase contact line, from a spherical shape. It results in the formation of a transition zone where the inﬂuence of the disjoining pressure is important and cannot be ignored. However, if we still want to consider a spherical droplet, then the existence of the transition region can be effectively taken into account by replacing the whole transition region by an additional free energy located on the three-phase contact line. This consideration, in a way, is similar to the introduction of an interfacial tension. If the line tension is taken into account, then the excess free energy of the droplet on the solid substrate, Φ, should be written as: Φ = γS + PeV + πr02 ( γ sl − γ svh ) + 2 πr0 τ ,

(2.188)

where r0 is the radius of the droplet base. Note that, according to Section 2.1, we used the interfacial tension of the solid substrate covered by an equilibrium liquid ﬁlm, γ svh , but not the corresponding interfacial tension of a bare solid © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

131

substrate, γ sv . In Section 2.1, we explained that the latter surface tension cannot be used at the equilibrium conditions, and these two interfacial tensions can differ considerably. Using the expression for the droplet proﬁle, hid(r), the latter equation can be rewritten as r0

 F = 2 π r  γ 1 + h ′i d  0

( )

∫

2

 + P ⋅ h + γ sl − γ svh  dr + 2 πrr0 τ , 

(2.189)

where τ is line tension, and hid(r) is the idealized droplet proﬁle: it has a spherical part up to the intersection with the surface of the equilibrium ﬂat ﬁlm. The ﬁrst two conditions of the minimum value of the excess free energy result in the identical equation for the spherical drop proﬁle (41), Section 2.3. However, the transversality condition (4) in Section 2.2 takes the following form:   γ dτ r0 + τ = 0. + γ sl − γ svh  +  2 dr 0  1 + h ′ r =r0

(2.190)

If we now introduce a new real equilibrium contact angle that takes into account the line tension as θe and use the previous deﬁnition of the contact angle, which is referred to now as θe∞ according to Equation 2.47, then Equation 2.190 can be rewritten as

(

)

γ cos θe − cos θe∞ +

dτ τ + = 0. dr0 r0

(2.191)

Note that the derivative in the preceding equation is usually neglected without any justiﬁcation. Neglecting the derivative of the line tension in Equation 2.191 results in: cos θe = cos θe∞ −

τ . γr0

(2.192)

If the line tension is negative, then the inﬂuence of line tension results in a bigger contact angle as compared with predictions according to Equation 2.192. In the general case, Equation 2.191 can be rewritten as  τ ∂τ  , m γ ⋅ cos θe − cos θe∞ = ±  +  r0 ∂r0 

(

)

(2.193)

where m and “plus” or “minus” depends on the system geometry: m = 1 and a minus sign corresponds to the drop on the solid substrate, m = 2 and a plus sign © 2007 by Taylor & Francis Group, LLC

132

Wetting and Spreading Dynamics

correspond to the two identical drops or bubbles in contact (Figure 2.37); θe∞ refer to the contact angle in the case of big cylindrical drops (see the following section). In the case of liquid drops on the ﬂat solid substrate, the positive values of τ cause an increase in the values of contact angles θe, whereas the negative value results in their decrease. In the case of ﬂat ﬁlms in contact with a concave meniscus, the inﬂuence of the line tension τ, is inversed because the minus sign should be used now in Equation 2.193. For water and aqueous electrolyte solutions, the line tension values are in the range of 10–6–10–5 dyn [23] and below. Thus, the terms on the right-hand side of Equation 2.193 become noticeable at r0 ≤ 10–4 cm. It is important to recall that the aforementioned value of the line tension can be used only at equilibrium. It is impossible to experimentally measure the equilibrium liquid droplets on solid substrates because they should be at equilibrium with oversaturated vapor, as was explained in Section 2.3. This would mean that usually everything in Equation 2.193 is either far from equilibrium or under a quasi-equilibrium condition (as caused by the hysteresis of contact angle — see Section 3.10). In that case, the value of line tension can be many orders of magnitude higher than 10–6–10–5 dyn. However, this line tension should be referred to as dynamic line tension. To the best of our knowledge, there has not been any attempt as yet to introduce or investigate the dynamic line tension. The values of the line tension, τ, for drops on solid substrates has been calculated as a difference between the values of γ ⋅ cos θe, and γ ⋅ cos θe∞ is calculated in two different ways: (1) neglecting the transition zone, (2) by taking it into account. As the line tension arises due to the existence of the transition zone, it is clear that this difference is just associated with the additional terms on the right-hand side of Equation 2.192. An expression for τ was obtained in the case of a model isotherm of disjoining pressure [23], and the line has been estimated as 10 −5 ÷ 10 −6 dyn and negative. De Feijter and Vrij [24] have considered the transition zone between a Newton black ﬁlm (a different name for α-ﬁlms) and bulk liquid. According to their estimations, the line tension value is also negative and has the same order of magnitude. Kolarov and Zorin [25] have measured the line tension value. They have used Sheludko’s cell for measurements of properties of free liquid ﬁlms. An aqueous solution of 0.1% NaCl with sodium dodecyl sulphate (SDS) 0.05% concentration (carboxymethylcellulose or CMC = 0.2%) has been used. They have calculated the line tension for this system using Young’s equation (Equation 2.193). The value of line tension has been found to be −1.7 ⋅ 10 −6 dyn. That is, it is negative and in good agreement with the theoretical predictions [23]. However, Platikanov et al. [26] have carried out experimental measurements of line tension dependency on salt concentration and presented this experimental evidence of line tension sign change. In the following text, we present a theory that is modiﬁed as compared with the presented theory [23], and it explains the experimentally discovered sign change in the line tension.

© 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

133

Let us consider two big (R >> R*) identical drops or bubbles in contact (Figure 2.37) under equilibrium conditions. It has been explained in Section 2.3 that it is important to properly select the reference state. This reference state is introduced as a ﬂat equilibrium liquid ﬁlm of thickness 2he. The reason for the same was presented in Section 2.1 and Section 2.3. From a mathematical point of view, it means addition or subtraction of a constant to or from the excess free energy. The latter constant does not inﬂuence the ﬁnal equation, which describes the proﬁle in the transition zone. However, as already seen in Section 2.3, this choice is essential for transversality conditions at the apparent three-phase contact line. In the following text, we see that this choice is also important for the deﬁnition of the line tension. This means that the choice of the reference state is very important, and we use the same choice of the reference state as in Section 2.1, which is the uniform ﬂat equilibrium ﬁlm of thickness 2he, where he is the half-thickness of the equilibrium ﬁlm. Using this choice, the excess free energy, Φ, of the system (curve 1 in Figure 2.34) has the following form: ∞ ∞   2  Φ = 2π r 2 γ ( 1 + h′ − 1) + Π (h) dh − Π (h) dh + 2 Pe ⋅ (h − h e )  dr ,   0  2h 2h e   (2.194) R

∫

∫

∫

where h(x) is the half-thickness of the liquid layer; he is the half-thickness of the ﬂat equilibrium thin liquid ﬁlm; Pe = 2γ/R is the excess pressure; Π(h) is the disjoining pressure; and h(R) = H is the position of the end of the drop. The lower limit of integration corresponds to the end of the transition zone where h = he. We can use inﬁnity as the upper limit of integration instead of R because, at this stage, we are not interested in the upper part of the drop or bubble. Note that according to the deﬁnition given by Equation 2.162 in Section 2.9, the excess pressure, Pe, is positive. Under the equilibrium condition, the system is at the minimum free energy state, that is, conditions (1) through (4) should be satisﬁed (Section 2.2). This results in the equation for the determination of the liquid proﬁle in the transition zone:  γ d  ⋅ h′′ r + Π 2h = Pe . 1   r dr  1 + h′ 2  2      

( )

( )

(2.195)

The transversality condition (4) results in (see Appendix 1) h ′ → 0,

© 2007 by Taylor & Francis Group, LLC

(2.196)

134

Wetting and Spreading Dynamics

at the end of the transition zone, which means a smooth transition from the transition zone to the ﬂat thin ﬁlm. Let us introduce an ideal proﬁle of the liquid interlayer in the transition zone, 2hid , which is a spherical part up to the intersection with the equilibrium liquid interlayer of thickness 2he (Curve 2 in Figure 2.34). The excess free energy of such an ideal proﬁle differs from the exact excess free energy given by Equation 2.194 because the presence of the transition zone is ignored in the case of the ideal proﬁle. This means that the line tension, τ, should be introduced to compensate the difference:   Φ = 2 π r 2 γ  1 + h ′i d   r0   ∞

( )

∫

2

 − 1 − 

 Π h dh + 2 Pe ⋅ h i d − h e  d r + 2 πr0 τ  2he  (2.197) ∞

∫ ()

(

)

Under equilibrium conditions, the same equilibrium conditions (1–4) should be satisﬁed, which gives an equation for the ideal liquid proﬁle in the transition zone  γ d  hid ′ = P, r 1  r dr   2 2    1 + hid ′       

( )

(2.198)

and the transversality condition in the case of ideal liquid proﬁle with excess free energy given by Equation 2.197 is as follows:  ∂f  d (r0 τ) −  fid − hid′ id  + = 0, ′ dr0  ∂hid r =r 0

(2.199)

where ∞     fid = r  2 γ  1 + hid ′ 2 − 1 − Π (h) dh + 2 P ⋅ (hid − h e )  .     2h e  

∫

Substitution of the latter expression into the condition (2.199) results in the following equation at r = r0 and h = he:  2 γ  

(

) ∫ ∞

1 + hid − 1 − 2

© 2007 by Taylor & Francis Group, LLC

2he

Π ( h ) dh −

  dτ τ  2 γhid ′ 2  − + =0.   dr0 r0  1 + hid ′ 2  r = r0

Equilibrium Wetting Phenomena

135

After a rearrangement, the latter condition becomes:  τ ∂τ  2 γ ⋅ cos θe − cos θe∞ =  + ,  r0 ∂r0 

(

)

(2.200)

where ∞

2 γ cos θe∞ = 2 γ +

∫ Π (h) dh .

(2.201)

2he

Equation 2.200 coincides with Equation 2.193 if we select m = 2 and a corresponding plus sign. Equation 2.201 gives the expression for the contact angle of a big cylindrical drop (see the following text). Excess free energy given by Equation 2.194 and Equation 2.197 should be equal. The latter gives the following deﬁnition of the line tension, τ:   r  2 γ  1 + h′i d   r0   ∞

( )

∫

 r  2γ  0  

∞

∫

(

2

 − 1 − 

 Π h dh + 2 P ⋅ h i d − h e  dr + r0 τ =  2h e  ∞

(

∫ ()

)

 1 + h′ 2 − 1 + Π (h) dh − Π (h) dh + 2 P ⋅ (h − h e )  dr.  2h 2h e 

)∫ ∞

(2.202)

∞

∫

The preceding equation presents an exact deﬁnition of the line tension, τ, in contrast to Equation 2.200, where the value of the line tension is unknown. In Equation 2.202, the real liquid proﬁle, h(r), is the solution of Equation 2.195, and the ideal liquid proﬁle, hid(r), is the solution of Equation 2.198. Dependency of the line tension on the radius, r0, has been investigated [23] in the case of a model disjoining pressure isotherm. Here, we focus on the absolute value of the line tension and a possible comparison with experimental data. For this purpose, let us consider the line tension in the simplest possible case: contact of two identical cylindrical drops or bubbles. In this case, the corresponding excess free energies given by Equation 2.194 and Equation 2.197 take the following form: ∞

Φ=

∫ 0

  2γ  

(

)∫ ∞

1 + h′ 2 − 1 + Π (h) dh −

© 2007 by Taylor & Francis Group, LLC

2h

 Π (h) dh + 2 Pe ⋅ (h − h e )  dx , (2.203)  2 he  ∞

∫

136

Wetting and Spreading Dynamics

and   Φ = 2 γ  1 + h ′i d   x0   ∞

∫

( )

2

 − 1 − 

 Π h dh + 2 Pe ⋅ h i d − h e  dx + τ , (2.204)  2he  ∞

(

∫ ()

)

where Φ is an excess free energy per unit length. From Equation 2.203, we conclude that γ ⋅ h ′′

( )

1 + h ′    2

3 2

( )

+ Π 2h = P .

(2.205)

Equation 2.205 describes the whole range of the liquid proﬁle, including the lower bulk part of the drop or bubble, the thin ﬂat liquid interlayer in front of it, and the transition zone in between. The boundary conditions for Equation 2.205 are

( )

( )

h R = H, h → he , h′

h = he

h ′ R = −∞ ,

(2.206)

x→∞ =0,

x → ∞.

(2.207)

We can integrate Equation 2.205 using the boundary condition (2.206), which yields: γ 1

( )

= Le ( h ) ,

(2.208)

1 + h ′ 2  2   where

()

(

)

Le h = Pe ⋅ H − h −

∞

∫ Π (h)dh .

2h

Equation 2.208 can now be rewritten as: h′ = −

© 2007 by Taylor & Francis Group, LLC

γ2 −1. L2e ( h )

(2.209)

Equilibrium Wetting Phenomena

137

The left-hand side of Equation 2.208 is always positive and less than γ. That means, the same should be true for the right-hand side of Equation 2.208:

()

0 ≤ L h ≤ γ,

(2.210)

where L (h) = γ , if h = h e and L (h) = 0, if h = H . Condition (2.207) results in:

(

∞

)

∫ Π (h)dh.

Pe ⋅ H − he = γ +

(2.211)

2 he

The capillary pressure can be expressed as before: Pe =

γ , R

(2.212)

where R is the radius of the curvature of the cylindrical drop. Simple geometrical considerations show that

R=

H . cos θe

(2.213)

With the help of the preceding condition, we can conclude that

Pe =

γ ⋅ cos θe . H

(2.214)

Using Equation 2.211 and Equation 2.214, we conclude:

1+ cos θe =

1 ⋅ 2γ

∞

∫ Π (h)dh

2 he

1−

he

(2.215)

H

γ ⋅ cos θe = Π 2 he H

( )

© 2007 by Taylor & Francis Group, LLC

,

(2.216)

138

Wetting and Spreading Dynamics

The two latter equations are modiﬁcations of our previous consideration in the case of drops/menisci (see Section 2.3). If he /H << 1, the contact angle in Equation 2.215 is referred to as θe∞ and coincides with that given by Equation 2.201. In the case of partial wetting, the contact angle is in the following range: 0 ≤ θe ≤

π , 2

or 0 < cos θe < 1. Using condition (2.210) the latter inequality can be rewritten as ∞

−γ <

∫ Π ( h) dh < −

2 he

γ ⋅ he . H

Hence, the integral, ∞

∫ Π ( h ) dh ,

2 he

should be negative in the case of partial wetting. In the case of the ideal proﬁle, we should use Equation 2.204, which results in γ ⋅ hi′′d

( )

1 + h ′ id 

3

2

= Pe .

(2.217)

2 

The boundary conditions for Equation 2.217 are

( )

hid R = H,

( )

h ′i d R = −∞.

(2.218)

Equation 2.217 can be integrated using boundary conditions (2.218), which yields:

h ′i d = −

( )

where Lid hid = P ( H − hid ). © 2007 by Taylor & Francis Group, LLC

γ2 − 1, L2id

(2.219)

Equilibrium Wetting Phenomena

139

Line tension, τ, can be expressed using Equation 2.203 and Equation 2.204 as ∞

τ=

∫ 0

  2γ  

(

)∫ ∞

2h

∞   −  2 γ  1 + h′i d   x0  

( )

∫

2

 Π (h) dh + 2 P ⋅ (h − h e )  dx  2h e  ∞

1 + h′ 2 − 1 + Π (h) dh −

 − 1 − 

∫

 Π h dh + 2 P ⋅ h i d − h e  dx  2h e  ∞

(

∫ ()

)

Using Equation 2.209 and Equation 2.219, we can rewrite the preceding equation as ∞

∫

τ = 2 γ  0

∞

 1 + h ′ 2 − L ( h )  dx − 2  γ 1 + h ′i d   x

( )

∫

2

0

 − Lid ( hid )  dx. (2.220) 

Using Equation 2.209 and Equation 2.219, we can switch from the integration over x to integration over the thickness, h. The latter transformation of Equation 2.220 gives

∫(

)

∞

τ=2

γ 2 − L 2 (h) − γ 2 − Lid2 (h) dh.

he

(2.221)

Note that the integration on the right-hand side is over h from he to inﬁnity, which does not depend on the proﬁle (real or ideal proﬁle) but only on the integration limits. Equation 2.221 can be rewritten as ∞  2 P( H − h) Π(h)dh −  Π(h)dh    2h 2h ∞

2 τ= γ

∞

∫

∫

∫

1 − L 2 (h) /γ 2 + 1 − Lid2 (h) /γ

he

2

2

dh.

(2.222)

In the case of h << H inside the whole transition zone, expressions for L(h) and Lid (h) can be rewritten as

(

)

1 L (h) = γ cos θe∞ − ε(h) , ε(h) = γ Lid (h) = γ cos θe∞ © 2007 by Taylor & Francis Group, LLC

∞

∫ Π(h)dh

2h

140

Wetting and Spreading Dynamics

Using the latter expression, Equation 2.222 can be rewritten as

τ=

2γ sin θe∞

∞

2 cos θe∞ ε( h ) − ε 2 ( h )

∫

2 cos θe∞ ε( h ) − ε 2 ( h ) he 1+ 1+ sin 2 θe∞

dh.

(2.223)

In the case of small contact angles, ε(h) << 1, and the latter equation takes the following form: 4γ τ= tan θe∞

∞

∫ 1+

he

ε(h) 2 cos θe∞ ε(h) 1+ sin 2 θe∞

dh.

(2.224)

It is possible to show that 0.5 <

1 2 cos θe∞ ε( h ) 1+ 1+ sin 2 θe∞

<1

in the case under consideration. Let the mean value of the latter expression be ω, where 0.5 < ω < 1; then Equation 2.224 takes the following form:  ∞ 4ω  Π( h ) dh  dh ≈ θe∞   2 he  2 h ∞

4ω τ= tan θe∞

∫ ∫

 ∞  Π( h ) dh  dh.   2 he  2 h ∞

∫ ∫

(2.225)

Equation 2.201 can be rewritten as cos θe∞ = 1 + ε( he ) , or

sin θe∞ ≈ θe∞

1 = 2 − ε( he ) = 2 − γ

∞

∫ Π(h)dh .

(2.226)

2 he

Combining the preceding expression and Equation 2.225 results in 2ω

τ= 1 − γ

∞

∫ Π(h)dh

∞   Π( h ) dh  dh =   2 he  2 h

2 he

∞

∫ ∫

2ω γ ∞

−

∫ Π(h)dh

∞   Π( h ) dh  dh .   2 he  2 h ∞

∫ ∫

2 he

(2.227)

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Equilibrium Wetting Phenomena

141

We can compare the experimental data by Platikanov et al. [26] to the theory predictions according to Equation 2.227 (at ω = 1). We have used only the dispersion and electrostatic components of the disjoining pressure. The following expressions for the different components of the disjoining pressure are used: for the dispersion component:

Πd = −

A , h3

(2.228)

where A ≈ 10–14 dyn·cm is the Hamaker constant and for the electrostatic component: Πel = 64 ⋅ c ⋅ R ⋅ T ⋅ tanh(ψ / 4) ⋅ exp(−κ ⋅ h),

(2.229)

where R, T, F, ψ=

κ=

Fψ , RT

8πF 2 c ε w RT

are the concentration of the electrolyte, the universal gas constant, temperature in °K, Faraday number, dimensionless zeta potential of the ﬁlm surfaces, and the inverse Debye length, respectively; εw is the dielectric constant of water. Hence, the total disjoining pressure is Π(h) = −

A + 64 ⋅ c ⋅ R ⋅ T ⋅ tanh(ψ / 4) ⋅ exp(−κ ⋅ h). h3

(2.230)

Unfortunately, the disjoining pressure in the form given by Equation 2.230 does not allow any equilibrium liquid ﬁlms at low thickness (α-ﬁlms). To overcome this problem, we introduce a cutoff thickness, t* (Figure 2.39). According to this choice, the equilibrium thickness 2he does not depend on the pressure inside the drops or bubbles and is always equal to t*. Platikanov et al. [26] have determined both contact angle and line tension on the NaCl concentration in the range of concentrations 0.2–0.45 mol/l. These two dependencies are discussed in the following section.

© 2007 by Taylor & Francis Group, LLC

142

Wetting and Spreading Dynamics Π

t∗

h

FIGURE 2.39 Model disjoining pressure isotherm according to Equation 2.230 with cutoff thickness t*.

COMPARISON

WITH

EXPERIMENTAL DATA

AND

DISCUSSION

Details of the experimental measurement and system under consideration are given in Reference 26. Line tension of the free liquid ﬁlm between two bubbles was investigated in the range of NaCl concentration 0.2–0.45 mol/l. The ﬁlm and bubble surfaces were stabilized by surfactants [26]. The zeta potential of the ﬁlm surface was ψ = 17 mV or ψ = 0.68 according to Reference 26. The cutoff thickness, t*, was used as a ﬁtting parameter. We used experimental values of the contact angle from Reference 26 on salt concentration to determine the cutoff thickness t*, according to Equation 2.226. A reasonable agreement between experimental dependency of the contact angle on the salt concentration and the calculated one according to Equation 2.226 has been attained. The ﬁtted dependency of the contact angle on the salt concentration was much weaker than the original experimental data [26]. However, we tried to compare our calculation of line tension according to Equation 2.227 and the corresponding experimental data of line tension from Reference 26, using the already calculated cutoff thickness, t*. The calculated dependency of line tension on the electrolyte concentration is shown in Figure 2.40. The following conclusion can be made, based on the consideration of the dependency presented in Figure 2.40: •

•

The line tension dependency on the salt concentration is in a qualitative agreement with experimental dependency in Reference 26, that is, line tension goes from positive to negative values with an increase in the electrolyte concentration. The absolute values of the calculated line tension were found considerably different from the corresponding experimental values; the calculated electrolyte concentration (0.022 mol/l) at which line tension

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Equilibrium Wetting Phenomena

143

10

Line tension τ ∗ 107, dyn

8 6 4 2 0 0.1 –2

0.2

0.3

0.4

0.5

Electrolyte concentration, mol/l

FIGURE 2.40 Calculated dependency of line tension in electrolyte concentration.

• •

switches from positive to negative values does not match the experimental value (0.36 mol/l). The calculated line tension remains almost constant in the range of electrolyte concentrations above 0.1 mol/l used in Reference 26. The line tension decreases much faster with the electrolyte concentration (Figure 2.40) than experimental values [26].

The discrepancy between the measured and calculated line tension dependencies can be caused by one or both of the following reasons: •

•

Only the dispersion and electrostatic components of disjoining pressure have been used to compare with the experimental data. It looks like these two components are not enough to adequately describe the behavior of thin liquid ﬁlms and the transition region in the system under consideration. The inﬂuence of both the structural (caused by the orientation of water dipoles in the vicinity of free ﬁlm surfaces) and steric (caused by the direct interaction of the head of the surfactant molecules on the ﬁlm surfaces) components cannot be ignored. The theory of these components of disjoining pressure is to be developed. According to the deﬁnition of line tension (2.222), it is determined by the equilibrium liquid proﬁle in the transition region from the thin ﬂat liquid interlayer to the bulk surface of bubbles. Note that in the case of partial wetting, which is under consideration, the static hysteresis of contact angle (see Chapter 3, Section 3.10) is unavoidable. The latter phenomenon can substantially inﬂuence the comparison.

In Reference 26, considerable efforts have been made to reach the equilibrium state. We would like to emphasize again: if the liquid proﬁle in the transition region from a thin liquid interlayer to the bulk drop or bubble interface is not at © 2007 by Taylor & Francis Group, LLC

144

Wetting and Spreading Dynamics

equilibrium, then values of line tensions can differ considerably from the theoretically predicted equilibrium values. In the case of quasi-equilibrium (no macroscopic motion but possible microscopic motion inside the transition zone, see Chapter 3, Section 3.10), it is necessary to introduce a new dynamic line tension. We believe that this is a challenging area that requires further research.

2.11 CAPILLARY INTERACTION BETWEEN SOLID BODIES In this section, we shall consider the capillary interaction between two solid plates partly immersed in liquid, which can be, one or both, completely wetted, partially wetted, or nonwetted by the liquid. We shall derive an expression for the force of interaction between them at large separation distances. We shall show that if one of the plates is wetted and the other is not, then a critical separation exists such that below this separation, the plates attract each other, whereas there is a repulsion at larger separations. Similarly, we shall see that there is a critical angle of relative inclination that also delineates the regions of attraction and repulsion [37]. We shall not specify the kind of contact angle as it can be either static advancing or static receding, depending on the way the system is placed. First we calculate the force of interaction between the plates at large separations, L >> a, where a = γ /ρg is the capillary length, γ is the surface tension, ρ is the density of the liquid, and g is the gravity acceleration. We consider only the case of two partially (or complete) wettable plates, that is, θ1 < π/2, θ2 < π/2, as the case of two nonwettable plates, θ1 > π/2, θ2 > π/2, can be treated in a similar manner. Note that in the case of partial or complete wetting, the height of the liquid between the plates is higher than the height outside the plates, that is, h1 > H1 and h2 > H2 (Figure 2.41).

1

2

h1 H1

θ1

h2 θ2

θ1 0

xm

H2 θ2 L

x

FIGURE 2.41 Liquid proﬁle between two partially wettable plates that are partially immersed into the liquid. Different contact angles, θ1 ≠ θ2. © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

145

We use the principle of the frozen state [27] to calculate the force of interaction between the plates. We imagine that the liquid between plates 1 and 2 (Figure 2.41) has solidiﬁed above the level of the free liquid surface. We divide the resulting solid body by a plane parallel to plates 1 and 2, passing through x = xm, where dh dx

= 0. x = xm

Displacement of plate 2 through dL while plate 1 is ﬁxed, changes the excess free energy of the system, Φ, by ρgl 2 hm dL , 2

dΦ =

where l is the width of the plates. As dΦ = FdL, where F is the force of attraction between the plates, we get ρg 2 lhm . 2

F=

(2.231)

According to the previous consideration, the shape of a liquid surface plates in the gravity ﬁeld can be described by the following equation: γh′′

(

1 + h′ 2

)

3/ 2

= ρgh,

or h′′

(1 + h′ )

3/ 2

2

=

h . a2

Close to the position xm, the liquid proﬁle has a low slope, that is, h′2 << 1, and hence, in the vicinity of this position, the shape of the liquid surface can be described by the linearized equation h′′ = h /a 2 , when close to x = xm,

()

(

((

)

) )

h x = c1 exp − x /a + c2 xp − L − x a . The condition dh dx © 2007 by Taylor & Francis Group, LLC

=0 x = xm

(2.232)

146

Wetting and Spreading Dynamics

results in

(

)

hm2 = 4c1c2 exp − L /a ,

(2.233)

where hm = h(xm). The exact solution for the shape of a liquid surface close to an isolated plate is deduced in Appendix 2:

a x = ln 2

( (

 h2  2 − 1 + sin θ  1 + 1 − 2   4 a   h2  2 + 1 + sin θ  1 − 1 − 2   4 a 

) )

 h2 + a 2  1 + sin θ − 2 − 2  2a

(A2.8)

  

Using the latter expression at x >> a, we get h( x ) ≈ c∞ (θ) exp(− x /a), where

c∞ (θ) = 4 a

2 − 1 + sin θ 2 + 1 + sin θ

.

Substituting ci = c∞(θi), i = 1,2 from the preceding equation into Equation 2.232 and Equation 2.233 gives an expression for the force of interaction between the two plates at L >> a:

F = 32ρga 2l

2 − 1 + sin θ1 2 + 1 + sin θ1

·

2 − 1 + sin θ2 2 + 1 + sinn θ2

e− L /a ,

(2.234)

In the case of two identical plates (θ1 = θ2 = θ), this expression results in F = 32ρga 2l

2 − 1 + sin θ 2 + 1 + sin θ

e− L /a

(2.234’)

and in the case of complete wetting (θ = 0), F = 5.5 ρga 2 le − L / a © 2007 by Taylor & Francis Group, LLC

(2.234’’)

Equilibrium Wetting Phenomena

147

Equation 2.234’ and Equation 2.234’’ show that two partially wettable plates attract each other, and the force of attraction between these plates decays exponentially at L >> a. We can similarly calculate the force of interaction between any two bodies for which the surface between the bodies, along which the height of capillary rise of the liquid has a minimum value, is a plane. Examples are: two identical spherical or cylindrical particles. In Reference 28, the proﬁle of a liquid close to partially immersed cylindrical plates has been calculated. Using this expression for the shape of the surface of the meniscus at a vertical cylinder of radius R at large separations, we get the following expression: ∞

∫ K (x)d

F = 4 ρga 2 R cos 2 θ

x 2 − L2 / 4 a 2

2 0

L /2a

≈

π ρga R cos θ 2

3/ 2

2

L /a

(2.235)

{

}

·exp − L /a ,

where K0 is a cylindrical function. The latter expression shows that the force between two partially wetted cylinders is also that of attraction, and this force decays faster compared to the case of the two plates. In the following text is an expression deduced in Appendix 2 for the height of capillary rise of the meniscus at the plates on the free liquid side at θ = θi, i = 1, 2:

(

)

H i = a 2 1 − sin θi .

(A2.5)

We now consider the interaction between wettable and nonwettable plates. Let there be an isolated plate 1 with contact angle θ1 < π/2. At some distances Lk , we draw an intersecting plate 2 (Figure 2.42). 2

1

θ1 0

θ1

θ2

Lk

θ2

x

FIGURE 2.42 Capillary interaction between wettable and nonwettable plates. © 2007 by Taylor & Francis Group, LLC

148

Wetting and Spreading Dynamics

The angle between the surface of the unperturbed meniscus and the intersecting plate θ2 > πI2 (Figure 2.42). If we replace the intersecting plate by a real plate with contact angle θ2, the surface of the meniscus between the plates will obviously be unchanged; the meniscus at the second plate on the side of the free liquid surface will sink to a height H2, according to Equation A2.5. Thus, there are no forces acting along the x-axis at plates l and 2. We denote the distance at which the angle between the intersecting plate and the meniscus is equal to the wetting angle θ2 as Lk (θ1, θ2). It is easy to check whether Lk(θ1, θ2) exists only if θ1 + θ2 < π, θ1 <

π π , θ2 > . 2 2

(2.236)

If θ1 + θ2 → π , Lk → 0, and if θ2 → π/2, Lk → ∞. If the condition θ1 + θ2 < π holds, we can easily determine Lk(θ1, θ2) by substituting

(

H 2 = a 2 1 − sin θ2

)

into the equation for the unperturbed meniscus at plate 1 (A2.8), which results in

(

)

Lk θ1 , θ2 =

a ln 2

( (

+a 2

)( 1 + sin θ ) (

2 − 1 + sin θ1 2+

(

1 + sin θ1 −

1

) 2 − 1 + sin θ ) 1 + sin θ ). 2 + 1 + sin θ2

2

(2.237)

2

The angle between the meniscus and the intersecting plate when L < Lk , is greater than θ2, and a consequent displacement of plate 2 from the position L = Lk toward L < Lk , causes the liquid to rise between the plates and thus sets up an attraction between the two plates. Similarly, a displacement of plate 2 from the position L = Lk toward L > Lk causes the liquid to fall between the plates and consequently sets up a repulsion. Thus, L = Lk (θ1, θ2) delineates two regions of interaction between the plates: attraction when L < Lk , and repulsion when L > Lk , i.e., L = Lk is a state of unstable equilibrium. Important conclusion: according to condition (2.236), any completely wettable plate on a liquid surface is attracted to a nonwettable plate at sufﬁciently small distances. At separations L >> a (when L > Lk ), between plates, one of which is partially wetted whereas the other is not, the liquid surface between them must intersect at the level h = 0. Using the principle of the frozen state, the force of interaction between the plates in this case is

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Equilibrium Wetting Phenomena

149

F = − γl (1 − cos α), where α = arctan h′(x0) and h(x0) = 0. Close to x = x0, the liquid proﬁle, h(x), as in the preceding case becomes h = c1 exp(− x /a) − c2 exp(−( L − x ) /a), when cos α =

1

( )

1 + h′ x 0 2

≈ 1−

( )

( )

2c1c22 1 2 exp L/aa , h′ x 0 = 1 − 2 a2

and for the interaction force, F, we get the following expression: F =−

2 γlc1c2 − L / a . e a2

As before, by expressing c1, c2 from the equations for unperturbed surfaces, we ﬁnally get 2 − 1 + sin θ1

F = −32 γl

2 + 1 + sin θ1

2 − 1 + sin θ2

⋅

2 + 1 + sin θ2

e− L / a .

(2.238)

Note that the latter force is a repulsive force, which is completely different from the case of two partially wettable plates, according to Equation 2.234. Now let us place a nonwettable plate 2 at an arbitrary position L > Lk at an angle α(L) > 0 (Figure 2.43), choosing α(L) so that the meniscus at the point x = L – ∆ would not be disturbed by the presence of plate 2. 1

2

θ2

α x

L–∆ θ2

FIGURE 2.43 Critical inclination of plates.

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Wetting and Spreading Dynamics

Figure 2.43 shows that the angle α(L) has to be selected as

( )

(

)

α L = π − θ2 − arctan h′ L − ∆ ,

(2.239)

where ∆, according to Figure 2.43, is given by  π ∆ = cotanα L a 2 1 + sin  − arctan h′ L − ∆ 2 

( )

(

 1 = cotanα L a 2 1 + 2  1 + h′ L − ∆ 

( )

(

)

   



)  . 

(2.240)

At L > Lk and L >> a, h′ → 0, and hence, according to Equation 2.239, α ( ∞) = π − θ 2 .

(2.241)

Thus, there are no forces acting on plate 1 when plate 2 is positioned at an angle αk; this applies equally to plate 2. Inclination of plate 2 at an angle smaller than α(L) causes the liquid to rise between the plates, i.e., sets up an attraction between them, but inclination at an angle bigger than α(L) causes the liquid to fall between plates, i.e., sets up a repulsion between the two plates. Thus, the angle α(L) delineates the regions of attraction and repulsion between the plates, deﬁning a state of unstable equilibrium. Equation 2.239 and Equation 2.240 can be used for numerical calculation of the unknown wetting angle of plate 2 at all separations, whereas Equation 2.241 can be used for large separations only. In conclusion, we note that positioning two nonwettable plates at an angle β = θ2 – π /2 from the vertical (Figure 2.44a) will constrain the liquid to a horizontal position between them, i.e., the liquid between the plates will form a convex meniscus when the inclination angle is smaller than β, and a concave meniscus when the inclination angle is bigger than βk . That is, in the latter case, the liquid will rise between the plates, although it does not wet either plate. Similarly, positioning wettable plates at an angle δ = π/2 – θ1 (Figure 2.44b) will constrain the liquid to a horizontal surface between them, i.e., when the inclination angle is bigger than δ, the liquid will sink between wettable plates, forming a convex meniscus, although it wets both plates. The latter consideration along with Figure 2.44a and Figure 2.44b exemplify the fact that capillary imbibition of a liquid into a porous body signiﬁcantly depends on the angle of opening of the pores and its change along the pore axis, i.e., the pore distribution function with respect to a derivative of the radius [29].

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Equilibrium Wetting Phenomena

151

β

x θ2

θ2 θ2

θ2 (a)

θ1

θ1 x θ1

θ1

γ

(b)

FIGURE 2.44 (a) Capillary rise between nonwettable plates. (b) Capillary interaction between wettable plates at inclination.

The liquid cannot move beyond those places where the meniscus in the pore becomes ﬂat, i.e, where the effective pore radius is equal to inﬁnity. In particular, the latter phenomenon is one of the reasons of the capillary hysteresis in porous bodies and the presence of trapped air. In summary, we have shown that the capillary attraction or repulsion between solid bodies depends not only on their wetting features but also on their separation and on the mutual angle of relative inclination.

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APPENDIX 2 Equilibrium Liquid Shape Close to a Vertical Plate Let us consider a vertical plate partially immersed in a liquid of density, ρ, and surface tension, γ. In this case, only the capillary and gravity forces act, and the equation that describes the liquid proﬁle is as follows: γh ′′

(

1 + h ′2

)

3/ 2

= ρgh ,

(A2.1)

where g is the gravity acceleration. Equation A2.1 is the differential equation of the second order; hence, the two boundary conditions should be speciﬁed. These conditions are h ′(0 ) = − cot anθ,

(A2.2)

h ( x ) → 0, x → ∞.

(A2.3)

Condition (A2.3) is used as h′

h =0

=0 .

(A2.3’)

Using the capillary length γ , ρg

a= Equation A2.1 can be rewritten as h ′′

(

1 + h ′2

)

3/ 2

=

h . a2

(A2.1’)

Multiplication of the latter equation by h ′ and integration with x results in 1

(1 + h ′ ) 2

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1/ 2

=C −

h2 , 2 a2

Equilibrium Wetting Phenomena

153

H θ

h(x) x

FIGURE 2.45 Liquid proﬁle close to the vertical wall. θ is the contact angle, H is the maximum elevation.

where C is an integration constant. Using the boundary condition (A2.3’), we conclude that C = 1 and 1

(1 + h ′ ) 2

1/ 2

= 1−

h2 . 2 a2

(A2.4)

The left-hand side of the preceding equation is positive and so should be the right-hand side, which results in the restriction h ≤ a 2 . We further see that H = a 2 corresponds to the maximum possible elevation (Figure 2.45) in the case of complete wetting, that is, at θ = 0. Using the boundary condition (A2.2), we conclude from Equation A2.4 that H = a 2 (1 − sin θ) ,

(A2.5)

which gives H = a 2 in the case of complete wetting. From Equation A2.4, we conclude: h2 2 a2 , h′ = −  h2  a 2 1− 2  2a  

(A2.6)

h (0 ) = H = a 2 (1 − sin θ).

(A2.7)

h 2−

with boundary condition

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Wetting and Spreading Dynamics

Integration of Equation A2.6 with the boundary condition (A2.7) results in

−

x = a

h

 h2  1− 2  2a  

H

h2 h 1− 2 4a

∫

dh ,

or

x 1 = ln a 2

+

( (

 h2  2 − 1 + sin θ  1 + 1 − 2   4 a   h2  2 + 1 + sin θ  1 − 1 − 2  .  4 a 

)

)

 h2 2  1 + sin θ − 2 − 2  2a

(A2.8)

  

Equation A2.8 gives an implicit dependence of the liquid proﬁle h on x.

2.12 LIQUID PROFILES ON CURVED INTERFACES, EFFECTIVE DISJOINING PRESSURE. EQUILIBRIUM CONTACT ANGLES OF DROPLETS ON OUTER/INNER CYLINDRICAL SURFACES AND MENISCI INSIDE CYLINDRICAL CAPILLARY In this section, we shall deduce effective disjoining pressure isotherms for liquid ﬁlms of uniform thickness on inner and outer cylindrical surfaces and on the surface of spherical particles. This effective disjoining pressure is expected to depend on the surface curvature. From its expression, we shall be able to calculate the equilibrium contact angles of drops on the outer surface of a cylinder and menisci inside cylindrical capillaries. We shall see that the contact angle is almost independent of liquid geometry. However, there are differences in the expressions for equilibrium contact angles according to geometry, in view of the difference in thickness of ﬁlms of uniform thickness with which the bulk liquid (drops or menisci) is at equilibrium. The latter thickness determines the lower limit of the integral in the expression for the equilibrium contact angle.

LIQUID PROFILES ON CURVED SURFACE: DERIVATION OF GOVERNING EQUATIONS Excess free energy, Φ, of the liquid droplet on an outer surface of a cylindrical capillary of radius a is as follows: © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

∫ {2πγ ( a + h) ∞

Φ=

0

155

(

(

)

) (

)

1 + h ′ 2 − a + he  + πPe  (a + h)2 − a 2 − (a + he )2 − a 2    

∞  ∞   + 2πa Π(h)dh − Π((h)dh   dx   he  h

∫

∫

(2.242)

where, as before, we selected a reference state as the outer surface of the cylindrical capillary covered by a the equilibrium liquid ﬁlm of the thickness he; x is in the direction parallel to the cylinder axis (Figure 2.46). The condition of equilibrium according to Section 2.2 (condition (1)) results in the following equation describing the liquid proﬁle on the surface of the cylindrical capillary:

(

γh ′′

1 + h ′2

)

3/ 2

−

(a + h)

γ 1 + h ′2

+

a Π( h ) = Pe . a+h

(2.243)

Note that the latter equation is different from Equation 2.23 in Section 2.2 not only due to the presence of the second curvature, −

(a + h)

γ 1 + h ′2

,

but also due to a difference in the deﬁnition of the disjoining pressure, which is now a Π( h ) a+h instead of Π( h ) in Equation 2.23, Section 2.2. This difference results in substantial consequences as shown in Section 2.7.

he a

θe

H

x

FIGURE 2.46 Cross section of an axisymmetric liquid droplet on the outer surface of a cylinder of radius a. H is the maximum height of the droplet, he is the thickness of an equilibrium ﬁlm of uniform thickness. © 2007 by Taylor & Francis Group, LLC

156

Wetting and Spreading Dynamics

Note that the excess pressure, Pe, is determined by the vapor pressure in the surrounding air and given by Equation 2.2 (Section 2.1). In the case of droplets on the cylindrical capillary, as in the case of droplets on ﬂat solid substrates, the equilibrium is possible only at oversaturation, that is, at Pe < 0. For the equilibrium ﬁlm of uniform thickness, he, we conclude from Equation 2.243: −

(

γ a + Π( he ) = Pe . a + he a + he

)

(2.244)

Let us introduce the effective disjoining pressure as Πeff ( h ) = −

(

γ a + Π( h ). a+h a+h

)

(2.245)

In the following text, we show that the introduced effective disjoining pressure provides the correct stability condition. For that purpose, we consider the excess free energy per unit length of the capillary, Φe, of the equilibrium ﬁlm of a uniform thickness, he, which is,

(

Φe = 2 πγ ( a + he ) + πPe  a + he 

)

2

∞

∫

− a  + 2 πa Π( h ) dh + 2 πa ( γ sl − γ sv ) .  2

he

(2.246) According to the requirements of equilibrium, the following conditions should be satisﬁed: d Φe = 0, dhe

(2.247)

d 2 Φe > 0. dhe2

(2.248)

The ﬁrst condition (2.247) of the equilibrium results in γ + Pe ( a + he ) − aΠ( he ) = 0. The latter equation can be rewritten using the deﬁnition of the effective disjoining pressure given by Equation 2.245 as © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

157

Πeff ( he ) = Pe

(2.249)

The second condition (2.248) gives Pe − aΠ′( he ) > 0 .

(2.250)

Let us check whether the effective disjoining pressure isotherm that is introduced according to Equation 2.245 satisﬁes the stability condition given by Equation 2.250. Indeed: d Πeff dhe

=−

a

(a + h )

2

e

 γ a Π( he ) − a  + a + h Π′( he ) . e  

Substituting the expression for Π(he) from Equation 2.249, we obtain: d Πeff dhe

=−

 γ a + he  γ  a  + Pe  − a  + a + h Π′(he ) a a  e  

a

(a + h )

2

e

=

1  aΠ′(he ) − Pe  < 0 a + he 

according to condition (2.250). Hence, we conclude d Πeff dhe

< 0,

(2.251)

which means that the effective disjoining pressure introduced according to Equation 2.245 will possess all the necessary properties according to Equation 2.249 and Equation 2.251. Just this effective disjoining pressure isotherm is used in Section 2.7 for the investigation of stability of uniform liquid ﬁlms on cylindrical surfaces. In the case of a uniform ﬁlm on a spherical particle of radius, a, we get an expression for the excess free energy similar to the one given by Equation 2.244: Φe = 4 πγ (a + he )2 +

4π  Pe a + he  3

(

)

3

− a3   (2.252)

∞

+ 4 πa

2

∫ Π(h)dh + 4πa (γ 2

he

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sl

− γ sv ).

158

Wetting and Spreading Dynamics

Let us introduce the effective disjoining pressure isotherm in this case as Πeff ( h ) =

a2

(a + h)

2

Π( h ) −

2γ . a+h

(2.253)

The same procedure as shown in the preceding section results in two equilibrium conditions (2.249) and (2.251). This means that the effective disjoining pressure isotherm deﬁned according to Equation 2.253 really describes the stability of uniform ﬁlms on spherical particles. In the case of liquid layers inside the inner part of the capillary of radius a, we have the following expression for the excess free energy:

∫ {2πγ ( a − h) ∞

Φ=

0

(

(

)

) (

)

1 + h ′ 2 − a − he  + πPe  a 2 − (a − h)2 − a 2 − (a − he )2    

∞ ∞   + 2πa  Π(h)dh − Π(h)dh   dx ,   he h 

∫

∫

(2.254)

which is similar to the expression for the excess free energy on the outer cylindrical surface Equation 2.242 (Figure 2.47). Exactly in the same way as in the case of the outer cylindrical surface, we deduce the following equation for the liquid proﬁle:

(

γh ′′

1 + h ′2

)

3/ 2

+

(a − h)

γ 1 + h ′2

+

a Π( h ) = Pe . a−h

(2.255)

Note again that the resulting equation (Equation 2.255) is different from both the corresponding Equation 2.243 (liquid on the outer cylindrical surface) and Equation 2.23 in Section 2.2 (for a ﬂat surface).

Pe

θe

2a

1 2

3

he x

FIGURE 2.47 Proﬁle of a meniscus in a cylindrical capillary of radius a. 1 — a spherical part of the meniscus of curvature Pe, 2 — transition zone between the spherical meniscus and ﬂat ﬁlms in front, 3 — ﬂat equilibrium liquid ﬁlm of thickness he. © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

159

Let us introduce the effective disjoining isotherm in the latter case as Πeff ( h ) =

γ a Π( h ) + . a−h a−h

(

)

(2.256)

The corresponding expression for the excess free energy per unit length of a uniform ﬁlm on the inner cylindrical surface is

(

∞

)

∫

2 Φe = 2 πγ ( a − he ) + πPe  a 2 − a − he  + 2 πa Π( h ) dh + 2 πa ( γ sl − γ sv )   he

(2.257) The latter expression and the deﬁnition given by Equation 2.256 result in the conditions (2.249) and (2.251), which describe the stability of a ﬁlm of uniform thickness on the inner surface of a cylindrical capillary.

EQUILIBRIUM CONTACT ANGLE OF A DROPLET SURFACE OF CYLINDRICAL CAPILLARIES

ON AN

OUTER

The droplet proﬁle is described by Equation 2.243. Let H be the maximum height of the droplet in the center, that is, h (0 ) = H . Let us introduce a new unknown function u=

1 1 + h ′2

in this equation and integrate Equation 2.243, which results in ∞

1 1 + h ′2

= 1+

 H2 h2  Pe  + aH − ah −  − a Π( h ) dh 2   2 h

∫

γ (a + h)

,

(2.258)

where condition h ′( H ) = 0 is taken into account. If we neglect the disjoining pressure on the right-hand side of Equation 2.258, we get the “outer solution,” which describes the drop proﬁle not distorted by the disjoining pressure action:  H2 h2  Pe  + aH − ah −  2   2 1 . = 1+ γ (a + h) 1 + h ′2 © 2007 by Taylor & Francis Group, LLC

(2.259)

160

Wetting and Spreading Dynamics

If we continue the outer solution to the intersection with the surface of the cylinder, we get h′(0) = –tan θe, where θe is the equilibrium contact angle to be determined. Using this condition in the outer solution obtained from Equation 2.259, we conclude:  H2  Pe  + aH  2   cos θe = 1 + , γa or Pe =

(cos θ

e

)

− 1 2 γa

H + 2 aH 2

< 0.

(2.260)

The preceding expression shows that the equilibrium droplets on the outer surface of a cylinder can be at equilibrium only at oversaturation as droplets on a ﬂat substrate. From the whole of Equation 2.258, we conclude that the local proﬁle tends asymptotically to the ﬁlm of the uniform thickness, he. Therefore, locally, the proﬁle satisﬁes the condition h ′( he ) = 0 . Using this condition, we conclude from Equation 2.258: ∞

0=

 H2 h2  Pe  + aH − ahe − e  − a Π( h ) dh 2   2 h

∫ e

γ ( a + he )

,

or ∞

 H2 h2  + aH − ahe − e  − a Π( h ) dh = 0 , Pe  2   2 h

∫

(2.261)

e

where the equilibrium thickness of the uniform ﬁlm is determined from the following equation: Πeff ( he ) = −

(

γ a + Π( h e ) = Pe . a + he a + he

)

(2.262)

Substitution of Equation 2.260 into Equation 2.261 results in the following equation for the determination of the equilibrium contact angle, © 2007 by Taylor & Francis Group, LLC

Equilibrium Wetting Phenomena

161

1 1 cos θe = 1 + 2 2 ah + he γ 1− 2 e H + 2 aH

∞

∫ Π(h)dh .

(2.263)

he

If we omit the small terms, 2 ahe + he2 , H 2 + 2 aH in Equation 2.263, we arrive at 1 cos θe ≈ 1 + γ

∞

∫ Π(h)dh .

(2.264)

he

This form of the preceding equation is identical to Equation 2.47 (meniscus in a ﬂat capillary) and Equation 2.55 (droplet on a ﬂat substrate) deduced in Section 2.3. However, there are substantial differences between Equation 2.47, Equation 2.55, and Equation 2.264: the lower limit of integration in these equations, which corresponds to the thickness of the uniform ﬁlm, is substantially different in each of them.

EQUILIBRIUM CONTACT ANGLE CYLINDRICAL CAPILLARIES

OF A

MENISCUS

INSIDE

In this case, the meniscus proﬁle is described by Equation 2.255. We introduce a new unknown function, u=

1 1 + h ′2

as in the case of a droplet on a cylindrical surface. After an integration, Equation 2.255 takes the form

1 1 + h ′2

(

=

Pe a−h 2

)

2

∞

∫

− a Π( h ) dh h

γ (a − h)

,

(2.265)

where we already take into account the condition in the center of the capillary, h ′( a ) = −∞ . If we neglect the disjoining pressure in the latter equation, we arrive at © 2007 by Taylor & Francis Group, LLC

162

Wetting and Spreading Dynamics

1 1 + h′

2

=

(

Pe a − h 2γ

),

(2.266)

which describes the spherical meniscus proﬁle. Continuation of this proﬁle to the intersection with the capillary surface results in a manner similar to the previous case of droplets: cos θe =

Pe a , 2γ

or Pe =

2γ cos θe , a

(2.267)

as expected. Now, from the whole of Equation 2.265, we conclude that the local proﬁle tends asymptotically to the ﬁlm of uniform thickness, he. That is, locally, the proﬁle satisﬁes the condition h′(he) = 0. Using this condition and Equation 2.267, we conclude from Equation 2.265 that 1 1 1 cos θe = + 2 h γ 1 − e  1 − he   a  a

∞

∫ Π(h)dh ,

(2.268)

he

where the thickness of the uniform ﬁlm is determined from Πeff ( he ) =

a γ Π( he ) + = Pe . a − he a − he

(

)

(2.269)

If we omit small terms such as he << 1 a in Equation 2.268, we arrive at the same functional dependency of cosθe as earlier (Equation 2.264, Equation 2.47, and Equation 2.55 in Section 2.3). Let us insist on the signiﬁcant difference (discussed in Section 2.3) between the equilibrium of drops and menisci: in the case of drops (no difference with a

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Equilibrium Wetting Phenomena

163

ﬂat surface or outer cylindrical surface), the external supersaturated vapor pressure in the ambient air may be arbitrary inside the narrow limits determined in Section 2.3, Equation 2.52. The drop size adjusts to the imposed pressure at equilibrium. The situation is very different in the case of an equilibrium meniscus on a capillary (no difference between a ﬂat chamber or a cylindrical capillary) Here, there is only a single vapor pressure allowing the equilibrium of the meniscus. At all other vapor pressures, the meniscus cannot be at equilibrium.

REFERENCES 1. Derjaguin, B. V., Churaev, N. V., and Muller, V. M. Surface Forces, Plenum Press, New York, 1987. 2. Deryagin, B. V., Starov, V. M., and Churaev, N. V. Colloid J., Russian Academy of Sciences, 38, 789 (1976). 3. Deryagin, B. V. and Churaev, N. V. Dokl. Akad. Nauk USSR, 207, 572 (1972). 4. Deryagin, B. V. and Zorin, Z. M. Sov. J. Phys. Chem., 29, 1755 (1955). 5. Exerowa, D. Adv. Colloid Interface Sci, 96, 75 (2002). 6. Exerowa, D. and Kruglyakov, P. M. Foam and Foam Films, Elsevier, 1998. 7. Cohen, R., Exerowa, D., Kolarov, T., Yamanaka, T., and Muller, V. Colloids & Surfaces, 65, 201 (1992). 8. Miller, C. A. and Neogi, P. Interfacial Phenomena, New York, Marcel Dekker, 1985; Miller, C.A. and Ruckenstein, E. J. Colloid Interface. Sci., 48, 368 (1974). 9. Churaev, N. V. Colloid J., Russian Academy of Sciences, 34, 988 (1972); 36, 318 (1974). 10. Parsegian, V.A. Mol. Phys., 37, 1503 (1974). 11. Dzyaloshinskii, I. E., Lifshitz, E. M., and. Pitaevskii, L. P. Zh. Eksp. Teor. Fiz., 37, 229 (1959). 12. Shishin,V. A., Zorin, Z. M., and Churaev, N. V. Colloid J., Russian Academy of Sciences, 39, 520 (1977). 13. Slavchov, R., Radoev, B., and Stöckelhuber, K.W. Colloids & Surfaces A: Physicochemical and Engineering Aspects, 261 (1-3), 135, (2005). Mahnke, J., Schulze, H. J., Stöckelhuber, K. W., and Radoev, B. Colloids & Surfaces A: Physicochemical and Engineering Aspects, 157 (1–3), 1–9 (1999); Mahnke, J., Schulze, H. J., Stöckelhuber, K. W., and Radoev, B. Ibid. 11–20. 14. Churaev, N. V., Colloid J., Russian Academy of Sciences, 36, 318 (1974). 15. Derjaguin, B.V., Starov, V.M., and Churaev, N.V. Colloid J., Russian Academy of Sciences, 37(2), 219, (1975). 16. Us’yarov, O. G. “Research in the Field of Stability of Disperse Systems and Wetting Films” [in Russian]. Doctor of Sciences Thesis. SanktPetersburg (Leningrad) University, Department of Colloid Science, (1976). 17. Zorin, Z. M. and Sobolev, V. D. In: “Research on Surface Forces,” 3, Consultants Bureau. New York, p. 29 (1971). 18. Rusanov, A. I. Colloid J., Russian Academy of Sciences, 37 (4), 678 (1975). 19. Lester, G. R. J. Colloid Sci., 16, No. 4, 315 (1961). 20. Derjaguin, B. V., Muller, V. M., and Toporov, Yu. P. Colloid J., Russian Academy of Sciences, 37, 455 (1975); 37, 1066 (1975).

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21. Muller, V. M. and Yushchenko, Y. S. Colloid J., Russian Academy of Sciences, 42, 500 (1980). 22. Rowlinson, J. F. and Widom, B. Molecular Theory of Capillarity, Clarendon, Oxford, 1984. 23. Starov, V. M. and Churaev, N. V. Colloid J., Russian Academy of Sciences, 42, 703 (1980). 24. de Feijter, J. A. and Vrij, A. J. Electroanal. Chem., 37, 9 (1972). 25. Kolarov, T. and Zorin, Z. M. Colloid Polym. Sci., 257, 1292 (1979). 26. Platikanov, D, Nedyalkov, M., and Nasteva, V. J. Colloid Interface Sci., 75, 620 (1980). 27. Stevin, S. De Beghinselen der Weeghconst. The Elements of the Art of Weighing, Francois van Raphelinghen, Leyden (1586). 28. Deryagin, B. V. Dokl. Akad. Nauk SSSR, 51, 517 (1946). 29. Eremeev, G. G. and Starov, V. M. J. Phys. Chem. of the USSR, (in Russian), 47(11), 2921 (1973). 30. Starov, V. J. Colloid Interface Sci., 269, 432 (2003). 31. Starov, V. M. and Churaev, N.V. Colloid J., Russian Academy of Sciences, 60 (6), 770 (1998). 32. Starov, V. M. and Churaev, N. V. Colloids and Surfaces A: Physicochemical & Engineering Aspects, 156, 243-248 (1999). 33. Starov, V. M., Churaev, N. V., and Derjaguin, B. V. Colloid J., Russian Academy of Sciences, 44(5), 770 (1982). 34. Starov, V. M. and Churaev, N. V. Colloid J., Russian Academy of Sciences, 45(5), 852 (1983). 35. Churaev, N. V. and Starov, V. M. J. Colloid Interface Sci., 103(2) 301 (1985). 36. V.Starov. In Emulsions: structure, stability and interactions, Edited by D. N. Petsev. Elsevier Academic Press, Amsterdam-Boston-Heidelberg-London-New York-Oxford-Paris-San Diego-San Francisco-Singapore-Sydney-Tokyo, 183–214 (2004). 37. Derjaguin, B. V. and Starov, V. M. Colloid J., Russian Academy of Sciences, 39(3), 383 (1977).

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3

Kinetics of Wetting

INTRODUCTION In Chapter 2, the importance of surface forces in the vicinity of the three-phase contact line has been shown and investigated in the case of equilibrium. It is obvious that the same forces are equally important in the kinetics of spreading. In this chapter we consider separately two cases that are drastically different: the complete wetting that we begin with and then the partial wetting. The discussion in this chapter shows that, in the case of spreading, the whole drop proﬁle should be divided into a number of regions, which are brieﬂy discussed below. Complete wetting. We should decide which dimensionless parameters are important in the case of spreading, and also which drops can be considered as “small drops” and which of them are “big drops.” The latter is determined by gravity action. Let us consider the simplest possible example of spreading of two-dimensional (cylindrical) droplets (Figure 3.1a). The capillary regime is an initial stage of spreading of small drops, whereas the gravitational regime is the ﬁnal stage of spreading of small drops or the regime of spreading of big drops. Transition from the capillary regime of spreading to the gravitational regime takes place at the moment tc, when R(tc) ~ a =

γ , ρg

where R(t) is the radius of the drop base at time t, which is referred to below as the radius of spreading; γ and ρ are the interfacial tension and density of the liquid, respectively; g is the gravity acceleration. In the case of water, (γ = 72.5 dyn/cm, ρ = 1 g/cm3), and hence, a ~ 0.27 cm. The next two important parameters are the Reynolds number and the capillary number. The Reynolds number characterizes the importance of inertial forces as compared with viscose forces. To deduce the relevant expression for the Reynolds number, let us consider the spreading of a two-dimensional droplet over a solid surface (gravity action is neglected). In this case, the Navier–Stokes equation with the incompressibility condition takes the following form:  ∂ 2u ∂ 2u   ∂u ∂u  ∂p ρ u + v  = − + η 2 + 2  ∂y  ∂x  ∂x ∂y   ∂x 165 © 2007 by Taylor & Francis Group, LLC

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a

H θ(t)

1

R(t) x

b 1

2 3

θ

4

FIGURE 3.1 (a) Spreading of liquid droplet over ﬂat solid substrate. R(t) – radius of spreading, which is the position of the apparent three-phase contact line; θ(t) – dynamic contact angle; 1 – vicinity of the apparent three-phase contact line. (b) A magniﬁcation of the vicinity of the moving apparent three-phase contact line in the case of complete wetting: (1) spherical part of the drop, which forms a dynamic contact angle, θ, with the solid substrate; (2) a region where a spherical shape is distorted by the hydrodynamic force; (3) a region where disjoining pressure comes into play and become increasingly important towards the end of the region 3; and (4) a region where a macroscopic description is not valid any more and surface diffusion takes place.

 ∂2v ∂2v   ∂v ∂v  ∂p ρ u + v  = − + η 2 + 2  ∂y  ∂y  ∂x ∂y   ∂x ∂u ∂v + =0 ∂x ∂y where v = (u, v) is the velocity vector, and the gravity action is neglected. Let U* and v* be scales of the velocity components in the tangential and the vertical directions, respectively. Using the incompressibility condition we conclude that U* v* = , r* h* or v* = εU* , ε = © 2007 by Taylor & Francis Group, LLC

h* . r*

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167

If the droplet has a low slope, then ε << 1, and hence, the velocity scale in the vertical direction is much smaller than the velocity scale in the tangential direction. Now, using the ﬁrst Navier–Stokes equation, we can estimate

ρu

∂u ∂u ρU*2 ∂ 2u ∂ 2u ∂ 2u ∂2u ηU ~ ρv ~ , η 2 ~ ε 2 η 2 << η 2 , η 2 ~ 2 * , ∂x ∂y r* ∂x ∂y ∂y ∂y h*

and hence, all derivatives in the low-slope approximation in the x direction can be neglected as compared with derivatives in the axial direction y. Now we can estimate the Reynolds number: 2 ∂u ρU* 2 ∂x ~ r* = ρU* h* = ε 2 ρU* r* Re ~ 2 ηU* ηr* η ∂u η 2 2 h* ∂y

ρu

or Re = ε 2

ρUr* η

(3.1)

The latter expression shows that the Reynolds number under the low slope approximation is proportional to ε2. Hence, during the initial stage of spreading, when ε ~ 1, the Reynolds number is not small, but as soon as the low slope approximation is valid, Re becomes small even if ρUr* η is not small enough. This means that, during the short initial stage of spreading, both the low-slope approximation and low Reynolds number approximations are not valid. However, we are interested only in the main part of the spreading process when the short initial stage is over. In the following text, we see that the Re number should be calculated only in the close vicinity of the moving contact line, where the low slope approximation is valid (see the following text), because in the main part of the spreading droplet the liquid is moving much slower than it does close to the edges. Hence, the inertial terms in Navier–Stokes equations can be safely omitted and only Stokes equations should be used instead:

0=−

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 ∂ 2u ∂ 2u  ∂p + η 2 + 2  ∂x ∂y   ∂x

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Wetting and Spreading Dynamics

0=−

 ∂2v ∂2v  ∂p + η 2 + 2  ∂y ∂y   ∂x

.

∂u ∂v + =0 ∂x ∂y The tangential stress on the free drop surface at y = h( x ) is    ∂u ∂v    ∂u ∂v   ∂u ∂v   η  h′ 2  h′  +   + 2h′ 2  + −  + h′  =0  ∂x ∂y   ∂y ∂x      ∂y ∂x   Under the low slope approximation, ε << 1, we can easily check that h′ << 1. Using this estimation, the aforementioned condition can be rewritten as  ∂u ∂v  =0 η +  ∂y ∂x  Using the previous estimations we conclude: ∂u U ~ ; ∂y h*

∂v εU U ∂u ~ = ε 2 << ∂x r* h* ∂y

This estimation shows that, under the low slope approximation, the tangential stress on the free liquid interface is η

∂u = 0. ∂y

This boundary condition is used below in Chapter 3. The capillary number, Ca =

Uη , γ

characterizes the relative inﬂuence of the viscose forces as compared with the capillary force. Let us estimate possible values of the Ca. Let us consider an oil droplet with r* ~ 0.1 cm, γ ~ 30 dyn/cm and η ~ 10–2 P. Let the droplet move forward on the distance equal to its radius over 1 sec, which can be considered as a very high velocity of spreading. This gives the following estimation: Ca ~ 3⋅10–5 << 1. That means, we should expect Ca to be even less than 10–5 over the duration of spreading. According to the previous estimation, we assume below that both the capillary and Reynolds numbers are very small except for a very © 2007 by Taylor & Francis Group, LLC

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169

short initial stage of spreading. We estimate the duration of the initial stage of spreading in Chapter 4 (Section 4.1) immediately after the drop is deposited on the solid substrate. Let us consider the consequence of the smallness of the capillary number, Ca << 1, using the simplest possible example of spreading of two-dimensional (cylindrical) droplets (Figure 3.1a). Let the length scales in both x and y direction in the main part of the spreading drop (Figure 3.1a) be r*, then the pressure inside the main part of the droplet has the order of magnitude of the capillary pressure, that is p~

γ . r*

Using the incompressibility condition, we immediately conclude that velocities in both directions, u and v, have the same order of magnitude U*. Let us introduce the following dimensionless variables, which are marked by an over-bar p=

p x y u v , x= , y= , u= , v= . γ / r* r* r* U* U*

Using these variables, the Stokes equations can be rewritten as  ∂ 2u ∂ 2u  ∂p = Ca  2 + 2  ∂x ∂y   ∂x  ∂2v ∂2v  ∂p = Ca  2 + 2  . ∂y ∂y   ∂x We already concluded that Ca << 1; that means the right-hand side of both foregoing equations is very small. Hence, these equations can be rewritten as ∂p = 0, ∂x ∂p = 0, ∂y which means that the pressure remains constant inside the main part of the spreading droplet. If we now write down the normal stress balance on the main part of the spreading droplet, we get

p=

 2 h′′  2 3/2 + Ca − 2 (1 + h′ )  1 + h′

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(

)

  ∂u ∂v  ∂v ∂u   + − − h′ 2  .  − h′  ∂x     ∂y ∂x  ∂y 

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Wetting and Spreading Dynamics

Using the condition Ca << 1, we conclude from the latter equation that p=

h′′ = const , (1 + h′ 2 )3/2

even in the case when the droplet proﬁle does not have the low slope approximation, that is, even if h′ 2 ~ 1, that is not small. This shows that the spreading droplet retains its spherical shape over the main part of the droplet. Note that the radius of the droplet base, R(t), changes over time, and this change results in a quasi-steady-state change of the droplet proﬁle. In the case of moving meniscus in a capillary, a similar estimation shows that Ca << 1 results in a spherical shape of the meniscus in the main part of the capillary. Once again, the smallness of the Ca means that the surface tension is much more powerful over the most part of the droplet/meniscus, and hence, the droplet/ meniscus has a spherical shape everywhere except for a vicinity of the apparent three-phase contact line. The size of this region, l*, is estimated in this chapter in Section 3.2. It will be shown that the following inequality is satisﬁed: h* << l* << r* . Hence, δ=

h* << 1 l*

is a small parameter inside the vicinity of the moving contact line. The latter means that the curvature of the liquid interface inside the vicinity of the moving contact line can be estimated as γ

γh′′

(1 + h′ ) 2

3/ 2

~

h* l*2

 h*2 2   1 + l 2 h′    *

γ 3/ 2

=

h* l*2

(1+ δ h′ ) 2

2

3/ 2

≈γ

h* . l*2

This provides a very important conclusion: the low slope approximation is valid inside the vicinity of the moving contact line even if the drop proﬁle is not very low sloped, that is, even if h′ 2 ~ 1 is not small. Hence, we can always use the low slope approximation inside the vicinity of the moving contact line except in the case when the slope is close to π/2 (see Section 3.10). Let us estimate a possible range of capillary numbers. If Ca ~ 1, then, in the case of water, we conclude that Ca ~ © 2007 by Taylor & Francis Group, LLC

U* η U* ⋅ 10 −2 P ~ ~1 . γ 72 dyn/cm

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171

This results in U* ~ 72 m/sec. The velocity here is so high that it probably can be achieved only under very special conditions. In the case of r* ~ 0.1–1 cm, the velocity results in Re ~ 7.2·104 – 7.2·105, which is a turbulent ﬂow, beyond the scope of this book. This signiﬁes that the possible range of Ca is between 0 and 1. Low Ca << 1 denotes a relatively low rate of spreading, whereas Ca ~ 1 denotes a very high velocity of motion. In its turn, the case Ca << 1 includes a high capillary number limit (see Section 3.5 through Section 3.7). Therefore, the case of low capillary numbers, Ca << 1, and intermediate capillary numbers, Ca ~ 1, should be considered in a completely different way. The situation is similar to the case of the Reynolds number: consideration of ﬂows at low Reynolds numbers is very much different from that at high Reynolds numbers. Now we are ready to consider in more detail the vicinity of the moving contact line (region 1 in Figure 3.1a), which is magniﬁed in Figure 3.1b. The whole vicinity of the three-phase contact line can be subdivided into four regions (Figure 3.1b). Region 1 is a spherical meniscus in the main part of the spreading droplet. This region is included to show the dynamic contact angle, θ(t), which is deﬁned at the intersection of the tangent to the spherical part of the droplet with the solid substrate. The dynamic contact angle is unknown and should be determined by matching all regions presented in Figure 3.1b. Inside the next region, 2, the spherical shape is distorted by the hydrodynamic ﬂow. This region is followed by region 3, where disjoining pressure comes into play. Over region 3, the disjoining pressure action becomes increasingly important as compared to the capillary force. Toward the end of region 3, the disjoining pressure overcomes the capillary force and becomes the only driving force of the spreading process. Region 3 is followed by region 4, where a macroscopic description of the spreading process becomes impossible because the characteristic scale in the vertical direction is of the order of the molecular size. We refer to region 4 as the region of surface diffusion. The picture of a spreading drop proﬁle in a vicinity of the three-phase contact line, presented in Figure 3.1b, has been understood only recently. A number of simpliﬁed physical mechanisms have been introduced previously, based on a simpliﬁcation of the above picture. For a long time the so-called singularity on a three-phase contact line [1] has been considered as a major problem in the consideration of the kinetics of spreading. We explain in the following text the source of this singularity and why it is removed by the disjoining pressure acting in a vicinity of the apparent threephase contact line. It is easy to see that the viscose stress in the tangential direction close to the three-phase contact line (Figure 3.1b) is η

∂vr ηU* ~ →∞, ∂y h

as h → 0. This means that the drop cannot spread out because the friction force at the moving front becomes inﬁnite. The way to overcome this problem has been © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

suggested in Reference 2. The concept is as follows: the very ﬁrst layer of the liquid molecules on the liquid–solid interface is attached to the solid substrate by a force of adhesion. However, the adhesion force is ﬁnite, not inﬁnite. If the tangential stress has become large enough, then the ﬁrst layer of the liquid molecules will be swept away by the tangential stress. The result is a “slippage velocity,” which is introduced as follows:  ∂vr   η ∂y  = [ αvr ]y= 0 ,   y= 0 where α is a proportionality coefﬁcient. That is, the slippage velocity is proportional to the applied shear stress on the solid substrate. This deﬁnition can be rewritten as  ∂vr  η  vr   ∂y  =  λ  , λ = α ,  y= 0   y= 0 

(3.2)

where λ has a dimension of length and can be referred to as a slippage length. However, it turns out [3] that λ ~ 10–6 cm, which is located just in the range where surface forces are the most powerful. Therefore, Condition 3.2 cannot be used as a macroscopic condition because the thickness is in the range of surface forces action. Hence, a modiﬁed consideration in this region, which takes into account surface forces action, should be used instead. Note that, in the case of complete wetting, the disjoining pressure is equal to Π(h ) =

A →∞ h3

as h → 0, that is, even faster than the tangential stress. Hence, the disjoining pressure is the driving force of spreading in a vicinity of the three-phase contact line. Let us estimate the thickness at which the disjoining pressure overcomes the increasing tangential stress: ηU* A < 3, h h or  A  h<  ηU* 

1/2

= ht .

That is, the lower the velocity of spreading, U, the higher the thickness below which the disjoining pressure overcomes the tangential stress. If, as before, we adopt at © 2007 by Taylor & Francis Group, LLC

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173

the initial stage of spreading a very high spreading velocity, U* ~ 10–1 cm/sec, A ~ 10–14 erg, η ~ 10–2 P, then ht ~ 0.3⋅10–6 cm, that is, in the range where the disjoining pressure is the most powerful. A simple way to overcome the problem of “singularity on the moving contact line” has been suggested in Reference 4: a cutting length was introduced in a vicinity of the three-phase moving contact line. However, the introduction of cutting length is similar to the introduction of slippage velocity. A simplifying approach has been suggested in Reference 5. According to this approach, the hydrodynamic ﬂow in region 2 and region 3 is ignored, as well as the disjoining pressure action in region 3 (Figure 3.1b). According to this approach, a spherical meniscus is followed directly by region 4, where surface diffusion takes place. This approach results in the following equation for the velocity of spreading: R = const ⋅  cos θe − cos θ(t )  , that is, the velocity of spreading is proportional to the difference between cosθe, where θe is the equilibrium contact angle (θe is a ﬁtting parameter in the theory [5]) and cosθ(t), where θ(t) is the instantaneous dynamic contact angle. In the case of complete wetting, that is at cosθe = 1, the latter equation results in R = const ⋅θ2 (t ). It is well established that, in the case of complete wetting (see Section 3.1 and Section 3.2), the law is R = const ⋅θ3 (t ). Comparison of these last two equations shows that the approach suggested in Reference 5 does not agree with well-established theoretical predictions. The next approach we mention was tried long ago and is based on the consideration of dynamic surface tension in the vicinity of the apparent moving three-phase contact line. It has been assumed that the surface tension of “the fresh interface” (which appears close to the apparent three-phase contact line) is higher than the surface tension behind the apparent three-phase contact line on “the old interface.” This surface difference could be the driving force behind spreading. However, both experimental investigations [6] and theoretical estimations [7] showed that the relaxation time of the surface tension on a fresh liquid–air interface of pure liquids is too small, and hence, cannot inﬂuence the spreading process, which proceeds on much larger time scales. However, recently, an attempt has been made to revive the same idea of a high surface tension on a fresh liquid–air interface [8]. The approach suggested in Reference 8 also completely ignores the disjoining pressure action in the vicinity of the moving threephase contact line. This approach was criticized in Reference 7. © 2007 by Taylor & Francis Group, LLC

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Surface diffusion (region 4 in Figure 3.1) results in an effective slippage [9]. The ﬁrst attempt to introduce surface slippage based on the consideration of surface diffusion was undertaken in Reference 9. This approach is to be developed further.

3.1 SPREADING OF DROPLETS OF NONVOLATILE LIQUIDS OVER FLAT SOLID SUBSTRATES: QUALITATIVE CONSIDERATION In this section we shall discuss the viscous spreading of drops over solid surfaces when there is complete wetting, and spreading proceeds completely down to the molecular level. We shall disregard the inﬂuence of surface forces in the vicinity of the apparent three-phase contact line. Thus, we know there is a singularity at the moving three-phase contact line (as shown in the Introduction to this chapter). In spite of that, we shall ﬁnd the correct time dependence of spreading (spreading laws in the case of capillary and gravitational spreading). Subsequently, we shall consider the spreading of microdrops, that is, very small drops, entirely subjected to the action of the surface forces. As expected, we shall see that the disjoining pressure action removes the singularity on the moving three-phase contact line. Also to be expected is that, giving due consideration to the disjoining pressure, the liquid proﬁle does not end at any particular point but vanishes asymptotically. Hence, only an apparent moving front can be identiﬁed. Considerable experimental and theoretical material dealing with the spreading of droplets over a solid, nondeformable, dry surface has been accumulated for decades [10–13]. In the following text we present a derivation of equations that describe the kinetics of spreading in different situations. Note that all considerations below are undertaken in the case of complete wetting. Partial wetting is considered in Section 3.10, which shows that the partial wetting case is far more complicated compared to complete wetting. This explains why the spreading law in the case of partial wetting is to be deduced. Let us consider an axisymmetric spreading of liquid droplet over dry solid substrate (Figure 3.1). Let R(t) be the radius of spreading, θ(t) the dynamic contact angle, and h(t,r) the unknown liquid. Our objective is to deduce the spreading law, R(t) (Figure 3.1). As discussed in the Introduction to this chapter, the Reynolds number is small over the main duration of the spreading process; hence, Re << 1. We also assume that the low slope proﬁle approximation  ∂h  ε ~   << 1  ∂r  is valid, which means the scale in the vertical direction, h*, is much smaller than the scale in the radial direction, r*. The cylindrical coordinate system (r,ϕ, z) should be used in the case of axisymmetric spreading. Because of symmetry, vϕ = 0, and all unknown values are independent of the angle ϕ. © 2007 by Taylor & Francis Group, LLC

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Let U* and v* be the scales of the velocity components in the tangential and vertical directions, respectively. Using the incompressibility condition, we conclude that U* v* h = , or v* = εU* << U , ε = * << 1. r* h* r*

(3.3)

Hence, the velocity scale in the vertical direction is much smaller than the velocity scale in the tangential direction. This means: vz << vr . Using the same small parameter ε, we can conclude that all derivatives in the radial direction, r, are much smaller than derivatives in the vertical direction z: ∂f ∂f << , ∂r ∂z

(3.4)

where f(r,z) is any function. Now we can easily conclude from the Stokes equation for the vertical velocity component that ∂p = 0; ∂z hence, the pressure depends only on the radial coordinate, r, that is p = p(r) and remains constant through the cross section of the spreading drop. Therefore the pressure can be presented as p = pa –γK – Π(h) + ρgh, where pa is the pressure in the ambient air, γK is the capillary pressure, K is the mean curvature of the interface (negative in the case of droplets of a ﬂat solid substrate and positive in the case of menisci in capillaries), g is the gravity acceleration, and ρ is the liquid density, Π(h) is the isotherm of the disjoining pressure. In the low slope case (see Equation 3.3) we conclude K=

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1 ∂  ∂h  . r r ∂r  ∂r 

176

Wetting and Spreading Dynamics

Hence, p = pa − γ

1 ∂  ∂h  − Π(h) + ρgh . r r ∂r  ∂r 

(3.5)

After that the only equation left is that for the radial velocity component, vr , which now can be written as ∂p ∂2v = η 2r , ∂r ∂z

(3.6)

with nonslip boundary conditions at z = 0: vr (t,z) = 0,

(3.7)

and no-tangential stress condition on the liquid–air interface surface: η

∂v r = 0 at z = h(t , r ) . ∂z

(3.8)

To deduce this condition we should take into account both Equation 3.3 and Equation 3.4; after that, all terms, except for the one shown in Equation 3.8, disappear (see Introduction to this chapter). Integration of Equation 3.6 with boundary condition (3.7) and (3.8) results in the following expression for the velocity proﬁle:

vr = −

1 ∂p  z2  hz −  .  2 η ∂r 

(3.9)

This equation allows calculation of the ﬂow rate, Q: h

∫

Q = 2π rvr dz = − 0

2π 3 ∂p . rh 3η ∂r

The conservation of mass reads: 2πr

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∂h ∂Q + = 0. ∂t ∂r

(3.10)

Kinetics of Wetting

177

Substitution of Equation 3.10 into the preceding one results in 1 ∂  3 ∂p  ∂h rh = , ∂t 3ηr ∂r  ∂r 

(3.11)

which is referred to in the following text as the equation of spreading. Nonvolatability of the liquid results in the conservation of the total liquid volume, V: R (t )

2π

∫ r h dr = V = const ,

(3.12)

0

where R(t) is the radius of the drop base, which is frequently referred to as the radius of spreading. Finally, we have two equations (3.11 and 3.12) with two unknowns: the liquid proﬁle, h(t,r), and the radius of spreading, R(t). Note that the pressure, p(t,r), in Equation 3.9 is speciﬁed according to Equation 3.5. All the thicknesses of a spreading droplet should be divided into the following parts: •

Big thickness, h ≥ 10–5 cm ~ ts; the effects of the disjoining pressure may be neglected. According to Equation 3.5, the hydrodynamic pressure within this region is equal to p = pa − γ

1 ∂  ∂h  + ρgh . r r ∂r  ∂r 

(3.13)

Let us introduce the following dimensionless variables, which are marked with an overbar: r=

r h , h= r* h*

or r = rr* , h = hh* .

Note that all dimensionless values have the same order of magnitude of 1. Using the dimensionless variable we conclude from Equation 3.13 that p = pa −

γ h* 1 ∂ r*2 r ∂r

 ∂h   r ∂r  + ρgh* h .

This expression includes two dimensionless parameters: γ h* r*2 © 2007 by Taylor & Francis Group, LLC

and ρgh* .

(3.14)

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Wetting and Spreading Dynamics

The ﬁrst one gives the intensity of capillary forces, and the second one the intensity of the gravitational force. If capillary forces prevail, then the capillary regime of spreading takes place; that is, if γ h* >> ρgh* r*2

or r* <<

γ = a, ρg

where a is the capillary length. If gravity prevails, then the gravitational regime of spreading takes place; that is, if γ h* << ρgh* r*2

•

γ = a. ρg

In the case of water, the capillary length is equal to a ≈ 0.27 cm. The capillary regime is an initial stage of spreading of small drops, whereas the gravitational regime is the ﬁnal stage of spreading of small drops or the regime of spreading of big drops. The transition from the capillary regime of spreading to the gravitational regime should take place at the moment tc , when Rtc) ~ a, where R(t) is the radius of the drop base at time t. Intermediate thickness 10–6 cm ≤ h ≤ 10–5 cm. In this region both the capillary and the disjoining pressures act simultaneously, i.e., the hydrodynamic pressure takes on the form p = pa −

•

or r* >>

γ ∂  ∂h ( t , r )  r − Π (h) . r ∂r  ∂r 

(3.15)

Small thickness h ≤ 10–6 cm. Here, the value of the capillary pressure is negligible in comparison to the disjoining pressure: p = pa − Π ( h ) .

(3.16)

Accordingly, the following stages of the spreading of a drop can be identiﬁed: 1. Inertial, when both Re and Ca are not small. We will not consider this stage in this book; however, an estimation of the duration of this period is given in Chapter 4, Section 4.1. 2. Gravitational, in the case of big droplets.

© 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

179 Z H(t) h1

θ(t)

ts 0

R(t) r0(t)

r

ℜ

FIGURE 3.2 Spreading of a spherical droplet. At h > h1, the spherical droplet proﬁle is not distorted by the hydrodynamic ﬂow; ts < h1 is the radius of action of the disjoining pressure; r0(t) is the macroscopic wetting perimeter (the apparent three-phase contact line); R(t) is the true microscopic wetting perimeter; θ(t) is the dynamic contact angle; and H(t) is the drop apex.

3. In the case of small droplets (which is considered in the following text), the gravitational stage is preceded by the capillary stage (3). 4. Disjoining pressure action, when the complete droplet is in the range of disjoining pressure action. Only during the ﬁrst stage does the spreading process proceed without regard to the form of the isotherm of the disjoining pressure, and all other stages are determined to different degrees by the disjoining pressure. The apparent macroscopic wetting perimeter r0(t) (Figure 3.2) and the true microscopic wetting perimeter, R(t), are different because of the formation of the precursor ﬁlm, caused by the action of both hydrodynamic and surface forces. In the vicinity of the point r0(t) there is a transitional region, where both the capillary and disjoining pressures act. In the following text we consider so-called similarity solutions of the equation of spreading (3.11) in the case of capillary and gravitation regimes of spreading.

CAPILLARY REGIME

OF

SPREADING

Small drops with an initial characteristic size smaller than the capillary length, a, are considered here. That means, after a short inertial period, the capillary regime of spreading begins. In this case the pressure is given by p = pa − γ

1 ∂  ∂h  r , r ∂r  ∂r 

and Equation 3.11 takes the following form:

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Wetting and Spreading Dynamics

∂h γ ∂  3 ∂ =− r h ∂t ∂r 3ηr ∂r 

 1 ∂  ∂h      r  ,  r ∂r  ∂r   

(3.17)

with conservation law (3.12) and the boundary conditions by virtue of the symmetry at the center of the droplet: ∂h ∂ 3h = 3 =0 . ∂r r = 0 ∂r r = 0

(3.17’)

First, let us introduce dimensionless values using the following characteristic scales h*, r*, and t*. From Equation 3.17 we conclude that γh 4 1 ∂  3 ∂ h* ∂h = − *4 r h t* ∂ t ∂r 3ηr* r ∂r 

 1 ∂  ∂h       . r  r ∂r  ∂r   

Hence, 3ηr*4 h* γh 4 . = * 4 , or t* = t* 3ηr* γh*3 In the same way, from Equation 3.12, R (t )

2πr*2h*

∫ r h dr = V . 0

Hence, 2πr*2h* = V , or h* =

V 3(2π)3 ηr*10 . , t* = 2 2πr* γV 3

In Section 3.2 we will see that the estimation of the characteristic time scale is unrealistically small. However, for a moment we ignore this because, as we will see in the following text, the spreading problem cannot be solved precisely in this way. Now, Equation 3.17 and Equation 3.12 can be rewritten as 1 ∂  3 ∂ ∂h =− r h r ∂r  ∂t ∂r

 1 ∂  ∂h     r   ,  r ∂r  ∂r   

(3.18)

R (t )

∫ r h dr = 1. 0

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(3.19)

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181

SIMILARITY SOLUTION

OF

EQUATION 3.18

AND

EQUATION 3.19

Let ξ be a new variable, which is a combination of r and t . The dimensionless thickness, h ( t , r ), should depend on this new single variable. Such a solution is referred to as a similarity solution. Let ξ = rf ( t ) , where f ( t ) is a new unknown function. From Equation 3.19 we conclude that R( t )

∫ rf (t ) f (t ) h drf (t ) f (t ) = 1 , 1

1

0

or R( t ) f ( t )

∫ 0

ξ

h dξ = 1 . f (t )

(3.20)

2

This equation must be independent of t . This gives two conditions: R( t ) f ( t ) = λ c = const , R( t ) =

λc , f (t )

(3.21)

where λc is an unknown constant. Equation 3.18 and Equation 3.19 do not include any dimensionless parameters; it means that λc ~ 1. As we see in the following example, the latter constant cannot be determined without consideration of the disjoining pressure, and it will be determined only in Section 3.2. The second condition, which immediately follows from Equation 3.20, is h = A(ξ) f 2 ( t ) .

(3.22)

According to Equation 3.21, the spreading law R( t ) is known if the function, f ( t ), is determined. Similarity solution (3.22) should be substituted in Equation 3.18. Let us calculate ∂h = A′(ξ)  rf ′( t )  f 2 ( t ) + A(ξ)2 f ( t ) f ′( t ) ∂t = A′(ξ)  rf ( t )  f ′( t ) f ( t ) + 2 A(ξ) f ( t ) f ′( t ) = f ( t ) f ′( t )  A′(ξ)ξ + 2 A(ξ)  © 2007 by Taylor & Francis Group, LLC

182

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Using the latter equation and r=

ξ , f (t )

we conclude from Equation 3.18 f ( t ) f ′( t )  A′(ξ)ξ + 2 A(ξ)  = − f 8 (t ) f 4 (t )

. 1 d  3 d  1 d  dA    ξ  ξA   ξ d ξ  d ξ  ξ d ξ  d ξ   

(3.23)

This equation should not depend on time but on variable ξ; hence, f ( t ) f ′( t ) = − f 8 ( t ) f 4 ( t ) , or f ′( t ) = − f 11 ( t ) .

(3.24)

The solution of this equation is f (t ) =

1

(

)

10 t + C   

0.1

,

where C is an integration constant. From this equation and Equation 3.21,

(

)

0.1

R( t ) = λ c 10 t + C  . Using the initial condition R(0) = R0, where R0 is different from the initial radius of the droplet but equal to the radius of the droplet in the end of the inertial period of spreading (see further discussion in Chapter 4, Section 4.1). Therefore, we should choose the scale of the radius of the droplet as follows: r* = R0 . According to this choice, R(0 ) = 1. Hence, C= © 2007 by Taylor & Francis Group, LLC

1 . 10 λ10 c

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183

It is reasonable to assume (it will be justiﬁed partially in Chapter 4, Section 4.1), that the constant is equal to the duration of the inertial stage of spreading. Hence, we adopt C=

tin = tin , t*

where tin is the duration of the inertial stage of spreading. Using this notation we can write the spreading law as R( t ) = (λ c10 0.1 )( t + tin )0.1 . Now back to dimensional variables:  t + tin  R(t ) = (λ c10 0.1 )  R0  t* 

0.1

(λ c10 0.1 )  γV 3   R(t ) = 0.1  3 2π 3   η   

( )

0.1

(t + t in )

0.1

 3 γV  = 0.65λ c   η 

0.1 0.1

(t + t in ) .

Finally,  γV 3  R(t ) = 0.65λ c    η 

0.1

(t + tin )0.1 .

Using Equation 3.25 and the deﬁnition (3.22) we conclude that  ηV 2  H (t ) ≈   γ 

1/ 5

(t + tin )−1/5 ,

and  η3V  θ (t ) ≈  3   γ 

1/10

(t + tin )−0,3 ,

where the dynamic contact angle is determined as θ(t ) ~

© 2007 by Taylor & Francis Group, LLC

H (t ) . R(t )

(3.25)

184

Wetting and Spreading Dynamics

The spreading rate according to Equation 3.25 is  γV 3  dR(t ) U= = 0.065λ c   dt  η 

0.1

(t + tin )−0.9 .

If we express time in this equation via the dynamic contact angle θ(t), we arrive at U ~ 0.065 λ C

γ 3 θ , η

which is the well-known and well-veriﬁed Tanner’s law of dynamics of spreading in the case of complete spreading. This law has already been mentioned in the Introduction to this chapter. As we already commented, the unknown constant, λc, is close to 1. According to Equation 3.25, the exponent 0.1 and R(t)/V 0.3 are independent of the drop volume. These two conclusions have been well conﬁrmed in the case of complete wetting by numerous experimental data. Similar calculations can be carried out for the dynamics of spreading of a one-dimensional droplet (cylinder), which results in γ  R (t ) ≈  V 3  η 

1/ 7

(t + tin )1/ 7 .

However, there is one very substantial drawback to the obtained solution (see the following text). Let us go back to the Equation 3.23, which takes the following form after selection (3.24): 1 d  3 d  1 d  dA     A′(ξ)ξ + 2 A(ξ)  = ξ ξA   . ξ d ξ  d ξ  ξ d ξ  d ξ    The equation can be transformed as follows:       A′(ξ)ξ 2 + 2ξA(ξ)  = d ξA3 d  1 d ξ dA       dξ d ξ  ξ d ξ  d ξ           A(ξ)ξ 2 ′ = d ξA3 d  1 d ξ dA      dξ d ξ  ξ d ξ  d ξ     © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

185

A(ξ)ξ 2 = ξA3

d  1 d  dA   ξ   d ξ  ξ d ξ  d ξ  

 d  1 d  dA    A  ξ − A2 ξ    = 0. d ξ  ξ d ξ  d ξ     The preceding equation shows that either A(ξ) = 0, or   1 d  dA    2 d ξ ξ − A   = 0 . d ξ  ξ d ξ  d ξ     This equation describes the spreading droplet proﬁle up to the point of intersection with the axis ξ; that is, to the point where A(ξ0) = 0. Hence, the following problem should be solved: A′′ A′  ξ  A 2 = A′′′ + ξ − ξ 2   .  A( 0 ) = a   A′ ( 0 ) = 0   A′′(0 ) = −b

(3.25′)

The equation should be solved with two additional conditions ξ0

∫ ξA(ξ)d ξ = 1,

where

A(ξ 0 ) = 0 ,

0

which determine the unknown constants a and b. Note that the ﬁrst condition is simply the conservation law (3.21). Hence, we can now determine λ C = ξ 0 . It could be the complete solution to the problem of the capillary regime of spreading. Unfortunately, no such solution of the above problem exists: at any choice of constants a and b, the solution behaves as shown in Figure 3.3: the similarity drop proﬁle never intersects the axis (curve 1, calculated according to the above system). Curve 2 in Figure 3.3 presents the parabolic proﬁle a−

b 2 ξ 2

for comparison. This is the manifestation of the so-called “singularity” at the moving apparent three-phase contact line. © 2007 by Taylor & Francis Group, LLC

186

Wetting and Spreading Dynamics A

a 1 2 c∗ ξ∗

ξ

FIGURE 3.3 Dependency of a similarity proﬁle of the spreading droplet, A(ξ): it never intersects the axis. (1) calculated according to Equation 3.25′; (2) the spherical (parabolic proﬁle).

This contradiction is resolved in Section 3.2. Surprisingly, the spreading law (3.25) remains almost untouched except for the determination of the unknown constant λc .

GRAVITATIONAL SPREADING In this case the radius of spreading is bigger than the capillary length. Hence, according to Equation 3.14, we can omit the capillary pressure. This results in the following expression for the pressure inside the drop: p = pa + ρgh . Substitution of this expression into the equation of spreading (3.11) results in ∂h ρg ∂  3 ∂h  = r h , ∂t 3ηr ∂r  ∂r 

(3.26)

with the same conservation law (3.12) and the boundary condition ∂h =0 . ∂r r = 0 Let us introduce the following dimensionless values using characteristic scales h*, a, and t* . The end of the capillary stage of spreading is when the radius of spreading reaches a. It is the reason why the latter length is selected as the characteristic length scale. From Equation 3.26 we conclude: h* ∂h ρgh*4 1 ∂  3 ∂h  = r h . t* ∂ t 3ηa 2 r ∂r  ∂r  © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

187

Hence, 3ηa 2 h* ρgh*4 or = , t = . * t* 3ηa 2 ρgh*3 Now, from conservation law (3.12) we conclude: R( t )

2πa h* 2

∫ r h dr = V . 0

Hence, V 3(2π)3 ηa8 2πa 2h* = V , or h = , t* = . 2 * 2πa ρgV 3 Using these notations we can conclude from Equation 3.26 and conservation law (3.12): ∂h 1 ∂  3 ∂h  = r h , ∂ t r ∂r  ∂r 

(3.27)

R( t )

∫ r h dr = 1.

(3.28)

0

SIMILARITY SOLUTION Let ξ be a new variable, which is a combination of r and t as in the previous case of capillary spreading; h(t,r) should depend on this new single variable. Such solution is referred to as a similarity solution. Let ξ = r f(t), where f(t) is a new unknown function. Substituting the latter deﬁnitions into Equation 3.28, we ﬁnd that R( t ) f ( t )

∫ 0

ξ

h dξ = 1 . f (t ) 2

This equation must be independent of t; this gives two conditions: R( t ) f ( t ) = λ g = const , R( t ) =

© 2007 by Taylor & Francis Group, LLC

λg , f (t )

(3.29)

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Wetting and Spreading Dynamics

and h = A(ξ) f 2 ( t ) ,

(3.30)

where λg should be close to 1. Equation 3.29 shows that the spreading law, R(t), is known if the unknown function, f(t), is determined. Similarity solution (3.30) should be substituted in Equation 3.27. Calculations similar to the previous case results in an identical expression for the time derivative. Using the deﬁnition r=

ξ f (t )

we conclude from Equation 3.27 that f ( t ) f ′( t )  A′(ξ)ξ + 2 A(ξ)  = f 8 ( t ) f 2 ( t )

1 d  3 dA  ξA . dξ  ξ dξ 

(3.31)

Equation 3.31 should not depend on time but on variable ξ only; hence, f ( t ) f ′( t ) = − f 8 ( t ) f 2 ( t )

(3.32)

or f ′( t ) = − f 9 ( t ). Note the “minus” in the latter equation because the radius of spreading, R(t), should be an increasing function of time. Solution of Equation 3.32 is f (t ) =

1

(

)

1/ 8

8 t + C   

,

where C is an integration constant. From the latter equation and Equation 3.29 we conclude:

(

)

1/ 8

R( t ) = λ g 8 t + C  .

(3.33)

The initial condition can be speciﬁed as R(0) = a or in dimensionless units as R(0 ) = 1. Using the latter initial condition we conclude from Equation 3.33: © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

189

C=

1 , 8 λ 8g

which we can refer to as the duration of the capillary stage of spreading, tc. – – Therefore. Equation 3.33 can be rewritten as R(t) = (λg81/8) (t– + t–c)1/8. After we go back to dimensional variables, the equation becomes 1/ 8

 t + tc  R(t ) = (λ g 81/8 )  , a  t*  or  ρgV 3  R(t ) = 0.57λ g    η 

1/ 8

(t + tc )1/8 ,

(3.34)

where λg is close to 1. Exponent 1/8 agrees very well with a number of experimental observations. Now we can consider the shape of the spreading droplet over the duration of the gravitational regime of spreading. Equation 3.31, taking into account Equation 3.32, now takes the following form: 1 d  3 dA   A′(ξ)ξ + 2 A(ξ)  = − ξA . ξ dξ  dξ  After simple rearrangements (similar to those in the capillary case), we arrive at A′ = −

ξ . A2

Solution of this equation is  3 A( ξ ) =    2

1/ 3

(λ

2 g

− ξ2

)

1/ 3

,

(3.35)

where the integration constant ξ0 should be determined from the conservation law λg

∫ ξA(ξ)dξ = 1 , 0

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190

Wetting and Spreading Dynamics ᐉn R 2 3

ᐉn a

ᐉn tc

ᐉn t

1

FIGURE 3.4 Time evolution of the radius of spreading in log–log coordinate system (3). (1) capillary spreading; (2) gravitational regime of spreading.

which results in λg ≈ 1.37, which is close to 1 as predicted earlier. Using this value, we can rewrite Equation 3.34 as  ρgV 3  R(t ) = 0.78    η 

1/ 8

(t + tc )1/8 .

(3.36)

That is, in the case of gravitational spreading, the spreading law can be determined completely according to Equation 3.36. However, we still have one substantial problem in the case of gravitational spreading. The proﬁle of the droplet in this case is shown in Figure 5.25 (the case n = 1). This ﬁgure shows that the low slope approximation used in our consideration is severely violated in a vicinity of the edge of the spreading droplet. In Figure 3.4 the time evolution of radius of spreading is schematically presented in log–log coordinates. Capillary and gravitational regimes are shown by line 1 (according to Equation 3.25, with λc still unknown) and line 2 (according to Equation 3.36), respectively. The capillary regime of spreading switches to the gravitational regime of spreading at the moment marked as tc, when the radius of spreading reaches the value of the capillary length, a. The real time evolution is shown by curve 3 in Figure 3.4. The experimental evidence of such dependency was obtained in Reference 15, where a transition from the capillary regime of spreading to the gravitation regime has been demonstrated similar to that shown in Figure 3.4.

SPREADING

OF

VERY THIN DROPLETS

In the earlier part of this section we showed that pure capillary spreading results in inconsistency of the mathematical treatment in the vicinity of the apparent moving contact line of the spreading droplet. This inconsistency is usually referred to as a singularity at the three-phase contact line. As we already mentioned in the Introduction to this chapter, this inconsistency is the result of

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191

neglecting the disjoining pressure action in the vicinity of the moving apparent three-phase contact line. Remember that as the drop thickness tends to zero, the disjoining pressure overcomes the capillary pressure and dominates the spreading process. This is why we consider here the spreading of the microdroplet, whose apex is located in the range of action of the surface forces. In the following text, we neglect the effect of capillary pressure. The hydrodynamic pressure in the liquid is described by Equation 3.16. Thus, the case under consideration refers to the ﬁnal stage of the spreading of droplets, which completely wets the substrate. The equation describing the droplet proﬁle, h(t, r), of a spreading droplet has the following form: 1 ∂  3 ∂h  ∂h =− r h Π′(h)  . 3ηr ∂r  ∂r  ∂t

(3.37)

This is obtained by a substitution of Equation 3.16 into the equation of spreading (3.11). The boundary conditions for second-order differential equation (3.37) are as follows: ∂h = 0, r = 0 , ∂r

(3.38)

h ( t, ∞ ) = 0 ,

(3.39)

and the conservation law (3.12). Note that Equation 3.37 has the form of a nonlinear equation of thermal conductivity. As we know, an equation of thermal conductivity often (for example, in the linear case) results in an inﬁnite rate of propagation of perturbations. Therefore, features similar to that of Equation 3.37 should be expected in the case under consideration. Let us ﬁrst consider the simplest form of the isotherm of the disjoining pressure, Π(h), in the case of complete wetting:

(

)

 b ts − h , 0 ≤ h ≤ ts  Π h = . 0, h ≥ ts

()

(3.40)

In this case, Equation 3.37 takes the form ∂h b ∂  3 ∂h  = r h . ∂t 3ηr ∂r  ∂r 

© 2007 by Taylor & Francis Group, LLC

(3.41)

192

Wetting and Spreading Dynamics

Introducing dimensionless values as before in the equation, we conclude: 2πr*2h* = V , or h* =

V 3(2π)3 ηr*8 , t = . * 2πr*2 bV 3

Using these notations we can conclude from Equation 3.41 and conservation law (3.12) that ∂h 1 ∂  3 ∂h  = r h , ∂ t r ∂r  ∂r 

(3.42)

R( t )

∫ r h d r = 1.

(3.43)

0

Surprisingly, Equation 3.42 and Equation 3.43 are identical to Equation 3.27 and Equation 3.28 in the case of gravitational spreading. Hence, we can use the already deduced similarity solution (3.35) for the droplet proﬁle with λg ≈ 1.37. After that, the spreading law becomes, as in (3.36),  bV 3  R(t ) = 0.78    η 

1/ 8

(t + t0 )1/8 ,

(3.44)

where t0 is the time when this stage of spreading starts. The only difference between the expression from (3.36) and (3.44) is that ρg in Equation 3.36 is replaced by b in Equation 3.44. At any rate, the disjoining pressure action, even in the simplest possible form (3.40), removed the artiﬁcial singularity on the moving apparent contact line. Let us consider an isotherm of the disjoining pressure of the form

()

Π h = A/h n ,

(3.45)

which, as we already know, is relevant in the case of complete wetting (for example, spreading of oils over glass and metals). Equation 3.37 now becomes ∂h nA ∂  2− n ∂h  = . r h ∂t 3ηr ∂r  ∂r 

(3.46)

Introducing dimensionless values using the following characteristic scales h* , r* , t* in Equation 3.46, we conclude: © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

193

h* ∂h nAh*3− n 1 ∂  2− n ∂h  =− r h . t* ∂ t ∂r  3ηr*2 r ∂r  Hence, we can choose

t* =

3ηr*2 . nAh*2− n

Using the conservation law (3.12) we conclude as before:

h* =

3ηV n− 2 V . , t* = 2 2πr* (2π)n− 2 r*2 n−6

Now Equation 3.46 and Equation 3.12 can be rewritten as ∂h 1 ∂  2− n ∂h  = r h , ∂ t r ∂r  ∂r 

(3.47)

and (3.19). Let ξ = rf ( t ), where f ( t ) is a new unknown function. Substitution of the foregoing deﬁnitions into Equation 3.19, we ﬁnd that R( t ) f ( t )

∫ 0

ξ

h dξ = 1. f (t ) 2

This equation must be independent of t; this gives two conditions: R( t ) f ( t ) = λ n = const , R( t ) =

λn , f (t )

(3.48)

and h = A(ξ) f 2 ( t ) ,

(3.49)

where λn should be close to 1 and depends on the exponent n of the disjoining pressure isotherm. Equation 3.48 shows that the spreading law, R(t), is known if the unknown function, f(t), is determined.

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Wetting and Spreading Dynamics

Similarity solution (3.49) should be substituted in Equation 3.47. Calculations similar to the previous cases result in an identical expression for the time derivative. Using the deﬁnition r=

ξ , f (t )

we conclude from Equation 3.47 that f ( t ) f ′( t )  A′(ξ)ξ + 2 A(ξ)  = f 8− 2 n ( t )

1 d  2− n dA  ξA . ξ dξ  dξ 

(3.50)

Equation 3.50 should not depend on time but on variable ξ only, hence, f ( t ) f ′ ( t ) = − f 8− 2 n ( t )

(3.51)

or f ′( t ) = − f 7− 2 n ( t ). Note the minus in the latter equation because the radius of spreading, R(t), should be an increasing function of time. Now Equation 3.50 takes the following form: 1 d  2− n dA   A′(ξ)ξ + 2 A(ξ)  = − ξA , ξ dξ  dξ  or, after rearrangement similar to the previous cases, ξ = − A1− n

dA . dξ

(3.52)

Let us consider two cases, n = 3 and n = 2, in the Equations 3.51 and 3.52. In the case n = 2 we conclude that f ′( t ) = − f 3 ( t ) and ξ = − A−1

dA . dξ

Taking into account Equation 3.48, we conclude that R( t ) = λ 2 2 ( t + t2 ) , © 2007 by Taylor & Francis Group, LLC

(3.53)

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and  ξ2  A(ξ ) = C exp  −  ,  2

(3.54)

where t2 is the dimensionless time of the beginning of this stage of spreading, and C is the integration constant to be determined from the conservation law. According to Equation 3.54, the droplet proﬁle does not vanish anywhere but tends asymptotically to zero. That is, we can determine only the effective apparent three-phase contact line as follows: According to conservation law, λ2

∫ 0

 ξ2  ξC exp  −  d ξ = 1 ;  2

hence, after integration.   λ2   C  1 − exp  − 2   = 1 .  2   If we select the apparent contact line as the point where the proﬁle is close to zero, that is,  λs  exp  − s  = 0.1 ,  2 then λ 2 = 2 ln 10 ≈ 2.15 . In this case, C ≈ 0.9, and we conclude that R( t ) = 3.04

(t

+ t2 ) ,

(3.55)

 ξ2  A(ξ ) = 0.9 exp  −  .  2 Note that the spreading law (3.55) with exponent 0.5 is the fastest yet that we have found. Let us emphasize again that there is no distinct three-phase contact line in the case under consideration; the droplet proﬁle tends to zero only asymptotically. This is the consequence of neglecting the surface diffusion in front of the moving contact line. It is still an open problem as to how to connect the macroscopic description based on the consideration of the disjoining pressure with surface diffusion, which is a microscopic description. © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

In the case n = 3, we conclude from Equation 3.51 and Equation 3.52 that f ′( t ) = − f ( t ) and ξ = − A−2

dA . dξ

The solution of the latter equations is f ( t ) = exp ( −( t + t3 ) and A(ξ ) =

1 C+

ξ2 2

,

where t3 is the time when this stage of spreading begins, and C is an integration constant. As in the previous case, there is no distinct three-phase contact line because the droplet proﬁle tends to zero only asymptotically. We select an apparent contact line as follows from the conservation law: λ3

ξd ξ

∫C+ ξ 0

2

= 1,

2

which gives  λ2  ln  1 + 3  = 1 . 2C   The latter equation has the following solution: 1+

λ 23 = e, 2C

or λ 23 = e − 1. 2C According to the previous consideration, λ3 should be around 1. Hence, if we select C~

1 = 0.29 , 2(e − 1)

then R( t ) = λ 3 exp ( t + t3 ) , © 2007 by Taylor & Francis Group, LLC

(3.56)

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197

and A(ξ ) ≈

1 ξ2 0.29 + 2

.

The latter Equation 3.56 gives the highest possible rate of spreading.

3.2 THE SPREADING OF LIQUID DROPS OVER DRY SURFACES: INFLUENCE OF SURFACE FORCES In this section we consider the spreading of an axisymmetric liquid drop on a plane solid substrate in the case of complete wetting. Both capillary and disjoining pressure are taken into account [17]. As we already concluded in Section 3.1 and the Introduction to this chapter, neglecting of the disjoining pressure in the vicinity of the moving apparent three-phase contact line results in a contradiction. On the other hand, we already showed at the end of Section 3.1 that the disjoining pressure action removes the singularity on the moving contact line. A cylindrical coordinate system is used in the following text. The initial size of the droplet is assumed to be smaller than the capillary length a=

γ ; ρg

that is, the gravity is neglected. We already estimated in the Introduction to this chapter that 1. The capillary number, Ca ~ 10–5 << 1, that is, the main part of the spreading droplet remains the spherical shape. 2. A low slope approximation is valid in the vicinity of the moving contact line. However, we should still estimate the scale of the narrow zone, where the latter approximation is satisﬁed. This means that the liquid proﬁle in the main part of the spreading droplet is h(r ) = ℜ 2 − r 2 − (ℜ − H ) ,

(3.57)

(see Figure 3.5). Using simple geometrical consideration, we conclude from Figure 3.2 that H and ℜ are expressed via the dynamic contact and the radius of spreading as ℜ= © 2007 by Taylor & Francis Group, LLC

R R(1 − cos θ) . , H= sin θ sin θ

198

Wetting and Spreading Dynamics z H θ(t)

1

R(t) r

ℜ

FIGURE 3.5 Spreading of axisymmetric droplet. (1) vicinity of the moving contact line.

If we assume that the liquid is mostly located in the spherical part of the droplet, then the volume, V, is V=

π 3 θ R tan  3 + tan 2 6 2

θ , 2

(3.58)

or  6V  R(t ) =   π 

1/ 3

1 1/ 3

 θ(t )  2 θ(t )    tan 2  3 + tan 2    

.

(3.59)

In the case of small contact angles, the latter equation results in 1/ 3

 4V  R(t ) =  .  πθ 

(3.60)

Equation 3.60 gives a good approximation of the right-hand side of Equation 3.59 in the range of dynamic contact angles 0 < θ(t) < π/4. Using the low slope approximation we already deduced the equation of spreading, which describes the shape of the liquid proﬁle (3.11). Now, this equation is used only in the vicinity of the moving contact line and will be matched with the solution (3.57). Inside the same vicinity of the moving contact line the pressure is given by expression (3.15), which should be substituted into Equation 3.11. In the case of complete wetting, which is discussed in this section, we adopt the disjoining pressure isotherm, Π(h), as Π (h) = © 2007 by Taylor & Francis Group, LLC

A , hn

(3.61)

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199

with A > 0 being the Hamaker constant and n = 2 or 3. Substituting the latter equations into Equation 3.11 and Equation 3.15 results in ∂h γ 1 ∂  3  ∂  1 ∂  ∂h  nA ∂ h  .  rh   =− r −  n +1   ∂r  ∂t 3η r ∂ r   ∂ r  r ∂ r  ∂ r    γ h  

(3.62)

The ﬁrst and the second terms in brackets on the right-hand side of Equation 3.62 describe the effects of capillarity and disjoining pressure, respectively. The drop thickness decreases from the center of the drop to its spreading edge. In this section we are interested in the three regions of the spreading drop (Figure 3.1b). The spherical part of the spreading droplet (that is, the region from the center of the drop to a point where the ﬁlm thickness is still large, such that in this region both viscous and disjoining pressure effects are negligible compared to capillary effects) is referred to as the outer region. Further outward radially, there is a region where both capillary and viscous effects dominate (region 2 in Figure 3.1b). The next region is further outward, where the ﬁlm is thin and disjoining pressure effects are of comparable importance to viscous and capillary effects (see Figure 3.1b). Region 2 and Region 3 are referred to as the inner region. Equation 3.62 can be made dimensionless by introducing appropriate scales for thickness, radius, and time,

−ε

∂ h 1 ∂  3  ∂  1 ∂  ∂h  λ ∂h rh   , r =  − n +1    ∂ t r ∂r  ∂ r r ∂ r ∂ r h ∂ r          

(3.63)

where we use the same notations for dimensionless values as for dimensional: h → h/h*, r → r/r*, t = t/t*, ε = 3η r*4/γ h*3t*, λ = nAr*2/h*n+1. The scales, h*, r*, and t* are determined in the following text. According to Equation 3.12, 2π r*2 h* = V, where V is the drop volume. Hence, from this equation, h* is determined as h* = V/(2πr*2 ), and there are now only two unknown scales, r* and t*. The volume of the liquid drop is constant during spreading; that is, Equation 3.12 should be used. If the whole liquid proﬁle satisﬁes the condition h′2 << 1, then Equation 3.63 describes the liquid proﬁle in all regions. In this case the following symmetry conditions are valid at r = 0: ∂h ∂ 3 h = = 0. ∂r ∂r3

(3.64)

The preceding case does not differ substantially from the more general case, when the dynamic contact angle is not sufﬁciently small. This is why only the low slope approximation, which is valid over the whole droplet, is under consideration here. © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

The initial solid surface is dry in front of the spreading droplet; that is: h → 0,

at

r → R(t ) .

(3.65)

It is necessary to comment on the latter boundary condition. As we already established in Section 3.1, the liquid proﬁle may tend asymptotically to zero; for this case the boundary condition will be speciﬁed in the following text. It is shown in the following text that both dimensionless constants in Equation 3.63 are small: ε << 1 ,

(3.66)

λ << 1 .

(3.67)

and

Condition (3.66) expresses the smallness of viscous forces, Fη, as compared with capillary forces, Fγ : Fη ∼ η∂2vr /∂z2 ~ η r*/t* h*2, and

Fγ ~

h ∂  γ ∂  ∂h ~ γ 3∗ , r     ∂r  r ∂r  ∂r  r∗

where z is the vertical coordinate and vr the radial component of velocity. Hence, Fη /Fγ ~ η r*4/(γ t* h *3) = ε/3 << 1. This consideration shows that, according to its physical meaning, ε is the modiﬁed capillary number, Ca. Condition (3.67) expresses the smallness of the disjoining pressure as compared with capillary forces at large thickness. Letting ε = λ = 0, we obtain the outer solution of Equation 3.63 in the region of the spherical part of the droplet as

( )

( )(

)

h r ,t = 4 f 2 t 1 − ξ 2 ,

0 < ξ < 1,

(3.68)

where ξ = r f(t), and an unknown function f(t) is determined as follows. Equation 3.68 determines the parabolic proﬁle of the drop away from the drop edge. The apparent contact line corresponds to the condition ξ = 1, i.e.,

()

()

r0 t = 1 f t ,

(3.69)

where r0 (t) is the macroscopic apparent radius of the spreading drop (Figure 3.2). In deriving Equation 3.68, the conservation law (3.12) was used (see Appendix 1). It was supposed that almost the whole volume of the spreading droplet is located in the spherical part. © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

201

To derive the inner solution of Equation 3.63 in the region, where the drop proﬁle is distorted by the action hydrodynamic ﬂow and disjoining pressure, we introduce new variables:

(

) ()

(3.70)

() ( )

(3.71)

µ= ξ−1 χ t , and h = h0 t ψ µ ,

where χ(t), ψ(µ), and h0 (t) are new unknown functions. Letting χ << 1, h0 << 1 and focusing only on the largest terms (see Appendix 1), from Equation 3.63 we obtain that −

ε h 0 f d ψ h 40 f 4 d  3  d 3ψ λχ2 dψ   =  ψ  3 − 2 n +1 n +1 , 4 f χ dµ χ dµ  dµ   f h0 ψ  dµ

or −

dψ   ε χ 3 f d ψ d  3  d 3ψ λχ2 =  ψ  3 − 2 n +1 n +1  , 3 5 h 0 f dµ dµ  f h0 ψ dµ    dµ

(3.72)

where the overdot indicates the ﬁrst derivative with respect to dimensionless time, t. For both sides of Equation 3.72 to be explicit functions of µ but not time, we require ε χ 3 f = −1 , h 03 f 5

(3.73)

λχ2 =1. f 2 h n0 +1

(3.74)

Equation 3.73 and Equation 3.74 are two ordinary differential equations for three unknown functions f(t), χ(t), and h0 (t). Integrating Equation 3.72 and using condition (3.64), which in a dimensionless form becomes ψ(µ) → 0, as η → ∞, we obtain  ψ′  ψ 2  ψ ′′′ − n +1  = 1 , ψ  

(3.75)

where ′ and indicate ﬁrst and third derivatives in regard to µ, respectively. The ﬁrst and second terms in the brackets on the left-hand side of the equation are due to the capillary and disjoining pressure effects, respectively. © 2007 by Taylor & Francis Group, LLC

202

Wetting and Spreading Dynamics

The condition for matching the inner and outer solutions is dψ f 2χ = −8 = − B, dµ h0

µ = −∞ ,

(3.76)

where B is a positive constant to be determined. The matching condition (3.76) is the third equation, which is required to determine the three unknown functions f(t), χ(t), and h0 (t). From Equation 3.73, Equation 3.74, and Equation 3.76 we ﬁnd (see Appendix 2)  10 E 3t  f ( t ) = 0.5  10 + 1   2 ε  2 2( n + 2 ) λ  χ (t ) =  n +1   E 

 26 λ  h 0 (t ) =  2   E 

1 n −1

1 n −1

− 0.1

,

(3.77)

n +1

 10 E 3t  5( n−1) ,  210 ε + 1

(3.78)

3

 10 E 3t  5( n−1) ,  210 ε + 1

(3.79)

where E = 8/B. It follows from Equation 3.69 and Equation 3.77 that the dimensionless macroscopic radius of the spreading drop r0 (t) has the following time-dependence:  10 E 3t  r 0 ( t ) = 2  10 + 1  2 ε 

0 .1

.

(3.80)

The constant B in Equation 3.77 and Equation 3.79 can be determined using the matching condition according to Equation 3.76: dψ = − B, dµ

µ = −∞ .

(3.81)

Unfortunately, Equation 3.75 (see Appendix 3) does not have the required asymptotic behavior at µ → −∞ to satisfy the condition (3.81). This suggests that in order for both the outer, Equation 3.68, and the inner, Equation 3.75, solutions to be meaningful in some common range of variables ξ and µ, it is necessary to replace the matching condition (3.81) with an approximate condition © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

203

dψ = − B, dµ

µ = −µ ∗ ,

(3.82)

where µ* > 0 is an unknown constant. According to the physical meaning of the point µ* the requirement µ* >> 1 should be satisﬁed, which is checked in the following text. Let us ﬁnd an expression for µ*. The outer solution, Equation 3.68, obtained from Equation 3.63 at ε = λ = 0, becomes meaningless when h3 ∼ ε. Changing the variable according to Equation 3.71 gives h03 ψ*3 ∼ ε, where ψ* = ψ(–µ*) >> 1. Substituting Equation 3.75 into this expression and omitting the time-dependent term, we have

(

26 λ /E 2

)

3 n −1

ψ ∗3 ∼ ε.

The above-mentioned approximate condition is chosen from the latter equation, where ∼ is replaced by equality; that is,

( 2 λ /E ) 6

2

3 n −1

ψ ∗3 = ε ,

and, at last, the necessary equation for µ ∗ is

(

ψ ∗ = ε 1/ 3 E 2 / 64 λ

)

1 n −1

.

(3.83)

Note that the condition ψ* >> 1 should be satisﬁed. Now we are ready to calculate parameter ε. In terms of dimensional variables, Equation 3.80 becomes  10 E 3t  r 0 ( t ) = 2 r∗  10 + 1  2 ε t∗ 

0 .1

.

Differentiating the latter equation we conclude that  d r 0 (t ) E 3r ∗  10 E 3t = 9 +1 dt 2 ε t ∗  210 ε t ∗ 

−0.9

.

Based on the foregoing two equations, we choose r∗ = r0 (0)/ 2 and r∗ d r0 ( 0 ) = − 0.5 . t∗ dt © 2007 by Taylor & Francis Group, LLC

204

Wetting and Spreading Dynamics

The two aforementioned equations determine the unknown scales, r∗ and t ∗ , and, hence, ε ε=

1 E3 = . 210 2 B 3

(3.84)

From deﬁnitions of ε, h*, r*, and Equation 3.84, we conclude t∗ =

10 10 3π 3η r00 B3 B3 η r00 = 1.45 , 6 3 2 γ V γ V3

where r00 is the initial macroscopic drop radius: r00 = r0 ( 0 ) .

(3.85)

We show in Appendix 3 that as µ → −∞ ψ ′ ≈ −3 1/ 3 ln 1/ 3 µ ,

(3.86)

ψ ≈ 3 1/ 3 ln 1/ 3 µ .

(3.87)

From Equation 3.82, Equation 3.83, Equation 3.86, and Equation 3.87 we deduce that

ψ∗ = ε

1/ 3

 1   λ B 2 

1 n −1

(

)

= 3 1/ 3 µ ∗ ln 1/ 3 µ ∗ = B µ ∗ = B exp B 3 / 3 .

(3.88)

From Equation 3.84 and Equation 3.88 we have an equation for parameter B:  1   λ B 2 

1 n −1

(

)

= 2 1/ 3 B 2 exp B 3 / 3 .

(3.89)

Now, from Equation 3.84, Equation 3.88, and Equation 3.89 we can determine B, µ*, and ε. Neglecting the time-dependent term in deriving Equation 3.83 and Equation 3.89 is justiﬁed, as discussed in Appendix 4, where the dependence of B on time is shown to be weak. Let us consider the solution of Equation 3.89 for the two particular cases of disjoining pressure isotherm mentioned earlier. © 2007 by Taylor & Francis Group, LLC

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205

CASE n = 2 From Equation 3.89 we conclude in this case that

(

)

B 4 exp B 3 / 3 =

1 2 λ 1/ 3

.

(3.90)

Let us examine the magnitude of the parameter λ for typical values of h*, r*, A, and γ. For h* = 0.01 cm, r* = 0.05 cm, A = 10–7 dyn, and γ = 20 dyn/cm we conclude from λ deﬁnition: λ = 2.5·10–5 << 1. Substituting the value of λ into Equation 3.90, we have B4 exp(B3/3) = 3.18·104. This gives B = 2.68. Substituting this value of B into Equation 3.84 and Equation 3.88, we ﬁnd ε = 0.026 << 1, ψ* = 1648 >> 1, and µ* = 615 >> 1. As λ changes from 1.25·10–5 to 5·10–5, B and ε change from 2.76 to 2.60 and from 0.024 to 0.029, respectively. For the same range of λ, ψ* and µ* change from 3025 to 904 and from 1096 to 348, respectively. This means that the approximate conditions (Equation 3.82 and Equation 3.83) are practically equivalent to the matching condition (Equation 3.81) over some range containing the point µ = –µ*. Also note that the values of parameters satisfy the conditions mentioned before: ε << 1, λ << 1, µ* >> 1, and ψ* >> 1.

CASE n = 3 From Equation 3.89 we conclude in this case that

(

)

B 3 exp B 3 / 3 =

1 2

1/ 3

.

λ

(3.91)

Let us estimate the magnitude of λ for typical values of h*, r*, A, and γ. For h* = 0.01 cm, r* = 0.05 cm, A = 10–14 erg, and γ = 20 dyn/cm: λ = 3.75·10–10 << 1. Substituting the value of λ into Equation 3.91, we have B3 exp(B3/3) = 4.10·104. This gives B = 2.82. Substituting this value of B into Equation 3.84 and Equation 3.88, we ﬁnd ε = 0.022 << 1, ψ* = 5138 >> 1, and µ* = 1819 >> 1. As λ changes from 1.875·10–10 to 7.5·10–10, B and ε change from 2.86 to 2.79 and from 0.0213 to 0.0231, respectively. For the same range of λ, ψ* and µ* change from 7073 to 3735 and from 2472 to 1341, respectively. Similar to the previous case, this means that the approximate conditions (3.82, 3.83) are practically equivalent to the matching condition (3.81) over some range containing the match point µ = –µ*. Also note that the values of parameters satisfy the conditions mentioned earlier: ε << 1, λ << 1, µ* >> 1, and ψ* >> 1. Using Equation 3.84 and Equation 3.85 we can simplify the equation for r0 (t) dependency as  10 t  r 0 t = r00  + 1  t∗ 

()

© 2007 by Taylor & Francis Group, LLC

0 .1

.

(3.92)

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Wetting and Spreading Dynamics

From Equation 3.68, Equation 3.77, and Equation 3.84 we can derive the droplet proﬁle as a function of radial position and time, h(r, t) = 4 f 2(t) (1 – r 2 f 2(t)), or, in dimensional variables,

(

( )

)

h r , t = h∗ 10 t /t ∗ + 1

− 0.2

(

)(

)

(

)

1 − r 2 / 4r 2 10 t /t + 1 ∗ ∗ 

−0.2

. 

(3.93)

,

(3.94)

The apex height of the drop H ( t ) = h ( 0, t ) is

(

() ( )

)

H t = h 0, t = h∗ 10 t /t ∗ + 1

−0.2

= h00 10 t /t ∗ + 1

−0.2

where h00 is the initial apex height of the drop. The error of assigning h∗ = h00 is negligible, as discussed in Appendix 2. For a small angle, the advancing dynamic contact angle θ(t) can be derived from Equation 3.93 as θ(t = –∂h/∂rR, at r = r0(t), or

() (

)(

)

θ t = h ∗ /r ∗ 10 t /t ∗ + 1

−0.3

)(

(

)

= 2 h00 /r00 10 t /t ∗ + 1

−0.3

.

(3.95)

Assuming the spreading drop has the shape of a spherical cap, and the angle is small, another advancing dynamic contact angle, θRH (t), can be derived from Equation 3.92 and Equation 3.94:

()

(

)

)(

(

)

θ RH t = 2 tan −1 H (t ) /r0 (t ) = 2 H (t ) /r0 (t ) = 2 r00 /rr00 10 t /t ∗ + 1

−0.3

. (3.95’)

The right-hand site in Equation 3.95’ is identical to that in Equation 3.95 as it should be in the case of low slopes. We now derive the relationship between the advancing dynamic contact angle and the capillary number Ca, which is deﬁned as Ca = ηU/γ, where the spreading speed at the drop edge, U, is deﬁned as

(

)(

)

U = d r0 /d t = r00 /t ∗ 10 t /t ∗ + 1

−0.9

.

(3.96)

From Equation 3.95 we conclude that

()

θ t = 21/ 3 B Ca1/ 3 = 82.64 B Ca1/ 3

© 2007 by Taylor & Francis Group, LLC

( in radians) . in degr r ees ( )

(3.97)

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207

We can rewrite the power laws in (10 t/t* + 1) of Equation 3.92, Equation 3.94, and Equation 3.95 to power laws in te, where te is the experimentally measured time. Then,

( ) (

r0 t e = 640 γ V 3 / 3π 2 η B 3

(

)

= K γ V /η 3

0 .1

t

)

0 .1

(

t e0.1 = 1.21 γ V 3/ η B 3

)

0 .1

t e0.1 (3.98)

0.1 e

where K = 1.21·B 0.3. Using deﬁnitions of h*, r*, t*, and Equation 3.85, we can reduce Equation 3.94 to

( ) (

H te = 2 B3

) (3η V 0 .2

3

/ 40 π 2 γ

)

0 .1

t e− 0.2 ,

(3.99)

and Equation 3.95 to

( ) (

θ te = 2 B3

) ( 27η 0 .3

3

V / 2000 π γ

)

3

0 .1

( in raddians) .

t e− 0.3

(3.100)

Equation 3.98, Equation 3.99, and Equation 3.100 are the same as those derived in Section 3.1, except for the prefactors, which could not be deduced in that section. From Equation 3.89 we know that B is a function of λ: 2 n + 4 n +1 λ = n Ar00 π / 2 n +3 γ V

n +1

,

and, in turn, depends on the parameters n and A of the disjoining pressure isotherm. For the n = 2 case, with typical values of h*, r*, A, and γ given above, we have K = 0.900, and Equation 3.98 becomes

( )

(

r 0 t e = 0.900 γ V 3/ η

)

0 .1

t e0.1 .

(3.101)

Similarly, for the n = 3 case, we have K = 0.887, and

( )

(

)

r0 te = 0.887 γV 3/ η

0.1

te0.1 .

(3.102)

Let us now examine the inﬂuence of the constant A (Hamaker constant in the n = 3 case) on K in the spreading laws (3.102) and (3.103). For the n = 2 case,

© 2007 by Taylor & Francis Group, LLC

208

Wetting and Spreading Dynamics

the value of B changes from 2.76 to 2.60, and hence, K changes from 0.892 to 0.909. For the n = 3 case, the value of B changes from 2.86 to 2.79, and therefore, K changes from 0.883 to 0.890. It is obvious that although the value of K is somewhat larger in the n = 2 case than in the n = 3 case, the difference is not signiﬁcant between the two cases. Thus, K is only slightly dependent on n and the constant A of the disjoining pressure isotherm in the case of complete wetting. From Equation 3.92 and Equation 3.98 the wetted area predicted by our theory, St , is

(

)

0.2 St = π r02 (t ) = π r00 10 t /t ∗ + 1

(

= K 2 γ V 3 /η

)

0 .2

0.2

(

= 640 γ V 3π 2 / 3η B 3

)

0 .2

t e0.2

t e0.2

We now brieﬂy discuss the applicable conditions of the results: 1. The solid surface must be ﬂat and smooth, and the liquid is Newtonian and nonvolatile. 2. The gravity and inertia effects must be negligible compared to capillary effects. That means the Bond number Bo =

ρ gH 2 << 1 γ

and Weber number We =

ρ U 2H << 1 , γ

where ρ is the liquid density. 3. From the solution it follows that H → 0, as t/t* → ∞. However, as the droplet apex is in the range of surface forces action, the inﬂuence of disjoining pressure is in the drop center, and even a weak volatility of the liquid becomes signiﬁcant. The equilibrium ﬁlm thickness, he, is determined by the vapor pressure in the surrounding air and, according to Chapter 2, is

(

) (

)

A = RT /vm ln ps /p . hen This means that H → he, as t/t* → ∞, and hence, our results are valid only when the drop apex is much bigger than the ﬁnal equilibrium ﬁlm thickness, H >> he;

© 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

209

(4) as we only discuss the macroscopic aspect of the spreading problem, the condition h(r,t) → 0, as r → ∞, is the macroscopic condition. Macroscopic description (including the disjoining pressure) is meaningless if the scale of the thickness is comparable with the molecular length scale, m. Thus, our results are valid in the wetted area within the radius rm, where h(rm, t) > m. In conclusion, we compare the calculated radius of time dependency according to Equation 3.92 and Equation 3.25 deduced in Section 3.1. We rewrite Equation 3.92 as  γV 3  r0 (t ) = 0.89   η 

0.1

(t + tin )0.1 ,

where tin = t*/10. This comparison results in λC = 1.34; that is, λC ~ 1, as predicted in Section 3.1.

COMPARISON

WITH

EXPERIMENTS

Chen [16,17] has reported a series of experiments on spreading drops of polydimethylsiloxane. His data allow us to compare the deduced Equation 3.92, Equation 3.94, Equation 3.95, and Equation 3.97. In those experiments the conditions described in the theory are satisﬁed. The experiments involve depositing a liquid drop on a glass surface and monitoring the silhouette of the spreading drop: for the drop radius, r0(t), the drop apex height, H(t), and the advancing dynamic contact angle, θ(t). All experiments are run at room temperature. The errors of measurement are within 0.001 cm, 0.0002 cm, and 0.75˚, respectively. These errors are estimated from three repeated measurements for each quantity. The liquid used is a silicone liquid (a polydimethylsiloxane, Dow Corning 200 ﬂuid; Dow Corning Corporation, Midland, MI). It has a number-average molecular weight of 7500. At the room temperature of 22.5 to 24.0˚C, its viscosity ranges from 1.98 to 1.93 P, its density from 0.970 to 0.969 g/cm3, and its surface tension against air from 20.9 to 20.8 dyn/cm. The glass sample used is a sodalime glass plate and a borosilicate microscope slide. An example of the comparison is shown in Figure 3.6. The straight lines are the results of least-square-ﬁt to data in power laws. We ﬁrst compare the leastsquare-ﬁt results in power laws of

(

)

N

r0 , H, θ and θ RH = M ⋅ 10 t /t ∗ + 1 ,

(3.103)

with the theory, according to Equation 3.92, Equation 3.94, and Equation 3.95. The values of M and N obtained from least-square-ﬁt of experimental data show excellent agreement with the theory prediction not only in exponents, N, but also in prefactors, M.

© 2007 by Taylor & Francis Group, LLC

210

Wetting and Spreading Dynamics 0.1 0.05

∗

∗

12

∗∗∗∗ ∗ ∗ ∗ ∗ ∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

10

r0, H cm

∗ radius θ θRH apex

0.02 0.01

6

θ, deg.

8 0.03

4

0.005 0.003 0.002

2

1

3 10 t/t∗ + 1

4

5

6

7

8

2

FIGURE 3.6 Experimental dependencies of radius of spreading, dynamic contact angle, and the drop apex height on time ﬁtted according to Equation 3.103.

0.15 0.145 0.14 r0, cm

0.135

∗

∗

∗

∗

∗

∗ ∗

∗

0.13 0.125

∗

0.12 0.115

∗

0.11

3

4 5 10 t/t∗ + 1

6

7

8 9 10 12

FIGURE 3.7 Comparison between theoretical prediction (according to Equation 3.92) and the same experimental data as in Figure 3.6. Comparison of all other dependences (contact angle, the drop apex) show the similar excellent agreement between theoretical equations and experimental data.

Figure 3.7 shows a typical comparison between our theory predictions and experimental data. These comparisons show that the prefactors predicted by the theory agree well with the experimental values. The least-square-ﬁt results for data points of θ(Ca) dependency and for the same number of data points of θRH (Ca) from all experiments results in © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

211

θ = 222 ⋅ Ca 0 . 353

( in degrees) ,

θ RH = 242 ⋅ Ca 0 . 353

( in degrees) .

(3.104) (3.105)

The value of Ca ranges from 1.8·10–6 to 3.3·10–4 for experiments. The prefactors and exponents in Equation 3.104 and Equation 3.105 are close to those predicted by Equation 3.97, which can be written as θ = θ RH = 238 ⋅ Ca 0 . 333

( in degrees) .

(3.106)

Ausserre et al. [18,19] have measured the time-dependence of the radius and advancing dynamic contact angle for a number of spreading drops of polydimethylsiloxane with different molecular weights. They found that the radius and contact angle show a power law dependence on time with an exponent of 0.100 ± 0.010 and 0.3 ± 0.015, respectively. These values agree with our predictions, 0.1 and 0.3. For similar experiments, Tanner [20] found that the dynamic contact angle follows a power law in time with an exponent ranging from 0.317 to 0.335, which is close to our predicted value, 0.3.

CONCLUSIONS The combined effect of viscosity, surface tension, and disjoining pressure is included in the theory for an axisymmetric, nonvolatile, Newtonian liquid drop spreading over a horizontal, dry, smooth, ﬂat solid surface in the case of complete wetting. It is shown that the disjoining pressure action removes the singularity on the moving three-phase contact line. The drop proﬁle is calculated as a function of time and radial position. The spreading radius, the apex height of the drop, and the advancing dynamic contact angle are found to follow different power laws in time. The dynamic contact angle is found to follow a power law in capillary number. Both the prefactors and exponents in the power laws are predicted, and the predicted power laws agree with known experimental data.

APPENDIX 1 This appendix shows the derivation of Equation 3.68 and Equation 3.72. Let ε = λ = 0; Equation 3.63 becomes, in this case, ∂  3  ∂  1 ∂  ∂h      rh   r   = 0 . ∂r   ∂r  r ∂r  ∂r     Integrating once with respect to r results in  ∂  1 ∂  ∂h    C1 h3   . r   =  ∂r  r ∂r  ∂r    r © 2007 by Taylor & Francis Group, LLC

212

Wetting and Spreading Dynamics

As h is ﬁnite at r = 0, C1 must be 0. Integrating with respect to r three times, we have h = C2r 2/4 + C3 ln r +C4 . For the same reason, C3 must be 0. Recognizing h(r,t) = 0 at r = r0, we have h = (C2/4)(r 2 – r 02 ). Now taking into account conservation law (3.12) and deﬁning r0 = 1/f(t), we arrive at Equation 3.68. From Equation 3.71 we conclude that

(

)

rf χ − rf − 1 χ ∂h d ψ ∂µ = h0 t ψ µ + h0 t = h0 ψ + h0 ψ ′ ∂t d µ ∂t χ2

() ( )

()

 1  f χ  µ + χ  f − µχ h ψ + h ψ ′ , 0 0 χ where the overhead dot and the superscript indicate the ﬁrst derivative with respect to dimensionless time, t, and coordinate, µ, respectively. As µ = rf → 1 and χ << 1, then µ << 1/ χ , hence,  1  f f f  µ + χ  f χ − µχ ≈ f − µχ ≈ f . In view of this we conclude that f ∂h = h0 ψ + h0 ψ ′ . fχ ∂t In the following text we use the estimations h f ψ h0 ≈ 0 , ψ ′ ≈ , f ≈ . t µ t These estimations give h0 ψ

h0 ψη = ηχ << 1. f ≈ f h0 ψ h0 ψ ′ fχ fχ Hence, combining all previous estimations, we get the following: ∂h f = h0 ψ ′ . ∂t fχ

© 2007 by Taylor & Francis Group, LLC

(A1.1)

Kinetics of Wetting

213

Now, for µχ << 1, the right-hand side of Equation 3.63 can be simpliﬁed as 1 ∂  3  ∂  1 ∂  ∂h   λ ∂h   rh   r   − n+1   r ∂r   ∂r  r ∂r  ∂r   h ∂r     = h04

(A1.2)

f4 1 d 4 χ 1 + µχ d µ   d  1 dψ  λχ 2 d  dψ    1 + µχ ψ 3  1 + µχ − 2 n+1 n+1      d µ   d µ   f h0 ψ   d µ  1 + µχ d µ  

(

= h04

)

(

)

λχ 2 f 4 d  3  d 3ψ dψ  − ψ    4 3 2 n +1 n +1 d µ   χ d µ   d µ f h0 ψ

Now, from Equation A1.1 and Equation A1.2, we arrive at Equation 3.72.

APPENDIX 2 This appendix shows the derivation of Equation 3.77 through Equation 3.79 and discusses the negligible error in assigning H(0,0) = H* in the derivation. From Equation 3.73 and Equation 3.76 we have f f

11

=−

E3 . ε

(A2.1)

Here, the overhead dot indicates the ﬁrst derivative with respect to the dimensionless time, t. Integrating Equation A2.1 with respect to t once, we have f −10 = 10 E 3t /ε + const.

(A2.2)

To determine the integration constant, we make use of the following initial condition at r0 = r0 (0 ) and t = 0: H(0,0) = H*‚ which in a dimensionless form is h(0, 0) = 1, at r = 0 and t = 0 ,

(A2.3)

where H(0,0) is the initial apex height of the drop. Substituting Equation A2.3 into Equation 3.68, we ﬁnd that f(0) = 1/2.

© 2007 by Taylor & Francis Group, LLC

(A2.4)

214

Wetting and Spreading Dynamics

Substituting Equation A2.4 into Equation A2.2, we ﬁnd that the constant = 210 and

(

()

)

f t = 0.5 10 E 3t / 210 ε + 1

−0.1

,

which coincides with Equation 3.77. From Equation 3.74, Equation 3.76, and Equation 3.77 we can derive Equation 3.78 and, at last, from Equation 3.76, Equation 3.77, and Equation 3.78 we can derive Equation 3.79. We now discuss the error in assigning H* = H(0,0) instead of using H* = V/(2πR2*). In view of r* = r0(0)/2 and Equation 3.85, H* can be written as

(

)

( )

H∗ = V 2πR∗2 = 2V πr002 .

(A2.5)

If the drop shape is a spherical cap, then

(

)(

)

2 V = πH 00 / 6 3r002 + H 00 ,

(A2.6)

where H00 = H(0,0) is the initial apex height of the drop, and r00 = r0(0) is the initial macroscopic drop radius. From Equation A2.5 and Equation A2.6, we conclude that

(

)(

)

(

)

2 2 H* = H 00 / 3r002 3r002 + H 00 = H 00 1 + H 00 / 3r002 .

(A2.7)

From Equation A2.7, we estimate the error in assigning H* = H(0,0) is 0.75% when H00/r00 is 0.15. Thus, the error is negligibly small.

APPENDIX 3 This appendix shows the derivation for Equation 3.86 and Equation 3.87. As µ → −∞, ψ → ∞ , Equation 3.75 becomes ψ 2ψ ′′′ = 1.

(A3.1)

( )

Let us deﬁne ϕ ψ = ψ ′ = d ψ /d µ . According to the chain rule, we have ψ ′′ = d 2ψ /d µ 2 = d ϕ /d µ = d ϕ /d ψ ⋅ d ψ /d µ = ϕ′ϕ

( )

( )

2

ψ ′′′ = d ϕ′ϕ d µ = d ϕ′ /d µ ϕ + ϕ′d ϕ /d µ = ϕ′′ϕ 2 + ϕ′ ϕ .

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Kinetics of Wetting

215

Equation A3.1 now becomes

(

( )

)

2

ψ 2 ϕ′′ϕ 2 + ϕ′ ϕ = 1 .

(A3.2)

Let y = ln ϕ and applying the chain rule once again, we have ϕ ′ = d ϕ /d ψ = d ϕ /dy ⋅ dy /d ψ = ϕ y / ψ

(

(

)

ϕ ′′ = d 2 ϕ /d ψ 2 = dy /d ψ d ϕ y / ψ dy =  d ϕ y e − y 

(

)

(

) dy 

ψ

)

= e − y ϕ yy e − y − ϕ y e − y = e −2 y ϕ yy − ϕ y . Substituting these results into Equation A2.2, we have ϕ 2yϕ + ( ϕ yy − ϕ y ) ϕ 2 = 1 .

(A3.3)

Assuming the solution to Equation A3.3 has the form ϕ = DyG , where D and G are constants to be determined,

(

)

ϕ 2yϕ + ϕ yyϕ 2 = 2G − 1 GD3 y3G − 2 .

ϕ yϕ 2 = GD3 y3G −1

If G = 2/3, then ϕ yϕ 2 = D3y1/3 2/3 → ∞ , at y → ∞ . Hence, G = 2/3 cannot satisfy Equation A3.3 at y → ∞. However, if G = 1/3, then ϕ 2yϕ + ϕ yyϕ 2 = D

3

(9 y ) → 0, at

y → ∞.

Hence, at y → ∞ , for G = 1/3, we have from Equation A3.3 that D = –31/3 and ϕ = Dy1/ 3 = ψ ′ = −31/ 3 y1/ 3 = −31/ 3 ln1/ 3 ψ .

(A3.4)

From Equation A3.4 we have Equation 3.86 and d ψ / ln1/ 3 ψ = −31/ 3 d µ . Introducing z = ln ψ and integrating the above equation by parts, we conclude that

∫ ( ln

−1/ 3

)

ψ dψ =

∫e z

z −1/ 3

( )∫e z

dz = e z z −1/ 3  + 1 / 3

( ) ∫ ln

=  ψ ln −1/ 3 ψ  + 1/ 3

© 2007 by Taylor & Francis Group, LLC

−4 / 3

z −4 / 3

dz

ψd ψ ≅ ψ ln −1/ 3 ψ ,

216

Wetting and Spreading Dynamics

as ln 4 / 3 ψ >> ln1/ 3 ψ ′, at ψ → ∞ . Hence, ψ / ln1/ 3 ψ = −31/ 3 µ .

(A3.5)

Solving Equation A3.5 in the limits µ >> 1 and ψ >> 1 , we conclude that

( (

))

ψ = 31/ 3 µ ln1/ 3 ψ = 31/ 3 µ ln 31/ 3 µ ln1/ 3 ψ ≈ 31/ 3 µ lnn1/ 3 µ This expression coincides with Equation 3.87.

APPENDIX 4 This appendix discusses the weak dependence of B on dimensionless time, t. In deriving Equation 3.83 and Equation 3.89, we have neglected the timedependence term in h03 . Here, we include the time-dependence term for an estimation. We determine match point µ = −µ∗ as the point where the following condition is satisﬁed: h03ψ ∗3 = ε,

(A4.1)

( )

where ψ ∗ = ψ −µ∗ >> 1, and, in view of Equation 3.79, Equation A4.1 can be rewritten as 9 / 5( n −1) 3/ n −1 (26 λ /E 2 ) ( ) 10 E 3τ / 210 ε + 1 ψ ∗3 = ε.

(

)

(A4.2)

From Equation 3.84 and Equation A4.2 we conclude that

(

ψ ∗ = ε1/ 3 E 2 / 64 λ

) ( ) (10t + 1) 1/ n −1

( )

−3/ 5 n −1

(

= ε1/ 3 E 2 / 64 Λ

)(

)

1/ n −1

,

where Λ = λ (10t + 1) . Following the same derivation for Equation 3.89, we have 3/ 5

( ΛB ) 2

( )

−1 n −1

( )

(

)

= 21/ 3 B 2 exp B3 / 3 .

(

)

n −1 / 3 Let us deﬁne F B = 1/Λ = 2( ) B 2 n exp  n − 1 B3 / 3 ; then,

(

(

) )

dF /dB = 2n /B + n − 1 B 2 © 2007 by Taylor & Francis Group, LLC

(

)

Λ and dF /dB ⋅ dB /dt = − d Λ /dt Λ 2 .

Kinetics of Wetting

217

As (dΛ/dt)/Λ = 6/(10t + 1), we have dB/dt = –6 B/[(10t + 1)(2n + (n – 1) B3)]. For n = 3 and B ≈ 3, we conclude from the preceding equation that

(

)

dB /dt = −0.03 t + 0.1 Hence,

B = 3 − 0.03 ln ( t + 0.1) . The latter equation proves that B is weakly dependent upon time and justiﬁes the omission of the time-dependent term in deriving Equation 3.83 and Equation 3.89.

3.3 SPREADING OF DROPS OVER A SURFACE COVERED WITH A THIN LAYER OF THE SAME LIQUID In this section, a solution is obtained for the problem of viscous spreading of a liquid drop on a plane solid surface that has been prewetted with a ﬁlm of the same liquid. The ﬁlm thickness is assumed thick enough, that is, the thickness is bigger than the radius of disjoining pressure action. Appearing in the spreading equation is a universal small parameter, which is independent of the nature of the liquid–substrate system and which is a characteristic of the viscous spreading regime. By an expansion in terms of this small parameter, a solution has been obtained for the problem, through which the proﬁle of the spreading drop and the velocity of motion of the drop boundary can be determined [11]. Let us examine a drop of a viscous liquid on a planar horizontal solid surface covered with a layer of the same liquid with thickness h0 (Figure 3.8). The thickness h0 is assumed to be outside the range of disjoining pressure action. However, the droplet is still small enough to neglect the gravity action. As the spreading process is axisymmetric in the case under consideration, the height of the drop, h(r,t), is a function of both the distance, r, from the coordinate origin (which is located on the plane of the solid surface in the center of the drop)

H(0, t)

h(r, t) r0(t) 0

h0 L∗

r

FIGURE 3.8 Spreading of liquid droplets over the solid surface covered with a ﬁlm of the same liquid of uniform thickness h0. © 2007 by Taylor & Francis Group, LLC

218

Wetting and Spreading Dynamics

(Figure 3.8), and the time, t. For a sufﬁciently low-sloped drop, for which gravitational and inertial forces can be neglected, the equation describing the process of viscous spreading of the drop is deduced in Section 3.1 (Equation 3.17). This assumption means h* << r*, that is, the length scale in the vertical direction is much smaller than in the horizontal direction. In view of the symmetry of the drop, the boundary conditions in the droplet center (Equation 3.17’) are satisﬁed. As ﬂow occurs only where the liquid surface is curved, so that there is no ﬂow going out into the ﬁlm to inﬁnity, the excess volume of liquid above the ﬁlm remains constant: ∞

∫

2 π r  h ( r, t ) − h0  drR = V = const .

(3.107)

0

Far from the droplet, its proﬁle tends to that of the ﬁlm, and hence h(r, t ) = h0 , r → ∞ .

(3.108)

In addition to the four boundary conditions (3.17’), (3.107), and (3.108), Equation 3.17, which is the fourth order partial differential equation, requires the assignment of an initial condition, which is formulated in the following text. Let us introduce the dimensionless quantities using the characteristic scales r*, h*, t*, and U* for the horizontal dimension, the height of the drop, the time, and the spreading velocity, respectively, with U* = r*/t* and V = 2πr*2 h *, the latter relationship determining the horizontal scale of r* with a given drop volume and initial height following from the conservation law (3.107) in the same way as in Sections 3.1 and 3.2. Using the following dimensionless variables r → r/r*, t → t/t* , h→ h/h* , and h0 → h0 /h* ‚ keeping the same symbols for dimensionless values as for dimensional, we arrive at the equation of spreading: ∂h 1 ∂  3 ∂  1 ∂  ∂h     rh  r   = −ε , ∂t r ∂r  ∂r  r ∂r  ∂r   

(3.109)

where the dimensionless parameter ε: ε=

3η U*r*3 γ h*3

(3.110)

which, as before, represents the ratio of characteristic values of the force of viscous friction in the drop Fη = ηU*/h*2 and the horizontal component of the gradient of capillary pressure Fγ = γh*/r*3. The conservation law (3.107) now becomes © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

219 ∞

∫ r h ( r, t ) − h  dr = 1 ,

(3.111)

0

0

and the condition (3.108) has an identical form. It is easy to check that ε << 1.

(3.112)

We also assume that the ﬁlm thickness is much smaller than the characteristic size of the droplet; that is, (3.113) h0 << 1. In solving this problem, we use the method of matching of asymptotic expansions, which we already used in Section 3.2. In view of the smallness of the parameter ε, the entire drop can be subdivided into two zones: the outer zone 0 ≤ r < r0(t) that determines the macroscopic boundary of the drop, i.e., the coordinate at which h(r0(t),t) ≈ h0, and the inner zone r > r0(t) with the characteristic scale, L*, much smaller than the scale of the outer region (Figure 3.8): L* << r*. Thus, the integral in Equation 3.111 can be rewritten as follows:

()

∞

∞

r0 t

∫ r ( h − h ) dr = ∫ r ( h − h ) dr − ∫ r (h − h ) dr. 0

0

0

0

0

0

These integrals can be estimated using dimensional variables as ∞

r0

∫ r ( h − h ) dr ≈ h r 0

and

* *

0

∫ r ( h − h ) dr ≈ h L . 0

0 *

r0

This estimation shows that ∞

r0

∫ r ( h − h ) dr ∫ r ( h − h ) dr ≈ h 0

0

0

r0

h0 L* << 1 ; * r*

hence, the entire volume of the drop, with the accuracy with which the problem is being solved, is actually concentrated in the outer zone; i.e., we can set

()

r0 t

∫ r ( h − h ) dr ≈ 1. 0

0

© 2007 by Taylor & Francis Group, LLC

(3.114)

220

Wetting and Spreading Dynamics

In the zeroth approximation we conclude from Equation 3.109, setting ε = 0, we obtain the equation for the outer solution: ∂  1 ∂  ∂h   r  = 0. ∂r  r ∂r  ∂r  

(3.115)

The boundary condition (3.108) should now be written as h ( r0 , t ) ≈ h0 .

(3.116)

We obtain the solution of the problem (3.115), (3.111), (3.114), and (3.116), in the following form:

(

)

h ( r, t ) = 4 f 2 ( t ) 1 − ξ 2 + h0 ,

(3.117)

ξ = rf ( t ) ,

(3.118)

where

and f(t) is a new unknown function of time, which is determined in the following text; 0 ≤ ξ < 1. It follows from Equation 3.117 that the drop has the parabolic shape; the apparent radius of the spreading drop corresponds to ξ = 1, or

()

()

r0 t = 1 f t .

(3.119)

In order to obtain the inner solution of the problem (3.109), (3.108), which is valid close to the boundary of the spreading drop, we introduce new scales for the space variable and a new form of the unknown h(r, t): ξ − 1 = χ (t ) µ ,

(3.120)

h = h0 ψ ( µ ) , where χ(t) and ψ(µ) are new unknown functions subject to determination; µ is the new local variable inside the inner zone; χ(t) << 1. Note, in this section, h0 = const, which is different from Section 3.2. In the following text we present a slightly modiﬁed way of deducing the inner equation for the droplet proﬁle as compared with that in the previous section. This method is more appropriate for the problem under consideration. To do this we integrate Equation 3.109 with respect to the variable r: r

rh 3

∂  1 ∂  ∂h   ∂h  r  = − ε r dr . ∂r  r ∂r  ∂r   ∂t

∫ 0

© 2007 by Taylor & Francis Group, LLC

(3.121)

Kinetics of Wetting

221

We now calculate the integral on the right side of Equation 3.121, using the outer solution, as the integration takes place mostly across this region: r

∫

−ε r 0

∂r dr = − ε ∂t

ξ

ξ

∫f () 2

0

= −8ε

f ′′ f

ξ

 2 ff ′ 1 − ξ − 2 f ξrf ′  ξd ξ  ( )∫ ( )

∂h 4ε dξ = − 2 f t t ∂t

2

2

0

ξ

f ′′ ∫ ξ (1 − 2ξ ) d ξ = −4ε f (ξ 2

24

)

− ξ4 = −ε

0

f′ 2 ξ h − h0 . f3

(

)

Substituting the resulting expression, together with the transformation (3.120), into Equation 3.121, we obtain

ψ3

( ) ( )( ()

3 d 3ψ ε χ t f′ t = − ψ −1 . dµ3 h03 f5 t

)

(3.122)

Equation 3.122 should include only the variable µ; the latter gives the following requirement: ε χ2 f ′ = 1. h03 f 5

(3.123)

Hence, Equation 3.122 can be rewritten as ψ3

d 3ψ = ψ − 1. dµ 3

(3.124)

The matching of the inner (3.120) and outer (3.118) solutions is determined by condition

() ()

8f2 t χ t dψ =− . dµ h0

(3.125)

This means that the following condition must be fulﬁlled:

() ()

8 f 2 t χ t h0 = B = const ,

(3.126)

where B can no longer be selected arbitrarily but is determined by the matching condition. © 2007 by Taylor & Francis Group, LLC

222

Wetting and Spreading Dynamics

The two equations (3.123) and (3.126) determine the two unknown functions f(t) and χ(t) that have been introduced in the preceding text. The expressions (3.125) and (3.126), together with the boundary condition (3.108), give the following boundary conditions for Equation 3.124: dψ = −B , dµ

(3.127)

ψ

(3.128)

µ→∞

= 1.

The problem (3.124), (3.127), and (3.128) is investigated in Section 3.5. It is shown that the function ψ(µ) has the form of damped oscillations as µ → +∞,  3  ψ µ = 1 + C exp −µ / 2 sin  µ .  2 

( )

(

)

(3.129)

When we take conditions (3.123) and (3.126) into account, we obtain 3

 B f′  8  ε f 11 = −1 .

(3.130)

Now we can return to the question of the initial condition for Equation 3.109. The analysis that has been performed is valid only after the passage of a certain initial period of time, tin, at which the spreading regime described in this section is established. As we already mentioned, the duration of the initial stage of spreading when both Re and Ca numbers are not small is estimated in Section 4.1. The moment t = tin is taken as the initial moment of time. At this time, a parabolic proﬁle has been formed in the center of the drop; for the assignment of this proﬁle at the initial moment, all that must be known (apart from the condition of constancy of volume and initial radius r0(0)) is the height at the center, h(0,0). Selecting h(0,0)-h0 = h* at the scale on the z axis, we obtain the missing initial condition for Equation 3.109: h ( 0, 0 ) = 1 + h0 .

(3.131)

This condition determines the initial condition for the function f(t) that is the solution of Equation 3.130: f(0) = 1/2. This makes it possible to determine from Equation 3.130 and Equation 3.126 the functions f(t) and χ(t):  1 5 f t =  3 t /ε + 1 2 B 

()

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−1/10

,

(3.132)

Kinetics of Wetting

223 1/ 5

 B  5 χ t = h0  3 t /ε + 1 . 2 B 

()

(3.133)

The expression for χ(t) gives the scale of the inner zone, which should be small, that is, χ(t) << 1. It follows from Equation 3.133 that this condition is satisﬁed if  B  5 h0 t / ε + 1 2  B3 

1/ 5

25 ε . h05 B 2 5

<< 1, or t <<

(3.134)

This provides the required restriction on the duration of the spreading process. After that the droplet becomes too small and undistinguished from the ﬁlm. From Equation 3.119 and Equation 3.132 we ﬁnd an expression for the radius of the spreading drop:  5  r0 t = 2  3 t /ε + 1 B 

()

1/10

,

or in dimensional variables,   5 γh*3 r0 ( t ) = 2 r*  3 4 t + 1   B 3ηr*

0.1

.

(3.135)

From these expressions we obtain the time dependence of the dynamic contact angle at the drop boundary as  ∂h h  5 tgθ = − = *  3 t /ε + 1 ∂r r =r r*  B 

−3/10

.

0

Expressing the right side of the last equation in terms of the spreading velocity, U(t) = dr0 /dt, we arrive at  3ηU  tgθ = B   γ 

1/ 3

= B ( 3Ca ) . 1/ 3

(3.136)

It is possible to check that the dimensional combination appearing in Equation 3.110 for the parameter ε does not depend upon the selection of the initial moment of time. Indeed, as we already established previously, the characteristic values of © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

the quantities appearing in this combination vary with time in accordance with the laws r* ≈ r0(t) ≈ t 1/10, h* ≈ h(0, t) ≈ t –1/5, t* ≈ t, i.e., it remains constant with time: r*4/t *h *3 = const. Differentiating both sides of Equation 3.135 with respect to time, we ﬁnd that, at the initial moment, t = 0, dr0 1 γh 3 = 3 *3 . dt 3B ηr*

(3.137)

As r0 (0) = 2r*, taking as the characteristic value of velocity 2U* =

dr0 2r* = , dt t*

from Equation 3.137 we ﬁnd the characteristic value of spreading time: t* = (6Bηr*4/(γh *3 )). In Section 3.2 we showed that Equation 3.124 does not have any solutions satisfying the matching condition (3.127); therefore, in place of the matching of the outer (3.118) and inner (3.120) solutions, we require, as in Section 3.2, that they must be patched at a certain point: µ = –µ* :dψ/dµ = –B. We determine the patching point, µ*, in precisely the same way as in Section 3.2. The outer solution (3.118) obtained from Equation 3.109 with ε = 0 loses its meaning if its left-hand side becomes of the same order of magnitude as the right-hand side, i.e., when h3 = ε. We set as the patching point h3 = ε or, in view of (3.120), h03 ψ3(–µ*) = ε. Equation 3.124 as µ → –∞ has the asymptotic representation

()

ψ ξ ≈ 3 3 ξ ln1/ 3 ξ,

dψ 3 dψ ≈ − 3 3 ln1/ 3 µ ≈ 3 ln1/ 3 ξ ψ µ ≈ 3 3 µ ln1/ 3 µ , dξ dµ

( )

(see Section 3.2 for details), which gives the following equation for determining of the quantity B. All details are identical to those presented in Section 3.2. This results in

( )

B 3 exp B3 = 1/ 2h03 and ε = 1/ 2 B3 .

(3.138)

If we now set h* ≈ 0.1 cm and h0 ≈ 10–5 cm, then h0 ≈ 10–4; for B and ε we obtain the following values: B ≈ 2.753, ε ≈ 0.024. Correspondingly, µ* ≈ 1048 and ψ(–µ*) ≈ 2886. The calculated values of B and ε are very close to those found in the previous section for the spreading over a dry substrate. Therefore, in the case of complete

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Kinetics of Wetting

225

wetting, the preexponential constant is almost insensitive to the conditions in front of the spreading droplet. We will see a further evidence of this insensitivity in Section 4.1.

3.4 QUASI-STEADY-STATE APPROACH TO THE KINETICS OF SPREADING A simpliﬁed approach is suggested here to the solution of problems of liquid drop spreading. The approach consists essentially of assuming constancy of the drop spreading velocity, U, at each ﬁxed moment of time, t. This gives the possibility of determining the relationship U = f(r0), where r0 is the radius of the drop base. As U = dr0/dt, this equation can be used to obtain an expression for the spreading radius as a function of time. The applicability of the method has been demonstrated using examples of spreading on a solid substrate (with disjoining pressure acting in the vicinity of the apparent three-phase moving contact line), spreading on a prewetted solid substrate, and spreading under the inﬂuence of gravity and under the inﬂuence of applied temperature gradient [52]. In Section 3.1 through Section 3.3 a number of spreading problems have been solved. In the case of spreading of small drops over dry solid substrate (Section 3.2), the explicit expressions obtained for the drop radius and height and the dynamic contact angle as functions of time. These expressions do not include any ﬁtting parameters and depend solely on the hydrodynamic characteristics of the liquid, including the disjoining pressure isotherm, proved to be in good agreement with experimental data. This solution was limited to the case of complete wetting for disjoining pressure isotherms of the type Π(h) = A/hn, where A > 0 and n ≥ 2. Similarity solution of gravitational regime of spreading and spreading of very small liquid drops were obtained in Section 3.1, and spreading over prewetted solid substrate was solved in Section 3.3. In all cases, the solution included more or less sophisticated mathematical treatment. From an analysis of all of the relationships that have been found between the drop radius and time, we arrive at a hypothesis of a “quasi-steady-state nature” of the spreading process. According to this hypothesis, the characteristics of spreading are determined for the most part only by the instantaneous value of the spreading velocity. Let us ﬁrst examine the problem of spreading of a low sloped drop of a viscous liquid on a horizontal surface covered with a layer of the same liquid with a thickness h0 — that is, the same problem as in Section 3.3. The same notations as in Section 3.3 are used here. Let h(r,t) be the equation of the drop proﬁle; r is the radial coordinate, and t is the time. We use the characteristic scales; h* , r* , and t* , respectively. The following relationships are satisﬁed: h* >> h0 ; r* = r0(0)/2, where r0(t) is the radius of drop spreading; h* << r*. Then, as shown in Section 3.3, in dimensionless variables h → h/h*, r →r/r* , t → t/t* , and h0 → h0/h* , the spreading process is described by the differential equation (3.109), with conservation law (3.107) and boundary conditions (Equation 3.17’) and (3.108). © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

As was shown in Section 3.3, the condition ε << 1 is usually met; i.e., viscous forces are small in comparison with capillary forces. Now, in Equation 3.109, we use a new quasi-steady-state variable: ξ = r − r0 ( t ) ,

(3.139)

where r0(t) is the spreading radius. Let U = r0 (t ) be the spreading velocity; the overdot denotes the derivative with respect to time t. Using these notations and assuming that the liquid proﬁle depends on the new variable only — that is, h = h(ξ) — we obtain from Equation 3.39 that

εU

dh 1 d  3 d  1 d  dh    = ξ  ξh . d ξ ξ d ξ  d ξ  ξ d ξ  d ξ   

(3.140)

Setting ε = 0 in the equation, we obtain, in the same way as in Section 3.3, the outer solution of the problem:

(

)

h = C 1 − ξ 2 /r02 + h0 , where C is the integration constant, determined from conservation law (3.107). That gives C = 4/r02, whereas the outer solution is h=

(

)

4 1 − ξ 2 /r02 + h0 , r02

(3.141)

which coincides with the outer solution deduced in Section 3.3 (Equation 3.117). Equation 3.141 describes a parabolic proﬁle of the drop and is valid far from the apparent moving three-phase contact line. In the vicinity of the moving apparent three-phase contact line, ξ = 0, where the proﬁle of the outer solution (3.141) intersects the surface of the liquid ﬁlm, we introduce the inner variable as before: µ=

ξ − r0 , χ

( )

χ << 1 (3.142)

h = h0 ψ µ . Using the new inner variable and retaining in Equation 3.140 only the leading terms, we obtain

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Kinetics of Wetting

227

εUr03χ 3 d ψ d  3 d 3ψ  = ψ . 3 h0 dµ dµ  dµ 3 

(3.143)

The requirement of “self-similarity” with the “frozen” time t results in εUr03χ3 =1 h03 dψ d  3 d 3ψ  = ψ . dµ dµ  dµ3 

(3.144)

Integration of Equation 3.144 with the condition ψ → 1, µ → ∞ yields Equation 3.124. The condition of “matching” the outer solution (3.141) and the inner solution (3.124) at a certain point µ = –µ* has the form similar to that deduced previously (3.127): dψ dη

η=− η*

=−

8χ = − B = const . r02 h0

(3.145)

Now, using the ﬁrst equation (3.144) and (3.145), we obtain 3

 8 εU =   r0−9 .  B

(3.146)

We now take into account the fact that U = dr0 /dt. Then, from Equation 3.146 with the initial condition r0(0) = 2, we obtain 0 ,1

 5 t  r0 = 2  3 + 1 . B ε 

(3.147)

The law of spreading (3.147) coincides with that obtained previously in Section 3.3 by a more rigorous method. However (see the following text), the constant B that appears in the spreading law (3.147), determining the point of matching of the outer and inner solutions differs from that found in Section 3.3. We assume, as in Section 3.3, that at the point of matching, h3 = ε. In this case, the right and left sides of Equation 3.140 have identical orders. Using this condition we conclude that h03ψ 3 ( −µ* ) = h03ψ *3 = ε .

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(3.148)

228

Wetting and Spreading Dynamics

Taking as the initial condition for the velocity U(0) = 2 as in Section 3.3, we conclude from Equation 3.148 that the relation is identical to relation (3.138) in the previous section: ε = 1/(2B3). The solution of Equation 3.124 has the asymptotic form (see Section 3.2) 3

ψ 3 ≈ 3 η ln η (3.149)

3

 dψ   d η  ≈ −3 ln η , η → −∞.

Now, using Equation 3.145, Equation 3.148, and Equation 3.149, we conclude that

( )

ψ *3 = εU /h03 = 3µ*3 ln η* = B3µ*3 = B3 exp B3 . For the determination of the constant B, we can limit ourselves in Equation 3.148 to the initial moment of time, i.e., we can set U = U(0) = 2. Then the value of B satisﬁes the relation ship

( )

B6 exp B3 = 1/h03 .

(3.150)

Equation 3.150 is very similar to Equation 3.138, which determines the constant B. The latter equation was obtained by a rigorous method in Section 3.3, but it has a right side that is exactly twice as large. Taking as an example the same values as in Section 3.3: h* = 10–1 cm, h0 = 10–5 cm, and, consequently, h0 = 10–4, we obtain from Equation 3.150 the following values: B ≈ 2.781 and ε ≈ 0.023, whereas the more rigorous method gave B ≈ 2.753 and ε ≈ 0.024. As can be seen, the approximate method under consideration, which is based on a quasi-steady-state approach, gives results in this particular problem that are very little different from those obtained by the rigorous method. Let us pass on now to the case of spreading of a liquid drop over a dry surface in the case of complete wetting — that is, with the disjoining pressure in the following form: Π(h) = A/hn (A > 0, n ≥ 2). In this case, as was shown in Section 3.2, the spreading equation is given by Equation 3.63. We now change over, the same way as done previously, to a quasi-steadystate variable (3.139). This is substituted into Equation 3.63, which gives

εU

dh 1 d  3  d  1 d  dh   λ dh    ξh  = ξ − . n +1    dξ ξ dξ   d ξ  ξ d ξ  d ξ   h d ξ   

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(3.151)

Kinetics of Wetting

229

With ε = λ = 0 , we obtain the outer solution of Equation 3.151 similar to Equation 3.68, which is valid far from the moving three-phase contact line: h=

(

)

4 1 − ξ 2 /r02 . r02

(3.152)

In the vicinity of the point ξ = 0, which corresponds to the apparent threephase contact line, we carry out a replacement of variables in Equation 3.142, in which h0 is now a new unknown function to be determined. The requirement of self-similarity of Equation 3.151 results in the following conditions: εUr03χ = 1, h03 λr02χ =1 h0n+1

(3.153)

and the equation itself for the inner solution ψ(µ) describing the drop surface proﬁle in the vicinity of the moving apparent three-phase contact line takes the form dψ d  3  d 3ψ 1 dψ   = ψ  3 − n+1  dµ dµ   dµ ψ dµ   or, after integration, considering that ψ → 0, µ → +∞ ,  d 3ψ 1 dψ  ψ 2  3 − n+1 = 1. ψ dµ   dµ

(3.154)

Equation 3.154 was investigated in detail in Section 3.2 (see also the end of Section 3.1). The condition of matching of the inner and outer solutions, as discussed, has the form (3.145), and it leads, after taking into account that U = dr0/dt, to the expression (3.147) for the spreading radius r0 . Expressions for the unknown functions h0(t) and χ(t), determining the scale of the zone of the inner solution, can be found easily using Equation 3.153 and Equation 3.147 (see Section 3.2 for details). We focus below on determining the constant B, which, according to Equation 3.148, determines the point of matching of the solutions and, in view of (3.138), determines the magnitude of the small parameter ε.

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Wetting and Spreading Dynamics

Equation 3.154, as µ → –∞, has the previous asymptotic behavior (3.149). Hence, in the same way as previously, we deduce that

(

( )

ψ * = εU

1/ 3

)

h0 = B exp B3 / 3 .

(3.155)

Substituting into Equation 3.153 r0 = 2 and U = 2 (values corresponding to the initial moment of spreading), we now obtain

( )

h0n−1 = λ εU

2/ 3

= λB 2 .

After that, using Equation 3.155 and taking into account that ε = 1/B3, we obtain

( )

1 λB 2 B

−

1 n −1

(

)

= B exp B3 / 3 .

(3.156)

Equation 3.156 for the determination of the constant B differs from Equation 3.89, obtained by the more rigorous method in Section 3.2 only in the absence of the factor 1/ 3 2 on the left-hand side. Comparison of numerical results obtained using the quasi-steady-state approach, Equation 3.156, with those obtained in Section 3.2, Equation 3.89, for the examples that were considered in Section 3.2, give almost identical numerical values for both n = 2 and n = 3 for both B and ε. Thus, in this particular problem as well, the proposed quasi-steady-state method of investigation of spreading has led to results that are essentially not different from those obtained previously by a more rigorous method. Now let us apply the proposed method to the solution of the problem of liquiddrop spreading on a horizontal substrate under the inﬂuence of gravitational forces. Neglecting capillary forces and considering the spreading of a two-dimensional (cylindrical) drop (along the OX axis), we write the spreading equation in dimensionless form according to Section 3.1 as ∂h ∂  ∂h  = β  h3  , ∂t ∂x  ∂x 

(3.157)

where β = ρgh*3t* ηr*2; ρ is the density of the liquid; g is the acceleration of gravity. The conservation of volume, V, in this case has the form x0

∫

2 hdx = V , 0

© 2007 by Taylor & Francis Group, LLC

(3.158)

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231

where x0 is the spreading radius, and V is the volume per unit length of the spreading droplet. Selection h* =

V , 2 r*

results in x0

∫ hdx = 1,

(3.159)

0

and β = ρgV 3t* 8ηr*8 , x0(t) is now dimensionless radius of spreading. Now, performing the replacement of variables according to Equation 3.139, we obtain from Equation 3.157:

−U

dh d  dh  = β  h3  , dξ dξ  dξ 

or, after integration, h2

dh U =− . dξ β

Hence,  3U  h = −  β 

1/ 3

ξ1/ 3 .

Using the condition of conservation of volume, we obtain 0

1=

∫

− x0

3  3U  hd ξ =  4  β 

1/ 3

x04 / 3 .

Therefore, from this equation we conclude that U = x0 =

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3 β 4

( 3x ) , 4 0

232

Wetting and Spreading Dynamics

and the drop-spreading law has the following form: 1/ s  320  x0 = k (βt ) , k =   81 

1/ s

≈ 1.316 .

(3.160)

The shape of the drop surface is determined here by the equation h = 4 ( x0 − x )

1/ 3

( 3x ) . 4 /3 0

(3.161)

The spreading law (3.160) differs from the self-similar solution obtained in Section 5.5 (see also Reference 22) only in the value of the constant coefﬁcient: k = 1.411 instead of k = 1.316. Now, let the drop spreading take place under the inﬂuence of a temperature gradient only. We consider the spreading of a nonvolatile liquid droplet using a low slope approximation — that is, h* << r* — and only spreading of a twodimensional droplet is considered. The Navier–Stokes equation in the case under consideration takes the following form: ∂p ∂2 v = η 2x ∂x ∂z ∂p =0 ∂z

.

These equations give p = p( x ), vx =

1 ∂p 2 z + C1z + C2 , 2 η ∂x

(3.162)

with the following two boundary conditions: a nonslip condition on the solid substrate vx(0) = 0,

(3.163)

and the condition of applied tangential stress on the droplet surface, caused by the surface tension gradient, which in turn is caused by the applied temperature gradient: η

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∂vx ∂z

= z=h

∂γ d γ ∂T = = Λ. ∂x dT ∂x

(3.164)

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233

Note that, in the latter condition we assume, the thickness of the droplet is small enough, to neglect the temperature variation through the vertical cross section of the droplet. Hence, the temperature is equal to the temperature of the solid support. Note that, usually, dγ < 0; dT that is, the liquid–air interfacial tension decreases with temperature. In the following text we consider the spreading of the droplet from the cold side in the center of the droplet to the hotter part of the solid substrate under the action of the constant temperature gradient. It is also assumed that in the range of temperature under consideration, the derivative, dγ/dT, is a negative constant. Hence, under the above assumption, Λ < 0 and remains constant. Using the latter two boundary conditions, we conclude from Equation 3.162 that vx =

 Λ 1 ∂p  z 2 − zh  + z .  η ∂x  2  η

(3.165)

The governing equation results from the integration of the continuity equation is ∂h ∂ =− ∂t ∂x

h

∫ v dz . x

(3.166)

0

Substitution of expression for the velocity according to Equation 3.165 into the governing equation (3.166) yields ∂h 1 ∂  h 3 ∂p Λh 2  = − , ∂t η ∂x  3 ∂x 2 

(3.167)

with the conservation law according to Equation 3.158. The pressure inside the spreading droplet is given by Equation 3.5. However, in the following text we concentrate on the spreading under the action of the temperature gradient only. That is, we assume that 2h* p* << Λ . r* Here we consider the negative values of the constant Λ. © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

In this case Equation 3.167 becomes ∂h ∂h 2 , =α ∂t ∂x

(3.168)

where α=

Λ h*t* . 2 ηr*

After the replacement of the variable according to Equation 3.139, we obtain dh dh 2 , =α dξ dx

(3.169)

Uh = αh 2 + C .

(3.170)

U or, after integration,

In the situation under consideration, the drop surface has an abrupt change at the point x = x0 (ξ = 0) corresponding to the liquid propagation front. Following the method described in Chapter 5, Section 5.6, we obtain the condition that must be satisﬁed by the solution of Equation 3.168 at the moving front. We integrate Equation 3.168 over x from x– = x0 – δ to x+ = x0 + δ, where δ is a small value: x+

∂h

∫ ∂t dx = α ( h

2 +

)

− h−2 .

x−

Let us calculate d dt

x+

x+

∂h

∫ hdx = x ( h − h ) + ∫ ∂t dx . +

0

−

x−

x−

Hence, x+

∫

x−

∂h d dx = − ∂t dt

x+

∫ hdx + x (h − h ) , 0

+

−

x−

where h+ = h( x + ), h− = h( x − ) are thickness of the droplet just in front and behind the moving front. © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

235

Taking the limit in the equation at x− → x0 , x+ → x0 (i.e., δ → 0), we obtain

(

)

x0 ( h+ − h− ) = α h+2 − h−2 . In front of the moving droplet the solid substrate is dry; that is, h+ = 0. Hence, from the preceding equation we conclude that x0 = αh− .

(3.171)

Equation 3.171, upon substitution into Equation 3.170, gives C = 0, and thus the drop has the form of a ﬂat step (pancake) with a height h = U /α ,

(3.172)

and a radius of the moving front x0(t). Equation 3.172 shows that the thickness of the ﬂat drop is a function of time only. From conservation law (3.159) we conclude now that

1=

x0

0

0

− x0

∫ hdx = ∫ hd ξ = Ux /α . 0

Taking into account U = x0 = α/x 0 and h = 1/x0, we ﬁnally get the spreading law as x0 = 2αt .

(3.173)

Summarizing the results obtained in this work, we can say that the proposed approximate quasi-steady-state method for solving spreading problems gives a satisfactory accuracy of solution in all of the cases examined in this section. The method itself is mathematically simple, and it offers a means for reducing a problem of complex nonlinear partial differential equations to the solution of ordinary differential equations. The quasi-steady-state approach that we have examined in this section can be used for the investigation of more complex spreading problems.

3.5 DYNAMIC ADVANCING CONTACT ANGLE AND THE FORM OF THE MOVING MENISCUS IN FLAT CAPILLARIES IN THE CASE OF COMPLETE WETTING In this section we shall analyze how both the disjoining pressure and the width of the capillary inﬂuence the dynamic advancing contact angle of the meniscus © 2007 by Taylor & Francis Group, LLC

236

Wetting and Spreading Dynamics

2H

re U = 0 1

2

ξ1

he x he 3

ξ1

he

r U1

he

ξ2

x

U2

θd

(a)

r

x

θd (b)

(c)

FIGURE 3.9 Schematic presentation of the proﬁle of the meniscus in a ﬂat capillary. (a) at equilibrium, (b) velocity of motion below the critical velocity; ξ1, the only minimum on the liquid proﬁle; (c) velocity of motion is above the critical velocity, ξ1 and ξ2 are thickness of the ﬁrst minimum and maximum on the liquid proﬁle; (1) the spherical meniscus, (2) the transition zone, (3) ﬂat equilibrium ﬁlm.

of a liquid, completely wetting the substrate. For simplicity we shall consider that the meniscus moves from an equilibrium position in a ﬂat capillary. The effect of both the disjoining pressure and the width of the capillary on the dynamic contact angle is investigated, following Reference 53. The problem of the form of the advancing meniscus and the dynamic contact angles have been considered earlier [23–25], neglecting the disjoining pressure of the thin layer of liquids and limiting the discussion only to capillary forces. However, in the close vicinity of the moving apparent three-phase contact line, the thickness on the liquid is so thin that the inﬂuence of disjoining pressure becomes signiﬁcant. The thickness of the ﬁlm on the walls of the capillary ahead of the advancing meniscus was assumed to be arbitrary in Reference 23 to Reference 25. This means that, up to the start of the ﬂow, the system is not in equilibrium, and that, consequently, there is ﬂow from the meniscus to the ﬁlm or the contrary, not connected with the dynamics of the meniscus. These ﬂows were also neglected in solution of the problems in Reference 23 to Reference 25. As has been shown earlier (Chapter 2, Section 2.2), in a state of equilibrium (Figure 3.9a) the capillary pressure of the meniscus, Pe , is connected with the isotherm of the disjoining pressure of ﬂat wetting ﬁlms, Π(h), by the following relationship: ∞   Pe =  γ + Πdh      he

∫

(H − h ) = Π , e

e

(3.174)

where γ is the surface tension of the liquid, H is the half-width of the capillary, and he is the thickness of the equilibrium ﬁlm on the solid surfaces corresponding to Π(he) = Pe. The relative pressure of the vapor, p/ps , above the ﬁlms and the meniscus related to the equilibrium pressure, Pe , according to Equation 1.7 in Chapter 1. © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

237

In sufﬁciently wide capillaries (H >> he), the meniscus in a central region has a spherical shape with a constant radius of curvature re = γ/Pe . Below we solve the problem for the case of complete wetting, that is, for isotherms of the disjoining pressure of the type Π = A/h n > 0 ,

(3.175)

where n ≥ 2, and A is the constant of the surface forces (Hamaker constant in the case n = 3). In Figure 3.9, schemes illustrating the proﬁles of the meniscus in a ﬂat capillary in a state of equilibrium (a) and with different rates of motion (b, c) are presented. In Chapter 2, Section 2.4, we showed that the radius of curvature of an equilibrium meniscus re < H, and in this case it is equal to re =

γ   n   = H −  h .  h − 1  e  Pe 

The form of the equilibrium proﬁle of the liquid in the transitional zone between the meniscus and the ﬁlm for isotherms of the type (175) was investigated in Chapter 2, Section 2.4. Let us consider the motion of the advancing meniscus in a ﬂat capillary (Figure 3.9b and Figure 3.9c) from a state of equilibrium (Figure 3.9a). The zone of the ﬂow in which the main hydrodynamic resistance is exerted takes in a region of the thicknesses of the layer h(x) above the surface of the substrate on the order of he. x-axis is directed along the capillary axis (Figure 3.9). According to the Introduction to this chapter, the proﬁle of the moving meniscus (Figure 3.9b and Figure 3.9c) can be subdivided at small capillary number, Ca << 1, in two regions: the outer region, where the meniscus has a spherical shape of an unknown radius of curvature, r, which is the function of the velocity of motion (or the same, capillary number, Ca), and the inner region in between the spherical meniscus and the initial equilibrium ﬂat ﬁlm. In the latter region, according to our estimations in the Introduction to this chapter, the curvature of the liquid proﬁle is small and a low slope approximation can be used. In the case under consideration, the equation of spreading (3.62) from Section 3.2 should be rewritten as ∂h γ ∂  3 ∂h nA ∂ h  =− h − ,  ∂t 3η ∂ x  ∂x γ h n − 2 ∂ r  with the boundary condition h → he , x → ∞ . © 2007 by Taylor & Francis Group, LLC

238

Wetting and Spreading Dynamics

At the steady-state motion we can introduce a new coordinate system moving with the meniscus y = x – Ut, which results in

U

dh γ d  3 dh nA d h  = h − , dy 3η d y  dy γ h n − 2 dy 

(3.176)

with the boundary condition h → he , y → ∞ .

(3.177)

Integration of Equation 3.176 with boundary condition (3.177) yields γh 3  nA  h′′′ − n+1 h′  = U ( h − he ) ,  γh 3η  

(3.178)

where ′ means differentiation with y. Let us introduce the following dimensionless variables ξ = h /he ; z = y/y*, ξ′ and ξ′′′ means differentiation with z,

(

y* = he γ /3ηU

)

1/ 3

=

he (3Ca)1/ 3

and the only one dimensionless parameter, which characterizes the intensity of the disjoining pressure action:

α=

nA1/n Pe( n−1)/n 2/3 . γ 1/ 3 ( 3ηU )

(3.179)

It is important to note that using these scales we get the following estimation of the derivative in the ﬂow zone: h′ ~

he = ( 3Ca )1/ 3 << 1. y*

The low slope condition is really satisﬁed in the ﬂow zone in a vicinity of the apparent moving three-phase contact line as we predicted in the Introduction to this chapter. It should be remembered that in the case under consideration, Ca ~ 10–6; hence, (Ca)1/3 ~ 0.01 << 1. Substituting the aforementioned dimensionless variables into Equation 3.178 results in © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

239

ξ 3ξ′′′ − αξ′ ξ 2−n = ξ − 1.

(3.180)

In Appendix 5 we show (see Equation A5.9) that Equation 3.180 has the proper asymptotic behavior at z → –∞. In this sense, the problem under consideration is completely mathematically correct: it is possible to match asymptotic expansions in the outer region (meniscus) and inner region in a vicinity of the moving apparent three-phase contact line. In the following text we undertake the matching procedure numerically. Before passing on to the results of a numerical calculation according to Equation 3.180, let us analyze the equation. Considering the behavior of the dimensionless thickness, ξ, far from the meniscus (i.e., at 1 – ξ << 1), introducing υ = ξ – 1 and linearizing Equation 3.180, we conclude that υ′′′ − αυ′ = 0 .

(3.181)

We seek the solution of Equation 3.181 in the form υ = exp(λx ). Substitution of this expression into Equation 3.181 yields λ 3 − αλ − 1 = 0 .

(3.182)

The discriminant of Equation 3.182, Q = (1/4) – (α/3)3, can be either positive or negative. If Q > 0 (i.e., with α < αc ≅ 1.89), Equation 3.182 has one real positive root and two conjugate complex roots, whose real parts are negative. This corresponds to the presence of damped waves ahead of the moving meniscus. Such a situation (Figure 3.9c) is possible with U > Uc (with A and H = const), or with A < Ac (with U and H = const) or, ﬁnally, with H > Hc (with U and A = const). With a decrease in α (with α < αc), the imaginary part of the roots rises monotonically, and the real part decreases, which corresponds to an increase in the amplitude and a decrease in the length of the surface waves. At Q < 0 (i.e., with α > αc), Equation 3.182 has two negative real roots: –λ1 and –λ2 (λ1 > λ > 0) and one positive real root, λ3 > 0. These roots, as a function of α, have the following properties: λ1 < α , λ2 < α . At α → ∞, λ2 → α and λ2 ≈ 1/α. It follows from this that at α > αc, the function ξ(x) has a single minimum at the point z ≅ (2/ α ) ln α , which corresponds to the situation illustrated in Figure 3.9b. Thus, proﬁles of this type, as in Figure 3.9b, are realized with α > αc, i.e., with rather large values of A, and/or at a small velocity of motion of the meniscus U, or in narrow capillaries. With an increase in the velocity of the ﬂow or a decrease in the value of A such that α becomes less than αc, wavy ﬁlms are formed ahead of the moving meniscus, as in Figure 3.9c. Thus, if the effect of surface forces is neglected, (A = 0) — that is, α = 0 — the solution of Equation 3.8 can only have wavy proﬁles of the ﬁlm in front of the moving meniscus, as has been obtained earlier in Reference 24. The effect of surface forces results (at α < αc) in damping of the © 2007 by Taylor & Francis Group, LLC

240

Wetting and Spreading Dynamics 1.0 3 2

4

4 3 0.9 H/r

H/r

1.0 0.9

0.8

1

–4

1 0.8

–6

–8

2

–2

–3

ᐉn (Ca)2/3

ᐉn (Ca)2/3

(a)

(b)

FIGURE 3.10 Dependence of the ratio H/r on the capillary number, Ca = Uη/γ. (a) Disjoining pressure isotherm Π(h) = 10–7/h2, H = 10–2 cm, he = 360 Å (1); H = 1.25.10–3, he = 125 Å (2); H = 10–5 cm, he = 11 Å (3). (b) Disjoining pressure isotherm Π(h) = 10–14/h3, H = 10–2 cm, he = 150 Å (1); H = 1.25.10–3 cm , he = 74 Å (2); H = 1.25.10–5 cm, he = 16 Å (3). The dotted line 4 relates to the Friz equation [24].

waves and to an increase in the wavelength (still at α < αc) ahead of the moving meniscus. The latter consideration shows the qualitative effect of the inﬂuence of disjoining pressure of thin layers of liquids on hydrodynamics of moving of menisci. Figure 3.10a gives the numerically calculated values of H/r (calculated according to Equation 3.180) as a function of the parameter (Ca)2/3 for different values of the width of the capillaries H. The values of r are located along a section of constant curvature, adjacent to the zone of the ﬂow, appearing with a rise in the value of the thickness, h, but still in the region of the low sloped proﬁle. The transition to this section corresponds to the condition d 2 h/dy2 = γ/r = const. At H/r < 1, the values of H/r = cos θd, where θd is the dynamic contact angle (Figure 3.9b, c). Calculations were made for the following parameters A = 10–7 dyn and n = 2, characteristic for the value of β ﬁlms of water on the surface of quartz [26]. The corresponding value of γ was taken as equal to 72 dyn/cm, and the viscosity η = 0.01P. Figure 3.10b gives similar results for another isotherm Π(h), characteristic for wetting ﬁlms of nonpolar liquids on a solid dielectric [26]: A = 10–4 erg, n = 3, γ = 30 dyn/cm, η = 0.01P. Figure 3.10 shows that the calculated dependences of cos θd on Ca (curves 1–3) are similar to those calculated using the approximate Friz equation [24]: tanθd ≅ 2.36 Ca (curve 4). However, as distinct from the Friz calculations, our calculations show a dependence of θd on the width of the capillary. This becomes particularly noticeable in “narrow capillaries,” when H < 103cm, where the © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

241 ξ2

ξ1

1.04

1.0 0.9

3

2

1

1.02

3

1

2

1.00

0.8 –4

–6

–8

–10

–4

–6

ᐉn (Ca)2/3/ᐉ (a)

(b)

FIGURE 3.11 Dependences of the thickness of the ﬁrst minimum, ξ1 = h1/he (Figure 3.9c), (a), and the ﬁrst maximum, ξ2 = h2/he (Figure 3.9c), (b) on Ca, obtained for values of H = 10–2 (1), 10–3 (2), and 10–5 cm (3). The parameters of the isotherm: A = 10–5 dyn, n = 2.

thickness of the surface ﬁlm becomes less than 100 Å. In the general case, an increase in H, as can be seen from Figure 3.10, leads to a decrease in cosθd , i.e., to a rise in θd. Figure 3.11 presents the dependence on Ca of the characteristic thickness inside the ﬂowing zone: ξ1 = h1/he is the point of the ﬁrst minimum, and ξ2 = h2/he is the point of the ﬁrst maximum. As can be seen from these curves, the values of ξ1 fall with an increase in the rate of motion of the meniscus. Under these circumstances a lowering of the thickness of the ﬁlm near the moving meniscus is observed at lower rates of motion; the lower the rate, the wider the capillary. Thus, in a capillary H = 10–2 cm, the thinning of the ﬁlm near the meniscus starts with Ca > 10–6 (for water, at U > 10–2 cm/sec). The relative height of the ﬁrst maximum of ξ2, on the contrary, rises with an increase in Ca, i.e., with a rise in the rate of motion of the meniscus. However, the appearance of convexity of the ﬁlm is observed with considerably greater velocity U than the appearance of concavity. It is important to note that, with a further rise in the value of the velocity of the meniscus, U, the values of ξ1 and ξ2 are stabilized, i.e., the ﬁrst wave practically does not change its form; an increase in U is accompanied by a change in the form of the second, third, etc. waves, propagating ahead of the meniscus. Under these circumstances, the value of α, corresponding to a transition to a wavy proﬁle, found from the values of (3Ca)2/3 with ξ2 = 0 by extrapolation of the curve of ξ2 (Ca) in Figure 3.11b, were found to lie in the interval from 1.5 to 2.0, i.e., they were close to the values of αc ≅ 1.89, obtained with a linearized preliminary analysis of Equation 3.180. As can be seen from Figure 3.11, the form of the wavy ﬁlm ahead of a moving meniscus depends more strongly than cosθd on the width of the capillary, H, and the thickness of the equilibrium ﬁlm, he, in front of the moving meniscus. An increase in the value of the constant A, signifying an increase in the effect of the surface forces, leads (with H = const) to a rise in ξ1 and to a lowering of ξ2, with identical values of Ca. Thus, the effect of surface forces, as followed from the © 2007 by Taylor & Francis Group, LLC

242

Wetting and Spreading Dynamics

preliminary analysis of Equation 3.180, leads to damping of the waves and makes the proﬁle of the ﬁlm of the moving meniscus smoother. The calculations made offered the possibility of evaluating the effect of surface forces on the dynamic contact angles θd and the proﬁle of the ﬁlm ahead of the moving meniscus for isotherms of the disjoining pressure, corresponding to complete wetting.

APPENDIX 5 Asymptotic behavior of solution of y3

d 3y = y − 1 at y → ∞ . dx 3

At sufﬁciently big thickness we can neglect the disjoining pressure action in Equation 3.180. Hence, we come to the following differential equation

y3

d 3y = y − 1, dx 3

(A5.1)

with the following boundary conditions y → +∞, x → –∞.

(A5.2)

According to the boundary condition (A5.2), Equation A5.1 asymptotically can be written as

y2

d 3y = 1. dx 3

(A5.3)

Let us introduce a new unknown function w(y) in Equation A5.3 w ( y) =

2

1  dy    . 2  dx 

(A5.4)

Hence, dw d 2 y = , dy d x 2

d 2w 1 d 3y . 2 = − dy 2w d x 3

Substitution of this expression into Equation A5.3 results in

y2

© 2007 by Taylor & Francis Group, LLC

d 2w 1 . 2 = − dy 2w

(A5.5)

Kinetics of Wetting

243

Let us introduce a new variable in this equation: dw = p(w ), dξ

ξ = ln y .

(A5.6)

Using the new variable in Equation A5.6, we arrive at

p

dp 1 −p=− . dw 2w

(A5.7)

Solution of Equation A5.7 can be expressed via cylindrical functions. However, for the investigation of the asymptotic behavior of Equation A5.2, we limit ourselves by considering only the main power terms in asymptotic expansion: p = Cw k .

(A5.8)

Substitution of this expression into Equation A5.7 results in C 2 w 2 k −1 = Ckw k −

w −1/2 . 2

Equalizing different exponents in the preceding equation, we conclude that only three options are available for the exponent k: (1) k = 1, (2) k = 1/4, (3) k = 1/2. At k = 1 we conclude from Equation A5.7: C = 1, hence, p = w. Substitution of this expression into Equation A5.6 gives w′ = w, or w = 2C1y, where C1 > 0 is the integration constant. The next step is the substitution into Equation A5.4, which results in y′ = 4C 1 y . The solution of this equation is y = C1 ( x + C2 )2 ,

(A5.9)

where C2 is a new integration constant. Note that the constant C1 can be only positive. We use the asymptotic solution (A5.9) in the current section 3.5 and in the next section 3.6. The second solution (at k = 1/4) does not satisfy the requirement (A5.2). Let us consider now the last, third exponent k = 1/2 , which results according to Equation A5.8 in p=

© 2007 by Taylor & Francis Group, LLC

1 −1/2 w . 2

(A5.10)

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Wetting and Spreading Dynamics

Substitution of this expression into Equation A5.10 and Equation A5.6 results in w=

1 ( 3 ln y )2/3 . 2

Casting the latter expression into Equation A5.4 yields: dy = −31/ 3 ln1/ 3 y dx

(A5.11)

Integration of the equation results in

∫ ln

dy 1/ 3

y

= −31/ 3 x .

Integration by parts gives

∫ ln

dy 1/ 3

y

=

y 1/ 3

ln

y

+

1 3

∫ ln

dy 4 /3

y

.

At y → + ∞ the following inequality holds: ln 4 / 3 y >> ln1/ 3 y . Using this the equation can be simpliﬁed as y = −31/ 3 x . ln1/ 3 y

(A5.12)

Equation A5.12 provides a possibility for deducing an explicit asymptotic behavior of function y(x):

(

)

y = −31/ 3 x ln1/ 3 y = 31/ 3 x ln1/ 3 31/ 3 x ln1/ 3 y . Taking into account that x >> ln1/ 3 y , we arrive at y = 31/ 3 x ln1/ 3 x ,

x → − ∞.

(A5.13)

Direct differentiation of Equation A5.13 gives dy = 31/ 3 ln1/ 3 x , dx

x→ −∞.

We used this asymptotic behavior in Section 3.2. © 2007 by Taylor & Francis Group, LLC

(A5.14)

Kinetics of Wetting

245

3.6 MOTION OF LONG DROPS IN THIN CAPILLARIES IN THE CASE OF COMPLETE WETTING By now the reader is expected to be familiar with the fact that for nonpolar liquids we can use disjoining pressure isotherms Π(h) = A/hn, A > 0, n ≥ 2 (h is the thickness of the ﬁlm). Such isotherms pertain to the case of complete wetting. In this section we consider the motion of long oil drops or air bubbles in thin capillaries [54–56]. Let us consider the motion of a long drop or bubble in a thin capillary (gravity action is neglected) (Figure 3.12). Under the action of applied pressure difference p– – p+ > 0, the drop or bubble moves from left to right with velocity U to be determined as a function of the applied pressure difference. Note that the velocity U is different from the average Poiseulle velocity because the drag force in the system presented in Figure 3.12 is different from the drag force in the same capillary completely ﬁlled with liquid 1. We consider below a relatively slow motion, when the capillary number, Ca =

U η1 << 1 , γ

where η1 is the viscosity of liquid 1 in the capillary, and γ is an interfacial tension, which can be substantially different from the liquid–air interfacial tension. The latter interfacial tension can be used only in the case of the motion of an air bubble. In the following text we show that the viscosity of liquid 2 inside the drop or bubble, η2, doe not play any signiﬁcant role and can usually be omitted. However, the interfacial tension in the case of the liquid bubble is still very important as well as its difference from the liquid–air interfacial tension. r L ι

L– p–

x

0 2

1 F′

L+

F

E′ E D

p+

h0 C

B B′ A

A′

FIGURE 3.12 Schematic presentation of a motion of a drop or bubble in a capillary under the action of applied pressure difference p– – p+ > 0, that is, the motion from left to right. (1) liquid in a capillary of length L, (2) drop or bubble of length . L– and L+ are parts of the capillary without the drop or bubble, F′F and AA′ are parts of the capillary where Poiseulle ﬂow takes place, E and B are positions of the end of spherical menisci (curvature of meniscus EE′ is bigger than the curvature of the meniscus BB′). ED and CB are transition zones from menisci to the region of DC of the ﬁlm of constant thickness, h0. © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

Let us estimate the velocity of the motion in the case Ca ~ 1: if η1 ≈ 10–2P, γ ≈ 72 dyn/cm, then the corresponding velocity U ≈ 7200 cm/sec = 72 m/sec. Such huge velocity can be achieved in a sufﬁciently thin capillary with radius less than 0.1 cm only under very special conditions. That means we can safely consider Ca << 1. As we already discussed in the Introduction to this chapter, 1. The advancing meniscus EE′ has a constant curvature up to the zone of the ﬂow, ED, where the low slope approximation is valid. The curvature of the advancing meniscus EE′ is a function of the capillary number Ca to be determined, 2. The same is valid for the receding meniscus BB′ and the zone of ﬂow CB. It is obvious that the curvature of the receding meniscus BB′ is smaller than the curvature of the advancing meniscus EE′. The author of reference 23 had studied the motion of a long drop or bubble in a capillary without allowance for the effect of disjoining pressure. The following dependence of ﬁlm thickness, h0, on drop velocity U has been deduced:

( )

h 0 = 1.337 R Ca

2/ 3

,

(3.183)

where R is the radius of the capillary. Relation (3.183) yields a zero ﬁlm thickness at Ca → 0. However, as we already understood in Chapter 2, this is impossible because the ﬁlm thickness should tend to the equilibrium thickness, he , at Ca → 0. This equilibrium value of the ﬁlm thickness should be found from the following condition (see Chapter 2) Π(he) = Pe, where Pe is the excess pressure in the drop. Hence, ignoring the action of disjoining pressure results in a wrong prediction of the ﬁlm thickness at low capillary numbers. However, according to Equation 3.183, the ﬁlm thickness increases unboundedly at high capillary numbers. If h0 > ts, where ts is the radius of surface forces action, then Equation 3.183 should become valid. Hence, at low capillary numbers we should expect a substantial deviation from the prediction according to Equation 3.183, and at high capillary numbers, Equation 3.183 should hold asymptotically. Our calculations (see Figure 3.13) conﬁrm the suggested dependency of ﬁlm thickness on capillary number. In the present section, we obtain the main characteristics of the motion of a drop under an applied pressure gradient, taking into account the disjoining pressure action. We examine the motion of a long drop of an immiscible ﬂuid 2 with a length (Figure 3.12) inside the cylindrical capillary of radius R ﬁlled with ﬂuid 1. The motion is axisymmetric, and we introduce a coordinate system connected with the drop. The x-axis coincides with the axis of the capillary. Let be the length of the drop, L+ and L– the lengths of the drop-free sections of the capillary, and L the overall length of the capillary, (L = L+ + L– + ). The pressure difference, p+ – p– is applied at the ends of the capillary, where p– > p+. We examine the © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

247

h0/R . 104

10

3 5

2 1 0 1

2

3

(3Ca)1/3 . 102

FIGURE 3.13 Dependency of the constant ﬁlm thickness, h0, inside the zone DC (Figure 3.12) on capillary number, Ca. (1) according to Reference 23, when the disjoining pressure action was ignored, (2) disjoining pressure A/h3, (3) disjoining pressure A/h2.

steady motion of the drop along the capillary in the positive direction of the x-axis at the velocity U to be determined. As before, we restrict ourselves to the cases of low Reynolds and capillary numbers. We also assume that the drop is long, that is, R/ << 1. Let η1 and η2 be the viscosities of ﬂuids 1 and 2, respectively. At η1 ≈ 10–2 P, γ ≈ 30 dyn/cm, U ≈ 1 cm/sec, we have Ca ≈ 3·10–4 << 1. As the value of U ∼ 1 cm/sec can be taken as the upper bound of the velocity of the drop, we will assume that the condition Ca << 1 is always satisﬁed. We will divide the ﬂow ﬁeld in the capillary into regions following (A′A and FF′) those parts of the capillary not containing the drop, where Poiseuille ﬂow is realized; AB and EF are spherical menisci at the ends of the drop; CD is the region with a constant ﬁlm thickness, h0 , to be determined; and BC and DE are transitional regions from the constant-thickness ﬁlm to menisci. In Appendix 6 we show that the ﬂow in both transition zones, ED and CB, and the zone of the constant-ﬁlm thickness, DC, almost coincides with Equation 3.178 deduced in the Section 3.5. It is assumed that the viscosity ratio of liquids – 1 and 2 is of the order of 1, i.e., η = (η2/η1) ~ 1. Equation 3.178, however, does not include the viscosity of the liquid inside the drop or bubble, η2. The proof of that is given in Appendix 6. Here, we give a qualitative explanation. As we already showed in the Introduction to this chapter, in zone EB the low slope approximation is valid. Hence, the equality of tangential stress at the liquid–liquid interface, h(x), has the following form η1

© 2007 by Taylor & Francis Group, LLC

∂v1 ∂x

= η2 h( x )

∂v2 ∂x

, h( x )

248

Wetting and Spreading Dynamics

where v1 = v2 ~ U are velocities at the liquid–liquid interface, h(x). Let us estimate the ratio of the tangential stress from the drop or bubble side to the tangential stress from the ﬁlm side η2

∂v2 ∂x

η1 h( x )

 U ∂v1 ~  η2  ∂x h ( x )  R 

 U h  η1 h  = η R << 1 .

This estimation shows that the ﬂow inside the drop or bubble can be safely ignored. Using the low slope approximation, the curvature of the interface, h(x), inside zone EB (Figure 3.12) can be written as K (x) =

1 d2 h . + d x2 R − h

(3.184)

Using this expression, we conclude from Equation 3.178 that

()

 d 3h 1 d h Π′ h d h  + h3  3 +  = 3Ca h − h 0 . 2 γ dx ( R − h) d x  dx

(

)

In this equation we neglect the small difference between the actual effective disjoining pressure, Πef =

R Π(h ) , R−h

where Π(h) is the disjoining pressure of the corresponding ﬂat ﬁlms. See the justiﬁcation in the following text. In the ﬂow zone EB (Figure 3.12), the thickness of the ﬁlm is much smaller than the capillary radius; hence, 2

h  ( R − h )2 = R 2  1 −  ≈ R 2 ,  R and the preceding equation can be rewritten as

()

 d 3h 1 d h Π′ h d h  h3  3 + 2 +  = 3Ca h − h 0 . γ dx R dx  dx

© 2007 by Taylor & Francis Group, LLC

(

)

(3.185)

Kinetics of Wetting

249

In the following text we consider only the case of complete wetting, where we use the following isotherms of disjoining pressure: Π (h) =

A , n = 2, 3, 4 . hn

Let us introduce the following dimensionless values in Equation 3.185: z = x/x*, ξ = h/h0, where the characteristic scale x* is the characteristic dimension of the regions BC and DE. In this section, our choice of this scale is different from the choice in the previous section because we are going to concentrate on a transitional regime of ﬂow from equilibrium to a relatively high velocity. That is why we select this scale equal to the length of the transition zone at the equilibrium (see Chapter 2, Section 2.3, Equation 2.49): x* = h e R . It is important to notice that using these scales we get the estimation of the derivative in the ﬂow zone, h′ ~

he = x*

he = he R

he << 1, R

and the low slope condition is really satisﬁed in the ﬂow zone in a vicinity of both advancing and receding menisci. The choice also shows that h′′′ ~

he = h ( e R )3/2

1 he R

3

>>

1 h h = e h′ ~ 2 e R2 R he R R

1 he R 3

,

and, hence, the second term on the left-hand side of Equation 3.185 can be omitted. Therefore, Equation 3.185 can be rewritten as d 3ξ 1 dξ ξ −1 − β n +1 =ε 3 , d z3 ξ dz ξ

(3.186)

where β=

nR A , γ h0n

ε = 3Ca ( R / h 0 )

3

2

.

The case β = 0 corresponds to the case of ﬁltrative motion at a high velocity, examined in Reference 23, where the effect of disjoining pressure was ignored. Equation 3.186 has the solution ξ ≡ 1 corresponding to a ﬁlm of constant thickness in the zone CD. This solution should ensure matching of this zone with the surfaces of the spherical menisci in regions BB′ and EE′ at the ends of the drop. © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

The matching conditions for the leading end of the drop have the form d 2ξ R , z → +∞ 2 → dz R+ ξ → 1,

(3.187)

z → −∞

while for the trailing end d 2ξ R → − , z → −∞ 2 dz R ξ → 1,

(3.188)

z → +∞.

Satisfaction of conditions (3.187) and (3.188) is assured by the existence of asymptotic solutions of Equation 3.186 having the form (see Appendix 5): η≈

A ± ( Ca ) 2 z + B ± ( Ca ) , 2

z → ±∞ ,

(3.189)

(the + sign denotes the leading meniscus, while the – sign denotes the trailing meniscus). The parameters A± and B±, dependent on the capillary number, Ca, are determined in the following using the numerical solution of Equation 3.186. Comparing Equation 3.189 with Equation 3.187 and Equation 3.188, we conclude that R ± = R /A± h0 =

(

) = R ( A − 1) .

R A+ − 1

−

A+ B +

A− B −

(3.190)

Assuming in Equation 3.186 that ε = 0 (or Ca = 0), we obtained an equation for the transitional zone between the ﬁlm and the meniscus for the equilibrium case, which was examined in Chapter 2, Section 2.4. In the case of complete wetting, the meniscus has the radius Re± = R − (n /n − 1)he , where he is the thickness of the equilibrium ﬁlm determined from the condition A = 2γ /Re . hen Comparing these relations with those in Equation 3.190, we conclude that © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

251

lim A± (ε) = 1 + ε→0

n he n −1 R

(3.191)

n lim B (ε) = . ε→0 n −1 ±

Figure 3.13 shows the dependence of ﬁlm thickness h0 on the dimensionless velocity (3Ca)1/3, whereas Figure 3.14 shows the dependences of the parameters A±, B±, and the radii of the leading R+ and trailing R– menisci on dimensionless velocity (3Ca)1/3. In each case, we adopted the radius of the capillary and the interfacial tension R = 10–2 cm, γ = 30 dyn/cm, respectively. Let us study the solution of Equation 3.186 at ε << 1, having represented it in the form ξ = ξ e + 3 ξ1 .

(3.192)

Using this representation we arrive at the following equation for the zeroth approximation: d 3ξ e β dξe − = 0. d z 3 ξ ne +1 d z

(3.193)

Having integrated Equation 3.193, we obtain d 2ξe β R + = const = . d z 2 n ξ ne Re

A±

(3.194)

B± 3 2.79

3

2 R ±/R

1.01

1 6

1.0

1

0.99

0

2 5 1

2

3 4

0.7 (3Ca)

1/3 .

102

FIGURE 3.14 Dependency of parameters A+ (curve 1), A– (curve 2), B+ (curve 3), B– (curve 4), and the radii of the leading meniscus, R+ (curve 5) and the trailing meniscus, R– (curve 6) on the capillary number, Ca. © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

Equation 3.194 describes the proﬁle of the transitional zone between the ﬁlm and the equilibrium meniscus and, as was shown in Chapter 2, Section 2.4, it has the asymptotic solution ηe ∼ Ae

ξ2 + Be , 2

(3.195)

where Ae =

R n = Ae+ = Ae− , Be = = Be+ = Be− . Re n −1

(3.196)

The equation of the ﬁrst approximation for (3.186) using the representation (3.192) takes the following form:

ξ 3e

d  d 2 ξ1 β ξ1  − = ξe − 1 .  2 dz  d z n ηen +1 

(3.197)

Having integrated the equation, we conclude that d 2 ξ1 β ξ 1 − = C± − d z 2 n ξ ne +1

±∞

∫ ξ

ξe − 1 d z, ξ e3

(3.198)

where C± = const. At z → –∞ for the leading meniscus and z → +∞ for the trailing meniscus, we conclude that ξ1 = 0. Then, we conclude from Equation 3.198 that ±∞ ±

C =

∫

∓∞

ξe − 1 d z. ξ e3

(3.199)

As the solution of equation (3.198) satisﬁes the asymptotic conditions

ξ1 =

A1+ 2 z + B1+ , 2

z → +∞ ,

(3.200)

A1− 2 z + B1− , 2

z → −∞ ,

(3.201)

for the leading meniscus and

ξ1 = © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

253

for the trailing meniscus, we ﬁnd from Equation 3.198 that A± = C ± .

(3.202)

Equation 3.190 can be rewritten in the following form: 1   h 0 B± = R  1 − ±   A  or, in a ﬁrst approximation with respect to the parameter ε: he B1± =

RA1± . A 2e

(3.203)

Hence, B1± =

RA1± R C ± = . he A 2e he A 2e

(3.204)

As was shown in Section 2.4, n he . n −1 R

Ae = 1 +

(3.205)

To within the leading terms in ε we obtain the following relation using Equation 3.204 and Equation 3.205: B1± =

R ± C . he

(3.206)

The ﬁnal asymptotic representation of the parameters A± and B± takes the form A ± = A e + ε A1± = 1 +

B± = B e + ε B± =

n he + ε C± , n −1 R

n R ± +ε C . n −1 he

(3.207)

(3.208)

The constants C± were found from the numerical solution of Equation 3.194. It follows from Equation 3.207 and Equation 3.208 that at ε << 1 the parameters © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

A± and B± are linear functions of ε (or on Ca). At ε = 0 (Ca = 0), we recover the equilibrium values. At high velocities (Figure 3.13), the values of the parameters B± are close to the values of the constants obtained in Reference 23 as expected: B± ~ 2.79, B – ~ –0.7. It can be seen from Figure 3.12 that the thickness of the ﬁlm in the region CD can be considered equal to the equilibrium thickness he at (3Ca)1/3 ≤ 8·10–3 for a disjoining-pressure isotherm of the form Π(h) = A/h3, and at (3Ca)1/3 ≤ 10–2 for Π(h) = A/h2. In the case of high velocities, the plot of the relation h0(Ca) tends asymptotically to that given by Equation 3.183, which is valid at a sufﬁciently high velocity of the drop or bubble. To close the problem, we need to establish the relationship between the applied pressure gradient, ∆p = p– – p+, at the ends of the capillary and the velocity of the drop, U. This relation can be represented in the following form (see Appendix 6 for details):

∆p=

2γ  R R L+ + L− + η l   + − − + 4 Ca . R R R R 

(3.209)

With allowance for Equation 3.190, we conclude from Equation 3.209 that

∆p=

2γ  + L+ + L− + η l  −  A Ca − A Ca + 4 Ca . R  R 

( )

( )

(3.210)

The plot of the velocity, U, according to of Equation 3.210, obtained using the previously numerically calculated dependencies, is shown in Figure 3.14, – where we used R = 10–2 cm, γ = 30 dyn/cm, η = 0, L– + L+ = 4 cm, = 1 cm, η1 = 10–2 P. For small ε, Equation 3.210 takes the following form:

∆p=

2γ  L+ + L− + η l  + −  ε (C − C ) + 4 Ca , R  R 

(3.211)

or 3    R 2 + 2γ L+ + L− + η l  −  ∆p= Ca 3   (C − C ) + 4 .    h0  R R  

(3.212)

Equation 3.212 shows that at small ε, the velocity U of the drop or bubble motion depends linearly on the applied pressure difference, ∆p.

© 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

255

U . 105, m/sec

3

2

2

1

1

0.1

0.3 Δp, Pa

0.5

FIGURE 3.15 Dependence of velocity of motion of a single drop or bubble, U, on the applied pressure difference, ∆p. (1) Π(h) ≡ 0; (2) Π(h) = A/h2, according to Equation 3.210.

Let us examine the motion of a sequence k of drops or bubbles in the capillary. Let i, i = 1, 2, …, k be the lengths of individual drops. We assume that the inﬂuence of drops on one another can be neglected if the distance between their ends exceeds 2R. Then, for a chain of k drops, Equation 3.210 can be rewritten as   L+ + L− + µ 2γ  ∆p= k A+ − A− + 4 Ca R R 

(

)

k

∑ i =1

 li   . 

(3.213)

Figure 3.15 and Figure 3.16 show the plot of the velocity of motion, U, of the chain of bubbles according to Equation 3.213 at k

k = 10,

∑ l = 9 cm, i

i =1

and L+ + L– = 1 cm, γ = 30 dyn/cm, R = 10–2 cm. In numerical calculations for the isotherm of disjoining pressure Π(h) = A/h3, we used A = 10–14 J/m at n = 2 and A = 10–21 J·m at n = 3.

APPENDIX 6 The low slope approximation is valid two ﬂow zones ED and CB, and the zone of the ﬂat ﬁlm DC (Figure 3.12). At a low Reynolds number and taking into account that all unknown functions do not depend on the angle, we get the following system of equation, which describes the ﬂow in both liquid 1 and 2:

© 2007 by Taylor & Francis Group, LLC

Wetting and Spreading Dynamics

U . 105, m/sec

256

5

2 2

1

0 10

20 Δp, Pa

FIGURE 3.16 Dependence of velocity of motion of a chain of drops on the applied pressure difference, ∆p: (1) Π(h) ≡ 0; (2) Π(h) = A/h3, according to Equation 3.213.

0=−

 ∂2v ∂2v 1 ∂vr vr  ∂p + η  2r + 2r + − , r ∂r r 2  ∂r ∂x  ∂r

(A6.1)

 ∂2v ∂p ∂2v 1 ∂vx  + η  2x + 2x + , r ∂r  ∂x ∂x  ∂r

(A6.2)

0=− and the continuity equation

∂vr ∂vx vr + + = 0. ∂r ∂x r This equation can be rewritten as ∂rvr ∂rvx + = 0. ∂r ∂x

(A6.3)

The low slope in the zone EB results in vr << vx ,

∂ ∂ . << ∂x ∂r

Using that, we conclude from Equation A6.1 and Equation A6.2 that ∂p , ∂r

(A6.4)

 ∂2 v  ∂p x + 1 ∂vx  . + η ∂x  ∂r 2 r ∂r 

(A6.5)

0=−

0=−

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Kinetics of Wetting

257

Equation A6.4 means that the pressure in both liquids 1 and 2 is a function of the axial co-ordinate x only. Integrating Equation A6.3 over radius in liquids 1 and 2, we conclude that R− h

∫

∂rvx dr + ( R − h )vrh = 0 , ∂x

(A6.6)

R

∂rvx dr − ( R − h )vrh = 0 , ∂x

(A6.7)

0

∫

R− h

where vrh, vxh are velocities on the liquid–liquid interface. The integral on the righthand side of the preceding equations can be transformed as ∂ ∂x

R− h

∫ 0

R− h

∂h rvx dr = − ( R − h )vxh + ∂x

∫ 0

∂rvx dr , ∂x

or R− h

∫ 0

∂rvx ∂ dr = ∂x ∂x

R− h

∂h

∫ rv dr + ∂x (R − h)v . h x

x

0

Substitution of this expression into Equation A6.6, and taking into account that ∂h ∂h = vxh + vrh , ∂t ∂x results in R− h

∂h ∂ ( R − h) =− ∂t ∂x

∫ rv dr , x

0

or ∂( R − h )2 ∂ = ∂t ∂x

R− h

∫ 2rv dr . x

(A6.8)

0

Similar transformations of Equation A6.7 results in ∂( R − h )2 ∂ =− ∂t ∂x

© 2007 by Taylor & Francis Group, LLC

R

∫ 2rv dr . x

R− h

(A6.9)

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Wetting and Spreading Dynamics

Integration of Equation A6.4 and Equation A6.5 using (1) a nonslip boundary condition on the capillary surface at r = R, (2) a symmetry condition in the capillary center at r = 0, and (3) equality of velocities and tangential stresses at the liquid–liquid interface at r = R – h, we can deduce expressions for axial velocities, vx, in both liquids 1 and 2. Substitution of those expressions in Equation A6.8 and Equation A6.9 results in

(

∂ R−h ∂t

)

2

(A6.10)

=

(

 ∂  p2′ R − h  ∂x 8η 2 

(

∂ R−h ∂t

)

2

)

4

(

)(

)

4  R − h p2′ − p1′  2 2 h p1′ 2 ln  1 −   − R−h R−h − R −   R  4 η1 2η1  

(

) (

(

 ∂  p2′ R − h =  8η1 ∂x 

)

)

4

−

2 p1′  R − h − R2    8η1 

(

)

( R − h ) ( p′ − p′ )  ( R − h ) − 2

2

2

1

 

8η1

− R2

2

(A6.11)

 2  h  − R − h ln  1 −    , R    

(

)

where p1(x) and p2(x) are pressures in liquids 1 and 2, respectively; ′ means a derivation with x. Subtracting Equation A6.11 from Equation A6.10 and integrating over x results in −

4 4 p2′ p′ R − h − 1  R4 − R − h    8η 2 8 η1 

(

)

(

( p′ − p′ ) ( R − h )  ( R − h ) +  4η 2

1

2

1

)

(A6.12) 2

()

− R  = 2A t ,  2

where A(t) is an integration function, which can dependent only on time. We consider the steady motion of the drop with the velocity U. Changing over to the variable z = x – U t in Equations A6.10 and A6.11, we conclude that −

2 2 p2′ p′ R − h − 1  R2 − R − h   8η 2 4 η 1 

(

)

(

( p′ − p′ ) ( R − h ) + 2

2 η1

© 2007 by Taylor & Francis Group, LLC

1

2

)

 h ln  1 −  = U . R 

(A6.13)

Kinetics of Wetting

−

259

2 2 p1′  2 R − R−h  +  8 η 1 

(

)

( p′ − p′ ) ( R − h ) 2

1

(

 R2 − R − h  R − h 2 ln  1 + h  +   R  2 

(

)

2

2 η1

)

2

  = 2A −U R − h 2 .  

(

(A6.14)

)

In the steady-state case being examined, the integration constant A should be independent of time. The ﬁlm has a constant thickness h0 far from the ends of the drop; hence, from Equation A6.13 and Equation A6.14, we obtained the solutions p1′ = p2′ = −

8 η2 U , 2 ( R − h 0 )2 + 2 η  R 2 − R − h 0   

(

)

4 4 4 U ( R − h0 ) + η  R − ( R − h0 )  , A= 2 ( R − h0 )4 + 2 η  R 2 − ( R − h0 )2   

(A6.15)

(A6.16)

–

where we introduced η = η1/η2. Equation A6.16 gives the dependence of the ﬂow rate in the capillary on the thickness of the ﬁlm, h0, and the velocity of the drop, U. Equation A6.13 to Equation A6.16 are used to obtain an equation for the thickness of a ﬁlm of liquid under a drop. Limiting ourselves to the leading terms of the expansion in h/R << 1, we move from Equation A6.16 to  dK 1 dh  + Π′(h )  = 3Ca(h − h0 )P , h3  dx   dx γ

(A6.17)

where h − h0 hh   h  + 8 η2 20   1 + 4 η   1 + 2 η   R R R P= . h − h0 h  2 hh0  η η η 1 + 4 + 16 1 +    R R2  R  If we assume that η ~ 1, then P = 1, and Equation A6.17 takes the form of Equation 3.185. –

3.7 COATING OF A LIQUID FILM ON A MOVING THIN CYLINDRICAL FIBER In this section we shall study how the disjoining pressure affects the thin ﬁlm adhering on the surface of a ﬁber being pulled out of a liquid. The problem of coating © 2007 by Taylor & Francis Group, LLC

260

Wetting and Spreading Dynamics

2

1

r h0 a U

0 III

x II

I

FIGURE 3.17 System under consideration: moving cylinder of radius a with velocity U through the interface between liquids 1 and 2. (I) zone of the meniscus, (II) the ﬂow zone, (III) zone of the ﬁlm of the uniform thickness, h0, to be determined as a function of Ca.

by liquid ﬁlms of a moving support is frequently encountered in technological processes. The problem of coating by a liquid ﬁlm of a ﬂat vertical support pulled out of the liquid in a gravity ﬁeld has been investigated earlier in Reference 27 and Reference 28, and the ﬁlm thickness vs. support velocity was deduced for a relatively high velocity of coating. In Section 3.6 we found that the effect of the disjoining pressure is predominant at low-support velocities. The problem of coating from this point of view is considered in this section. The effect of the disjoining pressure in the thin ﬁlm deposited on the surface of a ﬁber taken out of a liquid is taken into consideration in this section. Let us examine the problem of pulling out a thin cylindrical ﬁber of radius a through an interface between two immiscible liquids with a constant velocity U. The schematic presentation of the experimental system is shown in Figure 3.17. The ﬁber moves along its own axis perpendicular to the interface of the liquids. We neglect the forces of gravity in comparison to viscous and capillary forces.

STATEMENT

OF THE

PROBLEM

Let us introduce a cylindrical system of coordinates whose x-axis coincides with the axis of the ﬁber, and r-axis is perpendicular to it. In this system of coordinates, the ﬁber moves with velocity U in the negative direction of x-axis (Figure 3.17). It is assumed that liquid 1 wets completely the ﬁber surface. Both the Reynolds number, Re = Ua/η, and the capillary number, Ca = ηU/γ, are small, Re << 1, Ca << 1, and the ﬁlm thickness, h0, is much smaller than the radius of the ﬁber, a, h/a << 1, where η is the dynamic viscosity of the liquid 1, and γ is the liquid–liquid interfacial tension on the interface of liquids 1 and 2. The ﬁeld of ﬂow is subdivided as in Section 3.6 into the following three zones (Figure 3.17): © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

261

Zone I, the steady-state meniscus of the liquid. The movement of the liquid in this zone can be neglected. Zone II, the ﬂow zone in between the ﬁlm of constant thickness and the meniscus of the liquid. Zone III, a ﬁlm of constant thickness, h0, to be determined as a function of the Ca. According to the previous consideration in zones II and III, the low slope approximation is valid. Hence, the equations of ﬂow taking into account the previous assumptions in zones II and III can be written in the following form: ∂p1 = 0, ∂r

(3.214)

∂p1 1 ∂  ∂u  =η r  . r ∂r  ∂r  ∂x

(3.215)

p2 = const .

(3.216)

∂ 1 ∂ u+ ( rv ) = 0 . ∂x r ∂r

(3.217)

The continuity equation is

The boundary conditions are: a nonslip boundary condition on the ﬁber surface, u = −U , r = a ;

(3.218)

absence of tangential stress on the liquid–liquid interface (see the discussion in Section 3.6), η

∂u = 0, r = a + h ; ∂r

(3.219)

and normal stress jump on the same interface results in p 2 − p1 = γK ( x ) +

a Π (h), r = a + h , a+h

(3.220)

where u and v are the velocity along axes x and r, respectively; p1 and p2 are the pressures in liquids 1 and 2; K(x) is the curvature of the surface of the ﬁlm; and © 2007 by Taylor & Francis Group, LLC

262

Wetting and Spreading Dynamics

Π(h) is the isotherm of the disjoining pressure of ﬂat ﬁlms. Note the presence of the term a a+h in the boundary condition (3.220) (see Chapter 2, Section 2.12 for the derivation).

DERIVATION OF THE EQUATION INTERFACE PROFILE

LIQUID–LIQUID

FOR THE

Integration of Equation 3.214 and Equation 3.215 with consideration of boundary conditions (3.218) and (3.219) gives the following equation for the velocity of the liquid:

(

)

(

)

2 u = p1′ / 4 η r 2 − a 2 − 2 a + h ln r /a  − U ,  

(3.221)

where dp1 /dx = p1′ . The ﬂow rate of the liquid in the liquid layer is: a+ h

Q = 2π

∫ urdr .

(3.222)

a

Substituting Equation 3.221 for Equation 3.222, we obtain Q /2π =

(3.223)

((

2   2 2  a + h − a  4 − a + h   

(

)

(

)

)

2

)(

2  a+h 

) ( 2 ln ( a + h ) 2

)

 a  − 1 + a 2     

p1′ / 4 η − U  a + h − a 2  2.   2

The condition of constancy of ﬂow in any section of the ﬁlm, Q = const, follows from the equation of continuity (3.217). For a ﬁlm of constant thickness ho in zone III, we deduce from boundary condition (3.220) and Equation 3.216 that p1′ = 0 . The latter allows rewriting Equation 3.223 as

(

Q = −V  a + h0 

© 2007 by Taylor & Francis Group, LLC

)

2

− a2  2 . 

(3.224)

Kinetics of Wetting

263

Neglecting terms higher than the ﬁrst order in zones II and III with respect to h/a << 1, we obtain from Equation 3.223 and Equation 3.224 that

(

)

− p1′h 3 / 3η = U h − h0 .

(3.225)

Boundary condition (3.220) with consideration of (3.216) can be used to obtain the equation for the proﬁle of the liquid–liquid proﬁle in zones II and III from Equation 3.225: d  a h−h  γK ( x ) + Π ( h )  = ( 3ηU ) 3 0 . dx  a+h h 

(3.226)

The curvature of the surface of the liquid–liquid interface in the cylindrical system of coordinates is

()

(

K x = h′′ 1 + h′ 2

)

3/ 2

)(

(

 − 1  a + h 1 + h′ 2 

)

1/ 2

  .

Let us introduce as before the following dimensionless variables z = x/ and H = h/h0, where is the scale of the ﬂow zone. We are interested in low and intermediate capillary numbers; that is, the thickness of the ﬁlm, h0, does not differ very much from the equilibrium thickness, he. This allows us, as in Section 3.6, to select this scale as = ah 0 . According to that choice, h0 =

h0 << 1 . a

Using the new variables, we can rewrite the expression for the curvature as

() ( )

K z = 1 / a  H ′′ − 1 .

(3.227)

Using the same dimensionless variables in Equation 3.226 in combination with the latter Equation 3.227, we obtain   d  H ′′ − 1 + dz   

© 2007 by Taylor & Francis Group, LLC

 3/ 2   a  H −1 a . Π h0 H  = 3Ca     h0   h0  H3 γ 1+ H   a   

(

)

(3.228)

264

Wetting and Spreading Dynamics

According to Section 3.6, the dependency of the ﬁlm thickness, h0, on the capillary number is given by Equation 3.183. Therefore, our choice of the scale of the transition zone = ah 0 tends to L = ah e as Ca tends to zero, that is, to the length of the transition zone at the equilibrium, and = ah0 ~ a ⋅ a ⋅ Ca 2 / 3 = aCa1/ 3 at sufﬁciently high Ca. This corresponds to the choice adopted in Reference 23, which is valid only at sufﬁciently high Ca.

IMMOBILE MENISCUS In zone 1, the curvature of the interface should be constant, as the movement of the liquid in this region can be neglect (see for details Introduction to this chapter). However, away from the ﬁber, the interface is ﬂat, and the curvature consequently vanishes. It is evident that h ≈ a and the scale of this zone has the same order of magnitude, a. Hence, in this zone, h′2 ≈ 1. That is, from the expression for the curvature we conclude that

(

h′′ 1 + h′ 2

)

3/ 2

(

−1  a + h 

)

1 + h′ 2  = 0 . 

(3.229)

Integrating Equation 3.229 once results in h′ = C 2 ( a + h ) − 1 , 2

(3.230)

where C is the integration constant. The solution of Equation 3.230 is

( )

h = 1/C cosh C ( x + C1 )  − a ,

(3.231)

where C1 is the new integration constant. This equation has a stationary point h = hs, h′s = 0. Then we can determine, using Equation 3.231, that

(

)

C = 1 a + hs .

(3.232)

Let us shift the origin for x-axis to the stationary point, i.e., we select C1 = 0 and expand the solution (3.231) in the Taylor series, which results in h = hs + © 2007 by Taylor & Francis Group, LLC

x2 + ⋅⋅⋅ 2(a + hs )

(3.233)

Kinetics of Wetting

265

MATCHING OF ASYMPTOTIC SOLUTIONS (FIGURE 3.17)

ZONES I

IN

AND

II

Now we should match solution of the stationary meniscus according to Equation 3.233 with the solution of Equation 3.228, which describes the liquid proﬁle in zones II and III. Let us examine the asymptotic solution of Equation 3.228 for z → +∞ (or H → ∞), then we obtain as in the previous case (Equation 3.228). As we already mentioned, this equation has the following asymptotic: H = A0 z 2 / 2 + B0 ,

(3.234)

where the origin is transferred to the stationary point selected earlier to get rid of the term proportional to z; A0 and B0 are some functions of the capillary number Ca determined from the solution of Equation 3.228 with the boundary condition H → 1 for z → –∞ and from the matching conditions, which is obtained in the text below. Let us rewrite solution (3.233) using the inner variables in Equation 3.228:

H=

hs 1 z2 + + ⋅⋅⋅ . h0 1 + hs /a 2

(3.235)

Comparing Equation 3.234 and Equation 3.235, and as these expansions should coincide, we obtain two matching conditions for the solutions of the ﬁlm proﬁle equations in zones I and II: hs /h0 = B0 ,

(

(3.236)

)

1 1 + hs /a = A0 ,

(3.237)

where both thickness h0 and hs are still unknown. The two equations can be solved as follows:

(

h0 = a 1 − A0

(

)

hs = a 1 − A0

)

A0 B0 ,

(3.238)

A0 .

(3.239)

The constants A0 and B0 and their dependency on the capillary number, Ca, can be determined only numerically. The results of the numerical calculations are presented in Figure 3.18. Only the case of complete wetting is under consideration in this section; that is why the isotherm of the disjoining pressure used in the problem was selected as © 2007 by Taylor & Francis Group, LLC

266

Wetting and Spreading Dynamics B

A

3 1.00 (2.79) 3

2

1

(1.5) 1

0.99

0 –16

–14

–12 ᐉn Ca

–10

–8

ᐉn A∗

4 3 2 1

2

0 –1 –16

–14

–12

–10

–8

ᐉn Ca

FIGURE 3.18 Calculated parameters A0 (1), A* (2), and B0 (3) on capillary number, Ca.

Π (h) =

A , h3

(3.240)

where A is the Hamaker constant. A + Ca → 0, h0 → he , and B0 → 3/2 (as we show in the following text). Hence, A0 →

1 3h ≈ 1 − e , Ca → 0 , 3he 2a 1+ 2a

where he is the equilibrium thickness of the ﬁlm. © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

267

EQUILIBRIUM CASE (Ca = 0) In the equilibrium case (Ca = 0), the ﬁlm proﬁle equation will have the following form according to Equation 3.226:  d  h′′ dx  1 + h′ 2 

(

)

3/ 2

−

(a + h)

1 1 + h′ 2

+

 a Π h  = 0 . a+h 

()

Integrating once, we obtain

(

h′′

1 + h′

2

−

)

3/ 2

1 (a + h) 1 + h′

2

()

a Π h = const = 0 . a+h

+

(3.241)

The integration constant is equal to zero because the interface is ﬂat away from the ﬁber. In zone III we obtain the equation for determining the ﬁlm thickness from Equation 3.241, assuming h = he , h′ = 0, and h″ = 0:

( )

γ /a = Π he ,

(3.242)

or, using the disjoining pressure isotherm in the case of complete wetting (3.240), Equation 3.242 is rewritten as follows: γ/a = A/h3e . This equation results in  Aa  he =   γ 

1/ 3

.

(3.243)

Equation 3.241 in the case of complete wetting takes the following form:

(

h′′

1 + h′

2

)

3/ 2

−

(a + h)

1 1 + h′

2

+

Aa = 0. γ a + h h3

(

)

(3.244)

In zone III, the ﬁlm proﬁle according to Equation 3.244 becomes ﬂat and coincides with Equation 3.243; in zone I, where the disjoining pressure can be neglected, we obtain the meniscus proﬁle, which is similar to that given by Equation 3.231. All terms of Equation 3.244 are important in zone II. Integrating Equation 3.244 once we obtain

1

© 2007 by Taylor & Francis Group, LLC

  Aa   1 + h′ 2 = Ce −    2 γh 2   

(a + h) ,

268

Wetting and Spreading Dynamics

where Ce is the integration constant. At Ca → 0: h → he and h′ → 0, hence, we conclude from the latter equation:  Aa  Ce = a + he +  .  2 γhe2  Hence, ﬁnally, the liquid proﬁle is

1

  Aa   Aa   1 + h '2 =  a + he +   −   2 γhe2   2 γh 2   

(a + h) .

(3.245)

The matching conditions in solutions of Equation 3.245 in zone II and the equation for the steady-state meniscus in zone I can be written as follows: lim hII′ = lim hI′ ,

hII →∞

(3.246)

hI → 0

where hI′ and hII′ refer to zones I and II, respectively. The solution of Equation 3.229, which describes the ﬁlm proﬁle in zone I, is 1

(

)

1 + hI′ 2 = C2 a + hI ,

(3.247)

where C2 is the integration constant. Using Equation 3.245 and Equation 3.247, we ﬁnd that, using the matching condition (3.247), C2 = Ce .

(3.248)

At the stationary point, hI = hse, where h′I = 0, and we conclude from Equation 3.247 and Equation 3.248 that  Aa  hse = C2 − a = he +  2 ,  2γhe  or hse = 1.5 he ,

(3.249)

which is in agreement with the conclusion obtained in Section 2.4 (see Figure 2.19).

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Kinetics of Wetting

269

Hence, we obtain the following expression for coefﬁcient Be in the equilibrium case (see Equation 3.234): Be = lim B0 = 1.5 .

(3.250)

Ca→0

NUMERICAL RESULTS The coefﬁcients A* = (1 – A0)/B0 (3 Ca)2/3, A0, and B0 as functions of the capillary number according to Equation 3.237 and Equation 3.236 obtained from the numerical calculation and are presented in Figure 3.18. The isotherm of the disjoining pressure is deﬁned by Equation 3.240, that is, the case of complete wetting is under consideration. Figure 3.18 shows that in this case at Ca → 0, B → 3/2, which corresponds to the prediction (3.249). At the same time, A0 → 1 – (3/2) He as shown above, and A* → ∞. For relatively large capillary numbers, Ca, the ﬁlm thickness h0 becomes thick enough so that the effect of the disjoining pressure can be neglected. At the same time, A* → 0.643, B → 2.79 (these values were obtained in Reference 23) and A0 ≈ 1 – 0, 643 (3Ca)2/3 · 3/2. The dimensionless thickness H0 of the ﬁlm remaining on the ﬁber as a function of the capillary number Ca (dimensionless velocity) is presented in Figure 3.19. At Ca → 0, the graph has a plateau, h0 → he. For large Ca H 0 ≈ 1.33 ( Ca ) 2 / 3 ,

(3.251)

as was ﬁrst deduced in Reference 23 earlier.

ᐉn H0 –7 2 ᐉn Hε

–9 1 –15

–14

ᐉn Ca∗

–12

–11

ᐉn Ca

FIGURE 3.19 Dimensionless ﬁlm thickness H0 = h0 /a on the capillary number, Ca. (1) without disjoining pressure, according to Equation 3.251 and (2) with consideration of the disjoining pressure, according to Equation 3.238. Experimental point from Reference 29.

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270

Wetting and Spreading Dynamics

Let us determine the critical velocity, Ca*, characteristic of the transitional velocity from small to large capillary numbers as the point of intersection of the lines ln he = const and ln 1.33 + 2/3 ln Ca. In our case, lnCa* = 12.85. The critical velocity obtained from the experimental data in Reference 29 is equal to ln Ca* ≈ –13.26, that is, close to the above theoretical prediction.

3.8 BLOW-OFF METHOD FOR INVESTIGATION OF BOUNDARY VISCOSITY OF VOLATILE LIQUIDS A blow-off method allows determining the boundary viscosity as a function of the distance to a solid substrate. A theory is suggested, taking into account not only the ﬂow of a liquid ﬁlm but also its evaporation, with gas being blown through a plane-parallel channel over the ﬁlm. The theory allows one to ﬁnd the dependence of the dynamic viscosity, η, on the distance to the substrate, h, with η being a continuous function of h. A procedure is outlined for calculation of the viscosity, η, on h dependency based on experimental data. The theory is applied to the calculation of the boundary viscosity of hexadecane. It turns out that the viscosity in thin layers (40–200 Å) is lower than that in the bulk [35,57].

BOUNDARY VISCOSITY The measurements of the boundary viscosity are of substantial interest. Attempts have been made to apply the conventional methods adapted to the measurement of boundary viscosity. These methods may be divided into three groups: • • •

Methods based on measurement of the rate of liquid ﬂow through capillaries [62] Methods based on the rotation of coaxial cylinders or discs Methods based on measurement of the velocity of a falling sphere

These methods afford the possibility of determining only the mean value of viscosity for a sufﬁciently thick layer. The blow-off method [30] allows measuring the boundary viscosity. The method consists of the following: One of the walls of a channel formed by two plane-parallel surfaces is coated with a layer of the tested liquid. If a current of air (or an inert gas) is then passed through the channel, a ﬂow is generated in the ﬁlm due to a tangential force induced by the ﬂowing gas. In this case the ﬁlm proﬁle becomes wedge shaped, the slope of this wedge being dependent on the viscosity of the liquid. If the liquid viscosity over the whole distance to the wall is invariable, the slope remains constant until the wetting boundary is reached. In the case of an increased viscosity close to the wall, the slope increases; in the case of a decreased viscosity, the slope decreases. The viscosity is calculated from the slope at a given point of the proﬁle. The blow-off method was used to examine the boundary viscosity of some organic liquids [31]. It has been established that the proﬁle of a ﬁlm of very pure © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

271

vaseline oil is rectilinear. In this case, no anomaly of viscosity is observed, and vaseline oil preserves its bulk viscosity value up to the layer thickness of about 10–7 cm. The viscosity of incompletely hydrated benzontron was found to decrease in the boundary layer, whereas in a layer of 300 Å, the viscosity of chloroderivatives of saturated hydrocarbons was found to increase abruptly (jumpwise). An increase in the viscosity of some organic liquids and some vinyl polymers and their solutions in ﬁlms up to 50,000 Å thick as well as the instability of the ﬁlms of the solutions of vinyl polymers at certain thicknesses were observed in Reference 32 to Reference 34. Up to now, a limitation of the blow-off method has been its inapplicability to the measurement of the boundary viscosity of volatile liquids. In this section, the blow-off method has been modiﬁed so as to be applicable to studies of the boundary viscosity of volatile liquids. It is necessary to take into account the fact that when gas is blown through a plane-parallel channel, the ﬂow of a ﬁlm of liquid occurs (as in the usual variant of the blow-off method) simultaneously with its evaporation. The theory of the blow-off method, allowing determining the boundary viscosity of volatile liquids, is developed in the following text.

THEORY

OF THE

METHOD

Let us consider the proﬁle of a ﬁlm at instant of time t, which is characterized by the relationship x = x ( h, t ) ,

(3.252)

where x is the distance from the wetting boundary to the point at the surface of the ﬁlm area having local thickness h. An initial condition is adopted as follows: x ( h, 0 ) = 0 .

(3.253)

An increment, dx, per elementary increment of time, dt, is the sum of an increment, dx1, resulting from the ﬂow under the action of the tangential shear stress per unit area of the blow-off current, τ, and an increment, dx2, due to the effect of evaporation. It is obvious that the ﬁrst increment due to the gas ﬂow is h

dx1 = τ

∫ η( h) ⋅ dt , dh

(3.254)

0

where η(h) is the local viscosity in the boundary layer at distance h from the substrate. Evaporation results in dx2 = i © 2007 by Taylor & Francis Group, LLC

dt , sin α

(3.255)

272

Wetting and Spreading Dynamics

where i is the linear evaporation rate and sinα is the slope of the ﬁlm surface at a given point. With the exception of a very short initial period, the ﬁlm has a very low slope proﬁle, which allows one to substitute  dx    dh

−1

for sinα. Summing Equation 3.254 and Equation 3.255 and transforming them, we eventually obtain h

∂x dx dh −i =τ . ∂t ∂h η(h)

∫

(3.256)

0

The general solution of partial differential Equation 3.256 is τ x = f ( h + it ) − i

h

h1

∫ dh ∫ η( h ) . dh2

1

0

0

(3.257)

2

The initial condition (3.253) results in τ x=  i 

h + it

h1

∫ dh ∫ 1

0

0

h h1 h + it h1 dh2 dh2 dh2  τ  . − dh1 = × dh1 η ( h2 ) η ( h2 )  i η ( h2 ) 0 0 h 0 

∫ ∫

∫ ∫

(3.258)

Consider consequences of the solution obtained. If there is no special boundary viscosity, i.e., η(h) = η∞ = const, then x=

τt ( 2h + it ) . 2 η∞

(3.259)

Thus, having experimentally determined the relationship x = x(t, h) at constant thickness, h, and having revealed its strictly parabolic form on time, one may ascertain that the viscosity has a constant value. Therefore, using the parameters of the parabolic relationship x(t,h), at any h = const ≠ 0, one may determine both η∞ and the evaporation rate i. It is more convenient to attain the same objective by determining from the experiment the dependence of h on t for a given value x and by checking its hyperbolic form.

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Kinetics of Wetting

273

If one determines only the velocity at which the wetting boundary recedes (which is the simplest of all), then Equation 3.259 becomes i x = 2, 2 η∞ τt

(3.260)

which is of interest for its method, thus allowing one to determine the ratio i/η∞. From Equation 3.259 we conclude that h=

x η∞ it − . τt 2

If η(h) = η∞ = const, the equation gives η∞ and i at any constant x. From Equation 3.258 we conclude that  ∂x    = τ ∂t h

h + it

∫ η(h) , dt

(3.261)

0

 ∂2 x  1  ∂t 2  = τi η ( h + it ) ,

(3.262)

∂2 x 1 =τ . ∂h∂t η ( h + it )

(3.263)

h

Using Equation 3.262 and Equation 3.263 as the basis, it is difﬁcult to obtain precise data on the viscosity of boundary layers as it is difﬁcult to measure x at a given h as a function of t. If we conﬁne ourselves to the case where h = const ≠ 0, then the problem is rendered somewhat easier. Yet, we have to determine (∂x/∂h)t within the time interval it < HN , where HN is the thickness of the boundary layer, and where the viscosity is different from its bulk value. Measuring the values of h as a function of t at the ﬁxed coordinate x, it is possible to derive the experimental dependence h(t) or, conversely, t(h). Let us explore the possibility of obtaining η(h) using such an approach. For convenience, subsequently we use the following notation: C=

© 2007 by Taylor & Francis Group, LLC

xi , ϕ ( h ) = it ( h ) , τ

(3.264)

274

Wetting and Spreading Dynamics

f (h) =

h

h1

∫ dh ∫ η( h ) , 0

()

dh2

1

(3.265)

2

0

()

f 0 = f ′ 0 = 0, h ≥ 0 . It follows from Equation 3.258 that f(h) satisﬁes the following functional equation: f  h + ϕ ( h )  = f ( h ) + C .

(3.266)

As the viscosity is always positive, the deﬁnition (3.265) gives for all thicknesses h

()

f h > 0,

()

()

f ′ h > 0,

f ′′ h > 0 .

According to the deﬁnition, ϕ(h) = it(h) does not vanish anywhere, except at h = ∞. Note that ϕ(h) has the dimension of length. Using Equation 3.266 we determine the dependence η(h). We proceed according to the following procedure: (i) Let us deﬁne a sequence of ﬁlm thicknesses, Hm, m = 0, 1, 2, 3, … such that Hm

h1

∫ dh ∫ η( h ) = C, dh2

1

H m −1

0

m = 0, 1, 2, ….

2

Having determined the dependence η ( h ) at h = H m, we ﬁnd the value of i. (ii) We show that the function f(h) can be determined at h ≥ HN where h = HN is a ﬁlm thickness such that η(h) = η∞ = const for h ≥ HN , i.e.,

()

f ′′ h =

1 = const . η∞

(iii) The function f(h) for h ≥ HN , constructed according to (ii), will be extended to include values h < HN up to h = 0, which allows one to ﬁnd the dependence of viscosity on thickness for all h > 0, because

()

η h =

© 2007 by Taylor & Francis Group, LLC

1 . f ′′ h

()

Kinetics of Wetting

275

(iv) We develop a method for determining the H N thickness of the boundary layer of liquid. Everywhere in (i)–(iv), the viscosity is considered as a continuous function of thickness h. The dependency t(h) is considered to be determined experimentally with an accuracy sufﬁcient for further consideration. We now turn to the realization of the program indicated in the preceding discussion. (i) Let us introduce the following notations:

()

( )

(

)

ϕ −1 = ϕ 0 , ϕ 0 = ϕ ϕ −1 , …, ϕ m = ϕ ϕ −1 + ϕ 0 +…+ ϕ m−1 , …, H −1 = 0, H 0 = ϕ −1 , …, H m = ϕ −1 + ϕ 0 +…+ ϕ m−1 , …,

(3.267)

H m+1 = H m + ϕ m , where m = 0, 1, …. At h = 0, from Equation 3.267 we conclude that f(ϕ–1) = f(0) + C = C, or f(H0) = C. Let us assume that h = Hm f ( H m ) = ( m + 1) ⋅ C ,

(3.268)

and prove Equation 3.268 by induction. We assume that it is valid at m = l and show then that Equation 3.268 holds identically for m = l + 1. Equation 3.256 at h = Hl gives f (Hl + ϕl) = f(Hl) + c = (l + 1)C + C = (l + 2)· C, or f(Hl+1) = (l + 2) C. This relation proves the validity of Equation 3.268. As Equation 3.268 is valid at m = 0, 1, it is valid for any m. It follows from deﬁnition (3.267) that H m ≤ H m+1, i.e., H m is an increasing sequence of thicknesses; hence, it has a ﬁnite or an inﬁnite limit H = lim m→∞ H m. Passing over to the limit in the relationship Hm+1 = Hm + ϕ(Hm), we obtain H = H + ϕ(H). Hence, if H < ∞, then ϕ(H) = 0; however, as has already been stated previously, this is impossible as ϕ(h) = it(h) and does not vanish anywhere, except at h = ∞. In the following text, we use this notation:

( )

f H m = fm ,

( )

( )

f ′ H m = fm′ ,

( )

f ′′ H m = fm′′,

( )

ϕ′ H m = ϕ′m , ϕ′′ H m = ϕ′′m . © 2007 by Taylor & Francis Group, LLC

276

Wetting and Spreading Dynamics

By differentiating Equation 3.266 with respect to h we conclude that

()

()

()

f ′  h + ϕ h  ⋅ 1 + ϕ′ h  = f ′ h .

(3.269)

At h = 0, we obtain from Equation 3.269 the following expression:

( )(

)

()

f ′ ϕ 0 1 + ϕ′−1 = f ′ 0 = 0 .

(3.270)

As f ′(h) > 0 at h > 0, then ϕ′–1 = –1. We show below that the point at which ϕ′ (h) = –1 is unique. Let us assume that ϕ′ (h0) = –1 when h0 > 0. In this case, from Equation 3.269 we have

( )

( )

( )

f ′ h0 = f ′  h0 + ϕ h0  ⋅ 1 + ϕ′ h0  = 0 , i.e., f ′(h0 ) = 0, which is a contradiction, because f ′(h0 ) > 0. Bearing in mind Equation 3.265, the deﬁnition ϕ(h) gives it′(0) = −1, or 1 . t′ 0

i=−

()

(3.271)

We show below that derivatives of the function f ( h ) at h = H m < H N satisfy the following relation: N −1

fm′ = f N′ ⋅

∏ (1 + ϕ′ ) .

(3.272)

i

i=m

Equation 3.269 at h = H N −1 yields

(

)

(

)

f ′ H N −1 = f N′ ⋅ 1 + ϕ′N −1 .

(3.273)

Assuming that Equation 3.272 is valid at m = l, we show its validity at m = l – 1. From Equation 3.269 at h = H l −1 we have  fl′−1 = f ′ ⋅ 1 + ϕ′l −1 =  f N′ ⋅ 

(

)

N −1   = f N′ ⋅ 1 + ϕ′i  i = l −1

∏(

which proves the relation (3.272). © 2007 by Taylor & Francis Group, LLC



N −1

∏ (1 + ϕ′ ) ⋅ (1 + ϕ′ ) l −i

i

i =l

N −1   ⋅ 1 + ϕ′l −1 = f N′ ⋅ 1 + ϕ′i ,  i = l −1

) (

)

∏(

)

(3.274)

Kinetics of Wetting

277

Now we show by induction that the second derivative with respect to h of the function f(h) at H m < H N satisﬁes the following relation: N −1

f N′′− m = f N′′ ⋅

∏ (1 + ϕ′ )

2

i

m−2

+ f N′ − m+1 ⋅ ϕ′′N − m +

i= N −m

∑ f′

N −k

⋅ ϕ′N′ − k +1

k =0

(3.275)

m −1

×

∏ (1 + ϕ′ ). N − i −1

i = k +1

By differentiating Equation 3.266 twice with respect to h we obtain

()

()

()

2

()

()

f ′′  h + ϕ h  ⋅ 1 + ϕ′ h  + f ′  h + ϕ h  ⋅ ϕ′′ h = ϕ′′ h .

(3.276)

At h = H N −1 , Equation 3.276 changes to Equation 3.275. Assuming Equation 3.275 to hold at m = l, we show its validity at m = l + 1. From Equation 3.275 at H N −l −1 we get

(

)

2

f N′′−l −1 = f N′′−l ⋅ 1 + ϕ′N −l −1 + f N′ −l ⋅ ϕ′′N −l −1 N −1  2 1 + ϕ′i + f N′ −l +1 ⋅ ϕ′′N −l + =  f N′′ ⋅  i = N −l

∏(

(

)

)

l −2

∑

N −1

2

l −1

∑

2

N − i −1

∏ (1 + ϕ′ ) + f ′ 2

i



∏ (1 + ϕ′ )  i = k +1

k =0

× 1 + ϕ′N −l −1 + f N′ −l ⋅ ϕ′′N −l −1 = f N′ ⋅

+

l −1

f N′ − k ⋅ ϕ′′N − k −1 ⋅

N −l



⋅ ϕ′′N −l −1

i = N − l −1 l

f N′ − k ⋅ ϕ′′N − k −1 ⋅

k =0

∏ (1 + ϕ′ ) . 2

N − i −1

i = k +1

It follows from Equation 3.275 that when h = 0,

()

( )

f ′′ 0 = f ′ ϕ −1 ⋅ ϕ′′−1 ,

(3.277)

here f ′(ϕ −1 ) being determined from Equation 3.272 at m = 0. By differentiating Equation 3.266 three times with respect to h and putting h = 0, we obtain

()

( )

f ′′′ 0 = f ′ ϕ −1 ⋅ ϕ′′′ −1 . © 2007 by Taylor & Francis Group, LLC

278

Wetting and Spreading Dynamics

but, on the other hand,

()

f ′′′ 0 = −

( ) = − η′ ( 0 ) ⋅ f ′′ ( 0 ) . η (0) η′ 0

2

2

Consequently,

()

η′ 0 = −

()

ϕ′′′ 0

( )

()

f ′ ϕ −1 ⋅ ϕ′′ 2 0

.

(3.278)

Thus, the sign of η′(0) is opposite to that of the third derivative, ϕ′′′(h), at h = 0. It is shown in the following text that, in the case η(h) = η∞ = const,

()

()

ϕ′′′ 0 = 0, i.e., ϕ′ 0 = η′∞ = 0 . The relations (3.268), (3.272), and (3.275) allow determining the dependence η(h) at thicknesses h = Hm

( )

η Hm =

1 , f ′′ H m

( )

(3.279)

whereas the relation (3.278) gives a criterion for the behavior of viscosity at small h. Thus, we have fully completed the ﬁrst part of the program of determining the viscosity as a function of h.

()

(ii) As f ′′ h =

1 = const at h ≥ H N , then at such h η∞ f (h) =

h2 + Ah + B , 2η∞

(3.280)

where A and B are integration constants, which are as yet unknown. Let us set HN

λN =

∫ dh/ η( h); 0

then, at h ≥ H N , we derive from Equation 3.266 that © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

279

()

h+ϕ

()

f  h + ϕ h  − f h =

h1

1

0

h

=

h+ϕ



∫ dh ∫ ( ) ∫  λ

(h + ϕ) 2 η∞

dh2 = η h2

2

−

N

+

h

h1 − H N  dh1 η∞  (3.281)

h  H  +  λ N − N  ϕ = C. 2∞  η∞  2

On the other hand, from Equation 3.280 we ﬁnd that f (h + ϕ) − f (h) =

( h + ϕ )2 − 2 η∞

h2 + A ⋅ϕ = C . 2 η∞

(3.282)

Comparing Equation 3.281 and Equation 3.282, we infer that A = λN −

HN . η∞

(3.283)

Thus, A speciﬁes an integral deviation of η(h) from η∞ at h < HN . At η(h) ≡ η∞ for all h, we have from Equation 3.283: λN = HN /η∞ and A = 0. At this point, we assume λN and HN to be known, and they are deﬁned in (iv). From the second part of equality (3.281) we get  ϕ2 h − HN  + ϕ λ N − −C = 0; η∞  2 η∞ 

(3.284)

hence, λN =

ϕ (h) h − H N C . − − ϕ ( h ) 2 η∞ η∞

(3.285)

The condition λN = const at h ≥ HN imposes some restriction on the function ϕ(h). From Equation 3.284 ϕ(h) is determined by the following relation: ϕ (h) =

( λ N ⋅ η∞ + h − H N )2 + 2C η∞ − ( λ N ⋅ η∞ + h − H N ) .

(

For λ N ⋅ η∞ + h − H N

)

2

>> 2C η∞ ,we get from Equation 3.286 that ϕ (h) =

© 2007 by Taylor & Francis Group, LLC

(3.286)

C η∞ . λ N η∞ + h − H N

(3.287)

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Wetting and Spreading Dynamics

This expression at h >> [ H N − λ N ⋅ η∞ ] results in ϕ (h) =

C η∞ . h

(3.288)

Multiplying Equation 3.259 by iη∞ and dividing it by τ yields 2 η∞

x ⋅i 2 = 2 hit + ( it ) , τ

or, 2 η∞C = 2hϕ + ϕ 2 . Hence, ϕ ( h ) = h 2 + 2C η∞ − h, h ≥ 0 ,

(3.289)

C ⋅ η∇∞ , h >> 2C η∞ . h

(3.290)

ϕ (h) =

As has been pointed out previously, at

()

HN , η∞

η h ≡ η∞ , λ N =

and, hence, Equation 3.286 transforms to Equation 3.289. From Equation 3.289 we conclude that

()

h

0 > ϕ′ h = −1 +

()

ϕ′′ h =

(

h + 2C η∞ 2

2C η∞ h + 2C η∞

()

ϕ′′′ h = −

2

(h

)

3/ 2

6C η∞ h 2

+ 2C η∞

)

> −1,

> 0,

5/ 2

< 0,

()

ϕ′′′ 0 = 0. These properties mean that the inverse to ϕ(h) = it (h) function, which is h(t), decreases monotonically and is concave.

© 2007 by Taylor & Francis Group, LLC

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Using Equation 3.268, from Equation 3.280 at h = H N we obtain H N2 + AH N + B = ( N + 1) C . 2 η∞

(3.291)

As a result, we ﬁnd that B = ( N + 1) C −

H N2 − A⋅ HN . 2 η∞

(3.292)

Thus, Equation 3.280 determines f (h) at h ≥ H N (with λ N and H N assumed to be known constants). (iii) Let us set f ( h ) = fm ( h )

(3.293)

at H m ≤ h ≤ H m+1 , and

()

()

f h = fN h =

h2 + Ah + B, 2η∞

(3.294)

at h ≥ HN. Let h now vary from HN–1 to HN. In this case, by deﬁnition, h + ϕ(h) varies from HN to HN+1. However, at h ≥ HN , the unknown function, f(h), is already deﬁned and given by Equation 3.294. Hence, in accordance with Equation 3.266, we can determine f(h) at smaller thickness in the interval, HN–1 ≤ h ≤ HN, as

()

()

f N −1 h = f N  h + ϕ h  − C

()

2

h + ϕ h   + A  h + ϕ h  + B − C. =   2 η∞

()

(3.295)

Let us show that f(h) constructed according to Equation 3.295 is continuous at h = HN. Indeed, fN–1(HN = fN (HN+1) – C, but, by deﬁnition of HN+1: fN (HN+1) = fN (HN) + C. From the two last equalities we deduce that fN–1 (HN) = fN (HN). That is, the constructed function is continuous at h = HN. Let us assume now that we have already determined the function fm+1(h), i.e., the value of f(h) in the interval of thickness Hm+1 ≤ h ≤ Hm+2. Now, let h vary

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from Hm to Hm+1. According to the deﬁnition (3.267), h = ϕ(h) varies within Hm+1 to Hm+2. In this range, however, f(h) has already been determined and is equal to fm+1(h). According to Equation 3.266, we deﬁne fm ( h ) = fm+1  h + ϕ ( h )  − C .

(3.296)

By construction the function f(h) satisﬁes the functional Equation 3.266. The uniqueness of the function obtained follows from the additional condition: at h ≥ HN:

()

f ′′ h =

1 = const . η∞

Equation 3.295 and Equation 3.296 result in

( )

(

)

( ) ( ) = ( N + 1) ⋅ C − ( N − m ) ⋅ C = ( m + 1) ⋅ C ,

fm H m = fm+1 H m+1 − C = = f N H N − N − m ⋅ C

which coincides with Equation 3.268. Let us show that the deduced function is continuous at the joining thicknesses, that is, fm ( H m+1 ) = fm+1 ( H m+1 ). We do it for the case of fN–2(h) as an example because other joining thickness can be considered in the same way: f N −1 ( H N −1 ) = f N ( H N ) − C = ( N + 1) ⋅ C − C = N ⋅ C , f N − 2 ( H N −1 ) = f N −1 ( H N ) − C = f N ( H N ) − C = ( N + 1) ⋅ C − C = N ⋅ C . In the latter derivation we used the continuity of f (h) at the thickness h = HN . The two last equalities give f N −1 ( H N −1 ) = f N − 2 ( H N −1 ) = N ⋅ C . Similarly, assuming the continuity of fm+1 (h), we can show the continuity of fm(h) at thickness h = Hm+1. At all other thicknesses (different from the joining thicknesses), the continuity of f(h) follows from the continuity ϕ(h) and relation (3.296). Differentiating relation (3.296) with respect to h yields

()

()

()

fm′ h = fm′ +1  h + ϕ h  ⋅ 1 + ϕ′ h  > 0 . © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

283

As 0 > ϕ′ > –1 and f N′ (h) > 0 at h = Hm, we derive from the last equality

( )

(

)

(

)(

)(

fm′ H m = fm′ +1 H m+1 1 + ϕ′m  = fm′ + 2 H m+ 2 ⋅ 1 + ϕ′m ⋅ 1 + ϕ′m+1

)

N −1

( ) ∏ (1 + ϕ′ ),

= ⋅⋅⋅ = f N′ H N ⋅

i

i=m

which coincides with Equation 3.272. Differentiating relation (3.296) twice with respect to h, we obtain

()

(

)(

)

2

(

)

fm′′ h = fm′′+1 h + ϕ 1 + ϕ′ + fm′ +1 h + ϕ ⋅ ϕ′′ .

(3.297)

From Equation 3.297 at h = Hm, it is possible likewise to derive formula (3.275) by induction. Thus, the function f(h) obtained is a unique solution of Equation 3.266 and satisﬁes all the previous relations (3.268), (3.272), and (3.275). At any value of h, the viscosity is now given by

()

η h =

1 , H m ≤ h ≤ H m+1. fm′′ h

()

(3.298)

We now turn to the determination of the quantities HN

λN =

∫ η(h) dh

0

and HN, which are as yet unknown. Let N0 be some value of the number that we provisionally take as the N sought. In this case, according to Equation 3.285, at h = H N 0 we conclude that λ N0 =

C ϕ − N0 . ϕ N 0 2 η∞

Let ϕ ( N 0 ) (h) denote function (3.286) at λ N 0 , i.e., ϕ( N 0 ) ( h ) =

(λ N

⋅ η∞ + h − H N 0 ) + 2C η∞ − ( λ N 0 η∞ + h − H N 0 ) , 2

0

at h ≥ H N 0 . Determine N as the smallest number N 0 , at which © 2007 by Taylor & Francis Group, LLC

(3.299)

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Wetting and Spreading Dynamics

ϕ ( h ) − ϕ( N 0 ) ( h ) ≤ 0.01 , ϕ (h)

(3.300)

at all h ≥ H N 0 . Relations (3.299) and (3.300) determine the values of the thickness of a boundary layer and λN. Experimental Part [35] The experimental device used for studying the boundary viscosity of volatile substances was substantially the same as that for nonvolatile liquids (Figure 3.20). The body of device 1, made of brass, has a rectangular cutout, which receives a steel insert 3, with its upper surface polished to the ﬁnish quality 14-b and then subjected to additional optical polishing. Electron microscope examination revealed that the surface of the insert was fairly smooth. There are separate randomly oriented scratches with rounded-off edges. Their depth is so small that shading failed to be obtained. Their horizontal size varies from 20–160 Å, and their number is small. In the horizontal scale, the main working surface exhibits roughness of the size smaller than 80 Å; hence, their depth is of the order of 40 Å. At the top, the instrument is covered with prism 5. A plane-parallel channel is formed between the base of the prism and the upper surface of the insert. A current of gas (air or nitrogen) is passed through the channel to blow off the liquid being tested. The thickness of the channel is adjusted by putting calibrated shims 4 between insert 3 and the base 2. For measuring the pressure gradient, two narrow openings 6 with connections to join a pressure gauge are provided in the channel (the connections are not shown in Figure 3.20). In the device, a blow-off current is jetted up by a water-jet pump installed in an exhaust cabinet. This is done to prevent the vapor of volatile liquids from escaping into the air of laboratory rooms. The pressure differential is checked against a liquid differential pressure gauge, using a reading microscope. 6 5 7

7 3 1

1 4

2

FIGURE 3.20 Schematic diagram of the experimental device for investigations of the boundary viscosity of liquids. © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

285

The accuracy in the determination of differential pressure is 1%. Such an accuracy is ensured because an additional capillary tube is connected to the system in series with the instrument channel, the pressure drop being measured at the ends of the capillary tube. The capillary tube is chosen such that the pressure drop in the channel equal to 0.25 mm H2O corresponds to the pressure drop of 200 mm H2O in the capillary tube. This facilitates measurement and ensures the measurement accuracy required. In the experiments, the pressure drop was equal to 0.25 mm of water over the length of the working portion of the ﬁlm, which was equal to 3 cm. This corresponds to the shear stress of about τ = 0.4 dyn/cm2. Experiments were also conducted at other pressure differentials, namely, at 0.06, 0.125, and 0.5 mm of water. A saturated hydrocarbon, hexadecane, was used as a volatile liquid to be tested. Hexadecane of various degrees of puriﬁcation was used: industrial grades qualiﬁed as “chemically pure,” a grade that has been additionally puriﬁed on silica gel using the displacement chromatography method, and a chemically pure grade that has been puriﬁed in a chromatography unit. The content of organic admixtures in hexadecane, which has been qualiﬁed as chemically pure, did not exceed 0.5%; whereas that in the chromatographically puriﬁed grade was below 0.3%. The admixtures in such amounts certainly could not affect the measurement results. As a result of testing hexadecane, interference bands were found to occur, as before, on the insert. The movement of the wetting boundary was recoded. The gap width was chosen to be 1 mm to ensure complete evaporation. It is difﬁcult to blow off substances of very high volatility. In this case, evaporation was so prevalent over ﬂow that during an experiment the spacing between the interference bands would remain extremely narrow and measurements would become difﬁcult. The experiment is carried out as follows: The insert 3 (Figure 3.20) is coated with a ﬁlm of the liquid to be tested. Then, the insert coated with the ﬁlm is placed in the device, and the ﬁlm thickness is measured in the course of blowing off at different distances l from the wetting boundary (l = 10 mm, l = 20 mm, and l = 30 mm) and at different shear stresses (0.1, 0.2, and 0.4 dyne/cm2). The thickness of ﬁlms is measured by the ellipsometric method. Figure 3.21 gives experimental curves representing a variation in the ﬁlm thickness vs. the blow-off time. The curves have been obtained at ∆p = 0.25 mm of water (τ = 0.4 dyne/cm2 and l = 2 cm). The curves that have been obtained at other pressure differentials differ from the given curves in their slope as an increase or a decrease in the pressure differential is equivalent to a decrease or an increase in the blow-off time. The different curves in Figure 3.21 correspond to the virtually identical experimental conditions. However, curves are shifted from each other along the time axis. It may be supposed that this shift depends on the difference in deposition of the original ﬁlm. The initial condition (3.253) means that the initial contact angle of the ﬁlm is 90˚ = π/2. As the real initial state of the ﬁlm being blown off is different from the aforementioned one, this determines certain scattering in the © 2007 by Taylor & Francis Group, LLC

286

Wetting and Spreading Dynamics 700 3 600

500 Film thickness h, Å

4

2 1× ×

400

×

300

× ×

200

×

100

× × 0 2.8

2.9

3.0

3.1

Time t, 103 sec

FIGURE 3.21 Thickness of the hexadecane ﬁlm versus the blow-off time, t.

effective blow-off times. At considerable blow-off times, the inﬂuence of the initial state x(h) results only in a shift of experimental curves from each other, the curves themselves remaining similar. In the case under consideration, the dependences η(h) calculated by the above method prove to be practically the same. Certain minor differences are observed for larger thicknesses. This is quite understandable because the inﬂuence of the deposition of the initial ﬁlm prior to blowing off can inﬂuence only the initial stage of the process (at high-enough thicknesses). The experimental data were processed according to Equation 3.298, which gives the dependence of the boundary viscosity on ﬁlm thickness. Figure 3.22 represents the deduced dependences of the boundary viscosity on ﬁlm thickness. Figure 3.22 shows that in the thickness range from 50 to 200 Å, the viscosity of hexadecane is lower than the bulk viscosity of liquid. Note, the same decreased value of hexadecane viscosity in the boundary layer was obtained for two different pressure differences, ∆p = 0.25 mm, and ∆p = 0.06, 0.125, and 0.5 mm of water when the evaporation rates differ. This shows the independence of the procedure from evaporation rates. Measurements of viscosity at distances l = 1 cm and l = 3 cm from the wetting boundary also yielded the same decreased viscosity values within the indicated thickness range. Therefore, the phenomenon of ﬁlm spreading at the solid substrate (in the opposite direction to the blow-off gas) can be neglected. Otherwise, the spreading phenomenon would be most pronounced at l = 1 cm. © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

287 4

Viscosity η, P

3

12

0.03

0.02

0.01

0

100

200

300

400

500

Film thickness h, Å

FIGURE 3.22 Calculated dependencies of the boundary viscosity of hexadecane on the ﬁlm thickness. Curves from 1 to 4 corresponds to curves from 1 to 4 in Figure 3.21.

A hexadecane molecule is composed of 14 methylene groups that interact with both the substrate and each other, and of two end methyl groups. The number of methyl groups being small, they do not play an important part. In the layer closest to the wall, one may expect a horizontal orientation of molecules owing to their sufﬁcient rigidity and length. The observed decrease in the viscosity of hexadecane seems to be attributable to the horizontal orientation of molecules.

CONCLUSIONS The theory thus developed allows us to determine the boundary viscosity as a function of the distance to the substrate, η(h). The dependency η(h) is determined according to Equation 3.298, using the extension procedure set forth in (iii). In the case of constant viscosity, the thickness of a layer at a given point varies with time according to the hyperbolic law h∞ ( t ) = −

it x η∞ . + 2 τ ⋅t

3.9 COMBINED HEAT AND MASS TRANSFER IN TAPERED CAPILLARIES WITH BUBBLES UNDER THE ACTION OF A TEMPERATURE GRADIENT Simultaneous ﬂows of vapor and a liquid in a thin liquid ﬁlm under the action of both temperature and pressure gradients are investigated as a function of the radius and taper of capillaries for decane and hexane. The regions of the greatest effect of ﬁlm ﬂow are established [58]. © 2007 by Taylor & Francis Group, LLC

288

Wetting and Spreading Dynamics y α

T0

h(x)

r0

P0

0

P0

rL L

x TL PL

Ps0

PL PsL L

FIGURE 3.23 Combined heat and mass transfer in a tapered capillary of the length L. Radius of the capillary at cross section x is r(x) = r0 + x·tanα, imposed temperature gradient is constant,

dT dx

=

TL − T0 L

= const.

Combined heat and mass transfer in porous media is of great interest in a number of areas. Porous space inside any porous material has a sophisticated structure and is difﬁcult to model even using computer simulations. It is even more difﬁcult to model a combined heat and mass transfer in such complicated structures. In the following text we investigate a mass transfer of oils in a model porous system, which is a tapered capillary. The mass transfer takes place between two menisci of oil at x = 0 and x = L under the action of an imposed constant temperature gradient (Figure 3.23). We assume that the characteristic scale in the axial direction is much bigger than the capillary radius — that is, r(x) << L. As a result of vapor adsorption and the liquid ﬂow on the capillary walls, a ﬁlm of adsorbed liquid, h(x), forms on the walls. It is assumed that in each cross section there is a local equilibrium between the vapor and the adsorbed liquid. According to Chapter 2, the hydrodynamic pressure in the adsorbed liquid ﬁlm, Pl, is equal to Pl ( x ) = Pa −

γ [T ( x )] r( x) Π [ h( x ) ] , − r ( x ) − h( x ) r ( x ) − h( x )

(3.301)

where Pa is the pressure in the ambient air, and γ(T) and Π(h) are the liquid–vapor interfacial tension, which is temperature dependent, and the disjoining pressure of the ﬂat oil ﬁlms, respectively. A bubble of air separates two menisci of the wetting liquid having a different curvature. The surface of the capillary in between the two menisci is covered by © 2007 by Taylor & Francis Group, LLC

Kinetics of Wetting

289

a wetting ﬁlm whose thickness, h(x), is a function of the coordinate x; boundary values of the ﬁlm thickness are h(0) = h0 and h(L) = hL. The excess pressure in the liquid ﬁlm as before is P = Pa − Pl .

(3.302)

Note that now the excess pressure is a function of the position. Equality of chemical potentials of the liquid molecules in the vapor phase and in the liquid ﬁlms results in the same equation as in Chapter 2, which applied however locally in each cross-section: P=

RT ps (T ) , ln vm p( x )

(3.303)

where T(x), ps(T), and p(x) are the local temperature, the local pressure of the saturated vapor, and the local vapor pressure, respectively; R is the gas constant, and vm is the molar volume of the liquid. In the following text we assume that the liquid under investigation is an oil, which completely wet the capillary walls. Disjoining pressure in this case is determined by dispersion forces only — that is, Π(h) =

A , h3

where A is the Hamaker constant. The combination of Equation 3.301 and Equation 3.303 results in Pl ( x ) = Pa −

=

γ T ( x ) 

r ( x ) − h( x )

−

A 3 (x) h r ( x ) − h( x )  r (x)

(3.304)

RT ( x ) ps T ( x )  ln . vm p( x )

See Chapter 2, Section 2.7 and Section 2.12 for explanations on why the modiﬁed disjoining pressure r (x) A 3 r ( x ) − h( x )  h ( x ) should be used. Now we should deduce an equation for a ﬂow in thin liquid ﬁlms under the action of both the pressure gradient and the surface tension gradient. We consider a ﬂow in thin ﬁlms both at low Reynolds number Re << 1 and capillary numbers Ca << 1 because, in the following text, we consider only slow

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steady-state ﬂow. This means that both front and rear menisci (Figure 3.23) have an equilibrium proﬁle, and we ignore any deviation from the equilibrium shapes. Taking into account the condition r(x) << L, we can conclude as we did earlier in this chapter that (1) the velocity in the radial direction is much smaller than the velocity in the axial direction, (2) the velocity in the axial direction, v, depends only on the radial position r, and (3) the pressure in the ﬁlms depends only on the axial component x. After that, the remaining Stokes equation, taking into account the deﬁnition (3.302), becomes  d 2v 1 dv  − P′ = η  2 + , r dr   dr with nonslip condition on the capillary walls v(r ( x )) = 0 , and the condition on the free ﬁlm surface

η

dv dγ dγ = = T′ , dr r = r ( x )− h dx dT

where ′ marks the derivative with respect to x. This condition shows the tangential stress on the free ﬁlm surface caused by the presence of the surface tension gradient, which, in its turn, is caused by the temperature gradient. Solution of the preceding equation with two boundary conditions and Equation 3.304 results in the following expression for the mass ﬂow rate, Q: Q = Q1 + Q2 + Q3 ,

(3.305)

where the following expressions for mass ﬂow rates in vapor, Q1; in the thin liquid ﬁlm under the action of the temperature gradient (thermocapillary ﬂow), Q2; and under the action of the pressure gradient, Q3, are: Q1 = − λ1

π(r − h)2 Dµ p′ RT

πr ρh 2 d γ Q2 = λ 2 T′ η dT

© 2007 by Taylor & Francis Group, LLC

(3.306)

Kinetics of Wetting

291

Q3 = − λ 3

2πr ρRTh 3  p′ ps′ T ′ ps  − + ln  3ηvm  p ps T p

where D is the diffusion coefﬁcient of the vapor; ρ, η, and µ are density, viscosity and mass of the molecule of the liquid, respectively. We introduced additional coefﬁcients λ1 , λ2 , and λ3 . Each of these coefﬁcients can be 1 or zero to “switch” on or off the corresponding part of the mass ﬂow rate. The dependency of the ﬁlm thickness on x, h(x), is determined by Equation 3.304. The boundary conditions for the ﬁrst Equation 3.305 are as follows:  2 γ [ T ( 0 ) ] vm  p(0 ) = ps [T (0 )] exp  −  r (0 )RT (0 )   2γ [T ( L )] vm  p( L ) = ps [T ( L )] exp  −  r ( L )RT ( L ) 

(3.307)

where we took into account that the saturated pressure depends only on the temperature, T. Equation 3.305 was differentiated with respect to x, and after that, was solved numerically using a quasi-linearization method [36] combined with the method of iterations:

(Q

n

)

− Qn+1 Qn < ε ,

(3.308)

where n is the number of iterations and ε = 0.01 is the given relative error. Even in the case where condition (3.308) is satisﬁed, not less than 10 iterations were made. The calculations were made using mean values of ρ, η, and D inside the bubble length, L, corresponding to the mean temperature Tm = (T1 + T2)/2. It was assumed that a constant temperature gradient is imposed: dT/dx = const, i.e., that the distribution of the temperature, T, was linear inside the bubble. The distribution of the vapor pressure, p(x), corresponding to the imposed value of dT/dx = const, and the value of the total mass ﬂow rate, Q, were calculated. Tabulated values of the liquid–air interfacial tension, γ(T), and the saturated vapor pressure, ps(T), were used in the calculations. The intermediate values were found by linear (for the interfacial tension, γ) and logarithmic (for the saturated pressure, ps) interpolations. Calculations were made for two nonpolar liquids with a different volatility: decane and hexane. The well-known equation of the molecular component of the disjoining pressure, Π(h) = A/h3, where A ~ 10–13 erg [37], was used as the isotherm Π(h). The value of the Hamaker constant, A, was regarded as independent of the temperature [38]. Dependences of the total mass ﬂow rate, Q, and of

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the individual components of the ﬂow rates on the radius and the taper of the capillaries were obtained for dT/dx = const from 1 to 100 deg/cm with a constant mean temperature Tm = 300˚K. The total mass ﬂow rate was calculated using Equation 3.305, setting λ1 = λ2 = λ3 = 1. The vapor component, Q1, corresponding only to diffusion of the vapor (in the presence of liquid ﬁlms on the solid surface results in decreasing of the cross section of the capillary), was calculated with λ1 = 1 and λ2 = λ3 = 0. An analogous method was used to calculate the components Q2 (with λ2 = 1 and λ1 = λ3 = 0) and Q3 (with λ3 = 1 and λ1 = λ2 = 0), corresponding, respectively, to the thermocapillary ﬂow of the ﬁlm and the ﬁlm ﬂow under the action of the pressure gradient dP/dx, due to the gradients of the disjoining and capillary pressures (see Equation 3.304). The instantaneous radius of the conical capillary is given by the following equation: r = r1 + αx ,

(3.309)

where α = [r(L)–r(0)]/L = ∆r/L is the angle of taper of the capillary. Note that we are using the small angle approximation. For hexane, the following values were given: D = 0.075 cm2/sec; µ = 86.17 g/mol; ρ = 0.6534 g/cm3; η = 2.95 · 10–3 P; v = 132 cm3/mol. For decane, the corresponding values are: D = 0.046 cm2/sec; µ = 142.28 g/mol; ρ = 0.7245 g/cm3; η = 8.2 · 10–3 P, v = 196 cm3/mol. The ratio L/r varied from 10 (for large capillary radius, r) to 1000 (for small capillary radius, r). The absolute values of L were from the following range: from 10–1 to 10–3 cm. In view of this choice, with high temperature gradient, dT/dx = const, the temperature difference at the boundaries of a bubble was very small. This justiﬁed the possibility of using constant mean values of ρ,η, and D.

CYLINDRICAL CAPILLARIES We consider ﬁrst the effect of the radius of the capillaries on ﬁlm ﬂow. To this end, the values of the ratio Q/Q1 were calculated for cylindrical capillaries of equal radius r (Figure 3.24). At Q/Q1 = 1, the principal mechanism of mass transfer is diffusion of the vapor: Q1 >> Q2 + Q3. Figure 3.24 shows that, with an increase in the radius of the capillaries, the effect of ﬁlm ﬂow decreases. However, in narrow capillaries (r ∼ 10–6 cm), ﬁlm ﬂow is the principal mechanism of transfer. The ﬂow of vapor, Q1 in such thin capillaries makes a contribution to the total ﬂow, which is an order of magnitude less, although the thickness of the ﬁlms still remains considerably less than the radius of the capillaries (h/r = 0.14). The effect of ﬁlm transfer is more pronounced because the less volatile decane: curve 1 passes above curve 2 for hexane. The results of the calculations do not disclose a dependence of Q/Q1 on the value of the temperature gradient, dT/dx = const, which is explained by the smallness of the absolute values of T1 – T2, leading to a practically linear dependence of all the ﬂows on the temperature gradient, dT/dx = const. © 2007 by Taylor & Francis Group, LLC

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293 h(x)

r(x) 12

x

0

1

2

8

L b

1

2 4

4

0 10–2

Q/Q1

a

10–4 r, cm

10–6

0

FIGURE 3.24 (a) Calculating scheme, which was used both in the case of tapered (r(0) < r(L)) and cylindrical capillaries (r(0) = r(L)); (b) dependences of the ratio of the total mass ﬂow rate, Q, to the mass ﬂow rate of vapor, Q1, on the radius of cylindrical capillaries for decane (1) and hexane (2): r = 10–2 cm, L = 10–1 cm; r = 10–3 – 10–5, L = 10–2; r = 10–6, L = 10–3.

Flow in thin ﬁlms in cylindrical capillaries is mainly determined by thermocapillary ﬂow. The pressure-driven ﬂow, Q3, is directed to the opposite side as compared with the ﬂows of vapor, Q1, and thermocapillary ﬂow, Q2, and is always much less as compared with thermocapillary ﬂow, Q2. This can be shown also by obtaining analytical dependences for the individual components of the ﬂow. Thus, in a cylindrical capillary, the ratio Q3 /Q2 is equal to

(

Q3 /Q2 ≈ 4 / 3 A/ γr 2

)

1/ 3

.

(3.310)

At r → ∞, (Q3/Q2) → 0. Therefore, we evaluate the ratio (3.310) for the thinnest capillaries, setting A = 10–13 erg and γ = 30 dyn/cm. Calculations show that, for r ≥ 10–6 cm, the contribution of the ﬂow Q3 does not exceed 3% in comparison with the ﬂow Q2.

TAPERED CAPILLARIES The picture of the ﬂows in conical (tapered) capillaries has quite different features as compared with the ﬂow in cylindrical capillaries. With an increase in the angle of taper of the capillary, α, the contribution of the ﬂow, Q3, due to the differences in the capillary pressures of the menisci, rises and, at sufﬁciently large values of © 2007 by Taylor & Francis Group, LLC

294

Wetting and Spreading Dynamics 10 5 0

α∗ 10–3

10–4

10–5

10–6

Q/Q1

α 1 –10

2

–20

FIGURE 3.25 Dependences of Q/Q1 on the taper of the capillaries, α, for decane (1) and hexane (2): r0 = 10–4 cm, L = 10–2 cm, dT/dx = –1 deg/cm.

α, becomes dominating. As an example, Figure 3.25 shows dependences of Q/Q1 on α for r0 = 10–4 cm. As can be seen from Figure 3.25, at α ≤ 10–5, the ratio of Q/Q1 remains constant and has exactly the same value as in a cylindrical capillary of radius r = r(0). The dashed lines show the course of the dependences of (Q1 + Q2)/Q1 on α. The ﬂow rates Q1 and Q2 depend only weakly on α. Thus, the deviation of curves 1 and 2 downward with α ≥ 10–5 is connected only with the effect of the ﬂow Q3 directed toward the side of the narrowing of the capillary. At α ≥ 10–4, its effect becomes dominating, as a result of which the total ﬂow rate, Q, changes sign. At some value of the taper α = α*, the total ﬂow rate vanishes, Q = 0. Hence, the ﬂow of vapor, Q1, and the thermocapillary ﬂow, Q2, equalize out in a direction opposite to the direction of the ﬁlm ﬂow, Q3. Such circulating ﬂows are realized, for example, in heat pipes as well as in porous media completely saturated by a liquid. The local taper of a pore, corresponding to the condition Q = 0, depends on the temperature gradient and the radius of the capillaries. It can be found using the calculating procedure suggested in this section. In porous bodies, where the capillaries have a variable radius, bubbles of air, displaced toward the hot side, can be held in expanded pores if the values of r, α, and the imposed temperature gradient, ∆T, are such that the condition Q = 0 is satisﬁed. This hold-up means that the air- and, consequently, moisture-content cannot vary with time, in spite of the existence of the imposed temperature gradient, ∆T, as the ﬂow under the action of the temperature gradient, ∆T, is compensated by a reverse ﬂow under the action of the difference arising in the capillary pressures. © 2007 by Taylor & Francis Group, LLC

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1

2

100

Q/Q1

50

10

–50

α∗

α0

0

–5

10

–6

10

–7

10

α –8

–9

10

2

1 –100

FIGURE 3.26 Dependences of Q/Q1 on the taper of the capillaries, α, for decane (1) and hexane (2): r = 10–6 cm, L = 10–2 cm, dT/dx = –1 deg/cm.

With a transition to thinner capillaries, the picture of the mass transfer is becoming even more complicated. Figure 3.26 gives the results of calculations for r(0) = 10–6 cm with the same values of L and ∆T as in Figure 3.25. Whereas, at a small taper (α < 10–7), the dependences of Q/Q1 have qualitatively the same shape as in Figure 3.25, in the region of larger values of α, the dependency of Q/Q1 on α undergoes a discontinuity and changes the sign. The reasons for such a course of the curves can be established by analyzing the dependences of the individual components of the ﬂow on α. It follows from the calculations that the thermocapillary ﬂow, Q2, is relatively small and almost does not depend on α. The ﬂow of vapor, Q1, directed at small angles α toward the cold side, starts to be retarded with a rise in α, as a result of an increase in the pressure of the vapor above the meniscus r(L). Note that in the calculations it was assumed that r(0) = const, r(L) = r(0)+αL, that is, it increases with α. At α = α0 = 5⋅10–7, the difference ∆p, due to the different temperatures of the meniscuses, is compensated by the Kelvin’s difference in the pressures ∆p, connected with the different curvature of the meniscuses. At α > 5⋅10–7, the ﬂow of vapor changes direction: the Kelvin’s difference in the pressures of the vapor exceeds the thermal difference. An increase in the taper, α (at L = const), leads to a sharp increase in the rate of the back ﬂow of vapor (Q1 < 0). © 2007 by Taylor & Francis Group, LLC

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Distinct from the vapor ﬂow, Q1, the pressure ﬂow in thin liquid ﬁlms, Q3 , is always directed toward the side of the narrower part of the capillary, and Q3 < 0. With an increase in the taper, α, the absolute values of Q3 rise. At α > 5⋅10–8, Q3 makes the principal contribution to the total ﬂow; hence, here it can be assumed that Q ∼ Q3. As the ﬂow Q1 approaches zero, the ratio Q/Q1 = Q3/Q1 → ±∞, where Q/Q1 → −∞ at Q1 > 0 and Q3/Q1 → + ∞ at Q1 < 0, which explains the results given in Figure 3.26. As can be seen from Figure 3.26, at α > α0 , the values of Q/Q1 fall with an increase in the value of α. This is explained by the decreasing contribution of the ﬁlm ﬂow, Q3 , with an increase in the mean radius of the capillary rm = r(0) + (α/2). In thin pores (r ∼ 10–6 cm) and for small bubbles (L = 10–2 cm), even a small taper leads to a sharp increase in the rate of mass transfer due to the ﬂow of the liquid ﬁlms. In the region of values of the taper α ~ α0 , the ﬂow Q3 ~ Q can exceed the ﬂow of vapor. At α = α0 , the ﬂow in the liquid phase is the sole mass transfer mechanism in the system. For the less volatile decane, the effect of ﬁlm transfer is more strongly expressed (curves 1) than for hexane (curves 2) (Figure 3.26). The smaller the radius of the capillary, the greater the role of the taper; even the absolute value of the taper is insigniﬁcant. Whereas at r(0) = 10–4 cm, the effect of the taper starts to be appreciable at α > 10–5, with r = 10–6 cm, it is already appreciable at α > 10–8.

3.10 STATIC HYSTERESIS OF CONTACT ANGLE We have argued in Chapter 1, Section 1.3, that the hysteresis of the static contact angle is usually related to the heterogeneity of the surface, either geometric (roughness) or chemical. The underlying assumption was that at each point of the surface there is an equilibrium value of the contact angle, depending only on the local properties of the substrate. As a result, a whole series of local thermodynamic equilibrium states can be realized, corresponding to a certain interval of values of the contact angle. The maximum possible value corresponds to the value of the static advancing contact angle of wetting, θa, and the minimum corresponds to the static receding contact angle, θr . In this section we present a theory of static hysteresis of the contact angle on solid homogeneous substrates developed in terms of quasi-equilibrium phenomena. The values of the static receding, θr , and static advancing, θa , contact angles are calculated, based on the shape of disjoining pressure isotherm. It is shown that all contact angles, θ, in the range θr < θ < θa, which are different from the unique equilibrium contact angle, θe, correspond to the state of slow microscopic advancing at all contact angles, θ, in the range θe < θ < θa or receding motion at all contact angles, θ, in the range θr < θ < θe, respectively. This microscopic motion, step-wise, becomes fast macroscopic advancing or receding motion after the contact angle reaches the critical values θ = θa or θr = θ, correspondingly [59].

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There is no doubt that heterogeneity affects the wetting properties of any solid substrate. However, heterogeneity of the surface is apparently not the sole reason for static hysteresis of the contact angle. This follows from the fact that not all the predictions made on the basis of this theory have turned out to be true [39,40]. Besides that, static hysteresis of the contact angle has been observed in cases of quite smooth and uniform surfaces [41–45]. Further, it is present even on surfaces that are deﬁnitely molecularly smooth: free liquid ﬁlms [60,61]. Evidently, only a single unique value of an equilibrium contact angle, θe, is possible on a smooth, homogeneous surface. Hence, the static hysteresis contact angles θa ≠ θe, θr ≠ θe, and all contact angles in between, observed experimentally on such surfaces, can correspond only to a nonequilibrium or quasi-equilibrium state of the system. However, if the relaxation time of the system is long, a local equilibrium that is not in equilibrium with the surrounding medium can be established at the meniscus or droplet. It is shown in the following text that there are certain critical values of contact angles beyond which such a local equilibrium is not possible because the relaxation time becomes very small. We relate these critical values of the angles to static advancing, θa, and static receding, θr , contact angles. Thus, the discussion of the static hysteresis phenomenon in this section is based on the analysis of nonequilibrium states of a system and conditions of violation of local equilibrium of menisci or drops.

EQUILIBRIUM CONTACT ANGLES In Chapter 2, Section 2.3, we considered an equilibrium state of the wetting meniscus of a liquid (0 ≤ θe < 90˚) in a capillary with a ﬂat slit of breadth 2H >> he, where he is the thickness of the equilibrium wetting ﬁlm covering the surface of the capillary (Figure 2.13). If H >> he, the liquid in the central part of the slit is far from the range of the surface forces action. Neglecting the effect of gravitational forces, the radius of curvature of the surface of the meniscus in the central part of the slot, ρe, is constant. We denote by h the distance along the normal between the substrate and the surface of the liquid. Then h(x), as before, is an equation of the proﬁle of the liquid layer, where x is the coordinate in the direction of the plane of symmetry of the capillary. Between the meniscus of constant curvature, ρe, and the ﬂat ﬁlm of thickness, he = const, there is a transition zone. Only capillary forces act in the region of the unperturbed meniscus, and only surface forces act in the ﬂat ﬁlm. However, both these forces act simultaneously within the boundaries of the transition zone (see Chapter 2, Section 2.3, for details). When applied to a layer of liquid h(x), the equilibrium condition corresponds to a constant hydrostatic pressure, which can be written as follows (see Chapter 2, Section 2.1 and Section 2.3):

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298

Wetting and Spreading Dynamics

γh′′ + Π(h ) = Pe , (1 + h′ 2 )3/2

(3.311)

where the ﬁrst term characterizes the contribution of the capillary pressure caused by the local surface curvature. The second term of Equation 3.311 characterizes the contribution of the disjoining pressure (surface forces) acting from the side of the substrate. Because of the symmetry of the meniscus, it is sufﬁcient to consider the equilibrium of a liquid layer h(x) over the thickness range from h = he at x = ∞ to h = H at x = 0. We solved Equation 3.311 in Section 2.3. Solution of Equation 3.311 includes three constants, namely, the excess pressure Pe = const, and two integration constants. For determining these constants, we used in Section 2.3 three boundary conditions, with two at the center of the meniscus: h = H, h′ = −∞, x = 0 ,

(3.312)

and the condition of conjugation of the meniscus with the ﬂat ﬁlm: h = he, h′ = 0, x → ∞ .

(3.313)

We multiply both sides of Equation 3.311 by h′ and integrate it with respect to x from 0 to x. After that we arrived in Section 2.3 at the following solution, γ 1 + h′ 2

= ψ (h, Pe ) ,

(3.314)

where ∞

∫

ψ ( h, Pe ) = Pe ( H − h ) − Πdh .

(3.315)

h

By solving Equation 3.315 with respect to h′, we obtain 1/ 2

 γ2  − 1 . h′ = −  2  ψ (h, Pe ) 

(3.316)

From this follows in particular the condition of equilibrium of the meniscus with the ﬁlm (see Section 2.3). © 2007 by Taylor & Francis Group, LLC

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Substituting the condition (3.313) in (3.316), we conclude: ∞

∫

γ = ψ ( he , Pe ) = Pe ( H − he ) − Πdh .

(3.317)

he

In Section 2.3, using this equation, we deduced the expression (2.47) for the equilibrium contact angle, θe. For each value of H, the unique values of Pe can be found from Section 2.3. When the whole isotherm of the disjoining pressure is within the region Π > 0, i.e., in the case where complete wetting is realized, the concept of the contact angle cannot be introduced, as the radius of the meniscus, ρe, of the circumference of the existent transition zone does not intersect the surface of the capillary (see Chapter 2, Section 2.4, where the proﬁle of the transition zone was calculated analytically). Figure 3.27a shows two types of isotherms (curves 1 and 2) when partial wetting is possible. The ﬁrst one has two branches, α and β, corresponding to Π

a

Pe > Πmax α β

Πmax

Pe < Πmax

0 h

1 2

Πmin

b

he

ψ

hu

h3

hβ

γ 5

h4

3 4 0

h1

h2

h

FIGURE 3.27 (a) Shapes of the disjoining pressure isotherms (curves 1, 2) in the case of partial wetting; (b) corresponding dependences of ψ(h, P) (curves 3–5), which determine the condition of the local equilibrium of the meniscus. See explanation in the text. © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

stable states of the ﬁlm. Systems that possess isotherms of the second type (curve 2, Figure 3.27a) have large equilibrium contact angles, θe. For example, isotherms of the ﬁrst type are inherent to water ﬁlms on quartz surfaces [26]. Only α ﬁlms are observed on the surface of the capillary if Pe > Πmax, where Πmax is the stability limit of β ﬁlms (Figure 3.27a). This happens in sufﬁciently narrow capillaries. If Pe > Πmax, the formation of metastable β ﬁlms is possible, as observed experimentally in Reference 26. According to the data in Reference 46, this condition is fulﬁlled, e.g., in the case of water in quartz capillaries with a radius bigger than 0.25 µm. For analyzing the equilibrium proﬁle of a liquid in a ﬂat capillary, we rewrite Equation 3.311 in the following form:

h′′ =

( )

()

3/ 2 2 1 1 + h′   Pe − Π h  .  γ 

(3.318)

Equation 3.318 shows that the sign of h″, and therefore the curvature of the surface, is determined by the sign of the difference, Pe – Π(h). Note that Pe = const, but the value of Π(h) varies from Π(he) = Pe to Π = 0 at h → H. If Pe > Πmax, all the values of the difference are positive and the liquid proﬁle inside the transition zone is concave everywhere (h″ > 0). This condition is always fulﬁlled in the case of an isotherm of type 2. However, in sufﬁciently thick capillaries, Pe < Πmax in the case of disjoining pressure isotherms of the ﬁrst type. Then, h″ < 0 over the range of thickness from hu to hβ (Figure 3.27), and the surface of the liquid should possess a convex segment. According to Equation 3.314, the following inequality should hold: ψ(h, Pe) ≥ 0. On the other hand, Equation 3.316 shows that the second inequality should be satisﬁed: ψ(h, Pe) ≤ γ. Hence, the function ψ(h, Pe) should be located within the following limits: 0 ≤ ψ ( h, Pe ) ≤ γ .

(3.319)

Note: ψ(h, Pe) = 0 means the inﬁnite derivative at the corresponding thickness h, which is possible only in the center of the capillary at h = H. The equality ψ(h, Pe) = γ means the zero derivative at the corresponding thickness h. This is satisﬁed at h = he , and, based on this condition, we deduced an expression (2.47) for the equilibrium contact angle in Chapter 2, Section 2.3. As an example of the equilibrium function ψ(h, Pe), curve 3 is shown in Figure 3.27. The positions of the extrema of curve 3 can be found by differentiation of Equation 3.315 with respect to h, which results in d ψ(h, Pe ) = − Pe + Π ( h ) . dh

© 2007 by Taylor & Francis Group, LLC

(3.320)

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301

Hence, as the liquid thickness varies between he < h < H, the positions of the maximum and minimum on curve 3 correspond to the thicknesses hu and hβ, at which the straight line Pe = const intersects the disjoining pressure isotherm 1 and Pe = Π(h) (Figure 3.27).

STATIC HYSTERESIS

OF THE

CONTACT ANGLE

OF

MENISCI

We now assume that the pressure over the meniscus is changed by a value ∆P relative to the value of its equilibrium pressure Pe (Figure 3.28b), whereas the ﬁlm in front and the gas maintain the initial equilibrium state. Further consideration shows that the initial state as well as the presence or absence of the ﬁlm in front (zone 3 in Figure 3.28b) does not inﬂuence our consideration. That is, the same consideration as below can be applied to the static hysteresis of contact angle on initially dry surface. A transport process starts at once under the action of the pressure difference created, which can be separated provisionally into a rapid and a slow one. As the resistance of the liquid–gas interface to changes of its form is small at low capillary numbers, Ca << 1, the most rapid change is that of the curvature of a

h θe

he

ρe

2H

Pe

h(x)

0

x

b h

hc θ

P = Pe + ∆P

0

ρ

he 2H

h(x)

1

2

3

x

FIGURE 3.28 Schematics of the proﬁle of the menisci in a ﬂat capillary of the thickness 2H. (a) equilibrium state, no ﬂow; (b) a state of local equilibrium, when the ﬂow is located in zone 2 ((1) the new quasi-equilibrium spherical menisci and quasi-equilibrium transition zone, (3) the equilibrium liquid ﬁlm, which cannot be at the equilibrium with zone 1). © 2007 by Taylor & Francis Group, LLC

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the meniscus. As a result of this, a new quasi-equilibrium state of the meniscus is formed with the pressure drop: P = Pe + ∆P =

γh′′

(1 + h′ )

2 3/2

=

γ = const , ρ

(3.321)

outside the boundaries of the zone of action of surface forces, where ρ = cosθ/H is the radius of curvature of the main part of the meniscus, and θ is a new value of the contact angle, corresponding to the local equilibrium state (Figure 3.28b), which cannot be equilibrium any more. The new state of the whole system is not in equilibrium. The change in the curvature of the meniscus causes a change in the vapor pressure over it, as result of which the liquid starts to transfer by evaporation or condensation from the surface of the meniscus on the surface of the equilibrium ﬁlm in front (zone 3 in Figure 3.28b). Besides, the increase in pressure inside the liquid causes it to ﬂow to those parts of the ﬁlm at which the initial equilibrium pressure Π(he) = Pe is still maintained. All of this remains valid when the sign of ∆P is opposite. However, in the latter case, the direction of the transfer processes is inverted. In the following text we limit the discussion to liquids with low volatility whose rate of evaporation and condensation are small. The main assumption is that the liquid ﬂow from the quasi-equilibrium meniscus to the equilibrium ﬁlm in front is very slow until some critical pressure difference, ∆Pa (in the case of advancing meniscus) or ∆Pr (in the case of receding meniscus) is reached. These conditions may not exist in the case of complete wetting, when the equilibrium ﬁlm is sufﬁciently thick. However, it is known that in the case of complete wetting there is no static hysteresis of the contact angle. Static hysteresis is usually observed in cases of partial wetting (at 0 < θ < 90˚), when the surface of the solid body is covered with substantially thinner ﬁlms, where the viscose resistance is very high. At ∆P ≠ 0, i.e., in a nonequilibrium system, we subdivide the whole system into the following regions (Figure 3.28b): a region 1 with a state of quasiequilibrium inside where the hydrodynamic pressure is constant everywhere and equals P = Pe + ∆P; a transport region 2 where a viscous ﬂow of liquid occurs and in which the pressure gradually changes from the value P to Pe; and a region 3 of a thin ﬂat ﬁlm, where the pressure equals the initial equilibrium Pe (Figure 3.28b). As the largest pressure drop in the transition region occurs in the thinnest part 2, it is evident that region 2 covers part of the transition region that immediately adjoins the equilibrium ﬁlm he in region 3. We write the conditions of quasi-equilibrium of the meniscus in region 1, in the boundaries of which ﬂuxes can be neglected and where the excess pressure can be considered to be constant at all points and equal to P = Pe + ∆P = const. We assume that Equation 3.311 still describes the quasi-equilibrium of proﬁle of the liquid, h(x), in region 1 in the absence of full thermodynamic equilibrium in the whole system. That is, we adopt: © 2007 by Taylor & Francis Group, LLC

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303

γh′′ + Π(h ) = P . (1 + h′ 2 )3/2

(3.322)

The boundary conditions (3.312) at the center of the meniscus are maintained. Instead of (3.313), another condition should be used, as region 1 cannot now be connected by the equilibrium or quasi-equilibrium liquid proﬁle with the ﬂat equilibrium ﬁlm in front; there is a ﬂow zone 2 in between (Figure 3.28b). This third condition is P = const, as the value of the excess pressure P is now ﬁxed independently and is not determined by the thickness of the slit and the isotherm of the disjoining pressure, as in the case of the equilibrium. The region of solution of Equation 3.322 is limited from below by a certain thickness hc ≥ he, corresponding to the beginning of the ﬂow zone (Figure 3.28b). The condition h′ = 0 is not fulﬁlled at h = hc, and a micro-contact angle, tanθ ≈ θ = –h′(he), is formed here, the values of which can be found by solving Equation 3.322. As previously, by integrating Equation 3.322 once, we obtain once more Equation 3.314 to Equation 3.316, but with the difference that here ∞

∫

ψ ( h, P ) = P ( H − h ) − Πdh ,

(3.323)

h

where the equilibrium pressure, Pe, is replaced by the new nonequilibrium pressure, P. The region in which a solution of Equation 3.322 exists is determined by the real values of h′, which exist only under the same conditions as in the case of equilibrium: 0 ≤ ψ ( h, P ) ≤ γ .

(3.324)

There is no solution if ψ > γ or ψ < 0: if any of these conditions is violated, then the boundary of the ﬂow zone, hc, and the center of the meniscus cannot be connected by a continuous proﬁle. This implies that quasi-equilibrium becomes impossible, i.e., the meniscus cannot be at rest and must start the motion. We show below that the violation of one of the conditions (3.324) determines an advancing contact angle, θa, and the violation of the other condition, the receding contact angle, θr. We refer to these two angles as static advancing and receding contact angles, respectively. First, we determine the value of the static advancing contact angle, θa. For the motion of the meniscus in front of the capillary, the pressure in the liquid phase must be increased, which diminishes the capillary pressure drop at the meniscus–gas interface. Consequently, in this case, P < Pe and ∆P < 0. This means that curve ψ(h, P) (curve 4) is located below curve ψ(h, Pe) (curve 3) in © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics a

b h

H

H

ρa

ρr

Pr

Pa

h4 h1 0

θa

θr x

FIGURE 3.29 Determination of the critical proﬁles and corresponding static advancing and receding contact angles: (a) static advancing contact angle with a vertical tangent at the thickness h1; (b) static receding contact angle with violation of the quasi-equilibrium condition in the region of thick β-ﬁlm, h4. See explanation in the text.

Figure 3.27. As P < Pe, the interval of the thickness between the positions of the minimum and the maximum broadens in the case of an isotherm of disjoining pressure in the case of partial wetting (curve 1 in Figure 3.27). Under the condition P < Πmax, h1 < hu and h2 > hβ (Figure 3.27). When the absolute value of ∆P increases to a certain critical value ∆Pa, the curve ψ(h, P) can touch, at its minimum, the h-axis, as shown by curve 4 in Figure 3.27b. This means that a point with a vertical tangent appears on the proﬁle at h = h1, where h′ = –∞ (Figure 3.29a). At |∆P| > |∆Pa| the curve ψ(h, P) intersects the h-axis, and the value of ψ becomes negative in a certain region of thickness. Vertical tangents appear at the upper and lower parts of the proﬁle, which become discontinuous (dashed lines in Figure 3.29a). Note that in the region of the proﬁle marked by the dashed line, the different disjoining pressure acts, which is always the attraction between the identical phases. This means that the dashed part of the proﬁle is unstable, the proﬁle loses stability, and a transfusion of liquid starts towards the front of the spreading ﬁlm. It should be noted that a similar mechanism was proposed previously by Frenkel [47] for explaining the static hysteresis of a sliding drop over an inclined plane. We wish to point out that Equation 3.322 does not give the proﬁle of the ﬂowing ﬁlm but only determines the limiting positions of the static proﬁle before the ﬂow has set in. This consideration shows the rather complicated picture of ﬂow after the meniscus starts to advance macroscopically. This is the reason why the theory of spreading is well developed in the case of complete wetting and is still to be developed in the case of partial wetting. Although the mechanism of violation of the equilibrium is understood physically, the value of the static advancing contact angle, θa, cannot be calculated precisely at present, as the point h = h1 belongs to a region in which the condition h′2 << 1 is violated and the disjoining pressure, Π(h), of ﬂat ﬁlms cannot be used.

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We therefore limit ourselves to the estimation of values of the static advancing contact angles, θa. At θ = θa and P = Pa, the function ψ = 0 at h = h1. Hence, we conclude from Equation 3.323 that Pa ( H − h1 ) =

∞

∫ Π ( h) dh .

(3.325)

h1

Pressure Pa and contact angle θa are connected by the general relationship Pa = γ cos θa /H .

(3.326)

Combination of Equation 3.325 and Equation 3.326 results in

cos θa =

1 h   γ 1 − 1   H

∞

∫ h1

1 Π ( h ) dh ≈ γ

∞

∫ Π(h)dh .

(3.327)

h1

After subtracting the expression derived for the equilibrium contact angle, cosθe, according to Equation 2.47 (Section 2.3), we ﬁnd 1 cos θe − cos θa ≈ 1 + γ

h1

∫ Π ( h) dh .

(3.328)

he

The second term of the right-hand side of the equation is negative and smaller than one; hence: cos θe − cos θa < 1 and 90° > θa > θe .

(3.329)

In the case of the isotherm of disjoining pressure of the second type (Figure 3.27a, curve 2), violation of quasi-equilibrium occurs at Pa ≤ 0 because, in this case, ψ(h, Pa) becomes negative simultaneously for the whole range of large thicknesses, h. Hence, in this case, the limiting condition Pa = 0 corresponds to the contact angle θa = 90˚. However, at partial wetting, the condition θe < θa is always fulﬁlled. Values of the static advancing contact angle, θa, close to 90˚ have been observed experimentally in a number of systems [39]. The most important result of the above consideration is that the disjoining pressure isotherm determines not only the equilibrium values of the contact angles but also the static advancing contact angle, θa.

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We now consider the solution for a receding contact angle, θr . In this case, the value ∆P > 0, as the curvature of the meniscus increases with decreasing pressure in the liquid phase. It follows from Equation 3.316 and Equation 3.323 that, at P > Pe, the values of ψ(h, P) should be everywhere above the equilibrium curve ψ(h, Pe). The shape of the function ψ(h, P) in this case is shown by curve 5 in Figure 3.27. Violation of the conditions of quasi-equilibrium occurs in this case at ψ(h, P) = γ, i.e., upon an increase in ∆P to such a critical value ∆Pr , that the curve ψ(h, Pr) intersects the dashed line γ = const (Figure 3.27, curve 5). If the capillary is sufﬁciently narrow and Pe > Πmax, an intersection can occur only in the region of small thicknesses, e.g., at h = h3 (Figure 3.27), as in this case the curve ψ(h, P) possesses only one maximum. A formal intersection of the curve ψ(h, P) with the straight line γ = const should occur at an arbitrarily small difference between P and Pe, as the curve ψ(h, P) already touches the straight line γ = const at the point h = he. However, it should be remembered that there is a solution in region 1 (Figure 3.28b), limited from below by the value hc > he. Violation of the quasi-equilibrium may occur only if h3 > hc > he. Therefore, ∆Pr in this case has a small but ﬁnal value. Writing Equation 3.323 for ψ = γ, P = Pr , assuming h = h3 ≅ hc, and subtracting from it Equation 3.315 for the equilibrium state, we derive

(

)

(

)

Pr H − h3 − Pe H − he =

he

∫ Π (h ) dh.

(3.330)

h3

As the α branch of the disjoining pressure isotherm is generally quite steep, thicknesses he and h3 do not differ much from one another. Hence, the integral on the right-hand side of the equation is positive. Bearing in mind that H >> h3 ≅ he, it means that Pr > Pe , and consequently, θr < θe, which is indeed the case. In addition, it follows from this discussion that the value of θr should differ only a little from that of θe. This theoretical conclusion agrees with experimental data in the case of water on hydrophobized glass [49]. The critical proﬁle of the meniscus before receding sets in has such a form that a thickness with a horizontal tangent appears at h = h3. Upon further increase in pressure, the proﬁle of the meniscus can no longer be combined by a continuous line with the point h = h3. Consequently, the meniscus should start to recede at ∆P > ∆Pr . Things look differently in the case of violation of the quasi-equilibrium with receding meniscus in thicker capillaries, when Pe < Πmax. Under certain conditions, which are determined by the actual form of the disjoining pressure isotherm, Π(h), with increasing ∆P, the curve ψ(h,P) can touch the straight line γ = const with the right-hand rising maximum (at h = h4) earlier than thickness h3 comes out of the transfer zone (h3 < hc) (Figure 3.27b, curve 5). Then, a point with a horizontal tangent h′ = 0 appears on the convex part of the meniscus (Figure 3.29b). Violation of the quasi-equilibrium occurs at the point of the proﬁle at h = h4. Note, that, at this particular moment the stability condition C (Chapter 2, Section 2.2) is © 2007 by Taylor & Francis Group, LLC

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violated. The quasi-equilibrium liquid proﬁle becomes unstable and the ﬂow has to set in. At P > Pr , the part of the proﬁle indicated in Figure 3.29b by dashed lines starts sliding. It can be seen from Figure 3.27 that thickness h4 belongs to the β part of the disjoining pressure isotherm 1. Thus, when the meniscus is displaced from it, a thick metastable β ﬁlm should remain behind. This phenomenon (thick β-ﬁlms behind the receding meniscus in the case of S-shaped aqueous isotherm of disjoining pressure) was conﬁrmed in a number of experimental investigations [49–51]. As the proﬁle of the receding meniscus in the transition zone is low sloped, the value of the static receding contact angle, θr , in the case of sufﬁciently thick capillaries (that is, at Pe < Πmax) can be determined quite rigorously. For this purpose we use the following system of equations: ∞

∫

(3.331)

Pr = Π ( h4 ) ,

(3.332)

Pr ( H − h4 ) − Π ( h ) dh = γ , h4

where the ﬁrst equation was derived from expression (3.323), under the condition ψ = γ, for P = Pr and h = h4. The second equation results from the fact that h4 is an inﬂection point at which h″ = 0. It then follows from Equation 3.322 that Π(h4) = Pr . For the solution of the systems of Equation 3.331 and Equation 3.332, we must know the form of the isotherm Π(h). If we choose for the β branch of the disjoining pressure isotherm a relationship in the following form Π = B/h2, as for β-ﬁlms of water [26], we obtain, after integration in Equation 3.331,

h4 =

 γH B − 1  1+ γ B 

and Pr =

B , h42

(3.333)

where B is a constant of the surface forces [26]. Under the condition H >> h4, these expressions transform into h4 ≅ BH ⁄ γ and Pr ≈ γ/H, respectively. Hence, θr ≈ 0. Thus, if thick β- ﬁlms are formed behind the receding meniscus, the static receding contact angle, θr , should be close to zero. This, again, is in good agreement with experimental observations in Reference 49 to Reference 51. For isotherms of type two (Figure 3.27) or for narrow capillaries, when Pe > Πmax, calculation of θr is further complicated by the fact that thickness hc, corresponding to the beginning of the zone of ﬂow, is not known. In order to determine it as a function of ∆P, the corresponding hydrodynamic equation must be solved. This is the objective of a future investigation. © 2007 by Taylor & Francis Group, LLC

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It should be noted that the solution obtained is valid also for the case of contact and ﬂow of two immiscible liquids. The only difference is that, instead of the isotherm of a wetting ﬁlm, Equation 3.322 includes in this case the disjoining pressure isotherm of thin layers of wetting liquids Π(h) enclosed between a solid substrate and a nonwetting liquid. For the existence of partial wetting, the form of the disjoining pressure isotherm, Π(h), of the layers must here also be of the same type as shown in Figure 3.27a (curves 1 or 2).

STATIC HYSTERESIS CONTACT ANGLES

OF

DROPS

A similar approach is used in the following example for drops of liquids that form equilibrium contact angles θe with a ﬂat substrate. For the sake of simplicity, we limit the discussion to drops of a cylindrical shape, where the liquid proﬁle depends on only one variable x. This permits the reduction of the problem to a two-dimensional one, as in the case of slit capillaries. In the case of an equilibrium drop for solving Equation 3.311 instead of condition (3.312), a different boundary condition should be used: x = 0, h = H , h′ = 0,

(3.334)

where H is the maximum height of the drop over the surface, and we assume that H >> he. By integration of Equation 3.311, we obtain for the derivative of the liquid proﬁle, h′, in the same expression (3.316) as in the case of a meniscus. However, the function ψ now has a different form:

(

)

ψ h, Pe = γ − ϕ(h, Pe ),

(

)

∞

∫

ϕ(h, Pe ) = − Pe H − h + Πdh.

(3.335)

he

Condition (3.313) for equilibrium of a drop with an equilibrium ﬂat ﬁlm of thickness he, h′ → 0 at h → he is written now in the form ψ = γ. Using these conditions, we derive from Equation 3.335 that ∞

∫

Pe ( H − he ) − Πdh = 0 .

(3.336)

he

The excess pressure Pe is negative in the case of a drop, and Pe = –γ/ℜe, where ℜe is the radius of curvature of the spherical part of the surface of the drop. It follows from simple geometrical considerations that cosθe = (ℜe–H)/ℜe . By substituting these expressions in Equation 3.336, we derive the condition of equilibrium of a drop with a ﬁlm that is identical with Equation 3.55 derived © 2007 by Taylor & Francis Group, LLC

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previously in Chapter 2, Section 2.3 and Equation 2.47 in the same section for the case of the menisci. This is not surprising; as in the case of H >> he, the conditions of equilibrium should not depend on the sign of the curvature of the surface beyond the limits of the zone of inﬂuence of surface forces. By differentiating Equation 3.336 with respect to Pe and bearing in mind that Pe < 0, we obtain ∂H ( H − he ) = − H − he > 0 . =− ∂Pe Pe − Π ( h ) Pe

(3.337)

This equation shows that the maximum thickness of equilibrium drops, H, decreases with increasing supersaturation, when the value of Pe diminishes. However, this decrease has certain limits, as drops can be at equilibrium with ﬂat ﬁlms only if Pe > Πmin. It can be seen from Figure 3.30, curve 1, that Πmin is the pressure corresponding to the minimum of the isotherm Π(h). At Pe < –Πmin there is neither a ﬁlm nor a drop on the surface at the equilibrium.

Π α 1 β

hu

0

h Pr > Pe

Pe Pa < Pe –Πmin

ϕ γ

he

4 2

3 0

h 3 h4

h

FIGURE 3.30 The disjoining pressure isotherm Π(h) (1) in the case of partial wetting, and the corresponding curves of the functions ϕ(h) (2–4) determining the conditions of quasi-equilibrium of the drop. See explanations in the text.

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At a sufﬁciently steep dependency of the α-branch of the disjoining pressure isotherm, as is usually the case, an increase in oversaturation causes a rise in the value of the contact angle θe, in accordance with Equation 2.47 from Section 2.3. Thus, the decrease in the size of the equilibrium drop causes an increase in θe; i.e., the value of θe is a function of the height of the drop H. When Pe decreases and approaches –Πmin, the drops whose dimensions diminish should be torn off the surface to pass into the gaseous phase. Maintaining the external conditions, we consider now nonequilibrium proﬁles of drops when their volume changes by charging or withdrawing liquid (see Figure 1.14). Advancing contact angles are formed at P < Pe, and receding ones at P > Pe. Expression (3.324) is a condition for the existence of a solution for (3.322), which is found by the method described previously and is written for a drop in the following form: γ ≥ ϕ ( h, P ) ≥ 0 ,

(3.338)

where now the function ϕ(h, P) is given by ∞

∫

ϕ ( h, P ) = − P ( H − h ) + Π ( h ) dh .

(3.339)

h

Examples of the form of the function ϕ(h, P) (curves 2–4) for the disjoining pressure isotherm Π(h) of the same type as curve 1 are shown in Figure 3.30. The extremum values of ϕ(h, P) are found from the condition P = Π(h) just as previously, i.e., from the points of intersection of the isotherm with the straight line P = const. Note that, in the case of drops, the form of function ϕ(h, P) according to Equation 3.335 and Equation 3.339 differs from that of the corresponding function ψ(h, P) for menisci in capillaries (Figure 3.27) because of the difference in the boundary conditions. It is shown in the following text that this fact causes different expressions for the advancing and receding contact angles. Thus, for the same liquid–solid system, the values of the static hysteresis angles depend on the conﬁguration of the surface of the liquid (meniscus or drop). There is also a difference in the equilibrium contact angles calculated according to Equation 2.47 and Equation 2.55 in Section 2.3, as the equilibrium excess pressure, Pe, differs in sign for a concave meniscus and a convex drop. It follows from Equation 3.336 that, for an equilibrium proﬁle of a drop, i.e., at P = Pe , the function ϕ(h, Pe) vanishes at h = he. On the other hand ϕ(h, Pe) vanishes also at h = H. As ϕ(h, Pe) > 0, the function ϕ(h, Pe) has a maximum at the thickness h = hu (curve 2 in Figure 3.30). With increased pressure, i.e., at P > Pe, when the curvature of the surface of the drop decreases, the curve ϕ(h, P) (curve 3 in Figure 3.30) is located below © 2007 by Taylor & Francis Group, LLC

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the equilibrium curve 2 (Figure 3.30). Figure 3.30 shows that, in this case, the condition for the existence of a solution is violated even at a small excess of P over Pe. The situation is similar to that considered previously for the case of receding contact angles in a capillary. The perimeter of the drop starts to recede under the condition h3 ≥ hc, where hc is the thickness corresponding to the beginning of the ﬂow zone. Thus, in the case of a drop, θr ≤ θe just as in the case of a capillary. At lowered pressures, i.e., at P < Pe, when the surface of the drops becomes more convex, curve ϕ(h, P) (curve 4 in Figure 3.30) is located above that of equilibrium curve 2 (Figure 3.30). The condition of a quasi-equilibrium is violated, and the perimeter of the drop starts to advance after the maximum of ϕ(h, P) reaches the dashed line γ = const (curve 4 in Figure 3.30). This condition corresponds to the appearance of a thickness with a vertical tangent h′ = ∞ on the proﬁle of the drop. Just as in a capillary, a ﬂow from the drop to the ﬁlm commences by the Frenkel’s “caterpillar” mechanism if ϕ(h, P) > γ. We calculate the value of the advancing contact angle θa, using the condition ϕ(h, P) = γ: ∞

∫

− Pa ( H − h4 ) + Π ( h ) dh = γ .

(3.340)

h4

Keeping in mind that Pa = γ (cos θa – 1)/Ha, we conclude from this expression that 1 cos θa = Pa h4 + γ

∞

∫ Π(h)dh .

(3.341)

h4

Let us calculate the difference cosθe – cosθa. Using Equation 2.55 from Section 2.3 for the equilibrium contact angle, θe, and equilibrium excess pressure, Pe, in the drop, and Equation 3.341, we obtain h4   1 cos θe − cos θa = 1 − Pehe − Pa h4 + Π(h )dh  .  γ he  

∫

(3.342)

The sign of the second term on the right-hand side is determined by the actual form of the disjoining pressure isotherm, Π(h). If (cos θe – cos θa) ≥ 1, then cosθa ≤ 0 and θa ≥ 90˚ > θe . If (cos θe – cos θa) ≤ 1, then cos θe > cos θa and θa > θe. Thus, in the case of a drop, the advancing angle is always larger than the equilibrium angle.

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CONCLUSIONS A theory of static hysteresis of the contact angle has been developed on the basis of the analysis of conditions of quasi-equilibrium of the system and their violation. The suggested theory agrees qualitatively with known experimental data. For more rigorous quantitative calculations, a theory of disjoining pressure should be developed, which is applicable to proﬁles of drops or menisci having sufﬁciently steep part of the proﬁles. It is also necessary to determine the thickness of the ﬁlm thickness, hc , that corresponds to the beginning of the ﬂow zone. The fundamental conclusion of this section is the relation of the mechanism of static hysteresis of contact angle of smooth homogeneous surfaces to the form of the disjoining pressure isotherm. It should be noted that static hysteresis of the nature considered appears not only on smooth homogeneous surfaces but also on heterogeneous ones. Thus, in actual cases, the possibility of the simultaneous appearance of static hysteresis phenomena of different natures must be taken into consideration. In order to prove the concepts developed in this section of static hysteresis and equilibrium angles, one must combine for these systems the derivation of the disjoining pressure isotherms, Π(h), which characterize the forces acting in thin layers and ﬁlms. It would also be of interest to observe the behavior of deformed menisci in a capillary or a drop at θe < θ < θa and at θe > θ > θr . If static hysteresis is related to the conditions of quasi-equilibrium, a slow motion of the position of the apparent three-phase contact line wetting perimeter should be observed. The latter slow motion should change dramatically upon attaining the static advancing or receding static hysteresis contact angles, θa and θr , respectively. The previous remarks and conclusion also apply when the solid surface ahead of the menisci or the drops is initially dry.

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4

Spreading over Porous Substrates

INTRODUCTION In this chapter, we consider the kinetics of spreading over porous substrates. This phenomenon is widely used in the industry: printing, painting, imbibition into soils, health care, and home care products, and so on. However, it is only recently that this process started to develop on a theoretical and not just on a purely empirical basis. We show in this chapter that the kinetics of spreading over porous substrates is substantially different from the corresponding kinetics of solid nonporous substrates. First (Section 4.1), we considered the kinetics of spreading over the porous substrate already saturated with the same liquid. This allows extraction of an important parameter, which we referred to as the effective lubrication coefficient. This coefﬁcient turns out to be very insensitive to the properties of the porous substrate. Its pattern allows us to use its average value for the consideration of the kinetics of spreading over thin porous layers (Section 4.2). We show that the kinetics of spreading over these layers is described by a universal dependency. The unusual ﬁnding is that, in the case of complete wetting, the hysteresis of contact angle is present at the spreading on porous substrates. We call this form of hysteresis hydrodynamic, and it is determined by the ﬂow in the porous substrate. However, in the case of spreading over thick porous substrate, we cannot provide a theory on the current stage and restrict ourselves by summarizing experimental results only.

4.1 SPREADING OF LIQUID DROPS OVER SATURATED POROUS LAYERS In Chapter 3, we have considered the kinetics of spreading over smooth homogeneous surfaces. We made it clear that the singularity at the three-phase contact line is removed by the action of surface forces (see Section 3.2). However, the vast majority of real solid surfaces are rough to a varying degree, and in many cases, surfaces are either porous or covered with a thin porous skin. These features affect the spreading process. Brinkman’s equations [1,17] are frequently used for the description of the ﬂow in porous media and have a reasonable semi-empirical background [2] with physically meaningful coefﬁcients: an effective viscosity and a permeability coefﬁcient. A new method of calculation of these coefﬁcients, as functions of the porosity of the porous media, has been suggested in Reference 3. 315 © 2007 by Taylor & Francis Group, LLC

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We shall use them to study the spreading of liquid drops over thin porous substrates ﬁlled with the same liquid, i.e., when the thickness of the porous substrate, ∆, is assumed to be much smaller than the drop height, H, ∆ << H∗ (H* sets the scale of the drop height). We shall follow the evolution of the liquid, both in the drop above the porous layer and inside it. Spreading of small liquid drops over thin porous layers saturated with the same liquid is investigated in this section, from both theoretical and experimental points of view. A theory is presented, which shows that spreading is governed by the same power law as in the case of spreading over dry solid substrate. Brinkman’s equations are used to model the liquid ﬂow inside the porous substrate. An equation of the drop spreading is deduced, which shows that both an effective lubrication and the liquid exchange between the drop and the porous substrates are equally important. Presence of these two phenomena removes the well-known singularity at the moving three-phase contact line, because it results in an effective “slippage” velocity on a moving three-phase contact line. Matching of the drop proﬁle in the vicinity of the three-phase contact line, with the main spherical part of the drop, makes it possible to calculate preexponential factors in the spreading law via the permeability and effective viscosity of the liquid in the porous layer. However, this dependency turns out to be very weak. Spreading of silicone oil over different microﬁltration membranes is carried out. Radii of spreading on time experimental dependencies conﬁrm the theory predictions. Experimentally found coefﬁcients agree with theoretical predictions. Spreading of liquids over solid surfaces is one of the fundamental processes, with a number of applications such as coating, printing, and painting. In the previous chapter, we considered the spreading over smooth homogeneous surfaces. It has been established that singularity at the three-phase contact line is removed by the action of surface forces (see Section 3.2). An attempt to use Brinkman’s equations for description of the ﬂow inside the porous layer coupled with the drop ﬂow over the layer has been undertaken in Reference 4. In the following text, the same approach is applied to investigate the spreading of liquid drops over thin porous substrates ﬁlled with the same liquid. Brinkman’s equations are used for description of the liquid ﬂow inside the porous substrate [5].

THEORY The kinetics of spreading of small liquid drops over thin porous layers saturated with the same liquid is investigated in this section. Theoretical consideration takes into account the kinetics of liquid motion, both in the drop above the porous layer and inside the porous layer itself. Consideration of the ﬂow inside the porous layer is based on Brinkman’s equations. Liquid inside the Drop (0 < z < h(t,r), Figure 4. 1) Let us consider the spreading of an axisymmetric liquid drop over a thin porous layer with thickness ∆, saturated with the same liquid. The thickness of the porous © 2007 by Taylor & Francis Group, LLC

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layer is assumed much smaller than the drop height, that is, ∆ << H*, where H* is the scale of the drop height. The drop proﬁle is assumed to have a low slope (H* /L* << 1, where L* is the length scale of the drop base). Inﬂuence of the gravity is neglected (small drops, Bond number << 1, or the size of the drop is smaller than the capillary length, a). Therefore, only capillary forces are taken into account. In the case under consideration, the liquid motion inside the drop is described by the following system of equations: ∂p ∂2 v =η 2 2 , ∂r ∂ z

(4.1)

∂p = 0, ∂z

(4.2)

( )

1 ∂ rv ∂u + = 0, r ∂r ∂z

(4.3)

v = v 0 , u = u 0 , z = +0 ,

(4.4)

∂v = 0, z = h (t, r ) , ∂z

(4.5)

and boundary conditions:

p = pa −

γ ∂  ∂h  , r r ∂r  ∂r 

(4.6)

where t is the time; r, z are radial and vertical coordinates, respectively; z > 0 and –∆ < z < 0 correspond to the drop and the porous layer, respectively; z = 0 is the drop–porous layer interface; p, v, u are the pressure, radial, and vertical velocity components, respectively; v0, u0 are velocity components at the drop–porous layer interface, which are determined below by coupling with the ﬂow inside the porous layer; h(t,r) is the drop proﬁle; γ is the liquid–air interfacial tension; and pa is the pressure in the ambient air. Equation 4.1 and Equation 4.2 are Stokes equations, in the low slope case; Equation 4.3 is the incompressibility condition; Equation 4.5 shows the absence of a tangential stress on the liquid–air interface; and Equation 4.6 presents the pressure jump on the same interface determined by capillary forces only. © 2007 by Taylor & Francis Group, LLC

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Integration of Equation 4.1 to Equation 4.3 with boundary conditions (4.4 to 4.6) results in the following equation, which describes the evolution of the drop proﬁle: ∂h 1 ∂ = u0 − ∂t r ∂r

     ∂  1 ∂  ∂h  3 γ r  + v0 h  . r  h   3 η ∂r  r ∂r  ∂r      

(4.7)

The liquid velocity components, v0, u0, on the drop–porous layer interface are calculated in the following text. Inside the Porous Layer beneath the Drop ∆ < z < 0, Figure 4. 1) (–∆ If the porous layer is not completely saturated, then the capillary pressure inside the saturated part of the porous layer, pc, can be estimated as pc ≈

γ , r*

where r* is the scale of capillary radius inside the porous layer. According to Equation 4.6, the capillary pressure inside the drop, p – pa can be estimated as pa − p ≈

γh* h* γ γ γ = << << ≈ pc . 2 L L L r L* * * * *

Therefore, the capillary pressure inside the spreading drop is substantially smaller (actually several orders of magnitude smaller) than the capillary pressure inside the porous layer, in the case of noncomplete saturation. This means that the drop pressure cannot disturb, in any way, the drop–porous layer interface in front of the spreading drop when the porous layer is completely saturated. Hence, this interface always coincides with the surface z = 0. It is worth mentioning that everything is happening on time scales much bigger than the initial period considered in Reference 6. The liquid motion inside the porous layer with thickness ∆ is assumed to obey Brinkman’s equations. In this case, the liquid motion inside the porous layer is described by the following system of equations: v ∂p ∂2 v = ηp 2 − , K ∂r ∂z p

© 2007 by Taylor & Francis Group, LLC

(4.8)

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319

1

A

2

ζ0 ∆

L(t)

FIGURE 4.1 Spreading of liquid drop over saturated porous layer of thickness ∆. L(t) — macroscopic radius of the drop base; (1) spherical cap (outer region); (2) vicinity of the three-phase contact line (inner region). Inﬂections point, A, separates inner and outer regions. Inside the outer region, liquid ﬂows from the drop into the porous layer; inside the inner region, the liquid ﬂows from the porous layer onto the drop edge.

∂p = 0, ∂z

(4.9)

( )

1 ∂ rv ∂u + = 0, r ∂r ∂z

(4.10)

v = v 0 , u = u 0 , z = −0

(4.11)

∂v = u = 0, z = − ∆ , ∂z

(4.12)

∂v ∂v =η , ∂z z =−0 ∂z z =+0

(4.13)

=p

(4.14)

and boundary conditions:

ηp

p

z =−0

z =+0

,

where ηp, Kp are the viscosity and the permeability of Brinkman’s medium, respectively. The boundary condition (4.12) corresponds to the absence of a tangential stress on the lower boundary of the porous layer, which corresponds to experimental conditions (see the following text).

© 2007 by Taylor & Francis Group, LLC

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Let us introduce Brinkman’s radius, as δ = ηp K p

(4.15)

The solution of Equation 4.8 to Equation 4.10 with boundary conditions (4.11 to 4.15) results in:

v0 = −

1 ηp

(

)

 ∆ 2 1 ∂  2  ∂p 0 2 2 ∂p   hδ coth δ + δ  ∂r , u = η r ∂r r hδ + ∆δ ∂r  (4.16) p  

Substitution of the latter expressions into Equation 4.7 results in the following equation, which describes the kinetics of spreading of the liquid drop over a porous substrate: ∂h = ∂t γ 1 ∂ − 3η r ∂r

 1 ∂  ∂h    ∆ 2   3 2 2  ∂ r r  h + 3α coth h δ + 6αhδ + 3αδ ∆   , δ  ∂r  r ∂r  ∂r     

(4.17)

where α = η/ηp is the viscosity ratio. According to Reference 3, effective viscosity, ηp, is always higher than the liquid viscosity, η, that is, α < 1. If, instead of Brinkman’s equation (Equation 4.8), a slip condition on the liquid–porous layer interface is used [7], then the vertical velocity component, u0, should be set to zero in Equation 4.7, and the following boundary condition should be adopted for the radial velocity component: ∂v ∂z

= z =0

(

)

β 0 v − vp , δ

(4.18)

where β is an empirical parameter; vp = −

K p ∂p η p ∂r

is the velocity inside the porous substrate; and δ/β is a slip length. If boundary conditions (4.13) and (4.18) are compared, then it is easy to see that the former condition can be directly obtained from condition (4.18) if we adopt © 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

ηp

321

v0 − v p ∂v = ηp . ∂z z =−0 δ

Combination of this expression and boundary condition (4.13) gives the following value of the empirical coefﬁcient, β, as β=

1 . α

(4.19)

This means that the slip length is equal to

αδ = η

Kp ηp

.

Kp and ηp dependencies on porosity can be calculated according to Reference 3. However, if the slip condition is used, then the omitted contribution of a vertical component, u0, gives the comparable contribution in the resulting equation (see the following text). Equation 4.17 should be solved with the symmetry condition in the drop center, ∂h ∂3 h = 0, r = 0 , = ∂r ∂r 3

(4.20)

and conservation of the drop volume condition, L

∫

2 π rhdr = V ,

(4.21)

0

where L(t) is the macroscopic radius of the drop base (Figure 4.1). Everywhere, at r < L(t), except for a narrow region, ξ, close to the threephase contact line, the following inequality holds: h >> δ. The size of this narrow region close to the three-phase contact line is calculated in the following text. The same consideration as in Chapter 3 (Section 3.2, Reference 8) shows that the solution of Equation 4.17 can be presented as “outer” and “inner” solutions (Figure 4.1). The outer solution can be deduced in the following way: the lefthand side of Equation 4.17 should be set to zero and solved with boundary conditions (4.20 and 4.21), and an additional new boundary condition, h(t, L – ξ) ≈ 0.

© 2007 by Taylor & Francis Group, LLC

(4.22)

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Wetting and Spreading Dynamics

In a similar manner, as in Chapter 3 (Section 3.2), the outer solution is deduced in the following form: h(t , r ) =

2V 2 2 ( L − r ), r < L (t ). πL4

(4.23)

Equation 4.23 shows that the drop proﬁle retains the spherical shape over the duration of spreading (except for a very short initial stage). Note that the time dependency of the macroscopic position of the apparent three-phase contact line, L(t), is to be determined. The drop slope at the macroscopic apparent three-phase contact line can be found from Equation 4.23 as: ∂h ∂r

=− r=L

4V , πL3

(4.24)

which is used below as a boundary condition for the inner solution. Inside the inner region (Figure 4.1), the solution can be represented in the following form, h (t, r ) = δ f (ζ), ζ =

r − L (t ) , χ(t )

(4.25)

where f is a new unknown function; ζ is a similarity variable; and χ(t) << L(t) is the scale of the inner region. This means that ξ ≈ χ(t). Substitution of the solution in the form (4.25) into Equation 4.17 results in the following equation for determination of f(ζ): γ δ3 d  3 df ∆ 2 ∆  d3 f  f f f L (t ) = + 3 α coth + 6 α + 3 α   , (4.26) dζ 3 η χ3 (t ) dζ  δ δ  dζ3  where over-dot means the differentiation with respect to time, t, and small terms are neglected in the same way as in Section 3.2. The latter equation should not depend on time. This gives two equations: γ δ3 L (t ) = 3 η χ3 (t )

(4.27)

df d  3 ∆ 2 ∆  d3 f  =  f + 3α coth f + 6 αf + 3α  3  . dζ dζ  δ δ  dζ 

(4.28)

and

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The solution of Equation 4.28 should be matched with the outer solution (4.23). Matching of asymptotic solutions gives the following condition: δ df δ df =– χ(t ) dζ ζ→−∞ χ(t ) dζ ζ→−∞

(4.29)

The latter condition should not depend on time, t, which gives df = –λ dζ ζ→−∞

(4.30)

4V χ(t ) , πL3 (t )δ

(4.31)

and λ=

where λ is a dimensionless constant (see the following text). This equation gives χ(t ) = δ

πL3 (t )λ << L (t ) . 4V

(4.32)

This means that the scale of the inner region (Figure 4.1) is proportional to δ and is very small as compared with the size of the drop base. The combination of Equation 4.31 and Equation 4.27 gives the following equation for the determination of the radius of spreading, L(t): γ  4V  L (t ) L (t ) = 3 η  πλ 

3

9

(4.33)

The solution of this equation with the initial condition L(0) = L0, where L0 is the initial drop radius, is  t L (t ) = L0  1 +  τ 

0.1

,

where 3 ηL0  πλL30  τ= 10 γ  4V  is the time scale of the spreading process. © 2007 by Taylor & Francis Group, LLC

3

(4.34)

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Wetting and Spreading Dynamics

Now back to the determination of the parameter λ. Integration of Equation 4.28 and setting an integration constant zero (because of conservation of the drop volume and vanishing of the drop proﬁle in front of the spreading drop) gives: d3 f = d ζ3

f ∆ ∆ f 3 + 3α coth f 2 + 6 αf + 3α δ δ

(4.35)

This equation should be solved with the boundary conditions f (ς 0 ) =

df (ς 0 ) = 0, dζ

(4.36)

where ζ0 corresponds to the inner variable to the edge ξ0 in Figure 4.1, and boundary condition (4.30). We have seen already in Chapter 3 (see Section 3.2) that this condition cannot be satisﬁed because Equation 4.35 does not have a proper asymptotic behavior. An approximate method (“patching” of asymptotic solutions) has been suggested in Section 3.2, which allows an approximate determination of parameter λ. Now, the unknown constant λ can be calculated in the same approximate way, as it has been done in Section 3.2. Estimations in the following text show, however, that it is not worthwhile. The second of the two boundary conditions (4.36) refers to a zero microcontact angle on the microscopic drop boundary. Equation 4.24 gives the following value of an apparent dynamic contact angle, θ, (tanθ ≈ θ): θ=

4V , πL3

(4.37)

or  4V  L=  πθ 

1/ 3

.

(4.38)

Combination of Equation 4.38 and Equation 4.33 results in dL γθ3 =ω , dt µ where ω=

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1 3λ 3

(4.39)

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325

TABLE 4.1 Calculated Effective Lubrication Coefficient Membrane Pore Size, µm

L0, cm

η, P

τ, s

V, cm3

0.2 0.2 0.2 0.2 0.2 3 3 3 Optical glass Optical glass

0.176 0.193 0.150 0.150 0.165 0.119 0.250 0.253 0.226 0.113

5.58 0.55 1.18 1.18 1.18 1.18 1.18 0.55 0.55 1.18

0.333 0.0592 0.00163 0.0026 0.0086 0.0546 0.461 0.609 0.0378 0.0103

0.0039 0.0040 0.003 0.0034 0.003 0.0055 0.005 0.005 0.0068 0.002

ω 0.017 0.018 0.014 0.014 0.016 0.015 0.016 0.018 0.012 0.010

± ± ± ± ± ± ± ± ± ±

0.004 0.004 0.005 0.003 0.005 0.009 0.009 0.008 0.003 0.005

is referred to in the following text as an effective lubrication coefficient. If the liquid spreads over the solid substrate, this effective lubrication is determined by the action of surface forces in the vicinity of the three-phase contact line (Section 3.2). In this case, the effective lubrication coefﬁcient has been calculated in Section 3.2, and its value is 1.36*10–2. It is reasonable to expect the ω value to be higher than the one in the case under consideration. Spreading of the liquid over a prewetted solid substrate has been considered in Section 3.3 (see also Reference 9). The effective lubrication coefﬁcient in this case has been calculated as 1.6*10–2. The latter shows: 1. An effective lubrication coefﬁcient is not very sensitive to experimental conditions. That is, we have chosen not to try to theoretically calculate though the procedure of its approximate determination is very similar to those presented in Section 3.2. 2. It is reasonable to expect the values of the effective lubrication coefﬁcient in between these two values. Experimentally determined values of this coefﬁcient agree with our estimations reasonably well (see Table 4.1). Materials and Methods [5] Silicone oils SO50 (viscosity 0.55 P), SO100 (viscosity 1.18 P), and SO500 (viscosity 5.58 P) purchased from PROLABO are used in these spreading experiments. The cellulose nitrate membrane ﬁlters purchased from Sartorius (type 113) with an average pore size of 0.2 and 3 µm, respectively, are used as porous layers. All membrane samples used are plane circular plates with a radius of 25 mm and thickness from 0.0130 to 0.0138 cm. The porosity of the membranes ranges between 0.65 and 0.87. Prior to spreading experiments, membranes were dried for 3–5 h at 95˚C and then stored in dry atmosphere. © 2007 by Taylor & Francis Group, LLC

326

Wetting and Spreading Dynamics 9 2

5

6

CCD 1

4

15 13

14

12

10

11

16

CCD 2 7

8 1 3

FIGURE 4.2 Experimental set-up for monitoring the time evolution of droplets. (1) wafer; (2) sample chamber; (3) tested drop; (4) dosator; (5), (10) CCD cameras; (6), (11) VCRs; (7), (13) illuminators; (8), (9), (12), (14) interferential light ﬁlters (with wave length 520 nm [8], [9], and with wavelength 640 nm [12], [14]).

Figure 4.2 shows the sample chamber for monitoring the spreading drop over porous layers and dynamic contact angles. A porous wafer 1 (Figure 4.2) is placed in a thermostated and hermetically closed chamber 2 with a ﬁxed humidity and temperature. The chamber was made of brass to prevent temperature and humidity ﬂuctuations. In the chamber walls, several channels were drilled to enable pumping of a thermostating liquid. The chamber is equipped with a fan. The temperature is monitored by a thermocouple. Droplets of liquid 3 are placed onto the wafer by a dosator 4 (Figure 4.2). The volume of drops is set by the diameter of the separable capillary of the dosator. The chamber is also equipped with optical glass windows for observation of both the spreading drop shape and size (side view and view from above). Two CCD cameras and tape recorders are used for storing the sequences of spreading. Different colors of monochromatic light are used for side views and viewing from above to eliminate spurious illumination on images. The optical circuit for viewing from above (illuminator 7 as well as the camera 5) is equipped with interferential light ﬁlters 8,9 with a wavelength 520 nm. The side view circuit (the illuminator 13 and camera 10) are equipped with ﬁlters 12 and 14 with a wavelength 640 nm. Such an arrangement suppresses the illumination of a CCD camera 2 by the diffused light from the membrane and hence increases the precision of measurements. The automatic processing of images is carried out using the (image processor) Scion Image. The time discretization in processing ranges from 0.1 to 1 sec in different experiments. The size of pixel on the image corresponds to 0.0125 mm. The experiments are organized in the following order: A membrane under investigation is placed in the chamber; a big drop of oil is deposited. The volume of the drop, V, is selected as V = πm ∆Rm2 , where Rm is the radius of the membrane sample. This choice corresponds to the complete saturation of the membrane by a tested liquid. © 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

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After the imbibition process is completed (100–500 sec, depending on the liquid viscosity), the next tiny drop of the same liquid is deposited on the saturated porous layer, and the spreading of this drop is monitored. Volumes of drops are measured by the direct evaluation of video images. The precision of measurement is around 0.0001–0.0005 cm3. Results and Discussion. Experimental Determination of Effective Lubrication Coefficient ω According to the experimental observation [5], in all spreading experiments, drops retain the spherical shape, and no disturbances or instabilities are detected. Experimental data are ﬁtted using the following dependency L = L0 (1 + t / τ) n

(4.40)

It is necessary to comment on the adopted ﬁtting procedure. If experimentally determined values of the exponent, n, are taken for review [10], it is easy to see that in most cases this exponent is higher (sometimes considerably) than 0.1. In the following text, we present a possible explanation of this phenomenon that we encountered in our experiments. In all our experiments (probably in a number of others, too), drops have been placed on the solid substrate using a syringe. That is, the drops actually fall from a certain height. This means that during a very short initial stage of spreading, both inertia and a relaxation of the drop shape could not be ignored. This means that during the initial stage of spreading, both the Reynolds number, Re, and the capillary number, Ca, are not small, and the capillary regime of spreading is not applicable. Inertia spreading has been considered in Reference 11, where the following spreading law has been predicted: L (t ) = v∞ t ,

(4.41)

where  24 γ  v∞ =   ρL0 

1/ 2

and ρ is the liquid density. The latter equation shows that during the inertia spreading regime, the drop spreads much faster than during the capillary regime of spreading. Derivation in Reference 11 is applicable only if the Reynolds number is high enough. Let us estimate the Reynolds number, which is Re = © 2007 by Taylor & Francis Group, LLC

v∞ H (t )ρ , η

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where H(t) is the maximum drop height. According to Reference 11, H (t ) =

V L2 (t )

during the inertia period of spreading. Condition Re >> 1 gives: V ρ  ρL0  η  24 γ 

1/ 2

1 >> 1 t2

or t << tRe , tRe

 Vρ =  η 

1/ 2

 ρL0   24 γ 

1/ 4

.

(4.42)

The time, tRe, values relevant for our experiments are calculated below. Let us estimate the Reynolds number during the capillary spreading stage. Equation 4.39 gives the velocity of spreading, which should be used for calculation of the Reynolds number. Simple rearrangement gives Re =

ωγθ3 L ≈ 10 −2 . η2

The latter estimation should be compared with Equation 4.42, which gives 2

 tRe  −2  t  ≈ 10 or t ~ 10tRe ~ 0.1 sec. This means that the capillary regime of spreading takes place only at t > 10 tRe. Equation 4.39 is used in the following text to determine the condition for the fulﬁllment of the second requirement, Ca << 1. This equation can be rewritten as Ca = ωθ3 . According to our experimental condition ω ≈ 10–2, θ ≈ 0.5. This gives the following estimation of the capillary number: Ca ~ 10–3. This means, U η Lη ≈ ≈ 10 −3 , γ tγ or t ~ 103 tCa ~ 1 sec, where tCa =

© 2007 by Taylor & Francis Group, LLC

Lη . γ

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329

It is necessary to emphasize that tRe is reversibly proportional to η1/2 (decreases with an increase in viscosity), but tCa is proportional to the viscosity (that is, increases with an increase in viscosity). This corresponds well to our experimental observations. In any experimental observation, only a limited number of experimental values of L(t) dependency are measured. If some of these measurements are taken at the initial regime of spreading, then a higher value results for the ﬁtted exponent than 0.1. For example, let us take the experimental curve in the case of spreading of SO50 over a saturated porous layer (Figure 4.3a and Figure 4.3b). In Figure 4.3a, this dependency is presented in a log–log coordinate system. It is easy to see that the whole spreading process consists of two stages corresponding to two different power laws. During the ﬁrst stage, the inertia/shape relaxation cannot be neglected, whereas in the second stage, capillary spreading (exponent close to 0.1) takes place. In Figure 4.3b, the ﬁtting results for this particular spreading experiment are presented. The broken line corresponds to the ﬁtting procedure when all experimental points are taken into account. In this case, the ﬁtted exponent is higher than 0.1 (0.13 ± 0.01). However, if we do not take into account the ﬁrst three points, located within the initial stage of spreading, then the ﬁtted exponent becomes 0.11 ± 0.01, that is, much closer to 0.1. Figure 4.3a shows that the initial stage of spreading continues approximately around 0.1 sec, which agrees reasonably well with the previously mentioned estimations. The following procedure for the deﬁnition of the parameters L0 and τ was adopted [5]. First, the points, which correspond to the capillary stage of spreading, are selected using the presentation of experimental points in a log–log coordinate system. After this, the ﬁtting procedure using Equation 4.40 is carried out using only the experimental points, which correspond to the capillary stage of spreading. This procedure gives the values of L0 and τ in each run. After the experimental deﬁnition of L0 and τ, the value of ω is calculated in the following manner. Equation 4.34 can be rewritten as 3    4  V 3γ L = L0  1 + 10   10 ω t    π  L0 η 

0.1

.

(4.43)

Comparison of Equation 4.43 and Equation 4.40 gives 3

 π  1 L10 µ 0 . ω=  3  4  τ V 10 γ

(4.44)

Determined values of ω, as well as other experimental parameters, are presented in Table 4.1. The last two rows of Table 4.1 consist of the results of the spreading over a dry glass (microscope optical glass). The data presented in Table 4.1 show that the effective lubrication coefﬁcient (1) is higher in the case of © 2007 by Taylor & Francis Group, LLC

330

Wetting and Spreading Dynamics –1.20 –1.25 –1.30

ᐉn L

–1.35 –1.40 –1.45 –1.50 –1.55 –1.60 –2.0

–1.5

–1.0

–0.5

0.5

0.0

1.0

ᐉn t 0.29 0.28 0.27 Data: 1–12 point Model: L0 (1 + t/τ)n Chi 2/DoF = 0.00002 R 2 = 09689 L0 0.18 ±0.08 0.06 ±0.01 τ n 0.13 ±0.01 Data: 3–12 point n Model: L0 (1 + t/τ) Chi 2/DoF = 52229 E-6 R 2 = 098274 L0 0.20 ±0.03 τ 0.13 ±0.07 0.11 ±0.01 n

0.25

˘

0.24 0.23 0.22

˘

L, cm

0.26

˘

˘

0.21 0.20 0.19 0.0

0.5

1.0

1.5

2.0

2.5

t, s

FIGURE 4.3 (a) Radius of the drop base on time in log–log coordinates. SO50 drop, volume 0.0039 cm3, on a porous membrane with average pore size 0.2 µm. (b) Radius of the drop base on time. SO50 drop with volume 0.0039 cm3, on a porous membrane with average pore size 0.2 µm. Broken line: ﬁtted using all experimental points, solid line: ﬁtted using only points that correspond to the capillary stage of spreading. Fitted parameters are given in the insert.

spreading over saturated porous substrate than in the case of “dry spreading,” (2) experimentally determined values of the effective lubrication coefﬁcient, ω, agree well with the aforementioned theoretical estimations. However, precision of experimental determination of this parameter does not allow us to extract more information about effective viscosity of the porous substrate. © 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

331

4.2 SPREADING OF LIQUID DROPS OVER DRY POROUS LAYERS: COMPLETE WETTING CASE In this section, we take up the problem treated in the previous section but now we shall consider the drop spreading over a dry porous layer. We shall make use of the lubrication approximation, and we shall restrict consideration to the case of complete wetting. Spreading of small liquid drops over thin dry porous layers is investigated in the following text [12]. Drop motion over a porous layer is caused by an interplay of two processes: (1) the spreading of the drop over already saturated parts of the porous layer, which results in an expanding of the drop base, (2) the imbibition of the liquid from the drop into the porous substrate, which results in a shrinkage of the drop base and an expansion of the wetted region inside the porous layer. As a result of these two competing processes, the radius of the drop goes through a maximum value over time. A system of two differential equations is derived to describe the evolution, with time of radii, of both the drop base and the wetted region inside the porous layer. This system includes two parameters; one accounts for the effective lubrication coefﬁcient of the liquid over the wetted porous substrate, and the other is a combination of permeability and effective capillary pressure inside the porous layer. Two additional experiments are used for an independent determination of these two parameters. This system of differential equations does not include any ﬁtting parameter after these two parameters are determined. Experiments were carried out on the spreading of silicone oil drops over various dry microﬁltration membranes (permeable in both normal and tangential directions). The time evolution of the radii of both the drop base and the wetted region inside the porous layer are monitored. All experimental data fell on two universal curves, if appropriate scales are used with a plot of the dimensionless radii of the drop base and of the wetted region inside the porous layer on dimensionless time. The predicted theoretical relationships are the two universal curves accounting for the experimental data quite satisfactorily. According to our theory prediction (1) the dynamic contact angle dependence on the same dimensionless time (as before) should be a universal function, (2) the dynamic contact angle should change rapidly over an initial short stage of spreading and remain a constant value over the duration of the rest of the spreading process. The constancy of the contact angle at this stage has nothing to do with hysteresis of the contact angle; there is no hysteresis in our system. These conclusions again are in good agreement with our experimental observations [12]. It has been shown in Section 4.1 that the presence of roughness or a porous sublayer changes the spreading conditions. In the same section, the spreading of small liquid drops over thin porous layers saturated with the same liquid has been investigated. Instead of the “slippage conditions,” Brinkman’s equations have been used in Section 4.1 for the description of the liquid ﬂow inside the porous substrate. In the present section, we take up the same problem, as is the case when a drop spreads over a dry porous layer. The problem is treated in the following text © 2007 by Taylor & Francis Group, LLC

332

Wetting and Spreading Dynamics

under the lubrication theory approximation and in the case of complete wetting. Spreading of big drops (but still small enough to neglect the gravity action) over thin porous layers is considered in the following text.

THEORY The kinetics of liquid motion both in the drop above the porous layer and inside the porous layer itself are taken into account here. The thickness of the porous layer, ∆, is assumed to be much smaller than the drop height, that is, ∆ << h*, where h* is the scale of the drop height. The drop proﬁle is assumed to have a low slope (h*/L* << 1, where L* is the scale of the drop base) and the inﬂuence of the gravity is neglected (small drops, Bond number ρgL2* / γ << 1, where ρ, g, and γ are the liquid density, gravity acceleration, and the liquid–air interfacial tension, respectively). That is, only capillary forces are taken into account. Under such assumptions, a system of two differential equations is obtained to describe the evolution with time of the radii of both the drop base, L(t), and the wetted region inside the porous layer, l(t), (Figure 4.4). Further assumptions made are justiﬁed in the Appendix 1. As in Section 4.2, the proﬁle of axisymmetric drops spreading over the porous substrate (whether dry or saturated with the same liquid) is governed by the following equation: ∂h 1 ∂ = u0 − r ∂r ∂t

     ∂  1 ∂  ∂h  3 γ r  + v0 h  , r  h   3 η ∂r  r ∂r  ∂r      

(4.45)

where h(t,r) is the proﬁle of the spreading drop; t and r are the time and the radial coordinate, respectively; z > 0 corresponds to the drop; –∆ < z < 0 correspond to the porous layer; z = 0 is the drop–porous layer interface (Figure 4.4); v, u are

z

1 3

∆

θ 2

h(t, r)

r

L(t)

0 2

3 l(t)

FIGURE 4.4 Cross section of the axisymmetric spreading drop over initially dry thin porous substrate with thickness: ∆; (1) liquid drop; (2) wetted region inside the porous substrate; (3) dry region inside the porous substrate; L(t) — radius of the drop base; l(t) — radius of the wetted area inside the porous substrate; ∆ — thickness of porous substrate; r, z — coordinate system; h(t, r) — proﬁle of the spreading drop. © 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

333

the radial and vertical velocity components, respectively; v0, u0 are velocity components at the drop–porous layer interface; η is the liquid viscosity. The liquid velocity components, v0, u0, on the drop–porous layer interface are calculated by matching the ﬂow in the drop with the ﬂow inside the porous layer. The porous layer is assumed to be very thin, and the time for saturation in the vertical direction can be neglected, relative to other time scales of the process. Let us calculate the time required for a complete saturation of the porous layer in the vertical direction. According to Darcy’s equation u=

K p pc dz , − ∆ < z < 0; u = , η z dt

where Kp and pc are the permeability of the porous layer and the effective capillary pressure, respectively; z is the position of the liquid front inside the porous layer. The solution of this equation results in: ∆ 2 = 2 K p pct ∆ / η , where t∆ is the time of the complete saturation for the porous layer in the vertical direction and hence,

t∆ =

∆2 η . 2 K p pc

The consideration in the following text is restricted to t > t∆. Estimations show that t∆ is less than t0, which is the duration of the initial stage of spreading (see Section 4.1 for details and an estimation of t0). The capillary spreading regime (the only one considered here) is not applicable at t < t0. Thus, we must consider those instances such that t > max(t0, t∆), when both the initial stage is over, and the porous layer is completely saturated in the vertical direction. Accordingly, the porous layer beneath the spreading drop (0 < r < L(t)) is always assumed to be completely saturated. The capillary pressure inside the porous layer, pc, can be estimated as pc ≈

2γ , r*

where r* is the scale of capillary radii inside the porous layer. The capillary pressure inside the drop, p – pa, can be estimated as p − pa ≈

γh* h* γ γ γ = << << ≈ pc . L* L* L* r* L2*

This means that the capillary pressure inside the pores of the porous layer is of several orders of magnitude higher than the capillary pressure in the drop itself. © 2007 by Taylor & Francis Group, LLC

334

Wetting and Spreading Dynamics

The boundary conditions for Equation 4.45 are as follows: symmetry condition in the drop center, ∂h ∂3 h = 0, r = 0 , = ∂r ∂r 3

(4.46)

conservation of drop volume, L

∫

2 π rhdr = V (t ) .

(4.47)

0

The drop volume changes over time because of the imbibition of the liquid into the porous layer, which means V (t ) = V0 − πm ∆ l (t )2 ,

(4.48)

where V0 is the initial volume of the drop; m is the porosity of the porous layer; and l (t ) is the radius of the wetted area inside the porous layer. The wetted region is a cylinder with radius l(t) and the height ∆. l(t) is referred to here as the radius of the wetted region inside the porous layer. Let tmax be the time instant when the drop is completely sucked by the porous 2 substrate V (tmax ) = 0 = V0 − πm ∆lmax , where lmax is the maximum radius of the wetted region in the porous layer. The preceding equation gives  V  lmax =  0   πm ∆ 

1/ 2

.

(4.49)

lmax is used in the following text to scale the radius of the wetted region in the porous layer, l(t). It is easy to check that the previous equation results in lmax > L* in our case. Combination of Equation 4.47 and Equation 4.48 results in L

∫

2 π rhdr = V0 − π m ∆ l 2 (t ) .

(4.50)

0

Everywhere at (r < L(t)) except for a narrow region, ξ, close to the moving three-phase contact line, we have h >> ∆, and the liquid motion inside the porous layer under the drop can be neglected both in the vertical and horizontal directions (see the Appendix 1 for details). The size of this narrow region close to the moving

© 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

335

three-phase contact line, where suction of the liquid from the drop into the porous substrate takes place, is estimated in the Appendix 1. The latter means that Equation 4.45 can be rewritten as ∂h γ 1 ∂ =− ∂t 3 η r ∂r

 3 ∂  1 ∂  ∂h    r rh  , r < L (t ) − ξ . ∂r  r ∂r  ∂r    

(4.51)

Equation 4.51 should be solved with boundary conditions (4.46 and 4.50) and leads to h(t, L – ξ) ≈ 0.

(4.52)

Following arguments developed in Chapter 3 (Section 3.2), the solution of Equation 4.51 can be obtained using outer and inner solutions. The outer solution can be deduced in the following way: the left-hand side of Equation 4.51 should be set to zero. After integration of the resulting equation with boundary conditions (4.46, 4.50, and 4.52) the outer solution becomes: h (t, r ) =

2V 2 ( L − r 2 ), r < L (t ) − ξ . πL4

(4.53)

Equation 4.53 shows that the drop surface proﬁle remains spherical during the spreading process, except for a short initial stage when the porous layer is not saturated and a ﬁnal stage, when condition ∆ << h is violated everywhere over the whole proﬁle of the drop. Equation 4.53 gives the following value of the dynamic contact angle, θ, (tanθ ≈ θ): θ=

4V , πL3

(4.54)

or else  4V  L=  πθ 

1/ 3

.

(4.55)

The drop motion is a superposition of two motions: (1) the spreading of the drop over the already saturated part of the porous layer, which results in an expansion of the drop base, and (2) a shrinkage of the drop base caused by the imbibition into the porous layer. Hence, we can write the following equation:

© 2007 by Taylor & Francis Group, LLC

336

Wetting and Spreading Dynamics

dL = v+ − v− , dt

(4.56)

where v+, v– are unknown velocities of the expansion and the shrinkage of the drop base, respectively. Let us take the time derivatives of both sides of Equation 4.55. It gives: dL 1  4V  =−  4 dt 3  πθ 

1/ 3

dθ 1  4  + dt 3  πV 2 θ 

1/ 3

dV . dt

(4.57)

Over the whole duration of the spreading over the porous layer, both the contact angle and the drop volume can only decrease with time. Accordingly, the ﬁrst term on the right-hand side of Equation 4.57 is positive and the second one is negative. Comparison of the two latter equations yields 1  4V  v+ = −  4  3  πθ 

1/ 3

1 4  v− = −  3  πV 2 θ 

dθ >0 dt (4.58)

1/ 3

dV > 0. dt

There are two substantially different characteristic time scales in our problem: t η* << tmax , where tη* and tmax are time scales of the viscous spreading and the imbibition into the porous layer, respectively; λ=

t η* << 1 tmax

is a smallness parameter (around 0.08 under our experimental conditions, see the following text). Both the time scales are calculated here. Then we have L = L(Tη, Tp) [13], where Tη is a fast time of the viscous spreading, and Tp is a slow time of the imbibition into the porous substrate. The time derivative of L(Tη, Tp) is ∂L dL ∂L +λ = . ∂T p dt ∂Tη

(4.59)

Comparison of Equation 4.56, Equation 4.58, and Equation 4.59 shows that v+ =

1  4V  ∂L =−  4 3  πθ  ∂Tη

© 2007 by Taylor & Francis Group, LLC

1/ 3

dθ ∂L 1 4  , v− = − λ =−  dt ∂T p 3  πV 2 θ 

1/ 3

dV dt

Spreading over Porous Substrates

337

The smallness of λ=

t η* << 1 tmax

means that in the case under consideration, two processes actually function independently: the spreading of the drop over the saturated part of the porous layer, and the shrinkage of the drop base caused by the imbibition of the liquid from the drop into the porous layer. The decrease of the drop volume, V, with time is determined solely by the imbibition into the porous substrate and hence the drop volume, V, only depends on the slow time scale. According to the previous consideration, the whole spreading process can be subdivided into two stages: (1) A ﬁrst fast stage, when the imbibition into the porous substrate can be neglected, and the drop spreads with approximately constant volume. This stage goes in the same way as the process of spreading over saturated porous layer, and the arguments developed in Section 4.1 can be used here again. (2) A second slow stage, when the spreading process is almost already over, and the evolution is determined by the imbibition into the porous substrate. During the ﬁrst stage, Equation 4.59 from Section 4.1 can be rewritten as  10 γω  4V  3   L (t ) =   η  π    

0.1

(t + t )

0.1

0

,

(4.60)

where t0 is the duration of the initial stage of spreading, when the capillary regime of spreading is not applicable; and ω is an effective lubrication parameter, which has been discussed and estimated in Section 4.1. It is important to emphasize that the effective lubrication parameter, ω, is independent of the drop volume and depends solely on the porous layer properties. According to Equation 4.60, the characteristic time scale of the ﬁrst stage of spreading is 3

tη* where L0 = L (t0 ) .

© 2007 by Taylor & Francis Group, LLC

ηL0  πL30  = , 10 γω  4V0 

(4.61)

338

Wetting and Spreading Dynamics

Combination of Equation 4.59 and Equation 4.60 gives:  4V  θ=  π 

0.1

 η   10 γω 

0.3

(t + t )

−0.3

0

.

Substitution of this expression into the ﬁrst Equation 4.58 gives the following expression for the velocity of the drop base expansion, v+:  4V  v+ = 0.1   π 

0.3

 10 γω   η 

0.1

1

(t + t )

0.9

.

(4.62)

0

Substitution of Equation 4.48 into the second Equation 4.58 gives the following expression for velocity of the drop base shrinkage, v:  2 π 2 / 3 m ∆l  4 v− =  3  V0 − πm ∆l 2

(

  2  θ 

1/ 3

dl . dt

)

(4.63)

Substitution of the two latter equations into Equation 4.57 results in:  4V  dL = 0.1  dt  π 

0.3

 10 γω   η 

0.1

1

(t + t )

0.9

0

 2 π m ∆l  4 −  3  V0 − πm ∆l 2 2 /3

(

  2  θ 

)

1/ 3

dl dt (4.64)

The only unknown function now is the radius of the wetted region inside the porous layer, l(t), which is determined in the following text. ∆ < z < 0, L < r < l) Inside the Porous Layer outside the Drop (–∆ The liquid ﬂow inside the porous layer obeys the Darcy equation K p ∂p 1 ∂  ∂p  = 0, v = − r . η ∂r r ∂r  ∂r  The solution of the preceding equations is p = −( Aη/K p ) ln r + B v=

© 2007 by Taylor & Francis Group, LLC

A r

(4.65)

Spreading over Porous Substrates

339

where A and B are integration constants that are determined using the boundary conditions for the pressure at the drop edge, r = L(t), and at the circular edge of the wetted region inside the porous layer, r = l(t). The latter boundary condition is: p = pa − pc , r = l (t ) ,

(4.66)

where pc ≈

2γ r*

is the capillary pressure inside the pores of the porous layer, and r* is a characteristic scale of the pore radii inside the porous layer. The boundary condition at the drop edge is p = pa + pd , r = L (t ) ,

(4.67)

where pd is an unknown pressure. It is further shown that pd << pc . However, we keep this small value for a future estimation. Taking into account the two preceding boundary conditions, both the integration constants, A and B, can be determined, which gives the following expression for the radial velocity according to Equation 4.65:

v=

(

K p pc + pd l ηr ln L

).

(4.68)

The velocity at the circular edge of the wetted region inside the porous layer is: dl =v . r =l dt Combination of the two preceding equations gives the evolution equation for l(t):

(

)

dl K p pc + pd = . l dt ηl ln L

© 2007 by Taylor & Francis Group, LLC

(4.69)

340

Wetting and Spreading Dynamics

An estimation of the time scale tmax can be made according to Equation 4.69 and taking into account Equation 4.49 as follows K p pc / η lmax ≈ , lmax tmax lmax ln L* or l lmax V0 ln max L* L* ≈ = . πm ∆K p pc /η η K p pc / η 2 ln lmax

tmax

(4.70)

Comparison of the estimated values of tη* according to Equation 4.61 and of tmax according to Equation 4.70 shows that under all our experimental conditions (see the following text), the following inequality tη* << tmax is satisﬁed. Omitting the small term, pd, and substitution of Equation 4.69 into Equation 4.64 gives the following system of differential equations for the evolution of both the radius of the drop base, L(t), and that of the wetted region inside the porous layer, l(t):

(

 4 V0 − πm ∆l 2 dL = 0.1  dt π 

) 

0.3

 10 γω   η 



0.1

1

(t + t )

0.9

0

−

2 3

πm ∆K p pc L / η

(

)

 l V0 − πm ∆l 2  ln   L

,

(4.71) dl K p pc /η . = l dt l ln L

(4.72)

Let us make the system of differential equations (Equation 4.71 and Equation 4.72) dimensionless using new scales L = L /Lmax , l = l /lmax , t = t /tmax , where Lmax is the maximum value of the drop base, which is reached at the time instant tm (that has to be determined). The same symbols are used for the dimensionless variable as for corresponding dimensional variables (marked with an overbar). The system of Equations (Equation 4.71 and Equation 4.72) transforms as:

(

)

0.9

tm + τ dL 2 = dt 3 1 − l 2 1.3 1 + χ ln l m m

(

) (

© 2007 by Taylor & Francis Group, LLC

(1 − l )

0.3

2

) (t + τ)

0.9

−

2 3

(1 − l ) 2

L , (4.73)  l   1 + χ ln L 

Spreading over Porous Substrates

341

dl = dt

1  l  l  1 + χ ln  L 

,

(4.74)

where τ = t0 /tmax << 1, χ = 1 / ln

lmax . Lmax

Thus, the latter system includes only two dimensionless parameters, τ , χ ; the ﬁrst one is very small, and the second one changed insigniﬁcantly under our experimental conditions because of a weak logarithmic dependence on lmax /Lmax. Accordingly, the two dimensionless dependencies L ( t ), l ( t ) should fall on two almost universal curves, which is in very good agreement with our experimental observations (see Results and Discussion). According to Equation 4.73, the dimensionless velocities of the expansion of the drop base, v+ , and the shrinkage, v− , are as follows:

(

)

0.9

tm + τ 2 v+ = 3 1 − l 2 1.3 1 + χ ln l m m

(

) (

(1 − l )

0.3

2

) (t + τ)

0.9

, v− =

2 3

L

(1 − l ) 2

 l   1 + χ ln L 

. (4.75)

Figure 4.5 shows dimensionless velocity v+ and v− calculated according to Equation 4.75. It appears that: (1) The ﬁrst stage is very short. Here, the capillary spreading prevails over the drop base shrinkage caused by the liquid imbibition into the porous substrate; (2) The spreading of the drop almost stops after the ﬁrst stage of spreading, and the shrinkage of the drop base is determined by the suction of the liquid from the drop into the porous substrate. Let us consider the asymptotic behavior of system (4.71 and 4.72) over the second stage of the spreading. According to Figure 4.5, over the second stage of the spreading, velocity of the expansion of the drop, v+, decreases. To understand the asymptotic behavior, the term corresponding to v+ on the left-hand site of Equation 4.71 is omitted. This gives: dL 2 =− dt 3

© 2007 by Taylor & Francis Group, LLC

πm ∆K p pc L /η ,  l V0 − πm ∆l 2  ln   L

(

)

(4.76)

342

Wetting and Spreading Dynamics – V

15

– V+ – V– 10

5

0 0.5 t

0.0

1.0

FIGURE 4.5 Dimensionless velocity of spreading (v– +, solid line) and velocity of drop shrinkage (v– – , dotted line) on dimensionless time, calculated according to Equation 4.75. – Intersection of these two dependences determines the value of the dimensionless time tmax – ≈ 0.08, when the radius of the drop base reaches its maximum value, Lmax = 1 (in dimensionless units).

whereas the second Equation 4.72 is left unchanged. The system of differential equations (Equation 4.72 and Equation 4.76) can be solved analytically. For this purpose, Equation 4.76 is divided by Equation 4.72, which gives

dL 2 πm ∆ L l =− . dl 3 V0 − πm ∆l 2

(

)

If V = V0 − πm ∆l 2 is used instead of l, the latter equation takes the following form: dL L = , dV 3V which can be easily integrated and the solution is V = C L3 ,

(4.77)

where C is an integration constant. Let us rewrite Equation 4.54 using the same dimensionless variables: © 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

343

V=

4 L3max θ L3 . πV0

(4.78)

Comparison of Equation 4.77 and Equation 4.78 shows that the dynamic contact angle asymptotically remains constant over the duration of the second stage. This constant value is marked in the following text as θf . Let us introduce θm =

πV0 (1 − lm2 ) , 4 L3max

which is the value of the dynamic contact angle at the time instant when the maximum value of the drop base is reached. Then Equation 4.78 can be rewritten as: θ (1 − l 2 ) / (1 − lm2 ) = θm L3

(4.79)

and the latter relationship should be a universal function of the dimensionless – time, t. This conclusion agrees well with our experimental observations (see Results and Discussion). It is necessary to emphasize that in the case under consideration, the constancy of the contact angle has nothing to do with the contact angle hysteresis; there is no hysteresis in our system here. θf is not a receding contact angle but forms as a result of a self-regulation of the ﬂow in the drop–porous layer system. The system of equations (Equation 4.71 and Equation 4.72) includes seven parameters, ﬁve of which can be measured directly (V0, γ, η, m, and ∆ are the initial volume of the drop, the liquid–air interfacial tension, the liquid viscosity, the porosity of the porous layer, and the thickness, respectively), and two additional parameters, ω and Kp pc, which should be determined independently. It is noteworthy that the porous layer permeability and the capillary pressure always enter as a product, that is, this product can be considered as a single parameter. A procedure for the independent determination of an effective lubrication coefﬁcient, ω, has been discussed in Section 4.1. Experimental Part Silicone oils SO20 (viscosity 0.218 P), SO50 (viscosity 0.554 P), SO100 (viscosity 1.18 P), and SO500 (viscosity 5.582 P) purchased from Prolabo, Productos para Laboratorios Quimicos, a Spanish company, were used in the spreading experiments. The viscosity of oils were measured using the capillary Engler Viscometer VPG-3 at 20 ± 0.5˚C. Cellulose nitrate membrane ﬁlters purchased from Sartorius (type 113), with pore size 0.2 and 3 µm (marked by the supplier), © 2007 by Taylor & Francis Group, LLC

344

Wetting and Spreading Dynamics

were used as porous layers. These membranes are referred to as the membrane, 0.2 µm and the membrane, 3 µm, respectively. Pore size distribution and permeability of membranes were tested using the Coulter Porometer II. The pore size of the membrane, 0.2 µm, falls in the range 0.2–0.38 µm, with the average pore size 0.34 µm. The permeability of the same membrane is equal to 12 l/min*cm2 (air ﬂux at the transmembrane pressure 5 bar). The permeability of the membrane, 3 µm, is 2.5 l/(min*cm2) at the transmembrane pressure 0.1 bar. All membrane samples used are plane parallel circles of radius 25 mm and thickness in the range from 0.0130 to 0.0138 cm. The porosity of the membranes ranges between 0.65 and 0.87. The porosity was measured using the difference in the weight of dry membranes and membranes saturated with oil. Membranes were dried for 3–5 hours at 95˚C and then stored in a dry atmosphere prior to spreading experiments. The same experimental device as described in Section 4.1 (Figure 4.2) was used for monitoring the spreading of drops over initially dry porous layers. The time evolution of the radius of the drop base, L(t), the dynamic contact angle, θ(t), and the radius of the wetted region inside the porous layer, l(t), were monitored. The porous wafer 1 (Figure 4.2) was placed in a thermostated and hermetically closed chamber 2, where zero humidity and ﬁxed temperature (20 ± 0.5˚C) were maintained. The distance from the wafer to the tip of the dosator ranged from 0.5 to 1 cm in different experiments. The volume of drops was set by the diameter of the separable replaceable capillary of the dosator in the range 1–15 µl. Experiments were carried out in the following order: • •

•

The membrane was placed in the chamber and left in a dry atmosphere for 15–30 minutes. A light pulse produced by a ﬂash gun was used to synchronize the time instant when the drop started to spread and recorded using both video-tape recorders (a side view and a view from above). A droplet of silicone oil was placed onto the membrane.

Each run was carried out until complete imbibition of the drop into the membrane took place. Independent Determination of Kp pc As mentioned previously, the permeability of the porous layer and the capillary pressure always enter as a product, i.e., as a single coefﬁcient. Additional experiments were carried out to determine this coefﬁcient. For this purpose, the horizontal imbibition of the liquid under investigation into the dry porous sheet was undertaken. Rectangular sheets, 1.5cm∗3cm, were used. These porous sheets were cut from the same membranes used in the spreading experiments. Each sheet was immersed to a length of 0.3–0.5 cm into a liquid container, and the position of the imbibition front was monitored over time. In the case under investigation, a © 2007 by Taylor & Francis Group, LLC

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345

TABLE 4.2 The Measured Values of Kp pc Membrane Pore Size, µm

Liquid

3 3 3 0.2 0.2

SO20 SO100 SO500 SO5 SO100

dyn

Kp p c ,

(1.2 ± 0.4)*10–4 (1.77 ± 0.03)*10–4 (1.6 ± 0.2)*10–4 (3.4 ± 0.3)*10–5 (3.1 ± 0.3)*10–5

unidirectional ﬂow of liquid inside the porous substrate took place. Using Darcy’s law, we can conclude that d 2 (t ) = 2 K p pct /η,

(4.80)

where d(t) is the position of the imbibition front inside the porous layer. It was found that all runs d2(t)/2 proceed along a straight line, whose slope gives us Kp pc value. According to Reference 3, Kp pc should be independent of the tested liquid viscosity. The measured values of Kp pc are presented in Table 4.2, which shows that the coefﬁcient, Kp pc, for each type of membrane is independent of the tested liquid within the experimental error. Average values of Kp pc for each membrane were used in the calculations. Results and Discussion According to our observations, the whole spreading process can be subdivided into two stages (see Figure 4.6 as an example): fast spreading over the ﬁrst several seconds until the maximum radius, Lmax, of the drop base is reached. During the ﬁrst stage, an imbibition front inside the membrane expands slightly ahead of the spreading drop. After that the drop base starts to shrink slowly, and the imbibition front expands until the drop completely disappears. An example of the time evolution of the radius of the drop base and the radius of the wetted region inside the porous layer is provided in Figure 4.7. In all our spreading experiments, the drops remain spherical over the duration of both the ﬁrst and the second stages of the spreading process. This was cross checked by reconstructing of the drop proﬁles at different time instants of spreading, ﬁtting those proﬁles by a spherical cap:

(

h = zcenter + R 2 − r − rcenter

© 2007 by Taylor & Francis Group, LLC

)

2

,

346

Wetting and Spreading Dynamics

(a)

(b)

(c)

(d)

(e)

FIGURE 4.6 Time sequence of spreading of SO500, volume 8.7 µl over the membrane with pore size 3 µm (side view): (a) t = 0.5 sec (after deposition); (b) 3 sec; (c) 12 sec; (d) 22 sec; (e) 36 sec.

where (rcenter , zcenter) is the position of the center of the sphere, R is the radius of the sphere. rcenter , zcenter , and R are used as ﬁtting parameters. The ﬁtting is based on the Levenberg–Marquardt algorithm. In all cases, the reduced Chi-square value is found less than 10–4. The ﬁtted parameter R gives the radius of curvature of the spreading drops at different times. The edge of the wetted region inside the porous layers was always circular. Drops remained in the center of this circle over the entire duration of the spreading process. No deviations from cylindrical symmetry or instabilities were detected. The spherical form of the spreading drop allows the measurement of the evolution of the dynamic contact angle of the drop. In all cases, the dynamic contact angle decreases very fast over the ﬁrst stage of spreading until a constant value is reached, which is referred to as θf . The dynamic contact angle θf , remains constant over the main part of the second stage. © 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

347

0.5

L, ᐉ, cm

0.4

0.3

0.2

0.1 10

100

1000

t, s

FIGURE 4.7 Development over time of the drop base, L, and the radius of the wetted region inside the porous layer, l. The same drop as in Figure 4.6: SO500 drop, placed onto the membrane with pore size 3 µm. Solid lines calculated according to Equation 4.71 and Equation 4.72.

80

θ, degree

60

40

20

0 0

100

200

300

400

500

600

700

t, s

FIGURE 4.8 Development (over time) of the dynamic contact angle. The same drop as in Figure 4.4 and Figure 4.5: SO500 drop, placed onto the membrane with pore size 3 µm. Solid line calculated according to Equation 4.96 and Equation 4.97.

An example of the evolution with time of the dynamic contact angle is presented in Figure 4.8 for the same drop as in Figure 4.6 and Figure 4.7. Figure 4.8 shows that the dynamic contact angle decreases fast over the duration of the © 2007 by Taylor & Francis Group, LLC

348

Wetting and Spreading Dynamics

ﬁrst stage of the spreading when the radius of the drop base expands to its maximum value, Lmax. The dynamic contact angle remains almost unchanged over the duration of the second stage of spreading. Note that this behavior has nothing to do with the hysteresis of the contact angle because it does not occur in our system, that is, silicone oils on nitrate cellulose membranes. Solid lines in Figure 4.7 and Figure 4.8 represent the results of numerical integration of the system of equations (Equation 4.71 and Equation 4.72). The short ﬁnal period (just before the drop disappears) is not covered by our calculations. Close to this ﬁnal stage the calculation errors increased, which was caused by a division by a very small quantity on the second term in the right-hand side of Equation 4.71. Table 4.3 shows that the ﬁnal value of the dynamic contact angle, θt (last column in Table 4.3), depends on the volume of the drop, as well as on the viscosity of the liquid, and hence, θt is determined solely by hydrodynamics. Figure 4.9a presents experimentally measured dependencies of the radius of the drop base and the wetted region inside the porous layer on time for different silicone oils, porous layers, and drop volumes. All relevant values are summarized in Table 4.3. The main result appears in Figure 4.9b, which shows that all experimental data (the same as in Figure 4.9a) fall on two universal curves if dimensionless coordinates are selected as follows: L = L /Lmax , l = l /lmax , t = t /tmax , where Lmax is the maximum value of the drop base, which is reached at the time instant tm. The same symbols (with and without overbar) are used for dimensionless values as for dimensional ones. The scale lmax is determined by Equation 4.49, and the time scale tmax is given by the Equation 4.70. The measured values of Lmax, lmax, and tmax for all experimental runs are given in Table 4.3. Figure 4.9b shows that the dimensionless time tm is about 0.08 ≈ λ << 1 as it was stated above. The dimensionless time 1 corresponds to the time instant when the drop is completely sucked by the porous substrate. Solid curves in Figure 4.9b represent the solution of the system of differential equations (Equation 4.73 and Equation 4.74). If the parameters τ and χ change, then both the theoretical curves remain inside the array of experimental points. In this sense they represent universal relationships. The twelfth column in Table 4.3 gives the experimental values of the dynamic contact angle, θm , which the drop has when the drop base reaches its maximum value, Lmax. These values were used for plotting the time evolution of the dynamic contact angle, θ/θm. Figure 4.10 shows that all experimental points fall on a single universal curve, as predicted by Equation 4.79. The solid line in Figure 4.10 is a result of calculations according to Equation 4.79, where dimensionless dependencies L ( t ), l ( t ) are taken from the previous Figure 4.9b.

© 2007 by Taylor & Francis Group, LLC

Liquid SO20 SO20 SO20 SO20 SO20 SO100 SO100 SO500 SO500 SO500

Notation on Figures

Membrane Pore Size, µm

V 0, ml*103

∆, mm

m-Porosity

tmax S

lmax cm

lmax Theory

Lmax cm

Lmax Theory

θm Degree

θt Degree

0.2 0.2 0.2 3 3 3 3 3 3 3

3.1 9.0 15.6 3.8 5.5 8.6 14.5 3.6 8.7 14.5

0.114 0.116 0.116 0.136 0.134 0.137 0.138 0.138 0.138 0.136

0.85 0.72 0.73 0.87 0.83 0.82 0.77 0.89 0.88 0.78

102 440 814 10.9 17.1 186 354 296 851 1660

0.318 0.585 0.77 0.345 0.428 0.493 0.659 0.306 0.477 0.66

0.317 0.584 0.766 0.319 0.398 0.494 0.659 0.306 0.478 0.660

0.179 0.314 0.387 0.196 0.223 0.257 0.332 0.174 0.264 0.339

0.184 0.31 0.39 0.198 0.234 0.274 0.34 0.179 0.286 0.34

20.0 12.6 14.2 25.9 20.3 18.5 15.5 22.6 19.1 15.8

12 11.4 12.1 22.3 18.6 11 17.2 19.4 17.3 16.5

× ▫

Spreading over Porous Substrates

TABLE 4.3 Notations Used and Calculated/Predicted Values

349

© 2007 by Taylor & Francis Group, LLC

350

Wetting and Spreading Dynamics 8 7 6

SO 20 V0 = 3.1 mm3, r = 0.2 μ SO 20 V0 = 9.0 mm3, r = 0.2 μ SO 20 V0 = 15.6 mm3, r = 0.2 μ SO 20 V0 = 3.8 mm3, r = 3 μ SO 20 V0 = 5.5 mm3, r = 3 μ SO 100 V0 = 8.6 mm3, r = 3 μ SO 100 V0 = 14.5 mm3, r = 3 μ SO 500 V0 = 3.6 mm3, r = 3 μ SO 500 V0 = 8.7 mm3, r = 3 μ SO 500 V0 = 14.5 mm3, r = 3 μ

L, ᐉ mm

5 4 3 2 1 0 0

500

1000

1500

t, s 1.0

SO 20 V0 = 3.1 mm3, r = 0.2 μ SO 20 V0 = 9.0 mm3, r = 0.2 μ SO 20 V0 = 15.6 mm3, r = 0.2 μ SO 20 V0 = 3.8 mm3, r = 3 μ SO 20 V0 = 5.5 mm3, r = 3 μ SO 100 V0 = 8.6 mm3, r = 3 μ SO 100 V0 = 14.5 mm3, r = 3 μ SO 500 V0 = 3.6 mm3, r = 3 μ SO 500 V0 = 8.7 mm3, r = 3 μ SO 500 V0 = 14.5 mm3, r = 3 μ Theory

0.5

0.5

0.0

ᐉ/ᐉmax

L/Lmax

1.0

0.0 0.0

0.5 t/tmax

1.0

FIGURE 4.9 (a) Measured dependences of radii of the drop base (L, mm) and radii of the wetted region inside the porous layer (l, mm) on time (t, s). All relevant values are summarized in Table 4.3. (b) The same as in Figure 4.13a but using dimensionless coordinates: L/Lmax , l/lmax , and t/tmax , where Lmax is the maximum value of the drop base, which is reached at the moment tm . The same symbols (with overbar) are used for dimensionless values as for dimensional ones. The scale lmax is determined by Equation 4.49 and the time scale tmax is given by Equation 4.70. Solid lines according to Equation 4.73 and Equation 4.74. © 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

351

5 SO 20 V0 = 3.1 mm3, r = 0.2 µ SO 20 V0 = 9.0 mm3, r = 0.2 µ SO 20 V0 = 15.6 mm3, r = 0.2 µ SO 20 V0 = 3.8 mm3, r = 3 µ SO 20 V0 = 5.5 mm3, r = 3 µ S100 V0 = 8.6 mm3, r = 3 µ SO 100 V0 = 14.5 mm3, r = 3 µ SO 500 V0 = 3.6 mm3, r = 3 µ SO 500 V0 = 8.7 mm3, r = 3 µ SO 500 V0 = 14.5 mm3, r = 3 µ Theory

4

θ/θm

3

2

1

0 0.0

0.2

0.4

0.6

0.8

1.0

t/tmax

FIGURE 4.10 Dynamic contact angle on the dimensionless time. Solid line according to Equation 4.79.

APPENDIX 1 The slip boundary condition is used in the following text for the sake of simplicity. The slippage coefﬁcient is taken according to the Section 4.1. The similar results can be deduced using Brinkman’s equations for the description of the liquid ﬂow inside porous layers. The slippage condition at the drop–porous layer interface is

η

v0 − v p ∂v = ηp ∂z z =+0 δ

(A1.1)

where

vp =

K p ∂p p η p ∂r

is the velocity inside the porous substrate; ηp is an effective viscosity inside the porous layer (see Reference 3), and pa – pp is the pressure inside the porous layer, which may be different from the pressure in the spreading drop. The porous layer thickness, ∆, is assumed to be much bigger than Brinkman’s length, δ = K p η p . Hence, both velocities v0 and u0 change stepwise at the drop–porous layer interface: the jump of the ﬁrst velocity is given by Equation A1.1, whereas the jump of the second velocity is © 2007 by Taylor & Francis Group, LLC

352

Wetting and Spreading Dynamics

u0 =

 K p  γ ∂  ∂h  − r  + pp  .   η pδ  r ∂r  ∂r  

(A1.2)

This means that the vertical velocity changes stepwise from the value given by Equation A1.2 on the drop–porous layer interface to zero inside the porous layer. Equation A1.1 becomes v0 =

K p ∂p p δ h γ ∂  1 ∂  ∂h   + r . η p ∂r η p ∂r  r ∂r  ∂r  

(A1.3)

Substitution of Equation A1.2 and Equation A1.3 into Equation 4.45 gives:  K ∂  ∂p p  ∂h K p  γ ∂  ∂h  = r  − pp  − p    rh  ∂t η pδ  r ∂r  ∂r   η pr ∂r  ∂r  γ 1 ∂ − 3η r ∂r

   1 ∂  ∂h     3 2 ∂ r , r  h + 3αδh ∂r  r ∂r  ∂r       

(

)

(A1.4)

where α=

η <1 ηp

according to Reference 3. The latter equation describes the evolution of the spreading proﬁle of the drop, both in space and time. The only unknown dependence left is the pressure inside the porous layer, pp. The conservation law inside the porous layer

( )

1 ∂ rv ∂u + =0 r ∂r ∂z is used to determine this pressure. The latter equation is integrated over z from ∆ to 0 inside the porous layer, using condition (A1.2) and Darcy’s law to express the velocity components, using the pressure gradient. After certain transformations, the ﬁnal equation becomes 1 ∂  ∂p p  p p γ ∂   h ∂  1 ∂  ∂h     1 γ ∂  ∂h  − =− , r r r  − r    r ∂r  ∂r  ∆δ r ∂r   δ ∂r  r ∂r  ∂r     ∆δ r ∂r  ∂r    (A1.5) © 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

353

which describes the radial distribution of the pressure inside the porous layer. Equation A1.5 shows that (1) Under the spherical part of the spreading drop, the pressure inside the porous layer remains constant and equal to the pressure inside the drop, that is p ∞p =

2γθ L

(A1.6)

and hence, the liquid does not ﬂow inside the porous layer under the spherical part of the drop. (2) The pressure inside the porous layer changes from the constant value (A1.6) to the pressure outside the drop close to r = L in a narrow region with a scale ξ = ∆δ << L

(A1.7)

as stated earlier (see Equation 4.51 through Equation 4.53). Let us introduce a new local dimensionless variable, x, as follows: x=

r−L , −∞ < x < 0 ξ

(A1.8)

After omitting certain small terms, Equation A1.5 becomes p′′p − p p = −

γδ   f f ′′′ ′ + f ′′  ξ 2  

(

)

(A1.9)

where f = h/δ is the dimensionless proﬁle of the drop in this narrow transition region. The latter scale is selected in the same way as in Section 4.1. Equation A1.9 can be directly integrated using the following boundary conditions: p p (−∞) = p ∞p .

(A1.10)

p p (0 ) = pd .

(A1.11)

The boundary condition (A1.10) follows from Equation A1.6, whereas condition (A1.11) still includes the unknown pressure, pd , which is determined below. © 2007 by Taylor & Francis Group, LLC

354

Wetting and Spreading Dynamics

The solution of Equation A1.9, which satisﬁes both boundary conditions (A1.10) and (A1.11) is γδ p p = pd e + 3 ξ x

0





∫ ( f f ′′′ ) + f ′′  sinh ( x − y ) dy. ′

(A1.12)

x

Let us assume that pd >> p ∞p .

(A1.13)

This assumption means that the pressure inside the porous layer in the narrow region under consideration is much higher than the capillary pressure in the drop. This is a reasonable pressure to expect because the pressure just inside the porous layer determines the drop suction by the porous layer. In this case Equation A1.12 reduces to p p = pd e x .

(A1.14)

In order to determine the unknown value, pd, the solutions (4.65) and (4.68) for the pressure distribution inside the porous layer outside the drop are used. The solutions (4.65) and (A1.14) should give the same radial velocity from both sides at r = L. Thus we have: pd =

pc ξ ≈ pc << pc . L l l 1 + ln L ln ξ L L

(A1.15)

The latter equation justiﬁes the neglect of pd, relative to pc, in the abovementioned main text.

4.3 SPREADING OF LIQUID DROPS OVER THICK POROUS SUBSTRATES: COMPLETE WETTING CASE Let us extend the study of Section 4.2 to the case of spreading of small silicone oil drops (capillary spreading regime) over porous solid substrates thicker than the drop size. The spreading of small silicone oil drops (capillary regime of spreading) over various dry thick porous substrates (permeable in both normal and tangential directions) was experimentally investigated in this section. The time evolution of the radii of both the drop base and the wetted region on the surface of the porous substrate were monitored. It was observed that the total © 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

355

duration of the spreading process can be divided into two stages: a ﬁrst stage, when the drop base expands until its maximum value is reached, and a subsequent second stage, when the drop base shrinks. It was found that the dynamic contact angle remains constant during the second stage of spreading. This fact has nothing to do with the contact angle hysteresis, as there is no hysteresis in the system. Appropriate scales are used, with a dimensionless time, to plot the dimensionless radii of the drop base and of the wetted circle on the surface of the porous substrate, the relative dynamic contact angle, and the effective contact angle inside the porous substrate. All these experimental data fall onto universal curves, when the spreading of different silicone oils is done on porous substrates of similar pore size and porosity [14]. The spreading of small liquid drops over thin porous layers saturated with the same liquid (Section 4.1) or a dry porous layer (Section 4.2) has been considered by appropriately matching ﬂows in both the spreading drop and the porous substrate. In Section 4.2, the spreading of silicone oil drops over various dry microﬁltration membranes (permeable in both normal and tangential directions) was discussed. Plotting the dimensionless radii of the drop base and of the wetted region inside the porous layer using a dimensionless time scale, all the experimental data fell on two universal curves. According to the theory presented in Section 4.2: (1) the dynamic contact angle dependence on the same dimensionless time should be a universal function, (2) the dynamic contact angle should change rapidly over an initial short spreading stage and remain constant over the remaining duration of the spreading process. This fact has nothing to do with the contact angle hysteresis, as there was no hysteresis in the system under consideration in Section 4.2. These conclusions were in good agreement with experimental observations (Section 4.2). In the present section, we extend our study in Section 4.2 to the spreading of small silicone oil drops (capillary spreading regime) over different porous substrates, whose thickness is much bigger than the drop size. A number of similarities with the case of the spreading over thin porous substrates (Section 4.2) is found.

THEORY As already mentioned in the introduction to Chapter 3, at small capillary numbers, Ca =

Uη << 1 , γ

the drop proﬁle remains spherical in shape over the main part of the spreading drop. We also concluded in Section 4.1 that at the spontaneous spreading, the capillary number is always small, except for a short initial stage. Based on this we concluded that during the spreading over porous substrate, the liquid drops © 2007 by Taylor & Francis Group, LLC

356

Wetting and Spreading Dynamics

Vd Vp

l

– – 1 – θ– –

– – – – – 0 –

L

ψ

2 3

FIGURE 4.11 Spreading of liquid drops over dry porous substrates: (1) spherical drop; (2) wetted region inside the porous substrate (modeled by a spherical cap); (3) dry part of the porous substrate; L(t) — radius of the drop base; l(t) — radius of the wetted circle on the surface of the porous substrate; θ(t) — dynamic contact angle of the spreading drop, and ψ(t) — effective contact angle inside the porous substrate.

remain spherical in shape (see Figure 4.11). Hence, the drop proﬁle over the spherical part (assuming the low slope proﬁle) is: h (t, r ) =

2V 2 ( L − r 2 ), r < L (t ) 4 πL

(4.81)

where V(t) is the drop volume, h(t, r) is the drop proﬁle, L is the radius of the drop basis, and r is the radial coordinate. Equation 4.81 gives the following value of the dynamic contact angle, θ, (tanθ ≈ θ): θ=

4V , πL3

(4.82)

hence,  4V  L=  πθ 

1/ 3

.

(4.83)

The spreading process is a superposition of two motions: (1) the spreading of the drop over the already saturated part of the surface of the porous substrate, which results in an expansion of the drop base, and (2) the shrinkage of the drop base caused by the imbibition into the porous substrate. Hence, we can write the following balance equation: © 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

357

dL = v+ − v− , dt

(4.84)

where v+ and v− denote unknown velocities of the expansion and the shrinkage of the drop base, respectively. Using Equation 4.83 we get: dL 1  4V  =−  4 dt 3  πθ 

1/ 3

dθ 1  4  + dt 3  πV 2 θ 

1/ 3

dV . dt

(4.85)

Over the whole duration of the spreading over the porous layer, both the contact angle and the drop volume can only decrease with time. Accordingly, the ﬁrst term on the right-hand side of Equation 4.85 is positive, and the second one is negative. Comparison of the Equation 4.84 and Equation 4.85 yields 1  4V  v+ ≡ −  4  3  πθ 

1/ 3

dθ >0 dt (4.86)

1 4  v− ≡ −  3  πV 2θ 

1/ 3

dV > 0. dt

Let l(t) be the radius of the wetted region on the surface of the porous substrate. This unknown quantity cannot be determined without the numerical integration of Brinkman–Darcy equations inside the porous substrate and coupling with the ﬂow in the drop. However, we can draw some conclusion based on the analysis given in Section 4.2. According to this analysis, the whole spreading process can be subdivided into two stages: during the ﬁrst stage, v+ prevails, and v– dominates during the second stage of spreading. Let us consider the second stage of spreading. During this stage, the ﬁrst term on the right-hand side of Equation 4.85 can be neglected, and this equation reduces to: dL 1  4  = dt 3  πV 2 θ 

1/ 3

dV . dt

(4.87)

The latter equation after inserting the expression of θ (4.82) takes the form: dL L = , dV 3V © 2007 by Taylor & Francis Group, LLC

358

Wetting and Spreading Dynamics

which can be easily integrated. Its solution is V = C L3 ,

(4.88)

where C is an integration constant. Comparison with Equation 4.82 yields θ(t ) = θ f = const

(4.89)

over the duration of the second stage of the spreading. This conclusion agrees well with experimental observations [14] (see Results and Discussion), as well as with observations in Section 4.2. Note that, in the case under consideration, as in Section 4.2 the constancy of the contact angle has nothing to do with the contact angle hysteresis; there is no hysteresis in the system under consideration. θf is not a receding contact angle but forms as a result of a self-regulation of the drop–porous layer system. Inside the Porous Substrate The analysis of experimental data shows that the radius of the wetted region inside the porous layer is proportional to l3. If the drop base (Figure 4.11) is assumed to be a point source of liquid, then the shape of the wetted area inside the porous substrate is a hemisphere with an increasing radius and a constant effective contact angle ψ(t). Obviously, “a point source” assumption is too simple an approximation, as both radii, l(t) and L(t), are of similar size. Yet, it would help understanding the essence of the process. Hence, let us assume that the wetted region inside the porous substrate is of a spherical cap form with changing effective contact angle ψ(t) (Figure 4.11). Let Vp = (V0 – V)/m, where Vp is the volume of the liquid inside the porous substrate at time t; V0 is the initial volume of the drop, and m is the porosity. Under the above assumption, the liquid volume in the porous substrate, Vp, can be expressed as (Figure 4.11)

(

)

2

1 − cos ψ (2 + cos ψ ) π Vp = l 3 , 3 sin 3 ψ

(4.90)

where l is the radius of the wetted circle on the outer surface of the porous substrate. Equation 4.90 enables us to calculate the time evolution of ψ using the experimental data. Experimental Part Silicone oils S5 (viscosity 0.05 P), S100 (viscosity 1.0 P), and S500 (viscosity 5.0 P) purchased from Brookﬁeld Engineering Laboratories Inc. (Middleboro, © 2007 by Taylor & Francis Group, LLC

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359

MA, USA) were used in the spreading experiments. Glass ﬁlters, J. Bibby Science Products, Ltd, and metal ﬁlters Sintered Products, Ltd, both purchased from Claremont Ltd (Broseley, UK) were used as porous substrates. The diameters of glass ﬁlters were 5.0 cm, 2.9 cm, and 2.9 cm, and their thicknesses were 2.5 mm, 1.9, and 2.2 mm, respectively. The diameter of metal ﬁlters was 5.6 cm, and their thickness was 1.9 mm. Pore size distribution and permeability of membranes were tested using the Coulter Porometer II (Coulter Electronic Ltd, Luton, UK). Glass ﬁlters with three different pore size distributions were used. The average pore sizes were 3.7 µm, 4.7 µm, and 26.8 µm, respectively; their porosities were 0.56, 0.53, and 0.31, respectively. The permeability of the same membranes were 1.8 l/min/cm2; 1.9 l/min/cm2, and 11.5 l/min/cm2 (air ﬂux, transmembrane pressure 0.1 bar). Metal ﬁlter (Cupro-Nickel): average pore size 26.1 µm, and porosity 0.32. The porosity of ﬁlters was measured using the difference in the weight of the ﬁlters saturated with oil and the dry ﬁlters. Filters were dried for 3–5 h at 95˚C and then stored in a dry atmosphere prior to the spreading experiments. All relevant information and values are summarized in the Table 4.4.

TABLE 4.4 Characteristics of Porous Substrates and Drops Used Material, Figure, Symbol Glass Figure 4.12 ∆ Glass Figure 4.12

Porosity

Average Pore Size, µm

η, P

V 0, µl

tmax, sec

Lmax, mm

Lmax, mm

θm, grad

ψmax, grad

0.53

4.7

0.05

5.0

0.64

2.42

3.10

25.8

38

0.53

4.7

1

5.9

12.0

2.30

3.20

24.4

39

0.53

4.7

5

8.2

60.0

2.58

3.50

23.6

42

0.32

26.1

5

8.2

15.7

2.38

3.20

35.0

52

0.31

26.8

5

6.8

18.52

2.40

3.20

25.5

44

0.56

3.7

5

8.0

36.0

2.53

3.20

21.5

30

0.31

26.8

5

6.8

18.52

2.40

3.20

25.5

44

▫ Glass Figure 4.12

Cupro-Nickel Figure 4.13

▫ Glass Figure 4.13

Glass Figure 4.14

▫ Glass Figure 4.14

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Figure 4.2 shows a schematic description of the set-up. The time evolution of the radius of the drop base, L(t), the dynamic contact angle, θ(t), and the radius of the wetted circle on the surface of the porous substrate, l(t), were monitored. All porous substrates, before the start of the experiment, were placed for half an hour in a KOH aqueous solution inside an ultrasonic bath; later, they were rinsed out by plenty of Milli-Q water (Milli-Q water system, made by MILLIPORE S.A., France) in an ultrasonic bath. After that, they were dried for 2 hours in a 110˚C dry atmosphere. Porous substrates were stored in a dry atmosphere before starting the experiments. Experiments were carried out in the following order: • • •

The dry porous substrate was placed in a dry atmospheric chamber and left there for 15–30 minutes, A light pulse produced by a ﬂash gun was used to synchronize the time instant in both video recorders when the drop started to spread, A droplet of silicone oil was placed onto the porous substrate.

Each run was carried out until the complete imbibition of the drop into the porous substrate took place. Results and Discussion According to the observations, the whole spreading process can be subdivided into two stages (Figure 4.12a and Figure 4.12b): the ﬁrst stage is when the drop spreads until the maximum radius, Lmax, of the drop base is reached, which is followed by the second stage, when the drop radius decreases. Over the duration of the ﬁrst stage, the imbibition front inside the membrane grows slightly ahead of the spreading drop front. Subsequently, the drop base starts to shrink until the drop completely disappears, and the imbibition front grows until the end of the process. Examples of the time evolution of the radius of the drop base and the radius of the wetted circle on the surface of the porous substrates are presented in Figure 4.12a, Figure 4.12b, Figure 4.13a, and Figure 4.14a. In all experiments, the drops remained spherical over the whole spreading process. This was cross checked by reconstructing the drop proﬁles at different time instants of spreading, ﬁtting those drop proﬁles by a spherical cap:

(

h = zcenter + R 2 − r − rcenter

)

2

,

where (rcenter , zcenter) is the position of the center of the sphere, and R is the radius of the sphere. rcenter , zcenter , and R are used as ﬁtting parameters. The Levenberg–Marquardt algorithm was used for ﬁtting. In all cases the reduced Chi-square value was found smaller than 10–4. The ﬁtted parameter R gives the radius of curvature of the spreading drops at different times.

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3.5

3.5

3.0

3.0

2.5

2.5

2.0

2.0

1.5

1.5

1.0

ᐉ, mm

L, mm

Spreading over Porous Substrates

1.0 –1 –2 –3

0.5

0.5

0.0

0.0 0.01

0.1

1 t, s

10

1.0

1.0 –1 –2 –3

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

ᐉ/ᐉmax

L/Lmax

0.8

0.0 0.0

0.2

0.4

0.6

0.8

1.0

t/tmax

FIGURE 4.12 (a) Spreading of different silicone oils over identical dry porous glass ﬁlters (in dimensional form). Glass porous ﬁlter: porosity 0.56, average pore size 4.7 µm, permeability 1.9 l/min · cm2; (1) silicone oil η = 0.05 P; (2) silicone oil η = 1 P; (3) silicone oil η = 5 P (see insets). The same symbols show the evolution of l(t) and L(t) with time. Upper parts: L(t), radius of the base of the spreading drop; lower parts: l(t), radius of the wetted circle on the surface of the porous glass ﬁlter. (b) Spreading of different silicone oils over identical dry porous glass ﬁlters. (The same data as in Figure 4.12a, using appropriate dimensionless coordinates.) Glass porous ﬁlter: porosity 0.53, average pore size 4.7 µm, permeability 1.9 l/min/cm2; (1) silicone oil η = 0.05 P; (2) silicone oil η = 1 P; (3) silicone oil η = 5 P (see insets); Lmax — maximum value of the radius of the drop base; lmax — maximum value of the radius of the wetted circle on the surface of the glass ﬁlter; tmax — total duration of the process (see Table 4.4).

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Wetting and Spreading Dynamics 5 –1 –2 –3

θ/θm

4 3 2 1 0 0.0

0.2

0.4

0.6

0.8

1.0

t/tmax 1.0 –1 –2 –3

ψ/ψmax

0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6 t/tmax

0.8

1.0

FIGURE 4.12 (continued) (c) Spreading of different silicone oils over identical dry porous glass ﬁlters. Dynamic contact angle vs. time in dimensionless units. Glass porous ﬁlter: porosity 0.53, average pore size 4.7 µm, permeability 1.9 l/min/cm2; (1) silicone oil η = 0.05 P; (2) silicone oil η = 1 P; (3) silicone oil η = 5 P (see insets); θm — dynamic contact angle value at the time when the maximum value of the radius, Lmax, of the drop base is reached (see Table 4.4). (d) Spreading of different silicone oils over the same dry porous glass ﬁlter. Evolution of the effective contact angle inside the porous glass filters with the dimensionless time. Glass porous ﬁlter: porosity 0.53, average pore size 4.7 µm, permeability 1.9 l/min/cm2; (1) silicone oil η = 0.05 P; (2) silicone oil η = 1 P; (3) silicone oil η = 5 P (see insets); ψmax — maximum value of the effective contact angle (see Table 4.4).

The wetted area on the surface of the porous substrate was always circular. Drops remained in the center of this circle over the whole duration of the spreading process. No deviations from cylindrical symmetry or instabilities were detected. The spherical form of the spreading drop allows measuring the evolution of the dynamic contact angle of the spreading drops. In all cases, the dynamic contact angle decreased very fast during the ﬁrst spreading stage and remained constant © 2007 by Taylor & Francis Group, LLC

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during the second spreading stage. This constant value of the contact angle is denoted below as θf . Examples of the of the dynamic contact angle with respect to the evolution with time are presented in Figure 4.12c, Figure 4.13b, and Figure 4.14b. These ﬁgures show that the dynamic contact angle decreases during the ﬁrst spreading stage when the radius of the drop base grows up to its maximum value, Lmax. Within experimental error, the dynamic contact angle remains unchanged during the second spreading stage. Spreading of Silicone Oil Drops of Different Viscosity over Identical Glass Filters (Figure 4.12a to Figure 4.12d) Figure 4.12a to Figure 4.12d present experimental results on the spreading of silicone oil drops of different viscosity over identical glass ﬁlters. Silicone oils S5, S100, and S500 were used in these spreading experiments. The diameter of the glass ﬁlter was 2.9 cm, its thickness 1.9 mm, its porosity 0.53, and its average pore size 4.7 µm. From Figure 4.12a to Figure 4.12d, experimental data marked by symbol ∆ correspond to the spreading of the silicone oil S5 (drop volume V0 = 5.0 µl; the maximum radius of the drop base Lmax = 2.42 mm; the maximum (ﬁnal) radius of the wetted circle on the outer surface of the glass ﬁlter lmax = 3.10 mm; and the total duration of the spreading tmax = 0.64 sec); experimental data marked by symbol correspond to the spreading of the silicone oil S100 (drop volume V0 = 5.9 µl; the maximum radius of the drop base Lmax = 2.30 mm; the maximum (ﬁnal) radius of the wetted circle on the outer surface of the glass ﬁlter lmax = 3.20 mm; and the total duration of the spreading tmax = 12.0 sec), and experimental data marked by symbol Ο correspond to the spreading of the silicone oil S500 (drop volume V0 = 8.2 µl; the maximum radius of the drop base Lmax = 2.58 mm; the maximum (ﬁnal) radius of the wetted circle on the outer surface of the glass ﬁlter lmax = 3.50 mm; and the total duration of the spreading tmax = 60.0 sec). In Figure 4.12a, the time evolution of both the radius of the base of the spreading drops and the radius of the wetted circle on the outer surface of the glass ﬁlter for silicone oils of different viscosity are presented using experimental data in dimensional form. Figure 4.12a shows that the kinetics of the spreading and imbibition varies for drops of different size and different viscosity. Consequently, the total duration of the spreading process, the maximum radius of the drop base, and the radius of the wetted circle on the outer surface of the glass ﬁlter vary considerably. However, if, as in Section 4.2, we rescale quantities as L/Lmax, l/lmax, and t/tmax, then all experimental data fall into two universal curves as shown by Figure 4.12b. According to Section 4.2 the evolution of reduced dynamic contact angle, θ/θm, with the dimensionless time, t/tmax, should be universal. Here, θm is the value of the dynamic contact angle that is reached when the radius of the drop base reaches its maximum value (the end of the ﬁrst stage of spreading). The same procedure is used in the case of spreading over thick porous substrates. In © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

Figure 4.12c, the reduced dynamic contact angle, θ/θm, is plotted versus the dimensionless time t/tmax. This plot shows that: (1) all three experimental curves fall into a single universal curve, and (2) the dynamic contact angle, as the theory predicts, remains constant during the second spreading stage. In Figure 4.12d, the evolution of the relative effective dynamic contact angle, ψ/ψmax, inside the porous glass ﬁlter with the dimensionless time is presented. It is noteworthy that in all three cases, all the data follow a single universal curve. These three experimental runs show that the spreading behavior of drops of different viscosities and volumes on the same thick porous substrate is identical, if appropriate dimensionless coordinates are used. Spreading of Silicone Oil Drops over Filters with Similar Properties but Made of Different Materials (Figure 4.13a–Figure 4.13c) In this series of experiments, the spreading of drops of the same silicone oil S500, over different substrates was studied. We wanted to check if the universal behavior found in the previous case remains valid even when different porous substrates made of different materials, glass, and metal ﬁlters, are used. In Figure 4.13a–Figure 4.13c: = the spreading of the silicone oil S500 (drop volume V0 = 8.2 µl; the maximum radius of the drop base Lmax = 2.38 mm; the maximum radius of the wetted circle on the outer surface of the porous substrate lmax = 3.20 mm; the total duration of the spreading tmax = 15.7 sec over the metal

1.0

1.0 –1 –2

0.8

0.6

0.6

0.4

0.4

0.2

0.2

ᐉ/ᐉmax

L/Lmax

0.8

0.0

0.0 0.0

0.2

0.4

0.6

0.8

1.0

t/tmax

FIGURE 4.13 (a) Spreading of silicone oil (η = 5 P) over dry porous glass and metal ﬁlters. Radii in reduced coordinates. (1) metal ﬁlter: porosity 0.32, average pore size 26.1 µm (see insets); (2) glass ﬁlter: porosity 0.31, average pore size 26.8 µm (see insets), Lmax — maximum value of the radius of the drop base; lmax — maximum value of the radius of the wetted circle on the surface of the glass ﬁlter; tmax — total duration of the process (see Table 4.4). © 2007 by Taylor & Francis Group, LLC

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365

4 –1 –2

θ/θm

3

2

1

0 0.0

0.2

0.4

0.6

0.8

1.0

t/tmax

1.0 –1 –2

ψ/ψmax

0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6 t/tmax

0.8

1.0

FIGURE 4.13 (continued) (b) Spreading of silicone oil (η = 5 P) over dry porous glass and metal ﬁlters. Dynamic contact angle vs. time in dimensionless units. (1) metal ﬁlter: porosity 0.32, average pore size 26.1 µm (see insets); (2) glass ﬁlter: porosity 0.31, average pore size 26.8 µm (see insets); θm — dynamic contact angle value at the time when the maximum value of the radius of the drop base, Lmax, is reached (see Table 4.4). (c) Spreading of silicone oil (η = 5 P) over dry porous glass and metal ﬁlters. Evolution of the effective contact angle inside the porous filters with relative time. (1) metal ﬁlter: porosity 0.32, average pore size 26.1 µm (see insets); (2) glass ﬁlter: porosity 0.31, average pore size 26.8 µm (see insets); ψmax — maximum value of the effective contact angle (see Table 4.4).

ﬁlter (Cupro-Nickel with diameter 5.6 cm, thickness 1.9 mm, average pore size 26.1 µm, and porosity 0.32); Ο – the spreading of the silicone oil S500 (drop volume V0 = 6.8 µl; maximum radius of the drop base Lmax = 2.40 mm; maximum radius of the wetted circle on the outer surface of the porous substrate lmax = 3.20 mm; and the total duration of the spreading tmax = 18.52 sec over the glass ﬁlter with diameter 2.9 cm, thickness 2.2 mm, porosity 0.31, and average pore size 26.8 µm). © 2007 by Taylor & Francis Group, LLC

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Figure 4.13a presents the dependence of the dimensionless radius of the drop base (left ordinate) and that of the dimensionless radius of the wetted circle on the outer surface of the porous substrate (right ordinate) on the dimensionless time. Figure 4.13b presents the dependence of the relative dynamic contact angle on the dimensionless time. Figure 4.13c presents the dependence of the effective dynamic contact angle inside the porous substrate on the same dimensionless time. The curves show that the spreading of drops of different size on porous substrates made of different materials with, however, similar porosity and average pore size, fall on universal curves if, as in the previous case, the same dimensionless coordinates are used. Thus, the universal spreading behavior over porous substrates does not depend on the material of the substrate. Spreading of Silicone Oil Drops with the Same Viscosity η = 5P) over Glass Filters with Different Porosity and Average (η Pore Size (Figure 4.14a to Figure 4.14c) In this section the spreading of silicone oil drops with the same viscosity over glass ﬁlters with different porosity and average pore size is investigated to check if the universal behavior found in the two previous sections is still applicable. From Figure 4.14a to Figure 4.14c: – the spreading of silicone oil S500 (drop volume V0 = 8.0 µl; maximum radius of the drop base Lmax = 2.53 mm; maximum radius of the wetted circle on the surface lmax = 3.20 mm; and the total

1.0

1.0

0.8

–1 –2

0.6

0.6

0.4

0.4

0.2

0.2

0

I/Imax

L/Lmax

0.8

0.0 0.0

0.2

0.4

0.6

0.8

1.0

t/tmax

FIGURE 4.14 (a) Spreading of silicone oil (η = 5 P) over different dry glass ﬁlters. Radii of spreading in reduced coordinates. (1) glass ﬁlter: porosity 0.56; pore size 3.7 µm (see insets); (2) glass ﬁlter: porosity 0.31; pore size 26.8 µm (see insets). Lmax — maximum value of the radius of the drop base; lmax — maximum value of the radius of the wetted circle on the surface of the glass ﬁlter; tmax — total duration of the process (see Table 4.4).

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367

5 –1 –2

4

θ/θm

3

2

1

0 0.0

0.2

0.4

0.6

0.8

1.0

t/tmax 1.0 –1 –2

ψ/ ψmax

0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

t/tmax

FIGURE 4.14 (continued) (b) Spreading of silicone oil (η = 5 P) over different dry glass ﬁlters. Dynamic contact angle vs. time in dimensionless units. (1) glass ﬁlter: porosity 0.56; pore size 3.7 µm (see insets); (2) glass ﬁlter: porosity 0.31; pore size 26.8 µm (see insets). θm — the dynamic contact angle value at the time when the maximum value of the radius of the drop base, Lmax, is reached (see Table 4.4). (c) Spreading of silicone oil (h=5 P) over different dry glass ﬁlters. Evolution of the effective contact angle inside the porous filters with time in dimensionless units. (1) glass ﬁlter: porosity 0.56; pore size 3.7 µm (see insets), (2) glass ﬁlter: porosity 0.31; pore size 26.8 µm (see insets). ψmax — the maximum value of the effective contact angle (see Table 4.4).

duration of the spreading tmax = 36.0 sec over the glass ﬁlter with diameter 5.0 cm thickness 2.5 mm, porosity 0.56, and average pore sizes 3.7 µm); Ο = the spreading of the silicone oil S500 (drop volume V0 = 6.8 µl; maximum radius of the drop base Lmax = 2.40 mm; maximum radius of the wetted circle on the outer © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

surface of the glass ﬁlter lmax = 3.20 mm; and the total duration of the spreading tmax = 18.52 sec over the glass ﬁlter with diameter 2.9 cm, thickness 2.2 mm, porosity 0.31, and average pore sizes 26.8 µm). Figure 4.14b and Figure 4.14c show that the relationships between the relative dynamic contact angle, θ/θm, and the effective dynamic contact angle inside the porous substrate, ψ/ψmax, on the dimensionless time, t/tmax, exhibit universal behavior again. However, the dependence of the dimensionless radius of the drop base, L/Lmax, and that of the dimensionless radius of the wetted circle on the surface of the porous substrate, l/lmax, on the dimensionless time, t/tmax, deviate from universal behavior during the ﬁrst spreading stage. The duration of the ﬁrst stage, when the drop base increases with time and reaches its maximum value, is shorter in the case of the spreading of silicone oil drops over the glass ﬁlter with a smaller average pore size than over the glass ﬁlter with a larger average pore size (Figure 4.14a). In conclusion, we can safely say that the spreading behavior over porous substrates is mostly, if not entirely, determined by the porosity and the average pore size of the porous substrate and differs if these two characteristics are different. Conclusions Experiments were carried out on the spreading of small silicone oil drops (capillary regime of spreading) over various dry thick porous substrates (permeable in both normal and tangential directions). The time evolution of the radii of both the drop base and the wetted region on the surface of the porous substrate were monitored. It has been shown that the overall duration of the spreading process can be divided into two stages: a ﬁrst stage, when the drop base grows until a maximum value is reached, and a second stage, when the drop base shrinks. It has been observed that the dynamic contact angle remained constant during the second spreading stage. This fact is supplied by a heuristic argument and has nothing to do with hysteresis of contact angle, as there is no hysteresis in the system. Using appropriate scales, the dimensionless radius of the drop base, the radius of the wetted circle on the surface of the porous substrate, the dynamic contact angle, and the effective contact angle inside the porous substrate have been plotted using a dimensionless time. Experimental data shows that the spreading of silicone oil drops over dry thick porous substrates exhibits a universal behavior if: (1) porous substrates made of different materials with, however, similar porosity and average pore size are used and (2) if appropriate dimensionless coordinates are introduced to depict the data. However, if porous substrates with different porosity and average pore size are used the dynamic of both the radius of the drop base, L/Lmax, and the radius of the wetted circle on the outer surface of the porous ﬁlter, l/lmax, behave differently during the spreading process. Yet, both the relative dynamic contact angle, θ/θm, and the effective contact angle inside the porous substrate, ψ/ψmax, show universal behavior. © 2007 by Taylor & Francis Group, LLC

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369

4.4 SPREADING OF LIQUID DROPS FROM A LIQUID SOURCE In this section, we shall consider a liquid drop being created and then spread over a solid substrate with a liquid source. We shall look at both cases, complete and partial wetting, and for small and large drops. Then, we expect to observe spreading and forced ﬂow caused by the liquid source in the drop center. Both capillary and gravitational regimes of spreading shall be considered [15]. For conditions of complete wetting, the spreading is an overlapping of two processes: a spontaneous spreading and a forced ﬂow caused by the liquid source in the center. Both capillary and gravitational regimes of spreading are considered, and power laws are deduced. In both cases of small and large droplets, the exponent is a sum of two terms: the ﬁrst term corresponds to the spontaneous spreading, and the second term is determined by the intensity of the liquid source. In the case of a constant ﬂow rate from the source, the radius of spreading is given by the following law, R(t) ~ t 0.4, in the case of the capillary spreading, and R(t) ~ t 0.5 in the case of gravitational spreading. In the case of partial wetting, droplets spread with a constant advancing contact angle (at small capillary numbers). This yields R(t) ~ t 1/3. Experimental data are in good agreement with the theoretical predictions. The spreading of liquid drops over solid nonporous substrates has been investigated in Section 3.1, Chapter 3; for conditions of complete wetting, the spreading of small droplets is governed by the capillary law of spreading (Equation 3.25), which is  γV 3  R(t ) = 0.65λ c    η 

0.1

(t + tin )0.1 ,

where R is the radius of the drop base, t is time, and λc, is a preexponential factor, which is determined by the disjoining pressure isotherm in Section 3.2. Similar to the case of complete wetting, the spreading of bigger drops is governed by gravity according to Equation 3.36 in Section 3.1:  ρgV 3  R(t ) = 0.78    η 

1/ 8

(t + tc )1/8 .

For small drops, the capillary law of spreading is in excellent agreement with experimental data, as was shown in Section 3.2. Over time, small droplets spread out, and the radius of the drop base increases with time and, hence, should be a transition from a capillary regime of spreading to the gravitational regime. This transition has been experimentally conﬁrmed [16] in the case of spontaneous spreading. Note that, in the case of complete wetting, the dynamic contact angle © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

tends to zero over time, and droplets spread out completely (see Section 3.1 and Section 3.2 for details). Spreading of liquid drops in the case of partial wetting has been less investigated, and it is less understood. The main problem in this case is the presence of contact angle hysteresis, which was considered in Section 1.3 and Section 3.10. The hysteresis phenomenon is usually associated with nonhomogeneity/roughness of the solid substrates. However, it has been shown in Section 3.10 that an S-shaped disjoining pressure isotherm leads to the presence of the contact angle hysteresis even on smooth homogeneous substrates. In the case of partial wetting, the droplet spreads out until a static advancing contact angle is reached. After that the droplet does not spread out on a macroscale but still spreads out on a microscale (see Section 3.10). If the droplet is “gently” pushed from inside by pumping liquid from the oriﬁce at its center, then the droplet spreads out with a constant advancing contact angle, which is equal to the static advancing contact angle. The term gently means that the capillary number remains small during the spreading process. The spreading of liquid drops for both cases, complete and partial wetting, when liquid is injected into the droplets from a small oriﬁce at their center is considered in the following text.

THEORY Let us consider the spreading of a small liquid droplet over a solid substrate in the presence of liquid source in the drop center (Figure 4.15). It is assumed that the shape of the drop remains axisymmetrical, and hence, a cylindrical coordinate system, (r, z), is used in the following text, where r is the radial distance from the center, and z is the vertical coordinate. Because of symmetry, the angular component of the velocity vanishes, and all other unknowns

θ

1

R(t)

2

3

FIGURE 4.15 Schematic presentation of the spreading in the presence of the liquid source in the drop center. R(t) — radius of the drop base; θ — contact angle; (1) liquid drop; (2) solid substrate with a small oriﬁce in the center; (3) liquid source (syringe). © 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

371

are independent of the angle. It is assumed in the following text that the Reynolds number is Re << 1 and that the drop proﬁle has a low slope; that is, 2

 ∂h  ε =   << 1 ,  ∂r  where h(t, r) is the drop proﬁle. Using these assumptions, we arrive at the equation (Equation 3.11, Section 3.1), which describes the proﬁle of the spreading liquid ∂h 1 ∂  3 ∂p  = , rh ∂t 3ηr ∂r  ∂r 

(4.91)

which is referred to in the following text as the equation of spreading. The liquid is assumed nonvolatile and injected from the center according to a prescribed time rate V′(t); hence, the drop volume obeys the following conservation law: R (t )

2π

∫ r h dr = V (t) ,

(4.92)

0

where now V(t) is ﬁxed by the pumping rate. Two unknown functions are to be determined: the liquid proﬁle, h(t,r) and the radius of the spreading drop base, R(t), which is referred to in the following text as the radius of spreading. The pressure inside the spreading drop, p(t,r), can be determined via the liquid proﬁle, h(t,r). This expression includes several components (see Section 3.1. and Section 3.2), where we now keep only two components: capillary and gravitational parts: p = pa – γK + ρgh,

(4.93)

where pa is the pressure in the ambient air; γ is the liquid–air interfacial tension; ρ and g are the liquid density and gravity acceleration, respectively; K is the curvature of the liquid–air interface. In the low slope approximation (ε << 1), the curvature is K=

1 ∂  ∂h  , r r ∂r  ∂r 

hence, Equation 4.93 becomes: p = pa − γ

© 2007 by Taylor & Francis Group, LLC

1 ∂  ∂h  r + ρgh . r ∂r  ∂r 

(4.94)

372

Wetting and Spreading Dynamics

Let us introduce scales in radial and vertical directions, r*, h*, respectively. Then the following dimensionless quantities are introduced r=

r h , h= r* h*

or r = rr* , h = hh* ,

where dimensionless values are marked by an overbar. In dimensionless form, Equation 4.94 can be rewritten as p = pa −

γ h* 1 ∂ r*2 r ∂r

 ∂h   r ∂r  + ρgh* h .

In this expression we have two parameters: γ h* r*2

and ρgh* .

The ﬁrst dimensionless parameter estimates the intensity of capillary forces and the second one that of the gravity force. If capillary forces prevail, then we have the capillary regime of spreading; that is if γ h* >> ρgh* r*2

or r* <<

γ . ρg

If the gravity dominates, then the gravitational regime of spreading takes place, therefore

if

γ h* << ρgh* r*2

or r* >>

γ . ρg

The length

a=

γ , ρg

which was already introduced in Section 3.1, is the capillary length. Hence, the capillary regime (R(t) < a) is the initial stage of spreading of small drops, whereas the gravitational regime is the ﬁnal stage of spreading of small drops (R(t) > a) © 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

373

or the overall regime of spreading of big drops. In the following text, we consider the spreading of small drops with a transition from the capillary to the gravitational regime of spreading over time. Note that, in Equation 4.93 and Equation 4.94, we ignored the role played by the disjoining pressure and hence, we cannot consider the drop proﬁle in the vicinity of the three-phase contact line. However, here we are not going to calculate the drop proﬁle. Instead we try similarity solutions of the equation of spreading (4.91) in the case of both the capillary and gravitational regime of spreading (that is, initial and ﬁnal stages of spreading of small drops). This method allows us to calculate time evolution of the radius of spreading, but the preexponential factor includes an unknown dimensionless integration constant. Based on the consideration presented in Section 3.2 this constant can be calculated; however, in this section, it is extracted from experimental data. If the initial drop size is small enough, then the effect of gravity can be ignored. Accordingly, the drop radius, R(t), has to be smaller than the capillary length,

()

R t ≤

γ . ρg

The liquid is injected through the oriﬁce in the drop center with the ﬂow rate V′(t), where V(t) is an imposed function of time. In the case of spontaneous spreading, V(t) = V0 = const, where V0 is the constant volume of the spreading drop. In the case of constant liquid ﬂow rate from the liquid source, V(t) = I t, where I is the intensity of the liquid source. Mass conservation of liquid (Equation 4.92) demands that V(0) = 0 if there was not a drop at the initial time. In Appendix 3, we deduce a condition when the drop spreads according to the power law, and the general possible form of V(t) compatible with the power law of spreading is deduced. In the same Appendix 3 we deduce dependences of the radius of spreading with time in the case of complete wetting (both capillary and gravitational regime of spreading) and in partial wetting (small capillary numbers, Ca = Uη/γ). In the case of the constant source of liquid I in the drop centre, the following dependencies for the radius of the drop base are deduced in Appendix 3. During the initial (capillary) stage of the spreading of small drops, the radius of the drop base should follow  γI 3  R (t ) = α c    η 

0.1

t 0.4 ,

(4.95)

whereas at the ﬁnal (gravitational) stage of the spreading of small drops, the radius of the drop base follows the law © 2007 by Taylor & Francis Group, LLC

374

Wetting and Spreading Dynamics

 ρgI 3  R (t ) = α g    η 

1/ 8

t 0.5 ,

(4.96)

where αc, αg are the dimensionless constants to be determined from experimental data. In the case of partial wetting, the static hysteresis of the contact angle determines the spreading behavior at low capillary numbers, Ca << 1, where U = R (t ) is the rate of spreading. This condition was always satisﬁed under our experimental conditions. Because of the contact angle hysteresis, the drop does not move if the contact angle, θ, is in the range θr < θ < θa, where θ a and θr are the static advancing and static receding contact angles, respectively (see Section 3.10). In our experimental procedure (Figure 4.15), we are interested in the static advancing contact angle only. If the capillary number, Ca, is very small, which is the case in our experiments, then the advancing contact angle does not vary signiﬁcantly. It is assumed that the contact angle, θ, does not vary over duration of the spreading experiment and remains equal to its static value θa. In this case, the radius of the drop base should follow  I  R(t ) =   f (θa ) 

1/ 3

t1/ 3.

(4.97)

In spite of the similarity between expressions for the radius of spreading (Equation 4.95 and Equation 4.96 in the case of complete wetting and Equation 4.97 in the case of partial wetting), there is one signiﬁcant difference between these two spreading processes: if the liquid source is closed, then in the case of complete wetting, the drop will continue to spread out according to the law R(t) ~ t 0.1 (in the case of capillary regime), or R(t) ~ t 1/8 (in the case of gravitational regime). However, in the case of partial wetting, the drop will stop spreading as soon as the liquid source is closed.

EXPERIMENTAL SET-UP

AND

RESULTS

Materials and Methods The spreading of silicone oil and aqueous droplets over glass substrates were investigated. Silicone oil was purchased from Brookﬁeld Engineering of Middleboro, MA. Its viscosity was measured using the rheometer AR1000 (TA Instruments) at 25˚C. Density was measured by the weight method and for measuring surface tension, the Tensiometer (White Electrical Instrument Co. of Worcestershire, U.K.) was used. The following values were found: dynamic viscosity η = 91.0 cP, density ρ = 0.96 g/cm3, surface tension γ = 22.5 dyn/cm. Microscope glass slides (76 x 26 mm; Menzel-Glaser GmbH of Braunschweig, Germany) were used for spreading experiments. Circular oriﬁces of © 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

375

diameters 0.5mm were drilled in their centers. A liquid was injected through those oriﬁces in the glass substrate with a constant ﬂow rate using a Harvard Apparatus syringe pump. This produced a liquid drop over the solid substrate. Constant ﬂow rate resulted in a linear increase of the drop volume with time. The time evolution of the radius of the base of the spreading drops was monitored. The glass slides were cleaned by immersing them in a chromic acid solution for 2 h followed by rinsing 10 times with distilled water and twice with ultra pure water, then drying in an oven at 70˚C for 30 min. Each cleaned and dried slide was used only once. At least three runs were conducted for each experimental condition, and average values are reported in the following text. The diagram of the experimental set-up is shown in Figure 4.16. All experiments were carried out at 25 ± 0.5˚C. The solid substrate, 1 (Figure 4.16), was ﬁxed in the ring; a syringe (4) was positioned in the center of the substrate and connected to the Harvard Apparatus syringe pump (18). The droplets of silicone oil or water (3), were formed due to the injection. The following ﬂow rates were used: 0.005 ml/min; 0.01 ml/min, and 0.02 ml/min. The spreading process was recorded using a CCD camera (5, 10) (Figure 4.16) and a VHS recorder (6) (11). The camera has been equipped with ﬁlters having a wavelength of 640 nm. Such an arrangement suppresses illumination of the CCD camera by the scattered light from the substrate and hence, results in a higher precision of the measurements. The source of light (13) was used during experiments. The camera and a VHS recorder were connected to a computer (8). Images were analyzed using Drop Shape Analysis FTA 32.

9

7 2

16

6

CCD 1

17 13

5

15 16

12 16

14

10

11

8

CCD 2 4

1 3

18

FIGURE 4.16 Schematic presentation of the experimental set-up for monitoring the advancing and receding contact angles on a smooth substrate. (1) porous substrate; (2) hermetically closed, thermostated chamber; (3) liquid drop; (4) glued in syringe needle, positioned in the center of the solid substrate 1 (connected to the Harvard Apparatus syringe pump 18); (5, 10) CCD cameras; (6, 11) VCRs; (7) mirror; (8) PC; (9,14) telephoto objectives; (12) collimating lens; (13) light source; (15) ﬂash gun; (16) optical windows; (17) upper syringe; (18) Harvard Apparatus syringe pump. © 2007 by Taylor & Francis Group, LLC

376

Wetting and Spreading Dynamics

Results and Discussion The kinetics of spreading of silicone oil (complete wetting) and aqueous droplets (partial wetting) over glass substrates was investigated. The spreading process was caused by both the spontaneous spreading and the injection of liquid through the liquid source in the center of drops. Complete wetting. Two stages of spreading of small silicone oil drops have been observed: the ﬁrst initial stage — a capillary regime, and the second ﬁnal stage — a gravitational regime. The experimental time dependences of radius of spreading of silicone oil drops over glass surface are presented in Figure 4.17. Three experiments with different injection velocities (0.005 ml/min, 0.01 ml/min, and 0.02 ml/min) were conducted. In each experiment, the two above-mentioned stages of spreading are observed in Figure 4.17. The data obtained from the initial stage of spreading correspond to the capillary stage (R(t) < a). Equation 4.95 can be rewritten,   3  0.1  γI  + 0.4 ⋅ lg t, lg R (t ) = lg α c    η    

(4.98)

which is a linear function of time in lg–lg coordinates. 15 12 8

R (mm)

6 4 3 2

1 2

4

6

10

20

40

60

100

300

Time (sec)

FIGURE 4.17 Radius of spreading vs. time for the spreading of silicone oil drops on glass surface (log–log plot), diameter of the oriﬁce 0.5 mm: capillary stage, gravitational stage, I = 0.005 ml/min, experiment 1; capillary stage, ▫ gravitational stage, I = 0.01 ml/min, experiment 2; ♦ capillary stage, ◊ gravitational stage, I = 0.02 ml/min, experiment 3; dashed line ﬁtted according to Equation 4.103; solid line ﬁtted according to Equation 4.104. © 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

377

TABLE 4.5 Spreading of Silicone Oil Drops over Glass Surface with the Orifice in the Center Injection Flow Rate

Capillary Stage

Gravitational Stage

I mm3/sec

αc

n

αg

n

0.0833 0.1667 0.3333

2.7589 2.5426 2.4040

0.4007 0.4005 0.4005

2.6402 2.1811 1.8833

0.4893 0.5000 0.4991

According to Equation 4.98, the experimental data were ﬁtted according to lg R(t) = X + n lg t,

(4.99)

where intercept, X, and slope, n, are the ﬁtting parameters. The ﬁtted exponents, n, (Table 4.5) show good agreement with the theoretically predicted exponent 0.4. The ﬁtted value of X was compared with Equation 4.98, and the unknown dimensionless constant, αc, was determined as follows:  η  αc =  3   γI 

0.1

exp( X )

(4.100)

Using this equation, the constant αc was calculated for each set of data (Table 4.5). The average value ± S.D. obtained was αc = 2.57 ± 0.18. The model predictions are shown in Figure 4.17 by dashed lines. In the case of the gravitational regime of spreading (R(t) > a), experimental data were compared with the theoretical predictions according to Equation 4.96. This equation can be rewritten as: 1/ 8    ρgI 3   lg R(t ) = lg α g  + 0.5 ⋅ lg t.   η    

(4.101)

According to Equation 4.101, experimental data were ﬁtted according to: lg R(t) = Y + n lg t,

(4.102)

where Y and n are the ﬁtting parameters. The theoretical predictions are shown in Figure 4.17 by solid lines. The ﬁtting of the data resulted in an average value © 2007 by Taylor & Francis Group, LLC

378

Wetting and Spreading Dynamics

of the exponent, n, in Equation 4.102 equal to 0.4961, which shows good agreement with the theoretically predicted exponent 0.5. Using the ﬁtted value Y and Equation 4.101, the unknown dimensionless constant, αg, was calculated as:  η  αg =    ρgI 3 

1/ 8

exp(Y ) .

(4.103)

The average value of the constant ± S.D. was αg = 2.64 ± 0.38. The average drop height is 3.6 ± 0.3 mm, which is in good agreement with the calculated value 3.9 mm. Partial wetting. The partial wetting case was investigated using the spreading of water droplets over the same glass surfaces. Three experiments with different injection velocities 0.005 ml/min, 0.01 ml/min, and 0.02 ml/min were presented in Figure 4.18. The spreading behavior was compared with the theoretical prediction according to Equation 4.97, which can be written as:  I  lg R (t ) = lg   f (θ a ) 

1/ 3

+

1 lg t . 3

(4.104)

During the spreading of water droplets over glass surfaces, the advancing contact angle does not vary signiﬁcantly. The average value of the contact angle was calculated for each experimental run, and the average advancing contact 4.0 3.0

R (mm)

2.0 1.4 1.0

0.6 0.5 4

6

10

20

40

60

100

Time (sec)

FIGURE 4.18 Radius of spreading vs. time for the spreading of water droplets on glass surface (log–log plot), diameter of the oriﬁce 0.5 mm: I = 0.005 ml/min, experiment 1; I = 0.01 ml/min, experiment 2;♦ I = 0.02 ml/min, experiment 3; solid line drawn according to Equation 4.105.

© 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

379

angle was determined from three experimental runs. The average advancing contact angle θa ± S.D. is 54 ± 2˚. This value was used in Equation A2.34 for the calculation of the function f(θa), which was substituted in Equation 4.104. The dashed line in Figure 4.18 was plotted according to Equation 4.104. Note that Equation 4.104 does not include any ﬁtting parameters. Figure 4.18 shows that our experimental data are in good agreement with the theoretically predicted law (4.104). Conclusions The spreading of liquid over solid substrates when there is liquid injection through an oriﬁce is investigated from both theoretical and experimental points of view. Two cases of spreading over a glass substrate with a diameter of the oriﬁce 0.5 mm were studied: spreading of silicone oil droplets (complete wetting) and spreading of water droplets (partial wetting). In the case of silicone oil spreading, two regimes of spreading were observed: capillary regime and gravitational regime. A theory has been developed for the cases of complete wetting and partial wetting at low capillary numbers. In all three cases, power laws of spreading were deduced: capillary regime of spreading according to Equation 4.95, gravitational regime of spreading according to Equation 4.96, and partial regime of spreading according to Equation 4.97. Experimental data validated our theoretical dependences of the radius of spreading on both time and injection velocity in both cases of complete and partial wetting.

APPENDIX 2 Let us introduce the following similarity coordinate and functions, ξ=

r h (t, r ) , ϕ(ξ ) = , R (t ) H (t )

(A2.1)

where ϕ(ξ), H(t) are two new unknown functions. Note that we are interested only in the time evolution of the radius of spreading, R(t). Substitution of (A2.1) into the conservation law, Equation 4.92, results in 1

∫

2 πR (t ) H (t ) ξϕ(ξ) d ξ = F (t ) . 2

(A2.2)

0

Let us select the unknown function H(t) as H (t ) =

© 2007 by Taylor & Francis Group, LLC

V (t ) . 2 πR 2 (t )

(A2.3)

380

Wetting and Spreading Dynamics

Then, Equation A2.2 can be reduced to 1

∫ ξϕ(ξ)d ξ = 1 .

(A2.4)

0

To move further it is necessary to specify the equation of spreading (4.5), which determines the drop proﬁle h(t, r) or ϕ(ξ). Two different cases are under consideration: complete and partial wetting cases. Capillary Regime, Complete Wetting In this case, according to Equation 4.94, the pressure inside the spreading drop can be written as p = pa − γ

1 ∂  ∂h  r . r ∂r  ∂r 

This expression should be substituted into Equation 4.91, which yields the time evolution of the drop proﬁle, h(t, r): ∂h 1 1 ∂  3 ∂  γ ∂  ∂h    =− r rh . ∂t ∂r  r ∂r  ∂r    3 η r ∂r 

(A2.5)

Note that the omission of the action of surface forces results in the wellknown singularity on the moving three-phase contact line (see Section 3.1). Substitution of the similarity coordinate and function using Equation A2.1 and Equation A2.3 into Equation A2.5 results in ′  ′ 4 (t )   1 1 H ( t ) R γ H ( t ) ′ 3  ξϕ (ξ)  H (t )ϕ(ξ) − ξϕ′(ξ) = −  ξ ξϕ′(ξ)   , R(t ) 3η R 4 (t ) ξ   

(

)

where an overdot denotes differentiation with time, whereas ′ means differentiation with respect to the similarity variable, ξ. This equation can be rewritten as  ′ ′  3ηH (t ) R 4 (t ) 3ηR3 (t ) R (t ) 1 3 1 ′ ϕ(ξ ) − ξϕ′(ξ) = − ξϕ (ξ)  ξϕ′(ξ)  . (A2.6) ξ ξ   γH 4 (t ) γH 3 (t )  

(

© 2007 by Taylor & Francis Group, LLC

)

Spreading over Porous Substrates

381

Equation A2.6 should depend on the similarity coordinate only, and it should not include any time dependence. This is possible only if, simultaneously, the following two relations are satisﬁed 3 ηH (t ) R 4 (t ) = B1, γH 4 (t )

3 ηR 3 (t ) R (t ) = B2 , γH 3 (t )

(A2.7)

where B1 and B2 are unknown constants. Both constants should be positive because H(t) and R(t) are both increasing functions of time. Let α = B1/B2 and divide the ﬁrst equation in (A2.7) by the second equation. It results in: H R =α , H R that upon integration yields: H (t ) = CR α (t ) ,

(A2.8)

where C is an integration constant, and α is still an unknown exponent. Substitution of Equation A2.8 into Equation A2.7 results in the following time evolution of the radius of spreading, R(t),  ( 4 − 3α) γB2C 3  R (t ) =   3η  

1/( 4− 3 α )

t 1/( 4−3α ) ,

(A2.9)

which shows that 4-3α should be positive, that is, α < 4/3. Equation A2.9 and Equation A2.3 allow determination of the unknown function H(t):  ( 4 − 3α) γB2C 3  H (t ) = C   3η  

α /( 4− 3 α )

t α /( 4−3α ) .

(A2.10)

Using Equation A2.10, Equation A2.9, and Equation A2.3, we can conclude that the following relation should be satisﬁed: 2 +α

 ( 4 − 3α) γB2C 3  4−3α 42−+3αα V (t ) = 2 πC  t .  3η  

© 2007 by Taylor & Francis Group, LLC

(A2.11)

382

Wetting and Spreading Dynamics

This relation shows that the similarity mechanism considered in the preceding text is possible only if the dependency V(t) is deﬁned by the power law (A2.11). Let us assume now that V (t ) = a ′ t b ,

(A2.12)

where a′ and b are constants imposed by the source. The exponents in Equation A2.11 and Equation A2.12 should be equal, and hence α=

4b − 2 . 1 + 3b

Substitution of this expression into Equation A2.9 gives the following spreading law: R (t ) = C ⋅ t 0.1+0.3b .

(A2.13)

Accordingly, the exponent is the sum of two terms: the ﬁrst term, 0.1, stems from the spontaneous spreading (see Chapter 3), and the second term, 0.3b, is determined by the liquid source. In the case of constant ﬂow rate of the liquid, v = I ·t

(A2.14)

where I is the intensity of the liquid source. The comparison of exponents in Equation A2.14 and Equation A2.9 yields 1=

2+α 4 − 3α

or α = 1/2. On the other hand, α = B1/B2 or B2 = 2B1 though B1 is the only unknown constant. The comparison of the preexponential factors in Equation A2.14 and Equation A2.9 gives  3 ηI  C =  10 πγB1 

1/ 4

.

(A2.15)

The spreading law according to Equation A2.13 now takes the following form:  5 γB1I 3  R (t ) =    24 ηπ 3  © 2007 by Taylor & Francis Group, LLC

0.1

t 0.4 .

(A2.16)

Spreading over Porous Substrates

383

The only unknown constant in Equation A2.16 is the dimensionless constant B1. The equation of spreading (A2.6) can be rewritten now as  1 1 B1ϕ(ξ) − 2 B1ξϕ′(ξ) = −  ξϕ 3 (ξ)  ξϕ′(ξ)  ξ ξ 

(

′ ′ ′  .   

)

(A2.17)

Remember, this equation describes the behavior of the drop proﬁle, ϕ(ξ), not too close to the moving three-phase contact line (valid only away from the range of the action of surface forces). Let us try to include the disjoining pressure action into Equation A2.17. In the case of complete wetting, the disjoining pressure isotherm is Π( h ) =

A , h3

where A is the Hamaker constant. Now Equation 4.8 should be rewritten as 1 ∂  ∂h  A , p= p −γ r  + ρgh −  a r ∂r  ∂r  h3 and this expression should be substituted in the equation of spreading (4.5). Using the same similarity coordinate, ξ, and the drop proﬁle, ϕ, according to Equation A2.1, we arrive at the following equations for the determination of the unknown function, ϕ: ′  ′ 1 3 1 χB1   ′ B1ϕ(ξ) − 2 B1ξϕ′(ξ) = − ξϕ (ξ)  ξϕ′(ξ) + 3  , ξ ϕ (ξ )   ξ  

(

)

(A2.18)

where χ=

10 πA 3 ηI

is a dimensionless constant, which characterizes the intensity of the action of surface forces. Equation A2.18 shows that the same similarity property is valid for the full Equation A2.18, when the action of surface forces is taken into account, as for Equation A2.17. © 2007 by Taylor & Francis Group, LLC

384

Wetting and Spreading Dynamics

Equation A2.18 must satisfy the following boundary conditions: ϕ ′(0 ) = ϕ ′′′(0 ) = 0 ,

(A2.19)

which are the symmetry conditions in the drop center. It should also satisfy ϕ(ξ) → 0, ξ → ∞ ,

(A2.20)

where inﬁnity means the drop edge. It has been shown in Section 3.2 that the drop proﬁle tends asymptotically to zero inside “the inner region,” which is the boundary condition (A2.20). The conservation law (A2.10) should also be taken into account. The unknown parameter, B1, should be small according to the previous consideration in Chapter 3. In this case, the whole drop can be subdivided into two regions: the outer region, which has a spherical shape, and the inner region, where the inner coordinate should be introduced. Matching of these two regions allows the determination of the unknown parameter, B1, via the dimensionless Hamaker constant, χ. It has been shown in Section 3.2 that the dependence of B1 on the parameter χ is a weak one. It has been shown also in Section 3.2 that the lack of proper asymptotic behavior of Equation A2.18 does not allow precise determination of the unknown constant B1. That is why we use Equation A2.16 in the following form  γI 3  R (t ) = α c    η 

0.1

 5 B1  t , αc =   24π 3  0.4

0.1

,

(A2.21)

where the dimensionless parameter αc is determined in the following text using experimental data. Gravitational Regime, Complete Wetting In this case, Equation 4.94 becomes p = pa + ρgh , and substitution of the latter equation into the equation of spreading (Equation 4.91) results in ∂h ρg ∂  3 ∂h  rh = . ∂r  ∂t 3ηr ∂r 

(A2.22)

Using the same similarity coordinate and function (A2.1), we conclude that relations (A2.2–A2.4) are still valid. Using the same procedure as the one previously mentioned, we can transform Equation A2.22 to H (t ) R (t ) ρg H 4 (t ) 1  3 ′ ξϕ (ξ)ϕ′(ξ)  . H (t )ϕ(ξ) − ξϕ′(ξ) = 2  R(t ) 3η R (t ) ξ © 2007 by Taylor & Francis Group, LLC

Spreading over Porous Substrates

385

This equation can be rewritten as 1 3ηH (t ) R 2 (t ) 3ηR(t ) R (t ) ′ ϕ(ξ ) − ξϕ′(ξ) =  ξϕ 3 (ξ)ϕ′(ξ)  . 4 3 ξ ρgH (t ) ρgH (t )

(A2.23)

Equation A2.23 should depend on the similarity coordinate only; that is, it should not include any time dependence. This is possible only if the following relations are satisﬁed simultaneously: 3 ηH (t ) R 2 (t ) = D1, ρgH 4 (t )

3 ηR (t ) R (t ) = D2 , ρgH 3 (t )

(A2.24)

where D1 and D2 are unknown dimensionless constants. Both constants should be positive (or zero) because H(t) and R(t) are both increasing functions of time. Let β = D1/D2 and divide the ﬁrst equation in (A2.24) by the second equation. That results in H R =β , H R which upon integration yields H (t ) = GR β (t ) ,

(A2.25)

where G is an integration constant, and β is still the unknown exponent. Substitution of Equation A2.25 into Equation A2.24 results in the following time evolution of the radius of spreading, R,  (2 − 3β)ρgD2 G 3  R (t ) =   3η  

1/( 2 − 3β )

t 1/(2 −3β ) ,

(A2.26)

which shows that 2-3β should be positive, that is, β < 2/3. Equation A2.26 and Equation A2.25 allow the determination of the unknown function H(t):  (2 − 3β)ρgD2 G 3  H (t ) = G   3η  

β /( 2 − 3β )

t β /(2 −3β ) .

(A2.27)

Using Equation A2.26, Equation A2.27, and Equation A2.3, we can conclude 2 +β

2 +β

 (2 − 3β)ρgD2 G 3  2 −3β 2 −3β . V (t ) = 2 πG  t  3η   © 2007 by Taylor & Francis Group, LLC

(A2.28)

386

Wetting and Spreading Dynamics

Thus, we see that the similarity mechanism considered in the preceding text is possible only if the dependency V(t) is deﬁned by the power law (A2.28). Let us assume now that the liquid source produced the liquid in the same way as in the case of the capillary regime; that is, according to Equation A2.12. The exponents in Equation A2.28 and Equation A2.12 should be equal, and hence in β=

2b − 2 . 1 + 3b

Substitution of this expression into Equation A2.26 gives the following spreading law: R (t ) = const ⋅ t 1/ 8+3b / 8 ,

(A2.29)

that is, the exponent is the sum of two terms: the ﬁrst term, 1/8, stems from the spontaneous gravitational spreading (see Section 3.1), and the second term, 3b/8, is determined by the liquid source. In the case of constant ﬂow rate of the liquid, V (t ) = I t

(A2.30)

and the comparison of exponents in Equation A2.30 and Equation A2.12 results in 1=

2+β 2 − 3β

or β = 0. On the other hand β = D1/D2, hence, D1 = 0. Then, D2 is the only unknown constant. According to the ﬁrst Equation A2.24 and Equation A2.25, D1 = β = 0 means that, in the case of gravitational spreading, the maximum height of the spreading drop remains constant when the source of liquid follows Equation A2.30. The comparison of the preexponential factors in Equation A2.30 and Equation A2.28 gives  3 ηI  G=  4πρgD2 

1/ 4

.

The spreading law according to Equation A2.26 takes the following form now:

R (t ) = 2

© 2007 by Taylor & Francis Group, LLC

1/ 6

 ρgD2 I 3     3 ηπ 3 

1/ 8

t 0.5 .

(A2.31)

Spreading over Porous Substrates

387

The only unknown quantity in Equation A2.31 is the dimensionless constant D2, which is left unknown and was determined experimentally. Note that the constant D2 can be, in general, determined in the following way: (1) a narrow zone close to the drop edge, where capillary forces become important should be considered, and (2) matching of asymptotic solutions (capillary zone as an inner zone and the gravitational zone as an outer zone) should be made, which allows the determination of the numerical constant D2. In order to determine the unknown constant, we rewrite Equation A2.31 as  ρgI 3  R (t ) = α g    η 

1/ 8

 D  t 0.5 , α g = 21/ 6  23   3π 

1/ 8

,

(A2.32)

and the constant αg is obtained using experimental data. The latter expression allows the determination of the constant thickness of the spreading drop during the gravitational regime of spreading. Combining Equation A2.3 and Equation A2.32 would result in 1  I η H= 2  α g  ρg 

1/ 4

,

(A2.33)

which is independent of time as predicted in the preceding text. It means that during the gravitational regime of spreading from a liquid source with constant ﬂow rate intensity, the drop spreads like a “pancake.” Partial Wetting The drop volume can be expressed in terms of the spreading radius and the contact angle as follows:

V (t ) = R 3 (t ) f (θ a ),

f (θ a ) =

π θ tan a 6 2

 2 θa   3 + tan 2  ,

(A2.34)

where θa is the static advancing contact angle. We assume that the capillary number is very small and hence, the contact angle does not change during spreading. Hence, f(θa) also remains constant. Combination of Equation A2.10 and Equation A2.34 results in  V (t )  R (t ) =   f (θ a ) 

© 2007 by Taylor & Francis Group, LLC

1/ 3

.

388

Wetting and Spreading Dynamics

In the case of the constant ﬂow rate from the liquid source (Figure 4.15), according to Equation A2.30, the preceding equation gives  I  R (t ) =   f (θ a ) 

1/ 3

t 1/ 3 ,

(A2.35)

which is compared with our experimental observations in the case of partial wetting.

REFERENCES 1. Brinkman, H., J. Chem. Phys., 20, 571, 1952. 2. Whitaker, S., The Method of Volume Averaging, Kluwer Academic Publishers, Dortrecht/Boston/London, 1999, p. 221. 3. Starov, V.M. and Zhdanov, V.G., Colloids Surf. A: Physicochem. Eng. Aspects, 192, 363, 2001. 4. Kalinin, V.V. and Starov, V.M., Colloid J. (USSR Academy of Sciences, English Translation), 51(5), 860, 1989. 5. Starov, V.M., Kosvintsev, S.R., Sobolev, V.D., Velarde, M.G., and Zhdanov, S.A., Spreading of liquid drops over dry porous layers: complete wetting case, J. Colloid Interface Sci., 252, 397–408, 2002. 6. Kornev, K.G. and Neimark, A.V., J. Colloid Interface Sci, 235, 101, 2001. 7. Neogi, P. and Miller, C.A., J. Colloid Interface Sci., 92(2), 338, 1983. 8. Starov, V.M., Kalinin, V.V., and Chen, J.-D., Adv. Colloid Interface Sci., 50, 187, 1994. 9. Kalinin, V. and Starov, V., Colloid J. (USSR Academy of Sciences, English Translation), 48(5), 767, 1986. 10. Marmur, A., Adv. Colloid Interface Sci., 19, 75, 1983. 11. Joanny, J.-F., J. Phys., 46, 807, 1985. 12. Starov, V.M., Kosvintsev, S.R., Sobolev, V.D., Velarde, M.G., and Zhdanov, S.A., J. Colloid Interface Sci., 252, 397, 202. 13. Nayfeh, A.H., Perturbation Methods, Wiley-Interscience, New York, 1973. 14. Starov, V.M., Zhdanov, S.A., and Velarde, M.G., Langmuir, 18, 9744, 2002. 15. Holdich, R., Starov, V., Prokopovich, P., Hjobuenwu, D., Rubio, R., Zhdanov, S., and Velarde, M., Colloids Surf., A: Physicochem. Eng. Aspects, 2005. 16. Cazabat, A.M. and Cohen-Stuart, M.A., J. Phys. Chem., 90, 5849, 1986. 17. Brinkman, H., Appl. Sci. Res., A1, 27, 1947.

© 2007 by Taylor & Francis Group, LLC

5

Dynamics of Wetting or Spreading in the Presence of Surfactants

INTRODUCTION In this chapter we consider the kinetics of spreading of surfactant solutions over hydrophobic and porous substrates. In spite of the wide use of these processes, currently we are not in the position to answer even basic questions in this area, such as how surfactant molecules are transferred in a vicinity of the three-phase contact line. In the case of aqueous surfactant solution, our knowledge of the behavior of the transition zone from the meniscus to thin ﬁlms in front is very limited. Disjoining pressure isotherms in the presence of surfactants are investigated in the case of free liquid ﬁlms [30], and much less is known in the case of liquid ﬁlms on solid support. At the moment we cannot present a clear physical picture of the equilibrium of liquids or menisci ion the presence of surfactants. The most important problem of surfactant transfer in a vicinity of the threephase contact line is to be investigated here. In Chapter 1 and Chapter 2 we showed that the Young’s equation does not have any theoretical basis and should be replaced by the Deriaguin–Frumkin equation for equilibrium contact angle. In this chapter, in view of our limited knowledge of surfactant behavior in the vicinity of the moving three-phase contact line, we decided to use the Young’s equation for the quasi-equilibrium contact angle for the description of slowly developing time-spreading processes over hydrophobic surfaces. We realize that any conclusion based on this semiempirical relation should be understood accordingly as semiempirical. However, our hypothesis on adsorption of surfactants on a bare hydrophobic substrate in front of the moving meniscus allows us to develop some theoretical predictions, which are in reasonable agreement with known experimental data (Section 5.2 to Section 5.4; Section 5.7). However, the main question how this transfer goes on is left unanswered. The situation is even less investigated in the case of simultaneous spreading and imbibition into porous substrate (Section 5.1). We present some theoretical and experimental investigations of the process (Section 5.1), which should be considered as a ﬁrst step in this direction.

389 © 2007 by Taylor & Francis Group, LLC

390

Wetting and Spreading Dynamics

In Section 5.6 we consider the much more theoretically understood process of ﬂow caused by surface tension gradient (Maramgoni ﬂow). We show that the ﬂow caused by the point source of surfactant on the surface of thin aqueous ﬁlm is caused by the surface tension gradient only. All other forces can be disregarded. This process recently became a powerful tool for investigation of the newly recognized phenomenon called superspreading.

5.1 SPREADING OF AQUEOUS SURFACTANT SOLUTIONS OVER POROUS LAYERS In this section we shall follow the track of Chapter 4 for the case of surfactant aqueous solutions like sodium dodecyl sulfate (SDS, an anionic surfactant). We shall start with the spreading problem of big drops, albeit small enough to allow neglecting gravity, over porous solid substrates [1]. The porous substrate should be thin in the sense used in Chapter 4. We shall be considering various SDS concentrations: zero (pure water) and concentrations below, near, and above the critical micelle concentration (CMC). The overall spreading process would be divided into three stages: in the ﬁrst stage, the drop base expands until its maximum value is attained; the contact angle decreases very fast. In a second stage, the radius of the drop base remains constant, whereas the contact angle decreases linearly with time. Finally, in the third stage, the drop base shrinks but the contact angle remains constant, while the wetted area inside the porous solid substrate expands all the time [1]. Appropriate scales were used with a plot of the dimensionless radii of the drop base, of the wetted area inside the porous substrate, and the dynamic contact angle on the dimensionless time. The experimental data show that the overall time of the spreading of drops of SDS solution over dry thin porous substrates decreases with the increase of surfactant concentration; the difference between advancing and hydrodynamic receding contact angles decreases with the surfactant concentration increase; and the constancy of the contact angle during the third stage of spreading has nothing to do with the hysteresis of contact angle but is determined by hydrodynamic reasons. It is shown using independent spreading experiments of the same drops on nonporous nitrocellulose substrate that the static receding contact angle is equal to zero, which supports our conclusion on the hydrodynamic nature of the receding contact angle on porous substrates. In Chapter 4 the spreading of liquid drops over thin porous layers saturated with the same liquid (Section 4.1) or dry (Section 4.1) was investigated in the case of complete wetting. Brinkman’s equations were used for the description of the liquid ﬂow inside the porous substrate. In this section we take up the same problem in the case where a drop spreads over a dry porous layer as in partial wetting. Spreading of a big drop (but still small enough to neglect the gravity action) of aqueous SDS solutions over “thin porous layers” (nitrocellulose membrane) is considered in the following text. © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

EXPERIMENTAL METHODS

AND

391

MATERIALS [1]

Spreading on Porous Substrates (Figure 4.4) Aqueous solutions of SDS were used in the spreading experiments in zero concentration (pure water) and concentrations below CMC, near CMC, and above CMC. SDS was purchased from Fisher Scientiﬁc UK Ltd. of Leicestershire, U.K. and used as obtained, without further puriﬁcation. Nitrocellulose membranes, purchased from Millipore of Billerica, MA, with average pore size a = 0.22, a = 0.45, and a = 3.0 µm (marked by the supplier) were used as a model of thin porous layers. The same experimental chamber as in Chapter 4 (Figure 4.2) was used for monitoring of the spreading of liquid drops over initially dry porous substrates. The time evolution of the radius of the drop base, L(t), the dynamic contact angle, θ(t), and the radius of the wetted area inside the porous substrate, l(t), was monitored (Figure 4.4). The porous substrate 1 (Figure 4.2) was placed in a thermostated and hermetically closed chamber 2, where 100% relative humidity and ﬁxed temperature were maintained. The initial volume of the droplets ranged between 1.47µl and 10.45 µl. Care was taken so that all interfaces in the syringe and the attached replaceable capillary had been saturated with surfactants before the actual solutions were used. The drop was applied on the surface by pumping manually the piston of the syringe. The distance between the forming drop and the surface was kept minimal to avoid collateral inertia effects. Experiments were carried out in the following order: •

• •

The dry membrane (initially stored in 0% humidity atmosphere) was placed in the chamber with a 100% humidity atmosphere and left for 15 to 30 min. A light pulse produced by a ﬂash gun was used to synchronize both videotape recorders (a side view and a view from above). A droplet of liquid was placed onto the membrane.

Each run was carried out until the complete imbibition of the drop into the porous substrate took place. Measurement of Static Advancing and Receding Contact Angles on Nonporous Substrates A modiﬁed experimental setup (as compared with the previous case) was used for measurements of static advancing and receding contact angles on a nonporous nitrocellulose substrate. Figure 4.19 shows the schematic presentation of the sample chamber for monitoring of the advancing and receding contact angles during droplet spreading over a smooth nonporous nitrocellulose substrate. The time evolution of the radius of the drop base, L(t,) and the dynamic contact angle, θ(t), was monitored. © 2007 by Taylor & Francis Group, LLC

392

Wetting and Spreading Dynamics

The nonporous substrate under investigation 1 (Figure 4.19) was attached by double-sided tape to a solid substrate and was placed into a thermostated and hermetically closed chamber 2, where controlled ﬁxed humidity and ﬁxed temperature were maintained. A glued-in syringe needle 4 was positioned in the center of the solid substrate and connected to the Harvard Apparatus syringe pump 18 (Figure 4.19). The nonporous nitrocellulose substrate was prepared as follows: •

•

•

•

The nitrocellulose substrate was attached to the solid substrate 1 (Figure 4.19) using double-sided sticky tape, ensuring that no air bubbles could be trapped. The nitrocellulose substrate attached to the solid substrate was placed into an acetone atmosphere. Acetone vapor was used to seal any pores that may have been open, and the nitrocellulose substrate was left in the acetone atmosphere for approximately 20 min until the surface became transparent. After that, the nitrocellulose substrate attached to the solid substrate 1 was placed into a sealed container ﬁlled with air, allowing the surface to reach equilibrium after the acetone treatment. A hole was made in the center of the nitrocellulose substrate, allowing penetration of the liquid through a glued-in syringe needle 4 connected to the Harvard Apparatus syringe pump 18 (Figure 4.19). This allowed drawing the ﬂuid in or out of the surface and monitoring the droplet volume according to a prescribed rate.

The experimental runs were carried out in the following order: •

• • • •

•

The speed at which the ﬂuid would be drawn out of the drop was ﬁxed in the Harvard Apparatus syringe pump 18 (Figure 4.19). The reﬁll rate was varied between 0.01 and 0.1 µl per minute. The nitrocellulose substrate 1 was placed into the experimental chamber 2, and the needle was connected to the pump 18 (Figure 4.19). The experimental chamber was closed, and the fan was switched on to equilibrate the atmosphere inside the chamber. After approximately 20 min the fan was switched off and video recorders were switched on. The droplet of the liquid was deposited onto the surface under investigation, using the upper syringe 17 into the center of syringe needle 4 (Figure 4.19). After 1 min of recording, the pump was turned on to draw off the liquid from the drop.

The volume of the spreading drops should remain constant in the case of spreading over nonporous nitrocellulose surface. The constancy of the volume of the spreading drops was monitored during these experiments, which conﬁrmed that the substrate used was nonporous. © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

393

Results and Discussion In all our spreading experiments, the drops remained as spherical shapes over the whole spreading process. This was cross-checked by reconstructing of the drop proﬁles at different time instants of spreading, ﬁtting those drop proﬁles by a spherical cap (see Chapter 4, Section 4.1 and Section 4.2). In all cases, the reduced chi-square value was found to be less than 104. The wetted area inside the porous layer was mostly circular. In some cases it was of an ellipsoidal form, and it was taken into account for the calculation of the drops volume. Drops remained in the center of this spot over the duration of the spreading process. In all cases, saturation of the membrane in the vertical direction was much faster than the whole duration of the spreading–imbibition process; that is, the membrane was assumed always saturated in the vertical direction (Section 4.2). A schematic presentation of the process is presented in Figure 4.4. According to our observations, the whole spreading process can be subdivided into three stages (see Figure 5.1). During the ﬁrst stage, the contact angle, θ, rapidly decreases, whereas the radius of the drop base, L, increases until its maximum value, Lmax, is reached. The ﬁrst stage was followed by the second stage, when the radius of the drop base, L = Lmax, was constant, and the contact angle, θ, decreased linearly with time. During the third stage, the radius of the drop base decreased and the contact angles remained constant. At the ﬁnal third stage, the drop base shrank until the drop completely disappeared, and the imbibition front expanded until the end of the process. The spherical form of the spreading drop allows measuring the evolution of the contact angle of the spreading drops. The contact angle, θ, during the ﬁrst stage, decreased very fast; during the second stage, the contact angle, θ, decreased much slower and linearly, and the contact angle remained a constant value over the duration of the third stage. This constant value of the contact angle is referred to in the following text as θm. All relevant experimental data are summarized in the Table 5.1. In this section we present a brief theoretical explanation why the dependency of the contact angle during the second stage of the spreading is a linear function of time. During the second stage, L = Lmax , t > tm ,

(5.1)

where tm corresponds to the beginning of the second stage. This means that according to Equation 4.72 from Chapter 4, Section 4.2, dl K p pc /η = , l (tm ) = lm , dt l ln l Lmax

(5.2)

where l is the radius of the circular edge of the wetted region inside the porous layer, Kp is the permeability of the porous layer in the tangential direction, pc is the capillary pressure on the wetting front inside the porous layer, lm is the radius © 2007 by Taylor & Francis Group, LLC

394

Wetting and Spreading Dynamics 1.0

1.0 2

1

L/Lmax

0.6

3

0.6

0.4

0.4 Nitrocellulose membranes, a = 0.22 μm. SDS = 0%

0.2

0.2

tmax = 20.72 sec V0 = 2.9 mm3 tmax = 35.28 sec V0 = 5.3 mm3

0.0 2.0

/max

0.8

0.8

0.0

1.5

θ/θm

1

θa

1.0 θr

2 0.5

3 0.0 0.0

0.2

0.4

0.6

0.8

1.0

t/tmax

FIGURE 5.1 Spreading of pure aqueous drops over nitrocellulose membrane, a = 0.22 µm. L/Lmax — dimensionless radius of the drop base; l/lmax — radius of the wetted area; θ/θm — dynamic contact angle; t/tmax — dimensionless time; (1) ﬁrst stage: θ rapidly decreases while L increases until the maximum value Lmax; (2) second stage: L is constant, L = Lmax, and θ decreases linearly with time; (3) third stage: L decreases and θ remains constant.

of the circular edge of the wetted region inside the porous layer at the moment tm, and η is the liquid dynamic viscosity. This equation can be easily integrated, which gives l

l 2 ln

eLmax

− lm2 ln

lm eLmax

=

2 K p pc η

(t − tm ) ,

or lm lm2 ln 2 K p eLmax p c l2 = (t − tm ) + . l l η ln ln eLmax eLmaxx © 2007 by Taylor & Francis Group, LLC

(5.3)

Dynamics of Wetting or Spreading in the Presence of Surfactants

395

TABLE 5.1 Experimental Data Used in Figure 5.1 through Figure 5.5 Pore Size, Figure, Symbols

Averaged Pore Size, a µm

Figure 5.1 squares Figure 5.1 circles Figure 5.2 squares Figure 5.2 triangles Figure 5.2 circles Figure 5.3 diamonds Figure 5.3 circles Figure 5.3 squares Figure 5.3 triangles Figure 5.4 diamonds Figure 5.4 circles Figure 5.4 squares Figure 5.4 triangles Figure 5.5 diamonds Figure 5.5 circles Figure 5.5 squares Figure 5.5 circles

V0, µl Initial Volume of Drop

SDS Concentration, %

tmax, sec

Lmax, mm

lmax, mm

θm, grad

tm, sec

0.22

0

2.9

20.72

1.49

3.38

51

0.36

0.22

0

5.3

35.28

1.85

4.13

50

0.36

3.0

0.1

2.1

2.96

1.47

2.30

31

0.36

3.0

0.1

3.7

4.92

2.01

3.40

24

0.64

3.0

0.1

9.7

13.12

2.59

5.28

35

0.48

0.22

0

2.9

20.72

1.49

3.38

51

0.36

0.22

0.1

10.5

75.2

2.78

5.79

32

0.4

0.22

0.2

7.4

33.84

2.45

4.67

27

0.44

0.22

0.5

2.6

8.2

2.03

2.90

20

0.12

0.45

0

6.1

8.44

2.83

4.35

44

0.52

0.45

0.1

9.5

10.28

2.46

4.92

38

0.35

0.45

0.2

7.2

8.44

2.83

4.35

21

0.36

0.45

0.5

7.1

6.84

2.75

4.80

23

0.08

3.0

0

5.9

108.04

1.89

4.21

46

3.0

0.1

9.7

13.12

2.59

5.28

35

0.48

3.0

0.2

6.8

3.96

2.65

4.43

21

0.28

3.0

0.5

5.1

2.12

2.39

3.97

22

0.12

The term ln

l eLmax

© 2007 by Taylor & Francis Group, LLC

21

396

Wetting and Spreading Dynamics

in the denominator of the equation is a slow changing function of l; that is, the right-hand side of the equation is almost indistinguishable from the linear function of time. Equation 5.3 can be rewritten as lm2 ln

l 2 = At + B, A =

lm

2 K p pc 2 K p pc eLmax , B= . (5.4) − tm l l l ln η ln η ln e Lmax eLmax eLmax

According to Equation 4.48 and Equation 4.54 from Chapter 4, Section 4.3, V (t ) = V0 − πm ∆ l 2 (t ) and θ=

4V V0 − πm ∆ l 2 (t ) V πm ∆ 2 l (t ) , = = 03 − 3 3 πLm πLm πLm πL3m

(5.5)

where ∆ is the thickness of the porous layer. Combining Equation 5.4 and Equation 5.5 results in

θ=

 V V0 πm ∆  πm ∆ πm ∆A t +  30 − B 3  . − 3 ( At + B) = − 3 3 πLmax  πLmax πLmax πLmax  πLmax

(5.6)

This equation shows that the contact angle decreases linearly with time during the second stage of spreading. Everywhere in the following text the time evolution is presented for both the radius of the base of the spreading drops and the radius of the wetted area using dimensionless coordinates. The drops were of different volumes and different SDS concentrations. The total duration of the spreading process, tmax, the maximum radius of the drop base, Lmax, and the ﬁnal radius of the wetted area, lmax, varies considerably depending on the drop volume, SDS concentration, the averaged pore size, and porosity of nitrocellulose membranes. It has been suggested in Section 4.2 that the following dimensionless values, L/Lmax, l/lmax, and t/tmax, be used. The same dimensionless values are used in the following text. According to Section 4.3, the reduced contact angle dependency, θ/θm, against dimensionless time, t/tmax, is used in the following text, where θm is the value of the dynamic contact angle, which is reached at the moment t = tm (the end of the ﬁrst stage of the spreading process). Figure 5.1 shows that spreading behavior of drops of different volumes over the same porous substrate has a universal character in dimensionless coordinates. © 2007 by Taylor & Francis Group, LLC

397

1.0

1.0

0.8

0.8

0.6

0.6

0.4

/max

L/Lmax

Dynamics of Wetting or Spreading in the Presence of Surfactants

0.4 Nitrocellulose membranes, a = 3.0 μm. SDS = 0.1% tmax = 2.96 sec V0 = 2.1 mm3 tmax = 4.92 sec V0 = 3.7 mm3 tmax = 13.12 sec V0 = 9.7 mm3

0.2 0.0 3

0.2

0.0

θ/θm

2

1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

t/tmax

FIGURE 5.2 Spreading of droplets of 0.1% SDS solution over nitrocellulose membrane, a = 3.0 µm; L/Lmax — dimensionless radius of the drop base; l/lmax — radius of the wetted area; θ/θm — dynamic contact angle; t/tmax — dimensionless time.

However, not all of the experimental data have shown such universal behavior. Some experimental runs differ during the ﬁrst stage of spreading in dimensionless coordinates (Figure 5.2). This difference becomes bigger if the overall time of spreading is shorter (droplets of different volumes). In Figure 5.3 to Figure 5.5 (0.22 µm, 0.45 µm, and 3.0 µm nitrocellulose membranes, respectively) the time evolution is presented of the radius of the drop base, the radius of the wetted area inside the porous substrates, and the contact angle at different SDS concentrations. Figure 5.3 to Figure 5.5 show that the second stage of spreading becomes shorter in dimensionless coordinates with the increase in SDS concentrations. Contact angles show the universal constant behavior during the third stage of spreading for each of the concentrations. © 2007 by Taylor & Francis Group, LLC

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4 Nitratecellulose membranes, a = 0.22 μm SDS = 0.0%; V0 = 2.9 mm3; tmax = 20.72 sec SDS = 0.1%; V0 = 10.5 mm3; tmax = 75.2 sec SDS = 0.2%; V0 = 7.4 mm3; tmax = 33.84 sec SDS = 0.5%; V0 = 2.6 mm3; tmax = 8.2 sec

0.2 0.0 90

/max

Wetting and Spreading Dynamics

L/Lmax

398

0.2 0.0

θ

60

30

0 0.0

0.2

0.4

0.6

0.8

1.0

t/tmax

FIGURE 5.3 Spreading of droplets of SDS solutions over nitrocellulose membrane, a = 0.22 µm; L/Lmax — dimensionless radius of the drop base; l/lmax — radius of the wetted area; θ — dynamic contact angle; t/tmax — dimensionless time.

Advancing and Hydrodynamic Receding Contact Angles on Porous Nitrocellulose Membranes Using presented experimental data on the spreading of drops of aqueous SDS solutions over dry porous substrates, the values of advancing, θa, and hydrodynamic receding, θrh, contact angles were extracted as a function of the SDS concentration. We are using in the following text the term hydrodynamic receding contact angle and the symbol θrh to distinguish it from the static receding contact angle, which is found equal to zero (see the following text on static hysteresis of the contact angle of SDS solution drops on smooth nonporous nitrocellulose substrate). The advancing contact angle, θa, was deﬁned at the end of the ﬁrst stage when the drop stopped spreading (the radius of the drop base reached its maximum value). The hydrodynamic receding contact angle, θrh, was deﬁned at the moment when the drop base started to shrink.

© 2007 by Taylor & Francis Group, LLC

399

1.0

1.0

0.8

0.8

0.6

0.6 /max

L/Lmax

Dynamics of Wetting or Spreading in the Presence of Surfactants

0.4

0.4

Nitratecellulose membranes, a = 0.45 μm SDS = 0.0%; V0 = 6.1 mm3; tmax = 8.44 s

0.2

0.2

SDS = 0.1%; V0 = 9.5 mm3; tmax = 10.28 s SDS = 0.2%; V0 = 7.2 mm3; tmax = 8.44 s

0.0 90

SDS = 0.5%; V0 = 7.1 mm3; tmax = 6.84 s

0.0

θ

60

30

0 0.0

0.2

0.4

0.6

0.8

1.0

t/tmax

FIGURE 5.4 Spreading of droplets of SDS solutions over nitrocellulose membrane, a = 0.45 µm; L/Lmax — dimensionless radius of the drop base; l/lmax — radius of the wetted area; θ — dynamic contact angle; t/tmax — dimensionless time.

In Figure 5.6, experimental data on the apparent contact angle hysteresis are summarized. This ﬁgure shows that the advancing contact angle, θa, decreases with SDS concentration; the hydrodynamic receding contact angle, θrh, on the contrary, slightly increases with SDS concentration. These experimental runs show that (1) the difference between advancing and receding contact angles becomes smaller with the increase in the SDS concentration (Figure 5.6), and (2) the dimensionless time interval when the drop base does not move also decreases with the increase in the SDS concentration.

© 2007 by Taylor & Francis Group, LLC

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4 Nitratecellulose membranes, a = 3.0 μm SDS = 0.0%; V0 = 5.9 mm3; tmax = 108.04 s SDS = 0.1%; V0 = 9.7 mm3; tmax = 13.12 s SDS = 0.2%; V0 = 6.8 mm3; tmax = 3.96 s SDS = 0.5%; V0 = 5.1 mm3; tmax = 2.12 s

0.2 0.0 90

/max

Wetting and Spreading Dynamics

L/Lmax

400

0.2

0.0

θ

60

30

0 0.0

0.2

0.4

0.6

0.8

1.0

t/tmax

FIGURE 5.5 Spreading of droplets of SDS solutions over nitrocellulose membrane, a = 3.0 µm; L/Lmax — dimensionless radius of the drop base; l/lmax — radius of the wetted area; θ — dynamic contact angle; t/tmax — dimensionless time.

Static Hysteresis of the Contact Angle of SDS Solution Drops on Smooth Nonporous Nitrocellulose Substrate In the previous parts of this section the spreading of drops at different SDS concentrations on nitrocellulose membranes of various pore sizes was considered. In all cases during the third stage of spreading, the radius of the drop base, L, shrank, and the hydrodynamic receding contact angle, θrh , remained constant. The duration of the third stage of spreading increases with the SDS concentration increase. It is necessary to note that the behavior of drops of aqueous SDS solutions during the third stage of spreading (partial wetting) is remarkably similar to the behavior during the second stage of spreading in the case of complete wetting (Chapter 4, Section 4.2). © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

401

60 Nitrocellulose membranes

θa, θrh (degree)

50

a = 0.22 µm a = 0.45 µm a = 3.0 µm

40 30 20 10 0 0.0

0.1

0.2 0.3 SDS concentration, %

0.4

0.5

FIGURE 5.6 Porous nitrocellulose substrates. Apparent contact angle hysteresis variation with SDS concentration. Nitrocellulose membranes of different average pore sizes. Open symbols correspond to the advancing contact angle, θa. The same ﬁlled symbols correspond to the hydrodynamic receding contact angle, θrh.

It was found previously in this section that the advancing and hydrodynamic receding contact angles are strongly dependent on the SDS concentration on porous nitrocellulose membranes. Next, results are presented on measurements of static advancing and static receding contact angles on smooth nonporous nitrocellulose substrate for different SDS concentrations. The idea is to compare hysteresis of contact angles on the smooth nonporous nitrocellulose substrate with the hysteresis contact angles obtained earlier on porous nitrocellulose substrates at corresponding SDS concentrations. Static advancing or receding contact angle values were obtained using the experimental procedure described in Chapter 4, Section 4.5 (see Figure 4.19). The droplet was pumped using the syringe, and dynamics of the droplet spreading were monitored. The ﬁnal value of the contact angle under this experimental condition was equal to the static value of the advancing contact angle. The static advancing contact angle of pure water on nonporous nitrocellulose substrate was found approximately to be equal to 70˚. The static advancing contact angle decreases with the increase of SDS concentration (Figure 5.7). This trend continues until the CMC is reached. At concentrations above the CMC, the advancing contact angle remains constant and approximately equals 35˚. The receding contact angle values were obtained using the same experimental setup (Figure 4.19). In this case the contact angle dynamics were investigated using linearly decreasing droplet volume. The nonzero value of the static receding contact angle was found only in the case of pure water droplets. In all other cases (even at the smallest SDS concentrations used 0.025%), the static receding contact angle was found equal to zero © 2007 by Taylor & Francis Group, LLC

402

Wetting and Spreading Dynamics 70 θa Advancing contact angle θr Receding contact angle

Contact angle, degree

60 50 40 30 20 10 0 0.0

0.2

0.4 0.6 SDS concentration, %

0.8

1.0

FIGURE 5.7 Nonporous nitrocellulose substrate. Advancing and receding contact angles variation with SDS concentration. Open symbols correspond to the static advancing contact angle, θa. Filled symbols correspond to the static receding contact angle, θr .

in all the concentration ranges used: from 0.025% (ten times smaller than CMC) to 1% (ﬁve times higher than CMC). Both static receding and static advancing contact angles on smooth nonporous nitrocellulose substrate against SDS concentration are presented in Figure 5.7. The results highlight a linear decline of the static advancing contact angle from 70o at 0% SDS (pure water) to approximately 35˚ at the value of the CMC (2.4%); after that, the static advancing contact angle reaches a steady value that remains constant irrespective of further increase in the SDS concentration. In contrast to this, the static receding contact angle is approximately equal to 45˚ for the pure water and is equal to zero in the presence of SDS even at concentrations as low as 0.025%. Comparison of Figure 5.6 and Figure 5.7 shows: •

•

The advancing contact angle dependence on SDS concentration on porous nitrocellulose substrates is signiﬁcantly different from the static advancing contact angle dependence on nonporous nitrocellulose substrates. This means that, in the case of porous substrates, the inﬂuence of both the hydrodynamic ﬂow caused by the imbibition into the porous substrate and the substrate roughness change signiﬁcantly the advancing contact angle. The hydrodynamic receding contact angle in the case of the porous substrates has nothing to do with the hysteresis of the contact angle and is completely determined by the hydrodynamic interactions in a way similar to the complete wetting case in Section 4.3.

© 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

403

Conclusions Experimental investigation were carried out on the spreading of small drops of aqueous SDS solutions (capillary regime of spreading) over dry nitrocellulose membranes (permeable in both normal and tangential directions) in the case of partial wetting. Nitrocellulose membranes were chosen because of their partial hydrophilicity. The time evolution was monitored for the radii of both the drop base and the wetted area inside the porous substrate. The total duration of the spreading process was subdivided into three stages: •

• •

The ﬁrst stage, when the drop base expands until the maximum value of the drop base is reached; the contact angle rapidly decreases during this stage. The second stage, when the radius of the drop base remains constant and the contact angle decreases linearly with time. The third stage, when the drop base shrinks and the contact angle remains constant.

The wetted area inside the porous substrate expands during the whole spreading process. Appropriate scales were used with a plot of the dimensionless radii of the drop base, the wetted area inside the porous substrate, and the contact angle on the dimensionless time. Experimental data presented in this section show: • • •

The overall time of the spreading of drops of SDS solution over dry, thin, porous substrates decreases with the increase of surfactant concentration. The difference between advancing and hydrodynamic receding contact angles decreases with the surfactant concentration increases. The constancy of the contact angle during the third stage of spreading has nothing to do with the hysteresis of contact angle but is determined by hydrodynamic reasons, and the spreading behavior becomes similar to the case of the complete wetting.

5.2 SPONTANEOUS CAPILLARY IMBIBITION OF SURFACTANT SOLUTIONS INTO HYDROPHOBIC CAPILLARIES In this section, a theory is developed to describe a spontaneous imbibition of surfactant solutions into hydrophobic capillaries, which takes into account the micelle disintegration and solution concentration reduction close to the moving meniscus as a result of adsorption, as well as the surface diffusion of surfactant molecules. The theory predictions are in good agreement with the experimental investigations on the spontaneous imbibition of the nonionic aqueous surfactant © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

solution, Syntamide-5, into hydrophobized quartz capillaries [2,3]. Also, a theory of the spontaneous capillary rise of surfactant solutions in hydrophobic capillaries is presented, which connects the experimental observations with an adsorption of surfactant molecules in front of the moving meniscus on the bare hydrophobic interface [4]. Pure water does not penetrate spontaneously into hydrophobized quartz capillaries; however, surfactant solutions penetrate spontaneously, and the penetration rate depends on the concentration of surfactant. Both the air–liquid interfacial tension, γ, and the contact angle of the moving meniscus, θa, are concentration dependant, where a subscript a indicates the advancing contact angle. It is obvious that adsorption of surfactant molecules behind the moving meniscus results in a decrease of the bulk surfactant concentration from the capillary inlet in the direction of the moving meniscus. However, as we show in this section, the major process, which determines penetration of surfactant solutions into hydrophobic capillaries or spreading of surfactant solutions over hydrophobic substrates, is the adsorption of surfactant molecules onto a bare hydrophobic substrate in front of the moving three-phase contact line. This process results in a partial hydrophilization of the hydrophobic surface in front of the meniscus or drop, which, in its turn, determines spontaneous imbibition or spreading. It is easy to understand why the adsorption in front of the moving meniscus on a hydrophobic substrate is so vital in the case of the hydrophobic substrate. Let us consider the very beginning of the imbibition process, when a meniscus of a surfactant solution touches, for the ﬁrst time, an inlet of the hydrophobic capillary. The contact angle, θa, at this moment, is bigger than π/2, and the liquid cannot penetrate into the hydrophobic capillary. Solid–liquid and liquid–air interfacial tensions, γsl and γ, respectively, do not vary with time on the initial stage because the adsorption of surfactant molecules onto these surfaces is a fast process as compared with the rate of imbibition. The only interfacial tension that can vary is the solid–air interfacial tension, γsv . If the adsorption on the solid–air interface does not occur, then the spontaneous imbibition into a hydrophobic capillary cannot take place spontaneously because the advancing contact angle remains above π/2. However, if the adsorption of surfactant molecules on the bare hydrophobic surface in a vicinity of the three-phase contact line takes place, then the solid–air interfacial tension, γsv , grows with time. After some critical surface adsorption, Γsvcr , is reached, the advancing contact angle reaches π/2. Only after that can the spontaneous imbibition process start. This consideration shows that there is a critical bulk concentration, C*, below which Γsv remains below its critical value, Γsvcr , and the spontaneous imbibition process does not take place. Let us consider expression (1.2) from Section 1.1 in Chapter 1 for the excess free energy, Φ, of the droplet on a solid substrate: 2 Φ = γS + PV e + πR ( γ sl − γ sv ),

© 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

405

where S is the area of the liquid–air interface; P = Pa – Pl is the excess pressure inside the liquid, Pa is the pressure in the ambient air, and Pl is the pressure inside the liquid; the last term on the right-hand side gives the difference between the energy of the part of the bare surface covered by the liquid drop as compared with the energy of the same solid surface without the droplet. This expression shows that the excess free energy decreases if (1) the liquid– vapor interfacial tension, γ, decreases, (2) if the liquid–solid interfacial tension, γsl , decreases, and (3) the solid–vapor interfacial tension, γsv , increases. Let us assume that, in the absence of the surfactant, the drop forms an equilibrium contact angle above π/2. If the water contains surfactants, then three transfer processes take place from the liquid onto all three interfaces: surfactant adsorption at both (1) the inner liquid–solid interface, which results in a decrease of the interfacial tension, γsl , (2) the liquid–vapor interface, which results in a decrease of the interfacial tension, γ, and (3) transfer from the drop onto the solid–vapor interface just in front of the drop. As we already noticed in the foregoing text, all three processes result in decrease of the excess free energy of the system. However, adsorption processes (1) and (2) result in a decrease of corresponding interfacial tensions, γsv and γ; but the transfer of surfactant molecules onto the solid–vapor interface in front of the drop results in an increase of local free energy; however, the total free energy of the system decreases. That is, surfactant molecule transfer (3) goes via a relatively high potential barrier and, hence, goes considerably slower than adsorption processes (1) and (2). Hence, they are fast processes as compared with the third process (3). In the case of partial wetting, the capillary imbibition in the horizontal direction proceeds according to the following dependency:

=

Rγ cos θa t, 2η

(5.7)

where l is the length of the part of the capillary ﬁlled with the liquid, R is the radius of the capillary, η is the liquid viscosity, the advancing contact angle, θa , is below π/2: 0 < θa < π/2, and t is time. Pure water does not penetrate spontaneously into hydrophobic capillaries, and shows the advancing contact angle, θa > π/2. This means that the liquid can only be forced into the capillary. However, the advancing contact angle is a decreasing function of the surfactant concentration, and at some critical concentration, C*, is equal to π/2. This means that above C*, the surfactant solution penetrates spontaneously into hydrophobic capillaries. In the case of the imbibition of surfactant solutions into hydrophobic capillaries, the penetration is controlled by both the surfactant molecules transfer and the liquid viscosity, according to Equation 5.7. In this case the aim is to reveal the mechanism of the penetration and to determine the concentration of surfactant molecules near the meniscus Cm < C0, where C0 is the surfactant concentration © 2007 by Taylor & Francis Group, LLC

Wetting and Spreading Dynamics 3

90 3 1

2

89

2

88

1

87 0

C∗

0.1

γ cos θa

Advancing contact angle, θa degree

406

0 0.2

C, %

FIGURE 5.8 The inﬂuence of concentration of aqueous surfactant solutions (Syntamide-5, molecular weight 420) on the advancing contact angle, θa (curve 1) and γ cos θa (curve 2) measured on a ﬂat hydrophobized quartz surface. C* marks the critical surfactant concentration, below which the surfactant solution does not spread. Broken line (3) according to Equation 5.12.

at the capillary inlet. Figure 5.8 shows that, in the case of Syntamid-5, the advancing contact angle, θa, exceeds 90o at concentration C*, which is slightly under 0.05%. In the following text we show that both the bulk diffusion and the surface diffusion of surfactant molecules (including that on the unwetted portion of the capillary in front of the moving meniscus) play an important role, and a theory is presented for this case.

THEORY As we already mentioned in the introduction to this chapter, we know surprisingly little about the behavior of surfactant solutions in the vicinity of the three-phase contact line. That is why in this chapter we are using Young’s equation for theoretical treatment. We showed in Chapter 2 that the equation does not have a ﬁrm theoretical basis. Let us consider a dependency of Ψ(Cm) = γ(Cm) cosθa(Cm) on the concentration of surfactant, Cm, on the moving meniscus. According to the triangle rule, it can be calculated as Ψ(Cm ) = γ sv (Cm ) − γ sl (Cm ),

(5.8)

where γsv and γsl are solid–vapor and solid–liquid interfacial tensions. According to Antonov’s rule, the dependency of two interfacial tensions on the concentration can be presented as   Γ  Γ  Γ ∞ Γ+ γ sv (Cm ) = γ 0sv  1 − ∞+  + γ sv ; γ sl (Cm ) = γ 0sl  1− ∞−  + γ sl∞ ∞− , ∞  Γ   Γ  Γ Γ

© 2007 by Taylor & Francis Group, LLC

(5.9)

Dynamics of Wetting or Spreading in the Presence of Surfactants

407

where superscripts 0 and ∞ mark zero and the complete coverage of hydrophobic adsorption sites, respectively; subscripts + and – mark adsorption just behind and just in front of the moving meniscus, respectively; and Γ is the surface adsorption of surfactant molecules. Note that adsorption of surfactant molecules results in a decreasing of solid–liquid interfacial tension, that is, γ 0sl − γ ∞sl > 0. However, adsorption of surfactant molecules on the bare hydrophobic interface in front of the moving meniscus results in a local increase of the solid–vapor interfacial tension, that is, ∞ γ 0sv − γ sv < 0. The initial contact angle on the bare hydrophobic interface is assumed to be bigger than π/2, that is, γ 0sv − γ sl0 < 0. It is assumed in the following text that both adsorption isotherms are linear functions of the surfactant concentration below CMC (which is the only case considered below) and remain constant above CMC. This means that Γ − = GslCm ,

(5.10)

at concentrations below CMC. Both a spontaneous imbibition and a spontaneous capillary rise into hydrophobic capillaries are sufﬁciently slow processes, that is, we assume in the following text a condition of local equilibrium on the moving three-phase contact line. According to this assumption, the equality of chemical potentials of adsorbed surfactant molecules should be satisﬁed across the three-phase contact line, that is, ln Γ − + Φ sl = ln Γ + + Φ sv , where Γ- and Γ+ are jumps of adsorption across the meniscus surface (Figure 5.9), and ΦSL, ΦSV (in kT units) are corresponding values of the energy of surfactant molecules at solid–water and solid–air interfaces, respectively. From the latter equation we conclude that Γ+ =

Γ− . exp Φ sv − Φ sl

(

)

It is obvious that Φsv is higher than Φsl, and hence, Γ+ < Γ– . The relation can be rewritten using Equation 5.10 ast Γ + = GsvCm , Gsv =

Gsl . exp Φ sv − Φ sl

(

)

(5.11)

Substitution of Equation 5.9 through Equation 5.11 into Equation 5.8, and having in mind the latter inequalities, we can conclude, after rearrangements, that

(

)

Ψ(Cm ) = α Cm − C* ,

© 2007 by Taylor & Francis Group, LLC

(5.12)

408

Wetting and Spreading Dynamics

c0

2

c(x) cm

a0 a(x)

a–

a+ x

0

(t)

FIGURE 5.9 Distribution of surfactant concentrations along the capillary length during a spontaneous imbibition process: volume concentration, C, and surface concentration, a = 2Γ/R.

where

(

)

(

)

 G α0 G ∞ 0 α =  sv∞ γ sv − γ sv + ∞sl γ 0sl − γ ∞sl  > 0,, C* = , α 0 = γ 0sl − γ 0sv > 0. α Γ  Γ According to Equation 5.12 Ψ(Cm), dependency should be a linear function of concentration at Cm > C*, which is in good agreement with experimental observations (line 3 in Figure 5.8 at concentrations C > C*) in a range of surfactant concentration under consideration in this section. We now try to solve theoretically the problem of a spontaneous imbibition of surfactant solutions into hydrophobic cylindrical capillaries, taking into account the transfer and the surface diffusion of surfactant molecules as well as adsorption on the bare hydrophobic surface in front of the moving meniscus. The location of the moving meniscus in the capillary is l(t) (Figure 5.9). The transfer of surfactant molecules in the ﬁlled portion of the capillary is described by the convective diffusion equation ∂C (t , x , r ) ∂2C (t , x.r ) 1 ∂  ∂C (t , x , r )  ∂ =D +D r  − ∂x v(r )C (t , x , r , ∂t r ∂r  ∂r ∂x 2

(

)

where C(t,x,r) is the local concentration of surfactant; D is the diffusion coefﬁcient; t, x, and r are time, axial, and radial coordinates, respectively; and v(r) is the axial velocity distribution. Integration of this equation over radius from 0 to R, where R is the capillary radius, results in © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

409

  R R  ∂C (t , x , r )  ∂2 ∂  rC (t , x , r )dr  = D 2  rC (t , x , r )dr  + R  D  ∂r ∂t   ∂ x    r=R   0 0

∫

∫

 R ∂  rv(r )C (t , x , r )dr  − ∂x   0

∫

The second term on the right-hand side of the equation is equal to −D

∂C (t, x, r ) ∂Γ ∂2 Γ = − Dsl 2 , ∂r ∂t ∂x r=R

where Dsl is the surface diffusion coefﬁcient over the ﬁlled portion of the capillary. Hence, the two equations result in R  R   ∂Γ ∂2Γ  ∂ ∂2  rC (t , x , r )dr  = D 2  rC (t , x , r )dr  − R  − Dsl 2  ∂t  ∂x  ∂x     ∂t 0 0

∫

∫

R  ∂  rv(r )C (t , x , r )dr  − ∂x   0

∫

A characteristic time scale of the equilibration of the surfactant concentration in a cross section of the capillary, τ ~ R2/D ≈ 0.1 sec, if we use for estimations R ~ 10 µm and D ~ 10–5 cm2/sec. A characteristic time scale of the spontaneous capillary imbibition or rise into hydrophobic capillaries is much bigger than 0.1 sec (see Figure 5.10 and Figure 5.11). This means that the surfactant concentration is constant in any cross section of the capillary and depends only on the position, x (Figure 5.9), that is, C = C(t, x). We also assume that the adsorption equilibrium in any cross section is also reached. Taking this into account, the equation can be rewritten after both sides are divided by R2/2 as: ∂(a + C ) ∂2C ∂2a ∂C , =D 2 +D −v 2 sl ∂t ∂x ∂x ∂x

0 < x < l(t),

(5.13)

where D and Dsl are diffusion coefﬁcients of surfactant molecules in the volume and over the wetted capillary surface; v = dl/dt is the meniscus velocity; and a( x , t ) = where Fsl = (2/R) Gsl . © 2007 by Taylor & Francis Group, LLC

2 GslC ( x , t ) = FslC ( x , t ), R

(5.14)

Wetting and Spreading Dynamics

5

ᐉ, mm

5 10 9 8 7 6 5 4 3 2 1

4

4 3

3

ᐉ, cm

410

2 1 2 0

1

2

1

3 t

min1/2

FIGURE 5.10 The time evolution of the imbibition length l (mm) with time, t (min) for aqueous solutions of Syntamide-5 in a horizontal hydrophobized quartz capillary, R = 16 µm. (1) C0 = 0.05%; (2) C0 = 0.1%; (3) C0 = 0.4%; (4) C0 = 0.5%; (5) C0 = 1%.

25

ᐉ, mm

20 15 10 5

0

15

30

45

60

75

90

t, min1/2

FIGURE 5.11 Spontaneous capillary rise in a vertical hydrophobized quartz capillary (R = 11 µm), Syntamid-5 surfactant solution (C0 = 0.1 %) [3]. Time evolution of the imbibition length, l (mm), on time, t(min).

Concentration below CMC Let C(x, t) and a(x, t) be the local surfactant concentrations in the bulk solution and in the adsorbed state on the capillary surface. A constant surfactant concentration C(0, t) = C0 < CMC is kept at the capillary inlet. In this case the surfactant transport in the ﬁlled portion of the capillary obeys Equation 5.13. © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

411

Substitution of expression (5.14) into Equation 5.13 results in

(1 + Fsl )

∂C ∂2C dl ∂C = Def 2 − , ∂t dt ∂x ∂x

0 < x < l(t),

(5.15)

where Def = (D + Fsl Dsl) is the effective diffusion coefﬁcient. On the nonwetted portion of the capillary surface, l(t) < x, only surface diffusion takes place, that is: ∂a ∂2a = Dsv 2 , ∂t ∂x

(5.16)

where Dsv is the diffusion coefﬁcient of surfactant molecules on the nonwetted hydrophobic capillary surface. Equation 5.15 and Equation 5.16 are solved in the following text using the following boundary and initial conditions: C(0,t) = C0; a(∞,t) = 0,

(5.17)

l(0) = 0; a(x, 0) = 0.

(5.18)

The following condition of the mass balance on the moving meniscus surface should be satisﬁed:  ∂C  ∂a  ∂a  dl  Dsv ∂x  −  D ∂x + Dsv ∂x  = (a− − a+ ) dt , l+ l−

(5.19)

where l– and l+ represent two points located on the opposite sides of the meniscus: on the liquid phase side, l–, and on the unwetted side in front of the moving meniscus, l+ (Figure 5.9). Condition (5.19) expresses the conservation of mass at the moving meniscus and the moving three-phase contact line. As we already mentioned, the imbibition process is the slow one, and the duration is shown in Figure 5.10 and Figure 5.11. It allows us to assume in the following text a condition of local equilibrium on the moving three-phase contact line. Hence, we conclude in the same way as in Equation 5.11: a+ =

a− . exp Φ + − Φ −

(

)

The energy of the surfactant molecule on a bare hydrophobic substrate in front of the moving meniscus, Φ+, is higher than the corresponding energy in the aqueous solution, Φ–, and, hence, a+ < a–. © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

Equation 5.15 and Equation 5.16 with boundary (5.19) and initial conditions (5.17) to (5.18) have a solution only if the concentration on the moving meniscus remains constant; that is, Cm = const. This corresponds to the experimentally observed law of the spontaneous imbibition, l = K t , where the constant K is to be determined. The solution of Equation 5.15 and Equation 5.16 is tried in the following text in the following form C = C(ξ), a = a(ξ), where ξ=

x

.

t

Transformations of Equation 5.15 and Equation 5.16 yield: −

d 2C K dC ξ dC (1 + Fsl ) = Def − , 2 dξ d ξ2 2 d ξ

(5.20)

d 2a ξ da = Dsv 2 . 2 dξ dξ

(5.21)

−

Solution of these equations has the following form: ξ

 (1 + Fsl )ξ 2 K ξ C (ξ) = A1 exp  − + 4 Def 2 Def  0

∫

  d ξ + B1. 

(5.22)

ξ

 ξ2  a(ξ) = A2 exp  − d ξ + B2 . 4 Dsv   K

∫

(5.23)

Integration constants A1, A2, B1, and B2 should be determined using initial conditions (5.19). This results in A1 = −

C0 − Cm K

∫ 0

 (1 + Fsl )ξ 2 K ξ exp  − + 4 Def 2 Def 

A2 =

© 2007 by Taylor & Francis Group, LLC

− FsvCm ∞

 ξ2  exp  −  dξ  4 Dsv  K

∫

  dξ 

; B1 = C0 ,

; B2 = FsvCm .

(5.24)

(5.25)

Dynamics of Wetting or Spreading in the Presence of Surfactants

413

Substitution into boundary condition Equation 5.19 expressions from Equation 5.24 and Equation 5.25 yields the following transcendental equation:  (1 − Fsl ) K 2   C0    C − 1 Def exp  4 D K   m ef ( Fsl − Fsv ) = K − 2 2  (1 + Fsl )ξ Kξ  exp  − +  dξ 4 Def 2 Def   0

∫

 K2  Fsv Dsv exp  −  4 Dsv   − ∞ .  ξ2  exp  −  dξ 4 Dsv   K

(5.26)

∫

K value governs the imbibition rate according to Equation 5.7 and can be rewritten using Equation 5.12 as

K=

r α(C m − C * ) . 2η

(5.27)

Substitution of this equation into Equation 5.26 gives the required equation for the determining of the unknown concentration on the moving meniscus, Cm. Equation 5.26 and Equation 5.27 give the solution of the problem under consideration. To solve Equation 5.26, the following coefﬁcients should be known: diffusion coefﬁcients, D, Dsl , and Dsv; adsorption constants, Gsl and Gsv; capillary radius, R; solution viscosity, η; concentration at the capillary inlet, C0; and the coefﬁcient α in Equation 5.12 or Equation 5.27. Numerical analysis of the ﬁnal equation for K shows that its values increases as R, C0, D, and Dsl increase, or Dsv decreases. This is due to the fact that the high surface mobility of the surfactant on the unwetted portion of the capillary reduces the concentration on the meniscus, Cm, thereby inhibiting the imbibition. Concentration above CMC In the case of the surfactant concentration at the capillary inlet, C0, above CMC, diffusion and adsorption are accompanied by the destruction of the micelles. If the surfactant concentration at the capillary inlet is higher than CMC, then the total surfactant concentration, C, can be presented as C = Cmol + CM, where Cmol and CM are concentrations of free surfactant molecules and molecules inside micelles, respectively. Concentration of free molecules, Cmol, remains approximately constant above CMC and equal to CMC (Cc below). This means that (1) decrease in the surfactant concentration goes through disintegration of micelles, © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

whereas the concentration of the free surfactant molecules remain constant, and (2) the wetting depends on the concentration of the free molecules, and hence, remains independent of the concentration unless the concentration is below CMC: γ(Cc)cosθ(Cc) = ϕc = const. This means that the K value, which determines the imbibition rate, is constant and is equal to   Kc = r α  Cc − C  2 η = const ,  *

(5.28)

until the concentration at the meniscus, Cm, is higher than CMC. The values of Kc for C0 > CMC~1% obtained in experiments with Syntamide5 [2,3] were of the order of 10–1, which corresponds to the contact angle (a = 89˚, that is, only slightly different from 90˚). Unlike the case when the concentration at the capillary inlet is below CMC, the meniscus moves with velocity l = Kc t , where Kc is given by Equation 5.28 and does not vary with time. In this case, the surfactant adsorption on the capillary surface is accompanied by a continuous decrease of concentration, Cm, near the meniscus from Cm = C0 > CMC at the beginning of the imbibition process at t = 0 to Cm = CMC or Cm = Cc at the end of this ﬁrst fast stage. After Cm reaches CMC, the condition Kc = const is no longer valid. The K value decreases below Kc, and the imbibition rate slows down. After Cm = CMC is reached on the meniscus, a further reduction of concentration, Cm, causes separation of the micelle front (where C = CMC) from the meniscus surface. The micelle front movement is governed, as shown in the following text, by the same law lM = KM t , where KM < K. The further stage of impregnation occurring when Cm < Cc is described in a similar way as in the previous section: the concentration no longer varies but remains const and smaller than CMC. The variations of the impregnation process associated with the abovementioned phenomenon is illustrated in Figure 5.10. A sharp change in the rate of impregnation at some distance l = lc is indeed observed for Syntamide-5 solutions at C0 > CMC. Values of lc, which correspond to the ﬁrst fast stage of the imbibition process, are estimated in the following text. As mentioned previously, the ﬁrst fast stage of the imbibition is determined by the dissociation of micelles close to the moving meniscus. Let us assume an adsorption of micelles on the meniscus according to Reference 8, which is adopted according to the linear law Γ = GM (Cm – Cc), where Cm – Cc is the micelles concentration at C0 > Cc, and GM is the corresponding adsorption constant. Diffusion of micelles is neglected in the following discussion because of the short duration of the ﬁrst stage. The mass balance on the moving meniscus during the ﬁrst fast stage of the imbibition is ( Fsl − Fsv )Cc

dl d (Cm − Cc ) = −G M , Cm (0) − Cc = C0 − Cc , dt dt

© 2007 by Taylor & Francis Group, LLC

(5.29)

Dynamics of Wetting or Spreading in the Presence of Surfactants

415

where (Fsl – Fsv)Cc = const is the limiting adsorption on solid surface (micelles do not adsorb); the difference Cm – Cc is equal to the micelle concentration. The left-hand side of Equation 5.29 characterizes the surfactant adsorption rate on the newly wetted surface, and the right-hand side the micelle disintegration rate from the adsorbed layer of micelles. The solution of Equation 5.29 under the initial condition t = 0, Cm – Cc = C0 – Cc has the following form:

(

)

Cm (t ) − Cc = C0 − Cc −

(

)

2Cc Gsl − Gsv Kc t R GM

.

(5.30)

The left-hand side of the equation vanishes as all the micelles near the meniscus disintegrate, i.e., the concentration Cm reduces to CMC. Equation 5.30 allows determining the instant t = tc when Cm = Cc, which is the end of the ﬁrst fast stage of the imbibition. Using this condition and Equation 5.30, the length of the fast imbibition can be determined as C  rGM lc = Kc tc =  0 − 1 .  Cc  2 Gsl − Gsv

(

)

(5.31)

For R ~ 2*103 cm, C0 = 2%, Cc = 0.1%, lc = 0.5 cm (in agreement with experimental observations), we get

(

GM ~ 25 Gsl − Gsv

)

in an agreement with Reference 8. The ﬁrst fast stage of the imbibition is followed by the second slower stage. Now, concentration on the moving meniscus, Cm, is below CMC. Two regions can now be identiﬁed inside the capillary: the ﬁrst region, from the capillary inlet to some position that we mark as lM (t), where concentration is above CMC and the solution includes both micelles and individual surfactant molecules; the second region, from lM (t) to l(t), where concentration is below CMC and only individual molecules are transferred. Concentration is equal to CMC at x = lM (t). Consideration in the second region, lM (t) < x < l(t), is similar to that at concentration below CMC. That is why only the transport in the ﬁrst region is considered in the following text. Inside the ﬁrst region, 0 < x < lM(t), concentration of free surfactant molecules is constant and equal to CMC [9]. Hence, the transfer is determined by the diffusion of micelles and convection of all molecules. As mentioned in the preceding text, total concentration, C = Cmol + CM , and Cmol remain constant and equal to CMC; hence, © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

∂C ∂2C dl ∂C = DM 2 − , ∂t dt ∂x ∂x

(5.32)

where DM is the diffusion coefﬁcient of micelles, and C = Cc +CM is the total concentration. Adsorption on membrane pores is determined by the concentration of the free molecules, which is constant in the ﬁrst region and so is the adsorption. It is the reason why the diffusion of adsorbed molecule in the ﬁrst region is omitted in Equation 5.32. Transfer of surfactant molecules in the second region (micelles-free region) is described by Equation 5.15. Boundary conditions on the moving boundary between the ﬁrst and second regions, lM(t), are as follows:  ∂C  DM   ∂x  x =l

M−

 ∂C  = Def   ∂x  x =l

, C (l M , t ) = Cc .

(5.33)

M+

As before, we assume that l (t ) = K t , l M (t ) = K M t ,

(5.34)

where K is given by Equation 5.27, that is, expressed via unknown concentration on the moving meniscus, Cm, and KM is a new unknown constant. Let a similarity variable be introduced now in the same way as in the case of concentration below CMC, that is, ξ = x/ t in Equation 5.32, Equation 5.15, and Equation 5.16. Using boundary conditions (5.33) and (5.19), the following system of two nonlinear algebraic equations can be deduced:  (1 − Fsl ) K 2   K2   Cc  1 D − exp F D exp   −  ef sv sv  C  4 Dsv  K  4 Def  m  ( Fsl − Fsv ) = K − ∞ . 2  (1 + Fsl )ξ 2 K ξ   ξ2  exp  − + exp  −  dξ  dξ 4 Def 2 Def  4 Dsv     K K

∫

∫

M

(5.35)

DM

 K2  (C 0 − C c ) exp  M   4DM  KM

∫ 0

  1  ξ2 − K M ξ  d ξ exp  −     2 DM  2

= Def

 (1 − Fsl ) K 2  (C c − C m ) exp     4 Def K

 (1 + Fsl )ξ 2 Kξ  exp  − +  dξ 4 Def 2 Def   KM

.

∫

Two unknown values in this system of equations are concentrations on the moving meniscus, Cm, and unknown constant, KM. It is necessary to remember that K is expressed via Cm according to Equation 5.27. © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

417

It is possible to show, using Equation 5.35, that KM < K; that is, the border between the ﬁrst and the second regions moves slower than the meniscus. The diffusion coefﬁcient of micelles is much smaller than that of individual molecules. Thus, actually, the system of Equation 5.35 includes a small parameter Dm /Def << 1, which means that the second equation can be omitted in the zero approximation, and the constant, K, is only slightly different from the same value in the case of concentration below CMC. That is, during the second stage, the rate of spontaneous imbibition is only slightly higher than in the case when concentration is below CMC. Experimental data in Figure 5.10 conﬁrm this conclusion. Now we can conclude that the two-stage process of the imbibition (as presented in Figure 5.10) takes place only if the concentration on the capillary inlet is above CMC. It is necessary to emphasize that the duration of the fast stage is the shorter the thinner the capillary is. K values during the second stage of the imbibition are only slightly higher than in the case where concentration is below CMC. Spontaneous Capillary Rise in Hydrophobic Capillaries A further conﬁrmation of the adsorption of surfactant molecules in front of the moving meniscus on the bare hydrophobic substrate is a phenomenon of the spontaneous capillary rise of surfactant solutions in hydrophobic capillaries. Let us consider a spontaneous capillary rise of surfactant solutions in hydrophobic capillaries. Figure 5.11 shows the results of one of the capillary rise experiments with Syntamide-5 solution in a vertical hydrophobized quartz capillary [10]. The observed time evolution of the imbibition length, l(t), follows l(t) = K t dependency at the initial stage of the process. The value of K determined from the slopes of the l/ t dependencies correspond to the advancing contact angle value, θa, being only a few seconds less than 90˚. At such θa values the capillary rise would be expected to stop as soon as the liquid reached a height of lmax = 10–3 cm. However, it does not stop at this height but goes up to a height of 3–4 cm with almost constant K. The only explanation of this phenomenon is that the meniscus rises in the capillary following the surface diffusion front of surfactant molecules, which hydrophilizes the bare hydrophobic capillary surface in front of the moving meniscus. At each position, l, the meniscus curvature must satisfy the following equilibrium condition: 2Ψ(Cm ) = ρgl (t ), R

(5.36)

where Ψ(Cm) = γ(Cm) cosθa(Cm), ρ is the density of the solution, and g is the gravity acceleration. As θa is very close to π/2 according to Equation 5.1, we can use a linear dependency (5.12): Ψ(Cm ) = α(C m − C* ).

© 2007 by Taylor & Francis Group, LLC

418

Wetting and Spreading Dynamics

Using Equation 5.12 we can rewrite Equation 5.36 as l (t ) =

2α (Cm − C* ). ρgR

(5.37)

During the initial stage of the capillary rise, l(t) = K t (see Figure 5.11). This is possible only if Cm = C* + B t , and B = K

ρgR , 2α

(5.38)

where the constant K is to be determined. Equation 5.38 shows that the case under consideration is governed by a completely different mechanism as compared with the case of the horizontal imbibition (where Cm remains constant over time). In the case of the spontaneous capillary rise in hydrophobic capillaries, Cm(t) does not remain constant but must increase as the capillary rise progresses. The comparison of Figure 5.11 and Figure 5.10 shows that the time scale of the spontaneous capillary rise is around 100 times bigger than the corresponding time scale in the case of the capillary imbibition into horizontal capillaries. The maximum height of the capillary rise, lmax, is reached after the concentration on the meniscus, Cm, becomes equal to the concentration at the capillary entrance, C0. After that the capillary rise stops. Using Equation 5.37, lmax is determined as lmax =

2α (C0 − C* ). ρgR

(5.39)

Thus, the experimental observation presented in Figure 5.11 corresponds to l(t) << lmax, that is, the initial stage of the capillary rise. In the following text the problem of the spontaneous capillary rise of surfactant solutions in hydrophobic capillaries is considered in the case when concentration at the capillary inlet is below CMC. In this case, the transport of surfactant molecules is described by Equation 5.15 and Equation 5.16, and boundary conditions (5.17) through (5.19). The substantial difference from the spontaneous capillary imbibition is that now the relation between l(t) and the concentration on the moving meniscus, Cm, is given by relation (5.37), which shows that Cm is an unknown function of time. Using these equations and boundary conditions, we show in the following text that l(t) dependency on time can be calculated, and it is proportional to the square root of time at the initial stage of the capillary rise (see Appendix 1 for details). The solution in Appendix 1 shows that, at the initial stage of capillary rise, l(t) develops as © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

l (t ) = K t , K = lmax

2κ , ω

419

(5.40)

where κ and ω are deﬁned in Appendix 1. This dependency agrees with experimental observations in Reference 3 (see Figure 5.11). At the ﬁnal stage of the capillary rise, l(t) levels off as  ω + 1 l (t ) = lmax  1 − , κt   exp According to Equation 5.39 and Figure 5.11, lmax ~ 35 mm in the experiment presented in Figure 5.11. The experimental value of Kexp in Equation 5.40, calculated according to Figure 5.11, is Kexp ~ 5·10–3 cm/sec1/2. In experiments presented in Figure 5.11, ω ~ 1. Using these estimations we conclude κ ~ 10–5 sec–1, which coincides with the value calculated in Appendix 1 according to Equation A1.11, if we assume that Fsv << Fsl, D ~ 10–5 cm2/sec, using estimation GSL ~ 10–5 cm [8], and the value of α is taken directly from Figure 5.8.

APPENDIX 1 To simplify a mathematical treatment of the problem, we disregard in the following discussion the diffusion of surfactant molecules in front of the moving meniscus. That means only the adsorption in front of the moving meniscus is taken into account. Under this simpliﬁcation, the process of the capillary rise in a hydrophobic capillary is governed by Equation 5.15 with boundary condition (5.19). The concentration of surfactant changes considerably only in close proximity to the moving meniscus. That means the surfactant concentration differs from the concentration at the capillary inlet, C0, in a narrow region between l(t)–δ(t) and l(t), where δ(t) is to be determined. At the boundary l(t)–δ(t), the concentration of surfactant is equal to the concentration at the capillary inlet, C0, and the derivative of the concentration at this point is zero (a smooth transition to the constant concentration). Let us introduce for the sake of convenience a new unknown function Z = C0 – C, which satisﬁes the following equation and boundary conditions: (1 + Fsl )

∂Z ∂2 Z dl ∂Z = Def 2 − , 0 < x < l (t ). ∂t dt ∂x ∂x

From Equation 5.37, C m = C* + Al (t ) , where A= © 2007 by Taylor & Francis Group, LLC

ρgR . 2α

(A1.1)

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Wetting and Spreading Dynamics

Boundary condition (5.19) takes the following form using the preceding equations:  ∂Z  dl Def   = Fsl − Fsv C* + Al . dt  ∂x  x =l

(

)(

)

(A1.2)

Other boundary conditions are: Z (l − δ) = 0, Z (l ) = C0 − C* − Al ,

(A1.3)

 ∂Z  = 0.  ∂x  x =l −δ

(A1.4)

Integration of Equation A1.1 after simple manipulations using boundary conditions (A1.3) and (A1.4) results in d dt

l

Def  ∂Z  Fsl dl  ∂x  + 1 + F C0 − C* − Al dt . sl sl x =l

(

∫ Zdx = 1 + F

l −δ

)

(A1.5)

In the following text we ﬁnd a solution that satisﬁes the integral balance Equation A1.5 and boundary conditions (A1.2) through (A1.4). The simplest solution, which satisﬁed boundary conditions (A1.3) through (A1.4), is as follows: 2

 l− x Z = C0 − C* − Al  1 − , δ  

(

)

(A1.6)

where both l(t) and δ(t) dependencies are to be determined. Substitution of this expression into boundary condition (A1.2) and Equation A1.5 gives, after some rearrangements, a system of two differential equations for the determination of two unknown dependencies l(t) and δ(t):

(

)

d 6 Def C0 − C* − Al 3Fsl dl + C0 − C* − Al   C0 − C* − Al δ  = 1 + Fsl δ 1 + Fsl dt  dt ,   dl 2 Def C0 − C* − Al (A1.7)  dt = ∆F δ C* + Al 

(

)

(

(

)

(

)

)

where ∆F = Fsl − Fsv > 0. The following initial conditions should be satisﬁed: l(0) = δ(0) = 0. © 2007 by Taylor & Francis Group, LLC

(A1.8)

Dynamics of Wetting or Spreading in the Presence of Surfactants

421

If the ﬁrst equation in (A1.7) is divided by the second equation, then an equation for δ(l) dependence can be obtained. This equation can be solved and the solution substituted into the second equation in system (A1.7), which gives the following equation for l(t) determination: du (1 − u)2 =λ ; u(0) = 0, dt u(ω + u)(2 + χ − u)

(A1.9)

where u = l /lmax , lmax = λ=

(

4 Def 1 + Fsl 2 3∆FFsvlmax

2α C* C0 − C* , ω = , ρgR C0 − C*

(

), χ =

)

2 ∆f C0 . Fsv C0 − C*

(

(A1.10)

)

These deﬁnitions show that χ >> 1; hence, Equation A1.9 can be rewritten as du (1 − u)2 λ =κ ; κ= , u ( 0 ) = 0. dt u(ω + u) 2+χ

(A1.11)

This equation can be easily solved, and the solution is as follows: u+

(ω + 1)u + (ω + 2) ln(1 − u) = κt. 1− u

(A1.12)

If u << 1 (initial stage of the process), then from Equation A1.12 we conclude that u=

2κ t. ω

(A1.13)

At the ﬁnal stage of the process, 1 – u << 1, Equation A1.12 gives u = 1−

ω +1 . κt

(A1.14)

5.3 CAPILLARY IMBIBITION OF SURFACTANT SOLUTIONS IN POROUS MEDIA AND THIN CAPILLARIES: PARTIAL WETTING CASE Let us now consider the imbibition of surfactant solutions into porous solid substrates, which are partially wetted by water. We shall see that this case is © 2007 by Taylor & Francis Group, LLC

422

Wetting and Spreading Dynamics

considerably different from that of hydrophobic porous media. To be concrete, we shall continue studying the imbibition in a cylindrical capillary whose walls are partially wetted by water, recalling that such a capillary can be used to model a porous medium. At variance with the case of hydrophobic capillaries, here, water can penetrate into the capillary even in the absence of surfactants on the moving meniscus. The presence of surfactant molecules on the moving meniscus lowers the contact angle, and hence a higher capillary pressure builds behind the meniscus. Consequently, the imbibition rate increases with the increase of concentration of surfactant molecules on the moving meniscus. The moving meniscus covers fresh parts of the capillary walls where surfactant molecules have not adsorbed yet. Thus, with the imbibition process, there is the simultaneous adsorption of surfactant molecules onto fresh parts of the capillary walls in a vicinity of the moving meniscus. The amount adsorbed is inversely proportional to the radius of the capillary, i.e., the thinner the capillary, the higher the adsorption is. On the other hand, the imbibition rate is lower in thinner capillaries (higher friction). This gives more time to diffusion to bring new surfactant molecules to cover the fresh part of the capillary walls. Thus, we have two competing, opposite, trends. Indeed, if the capillary radius is smaller than some critical value, then adsorption goes faster than the imbibition process, and all surfactant molecules are adsorbed on the capillary walls, leaving nothing for the meniscus where the concentration vanishes. In such circumstance, the imbibition rate of a surfactant solution becomes independent of the surfactant concentration in the feed solution with a value equal to that of pure water [11]. These theoretical conclusions are in agreement with experimental observations [11]. The kinetics of the capillary imbibition of aqueous surfactant solutions into hydrophobic capillaries has been investigated earlier in Section 5.2. It has been shown that the rate of imbibition is controlled by the adsorption of the surfactant molecules in front of the moving meniscus on the bare hydrophobic surface of the capillary. This process results in a partial hydrophilization of the surface of the capillary in front of the moving meniscus and provides the possibility for the aqueous surfactant solution to penetrate into the initially hydrophobic capillary. Therefore, no surfactant molecules on the meniscus, no imbibition. In the following text, the imbibition of surfactant solutions into the porous substrates, which are partially wetted by water, is considered. It is shown that the situation in this case is considerably different from the case of hydrophobic porous media.

THEORY The imbibition of aqueous surfactant solutions into a single cylindrical capillary with walls partially wetted by water is considered in this section. A single capillary is used as a model of a porous medium. The situation in this case is different from the case of hydrophobic capillaries (Section 5.2); water can penetrate into the capillary even in the absence of surfactant molecules on the moving meniscus. However, the presence of surfactant

© 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants R

C0 0

423

cm (t)

L

x

C0 Cm x

FIGURE 5.12 Imbibition of a surfactant solution in thin capillary; C0 — concentration at the capillary entrance; Cm — concentration on the moving meniscus; (t) — position of the moving meniscus.

molecules on the moving meniscus results in a lower contact angle (as compared with pure water), and hence, in a higher capillary pressure behind the meniscus. As a result, the imbibition rate increases with the concentration of surfactant molecules on the moving meniscus. The moving meniscus covers fresh parts of the capillary walls where surfactant molecules have not adsorbed yet. This means that the imbibition process is accompanied by the simultaneous adsorption of surfactant molecules onto fresh parts of the capillary walls in a vicinity of the moving meniscus. The adsorption is inversely proportional to the radius of the capillary; that is, the thinner the capillary, the higher the adsorption. On the other hand, the rate of imbibition is lower in thinner capillaries (higher friction). This gives more time for diffusion to bring in new surfactant molecules and cover the fresh part of the capillary walls. Therefore, there are two competing, opposite, trends. This means that if the capillary radius is smaller than some critical value, then adsorption goes faster than the imbibition process, and all surfactant molecules are adsorbed on the capillary walls; therefore, there is nothing left for the meniscus, and the concentration on the moving meniscus remains zero. This qualitative consideration shows that if the capillary radius is smaller than some critical value, then the rate of the imbibition of surfactant solutions remains independent of the surfactant concentration in the feed solution and equals that of pure water. This qualitative conclusion is justiﬁed in the following text using the theoretical consideration of the capillary imbibition of aqueous surfactant solutions into cylindrical capillaries whose walls are partially wetted by water. Let us consider an imbibition of surfactant solution from a reservoir with a ﬁxed surfactant concentration, C0 (feed solution), into a thin capillary with radius R << L (Figure 5.12), where L is the capillary length. The capillary walls are partially wetted by pure water (at zero concentration of surfactant), that is, γ (0) cos θa (0) = γ sv (0) − γ sl (0) > 0,

© 2007 by Taylor & Francis Group, LLC

(5.41)

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Wetting and Spreading Dynamics

where γ , γ sv , γ sl , θ a are the liquid–air, the solid substrate–vapor, the solid substrate– liquid interfacial tensions, and the advancing contact angle, respectively. All these values are concentration dependent. Let Cm be the bulk concentration of surfactant behind the moving meniscus; then, γ (Cm ) cos θa (Cm ) = γ sv (0) − γ sl (Cm ) > γ sv (0) − γ sl (0) = γ (0) cos θa (0).

(5.42)

It is assumed that the imbibition process goes sufﬁciently fast, and transfer of the surfactant molecules on the bare surface in front of the moving meniscus can be neglected because this process goes much slower (see Section 5.2). Hence, the solid–vapor interfacial tension, γsv, does not depend on the surfactant concentration and remains equal to its value at zero concentration. It is also taken into account in Equation 5.42 that γ sl (Cm ) is a decreasing function of the surfactant concentration. Equation 5.42 shows that Ψ(Cm ) = γ (Cm ) cos θa (Cm ) is an increasing function of the concentration, with the maximal value, Ψ max = Ψ (CCMC ), reached at CMC and the minimal value, Ψ min = γ sv (0) − γ sl (0), reached at zero surfactant concentration. Concentration below CMC Let the surfactant concentration at the capillary entrance be below CMC, C0 < CCMC. The transfer of surfactant molecules in the ﬁlled portion of the capillary is described by the convective diffusion equation as in Section 5.2, ∂C (t, x, r ) ∂2C (t, x.r ) 1 ∂  ∂C (t, x, r )  ∂ r +D =D  − ∂x v (r )C (t, x, r , ∂r r ∂r  ∂t ∂x 2

(

)

where C(t,x,r) is the local concentration of surfactant; D is the diffusion coefﬁcient; t, x, and r are time, axial, and radial coordinates, respectively; and v(r) is the axial velocity distribution. Integration of the preceding equation over radius from 0 to R, where R is the capillary radius, results in R  R   ∂C (t , x , r )  ∂2 ∂  rC (t , x , r )dr  = D 2  rC (t , x , r )dr  + R  D  ∂r ∂t   ∂ x    r=R 0  0 

∫

∫

R  ∂  rv(r )C (t , x , r )dr  . − ∂x   0

∫

© 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

425

The second term on the right-hand side of the equation is equal to −D

∂C (t, x, r ) ∂Γ ∂2 Γ = − Dsl 2 , ∂r ∂t ∂x r=R

where Dsl is the surface diffusion coefﬁcient over the ﬁlled portion of the capillary, and Γ is the surface concentration. Hence, the preceding two equations result in R  R   ∂Γ ∂2Γ  ∂ ∂2  rC (t , x , r )dr  = D 2  rC (t , x , r )dr  − R  − DSL 2  ∂t  ∂x  ∂x     ∂t 0 0

∫

∫

−

R  ∂  rv(rr )C (t , x , r )dr  . ∂x   0

∫

A characteristic time scale of the equilibration of the surfactant concentration in a cross section of the capillary is τ ~ R2/D ≈ 10–3 sec, if we use for estimations R~1 µm and D ~ 10–5 cm2/sec. A characteristic time scale of the spontaneous capillary rise into partially hydrophilic capillaries is around 10 sec (see Figure 5.15), which is much bigger than 10–3 sec. Hence, the surfactant concentration is constant in any cross section of the capillary and depends only on the position, x (Figure 5.12), that is, C = C(t, x). Taking this into account, the preceding equation can be rewritten, after both sides are divided by R2/2, as ∂2C ∂C 2 ∂Γ ∂C 2 ∂2Γ = D 2 + DSL −v , + 2 ∂x ∂t R ∂t R ∂x ∂x

0 < x < l(t),

(5.43)

where 2 v= 2 R

R

∫ rv(r )dr 0

is the averaged velocity (the same symbol is used for the averaged velocity as for the local one). The averaged velocity, v, is equal to the meniscus velocity, that is, v=

d , dt

(5.44)

where (t ) is the position of the moving meniscus. Substitution of Equation 5.44 into Equation 5.43 and neglecting the surface diffusion results in the following © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

equation, which describes the concentration proﬁle inside the ﬁlled portion of the capillary: ∂2C dl ∂C 2 ∂Γ ∂C + =D 2 − , R ∂t ∂t dt ∂x ∂x

0 < x < l(t).

(5.45)

As a result of adsorption on the capillary wall, the concentration of the surfactant molecules on the moving meniscus, Cm, is lower than that at the capillary entrance and should be determined in a self-consistent way. Solution of Equation 5.45 should be subjected to the following boundary conditions: C (t, 0 ) = C 0 ,

(5.46)

C (t, l (t )) = C m ,

(5.47)

and the last boundary condition on the moving front is 2Γ(Cm ) d ∂C = −D R dt ∂x

.

(5.48)

x = (t )

The preceding condition expresses the conservation of mass at the moving meniscus and the three-phase contact line. In order to deduce this boundary condition, the following procedure should be undertaken: (1) a local coordinate should be introduced in a narrow region close to the moving meniscus; (2) local transport equations should be integrated over this narrow region; and (3) limits from the ﬁlled portion of the capillary should be calculated, which gives condition (5.48). Because R << L, the liquid ﬂow inside the capillary is the simple Poiseuille ﬂow, which means that dl R 2 2 γ (Cm ) cos θa (Cm ) 1 Rγ (Cm ) cos θa (Cm ) = = , 4 ηl dt 8 η R l where η is the dynamic viscosity, assumed to be independent of surfactant concentration. It is also assumed in the following discussion that the surfactant concentration remains constant at the moving meniscus (similar to the case of hydrophobic capillaries in Section 5.2). Solution of the preceding equation using initial condition l(0) = 0 gives

l (t ) = K t , K =

© 2007 by Taylor & Francis Group, LLC

Rγ (Cm ) cos θa (Cm ) . 2η

(5.49)

Dynamics of Wetting or Spreading in the Presence of Surfactants

427

Substitution of Equation 5.49 into Equation 5.45 results in ∂2C K ∂C 2 ∂Γ ∂C + =D 2 − , r ∂t ∂t ∂x 2 t ∂x

(5.50)

with boundary conditions (5.46) and (5.47), and Γ(Cm ) K K t

= −D

∂C ∂x

,

(5.51)

x=K t

which is the consequence of (5.48). Let us introduce the following similarity coordinate ξ=

x K t

,

and solution of Equation 5.50 is assumed to depend on the similarity coordinate only. In this case, Equation 5.50 and boundary conditions (5.46), (5.47), and (5.51) take the following form:  2   λ 2C ′′ = C ′ 1 −  Γ ′(C ) + 1 ξ  , R    

(5.52)

C (0) = C0 ,

(5.53)

C (1) = Cm ,

(5.54)

C ′(1) = −

Γ (Cm )Ψ (Cm ) , 2 Dη

(5.55)

where λ2 =

2D 4 Dη = 2 RΨ(Cm ) K

is a dimensionless parameter. In the case of aqueous surfactant solutions D ~ 10–5 cm2 sec, η ~ 10–2 P, γ ~ 102 dyn/cm; therefore, the last parameter can be estimated as λ2 ~ © 2007 by Taylor & Francis Group, LLC

10 −9 cm . R

428

Wetting and Spreading Dynamics

In the following text we consider only capillaries with R > 0.1 µm = 10–5 cm. In this case, λ 2 < 10 −4 << 1 . The ﬁrst consequence of Equation 5.52 to Equation 5.55 is that the problem under consideration is really a similarity one. To further simplify the mathematical treatment of the problem under consideration, the simplest adsorption isotherm is adopted: Γ ∞ , C > 0 Γ (C ) =  . C=0 0,

(5.56)

In this case, Equation 5.52 and Equation 5.55 can be rewritten as follows if the concentration on the moving meniscus is above zero (the case of zero concentration is considered separately in the following discussion):

(

)

λ 2C ′′ = C ′ 1 − ξ , 0 < ξ < 1, C ′(1) = −

Γ ∞ Ψ max , 2 Dη

(5.57)

(5.58)

with boundary conditions (5.53) to (5.54). Note that according to Equation 5.42, Ψ, in the case under consideration (according to Equation 5.56), is independent of the concentration and equal to its maximal value, Ψmax. In the following discussion the smallness of the parameter λ2 is utilized. This small parameter is a multiplier at the highest derivative in Equation 5.57. This means that matching of asymptotic solutions can be used. Let us introduce the following local variable, z: z=

1− ξ . λ

(5.59)

Using the new variable, the inner solution of Equation 5.57 satisﬁes the following system: C ′′ = − zC ′, 0 < z < ∞ C ′(0) =

Γ ∞ Ψ max λ 2 Dη

(5.60)

C (0) = Cm and the boundary condition C (∞) = C0 . © 2007 by Taylor & Francis Group, LLC

(5.61)

Dynamics of Wetting or Spreading in the Presence of Surfactants

429

Solution of the problem (5.60) is Γ Ψ λ C ( z ) = Cm + ∞ max 2 Dη

z

∫ exp ( − z / 2) dz . 2

(5.62)

0

The following equation for the determination of the unknown concentration on the moving meniscus, Cm, yields, using boundary condition (5.21) and solution (5.62):

C0 = Cm +

Γ ∞ Ψ max λ π , 23/ 2 Dη

or

Cm = C0 − Γ ∞

πΨ max . 2 DηR

(5.63)

The concentration on the moving meniscus should be positive, Cm > 0; therefore, the following requirement should be satisﬁed:

C0 > Γ ∞

πΨ max , 2 DηR

(5.64)

or

R>

Γ 2∞ πΨ max . 2 DηC02

(5.65)

Let us introduce the following notation:

Rcr =

Γ 2∞ πΨ max . 2 2 DηCCMC

(5.66)

Two cases are considered here: (1) R < Rcr and (2) R > Rcr . In the ﬁrst case, (1), condition (5.64) is violated at any concentrations in the feed solution between zero and CMC. This means that concentration of surfactant molecules on the moving meniscus is equal to zero at any concentration from this range. Hence, there are two regions behind the moving meniscus. The ﬁrst © 2007 by Taylor & Francis Group, LLC

430

Wetting and Spreading Dynamics

region is close to the capillary entrance, where the concentration is changing from C0 in the feed solution to zero on the moving border between two regions. The ﬁrst region is followed by the second region where concentration remains zero over the duration of the whole process. Let the moving border between these two regions be 1 (t ) = Kβ t ,

(5.67)

where β < 1 is a value to be determined. In this case, the concentration on the meniscus remains zero, and the meniscus moves slowly, according to Equation 5.49: RΨ min t. 2η

l (t ) =

(5.68)

The concentration proﬁle in the ﬁrst region is a solution of the following problem (using the same similarity variable as before):

(

)

λ 2C ′′ = C ′ 1 − ξ , 0 < ξ < β, C ′(β) = −

Γ ∞ Ψ minβ , 2 Dη

(5.69)

(5.70)

C (0) = C0 ,

(5.71)

C(β) = 0.

(5.72)

Condition λ << 1 is assumed to be satisﬁed; this means that β = 1 − λχ,

(5.73)

where χ is a new unknown value. Solution of the problem (5.69) to (5.73) gives the following equation for the determination of an unknown value, χ: C0 = Γ ∞

( )

πΨ min exp − χ 2 , 2 DηR

or  Γ χ =  ln  ∞   C0 © 2007 by Taylor & Francis Group, LLC

2πΨ min DηR

1/ 2

  .  

(5.74)

Dynamics of Wetting or Spreading in the Presence of Surfactants

431

The main conclusion from this consideration is that the adsorption process in sufﬁciently thin capillaries consumes all surfactant, and the imbibition is not inﬂuenced by the presence of surfactants in the feed solution at any concentration. Let us consider the second case when the capillary radius is bigger than the critical value determined by Equation 5.66, that is, R > Rcr . If the concentration in the feed solution is low enough, condition (5.65) is violated when the concentration on the moving meniscus, Cm, is equal to zero and the meniscus moves slowly according to Equation 5.68. It is worth noting that the capillary radius is assumed to be bigger than that in the previous case. If, however, the concentration in the feed solution, C0, is high enough, condition (5.65) is satisﬁed, and concentration of the surfactant molecules is different from zero on the moving meniscus; the imbibition process goes faster according to

l (t ) =

RΨ max t. 2η

(5.75)

Hence, if the capillary radius is bigger than the critical value, then the whole concentration range in the feed solution can be subdivided into two parts: the low concentration range

C0 < Ccr = Γ ∞

πΨ max , 2 DηR

when the adsorption consumes all surfactant molecules, the concentration on the moving meniscus is equal to zero, and the meniscus moves slowly according to (5.68); and the high concentration range

C0 > Ccr = Γ ∞

πΨ max , 2 DηR

when the adsorption does not consume all surfactant molecules, the concentration on the moving meniscus is different from zero, and the meniscus moves faster according to (5.75). In Figure 5.13, 2 /t against the concentration in the feed solution, C0, is schematically plotted according to the simpliﬁed theoretical model discussed. If a more realistic case of adsorption isotherm (approximated by a Langmuir type isotherm) ω c, c < c* , Γ (c) =  ω c* = Γ ∞ © 2007 by Taylor & Francis Group, LLC

432

Wetting and Spreading Dynamics

ᐉ2/t

2

1

Ccr

C0

FIGURE 5.13 Dependency of permeability on surfactant concentration: theoretical predictions. (1) thin capillary, the radius is below critical value; (2) thick capillary, the radius is above critical value.

is adopted, then the dependency presented in Figure 5.13 changes continuously from lowest to the highest value instead of the stepwise change. Concentration above CMC If the concentration in the feed solution, C0, is above CMC, then, after some short initial period inside the capillary, two zones form (similar to Section 5.2): in the ﬁrst region, close to the capillary entrance, the concentration inside the capillary is higher than CMC. This region is followed by the second region where concentration is below CMC. The concentration is equal to CMC at the border between these two regions. The consideration similar to that in Section 5.2 and presented in the preceding discussion shows that the main conclusion remains unchanged in this case: there is a critical radius of the capillary, below which the concentration on the moving meniscus remains zero at any concentration in the feed solution. The concentration on the moving meniscus, Cm, is below CMC. Two regions can be identiﬁed inside the capillary: the ﬁrst region, from the capillary inlet to some position, lM(t), where concentration is above CMC, and the surfactant solution includes both micelles and individual surfactant molecules: and the second region, from lM(t) to l(t), where concentration is below CMC and only individual surfactant molecules are transferred. The concentration is equal to CMC at x = lM(t). Consideration in the second region, lM(t) < x < l(t), is similar to that at concentration below CMC. That is why only the transport in the ﬁrst region is considered in the following discussion. Inside the ﬁrst region, 0 < x < lM(t), concentration of free surfactant molecules is constant and equal to CMC (see Section 5.2). Hence, the transfer is determined by the diffusion of micelles and convection of all molecules. The total concentration, C = Cmol + CM , and Cmol remain constant and equal to CMC; hence, © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

∂C ∂2C dl ∂C = DM 2 − , ∂t dt ∂x ∂x

433

(5.76)

where DM is the diffusion coefﬁcient of micelles, and C = Cc + CM is the total concentration of surfactant molecules. Adsorption on membrane pores is determined by the concentration of the free molecules, which is constant in the ﬁrst region, and so is the adsorption. This is why the diffusion of adsorbed molecules in the ﬁrst region is omitted in Equation 5.76. Transfer of surfactant molecules in the second region (micelles-free region) is described by Equation 5.45. Boundary conditions on the moving boundary between the ﬁrst and second regions, lM(t), are as follows:  ∂C  DM   ∂x  x =l

M−

 ∂C  = D  ∂x  x =l

, C (l M , t ) = Cc .

(5.77)

M+

As before, we assume that l (t ) = K t , l M (t ) = K M t ,

(5.78)

where K is given by Equation 5.49; that is, it is expressed via unknown concentration on the moving meniscus, Cm, and KM is an unknown value to be determined. Let a similarity variable be introduced in the same way as in the case of a concentration below CMC; that is, ξ = x/K t in Equation 5.45 and Equation 5.76. Using boundary conditions (5.77), expressions (5.78), and boundary conditions (5.54) to (5.55), the system of two nonlinear algebraic equations can be deduced for determination of two unknown values: concentration on the moving meniscus, Cm, and the position of the boundary, KM. This system includes a small parameter ε=

DM << 1 , D

which is the ration of diffusion coefﬁcients of micelles and free surfactant molecules. Using this new small parameter, it is possible to show that solution of the mentioned system only slightly deviates from the solution in the previous case when concentration is below CMC. This means that the constant K and the expression for the critical radius (5.66) are only slightly different from the same values in the case of concentration below CMC. Hence, the previous conclusion concerning the existence of the critical radius remains valid even at concentrations above CMC, which is conﬁrmed in the following text by our experimental data. © 2007 by Taylor & Francis Group, LLC

434

Wetting and Spreading Dynamics

Experimental Part Figure 5.14 shows the sample chamber for monitoring the permeability of the initially dry porous layers. The time evolution of the permeability front was monitored. The membrane (1) was fastened on a lifting up/down device (2) and placed in a thermostated and hermetically closed chamber (3), where 100% humidity (to prevent evaporation from the wetted part of the membrane) and ﬁxed temperature (20 ± 0.5˚C) were maintained. To prevent temperature ﬂuctuations, the chamber was made from brass and, in the chamber walls, several channels were drilled that were used for the pumping of a thermostating liquid. The chamber was equipped with a fan. The temperature was monitored by a thermocouple. A box with water was used to keep absolute humidity inside the chamber. On the bottom of the chamber, a small Petri dish (4) with different water solutions of SDS was placed. The chamber was equipped with optical glass windows (5) for observation of the imbibition front of the surfactant solution. A CCD camera (6) and VCR (7) were used for storing the sequences of the imbibition. Automatic processing of images was carried out on a PC (8) using an image processor, Scion Image. The duration of each experimental run was in the range from 2.5 to 30 sec. The discretization of time in the processing ranged from 0.04 to 2 sec in different experimental runs; the size of pixel in an image was 0.01 mm. Experiments were carried out in the following order: • •

•

The membrane was placed in the chamber and left in atmosphere of 100% humidity for several minutes. The membrane was immersed vertically (0.1–0.2 cm) into a container with SDS solution. After that, the position of the imbibition front was monitored over time. Several runs for each membrane type and each concentration of SDS solution were carried out.

2 1 9

5

6

5

7

8

CCD

4

3

FIGURE 5.14 Schematic presentation of the experimental setup. (1) membrane; (2) lifting up and down device; (3) thermostated chamber; (4) Petri dish with an SDS solution; (5) optical glass windows 5; (6) CCD camera; (7) video tape recorder; (8) PC; (9) light source.

© 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

435

A rectangular membrane sample 1.5 cm∗3 cm was used. Porous samples were cut from the cellulose nitrate membranes purchased from Millipore of Billerica, MA. Three different membranes with averaged pore size were used: 0.22 µm, 0.45 µm, and 3.0 µm. Each membrane sample was immersed 0.1–0.2 cm into a liquid container, and the position of the imbibition front was monitored over time. Results and Discussions It was conﬁrmed that in all our experimental runs, gravity action could be disregarded. A unidirectional ﬂow of liquid inside the porous substrate took place. Using Darcy’s law, we can conclude that 2 (t ) = K p pct /η,

(5.79)

where (t ) now is the position of the imbibition front inside the porous layer, Kp is the permeability of the porous membrane, and pc is an effective capillary pressure inside the porous sample. The permeability of the porous layer and the capillary pressure enter as a product in Equation 5.79; that is, as a single coefﬁcient. Experiments were carried out to determine this coefﬁcient, and its dependency on the surfactant concentration if any. It was found that in all runs, l 2(t) proceeds along a straight line whose slope gives us the Kp pc value. According to our previous notations, Kp pc = K 2η. Figure 5.15 is the example of the imbibition of 0.1% SDS solution into 0.22 µm nitrocellulose membrane. This dependency is in good agreement with Equation 5.79. 5

, mm

4

3 Nitrocellulose membrane Pore size 0.22 μm SDS-0.1%

2

1

0 0

5

10

15 t, s

20

25

30

FIGURE 5.15 Example of the time evolution of the imbibition front. SDS concentration 0.1%, nitrocellulose membrane with averaged pore size 0.22 µm.

© 2007 by Taylor & Francis Group, LLC

436

Wetting and Spreading Dynamics 0.04 3

kpc

0.03 ‘Millipore’ 0.22 µm 0.45 µm 3.0 µm

0.02

2

0.01

1 0.00 0.0

0.2

0.4

0.6

0.8

1.0

SDS-concentrations, %

FIGURE 5.16 Dependency of kpc on concentration of SDS solutions for nitrocellulose membranes with different averaged pore sizes. Remains constant in the case of membranes with averaged pore size both 0.22 µm (line 1) and 0.45 µm (line 2). Increases with surfactant concentration in the case of membrane with averaged pore size 3 µm (curve 3 is drawn simply to guide the eyes).

Figure 5.16 presents Kp pc dependency on the concentration of the SDS in the feed solution for three different membranes. Kp pc in the case of membranes with 0.22-µm and 0.45-µm averaged pore size is independent of concentration. However, in the case of membranes with 3.0-µm averaged pore size, Kp pc increases with SDS concentration. This means that the critical radius, Rcr , is somewhere in between 0.45 and 3.0 µm. Figure 5.16 conﬁrms our conclusion concerning the existence of the critical pore radius below which permeability is independent of surfactant concentration.

5.4 SPREADING OF SURFACTANT SOLUTIONS OVER HYDROPHOBIC SUBSTRATES We shall now study the spreading of aqueous surfactant solutions over hydrophobic surfaces. The spreading of surfactant solutions over hydrophobic surfaces is considered in the following text from both theoretical and experimental points of view. Water droplets do not wet a virgin solid hydrophobic substrate. It is shown in this section that the transfer of surfactant molecules from the water droplet onto the hydrophobic surface changes the wetting characteristics in front of the drop on the three-phase contact line. The surfactant molecules increase the solid–vapor interfacial tension and hydrophilize the initially hydrophobic solid substrate just in front of the spreading drop. This process causes water drops to spread over time. The time of evolution of the spreading of aqueous surfactant solution droplets is predicted and compared with experimental observations. The assumption that © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

437

surfactant transfer from the drop surface onto the solid hydrophobic substrate controls the rate of spreading is conﬁrmed by our experimental observations. Surfactant adsorption on solid–liquid and liquid–vapor interfaces changes the corresponding interfacial tensions. Liquid motion caused by surface tension gradients on liquid–vapor interfaces (Marangoni effect) is the most investigated process (see Section 5.6). The phenomena produced by the presence of surfactant molecules on a solid–vapor interface have been studied less. In Section 5.2, the imbibition of surfactant solutions into thin quartz capillaries was investigated. Spreading and imbibition of surfactant solutions into both hydrophobic and hydrophilic surfaces (Section 5.2, Section 5.3, the current section, and Section 5.7) revealed various new and intriguing phenomena. In this section we address the problem of aqueous surfactant solutions spreading over hydrophobic surfaces from both the theoretical and experimental points of view [14].

THEORY Let a small water drop be placed on a hydrophobic surface. If the drop is small enough, then the effect of gravity can be ignored. Accordingly, the radius of the drop base, R(t), has to be smaller than the capillary length, a, and hence,

()

R t ≤a=

γ , ρg

where ρ and γ are the liquid density and liquid–vapor interfacial tension, respectively; g is the gravity acceleration. First of all, let us consider expression (1.2) from Chapter 1, Section 1.1 for the excess free energy, Φ, of the droplet on a solid substrate: 2 Φ = γS + PV e + πR ( γ sl − γ sv ),

where S is the area of the liquid–air interface; P = Pa − Pl is the excess pressure inside the liquid, Pa is the pressure in the ambient air, and Pl is the pressure inside the liquid; the last term on the right-hand side gives the difference between the energy of the part of the bare surface covered by the liquid drop as compared with the energy of the same solid surface without the droplet (Figure 5.17). The foregoing expression shows that the excess free energy decreases if (1) the liquid–vapor interfacial tension, γ, decreases, (2) if the liquid–solid interfacial tension, γsl , decreases, and (3) the solid–vapor interfacial tension, γsv , increases. Let us assume that, in the absence of surfactant, the drop forms an equilibrium contact angle above π/2. If the water contains surfactants, then three transfer processes take place from the liquid onto all three interfaces: surfactant adsorption at both (1) the inner liquid–solid interface, which results in a decrease of the solid–liquid interfacial tension, γse, (2) the liquid–vapor interface, which results © 2007 by Taylor & Francis Group, LLC

438

Wetting and Spreading Dynamics γ0

γsv(t)

θ R(t)

Γs(t) γ 0sl

FIGURE 5.17 Sketch of the geometry of a drop placed on a solid substrate.

in a decrease of the liquid–vapor interfacial tension, γ, and (3) transfer from the drop onto the solid–vapor interface just in front of the drop. As we already noticed previously, all three processes result in a decrease of the excess free energy of the system. However, adsorption processes (1) and (2) result in a decrease of corresponding interfacial tensions, γsv and γ ; but the transfer of surfactant molecules onto the solid–vapor interface in front of the drop results in an increase of a local free energy. However, the total free energy of the system decreases. That is, surfactant molecule transfer (3) moves via a relatively high potential barrier, and hence, considerably slower than adsorption processes (1) and (2). Therefore, they are fast processes as compared with the third process (3). The transfer of surfactant molecules onto the unwetted (hydrophobic) solid–vapor interface in front of the liquid has been shown in Section 5.2 to play an important role in the wetting of hydrophobic surfaces. All three surfactant transfer processes are favorable to spreading, as they result in both an increase of the spreading power, γsv – γ – γsl , and hence, a decrease in the contact angle. As mentioned previously, the transfer of surfactant molecules from the drop onto the solid–vapor interface in front of the drop results in an increase of local surface tension, γsv . Hence, it is the slowest process that will be the rate-determining step. Let us deﬁne the initial contact angle by

cos θ0 =

0 γ sv − γ sl0 π ≥ , 2 γ0

(5.80)

with γ 0sv , γ 0sl , γ 0 the initial values of solid–vapor, solid–liquid, and liquid–vapor interfaces, respectively. The term initial means that, although the adsorption process on the liquid–vapor and solid–liquid interfaces has been completed (they are fast processes), the solid–vapor interface still has its initial condition as a bare hydrophobic interface without any surfactant adsorption. At this initial instant of time, a process of slow transfer of surfactant molecules starts from the drop onto the solid–vapor interface. Let Γs (t) be the instantaneous value of surfactant adsorption onto the solid surface in front of the liquid drop on the three-phase contact line, and Γe be the equilibrium surface density of adsorbed

© 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

439

surfactant molecules that will eventually be reached. The driving force of the process is proportional to the difference Γs (t) – Γe . Hence, the surfactant adsorption behavior with time is described by

( ) = α Γ

dΓs t

()

(5.81)

Γ s (0 ) = 0 at t = 0,

(5.82)



dt

e

− Γ s t  ,

with the initial condition that

and τs = 1/α is the time scale of surfactant transfer from the drop onto the solid–vapor interface at the three-phase contact line. Let us assume that  − ∆E  , α = α T Ξ exp   kT 

(5.83)

where the prefactor αT is determined by thermal ﬂuctuations only; ∆E is an energy barrier for surfactant transfer from the liquid drop onto the solid–vapor interface; k and T are Boltzmann’s constant and absolute temperature, respectively; Ξ is a fraction of the drop liquid–vapor interface covered with surfactant molecules. We assume that the position of surfactant molecules on a hydrophobic solid interface is the hydrophobic “tails down.” We have assumed that transfer of surfactant molecules onto the hydrophobic solid interface takes place only from the liquid–vapor interface. It is difﬁcult to assess the contribution of surfactant molecule transfer along the solid surface from beneath the liquid. However, experimental data presented in the following text in this section support our assumption (although they do not prove it decisively). The drop surface coverage, Ξ, is an increasing function of the bulk surfactant concentration inside the drop, whose maximum is reached close to the CMC. It follows from Equation 5.83 that at low surfactant concentrations inside the drop, the characteristic time scale of the surfactant molecules transfer, τs, decreases with increased concentration, whereas above the CMC, τs levels off and reaches its lowest value. Both of these effects are observed in experimental results in the following discussion (compare Figure 5.20). As the drop adopts a position according to the triangle rule, the contact angle, θ(t), is determined by the relationship

()

cos θ t =

© 2007 by Taylor & Francis Group, LLC

()

γ sv t − γ sl0 γ0

,

(5.84)

440

Wetting and Spreading Dynamics

where γsv(t) is the instantaneous solid–vapor interfacial tension at the three-phase contact line. This dependency is determined by Γs (t). According to Antonov’s rule,

()

γ sv t = γ ∞sv

( )+γ

Γs t Γ

∞

0 sv

()

 Γs t  1− ∞  , Γ  

(5.85)

where γsv∞ is the solid–vapor interfacial tension of the surface completely covered by surfactants, and Γ∞ is the total number of sites available for adsorption. Hence, the ﬁnal value of the contact angle can be determined from Equation 5.84 as cos θ∞ =

γ sv∞ − γ 0sl . γ0

(5.86)

According to Equation 5.85, the solid–vapor interface in front of the spreading drop changes its wettability with time: from highly hydrophobic at the initial stage to partially hydrophilic at the ﬁnal stage. Using Equation 5.85 in Equation 5.84 yields the instantaneous contact angle

()

cos θ t = cos θ0 + λ

( ),

Γs t Γ∞

(5.87)

where cosθ0 is given by Equation 5.80, and the positive value of λ is λ=

0 γ ∞sv − γ sv . γ0

Equation 5.81 with initial condition (5.82) yields the solution

()

(

( ))

Γ s t = Γ e 1 − exp −αt .

(5.88)

Using (5.88) in Equation 5.87 gives the ﬁnal expression for the instantaneous contact angle

()

cos θ t = cos θ0 + λ

(

( ))

Γe 1 − exp −αt . Γ∞

(5.89)

A simple geometrical consideration (Figure 5.17) shows that the radius of the wetted spot, R(t), occupied by the drop can be expressed as

© 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

 6V  R t =  π 

()

1/ 3

1 1/ 3  θ θ   tan  3 + tan 2   2   2  

441

,

(5.90)

where V is the drop volume, which is supposed to remain constant, and the contact angle, θ, is given by Equation 5.89. Equation 5.89 and Equation 5.90 include two parameters: the dimensionless parameter β = λΓe /Γ∞ and the parameter α with dimension of inverse of time. It follows from Equation 5.89 that β = cosθ∞ – cosθ0 > 0, where θ ∞ is the contact angle after the spreading process is completed. If both values of the contact angle, θ0 and θ∞, have been measured, β can be determined. Hence, only α is used in the following discussion to ﬁt the experimental data. Let us introduce a dimensionless wetted area, S(t), as

S (t ) =

R 2 (t )  6V   π 

2/ 3

1

= tan

2/ 3

θ θ 3 + tan 2  2  2

2/ 3

=

1+ X

(1 − X ) ( 4 + 2 X ) 1/ 3

2/ 3

,

where X = cosθ, (cosθ0 ≤ X ≤ cosθ∞), which, using Equation 5.89, becomes X = cosθ∞ − βe-αt . It follows that both dS(t)/dt and dS(X)/dX are always positive, and the second time derivative is

(

)

d 2 S (t ) α2 dS ( X )  = cos θ∞ − X −2 X 2 + 3 X cos θ∞ − (2 − cos θ∞ )  2  ( 1 − )( 2 + ) X X dX dt (5.91) Two different situations are possible: (A) if the second derivative (5.91) changes sign, then the spreading rate can go via an inﬂection point, whereas (B) if the second derivative (5.91) is always negative, the spreading rate dS(t)/dt decreases with time. Case A corresponds to high surfactant activity, cos θ∞ ≥

4 9

(

)

10 − 1 ≈ 0.961,

whereas case B corresponds to “low surfactant activity,” cos θ∞ <

© 2007 by Taylor & Francis Group, LLC

4 9

(

)

10 − 1 ≈ 0.961.

442

Wetting and Spreading Dynamics

Using Equation 5.86, the preceding two conditions can be rewritten as γ sv∞ > 0.961 γ 0 + γ sl0 at a high surfactant activity, and γ sv∞ < 0.961 γ 0 + γ sl0 at low surfactant activity. Under experimental conditions described in the following text, only case B was observed and, hence, low surface activity surfactants were used, whereas in Reference 15, high surface activity surfactants (superspreaders) were apparently used. Experiment: Materials Two types of substrate were used: a PTFE ﬁlm and a polyethylene (PE) wafer. The latter substrate was prepared by crushing granules of polyethylene composition (softening point is 100°C) between two clean glass plates under an applied pressure 1 kg/cm2 at temperature 110°C. Transparent wafers of circular section with radius 1.5 cm and thickness 0.01 cm were used [14]. The cleaning procedure of PTFE and PE wafers was as follows: the surfaces were rinsed with alcohol and water, then the substrates were soaked in a sulfochromic acid from 30 to 60 min at temperature 50°C. The surfaces were then washed with distilled water and dried with a strong jet of nitrogen. The equilibrium macroscopic contact angles obtained were 105° and 90° for PTFE and PE substrates, respectively (for pure water droplets). Aqueous solutions of sodium dodecyl sulfate (SDS) from Merck with weight concentration from 0.005 up to 1% (the CMC of the SDS is 0.2%) were used in spreading experiments. Monitoring Method The time evolution of the contact line was monitored by following VCR images of drops. The images were stored using a CCD camera and a recorder at 25 frames per second. The automatic processing of images was carried out using the image processor Optimas. In the case of spreading over PE, the initial contact angle of the drop was less than 90°, and the drop was observed from above. The observed wetting area of the drop was monitored, and the wetting radius was calculated. For the PTFE substrate, we used a side view of the drop, and hence, the wetting radius was determined directly. Simple mass balance estimations show that time variation of surfactant concentration inside the spreading drops can be neglected in our experiments (though it may become important in experiments of longer duration). © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

443

Water adsorption in front of the spreading drops was neglected because of the hydrophobic nature of substrates used (in contrast to spreading of evaporating drops [16]). The Peclet number in all experiments was so small that surfactant diffusion in front of the drop was neglected. Results and Discussion According to observations in Reference 14, all drops were of spherical shape, and no disturbances or instabilities were detected. Immediately after deposition, the drops had a contact angle, which differed slightly from the equilibrium angle of pure water on the same substrate. After a very short initial time, the drops reached a position referred to in the following discussion as the initial position. After that, for 1–15 sec, depending on the SDS concentration, the drops remained at the initial position. Then the drops started to spread until a ﬁnal value of the contact angle was reached, and the spreading process was completed. In Figure 5.18, the evolution of the spreading radius of a drop over PTFE ﬁlm at 0.05% SDS concentration is plotted. In Figure 5.19, a similar plot is given for 0.1% SDS concentration. In both ﬁgures, the solid lines correspond to the ﬁtting of the experimental data by Equation 5.89 and Equation 5.90, with τs = 1/α used as a ﬁtting parameter. Figure 5.20 shows that, qualitatively, the τs dependency agrees with the theoretical prediction and tends to support our assumption concerning the mechanism of surfactant molecule transfer onto the hydrophobic surface in front of the drop. Similar results were obtained for the spreading over the polyethylene substrate for concentrations below CMC. However, in this case, the spreading behavior of 1.10

R, mm

1.08

1.06

1.04 0.1

1

10

100

t, s

FIGURE 5.18 Time evolution of the spreading of a water drop (aqueous solution C = 0.05% SDS; 2.5 ± 0.2 µl volume) over PTFE wafer. Error bars correspond to the error limits of video evaluation of images (pixel size). © 2007 by Taylor & Francis Group, LLC

444

Wetting and Spreading Dynamics 1,600

R, mm

1,575

1,550

0.1

1

10

100

t, s

FIGURE 5.19 Time evolution of the spreading of a water drop (aqueous solution C = 0.1% SDS; 2.5 ± 0.2 µl volume ) over PTFE wafer. Error bars are the same as in Figure 5.18.

5

τ, s

4 3 2 1

0.1

1 c, %

FIGURE 5.20 Fitted dependency of τs on surfactant concentration inside the drop (spreading over PTFE wafer). Error bars correspond to the experimental points scattering in different runs; squares are average values.

drops at concentrations above CMC is drastically different with increasing SDS concentration (Figure 5.21). The rate of spreading is increased so much that at 1% concentration, the power law with the exponent 0.1 (solid line) ﬁts experimental data reasonably well. This clearly shows a transition to a different mechanism of spreading, which can be understood in the following way. In our previous considerations, the inﬂuence of the viscous ﬂow inside the drop was completely ignored. This means that it was assumed that τs >> τvis , where τvis is a time scale © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

445

1.4 n = 0.09 n = 0.1

n = 0.091

R(t)/A

1.2

n = 0.053 1.0

R(t) = A . tn -C = 0.3% -C = 0.5% -C = 0.7% -C = 1.0%

0.8

0.6 10

20 t, s

30

40

FIGURE 5.21 Spreading of SDS solution over polyethylene substrate, concentration above CMC. Dependency of spreading radius on time R(t) = A·t n, where A and n are ﬁtted parameters. The case n = 0.1 is shown by a solid line.

of viscous relaxation. In this case, τs decreases so considerably that the mentioned inequality becomes invalid, and now, τs ~ τvis becomes valid.

5.5 SPREADING OF NON-NEWTONIAN LIQUIDS OVER SOLID SUBSTRATES In this section the spreading of drops of a non-Newtonian liquid (Ostwald– de Waele liquid) over horizontal solid substrates is theoretically investigated in the case of complete wetting and small dynamic contact angles. Both gravitational and capillary regimes of spreading are considered. The evolution equation deduced for the shape of the spreading drops has self-similar solutions, which allows obtaining spreading laws for both gravitational and capillary regimes of spreading. In the gravitational regime case of spreading, the proﬁle of the spreading drop is provided [17]. The spreading of liquids over solid surfaces has been studied from both theoretical and experimental points of view in Chapter 3 (Section 3.1 and Section 3.2), where investigations have dealt with the kinetics of spreading of Newtonian liquids. Both gravitational and capillary spreading regimes have been considered, and the spreading laws have been established. It has been shown in Section 3.1 and Section 3.2 that the singularity at the three-phase contact line is removed by the action of the surface forces (disjoining pressure). The theoretically predicted spreading laws for gravitational and capillary regimes have been deduced as R (t ) ~ t 1/ 8 and R (t ) ~ t 1/10 , respectively, where R(t) is the radius of the base of the spreading drop, and t is the time. Comparison of the predicted spreading laws with experimental data in Chapter 3 has shown excellent agreement. © 2007 by Taylor & Francis Group, LLC

446

Wetting and Spreading Dynamics

However, a number of liquids (polymer liquids and suspensions [18,19]) show a non-Newtonian behavior. The aim of this section is to extend the similarity solution method used in Chapter 3 to the case of spreading of non-Newtonian liquids (Ostwald–de Waele liquids) over solid surfaces and to deduce the corresponding spreading laws for both gravitational and capillary regimes of spreading.

GOVERNING EQUATION FOR OF THE SPREADING DROP

THE

EVOLUTION

OF THE

PROFILE

The problem is solved under the following assumptions: 1. 2. 3. 4.

Complete wetting case. The dynamic contact angle is low. Reynolds number is low, Re << 1. The rheological properties of the liquid are determined by the viscosity dependency, η(S), on the shear deformation rate, S [18].

The ﬁrst assumption allows us not to consider the ﬂow in the vicinity of the three-phase contact line, where the inﬂuence of the surface forces become important. These forces inﬂuence only a preexponential factor in the spreading law according to Section 3.1 and Section 3.2. In the case of Newtonian liquids, the preexponential factor has been found to be almost insensitive to the details of the surface forces (Section 3.2). This provides a justiﬁcation of the adopted procedure discussed in the following text. The second assumption means that R* >> H*, where R*, H* are characteristic scales in the radial and axial directions, respectively. In the case of the complete wetting assumptions, (2) and (3) are always satisﬁed at the ﬁnal stage of spreading. Let us consider a drop of an incompressible non-Newtonian liquid with density ρ and surface tension γ, which spreads over a horizontal solid substrate. The density, viscosity, and the pressure gradient in the surrounding air are neglected. The solid substrate is assumed to be rigid and nondeformable. Both the axisymmetric and cylindrical problems of spreading (Figure 5.22) are considered in the following discussion. z H(t) h(r, t)

R(t) r

FIGURE 5.22 Cross section of the spreading drop.

© 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

447

The liquid ﬂow inside the spreading drop obeys the following equations: Incompressibility condition ∂v 1 ∂ rvr + z = 0 , r ∂r ∂z

( )

(5.92)

and Navier–Stokes equations, which are considerably simpliﬁed using assumptions (1)–(4): ∂p − ρg = 0 , ∂z

(5.93)

∂vr  ∂p ∂  = 0, + η S ∂z  ∂r ∂z 

(5.94)

−

−

( )

where (r,ϕ, z) is the cylindrical coordinate system, the z-axis coincides with the axis of symmetry, and z = 0 corresponds to the solid substrate; all functions are independent of the angle, ϕ, because of symmetry; g is the gravity acceleration; p is the pressure; and vr and vz are radial and axial components of the velocity vector, respectively. Let Drr =

∂v 1  ∂v ∂v  ∂vr v , Dϕϕ = r , Drz = Dzr =  r + z  , Dzz = z 2  ∂z ∂r r ∂r  ∂z

be components of the deformation rate tensor. The parameter S is expressed in terms of the components of the deformation rate tensor as

(

)

2 S = 2 Drr2 + Dϕϕ + Dzz2 + 2 Drz2 .

Under assumption (2) this expression becomes 2

 ∂v  S= r ,  ∂z 

(5.95)

No-slip conditions are adopted on the solid substrate: vz = 0, z = 0; © 2007 by Taylor & Francis Group, LLC

(5.96)

448

Wetting and Spreading Dynamics

vr = 0, z = 0.

(5.97)

Let the proﬁle of the spreading drop be z = h(r,t), which is to be determined. Boundary conditions on the free liquid–air interface include the kinematic condition ∂h ∂h = vz − vr , z = h r,t ; ∂t ∂r

( )

(5.98)

and the conditions for the normal and tangential components of the stress tensor p = pa − γ

1 ∂  ∂h  r , z = h; r ∂r  ∂r 

( ) ∂∂vz

η S

r

= 0, z = h,

(5.99)

(5.100)

where pa is the pressure in the surrounding air and small terms are omitted based on assumption R* >> H*. This assumption means h ′ 2 << 1 ; that is, the low slope approximation is valid. Using the continuity Equation 5.92 and the kinematic condition (5.98), an equation describing the evolution of the drop proﬁle becomes  h  ∂h 1 ∂  r vr dz  = 0 . + ∂t r ∂r  

∫

(5.101)

0

Integration of Equation 5.93 over z with boundary condition (5.99) results in the following expression for the pressure distribution:

(

)

p = pa + ρg h − z − γ

1 ∂  ∂h  r . r ∂r  ∂r 

(5.102)

This equation shows that ∂p/∂r is independent of z. Integration of Equation 5.94 over z with boundary condition (5.100) gives   ∂v  2  ∂v ∂p η  r   r = − h−z .   ∂z   ∂z ∂r

(

© 2007 by Taylor & Francis Group, LLC

)

(5.103)

Dynamics of Wetting or Spreading in the Presence of Surfactants

449

Integration over z of this equation results in the following expression for the radial component of the velocity:   ∂p     ∂p v r = G  h−z  −G h  ∂ r    ∂r   

∂pp , ∂r

(5.104)

∫ F ( y ) dy , η( F ( x )) F ( x ) ≡ x .

(5.105)

(

)

where the function G(x) is determined as

()

x

G x =

2

0

Substitution of the expression for the radial velocity (5.104) into Equation 5.101 gives an equation that describes the proﬁle of the spreading drop:   ∂p h3   ∂h 1 ∂  ∂r , = r ∂t r ∂r   ∂p    3 ηef  h    ∂r   

(5.106)

where an effective viscosity, ηef (y), is determined as y

∫

1 −1 ηef ( y ) = y −2 G ( y ) − y −3 G ( z) dz . 3

(5.107)

0

In the case of the spreading of a cylindrical drop (plane symmetry, twodimensional drop), a similar consideration using a Cartesian coordinate system yields the following equation:   ∂p h3   ∂h ∂  ∂r , = ∂t ∂r   ∂p    3ηef  h    ∂r   

(5.108)

where

(

)

p = pa + ρg h − z − γ

© 2007 by Taylor & Francis Group, LLC

∂2 h . ∂r 2

(5.109)

450

Wetting and Spreading Dynamics

In Equation 5.108 and Equation 5.109, r is the Cartesian coordinate perpendicular to the z-axis. The effective viscosity, ηef (y), is determined by the same relations (5.105) and (5.107) as in the axisymmetric case. Equation 5.106 to Equation 5.108 and Equation 5.102 to Equation 5.109 can be rewritten in the following form:   ∂p h3   ∂h 1 ∂  m ∂r , = m r ∂t r ∂r   ∂p    3 ηef  h    ∂r   

(

)

p = pa + ρg h − z − γ

1 ∂  m ∂h  , r r m ∂r  ∂r 

(5.110)

(5.111)

where m = 0 and m = 1 correspond to the case of the cylindrical and axisymmetric drops, respectively. Substitution of Equation 5.111 into Equation 5.110 results in the following nonlinear differential equation, which describes the evolution of the proﬁle of the spreading drop:    ∂h ∂  1 ∂  m ∂h      r h 3 ρg −γ    ∂r  r m ∂r  ∂r     ∂h 1 ∂  m  ∂r  = r  , (5.112) ∂t r m ∂r    ∂h ∂  1 ∂  m ∂h      3 ηef  h ρg ∂r − γ ∂r  m ∂r  r ∂r            r  Conservation of the drop volume reads:

()

Rt

V = 2π

m

∫ h ( r, t ) r

m

( () )

dr , h R t , t = 0 ,

(5.113)

0

where V is the drop volume in the axisymmetric case and the cross section area in the case of cylindrical drops; R(t) is the location of the three-phase contact line. In the case of Ostwald–de Waele liquid [18]: η(S) = k S (n-1)/2, n > 0,

(5.114)

where n < 1 corresponds to pseudoplastic ﬂuids (can be as low as 0.1 for some natural rubbers [18]), and n > 1 corresponds to dilatant ﬂuids. © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

451

In this case, ηef (y) is 2n + 1 n nn−1 k y , 3n 1

()

ηef y =

(5.115)

and Equation 5.112 transforms into − 1 ∂ ∂h n = k n m ∂t 2n + 1 r ∂r 1

(5.116)

1  2 n +1 n        m n ∂h ∂ 1 ∂  m ∂h  ∂h ∂ 1 ∂  m ∂h   r h sign  ρg − γ  m  r ∂r   ρg ∂r − γ ∂r  r m ∂r  r ∂r    . ∂ r ∂ r ∂ r r         

The position of the three-phase contact line, R(t), is the time-dependent characteristic horizontal scale. Let us introduce the time-dependent characteristic thickness of the drop, H(t), using conservation law (5.113) as V = 2πm H(t) Rm+1 (t), or

()

H t =

V . 2 π m R m +1 t

()

(5.117)

Self-similar solutions of Equation 5.116 are tried in the following text in the following form:

( )

() ( )

()

h r, t = H t ζ rˆ , r = R t rˆ .

(5.118)

According to Equation 5.113 and Equation 5.117, the dimensionless drop ˆ satisﬁes the following conditions: proﬁle, ζ(r), 1

∫ ζ (rˆ ) rˆ drˆ = 1, ζ (1) = 0. m

(5.119)

0

In the following two sections, two limiting cases of spreading are considered, when either capillary or gravitational forces can be neglected. In the case of gravitational regime of spreading, the capillary forces can be neglected if

R (t ) >> a = © 2007 by Taylor & Francis Group, LLC

γ . ρg

452

Wetting and Spreading Dynamics

In the capillary regime of spreading, the capillary forces dominate, that is, γ . ρg

R (t ) << a =

GRAVITATIONAL REGIME

OF

SPREADING

In this case, the spreading Equation 5.116 transforms into ∂h n  ρg  = ∂t 2 n + 1  k 

1/ n

1/ n 2 n +1  ∂h  ∂h  1 ∂  m n r h sign   ∂r  ∂r  . r m ∂r   

Assuming the solution in the form (5.118), this equation yields −

V R (m + 1)ζ + rˆζ′  = m 2π R m + 2  1

n  ρg  n  V  2n + 1  k   2π m 

2 n+ 2 n

2 n +1 1 1 ∂  m n n . ˆ r ζ ζ ζ sign ′ ′   ( 2 m+3)( n+1) rˆ m ∂rˆ   n

( )

1

R

(5.120)

The preceding equation shows that the radius of spreading, R(t), should satisfy the following equation:

R = λ

n  ρg  2 n + 1  k 

1/ n

 V   2 π m 

( n + 2 )/ n R

(

)(

)

−  n + 2 m +1 +1 / n

,

(5.121)

where λ is a dimensionless constant. If R(t) is selected according to Equation ˆ 5.121, then Equation 5.120, which describes the dimensionless drop proﬁle, ζ(r), becomes

(

)

1/ n ′ 1 2 n +1 / n λ  m + 1 ζ + rˆζ′  = − m rˆ mζ( ) sign ζ′ ζ′ . rˆ

(

)

(5.122)

Equation 5.122 should be solved with the following boundary conditions: ζ′(0 ) = 0 , which is the symmetry condition in the drop center, and © 2007 by Taylor & Francis Group, LLC

(5.123)

Dynamics of Wetting or Spreading in the Presence of Surfactants

ζ(1) = 0 .

453

(5.124)

Conservation law (5.119) gives the third equation for the determination of the unknown parameter λ. The solution of Equation 5.121 with initial condition R(0) = R0 is α

1/ n ( n + 2 )/ n  1  n λ  ρg   V   t , R t = R0 1 +  2 n + 1 α  k   2 π m  R01/ α   

()

(5.125)

where the spreading exponent α is α=

(

n . m + 2 (n + 2) − 1

)

(5.126)

In the case of the Newtonian liquid (n = 1), this exponent is 1/(3m + 5). Thus, at n < 1: α < 1/(3m + 5), and the drop spreads slower than the Newtonian liquid; at n > 1: α > 1/(3m + 5), and the drop spreads faster than the Newtonian liquid. The dependence of the spreading exponent α on n, according to Equation 5.126, in the case of axisymmetric spreading (m = 1), is shown in Figure 5.23. Multiplying Equation 5.122 by rˆ m , and after integration, taking into account the symmetry condition (5.123), results in λrˆ m +1ζ = −rˆ m ζ(

) sign ζ′ ζ′ 1/ n ,

2 n +1 / n

0.4 1/3

α

0.3

0.2 1/8 0.1

0.0 0.01

0.1

1 n

10

100

FIGURE 5.23 Axisymmetric (m = 1) gravitational regime of spreading. Spreading exponent α (Equation 5.126) vs. n; n = 1 — Newtonian ﬂuid. © 2007 by Taylor & Francis Group, LLC

454

Wetting and Spreading Dynamics

The solution of this equation with boundary condition (5.124) is

ζ = sign λ λ

(

n / n +2

(

(

1/ n + 2

)

) n +2 n +1   n + 1 1 − rˆ   

) .

(5.127)

Substitution of solution (5.127) into conservation law (5.119) gives an equation for the determination of the unknown constant λ:

sign λ λ

(

n / n +2

)

(n + 2) ( ( n + 1)( ) (

1/ n + 2

)

n +3 / n +2

1

∫ (1 − x )

)

(

1/ n + 2

) (m − n )/( n +1) dx = 1 , x

0

or, in terms of the gamma function (5.112),

sign λ λ

(

n / n +2

)

(

n+2

( n + 1)

)

(

1/ n + 2

(

)

( n + 3 )/ n + 2

)

(

 n + 3   m + 1 Γ Γ  n + 2   n + 1 

)( ) ( )(

 n + 3 n + 1 + (m + 1)(nn + 2 )  Γ    n +1 n +2

= 1.

)

The preceding equation has the following solution:

(

)( ) ( )(

  n + 3 n + 1 + (m + 1)( n + 2 )   1 Γ    n +1 n +2  ( n + 1) n +3  n    λ=     n + 3   m + 1  n+2    Γ Γ    n + 2   n + 1   

)

n +2 n

.

(5.128)

It is possible to check that λ → m + 1, if n → ∞. If n → 0, then at m = 0:  9 λ ~ 1.0121989    8

1/ n

→ ∞;

at m = 1:  225  λ ~ 1.7635846 ⋅   32 

1/ n

→ ∞.

Dependence of λ on n at m = 1 is shown in Figure 5.24. Substitution of the expression (5.128) into the solution (5.127) gives the dimensionless drop proﬁle in the case of the gravitational regime of spreading: © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

455

10 –1 –2 –3

8

λ

6 4 2 0 0

10

20

30

40

50

n

FIGURE 5.24 Axisymmetric (m = 1) gravitational regime of spreading. Dimensionless constant λ (Equation 5.128) vs. n; n = 1 — Newtonian ﬂuid. Solid line = according to Equation 5.128; broken line = asymptotic dependence λ ~ 1.7635846·(225/32)1/n at n << 1.

(

)( ) ( )(

 n + 3 n + 1 + (m + 1)( n + 2 )  Γ    n +1 n +2 1/ ( n + 2 ) ζ rˆ = ( n + 1) . (5.129) 1 − rˆ n +1  n + 3   m + 1 Γ Γ  n + 2   n + 1 

()

)

(

)

In the case of axisymmetric spreading (m = 1) of a Newtonian liquid (n = 1), this equation gives ζ=

(

8 1 − rˆ 2 3

)

1/ 3

,

(5.130)

which coincides with the solution obtained in Chapter 3, Section 3.1. The proﬁles of axisymmetric spreading drops at different n according to Equation 5.129 are shown in Figure 5.25.

CAPILLARY REGIME

OF

SPREADING

In this case, Equation 5.116 becomes ∂h n γ =− ∂t 2n + 1  k 

1/ n

1 rm

1/ n  2 n +1  ∂  1 ∂  m ∂h   ∂  1 ∂  m ∂hh    ∂  m n r h sign   m  m r   r   . ∂r   ∂r  r ∂r  ∂r   ∂r  r ∂r  ∂r     

© 2007 by Taylor & Francis Group, LLC

(5.131)

456

Wetting and Spreading Dynamics ∧

ζ

4

2

1 ∧

r 0

1 n=1 n = 0.5 n=2

FIGURE 5.25 Axisymmetric (m = 1) gravitational regime of spreading. Drop proﬁles (Equation 5.129) at different values of n: - - - - -, n = 1; – - – - –, n = 0.5; ----------, n = 2.

Assuming the solution of Equation 5.131 in the form (5.117) and (5.118) yields V R n γ − m m+ 2 (m + 1)ζ + rˆζ′  = − 2 n + 1  k  2π R

1/n n

2 n+ 2 n

 V   2π m 

1 R

( 3+ 2 m / n + 5/ n + 2 m )

1/ n   ′ ′  1 m ′  1 m ′ ∂  m 2 n +1)/ n  ( ×  rˆ sign  m rˆ ζ′   m rˆ ζ′  ζ . ∂rˆ   rˆ  rˆ   

( )

( )

1 ˆr m

(5.132)

This equation shows that a self-similar solution exists if the radius of spreading, R(t), satisﬁes the following equation:

R = λ

n γ 2 n + 1  k 

1/ n

 V   2 π m 

( n + 2 )/ n R

−

5+ n + mn +2 m n

.

(5.133)

In this case, the dimensionless drop proﬁle is determined by the following ordinary differential equation: ′ 1/ n   ′ ′ ′  1 ′  1 1  2 n +1 / n  λ  m + 1 ζ + rˆζ′  = m rˆ m sign  m rˆ m ζ′   m rˆ m ζ′  ζ( )  , rˆ     rˆ  rˆ    (5.134)

(

)

© 2007 by Taylor & Francis Group, LLC

( )

( )

Dynamics of Wetting or Spreading in the Presence of Surfactants

457

0.4 1/3

α

0.3

0.2

0.1

0.0 0.01

0.1

10

1 n

100

FIGURE 5.26 Axisymmetric (m = 1) capillary regime of spreading. Spreading exponent α (Equation 5.136) vs. n.

where λ is a dimensionless constant, which is different from the previous case (gravitational regime of spreading). The solution of Equation 5.133, with the initial condition R(0) = R0, has the following form:  1/ n n +2 / n n λγ V( )  R t = R0 1 + ( n +2 )/ n 1/ α 2 n + 1 α  k  R0 2πm 

()

( )

α

  t , 

(5.135)

where the spreading exponent α is α=

n , mn + 2 m + 5 + 2 n

(5.136)

which gives 0.1 in the case of the axisymmetric spreading of a Newtonian liquid (m = n = 1) and 1/7 in the case of the cylindrical drop spreading (n = 1, m = 0). The dependence (5.136) in the case of axisymmetric spreading, m = 1, is shown in Figure 5.26. Comparison of Equation 5.126 and Equation 5.136 shows that capillary and gravitational regimes of spreading give the same dependence R(t ) ~ t1/( m+ 2) if n >> 1. Multiplying Equation 5.134 by rˆ m and integrating yields ′ ′  1 m ′  1 m ′ m +1 m ˆ ˆ ˆ ˆ λr ζ = r sign  m r ζ′   m r ζ′    rˆ   rˆ

( )

© 2007 by Taylor & Francis Group, LLC

( )

1/ n

ζ(

) +C ,

2 n +1 / n

458

Wetting and Spreading Dynamics

where C is an integration constant. Taking into account that the functions ζ(rˆ) and ζ′′(rˆ) should be ﬁnite and the symmetry condition in the drop center, ζ′(0) = ζ′′′(0) = 0, this equation becomes n n − n −1  1 m ′ ′  ˆ m rˆ ζ′  = signλ λ rˆ ζ . r 

( )

(5.137)

According to Section 3.2, an alternative way of solving Equation 5.131 is as follows. The whole drop is subdivided into two parts: the main spherical part (outer solution) and the narrow region close to the three-phase contact line (inner solution). The volume of the liquid in the narrow inner zone can be neglected as compared with the main spherical part. In order to determine the liquid ﬂow in the inner region, the surface forces action should be introduced in this narrow region. However, the solution in the inner region gives only a preexponential factor in the spreading law. Its dependence on the details of the ﬂow has been found insigniﬁcant in the case of Newtonian liquids and compete wetting. In the case of non-Newtonian liquids and the complete wetting case, surface forces in the vicinity of the three-phase contact line can be of a very complex nature. We assume that the inﬂuence of these complex and unknown surface forces gives only a correction of a preexponential factor as in the case of Newtonian liquids. That is why the ﬂow in the inner region is not considered in the following discussion. Accordingly, the right-hand side of Equation 5.137 is small everywhere except for a narrow vicinity of the three-phase contact line, where ζ approaches zero, and hence, the volume of the liquid in this small region can be neglected. Thus, the central part of the spreading drop is ζ=

(m + 1) (m + 3) 2

(1 − rˆ ) , 2

(5.138)

that is, the parabolic cap. The dynamic contact angle, θ, is determined by the following relationship (tanθ ≈ θ): θ=

(m + 1)(m + 3)V . 2 π m R m +2

From Equation 5.133, n ( m+ 2)

  5+ n+ mn+ 2 m U = R − ( m+ 2) , 1/ n  ( n + 2 )/ n  λ n γ  V    2n + 1  k   2π m   © 2007 by Taylor & Francis Group, LLC

(5.139)

Dynamics of Wetting or Spreading in the Presence of Surfactants

459

where U = R (t ) . Substitution of this equation into Equation 5.139 results in the following dependence of the dynamic contact angle of the spreading drop on the rate of spreading, U: n (m +2 )

1− n

m +2

 2 n + 1  5+ n +mn +2 m  V  5+ n +mn +2 m  k  5+ n +mn +2 m 5+ nn +(mmn+2+)2 m , U θ = (m + 1)(m + 3)   2 π m   γ   λn  (5.140) or 1

1− n

 λn  γ  n  2 π m  n (m +2 )  θ U=      2n + 1  k   V   (m + 1)((m + 3) 

5+ n + mn + 2 m n (m +2 )

.

(5.141)

For the case of an axisymmetric spreading of a drop of Newtonian liquid (n = m = 1), Equation 5.141 gives the Tanner’s law (see Chapter 3, Section 3.1 and Section 3.2). It is interesting to note that spreading law (5.141) is independent of the drop volume only in the case of Newtonian liquids. If n ≠ 1 then the righthand side of Equation 5.141 depends on the drop volume.

DISCUSSION The spreading of drops of non-Newtonian liquids (Ostwald–de Waele liquids) over horizontal solid substrates was theoretically investigated. An equation was deduced that describes the liquid proﬁles of axisymmetric and cylindrical spreading drops. The problem was solved under the following assumptions: (1) complete wetting, (2) low dynamic contact angle approximation, (3) low Reynolds number, Re << 1, and (4) the rheological properties of the liquid are determined by the viscosity dependency, η(S), on the shear deformation rate, S. In the case of complete wetting, the second and third assumptions are always valid at the ﬁnal stage of spreading and allow a considerable simpliﬁcation of the description of the spreading. Both gravitational and capillary spreading regimes of the spreading were considered. In the gravitational regime case, the spreading law and the proﬁle of the spreading drop have been completely determined and given by Equation 5.125 and Equation 5.129, respectively. In the case of the capillary regime, the spreading law and the apparent contact angle of the spreading drop have been calculated and given by Equation 5.135 and Equation 5.140, respectively. In the case of a Newtonian liquid (n = 1), the spreading laws for gravitational and capillary regimes, Equation 5.125 and Equation 5.135, coincide with those found earlier in Chapter 3. If n < 1, an axisymmetric drop spreads slower than a drop of a corresponding Newtonian liquid with the same volume. If n > 1, the spreading exponents for gravitational and capillary regimes are greater than those in the case of Newtonian liquids. © 2007 by Taylor & Francis Group, LLC

460

Wetting and Spreading Dynamics

It is interesting to note that both capillary and gravitational axisymmetric regimes of spreading give the same power law R (t ) ~ t 1/ 3 if n >> 1.

5.6 SPREADING OF AN INSOLUBLE SURFACTANT OVER THIN VISCOSE LIQUID LAYERS In this section we present the results of the theoretical and experimental study of the spreading of an insoluble surfactant over a thin liquid layer. Initial concentrations of surfactant above and below CMC have been considered. If the concentration is above CMC, two distinct stages of spreading are found: (1) the ﬁrst stage is the fast one, and it is connected with micelles dissolution, and (2) the second stage is the slower one, when the surfactant concentration becomes below CMC. In the second stage, the formation of a dry spott in the center of the ﬁlm is observed. A similarity solution of the corresponding equations for spreading gives good agreement with the experimental observations [13]. When a drop of a surfactant solution is deposited on a clean liquid–air interface, tangential stresses on the liquid surface develop. They are caused by the nonuniform distribution of surfactant concentration, Γ, over a part of the liquid surface covered by the surfactant molecules, leading to surface stresses and ﬂow (Marangoni effect) [12]: η

( ) = dγ

∂u r, h ∂z

∂Γ , d Γ ∂r

(5.142)

where η and u are the liquid dynamic shear viscosity and tangential velocity on the liquid surface located at height h, respectively; (r,z) are radial and vertical coordinates; and γ(Γ) is the liquid–air interfacial tension whose linear dependency on surfactant surface concentration we assume in the following discussion. The surface tension gradient-driven ﬂow induced by the Marangoni effect moves the surfactant along the surface, and a dramatic spreading process takes place. Then the liquid–air interface deviates from an initially ﬂat position to accommodate the normal stress also occurring in the course of motion. We restrict considerations in the following discussion to insoluble surfactants. Note that though a surfactant may be soluble, there are cases such that nonsolubility conditions can be used during a certain initial period in the spreading process. Let us consider two characteristic time scales associated with surfactant transfer: (1) τd accounts for the transfer from the liquid–air interface to the bulk, and (2) τa accounts for the transfer from the bulk back to the interface. In both cases these characteristic time scales depend on an energy difference between corresponding states. Let us consider an aqueous surfactant solution: the latter has both a hydrophilic head and a hydrophobic tail. Let Ehl, Eta, Etl be the energies (in kT units) of head–water, tail–air, and tail–water interactions. Using these notations, the energy of a molecule in an adsorbed state at the interface is Ead = Eta + Ehl, whereas, for the same molecule in the bulk, it is Eb = Etl + Ehl. Then, © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

461

τd = τ⋅exp(Eb – Ead) = τ⋅exp{Etl – Eta}, and τa = τ⋅ exp(Ead – Eb) = τ⋅exp{–(Etl – Eta)}, where τ is determined by thermal ﬂuctuations. Generally, the tail–water interaction energy is considerably higher than that of the tail–air interaction (Etl ≈ nEtl1 , where Etl1 is an interaction energy per hydrophobic unit, and n is a number of those units in each tail). Consequently, τd /τa = exp{2(Etl – Eta)} >> 1, and transfer from the interface to bulk is a much slower process than the reverse one. If the duration of a spreading experiment is shorter than τd, then, during that experiment, the surfactant can be considered as insoluble. Otherwise, if t > τd , the solubility of the surfactant in the liquid must be taken into account. In the latter case, surfactant transfer to the bulk liquid tends to make concentration uniform both in the bulk and at the interface, and the result is a substantial decrease of the surfactant inﬂuence of that type (Marangoni ﬂow). Usually, surface diffusion can be neglected as compared to convective transfer (5.163). Indeed, from Equation 5.142, we have as a characteristic scale of surface velocity: u* ≈

γ * h* , ηL*

where γ * , h* , L* are characteristic scales of interfacial tension, initial ﬁlm thickness, and length in a tangential direction, respectively. The diffusion process over the liquid surface scales is

Ds

Γ* , L2*

where Ds , Γ* are the surface diffusion coefﬁcient and a characteristic scale of surfactant concentration on the surface, respectively. Then, the ratio of diffusion to convective ﬂux can be estimated as 1/Pe = Ds η/γ * H* ∼ 10–8 << 1, for Ds ∼ 10–5 cm2/s (5.164) and γ* ≈ 102 dyn/cm. Here, Pe is the mean Peclet number. This estimation shows that surface diffusion can be neglected everywhere except for a small diffusion layer, which is disregarded in the following discussion. We now consider two different cases: (1) when concentration in a droplet of surfactant solution, which is placed in the center of a liquid ﬁlm, is above CMC, and (2) when such concentration is below CMC. In the ﬁrst case, the spreading process involves two stages: (1) the faster one when the surfactant concentration is determined by the dissolution of micelles. This stage yields the maximum attainable surfactant concentration in the ﬁlm center, and it is independent of time. The duration of that stage is ﬁxed by the initial amount of micelles in the drop. (2) The second slower stage takes place when the surfactant concentration changes in the ﬁlm center, but the total mass of surfactant remains constant. In both cases, a similarity solution provides a power law predicting the position of the moving front rf (t ) as time proceeds. © 2007 by Taylor & Francis Group, LLC

462

THEORY

Wetting and Spreading Dynamics AND

RELATION

TO

EXPERIMENT

The motion of a thin liquid layer with initial thickness H* is considered under the action of an insoluble surfactant on its open surface. For simplicity we assume that surface tension varies linearly with the surface concentration of surfactant, γ (Γ ) = γ * − αΓ ,

at

0 < Γ < Γm ,

(5.143)

where γ * is the interfacial tension of the pure water–air interface, and Γ m corresponds to the maximum attainable surface concentration (in equilibrium with a micelle solution). We use, in the following discussion, dimensionless parameters and variables, and we use the same symbols as for dimensional quantities. The subscript * is used to mark initial or characteristic values. We further assume that ε = H*/L* << 1; hence, neglect of the nonlinear part of the interface curvature. A dimensionless Bond number, ρgH*2 /αΓ* , accounts for the ratio of the gravitational force to the Marangoni forcing, where ρ is the liquid density, and g is the gravity acceleration. In our experiments, a water ﬁlm with H* ≈ 0.1 mm is used, the Bond number is about 5*10–3 << 1 and hence, the gravity action can be safely neglected. Under these conditions, the evolution equations for mass balance for water and surfactant concentration on the surface are as follows: ∂h 1 ∂ + ∂t r ∂r

h

∫ ru dz = 0,

(5.144)

0

( ( ) )

∂Γ 1 ∂ + ru t , h Γ = 0, ∂t r ∂r

(5.145)

where h(t,r) is the ﬁlm thickness at time t; r is the radial coordinate, and Γ(t,r) the surfactant concentration on the surface. Equation 5.144 and Equation 5.145, after performing the integration, become (see Appendix 2 for details): ∂h 1 ∂ =− ∂t r ∂r

  γh 3 ∂  γ ∂  ∂h   αh 2 ∂Γ   r − , r  3η ∂r  r ∂r  ∂r   2 η ∂r     

(5.146)

1 ∂ ∂Γ =− ∂t r ∂r

  γh 2 ∂  γ ∂  ∂h   αh ∂Γ   r − , r Γ  2 η ∂r  r ∂r  ∂r   η ∂r     

(5.147)

and are to be solved subject to the following boundary conditions: © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

∂h ∂3h = = 0, ∂r ∂r 3 h → 1,

r = 0,

at

(5.148)

at r → ∞,

∂Γ = 0, ∂r

at

(5.149)

r = 0,

(5.150)

r → ∞.

(5.151)

at

Γ → 0,

463

Let us introduce the following dimensionless variables and values: h→

h , H*

t* =

ηL2* , H*αΓ*

r→

r , L*

β = ε2

Γ→

Γ , Γ*

t→

t , t*

γ* << 1. αΓ*

The time scale t* deserves a comment. The capillary number for the spreading process under consideration is very small: Ca =

ηU ∼ 10 −2 10 −1 / 10 2 = 10 −5 << 1 . γ*

On the other hand, Ca =

ηU* η L* ; = γ* γ * τ*

hence, a new time scale can be introduced as τ* =

ηL* , γ *Ca

where τ* ,U* are, respectively, the time scale that actually governs the spreading process and the characteristic velocity scale. Using these estimations we conclude: t* γ Ca = * = δ ≈ 10 −3 << 1 for ε ≈ 10 −2 , τ* αΓ* ε © 2007 by Taylor & Francis Group, LLC

464

Wetting and Spreading Dynamics

as it is the case in the experiments discussed in the following text. If we now introduce the dimensionless time τ=

t , τ*

then in Equation 5.144 and Equation 5.145 we have δ

∂ ∂ = . ∂τ ∂t

We show in the following text that the time evolution of the position of the moving ﬁlm front is  τ rf (t ) ∼ t 0.5÷0.25 =    δ

0.5 ÷ 0.25

.

If we take into account the initial value, , of rf (t), this dependence takes the form τ  r f (τ) ≈  +  δ 

0.5 ÷ 0.25

,

where ∼ 1 represents the contribution of the initial condition. According to our choice τ ∼ 1; hence, τ >> , δ and thus,  τ rf (t ) ~    δ

0.5÷ 0.25

= t 0.5÷ 0.25.

Multiplying Equation 5.146 by r, we conclude, after integration, that ∞

∫ r ( h − 1) dr = 0, 0

© 2007 by Taylor & Francis Group, LLC

(5.152)

Dynamics of Wetting or Spreading in the Presence of Surfactants

465

which reﬂects the conservation law for the liquid. In our experiments, a droplet of surfactant solution with concentration above CMC was placed in the center of a water ﬁlm. Experimental observations show that two distinct stages of spreading take place: (1) a ﬁrst faster stage, and (2) a second, slower stage. During the ﬁrst stage, there is dissociation of micelles; hence, surface concentration of single molecules is kept constant during that stage, and Γ (t, 0 ) = Γ m . Choosing Γ* = Γ m as a characteristic scale for surfactant concentration, we have, in dimensionless form, the following boundary condition during the ﬁrst stage: Γ(t , 0) = 1.

(5.153)

The ﬁrst stage lasts until all micelles are dissolved. The duration t1* of that stage is considered in the following discussion. Past t1* , a second stage starts. During the second stage, the total mass of surfactant remains constant; hence, the following boundary conditions apply: ∂Γ = 0, ∂r

at

r = 0,

(5.154)

and ∞

∫ rΓdr = 1,

(5.155)

0

where the characteristic scale for Γ is now selected as Q* , 2 πL2* with Q * being the total amount of surfactant molecules in the droplet. The First Spreading Stage The spreading process in this case is described by Equation 5.146 and Equation 5.147 with boundary conditions (5.148–5.151, and 5.153). According to Appendix 3, the inﬂuence of capillary forces can be neglected for t >> β, and Equation 5.146 and Equation 5.147 become ∂h 1 ∂ = ∂t r ∂r

© 2007 by Taylor & Francis Group, LLC

  h 2 ∂Γ   r  ,   2 ∂r  

(5.156)

466

Wetting and Spreading Dynamics

∂Γ 1 ∂ = ∂t r ∂r

  ∂Γ   r Γ  h  .   ∂r  

(5.157)

Equation 5.156 and Equation 5.157 cannot satisfy the boundary conditions at r → ∞, and a shock-like spreading front forms (for the derivation of these conditions see Appendix 3). In our case, h+ = 1, Γ+ = 0; hence, ∂Γ + = 0. ∂r Then, using conditions (A3.3 and A3.4), we get rf (h− − 1) = −

1 2 ∂Γ − , h− ∂r 2

 ∂Γ  Γ −  rf + h− −  = 0. ∂r  

(5.158)

(5.159)

Equation 5.159 actually implies two conditions: Γ − = 0, rf = − h−

∂Γ − . ∂r

(5.160)

(5.161)

The matching of asymptotic expansions at the moving shock front (see Appendix 5) shows that both conditions (5.160, 5.161) must be satisﬁed. Let us now introduce a new variable ξ=

r . t1/ 2

Then we have h(t , r ) = f (ξ),

Γ (t , r ) = ϕ(ξ),

(5.162)

where 0 < r < rf (t ) = ν t ; the constant ν is determined in the following discussion. In condition (5.152), the upper limit of integration must be replaced by ν; hence, © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

467

ν

∫ ξ f (ξ)d ξ = ν2 , 2

(5.163)

0

which is compatible with deﬁnitions (5.162). Equation 5.156 and Equation 5.157, after using Equation 5.162, become: − ξ f ′(ξ ) =

(

)

1 ξf 2 (ξ)ϕ′(ξ) ′ , ξ

(

(5.164)

)

1 1 − ξϕ′(ξ) = ξf (ξ)ϕ(ξ)ϕ′(ξ) ′ , 2 ξ

(5.165)

with the corresponding boundary conditions that follow from Equation 5.158, Equation 5.160, and Equation 5.161. We have ν  f ( ν) − 1 = − f 2 ( ν) ϕ′(ν), ϕ( ν) = 0, ν = − f ( ν) ϕ′( ν). 2 These boundary conditions, after simple transformations, become ϕ( ν) = 0, f ( ν) = 2,

(5.166)

ν ϕ′( ν) = − . 4 If we again change the variables using µ=

ξ , ν

f = Ψ (µ), ϕ = ν2 G (µ) ,

(5.167)

the new functions Ψ and G satisfy the same equations (5.164, 5.165) with the variable µ, where 0 < µ < 1, and the following boundary conditions at µ = 1: Ψ(1) = 2, G(1) = 0, and G′(1) = π. Then the problem does not depend on the unknown parameter ν.

© 2007 by Taylor & Francis Group, LLC

(5.168)

468

Wetting and Spreading Dynamics

8 G

6 4 2

ψ

0.2

0.4

0.6

0.8

µ

FIGURE 5.27 Theoretical predictions for dimensionless proﬁle Ψ(µ) and surfactant surface concentration G(µ) during the ﬁrst spreading stage (calculated according to equations and boundary conditions (5.164, 5.165, 5.168)).

Calculated dependencies of dimensionless ﬁlm thickness Ψ(µ) and surface concentration G(µ) are presented in Figure 5.27, which shows that a substantial depression is formed in the ﬁlm center. During the second stage, the depression becomes a dry spot right in the middle of the ﬁlm. In order to determine the unknown constant, ν, let us consider the total mass of surfactant, Q(t), during the ﬁrst spreading stage. We have ∞

ν

∫

∫

Q(t ) = 2π r Γ (t , r )dr = 2πL2*Γ*t ξϕ(ξ)d ξ, 0

0

or ν

∫ ξϕ(ξ)d ξ = 2πL Γ t . Q(t ) 2 *

0

(5.169)

*

The left-hand side of Equation 5.169 does not depend on time t; hence, the same is true for the right-hand side. Let us denote by q a constant value to be experimentally determined from the duration of the ﬁrst stage. Then, Equation 5.169 takes the form ν

∫ ξϕ(ξ)d ξ = q. 0

© 2007 by Taylor & Francis Group, LLC

(5.170)

Dynamics of Wetting or Spreading in the Presence of Surfactants

469

On the other hand, according to deﬁnition (5.167), 1

ν

4

∫ ηG(η) d η = q, 0

that may ﬁx the ν value. Unfortunately, the integral 1

∫ ηG(η)d η 0

diverges due to the singularity of the dependence G(µ) at µ = 0. Hence, we can only write that ν4 ∼ q, or ν ≈ q1/ 4 .

(5.171)

Let t1* be a dimensional time scale of the duration of the ﬁrst spreading stage; t1*/t* is the corresponding dimensionless time. If we choose t* = t1*, then, using the deﬁnition of the time scale, we get the corresponding value of the tangential length scale  t H αΓ  L* =  1* * *  η  

1/ 2

.

From Equation 5.169 we conclude that q=

Q* , 2πL2*Γ*

(5.172)

where Q* is the total amount of surfactant that is initially placed on the ﬁlm surface, which is supposed to be known, Q* = V*C* , where V* and C* are the droplet volume and surfactant concentration in the droplet, respectively. Unfortunately, the derived similarity solution cannot satisfy boundary condition (5.153) at the origin as the concentration dependence on radial coordinate diverges in a vicinity of the origin. Thus, it is more convenient to redeﬁne a characteristic scale of surfactant surface concentration from Equation 5.172 using the condition q = 1 in Equation 5.172 (this choice gives the same characteristic scale during both stages of the spreading process). The ν value is still undetermined, but we shall show later on how to determine it. At time t1* , the second stage of the spreading process starts. © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

The Second Spreading Stage In this stage, the ﬁlm proﬁle and the surfactant concentration obey the same system of equations (5.146) and (5.147) with boundary conditions (5.148 to 5.151) and (5.154 and 5.155). Let us introduce ξ=

r t

1/ 4

;

then the solutions of Equation 5.146 and Equation 5.147 are

()

h(t , r ) = f ξ ,

Γ (t , r ) =

ϕ(ξ ) , t1/ 2

where two unknown functions f (ξ) and ϕ(ξ) obey the following system of equations:    4βf 3 (ξ)  1 2 ξ f ′(ξ ) =  ξ   ξ f ′(ξ )   3  ξ 

(

′  ′   ′ − 2 f 2 (ξ)ϕ ϕ′(ξ )   ,   

)

′    ′   1   ′ 2 (ξ ϕ(ξ))′ = ξϕ(ξ)  2β f (ξ)  ξf '(ξ)  − 4 f (ξ)ϕ′(ξ)   .   ξ     

(

2

)

(5.173)

(5.174)

Equation 5.174 can be integrated using condition (5.151), which gives ϕ(ξ) → 0, at ξ → ∞, that is,    ′ 1    ′ 2 ξϕ(ξ) ξ − 2β f (ξ)  ξ f ′(ξ)  − 4 f (ξ)ϕ′(ξ)   = 0.   ξ       

(

)

Thus, either ϕ(ξ ) = 0,

(5.175)

or ϕ′(ξ ) = − © 2007 by Taylor & Francis Group, LLC

1 ′ ξ β + f (ξ )  ξ f ′(ξ ) ′  . 4 f (ξ ) 2 ξ 

(

)

(5.176)

Dynamics of Wetting or Spreading in the Presence of Surfactants

471

In the ﬁrst case, from Equation 5.173 we conclude that  4β  3  1 ξ f ′(ξ ) =  ξ f (ξ )  ξ f ′(ξ ) 3  ξ 

(

2

′ ′ ′  ,   

)

(5.177)

which is valid at the periphery of the spreading part of the ﬁlm. Equation 5.177 describes decaying capillary waves on the ﬁlm surface. Indeed, if we introduce a new local variable ζ = (ξ – λ)/χ with  4β  χ=   3λ 

1/ 3

near the moving edge λ (to be deﬁned in the following discussion), we get from Equation 5.177 f ′′′(ς ) =

f (ς ) − 1 . f 3 (ς )

The asymptotic behavior of this equation, according to Chapter 3 (Section 3.5), yields −

ς

f (ς ) ≈ 1 + e 2 ( A1 cos

3ς 3ς + A2 sin ), 2 2

at

ς → ∞,

which describes decaying capillary waves ahead the advancing front. In the opposite case, when Equation 5.176 is valid, we obtain from Equation 5.173 using Equation 5.176 that ′   3   ′ f (ξ )ξ     β f (ξ )  1 ′ ξ f ′(ξ ) =  ξ  ξ f ′(ξ)  + 2   .   3  ξ   

(

2

)

(5.178)

The value of λ is determined as a point where ϕ(λ) = 0, or from Equation 5.176 and condition (5.155): ′  ξ 1 β ′   d ξ. f ( ) f ( ) ξ ξ ξ − ′  4 f (ξ) 2  ξ    0

λ

2=

∫

© 2007 by Taylor & Francis Group, LLC

(

)

(5.179)

472

Wetting and Spreading Dynamics

The solution of Equation 5.176 is λ

ϕ(ξ ) =

∫ ξ

  ξ − β f (ξ )  1 ξ f ′(ξ )  ξ  4 f (ξ ) 2 

(

)′ 

′  dξ,  

where the integration constant is vanishing in accordance with Equation 5.179. It is easy to ﬁnd a solution of the problem under consideration in the zeroth approximation by setting β = 0 in Equation 5.176, Equation 5.177, and Equation 5.179 (see Appendix 5). It is also possible to ﬁnd an exact expression for λ using a zeroth order solution from Appendix 6 and Equation 5.176, Equation 5.177, and Equation 5.179, but it is not our aim. In conclusion of the theoretical consideration, let us summarize the results obtained. For the dimensional radius of the moving axisymmetric front, we have for the ﬁrst stage of spreading  t  rf = νL*    t1* 

1/ 2

(cm),

t ≤ t1* ,

(5.180)

with ν still undetermined. For the second stage, we ﬁnd  t  rf = λL*    t1* 

1/ 4

λ ≈ 25/ 4 ,

(cm),

t ≥ t1* ,

(5.181)

where t1* (sec) is the duration of the ﬁrst stage of spreading,  H αΓ t  L* =  * * 1*  µ  

1/ 2

,

Γ* =

Q* . 2πL2*

As the front position must be the same at time t1* according to both Equation 5.180 and Equation 5.181, then ν = λ, and Equation 5.180 becomes  t  rf = λL*    t1* 

1/ 2

(cm), t ≤ t1*.

(5.182)

Our theory predicts that the layer thickness decreases in the center with a vanishing value at the origin, which is the dry spot. When comparing with experiments, we must take into account both the ﬁnite precision in the measurements of the ﬁlm thickness and the possibility of evaporation during the experiment that may lead to discrepancy between our theory predictions and experimental results. Indeed, under experimental conditions, we can measure the ﬁlm © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

473

thickness only for values higher than a certain thickness, h*; we can consider that thicknesses below h* constitute the dry spot. Using a zeroth order solution of Equation A6.3, we obtain that 2

 r  2 H*  d1/ 4  = h* ,  λt 

or

 h  rd (t ) = λ  *   2 H* 

1/ 2

t1/ 4 ,

(5.183)

where rd (t) is the dry spot radius, whose motion according to the preceding equation obeys the same power law as the front of the ﬁlm. Moreover, evaporation has more pronounced inﬂuence at smaller thicknesses (h < h*) and that inﬂuence progressively increases with time. Hence, evaporation in the liquid ﬁlm near the center of the layer helps further ﬁlm thinning during the second stage. Experimental Results Observations of the spreading of a surfactant on a thin layer of liquid have been performed using aqueous solutions of sodium dodecyl sulfate at a concentration c = 20 g/l above CMS (the critical micellar concentration of the SDS is 4 g/l). The thin layer of liquid is prepared by coating the bottom of a borosilicate glass Petri dish of diameter 20 cm with 10 ml of distilled water. The resulting thickness is then H = 0.32 ± 0.01 mm. If carefully washed, this layer does not dewet during the time of the experiment. With a syringe, a drop of the surfactant solution, volume 3 µl, is put on the surface of this water layer. When touching the water surface, the surfactant spreads on it, and this motion is followed using a small amount of talc powder as a marker and a 25-Hz video camera to record it. The spreading of surfactants makes the water ﬂow away from the initial location of the drop, thus creating a depression where only a thin ﬁlm of liquid subsists. The periphery of this depression, i.e., the liquid front, has a sharp increase in thickness. If the layer is horizontal and the drop is carefully placed, there is no preferred direction, and the edge is circular but some modulation may appear in the experiment after a few seconds. Note that the surfactant occupies more surface area than the depressed zone as the talc powder is pushed ahead of it. The dependence on time of the radius of the surfactant patch is given in Figure 5.28a using log–log plot. The two successive stages described earlier in our theory are clearly seen in Figure 5.28a. First, the short period when the surfactant spreads following the power law rf (t ) ∼ t 0.60 ± 0.15. The exponent is not determined with a high precision because this ﬁrst stage is too fast (about 0.1 sec), which is only three times the time resolution (0.04 sec). The value 0.60 ± 0.15 agrees well with the earlier given theoretical prediction 0.5, Equation 5.182. At the end of the ﬁrst stage, the motion abruptly slows down, and the moving front follows a different power law rf (t ) ∼ t 0.17 ± 0.02. The new exponent is smaller than the theoretical prediction of 0.25 given by Equation 5.181. In Figure 5.28b, the radius of the shallow region, i.e., the dry spot radius, is plotted for the same © 2007 by Taylor & Francis Group, LLC

474

Wetting and Spreading Dynamics

log rf (mm)

100

10 0.01

0.1

1

10

log t (s)

log rf (mm)

100

1/4

10 0.01

0.1

1

10

log t (s)

FIGURE 5.28 (a) Radius of the spreading front vs. time. Points correspond to six different experimental runs. (b) Radius of the dry spot vs. time. Points correspond to six different experimental runs.

six experimental runs. We observe the power law dependence rd (t ) ∼ t0.25 ± 0.05, which is the value predicted by the theory, Equation 5.183. According to our measurements, the ratio rd (t ) rf (t ) is about 1/3, although it slowly changes with time during the second stage. Thus, from Equation 5.183 we conclude that h* ≈ 0.07 cm, the lowest thickness that can be detected by our experimental method. © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

475

In conclusion, the radius of the dry spot moves with the speed predicted by the theory, whereas the front edge of the moving part of the layer proceeds slower than theoretically predicted during the second spreading stage. The discrepancy may be due to one or both of the following reasons: 1. Gravity action, which is much more pronounced at the front higher edge of the layer than at the lower edge (dry spot). Although the Bond number is very low in our experiments, ﬂow reversal onset cannot be ruled out during the second stage. 2. Our assumption that the surfactant used is insoluble during the whole duration of the experiment may not be fully correct. Unfortunately, we have been unable to estimate the desorption time, τd . If that time is reached during our experiments, dissolution of surfactant in the bulk may be signiﬁcant at the higher front edge.

APPENDIX 2 Derivation of Governing Equations for Time Evolution of Both Film Thickness and Surfactant Surface Concentration At ε << 1, as we already noticed in Chapter 3, the Navier–Stokes equations reduces to dp ∂ 2u =η 2, dr ∂z

(A2.1)

p = p(r),

(A2.2)

1 ∂ ∂v ru + = 0, r ∂r ∂z

(A2.3)

( )

where p(r), v(r,z), z are pressure, axial velocity, and axial coordinate, respectively. The following boundary condition must be satisﬁed: u(r,0) = v(r,0) = 0, η

(A2.4)

( ) = ∂γ = d γ ∂Γ = −α ∂Γ ,

∂u r , h ∂z

∂r

p = pa −

d Γ ∂r

γ ∂  ∂h  , r r ∂r  ∂r 

∂r

(A2.5)

(A2.6)

where pa is the pressure in the ambient air. Solution of Equation A2.1 with boundary conditions (A2.4), (A2.5) gives © 2007 by Taylor & Francis Group, LLC

476

Wetting and Spreading Dynamics

u=−

 α ∂Γ 1 ∂  γ ∂  ∂h    z 2 z. r − zh  − η ∂r  r ∂r  ∂r    2  η ∂r

(A2.7)

In order to derive Equation 5.144, we make use of a condition at the free liquid–air interface: ∂h ∂h + u (r, h ) = v (r, h ) . ∂t ∂r

(A2.8)

After integration of Equation A2.3 over z, from 0 to h, and using the result of integration into Equation A2.8, we get Equation 5.144. Substitution of Equation A2.7 into Equation 5.144 and Equation 5.145 yield Equation 5.146 and Equation 5.147.

APPENDIX 3 Influence of Capillary Forces during Initial Stage of Spreading Let us estimate the inﬂuence of capillary forces during a short initial stage of spreading when it is signiﬁcant. Later on it can be neglected everywhere except for thin boundary layers that we consider negligible. A solution of governing Equation 5.146 and Equation 5.147 is assumed in the following form:  r  h = f  ω , t 

 r  Γ = ϕ ω  , t 

(A3.1)

where f and ϕ are two new unknown functions; and the constant ω is to be determined. Substitution of expressions (A3.1) into Equation 5.146 and Equation 5.147 results in ′    ′ 1   βf 3 (ξ)  γ ω f 2 (ξ )  ′ ξ f ′(ξ ) =  ξ ξ f ′ ξ  − 2 ω ϕ′ ξ   ,  ξ   3t 4 ω  ξ t 2t     

(A3.2)

′    2 ′ f (ξ ) ω β f (ξ )  γ 1  ′   ξ ϕ′(ξ) = ξϕ(ξ) ξ f ′ ξ  − 2 ω ϕ′ ξ  ,   2t 4 ω  ξ ξ t t     

(A3.3)

( ( ))

( ( ))

where ξ = r/t ω. There are two ways to choose ω: © 2007 by Taylor & Francis Group, LLC

()

()

Dynamics of Wetting or Spreading in the Presence of Surfactants

477

(i) If we require t = t4ω or ω = 1/4, then Equation A3.1 and Equation A3.2 become ′    ′ 2 1 1   β f 3 (ξ )  γ  ′ 1/ 2 f (ξ )  ϕ′ ξ  , ξ f ′(ξ ) =  ξ ξ f′ ξ  −t  2 4 ξ   3  ξ     

( ( ))

()

(A3.4)

′    2 ′   f ( ) 1 1 β ξ γ  ′ 1 / 2 ξ ϕ′(ξ) = ξϕ(ξ)  ξ f ′ ξ  − t f (ξ)ϕ′ ξ   . (A3.5)   2  ξ 4 ξ     

( ( ))

()

Equation A3.4 and Equation A3.5 show that the inﬂuence of the surfactant grows with time. (ii) If we require t = t 2ω or ω = 1/2, then Equation A3.2 and Equation A3.3 become ′    ′ 1 1   β f 3 (ξ )  γ f 2 (ξ )  ′  ξ f ′(ξ ) =  ξ ξ f′ ξ  − ϕ′ ξ  ,  2 2 ξ   3t  ξ     

(A3.6)

′   2  ′   f ( ) 1 1 β ξ γ  ′ ξ ϕ′(ξ) = ξϕ(ξ)  ξ f ′ ξ  − f (ξ)ϕ′ ξ   .  2t  ξ  2 ξ     

(A3.7)

( ( ))

( ( ))

()

()

Equation A3.6 and Equation A3.7 show that the inﬂuence of capillary forces decay with time. According to Equation A3.1, the spreading law is rf (t) ≈ t1/4 during the time period when the capillary force inﬂuence is dominant, and rf (t) ≈ t1/2 during the time period when the inﬂuence of the surfactants is dominant. Here, rf (t) marks the location of the moving boundary of the spreading front. It follows from Equation A3.4 and Equation A3.5 that capillary force inﬂuence is dominant if β >> t 1/2, or, t << β2 .

(A3.8)

In the same way, from Equation A3.6 and Equation A3.7 we ﬁnd that the capillary force inﬂuence is negligible, and the inﬂuence of the surfactant is dominant if t >> β . © 2007 by Taylor & Francis Group, LLC

(A3.9)

478

Wetting and Spreading Dynamics

Thus, the capillary force inﬂuence is signiﬁcant during a very short time interval only. As β << 1, we can safely consider just the asymptotic behavior when the surfactant inﬂuence is dominant and condition (A3.9) is satisﬁed. Note that by omitting the capillary force action we neglect the highest derivatives in Equation 5.146 and Equation 5.147; hence, thin layers arise where the capillary force action is of the same order magnitude as the surfactant action.

APPENDIX 4 Derivation of Boundary Condition at the Moving Shock Front Multiplication of Equation 5.156 and Equation 5.157 by r and integration over r from r1 to r2, where r1 < rf (t) < r2, where r1, r2 are some constant values, yields d dt

d dt

r2

∫

rh (t, r ) dr =

r1

r2 h22 ∂Γ 2 r1h12 ∂Γ1 , − 2 ∂r 2 ∂r

r2

∫ rΓ(t, r )dr = r Γ h 2

2 2

r1

∂Γ 2 ∂Γ − r1Γ1h1 1 . ∂r ∂r

(A4.1)

(A4.2)

Where we use the following abbreviation: fi = f (ri). The left-hand side of Equation A4.1 and Equation A4.2 can be transformed in the following way: d dt

r2  r f (t )  d rf (t , r )dr = rf (t , r )dr + rf (t , r )dr  =  dt  r1 r f (t )  r1 

r2

∫

∫

r f (t )

rf rf f− +

∫

r1

∫

∂f (t , r ) r dr − r f r f f+ + ∂t

r2

∫r

r f (t )

∂f (t , r ) dr , ∂t

where f± = f (t, rf ±). If we now consider limits r1 tends to rf from below (↑), and r2 tends to rf from above (↓) when both integrals on the left-hand side of the equation vanish. Then, from Equation A4.1 and Equation A4.2, using the same limits r1 ↑ rf , r2 ↓ r f , we conclude that

(

)

(

)

rf h− − h+ =

1  2 ∂Γ + ∂Γ  − h−2 −  , h+  2 ∂r  ∂r

rf Γ − − Γ + = Γ + h+

∂Γ + ∂Γ − Γ − h− − , ∂r ∂r

which are the required boundary conditions at the shock front. © 2007 by Taylor & Francis Group, LLC

(A4.3)

(A4.4)

Dynamics of Wetting or Spreading in the Presence of Surfactants

479

APPENDIX 5 Matching of Asymptotic Solutions at the Moving Shock Front Let us introduce a new local variable ς=

ξ−ν , χ(t )

where χ(t) << 1 is a new unknown length scale to be determined. Neglecting the second curvature in Equation 5.146 and Equation 5.147, and introducing unknown functions in the following form Γ = χ(t)Φ(ξ), h = F(ς), we get 1/ 3

 β χ(t ) =   . t

(A5.1)

It follows from this equation that χ(t) → 0, at t → ∞. Unknown functions Φ(ς) and F(ς) obey the following equations: ′  2 F 3F ′′′ νF ′ =  − F 2Φ′  ,   3

(

)

′ νΦ′ = ΦF 2 F ′′′ − 2 F ΦΦ′ .

(A5.2)

(A5.3)

After integration of Equation A5.2 and Equation A5.3 with boundary conditions F → 1, Φ → 0, at ς → ∞, we get

(

)

ν F −1 =

2 3 F F ′′′ − F 2Φ′, 3

(

)

(A5.4)

Φ ν − F 2 F ′′′ + 2 F Φ′ = 0.

(A5.5)

Φ = 0,

(A5.6)

Thus, either

and F ′′′ = © 2007 by Taylor & Francis Group, LLC

3ν F − 1 , 2 F3

(A5.7)

480

Wetting and Spreading Dynamics

or F ′′′ =

(

3ν F − 2 F3

),

(A5.8)

and

(

ν 3− F

Φ′ =

F2

).

(A5.9)

From Equation A5.8 and Equation A5.9 we conclude that ν Φ′ → − , 4

F → 2,

ς → −∞.

at

(A5.10)

Equation A5.10 shows that boundary conditions (5.166) at the shock front are the only possible conditions that can be matched with the inner solution. From the above derivation we conclude that in the boundary layer  β Γ=  t

1/ 3

()

Φ ς ~ β1/ 3;

hence, vanishes from the point of view of the outer solution. Thus, both conditions (5.160) and (5.161) must be satisﬁed at the shock front.

APPENDIX 6 Solution of the Governing Equations for the Second Stage of Spreading Putting β = 0 in Equation 5.176 and Equation 5.177, we get f '(ξ) =

2 f (ξ ) , ξ

ϕ '(ξ) = −

ξ , 4 f (ξ )

(A6.1)

with boundary conditions that follow from Appendix 4, Equation A4.3, and Equation A4.4, f (λ) = 2,

ϕ(λ) = 0,

ϕ '(λ) = −

λ . 8

(A6.2)

Then, the solution of Equation A6.1 is 2

 ξ f ξ = 2  ,  λ

()

© 2007 by Taylor & Francis Group, LLC

()

ϕ ξ =−

λ2 ξ ln . 8 λ

(A6.3)

Dynamics of Wetting or Spreading in the Presence of Surfactants

481

Substitution of f(ξ) in Equation 5.178 gives λ = 2 5/ 4 .

(A6.4)

Note that the solution (A6.3) satisﬁes Equation 5.176 to Equation 5.178 at arbitrary β.

5.7 SPREADING OF AQUEOUS DROPLETS INDUCED BY OVERTURNING OF AMPHIPHILIC MOLECULES OR THEIR FRAGMENTS IN THE SURFACE LAYER OF AN INITIALLY HYDROPHOBIC SUBSTRATE Let us study in this ﬁnal section the spontaneous spreading of a drop of a polar liquid over a solid when the amphiphilic molecules (or their amphiphilic fragments) of the substrate surface layer may overturn, creating hydrophilic parts on the surface. Such a situation may occur, for example, during the contact of an aqueous drop with the surface of a polymer whose macromolecules have hydrophilic lateral groups capable of rotating around the backbone, or during the wetting of polymers containing surface-active additives or Langmuir–Blodgett ﬁlms composed of amphiphilic molecules. It is shown in the following discussion that drop spreading is possible only if there is lateral-side interaction between neighboring amphiphilic molecules (or groups). This interaction leads to tangential transfer of the “overturned state” to some distance ahead of the advancing three-phase contact line, making it partially hydrophilic. This kind of “selforganization” of its surface layer lowers the interfacial free energy due to the emergence (or adsorption) of polar groups at the surface. The quantitative theory describing the kinetics of droplet spreading is developed in the following text with allowance for this mechanism of self-organization of the surface layer of a substrate in contact with a droplet [20]. A number of researches have been published (see, for example, Reference 21 to Reference 24) demonstrating that interaction of a polymer with a polar liquid (ﬁrst of all, with water and aqueous solutions) may result in the spontaneous rearrangement (self-organization) of its surface layer, providing the minimization of the interfacial free energy due to the emergence (or adsorption) of polar groups at the surface. Depending on the structure of the polymer macromolecules, the physical state of the polymer, and other factors, such a rearrangement may involve various forms of molecular motion, from the reorientation of individual amphiphilic side groups (when their rotation around the backbone is allowed) to the diffusion of macromolecules as a whole. Each of these processes may occur on different time scales. If the characteristic time scale of self-organization is comparable with the time of measurement, this process may be observed while studying the contact angle (and size) of an aqueous droplet on time of contact with a polymer [25]. Such a prolonged spreading of aqueous droplets occurs also during the study of the wettability of model systems such as apolar polymers © 2007 by Taylor & Francis Group, LLC

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Wetting and Spreading Dynamics

containing low-molecular-weight amphiphilic additives (surfactants) capable of adsorption and reorientation at the polymer–liquid interface [25]. Langmuir was the ﬁrst to mention the possible reorientation (overturning) of amphiphilic molecules in contact with a polar liquid [26]. Later, experimental data were obtained that demonstrated the occurrence of such a process in mono layers and Langmuir–Blodgett ﬁlms composed of long-chain fatty acids [27] and at the surface of mixtures of such acids with parafﬁn [28] during contact with an aqueous droplet, resulting in gradual droplet spreading. To start, the rate of spreading was analyzed in terms of formal chemical kinetics [27,28], and the mechanism of the process (the dynamic situation in the vicinity of the three-phase contact line) was not considered at all. Recently, such a situation has also been observed for the spreading of a droplet over the surfaces of polymers, when the amphiphilic groups of their macromolecules are capable of reorientation by rotating around the macromolecule backbone. The aim of this section is to analyze the mechanism of the spontaneous spreading of a droplet of a polar liquid induced by the overturning of amphiphilic molecules (or their fragments) in the surface layer of a solid substrate and to develop a quantitative theory describing this process.

THEORY

AND

DERIVATION

OF

BASIC EQUATIONS

Let us consider the spreading of an aqueous droplet over a solid substrate, ignoring the evaporation of the liquid, i.e., assuming that the droplet volume remains constant during spreading. It is assumed also that the substrate is smooth, horizontal, and (which is important for further discussion) contains in the surface layer rotationally mobile amphiphilic molecules (or amphiphilic fragments of molecules; for brevity, we hereafter mention only molecules) capable of overturning in the plane perpendicular to the surface and incapable of lateral motions in the plane of the substrate. Hence, each of the amphiphilic molecules may be in one of two states: (1) nonoverturned (normal), i.e., when the hydrophilic head group of the molecule is oriented downward into the substrate whereas the hydrophobic tail is directed upward into the second phase (air or water) in contact with the substrate; and (2) overturned state, i.e., in the opposite (as compared to the previous case) orientation of hydrophilic and hydrophobic moieties of the molecule (Figure 5.29). Let N ∞ and N be the total number of amphiphilic molecules per surface area of the substrate capable of overturning and the number of overturned molecules, respectively; then p = N/N∞ is the probability to ﬁnd an overturned molecule. Let us consider the process of transition of a system to the equilibrium state during the contact of the substrate surface with the air, provided that initially the system was somehow disturbed from an equilibrium state. Let us consider for simplicity a linear chain of amphiphilic molecules, because a switch to the two-dimensional case is self evident. Let pi (t) be the probability of the occurrence of molecule i at the instant t in the overturned state; then, the same probability at the time t + ∆t will be equal to

© 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

483

Water or air V

W

2-overturned state

Substrate

1 normal state S

FIGURE 5.29 Polymeric substrate containing rotationally mobile amphiphilic chains in contact with air or water; v — air, w — water, s — polymeric substrate. 1 — amphiphilic chains in the normal state, 2 — amphiphilic chains in the overturned state.

(

)

()

pi t + ∆t = pi t + PSV ∆t − PV ∆t + Pd ∆t − Pb ∆t ,

(5.184)

where P with subscripts are the probabilities of direct and reverse overturnings per unit time: Pv is the overturning probability caused by the interaction with the air of the amphiphilic molecule that has not been overturned previously; Psv is the overturning probability caused by the interaction with underlying molecules the molecule that was overturned previously; and Pd and Pb are the probabilities corresponding to the direct and the reverse overturnings of a molecule due to its interaction with neighboring molecules. Only the interactions with the nearest neighbors are accounted for, i.e., with the molecules having numbers i + 1 and i – 1. All the probabilities in Equation 5.184 may be written in the following form:

(

)

Pv = α v 1 − pi ,

(5.185)

Psv = α sv pi ,

(5.186)

(

)

(

)

Pd = β  pi−1 1 − pi + pi+1 1 − pi  ,

(

)

(

)

Pb = β  1 − pi−1 pi + 1 − pi+1 pi  ,

(5.187)

(5.188)

where constants αv , αsv , and β entering into the deﬁnitions of probabilities in Equation 5.185 and Equation 5.188 have the dimensionality of reciprocal time and are determined using the energy of molecular interaction with surrounding phases in overturned and nonoverturned states:

© 2007 by Taylor & Francis Group, LLC

484

Wetting and Spreading Dynamics

α v = α exp  − Φ v  ,

(5.189)

α sv = α exp  Φ v  ,

(5.190)

()

β = α  exp χ − 1 ,

(5.191)

where χ=z

2U th − U tt U + U hv − U tv − U hs ; , Φ v = ts RT RT

subscripts t, h, v, and s correspond to the hydrophobic tail of a molecule, to its hydrophilic head group, to the air, and to the underlying molecules of the substrate, correspondingly; α denotes the corresponding values in the absence of any interaction, i.e., determined only by thermal ﬂuctuations; and z is the number of neighboring molecules, i.e., z = 2 for a one dimension, and z = 4 for the twodimensional case. It follows from deﬁnitions (5.189) and (5.190) that αv << αsv . Let us consider Equation 5.191 in more detail. If expressions (5.189) and (5.190) involve both overturnings due to the thermal ﬂuctuations and those caused by the interaction with the surrounding media, then, in contrast to these expressions, Equation 5.191 involves only the overturnings related to the interactions between the neighbors; hence, random overturnings caused by thermal ﬂuctuations should not be taken into account because they do not result in the transfer of the overturned state. This is why the unity is subtracted in Equation 5.191. Substituting expressions (5.185) and (5.188) into Equation 5.184 yields

(

)

( )=α

pi t + ∆t − pi t

{

∆t

()

v

()

()

1 − pi t  − α sv pi t  

()

()

( )}

()

+β  pi −1 t + pi +1 t  1 − pi t  −  2 − pi −1 t − pi +1(t )  pi t   Taking the limit in the last expression at ∆t → 0 results in

( )=α

dpi t dt

v

()

()

()

()

()

1 − pi t  − α sv pi t + β  pi +1 t + pi −1 t − 2 pi t  . (5.192)    

Let a be the mean distance between molecules capable of overturning. Then, Equation 5.192 may be rewritten in the following form: © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

( )=α

dpi t dt

v

()

()

485

()

 pi+1 t + pi−1 t − 2 pi t   . (5.193) 1 − pi t  − α sv pi t + βa 2    a2

()

()

The latter part of Equation 5.193 is a discrete analogy of the second-order spatial derivative. Hence, using the continuous coordinate x ≈ ia, we obtain the following second-order partial differential equation instead of Equation 5.193:

( )

∂ 2 p t, x ∂p = α v 1 − p − α sv p + D , ∂t ∂x 2

(

)

(5.194)

where p(t, x) is the probability of the occurrence of an overturned molecule at time t at point x, and D = β a2 is the effective diffusion coefﬁcient of the moleculeoverturned state along the surface. The extension of Equation 5.194 to the plane (two-dimensional) case of our further interest is, as was stated before, a straightforward procedure: it is reduced to the simple substitution of the partial second-order derivative with respect to one coordinate x for the sum of the partial second-order derivatives with respect to x and y, because, in the two dimensional case, p = p(t, x, y). According to our previous consideration, the transfer of the overturned state is determined only by the interactions between adjacent molecules and should vanish in two cases: (1) in the absence of interactions between adjacent amphiphilic molecules, i.e., at χ = 0 (in this case, β = 0 and hence D = 0); and (2) upon unlimited increase in the distance a between adjacent molecules, i.e., when the surface concentration of molecules capable of overturning tends to zero; in this case, interactions between adjacent molecules also vanish. Indeed, let us assume that Uth, Utt , and U hh are determined by dispersion interactions only. In this case, χ(a) ~ B/a6, where B is a constant expressed as usual via the polarizabilities of hydrophilic head groups and hydrophobic tails: B = (2Ath – Att – Ahh)z/3, where Ath, Att, and Ahh are the corresponding Hamaker constants. Consequently, D(a) ~ β[expB/a6 – 1]a2 ~ βB/a4 → 0, with an increase in distance a between adjacent amphiphilic molecules. In the equilibrium state, the probability p does not depend either on time or coordinate; this equilibrium state is further denoted by pv , which is readily determined from Equation 5.194: pv =

αv 1 = . α v + α sv 1 + α sv /α v

From deﬁnitions (5.189) and (5.190), we obtain that α sv = exp 2 Φ v >> 1 , αv

(

i.e., pv is a small value. © 2007 by Taylor & Francis Group, LLC

)

(5.195)

486

Wetting and Spreading Dynamics

Let us discuss now the events occurring underneath the aqueous droplet at the solid–water interface. In this case, instead of equation 5.184, we arrive at

(

)

()

pi t + ∆t = pi t + Pw ∆t − Psw ∆t + Pd ∆t − Pb ∆t ,

(5.196)

where the subscript w refers to water. Expressions for the probabilities per unit time Pw and Psw are similar to those occurring previously, with corresponding constants α w and α sw , which can be obtained from relationships (5.185) and (5.186), and (5.189) and (5.190), respectively, by substituting the subscript w instead of v. Considerations similar to those used for deriving Equation 5.194 results, in this case, in the following equation:

( )

∂ 2 p t, x ∂p = α w 1 − p − α sw p + D . ∂t ∂x 2

(

)

(5.197)

The equilibrium value of the probability of the occurrence of an amphiphilic molecule in the overturned state, pw =

αw , α w + α sw

(5.198)

is determined, as in the case of contact with the air, from equation 5.197. Unlike the case of contact with the air, the pw probability is not a small value; on the contrary, it is close to one. It is this difference in probabilities that provides for the possibility of the aqueous droplet spreading over the initially hydrophobic surface. Note that, in the absence of lateral interactions between adjacent amphiphilic molecules, the aqueous droplet may not spread over the surface under consideration despite the effect of overturning of molecules with the hydrophilic portions upward, underneath the water. Indeed, in the absence of interactions between adjacent molecules, let the necessary quantity of molecules be overturned with their hydrophilic portions upward (Figure 5.30a), and the substrate surface underneath the aqueous droplet become sufﬁciently hydrophilic so that the droplet edge can move into the new position presented in Figure 5.30b. However, in the absence of lateral transfer of the overturned state of the amphiphilic molecules in the substrate, the surface both in front of the droplet edge in Figure 5.30b and outside the edge are still in the initial hydrophobic state, thus forcing the droplet edge to return immediately to the initial position (Figure 5.30a). Thus, in the absence of the lateral transfer of the overturned state described by the diffusion term in Equation 5.194 and Equation 5.197, the spreading of the aqueous droplet over the surface becomes impossible. However, if the adjacent amphiphilic molecules © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

487

(a)

(b)

FIGURE 5.30 Impossibility of spreading of a water droplet on a hydrophobic substrate without lateral interaction between neighboring chains (explanation in the text).

interact with each other and the lateral transfer of the overturned state due to these interactions is possible, the droplet edge moves over the surface, and this motion is determined exactly by the rate of the lateral transfer of the overturned state of the substrate molecules.

BOUNDARY CONDITIONS According to the theory described above, the propagation of the overturned state of amphiphilic molecules along the surface under the droplet is described by Equation 5.197, and the propagation beyond the droplet by Equation 5.194. It is now required to formulate the boundary conditions for the probability, p(t, r), at the boundary of the spreading axisymmetric droplet, i.e., at r = r0 (t) (Figure 5.31). In view of the assumption of the absence of evaporation, the droplet volume, V, remains constant during the spreading, and it is assumed also that the droplet is small enough, that is, the gravity action may be neglected. The time scale of the spreading process is so big (see the following discussion) that the deformations of the drop proﬁle caused by the spreading or ﬂow can be neglected: this means that the capillary number is enormously low, Ca << 1. According to the Introduction to Chapter 3, the droplet retains the shape of a spherical segment during the spreading V=

(

)

π h 3r02 + h 2 = const , 6

(5.199)

where h and r0 are the maximum height and radius of the base of the spreading droplet, respectively (Figure 5.31). The droplet height, h, is determined by relationship h = r0 tan(θ/ 2), where θ(t ) is the current value of the contact angle of the spreading droplet. Substituting this relationship into equation 5.199, we express © 2007 by Taylor & Francis Group, LLC

488

Wetting and Spreading Dynamics h γ γsv

θ

γsw p pw pin pv

pout d

x

r0(t) dout

FIGURE 5.31 Spreading of a spherical droplet: r0 (t) — radius of the base of a drop, θ(t) — dynamic contact angle, h — height at center of a drop.

the radius of the base of the spreading droplet via the current value of the contact angle as  6V  r0 t =   π 

()

1/ 3

1   θ 2 θ  tan 2  3 + tan 2    

1/ 3

.

(5.200)

Let us denote surface (interfacial) tensions for the water–air (constant value), substrate–air, and substrate–water interfaces directly at the boundary of the spreading droplet by γ , γ sv (t ), and γ sw (t ), respectively. Note that the values of γ sv (t ) and γ sw (t ) near the droplet edge differ from the constant values of surface tensions at the polymer–air and polymer–water interfaces far from the edge of the spreading droplet and in the depth of the droplet, respectively. It is assumed that Young’s equation is satisﬁed at any moment at the boundary of the spreading droplet cos θ =

γ sv (t ) − γ sw (t ) = cos θ0 + ∆(t ) , γ

where

cos θ0 =

γ 0 sv − γ 0 sw , γ

© 2007 by Taylor & Francis Group, LLC

∆(t ) =

γ sv (t ) − γ 0 sv γ sw (t ) − γ 0 sw − . γ γ

(5.201)

Dynamics of Wetting or Spreading in the Presence of Surfactants

489

In this equation, the superscript 0 marks corresponding initial values of interfacial tensions. Note that the ∆(t) value is positive and increases with time because the γsv(t) value rises in time due to possible appearance of hydrophilic head groups of the amphiphilic molecules at the substrate surface; in contrast, the γsw(t) value decreases with time due to overturning of molecules under the droplet. Let us emphasize once more that 1. In view of the lateral interaction between adjacent amphiphilic molecules of the substrate, the overturned state may be extended beyond the boundary of the spreading droplet, resulting in an increase in surface tension of the substrate, γsv(t), in front of the moving droplet (Figure 5.31). 2. Interfacial tensions γsv(t) and γsw(t) do not remain constant near the droplet boundary but vary depending on the coordinate in a close vicinity of the boundary of the moving droplet (see the following discussion). Hence, interfacial tensions in the close vicinity of the edge of the moving droplet, or, in a more formal manner, the limiting values of these tensions at r → r0(t) from the inner and outer droplet sides, enter Young’s equation 5.201. The corresponding limits of the degree of overturning inside and outside the droplet are denoted by pin and pout, respectively (Figure 5.31). Evidently, the ∆(t) value is not an explicit function of time but depends on time in an implicit manner via the values of pin, pout, i.e., ∆ = ∆ (pin, pout). As shown, in view of the equality of chemical potentials of amphiphilic molecules between the inner and outer boundaries of a droplet, the pout value is expressed as a function of pin. Hence, in view of the latter dependence, actually ∆ = ∆ (pin). Let us use Antonov’s rule to determine unknown dependence ∆ (pin, pout), which means the additivity of the formation of the interfacial tensions. Let γ ∞sw and γ 0sw be the interfacial tensions under the droplet in the case when all molecules are overturned (all hydrophilic head groups are oriented upward) and when neither of these molecules is overturned (all hydrophilic head groups are oriented downward), respectively. Similar surface tensions outside the droplet are denoted by γ ∞sv and γ 0sv , respectively. According to the assumption of the additivity, the interfacial tensions in the closest vicinity of a droplet acquire the following form: ∞ γ sv = γ 0sv (1 − pout ) + γ sv pout , 0 γ sw = γ sw (1 − pin ) + γ ∞sw pin .

(5.202)

Substitution of relationships (5.202) into Young’s equation (5.201) yields the following expression for the dependence ∆(pin, pout) under consideration: © 2007 by Taylor & Francis Group, LLC

490

Wetting and Spreading Dynamics

∆( pin , pout ) =

∞ γ sv∞ − γ 0sv γ 0 − γ sw pout + sw pin . γ γ

(5.203)

The obtained dependence ∆(pin, pout) is a linear function with respect to both variables. As mentioned previously, the spreading process under consideration is very slow (the time scale is hours). This allows employing the principle of local equilibrium accepted in nonequilibrium thermodynamics. In accordance with this principle, chemical potentials of overturned and nonoverturned amphiphilic molecules remain equal from both sides of the droplet boundary, i.e., µ ( pin , χ) = µ ( pout , χ) + ℑ,

ℑ=

U hv − U hw . RT

(5.204)

The µ(p,χ) dependence may be given, for example, in accordance with the Flory–Huggins theory [29] in the following form: µ(p,χ) = ln p + χ (1 – p)2. In the following discussion we consider only two limiting cases of weak and strong lateral interactions between amphiphilic molecules. At this stage, it is enough to take into account that, according to the equality of chemical potentials (5.204), pout and pin are interrelated by the known dependence pout = ϕ(pin). In view of all that has been said previously, the value of ∆ is dependent on only one variable, i.e., ∆ = ∆(pin). In view of the equality θ  1 − cos θ  tan =  2  1 + cos θ 

1/ 2

from Equation 5.200, we obtain 1/ 3

 6V  r0 ( pin ) =  · G ( pin ),  π 

(5.205)

where G ( pin ) =

1  1− A   1 + A 

1/ 6

 1− A   3 + 1 + A 

A( pin ) = cos θ0 + ∆ ( pin ).

© 2007 by Taylor & Francis Group, LLC

1/ 3

, (5.206)

Dynamics of Wetting or Spreading in the Presence of Surfactants

491

Equation 5.205 allows determining the ﬁnal droplet radius, r∞, at the end of the spreading process using the known dependence G(pin). To this end, it is necessary to substitute the expression for the equilibrium fraction of overturned molecules under droplet pw from relationship (5.198) into Equation 5.205, which yields 1/ 3

 6V  r∞ =  · G ( pw ).  π 

(5.207)

It is possible to verify that G(p) is an increasing function of p; i.e., r0(t) increases, approaching its ﬁnal value determined by Equation 5.207. Let us now consider the formulation of boundary conditions on a moving droplet edge. Two boundary conditions are required because the edge itself moves, and the law of this movement should be determined. The ﬁrst boundary condition expresses the balance of the number of overturned molecules at the boundary of the moving droplet, and it has the following form: −D

∂p ∂p d r (tt ) +D = ( pin − pout ) 0 , ∂ r r =r (t ) − ∂ r r =r (t ) + dt 0

(5.208)

0

and it is necessary to set the second boundary condition relating the pin and pout values. The other boundary conditions are straightforward: the symmetry in the droplet center ∂p ∂r

=0,

(5.209)

r= 0

and the tendency of the fraction of overturned molecules far from the droplet to the equilibrium value at the substrate–air interface determined from Equation 5.195 as p → pv , r → ∞.

(5.210)

We use the condition of equality of chemical potentials of overturned molecules to the right- and left-hand sides of the moving boundary of a droplet, r0(t), which determines the dependence ∆µ in( pin) = ∆µ out ( pout ).

© 2007 by Taylor & Francis Group, LLC

(5.211)

492

Wetting and Spreading Dynamics

To analyze the dependences of chemical potentials, we employ the expressions resulting from the Flory–Huggins theory [29] modiﬁed to take into account the interactions of amphiphilic molecules with the environment and the underlying molecules of a substrate, namely, interactions of their hydrophilic head groups with water and their hydrophobic tails with the substrate molecules (under the droplet), and interactions of hydrophilic head groups with the air and hydrophobic tails with underlying substrate molecules (outside the droplet). This leads to the following expression: ln pin + χ (1 − pin )2 = ln pout + χ (1 − pout )2 + ℑ ,

(5.212)

where χ is the known parameter of interaction of amphiphilic molecules with each other according to expression (5.191); in the case under consideration, the value of z is equal to 4. In the case of weak interactions between adjacent molecules, i.e., at χ << ℑ , from expression (5.212) we conclude that ln pin = ln pout + ℑ . As a result, we arrive at the Boltzmann distribution pout = pin exp(−ℑ) .

(5.213)

In the case of strong interaction between the neighboring molecules, i.e., χ >> ℑ, we obtain ln pin + χ(1 − pin )2 = ln pout + χ(1 − pout ) .

(5.214)

Hence, it results in equality of overturned fractions: pout = pin. The value of pout is always smaller than or equal to pin, thus enabling us to solve Equation 5.212 for the arbitrary case and to express pout as a function of pin, which was stated before by Equation 5.211. Thus, the problem of droplet spreading acquires the following form: the dependence p(t, r) under the droplet at 0 < r < r0(t) is described by the equation ∂p 1 ∂ ∂p = α w (1 − p ) − α sw p + D r , ∂t r ∂ r ∂r

(5.215)

and the dependence p (t, r) outside the droplet, r > r0 (t), is described by the equation ∂p 1 ∂ ∂p = α v (1 − p ) − α sv p + D r , ∂t r ∂ r ∂r

© 2007 by Taylor & Francis Group, LLC

(5.216)

Dynamics of Wetting or Spreading in the Presence of Surfactants

493

with the following boundary conditions: condition of symmetry in the droplet center (5.209), condition (5.210) far from the droplet, condition (5.205) expressing the radius of the spreading droplet via pin, condition (5.208) expressing the equality of ﬂuxes at the moving boundary of the droplet r = r0(t), and relationship (5.211) expressing the equality of chemical potentials of overturned molecules near the moving edge of the droplet. Solution of the Problem We perform the solution of the preceding problem introducing a number of simplifying assumptions whose validity is checked in the following text. It is obvious that the value of p under the main part of the spreading droplet is independent of the coordinate but changes only with time due to the interaction of the amphiphilic molecules with the aqueous phase. Let us denote this coordinate-independent value by pd(t), which, according to Equation 5.215, satisﬁes the following equation: d pd = α w (1 − pd ) − α sw pd , dt with the initial condition p d (0 ) = pv . The solution of the problem is pd (t ) = pw + ( pv − pw ) exp(−(α w + α sw )t ).

(5.217)

It follows from Equation 5.217 that the characteristic time scale of molecule overturning ttr* is equal to ttr* = 1/(αw + αsw). In a narrow region with width δ near the droplet edge (from the inner side), the diffusion term in Equation 5.215 becomes of the same order of magnitude as the term describing the overturning of molecules due to their interaction with water. Let us introduce dimensionless values y = (r0(t)r)/δ, λ = αs/αw; τ = t/t*; ξ(t) = r(t)/r*, where r* = (6V/π)1/3, and new unknown function g (t, y) = p (t, r) – pd (t). The time scale t* is selected below. Rewriting Equation 5.215 in dimensionless form using the introduced variables, we obtain ∂g D ∂2 g = − α w + α sw g + 2 2 . ∂t δ ∂y

(

)

Accounting for the smallness of δ/r*, we conclude that ∂ g 1 ∂ g ∂ g r* ∂ g r* = + ξ≈ ξ. ∂ t t* ∂ τ ∂y δ t* ∂ y δ t*

© 2007 by Taylor & Francis Group, LLC

(5.215)

494

Wetting and Spreading Dynamics

In this case, Equation 5.215 acquires the following form:

( )

( )

2 r∗ ∂g τ, y D ∂ g τ, y . ξ = − α w + α sw g τ , y + 2 δt∗ ∂y ∂y 2 δ

(

) ( )

(5.218)

All terms in Equation 5.218 should be of the same order of magnitude; hence, the characteristic values are interrelated by the following relationships

(

)

δ = D α w + α sw , t∗ = r∗δ /D. Let us compare ttr* and t*: t∗ /ttr ∗ = t∗ (α w + α sw ) = r∗ (α w + α sw ) /D = r∗ /δ >> 1; i.e., t* is much larger than the characteristic time of overturning of amphiphilic molecules under the main portion of the droplet. This implies that under the droplet, p = pw , and changes occur only in the narrow region with the width δ and are described by the equation

( )

( )

∂g τ, y ∂2 g τ, y , ξ = − g τ, y + ∂y ∂y 2

( )

(5.219)

with boundary conditions g → 0, y → ∞ ,

(5.220)

g(0) = pin – pw .

(5.221)

and

As the desired function g(τ, y) depends on τ as a parameter, the solution of Equation 5.219 satisfying conditions (5.220) and (5.221) may be readily obtained, and the expression for p acquires the form  ξ − ξ 2 + 4 p = pw + pin − pw exp  2 

(

)

 y . 

(5.222)

Let us perform similar transformations in the narrow region from the outer side of the droplet front

(

)

(

)

δ out = D α v + α sv , yout = r − r0 (t ) δ out , q = p(t , r ) − pv . © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

495

In this case, the dimensionless equation describing the probability p in the δout region outside the droplet has the following form:

( )

( )

∂2q τ , y α w + α sw ∂q τ , y − , ξ = q τ, y + 2 α v + α sv ∂yout ∂yout

( )

(5.223)

with boundary conditions q → 0, yout → +∞ ,

(5.224)

q(0) = pout − pv .

(5.225)

and

Assuming that αw ∼ αsv >> αv ∼ αsw , we can conclude that δ = δout , and from Equation 5.223 we obtain  ξ + ξ 2 + 4  p = pv + pout − pv exp  yout  2  

(

)

(5.226)

Let us rewrite boundary condition (5.208) to dimensionless form: D ∂p D ∂p r + = * pin − pout ξ . δ ∂ y y= 0 δ ∂ y y= 0 t*

(

−

)

(5.208′)

+

Using expressions (5.222) and (5.226), we arrive at an equation describing the motion of the droplet boundary: ( pin − pw )

ξ −

ξ 2 + 4 ξ + ξ 2 + 4 − ( pout − pv ) = ( pin − pout ) ξ . 2 2

(5.227)

Equation 5.227 has the following solution: ξ =

pw − pin − pout + pv ( pw − pout ) ( pin − pv )

.

(5.228)

Using condition (5.205), we obtain d ξ 1 d r0 d d G d pin ξ = = = G ( pin ) = . d τ r* d τ d τ d pin d τ © 2007 by Taylor & Francis Group, LLC

(5.229)

496

Wetting and Spreading Dynamics

Let us rewrite Equation 5.228 taking into account representation (5.229): d pin pw − pin − pout + pv = dτ G ′ pin pw − pout pin − pv

(

(

)(

))

.

(5.230)

Assuming that pout ∼ pv << pin, and pw ∼ 1, we have the ﬁnal equation for pin

(

)

1 − pin t* d pin = . dt G ′ pin pin

( )

(5.230′)

Taking into account the smallness of pout as compared to one, we obtain from Equation 5.202 that the value of γsv changes slightly and remains close to the initial value γ 0sv. In this case, expression (5.203) may be rewritten in the following form:

∆=

∞ ∞  γ 0 − γ sw γ 0sw − γ sw γ 0 − γ 0sw  ∞ 0 pin =  sv − sv  pin = (cos θ − cos θ ) pin , (5.231) γ γ γ  

and the function A(pin) acquires the form A( pin ) = cos θ0 + (cos θ∞ − cos θ0 ) pin .

(5.232)

Thus, according to the proposed theory, the droplet spreads in a completely different manner from what was suggested in Reference 27; i.e., the equilibrium concentration of overturned amphiphilic molecules (or their fragments) is established rapidly (as compared to the characteristic time of spreading) under the main portion of the drop and retains its value over the course of the entire spreading process. All changes occur only within the narrow region in the vicinity of the perimeter of the spreading droplet.

COMPARISON

BETWEEN

THEORY

AND

EXPERIMENTAL DATA

Equation 5.230 contains one unknown parameter, t*, which is the characteristic time of the propagation of the overturned state, i.e., the characteristic time scale of droplet spreading. Parameter t* was used as an adjustment parameter for the comparison of the theoretical predictions according to the described theory and experimental data reported elsewhere [27,28]. In Figure 5.32, the time dependences of the contact angle of an aqueous droplet at the surface of parafﬁn containing stearic acid at various concentrations, C, are presented. Lines show the solutions of Equation 5.230, and the symbols denote © 2007 by Taylor & Francis Group, LLC

Dynamics of Wetting or Spreading in the Presence of Surfactants

497

120

1

110

Contact angle, deg

2 100

3

90

80

70

60 0

5

10

15

20

Time, h

FIGURE 5.32 Time dependences of contact angle, θ, of water droplets at the surface of parafﬁn containing stearic acid of different concentrations in wt% (experimental data from Reference 27): (1) C = 0.6; (2) C = 2.0; (3) C = 9.0. Solid lines according to Equation 5.230.

experimental data [28] for three different concentrations, C. The values of t* were found as follows: t* = 50.5 h at C = 0.6 wt %, t* = 27.5 h at C = 2.0 wt %, t* = 24.0 h at C = 9.0 wt %. These values show that, as the concentration of stearic acid increases, the characteristic time of propagation of the overturned state decreases. This is due to an increase in diffusion coefﬁcient, D, because of the decreasing average distance between acid molecules capable of overturning. In Figure 5.33, the time dependences of the contact angle of aqueous drops at a Langmuir–Blodgett ﬁlm composed of stearic acid at various temperatures are presented. Here, symbols represent experimental data from Reference 27, and lines drawn correspond to the solutions of Equation 5.230′. The deduced dependence of parameter t* on temperature is shown in Figure 5.34. Characteristic time t* of the propagation of the overturned state decreases with temperature, which may be explained by an increase in the rotational mobility of molecules capable of overturning, and simultaneously, the diffusion coefﬁcient, D, of the overturned state of stearic acid molecules increases. © 2007 by Taylor & Francis Group, LLC

498

Wetting and Spreading Dynamics 110 1 2

Contact angle, deg

100

3 4

90

5 80

70 0

10

20

30

40

50

60

Time, h

FIGURE 5.33 Time dependences of the contact angle, θ, of water droplets at the surface of a Langmuir–Blodget ﬁlm formed by a stearic acid at various temperatures in °C. Experimental data from Reference 28. (1) t* = 13.5 h; (2) t* = 15.5 h; (3) t* = 23.0 h; (4) t* = 25.0 h; (5) t* = 28.5 h. Solid lines according to Equation 5.230′.

200

t∗ , h

150

100

50

0 12

16

20 24 Temperature, °C

28

FIGURE 5.34 Dependence of characteristic time of a spreading of drops, t*, on temperature.

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499

REFERENCES 1. Zhdanov, S., Starov, V., Sobolev, V., and Velarde, M., J. Colloid Interface Sci., 264, 481–489, 2003. 2. Zolotarev, P.P., Starov, V.M., and Churaev, N.V., Colloid J. (Russian Academy of Sciences, English Translation), 38, 895, 1976. 3. Churaev, N.V., Martynov, G.A., Starov, V.M., and Zorin, Z.M., Colloid Polym. Sci., 259, 747, 1981. 4. Starov, V., J. Colloid Interface Sci., 270, 180, 2003. 5. Zorin, Z.M., Iskandaryan, G.A., and Churaev, N.V., Colloid J. (Russian Academy of Sciences, English Translation), 40, 671, 1978. 6. Berezkin,V.V., Deryagin, B.V., Zorin, Z.M., Frolova, N.V., and Churaev, N.V., Dokl. AN SSSR, 225,109, 1975. 7. Berezkin, V.V., Zorin, Z.M., Iskandaryan, G.A., and Churaev, N.V., Trans. VIIth Int. Congr. Surfactants, B2, 329, Moscow 1978. 8. Berezkin, V.V, Zorin, Z.M., Frolova, N.V., and Churaev, N.V., Colloid J. (Russian Academy of Sciences, English Translation), 37, 1040, 1975. 9. Shinoda, K., Nakagawa, T., Tamamushi, B., and Isemura, T., Colloidal Surfactants: Some Physico-Chemical Properties, Academic Press, New York, 1963. 10. Starov, V., Spontaneous rise of surfactant solutions into vertical hydrophobic capillaries, J. Colloid Interface Sci., 270, 180–186, 2003. 11. Starov, V.M., Zhdanov, S.A., and Velarde, M.G., Capillary imbibition of surfactant solutions in porous media and thin capillaries: partial wetting case, J. Colloid Interface Sci., 273(2), 589–595, 2004. 12. Levich, V.G., Physicochemical Hydrodynamics, Prentice Hall, Englewood Cliffs, NJ, 1962. 13. Starov, V.M., de Ryck, A., and Velarde, M.G., J. Colloid Interface Sci., 190, 104, 1997. 14. Starov, V.M., Kosvintsev, S.R., and Velarde, M.G., Spreading of surfactant solutions over hydrophobic substrates, J. Colloid Interface Sci., 227, 185–190, 2000. 15. Stoebe, T., Lin, Z., Hill, R.M., Ward, M.D., and Davis, H.T., Langmuir, 12, 337, 1996; Langmuir, 13, 7270, 7276, 1997. 16. Reyes, R. and Wayner, P., J. Heat Transfer, 118, 822–830, 1996. 17. Starov, V.M., Velarde, M.G., Tjatjushkin, A.N., and Zhdanov, S.A., On the spreading of generalized newtonian liquids over solid substrates, J. Colloid Interface Sci., 257, 284–290, 2003. 18. Pearson, J.R.A., Mechanics of Polymer Processing, Elsevier Applied Science Publishers, London and New York, 1985. 19. Carre, A. and Eustache, F., Langmuir, 16, 2936, 2000. 20. Starov, V.M., Rudoy, V.M., and Ivanov, V.I., Spreading of a droplet of polar liquid induced by the overturning of amphiphilic molecules or their fragments in the surface layer of a substrate, Colloid J. (Russian Academy of Sciences, English Translation), 61(3), 374–382, 1999. 21. Andrade, J.D. and Chen, W.-Y., Surf. Interface Anal., 8(6), 253, 1986. 22. Andrade, J.D., Ed., Polymer Surface Dynamics, Plenum Press, New York, 1988. 23. Lewis, B.K. and Ratner, B.D., J. Colloid Interface Sci., 159(1), 77, 1993. 24. Miyama, M., Yang, Y., Yasuda, T., Okuno, T., and Yasuda, H., Langmuir, 13(20), 5494, 1997.

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25. Rudoy, V.M., Stuchebryukov, S.D., and Ogarev, V.A., Colloid J. (English Translation), 50(1), 199, 1988. 26. Langmuir, I., Science, 87, 1938, p. 493. 27. Yiannos, P.N., J. Colloid Sci., 17(4), 334, 1962. 28. Rideal, E. and Tadayon, J., Proc. R. Soc. (London) A, 225(1162), 346, 1954. 29. Flory, P.J., Principles of Polymer Chemistry, Cornell University Press, Ithaca, New York, 1990. 30. Exerowa, D. and Kruglyakov, P., Foam and Foam Films: Theory, Experiment, Application, Vol. 5, Studies in Interface Science, Elsevier, New York, 1988.

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Conclusions In the ﬁrst three chapters of this book, we have shown that all equilibrium and kinetics properties of liquids in contact with solid substrates can be described using a unifying approach by considering the simultaneous action of disjoining and capillary pressure (save cases where gravity or other external ﬁelds apply). Using this approach the overall equilibrium liquid proﬁle can be subdivided into three parts: (i) the bulk of the liquid (meniscus or drop), where only capillary (and, eventually gravity) forces act, (ii) thin equilibrium ﬁlms in front, where the surface forces dominate (in the form of disjoining pressure action), and (iii) a transition region in-between, where both capillary and surface forces are equally important. The main conclusion of Chapter 1 through Chapter 3 is that true progress in the area demands the consideration of disjoining pressure action in a vicinity of the apparent three-phase contact line. Though roughness and heterogeneity (chemical or otherwise) of the solid substrate affects wetting conditions, further progress in understanding of wetting and spreading must be based on the inclusion of surface forces action into consideration. The major dominant quantity used to describe the liquid-solid substrate interaction is the disjoining pressure isotherm and its dependence on the thickness of the layer. The latter dependence has been experimentally investigated only for a limited range of liquid ﬁlm thicknesses and only for ﬂat liquid ﬁlms or layers. Further experimental work needed to get such dependence of the disjoining pressure isotherm on the thickness in the whole range of thickness (including oversaturation and the region of unstable ﬂat ﬁlms). Theory needed to really understand the structural component of the disjoining pressure isotherm, and to understand how the disjoining pressure is expressed in the case of non-flat liquid layers. We have shown that in the case of complete wetting there is a reasonable agreement between the theory predictions and the progress is related to the fact that in such a case the disjoining pressure isotherm is well understood. The situation is drastically different in the case of partial wetting. We believe that the lack of progress in this case is related to the lack of understanding of the importance of disjoining pressure in those circumstances. Note, in colloid and interface science substantial progress was achieved only after the importance of surface forces action was understood. Hence, consideration of wetting and spreading processes on real, rough, and heterogeneous (chemical or otherwise) surfaces taking into account surface forces action in the case of partial wetting appears as the most challenging problem. In Chapter 2 some problems in this area were considered. In Chapter 4 we have provided results about the kinetics of spreading and imbibition when liquids are in contact with porous solid materials. An important conclusion is that the behavior of liquids in contact with porous materials is 501 © 2007 by Taylor & Francis Group, LLC

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substantially different from the corresponding processes with non-porous materials. We also deduced the novel universal laws in this chapter. However, the latter universal dependencies have been deduced only in the case of spreading over thin porous layers and kinetics of wetting and imbibition in the case of thick porous layers is still a challenge. The latter is especially true if surfactants are involved. In Chapter 5 we have considered how the spreading of aqueous surfactant solutions is affected by the presence of surfactants. It is difﬁcult to imagine our present-day life without surfactants (soaps, shampoos, detergents, washing liquids, etc.). Although some understanding has been accumulated in the case of surfactants acting on liquid-air interfaces where the transport processes are determined by the Marangoni effect, much less is known about the behavior of surfactants in the vicinity of the three-phase contact line. Here we are very far from the level of understanding offered in Chapter 1 through Chapter 3. It is the reason why we used a semi-empirical model to understand the role of surfactants in a vicinity of the three-phase contact line. In some cases such an approach leads to predictions and explanations of available experimental data. However, understanding of the real mechanism of surfactants transfer in the vicinity of the threephase contact line demands further research efforts. Recently new surfactants like trisiloxanes have attracted the attention of scientists and industrialists due to their unusual properties that have led to call them “superspreaders.” We did not touch the subject in this book, however, as understanding the nature of superspreading behavior is still a challenging open problem.

FREQUENTLY USED EQUATIONS NAVIER–STOKES EQUATIONS Navier–Stokes equations in cylindrical coordinate system (r, ϕ, z) are 2  ∂v v ∂v ∂p ∂v v  ρ  vr r + ϕ r + vz r − ϕ  = − ∂r ∂ ∂ ∂ r r z r ϕ  

  2 ∂v 1 ∂2vr ∂2vr 1 ∂vr 2 ∂vϕ vr  + η  2r + 2 + + − − , r ∂ϕ 2 ∂z 2 r ∂r r 2 ∂ϕ r 2   ∂r  ∂v v ∂v vv  ∂v 1 ∂p ρ  vr ϕ + ϕ ϕ + vz ϕ + r ϕ  = − r ∂ϕ r  r ∂ϕ ∂z  ∂r  2  ∂ vϕ 1 ∂2vϕ ∂2vϕ 1 ∂vϕ 2 ∂vr vϕ  − , + η 2 + 2 + 2 + + r ∂r r 2 ∂ϕ r 2  ∂z r ∂ϕ 2  ∂r © 2007 by Taylor & Francis Group, LLC

Conclusions

503

 ∂v v ∂v ∂v  ∂p ρ  vr z + ϕ z + vz z  = − r ∂ϕ ∂z  ∂z  ∂r   2 ∂v 1 ∂2vz ∂2vz 1 ∂vz  + + + η  2z + 2 . r ∂ϕ 2 ∂z 2 r ∂r   ∂r ∂vr 1 ∂vϕ ∂vz vr + = 0. + + ∂r r ∂ϕ ∂z r If the Reynolds number is small, then the latter equations become the Stokes equations:

0=−

0=−

 ∂2v ∂p 1 ∂2vr ∂2vr 1 ∂vr 2 ∂vϕ vr  − , + η  2r + 2 + + − ∂r r ∂ϕ 2 ∂z 2 r ∂r r 2 ∂ϕ r 2   ∂r

2 2  ∂2v 1 ∂p 1 ∂ vϕ ∂ vϕ 1 ∂vϕ 2 ∂vr vϕ  + − , + η  2ϕ + 2 + + r ∂ϕ r ∂ϕ 2 ∂z 2 r ∂r r 2 ∂ϕ r 2   ∂r

0=−

 ∂2v ∂p 1 ∂2vz ∂2vz 1 ∂vz  + + + η  2z + 2 , ∂z r ∂ϕ 2 ∂z 2 r ∂r   ∂r ∂vr 1 ∂vϕ ∂vz vr + + + = 0. ∂r r ∂ϕ ∂z r

In any case, we have four equations for four unknown functions, vr (r , ϕ, z ), vϕ (r , ϕ, z ), vz (r , ϕ, z ),

p(r , ϕ, z ).

The viscose stress tensor is:

σ rr = − p + 2 η

 1 ∂vr ∂vϕ vϕ  ∂vr , σ rϕ = η  + −  ∂r r   r ∂ϕ ∂r

 1 ∂vϕ vr   ∂v 1 ∂vz  σ ϕϕ = − p + 2 η  +  , σ ϕz = η  ϕ + ∂ ϕ r r ∂ z r ∂ϕ    

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σ zz = − p + 2 η

 ∂v ∂v  ∂vz , σ zr = η  z + r  ∂z ∂z   ∂r

where σrϕ is the tangential stress in the case of pure rotational ﬂow, σzr is the tangential stress in the case of pure axial ﬂow, and σzz in the normal stress to the ﬂat surface.

NAVIER-STOKES EQUATIONS

IN THE CASE OF

TWO-DIMENSIONAL FLOW

 ∂ 2 vx ∂ 2 vx   ∂v ∂p ∂vx  = − + ρ  vx x + v η  ∂x 2 + ∂y 2  y ∂y  ∂x  ∂x    ∂2v  ∂v ∂v  ∂2v  ∂p ρ  vx y + vy y  = − + η  2y + 2y  ∂y  ∂y ∂y   ∂x  ∂x ∂vx ∂vy =0 + ∂y ∂x In the case of a low Reynolds number, the latter equations become the Stokes equations:

0=−

 ∂2v ∂p ∂2v  + η  2x + 2x  ∂x ∂y   ∂x

0=−

 ∂2v ∂2v  ∂p + η  2y + 2y  ∂y ∂y   ∂x

∂vx ∂vy + =0 ∂x ∂y In any case, these are three equations for three unknown functions, vx , vy , p. The viscose stress tensor appears as

σ xx = − p + 2 η

 ∂v ∂v y ∂v y  ∂v x , σ yy = − p + 2 η , σ xy = σ yx = η  x + ∂x ∂y ∂x   ∂y

The last component is the tangential stress in the case of a plane parallel ﬂow.

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Conclusions

505

CAPILLARY PRESSURE Capillary pressure is determined as γK, where γ is the interfacial tension (liquid–air or liquid–liquid), and K is the mean curvature of the interface. Let h be the equation that describes the interface. Following are expressions for the mean curvature: The general case in the Cartesian coordinate system, the interface proﬁle, is h(x,y):

K=

∂h 2 ∂2 x   ∂h  2  ∂h  2  1 +   +     ∂y     ∂x   

3 /2 2

+

∂h 2 ∂2 y   ∂h  2  ∂h  2  1 +   +     ∂y     ∂x   

3/ 2

.

The axisymmetric case on the inner or outer surface of the cylindrical capillary of the radius a, the interface proﬁle, is h = h(x), where x is the axial coordinate K=

h ′′

(1 + h ′ ) 2

3/ 2

∓

h′

(

(a ± h) 1 + h ′2

)

1/ 2

,

Here, the upper sign corresponds to the outer surface, and the lower sign corresponds to the inner surface; ′ means the differentiation with x. The axisymmetric droplet on the plane substrate, the interface proﬁle, is h(r), where r is the radial coordinate: K=

1 d rh ′ r dr 1 + h ′ 2

′ means the differentiation with r.

LIST OF MAIN SYMBOLS USED GREEK γ θ Φ η Π

Interfacial tension Contact angle Excess free energy Dynamic viscosity Disjoining pressure

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(

)

1/ 2

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LATIN g P p S T V

Gravity acceleration Excess pressure Pressure Surface Absolute temperature in °K Volume

SUBSCRIPTS v s l * a e s c

Vapor Solid Liquid Characteristic scale or initial value Ambient air Equilibrium Surface forces or saturated Capillary

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