Fundamentals of Interface and Colloid Science. Volume III: Liquid-Fluid Interfaces

FUNDAMENTALS OF INTERFACE AND COLLOID SCIENCE

This Page Intentionally Left Blank

FUNDAMENTALS OF INTERFACE AND COLLOID SCIENCE

VOLUME III LIQUID-FLUID INTERFACES

J. Lyklema With contributions from: G.J. Fleer. J.M. Kleijn, F.A.M. Leermakers. W. Norde, T. van Vliet

ACADEMIC PRESS A Harcourt Science and Technology Connpany

San Diego San Francisco New York Boston London Sydney Tokyo

This book is printed on acid-free paper. Copyright © 2000 by ACADEMIC PRESS

All Rights Reserved No part of this book may be reproduced or tramsmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.

Academic Press A Harcourt Science and Technology Company 32 Jamestown Road, London, NWI 7BY, UK http://www.academic press.com Academic Press A Harcourt Science and Technology Company 524 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www.academic press.com

ISBN 0-12-460523-0 A catalogue record for this book is avaiilable from the British Libreiry Library of Congress Card Number: 00-101585

Printed in Great Britain by MPG Books Ltd., Bodmin, Cornwall 00 0 1 0 2 0 3 0 4 0 5 M P 9 8 7 6 5 4 3 2 1

GENERAL PREFACE Fundamentals of Interface and Colloid Science (FICS) is motivated by three related but partly conflicting observations. First, interface and colloid science is an important and fascinating, though often undervalued, branch of science. It has applications and ramifications in domains as disparate as agriculture, mineral dressing, oil recovery, the chemicsd industry, biotechnology, medical science and many more provinces of the living and non-living world. The second observation is that proper application and integration of interface and colloid science requires, besides factued knowledge, insight into the many basic laws of physics £ind chemistry upon which it rests. In the third place, most textbooks of physics eind chemistry pay only limited attention to interface and colloid science. These observations give rise to the dilemma that it is an almost impossible task to simultaneously master specific domains of application and be proficient in interface and colloid science itself, along with its foundations. The problem is aggravated by the fact that interface and colloid science has a very wide scope: it uses parts of classical, irreversible and statistical thermodynamics, optics, rheology, electrochemistry and other brsinches of science. Nobody can be expected to commaind all of these simultaneously. The prime goal of FICS is to meet these demands systematically, treating the most important interfacial and colloidal phenomena starting from basic principles of physics and chemistry, whereby these principles are first reviewed. In doing so, it will become clear that common roots often underlie seemingly different phenomena, which is helpful in identifying and recognizing them. Given these objectives, a deductive approach is required .Throughout the work, systems of increasing complexity are treated gradually, with, as a broad division, in Volume I the fundamentals (F), in Volumes II and III isolated interfaces (I) and in Volumes IV and V interfaces in interaction and colloids (C). The chosen deductive set-up serves two purposes: the work is intended to become a standard reference but with parts that will be suitable for systematic use, either as a self-study guide or as a reference for courses. In view of these objectives, a certain style is more or less defined and has the following characteristics: - Topics are arranged by the main principle(s) and phenomena on which they rest. Since mauiy researchers are not immediately familiar with these principles £ind as more than one principle may determine a phenomenon met in practice, an extensive and detailed subject index is provided, which in some places has double or triple entries.

11

As FICS is a book of principles rather thcin a book of facts, no attempt is made to give it an encyclopaedic character, although important data are tabulated for easy reference. For factual information, references are made to the literature, in particular to reviews and books. Experimental observations that illustrate or enforce specific principles are emphasized, rather than given for their own sake. This also implies a certain preference for illustrations with model systems. Some arbitrariness Ccinnot be avoided; our choice is definitely not a 'beauty contest'. - In order to formulate physical principles properly, some mathematics is unavoidable as we cannot always ignore complex and abstract formalisms. To that end, specialized techniques that are sometimes particularly suitable for solving certain types of problems, will be introduced when needed; much of this in the appendices to Volume I. However, the reader is assumed to be familiar with elementary calculus. - Generally, the starting level of Volume I is such that it can be read without having an advanced command of physics and chemistry. In turn, for the later Volumes, the physical chemistry of Volume I is the starting point. - In view of the fact that much space is reserved for the explanation and elaboration of principles, we had to restrict the number of systems treated in order to keep the size of the work manageable. Given the importance of interfacial and colloid science for biology, medicine, pharmacy, agriculture, etc., wet' systems, aqueous ones in particular, are emphasized. 'Dry' subjects such as aerosols and solid state physics are given less attention. Experimentad techniques are not described in great detail except where these techniques have a typical interfacial or colloidal nature. Considering all these features, FICS may be thought of as a work containing parts that can also be found in more deteiil elsewhere but rarely in the present context. Moreover, it stands out by integrating all these parts. It is hoped that through this integration many readers will use the work to find their way in the expanding literature and, in doing so, will experience the relevance and beauty of interface and colloid science and become fascinated by it. Hans Lyklema Wageningen, The Netherlands 2000

PREFACE TO VOLUME n i : LigUID-FLUID INTERFACES Having laid down the physico-chemical basis of interface a n d colloid science in Volume 1, a n d t h e n presented a systematic t r e a t m e n t of solid-fluid interfaces in Volume II, we now conclude the interface science part of FICS with a t r e a t m e n t of liquid-fluid interfaces. Colloids will be discussed in Volumes FV a n d V. In line with the general set-up of FICS, we try to keep the t r e a t m e n t systematic and deductive. Recurrent features are that each chapter begins, a s m u c h a s possible, with t h e general thermodynamic a n d / o r statistical t h e r m o d y n a m i c foundations a n d the various p h e n o m e n a are presented in order of increasing complexity. The r e q u i r e m e n t t h a t t h e work be both a reference a n d a textbook is reflected in its being comprehensive a s far a s the fundamentals are concerned a n d in its didactic style. Completeness always remains a matter of dispute, b u t it is hoped t h a t m a n y r e a d e r s will find the information they are looking for, especially regarding basic issues a n d m a i n principles. No attempt is made to give the book a n encyclopaedic character a s far a s the facts are concerned; exceptions are selected d a t a collections, a s in appendices 1 a n d 4. Experiments are included w h e n they help to illustrate certain points; there may be some arbitrariness in the choice. Typically we s h u n , multivariable systems because their very complexity might obscure basic features. The level is about the same a s t h a t of papers in m o d e m j o u r n a l s dealing with the subject. It is to be hoped t h a t readers without the background c a n find m u c h of the desired information in FICS, going back to earlier c h a p t e r s or to Volume I where needed. To facilitate this process, extensive back-referencing h a s b e e n applied, although, where appropriate at the beginning of each chapter, the relevant background in the previous volumes is briefly reviewed, so t h a t the present book s t a n d s on its own. For more specialistic background information, reference is m a d e to the literature a n d we provide a very extensive cumulative subject index. In a n u m b e r of instances a decision h a d to be t a k e n a s to w h a t to call 'fundamentals' a n d w h a t advanced'. As a rule, we have avoided detailed descriptions of a s s u m p t i o n s made, a n d p a r a m e t e r s selected, in statistical-thermodynamic analyses, although it is appreciated t h a t the results depend heavily on these choices. In a n u m b e r of places we have elaborated some principles ourselves. Interfacial rheology is introduced in some detail; we feel t h a t its relevance is not generally appreciated. In order to streamline the systematics, all monolayer techniques c a n be found in the chapter on Langmuir monolayers, which, as a consequence, grew to become t h e largest in t h e book. Wetting is, in a s e n s e , a n application of t h e preceding four chapters, b u t because of its relevance a full chapter is devoted to it. Regarding symbols a n d u n i t s of quantities, lUPAC recommendations have, a s a rule, been heeded. However, the very wide scope of FICS leads to a n u m b e r of clashes t h a t forced u s to deviate from these rules now and then. For example, we prefer F for Helmholtz energy instead of A because of the confusion with A for area.

iv

It is also unavoidable that sometimes the same symbol is used for different physical quantities, because we wanted to avoid excessive use of subscripts and superscripts. Mostly this does not give rise to problems, but in those places where it might, we have added a caveat. For instance, cross-sectional areas per molecule have different meanings in lattice theories and in surface pressure isotherms for Langmuir monolayers; we explain this in sections 3.3 and 3.4. As for the spelling of names, we prefer that of the country of origin. So, if referring to people, we would write for instance van der Waals, Deryagin and d'Arcy. However, for phenomena or laws, capitals are used (Van der Waals equation'). For Slavic names, originally written in the Cyrillic alphabet, we adhere to the Chemical Abstracts transcription. However, many Slavic authors have the custom of transcribing their names differently in different languages, and as literature citations should be verbatim, some inconsistency is unavoidable. Where appropriate, identifications are made (Deryaguin, Derjagin, Derjaguin, etc. = Deryagin). In a systematic text the size of FICS, numerous cross-references are unavoidable. We prefer to refer to sections, rather than to pages or to facilitate consultation, to subsections, where possible. References to Volumes I or II are preceded by I or II, respectively. Co-operation and acknowledgements In writing this book I have benefitted from the help of many people, whose cooperation is gratefully acknowledged. In the first place I am particularly indebted to: Prof. G.J. Fleer (subsecs. 3.4i, J and 3.8f, g). Dr. J.M. Kleijn (sees. 3.7 and subsec. 3.8c), Dr. F.A.M. Leermakers (parts of sees. 3.5 and 4.7), Dr. W. Norde (sees. 3.1-3.3 and subsec. 3.8d), and Dr. T. van Vliet (sees. 3.6 and 4.5), who actually wrote (parts of) sections for me, or at least laid the foundations for them. These contributors are acknowledged on the title page. In addition, I thank Dr. T. Golub for her assistance in completing appendix 1 and Dr. H. Visser for setting up appendix 4. For chapter 4, I could adways rely on Prof. B.H. Bijsterbosch to act as a 'sparring partner', by critically reading and commenting on just-drafted sections. For the corresponding expert help with chapter 5, I thank Dr. T.D. Blake, Dr. E.M. Blokhuis and Prof. A.-M. Cazabat, who also assisted me greatly in providing course material, written for other purposes. Professor P. Walstra's careful and constructive screening of chapters 3 and 4 is wholeheartedly appreciated, and so is Prof. G.A. Barnes' critical reading of chapter 3. Prof. A. Marmur's comments to chapter 5 are also appreciated. I found the comments of staff members and students on completed chapters particularly useful, because it helped me to 'try them out'. Often they gave rise to

textual improvements. My thanks in this respect go to Dr. R. Barchini (chapters 4 and 5), Dr. K. Besseling (parts of chapters 1, 2 aind 3), Prof. B.H, Bijsterbosch (chapters 1 and 2), Dr. T.D. Blake (chapter 1), Dr. E.M. Blokhuis (chapter 2, parts of chapter 3), Dr. J.A.G. Buys. Prof. A.-M. Cazabat (chapters 1 and 2), Prof. J. de Coninck (part of chapter 2), Dr. E.P.K. Currie (parts of chapters 2 and 3), Prof. G.J. Fleer (peirts of chapter 3), J. van der Gucht (part of chapter 4), Dr. A. de Keizer (chapter 1, parts of chapter 3 and much other 'scientific-logistic' help), Dr. L.K. Koopal (chapters 1, 2 and 4), Prof. E. van der Linden (part of chapter 3), Dr. H. van Leeuwen (parts of chapters 3 and 4), Drs. J. Lucassen and E.H. Lucassen-Reynders (lively discussions on interfacial thermodynamics and rheology), Dr. F. MacRitchie (chapter 3), Dr. W. Norde (chapters 1 and 2), J. van den Oever (parts of chapter 4), Dr. S.M. Oversteegen (parts of chapters 1 and 2), Mrs. J. Slofstra (chapter 1), L.H. Tom (chapter 3). Dr. R. Tuinier (chapters 1-4) and Dr. T. van Vliet (chapter 1). Obviously, I am responsible for any errors and imperfections which may remain after all this screening. I am greatly indebted to Mrs. Josie Zeevat for the actual lay-out and typing, all of it done with much dedication and technical skill, largely in her own spare time. FIGS is, in a sense, also her book. The airt work was again in the capable hands of Mr. G. Buurman. Many people in the Depairtment assisted me in different ways of services. Of them 1 would like to mention Mr. A.W. Bouman. All this help and encouragement has been instrumental in completing this third volume. Last but not least, I wish to express my sincere gratitude to my wife for her understanding and tolerance.

Hans Lyklema, Wageningen, The Netherlcinds April 2000

VI

This Page Intentionally Left Blank

CONTENTS 1 Interfacial tension: measurement 1.1 General introduction to capillarity and the measurement of interfacial tensions 1.2 On the mathematics of curvature 1.3 Capillary rise 1.3a Vertical cylinder 1.3b Vertical plate 1.3c Other geometries 1.4 Shapes of drops and bubbles on surfaces 1.5 Free drops in a density gradient or electric field 1.6 Drop weight method 1.7 Maximum bubble pressure 1.8 Force required to hold objects at an interface or to pull them through it 1.8a Wilhelmy plate 1.8b DuNouyring 1.8c Other geometries 1.9 Spinning drops and bubbles 1.10 Surface light scattering 1.11 Miscellaneous other static methods 1.12 A case study: the surface tension of water 1.12a Room temperature 1.12b FromO°- 100°C 1.13 Measuring the surface tension of solids 1.14 Surface tensions under dynamic conditions 1.14a Pure liquids 1.14b Solutions 1.14c A note on the pristine state of LG and LL surfaces 1.15 Bending moduli 1.16 Applications 1.17 General References 1.17a lUPAC recommendations 1.17b Tabulations 1.17c Capillarity, shapes of fluid bodies etc. 1.17d Measuring interfacial and surface tensions

1.2 1.7 1.12 1.12 1.21 1.23 1.24 1.31 1.33 1.36 1.39 1.39 1.45 1.48 1.49 1.53 1.56 1.58 1.58 1.59 1.65 1.67 1.68 1.70 1.76 1.77 1.82 1.84 1.84 1.85 1.85 1.86

viii

2 Interfacial tension: molecular interpretation 2.1 Introductory considerations 2.2 Thermodynamic and statistical thermodynamic fundamentals. Flat interfaces 2.3 Interfacial tension and interfacial pressure tensor 2.4 Interfacial tensions and distribution functions 2.5 Van der Waals theory 2.5a Some elements of van der Waals' theory 2.5b Comments and consequences 2.5c Van der Waals theory in the Hamaker-de Boer approximation 2.6 Cahn-Hilliard theory 2.7 Interfacial tensions from simulations 2.8 The thickness of the interfacial region 2.9 Quasi-thermodynamic approaches. Effects of temperature and pressure. Corresponding states 2.9a Influence of temperature. Energetic and entropic contributions 2.9b Influence of pressure 2.9c Surface tensions as capillary waves 2.10 Lattice theories for the interpretation of interfacial tensions 2.11 Empirical relationships 2.11a Relations containing molar volumes and compressibilities 2.11b Relationships for interfacial tensions, containing geometric means 2.11c Other empirical relationships 2.12 Conclusions and applications 2.13 General references 3 Langmuir Monolayers 3.1 Langmuir- and Gibbs monolayers. Distinctions and analogies 3.2 How to make monolayers 3.3 Two-dimensional phases and surface pressure 3.3a Determination of the interfacial pressure. Film balances 3.3b Introduction to 2D phase behaviour 3.3c Some representative illustrations 3.4 Monolayer thermodynamics 3.4a General formalism and definitions 3.4b Some expressions of general validity 3.4c Application to a real system 3.4d The thermodynamics of phase transitions

2.3 2.5 2.9 2.11 2.17 2.18 2.25 2.31 2.34 2.37 2.43 2.48 2.48 2.57 2.59 2.60 2.64 2.64 2.68 2.72 2.73 2.76

3.2 3.6 3.13 3.14 3.18 3.23 3.28 3.28 3.33 3.36 3.38

IX

3.4e Two-dimensional equations of state 3.4f Mixed Langmuir monolayers 3.4g Half open Langmuir monolayers 3.4h Ionized monolayers 3.4i Polymer monolayers 3.4j Brushes 3.5 Monolayer molecular thermodynamics 3.5a Some general considerations 3.5b Strictly two-dimensioned approaches 3.5c Monte Carlo (MC) simulations 3.5d Molecular dynamics (MD) simulations 3.5e Mean field lattice (MFL) theories 3.6 Interfacial rheology 3.6a Some basic issues 3.6b A review of bulk rheology 3.6c Basic interfacial rheology 3.6d Intermezzo. Dispeirate definitions of interfacial tension 3.6e Coupling of interface and bulk motions. Marangoni effect 3.6f Experimental methods to determine surface rheological properties. Principles 3.6g Wave propagation and damping 3.6h Relaxation processes in Langmuir monolayers 3.6i Equivalent mechanical circuits 3.7 Measuring monolayer properties 3.7a Monolayer transfer on solid substrates: LemgmuirBlodgett films 3.7b Reflection and diffraction 3.7c Spectroscopic techniques 3.7d Scanning probe microscopy 3.7e Rheology 3.7f Volta potentials 3.8 Case studies 3.8a A note on equilibrium and reproducibility 3.8b Fatty acids, fatty alcohols and related compounds 3.8c Phospholipids 3.8d Cholesterol 3.8e PMA at the air/water interface 3.8f PEG brushes 3.9 Applications

3.39 3.46 3.48 3.49 3.53 3.58 3.62 3.62 3.65 3.67 3.69 3.75 3.79 3.82 3.84 3.91 3.95 3.96 3.102 3.109 3.120 3.125 3.131 3.132 3.141 3.156 3.174 3.180 3.190 3.194 3.194 3.200 3.215 3.223 3.227 3.231 3.235

3.10 General references

4

3.241

3.10a lUPAC Recommendations

3.241

3.10b Monographs, reviews

3.242

3.10c Classical a n d molecular thermodynamics

3.243

3.10d Interfacial rheology

3.243

3.10e C h a r a c t e r i z a t i o n

3.244

3.10f Langmuir-Blodgett layers

3.247

3,10g Data collection

3.247

Gibbs Monolayers 4.1

Introduction

4.1

4.2

The surface tension of miscible binary mixtures

4.2

4.2a General features a n d thermodynamics

4.2

4.2b From F^"^ to y

4.6

4.2c F r o m / t o F^"^

4.18

4.2d A note on surface excess entropy

4.20

4.3

Dilute solutions of simple molecules

4.20

4.3a Experimental techniques

4.21

4.3b Theory

4.22

4.3c A case study: alcohols

4.27

Simple electrolytes

4.34

4.4

4.4a The pristine surface of water

4.35

4.4b Electrolytes at the air-water interface

4.36

4.4c Electrolytes at water-oil interfaces)

4.41

4.4d Some practical implications

4.45

4.5

Rheology a n d kinetics

4.46

4.5a Identification of basic transport p h e n o m e n a

4.48

4.5b Basic mathematics

4.51

4.5c Experimental techniques

4.58

4.6

Surfactants

4.68

4.6a Surfactants in solution

4.69

4.6b Surfactant monolayers. General features

4.73

4.6c Non-ionics

4.76

4.6d Ionic surfactants

4.83

4.7

Curved interfaces

4.93

4.8

Applications

4.97

4.9

General references

4.99

XI

5

Wetting 5.1 General considerations 5.1a Principles and definitions 5. lb The laws of Young and Neumann 5.2 Thermodynamics of wetting and adhesion 5.3 The relation between adsorption and wetting. Wetting films 5.3a The disjoining pressure in wetting films 5.3b Film formation by adsorption 5.3c Case studies 5.3d A note on film thinning 5.4 Measuring contact angles 5.4a Factors affecting contact angles 5.4b Sessile drops and captive bubbles on a flat surface 5.4c Contact angles from forces on objects in interfaces 5.4d Tilted plates 5.4e Capillary rise or depression 5.4f Vertical plates and cylinders 5.4g Fibres 5.4h Individual colloidal particles 5.4i Powders and porous materials 5.4j Concluding remark 5.5

Contact angle hysteresis 5.5a The phenomenon 5.5b Origins 5.5c Interpretations 5.5d Hysteresis on model surfaces 5.6 Line tensions 5.7 Interpretation of static contact angles 5.8 Dynamics 5.9 Porous systems 5.10 Influence of surfactants 5.11 Applications 5.1 la Characterization of the wettability of solid surfaces 5.11b Flotation 5.11c Particles at interfaces 5.lid Various other applications 5.12 General references 5.12a lUPAC recommendation 5.12b Books, Reviews and Symposium Proceedings

5.3 5.3 5.10 5.12 5.22 5.23 5.28 5.33 5.38 5.39 5.39 5.41 5.47 5.47 5.48 5.50 5.53 5.54 5.56 5.58 5.59 5.59 5.60 5.63 5.66 5.68 5.73 5.77 5.83 5.89 5.93 5.94 5.96 5.98 5.101 5.103 5.103 5.103

xii

Appendices APPENDIX 1 Surface Tensions of Pure Liquids and Mixtures a. Surface tension of some inorganic fluids b. Surface tensions of some molten metals c. Surface tension of some molten halides d. Surface tensions of some low boiling point liquids e. Surface tensions of linear alkanes f. Surface tensions of linear aliphatic n-alcohols g. Surface tensions of linear ailiphatic n-sddehydes h. Surface tensions of linear n-amines i. Surface tensions of n-aliphatic acids J. Surface tensions of n-cdiphatic nitriles k. Surface tensions of n-aliphatic isomers compared 1. Surface tensions of some other common cdiphatic compounds m. Surface tensions of some triglycerides n. Surface tensions of benzene and some mono-substituted benzenes o. Surface tensions of some other cyclic compounds p. Surface tensions of some binary mixtures References, indicating methods where appropriate APPENDIX 2 a. Integral characteristic functions of flat interfaces b. Differential characteristic functions of flat interfaces

Al. 1 Al. 1 A 1.6 A 1.7 A 1.8 A 1.9 Al.l 1 A1.15 A 1.16 A 1.17 A 1.19 A1.22 A1.24 A1.27 A1.29 Al .34 A1.38 A 1.45 A2.1 A2.1

APPENDIX 3 Some principles of variational calculus

A3.1

APPENDIX 4 Contact angles

A4.1

a. Contact angles on metals b. Contact angles on polymers c. Contact angles on oxides, minerals and metalloids SUBJECT INDEX

A4.1 A4.5 A4.13

xiii

CONDENSED CONTENTS OF VOLUME I 1 Introduction to interfacial and colloid chemistry 1.1 First encounter 1.1 1.2 A first review of interface and colloid science 1.3 1.3 Further insights 1.8 1.4 Guiding principles 1.28 1.5 Structure of this book 1.30 2 Thermod3mamic foundations of interface and colloid science 2.1 Thermodynamic principles 2.1 2.2 Thermodynamic definitions 2.2 2.3 Equilibrium, reversibility and relaxation 2.4 2.4 The First Law 2.8 2.5 The excess character of interfacial energy. The Gibbs dividing plane 2.11 2.6 Enthalpy and interfacial enthalpy 2.15 2.7 Heat capacities of bulk phases and interfaces 2.17 2.8 The Second Law 2.20 2.9 Entropies of bulk phases and interfaces 2.23 2.10 Gibbs and Helmholtz energies of bulk phases and interfaces 2.24 2.11 Differential and integral characteristic functions of bulk phases and interfaces. 2.28 2.12 Thermodynamic conditions for equilibrium 2.30 2.13 Gibbs-Duhem relations, the Gibbs adsorption equation and the Phase Rule 2.35 2.14 Relations of general validity. Properties of toted differentials 2.37 2.15 Gibbs-Helmholtz relations 2.42 2.16 Homogeneous mixtures. General features 2.44 2.17 Ideal homogeneous mixtures 2.50 2.18 Non-ideal mixtures and solutions 2.58 2.19 Phase separation and demixing 2.64 2.20 The laws of dilute solutions 2.68 2.21 Chemical equilibrium 2.75 2.22 Further discussion of the Gibbs adsorption equation 2.78 2.23 Curved interfaces. Introduction to capillarity and nucleation 2.86 2.24 Thermodynamics of solid interfaces 2.100 2.25 Concluding remarks 2.102 2.26 General references 2.103 3 Statistical thermodjmamic foundations of interface and colloid science 3.1 Basic principles and definitions 3.1 3.2 Partition Functions 3.9 3.3 Statistical and classical thermodynamic equivgdence. Characteristic functions 3.15 3.4 Derivation of macroscopic quantities from partition functions 3.18 3.5 Partition functions of subsystems 3.19 3.6 Partition functions for systems with independent subsystems 3.26 3.7 Fluctuations 3.33 3.8 Dependent subsystems 3.39 3.9 Classical statistical thermod3namics 3.56 3.10 Concluding remarks 3.72 3.11 General references 3.72 4 Interactions in interface and colloid science 4.1 Types of interactions 4.1 4.2 Pair interaction curves. The disjoining pressure 4.3 4.3 Force fields and potentials 4.10 4.4 Van der Waals forces between isolated molecules 4.16

XIV

4.5 4.6

Molecular forces in a medium 4.41 Van der Waals forces between macrobodies. The Hamaker-De Boer approximation 4.57 4.7 Van der Waals forces between macrobodies. The Lifshits theory 4.75 4.8 Measuring Van der Waals interactions between macrobodies 4.82 4.9 References 4.84 5 Electrochemistiy and its applications to colloids and interfaces 5.1 Some basic issues 5.2 5.2 Ionic interactions in electrolyte solution 5.15 5.3 Ion-solvent interactions 5.33 5.4 Solvent structure-originated interactions 5.66 5.5 Interfacial potentials 5.71 5.6 Surface charge 5.89 5.7 The Gibbs energy of an electrical double layer 5.103 5.8 General references 5.109 6 Transport phenomena in interface and colloid science 6.1 Transport fundamentals 6.3 6.2 Elements of irreversible thermodynamics 6.12 6.3 Stochastic processes 6.17 6.4 Viscous flow phenomena 6.31 6.5 Diffusion 6.53 6.6 Conduction 6.73 6.7 Mixed transport modes 6.88 6.8 General References 6.94 7 Optics and its application to interface and colloid science 7.1 Electromagnetic waves in a vacuum 7.2 7.2 Introduction of the Maxwell equations and medium effects 7.9 7.3 Interaction of radiation with matter 7.13 7.4 Sources of electromagnetic waves 7.20 7.5 Particle-wave duality 7.23 7.6 Quasi-elastic light scattering (QELS) techniques 7.25 7.7 Quasi-elastic scattering of liquids 7.42 7.8 Quasi-elastic scattering by colloids 7.47 7.9 Neutron and X-ray scattering by colloids 7.67 7.10 Optics of interfaces 7.71 7.11 Surface spectroscopy and imaging techniques 7.84 7.12 Infrared and Raman spectroscopy 7.91 7.13 Magnetic resonance techniques 7.94 7.14 Anisotropic media and polarization phenomena 7.96 7.15 Measuring diffusion coefficients: a review 7.101 7.16 General references 7.104 Appendices 1 Fundamental constants 2 Common SI prefixes 3 Integral characteristic functions for homogeneous bulk phases and for two phases, separated by a flat interface 4 Differentials of characteristic functions for homogeneous bulk phases and for two phases, separated by a flat interface 5 Integral characteristic functions for flat interfaces and their differentials 6 Derivation of macroscopic quantities from partition functions 7 Vectors and vector fields. Introduction to tensors 8 Complex and imaginciry quantities 9 Hamaker constants 10 Laplace and Fourier transformations 11 Time correlation functions

CONDENSED CONTENTS OF VOLUME II 1 Adsorption at the solid-gas interface 1.1 Selected review of Volume 1 1.2 Characterization of solid surfaces 1.3 Thermodynamics of adsorption 1.4 Presentation of adsorption data 1.5 Adsorption on non-porous, homogeneous surfaces 1.6 Porous surfaces 1.7 Surface heterogeneity 1.8 Conclusion 1.9 General references 2 Adsorption from solution. Low molecular mass, uncharged molecules 2.1 Basic features 2.2 The interface between solids and pure liquids 2.3 The surface excess isotherm 2.4 Adsorption from binary fluid mixtures. Theory 2.5 Experimental techniques 2.6 Binary systems, a few illustrations 2.7 Adsorption from dilute solutions 2.8 Kinetics of adsorption 2.9 Conclusions and applications 2.10 General references 3 £lectric Double Layers 3.1 Some examples of double layers 3.2 Why do ionic double layers form? 3.3 Some definitions, sjmibols and general features 3.4 Thermodynamics. Application of the Gibbs equation 3.5 The diffuse part of the double layer 3.6 Beyond Poisson-Boltzmann. The Stem picture 3.7 Measuring double layer properties 3.8 Points of zero charge and isoelectric points 3.9 Interfacial polarization and the ;|f-potential 3.10 Case studies 3.11 Double layers in media of low polarity 3.12 Electrosorption 3.13 Charged particles in external fields 3.14 Applications 3.15 General references 4 Blectrokinetics and Related Phenomena 4.1 Basic principles 4.2 Survey of electrokinetic phenomena. Definitions of terms 4.3 Elementary theory 4.4 Interpretation of electrokinetic potenticds 4.5 Experimental techniques 4.6 Generalized theory of electrokinetic phenomena. Application to electrophoresis and experimental verification 4.7 Electrokinetics in plugs and capillaries 4.8 Electrokinetics in alternating fields. Dielectric dispersion 4.9 Less familiar types of electrokinetics 4.10 Applications 4.11 General References

1.1 1.7 1.19 1.36 1.42 1.81 1.102 1.109 1.111 2.1 2.5 2.19 2.28 2.46 2.57 2.64 2.83 2.86 2.89 3.2 3.4 3.6 3.11 3.17 3.44 3.84 3.101 3.118 3.127 3.186 3.188 3.206 3.220 3.225 4.2 4.4 4.8 4.37 4.44 4.64 4.104 4.111 4.122 4.126 4.133

XV i

5 Adsorption of Polymers and Polyelectrolytes 5.1 Introduction 5.2 Polymers in solution 5.3 Some general aspects of polymer adsorption 5.4 Polymer adsorption theories 5.5 The Self-Consistent-Field model of Scheutjens and Fleer 5.6 Experimental methods 5.7 Theoretical and experimental results for uncharged poljmiers 5.8 Theoretical and experimental results for polyelectrolytes 5.9 Applications 5.10 General references Appendices 1. Survey of adsorption isotherms and two-dimensional equations of state for homogeneous, nonporous surfaces. 2 Hyperbolic functions 3 Pristine points of zero charge

5.1 5.3 5.17 5.28 5.36 5.57 5.67 5.84 5.96 5.98

xvii

Contents planned for Volume IV Introduction to colloid science Characterization, size distributions Preparations lyophobic colloids, suspensions DLVO part of stability, application to thin films, particulate systems Coagulation kinetics Phase formation Rheology Contents planned for Volume V Poljnners £ind their effects on colloid stability Polyelectrolytes and their effects on colloid stability Proteins (colloidal aspects) Association colloids (including micro-emulsions, vesicles, bilayers) Emulsions, foams Porous systems, membranes

xviii This Page Intentionally Left Blank

SYMBOL LIST

xix

LIST OF FREgUENTLT USED SYMBOLS (Volumes I, II and W) S)mibols representing physical quantities are printed in italics. Thermodynamic functions: capital for macroscopic quantities, small for molecular or subsystem quantities (example: U = total energy, u = pair energy between molecules). Supers;cripts standard pressure t standard in genercd o * pure substance; complex conjugate interfacial (excess) a E excess, X (real) - X (idccd) in solid, liquid or gaseous state S,L,G 1 normal to surface parallel to surface // Subscripts m molar (sometimes molecular) a aread (per unit area) g per unit mass Reciurent special symbols 0( 1020) of the order of lO^o AX X(final) -X(initial). Subscript attached to A to denote type of process: ads (adsorption, diss (dissociation), hydr (hydration), mix (mixing), r (reaction), sol (dissolution), solv (solvation), subl (sublimation), trs (transfer), vap (vaporization, evaporation) Some mathematical signs and operators Vectors bold face. Example: F for force, but F^ for z-component of force Tensors bold face with tilde ( ) complex quantities bear a circumflex ( ), the corresponding conjugate is * IXI absolute value of x (x) averaged value of x Fourier or Laplace trsmsform of x (sometimes this bar is omitted) For vectorial signs and operators (V, V^, grad, rot, and x), see I.appendix 7.

XX

SYMBOL LIST

Latin a

activity (mol m-^) mean activity of an electrolyte (mol m*^) attraction parameter in Van der Waals equation of state (N m"^) two-dimensional attraction parameter in Van der Waals equation of state (N m^) radius (m) radius of gyration (m) Eirea per molecule (m^) area (m^) specific area (m^ kg"^) Hamaker constant (J) Hamaker constant for interaction of materials i and J across material k (J) volume correction parameter in Van der Waads equation of state (m^) magnetic induction (T = V nr^ s)

a+ a a^ a ag a^ A Ag A A y(y b B, B BgCT) B^ (T)

c c c C C^ C* C C^ C^ C^ Cp Cy C^

first

C^ Ca d d^^ d^ d^"

second virial coefficient (m^ mol"^ or m^ molecule" M interfacial second virial coefficient (m^ mol-^ m^ molecule"^

or-) velocity of electromagnetic radiation in a vacuum (m s"^) concentration (usually mol m"^, sometimes kg m"^) principal curvature (m"^) (differential) electric capacitance (C V-^ or, if per unit area, C m-2 V-i) (differential) electric capacitance of diffuse double layer (C m-2 V-i) (differential) electric capacitance of Stem layer (C m~^ V"^) BET-transformed (-) bending moment (N) second bending moment (J) (time-) correlation function of x (dim. x^] molar heat capacity at constant pressure (J K"^ mol~^) molair heat capacity at constant volume (J K~^ mol"^) interfacial excess molar heat capacity at constant airea per unit area (J K"^ m"^ mol"^) interfacial excess molar heat capacity at constaint interfacial tension per unit area (J K"^ m-^ mol"^) capillary number (-) layer thickness (m) electrokinetic thickness (m) hydrodynamic thickness (m) ellipsometric thickness (m)

SYMBOL LIST d®^

xxi

steric tJiickness (polymeric adsorbates) (m) diffusion coefficient (m^ s"^) D^ surface diffusion coefficient (m^ s"^) Dj. rotational diffusion coefficient (s"^) Dg self-diffusion coefficient (m^ s-^) D, D dielectric displacement (C m"^) De Deborah number (-) Du Dukhin number (-) e elementciry charge (C) E, E electric field strength (V m-^) Ejj^ irradiance (J m-^ s"^ = W m-^) E^^^ sedimentation potential (V m"^) E^^ streaming potential (V m^ N"^) / friction coefficient (kg s"^) / activity coefficient (mol fraction scale) (-) /y Mayer function for interaction between particles i and j (-) F Fciraday constant (C mol"M F Helmholtz energy (J) ^ i ' ^mi partial moleir Helmholtz energy (J mol~^) Fjjj molar Helmholtz energy (J mol~^) F^ interfacial (excess) Helmholtz energy (J) pa po^^ interfacial (excess) Helmholtz energy per unit area (J m-^) F, F force (N) g(r) radial distribution function (-) g^^^ hth order distribution function (-) g(q, t) time correlation function, if real (light scattering usage) (dimensions as C^^) g standard acceleration of free fall (m s"^) G Gibbs energy (J) ^ i' ^mi partial molar Gibbs energy (J mol"^) G^ molgir Gibbs energy (J mol"^) G^ interfacial (excess) Gibbs energy (J) G^, G^/A interfacial (excess) Gibbs energy per unit area (J m-2) G{z) segment weighting factor in polymer adsorption theory (-) G(z, s) endpoint distribution in a segment of s segments (polymer adsorption (-) h Planck's constant (J s) n h/2n (J s) h (shortest) distance between colloidal particles or macrobodies (m) h height (m) h(r) total correlation function (-) D

xxii

SYMBOL LIST

H

enthalpy (J) H^, H^

partial molar enthalpy (J mol"^)

H^

molar enthalpy (J mol"^)

H^

interfaclal (excess) enthalpy (J)

H^, H^/A

interfaclal (excess) enthalpy per unit area (J m-^)

H H

magnetic field strength (C m-i s"^)

H(p,q)

Hamiltonian (J)

i

intensity of radiation (V^ m~2) ii

incident intensity (V^ m"^)

IQ

i n t e n s i t y i n a v a c u u m (V^ m~2)

ig

s c a t t e r e d i n t e n s i t y (V^ m"^)

i

unit vector in x-direction (-) (not in chapter L7)

I^^^

streaming current (C m^ N"^ s"^)

/

ionic strength (mol m"^)

/

radiant intensity (J s"^ sr -^ = W sr -^)

IJCD)

spectral density of x (dim. x^ s)

j

. unit vector in y-direction (-) (not in chapter L7)

jj

(electric) current density (A m'^ = C nr^ s"^) j ^ , J^

J, J

flux

surface current density (C m-^ s"^) (mol m"2 s"^ or kg m"^ s"^)

J^

surface flux (mol m'^ s~^ or kg m"^ s"^)

J°

interfaclal compliance (mN"^)

J

first, or mean, curvature (m~i)

k

Boltzmann's constant (J K"^)

k

rate constant (dimensions depend on order of process)

fcj

bending modulus (J)

k

saddle splay modulus (J)

fc

unit vector in y-direction (-) (not in chapter L7)

k,

wave vector (m-^)

!?C

Optical constant (m^ kg"^ or m^ moH)

K(R)

optical constant (V^ C-2 m"^)

K(o))

absorption index (-)

K

chemical equilibrium constant (general) K K^

K

on pressure basis (-) on concentration basis (-) (integral) electric capacitance (C V-^ or C m-^ V-^)

K^

(integral) electric capacitance of diffuse layer (C m-^ V"^)

K^

(integrcd) electric capacitance of S t e m layer (C m-^ V-^)

K pj

Henry constant (m)

Kj

distribution (partition) coefficient (-)

KL

Langmuir constant (m^ mol"^)

SYMBOL LIST K

xxiii

conductivity (S mr^ = C V-^ nr^ s'^) surface conductivity (S = C V-^ s'^) interfacial dilational modulus (N m - ^ contour length (polymers) (m) cross coefficients in irreversible thermodjniamics (varying dimensions) bond length in a polymer chain (m) m a s s (kg) (relative) molecular m a s s (-) (M)^, M ibid., m a s s average (-) {M)^, M^ ibid.. Z-average (-) (M)^, M ibid., n u m b e r average (-) refractive index (-) n u m b e r of moles (- or moles) n^ n u m b e r (excess) of moles in interface (- or moles) unit vector in x-, y- or z-direction (-) (chapter L7 only) number of segments in a polymer chain number of molecules (-) N^y Avogadro constant (mol"^) Ng number of sites (-) bound fraction (of polymers) (-) pressure (N m"^) Ap capillcuy pressure (N m-^) stiffness (persistence) parameter (poljnners) (-) dipole moment (C m)

K"" K^ L Ljj^ ^ m M

n n n^yz N N

p p p p, p

^ind' ^ind p, p P P, P Pe q q(isost) q q q q q, q Q Q(N,V,T) Q^^ ^

Induced dipole moment (C m) {= mv) momentum (J m"^ s) probability (-) polarization (C m'^) Peclet number (-) heat exchanged (incl. sign) (J) isosteric heat of adsorption (J) generalized parameter indicating place coordinates in Hamiltonian subsystem canonical partition function (-) electric charge (on ions) (C) persistence length (polymers) (m) scattering vector (m-^) electric charge (on colloids, macrobodies) (C) canonical partition function (-) electro-osmotic volume flow per unit field strength (m^^ V"^ s"^)

xxiv Q J r r

SYMBOL LIST

electro-osmotic volume flow per unit current (m^ C"^) distance (m) fg Bjerrum length (m) r number of segments in a polymer (-) R gas constant (J K'^ mol'^) R (principal) radius of curvature (m) R Poynting vector (W m-^) Rg Rayleigh ratio (m'M Re Reynolds number (-) S entropy (J K'^) ^ V '^mi partial molar entropy (J K"^ mol'^) Sj„ molar entropy (J K"^ mol'^) S^ interfacial (excess) entropy (J K"^) S^, S^/A interfacial (excess) entropy per unit area (J K'^ m'^) S(q, R, Q) spectral density as a function oio)^- o\ = Q (V^ m'^ s) Slq, c) structure factor (-) S (s) ordering parameter of s (-) t time (s) t trainsport (or transference) number (-) T temperature (K) Ta Taylor number (-) u (internal) energy per subsystem (J) u (electric) mobility (m^ V-^ s'^) U (internal) energy, general (J) U ^, Uj^j partial molar energy (J mol"^) U^ molar energy (J mol"^) U^ interfacial (excess) energy (J) U^, U^/A interfacial (excess) energy per unit airea (J m-^) V excluded volume parameter (polymers) (= 1-2;)^) V, V velocity (m s"^) V electrophoretic velocity (m s'^) V electro-osmotic velocity (m s"^) V slip velocity (m s"M V volume (m^) Vp V^^ partial molar volume (m^ mol"^) V^ molar volume (m^ mol'M w work (incl. sign) (J) w interaction parameter in regular mixture theory (J mol-^) w interaction energy between pair of molecules or segments (landJ) (J) X mol fraction (-)

SYMBOL LIST X X. X y y z z z Z lN,p,T) Z^ Greek a a a a, a j3j2 /3 J3D /^K /^L P P 7 y Y y r 5 6A: 5(x) A AX "A^X "A^0 "AV "AVdiff e £ £ ^

xxv distance from surface generalized force in irreversible thermodynamics (varying units) activity coefficient (molar scale) (-) dimensionless potential {Fy//RT] (-) coordination number (-) distance from surface (m) valency (-) isobaric-isothermal pairtition function (-) configuration integral for N particles (-)

real potential (V) degree of dissociation (-) contact angle (-) polailzabmty (C V-i m^ = C^ J-^ m2) twice binsiry cluster integral (-) V a n d e r W a a l s c o n s t a n t (molecular) (J m"^); Debye-Van der W a a l s c o n s t a n t (molecular) (J m"^) Keesom-Van d e r W a a l s c o n s t a n t (molecular) (J m"^) London-Van d e r W a a l s c o n s t a n t (molecular) (J m"^) E s i n - M a r k o v coefficient (-) d a m p i n g coefficient (m"^) interfacial or surface tension (N m"^ or J m"^) activity coefficient (molal scale) (-) shccir s t r a i n (-) rate of s h e a r (s"^) surface (excess) c o n c e n t r a t i o n (mol m"^) diffusion layer thickness (m) small variation of x (dim. x) Dirac delta function of x (dim. x'M displacement (m) X (final) - X(initial) X(phase p) - X ( p h a s e a) (Galvani) potential difference (V) (Volta) potential difference (V) liquid Junction potential (V) relative dielectric permittivity (dielectric constant) (-) dielectric permittivity of vacuum (C^ N-^ m-^ or C m"^ V-^) porosity (-) electrokinetic potential (V)

xxvi

SYMBOL LIST

6

surface coverage = F/F (saturated monolayer) (-)

9

angle, aingle of rotation, loss angle (-)

K

reciprocal Debye length (m"^)

K

capillary length (m)

E

grand (c£inonical) partition function (-)

A

wavelength (m)

A

charging parameter (-)

A

ionic (or molar) conductivity (C V"^ m^ s~^ mol"i = S m^ mol"^)

A

molar conductivity (S m^ mol"^)

A

thermal wavelength (m)

A

penetration depth of evanescent waves (m)

A/Q

magnetic permeability in v a c u u m (V m"^ C"^ s^)

ju

magnetic dipole moment (C m^ s~^)

ju

chemical potential (J mol"^ or J molecule'^)

jj.

kinematic viscosity (m^ s'^)

rj

dynamic viscosity (N s m~2) T]^

interfacial s h e a r viscosity (N m~^ s)

r]^

interfacial dilational viscosity (N m"^ s)

V

frequency (s"^ = Hz)

^

coupling pcirameter (Kirkwood) (-)

^

grand (canonical) partition function of subsystem (-)

n

surface pressure (N m"^ or J m"^)

77

osmotic pressure (N m"^)

n(h)

disjoining pressure (N m"^)

p

density (kg m"^) p^

n u m b e r density [N/V) (m"^)

p

space charge density (C m-^)

a

distance of closest approach for h a r d spheres (m)

a

surface density of b r u s h e r (m-^)

G, (T°

surface charge density (C m"^) o;

contribution of ionic species i to surface charge (C m~^)

G^

surface charge density diffuse layer (C m"^)

a^

surface charge density S t e m layer (C m"^)

G^

s t a n d a r d deviation of X (dim. x)

T

characteristic time (s)

T

interfacial stress (N m~^)

r

line tension (N)

T

turbidity (m'M

T

stress tensor (N m"^) T^

interfacial stress tensor (N m"^)

SYMBOL LIST T^y T^ Tj. 0 (p 0 X X , (p XiX^^) Xe X^ Xl^^ If/ CO 0)^ (0^ (o Q D Q £2 £2^ £2^ Q(N, V, U)

xxvii flux

of x - m o m e n t u m in y-direction (kg m"^ s"^) = s h e a r s t r e s s (N m"2), one of the nine components of the stress t e n s o r rotational correlation time (s) rotational relaxation time (reorientation time) (s) osmotic coefficient (-) volume fraction (-) p h a s e (-) excess interaction energy p a r a m e t e r (-) critical values of x a n d (p a t p h a s e separation interfacial potential J u m p (between p h a s e s p a n d a) (V) electric susceptibility (-) adsorption energy p a r a m e t e r (-) critical value of x^ at the adsorption/desorption point electric potential (V) angular frequency (rad s"^ or s"M a n g u l a r frequency of incident radiation (rad s"^ or s"^) a n g u l a r frequency of scattered radiation (rad s"^ or s"^) degeneracy of subsystem (-) degeneracy of a system or n u m b e r of realizations (-) 0)^- (o^ (rad s"^ or s"^) solid angle (sr) g r a n d potential (J) interfacial (excess) g r a n d potential (J) interfacial (excess) grand potential p e r u n i t area (J m-^) n u m b e r of realizations = microcanonical pgirtition function (-)

This Page Intentionally Left Blank

1 Interfacial tension: measurement 1.1 General introduction to capillarity and the measurement of interfacial tensions 1.2 On the mathematics of curvature 1.3 Capillary rise 1.3a Vertical cylinder 1.3b Vertical plate 1.3c Other geometries 1.4 Shapes of drops and bubbles on surfaces 1.5 Free drops in a density gradient or electric field 1.6 Drop weight method 1.7 Maximum bubble pressure 1.8 Force required to hold objects at an interface or to pull them through it 1.8a Wilhelmy plate 1.8b DuNoiiyring 1.8c Other geometries 1.9 Spinning drops and bubbles 1.10 Surface light scattering 1.11 Miscellaneous other static methods 1.12 A case study: the surface tension of water 1.12a Room temperature 1.12b FromO°- lOOX 1.13 Measuring the surface tension of solids 1.14 Surface tensions under dynamic conditions 1.14a Pure liquids 1.14b Solutions 1.14c A note on the pristine state of LG and LL surfaces 1.15 Bending moduli 1.16 Applications 1.17 General References 1.17a lUPAC recommendations 1.17b Tabulations 1.17c Capillarity, shapes of fluid bodies etc. 1.17d Measuring interfacial and surface tensions

1.1 1.2 1.7 1.12 1.12 1.21 1.23 1.24 1.31 1.33 1.36 1.39 1.39 1.45 1.48 1.49 1.53 1.56 1.58 1.58 1.59 1.65 1.67 1.68 1.70 1.76 1.77 1.82 1.84 1.84 1.85 1.85 1.86

This Page Intentionally Left Blank

1

INTERFACIAL TENSION: MEASUREMENT

This Volume deals with various aspects of surface tensions and interfacial tensions. Together with the phenomenon of adsorption (enrichment of molecules at interfaces), these tensions constitute the basic characteristics of interfaces. The concept of surface tension is a very old one. Reportedly, Leonardo da Vinci had already observed and recorded the spontaneous rise of liquids in narrow, wetted capillaries, bores and plugs^^. From this rising the phenomenon acquired its name: capillus (lat.) = hair: the bores should be as narrow as a hair. Nowadays the term capiRary phenomena is used more widely (in this book also) to indicate not only capillary rise, involving curved interfaces, but also all phenomena determined by the tendency of interfaces to adopt a minimum area, such as drop shapes, bubble shapes, liquid bridges and wetting. An important historic development was the insight that surface tension is a surface- aind not a bulk-property. Newton^^ had already discriminated between cohesive and adhesive forces. A decisive discriminative experiment was carried out by Hawksbee^^ He investigated liquid rise in capillaries and between glass plates and found that the thickness of the glass did not matter. So the phenomenon of wetting was established as a surface phenomenon, although the depth of the interactions responsible for the various tensions still remains an issue today (chapter 2). In chapter I.l we introduced the notions of surface and interfacial tensions phenomenologically. Volume II mainly dealt with interfaces of solids. For such systems, the adsorption of molecules and ions is the primary phenomenon. Interfacial tensions Ccinnot generally be measured, although interfacial pressures are obtainable from adsorption isotherms using Gibbs' law. As a logical sequel, in this third Volume of FICS, liquid-fluid (LG or LL) interfaces will be treated. For such systems the interfacial tension y is the primary, and measurable, variable. Often the areas involved are so small that the analytical determination of adsorbed amounts (F' s) is difficult. In those cases Gibbs' law can be used to relate interfacial tension to the adsorption, provided adsorbate and 1) A footnote by C. Wolf, Pogg. Ann. 101 (1857) 551 refers to this. ^^ I. Newton, Optiks, for instance 3rd ed., 31st Query. Wand J. Innys, St. Paul's London (1721). ^^ F. Hawksbee, Physico-mechanical Experiments, Lx)ndon (1709) 139.

1.2

INTERFACIAL TENSION: MEASUREMENT

solution are in equilibrium with each other. For systems with large LG or LL areas, like foams a n d emulsions, adsorbed a m o u n t s are often more easily m e a s u r e d , b u t then it m u s t be realized t h a t the surface excesses are to a large extent determined by t h e history of t h e s a m p l e s , a n d therefore not necessarily representative of t h e adsorption equilibrium in systems with low a r e a s . Foams a n d emulsions will be dealt with in Volume V. In line with the systematics of FIGS, it is logical to s t a r t with the m e a s u r e m e n t of interfacial tensions. This will be done in t h e p r e s e n t c h a p t e r , e m p h a s i z i n g p u r e liquids. A selection of r e s u l t s will be t a b u l a t e d in Appendix 1. This chapter is followed by one on interpretations, two on

fluid-fluid

interfaces carrying adsorbates, a n d finally the Volume will be r o u n d e d off with a discussion of t h r e e - p h a s e contacts, i.e. with wetting p h e n o m e n a . In view of the scope of FIGS, results u n d e r ambient conditions of temperature a n d p r e s s u r e will be emphasized. Regarding the t e r m s interface and surface, we follow the convention adopted in Volume I. 'Interface' is the general term; the word surface' is restricted to p h a s e b o u n d a r i e s in which one of the p h a s e s is gaseous, or to indicate specifically t h e o u t m o s t p a r t of a condensed p h a s e u n d e r consideration ('the surface of a n oil droplet in a n oil-in-water emulsion'). In the literature this distinction is n o t always so systematically maintained. In other places the symbol for interfacial or surface tension is a or / . We prefer the latter because a is widely used for surface charge density, a n d the two do occur simultaneously in some equations. Much background information about the notion of interfacial tension, its thermodynamical a n d statistical interpretation a n d a n u m b e r of other a s p e c t s have already been dealt with in Volumes I a n d II ^^ We start by briefly reviewing these. 1.1 General i n t r o d u c t i o n t o capillarity and t h e m e a s u r e m e n t of interfacial t e n s i o n s For the m e a s u r e m e n t of interfacial tensions no molecular models are needed. It is e n o u g h to know t h a t the interfacial tension is a m e a s u r e of the tendency of all a r e a s to become a s small a s possible. Following chapter 1.2, this contractile action c a n be interpreted thermodynamically or mechanically. Thermodynamically,

the interfacial tension is interpreted a s the increase in the

Helmholtz or Gibbs energy of the system w h e n the area of the interface u n d e r consideration is increased reversibly by a n infinitesimal a m o u n t d A at c o n s t a n t t e m p e r a t u r e a n d composition, a n d at constant volume or c o n s t a n t p r e s s u r e , respectively. We can express this a s

^' Where expedient, some results of this discussion will briefly be repeated. When reference is made to equations, sections or chapters of Volumes I or II, these will be indicated by I or II, respectively; references not preceded by a roman number refer to the present Volume (III).

INTERFACIAL TENSION: MEASUREMENT y = OF/aA)^^ „

1.3

y = (aG/aAl^^ „

w h e r e n is a n abbreviation for the set of a m o u n t s n^, n^

I l . l a , b) t h a t define t h e

composition of the system. The dimensions of / are [energy/area]; we shadl express them in m J m~^. As F a n d G are in principle measurable, so is 7 . Mechanically,

the interfacial tension is the contractive force per u n i t length,

acting in t h e interface a n d parallel to it. The dimensions are [force/length], a n d o u r u s u a l u n i t is mN m"^ = m J m"^*^ Interfacial t e n s i o n s a r e m e c h a n i c a l l y m e a s u r a b l e a s the forces required to enlarge a n interface by a n infinitesimal a m o u n t . For isotropic interfaces this force is the same in each direction. At equilibrium the thermodynamical a n d mechanical interpretations s h o u l d b e equivalent. In sec. 1.2.3, in connection with fig. 1.2.1, this equivalence w a s a d d r e s s e d a n d found to be achieved provided the extension of the area is done reversibly, t h a t is, at low Deborah n u m b e r (De «

1). Only u n d e r this condition h a s

the interface enough time to come to equilibrium, t h a t is, to achieve full relaxation of all a d s o r p t i o n equilibria. The v a l u e s of 7 obtained in t h i s way a r e t h e equilibrium values 7(eq.), i.e. those tabulated in reference books. When we w a n t to distinguish t h e s e from the non-equilibrium, or dynamic, interfacial tensions, we call t h e m t h e static interfacial tension. Under non-equilibrium conditions t h e mechanical interpretation r e m a i n s valid, b u t the thermodynamical one becomes ill-defined. This b r i n g s u p the question of how this s c h e m e h a s to be modified w h e n equilibrium is not attained. The answer is t h a t the identity of thermodynamical a n d mechanical m e a s u r e m e n t persists, b u t t h a t the value obtained for y

differs

from 7 (eq.); in fact, often 7 (non-eq.) > 7 (eq.). Suppose a given interface is created very rapidly a n d t h e n s t a r t s to relax with a time scale r. At a n y time t = cos a. The result is w = 2Kaycosa

[1.3.11]

which confirms the mechanical interpretation of capillary rise: the upward force is /cosOf multiplied by the perimeter 2na, and independent of shape. Otherwise stated, the shape is not an independent parameter; once y, cos a and Ap are given, the shape is also fixed. For interpretational reasons this is a rewarding result, but not for determining interfacial tensions, because in practice it is not the weight of the column that is measured but rather the profile: no analyticcd solution of [1.3.8] for, say h in the lowest point of the meniscus, is available. Hence, numerical solutions are required. For axisymmetrical interfaces, as in cylindrical capillaries, pendent and sessile drops, this is often achieved by introducing the radius of curvature b at the apex or lowest point as the measure of the meniscus size, anticipated in table 1.1. At that point R^ = K = b, hence J = 2/b and at the top Ap = 2y/b. If z is now measured with respect to that apex (positive or negative, depending on the case under consideration), the pressure at any z can be given as 7 J = Apgz + 2y/b

[1.3.12]

For J = c^+ c^ one may substitute the sum of [1.2.7 and 8] or one of the expressions [1.2.14]. One way in which this equation is often rewritten involves the introduction of a dimensionless parameter

with which [1.3.12] and [1.2.11] converts into the, also dimensionless, expression

^

sin^^£z^2 x/b b

11.3.141

The parameter p is positive for oblate figures of revolution (for instance, a sessile drop or a meniscus in a capillary) but negative for prolate ones, like a pendent drop. In the absence of gravity /3 = 0 and the profile is spherical. Equation [1.3.14] is deceptively simple, but it must be realized that c^ and sin0/x = c^ (see [1.2.11]) are given by [1.2.7 and 8]; hence [1.3.14] is a second order differential equation. Only for very narrow capillaries (b/K —> 0) and complete wetting does it reduce to c^+c^=2/b, the Laplace pressure for a spherical

1.18

INTERFACIAL TENSION: MEASUREMENT

m e n i s c u s of r a d i u s b. In all other s i t u a t i o n s n u m e r i c a l analysis is required. Basically, t h e profile is m e a s u r e d , u s u a l l y by some optical t e c h n i q u e , a n d compared with the numerical d a t a to obtain y (and cos a ) for which t h e fit is maximized. The archetype numerical tabulations of x(z) profiles date back to Bashforth a n d Adams (almost to 1853^^). These tables were used for more t h a n a century, either in their original form or after modification, until their significance w a n e d with the advent of m o d e r n c o m p u t e r s . The tables do not only apply to liquid menisci in capillaries, b u t also to cross-sections of sessile drops, pendent drops, etc. They also contain s u c h information a s the diameters and heights of sessile drops a n d contact angles. They give x / b and zib a s a function of 0 for various closely-spaced ^ values. Their a p p l i c a t i o n r e q u i r e s successive a p p r o x i m a t i o n b e c a u s e /J c a n only be established if y is known. A starting value of y could, for instance, be obtained from one of the simpler equations, say from [1.3.2 or 6]. Several a u t h o r s have extended Bashforth a n d Adams' tables or rendered t h e m more useful. One of the earliest was Sugden-^^ who dealt with the problems t h a t b is sometimes difficult to m e a s u r e accurately a n d t h a t the tables do not extend into the region of t u b e s t h a t are so small t h a t b -^ a. He gave tables relating r/h to V/K a n d p a r t s of t h e s e tables have been reproduced by Adamson a n d Padday*^^. The Sugden tables are convenient for obtaining the capillary length from the r a d i u s of the t u b e a n d t h e capillary rise. Later elaborations have been provided by Padday a n d Pitt'*^ who compared different t a b u l a t i o n s a n d Lane^^ who gave a c c u r a t e polynomial r e p r e s e n t a t i o n s of t h e profile. Erikson^^ p u b l i s h e d

computer-

calculated a r e a s . See further the references in sees. 1.17c a n d d. Some of t h e s e contain useful tables. In practice, the capillary rise method was (and p e r h a p s still is) one of the best a n d most precise m e t h o d s , b e c a u s e the experimental variables c a n be well controlled. In order to g u a r a n t e e visibility of the meniscus, the capillaries are invariably m a d e of glass. One of the best situations for m e a s u r i n g is t h a t of complete wetting. To t h a t end the inner side of the capillary should be meticulously clean a n d it is advisable to m e a s u r e the receding contact angle by first sucking the liquid m e n i s c u s higher into the capillary a n d t h e n letting it sink to its equilibrium position. Care should be t a k e n that, if the surface tension of solutions is to be m e a s u r e d , the solute does not react with, or adsorb on the glass, so changing the ^^ F. Bashforth, J.C. Adams, see the reference in sec. 1.17c. 2) S.Sugden, J. Chem. Soc. 119 (1921) 1483. ^^ See the references in sees. 1.17c cind 17d, respectively. Note that their a equals our K-V2. 4) J.F. Padday, A. Pitt, J. Colloid Interface Set 38 (1972) 323. ^^ J.E. Lane, J. Colloid Interface Set 42 (1973) 145. 6) T.A. Erikson, J. Phys. Chem, 6 9 (1965) 1809.

INTERFACIAL TENSION: MEASUREMENT

1.19

contact angle. This may, for example, h a p p e n with catlonic surfactants t h a t could r e n d e r glass hydrophobic, or in the Lippmann electrometer (see below) u s e d to m e a s u r e the interfacial tension of mercury in solutions of sodium fluoride; t h i s solute is often studied because it exhibits no specific adsorption in the double layer, b u t it reacts with glass a n d may give rise to erosion^^. The capillary should have a circular cross-section. Its internal diameter can be obtsiined optically or by filling it, or p a r t s of it, with mercury and weighing. In situations where m e a s u r e m e n t of the height suffices, high-grade, vibration-free cathetometers should be used. Otherwise the full meniscus profile h a s to be measured, usually optically. Harkins a n d Brown^) have summarized a n u m b e r of precautions which need to be taken to optimize the performance of this method. These recommendations are old, b u t not obsolete a n d some have already been mentioned above. Another one involves the proper establishment of the zero-level (the horizontal liquid surface in fig. 1.4). To e n s u r e horizontality, the vessel with liquid should be large enough a n d t h e implication is t h a t the method requires relatively large a m o u n t s of liquid. Harkins a n d Brown suggested locating the exact position of the level by observing the reflection of a needle j u s t above the fluid. Measurement of the rise with respect to a horizontal reference level can be avoided by placing two capillaries of different r a d i u s next to each other, a n d determining the height differenced^ capillary

[differential

rise method). For instance, in the very simple case t h a t [1.3.1] is a

r e a s o n a b l e approximation

2K-^

P9

1 V 1

1

[1.3.15]

2J

Variations to improve accuracy, facilitate handling, or r e n d e r t h e m e t h o d applicable to special s y s t e m s have been proposed. For i n s t a n c e , Richards a n d Carver^) developed a capfllary with a reflush device (a wider tube, parallel to the vertical capillary) to facilitate rejuvenation of the liquid surface. This a p p a r a t u s w a s modified by Young a n d Gross^^. R a m a k r i s h n a n a n d Hartland^^ developed a p r o c e d u r e of m e a s u r i n g surface t e n s i o n s in t h e a n n u l a r ring b e t w e e n two concentric cylinders. This approach w a s duplicated by Agrawal a n d Menon^^. Long ago Sentis^^ experimented with a n isolated capillary on the lower end of which a 1^ A. de Battisti, R. Amadelli and S. Trasatti, J. Colloid Interface Set 6 3 (1978) 61. 2) W.D. Harkins, F.E. Brown. J. Am. Chem. Soc. 41 (1919) 499. ^^ N. Pilchikov. Zhur. Fiz. Khim, Obshch. Sen Fiz. 2 0 (1888) 65; S. Hartland, S. Ramakrishnan, Commun. Jom. Com. Esp. Deterg. 12th (1981) 419. 4) T.W. Richards. E.K. Carver. J. Am. Chem. Soc. 4 3 (1921) 827. ^^ Mentioned by W.D. Harkins. in Physical Methods of Organic Chemistry 1. 2nd ed. A. Weissberger. Ed., Interscience (1927) p. 367. ^J S. Ramakrishnan, S. Hartland, J. Colloid Interface Set 80 (1981) 497. "^^ D.C. Agrawal. V.J. Menon, Am. J. Phys. 52 (1984) 472. ^^ M.H. Sentis, J. Phys. Theor. Appl Ser. 2, 6 (1887) 571; Ser. 3. 6 (1897) 183.

INTERFACIAL TENSION: MEASUREMENT

1.20

hanging drop appeared (see sec. 1.4), so he virtually combined two techniques. Prigorodovl) modified this idea by working with a capillary t h a t w a s thin at t h e top a n d thick a t its lower end. The advantage over differential m e t h o d s is t h a t m u c h less liquid is needed. This brings to mind the familiar feature of the small a m o u n t of fluid remaining in a pipette after emptying (fig. 1.8). This left-over h a s a negligible influence on the accuracy of the dispensed a m o u n t s of liquid because it is constant a n d determined by the balance between the two Laplace pressures, the contact angle(s) a n d gravity. Even if the pipette is t u r n e d upside down (fig. 1.8a,c) the upward pull of the top m e n i s c u s against the concerted action of the downward pull of the lower m e n i s c u s a n d gravity k e e p s the liquid in the tip. Heller et al.^) have used this principle to develop a method for determining surface tensions. The method is suitable for very small a m o u n t s of liquid, b u t requires elaborate calibration if absolute values of the surface tension are required.

(a)

(b)

(c)

Figure 1.8 A tiny plug of liquid in the conical tip of a pipette. For p r a c t i c a l r e a s o n s t h e capillary rise t e c h n i q u e is rarely u s e d for t h e m e a s u r e m e n t of interfacial (rather t h a n surface) tensions; large a m o u n t s of the two liquids are needed a n d there are suitable a n d convenient alternatives. An exception to this is the m e a s u r e m e n t of the interfacial tension between m e r c u r y a n d (mostly) a q u e o u s solutions at various potential differences applied across the liquid-liquid interface. S u c h m e a s u r e m e n t s are done in a so-called capillary

electrometer,

Lippmann

already described in the chapter on electric double layers

(fig. II.3.47). Capillary rise is very high if the capillary is well-wetted and narrow. In a wetted capillary with a n internal r a d i u s of 0.1 or 0.01 mm, water a s c e n d s a b o u t 15 cm or 1.5 m, respectively. This not only a c c o u n t s for the possibility of plant growth far above the g r o u n d water table, it also indicates why the method is so sensitive; u n d e r favourable conditions surface tensions can be m e a s u r e d down to ± 0 . 0 5 mN m"^ Most of the surface tensions reported in the appendix have been obtained

1) V.N. Prigorodov, Koll Zhur. 3 3 (1971) 168 (transl. 139). 2) W. Heller. M-H. Cheng and B.W. Greene, J. Colloid Interface Set 22 (1966) 179-194.

INTERFACIAL TENSION: MEASUREMENT

1.21

by this technique^^. 1.3b

Vertical

plate

Consider a flat plate of length £ a n d width b(i

» b), immersed in the liquid to

be measured, a s sketched in fig. 1.9a. The geometry is now simple b e c a u s e in fig. 1.9b only t h e r a d i u s of curvature in the plane of the paper h a s to be considered^^ This sweeping s t a t e m e n t n e e d s clarification b e c a u s e it implies t h a t no end corrections are required. The problem h a s attracted m u c h discussion in the literature a n d h a s led to confusion. Whether there are s u c h things a s edge effects depends on the observations m a d e a n d on the deductions drawn from them. According to [1.1.3], at equilibrium t h e s u m Cj + c^ m u s t be the same everywhere. Let c refer to the curvature perpendicular to the plate. In the centre of the large flat part of the plate it is the only one;

(a)

b)

Figure 1.9. Capillary rise on a flat plate. Top: general view, bottom: meniscus details.

^^ We shall not give illustrations for each individual technique but shall later make a comparison (sec. 1.12). '^^ For a detailed mathematical treatment of such profiles see e.g. J.E. McNutt, C M . Andes, J. Chem, Phys. 30 (1959) 1300.

1.22

INTERFACIAL TENSION: MEASUREMENT

this is the c u r v a t u r e considered in fig. 1.9b. Near the e n d s of the plates the curvature c^, normal to the former, also plays a role. The result is t h a t near the end, h tends to decrease. If the end-effect is interpreted a s this downward trend, we speak of a real effect, t h a t h a s been observed by a variety of investigators. However, this end-effect h a s no consequences

for the capillary force exerted on

the plate. In this respect there is no end-effect in the measurement; adding 'correction terms' to [1.3.20] is wrong. Laplace^^ recognized this long ago a n d demons t r a t e d t h a t t h e a m o u n t of liquid, risen above t h e equilibrium level, m u s t be c o n s t a n t per u n i t length of t h e projection of the perimeter of t h e object on t h e equilibrium surface plane. The risen liquid at the edges h a s to come from a larger surface area t h a n in the central p a r t s of the plates, so there the capillary rise is lower. This also explains why the capillary rise outside thin t h r e a d s is so m u c h smaller t h a n against flat p l a t e s ^ l Laplace himself elaborated this principle for the capillary rise inside a capillary of non-circular cross-section a n d stated explicitly t h a t it remained valid for polygons. More recently, Loos^' extended this work to the o u t s i d e of cylinders a n d s y s t e m s with other geometries. This work h a s a t t r a c t e d little a t t e n t i o n , p e r h a p s b e c a u s e it w a s written in D u t c h , b u t t h e relevance is n o t easily overestimated; certainly it applies to t h e

often-used

Wilhelmy plate (sec. 1.8a). Let u s now consider the rise sdong the long sides of the plates. At any point on the m e n i s c u s the Laplace pressure is bcdanced by gravity pressure. For the former we have, according to [1.1.3] a n d [1.2.12], / d c o s ^ / d z , w h e r e t h e angle 0 is indicated

in fig. 1.9b, a n d for the latter Apgz, where z is counted with respect to

the horizontal level. Integration is straightforward: d c o s ^ / d z c a n be written a s - sin0d(/> (d/dz); the integration boundaries are z = 0, cos(/> = 1 a n d z = h, c o s 0 = sin a if a is the contact angle. The result is i-Ap^fh^ = y d - s i n a)

[1.3.16]

or 2 ( 1 - s i n a)

[1.3.17]

In t e r m s of t h e capillairy length A/2

h = ^^

(1-sina)

[1.3.18]

K

For the simple case of complete wetting ^^ See his Mecanique Celeste, Supplement au Livre X (1806) 440; or Oeuvres completes de Laplace, quatrieme tome, 2nd Supplement au Livre X, 439 ff, Gauthiers-Villars, Paris (1880). 2^ D.F. James, J. Fluid Mech. 6 3 (1974) 657. 3) R. Loos, Verhandelingen Koninkl Vlaamse Acad. Klasse Wetensch., XII (1950) 5.

INTERFACIAL TENSION: MEASUREMENT 7 = iApgh2

1.23 [1.3.191

These equations may be compared with those for cylinders, see for instance [1.3.2]. For flat plates one does not have to worry about complications of the details of the profile, but this advantage is offset by the much lower rise. Typically, h is of order K~^ , i.e. h = O (mm) and y is proportional to h^ whereas it scales with ah in capillaries. Over the last few decades laser-opticsd techniques for scanning the meniscus and establishing h down to about 10"^ mm have become available l'^'^'"^^. In a modem variant of the Wilhelmy plate technique, to be described in sec. 1.8a, the force needed to pull the plate out of the liquid is measured as a function of the height above the zero level. In this way the surface tension and contact angle can be determined simultaneously. Alternatively, the method can be used to obtain contact angles, i.e. from [1.3.16] after / has been measured by some other technique. The weight of the liquid pulled up along the plate, or rather the capillary force, can be obtained in a way similar to the derivation of [ 1.3.11]; the result can also be written down immediately ii;(plate) = 2U + b)y cos a

[1.3.20]

The factor 2 accounts for the two menisci, on both sides of the plate. Studying capillary rise at vertical plates in electrolyte solutions is important for a number of interfacial electrochemistry issues, such as electroplating and electrocapillarity. 1.3c Other geometries Capillarity equations have been elaborated for a variety of other geometries, but not always with the prime goal of measuring surface tensions. Often the attention is focussed on the force that keeps objects connected to a liquid if they are pulled out. The liquid bridge that, in the case of wetted materials, keeps object and fluid together is characterized by a concave meniscus, i.e. by an attractive Laplace pressure. Such studies are relevant for the interpretation of some adhesion problems. One other illustration is shown in fig. 1.10. Two vertical, wetted glass plates are placed in the liquid. The rod creates a separation, increasing proportionally with £ from left to right. To a first approximation, at any position the capillary rise is proportional to the gap width, b and hence to i, but in practice it seems more like that shown in the figure. This has to do with the change of the meniscus IJ I. Morcos, J. Chenh Phys. 55 (1971) 4125. 2) K. Rodel, P. Friese and G.-R. Wessler, Z. Phys. Chem. (Leipzig) 254 (1973) 289. ^^ K. Rodel, K. Lunkenhelmer, Tenside Deterg. 15 (1978) 135. ^^ S. Lahooti, O.I. del Rio, A.W. Neumcinn and P. Cheng, Axisymmetric Drop Shape Analysis (ADSA), ch. 10 in the book edited by Neumcinn and Spelt, mentioned in sec. 1.17c.

1.24

INTERFACIAL TENSION: MEASUREMENT

Figure 1.10. Capillary rise in a wedge. profile. To the left, where this profile approaches sphericity, h-- b, j u s t a s h ~ a for capillaries, [1.3.4], b u t to the right, where the two plates are further a p a r t t h a n a few times the capillary length, h is given rather by [1.3.18] and is independent of ^. The s e t - u p is suitable for verifying computed h(0

profiles a n d , if theory a n d

experiment m a t c h , the surface tension is obtainable. For narrow wedges W)

is a

hyperbola. However, it is also useful for analyzing h[i) for t h e case t h a t , in addition to the capillary pressure, a disjoining pressure /7(h) acts between the two s u r f a c e s . The m a t h e m a t i c a l generalization of t h i s s y s t e m to a n y m o n o t o n i c function

/7(h) h a s been given by Kagan and Pinczewski^^

P a d d a y et al.^^ analyzed the m e n i s c u s profile for a vertical cylindrical rod, either standing in the fluid, or being moved upward without detachment. From the pull on the rod the surface tension can be obtained. See also ref. ^\ For detailed analyses of s u c h capillary p h e n o m e n a we mostly refer to the references in sec. 1.17c. However, we shall consider a n u m b e r of these in connection with m e t h o d s in which the force is measured either to keep objects in a n interface or to detach them completely (sec. 1.8). 1.4

S h a p e s of drops and bubbles on surfaces'^)

A n u m b e r of systems such as sessile drop, captive bubble, hanging, or pendent drop in which a drop or bubble is kept in position on a surface belong to this category.

1^ M. Kagan, W.V. Pinczewski, J. Colloid Interface Set 180 (1996) 293. 2) J.F. Padday, A.R. Pitt and R.M. Pashley, J. Chem. Soc. Faraday Trans. (I) 7 1 (1975) 1919. ^^ C.J. Lyons, E. Elbing and I.R. Wilson, J. Colloid Interface Set 162 (1984) 292. ^^ This technique has been pioneered by A.M. Worthington, Phil. Mag. 19 (1885) 46.

INTERFACIAL TENSION: MEASUREMENT

1.25

The s h a p e s of these drops and bubbles are governed by the competition between the contractile surface tension a n d gravity, j u s t a s in capillary rise p h e n o m e n a . The p r o c e d u r e for obtaining interfacial tension normally c o n s i s t s of t h e following steps: (i) The profile m u s t be mathematically defined a n d elaborated into appropriate tables a n d / o r analytical solutions for certain limiting conditions. (ii) An experimental technique h a s to be developed to m e a s u r e the contours in sufficient detail. This is not a trivial problem: the drop is not a two-dimensional opaque screen b u t a three-dimensional, usually refracting, body. (iii) Theoretical a n d experimental profiles are matched, with / and, sometimes, cos a a s the parameters. Essentially, t h e s e s t e p s are similar to those for capillary rise. For (i) t h e Bashforth-Adams tables a n d their modern v a r i a n t s c a n be u s e d . Regarding (ii) l a s e r - o p t i c a l reflection t e c h n i q u e s c a n n o w a d a y s yield profiles w i t h

great

precision, so t h a t accurate 7 values can be obtained. We shall now discuss some of t h e m a i n features, leaving the n u m e r o u s technical details t o t h e specialized literature, except for noting t h a t m o d e r n image a n a l y s e s a n d a u t o m a t i z a t i o n render steps (i) and (ii) less tedious, if not obviating parts of them.

y///////AV///////////A 7

/ / \ V \V

\ \—

1

X

h

_ .^—• ^ " - "

Figure 1.11. Captive bubble, h is the vertical distance between the equatorial phase and the apex.

An example of a sessile drop was shown in fig. 1.3, together with the definitions of the two radii of curvature R^ a n d jR . The m a t h e m a t i c a l laws governing t h e c u r v a t u r e are general, b u t the geometry may be different for different situations. W h e n t h e surface on which t h e drop is sitting is homogeneous, t h e profile i s axisymmetric, so t h a t t h e cross-section of this profile suffices to describe t h e entire system. In t h e example the contact angle is o b t u s e , b u t t h i s i s n o t a requirement. Some drops do not have a n equator. The s h a p e also d e p e n d s on the volume (or weight) of the drop: very small drops (size 90° seem lighter because the capillary forces tend to p u s h them upward. From this additional force the surface tension can be obtained if the plate is fully wetted. The m e t h o d derives its n a m e from Wilhelmy who long ago^^ described t h e d e t a c h m e n t of a t h i n g l a s s plate. Although n o w a d a y s static, or s e m i - s t a t i c m e t h o d s are preferred, t h e n a m e of Wilhelmy r e m a i n s associated with t h e s e t e c h n i q u e s . This 'static Wilhelmy' approach s t e m s from Dognon a n d Abribat^^ who also introduced roughened platinum plates to improve wetting. S u c h plates r e m a i n popular. Quantitatively, the force F to be exerted to keep a plate of total height z, width b and length [ in the surface with h a s shown in fig. 1.19, is given by^^

^^2 i i

1^1

G

>

t

;^ ^ 6

.^0

L

—

1

—

_

-\

ts"-^

i

1

—[[2

i

1 4

-| 1

1

—

T

1

\~ — —"

\

\poi —

[■5

J- 1 —

—

-j L —

— —- J:6—1

Figure 1.19. Force exerted on a vertical plate, kept partly or completely immersed in a liquid. Side-view of fig. 1.9. One cycle of the oscillatory mode is shown.

1) L. Wilhelmy, Ann. Phys. Chem, Reihe 4 29 (1863) 177. 2) A. Dognon, M. Abribat, Con^jt Rend. Acad. Set (Paris) 208 (1939) 1881. ^^ Similar equations but only for ^ » b have been rigorously derived by F. Neumann, Vorlesungen uber die Theorie der Capillaritdt, Leipzig (1894), D.O. Jordan, J.E. Lane, Austr. J. Chem. 17 (1964) 7. £ind others.

INTERFACIAL TENSION: MEASUREMENT F(h) = pghib

-h (z - h)Apgib + 2{i + b]y cos a

1.41 [1.8.1]

The first term on the r.h.s. is the weight of the part of the plate above the liquid, with p the density of the plate material (density of the vapour p h a s e neglected). The second term is the weight of the submerged part, corrected for buoyancy a n d the third is the required capillary contribution. Note t h a t the buoyancy of the p a r t of the plate t h a t is in the meniscus above the fluid level (indicated by h in fig. 1.9) does not have to be accounted for. In practice b is kept a s smedl a s is practically possible. For m e a s u r e m e n t of the interfacial tension between two liquids t h e equation changes in that p is replaced by [ p(plate) - p{L^)] and Ap by [p(plate) - p(L^)] if L^ is the heavier of the two liquids L^ and L^. Generally these m e a s u r e m e n t s tend to be less precise because usually /^^ < y ^ . Let u s postpone t h e discussion of practical problems like establishing t h e zero level a n d now consider the various measuring modes, starting with the oscillatory mode. Let also b « ^, unless stated otherwise. Figure 1.19 illustrates one cycle. The forces F

F^, F^, etc. refer to heights h = h , h , h , etc., above the m e n i s c u s . In

position 0 the plate is completely submerged. F^ is determined by the second term on the r.h.s. of [1.8.1] with h = 0. Now the plate is raised to attain a certain height h above t h e zero level (position 1). The force is now higher, partly b e c a u s e t h e buoyancy is less a n d partly b e c a u s e of the capillary contribution / 1 cos a. Upon further increase F(h) increases linearly b e c a u s e of the steady reduction of t h e Archimedes term; the capillary contribution does not depend on h. This increase continues until a certain maximum (position 3) is reached, after which the process is reversed (positions 4, 5, ... until complete resubmersion). Under ideal conditions the forward cind backward traces coincide, b u t in practice the r e t u r n force is lower b e c a u s e of contact angle hysteresis:

the advancing contact angle cf(adv) always

exceeds the receding one, a(rec). We shall discuss this further in sees. 5.1 a n d 5.5. As a c o n s e q u e n c e t h e

trace F{h] t h a t , after a u t o m a t i o n , c a n be r e a d from a

c o m p u t e r screen may look like t h a t in fig. 1.20. Extrapolation of the two linear parts to h = 0 yields / cos a(adv) or / cos a(rec). It follows t h a t this oscillating mode gives a good insight into the extent of contact angle hysteresis, b u t t h a t y c a n only be obtained

F/i

^

^

1^ /cos a (rec)

" f rs

ycos a (adv)

-^

4

0,6 I

height

Figure 1.20. Example of an F[h) diagram for pulling a Wilhelmy plate out of a solution (->) followed by resubmersion ( is finite. Experimentally ^{q,t) can be measured; it is the Fourier transform of ^{z,t), and a complex quantity. The scattered intensity is proportional to the time-averaged amplitude, i.e. to 72 mN m-^ for t ^ 0 is not attainable. This experiment illustrates the wellk n o w n complication of the hydrolysis of NaDS to produce the strongly surface 50

h ^

s 45

•*•.

40 -I

1.1111

il.

10

-J

I

' I 111 ii

iiL

lO-^s 10^ bubble age

Figure 1.30. Surface tension of a 6.2 mM solution of purified sodium dodecyl sulfate. Maximum bubble pressure method. (Redrawn from K.J. Mysels, Colloids Surf. 4 3 (1990) 241.) Discussion of the regions I and II in the text.

^^ A. Passerone, L. Liggieri, N. Rando, F. Ravera and E. Ricci, J. Colloid Interface Set 146 (1991) 152, see also ibid.. 169 (1995) 226.; R. Nagarajan, D.T. Wasan. ibid. 159 (1993) 164; C.A. MacLeod, C.J. Radke, ibid.. 160 (1993) 435; X. Zhang, M.T. Harris and O.A. Basaran, ibid. 168 (1994) 47. ^^ R. Miller, P. Joos and V.B. Fainerman, Dynamic Surface and Interfacial Tensions of Surfactant and Polymer Solutions, Adv. Colloid Interface Set 49 (1994) 249. ^^ Recall that at the end of sec. 1.11a few dynamic methods were briefly mentioned.

1.75

INTERFACIAL TENSION: MEASUREMENT

active dodecyl alcohol (DOH). The initial sample was free of this admixture. The two regimes I and II may probably be attributed to adsorption of NaDS and DOH, respectively, and give some feeling for the rates of the processes involved. Figure 1.31 gives one example of a comparison between results from two different techniques, applied to one and the same system. In this case the (nonionic) surfactant was specially synthesized and well-defined: p-tert.butylphenol with 10 EO groups. It is seen that the results of the (faster) maximum bubble technique and the (slower) drop volume method, connect well. Many more 70 h

maximum bubble pressure ■■■■■■■

50i-

drop volume

A^ "^ ^Wv • ^"'""""""^""^^-"-^--^-A.A,^ . '""

°''°°°° 10"^ M ^-v,,,^5xlO-^M

40 ° ^oocooro

coocccoo

2.5x10"^ M

30

10r2

10"

10"

lO^s

10 time

Figure 1.31. Juxtaposition of the time dependence of the surface tension of a non-ionic surfactant (p-tert butylphenol-Ejo) as measured by two different techniques. (Redrawn from R. Miller, P. Joos and V.B. Fainerman, Progr. Colloid Polyrrt Set 97 (1994) 188.) 40 h cTco^

e S

35

---rH

30 CO

25h

^"^^-a^

c=10-^M

rH 10

15

20s

Figure 1.32. Non-equilibrium interfacial tension at the oil-water interface system: water + hexane, containing palmitic acid, of which the concentration c is indicated. The drawn curves relate to a model interpretation involving diffusion. (Redrawn from J. van Hunsel, G. Bleys and P. Joos, J. Colloid Interface Set 114 (1986) 432.)

1.76

INTERFACIAL TENSION: MEASUREMENT

examples of s u c h methodical comparisons can be found in the book by Dukhin et al., mentioned in sec. 1.17d. Disagreement between disparate studies can have a methodical (measuring technique), interpretational (computation of surface age) or chemical (purity of surfactant) background. Figure 1.32 deals with adsorption of palmitic acid from hexane to the oil-water interface, using t h e drop volume method. As the drop volume method is relatively slow, t h e initial decay from the pristine hexane-water interfacial tension to t h e first reported d a t a c a n n o t be given. Otherwise stated, t h e d a t a refer to t h e later stages of diffusion. The trend is t h a t equilibration is somewhat slower t h a n t h e adsorption of surfactants from aqueous solution. 1.14c

A note on the pristine

state

ofLG

and LL

surfaces

Surface or interfacial tension m e a s u r e m e n t s offer one possibility of verifying the absence of impurities at interfaces. First, measured values should not change with time (provided the fast surface r e a r r a n g e m e n t s discussed in sec. 1.14a are fully relaxed). Second, the absolute Vcilues should agree with s t a n d a r d data, where available. The latter a r g u m e n t is not fully u n a m b i g u o u s b e c a u s e the conditions u n d e r which t h e experiment w a s carried out may have played a role (the tension may depend on the n a t u r e of the gas applied to obtain a given pressure) a n d even s t a n d a r d d a t a may be subject to improvement. See the introductory notes to table Al. However, t h e dynamics of interfaces offer a n additional, a n d more sensitive option to ascertain virginity. Pristine surfaces should have tensions t h a t neither depend on the area nor chamge u p o n compression or expansion. When, on the other h a n d , impurities a r e p r e s e n t t h e t r e n d is t h a t their interfacial c o n c e n t r a t i o n s increase with compression and decrease with expansion, both to a rate-determined extent, with a concomitant change in the tension. Figure 1.33 gives a n illustration^^. In this case the interfacial tension of the commercial product is not only lower t h a n t h a t of the purified sample, b u t it also c h a n g e s u p o n a l t e r n a t i n g compression a n d expansion cycles. Generally, gradients of interfacial t e n s i o n s a n d their dependence on area changes are more sensitive indicators of interfacial impurities t h a n t h e absolute values. S u c h m e t h o d s c a n detect the presence of impurities, b u t c a n n o t identify their nature-^^ An illustrative example refers to the n - a l k a n e - w a t e r interface, a s reported by Goebel a n d Lunkenheimer*^). For a n u m b e r of 'as received' a l k a n e s the interfacial tension against water appeared to decrease a s a function of time; moreover it went u p upon expansion cind down u p o n compression. Apparently, the alkanes contained a slowly adsorbing surface-active 1) Basically the same idea was already suggested by Mysels and Florence, J. Colloid Interface Set 4 3 (1973) 577. 2^ K. Lunkenheimer, R. Miller, J. Colloid Interface Set 120 (1987) 176. 3) A. Goebel, K. Lunkenheimer, Langmair 13 (1997) 360.

INTERFACIAL TENSION: MEASUREMENT

1.77

52 ^.''lllll'BKavMM^IiriPillllUB^III ■B""%V'B'I''IL%1I""H"HI!''VVII"'I"""""WH"'IIIH' 1

(a)

-

s

51 -

50 1

1

ci'^DPD°n'a-n-D^ 49

\ 500

1 1 1

1 1 1

[JCn^DDn-n-D

\

\

1

1000

1500

2000 s time

Figure 1.33. Test for the pristine state of the water-decane interface. The interfacial tension is measured during compression and expansion cycles: (a) meticulously purified decane, (b) commercial decane sample. (Redrawn from R. Miller, P. Joos and V.B. Fainerman, Adv. Colloid Interface Set 49 (1994) 249.) component. This impurity could be removed by passing t h e a l k a n e s a b o u t five times over a column of alumina. After this cleaning procedure, t h e obtained tension w a s slightly higher t h a n those tabulated in t h e literature. Most interestingly, these critically-evaluated tensions showed indications of a n odd-even

alteration

as

a function of chain length, t h e tension being slightly, b u t significantly ( - 0 . 3 mN m"^) higher for t h e even hydrocarbons. So far, s u c h effects have commonly been reported for b u l k properties, like melting points, molar volumes a n d solubilities in water, b u t n o t yet for surface properties. P e r h a p s they a r e related to packing constraints. 1.15

Bending moduli

Basically, all t h e m e t h o d s for measuring interfacial tensions described so far have in common t h a t t h e Helmholtz energy for extending a n interface is determined. Upon this extension, t h e interfacial tension should not vary, otherwise t h e q u a n tity y would become ill-defined. One of the changes t h a t might be incurred could result from strong curving of t h e interface. In t h e present chapter this i s s u e w a s avoided b e c a u s e we have only considered macroscopic interfaces with radii of c u r v a t u r e s above 0 ( 1 0 - 1 0 0 nm). Already in sec. 1.2.23c we showed t h a t / is t h e n still independent of curvature. Another way of changing the Helmholtz energy of a n interface is by bending it. For simple fluids t h i s force will b e negligible, u n l e s s t h e r a d i u s of c u r v a t u r e becomes a s small a s t h e 'thickness' of t h e interface. Here we shall disregard this

1.78

INTERFACIAL TENSION: MEASUREMENT

situation, b e c a u s e t h e n the macroscopic n a t u r e of interface loses its meaning, a s d o e s t h a t of interfacial tension. However, interfaces carrying m o n o l a y e r s of s u r f a c t a n t s , lipids, etc. u s u a l l y display a significant r e s i s t a n c e to b e n d i n g a t weaker c u r v a t u r e s , say for radii in the colloidal range. Moreover, b e c a u s e of t h e packing of these surfactants, s u c h interfaces may exhibit a spontaneous

curvature

a n d work then h a s to be done to change this curvature. From the required force the so-called bending moduli (sometimes called bending elastic moduli) can be derived. Like

the

interfacial

tension,

bending

moduli

are

system-characteristic

parameters. Measurement of bending moduli is a n issue of growing interest. Obviously they m u s t play a role in the formation a n d stability of micro-emulsions, where t h e type of e m u l s i o n formed (o/w or w/o) will be p r e d o m i n a n t l y d e t e r m i n e d by t h e s p o n t a n e o u s curvature of the interface and by its resistance against further curving. (Macro-emulsions are thermodynamically unstable, for these this a r g u m e n t does not apply. Moreover, usually the curvature is not strong.) Vesicles provide a n o t h e r relevant example. More indirectly, bending moduli also play their roles in some properties of thin liquid films a n d m e m b r a n e s . The reason is t h a t each free liquid interface is subject to small thermal fluctuations, ripples or

undulations.

The interfacial tension opposes this rippling, b u t the extent to which this h a p p e n s is determined by the change of y with bending. For bilayers a n d vesicles y ~ 0 a n d resilience against c h a n g e s in curvature is entirely d u e to the resistance against bending. One of the consequences is t h a t physically observable p h e n o m e n a like (i) surface rugosity a n d (ii) undulation forces, depend on the moduli. This suggests m e t h o d s of m e a s u r e m e n t . Regarding the former, [1.10.2] may be recalled, where a bending modulus k^ a p p e a r s in the denominator, the k^q^ term accounting for t h e inhibition of surface u n d u l a t i o n s because of the restoring bending force. The l.h.s. is m e a s u r a b l e . Regarding (ii) there are two famous equations by Helfrich^). For the Helmholtz energy per unit area, caused by undulations F^^(bending) = i / C j ( j - j j ^ +k^K

[1.15.1]

a n d for the interfacial tension yiJ.K)

= 7(0,0) + ^k^J^- k^J^J-^k^K

[1.15.2]

Here, a s before, J is the first, or mean, curvature a n d K t h e second, or G a u s s , curvature, see [1.1.4 a n d 5]. J

is the s p o n t a n e o u s m e a n c u r v a t u r e . For sym-

metrical interfaces ^Q = 0. Here, we shall neglect the spontaneous G a u s s curvature. The interfacial tension for the u n b e n t surface is 7(0,0). The two bending moduli have dimensions of energy a n d are of the order of k T . We shall call k^ a n d k^ t h e ^^ W. Helfrich, Z. Naturjorsch. 28c (1973) 693; 33a (1978) 305.

INTERFACIAL TENSION: MEASUREMENT

(a)

(b)

1.79

(c)

Figure 1.34. Illustration of the bending types responsible for the moduli k^ (from (a) to (b)) and k^ (from (a) to (c)). first, or mean, a n d the second, or Gauss bending modulus, respectively^^. These two moduli a r e t h e r m o d y n a m i c , r a t h e r t h a n m e c h a n i c a l q u a n t i t i e s ; t h e y h a v e a n entropic a n d a n energetic pairt. Sometimes k^ is called the saddle modulus

splay

(bending)

b e c a u s e it expresses the resilience against saddle-like bending (fig. 1.34),

a t l e a s t if it is positive. For colloid science u s u a l l y k^ is t h e

determining

parameter. For instance, Helfrich derived from [1.15.1] a n 'undulation force' across thin liquid films originating from the correlations between the u n d u l a t i o n s on the two surfaces. The Helmholtz energy F(und) w a s inversely proportional to k^ a n d decreased a s h~^ (h is t h e film thickness), j u s t a s the Van der Waals energy, see [1.4.6.2], F(und) h a s to be added to the other components of the disjoining pressure. The second bending m o d u l u s plays a role in, for instance, p h a s e diagrams. Measuring k^ and k^ is subject to various pitfalls. One of t h e difficulties is t h a t bending is rarely the sole physical process taking place. Usually, extension of t h e area c a n n o t be avoided, so t h a t the bending term appears in equations next to the / d A t e r m , which is m u c h higher (see the denominator of [1.10.2]). Only for very low y c a n t h e bending term dominate. Brochard et al.-^^ argued t h a t this would b e t h e case for vesicles. Very low interfacial tensions are also needed to obtain microemulsions. Another issue is whether the bending takes place at constant n u m b e r of moles (n.'s) in the system or at constant chemical potentials (A^.'S). In t h e former case t h e chemical potentials c h a n g e u p o n bending; in t h e latter curving a n interface will lead to changes in the adsorbed a m o u n t s (r.'s). Still a n o t h e r problem is t h a t of interfacial rheology, discussed in sec. 1.10. The conclusion is t h a t we are dealing with a subtle feature t h a t rarely s t a n d s on its own a n d requires precise definition of the process conditions. The thermodynamics to lay t h e foundations for t h a t will be considered in sec. 4.7. Anticipating this, we shall now briefly indicate the line of reasoning. For a curved interface, the Gibbs equation requires two additional t e r m s , to

^ In_the literature, the nomenclature may vary. Some authors use the pair k and K, k and k, k and K, or k^ and k^. Sometimes only one modulus, called kor k , is used, tacitly identified with our k^, or with k.-\-k^/2. 2J F. Brochard, P.G. de Gennes and P. Pfenty, J. Phys. 37 (1976) 1099.

INTERFACIAL TENSION: MEASUREMENT

1.80

account for the two curvatures. Formally, dy = - S^dT - ^

[1.15.3]

rdju^ + C^dJ + C^dK

The coefficients C^ and C^ are the first a n d second bending momenU respectively. Formally their definitions are

'='-m

0 is obtained by integration over x^ from 0 to 00, a n d over y^ a n d z^ from -00 to +00. We note in passing t h a t this m e a n s t h a t all forces aire taken to be additive. Let t h e result b e called -G(Xj, z^):

^) T.L. Hill, Introduction to Statistical Thermodynamics, Addison-Wesley (1960), sec. 17-5. 2) J.G. Kirkwood, F.P. Buff, J. Chem Phys. 17 (1949) 338.

2.14

INTERFACIAL TENSION: MOLECULAR INTERPRETATION

- G ( x , z.) = - J I J

^

^

^ ^ . J - 2 ) 9 ' ^ ' ( - . -r r) d x , dy, d z .

The horizontal component of the force exerted by the molecules in an infinitesimal layer of width £, thickness dz^ and extending to x^ = -«> (indicated in fig. 2.3) is o

-MZj I G(Xj, Zj)p^(2j)dXj and if this expression is divided by the area, ^dz^, we obtain the corresponding pressure experienced at a certain location z^. TTiis is Just the interaction part of the tangential pressure tensor occurring in [2.3.4] gind [2.3.5]. Adding the contribution of the kinetic energy, i.e. the momentum transport part, p^[z^)kT, we obtain o

p^iz^] = p^(z^)kT - p^iz^) J G(x^, z^)dx^

[2.4.3]

which may be considered the 'interfacial' equivalent of [2.4.2]. Some further elaboration is possible by changing variables from x^, y^, z^ to x._ = x_ - X,, y,o- Un' y^^ 2._ = z_ - z,, i.e. to intermolecular distances thereby eliminating the position of the vertical plane in fig. 2.4. The distance r equals (^fg +yf2 '^^12^^^^ ^^^ ^ ^ lower integration boundary of x^^ equals -x^. After these substitutions Gi.,z^)

= J 7 J

M ^1

^

P . ( z . -zj^^^)(z,z,,,r)dx,,dy,,dz^.

'

[2.4.4] The integration over x^ in [2.4.3] can be done by parts. Basically, 0

0

0 UK.

J G ^ . z , ) dx. = G[x^,z^) x j ^ - J x^ ^ ^ X —oo ax.

— oo

dx,

1

where the first term on the r.h.s. is zero (the product is zero for x^ = 0 and also for Xj = -oo because G decays more rapidly than x^). The derivative of G with respect to Xj is, according to [2.4.4], a differentiation with respect to an integration boundary which simply yields the integrand, x^, and leads to an x^^, rather than to an x^^» term. This can be verified by temporarily rewriting [2.4.4], everywhere replacing Xj2' by Xj throughout, also in r; an x^ is then obtained. By symmetry, we can extend the range of integration from -