Hans-Jürgen Butt, Karlheinz Graf, Michael Kappl

Physics and Chemistry of Interfaces

WILEY-VCH GmbH & Co. KGaA

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www.pdfgrip.com


Hans-Jürgen Butt, Karlheinz Graf, Michael Kappl
Physics and Chemistry of Interfaces

www.pdfgrip.com


www.pdfgrip.com


Hans-Jürgen Butt, Karlheinz Graf, Michael Kappl

Physics and Chemistry of Interfaces

WILEY-VCH GmbH & Co. KGaA

www.pdfgrip.com

Authors
Hans-Jürgen Butt
MPI für Polymerforschung Mainz
e-mail:
Karlheinz Graf
MPI für Polymerforschung Mainz
e-mail:

This book was carefully produced. Nevertheless, authors and publisher do not warrant
the information contained therein to be free
of errors. Readers are advised to keep in
mind that statements, data, illustrations,
procedural details or other items may
inadvertently be inaccurate

Library of Congress Card No. applied for
Michael Kappl
MPI für Polymerforschung Mainz
e-mail:

Cover Pictures
The left picture shows aggregates of silicon
oxide particles with a diameter of 0.9 μm (see
example 1.1). At the bottom an atomic force
microscope image of cylindrical CTAB micelles adsorbed to gold(111) is shown (see
example 12.3, width: 200 nm). The right
image was also obtained by atomic force
microscopy. It shows the surface of a selfassembled monolayer of long-chain alkylthiols
on gold(111) (see fig. 10.2, width: 3.2 nm).


British Library Cataloguing-in-Publication
Data: A catalogue record for this book is
available from the British Library.

Bibliographic information published by
Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication
in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the
Internet at <>.
© 2003 WILEY-VCH Verlag GmbH & Co.
KGaA, Weinheim
All rights reserved (including those of translation into other languages). No part of this
book may be reproduced in any form – by
photoprinting, microfilm, or any other
means – nor transmitted or translated into
machine language without written permission from the publishers. Registered names,
trademarks, etc. used in this book, even
when not specifically marked as such, are
not be considered unprotected by law.
printed in the Federal Republic of Germany
printed on acid-free paper
Composition Uwe Krieg, Berlin
Printing betz-druck GmbH, Darmstadt
Bookbinding Litges&Dopf Buchbinderei
GmbH, Heppenheim
ISBN 3-527-40413-9

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Preface

Interface science has changed significantly during the last 10–15 years. This is partially due to
scientific breakthroughs. For example, the invention of scanning probe microscopy and refined
diffraction methods allow us to look at interfaces under “wet” conditions with unprecedented
accuracy. This change is also due to the greatly increased community of interfacial scientists. One reason is certainly the increased relevance of micro- and nanotechnology, including
lab-on-chip technology, microfluids, and biochips. Objects in the micro- and nanoworld are
dominated by surface effects rather than gravitation or inertia. Therefore, surface science is
the basis for nanotechnology.
The expansion of the community is correlated with an increased interdisciplinarity. Traditionally the community tended to be split into “dry” surface scientists (mainly physicists working under ultrahigh vacuum conditions) and “wet” surface scientists (mainly colloid chemists).
In addition, engineers dealing with applications like coatings, adhesion, or lubrication, formed
a third community. This differentiation is significantly less pronounced and interface science
has become a really interdisciplinary field of research including, for example, chemical engineering and biology.
This development motivated us to write this textbook. It is a general introduction to surface
and interface science. It focuses on basic concepts rather than specific details, on understanding 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 scientific or engineering disciplines are involved. It contains basic science and applied topics such
as wetting, friction, and lubrication. Many textbooks concentrate on certain aspects of surface
science 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, a
comprehensive introduction is required.
Our manuscript is based on lectures given at the universities of Siegen and Mainz. It addresses (1) advanced students of engineering, chemistry, physics, biology, and related subjects
and (2) scientists in academia or industry, who are not yet specialists in surface science but
want to get a solid background knowledge of the subject. The level is introductory for scientists 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 of
this book you will see a substantial number of equations. Please do not be scared! We preferred to give all transformations explicitly rather than writing “as can easily be seen” and
stating the result. Chapter “Thermodynamics of Interfaces” is the only exception. To appreciate it a basic knowledge of thermodynamics is required. You can skip and still be able
to follow the rest. In this case please read and try to get an intuitive understanding of what
surface excess is (Section 3.1) and what the Gibbs adsorption equation implies (Section 3.4.2).


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VI

Preface

A number of problems with solutions were included to allow for private studies. If not
mentioned otherwise, the temperature was assumed to be 25◦ C. At the end of each chapter the
most important equations, facts, and phenomena are summarized to given students a chance
to 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 to
keep the volume within the scope of one advanced course of roughly 15 weeks, one day per
week. Unfortunately, this means that certain topics had to be cut short or even left out completely. Statistical mechanics, heterogeneous catalysis, and polymers at surfaces are issues
which could have been expanded.
This book certainly contains errors. Even after having it read by different people independently, this is unavoidable. If you find an error, please write us a letter (Max-Planck-Institute
for Polymer Research, Ackermannweg, 55128 Mainz, Germany) or an e-mail () so that we can correct it and do not confuse more students.
We are indebted to several people who helped us collecting information, preparing, and
critically 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. Bonaccurso, P. Broekmann, G. Glasser, G. Gompper, M. Grunze, J. Gutmann, L. Heim, M. Hillebrand, 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

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Contents


Preface

V

1

1

Introduction

2 Liquid surfaces
2.1 Microscopic picture of the liquid surface . . . . . .
2.2 Surface tension . . . . . . . . . . . . . . . . . . .
2.3 Equation of Young and Laplace . . . . . . . . . . .
2.3.1 Curved liquid surfaces . . . . . . . . . . .
2.3.2 Derivation of the Young–Laplace equation
2.3.3 Applying the Young–Laplace equation . . .
2.4 Techniques to measure the surface tension . . . . .
2.5 The Kelvin equation . . . . . . . . . . . . . . . . .
2.6 Capillary condensation . . . . . . . . . . . . . . .
2.7 Nucleation theory . . . . . . . . . . . . . . . . . .
2.8 Summary . . . . . . . . . . . . . . . . . . . . . .
2.9 Exercises . . . . . . . . . . . . . . . . . . . . . .

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4
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5
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23
24

3 Thermodynamics of interfaces
3.1 The surface excess . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Fundamental thermodynamic relations . . . . . . . . . . . . . . . . . . .
3.2.1 Internal energy and Helmholtz energy . . . . . . . . . . . . . . .
3.2.2 Equilibrium conditions . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Location of the interface . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Gibbs energy and definition of the surface tension . . . . . . . . .
3.2.5 Free surface energy, interfacial enthalpy and Gibbs surface energy
3.3 The surface tension of pure liquids . . . . . . . . . . . . . . . . . . . . .
3.4 Gibbs adsorption isotherm . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 System of two components . . . . . . . . . . . . . . . . . . . . .

3.4.3 Experimental aspects . . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 The Marangoni effect . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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41

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VIII

4

5

6

Contents


The electric double layer
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
4.2 Poisson–Boltzmann theory of the diffuse double layer
4.2.1 The Poisson–Boltzmann equation . . . . . .
4.2.2 Planar surfaces . . . . . . . . . . . . . . . .
4.2.3 The full one-dimensional case . . . . . . . .
4.2.4 The Grahame equation . . . . . . . . . . . .
4.2.5 Capacity of the diffuse electric double layer .
4.3 Beyond Poisson–Boltzmann theory . . . . . . . . . .
4.3.1 Limitations of the Poisson–Boltzmann theory
4.3.2 The Stern layer . . . . . . . . . . . . . . . .
4.4 The Gibbs free energy of the electric double layer . .
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . .
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . .

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42
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49
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55

56

Effects at charged interfaces
5.1 Electrocapillarity . . . . . . . . . . . . . . . . . .
5.1.1 Theory . . . . . . . . . . . . . . . . . . .
5.1.2 Measurement of electrocapillarity . . . . .
5.2 Examples of charged surfaces . . . . . . . . . . . .
5.2.1 Mercury . . . . . . . . . . . . . . . . . . .
5.2.2 Silver iodide . . . . . . . . . . . . . . . .
5.2.3 Oxides . . . . . . . . . . . . . . . . . . .
5.2.4 Mica . . . . . . . . . . . . . . . . . . . .
5.2.5 Semiconductors . . . . . . . . . . . . . . .
5.3 Measuring surface charge densities . . . . . . . . .
5.3.1 Potentiometric colloid titration . . . . . . .
5.3.2 Capacitances . . . . . . . . . . . . . . . .
5.4 Electrokinetic phenomena: The zeta potential . . .
5.4.1 The Navier–Stokes equation . . . . . . . .
5.4.2 Electro-osmosis and streaming potential . .
5.4.3 Electrophoresis and sedimentation potential
5.5 Types of potentials . . . . . . . . . . . . . . . . .
5.6 Summary . . . . . . . . . . . . . . . . . . . . . .
5.7 Exercises . . . . . . . . . . . . . . . . . . . . . .

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57
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76
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79
79

Surface forces
6.1 Van der Waals forces between molecules . . . . . . .
6.2 The van der Waals force between macroscopic solids
6.2.1 Microscopic approach . . . . . . . . . . . .
6.2.2 Macroscopic calculation — Lifshitz theory .
6.2.3 Surface energy and Hamaker constant . . . .

6.3 Concepts for the description of surface forces . . . .
6.3.1 The Derjaguin approximation . . . . . . . .
6.3.2 The disjoining pressure . . . . . . . . . . . .

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80
80
84
84
87
92
93
93
95

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Contents

IX

6.4
6.5

Measurement of surface forces . . . . . . . . . . . . . . . . .
The electrostatic double-layer force . . . . . . . . . . . . . .
6.5.1 General equations . . . . . . . . . . . . . . . . . . . .
6.5.2 Electrostatic interaction between two identical surfaces
6.5.3 The DLVO theory . . . . . . . . . . . . . . . . . . . .
6.6 Beyond DLVO theory . . . . . . . . . . . . . . . . . . . . . .
6.6.1 The solvation force and confined liquids . . . . . . . .
6.6.2 Non DLVO forces in an aqueous medium . . . . . . .
6.7 Steric interaction . . . . . . . . . . . . . . . . . . . . . . . .

6.7.1 Properties of polymers . . . . . . . . . . . . . . . . .
6.7.2 Force between polymer coated surfaces . . . . . . . .
6.8 Spherical particles in contact . . . . . . . . . . . . . . . . . .
6.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Contact angle phenomena and wetting
7.1 Young’s equation . . . . . . . . . . . . . . . . .
7.1.1 The contact angle . . . . . . . . . . . . .
7.1.2 Derivation . . . . . . . . . . . . . . . . .
7.1.3 The line tension . . . . . . . . . . . . . .
7.1.4 Complete wetting . . . . . . . . . . . . .
7.2 Important wetting geometries . . . . . . . . . . .
7.2.1 Capillary rise . . . . . . . . . . . . . . .
7.2.2 Particles in the liquid–gas interface . . .
7.2.3 Network of fibres . . . . . . . . . . . . .
7.3 Measurement of the contact angle . . . . . . . .
7.3.1 Experimental methods . . . . . . . . . .
7.3.2 Hysteresis in contact angle measurements
7.3.3 Surface roughness and heterogeneity . . .
7.4 Theoretical aspects of contact angle phenomena .
7.5 Dynamics of wetting and dewetting . . . . . . .
7.5.1 Wetting . . . . . . . . . . . . . . . . . .
7.5.2 Dewetting . . . . . . . . . . . . . . . . .
7.6 Applications . . . . . . . . . . . . . . . . . . . .
7.6.1 Flotation . . . . . . . . . . . . . . . . .
7.6.2 Detergency . . . . . . . . . . . . . . . .
7.6.3 Microfluidics . . . . . . . . . . . . . . .
7.6.4 Adjustable wetting . . . . . . . . . . . .
7.7 Summary . . . . . . . . . . . . . . . . . . . . .
7.8 Exercises . . . . . . . . . . . . . . . . . . . . .


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8 Solid surfaces
145
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8.2 Description of crystalline surfaces . . . . . . . . . . . . . . . . . . . . . . . 146
8.2.1 The substrate structure . . . . . . . . . . . . . . . . . . . . . . . . . 146

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X

Contents

8.2.2 Surface relaxation and reconstruction . . . . . . . . . . .
8.2.3 Description of adsorbate structures . . . . . . . . . . . . .
8.3 Preparation of clean surfaces . . . . . . . . . . . . . . . . . . . .
8.4 Thermodynamics of solid surfaces . . . . . . . . . . . . . . . . .
8.4.1 Surface stress and surface tension . . . . . . . . . . . . .
8.4.2 Determination of the surface energy . . . . . . . . . . . .
8.4.3 Surface steps and defects . . . . . . . . . . . . . . . . . .
8.5 Solid–solid boundaries . . . . . . . . . . . . . . . . . . . . . . .
8.6 Microscopy of solid surfaces . . . . . . . . . . . . . . . . . . . .
8.6.1 Optical microscopy . . . . . . . . . . . . . . . . . . . . .
8.6.2 Electron microscopy . . . . . . . . . . . . . . . . . . . .
8.6.3 Scanning probe microscopy . . . . . . . . . . . . . . . .
8.7 Diffraction methods . . . . . . . . . . . . . . . . . . . . . . . . .
8.7.1 Diffraction patterns of two-dimensional periodic structures
8.7.2 Diffraction with electrons, X-rays, and atoms . . . . . . .
8.8 Spectroscopic methods . . . . . . . . . . . . . . . . . . . . . . .
8.8.1 Spectroscopy using mainly inner electrons . . . . . . . . .
8.8.2 Spectroscopy with outer electrons . . . . . . . . . . . . .
8.8.3 Secondary ion mass spectrometry . . . . . . . . . . . . .
8.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Adsorption
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.2 The adsorption time . . . . . . . . . . . . . . . . . . . . .
9.1.3 Classification of adsorption isotherms . . . . . . . . . . . .
9.1.4 Presentation of adsorption isotherms . . . . . . . . . . . . .
9.2 Thermodynamics of adsorption . . . . . . . . . . . . . . . . . . . .
9.2.1 Heats of adsorption . . . . . . . . . . . . . . . . . . . . . .
9.2.2 Differential quantities of adsorption and experimental results
9.3 Adsorption models . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.1 The Langmuir adsorption isotherm . . . . . . . . . . . . . .
9.3.2 The Langmuir constant and the Gibbs energy of adsorption .
9.3.3 Langmuir adsorption with lateral interactions . . . . . . . .
9.3.4 The BET adsorption isotherm . . . . . . . . . . . . . . . .
9.3.5 Adsorption on heterogeneous surfaces . . . . . . . . . . . .
9.3.6 The potential theory of Polanyi . . . . . . . . . . . . . . . .
9.4 Experimental aspects of adsorption from the gas phase . . . . . . .
9.4.1 Measurement of adsorption isotherms . . . . . . . . . . . .
9.4.2 Procedures to measure the specific surface area . . . . . . .
9.4.3 Adsorption on porous solids — hysteresis . . . . . . . . . .
9.4.4 Special aspects of chemisorption . . . . . . . . . . . . . . .
9.5 Adsorption from solution . . . . . . . . . . . . . . . . . . . . . . .
9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

XI

10 Surface modification
10.1 Introduction . . . . . . . . . . . . .
10.2 Chemical vapor deposition . . . . .
10.3 Soft matter deposition . . . . . . . .
10.3.1 Self-assembled monolayers
10.3.2 Physisorption of Polymers .
10.3.3 Polymerization on surfaces .
10.4 Etching techniques . . . . . . . . .
10.5 Summary . . . . . . . . . . . . . .
10.6 Exercises . . . . . . . . . . . . . .

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11 Friction, lubrication, and wear
11.1 Friction . . . . . . . . . . . . . . . . . . .
11.1.1 Introduction . . . . . . . . . . . . .
11.1.2 Amontons’ and Coulomb’s Law . .
11.1.3 Static, kinetic, and stick-slip friction
11.1.4 Rolling friction . . . . . . . . . . .
11.1.5 Friction and adhesion . . . . . . . .
11.1.6 Experimental Aspects . . . . . . .
11.1.7 Techniques to measure friction . . .
11.1.8 Macroscopic friction . . . . . . . .
11.1.9 Microscopic friction . . . . . . . .
11.2 Lubrication . . . . . . . . . . . . . . . . .
11.2.1 Hydrodynamic lubrication . . . . .
11.2.2 Boundary lubrication . . . . . . . .
11.2.3 Thin film lubrication . . . . . . . .
11.2.4 Lubricants . . . . . . . . . . . . .
11.3 Wear . . . . . . . . . . . . . . . . . . . . .
11.4 Summary . . . . . . . . . . . . . . . . . .
11.5 Exercises . . . . . . . . . . . . . . . . . .

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12 Surfactants, micelles, emulsions, and foams
12.1 Surfactants . . . . . . . . . . . . . . . .

12.2 Spherical micelles, cylinders, and bilayers
12.2.1 The critical micelle concentration
12.2.2 Influence of temperature . . . . .
12.2.3 Thermodynamics of micellization
12.2.4 Structure of surfactant aggregates
12.2.5 Biological membranes . . . . . .
12.3 Macroemulsions . . . . . . . . . . . . . .
12.3.1 General properties . . . . . . . .
12.3.2 Formation . . . . . . . . . . . . .
12.3.3 Stabilization . . . . . . . . . . .
12.3.4 Evolution and aging . . . . . . .
12.3.5 Coalescence and demulsification .

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XII

Contents


12.4 Microemulsions . . . . . . . . . . . . . . . . . . . . . . .
12.4.1 Size of droplets . . . . . . . . . . . . . . . . . . .
12.4.2 Elastic properties of surfactant films . . . . . . . .
12.4.3 Factors influencing the structure of microemulsions
12.5 Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5.1 Classification, application and formation . . . . .
12.5.2 Structure of foams . . . . . . . . . . . . . . . . .
12.5.3 Soap films . . . . . . . . . . . . . . . . . . . . .
12.5.4 Evolution of foams . . . . . . . . . . . . . . . . .
12.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
12.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .
13 Thin films on surfaces of liquids
13.1 Introduction . . . . . . . . . . . . . . . . . . .
13.2 Phases of monomolecular films . . . . . . . . .
13.3 Experimental techniques to study monolayers .
13.3.1 Optical methods . . . . . . . . . . . .
13.3.2 X-ray reflection and diffraction . . . . .
13.3.3 The surface potential . . . . . . . . . .
13.3.4 Surface elasticity and viscosity . . . . .
13.4 Langmuir–Blodgett transfer . . . . . . . . . . .
13.5 Thick films – spreading of one liquid on another
13.6 Summary . . . . . . . . . . . . . . . . . . . .
13.7 Exercises . . . . . . . . . . . . . . . . . . . .

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297

14 Solutions to exercises

299

Appendix
A Analysis of diffraction patterns
A.1 Diffraction at three dimensional crystals
A.1.1 Bragg condition . . . . . . . . .
A.1.2 Laue condition . . . . . . . . .
A.1.3 The reciprocal lattice . . . . . .
A.1.4 Ewald construction . . . . . . .
A.2 Diffraction at Surfaces . . . . . . . . .
A.3 Intensity of diffraction peaks . . . . . .

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323
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327

B Symbols and abbreviations

331

Bibliography

335

Index

355

www.pdfgrip.com


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. Interface
is, however, a more general term than surface. Interfaces can also separate two immiscible
liquids such as water and oil. These are called liquid–liquid interfaces. Solid–solid interfaces
separate 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, in

which 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 by
Graham1 . He applied it to materials which seemed to dissolve but were not able to penetrate a
membrane, such as albumin, starch, and dextrin. A dispersion is a two-phase system which is
uniform on the macroscopic but not on the microscopic scale. It consists of grains or droplets
of one phase in a matrix of the other phase.
Different kinds of dispersions can be formed. Most of them have important applications
and have special names (Table 1.1). While there are only five types of interface, we can distinguish 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 is
obvious. Nobody will, for instance, mix up fog with a foam although in both cases a liquid and
a gas are involved. In other cases the distinction between continuous and inner phase cannot
be made because both phases might form connected networks. Some emulsions for instance
tend to form a bicontinuous phase, in which both phases form an interwoven network.
Continuous
phase

1 nm - 1 μm

Dispersed
phase
Figure 1.1: Schematic of a dispersion.

Colloids and interfaces are intimately related. This is a direct consequence of their enormous specific surface area. More precisely: their interface-to-volume relation is so large, that
their behavior is completely determined by surface properties. Gravity is negligible in the
1

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

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2

1 Introduction

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

Continuous
phase

Dispersed
phase

Term

Example

Gas

liquid
solid

aerosol
aerosol

clouds, fog, smog, hairspray
smoke, dust, pollen

Liquid


gas
liquid
solid

foam
emulsion
sol

lather, whipped cream, foam on beer
milk
ink, muddy water, dispersion paint

Solid

gas

porous solids*
foam
solid emulsion
solid suspension

styrofoam, soufflés
butter
concrete

liquid
solid

majority of cases. For this reason we could define colloidal systems as systems which are

dominated by interfacial effects rather than bulk properties. This is also the reason why interfacial science is the basis for nanoscience and technology and many inventions in this new
field originate from surface science.
Q Example 1.1. A system which is dominated by surface effects is shown on the left side of
the 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 chamber
filled with gas. While sedimenting they formed fractal aggregates due to attractive van der
Waals forces. On the bottom they were collected. These aggregates are stable for weeks
and months and even shaking does not change their structure. Gravity and inertia, which
rule the macroscopic world, are not able to bend down the particle chains. Surface forces
are much stronger.
In the recent literature the terms nanoparticles and nanosystems are used, in analogy to colloid
and colloidal systems. The prefix “nano” indicates dimensions in the 1 to 100 nm range. This
is above the atomic scale and, unless highly refined methods are used, below the resolution of
a 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 of
natural processes. For example, in biology the surface tension of water allows to form lipid
membranes. This is a prerequisite for the formation of compartments and thus any form of
life. 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 is
dominated 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 technological applications. One such example is flotation in mineral processing or the bleaching
of scrap paper. Washing and detergency are examples which any person encounters every day.

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1 Introduction

3


Often the production of new materials such as composite materials heavily involves processes
at interfaces. Thin films on surfaces are often dominated by surface effects. Examples are
latex-films, coatings, and paints. The flow behavior of powders and granular media is determined by surface forces. In tribology, wear is reduced by lubrication which again is a surface
phenomenon.
Typical for many of the industrial applications is a very refined and highly developed
technology, but only a limited understanding of the underlying fundamental processes. A
better understanding is, however, required to further improve the efficiency or reduce dangers
to the environment.
Introductory books on interface science are Refs. [1–6]. For a deeper understanding we
recommend the series of books of Lyklema [7–9].

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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 a
certain 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 liquid
to that of its vapor [10].

Figure 2.1: Density of a liquid versus the coordinate normal to its surface: (a) is a schematic
plot; (b) results from molecular dynamics simulations of a n-tridecane (C13 H28 ) at 27◦ C
adapted from Ref. [11]. Tridecane is practically not volatile. For this reason the density in
the vapor phase is negligible.

The density is only one criterion to define the thickness of an interface. Another possible
parameter is the orientation of the molecules. For example, water molecules at the surface
prefer to be oriented with their negative sides “out” towards the vapor phase. This orientation

fades with increasing distance from the surface. At a distance of 1–2 nm the molecules are
again randomly oriented.
Which thickness do we have to use? This depends on the relevant parameter. If we are
for instance, interested in the density of a water surface, a realistic thickness is in the order of
1 nm. Let us assume that a salt is dissolved in the water. Then the concentration of ions might
vary over a much larger distance (characterized by the Debye length, see Section 4.2.2). With
respect to the ion concentration, the thickness is thus much larger. In case of doubt, it is safer
to choose a large value for the thickness.
The surface of a liquid is a very turbulent place. Molecules evaporate from the liquid into
the vapor phase and vice versa. In addition, they diffuse into the bulk phase and molecules
from the bulk diffuse to the surface.

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2.2

Surface tension

5

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

PA
2πmkB T

(2.1)


Here, A is the cross-sectional area of the hole and m is the molecular mass. This is equal
to the number of water molecules hitting a surface area A per second. Water at 25◦ C has
a 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 10
Å2 . 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 water
molecule 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 surfaces
is large enough to avoid overlapping of the two interfacial regions. Practically, this experiment
might be tricky even in the absence of gravity but it does not violate a physical law so that it
is in principle feasible. If we increase the surface area by moving the slider a distance dx to
the right, work has to be done. This work dW is proportional to the increase in surface area
dA. 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.
dx

liquid film

b

dA = 2bdx


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

Equation (2.2) is an empirical law and a definition at the same time. The empirical law is
that the work is proportional to the change in surface area. This is not only true for infinitesimal 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 liquid and the vapor, temperature, and pressure, but it is independent of the area. The definition
is that we call the proportionality constant “surface tension”.

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6

2 Liquid surfaces

The surface tension can also be defined by the force F that is required to hold the slider in
place 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 can
write
F =−

dW
= −2γb
dx


(2.4)

The force is directed to the left while x increases to the right. Therefore we have a negative
sign.
The unit of surface tension is either J/m2 or N/m. Surface tensions of liquids are of the
order 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 the
surface area of its rubber membrane against a tension. There is, however, a difference: while
the expansion of a liquid surface is a plastic process the stretching of a rubber membrane is
usually elastic.
Table 2.1: Surface tensions γ of some liquids at different temperatures T .

Substance
Water

Argon
Methanol
Ethanol

1-propanol
1-butanol
2-butanol
Acetone

T

γ

( mNm−1 )

10◦ C
25◦ C
50◦ C
75◦ C
100◦ C
90 K
25◦ C
10◦ C
25◦ C
50◦ C
25◦ C
25◦ C
25◦ C
25◦ C

74.23
71.99
67.94
63.57
58.91
11.90
22.07
23.22
21.97
19.89
23.32
24.93
22.54

23.46

Substance
Mercury
Phenol
Benzene
Toluene
Dichloromethane
n-pentane
n-hexane
n-heptane
n-octane

Formamide

T

γ
(mNm−1 )

25◦ C
50◦ C
25◦ C
25◦ C
25◦ C
25◦ C
25◦ C
25◦ C
10◦ C
25◦ C

50◦ C
75◦ C
100◦ C
25◦ C

485.48
38.20
28.22
27.93
27.20
15.49
17.89
19.65
22.57
21.14
18.77
16.39
14.01
57.03

Q Example 2.2. If a water film is formed on a frame with a slider length of 1 cm, then the
film pulls on the slider with a force of
2 × 0.01 m × 0.072 Jm−2 = 1.44 × 10−3 N
That corresponds to a weight of 0.15 g.

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2.2


Surface tension

7

Figure 2.3: Schematic molecular structure
of a liquid–vapor interface.

How can we interpret the concept of surface tension on the molecular level? For molecules
it is energetically favorable to be surrounded by other molecules. Molecules attract each
other by different interactions such as van der Waals forces or hydrogen bonds (for details see
Chapter 6). Without this attraction there would not be a condensed phase at all, there would
only be a vapor phase. The sheer existence of a condensed phase is evidence for an attractive
interaction between the molecules. At the surface, molecules are only partially surrounded by
other 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 bring
molecules from inside the liquid to the surface and to create new surface area. Therefore often
the term “surface energy” is used for γ. As we shall see in the next chapter this might lead to
some confusion. To avoid this we use the term surface tension.
With this interpretation of the surface tension in mind we immediately realize that γ has
to be positive. Otherwise the Gibbs free energy of interaction would be repulsive and all
molecules would immediately evaporate into the gas phase.
Q Example 2.3. Estimate the surface tension of cyclohexane from the energy of vaporization Δvap U = 30.5 kJ/mol at 25◦ C. The density of cyclohexane is ρ = 773 kg/m3 , its
molecular 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 roughly
Δvap U/6 = 5.08 kJ/mol. At the surface one neighbor and hence one bond is missing. Per
mole 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 one
molecule. If the molecules form a cubic structure, the volume of one unit cell is a3 , where

a is the distance between nearest neighbors. This distance can be calculated from the
density:
M
0.08416 kg/mol
=
= 1.81 × 10−28 m3 ⇒
a3 =
3
ρNA
773 kg/m · 6.02 × 1023 mol−1
a

= 0.565 nm

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8

2 Liquid surfaces

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

Δvap U
5080 Jmol−1
J
=
= 0.0264 2
−1

2
23
−9
2
6NA a
m
6.02 × 10 mol · (0.565 × 10 m)

For such a rough estimate the result is surprisingly close to the experimental value of
0.0247 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 the
surface. The surface tension tends to minimize the area. This results in a planar geometry of
the surface. In order to curve the surface, the pressure on one side must be larger than on the
other side. The situation is much like that of a rubber membrane. If we, for instance, take a
tube and close one end with a rubber membrane, the membrane will be planar (provided the
membrane is under some tension) (Fig. 2.4). It will remain planar as long as the tube is open at
the other end and the pressure inside the tube is equal to the outside pressure. If we now blow
carefully into the tube, the membrane bulges out and becomes curved due to the increased
pressure inside the tube. If we suck on the tube, the membrane bulges inside the tube because
now the outside pressure is higher than the pressure inside the tube.

Pa

Pi

Pa = Pi


Pa < Pi

Pa > Pi

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

The Young1 –Laplace2 equation relates the pressure difference between the two phases ΔP
and the curvature of the surface:
ΔP = γ ·

1
1
+
R1
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.


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2.3

Equation of Young and Laplace

9

It is perhaps worthwhile to describe the principal radii of curvature in a little bit more
detail. The curvature 1/R1 + 1/R2 at a point on an arbitrarily curved surface is obtained as
follows. At the point of interest we draw a normal through the surface and then pass a plane
through this line and the intersection of this line with the surface. The line of intersection
will, in general, be curved at the point of interest. The radius of curvature R1 is the radius of
a circle inscribed to the intersection at the point of interest. The second radius of curvature
is obtained by passing a second plane through the surface also containing the normal, but
perpendicular to the first plane. This gives the second intersection and leads to the second
radius of curvature R2 . So the planes defining the radii of curvature must be perpendicular
to each other and contain the surface normal. Otherwise their orientation is arbitrary. A law
of differential geometry says that the value 1/R1 + 1/R2 for an arbitrary surface does not
depend on the orientation, as long as the radii are determined in perpendicular directions.

Figure 2.5: Illustration
of the curvature of a
cylinder and a sphere.

Let us illustrate the curvature for two examples. For a cylinder of radius r a convenient
choice is R1 = r and R2 = ∞ so that the curvature is 1/r + 1/∞ = 1/r. For a sphere with
radius R we have R1 = R2 and the curvature is 1/R + 1/R = 2/R (Fig. 2.5).
Q Example 2.4. How large is the pressure in a spherical bubble with a diameter of 2 mm

and a bubble of 20 nm diameter in pure water, compared with the pressure outside? For a
bubble the curvature is identical to that of a sphere: R1 = R2 = R. Therefore
ΔP =

2γ
R

(2.6)

With R = 1 mm we get
ΔP = 0.072

J
2
× −3 = 144 Pa
m2
10 m
2

With R = 10 nm the pressure is ΔP = 0.072 J/m × 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 the
pressure difference.
• In the absence of external fields (e.g. gravity), the pressure is the same everywhere in the
liquid; otherwise there would be a flow of liquid to regions of low pressure. Thus, ΔP
is constant and Young–Laplace equation tells us that in this case the surface of the liquid
has the same curvature everywhere.


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10

2 Liquid surfaces

• With the help of the Young–Laplace Eq. (2.5) it is possible to calculate the equilibrium
shape of a liquid surface. If we know the pressure difference and some boundary conditions (such as the volume of the liquid and its contact line) we can calculate the geometry
of 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 curvature
involves the second derivative. As a result, calculating the shape of a liquid surface involves
solving 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 of
symmetry is identical to the y axis of an orthogonal cartesian coordinate system, then it is
convenient to put one radius of curvature in the plane of the xy coordinate. This radius is
given by
1
=
R1

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 second
bending radius is perpendicular to the xy plane. It is
y

1
=
R2
x

(2.8)

1+y2

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 all
points on that line are the same distance d away from X (Fig. 2.6). If the liquid surface is
planar, this would be a flat circle. On this line we take two cuts that are perpendicular to each
other (AXB and CXD). Consider in B a small segment on the line of length dl. The surface
tension pulls with a force γ dl. The vertical force on that segment is γ dl sin α. For small
surface areas (and small α) we have sin α ≈ d/R1 where R1 is the radius of curvature along
AXB. The vertical force component is
γ · dl ·

d
R1

(2.9)


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

2d
2d
+
R1
R2

= γ · dl · 2d ·

1
1
+
R1
R2

(2.10)

This expression is independent of the absolute orientation of AB and CD. Integration over the
borderline (only 90◦ rotation of the four segments) gives the total vertical force, caused by the
surface tension:
πd2 · γ ·

1
1
+
R1
R2


(2.11)

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