Physics and Chemistry of Interfaces · 2013. 7. 16. · Karlheinz Graf MPI für Polymerforschung...

Hans-Jürgen Butt, Karlheinz Graf, Michael Kappl

Physics and Chemistry of Interfaces

WILEY-VCH GmbH & Co. KGaA

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WILEY-VCH GmbH & Co. KGaA

Authors

Hans-Jürgen ButtMPI für Polymerforschung Mainze-mail: [email protected]

Karlheinz GrafMPI für Polymerforschung Mainze-mail: [email protected]

Michael KapplMPI für Polymerforschung Mainze-mail: [email protected]

Cover PicturesThe left picture shows aggregates of siliconoxide particles with a diameter of 0.9 μm (seeexample 1.1). At the bottom an atomic forcemicroscope image of cylindrical CTAB mi-celles adsorbed to gold(111) is shown (seeexample 12.3, width: 200 nm). The rightimage was also obtained by atomic force microscopy. It shows the surface of a self-assembled monolayer of long-chain alkylthiolson gold(111) (see fig. 10.2, width: 3.2 nm).

This book was carefully produced. Never-theless, authors and publisher do not warrantthe information contained therein to be freeof errors. Readers are advised to keep inmind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate

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Bibliographic information published by Die Deutsche BibliothekDie Deutsche Bibliothek lists this publicationin the Deutsche Nationalbibliografie; detailed bibliographic data is available in theInternet at .

© 2003 WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

All rights reserved (including those of trans-lation into other languages). No part of thisbook may be reproduced in any form – byphotoprinting, microfilm, or any othermeans – nor transmitted or translated intomachine language without written permis-sion from the publishers. Registered names,trademarks, etc. used in this book, evenwhen not specifically marked as such, are not be considered unprotected by law.

printed in the Federal Republic of Germanyprinted on acid-free paper

Composition Uwe Krieg, BerlinPrinting betz-druck GmbH, DarmstadtBookbinding Litges&Dopf BuchbindereiGmbH, Heppenheim

ISBN 3-527-40413-9

Preface

Interface science has changed significantly during the last 10–15 years. This is partially due toscientific breakthroughs. For example, the invention of scanning probe microscopy and refineddiffraction methods allow us to look at interfaces under “wet” conditions with unprecedentedaccuracy. This change is also due to the greatly increased community of interfacial scien-tists. One reason is certainly the increased relevance of micro- and nanotechnology, includinglab-on-chip technology, microfluids, and biochips. Objects in the micro- and nanoworld aredominated by surface effects rather than gravitation or inertia. Therefore, surface science isthe basis for nanotechnology.

The expansion of the community is correlated with an increased interdisciplinarity. Tradi-tionally the community tended to be split into “dry” surface scientists (mainly physicists work-ing under ultrahigh vacuum conditions) and “wet” surface scientists (mainly colloid chemists).In addition, engineers dealing with applications like coatings, adhesion, or lubrication, formeda third community. This differentiation is significantly less pronounced and interface sciencehas become a really interdisciplinary field of research including, for example, chemical engi-neering and biology.

This development motivated us to write this textbook. It is a general introduction to surfaceand interface science. It focuses on basic concepts rather than specific details, on understand-ing rather than learning facts. The most important techniques and methods are introduced.The book reflects that interfacial science is a diverse field of research. Several classical scien-tific or engineering disciplines are involved. It contains basic science and applied topics suchas wetting, friction, and lubrication. Many textbooks concentrate on certain aspects of surfacescience such as techniques involving ultrahigh vacuum or classical “wet” colloid chemistry.We tried to include all aspects because we feel that for a good understanding of interfaces, acomprehensive introduction is required.

Our manuscript is based on lectures given at the universities of Siegen and Mainz. It ad-dresses (1) advanced students of engineering, chemistry, physics, biology, and related subjectsand (2) scientists in academia or industry, who are not yet specialists in surface science butwant to get a solid background knowledge of the subject. The level is introductory for sci-entists and engineers who have a basic knowledge of the natural sciences and mathematics.Certainly no advanced level of mathematics is required. When looking through the pages ofthis book you will see a substantial number of equations. Please do not be scared! We pre-ferred to give all transformations explicitly rather than writing “as can easily be seen” andstating the result. Chapter “Thermodynamics of Interfaces” is the only exception. To ap-preciate it a basic knowledge of thermodynamics is required. You can skip and still be ableto follow the rest. In this case please read and try to get an intuitive understanding of whatsurface excess is (Section 3.1) and what the Gibbs adsorption equation implies (Section 3.4.2).

VI Preface

A number of problems with solutions were included to allow for private studies. If notmentioned otherwise, the temperature was assumed to be 25◦C. At the end of each chapter themost important equations, facts, and phenomena are summarized to given students a chanceto concentrate on important issues and help instructors preparing exams.

One of the main problems when writing a textbook is to limit its content. We tried hard tokeep the volume within the scope of one advanced course of roughly 15 weeks, one day perweek. Unfortunately, this means that certain topics had to be cut short or even left out com-pletely. Statistical mechanics, heterogeneous catalysis, and polymers at surfaces are issueswhich could have been expanded.

This book certainly contains errors. Even after having it read by different people indepen-dently, this is unavoidable. If you find an error, please write us a letter (Max-Planck-Institutefor Polymer Research, Ackermannweg, 55128 Mainz, Germany) or an e-mail ([email protected]) so that we can correct it and do not confuse more students.

We are indebted to several people who helped us collecting information, preparing, andcritically reading this manuscript. In particular we would like thank R. von Klitzing, C. Lorenz,C. Stubenrauch, D. Vollmer, J. Wölk, R. Wolff, K. Beneke, J. Blum, M. Böhm, E. Bonac-curso, P. Broekmann, G. Glasser, G. Gompper, M. Grunze, J. Gutmann, L. Heim, M. Hille-brand, T. Jenkins, X. Jiang, U. Jonas, R. Jordan, I. Lieberwirth, G. Liger-Belair, M. Lösche,E. Meyer, P. Müller-Buschbaum, T. Nagel, D. Quéré, J. Rabe, H. Schäfer, J. Schreiber,M. Stamm, M. Steinhart, G. Subklew, J. Tomas, K. Vasilev, K. Wandelt, B. Wenclawiak,R. Wepf, R. Wiesendanger, D.Y. Yoon, M. Zharnikov, and U. Zimmermann.

Hans-Jürgen Butt, Karlheinz Graf, and Michael Kappl

Mainz, August 2003

Contents

Preface V

1 Introduction 1

2 Liquid surfaces 42.1 Microscopic picture of the liquid surface . . . . . . . . . . . . . . . . . . . . 42.2 Surface tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Equation of Young and Laplace . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Curved liquid surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.2 Derivation of the Young–Laplace equation . . . . . . . . . . . . . . 102.3.3 Applying the Young–Laplace equation . . . . . . . . . . . . . . . . . 11

2.4 Techniques to measure the surface tension . . . . . . . . . . . . . . . . . . . 122.5 The Kelvin equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Capillary condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.7 Nucleation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Thermodynamics of interfaces 263.1 The surface excess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Fundamental thermodynamic relations . . . . . . . . . . . . . . . . . . . . . 29

3.2.1 Internal energy and Helmholtz energy . . . . . . . . . . . . . . . . . 293.2.2 Equilibrium conditions . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.3 Location of the interface . . . . . . . . . . . . . . . . . . . . . . . . 313.2.4 Gibbs energy and definition of the surface tension . . . . . . . . . . . 323.2.5 Free surface energy, interfacial enthalpy and Gibbs surface energy . . 32

3.3 The surface tension of pure liquids . . . . . . . . . . . . . . . . . . . . . . . 343.4 Gibbs adsorption isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4.2 System of two components . . . . . . . . . . . . . . . . . . . . . . . 373.4.3 Experimental aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4.4 The Marangoni effect . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

VIII Contents

4 The electric double layer 424.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Poisson–Boltzmann theory of the diffuse double layer . . . . . . . . . . . . . 43

4.2.1 The Poisson–Boltzmann equation . . . . . . . . . . . . . . . . . . . 434.2.2 Planar surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2.3 The full one-dimensional case . . . . . . . . . . . . . . . . . . . . . 464.2.4 The Grahame equation . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.5 Capacity of the diffuse electric double layer . . . . . . . . . . . . . . 50

4.3 Beyond Poisson–Boltzmann theory . . . . . . . . . . . . . . . . . . . . . . . 504.3.1 Limitations of the Poisson–Boltzmann theory . . . . . . . . . . . . . 504.3.2 The Stern layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 The Gibbs free energy of the electric double layer . . . . . . . . . . . . . . . 544.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Effects at charged interfaces 575.1 Electrocapillarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.1.2 Measurement of electrocapillarity . . . . . . . . . . . . . . . . . . . 60

5.2 Examples of charged surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2.1 Mercury . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2.2 Silver iodide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2.3 Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2.4 Mica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2.5 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3 Measuring surface charge densities . . . . . . . . . . . . . . . . . . . . . . . 685.3.1 Potentiometric colloid titration . . . . . . . . . . . . . . . . . . . . . 685.3.2 Capacitances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.4 Electrokinetic phenomena: The zeta potential . . . . . . . . . . . . . . . . . 725.4.1 The Navier–Stokes equation . . . . . . . . . . . . . . . . . . . . . . 725.4.2 Electro-osmosis and streaming potential . . . . . . . . . . . . . . . . 735.4.3 Electrophoresis and sedimentation potential . . . . . . . . . . . . . . 76

5.5 Types of potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 Surface forces 806.1 Van der Waals forces between molecules . . . . . . . . . . . . . . . . . . . . 806.2 The van der Waals force between macroscopic solids . . . . . . . . . . . . . 84

6.2.1 Microscopic approach . . . . . . . . . . . . . . . . . . . . . . . . . 846.2.2 Macroscopic calculation — Lifshitz theory . . . . . . . . . . . . . . 876.2.3 Surface energy and Hamaker constant . . . . . . . . . . . . . . . . . 92

6.3 Concepts for the description of surface forces . . . . . . . . . . . . . . . . . 936.3.1 The Derjaguin approximation . . . . . . . . . . . . . . . . . . . . . 936.3.2 The disjoining pressure . . . . . . . . . . . . . . . . . . . . . . . . . 95

Contents IX

6.4 Measurement of surface forces . . . . . . . . . . . . . . . . . . . . . . . . . 966.5 The electrostatic double-layer force . . . . . . . . . . . . . . . . . . . . . . 98

6.5.1 General equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.5.2 Electrostatic interaction between two identical surfaces . . . . . . . . 1016.5.3 The DLVO theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.6 Beyond DLVO theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.6.1 The solvation force and confined liquids . . . . . . . . . . . . . . . . 1046.6.2 Non DLVO forces in an aqueous medium . . . . . . . . . . . . . . . 106

6.7 Steric interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.7.1 Properties of polymers . . . . . . . . . . . . . . . . . . . . . . . . . 1076.7.2 Force between polymer coated surfaces . . . . . . . . . . . . . . . . 108

6.8 Spherical particles in contact . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7 Contact angle phenomena and wetting 1187.1 Young’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.1.1 The contact angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.1.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.1.3 The line tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217.1.4 Complete wetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.2 Important wetting geometries . . . . . . . . . . . . . . . . . . . . . . . . . . 1227.2.1 Capillary rise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1227.2.2 Particles in the liquid–gas interface . . . . . . . . . . . . . . . . . . 1237.2.3 Network of fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.3 Measurement of the contact angle . . . . . . . . . . . . . . . . . . . . . . . 1267.3.1 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . 1267.3.2 Hysteresis in contact angle measurements . . . . . . . . . . . . . . . 1287.3.3 Surface roughness and heterogeneity . . . . . . . . . . . . . . . . . . 129

7.4 Theoretical aspects of contact angle phenomena . . . . . . . . . . . . . . . . 1317.5 Dynamics of wetting and dewetting . . . . . . . . . . . . . . . . . . . . . . 133

7.5.1 Wetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.5.2 Dewetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387.6.1 Flotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387.6.2 Detergency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.6.3 Microfluidics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.6.4 Adjustable wetting . . . . . . . . . . . . . . . . . . . . . . . . . . . 142


8 Solid surfaces 1458.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.2 Description of crystalline surfaces . . . . . . . . . . . . . . . . . . . . . . . 146

8.2.1 The substrate structure . . . . . . . . . . . . . . . . . . . . . . . . . 146

X Contents

8.2.2 Surface relaxation and reconstruction . . . . . . . . . . . . . . . . . 1478.2.3 Description of adsorbate structures . . . . . . . . . . . . . . . . . . . 149

8.3 Preparation of clean surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 1508.4 Thermodynamics of solid surfaces . . . . . . . . . . . . . . . . . . . . . . . 153

8.4.1 Surface stress and surface tension . . . . . . . . . . . . . . . . . . . 1538.4.2 Determination of the surface energy . . . . . . . . . . . . . . . . . . 1548.4.3 Surface steps and defects . . . . . . . . . . . . . . . . . . . . . . . . 157

8.5 Solid–solid boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1598.6 Microscopy of solid surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8.6.1 Optical microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 1628.6.2 Electron microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 1628.6.3 Scanning probe microscopy . . . . . . . . . . . . . . . . . . . . . . 164

8.7 Diffraction methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1678.7.1 Diffraction patterns of two-dimensional periodic structures . . . . . . 1688.7.2 Diffraction with electrons, X-rays, and atoms . . . . . . . . . . . . . 168

8.8 Spectroscopic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.8.1 Spectroscopy using mainly inner electrons . . . . . . . . . . . . . . . 1718.8.2 Spectroscopy with outer electrons . . . . . . . . . . . . . . . . . . . 1738.8.3 Secondary ion mass spectrometry . . . . . . . . . . . . . . . . . . . 174


9 Adsorption 1779.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

9.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1779.1.2 The adsorption time . . . . . . . . . . . . . . . . . . . . . . . . . . 1789.1.3 Classification of adsorption isotherms . . . . . . . . . . . . . . . . . 1799.1.4 Presentation of adsorption isotherms . . . . . . . . . . . . . . . . . . 181

9.2 Thermodynamics of adsorption . . . . . . . . . . . . . . . . . . . . . . . . . 1829.2.1 Heats of adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 1829.2.2 Differential quantities of adsorption and experimental results . . . . . 184

9.3 Adsorption models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1859.3.1 The Langmuir adsorption isotherm . . . . . . . . . . . . . . . . . . . 1859.3.2 The Langmuir constant and the Gibbs energy of adsorption . . . . . . 1889.3.3 Langmuir adsorption with lateral interactions . . . . . . . . . . . . . 1899.3.4 The BET adsorption isotherm . . . . . . . . . . . . . . . . . . . . . 1899.3.5 Adsorption on heterogeneous surfaces . . . . . . . . . . . . . . . . . 1929.3.6 The potential theory of Polanyi . . . . . . . . . . . . . . . . . . . . . 193

9.4 Experimental aspects of adsorption from the gas phase . . . . . . . . . . . . 1959.4.1 Measurement of adsorption isotherms . . . . . . . . . . . . . . . . . 1959.4.2 Procedures to measure the specific surface area . . . . . . . . . . . . 1989.4.3 Adsorption on porous solids — hysteresis . . . . . . . . . . . . . . . 1999.4.4 Special aspects of chemisorption . . . . . . . . . . . . . . . . . . . . 201

9.5 Adsorption from solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2029.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2039.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Contents XI

10 Surface modification 20610.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20610.2 Chemical vapor deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 20710.3 Soft matter deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

10.3.1 Self-assembled monolayers . . . . . . . . . . . . . . . . . . . . . . 20910.3.2 Physisorption of Polymers . . . . . . . . . . . . . . . . . . . . . . . 21210.3.3 Polymerization on surfaces . . . . . . . . . . . . . . . . . . . . . . . 215

10.4 Etching techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21710.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22110.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

11 Friction, lubrication, and wear 22311.1 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

11.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22311.1.2 Amontons’ and Coulomb’s Law . . . . . . . . . . . . . . . . . . . . 22411.1.3 Static, kinetic, and stick-slip friction . . . . . . . . . . . . . . . . . . 22611.1.4 Rolling friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22811.1.5 Friction and adhesion . . . . . . . . . . . . . . . . . . . . . . . . . . 22911.1.6 Experimental Aspects . . . . . . . . . . . . . . . . . . . . . . . . . 23011.1.7 Techniques to measure friction . . . . . . . . . . . . . . . . . . . . . 23011.1.8 Macroscopic friction . . . . . . . . . . . . . . . . . . . . . . . . . . 23211.1.9 Microscopic friction . . . . . . . . . . . . . . . . . . . . . . . . . . 232

11.2 Lubrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23611.2.1 Hydrodynamic lubrication . . . . . . . . . . . . . . . . . . . . . . . 23611.2.2 Boundary lubrication . . . . . . . . . . . . . . . . . . . . . . . . . . 23811.2.3 Thin film lubrication . . . . . . . . . . . . . . . . . . . . . . . . . . 23911.2.4 Lubricants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

11.3 Wear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24111.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24411.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

12 Surfactants, micelles, emulsions, and foams 24612.1 Surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24612.2 Spherical micelles, cylinders, and bilayers . . . . . . . . . . . . . . . . . . . 250

12.2.1 The critical micelle concentration . . . . . . . . . . . . . . . . . . . 25012.2.2 Influence of temperature . . . . . . . . . . . . . . . . . . . . . . . . 25212.2.3 Thermodynamics of micellization . . . . . . . . . . . . . . . . . . . 25312.2.4 Structure of surfactant aggregates . . . . . . . . . . . . . . . . . . . 25512.2.5 Biological membranes . . . . . . . . . . . . . . . . . . . . . . . . . 258

12.3 Macroemulsions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25912.3.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 25912.3.2 Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26112.3.3 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26212.3.4 Evolution and aging . . . . . . . . . . . . . . . . . . . . . . . . . . 26512.3.5 Coalescence and demulsification . . . . . . . . . . . . . . . . . . . . 267

XII Contents

12.4 Microemulsions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26812.4.1 Size of droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26812.4.2 Elastic properties of surfactant films . . . . . . . . . . . . . . . . . . 26912.4.3 Factors influencing the structure of microemulsions . . . . . . . . . . 270

12.5 Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27212.5.1 Classification, application and formation . . . . . . . . . . . . . . . 27212.5.2 Structure of foams . . . . . . . . . . . . . . . . . . . . . . . . . . . 27412.5.3 Soap films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27412.5.4 Evolution of foams . . . . . . . . . . . . . . . . . . . . . . . . . . . 277


13 Thin films on surfaces of liquids 28013.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28013.2 Phases of monomolecular films . . . . . . . . . . . . . . . . . . . . . . . . . 28313.3 Experimental techniques to study monolayers . . . . . . . . . . . . . . . . . 286

13.3.1 Optical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28613.3.2 X-ray reflection and diffraction . . . . . . . . . . . . . . . . . . . . . 28713.3.3 The surface potential . . . . . . . . . . . . . . . . . . . . . . . . . . 29013.3.4 Surface elasticity and viscosity . . . . . . . . . . . . . . . . . . . . . 292

13.4 Langmuir–Blodgett transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 29313.5 Thick films – spreading of one liquid on another . . . . . . . . . . . . . . . . 29513.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29713.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

14 Solutions to exercises 299

Appendix

A Analysis of diffraction patterns 321A.1 Diffraction at three dimensional crystals . . . . . . . . . . . . . . . . . . . . 321

A.1.1 Bragg condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321A.1.2 Laue condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321A.1.3 The reciprocal lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 323A.1.4 Ewald construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

A.2 Diffraction at Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325A.3 Intensity of diffraction peaks . . . . . . . . . . . . . . . . . . . . . . . . . . 327

B Symbols and abbreviations 331

Bibliography 335

Index 355

1 Introduction

An interface is the area which separates two phases from each other. If we consider the solid,liquid, and gas phase we immediately get three combinations of interfaces: the solid–liquid,the solid–gas, and the liquid–gas interface. These interfaces are also called surfaces. Interfaceis, however, a more general term than surface. Interfaces can also separate two immiscibleliquids such as water and oil. These are called liquid–liquid interfaces. Solid–solid interfacesseparate two solid phases. They are important for the mechanical behavior of solid materials.Gas–gas interfaces do not exist because gases mix.

Often interfaces and colloids are discussed together. Colloids are disperse systems, inwhich one phase has dimensions in the order of 1 nm to 1 μm (see Fig. 1.1). The word“colloid” comes from the Greek word for glue and has been used the first time in 1861 byGraham1. He applied it to materials which seemed to dissolve but were not able to penetrate amembrane, such as albumin, starch, and dextrin. A dispersion is a two-phase system which isuniform on the macroscopic but not on the microscopic scale. It consists of grains or dropletsof one phase in a matrix of the other phase.

Different kinds of dispersions can be formed. Most of them have important applicationsand have special names (Table 1.1). While there are only five types of interface, we can distin-guish ten types of disperse system because we have to discriminate between the continuous,dispersing (external) phase and the dispersed (inner) phase. In some cases this distinction isobvious. Nobody will, for instance, mix up fog with a foam although in both cases a liquid anda gas are involved. In other cases the distinction between continuous and inner phase cannotbe made because both phases might form connected networks. Some emulsions for instancetend to form a bicontinuous phase, in which both phases form an interwoven network.

Continuousphase

Dispersedphase

1 nm - 1 μm

Figure 1.1: Schematic of a dispersion.

Colloids and interfaces are intimately related. This is a direct consequence of their enor-mous specific surface area. More precisely: their interface-to-volume relation is so large, thattheir behavior is completely determined by surface properties. Gravity is negligible in the

1 Thomas Graham, 1805–1869. British chemist, professor in Glasgow and London.

2 1 Introduction

Table 1.1: Types of dispersions. *Porous solids have a bicontinuous structure while in a solidfoam the gas phase is clearly dispersed.

Continuous Dispersed Term Examplephase phase

Gas liquid aerosol clouds, fog, smog, hairspraysolid aerosol smoke, dust, pollen

Liquid gas foam lather, whipped cream, foam on beerliquid emulsion milksolid sol ink, muddy water, dispersion paint

Solid gas porous solids*foam styrofoam, soufflés

liquid solid emulsion buttersolid solid suspension concrete

majority of cases. For this reason we could define colloidal systems as systems which aredominated by interfacial effects rather than bulk properties. This is also the reason why in-terfacial science is the basis for nanoscience and technology and many inventions in this newfield originate from surface science.

Example 1.1. A system which is dominated by surface effects is shown on the left side ofthe cover. The scanning electronic microscope (SEM) image shows aggregates of quartz(SiO2) particles (diameter 0.9 μm). These particles were blown as dust into a chamberfilled with gas. While sedimenting they formed fractal aggregates due to attractive van derWaals forces. On the bottom they were collected. These aggregates are stable for weeksand months and even shaking does not change their structure. Gravity and inertia, whichrule the macroscopic world, are not able to bend down the particle chains. Surface forcesare much stronger.

In the recent literature the terms nanoparticles and nanosystems are used, in analogy to colloidand colloidal systems. The prefix “nano” indicates dimensions in the 1 to 100 nm range. Thisis above the atomic scale and, unless highly refined methods are used, below the resolution ofa light microscope and thus also below the accuracy of optical microstructuring techniques.

Why is there an interest in interfaces and colloids? First, for a better understanding ofnatural processes. For example, in biology the surface tension of water allows to form lipidmembranes. This is a prerequisite for the formation of compartments and thus any form oflife. In geology the swelling of clay or soil in the presence of water is an important process.The formation of clouds and rain due to nucleation of water around small dust particles isdominated by surface effects. Many foods, like butter, milk, or mayonnaise are emulsions.Their properties are determined by the liquid–liquid interface. Second, there are many tech-nological applications. One such example is flotation in mineral processing or the bleachingof scrap paper. Washing and detergency are examples which any person encounters every day.

1 Introduction 3

Often the production of new materials such as composite materials heavily involves processesat interfaces. Thin films on surfaces are often dominated by surface effects. Examples arelatex-films, coatings, and paints. The flow behavior of powders and granular media is deter-mined by surface forces. In tribology, wear is reduced by lubrication which again is a surfacephenomenon.

Typical for many of the industrial applications is a very refined and highly developedtechnology, but only a limited understanding of the underlying fundamental processes. Abetter understanding is, however, required to further improve the efficiency or reduce dangersto the environment.

Introductory books on interface science are Refs. [1–6]. For a deeper understanding werecommend the series of books of Lyklema [7–9].

2 Liquid surfaces

2.1 Microscopic picture of the liquid surface

A surface is not an infinitesimal sharp boundary in the direction of its normal, but it has acertain thickness. For example, if we consider the density ρ normal to the surface (Fig. 2.1),we can observe that, within a few molecules, the density decreases from that of the bulk liquidto that of its vapor [10].

Figure 2.1: Density of a liquid versus the coordinate normal to its surface: (a) is a schematicplot; (b) results from molecular dynamics simulations of a n-tridecane (C13H28) at 27◦Cadapted from Ref. [11]. Tridecane is practically not volatile. For this reason the density inthe vapor phase is negligible.

The density is only one criterion to define the thickness of an interface. Another possibleparameter is the orientation of the molecules. For example, water molecules at the surfaceprefer to be oriented with their negative sides “out” towards the vapor phase. This orientationfades with increasing distance from the surface. At a distance of 1–2 nm the molecules areagain randomly oriented.

Which thickness do we have to use? This depends on the relevant parameter. If we arefor instance, interested in the density of a water surface, a realistic thickness is in the order of1 nm. Let us assume that a salt is dissolved in the water. Then the concentration of ions mightvary over a much larger distance (characterized by the Debye length, see Section 4.2.2). Withrespect to the ion concentration, the thickness is thus much larger. In case of doubt, it is saferto choose a large value for the thickness.

The surface of a liquid is a very turbulent place. Molecules evaporate from the liquid intothe vapor phase and vice versa. In addition, they diffuse into the bulk phase and moleculesfrom the bulk diffuse to the surface.

2.2 Surface tension 5

Example 2.1. To estimate the number of gas molecules hitting the liquid surface per sec-ond, we recall the kinetic theory of ideal gases. In textbooks of physical chemistry the rateof effusion of an ideal gas through a small hole is given by [12]

PA√2πmkBT

(2.1)

Here, A is the cross-sectional area of the hole and m is the molecular mass. This is equalto the number of water molecules hitting a surface area A per second. Water at 25◦C hasa vapor pressure P of 3168 Pa. With a molecular mass m of 0.018 kgmol−1/6.02 ×1023 mol−1 ≈ 3 × 10−26 kg, 107 water molecules per second hit a surface area of 10Å2. In equilibrium the same number of molecules escape from the liquid phase. 10 Å2

is approximately the area covered by one water molecule. Thus, the average time a watermolecule remains on the surface is of the order of 0.1 μs.

2.2 Surface tension

The following experiment helps us to define the most fundamental quantity in surface science:the surface tension. A liquid film is spanned over a frame, which has a mobile slider (Fig. 2.2).The film is relatively thick, say 1μm, so that the distance between the back and front surfacesis large enough to avoid overlapping of the two interfacial regions. Practically, this experimentmight be tricky even in the absence of gravity but it does not violate a physical law so that itis in principle feasible. If we increase the surface area by moving the slider a distance dx tothe right, work has to be done. This work dW is proportional to the increase in surface areadA. The surface area increases by twice b · dx because the film has a front and back side.Introducing the proportionality constant γ we get

dW = γ · dA (2.2)The constant γ is called surface tension.

liquid film

dA = bdx2

dx

b

Figure 2.2: Schematic set-up to verifyEq. (2.2) and define the surface tension.

Equation (2.2) is an empirical law and a definition at the same time. The empirical law isthat the work is proportional to the change in surface area. This is not only true for infinitesi-mal small changes of A (which is trivial) but also for significant increases of the surface area:ΔW = γ ·ΔA. In general, the proportionality constant depends on the composition of the liq-uid and the vapor, temperature, and pressure, but it is independent of the area. The definitionis that we call the proportionality constant “surface tension”.

6 2 Liquid surfaces

The surface tension can also be defined by the force F that is required to hold the slider inplace and to balance the surface tensional force:

|F | = 2γb (2.3)Both forms of the law are equivalent, provided that the process is reversible. Then we canwrite

F = −dWdx

= −2γb (2.4)

The force is directed to the left while x increases to the right. Therefore we have a negativesign.

The unit of surface tension is either J/m2 or N/m. Surface tensions of liquids are of theorder of 0.02–0.08 N/m (Table 2.1). For convenience they are usually given in mN/m (or 10−3

N/m), where the first “m” stands for “milli”.The term “surface tension” is tied to the concept that the surface stays under a tension.

In a way, this is similar to a rubber balloon, where also a force is required to increase thesurface area of its rubber membrane against a tension. There is, however, a difference: whilethe expansion of a liquid surface is a plastic process the stretching of a rubber membrane isusually elastic.

Table 2.1: Surface tensions γ of some liquids at different temperatures T .

Substance T γ Substance T γ( mNm−1) (mNm−1)

Water 10◦C 74.23 Mercury 25◦C 485.4825◦C 71.99 Phenol 50◦C 38.2050◦C 67.94 Benzene 25◦C 28.2275◦C 63.57 Toluene 25◦C 27.93

100◦C 58.91 Dichloromethane 25◦C 27.20Argon 90 K 11.90 n-pentane 25◦C 15.49Methanol 25◦C 22.07 n-hexane 25◦C 17.89Ethanol 10◦C 23.22 n-heptane 25◦C 19.65

25◦C 21.97 n-octane 10◦C 22.5750◦C 19.89 25◦C 21.14

1-propanol 25◦C 23.32 50◦C 18.771-butanol 25◦C 24.93 75◦C 16.392-butanol 25◦C 22.54 100◦C 14.01Acetone 25◦C 23.46 Formamide 25◦C 57.03

Example 2.2. If a water film is formed on a frame with a slider length of 1 cm, then thefilm pulls on the slider with a force of

2 × 0.01 m × 0.072 Jm−2 = 1.44 × 10−3NThat corresponds to a weight of 0.15 g.

2.2 Surface tension 7

Figure 2.3: Schematic molecular structureof a liquid–vapor interface.

How can we interpret the concept of surface tension on the molecular level? For moleculesit is energetically favorable to be surrounded by other molecules. Molecules attract eachother by different interactions such as van der Waals forces or hydrogen bonds (for details seeChapter 6). Without this attraction there would not be a condensed phase at all, there wouldonly be a vapor phase. The sheer existence of a condensed phase is evidence for an attractiveinteraction between the molecules. At the surface, molecules are only partially surrounded byother molecules and the number of adjacent molecules is smaller than in the bulk (Fig. 2.3).This is energetically unfavorable. In order to bring a molecule from the bulk to the surface,work has to be done. With this view γ can be interpreted as the energy required to bringmolecules from inside the liquid to the surface and to create new surface area. Therefore oftenthe term “surface energy” is used for γ. As we shall see in the next chapter this might lead tosome confusion. To avoid this we use the term surface tension.

With this interpretation of the surface tension in mind we immediately realize that γ hasto be positive. Otherwise the Gibbs free energy of interaction would be repulsive and allmolecules would immediately evaporate into the gas phase.

Example 2.3. Estimate the surface tension of cyclohexane from the energy of vaporiza-tion ΔvapU = 30.5 kJ/mol at 25◦C. The density of cyclohexane is ρ = 773 kg/m3, itsmolecular weight is M = 84.16 g/mol.

For a rough estimate we picture the liquid as being arranged in a cubic structure.Each molecule is surrounded by 6 nearest neighbors. Thus each bond contributes roughlyΔvapU/6 = 5.08 kJ/mol. At the surface one neighbor and hence one bond is missing. Permole we therefore estimate a surface tension of 5.08 kJ/mol.

To estimate the surface tension we need to know the surface area occupied by onemolecule. If the molecules form a cubic structure, the volume of one unit cell is a3, wherea is the distance between nearest neighbors. This distance can be calculated from thedensity:

a3 =M

ρNA=

0.08416 kg/mol773 kg/m3 · 6.02 × 1023 mol−1 = 1.81 × 10

−28 m3 ⇒

a = 0.565 nm

8 2 Liquid surfaces

The surface area per molecule is a2. For the surface energy we estimate

γ =ΔvapU6NAa2

=5080 Jmol−1

6.02 × 1023 mol−1 · (0.565 × 10−9 m)2 = 0.0264J

m2

For such a rough estimate the result is surprisingly close to the experimental value of0.0247 J/m2.

2.3 Equation of Young and Laplace

2.3.1 Curved liquid surfaces

We start by describing an important phenomenon: If in equilibrium a liquid surface is curved,there is a pressure difference across it. To illustrate this let us consider a circular part of thesurface. The surface tension tends to minimize the area. This results in a planar geometry ofthe surface. In order to curve the surface, the pressure on one side must be larger than on theother side. The situation is much like that of a rubber membrane. If we, for instance, take atube and close one end with a rubber membrane, the membrane will be planar (provided themembrane is under some tension) (Fig. 2.4). It will remain planar as long as the tube is open atthe other end and the pressure inside the tube is equal to the outside pressure. If we now blowcarefully into the tube, the membrane bulges out and becomes curved due to the increasedpressure inside the tube. If we suck on the tube, the membrane bulges inside the tube becausenow the outside pressure is higher than the pressure inside the tube.

Pi

Pa

P = Pa i P < Pa i P > Pa i

Figure 2.4: Rubber membrane at the end of a cylindrical tube. An inner pressure Pi can beapplied, which is different than the outside pressure Pa.

The Young1–Laplace2 equation relates the pressure difference between the two phases ΔPand the curvature of the surface:

ΔP = γ ·(

1R1

+1

R2

)(2.5)

R1 and R2 are the two principal radii of curvature. ΔP is also called Laplace pressure.Equation (2.5) is also referred to as the Laplace equation.

1 Thomas Young, 1773–1829. English physician and physicist, professor in Cambridge.2 Pierre-Simon Laplace, Marquis de Laplace, 1749–1827. French natural scientist.

2.3 Equation of Young and Laplace 9

It is perhaps worthwhile to describe the principal radii of curvature in a little bit moredetail. The curvature 1/R1 + 1/R2 at a point on an arbitrarily curved surface is obtained asfollows. At the point of interest we draw a normal through the surface and then pass a planethrough this line and the intersection of this line with the surface. The line of intersectionwill, in general, be curved at the point of interest. The radius of curvature R1 is the radius ofa circle inscribed to the intersection at the point of interest. The second radius of curvatureis obtained by passing a second plane through the surface also containing the normal, butperpendicular to the first plane. This gives the second intersection and leads to the secondradius of curvature R2. So the planes defining the radii of curvature must be perpendicularto each other and contain the surface normal. Otherwise their orientation is arbitrary. A lawof differential geometry says that the value 1/R1 + 1/R2 for an arbitrary surface does notdepend on the orientation, as long as the radii are determined in perpendicular directions.

Figure 2.5: Illustrationof the curvature of acylinder and a sphere.

Let us illustrate the curvature for two examples. For a cylinder of radius r a convenientchoice is R1 = r and R2 = ∞ so that the curvature is 1/r + 1/∞ = 1/r. For a sphere withradius R we have R1 = R2 and the curvature is 1/R + 1/R = 2/R (Fig. 2.5).

Example 2.4. How large is the pressure in a spherical bubble with a diameter of 2 mmand a bubble of 20 nm diameter in pure water, compared with the pressure outside? For abubble the curvature is identical to that of a sphere: R1 = R2 = R. Therefore

ΔP =2γR

(2.6)

With R = 1 mm we get

ΔP = 0.072J

m2× 2

10−3m= 144 Pa

With R = 10 nm the pressure is ΔP = 0.072 J/m2 × 2/10−8 m = 1.44 × 107 Pa =144 bar. The pressure inside the bubbles is therefore 144 Pa and 1.44×107 Pa, respectively,higher than the outside pressure.

The Young–Laplace equation has several fundamental implications:

• If we know the shape of a liquid surface we know its curvature and we can calculate thepressure difference.

• In the absence of external fields (e.g. gravity), the pressure is the same everywhere in theliquid; otherwise there would be a flow of liquid to regions of low pressure. Thus, ΔPis constant and Young–Laplace equation tells us that in this case the surface of the liquidhas the same curvature everywhere.

10 2 Liquid surfaces

• With the help of the Young–Laplace Eq. (2.5) it is possible to calculate the equilibriumshape of a liquid surface. If we know the pressure difference and some boundary condi-tions (such as the volume of the liquid and its contact line) we can calculate the geometryof the liquid surface.

In practice, it is usually not trivial to calculate the geometry of a liquid surface with Eq. (2.5).The shape of the liquid surface can mathematically be described by a function z = z(x, y).The z coordinate of the surface is given as a function of its x and y coordinate. The curvatureinvolves the second derivative. As a result, calculating the shape of a liquid surface involvessolving a partial differential equation of second order, which is certainly not a simple task.

In many cases we deal with rotational symmetric structures. Assuming that the axis ofsymmetry is identical to the y axis of an orthogonal cartesian coordinate system, then it isconvenient to put one radius of curvature in the plane of the xy coordinate. This radius isgiven by

1R1

=y′′√

(1 + y′2)3, (2.7)

where y′ and y′′ are the first and second derivatives with respect to x. The plane for the secondbending radius is perpendicular to the xy plane. It is

1R2

=y′

x√

1 + y′2(2.8)

2.3.2 Derivation of the Young–Laplace equation

To derive the equation of Young and Laplace we consider a small part of a liquid surface.First, we pick a point X and draw a line around it which is characterized by the fact that allpoints on that line are the same distance d away from X (Fig. 2.6). If the liquid surface isplanar, this would be a flat circle. On this line we take two cuts that are perpendicular to eachother (AXB and CXD). Consider in B a small segment on the line of length dl. The surfacetension pulls with a force γ dl. The vertical force on that segment is γ dl sin α. For smallsurface areas (and small α) we have sin α ≈ d/R1 where R1 is the radius of curvature alongAXB. The vertical force component is

γ · dl · dR1

(2.9)

The sum of the four vertical components at points A, B, C, and D is

γ · dl ·(

2dR1

+2dR2

)= γ · dl · 2d ·

(1

R1+

1R2

)(2.10)

This expression is independent of the absolute orientation of AB and CD. Integration over theborderline (only 90◦ rotation of the four segments) gives the total vertical force, caused by thesurface tension:

πd2 · γ ·(

1R1

+1

R2

)(2.11)

2.3 Equation of Young and Laplace 11

In equilibrium, this downward force must be compensated by an equal force in the oppositedirection. This upward force is caused by an increased pressure ΔP on the concave side ofπd2ΔP . Equating both forces leads to

ΔP · πd2 = πd2 · γ ·(

1R1

+1

R2

)⇒ ΔP = γ ·

(1

R1+

1R2

)(2.12)

These considerations are valid for any small part of the liquid surface. Since the part is arbi-trary the Young–Laplace equation must be valid everywhere.

A

X

C

B

Dd

R1 R1

R2R2

�dl

� Figure 2.6: Diagram used for deriving theYoung–Laplace equation.

2.3.3 Applying the Young–Laplace equation

When applying the equation of Young and Laplace to simple geometries it is usually obviousat which side the pressure is higher. For example, both inside a bubble and inside a drop, thepressure is higher than outside (Fig. 2.7). In other cases this is not so obvious because thecurvature can have an opposite sign. One example is a drop hanging between the planar endsof two cylinders (Fig. 2.7). Then the two principal curvatures, defined by

C1 =1

R1and C2 =

1R2

, (2.13)

can have a different sign. We count it positive if the interface is curved towards the liquid.The pressure difference is defined as ΔP = Pliquid − Pgas.

Example 2.5. For a drop in a gaseous environment, the two principal curvatures are posi-tive and given by C1 = C2 = 1/R. The pressure difference is positive, which implies thatthe pressure inside the liquid is higher than outside.

For a bubble in a liquid environment the two principal curvatures are negative: C1 =C2 = −1/R. The pressure difference is negative and the pressure inside the liquid islower than inside the bubble.

For a drop hanging between the ends of two cylinders (Fig. 2.7B) in a gaseous envi-ronment, one curvature is conveniently chosen to be C1 = 1/R1. The other curvatureis negative, C2 = −1/R2. The pressure difference depends on the specific values of R1and R2.


Liquid

Gas

Gas

Liquid

A B

R1 R2

Figure 2.7: (A) A gas bubble in liq-uid and a drop in a gaseous environ-ment. (B) A liquid meniscus withradii of curvature of opposite sign be-tween two solid cylinders.

The shape of a liquid surface is determined by the Young–Laplace equation. In large structureswe have to consider also the hydrostatic pressure. Then the equation of Young and Laplacebecomes

ΔP = γ ·(

1R1

+1

R2

)+ ρgh (2.14)

Here, g is the acceleration of free fall and h is the height coordinate.What is a large and what is a small structure? In practice this is a relevant question because

for small structures we can neglect ρgh and use the simpler equation. Several authors definethe capillary constant

√2γ/ρg (as a source of confusion other authors have defined

√γ/ρg

as the capillary constant). For liquid structures whose curvature is much smaller than thecapillary constant the influence of gravitation can be neglected. At 25◦C the capillary constantis 3.8 mm for water and 2.4 mm for hexane.

2.4 Techniques to measure the surface tension

Before we can discuss the experimental techniques used to measure the surface tension, weneed to introduce the so called contact angle Θ. When we put a drop of liquid on a solidsurface the edge usually forms a defined angle which depends only on the material propertiesof the liquid and the solid (Fig. 2.8). This is the contact angle. Here we only need to knowwhat it is. In Chapter 8, contact angle phenomena are discussed in more detail. For a wettingsurface we have Θ = 0.

�

Liquid

Solid

Three phasecontact line

Figure 2.8: Rim of a liquid drop on aplanar solid surface with its contact an-gle Θ.

There are several techniques used to measure the surface tension of liquids. The mostcommon technique is to measure optically the contour of a sessile or pendant drop. Themeasured contour is then fitted with a contour calculated using the Young–Laplace Eq. (2.5).

2.4 Techniques to measure the surface tension 13

From this fit the surface tension is obtained. The same method is applied with a pendant orsessile bubble. Using a bubble ensures that the vapor pressure is 100%, a requirement fordoing experiments in thermodynamic equilibrium. Often problems caused by contaminationare reduced.

In the Maximum-bubble-pressure method the surface tension is determined from thevalue of the pressure which is necessary to push a bubble out of a capillary against theLaplace pressure. Therefore a capillary tube, with inner radius rC , is immersed into the liquid(Fig. 2.9). A gas is pressed through the tube, so that a bubble is formed at its end. If thepressure in the bubble increases, the bubble is pushed out of the capillary more and more. Inthat way, the curvature of the gas–liquid interface increases according to the Young–Laplaceequation. The maximum pressure is reached when the bubble forms a half-sphere with a ra-dius rB = rC . This maximum pressure is related to the surface tension by γ = rCΔP/2.If the volume of the bubble is further increased, the radius of the bubble would also have tobecome larger. A larger radius corresponds to a smaller pressure. The bubble would thusbecome unstable and detach from the capillary tube.

�P

rC

rb

not wetting

wetting

Figure 2.9: Maximal bub-ble pressure and drop-weightmethod to measure the surfacetension of liquids.

Drop-weight method. Here, the liquid is allowed to flow out from the bottom of a capil-lary tube. Drops are formed which detach when they reach a critical dimension. The weightof a drop falling out of a capillary is measured. To get a precise measure, this is done for anumber of drops and the total weight is divided by this number.

As long as the drop is still hanging at the end of the capillary, its weight is more thanbalanced by the surface tension. A drop falls off when the gravitational force mg, determinedby the mass m of the drop, is no longer balanced by the surface tension. The surface tensionalforce is equal to the surface tension multiplied by the circumference. This leads to

mg = 2πrCγ (2.15)

Thus, the mass is determined by the radius of the capillary. Here, we have to distinguishbetween the inner and outer diameter of the capillary. If the material of which the capillaryis formed is not wetted by the liquid, the inner diameter enters into Eq. (2.15). If the surfaceof the capillary tube is wetted by the liquid, the external radius of the capillary has to betaken. For completely nonwetting surfaces (contact angle higher than 90◦) the internal radiusdetermines the drop weight. Experimentally, Tate already observed in 1864 that “other thingsbeing the same, the weight of a drop of a liquid is proportional to the diameter of the tube inwhich it is formed” [13].


Figure 2.10: Release of a liq-uid drop from a capillary.

In practice, the equation is only approximately valid, and a weight less than the ideal valueis measured. The reason becomes evident when the process of drop formation is observedclosely (Fig. 2.10): A thin neck is formed before the drop is released. Correction factors f aretherefore used and Eq. (2.15) becomes: mg = 2πfrCγ.

We have to admit that the maximum bubble pressure and the drop-weight methods are notvery common for measuring the surface tension. Nevertheless we described them because theunderlying phenomena, that is bubbling a gas into liquid and pressing a liquid out of a capil-lary, are very important. A common device used to measure γ is the ring tensiometer, calledalso the Du-Noüy3 tensiometer [14]. In a ring tensiometer the force necessary to detach a ringfrom the surface of a liquid is measured (Fig. 2.11). The force required for the detachment is

2π · (ri + ra) · γ (2.16)

A necessary condition is that the ring surface must be completely wetting. A platinum wireis often used which can be annealed for cleaning before the measurement. Even in the earlymeasurements it turned out that Eq. (2.16) was generally in serious error and that an empiricalcorrection function is required [15].

A widely used technique is the Wilhelmy 4-plate method. A thin plate of glass, platinum,or filter paper is vertically placed halfway into the liquid. In fact, the specific material isnot important, as long as it is wetted by the liquid. Close to the three-phase contact line theliquid surface is oriented almost vertically (provided the contact angle is 0◦). Thus the surfacetension can exert a downward force. One measures the force required to prevent the plate frombeing drawn into the liquid. After subtracting the gravitational force this force is 2lγ, wherel is the length of the plate. In honor of Ludwig Wilhelmy, who studied the force on a platein detail, the method was named after him [16]. The Wilhelmy-plate method is simple andno correction factors are required. Care has to be taken to keep the plates clean and preventcontamination in air.

Finally there are dynamic methods to measure the surface tension. For example, a liquidjet is pushed out from a nozzle, which has an elliptic cross-section. The relaxation to a circularcross-section is observed. An advantage of this method is that we can measure changes of thesurface tension, which might be caused by diffusion of amphiphilic substances to the surface.

3 Pierre Lecomte du Noüy, 1883–1974. French scientist who worked in New York and Paris.

4 Ludwig Ferdinand Wilhelmy, 1812–1864. German physicochemist.

2.5 The Kelvin equation 15

ir

Du-Noüy ring tensiometer

ra

l

2�l

Wilhelmy plate

Figure 2.11: Du-Noüy ring tensiometer and Wilhelmy-plate method.

2.5 The Kelvin equation

In this chapter we get to know the second essential equation of surface science — the Kelvin5

equation. Like the Young–Laplace equation it is based on thermodynamic principles and doesnot refer to a special material or special conditions. The subject of the Kelvin equation isthe vapor pressure of a liquid. Tables of vapor pressures for various liquids and differenttemperatures can be found in common textbooks or handbooks of physical chemistry. Thesevapor pressures are reported for vapors which are in thermodynamic equilibrium with liquidshaving planar surfaces. When the liquid surface is curved, the vapor pressure changes. Thevapor pressure of a drop is higher than that of a flat, planar surface. In a bubble the vaporpressure is reduced. The Kelvin equation tells us how the vapor pressure depends on thecurvature of the liquid.

The cause for this change in vapor pressure is the Laplace pressure . The raised Laplacepressure in a drop causes the molecules to evaporate more easily. In the liquid, which sur-rounds a bubble, the pressure with respect to the inner part of the bubble is reduced. Thismakes it more difficult for molecules to evaporate. Quantitatively the change of vapor pres-sure for curved liquid surfaces is described by the Kelvin equation:

RT · ln PK0

P0= γVm ·

(1

R1+

1R2

)(2.17)

PK0 is the vapor pressure of the curved, P0 that of the flat surface. The index “0” indicates thateverything is only valid in thermodynamic equilibrium. Please keep in mind: in equilibriumthe curvature of a liquid surface is constant everywhere. Vm is the molar volume of the liquid.For a sphere-like volume of radius r, the Kelvin equation can be simplified:

RT · ln PK0

P0=

2γVmr

or PK0 = P0 · e2γVmRT r (2.18)

The constant 2γVm/RT is 1.03 nm for Ethanol (γ = 0.022 N/m, Vm = 58 cm3/mol) and1.05 nm for Water (γ = 0.072 N/m, Vm = 18 cm3/mol) at 25◦C.

To derive the Kelvin equation we consider the Gibbs free energy of the liquid. The molarGibbs free energy changes when the surface is being curved, because the pressure increases

5 William Thomson, later Lord Kelvin, 1824–1907. Physics professor at the University of Glasgow.


due to the Laplace pressure. In general, any change in the Gibbs free energy is given by thefundamental equation dG = V dP − SdT . The increase of the Gibbs free energy per mole ofliquid, upon curving, at constant temperature is

ΔGm =∫ ΔP

0

VmdP = γVm ·(

1R1

+1

R2

), (2.19)

We have assumed that the molar volume remains constant, which is certainly a reasonableassumption because most liquids are practically incompressible for the pressures considered.For a spherical drop in its vapor, we simply have ΔGm = 2γVm/r. The molar Gibbs freeenergy of the vapor depends on the vapor pressure P0 according to

Gm = G0m + RT · ln P0 (2.20)For a liquid with a curved surface we have

GKm = G0m + RT · ln PK0 (2.21)

The change of the molar Gibbs free energy inside the vapor due to curving the interface istherefore

ΔGm = GKm − Gm = RT · lnPK0P0

(2.22)

Since the liquid and vapor are supposed to be in equilibrium, the two expressions must beequal. This immediately leads to the Kelvin equation.

When applying the Kelvin equation, it is instructive to distinguish two cases: A drop inits vapor (or more generally: a positively curved liquid surface) and a bubble in liquid (anegatively curved liquid surface).

Drop in its vapor: The vapor pressure of a drop is higher than that of a liquid with a planarsurface. One consequence is that an aerosol of drops (fog) should be unstable. To see this, letus assume that we have a box filled with many drops in a gaseous environment. Some dropsare larger than others. The small drops have a higher vapor pressure than the large drops.Hence, more liquid evaporates from their surface. This tends to condense into large drops.Within a population of drops of different sizes, the bigger drops will grow at the expense ofthe smaller ones — a process called Ostwald ripening6. These drops will sink down and, atthe end, bulk liquid fills the bottom of the box.

For a given vapor pressure, there is a critical drop size. Every drop bigger than this sizewill grow. Drops at a smaller size will evaporate. If a vapor is cooled to reach over-saturation,it cannot condense (because every drop would instantly evaporate again), unless nucleationsites are present. In that way it is possible to explain the existence of over-saturated vaporsand also the undeniable existence of fog.

Bubble in a liquid: From Eq. (2.19) we see that a negative sign has to be used for a bubblebecause of the negative curvature of the liquid surface. As a result we get

RT · ln PK0

P0= −2γVm

r(2.23)

6 In general, Ostwald ripening is the growth of long objects at the expense of smaller ones. Wilhelm Ostwald,1853–1932. German physicochemist, professor in Leipzig, Nobel price for chemistry 1909.

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Physics and Chemistry of Interfaces · 2013. 7. 16. · Karlheinz Graf MPI für Polymerforschung...

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