Physics and Chemistry of Interfaces [4 ed.] 3527414053, 9783527414055

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ISTUDY

Physics and Chemistry of Interfaces

Physics and Chemistry of Interfaces Fourth Edition

Hans-Jürgen Butt Karlheinz Graf Michael Kappl

Authors Prof. Hans-Jürgen Butt

MPI for Polymer Research Ackermannweg 10 55128 Mainz Germany

All books published by WILEY-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, 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.

Prof. Dr. Karlheinz Graf

Hochschule Niederrhein Physikalische Chemie Adlerstr. 32 47798 Krefeld Germany

Library of Congress Card No.: applied for

Dr. Michael Kappl

Bibliographic information published by the Deutsche Nationalbibliothek

MPI for Polymer Research Ackermannweg 10 55128 Mainz Germany Cover image: © Getty Images

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at . © 2023 WILEY-VCH GmbH, Boschstraße 12, 69469 Weinheim, Germany 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 a 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 to be considered unprotected by law. Print ISBN: 978-3-527-41405-5 ePDF ISBN: 978-3-527-83617-8 ePub ISBN: 978-3-527-83616-1 Cover Design

Adam Design, Weinheim,

Germany Typesetting

Straive, Chennai, India

v

Contents Preface

xiii

1

Introduction 1

2 2.1 2.2 2.3 2.3.1 2.3.2 2.3.3 2.4 2.5 2.6 2.7 2.8 2.9

Liquid Surfaces 5 Microscopic Picture of a Liquid Surface 5 Surface Tension 6 Equation of Young and Laplace 10 Curved Liquid Surfaces 10 Derivation of Young–Laplace Equation 13 Applying the Young–Laplace Equation 14 Techniques to Measure Surface Tension 15 Kelvin Equation 20 Capillary Condensation 23 Nucleation Theory 26 Summary 31 Exercises 31

3 3.1 3.2 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.4 3.5 3.5.1 3.5.2 3.5.3 3.5.4

Thermodynamics of Interfaces 33 Thermodynamic Functions for Bulk Systems 33 Surface Excess 34 Thermodynamic Relations for Systems with an Interface Internal Energy and Helmholtz Energy 38 Equilibrium Conditions 39 Location of Interface 40 Gibbs Energy and Enthalpy 41 Interfacial Excess Energies 41 Pure Liquids 43 Gibbs Adsorption Isotherm 45 Derivation 45 System of Two Components 46 Experimental Aspects 48 Marangoni Effect 49

38

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Contents

3.6 3.7

Summary 51 Exercises 51

4 4.1 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 4.2.6 4.2.7 4.2.8 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 4.4 4.4.1 4.4.2 4.4.3 4.5 4.6

Charged Interfaces and the Electric Double Layer 53 Introduction 53 Mathematical Description of the Electric Double Layer 55 The Poisson–Boltzmann Equation 56 Gouy–Chapman Model for Planar Surfaces 57 The Full One-Dimensional Case 59 The Electric Double Layer around a Sphere 61 The Grahame Equation 62 Gibbs Energy of the Electric Double Layer 64 Limitations of the Poisson–Boltzmann Theory 65 Stern Model 67 Experimental Characterization of Charged Interfaces 68 Types of Potentials 68 Electrocapillarity 70 Examples of Charged Surfaces 73 Potentiometric Colloid Titration 80 Capacitances 82 Electrokinetic Phenomena: The Zeta Potential 83 The Navier–Stokes Equation 84 Electro-osmosis and Streaming Potential 86 Electrophoresis and Sedimentation Potential 88 Summary 91 Exercises 91

5 5.1 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.3 5.3.1 5.3.2 5.4 5.5 5.5.1 5.5.2 5.6 5.6.1 5.6.2

Surface Forces 93 Van der Waals Forces between Molecules 93 Van der Waals Force Between Macroscopic Solids 97 Microscopic Approach 97 Macroscopic Calculation – Lifshitz Theory 100 Retarded Van der Waals Forces 105 Surface Energy and the Hamaker Constant 106 Concepts for the Description of Surface Forces 107 The Derjaguin Approximation 107 Disjoining Pressure 109 Measurement of Surface Forces 110 Electrostatic Double-Layer Force 112 Electrostatic Interaction Between Two Identical Surfaces 112 DLVO Theory 116 Beyond DLVO Theory 119 Solvation Force and Confined Liquids 119 Non-DLVO Forces in Aqueous Medium 121

Contents

5.7 5.7.1 5.7.2 5.7.3 5.8 5.9 5.10

Steric and Depletion Interaction 122 Properties of Polymers 122 Force Between Polymer-coated Surfaces 123 Depletion Forces 126 Spherical Particles in Contact 127 Summary 131 Exercises 132

6 6.1 6.1.1 6.1.2 6.1.3 6.1.4 6.2 6.2.1 6.2.2 6.2.3 6.2.4 6.2.5 6.2.6 6.3 6.3.1 6.3.2 6.3.3 6.4 6.4.1 6.4.2 6.4.3 6.5 6.5.1 6.5.2 6.5.3 6.5.4 6.6 6.7 6.8

Contact Angle Phenomena and Wetting 135 Young’s Equation 135 Equilibrium Contact Angle 135 Derivation 136 Complete Wetting, Surface Forces, and the Core Region 140 Line Tension, Wetting Transitions, Estimation of Interfacial Energies 142 Wetting of Real Surfaces 145 Advancing and Receding Contact Angles 145 Measurement of Contact Angles 146 Causes of Contact Angle Hysteresis 147 Surface Roughness and Heterogeneity 149 Superhydrophobic Surfaces 150 Surfaces with Low Sliding Angle 152 Important Wetting Geometries 153 Capillary Rise 153 Particles at Interfaces 155 Network of Fibers 157 Dynamics of Wetting and Dewetting 158 Spontaneous Spreading 158 Dynamic Contact Angles 160 Coating and Dewetting 163 Applications 164 Flotation 164 Detergency 166 Microfluidics 167 Electrowetting 168 Thick Films: Spreading of One Liquid on Another 169 Summary 172 Exercises 173

7 7.1 7.2 7.2.1 7.2.2 7.2.3

Solid Surfaces 175 Introduction 175 Description of Crystalline Surfaces 176 Substrate Structure 176 Surface Relaxation and Reconstruction 177 Description of Adsorbate Structures 179

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Contents

7.3 7.3.1 7.3.2 7.3.3 7.3.4 7.4 7.4.1 7.4.2 7.4.3 7.5 7.5.1 7.5.2 7.6 7.7 7.7.1 7.7.2 7.7.3 7.8 7.8.1 7.8.2 7.9 7.9.1 7.9.2 7.9.3 7.9.4 7.10 7.11

Preparation of Clean Surfaces 180 Thermal Treatment 181 Plasma or Sputter Cleaning 181 Cleavage 182 Deposition of Thin Films 183 Thermodynamics of Solid Surfaces 183 Surface Energy, Surface Tension, and Surface Stress 183 Determining Surface Energy 186 Surface Steps and Defects 189 Surface Diffusion 191 Theoretical Description of Surface Diffusion 192 Measurement of Surface Diffusion 195 Solid–Solid Interfaces 198 Microscopy 200 Optical Microscopy 200 Electron Microscopy 201 Scanning Probe Microscopy 203 Diffraction Methods 206 Diffraction Patterns of Two-Dimensional Periodic Structures Diffraction with Electrons, X-Rays, and Atoms 207 Spectroscopy 209 Optical Spectroscopy of Surfaces 209 Spectroscopy Using Inner Electrons 213 Spectroscopy with Outer Electrons 214 Secondary Ion Mass Spectrometry 215 Summary 217 Exercises 218

8 8.1 8.1.1 8.1.2 8.1.3 8.1.4 8.2 8.2.1 8.2.2 8.3 8.3.1 8.3.2 8.3.3 8.3.4 8.3.5 8.4 8.4.1

Adsorption 219 Introduction 219 Definitions 219 The Adsorption Time 221 Classification of Adsorption Isotherms 222 Presentation of Adsorption Isotherms 224 Thermodynamics of Adsorption 224 Heats of Adsorption 224 Differential Quantities of Adsorption 226 Adsorption Models 228 The Langmuir Adsorption Isotherm 228 The Langmuir Constant and Gibbs Energy of Adsorption 230 Langmuir Adsorption with Lateral Interactions 231 The BET Adsorption Isotherm 232 Adsorption on Heterogeneous Surfaces 235 Experimental Aspects of Adsorption from the Gas Phase 236 Measuring Adsorption to Planar Surfaces 236

206

Contents

8.4.2 8.4.3 8.4.4 8.5 8.6 8.7

Measuring Adsorption to Powders and Textured Materials 238 Adsorption to Porous Materials 239 Chemisorption and Temperature-programmed Desorption 248 Adsorption from Solution 249 Summary 251 Exercises 252

9 9.1 9.2 9.2.1 9.2.2 9.3 9.3.1 9.3.2 9.3.3 9.3.4 9.4 9.5 9.6 9.7

Surface Modification 255 Introduction 255 Physical and Chemical Vapor Deposition Physical Vapor Deposition 256 Chemical Vapor Deposition 259 Soft Matter Deposition 262 Self-assembled Monolayers 262 Physisorption of Polymers 266 Polymerization on Surfaces 268 Plasma Polymerization 272 Etching Techniques 274 Lithography 278 Summary 280 Exercises 281

10 10.1 10.1.1 10.1.2 10.1.3 10.1.4 10.1.5 10.1.6 10.1.7 10.1.8 10.2 10.2.1 10.2.2 10.2.3 10.2.4 10.2.5 10.3 10.4 10.5

Friction, Lubrication, and Wear 283 Friction 283 Introduction 283 Amontons’ and Coulomb’s Law 284 Static, Kinetic, and Stick-Slip Friction 286 Rolling Friction 288 Friction and Adhesion 289 Techniques to Measure Friction 290 Macroscopic Friction 292 Microscopic Friction 293 Lubrication 296 Hydrodynamic Lubrication 296 Boundary Lubrication 299 Thin-film Lubrication 300 Superlubricity 301 Lubricants 303 Wear 305 Summary 306 Exercises 307

11 11.1 11.2

Surfactants, Micelles, Emulsions, and Foams 309 Surfactants 309 Spherical Micelles, Cylinders, and Bilayers 314

256

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Contents

11.2.1 11.2.2 11.2.3 11.2.4 11.2.5 11.3 11.3.1 11.3.2 11.3.3 11.3.4 11.3.5 11.4 11.4.1 11.4.2 11.4.3 11.5 11.5.1 11.5.2 11.5.3 11.5.4 11.6 11.7

Critical Micelle Concentration 314 Influence of Temperature 316 Thermodynamics of Micellization 317 Structure of Surfactant Aggregates 319 Biological Membranes 322 Macroemulsions 323 General Properties 323 Formation 326 Stabilization 328 Evolution and Aging 330 Coalescence and Demulsification 332 Microemulsions 333 Size of Droplets 334 Elastic Properties of Surfactant Films 335 Factors Influencing the Structure of Microemulsions Foams 338 Classification, Application, and Formation 338 Structure of Foams 339 Soap Films 341 Evolution of Foams 343 Summary 344 Exercises 345

12 12.1 12.2 12.3 12.3.1 12.3.2 12.3.3 12.3.4 12.3.5 12.4 12.5 12.6

Thin Films on Surfaces of Liquids 347 Introduction 347 Phases of Monomolecular Films 350 Experimental Techniques to Study Monolayers 353 Optical Microscopy 353 Infrared and Sum Frequency Generation Spectroscopy X-Ray Reflection and Diffraction 356 Surface Potential 359 Rheologic Properties of Liquid Surfaces 361 Langmuir–Blodgett Transfer 366 Summary 368 Exercises 369

13

Solutions to Exercises 371 Chapter 2: Liquid Surfaces 371 Chapter 3: Thermodynamics of Surfaces 373 Chapter 4: Charged Interfaces and the Electric Double Layer Chapter 5: Surface Forces 377 Chapter 6: Contact Angle Phenomena and Wetting 380 Chapter 7: Solid Surfaces 382 Chapter 8: Adsorption 384 Chapter 9: Surface Modification 388

336

355

375

Contents

Chapter 10: Friction, Lubrication, and Wear 389 Chapter 11: Surfactants, Micelles, Emulsions, and Foams 390 Chapter 12: Thin Films on Surfaces of Liquids 391 14 14.1 14.1.1 14.1.2 14.1.3 14.1.4 14.2 14.3

Analysis of Diffraction Patterns 395 Diffraction at Three-Dimensional Crystals Bragg Condition 395 Laue Condition 395 Reciprocal Lattice 397 Ewald Construction 398 Diffraction at Surfaces 399 Intensity of Diffraction Peaks 400

Appendix A Symbols and Abbreviations Bibliography 411 Index 441

405

395

xi

xiii

Preface This textbook serves as a general introduction to surface and interface science. It focuses on basic concepts rather than specific details, on understanding rather than on learning facts. The most important techniques and methods are introduced. The book reflects that interfacial science is a diverse and interdisciplinary field of research. Several classic 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 classic “wet” colloid chemistry. We tried to include all aspects because we feel that for a good understanding of interfaces, a comprehensive introduction is helpful. Our book is based on lectures given at the universities of Siegen and Mainz. It addresses advanced students of engineering, chemistry, physics, biology, and related subjects and scientists in academia or industry who are not yet specialists in surface science but desire 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. An advanced level of mathematics is not 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 3 is the only exception; to appreciate that chapter, a basic knowledge of thermodynamics is required. However, you can skip it and still be able to follow most of the rest of the book. If you do decide to skip it, please at least read and try to gain an intuitive grasp of surface excess (Section 3.2) and the Gibbs adsorption equation (Section 3.5.2). A number of problems with solutions are included to enable for self-study. Unless noted otherwise, the temperature was assumed to be 25 ∘ C. At the end of each chapter, the most important equations, facts, and phenomena are summarized. One of the main problems with writing a textbook is trying to limit its content. We tried hard to keep the volume within the scope of one advanced course lasting roughly 15 weeks, two days per week. Facing the growth in knowledge, this meant cutting short or leaving out altogether certain topics. Statistical mechanics, heterogeneous catalysis, and polymers at surfaces are issues that could be expanded. This book no doubt contains errors. Even after several readings by various people, this is unavoidable. If you find any mistakes, please let us know about them by

xiv

Preface

sending an e-mail ([email protected]) so that they can be corrected and do not confuse more readers. We are indebted to several people who helped us in collecting information and preparing and critically reading the manuscript. In particular, we would like to thank K. Amann-Winkel, E. Backus, T. Balgar, C. Bayer, K. Beneke, T. Blake, J. Blum, M. Böhm, E. Bonaccurso, P. Broekmann, R. de Hoogh, A. del Campo, A. de los Santos, M. Deserno, W. Drenckhan, K. Drese, J. Elliott, G. Ertl, R. Förch, S. Geiter, G. Glasser, G. Gompper, M. Grunze, J. Gutmann, L. Heim, M. Hietschold, M. Hillebrand, T. Jenkins, X. Jiang, U. Jonas, J. Krägel, R. Jordan, Krüss GmbH, J. Laven, I. Lieberwirth, G. Liger-Belair, C. Lorenz, M. Lösche, S. Luding, T. Makulik, E. Meyer, R. Miller, A. Müller, P. Müller-Buschbaum, T. Nagel, D. Quéré, J. Rabe, H. Schäfer, P. Schmiedel, J. Schreiber, M. Stamm, M. Steinhart, C. Stubenrauch, G. Subklew, F. Thielmann, J. Tomas, K. Vasilev, J. Venzmer, D. Vollmer, R. von Klitzing, K. Wandelt, B. Wenclawiak, R. Wepf, R. Wiesendanger, J. Wintterlein, G. de With, J. Wölk, D.Y. Yoon, M. Zharnikov, and U. Zimmermann. Hans-Jürgen Butt, Karlheinz Graf, and Michael Kappl Mainz, June 2022

1

1 Introduction An interface is an area that separates two phases from each other. If we consider the solid, liquid, and gas phases, we immediately get three combinations of interfaces: solid–liquid, solid–gas, and liquid–gas interfaces. The term surface is often used synonymously with interface, although interface is preferred to indicate the boundary between two condensed phases and in cases where the two phases are named explicitly. For example, we talk about a solid–gas interface and a solid surface. Surface is used for a condensed phase in contact with a gas or a vacuum. Interfaces can also separate two immiscible liquids such as water and oil. These are called liquid–liquid interfaces. Interfaces may even separate two different phases within one component. In a liquid crystal, for example an ordered phase may coexist with an isotropic phase. Solid–solid interfaces separate two solid phases. They are important for the mechanical behavior of solid materials such as concrete. Gas–gas interfaces do not exist because gases mix. Often, interfaces and colloids are discussed together. Colloid is an abbreviated synonym for colloidal system. Colloidal systems are disperse systems where one phase has dimensions on the order of 1 nm to 1 μm (Figure 1.1). The word colloid comes from the Greek word for glue and was first used in 1861 by Thomas Graham.1 He used the word to refer to materials that seemed to dissolve but were unable to penetrate membranes such as albumin, starch, and dextrin. A colloidal dispersion is a two-phase system that 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 have important applications and have special names (Table 1.1). While there are only five types of interfaces, we can distinguish ten types of disperse systems because we must discriminate between the continuous, dispersing (external) phase and the dispersed (inner) phase. In some cases, this distinction is obvious. Nobody will, for instance, confuse fog with 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.

1 Thomas Graham, 1805–1869; British chemist, professor in Glasgow and London. Physics and Chemistry of Interfaces, Fourth Edition. Hans-Jürgen Butt, Karlheinz Graf, and Michael Kappl. © 2023 WILEY-VCH GmbH. Published 2023 by WILEY-VCH

2

1 Introduction

Continuous phase

Figure 1.1 Schematic of a dispersion.

Dispersed phase

1 nm–1 μm

Table 1.1 Types of dispersions. Continuous phase

Dispersed phase

Term

Example

Gas

Liquid

Aerosol

Clouds, fog, smog, hairspray

Solid

Aerosol

Smoke, dust, pollen

Gas

Foam

Lather, whipped cream, foam on beer

Liquid

Emulsion

Milk

Solid

Suspension

Ink, muddy water, dispersion paint

Gas

Porous solids

Liquid

Solid

Solid foam

Styrofoam, soufflé

Liquid

Solid emulsion

Butter

Solid

Solid suspension

Concrete

Colloids and interfaces are closely related. This is a direct consequence of the enormous specific surface area of colloids. Their interface-to-volume ratio is so large that their behavior is determined mainly by interfacial properties. Gravity and inertia are negligible in the majority of cases. For this reason, colloidal systems are often dominated by interfacial effects rather than bulk properties. For the same reason, interfacial science is essential in nanoscience and technology. Colloidal systems are often influenced by thermal fluctuation and colloidal particles move due to Brownian motion. This distinguishes them from granular matter, a material composed of macroscopic elements. The constituents of granular matter must be large enough so that they are not subject to thermal motion. Usually, the structure of granular matter depends on its history; it is generally far from being in thermodynamic equilibrium, and gravitation can play a significant role. Example 1.1 A granular system that is dominated by surface effects is shown in Figure 1.2. A scanning electron microscope (SEM) image shows aggregates of 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 and collected on the bottom. These aggregates are stable for weeks or months, and even shaking does not change their structure. Thermal fluctuations are

1 Introduction

Figure 1.2 Agglomerate of silicon oxide particles.

20 μm

completely insignificant, and the material would count as granular matter. Still, it is not a typical granular material because once it has formed, gravity and inertia, which rule the macroscopic world, are not able to bend down the particle chains. Surface forces are much stronger. Sometimes a distinction is made between colloidal particles, which are in the size range of 1–1000 nm, and nanoparticles, which are 1–100 nm in diameter. For particles smaller than ≈100 nm, sedimentation is usually negligible. It is useful to introduce the characteristic length scale of a system. The characteristic length scale can often be given intuitively. For example, for a spherical particle, one would use the radius or the diameter. For more complex systems, however, intuition leads to ambiguous results. We suggest using the ratio of the total volume V divided by the total interfacial area A of a system as the characteristic length scale: λc = V∕A. For a sphere of radius Rp , the characteristic length scale is λc = Rp ∕3. For a thin film λc is equal to the thickness. For a dispersion of spherical particles with a volume fraction 𝜙, the total volume of the system is V = N4𝜋R3p ∕(3𝜙), where N is the number of particles. With a total surface area of A = N4𝜋R2p , we get a characteristic length scale of λc = Rp ∕(3𝜙). Why is there an interest in interfaces and colloids? First, to gain a better understanding of natural processes, for example, in biology. The interfacial tension between water and lipids allows for the formation of 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 the nucleation of water around small dust particles is dominated by surface effects. Many foods, such as butter, milk, or mayonnaise, are emulsions; their properties are determined by the liquid–liquid interface. Second, interfaces and colloids have many technological applications. An example is flotation in mineral processing or the bleaching of scrap paper. Washing and detergency are examples of applications encountered in everyday life. Often, the production of new materials such as composite materials involves intensive processes at interfaces. Thin films on surfaces are often dominated by surface effects. Examples include latex films, coatings, and paints. The flow behavior of powders and granular media is

3

4

1 Introduction

determined by surface forces. In tribology, wear is reduced by lubrication, which, again, is a surface phenomenon. Typical of many industrial applications is a very refined and highly developed technology, but only a limited understanding of the underlying fundamental processes of that technology. A better understanding is, however, required to further improve the efficiency or reduce dangers to the environment. Introductory books on interface science are [1–3]. For a deeper understanding, we recommend the series of books by Lyklema [4–7].

5

2 Liquid Surfaces 2.1 Microscopic Picture of a Liquid Surface A liquid surface is not an infinitesimal sharp boundary in the direction of its normal; it has a certain thickness. For example, if we consider the density 𝜌 normal to a surface (Figure 2.1), we can observe that, within a few molecules, the density decreases from that of the bulk liquid to that of its vapor [8]. The density is only one criterion by which one may 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 hydrogen atoms “out” toward the vapor phase. This orientation fades with increasing distance from the surface. At a distance of 0.5-1 nm, the molecules are again randomly oriented like in the bulk. Which thickness do we have to use? This depends on the relevant parameter. If we are interested, for instance, in the density of a water surface, a realistic thickness is on the order of 1 nm. Let us assume that a salt is dissolved in water. Then the concentration of ions might vary over a larger distance (characterized by the Debye length, Section 4.2.2). With respect to the ion concentration, the thickness is thus much larger. When in doubt, it is safer to choose a large value for the thickness. The surface of a liquid is a very turbulent place. Molecules may 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. Example 2.1 To estimate the number of gas molecules hitting a liquid surface per second, we recall the kinetic theory of ideal gases. In textbooks on physical chemistry, the rate of effusion of an ideal gas through a small hole is given by √

PA

(2.1)

2πmkB T

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 the 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 kg mol−1 ∕6.02 × 1023 mol−1 ≈ 3 × 10−26 kg, 107 water molecules per 2 second hit a surface area of 10 Å . In equilibrium, the same number of molecules Physics and Chemistry of Interfaces, Fourth Edition. Hans-Jürgen Butt, Karlheinz Graf, and Michael Kappl. © 2023 WILEY-VCH GmbH. Published 2023 by WILEY-VCH

2 Liquid Surfaces

0.2 0.1 Position (nm)

6

0.0 –0.1 –0.2 0.0

0.2

0.4 Density

0.6

0.8

1.0

(g/cm3)

Figure 2.1 Snapshot of molecular structure of water as obtained by computer simulation and density of water versus coordinate normal to its surface [9]. The density in water vapor at saturation and 25 ∘ C is only 0.02 g/cm3 . Therefore, it is negligible on the scale plotted (kindly provided by D. Horinek).

escapes from the liquid phase. The area covered by one water molecule is approxi2 mately 10 Å . Thus, the average time a water molecule remains on the surface is in 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 with a mobile slider (Figure 2.2). The film is relatively thick, say 1 μm, 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 violates no physical laws, so in principle, it is feasible. If we increase the surface area by moving the slider a distance dx to the right, work must 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. Equation (2.2) is an empirical law and a definition at the same time. The empirical law states that the work is proportional to the change in surface area. This is not only true for infinitesimally small changes in A (which is trivial) but also for significant increases in 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. By definition, we call the proportionality constant surface tension.

2.2 Surface Tension

Figure 2.2 Schematic setup to verify Eq. (2.2) and define the surface tension.

dx

b

Liquid film

Slider Frame

dA = 2bdx

The surface tension can also be defined by the force F 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 dW = −2𝛾b (2.4) F=− dx 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 joule per square meter or newton per meter. Surface tensions of liquids are on the order of 0.02–0.08 N/m (Table 2.1). For convenience, they are usually given in millinewtons per meter (or 10−3 N∕m). Empirically one finds that the surface tension of liquids decreases linearly with temperature. Thus, if we know the surface tension at a given temperature T0 , then we can approximate the surface tension at a temperature T according to 𝜕𝛾 || 𝛾(T) = 𝛾(T0 ) + (T − T0 ) (2.5) 𝜕T ||T=T0 The coefficient 𝜕𝛾∕𝜕T is negative. As we will see in Chapter 3, −𝜕𝛾∕𝜕T is equal to the surface entropy. 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 J∕m2 = 1.44 × 10−3 N That corresponds to a weight of 0.15 g. Example 2.3 Calculate the surface tension of water at 50 ∘ C. With 𝛾 = 0.0720 N∕m and 𝜕𝛾∕𝜕T = −15.6 × 10−5 N/(K m) at 25 ∘ C (Table 2.1), we get 𝛾(50 ∘ C) = 0.0720 N∕m − 15.6 × 10−5 N∕K m ⋅ 25 K = 0.0681 N∕m This is close to the experimental value of 67.9 mN/m. 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 a force is required as well to

7

8

2 Liquid Surfaces

Table 2.1 Surface tensions 𝛾 and 𝜕𝛾∕𝜕T of some liquids at different temperatures T [10]. Substance

T (∘ C)

𝜸 (mN∕m)

𝝏𝜸∕𝝏T (N m−𝟏 ∕K−𝟏 )

−15.6 × 10−5

Water

H2 O

25

71.99

Methanol

CH3 OH

25

22.07

−7.73 × 10−5

Ethanol

C2 H5 OH

25

21.97

−8.33 × 10−5

1-Propanol

C3 H7 OH

25

23.32

−7.75 × 10−5

1-Butanol

C4 H9 OH

25

24.93

−8.98 × 10−5

2-Butanol

CH3 CHOHC2 H5

25

22.54

−7.95 × 10−5

Phenol

C6 H5 OH

50

38.20

Glycerol

C3 H5 (OH)3

25

63.70

Cyclohexane

C6 H12

25

24.65

−11.9 × 10−5

Benzene

C6 H6

25

28.22

−12.9 × 10−5

Toluene

C6 H5 CH3

25

27.93

−11.8 × 10−5

n-Pentane

C5 H12

25

15.49

−11.1 × 10−5

n-Hexane

C6 H14

25

17.89

−10.2 × 10−5

n-Heptane

C7 H16

25

19.65

−9.80 × 10−5

n-Octane

C8 H18

25

21.14

−9.51 × 10−5

n-Nonane

C9 H20

25

22.38

−9.36 × 10−5

n-Decane

C10 H22

25

23.37

−9.20 × 10−5

Acetone

CO(CH3 )2

25

23.46

Formamide

CHONH2

25

57.03

Dichloromethane

CCl2 H2

25

27.20

−12.8 × 10−5

Chloroform

CCl3 H

25

26.67

−12.9 × 10−5

Decaline

C10 H18

25

31.0

−10.3 × 10−5

25

19.0–20.4

−3.65 × 10−5

PDMS

−10.7 × 10−5 −5.98 × 10−5

−11.2 × 10−5 −8.44 × 10−5

Hexamethyldisiloxane

C6 H18 OSi2

25

15.70

−8.77 × 10−5

Octamethylcyclotetrasiloxane

C8 H24 O4 Si4

25

17.61

−6.60 × 10−5

Perfluorohexane

C6 F14

25

12.03

Perfluoroheptane

C7 F16

25

12.70

−9.51 × 10−5

Perfluorooctane

C8 F18

25

13.83

−8.94 × 10−5

25

45.9

−7.83 × 10−5

25

40.6

−5.72 × 10−5

+

bmin PF−6 bmin+ BF−4

−10.5 × 10−5

Sodium chloride

NaCl

801

192

−7.2 × 10−5

Potassium chloride

KCl

771

176

−7.4 × 10−5

Calcium chloride

CaCl2

775

196

−4.5 × 10−5

Argon

Ar

−186

11.90

−25.1 × 10−5

Nitrogen

N2

−196

8.85

−22.5 × 10−5

Mercury

Hg

25

485.48

−20.5 × 10−5

Gallium

Ga

29.8

715.3

−9.0 × 10−5

Silver

Ag

961

966

Gold

Au

1064

1120

−24.5 × 10−5 −14 × 10−5

Notes: PDMS stands for poly(dimethylsiloxane) or Si(CH3 )3 (OSi(CH3 )2 )n OSi(CH3 )3 . The surface tension of polymer melts increases slightly with the number of monomers and saturates for long chains [11]. 1-C4 H9 -3-methylimidazolium+ (bmin) with PF−6 or BF−4 are ionic

2.2 Surface Tension

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

Vapor

Liquid

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 and the surface tension remains constant, the stretching of a rubber membrane is usually elastic, and the tension increases with increasing surface area. 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 5). 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 attractive interactions between molecules. At the surface, molecules are only partially surrounded by other molecules, and the number of adjacent molecules is smaller than in the bulk (Figure 2.3). This is energetically unfavorable. To bring a molecule from the bulk to the surface, work must be carried out. 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. With this interpretation of surface tension in mind, we immediately realize that 𝛾 must be positive. Otherwise, the Gibbs free energy of interaction between molecules would be repulsive, and all molecules would immediately evaporate into the gas phase. Example 2.4 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 , and 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 its six 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 miss an energy of 5.08 kJ/mol.

9

10

2 Liquid Surfaces

To estimate the surface tension, we need to know the surface area occupied by one molecule. If molecules form a cubic structure, then the volume of one unit cell is a3M , where aM is the distance between nearest neighbors. This distance can be calculated from the density: 0.084 16 kg∕mol M a3M = = = 1.81 × 10−28 m3 ⇒ 𝜌NA 773 kg∕m3 ⋅ 6.02 × 1023 mol−1 aM = 0.565 nm The surface area per molecule is a2M . For the surface energy, we estimate 𝛾=

Δvap U 6NA a2M

=

5080 J∕mol−1 23

6.02 × 10

mol−1 ⋅ (0.565 × 10−9 m)2

= 0.0264 J∕m2

For such a rough estimate, the result is surprisingly close to the experimental value of 0.0247 J/m2 . The concept of surface tension can be generalized to liquid–liquid interfaces. For example, n-octane and water form two phases that are separated by an interface. The associated interfacial tension is 51.2 N/m at 25 ∘ C. This is higher than the surface tension of octane. Example 2.5 The range of interfacial tensions is large. At room temperature, mercury has the highest surface tension with 485 mN/m. Above 29.8 ∘ C, gallium melts and has an even higher surface tension of 708 mN/m. At even higher temperatures, other molten metals have higher surface tensions. Examples are nickel, with 1780 mN/m (at a melting temperature Tm of 1455 ∘ C), and iron, with 1940 mN/m (Tm = 1538 ∘ C) [12]. Very low interfacial tensions are measured between the different phases of liquid crystals. For example, at high temperature, ′ 4-octyl-4 -cyanobiphenyl (8CB, H17 C8 (C6 H4 )2 CN) forms an isotropic liquid. When it is slowly cooled to 40.5 ∘ C, the material becomes a nematic liquid crystal, where the molecules align and show a preferred orientation. The tension of the interface between the isotropic and the nematic phase is only 9.5 μN/m [13]. Very low interfacial tensions were measured between phase-separated polymer mixtures in solutions. For example, the interfacial tension between aqueous gelatine and dextrane solutions can be as low as 0.5 μN/m [14].

2.3 Equation of Young and Laplace 2.3.1

Curved Liquid Surfaces

If in equilibrium a liquid surface is curved, then 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. 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

2.3 Equation of Young and Laplace

Pa

Pi

Pa = Pi

Pa < Pi

Pa > Pi

Figure 2.4 Rubber membrane at the end of a cylindrical tube to illustrate the Laplace pressure. An inner pressure Pi that is different than the outside pressure Pa can be applied.

the membrane is under some tension) (Figure 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. The Young1 –Laplace2 equation, also simply called Laplace equation, relates the pressure difference between the two phases ΔP and the curvature of the surface [15, 16]: ) ( 1 1 ΔP = 𝛾 + (2.6) R1 R2 Here, R1 and R2 are the two principal radii of curvature. ΔP is called the Laplace or capillary pressure. Equation (2.6) is also referred to as Laplace equation. It is valid in the absence of gravitation. 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. One angle of orientation of this plane is not defined and can be chosen conveniently. 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 in the intersection at the point of interest. The second radius of curvature is obtained by passing a second plane through the surface; this second plane also contains the normal but is 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 is invariant and does not depend on the orientation as long as the radii are determined in perpendicular directions. 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 (Figure 2.5). 1 Thomas Young, 1773–1829; English physician and physicist, professor at Cambridge. 2 Pierre-Simon Laplace, Marquis de Laplace, 1749–1827; French natural scientist.

11

12

2 Liquid Surfaces

R1 = radius of cylinder

R1 = radius of sphere

R2 = ∞

R2 = R1

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

Example 2.6 How large is the pressure in a spherical bubble with a diameter of 2 mm and a bubble 20 nm in diameter in pure water compared with the pressure outside? The curvature of a bubble is identical to that of a sphere: R1 = R2 = R. Therefore, 2𝛾 R With R = 1 mm, we get

(2.7)

ΔP =

ΔP = 0.072 J∕m2 ⋅

2 10

−3

m

= 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, then 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 the Young–Laplace equation tells us that in this case, the surface of the liquid has the same curvature everywhere. With the help of the Young–Laplace equation (2.6), it is possible to calculate the equilibrium shape of a liquid surface [17, 18]. If we know the pressure difference and some boundary conditions (such as the volume of the liquid and its contact line), then we can calculate the geometry of the liquid surface. For axially symmetric situations, interfaces of constant curvature are spheres, cylinders, nodoids, or unduloids.

In practice, it is usually not trivial to calculate the geometry of a liquid surface with Eq. (2.6). 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-coordinates. 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 usually not an easy task.

2.3 Equation of Young and Laplace

In many cases, we deal with rotational symmetric structures. Assuming that the axis of symmetry is identical to the z-axis, z = z(r) describes a liquid surface, where r is the radial coordinate. Let us assume the z-axis is vertical and in the plane of the paper. Then it is convenient to put one radius of curvature also in the plane of the paper. This radius is given by [19] 1 z′ = √ (2.8) R2 r 1 + z′2 The other principal radius of curvature is in a plane perpendicular to the plane of the paper and oriented parallel to R2 . It is given by 1 z′′ = √ R1 (1 + z′2 )3

(2.9)

where z′ and z′′ are the first and second derivatives with respect to r, respectively. Mathematically, surfaces of constant curvature are spheres, cylinders, or parts of nodoids or unduloids [18].

2.3.2

Derivation of Young–Laplace Equation

To derive the equation of Young and Laplace, we consider a small part of a liquid surface. This part should be so small that the curvature does not change significantly. First, we pick a point X and draw a line around it that is characterized by the fact that all points on that line are the same distance d away from X (Figure 2.6). This line is the cross section that the studied surface will have with a sphere with its center in X and with a radius d. If the liquid surface is planar, the line 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 a line of length dl. The surface tension pulls with a force 𝛾dl. The vertical force on that segment is 𝛾 dl sin α. For small surface areas (d ≪ R1 , R2 ) and small α, we have sin α ≈ d∕R1 , where R1 is the radius of curvature along AXB. The vertical force component is d 𝛾 dl R1 Figure 2.6 Diagram used to derive the Young–Laplace equation.

X D

d

B

A C R2

γdl

R2

R1

R1

α

13

14

2 Liquid Surfaces

The sum of the four vertical components of the forces related to the small line segments dl at points A, B, C, and D is ( ) ( ) 2d 2d 1 1 𝛾 dl + = 2d𝛾 dl + (2.10) R1 R2 R1 R2 The value of 1∕R1 + 1∕R2 is independent of the particular orientation. This means that, although it was developed for the ABCD orientation, Eq. (2.10) is valid for any orientation of the cross-sectional planes as long as they are orthogonal to the surface at X and mutually orthogonal. Integration over the borderline (only 90∘ rotation of the four segments) gives the total vertical force, caused by the surface tension: ( ) 1 1 2 πd 𝛾 + (2.11) R1 R2 In equilibrium, this downward force must be compensated by an equal force in the opposite direction. This upward force is caused by an increased pressure difference ΔP on the concave side of πd2 ΔP. Equating both forces leads to ( ) ( ) 1 1 1 1 2 2 ΔPπd = πd 𝛾 + ⇒ ΔP = 𝛾 + (2.12) R1 R2 R1 R2 These considerations are valid for any small part of the liquid surface. Since the part is arbitrary, the Young–Laplace equation must be valid everywhere.

2.3.3

Applying the Young–Laplace Equation

When applying the equation of Young and Laplace to simple geometries, it is usually obvious at which side the pressure is higher. For example, both inside a bubble and inside a drop, the pressure is higher than outside (Figure 2.7). In other cases, this is not so obvious because the curvatures can have an opposite sign. One example is a drop hanging between the planar ends of two cylinders (Figure 2.7b). Then the two principal curvatures, defined by 1 1 C1 = and C2 = (2.13) R1 R2 can have a different sign. We count it positive if the interface is curved toward the liquid. The pressure difference is defined as ΔP = Pliquid − Pgas . Figure 2.7 (a) A gas bubble in liquid and a drop in a gaseous environment. (b) A liquid meniscus with radii of curvature of opposite sign between two solid cylinders.

Liquid Gas R1 Gas Liquid

(a)

(b)

R2

2.4 Techniques to Measure Surface Tension

Example 2.7 For a drop in a gaseous environment, the two principal curvatures are positive and given by C1 = C2 = 1∕R. The pressure difference is positive, which implies that the 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 is lower than inside the bubble. For a drop hanging between the ends of two cylinders (Figure 2.7b) in a gaseous environment, one curvature is conveniently chosen to be C1 = 1∕R1 ; the other curvature is negative, C2 = −1∕R2 . The pressure difference depends on the specific values of R1 and R2 . Neglecting gravitation, the shape of a liquid surface is determined by the Young– Laplace equation (2.6). In large structures, we must consider also the hydrostatic pressure due to gravitation. Then the equation of Young and Laplace becomes ) ( 1 1 ΔP = 𝛾 + + 𝜌gh (2.14) R1 R2 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. As a measure for the characteristic length scale, we define the capillary constant √ 𝛾 𝜅= (2.15) g𝜌 also called capillary length. For liquid structures, whose curvature is much smaller than the capillary constant, the influence of gravitation can be neglected. At 25 ∘ C, the capillary length is 2.71 mm for water and 1.67 mm for hexane. The capillary constant has a fundamental significance. When dividing the Laplace equation (2.14) by g𝜌, we see that the only independent parameter determining the shape of a liquid surface under gravity is 𝛾∕g𝜌. As we shall see in Sections 2.4 and 6.2.3, in most measurements of surface tension, it is in fact the capillary length that is determined. The surface tension is then calculated from the known density.

2.4 Techniques to Measure Surface Tension Before we discuss capillary phenomena and experimental techniques used to measure the surface tension, we need to introduce the so-called contact angle Θ. When we put a drop of liquid on a solid surface, the edge usually forms a defined angle, which depends only on the material properties of the liquid and the solid (Figure 2.8). This is the contact angle. Here we only need to know what it is. Contact angle phenomena are discussed in more detail in Chapter 6. For a wetting surface, we have Θ = 0∘ . Several techniques are used to measure the surface tension of liquids [1]. A common technique is to measure optically the contour of a sessile drop resting on a table. Its shape will depend on the size. A tiny droplet forms a spherical cap (Figure 2.9a).

15

16

2 Liquid Surfaces

Liquid

Three-phase contact line

Figure 2.8 Rim of a liquid drop on a planar solid surface with its contact angle Θ.

Solid Θ

2 mm

2 mm

(a)

(b)

3 mm

(c)

Figure 2.9 Shape of a drop of water resting on a flat surface. The size of the amount of water increases from 0.5 μl (a) over 34 μl (b) to 1 ml (c).

For a given volume and contact angle, a spherical cap is the shape with the least surface area. It is independent of the surface tension of the specific liquid. When the drop size increases, hydrostatic pressure comes into play (Figure 2.9b). The width of the drop grows more than its height, and the shape is determined by an interplay of gravity and surface tension. Assuming that the drop shape is rotationally symmetric around a central vertical axis, the contour is fitted with the Laplace equation (2.14). If the density of the liquid is known, then the surface tension can be calculated [20]. When further increasing the amount of liquid, the drop increases in height and width, its shape flattens, and eventually a flat top is formed (Figure 2.9c). The height of the resulting liquid film [21] Θ (2.16) 2 is again given by an interplay between gravity, which tends to flatten the film and surface tension [22]. Thus, from a measurement of the height of a liquid puddle one can determine the surface tension of the liquid, provided its density is known. The sessile drop method is not the only technique in which the drop shape is analyzed to measure the surface tension. Other geometries are h = 2𝜅 sin

●

pendant drops, in which the shape of a hanging drop with its familiar teardrop shape hanging down from a solid (typically the end of a needle) is analyzed [23];

2.4 Techniques to Measure Surface Tension ●

●

pendant bubbles, in which the shape of a bubble at the end of a needle in the liquid is analyzed; and sessile bubbles, in which a bubble is floated against the top of a container [24].

For all drop (and bubble) shape methods, the shape is fitted with a solution of the Young–Laplace equation. An implicit assumption is that the drop is not in motion, and viscosity and inertia do not play a role. Surface tension and gravity are the only forces shaping the drop. Using a bubble ensures that the vapor pressure will be 100%, which is a requirement for conducting experiments in thermodynamic equilibrium. Often, problems caused by contamination are reduced. In the maximum bubble pressure method, the surface tension is determined from the value of the pressure necessary to push a bubble out of a capillary against the Laplace pressure [25, 26]. Therefore, a capillary tube, with inner radius rc , is immersed into the liquid (Figure 2.10). A gas is pressed through the tube so that a bubble forms at its end. If the pressure in the bubble increases, then the bubble is pushed out of the capillary more and more. In that way, the curvature of the gas–liquid interface increases according to the Young–Laplace equation. The maximum pressure is reached when the bubble forms a half-sphere with a radius Rb = rc . This maximum pressure is related to the surface tension by 𝛾 = rc ΔP∕2. If the volume of the bubble is further increased, then the radius of the bubble must also increase. A larger radius corresponds to a smaller pressure. The bubble would thus become unstable and detach from the capillary tube. An advantage of the maximum bubble pressure method is that the vapor inside the bubble will most likely be saturated and evaporation effects can be neglected [27]. Before computers became widely available, two other methods were quite popular: the capillary rise method and the drop-weight method. Capillary rise is described in detail in Chapter 6. In the drop-weight method (for a review see [28]), the liquid is allowed to flow out from the bottom of a capillary tube. Drops are formed that detach when they reach a critical size. The weight of a drop falling out of a capillary is measured. To obtain a precise measure, this is done for a number 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 than balanced by the surface tension. A drop falls off when the gravitational force mg, determined by the mass m of the drop, is no longer balanced by the surface tension. The surface tensional force is equal to the surface tension multiplied by the circumference. This leads to mg = 2πrc 𝛾

(2.17)

Thus, the mass is determined by the radius of the capillary. We must distinguish between the inner and outer diameters of the capillary. If the material that the capillary is made of is not wetted by the liquid, then the inner diameter enters into Eq. (2.17). If the surface of the capillary tube is wetted by the liquid, then the external radius of the capillary must be taken. For completely nonwetting surfaces (where the contact angle higher than 90∘ ), the inner radius determines the drop weight [16]. Experimentally, Tate already observed in 1864 that “other things being the same, the weight of a drop of a liquid is proportional to the diameter of the tube in which it is formed” [29].

17

18

2 Liquid Surfaces

ΔP

rc

Not wetting

Rb

Wetting (a)

(b)

Figure 2.10 Maximum bubble pressure (a) and drop-weight method (b) to measure the surface tension of liquids.

0 ms

123 ms

205 ms

266 ms

319 ms

327 ms

330 ms

346 ms

Figure 2.11 Release of a water drop from a capillary of 2 mm inner diameter imaged with a high-speed camera.

In practice, the equation is only approximately valid, and a weight less than the ideal value is measured [30, 31]. The reason becomes evident when the process of drop formation is observed closely (Figure 2.11): once the drop grows larger than a critical size, the drop rapidly elongates, thins in the middle, and a large portion falls off the capillary. A small portion of the drop remains hanging on the capillary. In addition, often satellite drops are formed. A correction factor f is introduced, and Eq. (2.17) becomes mg = 2πf rc 𝛾 (Exercise 2.6). A common device used to measure 𝛾 is the ring tensiometer, also called the du Noüy3 tensiometer [32]. In a ring tensiometer, the force necessary to detach a ring from the surface of a liquid is measured (Figure 2.12). The force required for the detachment is F = 2π(ri + ra )𝛾

(2.18)

A necessary condition is that the ring surface must be completely wetted. A platinum wire is often used; it can be annealed for cleaning before the measurement. Even in early measurements, it turned out that Eq. (2.18) was generally in serious error and that an empirical correction function is required [33, 34]. 3 Pierre Lecomte du Noüy, 1883–1947; French scientist, worked in New York and Paris.

2.4 Techniques to Measure Surface Tension

ra

2γl

ri l Du-Noüy ring tensiometer

Wilhelmy plate

Figure 2.12 Du Noüy ring tensiometer and Wilhelmy plate method.

A widely used technique is the Wilhelmy4 plate method (Figure 2.12). A thin plate of glass, platinum, or filter paper is vertically placed halfway into the liquid. In fact, the specific material is not important, as long as it is wetted by the liquid. Close to the three-phase contact line, the liquid surface is oriented almost vertically (provided the contact angle is 0∘ ). Thus, the surface tension can exert a downward force. One measures the force required to prevent the plate from being drawn into the liquid. After subtracting the gravitational force, this force is 2l𝛾, where l is the width of the plate; the plate is assumed to be so thin that rim effects can be neglected. In honor of Ludwig Wilhelmy, who studied the force on a plate in detail, the method was named after him [35]. The Wilhelmy plate method is simple, and no correction factors are required. Care must be taken to keep the plates clean, prevent contamination in air, and avoid evaporation of the liquid at the three-phase contact line [27]. Finally, dynamic methods can be used to measure the surface tension. Lord Rayleigh5 pushed a liquid jet out of a nozzle having an elliptic cross section [36] (Figure 2.13). The relaxation of the jet to a circular cross section depends on the surface tension of the liquid. An advantage of this method is that we can measure changes in the surface tension, which might be caused by the diffusion of amphiphilic substances to the surface. Another method is to observe oscillations of drops, either levitated or in free space [37]. From the angular resonance frequency of the first vibration mode 𝜔, the surface tension can be calculated according to √ 8𝛾 𝜔= 𝜌rd3 Figure 2.13 Jet of water ejected from an elliptical orifice.

4 Ludwig Ferdinand Wilhelmy, 1812–1864; German physicochemist. 5 John William Strutt, Lord Rayleigh, 1842–1919; English physicist, professor at Cambridge; Nobel Prize in Physics, 1904.

19

20

2 Liquid Surfaces

assuming the viscosity of the liquid is low [36]. One only needs to know the density of the liquid 𝜌 and the drop radius rd . The oscillating-drop method is often used to measure the surface tension of materials with high melting temperatures, for example metals [12, 38].

2.5 Kelvin Equation In this chapter, we get to know the second essential equation of surface science – the Kelvin6 equation. Like the Young–Laplace equation, it is based on thermodynamic principles and does not refer to a specific material or special conditions [39]. The subject of the Kelvin equation is the vapor pressure of a liquid. Tables of vapor pressures for various liquids and different temperatures can be found in common textbooks or handbooks of physical chemistry. These vapor pressures are reported for vapors that are in thermodynamic equilibrium with liquids having planar surfaces. When the liquid surface is curved, the vapor pressure changes. The vapor pressure of a drop is higher than that of a flat, planar surface. In a bubble, the vapor pressure is reduced. The Kelvin equation tells us how the vapor pressure depends on the curvature of the liquid surface. The cause for this change in vapor pressure is the Laplace pressure. The raised Laplace pressure in a drop causes the molecules to evaporate more easily. In the liquid that surrounds a bubble, the pressure with respect to the inner part of the bubble is reduced. This makes it more difficult for molecules to evaporate. Quantitatively, the change in vapor pressure for curved liquid surfaces is described by the Kelvin equation: ( ) PK 1 1 + (2.19) RT ln 0 = 𝛾Vm P0 R1 R2 Here, P0K is the vapor pressure of the curved surface and P0 that of the flat surface. The index “0” indicates that everything is only valid in thermodynamic equilibrium. Please keep in mind that in equilibrium, the 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 to RT ln

P0K P0

=

2𝛾Vm r

or P0K = P0 e

2𝛾Vm RTr

(2.20)

The constant 2𝛾Vm ∕(RT) characterizes the curvature for which the vapor pressure changes by a factor e. For convenience, we call 𝛾Vm λK = (2.21) RT the Kelvin length. Example 2.8 Calculate the Kelvin lengths for water, ethanol, and 1,2-propanediol ( ) C3 H8 O2 , γ = 35, 6 mN∕m at 25 ∘ C. The molar masses of the three substances are M = 18.02, 46.07, 76.09 g/mol, and the densities are 𝜌 = 997, 789, and 1036 kg/m3 , respectively. 6 William Thomson, later Lord Kelvin, 1824–1907; physics professor at the University of Glasgow.

2.5 Kelvin Equation

For water, we get 0.072 N∕m ⋅ 0.018 kg∕mol λK = = 0.52 nm 997 kg∕m3 ⋅ 2479 J∕mol For ethanol and 1,2-propanediol, the Kelvin lengths are 0.52 and 1.05 nm. To derive the Kelvin equation, we consider the Gibbs energy of the liquid. The molar Gibbs energy changes when the surface is being curved because the pressure increases due to the Laplace pressure. In general, any change in the Gibbs energy is given by the fundamental equation dG = VdP − SdT. Here, the influence of external forces such as gravitation, electrostatic, or magnetic forces is not considered. The increase in the Gibbs energy per mole of liquid, upon curving, at constant temperature is ( ) ΔP 1 1 ΔGm = Vm dP = 𝛾Vm + (2.22) ∫0 R1 R2 We have assumed that the molar volume remains constant, which is certainly a reasonable assumption 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 energy of the vapor depends on the vapor pressure P0 according to Gm = G0m + RT ln P0

(2.23)

assuming that the vapor behaves like an ideal gas. For a liquid with a curved surface, we have GKm = G0m + RT ln P0K

(2.24)

The change in the molar Gibbs energy inside the vapor due to the curving of the interface is therefore PK ΔGm = GKm − Gm = RT ln 0 (2.25) P0 Since the liquid and vapor are supposed to be in equilibrium, the two expressions must be equal. This immediately leads to the Kelvin equation. When applying the Kelvin equation, it is instructive to distinguish two cases: a drop in its vapor (or, more generally, a positively curved liquid surface) and a bubble in liquid (a negatively curved liquid surface). Drop in vapor. The vapor pressure of a drop is higher than that of a liquid with a planar surface. One consequence is that an aerosol of drops (fog) should be unstable. To see this, let us assume that we have a box filled with many drops in its vapor. Some drops are larger than others. The small drops have a higher vapor pressure than the large drops. Hence, more liquid evaporates from their surface. The resulting vapor tends to condense into large drops. Within a population of drops of different sizes, the bigger drops will grow at the expense of the smaller ones – a process called Ostwald ripening.7 These drops will sediment down, and in the end, bulk liquid fills the bottom of the box. 7 In general, Ostwald ripening is the growth of large objects at the expense of smaller ones. Friedrich Wilhelm Ostwald, 1853–1932; German physicochemist, professor in Leipzig; Nobel Prize in Chemistry, 1909.

21

22

2 Liquid Surfaces

For a given vapor pressure, there exists a critical drop size. Every drop larger than this size will grow; smaller drops will evaporate. If a vapor is cooled to reach oversaturation, it cannot condense (because every drop would instantly evaporate again), unless nucleation sites are present. Thus, it is possible to explain the existence of oversaturated vapors in this way. Bubble in a liquid. From Eq. (2.22), we see that a negative sign must be used for a bubble because of the negative curvature of the liquid surface. As a result, we get RT ln

P0K P0

=−

2𝛾Vm r

(2.26)

Here, r is the radius of the bubble. The vapor pressure inside a bubble is therefore reduced. This explains why it is possible to overheat liquids: when the temperature increases above the boiling point (at a given external pressure) occasionally, tiny bubbles are formed. Inside the bubble, the vapor pressure is reduced, the vapor condenses, and the bubble collapses. Only if a bubble forms that is larger than a certain critical size it is more likely to increase in size than to collapse. As an example, vapor pressures for water drops and bubbles in water are given in Table 2.2. At this point, it may be instructive to clarify the following question: When do we use the term vapor instead of gas? Vapor is used when a liquid is present in the system and liquid evaporation and vapor condensation take place. This distinction is not always clear cut because when dealing with adsorption (Chapter 8), we do take the two corresponding processes – adsorption and desorption – into account but still talk about gas. How does the presence of an additional background gas change the properties of a vapor? For example, does pure water vapor behave differently from water vapor at the same partial pressure in air (in the presence of nitrogen and oxygen)? Answer: to a first approximation, there is no difference as long as phenomena in thermodynamic equilibrium are concerned. First approximation means as long as interactions between the vapor molecules and the molecules of the background gas are negligible. However, time-dependent processes and kinetic phenomena such as condensation and evaporation can be completely different and depend on the background gas. This is, for instance, why drying in a vacuum is much faster than drying in air. Table 2.2 Relative equilibrium vapor pressure of a curved water surface at 25 ∘ C for spherical drops and bubbles of radius r. r(nm)

PK𝟎 ∕P𝟎 drop

PK𝟎 ∕P𝟎 bubble

1000

1.001

0.999

100

1.011

0.989

10

1.114

0.898

1

2.950

0.339

2.6 Capillary Condensation

2.6 Capillary Condensation An important application of the Kelvin equation is the description of capillary condensation. This is the condensation of vapor into capillaries or fine pores even for vapor pressures below P0 ; P0 is the equilibrium vapor pressure of the liquid with a planar surface [40, 41]. Lord Kelvin was the first to realize that the vapor pressure of a liquid depends on the curvature of its surface. In his words, this explains why “moisture is retained by vegetable substances, such as cotton cloth or oatmeal, or wheat-flour biscuits, at temperatures far above the dew point of the surrounding atmosphere” [42]. Capillary condensation will also be discussed in Section 8.4.3. Capillary condensation can be illustrated by the model of a conical pore with a totally wetted surface (Figure 2.14a). At a given vapor pressure P0 , liquid will condense in the tip of the pore. Condensation continues until the radius of curvature of the liquid has reached the value given by the Kelvin equation. Since the liquid surface is that of a spherical cap, the situation is analogous to that of a bubble, and we can apply Eq. (2.26). The vapor pressure of the liquid inside the pore decreases to P0K , with r being the radius of the pore at the point where the meniscus is in equilibrium. The curvature of the liquid surface is −2∕r. Many surfaces are not totally wetted, but they form a certain contact angle Θ with the liquid. In this case, the radius of curvature increases. It is no longer equal to the radius of the pore, but to r∕ cos Θ (Figure 2.14b). Example 2.9 We consider a porous solid with pores of all dimensions at a humidity of 90% and at 20 ∘ C. What size pores fill up with water if the surfaces are fully wetted? 2𝛾Vm 2 ⋅ 0.072 J∕m2 ⋅ 18 × 10−6 m3 rc = − =− = 10 nm RT ln 0.9 8.31 J∕K ⋅ 293 K ⋅ ln 0.9 Attention must be paid as to which radius is inserted into the Kelvin equation. For a cylindrical pore of radius rc , the radius of curvature is 2∕rc . For a nonrotational symmetric geometry, 2∕rc must be substituted by 1∕R1 + 1∕R2 . For example, in a

r >0

=0

=0

r

2r

Condensed liquid (a)

(b)

(c)

Figure 2.14 Capillary condensation into a narrow conical pore and a slit pore at a given vapor pressure P0 . For the conical pore, the perfectly wetted case (a, c) and the case of finite contact angle (b) are shown.

23

24

2 Liquid Surfaces

fissure or crack (Figure 2.14c), one radius of curvature is infinitely large. Instead of 2∕r, there should be 1∕r in the equation, with r being the radius of curvature vertical to the crack direction. Experimentally, it has been found that for gap sizes below 2–4 nm, adsorption effects and the molecular nature of the liquids need to be considered [43–45]. For a more detailed discussion of adsorption, see Section 8.4.3. Taking such effects into account, very good agreement has been achieved between theory and experiments [46, 47]. An important consequence of capillary condensation is the existence of capillary forces, also called meniscus forces (review [48]). Capillary condensation often strengthens the adhesion between fine particles and, in many cases, determines the flow behavior of powders [49]. When two hydrophilic particles come into contact, liquid (usually water) will condense into the gap of the contact zone and form a meniscus. The meniscus is curved in such a way that the Laplace pressure in the liquid is reduced as compared to the outside pressure. We consider it negative. This negative Laplace pressure leads to an attraction between the particles. In addition, the direct action of the surface tension of the liquid around the periphery of the meniscus pulls the particles together [50–52]. Calculating the capillary force is usually not an easy task. The main difficulty is in deriving the shape of the liquid meniscus for a given vapor pressure and contact geometry. To demonstrate how in principle capillary forces are calculated, we consider the contact between two spherical particles of identical radius Rp (Figure 2.15). To calculate the capillary force between two macroscopic, perfectly smooth spheres, we assume that the liquid perfectly wets the surface of the particles. This is the case for many minerals with water. We further assume that the meniscus is small and that we can neglect gravitation. Practically, the menisci considered should be much smaller than the capillary constant. The capillary force is given by the pressure term plus the direct action of the surface tension: F = πx2 ΔP + 2πx𝛾

(2.27)

r

x Rp

Figure 2.15 Two spherical particles with liquid meniscus.

2.6 Capillary Condensation

To obtain ΔP, we need to know the curvature of the liquid surface. The total radius of curvature 1∕R1 + 1∕R2 is 1 1 1 − ≈− (2.28) x r r In most practical cases, we can safely assume that x ≫ r. The pressure is therefore ΔP = 𝛾∕r lower in the liquid than in the outer vapor phase. It acts upon a cross-sectional area πx2 leading to an attractive force of πx2 ΔP. We use Pythagoras’ theorem to express x2 by r: (Rp + r)2 = (x + r)2 + R2p ⇒ R2p + 2rRp + r 2 = x2 + 2xr + r 2 + R2p ⇒ 2rRp = x2 + 2xr ≈ x2

(2.29)

For the last approximation, we assumed that x ≫ 2r. It follows x2 = 2rRp . As a result, the attractive capillary force is F = 2π𝛾Rp + 2πx𝛾

(2.30)

For macroscopic spheres and not too high vapor pressure, Rp ≫ x. The second term can be neglected, and we obtain [18, 53, 54] F = 2π𝛾Rp

(2.31)

The force depends only on the radius of the particles and the surface tension of the liquid. It does not depend on the actual radius of curvature of the liquid surface or on the vapor pressure. This at-first-sight-surprising result is due to the fact that, with decreasing vapor pressure, the radius of curvature, and therefore also x, decreases. At the same time, the Laplace pressure increases as 1∕r increases by the same amount. When the two particles are not in contact but separated by a certain distance, the capillary attraction decreases [53, 55, 56]. Example 2.10 A quartz sphere hangs on a second similar sphere. Some water vapor is in the room, which leads to a capillary force. Small particles are held by the capillary force, and large particles fall due to the dominating gravitational force. Beyond which particle radius is gravity strong enough to separate the two spheres? For silicon oxide, we assume a density of 𝜌 = 3000 kg∕m3 . The weight of the sphere is 4 3 πR 𝜌g = 1.23 × 105 kg∕m2 s2 ⋅ R3p 3 p The capillary force is 2π ⋅ 0.072 J∕m2 ⋅ Rp = 0.45 kg∕s2 ⋅ Rp Both are equal for √ Rp =

0.45 m2 = 1.9 mm 1.23 × 105

In reality, the capillary force is often weaker than the calculated value. As it turns out, perfect macroscopic spheres are a very special case. In general, the capillary force depends critically on the relative vapor pressure [48]. One reason is, that the

25

26

2 Liquid Surfaces

Low P

Intermediate P

High P

Capillary bridge Rp

Figure 2.16 Schematic of two particles in contact at different vapor pressures P of a condensing liquid. Macroscopically, both particles are assumed to be spherical and described by the apparent radius Rp . On the nanometer scale, they are rough.

particle surfaces are usually rough and touch only at some points, as illustrated in Figure 2.16 [49, 57]. At low vapor pressure, capillary condensation takes place only at these points. With increasing roughness, the menisci grow and possibly new menisci form. Only at a relative vapor pressure close to one do the menisci merge and form one liquid bridge, which is determined by the macroscopic shape of the particles. The relevant length scale is the Kelvin length. Thus, roughness even on the 1 nm scale is relevant [58]. For this reason, it is difficult to predict capillary forces. Is it possible that the capillary pressure leads to an absolute pressure below zero? Typically, a pressure is associated with molecules bombarding a surface and transferring momentum. The lowest pressure in that picture occurs in the absence of molecules (vacuum). Then no force is applied to the surface since no momentum is transferred. However, a capillary pressure can indeed be negative because it originates from the cohesion of liquid molecules (Example 6.6).

2.7

Nucleation Theory

The change in vapor pressure with the curvature of a liquid surface has important consequences for condensation. The formation of a new phase in the absence of an external surface is called homogeneous nucleation. In homogeneous nucleation, first small clusters of molecules are formed. These clusters grow due to the condensation of other molecules. In addition, they aggregate to form larger clusters. Finally, macroscopic drops form. Usually, this happens only if the vapor pressure is significantly above the saturation vapor pressure. Also, the formation of bubbles in a liquid above the boiling point is a nucleation process. In most practical situations, we encounter heterogeneous nucleation, where a vapor condenses onto a surface such as a dust particle. The formation of clouds

2.7 Nucleation Theory

in the atmosphere, for example, is usually caused by water condensing onto dust particles. A well-known example of heterogeneous nucleation is the formation of bubbles when pouring sparkling water (or, if you prefer, beer) into a glass. Bubbles nucleate at the glass surface, grow in size, and eventually rise. Here we only discuss homogeneous nucleation of liquid droplets from an oversaturated vapor. Though homogeneous nucleation is less common, the mathematical treatment and concepts developed are important and used for other applications as well. The classical theory of homogeneous nucleation was developed between 1920 and 1940 [59–61]. Let us consider a vapor at a constant pressure P and constant temperature. Please note that in this chapter, P is not the total pressure. The total pressure might be higher than the vapor pressure due to the presence of other gases. Since the actual vapor pressure is supposed to be higher than the saturation vapor pressure, P ≥ P0 , the vapor will eventually condense. A liquid phase is formed because attractive forces between the molecules overcome thermal fluctuations. Since the process is spontaneous, the Gibbs free energy of condensation must be negative. Typically, it is the inverse process, vaporization, that is tabulated. The molar Gibbs energy of vaporization, Δv Gm , can be calculated from Δv Gm = RT ln

P P0

(2.32)

Δv Gm depends on the vapor pressure and on the temperature. If the actual vapor pressure is equal to the saturation vapor pressure, P = P0 , then the molar Gibbs energy of vaporization is zero (Δv Gm = 0). If the actual vapor pressure is higher than the saturation vapor pressure, P > P0 , then Δv Gm > 0, and we gain Gibbs free energy by condensation. If the actual vapor pressure is lower than the saturation vapor pressure, P < P0 , then Δv Gm < 0 and liquid evaporates spontaneously. For example, at 25 ∘ C water has a saturation vapor pressure of P0 = 3169 Pa. If we have an oversaturated vapor at 6338 Pa with P∕P0 = 2, then Δv Gm = RT ln(P∕P0 ) = 1.72 kJ∕mol. Upon condensation and with Δcon Gm = −Δv Gm , we gain 1.72 kJ/mol. To derive Eq. (2.32), we first consider a pool of liquid in a vessel in equilibrium with its vapor. The liquid is supposed to have a planar surface. The molar Gibbs energy (equal to the chemical potential) of the vapor is Gm = G0 + RT ln P0

(2.33)

assuming that the vapor behaves as an ideal gas. Here, G0 is the chemical potential of a reference state. In equilibrium, the chemical potential of the liquid is the same. Now, we increase the vapor pressure of the vapor above the liquid. Let us assume that for a brief time, we have a vapor at vapor pressure P > P0 above the liquid. The molar Gibbs energy of the vapor is simply Gm = G0 + RT ln P

(2.34)

27

28

2 Liquid Surfaces

The molar Gibbs energy of the liquid is still the same and given by Eq. (2.33).8 Thus, the difference in Gibbs energy between one mole of vapor at a pressure P and the liquid is P Δv Gm = RT ln P − RT ln P0 = RT ln (2.35) P0 which agrees with Eq. (2.32). To describe nucleation, we calculate the change in the Gibbs energy for the condensation of n moles of vapor into a droplet. Please note that n is much smaller than one mole. We must take two contributions into account. First, the Gibbs energy changes due to condensation. Second, work must be carried out to create the surface of the drop A. For a spherical drop, the total change in Gibbs energy is ΔG = −nΔv Gm + 𝛾A

(2.36) 4πr 3 ∕(3Vm )

In a drop of radius r, there are n = moles of molecules, where Vm is the molar volume of the liquid phase. The surface area is 4πr 2 . Inserting these expressions and with Eq. (2.32), we obtain ΔG = −

4πRTr 3 P ln + 4π𝛾r 2 3Vm P0

(2.37)

This is the change in Gibbs free energy upon condensation of a drop from a vapor phase with partial pressure P. Let us analyze Eq. (2.37) in more detail. For P < P0 the first term is positive, and therefore ΔG is positive. Any drop formed by randomly clustering molecules will evaporate again; no condensation can occur. For P > P0 , ΔG increases with increasing radius, has a maximum at the so-called critical radius r ∗ , and then decreases again. At the maximum, we have dΔG∕dr = 0, which leads to a critical radius of 2Vm 𝛾 r∗ = (2.38) RT ln(P∕P0 ) One (at first sight) surprising fact is that Eq. (2.38) is equal to the Kelvin equation (2.20). The Kelvin equation applies to systems in thermodynamic equilibrium, for which also dΔG∕dr = 0. For the nucleated droplet, however, any small fluctuation will bring the system either to a larger and growing droplet or to a smaller and evaporating droplet. As an example, Figure 2.17 shows a plot of ΔG versus the drop radius for water at different supersaturations. Supersaturation is the actual vapor pressure P divided by the vapor pressure P0 of a vapor that is in equilibrium with a liquid having a planar surface. Example 2.11 For water at T = 0 ∘ C and a supersaturation P∕P0 = 4, the critical radius is r ∗ = 8 Å. This corresponds approximately to 70 molecules. The maximum Gibbs energy to form such a cluster is ΔGmax = 1.9 × 10−19 J. 8 One might argue that the molar Gibbs energy of the liquid should also change because it is now exposed to a higher pressure, and we should add Vm (P − P0 ). This effect is, however, small. For example, for water at 25 ∘ C with P0 = 3169 Pa and P = 2 P0 , we have Vm (P − P0 ) = 0.057 J∕mol. Compared to RT ln(P∕P0 ) = 1720 J∕mol this is negligible.

2.7 Nucleation Theory

γ = 0.072 J/m2 150

Vm = 18 cm3

ΔG (kBT)

T = 20 °C

P/P0 = 2

100

50

P/P0 = 3 P/P0 = 4

0 0.0

0.5

1.0

1.5

2.0

2.5

Radius (nm)

Figure 2.17 Change in Gibbs free energy in units of kB T for the condensation of water vapor to a drop of a certain radius. 3kB T∕2 is the mean translational kinetic energy of one gas molecule.

How does nucleation proceed? In a vapor, there are always a certain number of clusters. Most of them are very small and consist only of a few molecules. Others are slightly larger. When the actual partial pressure P exceeds the equilibrium vapor pressure P0 , large clusters occur more frequently. If a cluster exceeds the critical size, thermal fluctuations tend to enlarge it even more, until it becomes infinitely large and the liquid condenses. The aim of any theory of nucleation is to find a rate J with which clusters of critical size are formed. This number is proportional to the Boltzmann factor exp(−ΔGmax ∕kB T). A complete description of the classical nucleation theory is not possible within the scope of this book. The result is [ ] √ ( )2 16πv2m 𝛾 3 2𝛾 P J= v exp − (2.39) πm m kB T 3(kB T)3 ln2 (P∕P0 ) Here, m is the mass and vm = Vm ∕NA the volume of one molecule. Classical nucleation theory is the basis for understanding condensation, and it predicts the dependencies correctly. Unfortunately, quantitatively the predictions often do not agree with experimental results [62, 63]. Theory predicts too low nucleation rates at low temperatures. At high temperatures, the calculated rates are too high. Empirical correction functions can be used, and then very good agreement is achieved [64]. Reference [65] reviews experimental methods. General overviews are [66]. Experimentally, nucleation rates can be determined in expansion chambers [64]. The vapor is expanded in a fast and practically adiabatic process. Then it cools down. Since at low temperatures, the equilibrium vapor pressure is much lower,

29

30

2 Liquid Surfaces

supersaturation is reached. This is partially compensated by the pressure reduction during the expansion, but the temperature effect dominates. The density of nuclei can be measured by light scattering. Example 2.12 The nucleation of water is analyzed in an expansion chamber. A vapor at an initial pressure of 2330 Pa at 303 K is expanded to a final pressure of 1575 Pa. In this process, it cools down to 260 K. At 260 K, the equilibrium vapor pressure is 219 Pa. Thus, the supersaturation reaches P∕P0 = 7.2. What is the nucleation rate? At 260 K, the surface tension extrapolated from values above 0 ∘ C is 𝛾 ≈ 77 mN/m. The molecular volume is vm = m∕𝜌 = 2.99 × 10−26 kg∕1000 kg∕m3 = 2.99 × 10−29 m3 , where m is the mass of a water molecule. Inserting these values into Eq. (2.39) leads to a nucleation rate of √ )2 ( 2 ⋅ 0.077 N∕m 1575 Pa −29 3 J= ⋅ 2.99 × 10 m ⋅ π ⋅ 2.99 × 10−26 kg 3.59 × 10−21 J { } −29 3 2 16π ⋅ (2.99 × 10 m ) (0.077 N∕m)3 ⋅ exp − 3 ⋅ (3.59 × 10−21 J)3 ⋅ ln2 7.2 = 1.28 × 1012 s−1 ⋅ 2.99 × 10−29 m3 ⋅ 1.92 × 1047 m−6 ⋅ e−37.9 = 2.54 × 1014 s−1 ∕m3 In most practical situations, nucleation occurs at certain nucleation sites [67]. One example is the formation of bubbles in champagne [68]. At the end of the fermentation process, the CO2 pressure in a bottle of champagne is around 6 atm. When the bottle is opened, the pressure in the vapor phase suddenly drops and an oversaturation of 5 is typically reached. After the champagne is poured into a Figure 2.18 CO2 bubbles nucleating from champagne at the bottom of a glass. Often, gas pockets entrapped inside cellulose particles serve as nucleation sites. Source: Liger-Belair et al. [68], American Chemical Society.

100 μm

2.9 Exercises

glass, the dissolved CO2 molecules escape by forming bubbles (only a small part escapes by diffusion to the surface). Several kinds of particles, stuck on the glass wall, are able to entrap gas pockets while the glass is filled. These particles are responsible for the repetitive production of bubbles rising in the form of bubble trains (Figure 2.18). Most of these particles are cellulose fibers coming from the surrounding air or remaining from the wiping process.

2.8 Summary ●

The surface tension of a liquid is defined as the work required to produce a new surface per unit area: dW = 𝛾dA

● ●

●

●

●

Surface tensions of common liquids are 20–80 mN/m. In equilibrium and neglecting gravity, the curvature of a liquid surface is constant and given by the Young–Laplace equation ( ) 1 1 ΔP = 𝛾 + R1 R2 For a liquid surface with a net curvature, there is always a pressure difference across the interface. The pressure on the concave side is higher. Important techniques to measure the surface tension of liquids are the sessile drop method, the pendant or sessile bubble method, the du Noüy ring tensiometer, and the Wilhelmy plate method. The vapor pressure of a liquid depends on the curvature of its surface. For drops, it is increased compared to the vapor pressure of a planar surface under the same conditions. For bubbles, it is reduced. Quantitatively, this is described by the Kelvin equation. One consequence of the curvature dependence of the vapor pressure is capillary condensation, that is the spontaneous condensation of liquids into pores and capillaries. Capillary condensation plays an important role for the adsorption of liquids into porous materials and powders. It also causes the adhesion of particles. The condensing liquid forms a meniscus around the contact area of two particles, which cause the capillary attraction.

2.9 Exercises 2.1

We would like to study a clean solid surface. Let us assume we have produced a pure, clean surface in an ultrahigh vacuum (UHV). We would like to analyze it for one hour and can tolerate a contamination of 10% of a monolayer. Estimate the value to which we must reduce the pressure in the UHV chamber. Assume that on a clean solid surface, most of the gas molecules that hit the surface are adsorbed.

31

32

2 Liquid Surfaces

2.2

Consider a spherical water drop of radius Rd = 1, 10, and 100 nm. How many molecules are in the drop? What is the percentage of molecules at the surface as compared to the total number of molecules? For simplicity, assume that the molecules are distributed isotropically and show no specific orientation at the surface.

2.3

A plastic box is filled with water to a height h = 1 m. A hole of radius 0.1 mm is drilled into the bottom. Does all the water run out? The plastic is not wetted.

2.4

Figure 2.9c shows a large, spread-out drop of water. Use the figure to estimate the surface tension of water at 25 ∘ C.

2.5

Wilhelmy plate method. What is the force on a plate 1 cm wide having a contact angle of 45∘ in water?

2.6

Drop-weight method. To determine the surface tension of hexadecane (C16 H34 ), you let it drop out of a capillary with 4 mm outer and 40 μm inner diameter. Hexadecane wets the capillary. Its density is 773 kg/m3 . 100 drops weigh 2.2 g. Calculate the surface tension of hexadecane. With good accuracy, the correction factor f is a function of the parameter 𝜙 = rc ∕V 1∕3 , with V being the volume of the drop. Based on experimental tables, f can be described by [28] f = 1 − 0.9121 𝜙 − 2.109 𝜙2 + 13.38 𝜙3 − 27.29 𝜙4 + 27.53 𝜙5 − 13.58 𝜙6 + 2.593 𝜙7 for 𝜙 < 1.2.

2.7

A hydrophilic sphere of radius RP = 5 μm sits on a hydrophilic planar surface. Water from the surrounding atmosphere condenses into the gap. What is the circumference of the meniscus? Make a plot of radius of circumference x versus humidity. At equilibrium, the humidity is equal to P0K ∕P0 .

2.8

A platinum wire of rc = 0.3 mm radius and l = 4 cm length is shaped into a ring. It is carefully pulled out of 1-butanol, always keeping it in a horizontal position. The force required to pull the ring out of the wetting liquid is measured to be 2.08 mN. What is the surface tension of the liquid? The density of butanol is 𝜌Bu = 810 kg∕m3 .

33

3 Thermodynamics of Interfaces In this chapter, we introduce the basic thermodynamics of interfaces. The purpose is to provide a broader base of understanding and point out some of the difficulties. For further reading, we recommend [4, 69, 70].

3.1

Thermodynamic Functions for Bulk Systems

We start by recalling some of the basics of bulk thermodynamics. The internal energy U is the total energy that arises due to the motion of the molecules in a system and the interactions between them. A variation of the internal energy of a homogeneous single-phase system is, according to the first and second principles of thermodynamics, ∑ dU = TdS − PdV + 𝜇i dNi + dW (3.1) Here, TdS stands for the change in internal energy caused by an entropy change, for example a heat flow. Ni is the number of molecules of the ith substance, 𝜇i is its chemical potential, 𝜇i dNi is the energy change caused by a change in the composition, and dW is the work done on the system without expansion work PdV. Changes in the enthalphy H, Helmholtz energy F, and Gibbs energy G are, respectively, ∑ dH = TdS + VdP + 𝜇i dNi + dW (3.2) dF = −SdT − PdV + dG = −SdT + VdP +

∑ ∑

𝜇i dNi + dW

(3.3)

𝜇i dNi + dW

(3.4)

Extensive parameters are those that directly increase with the extent of the system, for example volume V, Ni , mass. Intensive parameters such as temperature T, pressure P, or density do not depend on the extent of the system. A homogeneous system is one for which all extensive functions increase linearly if we increase the size of the system. For homogeneous bulk phases, the integral characteristic functions are ∑ U = TS − PV + 𝜇i Ni (3.5) Physics and Chemistry of Interfaces, Fourth Edition. Hans-Jürgen Butt, Karlheinz Graf, and Michael Kappl. © 2023 WILEY-VCH GmbH. Published 2023 by WILEY-VCH

34

3 Thermodynamics of Interfaces

H = U + PV = TS +

∑

F = U − TS = −PV + G = U + PV − TS =

𝜇 i Ni

∑

∑

(3.6)

𝜇i Ni

(3.7)

𝜇i Ni

(3.8)

3.2 Surface Excess The presence of an interface generally influences all thermodynamic parameters of a system. To consider the thermodynamics of a system with an interface, we divide that system into three parts: the two bulk phases with volumes V 𝛼 and V 𝛽 and the interface 𝜎. The two phases may, for example be the liquid and vapor phases of a substance. They may also be two immiscible liquids such as oil and water. In this introduction, we adhere to the Gibbs1 convention [71] (p. 219). In this convention, the two phases 𝛼 and 𝛽 are thought to be separated by an infinitesimally thin boundary layer, the Gibbs dividing plane. This is, of course, an idealization, and the Gibbs dividing plane is also called an ideal interface. A good introduction into Gibbsian surface thermodynamics is Ref. [72]. There are alternative models. Guggenheim,2 for example took the extended interfacial region, including its volume, explicitly into account [73, 74] (Figure 3.1). We use the Gibbs model because for most applications, it is more practical. In the Gibbs model, the interface is ideally thin (V 𝜎 = 0), and the total volume of the system is V = V𝛼 + V𝛽

(3.9)

All other extensive parameters can be written as a sum of three components: one of bulk phase 𝛼, one of bulk phase 𝛽, and one of the interfacial region 𝜎. Examples are the internal energy U, the number of molecules of the ith substance Ni , and the entropy S: U = U𝛼 + U𝛽 + U𝜎

Va

(3.10)

Vα

Vβ

(a) Gibbs ideal interface

(b)

Vσ

Vβ

Guggenheim

Figure 3.1 (a) In the Gibbs convention, the two phases 𝛼 and 𝛽 are separated by an ideal interface 𝜎 that is infinitely thin. (b) Guggenheim explicitly treated an extended interphase with a finite volume. 1 Josiah Willard Gibbs, 1839–1903; American mathematician and physicist, Yale College. 2 Edward Armand Guggenheim, 1901–1970 English physicochemist, University of Reading.

3.2 Surface Excess

Ni = Ni𝛼 + Ni𝛽 + Ni𝜎

(3.11)

S = S𝛼 + S𝛽 + S𝜎

(3.12)

The contributions of the two phases and of the interface are derived as follows: let u𝛼 and u𝛽 be the internal energies per unit volume of the two phases. The internal energy densities u𝛼 and u𝛽 are determined from the homogeneous bulk regions of the two phases. Close to the interface, they might be different. Still, we take the contribution of the volume phases to the total energy of the system as u𝛼 V 𝛼 + u𝛽 V 𝛽 . We extrapolate the internal energy densities of the two phases up to the ideal interface. The internal energy of the interface is the total internal energy of the whole system minus the contributions of the two phases: U 𝜎 = U − u𝛼 V 𝛼 − u 𝛽 V 𝛽

(3.13)

At an interface, the molecular constitution changes. The concentration (number of molecules per unit volume) of the ith material is, in the two phases, respectively, c𝛼i and c𝛽i . The additional quantity present in the system due to the interface is Ni𝜎 = Ni − c𝛼i V 𝛼 − c𝛽i V 𝛽

(3.14)

With Eq. (3.14), it is possible to define something like a surface concentration, the so-called interfacial excess: Γi =

Ni𝜎 A

(3.15)

A is the interfacial area. The interfacial excess is given as a number of molecules per unit area (m−2 ) or in mol/m2 . In the Gibbs model of an ideal interface, there is one question: where precisely do we position the ideal interface? Let us therefore look more closely at a liquid–vapor interface of a pure liquid. The density decreases continuously from the high density of the bulk liquid to the low density of the bulk vapor (Figure 3.2). There could even be a density maximum in between since it should in principle be possible to have an increased density at the interface. It is natural to place the ideal interface in the middle of the interfacial region so that Γ = 0. In this case the two dashed regions, to the left and right of the ideal interface, are equal in size. If the ideal interface is placed more into the vapor phase, then the total number of molecules extrapolated from the bulk densities is higher than the real number of molecules, N < c𝛼 V 𝛼 + c𝛽 V 𝛽 . Therefore, the surface excess is negative. Conversely, if the ideal interface is placed more into the liquid phase, the total number of molecules extrapolated from the bulk densities is lower than the real number of molecules, N > c𝛼 V 𝛼 + c𝛽 V 𝛽 , and the surface excess is positive. Let us now turn to two-component or multicomponent liquids such as a solvent with dissolved substances. Substituting V 𝛼 = V − V 𝛽 , we can write ( ) N1𝜎 = N1 − c𝛼1 V + c𝛼1 − c𝛽1 V 𝛽 (3.16)

35

36

3 Thermodynamics of Interfaces

c

Gibbs dividing interface

Γ=0

Liquid phase α

Vapor phase β

x

c

Figure 3.2 Schematic plot of density of a liquid versus a coordinate normal to its surface. For three different positions of the Gibbs dividing plane, the extrapolated contributions of the two bulk phases are indicated by dashed areas. The surface excess Γ depends on the position of the Gibbs dividing plane.

Γ0

x

for the first component, which is taken to be the solvent. For all other components, we get similar equations: ( ) Ni𝜎 = Ni − c𝛼i V + c𝛼i − c𝛽i V 𝛽 (3.17) No quantities on the right-hand side of Eqs. (3.16) and (3.17), except V 𝛽 , depend on the position of the dividing plane, and all are measurable quantities. Only V 𝛽 depends on the choice of the dividing plane. We can eliminate V 𝛽 by multiplying Eq. (3.16) by (c𝛼i − c𝛽i )∕(c𝛼1 − c𝛽1 ) and subtracting Eq. (3.16) from Eq. (3.17): Ni𝜎 − N1𝜎

c𝛼i − c𝛽i

𝛽 ( ) c𝛼i − ci 𝛼 𝛼 = N − c V − N − c V i 1 1 i c𝛼1 − c𝛽1 c𝛼1 − c𝛽1

(3.18)

The right-hand side of the equation does not depend on the position of the Gibbs dividing plane, and thus the left-hand side is also invariant. We divide this quantity by the surface area and obtain the invariant quantity Γ(1) ≡ Γ𝜎i − Γ𝜎1 i

c𝛼i − c𝛽i c𝛼1 − c𝛽1

(3.19)

This is called the relative adsorption of component i with respect to component 1. It is an important quantity because it can be determined experimentally. As we will see later, it can be measured by determining the surface tension of a liquid versus the concentration of the solute.

3.2 Surface Excess

Example 3.1 To show how our choice of the position of the Gibbs dividing plane influences the surface excess, we consider an equimolar mixture of ethanol and water [75, p. 25]. If the position of the ideal interface is such that ΓH2 O = 0, then one finds experimentally that ΓEthanol = 9.5 × 10−7 mol∕m2 . If the interface is placed 1 nm outward, then we obtain ΓEthanol = −130 × 10−7 mol∕m2 . For the case where component 1 is a solvent in which all other components are dissolved and thus have a much lower concentration, we choose the position of the dividing plane such that Γ𝜎1 = 0. From Eq. (3.19), we get Γ(1) = Γ𝜎i i

(3.20)

In Figure 3.3a, the concentration profiles for solute 2 dissolved in liquid 1 are illustrated schematically. We assume that the solute is enriched at the surface. An example of a measured concentration profile is plotted in Figure 3.3b. Andersson et al. [76] dissolved tetrabutylphosphonium bromide (Bu4 PBr) in formamide (CHONH2 ) and measured the concentration profiles with neutral impact collision ion scattering spectroscopy (NICISS). Bu4 PBr is surface active and accumulates at the surface of the formamide. Formamide was chosen because it has a low vapor pressure and for NICISS a vacuum is required.

(a)

(b)

Figure 3.3 (a) Schematic concentration profile of a solute (2) dissolved in a liquid (1). The gray area corresponds to the surface excess Γ(1) of solute. (b) Concentration of 2 formamide and dissolved Bu4 PBr (bulk concentration 0.2 mol/l) at 6 ∘ C [76]. The continuous straight lines show the concentrations extrapolated from the bulk to the Gibbs dividing plane.

37

38

3 Thermodynamics of Interfaces

3.3 Thermodynamic Relations for Systems with an Interface 3.3.1

Internal Energy and Helmholtz Energy

Let us consider a process in a system with two phases, 𝛼 and 𝛽. The system is divided by an interface. We could, for instance, do work on that system. As a consequence, the state functions like the internal energy, the entropy, and so on change. How do they change and how can we describe this mathematically? In addition to the usual bulk thermodynamics, we must take the interface into account. We start our analysis with the internal energy. We first analyze the internal energy, not the enthalpy, the Helmholtz energy, or the Gibbs energy, because only the internal energy contains extensive quantities (S, V, Ni , A) as variables. A variation of the internal energy of a two-phase system is, according to the first and second principles of thermodynamics, ∑ dU = TdS − PdV + 𝜇i dNi + 𝛾dA + dW ′ (3.21) The mechanical work dW in Eq. (3.1) was split into surface work 𝛾dA and dW ′ , which is the work done on the system without expansion work PdV and surface work. For simplicity, we ignore the additive term dW ′ in the following discussion. The sum runs over all components, meaning over all substances that are chemi∑ cally different. We split the PdV and 𝜇i dNi terms into two bulk and interfacial components: dU = TdS − P𝛼 dV 𝛼 − P𝛽 dV 𝛽 ∑ ∑ 𝛽 𝛽 ∑ + 𝜇i𝛼 dNi𝛼 + 𝜇i dNi + 𝜇i𝜎 dNi𝜎 + 𝛾dA

(3.22)

Since the interface is infinitely thin, it cannot perform volume work. With dV = dV 𝛼 + dV 𝛽 ⇒ dV 𝛼 = dV − dV 𝛽 the equation simplifies to ( ) dU = TdS − P𝛼 dV − P𝛽 − P𝛼 dV 𝛽 ∑ ∑ 𝛽 𝛽 ∑ + 𝜇i𝛼 dNi𝛼 + 𝜇i dNi + 𝜇i𝜎 dNi𝜎 + 𝛾dA (3.23) This equation allows us to provide a thermodynamic definition of surface tension: 𝜕U || ≡𝛾 𝜕A ||S,V,V 𝛽 ,Ni

(3.24)

The surface tension tells us how the internal energy of the system changes when increasing the surface area while keeping constant the entropy, total volume, volume of phase 𝛽, and total amounts of all components. For an isolated system3 Eq. (3.24) is useful because in equilibrium the entropy is constant. We can also keep V and the amounts Ni constant. However, V 𝛽 might be difficult to keep constant. As we will see later, for planar surfaces (and practically 3 No heat or molecules are allowed to flow into or out of an isolated system. In a closed system, heat flow is allowed, but the number of molecules remains constant. In an open system, both molecules and heat can be exchanged.

3.3 Thermodynamic Relations for Systems with an Interface

those with small curvatures), the condition that V 𝛽 must be kept constant can be dropped. In many applications, the system is not isolated. Heat is allowed to flow into or out of the system. It is the temperature that is constant rather than the entropy. Then it is more useful to consider the Helmholtz energy. In general, the change in Helmholtz energy of a system ignoring dW ′ is ∑ dF = −SdT − PdV + 𝜇i dNi + 𝛾dA (3.25) For a two-phase system with one interface, it follows that ( ) dF = −SdT − P𝛼 dV − P𝛽 − P𝛼 dV 𝛽 ∑ ∑ 𝛽 𝛽 ∑ + 𝜇i𝛼 dNi𝛼 + 𝜇i dNi + 𝜇i𝜎 dNi𝜎 + 𝛾dA

(3.26)

When the temperature and volume are constant (dV = 0, dT = 0), the first two terms are zero. We can use Eq. (3.26) to formulate an alternative, equivalent definition of surface tension: 𝜕F || ≡𝛾 (3.27) 𝜕A ||T,V,V 𝛽 ,Ni The surface tension is the change in the Helmholtz energy of the system when the surface area is increased while the temperature, total volume, volume of phase 𝛽, and total amounts of all components are kept constant.

3.3.2

Equilibrium Conditions

At equilibrium, Eq. (3.26) can be simplified even further because the chemical potentials in the two bulk phases and at the interface are equal. This can easily be demonstrated. We assume that there is no exchange of material with the outside world (dNi = 0); we have a closed system. Then the three parameters Ni𝛼 , Ni𝛽 , and Ni𝜎 are not independent because Ni = Ni𝛼 + Ni𝛽 + Ni𝜎 is constant. Only two parameters at a time, for example Ni𝛼 and Ni𝛽 , can be varied independently. Ni𝜎 is then determined by the other two amounts because dNi𝜎 = −dNi𝛼 − dNi𝛽 . Therefore, we can write ∑ ∑ 𝛽 ( ) dF = − P𝛽 − P𝛼 dV 𝛽 + 𝛾dA + (𝜇i𝛼 − 𝜇i𝜎 )dNi𝛼 + (𝜇i − 𝜇i𝜎 )dNi𝛽 (3.28) At equilibrium, with constant volume, temperature, and constant amounts of material, the Helmholtz energy is minimal. At a minimum, the derivatives with respect to all independent variables must be zero: dF = 𝜇i𝛼 − 𝜇i𝜎 = 0, dNi𝛼

dF dNi𝛽

= 𝜇i𝛽 − 𝜇i𝜎 = 0

(3.29)

It follows that 𝜇i𝛼 = 𝜇i𝜎 = 𝜇i𝛽 = 𝜇i

(3.30)

Hence, in equilibrium, the chemical potentials are the same everywhere in the system. With this, we can further simplify Eq. (3.28) to dF = −(P𝛽 − P𝛼 )dV 𝛽 + 𝛾dA

(3.31)

39

40

3 Thermodynamics of Interfaces

Question: Why is it not possible, using the same argument, for example, dF∕dA = 0, to conclude that at equilibrium 𝛾 must be zero? Answer: The surface area A is not an independent parameter. Surface A and the volume V 𝛽 are related. If the volume of a body changes, then in general its surface area also changes. V 𝛽 and A can thus not be varied independently. In fact, a law of differential geometry says that in general, 𝜕V∕𝜕A = (1∕R1 + 1∕R2 )−1 . At this point we mention a simple, alternative way of deriving the Young–Laplace equation. In equilibrium, we have dF∕dA = 0. This leads to dF 𝜕F 𝜕F 𝜕V 𝛽 𝜕V 𝛽 = + = 𝛾 − (P𝛽 − P𝛼 ) =0 𝛽 dA 𝜕A 𝜕V 𝜕A 𝜕A

(3.32)

Inserting 𝜕V 𝛽 ∕𝜕A = (1∕R1 + 1∕R2 )−1 , taking ΔP = P𝛽 − P𝛼 , and rearranging the equation directly leads to the Young–Laplace equation.

3.3.3

Location of Interface

At this point we should note that, fixing the bending radii, we define the location of the interface. A possible choice for the ideal interface is the one that is defined by the Young–Laplace equation. If the choice for the interface is different, then the value of the surface tension must be changed accordingly. Otherwise, the Young–Laplace equation would no longer be valid. All this can be illustrated with the example of a spherical drop (Figure 3.4) [77]. We can, for instance, consider the evaporation or the condensation of liquid from, or to, a drop of radius r. There we have V𝛽 =

( ) r 4π 3 4π A 3∕2 𝜕V 𝛽 r , A = 4πr 2 ⇒ V 𝛽 = ⇒ = 3 3 4π 𝜕A 2

(3.33)

If the interface is chosen to be at a radius r ′ , then the corresponding value of 𝜕V 𝛽 ∕𝜕A is r ′ ∕2. The pressure difference P𝛽 − P𝛼 can in principle be measured. This implies that P𝛽 − P𝛼 = 2𝛾∕r and P𝛽 − P𝛼 = 2𝛾 ′ ∕r ′ are both valid at the same time. This is possible only if, depending on the radius, one accepts a different interfacial tension. Therefore, we used 𝛾 ′ in the second equation. In the case of a curved surface, the interfacial tension depends on the location of the Gibbs dividing plane! In the case of flat surfaces, this problem does not occur. In that case, the pressure difference is zero, and the surface tension is independent of the location of the ideal interface. One may object: the surface tension is measurable, and thus the Laplace equation assigns the location Vapor Liquid of the ideal interface! This argument is, however, α β not valid. The only quantity that can be measured is mechanical work and the forces acting during r the process. For curved surfaces, it is not possible to divide work into volume and surface work. Therefore, Figure 3.4 A drop in its vapor phase. it is not possible to measure only the surface tension.

3.3 Thermodynamic Relations for Systems with an Interface

3.3.4

Gibbs Energy and Enthalpy

As a third boundary condition, let us consider a system at constant temperature and pressure P. In this case, it is useful to define the surface tension via the Gibbs energy. In general, any change in the Gibbs energy for a system with one interface and ignoring W ′ can be written as follows: ∑ dG = −SdT + VdP + 𝜇i dNi + 𝛾dA ∑ = −SdT + V 𝛼 dP𝛼 + V 𝛽 dP𝛽 + 𝜇i dNi + 𝛾dA (3.34) Surface tension is defined as follows 𝜕G || ≡𝛾 𝜕A ||T,P𝛼 ,P𝛽 ,Ni

(3.35)

Again, the definition is equivalent to Eqs. (3.24) and (3.27). For flat (planar) interfaces, we have the same pressure in both phases P𝛼 = P𝛽 = P, and the definition simplifies to 𝜕G || ≡𝛾 (3.36) 𝜕A ||T,P,Ni The interfacial tension is the increase in the Gibbs energy per increase in surface area at constant T, P, and Ni . For completeness, we also report the enthalpy of the system. Extending Eq. (3.2) by 𝛾dA to take the interface into account and ignoring again all additional work dW ′ , the change in enthalpy of the system is ∑ H = TdS + VdP + 𝜇i dNi + 𝛾dA ∑ 𝛼 𝛼 = TdS + V dP + V 𝛽 dP𝛽 + 𝜇i dNi + 𝛾dA (3.37)

3.3.5

Interfacial Excess Energies

Until now we have considered the energy quantities of an entire system U, F, G, and H. Now, we turn to the interfacial excess quantities U 𝜎 , F 𝜎 , G𝜎 , and H 𝜎 . In full, F 𝜎 is called the interfacial excess Helmholtz energy. Usually we drop “excess” and just call it the interfacial Helmholtz energy. We start with the internal interfacial or internal surface energy ∑ dU 𝜎 = TdS𝜎 + 𝜇i dNi𝜎 + 𝛾dA (3.38) The term PdV 𝜎 disappears because the ideal interface has no volume. In Eq. (3.38), U 𝜎 is a homogeneous, linear function of the extensive properties S𝜎 , A, and all Ni𝜎 of the system. Therefore, it may be integrated, keeping the intensive properties T and 𝛾 and all 𝜇i constant. Physically, this means that it is possible to increase the size of the system by increasing the surface area and in proportion adding matter to the surface in such a way that the ratio dN1𝜎 ∶ dN2𝜎 ∶ dN3𝜎 ∶ … remains the same as in the original (and final) system. This can be realized by, for instance, tilting a sealed test tube that is partially filled with a liquid (Figure 3.5). Mathematically, the integration is possible because of Euler’s theorem. Euler’s

41

42

3 Thermodynamics of Interfaces

Figure 3.5 Increasing the surface area size by tilting a test tube.

theorem states that if f (x, y) is a homogeneous, linear function of the variables x and y, then f = x𝜕f ∕𝜕x|y + y𝜕f ∕𝜕y|x . Application of Euler’s theorem to U 𝜎 in Eq. (3.38) with S𝜎 , Ni𝜎 , and A as variables leads to the integral form of the internal interfacial energy function: ∑ U 𝜎 = TS𝜎 + 𝜇i Ni𝜎 + 𝛾A (3.39) For the interfacial Helmholtz energy, we use F 𝜎 = U 𝜎 − TS𝜎 and obtain the integral form of the interfacial Helmholtz energy function: ∑ F 𝜎 = 𝛾A + 𝜇i Ni𝜎 (3.40) Division by A leads to ∑ F𝜎 =𝛾+ 𝜇i Γi (3.41) A In what follows, we will use the symbol f 𝜎 ≡ F 𝜎 ∕A for the interfacial Helmholtz energy. The differential form of the Helmholtz energy function is obtained when applying Eq. (3.25) to the interface: ∑ 𝜇i dNi𝜎 + 𝛾dA (3.42) dF 𝜎 = −S𝜎 dT + Equation (3.42) allows us to derive an important relationship between the surface entropy and the temperature dependence of the surface tension. The interfacial Helmholtz energy is a state function. If we take the partial derivative with respect first to T and then to A, then the result should be the same as if we had first differentiated with respect to A and then to T. From Eq. (3.42), one can easily see that differentiation of F 𝜎 with respect to T or A leads to 𝜕F 𝜎 || 𝜕F 𝜎 || = −S𝜎 and =𝛾 (3.43) | 𝜕T |A,Ni𝜎 𝜕A ||T,Ni𝜎 If we differentiate the first expression at constant T with respect to A, then the result must be equal to the second expression differentiated at constant A with respect to T. This leads to an important equation for the surface entropy: 𝜕𝛾 || 𝜕S𝜎 || − = (3.44) | 𝜕A |T,Ni𝜎 𝜕T ||A,Ni𝜎 Equation (3.44) is one of the so-called Maxwell relations. Let us turn to interfacial enthalpy. Usually, the enthalpy is defined by subtracting the mechanical work term VP from the internal energy. If we turn to interfacial functions, then the question is whether or not 𝛾dA counts as a mechanical

3.4 Pure Liquids

work term. There are two definitions of the interfacial excess enthalpy. We could argue that enthalpy is equal to the internal energy minus the total mechanical work 𝛾A − PV 𝜎 . Since in the Gibbs convention PV 𝜎 = 0, we define H 𝜎 ≡ U 𝜎 − 𝛾A

(3.45)

This definition is recommended by the International Union of Pure and Applied Chemistry [4, 78]. One consequence of it is that H = H 𝛼 + H 𝛽 + H 𝜎 + 𝛾A. The differential is again easily obtained as follows: ∑ dH 𝜎 = TdS𝜎 + 𝜇i dNi𝜎 − Ad𝛾 (3.46) Alternatively, one could argue that the enthalpy is equal to the internal energy minus the volume work PV 𝜎 . Since the volume work is zero in the Gibbs convention, we simply obtain H ′𝜎 ≡ U 𝜎

(3.47)

To define the interfacial Gibbs energy, we argue that the difference between U 𝜎 and F 𝜎 should be the same as that between H 𝜎 and G𝜎 . Therefore, we define ∑ G𝜎 ≡ H 𝜎 − TS𝜎 = F 𝜎 − 𝛾A = 𝜇i Ni𝜎 (3.48) One consequence of this definition is that G = G𝛼 + G𝛽 + G𝜎 + 𝛾A. The differential is ∑ dG𝜎 = −S𝜎 dT + 𝜇i dNi𝜎 − Ad𝛾 (3.49) With the alternative definition of H ′𝜎 , we obtain G′𝜎 ≡ H ′𝜎 − TS𝜎 = F 𝜎

(3.50)

and G = G𝛼 + G𝛽 + G′𝜎 .

3.4 Pure Liquids For pure liquids, the description becomes much simpler. We start by asking how the surface tension is related to the surface excess quantities, in particular to the internal surface energy and the surface entropy. One important relationship can be derived directly from Eq. (3.41). For pure liquids, we choose the Gibbs dividing plane such that Γ = 0. Then the surface tension is equal to the interfacial Helmholtz energy per unit area f 𝜎 : F𝜎 =𝛾 (3.51) A Let us turn to the entropy. For pure liquids, the position of the interface is chosen such that N 𝜎 = 0. For homogeneous systems, we also know that S𝜎 ∕A = 𝜕S𝜎 ∕𝜕A. With the interfacial entropy per unit area defined as s𝜎 ≡ S𝜎 ∕A and with Eq. (3.44), we find 𝜕𝛾 || s𝜎 = − (3.52) 𝜕T ||P,A f𝜎 =

43

3 Thermodynamics of Interfaces

120

n-octane

uσ

100 (mN/m)

44

Water

80

γ

60

uσ

40

Tsσ

Tsσ

γ

20 0

Figure 3.6 Surface tension 𝛾, surface entropy per unit area multiplied by the temperature Ts𝜎 , and internal surface energy per unit area u𝜎 versus temperature for n-octane and water.

0

20 40 60 80 0 Temperature (°C)

20 40 60 80 100 Temperature (°C)

The surface entropy per unit area is given by the change in the surface tension with temperature. To determine the surface entropy, one needs to measure how the surface tension changes with temperature. Question: If the volume of the interface is zero, why is the condition important that P is constant? Answer: A change in pressure may change the structure of the interface. Since in the Gibbs approach, we view the surface as being collapsed to an ideal plane, its entropy may change, even though its volume is zero. Equation (3.52) is generally valid, not only within the Gibbs formalism. For the majority of liquids, the surface tension decreases with increasing temperature (Figure 3.6). This behavior was already observed by Eötvös, Ramsay, and Shields at the end of the nineteenth century [79, 80]. The entropy on the surface is thus increased. This is reasonable because surface molecules have fewer nearest neighbors and therefore have more room to move about. The internal energy of a pure liquid is U 𝜎 = TS𝜎 + 𝛾A. Division by A and, assuming that we have a homogeneous system, leads to u𝜎 ≡

U𝜎 = Ts𝜎 + 𝛾 A

(3.53)

or u𝜎 = 𝛾 − T

𝜕𝛾 || 𝜕T ||P,A

(3.54)

It is thus possible to determine the internal surface energy and the surface entropy by measuring the dependence of the surface tension on the temperature. How does heat flow during an increase in the surface area? In a reversible process, TdS is the heat δQ that a system absorbs. The heat absorption is proportional to the surface increase, and we can write δQ = qdA. Here, q is the heat per unit area that is taken up by the system. With dS = s𝜎 dA and s𝜎 = −𝜕𝛾∕𝜕T, we obtain qdA = δQ = TdS = Ts𝜎 dA = −T

𝜕𝛾 dA 𝜕T

(3.55)

or q = −T

𝜕𝛾 𝜕T

(3.56)

3.5 Gibbs Adsorption Isotherm

Table 3.1 Surface tension, surface entropy, and internal surface energy of some liquids at 25 ∘ C [10, 38]. 𝜸 = f 𝝈 (mN∕m)

Ts𝝈 (mN∕m)

u𝝈 (mN∕m)

Water

71.99

46.4

118.4

n-Hexane

17.89

30.5

48.4

n-Heptane

19.65

29.2

48.9

n-Octane

21.14

28.3

49.5

n-Nonane

22.38

27.9

50.3

n-Decane

23.37

27.4

50.8

Methanol

22.07

23.0

45.1

Ethanol

21.97

24.8

46.8

1-Propanol

23.32

23.1

46.4

1-Butanol

24.93

26.8

51.7

1-Hexanol

25.81

23.9

49.7

Benzene

28.22

38.4

66.6

Toluene

27.93

35.3

63.1

Acetone

23.46

33.4

56.9

Chloroform

26.67

38.5

65.2

Mercury Gold (1064 ∘ C)

485.48 1120

61.0 150

546.5 1270

Note: The surface entropy was calculated with Eq. (3.52) and Table 2.1.

This is the heat per unit area absorbed by the system during an isothermal increase of the surface area. Since 𝜕𝛾∕𝜕T is mostly negative, the system usually takes up heat when the surface area is increased. Table 3.1 lists the surface tension, surface entropy, and internal surface energy of some liquids at 25 ∘ C.

3.5 Gibbs Adsorption Isotherm It is well known that the surface tension of water decreases when a detergent is added. Detergents are strongly enriched at the surface, which lowers the surface tension. This change in surface tension upon adsorption of substances to the interface is described by the Gibbs adsorption isotherm.

3.5.1

Derivation

The Gibbs adsorption isotherm is a relationship between the surface tension and excess concentrations. To derive it, we start with Eq. (3.39). Differentiation of Eq. (3.39) leads to ∑ ∑ dU 𝜎 = TdS𝜎 + S𝜎 dT + 𝜇i dNi𝜎 + Ni𝜎 d𝜇i + 𝛾dA + Ad𝛾 (3.57)

45

46

3 Thermodynamics of Interfaces

Equating this to expression (3.38) results in ∑ 0 = S𝜎 dT + Ni𝜎 d𝜇i + Ad𝛾

(3.58)

At constant temperature, it can be simplified to ∑ d𝛾 = − Γi d𝜇i

(3.59)

Equations (3.58) and (3.59) are called Gibbs adsorption isotherms. In general, isotherms are state functions plotted versus pressure, concentration, and so on, at constant temperature. One word of caution: the given equation is only valid for those surfaces whose deformation is reversible and plastic, that is, liquid surfaces. In solids, changes in the surface are usually accompanied by elastic processes [70, 81]. To consider elastic tensions, an additional term must be added to Eq. (3.58). This will be discussed in Section 7.4.1.

3.5.2

System of Two Components

The simplest application of the Gibbs adsorption isotherm is a system of two components, for example, a solvent 1 and a solute 2. In this case, we have d𝛾 = −Γ1 d𝜇1 − Γ2 d𝜇2

(3.60)

The ideal interface is conveniently defined such that Γ1 = 0. Then we obtain d𝛾 = −Γ(1) d𝜇2 2

(3.61)

The superscript “(1)” should remind us of the special choice of the interface. The chemical potential of the solute is described by the equation: a (3.62) 𝜇2 = 𝜇20 + RT ln a0 Here, a is the activity and a0 is a standard activity (e.g. 1 mol/l). Differentiating with respect to a∕a0 at constant temperature leads to d𝜇2 = RT

d(a∕a0 ) da = RT a∕a0 a

(3.63)

Substituting this expression into Eq. (3.61) leads to Γ(1) =− 2 With

𝜕𝛾 𝜕a

=

a 𝜕𝛾 || RT 𝜕a ||T

𝜕𝛾 𝜕 ln a 𝜕 ln a 𝜕a

Γ(1) =− 2

=

𝜕𝛾 1 𝜕 ln a a

1 𝜕𝛾 || RT 𝜕 ln a ||T

(3.64) we can alternatively write (3.65)

Equation (3.64) is a very important! It directly tells us that when a solute is enriched at the interface, Γ(1) > 0, the surface tension decreases when the con2 centration is increased. Such solutes are said to be surface active and are called surfactants or surface active agents.

3.5 Gibbs Adsorption Isotherm

When a solute avoids the interface, Γ(1) < 0, the surface tension increases with 2 the addition of the substance. Experimentally, Eq. (3.64) can be used to determine the surface excess by measuring the surface tension versus the bulk concentration. If a decrease in the surface tension is observed, then the solute is enriched at the interface. If the surface tension increases upon addition of solute, then the solute is depleted at the interface. Example 3.2 The alkylethylene glycol C10 E4 (C10 H21 (OCH2 CH2 )4 OH) is a relatively strong surfactant in water. If 1.33 𝜇M is added to pure water at 25 ∘ C, then the surface tension decreases from 72.0 to 68.6 mJ/m2 . What is the surface excess of C10 E4 ? At such low activities and as an approximation, we replace activity a by concentration c and obtain (0.0686 − 0.0720) N∕m 𝜕𝛾 Δ𝛾 Nl ≈ = = −2550 −6 𝜕a Δc mol m (1.33 × 10 − 0) mol∕l It follows that Γ=−

1.33 × 10−6 mol∕l a 𝜕𝛾 Nl = ⋅ 2550 = 1.37 × 10−6 mol∕m2 RT 𝜕a 8.31 ⋅ 298 J∕mol mol m

Every surfactant molecule occupies an average surface area of 1.21 nm2 . The choice of the ideal interface in the Gibbs adsorption isotherm Eq. (3.64) for a two-component system is, in a certain view, arbitrary. It is, however, convenient. There are two reasons for this. First, on the right-hand side are physically measurable quantities (a, 𝛾, T) that are related in a simple way to the interfacial excess. Any other choice of the interface would lead to a more complicated expression. Second, the choice of the interface is intuitively evident, at least for c1 ≫ c2 . One should, however, keep in mind that different spatial distributions of the solute can lead to the same Γ(1) . Figure 3.7 shows two examples of the same interfacial excess 2 (1) concentration Γ2 . In the first case, the distribution of molecules 2 stretches out beyond the interface, but the concentration is nowhere increased. In the second case, the concentration of molecules 2 is actually increased locally.

c

Substance 1

c

Substance 2

Substance 2

x Gibbs dividing interface Figure 3.7 Examples of two different concentration profiles leading to the same interfacial excess concentration Γ2(1) .

x

47

3 Thermodynamics of Interfaces

3.5.3

Experimental Aspects

To verify Eq. (3.64), we must determine the two variables – concentration and surface tension – independently. One way to do this is to use radioactively labeled dissolved substances. The radioactivity close to the surface is measured. 𝛽 − emitters (3 H, 14 C, 35 S) are suitable because electrons only travel a short range, that is, any recorded radioactivity comes from molecules from the interface, or close below [82]. Neutron reflectivity has been applied to measure the amount of surfactant adsorbed to aqueous solutions [83, 84]. The whole concentration profile could be measured by NICISS [76]. In all cases, the measured amount agreed with the surface excess predicted by the Gibbs adsorption isotherm from a measurement of the surface tension. Plots of surface tension versus concentration for C10 E4 , SDS (sodium dodecylsulfate), n-pentanol, and LiCl in an aqueous medium at room temperature are shown in Figure 3.8. The four curves are typical for three different types of adsorption. C10 E4 and SDS are typical for amphiphilic substances. Often, the term amphiphilic molecule, or simply amphiphile, is used. An amphiphilic molecule consists of two well-defined regions: one that is oil-soluble (lyophilic or hydrophobic) and one that is water-soluble (hydrophilic). Due to the nonpolar hydrocarbon chain, they show a strong tendency to enrich at the air–water interface. In many cases, above a certain critical concentration defined aggregates called micelles are formed (Section 11.1). This concentration is called the critical micellar concentration (CMC). In the case of SDS (NaSO4 (CH2 )11 CH3 ) at 25 ∘ C, this is at 8.9 mM. For C10 E4 , the CMC is lower – at 0.79 mM. Above the CMC, the surface tension does not change significantly any further because any added substance goes into micelles, not to the air–water interface. The adsorption isotherm for pentanol is typical for lyophobic substances, that is, substances that do not like to stay in solution and for weakly amphiphilic substances. They become enriched in the interface and decrease the surface tension.

70 Surface tension (mJ/m2)

48

LiCl

60 50

C10E4

SDS

Pentanol

40 30 1E-3

0.01

0.1 1 10 Concentration (mM)

100

1000

Figure 3.8 Plots of surface tension versus concentration for aqueous solutions of C10 E4 [85], SDS, SDS before purification [86] (+), n-pentanol [87], and LiCl [88] at 25 ∘ C.

3.5 Gibbs Adsorption Isotherm

If water is the solvent, most organic substances show such a behavior. The LiCl adsorption isotherm is characteristic of lyophilic substances. Most ions in water show such behavior. Example 3.3 Charged surfactants. Historically, the measurement of the surface tension of charged surfactants, in particular of SDS, and its relation to the Gibbs adsorption isotherm was debated. According to Gibbs’ equation, the surface tension has to decrease monotonically when the surfactant concentration c increases. It was, however, frequently observed that graphs of 𝛾-VS-c showed a minimum around 7 mM (crosses in Figure 3.8). Miles and Shedlovsky [86] showed that this minimum was due to impurities (see [89] for an overview). By a thorough purification, they obtained a monotonically decreasing surface tension with increasing concentration. Keeping the solution pure is, however, difficult because dodecanol is produced spontaneously by hydrolysis of SDS. For charged surfactants a factor m = 2 must be included in the Gibbs equation: a 𝜕𝛾 (3.66) mRT 𝜕a The reason is that for a charged surfactant molecule at the surface, we must also take into account the counterion (Chapter 4). Therefore, it counts twice [84, 90]. =− Γ(1) 2

To describe the influence of a substance on the surface tension, one could specify the gradient of the adsorption isotherm for c → 0. A list of these values for some substances dissolved in water at room temperature is shown in Table 3.2. Example 3.4 Adding 1 mM NaCl to water results in a slight increase in the surface tension of Δ𝛾 = 1.82 × 10−3 N∕m ⋅ 0.001 = 1.82 × 10−6 N∕m. Upon addition of 1 mM CH3 COOH, the surface tension decreases by 3.8 × 10−5 N∕m.

3.5.4

Marangoni Effect

In 1855, James Thomson, elder brother of Lord Kelvin, described the following phenomenon [91]: “… if, in the middle of the surface of a glass of water, a small quantity of alcohol or strong spirituous liquor be gently introduced, a rapid rushing of the Table 3.2 Gradient of adsorption isotherm for c → 0 of different solutes in water at 25 ∘ C. Solute

d(𝜟𝜸)∕dc(10−𝟑 N∕m∕M)

HCl

−0.28

LiCl

1.81

NaCl

1.82

CsCl

1.54

CH3 COOH

−38

49

50

3 Thermodynamics of Interfaces

surface is found to occur outwards from the place where the spirit is introduced. It is made more apparent if fine powder be dusted on the surface of the water.” Carlo Marangoni4 observed that when throwing a sponge soaked with oil into a pond of water, the oil spreads fast in a radial direction to a very thin film [92]. Spreading was so fast that Marangoni did the experiments in a pond in the Tuileries Garden in Paris, which had a diameter of 70 m, to estimate the spreading velocity. Both phenomena are manifestations of what is now called the Marangoni effect: movement at a fluid interface caused by local variations in surface tension. In the first experiment, the ethanol locally reduced the surface tension. This horizontal gradient in surface tension caused the flow of the liquid. Ethanol spreads outward to reduce the surface tension of the surrounding liquid. In steady state, the hydrodynamic flow and the gradient in surface tension are coupled by 𝜂

dvx d𝛾 = dz dx

(3.67)

Here, dvx ∕dz is the vertical gradient of the horizontal velocity (Figure 3.9). The viscous drag is balanced by a gradient in surface tension. Or, conversely, a gradient in surface tension causes a hydrodynamic drag, which leads to a flow of the liquid. A gradient in surface tension can be caused by a local variation in temperature [93] or composition [94, 95]. Marangoni effects often occur in connection with evaporation since evaporation leads to local cooling and, in the case of mixtures of liquids with different boiling points, to local changes in composition [96]. For example, when a liquid film is heated from below or cooled from above (e.g. by evaporation), frequently convection sets in [93, 95]. This effect was first observed by Bénard [97]. The instability leading to convection is due to temperature-induced gradients in surface tension [93, 98]. It is referred to as the Bénard–Marangoni convection. Temperature-induced and solutal Marangoni effects play a crucial role in many processes such as the evaporation of drops, drying of paint films, dynamics of wetting, and heat and mass transfer. Example 3.5 Tears of wine are commonly observed when filling wine into a clean glass and gently shaking the glass so as to wet the inside of the glass above the filling height. Drops of wine form on the inside of the glass in a regular pattern [91, 99]. The classic explanation is as follows: when the wine is poured into the glass, it climbs up the walls of the glass to wet them, much like in capillary rise (Section 6.3.1). Liquid evaporates from the film. Preferentially, ethanol evaporates because it has a lower vapor pressure than water. Evaporation is more efficient near the contact x

Gas

Figure 3.9 A gradient in surface tension in the x-direction causes a flow of the liquid.

vx z

Liquid

4 Carlo Marangoni, 1840–1925 Italian physicist, professor at a lyceum in Florence.

3.7 Exercises

line. As a result, ethanol in the meniscus region becomes more depleted than in the bulk. A gradient in surface tension is established because water has a higher surface tension than ethanol. The surface tension at the bottom of the film is lower than further up on the wall. As a result, the higher regions of the film pull on the lower regions. In the higher regions, the combination of Marangoni stress, capillary flow, and gravity leads to an instability [100]. A horizontal, convective flow sets in so that liquid concentrates in tears.

3.6 Summary ●

●

●

To apply the thermodynamic formalism to surfaces, Gibbs defined the ideal dividing plane, which is infinitely thin. Excess quantities are defined with respect to a particular position of the dividing plane. The most important quantity is the interfacial excess, which is the amount of substance enriched or depleted at an interface per unit area. For a pure liquid, the Gibbs dividing plane is conveniently positioned so that the surface excess is zero. Then the surface tension is equal to the surface Helmholtz energy and the interfacial Gibbs energy: f 𝜎 = g𝜎 = 𝛾. For solutions, the Gibbs dividing plane is conveniently positioned so that the surface excess of the solvent is zero. Then the Gibbs adsorption isotherm Eq. (3.64) relates the surface tension to the amount of solute adsorbed at the interface: Γ=−

a 𝜕𝛾 RT 𝜕a

When the solute is enriched at the interface, the surface tension decreases upon addition of a solute. When the solute avoids the interface, the surface tension increases when the substance is added.

3.7 Exercises 3.1

Express the interfacial tension by the interfacial Helmholtz energy.

3.2

Calculate the surface entropy and the internal surface energy of ethyl acetate (C2 H5 OCOCH3 ) at 25 ∘ C [10]. The surface tensions at 10, 25, and 50 ∘ C are 25.13, 23.39, and 20.49 mN/m, respectively.

3.3

Estimate the orientational surface entropy per unit area for water (molar volume Vm = 18 cm3 ∕mol), ethanol (58 cm3 ∕mol), and toluene (107 cm3 ∕mol). Following Good [101], we assume that “on bringing a molecule from the interior to the surface, just half the possible orientation is lost.” Assume further that the oriented surface layer is just one molecule deep. Compare it to experimental results (Table 3.2).

51

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3 Thermodynamics of Interfaces

3.4

The surface tension of water with a mole fraction of 0.001, 0.002, 0.003, …, 0.007 of n-propanol is 67.4, 64.4, 61.9, 59.7, 57.7, 55.8, and 54.1 mN/m at 25 ∘ C, respectively. Estimate the surface excess of propanol at mole fractions of 0.002, 0.004, and 0.006. Does the surface excess increase linearly with the mole fraction?

3.5

Soap bubbles. To stabilize a bubble, surfactants are usually added to water. Assume we add a surfactant to a concentration of 2 mM. At this concentration, we have a positive surface excess. On average, each surfactant molecule occupies a surface area of 0.7 nm2 . Estimate the change in pressure inside a soap bubble with a radius of 1 cm compared to a hypothetical bubble formed from pure water.

3.6

Equation (3.65) can be used to describe the adsorption of gases to surfaces. Then it can be written as follows: 1 d𝛾 (3.68) Γ=− RT d ln P with P being the partial pressure of the adsorbing gas. Derive Eq. (3.68) from Eq. (3.65).

53

4 Charged Interfaces and the Electric Double Layer 4.1 Introduction This chapter deals with charged solid surfaces in liquids. Electric charges are important to stabilize suspension, emulsions, and foams. “Stabilize” means: electrostatic double-layer repulsion prevents dispersed particles, droplets, or bubbles to aggregate or coalesce. In physiology and biophysics, electric potentials at and across biological membranes are essential for transmission of nerve pulses, energy conversion, and transport of nutrients. In electrochemistry, the transfer of charge and chemical reactions depend on the ion distribution at interfaces. The concentration of charged species near interfaces is influenced by surface charges, which are relevant for heterogeneous catalysis, ion exchange, and the properties of soils. The most important liquid is water. Because of its high dielectric constant, water is a good solvent for ions. For this reason, most surfaces in water are charged, while in organic liquids with low dielectric permittivity, charging is negligible. Different processes can lead to charging. Ions adsorb to a surface or dissociate from a surface. A protein might, for instance, expose an amino group on its surface. This can become protonated and thus positively charged (∼NH2 + H+ ⇌ ∼NH+3 ). Oxides are often negatively charged in water due to the dissociation of a proton from a surface hydroxyl group (∼OH ⇌ ∼O− + H+ ). Surface potentials originating by spontaneous dissociation or adsorption of ions are typically below 100 mV. In contrast, in electrochemistry, electrode potentials of the order of 1 V are externally be applied. Surface charges cause an electric field. This electric field attracts counterions. The layer of surface charges and counterions is called electric double layer. To gain an impression of the structure and the potential distribution of an electrified interface, we consider a negatively charged surface in an aqueous electrolyte (Figure 4.1). Water molecules are plotted as gray spheres with an arrow; the arrow indicates the dipole moment. Cations are usually solvated by a shell of water molecules. Some ions adsorb directly, breaking up their solvation shell in the process. In this case, chemical interactions play an important role. Usually (although not in our figure), anions bind more easily to a surface, in particular metal surface, because they are not as strongly hydrated as cations. To balance the negative surface charges, some cations adsorb with their hydration shell intact. They are still separated by a water Physics and Chemistry of Interfaces, Fourth Edition. Hans-Jürgen Butt, Karlheinz Graf, and Michael Kappl. © 2023 WILEY-VCH GmbH. Published 2023 by WILEY-VCH

4 Charged Interfaces and the Electric Double Layer

Bulk water

Diffuse layer

54

Hydrated cation

OHP IHP Solid Surface charge

Adsorbed cation

Figure 4.1 Schematic structure of an electric double layer in aqueous electrolyte. The electric double layer can be divided into a compact layer of counterions, also called Stern layer, and a diffuse layer. Within the Stern layer, one can distinguish an inner Helmholtz plane (IHP) and outer Helmholtz plane (OHP).

layer from the surface. In honor of the work of Ludwig Helmholtz1 on electric double layers, the layers of directly adsorbed and hydrated counterions are called inner and outer Helmholtz layers,, respectively. Helmholtz originally suggested a simple model of an electric double layer, in which the counterions bind to the surface and neutralize the surface charges much like in a plate capacitor [102]. The inner layer is also called compact or Stern layer. Beyond the Helmholtz layer begins a diffuse layer. Thermal fluctuations tend to drive counterions away from a surface. The balance between electrostatic attraction and thermal fluctuations leads to the formation of a diffuse layer, which is more extended than a molecular layer. In the years 1910–1917, Gouy2 and Chapman3 derived a theory to describe the diffuse layer [103–105]. In 1924 Otto Stern4 combined both ideas to obtain a comprehensive model of the electric double layer [106], the so-called Gouy–Chapman–Stern model. It divides the double layer into two parts: a diffuse Gouy–Chapman part in which the ions are distributed according to the Poisson–Boltzmann distribution, and a compact, inner layer, which accounts for the finite size of the ions that are closest to the charged surface. While the compact layer is only few molecules thick, the diffuse layer can extend up to few 100 nm, depending on salt concentration. Beyond the diffuse layer, the concentrations of cations and anions are like in bulk electrolyte. At the top left of Figure 4.1, an anion is shown as a reminder that in aqueous electrolyte we 1 Hermann Ludwig Ferdinand von Helmholtz, 1821–1894; German physicist and physiologist, professor in Königsberg, Bonn, Heidelberg, and Berlin. 2 Louis Georges Gouy, 1854–1926; French physicist, professor in Lyon. 3 David Leonard Chapman, 1869–1958; English chemist, professor in Manchester and Oxford. 4 Otto Stern, 1888–1969; German physicist, professor in Hamburg; Nobel Prize in Physics, 1943.

4.2 Mathematical Description of the Electric Double Layer

always have ions present, either due to the presence of a background salt or the dissociation of water. How surface charges are generated, which ions adsorb, how strongly they adsorb, and how the electric potential decreases with distance depends on the specific surface material, the concentration, and the type of added salt and on the pH. In the following, we start by introducing mathematical models of the ion and potential distribution in the electric double layer. In the section of “electrocapillarity,” we link electric properties of the double layer to thermodynamic quantities such as surface tension, and we get to know an experimental method to measure these properties. Electrocapillary measurements are, however, practically restricted to liquid metals, such as mercury [107]. Other methods to measure surface charge and potential are described in 4.3.3. Then we discuss, how surface charges are formed for different materials. Finally, we discuss electrokinetic phenomena. They occur, when an electrolyte flows over a charged surfaces. Charging phenomena are particularly relevant in water. In nonpolar liquids, with low dielectric permittivity, ions are rare. For an ionizable group to dissociate or a salt to dissolve, a high permittivity is required. It is energetically unfavorable for a free charge to be in a medium with low dielectric permittivity. Quantitatively, the electrostatic energy required to increase the charge of a sphere of radius r deep within a medium of dielectric permittivity 𝜀 from zero up to Q is [108]: U=

Q2 8𝜋𝜀𝜀0 r

(4.1)

The energy U is also called “self-energy.” It can also be interpreted as the energy required to generate the electric field around a charge Q. A unit charge of r = 0.18 nm in vacuum at 20 ∘ C has an energy of 158 kB T. In water with 𝜀 = 80.1, the energy is reduced to 2 kB T. For this reason, charging is a less-common phenomenon in oils. One should, however, keep in mind that large potentials can build up because of the negligible dissociation of salts and the resulting low conductivity in oil (review [109, 110]).

4.2 Mathematical Description of the Electric Double Layer In this section, we introduce analytical models to describe the electric double layer. Guiding questions are the following: What is the structure of the electric double layer? How are the ions distributed and how does the electric potential decay with increasing distance from the surface? How is the surface potential related to the surface charge? We start with the Gouy–Chapman model of the electric double layer. It is based on the Poisson5 equation of electrostatics and Boltzmann6 statistics. Solvents are treated as continuous media with a certain dielectric constant, ignoring the molecular nature of the liquid. 5 Denis Poisson, 1781–1840; French mathematician and physicist, professor in Paris. 6 Ludwig Boltzmann, 1844–1906; Austrian physicist, professor in Vienna.

55

56

4 Charged Interfaces and the Electric Double Layer

4.2.1

The Poisson–Boltzmann Equation

To calculate, the potential distribution in a solution near a charged interface 𝜑(x, y, z), we start with the Poisson equation. In general, charge density and electric potential are related by the Poisson equation: ∇2 𝜑 =

𝜌 𝜕2 𝜑 𝜕2 𝜑 𝜕2 𝜑 + 2 + 2 =− e 2 𝜀𝜀0 𝜕x 𝜕y 𝜕z

(4.2)

Here, 𝜌e is the local electric charge density in C/m3 . The potential 𝜑 is the inner potential of the liquid phase with its bulk potential far away from the interface set to zero. With the Poisson equation, the potential distribution can be calculated once the exact charge distribution is known. The complication in our case is that the ions in solution are free to move. Before we can apply the Poisson equation, we need to know more about their spacial distribution. This information is provided by Boltzmann statistics. According to the Boltzmann equation, the local ion density is given by ci = c0i e−Wi ∕kB T

(4.3)

Here, Wi is the work required to bring an ion in solution from infinite distance to a certain position closer to the surface. Equation (4.3) relates the local ion concentration ci of the ith species to the electric potential at a certain position. For example, if the potential at a certain position in solution is positive, the chance of finding an anion at this position there is increased, while the cation concentration is reduced. Now, we assume that only electric work has to be done. We neglect for instance that the ion must displace other molecules. In addition, we assume that only a 1 : 1 salt is dissolved in the liquid. We further assume that the concentration of this background salt is much higher than the concentration of ions, which have dissociated from the surface to build up the surface charge (otherwise, the number of anions and cations would not be the same). The electric work required to bring a charged cation to a place with potential 𝜑 is W + = e𝜑. For an anion, it is W − = −e𝜑. The local anion and cation concentrations c− and c+ are related with the local potential 𝜑 by the Boltzmann factor: c− = c0 ee𝜑∕kB T and c+ = c0 e−e𝜑∕kB T [104]. Here, c0 is the bulk concentration of the salt in number of salt molecules per volume. The local charge density is ( e𝜑(x, y, z) ) e𝜑(x, y, z) − k T + − kB T B 𝜌e = e(c − c ) = c0 e e −e (4.4) As a reminder that the potential depends on the position we explicitly wrote 𝜑(x, y, z). Substituting the charge density into the Poisson Eq. (4.2) leads to ( e𝜑(x, y, z) ) ce − e𝜑(x, y, z) ∇2 𝜑 = 0 e kB T − e kB T (4.5) 𝜀𝜀0 Often, this equation is referred to as the Poisson–Boltzmann equation. It is a partial differential equation of second order. In most cases, it has to be solved numerically. Only for some simple geometries can it be solved analytically. One such geometry is a planar surface.

4.2 Mathematical Description of the Electric Double Layer

4.2.2

Gouy–Chapman Model for Planar Surfaces

The aim of this section is to calculate the electric potential 𝜑 near a homogeneously charged, planar, infinitely extended interface [103, 105]. The electric surface charge density is denoted by 𝜎. We assume that no Stern–layer is present. Then the diffuse layer potential is equal to the surface potential 𝜑0 = 𝜑(x = 0). This potential depends on the distance normal to the surface x. For the simple case of a planar surface, the potential cannot change in the y- and z-directions because of the symmetry. The derivatives with respect to y and z must be zero. We are left with the Poisson–Boltzmann equation, which contains only the coordinate normal to the plane x: ( e𝜑(x) ) c0 e d2 𝜑 − e𝜑(x) kB T kB T = e −e (4.6) 𝜀𝜀0 dx2 Before we solve this equation for the general case, it is illustrative and, for many applications, sufficient to treat the special case of low potentials. “Low” means, in a strict sense, e |𝜑| ≪ kB T. At room temperature, this is |𝜑| ≤ 25 mV. Fortunately, in many applications, the result is valid even for higher potentials, up to 50–80 mV. For low potentials, we can expand the exponential functions into a series and neglect all but the first (i.e. the linear) term: ( ) c0 e 2c0 e2 d2 𝜑 e𝜑 e𝜑 = 1 + − 1 + ± · · · ≈ 𝜑 (4.7) 2 𝜀𝜀0 kB T kB T 𝜀𝜀0 kB T dx This expression is called the linearized Poisson–Boltzmann equation or Debye–Hückel approximation. Debye7 and Hückel8 used the approximation to describe osmotic properties of strong electrolytes [111]. The general solution of the linearized Poisson–Boltzmann equation is 𝜑(x) = C1 e−𝜅x + C2 e𝜅x with

√ 𝜅=

(4.8)

2c0 e2 𝜀𝜀0 kB T

(4.9)

C1 and C2 are constants, which are defined by the boundary conditions. The boundary conditions require that, at the surface, the potential is equal to the surface potential, 𝜑(x = 0) = 𝜑0 , and that, for large distances from the surface, the potential should disappear 𝜑(x → ∞) = 0. The second boundary condition guarantees that, for very large distances, the potential becomes zero and does not grow infinitely. It directly leads to C2 = 0. From the first boundary condition, we get C1 = 𝜑0 . Hence, the potential is given by 𝜑 = 𝜑0 e−𝜅x

(4.10)

7 Peter Debye, 1884–1966; American physicist of Dutch origin, professor in Zürich, Utrecht, Göttingen, Leipzig, Berlin, and Ithaca; Nobel prize for chemistry, 1936. 8 Erich A.A.J. Hückel, 1886–1980; German physicist and chemist, professor in Marburg.

57

58

4 Charged Interfaces and the Electric Double Layer

The potential decreases exponentially. The decay length is given by λD = 𝜅 −1 . It is called the Debye length. The Debye length decreases with increasing salt concentration. This is plausible because the more ions are in solution, the more effective is the screening of surface charge. For water at 25 ∘ C and for a monovalent salt, the Debye length is 3.04 Å λD = √ c0

(4.11)

with the concentration c0 in mol/l. For example, the Debye length of a 0.1 M aqueous NaCl solution at 25 ∘ C is 0.96 nm. In water λD cannot be longer than 680 nm. Due to the self-dissociation of water (according to 2 H2 O ⇌ H3 O+ + OH− ), the ion concentration cannot decrease below 2 × 10−7 mol/l. Practically, the Debye length even in distilled water is only a few 100 nm due to ionic impurities or a deviation from neutral pH. The pH can, for example decrease after dissolution of carbon dioxide from the atmosphere and the formation of carbonic acid according to CO2 + H2 O ⇌ H2 CO3 ⇌ H+ + HCO−3 . Until now we considered only monovalent, so-called 1 : 1 salts. If ions of higher valency are also present, then the inverse Debye length is given by √ e2 ∑ 0 2 𝜅= cZ (4.12) 𝜀𝜀0 kB T i i i Here, Zi is the valency of the ith ion sort. Please keep in mind: The concentrations have to be given in particles per m3 . Example 4.1 Human blood plasma, that is, blood without red and white blood cells and without thrombocytes, contains 143 mM Na+ , 5 mM K+ , 2.5 mM Ca2+ , 2− 1 mM Mg2+ , 103 mM Cl− , 27 mM HCO−3 , 1 mM HPO2− 4 , and 0.5 mM SO4 . What is the Debye length? We insert c0Na = 861 × 1023 m−3 c0Ca

23

= 15 × 10

−3

m

c0Cl = 620 × 1023 m−3 c0HPO 4

23

= 6 × 10

−3

m

ZNa = 1

c0K = 30 × 1023 m−3

ZK = 1

ZCa = 2

c0Mg

ZMg = 2

ZCl = −1

c0HCO = 163 × 1023 m−3

23

= 6 × 10

−3

m

3

c0SO 4

23

ZHCO3 = −1

−3

ZHPO4 = −2 = 3 × 10 m ZSO4 = −2 ∘ into Eq. (4.12). Considering that at 36 C the dielectric constant of water is 𝜀 = 74.5, we get a Debye length of 0.78 nm. All ions come from the dissociation of salts according to AB → A− + B+ . For this ∑ reason, the sum coi Zi should always be zero (electroneutrality). Inserting the preceding values, we find a surplus of cations of 22 mM. These cations come from the dissociation of organic acids (6 mM) and proteins (16 mM). Figure 4.2 illustrates several features of the diffuse electric double layer. The potential decreases exponentially with increasing distance. This decrease becomes steeper with increasing salt concentration. Using Eq. (4.3), we can calculate the local ion concentration setting Wi = −e𝜑(x) for the counter- and Wi = e𝜑(x) for the coions. The concentration of counterions is strongly increased close to the surface. As a result, the total concentration of ions at the surface and thus the osmotic pressure

4.2 Mathematical Description of the Electric Double Layer

(a)

(b)

Figure 4.2 (a) Potential versus distance for a surface potential of 𝜑0 = 50 mV and different concentrations of a monovalent salt in water. (b) Local co- and counterion concentrations are shown for a monovalent salt at a bulk concentration of 0.1 M and a surface potential of 50 mV. In addition, the total concentration of ions, that is the sum of the co- and counterion concentrations, is plotted.

is increased. That the local ion concentration, and thus, the local pH, near-charged surfaces can be significantly different from the bulk concentration is important in heterogeneous catalysis.

4.2.3

The Full One-Dimensional Case

In many practical cases in colloid science, we can use the low-potential-assumption and it leads to realistic results. Equations are simple and dependencies like the one on the salt concentration can easily be seen. In some cases, however, we have high potentials, and we cannot linearize the Poisson–Boltzmann equation. Therefore, we now treat the general solution of the one-dimensional Poisson–Boltzmann equation and drop the assumption of low potentials. It is convenient to solve the equation with the dimensionless potential y ≡ e𝜑∕kB T. Please do not mix this up with the spacial coordinate y! In this section, we use the symbol “y” for the dimensionless potential. The Poisson–Boltzmann equation for a 1 : 1 salt becomes ) ) c e2 ( y 2c0 e2 1 ( y d2 y = 0 e − e−y = e − e−y = 𝜅 2 sinh y 2 𝜀𝜀0 kB T 𝜀𝜀0 kB T 2 dx

(4.13)

We used sinh y = (ey − e−y ) ∕2. To solve the differential equation, we multiply both sides by 2 dy∕dx: dy d2 y dy 2 =2 𝜅 sinh y dx dx2 dx ( )2 d dy The left side is equal to dx . We insert this expression and integrate: dx 2

( )2 dy dy d dx′ = 2𝜅 2 sinh y dx′ ⇔ ∫ dx′ dx′ ∫ dx′ ( )2 dy = 2𝜅 2 sinh y′ dy′ = 2𝜅 2 cosh y + C1 ∫ dx

(4.14)

(4.15)

59

60

4 Charged Interfaces and the Electric Double Layer

C1 is an integration constant. It is determined by the boundary conditions. At large distances, the dimensionless potential y and its derivative dy∕dx are zero. Since cosh y = 1 for y = 0, this constant is C1 = −2𝜅 2 . It follows that ( )2 √ dy dy = 2𝜅 2 (cosh y − 1) ⇒ = −𝜅 2 cosh y − 2 (4.16) dx dx A minus sign appears in front of the square root because y has to decrease for a positive potential with increasing distance, i.e. y > 0 ⇒ dy∕dx < 0. Now, we recall √ the mathematical identity sinh 2y = 12 (cosh y − 1). Thus, dy y = −2𝜅 sinh dx 2 Separation of variables and integration leads to dy dy′ = −2𝜅 dx′ ⇒ ′ y = −2𝜅 dx ⇒ ∫ ∫ sinh 2 sinh y2 ( y) 2 ln tanh = −2𝜅x + 2C2 4 C2 is another integration constant. Written explicitly, we get ( y∕4 ) e − e−y∕4 ln = −𝜅x + C2 ey∕4 + e−y∕4 Multiplying the denominator and numerator (in brackets) by ey∕4 leads to ( y∕2 ) e −1 ln = −𝜅x + C2 ey∕2 + 1

(4.17)

(4.18)

(4.19)

(4.20)

Using the dimensionless surface potential y0 = y(x = 0) = e𝜑0 ∕kB T, we can determine the integration constant ( y ∕2 ) e 0 −1 ln = C2 (4.21) ey0 ∕2 + 1 Substituting the results in Eq. (4.20) [( )( )] ( y∕2 ) ( y ∕2 ) ey∕2 − 1 ey0 ∕2 + 1 e −1 e 0 −1 ln − ln = ln ( )( ) = −𝜅x ey∕2 + 1 ey0 ∕2 + 1 ey∕2 + 1 ey0 ∕2 − 1 ( y∕2 )( ) e − 1 ey0 ∕2 + 1 −𝜅x ⇒e = ( (4.22) )( ) ey∕2 + 1 ey0 ∕2 − 1 Solving the equation for ey∕2 leads to the alternative expression ey∕2 =

ey0 ∕2 + 1 + (ey0 ∕2 − 1) e−𝜅x ey0 ∕2 + 1 − (ey0 ∕2 − 1) e−𝜅x

(4.23)

Let us compare the results obtained with the linearized Poisson–Boltzmann equation (4.10) with the full solution Eq. (4.23). Figure 4.3 shows the potential calculated for a monovalent salt at a concentration of 20 mM in water. The Debye length is 2.2 nm. For a low-surface potential of 50 mV, both results agree well. When increasing the surface potential to 100, 150, or even 200 mV, the full solution leads to lower potentials. At distances below ≈ λD ∕2, the decay is therefore steeper than just

4.2 Mathematical Description of the Electric Double Layer

Linear Full

100 Potential φ (mV)

Figure 4.3 Potential versus distance for surface potentials of 50, 100, 150, and 200 mV with 20 mM monovalent salt at 25 ∘ C. The full solution Eq. (4.23) and the solution of the linearized Poisson– Boltzmann equation (4.10) are shown.

10

0

1

2 3 4 Distance x (nm)

5

6

the exponential decay. This steep decay at small distances becomes progressively more effective at higher and higher surface potentials, which lead to saturation behavior.

4.2.4

The Electric Double Layer around a Sphere

In many applications, the electric double layer around spherical particles is studied. If the radius of the particle, Rp , is much larger than the Debye length, we can treat the double layer as planar. Otherwise, we have to consider the Poisson–Boltzmann equation for spherical symmetry: ( e𝜑(r) ) c0 e d2 𝜑 2 d𝜑 − e𝜑(r) kB T kB T + = e − e (4.24) r dr 𝜀𝜀0 dr 2 Here, r is the radial coordinate. For low potentials, we can linearize the differential equation: d2 𝜑 2 d𝜑 + = 𝜅2𝜑 r dr dr 2

(4.25)

The general solution of the linearized Poisson–Boltzmann equation is C1 −𝜅r C2 𝜅r e + e (4.26) r r The constants C1 and C2 are again determined by the boundary conditions. The constant C2 is zero because 𝜑 → 0 for r → ∞. The boundary condition 𝜑(r = Rp ) = 𝜑0 leads to C1 = 𝜑0 Rp e𝜅Rp , and we finally get [111, 112] 𝜑=

𝜑(r) = 𝜑0

Rp r

e−𝜅(r−Rp )

(4.27)

Since Debye and Hückel were the first to derive Eq. (4.27), the model is sometimes called Debye–Hückel model. For high potentials, one has to solve the full Poisson–Boltzmann equation in radial coordinates numerically or use approximations [113, 114].

61

62

4 Charged Interfaces and the Electric Double Layer

4.2.5

The Grahame Equation

Let us return to planar surfaces. In many cases, we have an idea about the number of charged groups on a surface and then we might want to know the potential. The question is: how are surface charge 𝜎 and surface potential 𝜑0 related? Grahame9 derived an equation between 𝜎 and 𝜑0 based on the Gouy–Chapman theory. We can deduce the equation easily from the so-called electroneutrality condition. This condition demands that the total charge, that is, the surface charge plus the charge of the ions in the whole double layer, must be zero. The total charge in ∞ the double layer is ∫0 𝜌e dx, and we get [115] ∞

𝜎=−

∫0

𝜌e dx

(4.28)

Using the one-dimensional Poisson equation and the fact that at large distances, the potential and thus its gradient are zero (d𝜑∕dx|z=∞ = 0), we get ∞

𝜎 = 𝜀𝜀0

∫0

d2 𝜑 d𝜑 || dx = −𝜀𝜀0 2 dx ||x=0 dx

With dy∕dx = −2𝜅 sinh (y∕2) and ( ) d e𝜑∕kB T dy e d𝜑 = = dx dx kB T dx we get the Grahame equation: ( ) √ e𝜑0 𝜎 = 8c0 𝜀𝜀0 kB T sinh 2kB T

(4.29)

(4.30)

(4.31)

In case of nonmonovalent electrolytes, the surface charge density is given by [115] √ ( Z e𝜑 ) ∑ − ki T0 0 B 𝜎 = ± 2𝜀𝜀0 kB T ci e −1 (4.32) i

The plus sign is for positive, and the minus sign for negative surface potentials. For low potentials, we can expand sinh into a series (sinh x = x + x3 ∕3! + · · · ) and ignore all but the first term. That leads to the simple relationship: 𝜀𝜀 𝜑 𝜎= 0 0 (4.33) λD Example 4.2 On the surface of a certain material, there is one ionized group per (4 nm)2 in aqueous solution containing 10 mM NaCl. What is the surface potential? With a Debye length of 3.04 nm at 25 ∘ C, the surface charge density in SI units is 𝜎 = 1.60 × 10−19 A s∕16 × 10−18 m2 = 0.01 A s∕m2 . With this we get 𝜑0 =

𝜎λD 0.01 A s∕m2 ⋅ 3.04 × 10−9 m = = 0.0438 V 𝜀𝜀0 78.4 ⋅ 8.85 × 10−12 A s∕V∕m

(4.34)

Using the Grahame equation (4.31) we get a surface potential of 39.7 mV. 9 David Caldwell Grahame, 1912–1958; American physical chemist, professor at Amherst College.

Figure 4.4 Surface potential versus surface charge calculated with the full Grahame equation (4.31, continuous line) and with the linearized version Eq. (4.33, dotted).

Surface potential φ0 (mV)

4.2 Mathematical Description of the Electric Double Layer

120 100 80 1 mM 60 40

10 mM 0.1 M

20 0 0,00

0,01

0,02

0,03

Surface charge σ (C/m2)

Figure 4.4 shows the relationship between the surface potential and surface charge for different concentrations of a monovalent salt. We see that for small potentials, the surface charge density is proportional to the surface potential. Depending on the salt concentration, the linear approximation (dashed) is valid till 𝜑0 ≈ 40–80 mV. At high salt concentration, more surface charge is required to reach the same surface potential than for a low salt concentration. One reason why the Grahame equation is so important is that we can use it to derive the capacitance of the diffuse layer. The capacitance can be measured. The measured capacitance can be compared to the theoretical result to verify the whole theory and to obtain the surface charge. In general, the differential capacitance between two regions of separated charges is defined as dQ∕dU. Here, Q is the charge on each electrode and U is the voltage. The capacity of an electric double layer per unit area is thus √ ( ) ( ) 2e2 c0 𝜀𝜀0 e𝜑0 𝜀𝜀 e𝜑0 d𝜎 A Cd = = cosh = 0 cosh (4.35) d𝜑0 λD kB T 2kB T 2kB T The index “d” is a reminder that we calculated the capacitance of the diffuse layer in the Gouy–Chapman model. For sufficiently small surface potentials, we can expand cosh into a series (cosh x = 1 + x2 ∕2! + x4 ∕4! + · · · ) and consider only the first term. Then we get CdA =

𝜀𝜀0 λD

(4.36)

It is instructive to compare this to the capacitance per unit area of a plate capacitor 𝜀𝜀0 ∕d. Here, d is the separation between the two plates. We see that the electric double layer behaves like a plate capacitor, in which the distance between the plates is given by the Debye length! The capacity of a double layer – that is the ability to store charge – rises with increasing salt concentration because the Debye length decreases. A To avoid confusion, we should point out that CGC as defined above is the differential capacitance. The integral capacitance per unit area is 𝜎∕𝜑0 . Experimentally, the differential capacitance is easier to measure.

63

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4 Charged Interfaces and the Electric Double Layer

4.2.6

Gibbs Energy of the Electric Double Layer

The energy of an electric double layer plays a central role in colloid science, for instance to describe the properties of charged polymers (polyelectrolytes) or the interaction between colloidal particles. In order to calculate the Gibbs energy of a Gouy–Chapman layer, we split its formation into three steps [116, 117]. In reality, it is not possible to do these steps separately but we can do the Gedanken experiment without violating any physical principle. First, the uncharged colloidal particle is brought into an infinitely large solution. If the dielectric constant of the solution is high enough, such as in water, a charge builds up on the surface of the particle. The charge builds up, for example by the dissociation of ions. For example, if a surface carboxyl group is brought in water, an H+ dissociates leaving a negative surface behind. Alternatively, ions from solution bind to the surface. For example, a gold surface binds a Cl− ion and the surface becomes negative. It is important to realize that the buildup of charge is spontaneous, driven by chemical forces. Thus, the Gibbs energy of the system decreases. To calculate the chemical energy, we note that the dissociation (or binding) of ions does not proceed forever. The more ions dissociate, the higher the electric potential becomes. This potential prevents further ions from dissociating from the surface. The process stops when the chemical energy is equal to the electrostatic energy. The electrostatic energy of one ion of charge Q at this point is simply Q𝜑0 . The chemical energy of this ion is −Q𝜑0 . Hence, the Gibbs energy per unit area for the formation of an electric double layer is −𝜎𝜑0 . Second, we bring the counterions to the surface. They are supposed to go directly to the surface at x = 0. The number of counterions is equal to the number of ions in the diffuse double layer. The first counterions are still attracted by the full surface potential. Their presence, however, reduces the surface potential and the following counterions only notice a reduced surface potential. To bring counterions to the surface, the work dG = 𝜑′0 d𝜎 has to be performed, where 𝜑′0 is the surface potential at a certain time of charging – or better discharging – process. The total energy we gain (and the double layer loses) is 𝜎

∫0

𝜑′0 d𝜎 ′

(4.37)

In a third step, the counterions are released from the surface. Stimulated by thermal fluctuations, they partially diffuse away from the surface and form the diffuse double layer. The entropy increases (which decreases Gibbs energy) and, at the same time, the energy increases (which increases Gibbs energy). One can show that both terms compensate, so that the third step does not contribute to the Gibbs energy. Summing up all contributions, we obtain the total Gibbs energy of the diffuse double layer per unit area: 𝜎

g = −𝜎𝜑0 +

∫0

𝜑′0 d𝜎 ′

Now, we apply the mathematical identity ) ( ) ( d 𝜑′0 𝜎 ′ = 𝜎 ′ d𝜑′0 + 𝜑′ d𝜎 ′ d 𝜑′0 𝜎 ′ = 𝜎 ′ d𝜑′0 + 𝜑′0 d𝜎 ′ ⇒ ∫ ∫ ∫ 0

(4.38)

4.2 Mathematical Description of the Electric Double Layer

and write 𝜎𝜑0

g = −𝜎𝜑0 +

∫0

d(𝜎 ′ 𝜑′0 ) −

𝜑0

∫0

𝜎 ′ d𝜑′0 = −

𝜑0

∫0

𝜎 ′ d𝜑′0

The integral can be solved with the help of Grahame’s equation (4.31): ( ) 𝜑0 𝜑0 √ e𝜑′0 ′ g=− 𝜎 d𝜑0 = − 8c0 𝜀𝜀0 kB T sinh d𝜑′0 ∫0 ∫0 2kB T [ ( )]𝜑0 √ e𝜑′0 2kB T = − 8c0 𝜀𝜀0 kB T cosh e 2kB T 0 [ ( ) ] e𝜑0 = −8c0 kB TλD cosh −1 2kB T

(4.39)

(4.40)

For low potentials, we can use the simpler relation (4.33) and get 𝜑0

g = −𝜅𝜀𝜀0

∫0

𝜑′0 d𝜑′0 = −

𝜅𝜀𝜀0 2 1 𝜑0 = − 𝜎𝜑0 2 2

(4.41)

The Gibbs energy of an electric double layer is negative because it forms spontaneously. Roughly, it increases in proportion to the square of the surface potential. Example 4.3 Estimate the energy per unit area of an electric double layer for a surface potential of 40 mV in an aqueous solution containing 0.01 M monovalent ions. Inserting 𝜎 = 𝜀𝜀0 𝜑0 ∕λD (Eq. (4.33)) into Eq. (4.41) leads to g=−

𝜀 𝜀0 2 𝜑 2λD 0

With a Debye length of 3.04 nm, we get g=−

78.4 ⋅ 8.85 × 10−12 A s/V∕m 2 ⋅ 3.04 × 10−9 m

⋅ (0.04 V)2 = −0.183 × 10−3 J∕m2

Compared to typical surface tensions of liquids, this is small.

4.2.7

Limitations of the Poisson–Boltzmann Theory

In the treatment of the diffuse electric double layer, several assumptions were made [118–120]: ●

The finite size of the ions was neglected [121–123]. In particular, close to the surface, this is a daring assumption because the ion concentration can get very high. For example, if we have a surface potential of 100 mV, the local counterion concentration is increased by a factor of ≈ 50. At a bulk concentration of 0.1 M, the Poisson–Boltzmann theory predicts a local concentration at the surface of roughly 5 M. Then the ions have an average distance less than 1 nm. Considering the diameter of an ion with its hydration shell of typically 3–6 Å, the detailed molecular structure should become significant. In general, all non-Coulombic interactions and thus ion-specific effects were disregarded, e.g. van der Waals attraction between the ions and the walls.

65

66

4 Charged Interfaces and the Electric Double Layer ●

●

●

●

●

●

Poisson–Boltzmann theory is a mean field theory. Each ion is supposed to interact only with the average electrostatic field of all its neighbors, not with the individual neighbors. However, under certain conditions, ions coordinate their mutual arrangements to lower the free energy even further [124–126]. The ions in a solution were considered as a continuous charge distribution. We ignored their discrete nature, namely that they can only carry a multiple of the unit charge. In particular for di- or trivalent ions, this can lead to strong deviations [123, 127, 128]. Also, the surface charge is assumed to be homogeneous and smeared out. In reality, it is formed by individual adsorbed ions or charged groups [119, 129, 130]. In general, the dielectric constants of the liquid and the solid are not the same. Then the normal component of the electric field is discontinuous at the interface. As a result, ions experience image charges [119, 125]. In particular, for metal surfaces, image charge effects can be strong. They are ignored in Poisson–Boltzmann theory. The solvent is supposed to be continuous, and we took the permittivity of the medium to be constant. This is certainly a rough approximation because polar molecules are hindered from rotating freely in the strong electric field at the surface. In addition, the high concentration of counterions in the proximity of the surface can change the permittivity drastically. For water, more realistic values are ϵ ≈ 6 for the first layer of molecules and ϵ ≈ 30 for the second [131]. Surfaces are assumed to be flat on the molecular scale. In many cases, this is not a reasonable assumption. If we, for instance, consider a biological membrane in a physiological buffer, the ion concentration is roughly 150 mM leading to a Debye length of 0.8 nm. Charges in phospholipids are distributed over a depth of up to 8 Å. In addition, the molecules thermally jump up and down so that the charges at the membrane surface are distributed over a depth of almost 1 nm. Ion-specific effects are not considered in the Poisson–Boltzmann theory. For example, metals tend to adsorb anions more strongly than cations. The reason is that anions are not as strongly hydrated as cations. Upon binding, they do not need to give up their hydration shell. A specific effect of ions was already discovered in 1888. Lewith and Hofmeister observed great differences between the minimum concentrations of various salts required to precipitate a given protein from solution [132, 133]. The precipitation capacity of a salt was largely given by the specific anion. They could arrange the anions in a series according to their precipitation capability. This series is called the Hofmeister series. Later it was discovered that the same capacity of a specific ion to precipitate proteins was also manifest in very different effects (review [134]). The effect is still not fully understood. Two contributions are the different abilities of ions to arrange the water molecules around them and their different van der Waals interactions [135, 136]; anions have stronger van der Waals interactions than cations because of the additional electron and their stronger polarizability [137].

Despite these strong assumptions, the Poisson–Boltzmann theory describes electric double layers surprisingly well. The reason is that errors partly compensate each other. For aqueous solutions, Poisson–Boltzmann theory provides relatively good

4.2 Mathematical Description of the Electric Double Layer

predictions for monovalent salts at concentrations below 0.1 M and for potentials below 50–80 mV. This is helpful in colloid science. The applicability in electrochemistry, where often potentials far above 0.1 V are applied and where metal surface tend to attract ions by image charge effects, is, however, limited.

4.2.8

Stern Model

Fortunately, there is a relatively simple extension of the Gouy–Chapman theory, which accounts for many experimental observations. This extension was proposed by Stern. Stern combined the ideas of Helmholtz and that of a diffuse layer [106]. In the Gouy–Chapman–Stern model, we divide the double layer into two parts: an inner part, the Stern layer, and an outer part, the diffuse layer. The Stern layer is an adsorbed layer of ions (Figure 4.5). Its thickness is equal to the distance of closest approach for the counterions δ. This distance can be identified with the ion radius, or, if more than one species is adsorbed, with the number of average radius of the adsorbed ions. In contrast, the Gouy–Chapman layer consists of mobile ions, which obey the Poisson–Boltzmann statistics. The total excess charge density on the solution side of the double layer is given by 𝜎i + 𝜎d . Here, 𝜎i includes all directly adsorbed charges from the inner Helmholtz layer (Figure 4.1). 𝜎d includes all charges from the diffuse layer and the ones from the outer Helmholtz layer [138]. These charges just compensate the surface charges to ensure electroneutrality: ( ) 𝜎 = − 𝜎i + 𝜎d (4.42) To obtain the potential distribution in the inner region, we picture the adsorbed ions as spheres with a point charge at their center. Thus, the region up to the ion radius (0 ≤ x ≤ δ) is practically charge free. As a result, the electric potential decreases linearly from 𝜑0 to 𝜑δ . The potential distribution in the Stern layer is like in a plate capacitor with capacitance per unit area 𝜀𝜀 A CSt = 0 (4.43) δ A Here, 𝜀 is the relative permittivity in the Stern layer. CSt is referred to as the Helmholtz capacitance. The surface potential is the sum of the potential across the diffuse layer, 𝜑δ , and the potential across the Stern layer. Figure 4.5 Illustration of the electric potential at a Stern double layer. In this case, we assumed a positively charged surface with adsorbed anions. The adsorbed anions are plotted larger than the other ions to better illustrate the thickness of the Stern layer.

φ0 Liquid

Solid

φδ

δ

x

67

68

4 Charged Interfaces and the Electric Double Layer

We can use all equations of Sections 4.2.2, 4.2.3, 4.2.4, 4.2.5, 4.2.6 and only need to replace 𝜑0 by 𝜑δ . The charge up to the beginning of the diffuse layer is, however, reduced to 𝜎 + 𝜎i . For example, the full Grahame equation (4.32) becomes √ ( Z e𝜑 ) ∑ − ki Tδ 0 B 𝜎 + 𝜎i = −𝜎d = ± 2𝜀𝜀0 kB T ci e −1 (4.44) i

The plus sign applies for positive, and the minus sign for negative surface potentials. An important quantity with respect to experimental verification is the differential capacitance of the electric double layer. In the Stern picture, it is composed of two A capacitors in series: the capacity of the Stern layer, CSt , and the capacitance of the diffuse Gouy–Chapman layer, CdA . The total capacitance per unit area, CA = d𝜎∕d𝜑0 , is given by 1 1 1 = A + A CA CSt Cd

(4.45)

A Let us estimate CSt using the simple equation for a plate capacitor. The two plates are formed by the interface and by the adsorbed ions. Denoting the radius of the adsorbed ions by Rion , the distance is in the order of Rion ≈ 1 Å. The capacitance A per unit area of the Stern layer is CSt = 𝜀𝜀0 ∕Rion . The permittivity at the surface is reduced [131] and typically of the order of 𝜀 ≈ 6 for water. Therefore, we estimate a A capacitance for the Stern layer of CSt = 0.53 F/m2 = 53μF/cm2 . Experimental values 2 are typically 10–100 μF/cm .

4.3

Experimental Characterization of Charged Interfaces

In this section, we describe methods to measure the properties of electrified interfaces. We start, however, with theory. The reason is that we need to define precisely what exactly is measured. Only after a clear definition, we turn to describe how the surface charge of metals and insulators can be measured, what are typical values for different materials, and which factors determine surface charges.

4.3.1

Types of Potentials

We start with a precise definition of different types of electrical potentials. Electrical potential differences between two points, say A and B, are defined as the electrical work done in transporting a point charge from A to B; the potential is the work divided by the charge. To each thermodynamically stable phase, we can assign an inner or Galvani potential 𝜑 [139]. The inner potential is defined by the work, which is required to bring a test charge from an infinite distance and from vacuum into the inner of the phase far from the phase boundary. The transport of the test charge can be pictured as occurring in two steps (Figure 4.6): first, the charge is brought close (≈ 1 μm) to the interface. This step is related to the Volta or outer potential 𝜓. In the second step, the charge arrives at the inner phase passing through the interface. The associated potential is known as the

4.3 Experimental Characterization of Charged Interfaces

Figure 4.6 Illustration of the inner (Galvani) 𝜑, outer (Volta) 𝜓, and surface potential 𝜒.

Vacuum ψ

∞

φ

≈1 µm

χ

Medium

surface potential jump 𝜒 = 𝜑 − 𝜓 (also called surface potential, surface electrical potential, etc.). It is determined by dipoles aligned at the interface and by surface charges. In particular in water, the aligned water molecules contribute substantially to the surface potential jump 𝜒. The outer potentials of individual phases and outer potential differences Δ𝜑 between two phases can be measured. In contrast, the inner potential and accordingly the surface potential of a single phase cannot be measured. Explanation: If we bring a test charge (an ion or an electron) into a medium, there is always some chemical work in addition to the electrical work. It can be due to van der Waals forces or image charge effects. There is no way of distinguishing the chemical from the electrical work as long as the chemical environment is different in the two phases. For the same reason, differences in the inner potential Δ𝜑 between two materials and differences in the surface potential Δ𝜒 cannot be measured. = μ𝛼i + Zi FA 𝜑 is defined as the work required to The electrochemical potential μ𝛼∗ i bring a particle of the sort i from vacuum into the phase 𝛼. As described above, it is the sum of the chemical and the electrostatic contribution. Experimentally it is, however, impossible to separate them. Only the combination of both is measurable. As an example, we consider the potential in an electrochemical cell (Figure 4.7). For the sake of simplicity, electrode A′ is made of the same metal A as the connection to the second electrode. We assume that the conductivity in the metals is so high that we have the same potential everywhere. The measured potential corresponds to the difference in the inner potentials: ′

U = 𝜑A − 𝜑A

(4.46)

Figure 4.7 Schematic of the electrochemical cell considered. The voltage meter is supposed to have infinite resistance.

U

A′

A

B C

69

70

4 Charged Interfaces and the Electric Double Layer

One could object that now differences of the inner potential can be measured, although we just stated that this is impossible. In fact, what is measured is the change in the potential difference due to the presence of the other materials. To illustrate this fact, we sum voltages for all transitions in the electrochemical cell. In accordance with Kirchhoffs law, this sum must be zero, leading to ) ( ) ( ) ( ′ ′ U = 𝜑A − 𝜑A = 𝜑A − 𝜑B + 𝜑B − 𝜑C + 𝜑C − 𝜑A (4.47) Thus, the voltage detected depends on the materials B and C. In this context, it is instructive to mention the work function Φ of a metal. Here, eΦ is the minimal work required to transport an electron from the Fermi level of the metal into vacuum just outside the surface. Here “just outside” corresponds to the preceding definition of an outer potential. It means that the electron is far away on the atomic scale, but not subject to outside electric fields; practically micrometers are the right distance [140]. The work function contains a surface term. Therefore, it may be different for different surfaces of a single crystal or for contaminated surfaces. The work function can be measured, for example by ultraviolet photon spectroscopy (for a discussion see Ref. [141]).

4.3.2

Electrocapillarity

Having clarified the definition of electric potentials, we can now discuss one of the classical experiments with respect to charges at interfaces. It was known for a long time that the shape of a mercury drop which is in contact with an electrolyte depends on the electric potential. Lippmann10 examined the electrocapillary effect in 1875 for the first time systematically [142]. He succeeded in calculating the interfacial tension as a function of applied potential, and he measured it with mercury. As it turned out: from plots of the interfacial tension versus applied potential, one can measure the surface charge density. Therefore, fundamental knowledge about the behavior of charged surfaces comes from experiments with mercury. Since mercury is a liquid, the interfacial tension can be measured precisely. Here, we ask the question: How does the interfacial tension of a metal–electrolyte interface depend on an applied electric potential U (Figure 4.8)? The change of the interfacial tension can be calculated with Gibbs adsorption isotherm Eq. (3.59) even when a potential is applied. In order to use the equation, we first need to Figure 4.8 Schematic of a metal–electrolyte interface with an applied potential U.

U

φα

φβ

Metal

Electrolyte

10 Gabriel Lippmann, 1845–1921; French physicist, Nobel Prize in physics 1908.

4.3 Experimental Characterization of Charged Interfaces

identify all mobile charge carriers present. In our case, the mobile charge carriers are the electrons in the metal and the ions in the electrolyte. Since in addition to the chemical potentials μi , also the electrical potential, 𝜑, affects the charged species, electrochemical potentials μ∗i must be used. The Gibbs adsorption isotherm (3.59) for changes at constant temperature is n ∑ dγ = − Γi dμ∗i − Γe dμ∗e

(4.48)

i=1

where dμ∗i = dμi + Zi FA d𝜑 and dμ∗e = dμe − FA d𝜑 The first term refers to the electrolyte. Accordingly, the sum runs over all ion types present in the electrolyte. The second term contains the contribution of the electrons in the metal. Γi and Γe are the interfacial excess concentrations of the ions in solution and of the electrons in the metal, respectively. μi is the chemical potential of the ion type i in the electrolyte, FA is Faradays constant, μ∗e is the electrochemical potential of the electrons with the chemical potential of the electrons in the metal μe . Substitution leads to n n ∑ ∑ dγ = − Γi dμi − FA Γi Zi d𝜑β − Γe dμe + FA Γe d𝜑𝛼 i=1

(4.49)

i=1

Here, 𝜑𝛼 and 𝜑β are the electric (Galvani) potentials in the two phases. What are the correct values of the potentials? In the metal, the potential is the same everywhere and therefore, 𝜑𝛼 has one clearly defined value. In the electrolyte, the potential close to the surface depends on the distance. Directly at the surface, it is different from the potential one Debye length away from it. Only at a large distance away from the surface is the potential constant. In contrast to the electric potential, the electrochemical potential is the same everywhere in the liquid phase assuming that the system is in equilibrium. For this reason, we use the potential and chemical potential far away from the interface. The concentrations of ions and electrons are not independent. Assuming that the whole system is electrically neutral (electroneutrality), one degree of freedom is lost. It is convenient to regard the electrons in the metal as the dependent component. ∑ With the help of the electroneutrality condition Γi Zi = Γe ,, the last term in Eq. ∑ (4.49) can be written as FA Γi Zi d𝜑𝛼 . Then it is possible to summarize the second and fourth term, and we get n n n ∑ ∑ ∑ dγ = − Γi dμi − FA Γi Zi d𝜑β − Γe dμe + FA Γi Zi d𝜑𝛼 i=1

i=1

n ∑ = − Γi dμi − Γe dμe − 𝜎 d(𝜑𝛼 − 𝜑β )

i=1

(4.50)

i=1

This is the fundamental equation for the description of electrocapillarity. In the ∑ last step, we identified 𝜎 = −FA Γi Zi with the surface charge density, which is produced by electrons in the metal and compensated by ions in solution.

71

72

4 Charged Interfaces and the Electric Double Layer

This identification is generally doubtful because the surface excesses Γi depend on the position of the interface. If, however, the electrode is totally polarizable (no electrons or ions are exchanged between the metal and the electrolyte), then the positioning of the interface is at the physical boundary and 𝜎 indeed represents the surface charge density. In general, 𝜑𝛼 − 𝜑β is not equal to the externally applied voltage U. Both may differ by a constant U0 so that U − U0 = 𝜑𝛼 − 𝜑β . A possible voltage drop at the electrode–electrolyte interface and a voltage difference at the contact between the wires connecting the power supply with the metal should be considered too. Since 𝜑𝛼 − 𝜑β and U differ only by a constant, we can replace d(𝜑𝛼 − 𝜑β ) = dU in Eq. (4.50). From the last expression, we obtain the Lippmann equation: 𝜕γ = −𝜎 (4.51) 𝜕U If the interfacial tension can be measured, the charge density is obtained from the slope of the graph γ-vs-U. For the point of zero charge (pzc), the curve has an extremum. A second differentiation leads to the differential capacitance of the electrical double layer per unit area: 𝜕2 γ 𝜕𝜎 =− = −CA (4.52) 2 𝜕U 𝜕U Since CA can only be positive, the extremum is a maximum. Thus, the graph γ-vs.-U has a maximum at the pzc. In the derivation, we assumed that no electrochemical reactions take place, i.e. the electrode should be totally polarizable. A more general derivation of the Lippmann equation, which explicitly considers electrochemical reactions, is found in electrochemistry text books or Refs. [115, 143]. The simplest electrocapillary curves are obtained under the assumption that CA is constant. Then we can integrate Eq. (4.52) from zero surface potential where the applied potential is U0 : U

𝜎=

∫U0

( ) CA dU ′ = CA U − U0

(4.53)

Inserting into Eq. (4.51) and one more integration leads to U

γ − γ0 = −CA

∫U0

( ′ ) )2 CA ( U − U0 dU ′ = − U − U0 2

(4.54)

Here, γ0 is the interfacial tension, where the interface is uncharged and the surface potential accordingly disappears. Thus, we obtain an inverted parabola with a maximum of γ0 at the pzc. The interfacial tension decreases with increasing amount of surface potential. The reason is the increased interfacial excess of counterions in the electric double layer. In accordance with the Gibbs adsorption isotherms, the interfacial tension decreases with increasing interfacial excess. At charged interfaces, ions have an effect similar to surfactants at liquid surfaces. For illustration, one curve calculated with Eq. (4.54) is plotted in Figure 4.10 as a dashed line.

4.3 Experimental Characterization of Charged Interfaces

Figure 4.9 Dropping mercury electrode.

Tunable voltage U Counter electrode

Mercury manometer

Hg

Electrolyte solution

An important instrument to measure electrocapillary curves is the so-called dropping mercury electrode (Figure 4.9) [138, 144]. A capillary of typically 50 μm inner diameter is fed by a head of mercury. It is immersed into the electrolyte. Mercury is pressed into the electrolyte solution under constant pressure. Mercury flows through the capillary and forms a drop at the end until the surface tension no longer supports the weight of the drop. Mercury emerges from the capillary at a typical rate of one drop every 2–10 seconds. From the weight of the released drops (or practically often the time interval between drops) and the radius of the capillary, the surface tension can be calculated. A voltage is applied between the mercury and a counterelectrode in the electrolyte solution to obtain surfaces tension vs. voltage curves.

4.3.3

Examples of Charged Surfaces

Charging of surfaces in contact with water can usually be traced to four sources: (i) The application of an external electric potential as in electrochemistry, (ii) the exchange of ions between surfaces and solution, (iii) the deprotonation or protonation of surface groups, and (iv) the spontaneous adsorption or desorption of ions from / into solution. As examples of these four mechanisms, we discuss the surfaces of mercury, silver iodide, oxides, and mica. Except for the first case, charging of surfaces in aqueous electrolyte is spontaneous. It is energetically favorable for an interface to either bind ions from solution or for surface groups to dissociate and leave a net surface charge behind. The presented mechanisms are not the only ones occurring. Air bubbles or oil drops in water, for instance, are negatively charged [145]. The precise charging mechanism is still debated. One possibility is the adsorption of hydroxyl ions. Many polymer surfaces acquire a negative surface charge in water. One reason is most likely that van der Waals forces between the extended electron shell of anions are stronger than with cations. In addition, the cations are stronger hydrated; upon adsorption to the polymer surface, they would need to give up part of their hydration shell.

73

4 Charged Interfaces and the Electric Double Layer

Mercury

The mercury surface is probably the best characterized surface with respect to its electric properties. The explanation is that mercury is one of the few metals that are liquid at room temperature. Since it is a metal, a voltage can easily be applied. Since it is a liquid, the surface tension can be measured simply and precisely. Then the surface charge can be calculated with the help of the Lippmann equation. Additionally, a fresh surface free of contamination can be continuously produced. If we measure electrocapillary curves of mercury in an aqueous medium which contains KF, NaF, or CsF, the maximum of the electrocapillarity curve, and thus the pzc, remains constant, i.e. neither the cations nor fluoride adsorb strongly to mercury. Another behavior is observed in solutions with KOH, KSCN, and KI (Figure 4.10). The pzc of KI is at more negative potentials than that of KCNS and that of KOH. Explanation: the anions bind specifically to mercury and shift the pzc. Iodide adsorbs more strongly than cyanide and cyanide more strongly than hydroxide. A negative potential must be applied in order to drive the anions away from the surface. Anions bind also to other metals, such as gold, platinum, or silver [146, 147]. The explanation is a strong hydration of cations. A cation would have to give up its hydration shell for an adsorption. This is energetically disadvantageous. Anions are barely hydrated and can therefore bind more easily to metals [148]. Another possible explanation is the stronger van der Waals force between anions and metals. The binding of ions to metallic surfaces is not yet understood and even the idea that cations are not directly bound to the metal, was questioned based on molecular-dynamics simulations [149].

Surface tension (mN/m)

74

400

KOH KCNS

350 KI 300

250 0,4 0,2 0,0 –0,2 –0,4 –0,6 –0,8 –1,0 –1,2 U (V) Figure 4.10 Electrocapillary curves of mercury measured in different aqueous electrolytes at 18 ∘ C. The zero of the applied electric potential was chosen to be at the maximum of the electrocapillary curve for electrolytes such as NaF, Na2 SO4 , and KNO3 , which do not strongly adsorb to mercury. Source: Redrawn after Ref. [115]. The dashed line was calculated with Eq. (4.54) with C A = 0.53 F/m2 (estimated above for the Helmholtz capacitance), γ0 = 400 mN/m and U0 = −0.337 V, which is where the pzc for the KI solution lies.

4.3 Experimental Characterization of Charged Interfaces

Silver Iodide

If AgI or AgCl is submerged in water, a certain number of molecules dissolve (review: Ref. [150]). An equilibrium between the ions in solution and the crystal is established: AgI⇌Ag+ + I−

(4.55)

The concentrations of Ag+ and I− in the solution are low because the solubility prod[ ] [ ] uct K = Ag+ [I− ] ≈ 10−16 M2 is small. Here, Ag+ and [I− ] are the concentrations of Ag+ and I− in water; to be more precise, the concentrations have to be replaced by the respective activities. The surface of AgI can be imagined as a regular lattice of Ag+ and I− ions. With the same number of Ag+ and I− , the surface would be uncharged. It is, however, clear that this does not agree with the case where equal ion concentrations are present also in solution. Iodide has a somewhat higher affinity for the AgI surface than Ag+ . [ ] In the case Ag+ = [I− ] = 10−8 M, silver iodide is negatively charged. The surface can be neutralized by increasing the Ag+ concentration in solution, e.g. by addition of AgNO3 . Already a small concentration of 10−5.5 M AgNO3 is sufficient to reach the pzc of AgI. We then have 10−10.5 M I− in solution, i.e. 100.000 times less than Ag+ . The surface charge of AgI can thus be changed by addition of Ag+ or I− . The two ions are called the “potential determining ions” of AgI. Without proof, we report the quantitative relation between the surface potential and the activity of the potential determining ions. It is called the Nernst equation: [ +] Ag RT 𝜑0 = ⋅ ln [ + ] (4.56) FA Ag pzc The concentration of Ag+ (and thus that of I− ) determines the surface potential. At [ ] the concentration of Ag+ pzc , the surface charge and thus the surface potential is zero (pzc). An increase in the concentration by a factor of 10 causes an increase in the surface potential of 𝜑0 = RT∕FA ⋅ ln 10 = 59 mV at 25 ∘ C. Example 4.4 In an uncharged, perfectly crystalline AgI surface, the distance between Ag+ ions is ≈ 0.4 nm. This corresponds to a surface area for each Ag+ ion of 0.16 nm2 . In every nm2 , there are 6.25 Ag+ ions. How much does the density of the Ag+ ions in an aqueous electrolyte containing 1 mM KNO3 at 25 ∘ C increase, if the potential increases by 100 mV? In order to estimate the surface charge, we use the Grahame equation (4.31). With e𝜑0 1.60 × 10−19 A s ⋅ 0.1 V = = 1.95 2kB T 2 ⋅ 1.38 × 10−23 J∕K ⋅ 298 K and c0 = 6.02 × 1020 l−1 = 6.02 × 1023 m−3 , we get √ 𝜎 = 8 ⋅ 6.02 × 1023 m−3 ⋅ 78.4 ⋅ 8.85 × 10−12 C2 ∕J m ⋅ 1.38 × 10−23 J ⋅ 298 ⋅ sinh 1.95 = 0.0128 C∕m2

75

76

4 Charged Interfaces and the Electric Double Layer

If we divide this value by unit charge, we get an ion density of 0.080 Ag+ ions per nm2 . The number of Ag+ ions at the surface changes only by 1.3%. In the previous section, the mercury electrode has been described. If no redox pairs (e.g. Fe2+ and Fe3+ ) are in solution and if we exclude gas reactions, the mercury electrode is completely polarizable. Polarizable means: If a potential is applied, a current flows only until the electric double layer has formed. No electrons are transferred from mercury to molecules in the solution and vice versa. The other extreme is a completely reversible electrode, for which the AgI electrode is an example. Each attempt to change the potential of an AgI electrode leads to a current because the equilibrium potential is fixed by the concentrations of Ag+ or I− according to the Nernst equation. Oxides

A third mechanism of surface charging dominates for oxides (e.g. SiO2 , TiO2 , Al2 O3 ) [151], proteins, and many water soluble polymers. At the surface of these substances, there are groups that can dissociate. They take up or release a proton depending on pH. Examples are hydroxyl, carboxyl, sulfate, and amino groups. The potential determining ions are OH− and H+ . To calculate the surface potential, we consider the simplest example of a surface with one dissociable group. Dissociation leads to a negatively charged group according to [ ] [A− ] H+ local − + ∼ AH⇌ ∼ A + H with KA = (4.57) [AH] The surface concentrations of the negatively charged dissociated groups and the neutral nondissociated groups are given in mol per surface area and not in mol per volume. [H+ ]local is the local proton concentration in the solution directly at the surface. This local concentration can differ from the concentration in the volume phase: if the surface is charged, it either attracts protons (by negative surface charge) or it repels protons (by positive surface charge). A relation between the local concentration and the bulk concentration can be found with the help of the Boltzmann factor: [ +] [ ] − e𝜑0 H local = H+ e kB T (4.58) Substituting Eq. (4.58) into Eq. (4.57) and taking the logarithm leads to log KA = log

[ ] e𝜑 log e [A− ] + log H+ − 0 [AH] kB T

(4.59)

Since the pK is the negative logarithm of the dissociation constant, we get −pKA = log

e𝜑0 [A− ] − pH − 0.434 [AH] kB T

(4.60)

or 𝜑0 = 2.30

) [A− ] RT ( RT pKA − pH + 2.30 log FA FA [AH]

(4.61)

4.3 Experimental Characterization of Charged Interfaces

Table 4.1

Points of zero charge for various oxides

Substance

pzc

SiO2

1.8–3.4

TiO2

2.9–6.4

Al2 O3

8.1–9.7

MnO2

1.8–7.3

Fe3 O4

6.0–6.9

𝛼-Fe2 O3

7.2–9.5

At 25 ∘ C, we have 𝜑0 = 59 mV

(

[A− ] pKA − pH + log [AH]

) (4.62)

At low pH, many oxides are positively charged. This is at least partly attributable to the process ∼ AOH+2 ⇌ ∼ AOH + H+

(4.63)

The model was further extended and the binding of other ions at certain binding sites on the solid was allowed [152–155]. These ions partially compete for binding sites with H+ or OH− . Table 4.1 shows the points of zero charge of several oxides taken from [5]. The pzc is the pH at which the surface charge is zero. In each case, a range is given and not a unique value because most oxides occur in different structures and different sorts and concentrations of background electrolytes were used. Surface charges also depend on the pretreatment of samples. A comprehensive list of points of zero charge for various oxides can be found in Ref. [156]. As an example, the variation of the surface charge with changing pH is shown in Figure 4.11 for silicon nitride. Silicon nitride is oxidized at its surface. Charging 0.1 0.0 Surface charge (C/m2)

Figure 4.11 Surface charge density of commercial silicon nitride in NaNO3 aqueous solutions as a function of pH. Source: Redrawn after Ref. [157].

–0.1 –0.2 –0.3 0.001 M NaNO3

–0.4

0.1

M NaNO3

–0.5 –0.6 4

5

6

7 pH

8

9

10

77

78

4 Charged Interfaces and the Electric Double Layer

is primarily caused by the binding and dissociation of H+ according to ∼ SiOH+2 ⇌ ∼ SiOH + H+ and ∼ SiOH ⇌ ∼ SiO− + H+ . For this reason, the surface charge becomes increasingly negative with increasing pH [157]. Mica

Clay minerals are formed by two building blocks [158]: tetrahedrons of oxygen with Si4+ ions in their centers or octahedrons of oxygen with Al3+ or Mg2+ in their centers. The tetrahedrons share oxygens and form hexagonal rings. Some oxygen atoms form hydroxyls, in particular when the clay is filled with Ca2+ . This pattern can be repeated ad infinitum to form flat tetrahedral sheets. Similarly, the octahedrons are linked to form octahedral layers. The tetrahedral and octahedral sheets can be stacked on top of each other in various forms to build the different kinds of clays. In mica each sheet is composed of three layers (Figure 4.12). The top and bottom layer are formed by hexagons filled with Si4+ . The intermediate layer is octahedral, and each octahedron is filled with Al3+ or Mg2+ . The sheets are held together by cations. Since this binding is relatively weak, mica can easily be cleaved. An atomically flat surface over huge areas can be obtained. This is the reason why mica is often used as a substrate in scanning probe microscopy and for surface force experiments. How does a surface of mica acquire its charge when it is put into a liquid? Here, we first have to distinguish between charging at the edges and charging of the plate. At the edges, often hydroxyl groups are responsible for surface charges as on an oxide surface. In water, this charge depends sensitively on the pH. Since mica cleaves easily, most of the surface will be plate. Charges are acquired by the dissolution of cations. Si4+ , Al3+ , and Mg2+ dissolve in the liquid leaving a negatively charged mineral behind. Other cations dissolved in the liquid might partially replace the original cations. The total charge is determined by the ion exchange properties of the clay. Charging is only partially due to processes at the surface, but more significant is a Mz+

Mz+

Mz+

2K 6O 3 Si + 1 Al 2 OH + 4 O 4 Al 2 OH + 4 O

Mz+

O Si OH Al

Mz+

Mz+

3 Si + 1 Al 6O 2K

Figure 4.12 Molecular structure of muscovite mica (KAl2 Si3 AlO10 (OH)2 ) in side view. A unit cell is indicated by the dashed line. The number of atoms of a certain species in a layer for two unit cells is indicated at the top right. As an average in the third molecular layer from the top one, silicon atom is replaced by an aluminum atom.

4.3 Experimental Characterization of Charged Interfaces

depletion of positive charge in the mineral. In water, this leads to a relatively constant charge, which is not very sensitive to pH [159]. Semiconductors

Up to this point, we have discussed the properties of metals and insulators. Metals have a high conductivity because the electrons are relatively free to move, and they can react to an applied potential. Each metal atom contributes one or two electrons so that the density of charge carriers is high. Typically, the density of charge carriers is 1029 m−3 . Due to their high concentration and mobility, the electrical conductivity is very high (106 –108 S/m). Insulators, with their tightly bound electrons, are at the other extreme, with conductivities typically lower than 10−6 S/m. Semiconductors like silicon or germanium are an intermediate case. Their electrons are not as tightly bound as in insulators so that at any given time, a small fraction of them will be mobile. In a perfect germanium crystal, for instance at 25 ∘ C, about 3 × 1019 electrons per m3 are free. This corresponds to a concentration of 5 × 10−8 M or 50 nM. It is much lower than the concentration of charge carriers (catand anions) in an aqueous electrolyte solution. Despite this small concentration, the conductivities are of the same order of magnitude, because the electrons in a semiconductor are typically 108 times more mobile than ions in a solution. In a semiconductor, we not only have free electrons but also holes (or vacancies left by electrons). The holes are positively charged and move much like the electrons. Therefore, the situation is similar to an electrolyte solution with negative and positive charge carriers. The density of electrons can be drastically increased by adding small amounts of P, As, or Sb. The density of holes can be increased by adding B, Al, or Ga. This process is called doping. If we place a semiconductor into an aqueous electrolyte, a relevant question is the following: How does the electric potential change with respect to distance from the interface? On the solution side, the potential decays as described before. We have a Stern and a diffuse layer. For a semiconductor, the potential variation at the solid side is also of interest. It also decays and this decay can be described as the decay of the diffuse layer. We only have to replace the salt concentration by the concentration of electrons ce and holes ch and we have to use the appropriate dielectric permittivity. Example 4.5 Estimate the Debye length of Germanium at 25 ∘ C. The dielectric permittivity is 𝜀 = 16. √ 𝜀𝜀0 kB T λD = 2ce e2 √ 16 ⋅ 8.85 × 10−12 A s∕V∕m ⋅ 4.12 × 10−21 J = 2 ⋅ 5 × 10−5 mol∕m3 ⋅ 6.02 × 1023 mol−1 ⋅ (1.60 × 10−19 As)2 = 615 nm Figure 4.13 shows schematically the potential across a semiconductor–electrolyte interface [160]. To understand it, we have to take two additional effects into account.

79

80

4 Charged Interfaces and the Electric Double Layer

Contributions due to dipoles and surface states

ψsc

Figure 4.13 Potential at a semiconductor–electrolyte interface. The index “sc” refers to semiconductor.

Contributions due to surface charges and adsorbed ions

Semiconductor

ψd OHP xsc

Aqueous electrolyte

0

x

First, the liquid molecules usually show a preferred orientation at the surface. Their dipole moment causes a jump of the potential. Second, on a solid surface, the electrons can occupy surface states. These extra electrons contribute to the potential.

4.3.4

Potentiometric Colloid Titration

In many applications, we want to know the charge density of dispersed systems. To determine the surface charge of dispersed particles, titration methods can be used. Before we can do a quantitative titration experiment, we need to know the specific surface area, that is the total surface area per gram of dispersed material and the potential-determining ions. If the particles have a known size and shape, the specific surface area can be calculated. For irregular or rough particles, specific surface area is typically determined from BET adsorption measurements (see Section 8.3.4). The potential determining ions are either obtained from additional experimental evidence or physicochemical reasoning. To illustrate the method, let us assume that an oxide is titrated in aqueous solution and that H+ and OH− are the potential-determining ions. A certain amount of the dispersed substance is filled into a cell, which contains a pH electrode. We start the titration at, say, high pH and low ionic strength. The solution could, for instance, contain 1 mM KNO3 . An indifferent salt should be used where the ions do not bind specifically to the particles surface. A tiny amount of KOH is added to reach a high pH (Figure 4.14 point A). Then a small aliquot HNO3 is added. Since HNO3 dissociates completely, we know that the number of protons added is equal to the amount of HNO3 added. The pH measurement reveals how many protons are in the solution. The difference must be bound to the surface of the oxide. Dividing this number by the total surface area of the particles and multiplying by unit charge, leads to the change in surface charge density (point B). By repeating this procedure many times we reach, step by step, a low pH (C). The pH range is limited by the condition that the amount of HNO3 added should not change the total ionic strength significantly. In addition, we need to make sure that the particles are stable and do not change their properties. This can be controlled by going up with the pH again by adding KOH (to A again). No hysteresis should be observed.

4.3 Experimental Characterization of Charged Interfaces

Figure 4.14 Schematic result of a potentiometric titration experiment.

F

σ

D A B

Negative Positive

C E pzc

pH

Then the salt concentration is increased to, say, 10 mM KNO3 (D). The surface charge increases because the capacitance of the electric double layer increases and so for a given surface potential, the charge increases. We repeat the titration with HNO3 (to E) and KOH (back to D). Afterward the salt concentration is increased to say 0.1 M (F), and the cycle is repeated. The three titration curves intersect at one point – the pzc – if the background ions (in our example K+ and NO−3 ) do not bind specifically to the surface [161]. This is essential because, from the titration curves alone, we would only get the shape but not the absolute position of the titration curve. The common intersection gives us the position with respect to the charge axis. To verify that the salt is really indifferent and no specific binding of the ions occurs, the experiment can be repeated with a different salt. If titration curves measured at the same ionic strength are identical, we have good evidence for the absence of specific binding. In potentiometric colloid titration, the amount of potential-determining ions in solution is measured by an appropriate electrode. For oxides, where mainly the pH determines the potential, a glass electrode is a suitable detector. For AgI, we could use an AgI electrode, etc. At low ionic strength conductometric titration is an alternative approach to determine the amount of ions in solution. It is determined by measuring the electrical conductivity versus the amount of added potential-determining ion. One example is shown in Figure 4.15. Different amounts of aqueous NaOH solution at a concentration of 5 mM were added to an aqueous dispersion of latex particles. Before adding NaOH, the pH of the dispersion is 3.3, and the conductance is 7.1 × 10−3 Ω−1 ∕m. When adding NaOH, the pH first increases gradually, then around an added amount of 2.1 ml, the increase becomes steep. Above 3 ml, the pH is not affected so much. This indicates that the latex particles have dissociable groups with a pK around 6 on their surface. Correlated with the change in pH is the specific conductance. When adding NaOH, one would expect an increase in conductivity because the number of charge carriers increases. This linear increase is indeed observed in the second half of the curve. In the first linearly descending part of the titration curve, protons bind to the surface of the latex particles. As a result, in solution, highly mobile protons are replaced by slower Na+ ions.

81

4 Charged Interfaces and the Electric Double Layer

10

0.0016 0.0015

6

pH

0.0014 Cond.

4

2

0.0013

0

1

2 3 mL NaOH 5 mM

4

5

0.0012

Specific conductance (Ω–1/m)

8 pH

82

Figure 4.15 Conductometric and potentiometric titration of latex particles dispersed in an aqueous medium. Source: Redrawn after Ref. [162].

4.3.5

Capacitances

For conducting and completely polarizable surfaces, the capacitance can be measured directly with great precision. In the most simple capacitance measurement, called chronoamperometry, a potential step ΔU is applied to an electrode (Figure 4.16). A current flows due to charging of the diffuse electrical double layer, assuming that no electrochemical reactions occur at the electrodes. The current flows until the capacitance is fully charged. By measuring the current as a function of time and integrating the curve with respect to time, we get the charge Q. The total capacitance C is easily obtained from C = CA A = Q∕ΔU. Example 4.6 A potential ΔU is applied as a step function from 0.1 V to 0.3 V to a platinum wire of 0.5 mm diameter in 0.1 M KCl versus an Ag/AgCl counterelectrode. The counterelectrode has a huge surface area compared to the platinum wire, and it is a reversible electrode. For the platinum, the applied potential range is too small to allow electrochemical reactions to take place and we have no Faradayic current. The measured current is shown in Figure 4.17. Integrating the charge (this corresponds

I

U

Electrode Counter electrode Electrolyte solution Figure 4.16 Schematic experimental set-up to measure capacitance.

4.4 Electrokinetic Phenomena: The Zeta Potential

Current (μA)

0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

Time (s) Figure 4.17 Chronoamperometric measurement as described in example 4.6.

to the area underneath the curve) leads to Q = 59.9 nC. Dividing by ΔU, we get the a total capacity of C = 0.299 μF. The electrode surface area was 0.00196 cm2 . This leads to a capacitance per unit area of CA = 153 μF/cm2 . The value is higher than expected from Gouy–Chapman theory. One reason is that surface roughness increases the effective surface area. The main reason is, however, that the Gouy–Chapman theory does not adequately describe the capacitance of the double layer, especially at high salt concentrations. A Stern layer has to be taken into account. The example was kindly provided by T. Jenkins. A standard method to determine capacitance is cyclic voltammetry (see Refs. [147, 163] for an introduction). One electrode, made of the material of interest (with a known surface area A), and a counter electrode, are introduced into the electrolyte solution. A reference electrode can be used in addition. Then a triangular potential is applied, and the electric current is measured. From the current, the capacitance can be calculated. What can we do for reversible electrodes, at which electrons are transferred between the electrode surface and ions in solution? In this case, we use the fact that different processes occur at different timescales. Instead of the relatively slow change of the voltage in cyclic voltammetry, AC potentials with varying frequencies are applied and the current is detected. The method is called impedance spectroscopy (e.g. [164, 165]). Using impedance spectroscopy, even semiconducting [160] or insulating materials can be analyzed by coating them onto metallic electrodes.

4.4

Electrokinetic Phenomena: The Zeta Potential

In this section, we deal with liquids, which flow along charged solid surfaces. In many cases, the surface binds one, two, or several layers of liquid molecules and possibly ions more or less tightly. As a result, the shear plane is often not directly at

83

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4 Charged Interfaces and the Electric Double Layer

the interface. Only at a distance δ away from the surface do the molecules start to move. The potential at this distance is called the zeta potential ζ. The concentration of potential-determining ions at which the zeta potential is zero (ζ = 0) is called the isoelectric point (iep). The isoelectric point is determined by electrokinetic measurements. We have to distinguish it from the pzc. At the pzc, the surface charge is zero. The zeta potential refers to the hydrodynamic interface, while the surface charge is defined for the solid–liquid interface. If a liquid moves tangential to a charged surface, then so-called electrokinetic phenomena arise [166, 167]. Electrokinetic phenomena can be divided into four categories: Electrophoresis, electro-osmosis, streaming potential, and sedimentation potential. In all these phenomena, the zeta potential plays a crucial role. First quantitative observations go back to the nineteenth century, e.g. by Quincke [168].11 The classic theory of electrokinetic effects was proposed by Smoluchowski [169].12

4.4.1

The Navier–Stokes Equation

If we slide one plate over another, parallel plate and the interstitial space is filled with a fluid (Figure 4.18), experiments showed that the shear force required is Δv F =𝜂 (4.64) A Δz Here, A is the area of the plates, Δv is the velocity difference tangential to the orientation of the plates, and Δz is the distance between the plates. Eq. (4.64) is strictly only valid as long as the flow is laminar and not turbular. F∕A is proportional to the difference in velocity divided by the distance. It is also proportional to the constant 𝜂 called dynamic viscosity or just viscosity.13 The viscosity of water at 25 ∘ C is 0.89 × 10−3 Pa s. The velocity gradient Δv∕Δz is called shear rate. It is in units of s−1 . If 𝜂 does not change with the shear rate, the fluid is called a Newtonian14 fluid. In continuum mechanics, the flow of a Newtonian fluid is described by the Navier–Stokes equation. In order to make the equation plausible, we consider an infinitesimal quantity of the liquid having a volume dV = dx dy dz and a mass dm. If we want to write Newton’s equation of motion for this volume element, we have to consider different forces. The first contribution is a viscous force caused by gradients in the shear stress of ( ) the fluid. The flow velocity is denoted by ⃗v = vx , vy , vz . To quantify the viscous A Δz

F

Figure 4.18 The viscous force required to slide a plane of area A over another, parallel plane at a distance Δz across a fluid of viscosity 𝜂 is F = 𝜂AΔv∕Δz.

11 Georg Hermann Quincke, 1834–1924; German physicist, professor in Würzburg, Berlin, and Heidelberg. 12 Marian von Smoluchowski, 1872–1917; Polish physicist, professor in Lemberg and Krakowia. 13 Sometimes also the kinematic viscosity defined by 𝜂k = 𝜂∕𝜌 in units of m2 ∕s is used. 14 Sir Isaac Newton, 1643–1727; British mathematician, physicist, and astronomer. Founder of mechanics and geometric optics.

4.4 Electrokinetic Phenomena: The Zeta Potential

Figure 4.19 Fluid volume element in a flow field. In this case, the flow is supposed to be in x direction, with a gradient in z direction. As a result, a shear force is acting between the upper and the lower surface elements. The pressure acting on the right surface, P ′ , and on the left surface, P ′′ , can be different.

v

Fxʺ

dy

P′

Pʺ dz Fx′ dx

force, let us suppose that the fluid is flowing in x direction (Figure 4.19). Also, suppose that there is a velocity gradient in the flow in z direction; the shear rate 𝜕vx ∕𝜕z is not zero. This implies that a tangential force that is acting on the upper face of the volume element in x direction is Fx′′ . It is different from the tangential force on the lower face Fx′ . The net force acting in x direction is Fx′′ − Fx′ . Since the volume element is infinitesimally small, the force difference is also infinitesimally small, and we write dFx = Fx′′ − Fx′ . Applying Eq. (4.64) to our volume element, we can express the force difference by dFx = 𝜂 dx dy

𝜕vx 𝜕2v = 𝜂 2x dx dy dz 𝜕z 𝜕z

In three dimensions, the viscous force caused by gradients in the shear rate of the fluid is a vector with components in all directions. It is given by 𝜂∇2 ⃗v ⋅ dV. Here, ∇2 is the Laplace operator and ∇2 ⃗v is a vector, which is written in full in Eq. (4.67). The second force acting on dV is caused by a pressure gradient. The pressure acting on the left area, P′′ , is different from the pressure P′ on the right side of the volume element. The resulting force in x direction is ( ) 𝜕P dFx = P′′ − P′ dy dz = − dx dy dz 𝜕x In three dimensions, a pressure gradient causes the force − (∇P) dV. Here, ∇ is the Nabla operator and ∇P is the vector (𝜕P∕𝜕x, 𝜕P∕𝜕y, 𝜕P∕𝜕z). As a third contribution, external forces might influence the movement of the volume element. For example, gravity causes a hydrostatic pressure. To describe electrokinetic phenomena, we need to consider electrostatic forces caused by the action ⃗ dV, where of an electric field on the ions in solution. The electrostatic force is 𝜌e E ⃗ is the field strength. 𝜌e is the charge density and E According to Newton’s law, the sum of these forces is equal to the mass dm times its acceleration: ) ( d⃗v ⃗ dV dm = 𝜂∇2 ⃗v − ∇P + 𝜌e E (4.65) dt In many cases, a steady-state flow is considered. Then the fluid velocity is constant (d⃗v∕dt = 0), and we get the so-called Navier–Stokes equation: ⃗ =0 𝜂∇2 ⃗v − ∇P + 𝜌e E

(4.66)

85

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4 Charged Interfaces and the Electric Double Layer

It is a vector equation. Written in full and in Cartesian coordinates, it reads: ( ) 𝜕 2 vx 𝜕 2 vx 𝜕 2 vx 𝜕P 𝜂 + 2 + 2 − + 𝜌e E x = 0 𝜕x 𝜕x2 𝜕y 𝜕z ( 2 ) 𝜕 vy 𝜕 2 vy 𝜕 2 vy 𝜕P 𝜂 + 2 + 2 − + 𝜌e E y = 0 (4.67) 𝜕y 𝜕x2 𝜕y 𝜕z ( ) 𝜕 2 vz 𝜕 2 vz 𝜕 2 vz 𝜕P 𝜂 + 2 + 2 − + 𝜌e Ez = 0 𝜕z 𝜕x2 𝜕y 𝜕z We derive another important equation. Starting point is the fact that mass is not destroyed or created. This can be expressed in an equation for mass conservation: ( ) 𝜕𝜌 + ∇ 𝜌⃗v = 0 𝜕t

(4.68)

Many liquids are practically incompressible. Their mass density 𝜌 is constant over space, and in time, we obtain the continuity equation: ∇ ⋅ ⃗v =

𝜕vx 𝜕vy 𝜕vz + + =0 𝜕x 𝜕y 𝜕z

(4.69)

Equation (4.69) basically tells us that any liquid flowing into a volume element has to be compensated by the same amount of liquid flowing out of that volume element. The Navier–Stokes and the continuity equations are the basic equations describing the flow of an incompressible liquid.

4.4.2

Electro-osmosis and Streaming Potential

Let us start by considering a liquid on a planar, charged surface. If we apply an electric field parallel to the surface, the liquid begins to move (Figure 4.20). This phenomenon is called electro-osmosis. Why does the liquid start to move? The charged surface causes an increase in the concentration of counterions in the liquid close to the surface. This surplus of counterions is moved by the electric field U

Liquid flow z

x Figure 4.20 Electro-osmosis shown schematically for a solid surface with a negative surface charge.

4.4 Electrokinetic Phenomena: The Zeta Potential

toward the corresponding electrode. The counterions drag the surrounding liquid with them and the liquid starts to flow. When treating this simple case of electro-osmosis mathematically, we immediately realize that the y component of the Navier–Stokes equation disappears. All y derivatives are zero because no quantity can change with y due to the symmetry. We further assume that the liquid flows only parallel to the x coordinate that is parallel to the applied field and then vz = 0 and vy = 0. As a consequence, all derivatives of vy and vz are zero. From the equation of continuity, we can directly conclude that 𝜕vx ∕𝜕x = 0. The flow rate does not change with position along the x direction. Since this is valid at all x, it follows that 𝜕 2 vx ∕𝜕x2 = 0. From the three-component Navier–Stokes equation, the only two remaining equations are: 𝜕 2 vx 𝜕P 𝜕P − + 𝜌e Ex = 0 and − + 𝜌e Ez = 0 (4.70) 𝜕x 𝜕z 𝜕z2 The electric field in x direction is applied externally. In z direction, the field results from the surface charges. We also assume that no pressure in x direction is applied. Then 𝜕P∕𝜕x disappears and after rearranging the first equation, we get 𝜂

𝜌e Ex = −𝜂

𝜕 2 vx 𝜕z2

(4.71)

Now, we use the Poisson equation (4.2), 𝜕 2 𝜑∕𝜕z2 = −𝜌e ∕𝜀𝜀0 , for the charge density: 𝜕 2 vx 𝜕2 𝜑 = 𝜂 (4.72) 𝜕z2 𝜕z2 This equation can be integrated twice in z. The integration starts at a point far away from the surface, where 𝜑 = 0 and vx has a stationary value v0 , up to the shear plane at a distance δ from the surface, where vx = 0 and 𝜑 = ζ. Note that, far from the plane, 𝜕𝜑∕𝜕z = 0 and 𝜕vx ∕𝜕z = 0. Ex 𝜀𝜀0

z 2 𝜕 vx ′ 𝜕v 𝜕2 𝜑 ′ 𝜕𝜑 dz = E 𝜀𝜀 = 𝜂 dz = 𝜂 x x 0 ′2 ′2 ∫∞ 𝜕z ∫∞ 𝜕z 𝜕z 𝜕z δ δ 𝜕vx ′ 𝜕𝜑 ′ Second integration: Ex 𝜀𝜀0 dz = Ex 𝜀𝜀0 ζ = 𝜂 dz = −𝜂v0 ∫∞ 𝜕z′ ∫∞ 𝜕z′ z

First integration: Ex 𝜀𝜀0

It follows that ζEx (4.73) 𝜂 The flow velocity is proportional to the zeta potential and to the applied field. We can observe electro-osmosis directly with an optical microscope using liquids, which contain small, yet visible, particles as markers. Most measurements are made in capillaries. An electric field is tangentially applied and the quantity of liquid transported per unit time is measured (Figure 4.21). Capillaries have typical diameters from 10 μm up to 1 mm. The diameter is thus much larger than the Debye length. Then the flow rate will change only close to a solid–liquid interface. Some Debye lengths away from the boundary, the flow rate is constant. Neglecting the thickness of the electric double layer, the liquid volume transported per time is E𝜍 dV = 𝜋rc2 v0 = 𝜋rc2 𝜀𝜀0 (4.74) dt 𝜂 v0 = −𝜀𝜀0

87

88

4 Charged Interfaces and the Electric Double Layer

Figure 4.21 Electro-osmotic flow profile in a capillary.

rC

U v0

Electro-osmosis is used in microfluidics to drive aqueous media through narrow channels. Example 4.7 We apply a potential of 1 V across a capillary of 1 cm length and of 100 μm diameter. The field should decrease linearly, i.e. E = 100 V/m. The capillary is filled with water and the zeta potential is 0.05 V. How much electrolyte flows through the capillary per unit time? ( )2 100 V∕m ⋅ 0.05 V dV = 𝜋 50 × 10−6 m ⋅ 78.5 ⋅ 8.85 × 10−12 C2 ∕(V m) dt 0.001 kg∕s∕m = 2.73 × 10−14 m3 ∕s ( )2 With a volume of the capillary of 𝜋 50 × 10−6 m ⋅ 0.01 m = 7.85 × 10−11 m3 , the liquid in the capillary is exchanged in 52 minutes. The inverse effect to electro-osmosis is the build-up of a streaming potential. In this case, the liquid is pressed through a capillary (more generally along a charged wall). The liquid drags the charges of the electrical double layer with it. As a result, some counterions accumulate at the end of the capillary and generate a potential ΔU – the streaming potential – between the two ends of the capillary. If the radius is substantially larger than the Debye length, it is easily possible to calculate the streaming potential between the beginning and the end of the capillary (Ref. [3] p. 379): ΔU =

𝜀0 𝜀ζ ΔP 𝜂𝜅e

(4.75)

Here, ΔP is the applied pressure and 𝜅e is the electrical conductivity of the electrolyte (in Ω−1 ∕m). Streaming potential measurements are routinely used to characterize the electrical properties of surfaces (e.g. [170]).

4.4.3

Electrophoresis and Sedimentation Potential

In electrophoresis, we consider the motion of charged particles in electric fields (in contrast, the movement of ions is treated in electrochemistry under the

4.4 Electrokinetic Phenomena: The Zeta Potential

Figure 4.22 Spherical particle with a negative surface charge in a liquid. The shear plane at a distance δ and the Debye length λD are indicated. Only the counterions in the double layer are shown. All other ions of the background salt are omitted.

λD

R

δ

keyword “ionic conductivity”). Electrophoresis is of great practical importance. It is a relatively easy tool to assess the amount of charge on colloidal particles. In biochemistry, it is used to isolate proteins. As an example, we consider the motion of a spherical particle. The electric field and the resulting motion are supposed not to disturb the double layer around the sphere as given by the Poisson–Boltzmann equation. We further assume that the electric field is homogeneous and not disturbed by the presence of the particle. In reality, this assumption is not perfectly fulfilled but it leads to a good approximation. The hydrodynamic radius of the particle is R + δ (Figure 4.22). Bound ions and liquid molecules cause the shear plane to be a distance δ away from the real particle surface. ζ is the potential at a distance δ. All charges within the radius R + δ amount to a value Q, which is equivalent to a surface charge density 𝜎δ on a fictitious sphere with radius R + δ. Q and 𝜎δ are related by Q = 4𝜋(R + δ)2 𝜎δ ≈ 4𝜋R2 𝜎δ . An electric field of strength E will impose forces on the particle and on all liquid domains that have a local nonzero charge, i.e. where ||𝜌e || > 0. We approximate the locations of all double-layer charges to be collected at a radius R + λD . The total charge of the double layer is −Q. Thus, the electric field acts on an “inner” sphere (the particle) with radius R and charge Q and on an “outer” sphere with radius R + λD and charge −Q. Each sphere alone would move with a drift velocity given by Stokes law. Stokes law gives the friction force of a sphere moving at drift velocity v in a liquid (assuming lamellar flow): F = 6𝜋𝜂Rv

(4.76)

In equilibrium, the friction force is equal to the applied electric force. The applied forces are QE and −QE for the inner and outer sphere, respectively. This leads to drift velocities of vin =

QE 6𝜋𝜂R

and

vout = −

QE 6𝜋𝜂(R + λD )

(4.77)

As Stokes hydrodynamics only involves linear equations, the net particle velocity is the sum of both contributions: ( ) QEλD QE 1 v = vin + vout = 1− = (4.78) ( ) 2 6𝜋𝜂R 1 + λD ∕R 6𝜋𝜂R 1 + λD ∕R

89

90

4 Charged Interfaces and the Electric Double Layer

R > λD

Figure 4.23 Electric field around a particle with a thick (a) and thin (b) double layer.

R + λD Particle

R

E (a)

(b)

Mathematically it is as if the particle moves with vi within the outer liquid sphere, which itself moves with a drift velocity vo . For λD ∕R ≪ 1, this equation reduces to v=

QEλD 2 𝜎δ EλD = 2 3 𝜂 6𝜋𝜂R

(4.79)

In the last step, we replaced Q by 4𝜋R2 𝜎δ . Finally, we remove the explicit dependence on the Debye length by applying the Grahame equation (4.31) for small values of ζ (𝜎δ = 𝜀𝜀0 ζ∕λD ): v=

2 𝜀𝜀0 ζE 3 𝜂

(4.80)

The result is identical to the result obtained in an exact derivation, apart from the numerical factor 2∕3. This factor is due to the incorrect assumption of a uniform electric field. In fact, with thin double layers, the electric field is disturbed and partially guided around the particle due to the increased ion concentration (Figure 4.23). In the case of thick double layers, the electric field is undisturbed. The exact procedure here is to mimic the distribution of counter charges as a set of charged concentric rings around the particle and integrate the electrophoretic effects. The result is equal to our approximation derived for a thin double layer. Summarizing, exact solutions for electrophoresis are 𝜀𝜀0 ζE 𝜂 𝜀𝜀 2 0 ζE v= 3 𝜂 v=

for for

λD ≪ R λD ≫ R

(4.81) (4.82)

For the usual, intermediate cases, the numerical factor gradually changes from 1 to 2∕3, and it must be replaced by a monotonic function f (R∕λD ) [171, 172]. Note that the exact solution for electrophoresis with thin double layers is identical to the result obtained for electro-osmosis (Eq. (4.73)). Electrophoresis is still a matter of ongoing research, in particular if it comes to soft particles, nonspherical particles, or concentrated dispersions, where the influence of neighboring particles cannot be neglected. For a review see Ref. [173]. The “opposite” effect to electrophoresis is the generation of a sedimentation potential. If a charged particle moves in the gravitational field or in a centrifuge, an electric potential arises – the sedimentation potential. While the particle moves, the

4.6 Exercises

ions in the electric double layer lag somewhat behind due to the liquid flow. A dipole moment is generated. The sum of all dipoles causes the sedimentation potential.

4.5 Summary ●

●

Most solid surfaces in water are charged. Reason: Due to the high dielectric permittivity of water, ions are easily dissolved. The resulting electric double layer consist of an inner Stern or Helmholtz layer, which is in close contact with the solid surface, and a diffuse layer, also called the Gouy–Chapman layer. The electric potential in the diffuse layer of a planar surface decays approximately exponentially 𝜑 = 𝜑0 e−x∕λD ,

●

●

●

provided the potential does not exceed 50–80 mV. For a monovalent salt and at √ 25 ∘ C, the Debye length is given by λD = 3.04∕ c0 Å, with the concentration c0 in mol/l. Valuable information about the properties of electrical double layers can be obtained from electrocapillary experiments. In an electrocapillary experiment, the surface tension of a metal surface versus the applied electrical potential is measured. The capacitance and the pzc are obtained. Surface charge densities for disperse systems can be determined by potentiometric and conductometric titration. Different processes are responsible for the build-up of surface charge in different materials such as metals, weakly soluble salts, oxides, and clay minerals. The zeta potential is the potential of a solid surface at the shear plane of the surrounding liquid. The zeta potential is relevant in electrokinetic phenomena. In electro-osmosis, a liquid flow is induced by applying an electrical potential parallel to a charged surface. Streaming potentials are caused by a liquid flow tangential to a charged surface. Electrophoresis is the movement of a charged particle in an electric field.

4.6 Exercises 4.1

Compare the Debye lengths of 0.1 mM NaCl solution of water and ethanol (𝜀 = 25.3).

4.2

Plot the potential versus distance for surface potentials of 60, 100, and 140 mV using the solution of the linearized and the full Poisson–Boltzmann equation for an aqueous solution with 2 mM KCl.

4.3

For an electrophysiological experiment, you form an electrode from a 5 cm long platinum wire (0.4 mm diameter) by bending it in the shape of a spiral.

91

92

4 Charged Interfaces and the Electric Double Layer

Calculate the total capacitance of the diffuse electric double layer for aqueous solutions of a monovalent salt at concentrations of 0.1 and 0.001 M. Assume a low-surface potential. 4.4

Silicon oxide has a typical surface potential in an aqueous medium of −70 mV in 50 mM NaCl at pH 9. What concentration of cations do you roughly expect close to the surface? What is the average distance between two adjacent cations? What is the local pH at the surface?

4.5

We consider a spherical particle of radius R in a liquid electrolyte with a low-surface potential 𝜑0 . Verify that the total charge of the particle is given by Q = 4𝜋𝜀 𝜀0 R𝜑0 (1 + R∕λD ).

4.6

The differential capacitance of a mercury electrode in an aqueous medium containing NaF has been measured at the pzc. It is 6.0 μF∕cm2 at 1 mM, 13.1 μF∕cm2 at 10 mM, 20.7 μF∕cm2 at 100 mM and 25.7 μF∕cm2 at 1 M concentration. Compare this with the result of the Gouy–Chapman theory and draw conclusions.

4.7

What is the Gibbs energy of the electrostatic double layer around a sphere of radius R that has a surface charge density 𝜎? Derive an equation where the Gibbs energy of one sphere is given as a function of the total charge (Exercise 4.5). You can assume a low-surface potential.

4.8

For a microfluidic application, a capillary with a radius of 10 μm and 5 cm in length was fabricated in glass. The zeta potential of this glass in 0.01 M KCl aqueous solution at neutral pH is −30 mV. A potential of 5 V is applied along the capillary. How fast and in which direction does the liquid flow?

4.9

To observe the flow, fluorescently labeled small spherical polystyrene particles of 50 nm radius are added. To keep them dispersed, they have sulfate groups on their surface. This leads to a zeta potential of −20 mV. How fast and in which direction do these particles move? A good marker should move with the same speed as the liquid flow. How should these particles be used as markers?

93

5 Surface Forces A whole range of phenomena in interface science revolve around the effect of surface forces. Many practical applications in colloid science come down to the problem of controlling the force between colloidal particles, between particles and interfaces, and between two interfaces. For this reason, scientists have devoted considerable effort to understanding surface forces and being able to influence them. In talking about surface forces, the first association is that of two solid particles interacting in a fluid medium. Practically, this is of direct relevance when dealing with the stability of sols. If attractive interactions dominate, then the particles aggregate; if repulsive forces dominate, the dispersion is stable. The subject of surface forces is, however, more general. It includes, for example, the interaction between two liquid–gas interfaces of a liquid lamella, which is relevant for the stability of foams. Here, the two opposing liquid–vapor interfaces interact. If they are attractive, then the liquid lamella is unstable and ruptures. For a repulsive interaction, the lamella is stable. Another example is thin liquid films on solid surfaces. Here, the solid–liquid and liquid–vapor interfaces interact. A repulsive interaction leads to a stable film that tends to wet the solid surface while attractive interactions destabilize thin films. This will also be discussed in Chapter 6. Introductions to surface forces include [7, 174, 175]. Van der Waals forces are discussed comprehensively in [176].

5.1

Van der Waals Forces Between Molecules

Forces between macroscopic objects result from a complex interplay of the interaction between molecules in the two objects and the medium separating them. The basis for an understanding of intermolecular forces is the Coulomb1 force. The Coulomb force is the electrostatic force between two charges Q1 and Q2 : F=

Q1 Q2 4π𝜀𝜀0 D2

Q1

D

Q2

(5.1)

1 Charles Augustin Coulomb, 1736–1806; French physicist and engineer. Physics and Chemistry of Interfaces, Fourth Edition. Hans-Jürgen Butt, Karlheinz Graf, and Michael Kappl. © 2023 WILEY-VCH GmbH. Published 2023 by WILEY-VCH

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5 Surface Forces

If the two charges are in a medium, the permittivity 𝜀 is higher than one, and the electrostatic force is reduced accordingly. The potential energy between two electrical charges that are a distance D apart is Q1 Q2 W= (5.2) 4π𝜀𝜀0 D For charges with opposite signs, the potential energy is negative. They reduce their energy when they get closer. Example 5.1 The potential energy between Na+ and Cl− , being 1 nm apart, in a vacuum is (1.60 × 10−19 C)2 = −2.30 × 10−19 J W =− 4π ⋅ 8.85 × 10−12 A s∕V∕m ⋅ 10−9 m This is 56 times higher than the thermal energy kB T = 4.12 × 10−21 J at room temperature. Most molecules are not charged. Still, the electric charge is often not distributed evenly. A molecule can have a more negative side and a more positive side. In carbon monoxide, for example, the oxygen is more negative than the carbon atom. To first order, the electric properties of such molecules are described by the so-called “dipole moment”. For the simple case of two opposite charges Q and −Q that are a distance d apart, the dipole moment 𝜇 is given by 𝜇 = Qd. It is given in units of Coulomb meters. Often, the old unit Debye is used. One Debye is equal to a positive unit charge and a negative unit charge that are 0.21 Å apart; it is denoted by 1 D = 3.336 × 10−30 C m. The dipole moment is a vector that points from minus to plus. If we have more than two charges, we must integrate the charge density 𝜌e over the whole volume of the molecule, which leads to the general definition of the dipole moment: 𝜇=

∫

𝜌e (r)rdV

(5.3)

Let us now return to intermolecular interactions. If more than two charges are present, then the net potential energy of a charge can be calculated by summing up the contributions of all the other charges. This is called the superposition principle. Using this superposition principle, we can calculate the potential energy between a dipole and a single charge: W =−

Q𝜇 cos 𝜗 4π𝜀0 D2

µ

ϑ D

Q

(5.4)

Here, we have assumed that the distance D is large compared to the extension of the dipole. In practice, a molecule with a dipole moment is often mobile. If the dipole is free to rotate and close to a positive charge, then it tends to rotate until its negative pole points toward the positive charge. On the other hand, thermal fluctuations drive it away from a perfect orientation. On average, a net preferential orientation remains and the dipole is attracted by the monopole. The average potential energy is Q2 𝜇 2 W =− 6(4π𝜀0 )2 kB TD4

µ

D

Q

(5.5)

5.1 Van der Waals Forces Between Molecules

Example 5.2 Calculate the potential energy between Na+ and a water molecule (dipole moment 6.17 × 10−30 C m) that are 1 nm apart in vacuum at 25 ∘ C: W =−

(1.60 × 10−19 C)2 (6.17 × 10−30 C m)2 6 (4π ⋅ 8.85 × 10−12 A s∕V∕m)2 ⋅ 4.12 × 10−21 J ⋅ (10−9 m)4

= −3.20 × 10−21 J This is roughly equal to the thermal energy kB T. Two freely rotating dipoles attract each other because they preferentially orient with their opposite charges facing each other. This randomly oriented dipole–dipole interaction is often referred to as the Keesom energy2 : W =−

𝜇12 𝜇22 Corient = − D6 3(4π𝜀0 )2 kB TD6

µ1

D

µ2

(5.6)

The coefficient Corient is independent of the distance. For two water molecules 1 nm apart, the Keesom energy is, for example −9.5 × 10−24 J. All expressions reported so far give the Helmholtz free energies of interaction because they were derived under constant volume conditions. Until now, free energy and internal energy were identical. For a randomly oriented dipole–dipole interaction, entropic effects, namely, the ordering of one dipole by the field of the other dipole, contribute to the free energy. If one dipole approaches another, one-half of the internal energy is taken up in decreasing the rotational freedom of the dipoles as they become progressively more aligned. For this reason, the free energy as given in Eq. (5.6) is only half of the internal energy. The Gibbs free energy is in this case equal to the Helmholtz free energy. When a charge approaches a molecule without a static dipole moment, all energies considered so far would be zero. Nevertheless, there is an attractive force. The reason for this attraction is that the monopole induces a charge shift in the nonpolar molecule. An induced dipole moment arises and interacts with the charge. The Helmholtz free energy is W =−

Q2 𝛼 2(4π𝜀0 )2 D4

(5.7)

Here, 𝛼 is the polarizability in C2 m2 ∕J. The polarizability is defined by 𝜇ind = 𝛼E, where E is the electric field strength. Often, one still finds 𝛼 expressed in the old −3 CGS system as 𝛼∕(4π𝜀0 ) in units of Å . Example 5.3 The polarizability of a water molecule in the gas phase is 1.65 × 10−40 C2 m2 ∕J. Which dipole moment is induced by a unit charge that is 1 nm away and what is the potential energy between the two? The electric field of a point charge at a distance D is E=

Q = 1.44 × 109 V∕m 4π𝜀0 D2

2 Wilhelmus Hendrik Keesom, 1876–1956; Dutch physicist, professor in Utrecht and Leiden.

95

96

5 Surface Forces

The induced dipole moment is 𝜇ind = 1.65 × 10−40 C2 m2 ∕J ⋅ 1.44 × 109 V∕m = 2.38 × 10−31 C m and the potential energy is −1.71 × 10−22 J. Also, a molecule with a static dipole moment interacts with a polarizable molecule. If the dipole can freely rotate, then the Helmholtz free energy is W =−

Cind 𝜇2 𝛼 =− 6 D (4π𝜀0 )2 D6

(5.8)

for different molecules. If both molecules have the same static dipole moment, then a factor of two must be inserted. This randomly oriented induced dipole interaction is called Debye interaction. All energies considered so far can be calculated using classical physics. They fail to explain the attraction between nonpolar molecules. That such an attraction exists is evident because all gases – even the nonpolar noble gases – condense at some temperature. The force responsible for this attraction is the so-called London3 or the dispersion force. To calculate the dispersion force, quantum mechanical perturbation theory is required. An impression about the origin of dispersion forces can be obtained by considering an atom with its positively charged nucleus around which electrons circulate with a high frequency of typically 1015 –1016 Hz. At every instant, the atom is therefore polar. Only the direction of the polarity changes with this high frequency. When two such oscillators approach one another, they start to influence each other. Attractive orientations have higher probabilities than repulsive ones. On average, this leads to an attractive force. The Helmholtz free energy between two molecules with ionization energies h𝜈1 and h𝜈2 can be approximated by W =−

Cdisp D6

=−

h𝜈1 𝜈2 3 𝛼1 𝛼2 2 (4π𝜀0 )2 D6 (𝜈1 + 𝜈2 )

(5.9)

Dispersion interactions increase with the polarizability of the two molecules. The optical properties enter in the form of excitation frequencies. Expression (5.9) only considers one term of a series over all existing dipole transition moments. Usually, however, this term is by far the most dominant one. Keesom, Debye, and London contributed much to our understanding of forces between molecules [177–179]. For this reason, the three dipole interactions are named after them. The van der Waals4 force is the Keesom plus the Debye plus the London dispersion interactions – thus, all the terms that consider dipole–dipole interactions: Ctotal = Corient + Cind + Cdisp . All three terms contain the same distance dependency: the potential energy decreases with 1∕D6 . Usually the London dispersion term predominates. Note that not only do polar molecules interact via the Debye and Keesom forces but the dispersion forces are also present. Table 5.1 lists the contributions of the individual terms for some gases. 3 Fritz London, 1900–1954; American physicist of German origin, professor in Durham. 4 Johannes Diderik van der Waals, 1837–1923; Dutch physicist, professor in Amsterdam.

5.2 Van der Waals Force Between Macroscopic Solids

Table 5.1 Contributions of the Keesom, Debye, and London potential energies to the total van der Waals interaction between similar molecules as calculated with Eqs. (5.6), (5.8), and (5.9) using Ctotal = Corient + Cind + Cdisp .

He

𝝁 (D)

𝜶∕(𝟒𝝅𝜺𝟎 ) (10 −𝟑𝟎 m𝟑 )

h𝝂 (eV)

Corient

0.00

0.20

24.6

0.0

0.0

Cind

Cdisp

Ctotal

1.2

1.2

Cexp

0.86

Ne

0.00

0.40

21.6

0.0

0.0

4.1

4.1

3.6

Ar

0.00

1.64

15.8

0.0

0.0

50.9

50.9

45.3

CH4

0.00

2.59

12.5

0.0

0.0

101.1

101.1

103.3

HCl

1.04

2.70

12.8

9.5

5.8

111.7

127.0

156.8

HBr

0.79

3.61

11.7

3.2

4.5

182.6

190.2

207.4

HI

0.45

5.40

10.4

0.3

2.2

364.0

366.5

349.2

CHCl3

1.04

8.80

11.4

9.5

19.0

1058.0

1086.0

1632.0

CH3 OH

1.69

3.20

10.9

66.2

18.3

133.5

217.9

651.0

NH3

1.46

2.30

10.2

36.9

9.8

64.6

111.2

163.7

H2 O

1.85

1.46

12.6

95.8

10.0

32.3

138.2

176.2

CO

0.11

1.95

14.0

0.0012

0.047

64.0

64.1

60.7

CO2

0.00

2.91

13.8

0.0

0.0

140.1

140.1

163.6

N2

0.00

1.74

15.6

0.0

0.0

56.7

56.7

55.3

O2

0.00

1.58

12.1

0.0

0.0

36.2

36.2

46.0

They are given in units of 10−79 Jm6. For comparison, the van der Waals coefficient Cexp as derived from the van der Waals equation of state for a gas (P + a∕Vm2 )(Vm − b) = RT is tabulated. From the experimentally determined constants a and b, the van der Waals coefficient can be calculated with Cexp = 9ab∕(4π2 NA3 ) [174] assuming that at very short range, the molecules behave like hard-core particles. Dipole moments 𝜇, polarizabilities 𝛼, and the ionization energies h𝜈 of isolated molecules are also listed.

5.2

Van der Waals Force Between Macroscopic Solids

We now move from the interaction between two molecules to the interaction between two macroscopic solids. There are two approaches to calculating the van der Waals force between extended solids: the microscopic and the macroscopic [180].

5.2.1

Microscopic Approach

The potential energy of the interaction between molecules A and B is WAB (D) = −

CAB D6

(5.10)

The minus sign arises because it is an attractive interaction. CAB is equal to Ctotal and sums up contributions of all three dipole–dipole interactions.

97

98

5 Surface Forces

To determine the interaction between macroscopic solids, in the first step, we calculate the van der Waals energy between a molecule A and an infinitely extended body with a planar surface made of molecules B. This is also of direct relevance in understanding the adsorption of gas molecules to surfaces. We sum up the van der Waals energy between molecule A and all molecules in the solid B. Practically, this is done via an integration of the molecular density 𝜌B over the entire volume of the solid:

A

D

D′

r

x B

Figure 5.1 Calculating the van der Waals force between a macroscopic body and a molecule.

∞ ∞

WMol∕plane

𝜌B 2πrdrdx = −CAB dV = −CAB 𝜌B ∫ ∫ ∫ D′6 ∫ ∫ [(D + x)2 + r 2 ]3 0

(5.11)

0

We have used cylindrical coordinates (Figure 5.1) and assumed that the density of molecules B in the solid is constant. With 2rdr = d(r 2 ), we get ∞

∞

d(r 2 ) dx ∫0 ∫0 [(D + x)2 + r 2 ]3 }∞ ∞{ 1 = −π𝜌B CAB − dx ∫0 2[(D + x)2 + r 2 ]2 0 ∞ π𝜌 C 1 = − B AB dx ∫0 (D + x)4 2 [ ]∞ π𝜌 C 1 = − B AB − 2 3(D + x)3 0 π𝜌 C = − B 3AB (5.12) 6D Here we already see a remarkable property of van der Waals forces: the energy of a molecule and a macroscopic body decreases less steeply than the energy between two molecules. Instead of the D−6 dependence, the energy decreases proportional to D−3 . In the second step, we calculate the van der Waals energy between two infinitely extended solids that are separated by a parallel gap of thickness D. Therefore, we use Eq. (5.12) and integrate over all molecules in the solid A: WMol∕plane = −π𝜌B CAB

W =−

∞ ∞ ∞ 𝜌A πC 𝜌 𝜌A dzdydx πCAB 𝜌B dV = − AB B 3 ∫ ∫ ∫ (D + x) ∫0 ∫−∞ ∫−∞ (D + x)3 6 6 (5.13)

Here, y and z are the coordinates parallel to the gap. The integral is infinite because the solids are infinitely large. We must divide by the area. The van der Waals energy per unit area is ∞ π𝜌 𝜌 C dx W = − A B AB w= ∫0 (D + x)3 A 6 [ ]∞ π𝜌A 𝜌B CAB 1 =− − 6 2(D + x)2 0 π𝜌A 𝜌B CAB =− (5.14) 12D2

5.2 Van der Waals Force Between Macroscopic Solids

With the definition of the so-called “Hamaker constant” AH = π2 CAB 𝜌A 𝜌B

(5.15)

we get AH (5.16) 12πD2 The force per unit area is equal to the negative derivative of w versus distance: w=−

AH (5.17) 6πD3 In the same way, it is possible to calculate the van der Waals energy between solids having different geometries. One important case is the interaction between two spheres with radii R1 and R2 . The van der Waals energy is [181] [ ( 2 )] A 2R1 R2 2R1 R2 d − (R1 + R2 )2 W =− H + + ln 6 d2 − (R1 + R2 )2 d2 − (R1 − R2 )2 d2 − (R1 − R2 )2 (5.18) f =−

where d is the distance between the centers of the spheres. The distance between the surfaces is D = d − R1 − R2 . In Eq. (5.18), only attractive van der Waals forces were taken into account. At a very short range, the electron orbitals start to overlap and the molecules repel each other. Henderson considered this short-range repulsion and reports a modified version of Eq. (5.18) [182]. If the radii of the spheres are substantially larger than the distance (D ≪ R1 , R2 ), then Eq. (5.18) can be simplified [3, page 543] to W =−

AH R1 R2 6D R1 + R2

(5.19)

The van der Waals force is the negative derivative F=−

AH R1 R2 6D2 R1 + R2

(5.20)

For a sphere and a flat, planar surface, the energy and force can be obtained by allowing R2 go to infinity: W =−

AH R 6D

and F = −

AH R 6D2

(5.21)

Example 5.4 A spherical quartz (SiO2 ) particle hangs on a planar quartz surface caused by the van der Waals attraction of F = AH R∕6D2 . The van der Waals attraction increases linearly with the radius of the sphere. The gravitational force, which pulls the sphere down, is 4πR3 𝜌g∕3. It increases cubically with the radius. As a consequence, the behavior of small spheres is dominated by van der Waals forces, while for large spheres, gravity is more important. At which radius is the gravitational force so strong that the sphere detaches? The Hamaker constant is AH ≈ 6 × 10−20 J, and the density is 𝜌 = 3000 kg/m3 . For two solids in contact, we take the distance to be 1.7 Å, which corresponds to

99

100

5 Surface Forces

1 μm (a)

(b)

(c)

Figure 5.2 Picture of a gecko foot (a) and scanning electron microscope images of different resolutions of setae (b) and spatulae (c). Source: K. Autumn.

a typical interatomic spacing. We use the condition that, just before the sphere falls down, the van der Waals force is equal to the gravitational force: √ AH R 4 3 AH = πR 𝜌g ⇒ R = ⇒ 3 6D2 8πD2 𝜌g √ 6 × 10−20 J = 1.7 mm R= −10 8π ⋅ (1.7 × 10 m)2 ⋅ 3000 kg∕m3 ⋅ 9.81 m∕s In reality, the value is lower due to surface roughness and contamination on the surfaces. Roughness and contamination layers increase the effective contact distance. Example 5.5 Geckos are able to run up and down walls and even walk along the ceiling head down. They have evolved one of the most versatile and effective mechanisms to adhere to a surface. Geckos accomplish this by using van der Waals forces [183, 184]. Their feet are covered with millions of tiny foot hairs called setae; typically they have 14 000 setae per square millimeter. Each seta consists of many spatulae (Figure 5.2), which even out the surface roughness and in this way achieve a high total contact area.

5.2.2

Macroscopic Calculation – Lifshitz Theory

In the microscopic calculation pairwise additivity was assumed. We ignored the influence of neighboring molecules on the interaction between any pair of molecules. In reality, the van der Waals force between two molecules is changed by the presence of a third molecule. For example, the polarizability can change. This problem of additivity is completely avoided in the macroscopic theory developed by Lifshitz [185, 186]. Lifshitz theory neglects the discrete atomic structure, and solids are treated as continuous materials with bulk properties such as the permittivity and the refractive index. The main idea behind this approach is that the dipolar interaction between materials is based on the fluctuations of their charge distribution. As a general consequence of the fluctuation–dissipation theorem, these spontaneous charge fluctuations exhibit the same frequency dependence as the

5.2 Van der Waals Force Between Macroscopic Solids

response to an external electromagnetic perturbation. The strongest fluctuations are expected at the electromagnetic resonances of the materials, that is, at peaks in the absorption spectrum. Therefore, calculation of the respective van der Waals forces is possible on the basis of a precise knowledge of the absorption spectra of materials. For a detailed discussion of the calculation of van der Waals forces from optical material properties, we refer the reader to the comprehensive book by Parsegian [176]. The original work of Lifshitz is difficult to understand, and for some time, direct application of the Lifshitz theory was considered impractical. However, Parsegian and Ninham later demonstrated that it is indeed possible to calculate van der Waals forces from known dielectric properties of bulk materials [187]. Since then several reviews on the van der Waals interaction between macroscopic bodies have appeared [180, 188]. Fortunately, the same equations as derived previously for the microscopic calculation are obtained. In particular, the distance dependencies turn out to be correct. Only the Hamaker constant is calculated in a different way. The molecular polarizability and the ionization frequency used in the microscopic approach are replaced by the static and frequency-dependent dielectric permittivity. The Hamaker constant turns out to be the sum over many frequencies. The sum can be converted into an integral. For a material 1 interacting with material 2 across a medium 3, the Hamaker constant is )( ) 𝜀1 − 𝜀3 𝜀2 − 𝜀3 𝜀1 + 𝜀3 𝜀2 + 𝜀3 )( ) ∞( 𝜀1 (i𝜈) − 𝜀3 (i𝜈) 𝜀2 (i𝜈) − 𝜀3 (i𝜈) 3h + d𝜈 4π ∫𝜈1 𝜀1 (i𝜈) + 𝜀3 (i𝜈) 𝜀2 (i𝜈) + 𝜀3 (i𝜈)

3 AH = kB T 4

(

(5.22)

The first term, which contains the static permittivities of the three media 𝜀1 , 𝜀2 , and 𝜀3 , represents the Keesom plus the Debye energy. It plays an important role for forces in water since water molecules have a strong permanent dipole moment. Usually, however, the second term in Eq. (5.22) dominates. The dielectric permittivity is not a constant but depends on the frequency of the electric field. The static permittivities are the values of this dielectric function at zero frequency. 𝜀1 (i𝜈), 𝜀2 (i𝜈), and 𝜀3 (i𝜈) are the dielectric permittivities at imaginary frequencies i𝜈, and the lower integration boundary is given by 𝜈1 = 2πkB T∕h = 3.9 × 1013 Hz at 25 ∘ C. This corresponds to a wavelength of 760 nm, which is within the optical regime of the spectrum. This implies that mainly the dielectric response in the optical and ultraviolet (UV) or vacuum ultraviolet (VUV) spectral range contributes to the value of the Hamaker constant. Equation (5.22) is valid under the assumption that distances are short enough to exclude the effect of retardation, which is described in Section 5.2.3.5 To calculate the Hamaker constant, the dielectric permittivities of all three materials need to be known for all relevant frequencies. In a simple model assuming a

5 A complete equation of the van der Waals force, which also contains retarded parts, is described in [186].

101

102

5 Surface Forces

single absorption band at the ionization energy of the material, the dielectric permittivity can be described by 𝜀(i𝜈) = 1 +

n2 − 1 1 + 𝜈 2 ∕𝜈e2

(5.23)

Here, n is the refractive index and 𝜈e is the mean ionization frequency of the material. Typically, this is 𝜈e ≈ 3 × 1015 Hz. If the absorption frequencies of all three materials are assumed to be the same, we obtain the following approximation for the nonretarded Hamaker constant: ( )( ) 𝜀1 − 𝜀3 𝜀2 − 𝜀3 3 AH ≈ kB T 4 𝜀1 + 𝜀3 𝜀2 + 𝜀3 (n21 − n23 )(n22 − n23 ) 3h𝜈 + √e √ (5.24) (√ ) √ √ 8 2 n2 + n2 n2 + n2 2 2 2 2 n + n + n + n 1 1 3 2 3 3 2 3 This model is a strong simplification of the true spectrum of a material. However, due to the fact that the major contribution to the Hamaker constant comes from frequencies in the UV or VUV range, it usually leads to reasonable estimates. The refractive indices of the materials are n1 , n2 , and n3 . Values of 𝜀, n, and 𝜈e for different materials are listed in Table 5.2. Example 5.6 Calculate the Hamaker constant for the interaction of amorphous silicon oxide (SiO2 ) with silicon oxide across water at 20 ∘ C. According to Table 5.2, we insert 𝜀 = 78.5 and n = 1.33 for water, 𝜀 = 3.82 and n = 1.46 for silicon oxide, and 𝜈e = 3.4 × 1015 Hz for the mean absorption frequency, into Eq. (5.24): ( ) 3.82 − 78.5 2 AH = 3.04 × 10−21 J ⋅ 3.82 + 78.5 (1.462 − 1.332 )2 −20 + 59.7 × 10 J⋅ √ (1.462 + 1.332 ) ⋅ (2 1.462 + 1.332 ) = 2.50 × 10−21 J + 5.10 × 10−21 J = 7.60 × 10−21 J Using full spectroscopic results and Eq. (5.22) leads to somewhat lower Hamaker constants (Table 5.3). Equation (5.24) not only allows us to calculate the Hamaker constant. It also allows us to predict whether we can expect attraction or repulsion. An attractive van der Waals force corresponds to a positive sign of the Hamaker constant. Repulsion corresponds to a negative Hamaker constant. Van der Waals forces between similar materials are always attractive. This can easily be deduced from the last equation: for 𝜀1 = 𝜀2 and n1 = n2 , the Hamaker constant is positive, which corresponds to an attractive force. If two different media interact across vacuum (𝜀3 = n3 = 1) or a gas, the van der Waals force is also attractive. Van der Waals forces between different materials across a condensed phase can be repulsive for special material combinations. Repulsive van der Waals forces occur when medium 3 is more strongly attracted to medium 1 than medium 2. Repulsive forces were, for instance,

5.2 Van der Waals Force Between Macroscopic Solids

Table 5.2 Permittivity 𝜀, refractive index n, and main absorption frequency 𝜈e in the UV range for various solids, liquids, and polymers at 20 ∘ C (see [174, 191, 192], handbooks, and the authors’ own measurements). Material

𝜺

n

𝝂 e (10 𝟏𝟓 Hz)

Al2 O3 (alumina)

9.3–11.5

1.75

3.2

C (diamond)

5.7

2.40

2.7

CaCO3 (calcium carbonate, average)

8.2

1.59

3.0

CaF2 (fluorite)

6.7

1.43

3.8

KAl2 Si3 AlO10 (OH)2 (muscovite mica)

5.4

1.58

3.1

KCl (potassium chloride)

4.4

1.48

2.5

NaCl (sodium chloride)

5.9

1.53

2.5

Si3 N4 (silicon nitride, amorphous)

7.4

1.99

2.5

SiO2 (quartz)

4.3–4.8

1.54

3.2

SiO2 (silica, amorphous)

3.82

1.46

3.2

TiO2 (titania, average)

114

2.46

1.2

ZnO (zinc oxide)

11.8

1.91

1.4

Acetone

20.7

1.359

2.9

Chloroform

4.81

1.446

3.0

n-Hexane

1.89

1.38

4.1

n-Octane

1.97

1.41

3.0

n-Hexadecane

2.05

1.43

2.9

Ethanol

25.3

1.361

3.0

1-Propanol

20.8

1.385

3.1

1-Butanol

17.8

1.399

3.1

1-Octanol

10.3

1.430

3.1

Toluene

2.38

1.497

2.7

Water

78.5

1.333

3.6

Polyethylene

2.26–2.32

1.48–1.51

2.6

Polystyrene

2.49–2.61

1.59

2.3

Poly(vinyl chloride)

4.55

1.52–1.55

2.9

Poly(tetrafluoroethylene)

2.1

1.35

4.1

Poly(methyl methacrylate)

3.12

1.50

2.7

1.45

2.8

2.6–2.8

1.4

2.8

Nylon 6

3.8

1.53

Bovine serum albumin

4.0

Poly(ethylene oxide) Poly(dimethyl siloxane)

2.7 2.4–2.8

103

104

5 Surface Forces

Table 5.3 Hamaker constants for medium 1 interacting with medium 2 across medium 3.

Medium 1

Medium 3

Medium 2

AH calc. (10−𝟐𝟎 J)

Au/Ag/Cu

Vacuum

Au/Ag/Cu

20–50

Mica

Vacuum

Mica

7.0

Al2 O3

Vacuum

Al2 O3

14.5–15.2

SiO2

Vacuum

SiO2

6.4–6.6

Si3 N4

Vacuum

Si3 N4

16.2–17.4

TiO2

Vacuum

TiO2

14.3–17.3

Perfluorocarb.

Vacuum

Perfluorocarb.

3.4–6.0

Carbonhydr.

Vacuum

Carbonhydr.

2.6–3.0

AH exp. (10−𝟐𝟎 J)

10–13.5 5–6

Au/Ag/Cu

Water

Au/Ag/Cu

10–13

Mica

Water

Mica

0.29

2.2

Al2 O3

Water

Al2 O3

2.8–4.7

6.7

SiO2

Water

SiO2

0.16–1.51

Si3 N4

Water

Si3 N4

4.6–5.9

2–8

TiO2

Water

TiO2

5.4–6.0

4–8

Perfluorocarb.

Water

Perfluorocarb.

0.36–0.74

Carbonhydr.

Water

Carbonhydr.

0.39–0.44

Polystyrene

Water

Polystyrene

0.9–1.3

SiO2

Water

Air

–1.0

BSA (Albumin)

Water

SiO2

0.7

0.3–0.6

The variation in calculated Hamaker constants is due to the fact that different dielectric functions were used in Eq. (5.22). Results were partially taken from [191, 192, 194–196].

measured for the interaction of silicon nitride with silicon oxide in diiodomethane [189]. Repulsive van der Waals forces can also occur across thin films on solid surfaces. In the case of thin liquid films on solid surfaces, there is often a repulsive van der Waals force between the solid–liquid and the liquid–gas interface [190]. An important example of a repulsive van der Waals force is the force between a solid particle interacting in water with an air bubble. This is a typical situation in flotation, where air bubbles are used to extract mineral particles from an aqueous dispersion (Section 6.5.1). For some materials, the van der Waals force between the solid–liquid and the liquid–vapor interfaces is repulsive and a stable water film is formed (e.g. in [193]). This repulsion prevents flotation (Section 6.5.1). The preceding analysis applies to insulating materials. For electrically conductive materials such as metals, the static dielectric constant becomes infinity. The frequency-dependent dielectric permittivity of a metal can be derived within the framework of the Drude model: 𝜈2 (5.25) 𝜀(i𝜈) = 1 + e2 𝜈

5.2 Van der Waals Force Between Macroscopic Solids

where 𝜈e is the so-called “plasma frequency” of the free electron gas; it is typically 5 × 1015 Hz. Inserting Eq. (5.25) into Eq. (5.22) leads to the approximate Hamaker constant: 3 AH ≈ √ h𝜈e ≈ 4 × 10−19 J (5.26) 16 2 for two metals interacting across vacuum (Exercise 5.2). Thus, the Hamaker constants of metals and metal oxides can be up to an order of magnitude higher than those of nonconducting media. In Table 5.3, nonretarded Hamaker constants are listed for different material combinations. Hamaker constants, calculated from spectroscopic data, are found in many publications [191, 194–196]. Reviews are [192, 198]. In many applications, we are interested in the van der Waals force across an aqueous medium. Then, an important question concerns the effect of dissolved ions. Ions hinder the water molecules in their hydration shell from orienting in an external electric field. The first term in the equation of the Hamaker constant is affected. In addition, the salt concentration is often much higher on surfaces than in the bulk phase. As a consequence, the dielectric permittivity can be smaller than in the bulk phase. Example 5.7 For the interaction of lipid bilayers across a layer of water, a Hamaker constant of 7.5 × 10−21 J is calculated. A value of only 3 × 10−21 J was measured. One reason is probably a reduction of the first term in Eq. (5.22) by the presence of ions [199]. From Eq. (5.22), a useful approximation can be derived: √ A132 ≈ A131 A232

(5.27)

If we know the Hamaker constant of material 1 interacting across medium 3 with itself, A131 , and we know the Hamaker constant of material 2 interacting across medium 3 with itself, A232 , then we can estimate the Hamaker constant for the interaction between material 1 with material 2 across medium 3, A132 .

5.2.3

Retarded Van der Waals Forces

So far we have implicitly assumed that the surfaces are at such close distances to each other that the propagation of the electric field is instantaneous. This is, however, not necessarily true. To illustrate this effect, let us have a closer look at what happens when two molecules interact. In one molecule, a spontaneous random dipole moment arises, which generates an electric field. The electric field expands with the speed of light. It polarizes the second molecule, whose dipole moment in turn causes an electric field that reaches the first molecule with the speed of light. The process takes place as calculated only if the electric field has enough time to cover twice the distance D between the molecules before the dipole moment has completely changed again. This takes a time Δt = D∕c, where c is the speed of light. If the

105

106

5 Surface Forces

first dipole changes faster than Δt, the interaction becomes weaker. The time during which the dipole moment changes is in the order of 1/𝜈. Hence, only if 2D 1 < (5.28) c 𝜈 does the interaction take place as considered. The relevant frequencies are those corresponding to the ionization of the molecule, which are typically 3 × 1015 Hz. Thus, from this very simplistic picture, one obtains that for distances larger than D > c∕2𝜈 ≈ 3 × 108 ∕6 × 1015 m = 50 nm, the London contribution to the van der Waals energy should vanish. This effect is known as retardation. A more detailed analysis of the Lifshitz equations – which inherently contain the retardation effect – shows that retardation sets in already at 5–10 nm, leading to a faster decay (i.e. for molecules with 1∕D7 ) of the van der Waals interaction with distance. For distances larger than a few micrometers, the dispersion interaction completely vanishes due to retardation and only the Keesom and Debeye contributions remain, which again leads to a distance dependence of 1∕D6 as for short distances, however, with a lower value of the Hamaker constant. Retardation therefore implies that when distances larger than 5–10 nm are relevant, the Hamaker constants effectively become “Hamaker functions” that depend on distance.

5.2.4

Surface Energy and the Hamaker Constant

To calculate the surface energy of molecular crystals, we imagine the following thought experiment: a crystal is cleaved into two parts separated by an infinite distance (Figure 5.3). The work required per unit area is w = AH ∕(12πD20 ), where D0 is the distance between two atoms. Upon cleavage, two fresh surfaces are formed. With the surface energy 𝛾S , the work per unit area required is 2𝛾S . Equating the results leads to AH 𝛾S = (5.29) 24πD20 Example 5.8 Surface energy of helium [174]. As an interatomic distance, often a value of 1.6 Å is used. The Hamaker constant of helium–vacuum–helium is 5.7 × 10−22 J. Calculating the surface energy with Eq. (5.29) leads to 0.29 mJ/m2 . This value is in good agreement with measured values for liquid helium of 0.12–0.35 mJ/m2 . For Teflon, we calculate a surface energy of 16–28 mJ/m2 [196] using an atomic spacing of 1.7 Å and a Hamaker constant of 3.4–6.0 × 10−20 J.

D0

Figure 5.3 Cleaving a molecular crystal to calculate the surface energy of a solid.

5.3 Concepts for the Description of Surface Forces

5.3 Concepts for the Description of Surface Forces 5.3.1

The Derjaguin Approximation

In Section 5.1, we calculated the van der Waals force between two spheres and between two planar surfaces. What if the two interacting bodies do not have such a simple geometry? We could try to do an integration similar to what was described for the two spheres. This integration, however, might be very difficult and lead to long expressions. The Derjaguin6 approximation is a simple way to overcome this problem. The Derjaguin approximation relates the energy per unit area between two planar surfaces w that are separated by a gap of width x to the energy between two bodies of arbitrary shape W, which are at a distance D (Figure 5.4): ∞

W(D) =

(5.30)

w(x)dA

∫D

The integration runs over the entire surface of the solid. Note that here A is the cross-sectional area. Often, we deal with rotational-symmetric configurations. Then it is reasonable to integrate in cylindrical coordinates: ∞

W(D) = 2π

∫0

w(x(r))r dr

(5.31)

In many cases, the following expression is more useful: ∞

W(D) =

∫D

w(x)

dA dx dx

(5.32)

The approximation is only valid if the characteristic decay length of the surface force is small in comparison to the curvature of the surfaces. Approximation Eq. (5.30) is sometimes called the Derjaguin approximation in honor of Derjaguin’s work. He used this approach to calculate the interaction between two ellipsoids [53]. Example 5.9 Calculate the van der Waals force between a cone with an opening angle 𝛼 and a planar surface (Figure 5.5). The cross-sectional area is given by A = π[(x − D) tan 𝛼]2

for

x≥D

Figure 5.4 Schematic of Derjaguin’s approximation.

r

x D

6 Boris Vladimirovich Derjaguin, 1902–1994; Russian physicochemist, professor in Moscow.

107

108

5 Surface Forces

Figure 5.5 Interaction between a cone and a planar surface.

x

α D

r

which leads to dA = 2π tan2 𝛼 ⋅ (x − D) dx For the force, we can use an equation similar to that used for the energy: ∞ ∞ AH dA F(D) = f (x) dx = − 2π tan2 𝛼 ⋅ (x − D) dx ∫D ∫D 6πx3 dx A tan2 𝛼 ∞ x − D =− H dx ∫D 3 x3 ] ) 2 ( A tan2 𝛼 [ 1 D ∞ AH tan 𝛼 1 1 =− H − + 2 = − + 3 x 2x D 3 D 2D AH tan2 𝛼 =− 6D A special, but nevertheless important, case is the interaction between two identical spheres. It is important to understand the stability of dispersions. For the case of two spheres of equal radius R, the parameters x and r are related by (Figure 5.6): √ √ 2r x(r) = D + 2R − 2 R2 − r 2 ⇒ dx = √ dr ⇒ 2rdr = R2 − r 2 ⋅ dx R2 − r 2 If the range of the interaction is substantially smaller than R, then we only need to consider the outer caps of the two spheres, and only the contributions with a small r are effective. We can simplify the preceding equation to √ 2rdr = R2 − r 2 ⋅ dx ≈ R dx (5.33) and the integral in Eq. (5.32) becomes ∞

W(D) = πR

∫D

w(x)dx

Figure 5.6 Calculating the interaction between two spheres with Derjaguin’s approximation.

x(r) R

r

R D

(5.34)

5.3 Concepts for the Description of Surface Forces

From the potential energy, we can calculate the force between two spheres: ( ∞ ) d dW F=− = −πR w(x)dx = πRw(D) (5.35) dD D ∫D since w(∞) = 0. Thus, assuming that the range of the interaction is substantially smaller than R, there is a simple relationship between the potential energy per unit area and the force between two spheres. As one example we calculate the van der Waals force between two identical spheres from the van der Waals energy per unit area w = −AH ∕(12πx2 ) Eq. (5.16). Using Derjaguin’s approximation we can directly write AH R (5.36) 12D2 which agrees with Eq. (5.20) for R1 = R2 = R. Equation (5.35) refers to the interaction between two spheres. For a sphere that is at a distance D from a planar surface, we get a similar relation: F=−

F = 2πRw(D)

(5.37)

White extended the approximation to solids with arbitrary shape [200]. Derjaguin’s approximation has a fundamental consequence. In general, the force or energy between two bodies depends on the shape, on the material properties, and on the distance. Now, it is possible to divide the force (or energy) between two solids into a purely geometrical factor and into a material and distance-dependent term w(x). Thus, it is possible to describe the interaction independent of the geometry. This also gives us a common reference, which is w(x). In what follows, we only discuss w(x).

5.3.2

Disjoining Pressure

The term disjoining pressure was introduced in 1936 by Derjaguin [201]. It is defined as the change in Gibbs energy with distance and per unit area at constant cross-sectional area, temperature, and pressure: 1 𝜕G || Π=− (5.38) A 𝜕x ||A,T,P The film is in equilibrium with the bulk phase surrounding it. The disjoining pressure Π can be interpreted in two ways. On the one hand, it can be seen as the difference between the pressure within a film between two interfaces and the pressure in the bulk phase (Figure 5.7). On the other hand, it is simply given as surface force per unit area. Taking the example of Figure 5.7, the surface forces acting between the two parallel plates lead to attraction or repulsion between them. Figure 5.7 Disjoining pressure between two parallel plates.

Pressure in the film P+Π x

Pressure P of bulk liquid

109

110

5 Surface Forces

This net force acting on the plates (apart from gravitation) divided by the area of the plates is then identical to the disjoining pressure. To think in terms of disjoining pressure instead of surface forces may be more intuitive for some cases. For example, with respect to the stability of a liquid foam film, it is more appropriate to think in terms of an increased pressure in the film rather than of the fact that this pressure is caused by repulsive forces between two liquid–air interfaces.

5.4 Measurement of Surface Forces The development of the theory of van der Waals forces stimulated an interest in measuring forces between surfaces to verify the theory. Early direct force measurements were made between two polished glass bodies [202]. One glass body was fixed, and the other was mounted on a spring. The distance between the two glass surfaces and the deflection of the spring were measured. Multiplication of the deflection by the spring constant gave the force. Using these simple early devices it was possible to verify the theoretically predicted van der Waals force for glass interacting with glass across a gaseous medium [203, 204]. Derjaguin, Rabinovich, and Churaev measured the force between two thin metal wires. In this way, they could determine the van der Waals force for metals [205]. In these early measurements, several problems became obvious. The minimal distance accessible (≈ 20 nm) was rather limited, and experiments could not be done in a liquid environment. One problem inherent in all techniques used to measure surface forces is surface roughness. The surface roughness over the interacting areas limits the distance resolution and the accuracy of the definition of zero distance (contact). Practically, zero distance is the distance of closest approach because if a protrusion is sticking out of one surface, the other surface cannot get any closer. Two methods can be used to reduce this problem of surface roughness: either atomically smooth surfaces are chosen or the interacting areas are reduced. The first approach was realized with the development of the surface force apparatus (SFA), while the advantage of small interacting areas could be utilized with the invention of the atomic force microscope (AFM). The development of the SFA [206, 207] was a big step forward because it allowed one to measure directly the force law in liquids and vapors at the Ångström resolution level [208]; for recent developments, see [209–211]. The SFA contains two crossed, atomically smooth mica cylinders of roughly 1 cm radius between which the interaction forces are measured (Figure 5.8). One mica cylinder is mounted on a piezoelectric translator with which the distance is adjusted. The other mica surface Figure 5.8 Schematic of two crossed mica cylinders used in a surface force apparatus.

5.4 Measurement of Surface Forces

Photo detector

Laser light

Microfabricated tip or Particle fixed on the cantilever Sample

Piezoelectric scanner (a)

(b)

Figure 5.9 Schematic of an atomic force microscope (a) and an SEM image of a colloidal probe (b). The glass sphere is sintered to the end of an AFM cantilever and has a diameter of ≈ 10 𝜇m.

is mounted on a spring of known and adjustable spring constant. The separation between the two surfaces is measured by the use of an optical technique using multiple beam interference fringes. Knowing the position of one cylinder and the separation from the surface of the second cylinder, the deflection of the spring and the force can be calculated. The second device with which surface forces can be measured directly is the AFM, sometimes also called the scanning force microscope (Figure 5.9) [212, 213]. In the AFM, we measure the force between a sample surface and a microfabricated tip, placed at the end of an approximately 100 μm long and 0.4–10 μm thick cantilever. Alternatively, colloidal particles are fixed on the cantilever. This technique is called the colloidal probe technique. With the AFM, the forces between surfaces and colloidal particles can be directly measured in a liquid [214, 215]. The practical advantage is that measurements are quick and simple. Even better, the interacting surfaces are substantially smaller than in the SFA. Thus, the problem of surface roughness, deformation, and contamination is reduced. This again makes it possible to examine surfaces of different materials. Another method to measure force acting between solid surfaces in liquid is total internal reflection microscopy (TIRM) [216]. Using TIRM, the distance between a single microscopic sphere immersed in a liquid and a transparent plate can be monitored with a typical resolution of 1 nm in distance. The distance is calculated from the intensity of light scattered by the sphere when illuminated by an evanescent wave through the plate. From the equilibrium distribution of distances sampled by Brownian motions, the potential energy versus distance can be determined. TIRM complements force measurements with the AFM and the SFA because it covers a lower force range. With optical tweezers, the interaction between light and matter is used to trap and steer micrometer-sized beads in liquid with the help of a laser beam that is tightly focused by a high numerical aperture objective [217]. The effective spring constant of the laser trap depends on the intensity gradient of the laser focus. Calibration of the effective spring constant of the trap can be done by applying a known force,

111

112

5 Surface Forces

for example by applying a laminar flow of the liquid around the particle at a fixed velocity and measuring the resulting displacement. Displacement of the particle can be monitored either by video microscopy and digital image analysis or by analysis of the intensity pattern of the scattered light. A detailed description of the technique is given in [218]. Optical tweezers make it possible to resolve forces in the subpiconewton range but have a limited maximum force range of typically less than 100 pN. Therefore, they have mostly been applied to biological systems, where a trapped bead acts as a handle to apply forces on single molecules that bridge the gap between the bead and a solid surface or a cell, but they have also been used to study colloidal interactions [219]. Another important, although less direct, technique for measuring forces between macromolecules or lipid bilayers is the osmotic stress method [220, 221]. A dispersion of vesicles or macromolecules is equilibrated with a reservoir solution containing water and other small solutes, which can freely exchange with the dispersion phase. The reservoir also contains a polymer that cannot diffuse into the dispersion. The polymer concentration determines the osmotic stress acting on the dispersion. The spacing between the macromolecules or vesicles is measured by X-ray diffraction. In this way, one obtains pressure-versus-distance curves. The osmotic stress method is used to measure interactions between lipid bilayers, DNA, polysaccharides, proteins, and other macromolecules [222]. It was particularly successful in studying the hydration force (see below) between lipid bilayers and biological macromolecules. All techniques mentioned so far are mainly used to study the force between solid surfaces. In many applications, one is interested in the disjoining pressure between liquid–liquid or liquid–gas interfaces, such as those found in foams and emulsions. One such technique is described in Section 11.5.3.

5.5

Electrostatic Double-Layer Force

In Chapter 4, we learned that in water, most surfaces bear an electric charge. If two such surfaces approach each other and the electric double layers overlap, an electrostatic double-layer force arises. This electrostatic double-layer force is important in many natural phenomena and technical applications. For example, it stabilizes dispersions.7

5.5.1

Electrostatic Interaction Between Two Identical Surfaces

An important case is the interaction between two identical parallel surfaces of two infinitely extended solids. It is important, for instance, to understand the coagulation of sols. We can use the resulting symmetry of the electric potential to simplify the 7 The electrostatic double-layer force is fundamentally different from the Coulomb force. For example, if we consider two identical spherical particles of radius R, we cannot take Eq. (5.1), insert the total surface charge as Q1 and Q2 , use the permittivity of water, and expect to obtain a reasonable result. The main differences are the free charges (ions) in solution. They screen the electrostatic field emanating from the surfaces.

5.5 Electrostatic Double-Layer Force

x

x

φ0

φ

φ0

φm

ξ

Figure 5.10 Change in potential distribution when two parallel planar surfaces approach each other. The gap is filled with electrolyte solution.

calculation. For identical solids, the surface potential 𝜑0 on both surfaces is equal. In between, the potential decreases (Figure 5.10). In the middle, the gradient must be zero because of the symmetry, that is, d𝜑∕d𝜉 = 0 at 𝜉 = x∕2. For simplicity, we consider only monovalent salts. To calculate the Gibbs energy per unit area w(x), we follow Verwey and Overbeek [116] and consider how the Gibbs energy of two double layers changes when they approach each other. The Gibbs energy of one isolated electric double layer per unit area is Eq. (4.39) 𝜑0

−

∫0

𝜎 d𝜑′0

(5.39)

For two identical and homogeneous double layers that are infinitely separated, the Gibbs energy per unit area is twice this value: g∞ = −2

𝜑0

∫0

𝜎 d𝜑′0

(5.40)

If the two surfaces approach each other up to a distance x, the Gibbs energy changes. Now surface charge and potential depend on the distance: 𝜑0

g(x) = −2

∫0

𝜎(x)d𝜑′0

(5.41)

The Gibbs interaction energy per unit area is w(x) = g(x) − g∞

(5.42)

Because of electroneutrality in the double layer, surface charge and surface potential are related by (Eq. 4.29) d𝜑 || 𝜎 = −𝜀𝜀0 (5.43) d𝜉 ||𝜉=0 The surface charge is (except for two constants) equal to the potential gradient |d𝜑∕d𝜉| at the surface. Equation (5.41) allows us to gain some intuitive understanding of the electrostatic double-layer force. If two surfaces are brought close to each other, their double layers cannot develop fully. The charge on the surfaces is lower than that on the corresponding free double layers. The potential gradient at the surface decreases (Figure 5.10), and the surface charge density decreases accordingly. If, for instance,

113

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5 Surface Forces

AgI particles are brought together, then I− ions are removed from the surface and the surface charge density decreases. During the approach of SiO2 particles, previously dissociated H+ ions bind again. Neutral hydroxide groups are formed, and the negative surface charge becomes weaker. This reduction in the surface charge increases the Gibbs energy of the surface (otherwise, the double layer would not have formed). As a consequence, the surfaces repel each other. At this point, it is probably instructive to discuss the use of the symbols D, x, and 𝜉. D is the shortest distance between finite, macroscopic bodies with a specific geometry. Usually, we use x for the thickness of the gap between two infinitely extended solids. For example, it appears in the Derjaguin approximation because there we integrate over many such hypothetical gaps. 𝜉 is a coordinate describing a position within the gap. At a given gap thickness of x, the potential changes with 𝜉 (Figure 5.10). To calculate the double-layer force with Eq. (5.41), we need to know 𝜑(𝜉). Therefore, we solve the Poisson–Boltzmann equation. The one-dimensional Poisson–Boltzmann equation (4.6) is ( ) d2 𝜑 ec0 ee𝜑∕(kB T) − e−e𝜑∕(kB T) − 𝜀𝜀0 2 = 0 d𝜉

(5.44)

The first term is sometimes written as follows: 2ec0 sinh(e𝜑∕kB T). Integration of Eq. (5.44) leads to (Section 4.2.3) ( ) ( ) 𝜀𝜀 d𝜑 2 c0 kB T ee𝜑∕kB T + e−e𝜑∕kB T − 0 =P (5.45) 2 d𝜉 Both terms have an intuitive physical meaning: the first term in Eq. (5.45) describes the osmotic pressure caused by the increased number of particles (ions) in the gap. The second term, sometimes called the Maxwell stress term, corresponds to the electrostatic force caused by the electric field of one surface that affects charges on the other surface and vice versa. The sum of both terms is constant and equal to P. So far, P is only an integration constant. It is determined by the boundary conditions. The boundary conditions are conveniently chosen at the midpoint between the plates because there the slope of the potential is zero. Thus, at 𝜉 = x∕2, we have 𝜑 = 𝜑m

and

d𝜑 =0 d𝜉

(5.46)

The index m stands for midpoint. With these boundary conditions, we obtain ( ) P = c0 kB T ee𝜑m ∕(kB T) + e−e𝜑m ∕(kB T) (5.47) P is equal to the osmotic pressure at the midplane caused by the ions in solution. At the Midplane, the electric field is zero, and there is no direct force caused by the electric field. Now, we can derive an expression for the disjoining pressure. The solution in the infinitely extended gap is in contact with an infinitely large reservoir. As the force per unit area is Π, only the difference of the pressure inside the gap and the pressure in the reservoir is effective. Therefore, the osmotic pressure in the reservoir 2kB Tc0

5.5 Electrostatic Double-Layer Force

must be subtracted from P to get the disjoining pressure: Π = P − 2kB Tc0 . Finally, for the force per unit area, we obtain ( ) ( ) 𝜀𝜀 d𝜑 2 Π = c0 kB T ee𝜑∕(kB T) + e−e𝜑∕(kB T) − 2 − 0 (5.48) 2 d𝜉 Using x

w(x) = −

∫∞

Π(x′ )dx′

(5.49)

we can calculate the Gibbs energy per unit area. To calculate the disjoining pressure (Eq. 5.48) or the Gibbs energy per unit area (Eqs. 5.41 and 5.42), the potential distribution in the gap 𝜑(𝜉) must first be determined. This is done by solving the Poisson–Boltzmann equation (5.45). Unfortunately, it can only be solved numerically [116] since the solution involves elliptical integrals of the first kind. Equation (5.48) allows us an alternative approach. The disjoining pressure in the center is given only by the osmotic pressure. Toward the two surfaces, the osmotic pressure increases. This increase is, however, compensated by a decrease in the Maxwell stress term. The disjoining pressure must be independent of 𝜉. We are free to choose the most convenient value for 𝜉, which is at the midpoint. At 𝜉 = x∕2, the disjoining pressure is equal to ( ) Π = c0 kB T ee𝜑m ∕kB T + e−e𝜑m ∕kB T − 2 (5.50) If we know how the potential at midplane 𝜑m depends on the surface potential 𝜑0 and the distance x, we can calculate the disjoining pressure. Again, for the exact relation, we would need to solve the Poisson–Boltzmann equation. In some limiting cases, however, we can obtain simple approximations. For low potentials, we can simplify expression (5.50). Thus, we write the exponential functions in a series and neglect all terms higher than the quadratic one: [ ] ( ) ( ) e𝜑m 1 e𝜑m 2 e𝜑m 1 e𝜑m 2 Π = kB Tc0 1 + + +···+1− + ±···−2 kB T 2 kB T kB T 2 kB T ≈

c0 e2 2 𝜀𝜀 𝜑m = 20 𝜑2m kB T 2λD

(5.51)

It remains to find 𝜑m . If the electric double layers of the two opposing surfaces overlap only slightly (x ≫ λD ), then we can approximate ( ) x 𝜑m = 2𝜑′ (5.52) 2 where 𝜑′ is the potential of an isolated double layer. For 𝜑′ , we can use various exact functions. For low potentials, we can use 𝜑(𝜉) = 𝜑0 exp(−𝜉∕λD ), which leads to a repulsive force per area of Π(x) =

2𝜀𝜀0 λ2D

𝜑20 e−x∕λD

(5.53)

115

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5 Surface Forces

To calculate the Gibbs free interaction energy per unit area, we must still integrate: x

w(x) = −

∫∞

Π(x′ )dx′ = −

2𝜀𝜀0 𝜑20 λ2D

x

∫∞

′

e−x ∕λD dx′ =

2𝜀𝜀0 𝜑20 λD

e−x∕λD

(5.54)

If we use expression (4.23) for 𝜑′ , which is also valid at higher potentials, we get ( e𝜑 ∕2k T )2 e 0 B −1 w(x) = 64c0 kB TλD e𝜑 ∕2k T e−x∕λD (5.55) e 0 B +1 This is sometimes written as follows: ( ) e𝜑0 w(x) = 64c0 kB TλD tanh2 e−x∕λD 4kB T

(5.56)

Both expressions are identical. This identity is easily seen if we recall the definition of the tanh function: ez − e−z tanh z = z e + e−z Multiplication of both the numerator and denominator by ez leads to e2z − 1 e2z + 1 The approach described can be generalized to surfaces with different surface potentials. It can also be generalized with respect to the boundary condition at the surfaces. Two common types of boundary conditions are [223, 224] as follows: Constant potential:

Upon approach of the two surfaces the surface potentials remain constant: 𝜑(𝜉 = 0) = 𝜑1 and 𝜑(𝜉 = x) = 𝜑2 .

Constant charge:

During an approach, the surface charge densities 𝜎1 and 𝜎2 are constant.

For large distances (x ≫ λD ), both boundary conditions lead to identical forces. At small distances, the constant charge condition leads to more repulsive forces than the constant potential condition (Figure 5.11). The approach between two charged surfaces may lead to adsorption or desorption of ions at the surfaces. In this case, neither constant potential nor constant charge will describe the system correctly, but rather the chemical equilibrium of ionizable surface groups has to be considered. One possible approach is to use the linear charge regulation model [225], where a linear relationship between surface charge and surface potential is assumed.

5.5.2

DLVO Theory

It has been known for more than 100 years that many aqueous dispersions precipitate upon the addition of salt. Schulze and Hardy observed that most dispersions precipitate at concentrations of 25–150 mM of monovalent counterions [226, 227]. For divalent ions, they found far smaller precipitation concentrations of 0.5–2 mM. Trivalent counterions lead to precipitation at even lower concentrations of 0.01–0.1 mM. For example, gold colloids are stable in NaCl solution, as long as

5.5 Electrostatic Double-Layer Force

Figure 5.11 Electrostatic double-layer force between a sphere of R = 3 𝜇m radius and a flat surface in water containing 1 mM monovalent salt. The force was calculated using the nonlinear Poisson–Boltzmann equation and the Derjaguin approximation for constant potentials (𝜑1 = 80 mV, 𝜑2 = 50 mV) and for constant surface charge (𝜎1 = 0.0058 C∕m2 = 0.036 e∕nm2 , 𝜎2 = 0.0036 C∕m2 = 0.023 e∕nm2 ). The surface charge was adjusted by 𝜎1∕2 = 𝜀𝜀0 𝜑1∕2 ∕λD so that at large distances both lead to the same potential.

the NaCl concentration does not exceed 24 mM. If the solution contains more NaCl, then the gold particles aggregate and precipitate. The appropriate concentrations for KNO3 , CaCl2 , and BaCl2 are 23, 0.41, and 0.35 mM [228]. This coagulation can be understood as follows [229]: The gold particles are negatively charged and repel each other. With increasing salt concentration, the electrostatic repulsion decreases. The particles, which move around thermally, frequently have a higher chance of approaching each other to a few Ångströms. Then the van der Waals attraction causes them to aggregate. Since divalent counterions weaken the electrostatic repulsion more effectively than monovalent counterions, only small concentrations of CaCl2 and BaCl2 are necessary for coagulation. Roughly 80 years ago, Derjaguin, Landau, Verwey, and Overbeek developed a theory to explain the aggregation of aqueous dispersions quantitatively [116, 230, 231]. This theory is called the DLVO theory (recent review: [232]). In DLVO theory, coagulation of dispersed particles is explained by the interplay between two forces: the attractive van der Waals force and the repulsive electrostatic double-layer force. These forces are sometimes referred to as DLVO forces. Van der Waals forces promote coagulation while the double-layer force stabilizes dispersions. Taking into account both components, we can approximate the energy per unit area between two infinitely extended solids separated by a gap x: ( w(x) = 64c0 kB TλD

ee𝜑0∕2kB T − 1 ee𝜑0∕2kB T + 1

)2 e−x∕λD −

AH 12πx2

(5.57)

117

118

5 Surface Forces

This is sometimes written as follows: ( ) e𝜑0 AH 2 w(x) = 64c0 kB TλD tanh e−x∕λD − 4kB T 12πx2

(5.58)

Figure 5.12 shows the interaction energy between two identical spherical particles calculated with the DLVO theory. In general, it can be described by a very weak attraction at large distances (secondary energy minimum), an electrostatic repulsion at intermediate distances, and a strong attraction at short distances (primary energy minimum). At different salt concentrations, the three regimes are more or less pronounced and sometimes even completely missing. At low and intermediate salt concentrations, the repulsive electrostatic barrier prevents particles from aggregating. With increasing salt concentration, the repulsive energy barrier decreases. At low salt concentrations, the energy barrier is so high that particles in a dispersion have practically no chance of gaining enough thermal energy to overcome that barrier. At high salt concentrations, this energy barrier is reduced and the van der Waals attraction dominates. This leads to precipitation. In addition, the surface potential usually decreases with increasing salt concentration, thereby lowering the energy barrier even more (this effect was not taken into account in Figure 5.12). As mentioned above, the choice of boundary conditions will influence the calculated values of the electric double layer force and thus also the forces calculated by the DLVO theory. In addition, complexation of ions in solution or adsorption of multivalent counterions can lead to charge reversal at interfaces. If such effects and

Figure 5.12 Gibbs free interaction energy (in units of kB T) versus distance for two identical spherical particles of R = 100 nm radius in water, containing different concentrations of monovalent salt. The calculation is based on DLVO theory using Eqs. (5.57) and (5.32). The Hamaker constant was AH = 7 × 10−21 J and the surface potential was set to 𝜑0 = 30 mV. The insert shows the weak attractive interaction (secondary energy minimum) at very large distances.

5.6 Beyond DLVO Theory

charge regulation are taken into account, DLVO theory can successfully describe forces between surfaces in electrolyte solutions for a broad range of systems [232]. For small distances, DLVO theory predicts that the van der Waals attraction will always dominate. Note that the van der Waals force between identical media is always attractive irrespective of the medium in the gap. Thus, thermodynamically, or after long periods of time, we expect all dispersions to precipitate. Once in contact, particles should not separate again, unless they are strongly hit by a third object and gain a lot of energy. A closer look at an interaction at large distances shows the weak attractive energy. This secondary energy minimum can lead to a weak, reversible coagulation without leading to direct molecular contact between the particles. For many systems, this is indeed observed. There are, however, important exceptions, for example, the swelling of clay [233–235]. In the presence of water or even water vapor, clay swells even at high salt concentrations. This cannot be understood based on DLVO theory. To understand phenomena like the swelling of clay, we must consider the molecular nature of the solvent molecules involved.

5.6 Beyond DLVO Theory For large separations, the force between two solid surfaces in a fluid medium can usually be described by continuum theories such as the van der Waals and the electrostatic double-layer theory. The individual nature of the molecules involved and their discrete size, shape, and chemical nature is neglected. At surface separations approaching molecular dimensions, continuum theory breaks down and the discrete molecular nature of the liquid molecules must be taken into account.

5.6.1

Solvation Force and Confined Liquids

Often, the liquid structure close to an interface is different from that in the bulk. For many fluids, the density profile normal to a solid surface oscillates about the bulk density with a periodicity of about one molecular diameter close to the surface. This region typically extends over a few molecular diameters. In this range, the molecules are ordered in layers. Hints about this structure came from simulations and theory [236–239], while direct experimental proof is relatively recent [240–242]. When two such surfaces approach each other, layer after layer is squeezed out of the closing gap (Figure 5.13). Density fluctuations and the specific interactions then cause an exponentially decaying periodic force; the periodic length corresponds to the thickness of each layer. Such forces are termed solvation forces because they are a consequence of the adsorption of solvent molecules to solid surfaces [243]. Periodic solvation forces across confined liquids were first predicted by computer simulations and theory [243–246]. Experimental proof came few years later using the SFA [247, 248]. Solvation forces are not only an important factor in the stability of dispersions; they are also important for analyzing the structure of confined liquids.

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5 Surface Forces

1

2

3

4

5

6

7

Force per unit area

x Repulsive

2 4

6 5

3

1

7 Attractive

1 2 3 4 Distance/molecular diameters Figure 5.13 Schematic figure of the structure of a simple liquid confined between two parallel walls. The molecular order changes depending on distance, which results in a periodic force.

Solvation forces are often well described by an exponentially decaying cosine function of the form: ( ) −x 2πx ⋅ e x0 (5.59) f (x) = f0 ⋅ cos d0 Here, f is the force per unit area, f0 is the force extrapolated to x = 0, d0 is the layer thickness (which in the case of simple liquids is equal to the molecular diameter), and x0 is the characteristic decay length. Example 5.10 The force across confined propanol is shown in Figure 5.14. At molecular separations, the solvation force arising from the ordering of alcohol molecules dominates. The van der Waals force is plotted as a continuous line. Figure 5.14 Normalized force between a microfabricated silicon nitride tip of an atomic force microscope and a planar mica surface in 1-propanol at room temperature [250]. The tip had a radius of curvature of R ≈ 50 nm. The different symbols were recorded during approach (filled circles) and retraction (open circles) of the tip.

50 40 Force/radius (mN/m)

120

30 20 10 0 –10 –20 –30 0

1

2 Distance (nm)

3

4

5.6 Beyond DLVO Theory

Solvation or structural forces may be observed not only in simple liquids, where they arise from the molecular order of the solvent molecules, but also in solutions containing particles, micelles, or polymers [249]. In such cases, the oscillatory force will originate from the solute. For a pure hard sphere interaction, one expects an oscillation length that reflects the particle diameter. For particles in aqueous solution, usually an additional contribution from the electric double layer will lead to a larger effective size that could be approximated by deff = 2(R + λD ), where R is the particle radius and λD is the Debye length.

5.6.2

Non-DLVO Forces in Aqueous Medium

Non-DLVO forces in water deserve a special subchapter because they are important and far from being understood. They are important because water is the universal solvent in nature. Also, in more and more industrial processes, water is used instead of organic solvent since it is environmentally friendly. When two hydrophilic surfaces are brought into contact, repulsive forces in the 1 nm range have been measured in aqueous electrolyte between a variety of surfaces: clays, mica, silica, alumina, lipids, DNA, and surfactants. Because of the correlation with the low (or negative) energy of wetting of these solids with water, the repulsive force has been attributed to the energy required to remove the water of hydration from the surface, or the surface-adsorbed species, presumably because of strong charge–dipole, dipole–dipole, or H-bonding interactions. These forces are termed hydration forces [222, 251]. To date, the origin of hydration forces is not clear, and several effects are discussed [232, 252]. Certainly, the fact that one layer of water molecules is bound to solid surfaces is important. The hydration force, however, extends over more than only two water layers. In a classic paper, Marcelja and Radic proposed an elegant theory to explain the short-range repulsion by a modification of water structure near hydrophilic surfaces [253]. Modern theories take additional effects into account. In fact, short-range monotonically repulsive forces observed between inorganic surfaces are probably due not only to structured water layers propagated away from the surfaces but also to the osmotic effect of hydrated ions that are electrostatically trapped between two approaching surfaces [254]. This hypothesis is supported by the observation that the hydration force is strongly influenced by the ion concentration. In particular, cations, which are widely known to have a shell of water molecules around them, tend to increase the hydration repulsion. It is quite possible that several effects contribute to short-range repulsion. This is especially likely for the interaction between flexible surfaces such as lipid bilayers [255, 256]. Molecular-scale fluctuations of hydrocarbon chains and a steric repulsion between mobile head groups may significantly contribute [257]. In lipid bilayers, the individual lipid molecules are not fixed to a certain position but thermally jump up and down by several Ångströms. If two such bilayers approach each other, this fluctuation is hindered. As a result, the entropy of the lipid molecules decreases, their Gibbs free energy increases, and the two bilayers repel each other. Molecular-scale corrugations can cause a short-range repulsion as well [258].

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5 Surface Forces

Between hydrophobic surfaces, a completely different interaction is observed. Hydrophobic surfaces attract each other [259]. This attraction is called hydrophobic force. The first direct evidence that the interaction between solid hydrophobic surfaces is stronger than the van der Waals attraction was provided by Pashley and Israelachvili [260]. With an SFA they observed an exponentially decaying attractive force between two mica surfaces with an adsorbed monolayer of the cationic surfactant cetyltrimethylammonium bromide (CTAB). Since then the hydrophobic force has been investigated by different groups, and its existence is now generally accepted [259]. The origin of the hydrophobic force is, however, still under debate [261]. Usually two components of an attraction are observed [262]. One is short-range and decays roughly exponentially with a decay length of typically 1–2 nm. It is attributed to a change in the water structure when two surfaces approach each other. The second component is more surprising: it is very long-ranged and extends out to 100 nm in some cases. Its origin is not understood. One hypothesis is that this attraction is due to gas bubbles that form spontaneously [263]. This is called cavitation. Estimations of the rate of cavitation, however, result in much too low values. Another hypothesis is that there are always some gas bubbles residing on hydrophobic surfaces. Once these gas bubbles come into contact, they fuse and cause a strong attraction due to the meniscus force. An open question remains as to how these bubbles can be stable since the reduced vapor pressure inside a bubble and the surface tension should lead to immediate collapse. Non-DLVO forces also occur when the aqueous medium contains surfactants, which form micelles, or polyelectrolytes. A discussion of the complex interaction is, however, beyond the scope of this book. We recommend [264].

5.7

Steric and Depletion Interaction

5.7.1

Properties of Polymers

Many dispersions are stabilized by polymers. The underlying interaction is often called the steric force. l To understand steric interactions, it is necessary to know some of the fundamental principles of polymer physics (a good introduction is the book by Strobl [265]). Here we are mainly concerned with linear R polymers because these are commonly used for steric stabilization. Fortunately, in many applications, we do Figure 5.15 Picture of a not need to consider the detailed molecular chemical linear polymer in the ideal nature of the polymer such as the effects of bond freely jointed chain model. lengths, bond angles, and rotation energy. In many discussions, we can use simpler models to describe the polymer. One such model is the “ideal freely jointed chain segment” (Figure 5.15). In this model, the polymer is considered to consist of a chain of n links. We call each chain link a subunit. Each subunit has a length l. This parameter l can correspond to the length of a monomer, but it can also be shorter or longer, depending on how flexible

5.7 Steric and Depletion Interaction

the monomers are and how stiff the bonds between monomers are. The angle between adjacent chain links is taken to be arbitrary. The chain forms a random coil. To characterize the size and volume of such a coil, we use the mean square of the end-to-end distance R2 . The square root of this value – we call it the size of a polymer chain – is given by √ √ R0 = R2 = l n (5.60) In light-scattering experiments, a corresponding value, the radius of gyration, is determined: √ l n Rg = √ (5.61) 6 Example 5.11 A linear polymer with a total mass of M = 105 g∕mol, a monomer mass of M0 = 100 g∕mol, and l = 0.5 nm has n = M∕M0 = 103 segments, assuming that each √ monomer corresponds to one segment. The size of the polymer is R0 = 0.5 nm ⋅ 103 = 15.8 nm, and its radius of gyration is Rg = 6.5 nm. The polymer assumes this average radius only if the individual links can move freely. This is the case if we neglect the excluded volume effect due to the other segments and if we assume an ideal solvent. In an ideal solvent, the interaction between subunits is equal to the interaction of a subunit with the solvent. In a real solvent, the actual radius of gyration can be larger or smaller. In a good solvent, the interaction between subunits and solvent molecules is more favorable than that between the subunits themselves. This leads to an attraction of solvent molecules into the polymer coil. The polymer swells and Rg increases. In a bad solvent, the monomers have a stronger interaction with each other than to the solvent molecules. As a consequence, the polymer shrinks and Rg decreases. Often, a bad solvent becomes a good solvent if the temperature is increased. The temperature at which the polymer behaves ideally is called the theta temperature, TΘ . An ideal solvent is also called a theta solvent.

5.7.2

Force Between Polymer-coated Surfaces

The ancient Egyptians already knew that one could keep soot particles dispersed in water when they were incubated with either gum arabicum, an exudate from the stems of acacia trees, or egg white. In this way ink was made. The reason for the stabilizing effect is the steric repulsive force cause by adsorbed polymers. In the first case, these are a mixture of polysaccharides and glycoproteins; in the second case, it is mainly the protein albumin. Steric stabilization of dispersions is very important in many industrial applications. In nonpolar liquids electrostatic double-layer forces are often absent because surfaces are not charged. For this reason, one relies on steric stabilization. Direct quantitative measurements were made with an SFA [266–269] and the AFM [270–272]. Overviews on the topic are given in [7, 264, 273]. The force between surfaces coated with polymers is determined mainly by two factors. The first one is the quality of the solvent. In good solvents, the force tends

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5 Surface Forces

Grafted

Adsorbed

Brush Low density

High density

Figure 5.16 Structure of polymers on surfaces.

to be repulsive, and in bad solvents, attractive. Moreover, in good solvents, polymers tend to remain in solution rather than adsorbing to surfaces. The second important factor is how and how much polymer is bound to the surface (Figure 5.16). If the polymer is only weakly adsorbed, individual molecules can still diffuse laterally. Physisorbed polymers only reach a maximal density on the order of 1∕R2g and extend typically for a distance of 2Rg into the solution. Each polymer molecule might have several binding sites and might form loops. The adsorption time plays a crucial role since polymers at surfaces rearrange, a process that can take hours. When a polymer is chemically bound to a surface, the number of molecules on the surface – called grafting density Γ – can be much higher. When the molecules are closely packed (Γ ≫1∕R2g ), we talk about a polymer brush. In this case, the steric force acts over lengths that are substantially larger than the radius of gyration; the thickness is roughly L0 = nl5∕3 Γ1∕3 [274], where Γ is the number of molecules per unit area. Polymer brushes can be made by binding a polymer to a surface (grafting to) or by synthesis of the polymer directly on the surface (grafting from). How such polymer layers are formed is described in Section 9.3.3. There is no simple, comprehensive theory of steric forces; they are complex and difficult to describe. Different components contribute to the force and, depending on the situation, dominate the total force. The most important interaction is repulsive and of entropic origin. It is caused by the reduced configuration entropy of polymer chains [275] when their available volume is reduced. If the thermal movement of a polymer chain at a surface is limited by the approach of another surface, then the entropy of the individual polymer chain decreases. In addition, the concentration of monomers in the gap increases. This leads to an increased osmotic pressure and additional repulsion. This repulsion has been calculated by different authors [276–278]. For a low grafting density (Γ < 1∕R2g ), the repulsive force per unit area in a good solvent between two polymer-coated surfaces is [276] ( 2 2 ) √ kB TΓ 2π Rg Π(x) = −1 for x ≤ 3 2Rg 2 x x

Π(x) =

⎛ x ⎞ ⎜ ⎟ kB TΓx −⎜⎝ 2Rg ⎟⎠ e R2g

2

for

√ x > 3 2Rg

(5.62)

5.7 Steric and Depletion Interaction

For the other extreme, at high grafting density, the force per unit area can be approximated by [277] [( ) ( )3∕4 ] 2L0 9∕4 x 3∕2 Π(x) = kB TΓ − (5.63) x 2L0 for x < 2L0 . The strong increase of the interaction strength with increasing grafting density is demonstrated in Figure 5.17. To derive Eq. (5.62), a step profile for the segment density of the undisturbed brush was assumed. Milner, Witten, and Cates used a more realistic parabolic profile [278]. Another contribution to the steric interaction is the intersegment force. The intersegment force is caused by the direct interaction between segments of polymers with each other. This interaction depends strongly on the solvent. Below TΘ , the interaction among the monomers is stronger than the interaction of the monomers with the solvent, which results in an attractive force. Bridging forces, arising when a polymer binds to both surfaces, usually lead to an attraction at large separations. Bridging is only effective at low surface coverage.

Figure 5.17 Disjoining pressure between two similar parallel plates coated with grafted polymer in a good solvent. For the calculation, we assumed a monomer length of l = 0.4 nm and a chain length of n = 100 monomers. At low grafting density (Γ = 4 × 1016 m−2 ), the characteristic decay length of the force is determined by Rg = 1.6 nm. It was calculated using Eq. (5.62). For an intermediate grafting density (Γ = 2 × 1017 m−2 ), we used [277] Π = (kB T Γ∕RF )(RF ∕x)8∕3 , with RF = l n3∕5 = 6.3 nm. For the high grafting density (Γ = 1018 m−2 ), the force per unit area was calculated with Eq. (5.63) using L0 = nl5∕3 Γ1∕3 = 22 nm.

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5 Surface Forces

Figure 5.18 Steric force between an atomic force microscope tip made of silicon nitride and oxidized silicon onto which polystyrene was grafted [280]. The force was measured in toluene.

1.5

Force (nN)

126

1.0 52.5 °C 35 °C

0.5

0.0 Bare silicon oxide 0

100 200 Distance (nm)

300

400

Only then do the polymer segments have a chance to find an adsorption site on the opposite surface. In a dispersion, attractive forces lead to the aggregation of the dispersed particles. If this aggregation is caused by the addition of polymer, it is called flocculation [279]. Bridging is one cause on flocculation. Depletion, which is discussed in Section 5.7.3, is another. Example 5.12 The steric force caused by grafted polystyrene in toluene is repulsive (Figure 5.18). Toluene is a good solvent for polystyrene. With increasing temperature, the force becomes stronger and its range increases. This is a hint that it is caused by entropic effects.

5.7.3

Depletion Forces

In 1954, Asakura and Oosawa [281] realized that dissolved polymers could influence the interaction between particles in a dispersion, even if they do not interact at all with the particle surfaces. Asakura and Oosawa themselves described the interaction as follows: “Let us consider two parallel and large plates of the area A immersed in a solution of rigid spherical macromolecules. If the distance between the plates x is smaller than the diameter of solute molecules, none of these molecules can enter between the plates. Then this region becomes a phase of the pure solvent, while the solution outside the plates is little affected by them. Therefore, a force equivalent to the osmotic pressure of the solution of macromolecules acts inwards on each plane.” Though macromolecules are usually not rigid, the same effect occurs for soft molecules. In this case, the mean end-to-end distance of a linear, randomly coiled polymer R0 corresponds to the radius of the macromolecules (Figure 5.19). Depletion forces are effective over ranges of the size of the dissolved molecules. The force per unit area is in the order of the osmotic pressure of dissolved macromolecules. The Gibbs free energy for the depletion attraction between two spherical particles of radius Rp at a distance D in a solution containing

5.8 Spherical Particles in Contact

Figure 5.19 Schematic of two particles dispersed in a polymer solution. The reduced osmotic pressure in the zone depleted of polymer in the gap between the particles leads to an effective attractive force, the depletion force.

Depletion zone

Rp

R0

macromolecules of radius R0 at a number density c is roughly (see Exercise 5.7 and [282]) π (5.64) W(D) ≈ ckB TRp (2R0 − D)2 2 for D ≤ 2R0 and W = 0 for D > 2R0 . Practically, the addition of a nonadsorbing polymer to a dispersion can induce flocculation of dispersed particles due to the depletion attraction [283–285]. This leads to a different view of the depletion attraction, although the underlying physical principle is the same: particles aggregate because in this way, the total free volume available to the dissolved macromolecules increases. It increases so much that the gain in translational entropy of the macromolecules is higher than the loss of entropy of the particles. It has also been realized that the depletion force is not always attractive. It is attractive at short ranges, but it can be repulsive at larger distances [286, 287]. For a detailed discussion of the depletion force between colloids see [288]. How depletion forces were discovered is described in [289].

5.8 Spherical Particles in Contact So far we have assumed that interacting surfaces are not deformable. In reality, all solids have a finite elasticity. They deform upon contact. This has important consequences for the aggregation behavior and the adhesion of particles because the contact area is larger than one would expect from infinitely hard particles. Heinrich Hertz8 established the basis for the treatment of elastic solids in contact [290]. He considered two spheres with radii R1 and R2 in contact. The two spheres are from materials with Young’s moduli E1 and E2 and with Poisson ratios 𝜈1 and 𝜈2 . The two spheres are pressed together with a force F (Figure 5.20). This force is also called the load. The load can be the gravitational force or an applied force. Hertz supposed that no surface forces act between the solids. He showed that the pressure between the spheres decreased as a quadratic function with the distance 8 Heinrich Hertz, 1857–1894; German physicist, professor in Karlsruhe and Bonn.

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5 Surface Forces

F

Figure 5.20 Schematic diagram of two spherical particles in contact.

R1 δ 2a R2

to the contact center. At the outer rim of the contact, at the radial distance a, the pressure is zero. Integrating the pressure over the entire contact area, we get the load. In this way, he obtained a relationship between the contact radius a and the load F: 3R∗ F (5.65) a3 = 4E∗ ∗ ∗ R is the so-called reduced radius and E the reduced Young’s modulus as defined by 1 − 𝜈12 1 − 𝜈22 R1 R2 1 and = + (5.66) R1 + R2 E∗ E1 E2 The contact radius at the same time determines the indentation: a2 𝛿= ∗ (5.67) R √ As a force-versus-distance relationship, we thus get (with a = 𝛿R∗ ): 4 √ F = E∗ R∗ 𝛿 3 (5.68) 3 Since the Hertz model does not consider any attractive surface forces, the contact radius a and the indentation 𝛿 are both zero in the absence of an external load (F = 0), and no adhesion between the spheres is observed. In reality, surface forces such as the van der Waals force will attract two solids. This leads to an adhesion between the bodies and finite values of a and 𝛿, even in the absence of a load. Such attractive forces were taken into account by Johnson, Kendall, and Roberts [291]; their model is called the JKR model. They considered the following situation: if two solid surfaces come into contact, the formation of a contact area leads to a corresponding reduction of free surface. This process is associated with an energy gain per unit area, which we call the work of adhesion W. Since the work of adhesion is equivalent to a reduction in the energy of the system, the formation of the contact area should happen spontaneously. At the same time, work must be done for the elastic deformation of the solids. The resulting contact area is given by the balance between the work of adhesion and the elastic deformation energy. Johnson, Kendall, and Roberts calculated this elastic deformation energy using the Hertz theory. The equilibrium contact radius is given by [ ] √ 3 R∗ ∗ ∗ F + (3πWR∗ )2 F + 3πWR + 6πWR (5.69) a3 = 4 E∗ R∗ =

5.8 Spherical Particles in Contact

Figure 5.21 Schematic of neck formation as described by JKR and DMT theories.

hn

It is larger than in the Hertz model (for W = 0, we retrieve the Hertz result). Another consequence of attractive surface forces is that a kind of neck or meniscus forms at the contact line. As one example, the case of a hard sphere on a soft planar surface is shown in Figure 5.21. In the JKR model, a force is required to separate two solids, and separation occurs abruptly at a critical force and at a finite contact radius. This force is called the adhesion force. In the JKR model, the adhesion force is given by 3π WR∗ (5.70) 2 Usually, the work of adhesion is expressed in terms of the surface energy of the solid 𝛾S by Fadh =

W = 2𝛾S

(5.71)

At this point, however, we need to be careful because for solids, we must distinguish between surface tension and surface energy. The work required to form a new surface depends on how this surface is formed: plastically (as for a liquid) or elastically. In an experiment, usually both effects contribute. Therefore, we should consider the surface energy obtained from adhesion experiments as an “effective” surface energy. Inserting Eq. (5.71) into Eq. (5.70) leads to Fadh = 3π𝛾S R∗

(5.72)

The adhesion force increases linearly with the particle radius. Surprisingly, it is independent of the elasticity of the materials. This is because of two opposing effects. In a hard material, the deformation of the solid is small. As a result, the contact area and the total attractive surface energy are small. On the other hand, the repulsive elastic component is small. Both effects compensate each other. Soft materials are strongly deformed. Thus, both the attractive surface energy term and the repulsive elastic term are high. The JKR theory used the simplifying assumption that surface forces are acting only within the contact area. In reality, surface forces are active also outside of direct contact. This is, for instance, the case for van der Waals forces. Derjaguin, Muller, and Toporov took this effect into account and developed the so-called DMT theory [292]. Unfortunately, important results of the DMT theory cannot be expressed as convenient analytic expressions. There is, however, one simple result. For the adhesion force, they obtained Fadh = 4π𝛾S R∗

(5.73)

An exact analysis shows that the JKR and DMT models represent two extremes of a more general model [293–295]. For large, soft solids, the JKR model describes the

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5 Surface Forces

situation more realistically. For small, hard solids, it is appropriate to use the DMT model. A criterion by which one can determine which model should be used stems from the height of the neck (Figure 5.21): ( )1∕3 𝛾S2 R∗ hn ≈ (5.74) E∗2 If the neck height is larger than some atomic distances, then the JKR model is more favorable. With shorter neck heights, the DMT model is more suitable. Example 5.13 A silicon oxide sphere with 20 μm diameter sits on a silicon oxide surface. Estimate the contact radius at negligible external force and calculate the adhesive force. E = 5.4 × 1010 Pa, 𝜈 = 0.17, 𝛾S = 50 mN/m, density 𝜌 = 3000 kg∕m3 . With R1 = 10 μm and R2 = ∞, the effective particle radius is R∗ = R1 = 10 𝜇m. 1 1 − 0.172 = 2 ⋅ ⇒ E∗ = 2.8 × 1010 Pa E∗ 5.4 × 1010 Pa The contact radius is estimated using the JKR model. Without external forces, we have ( ) 3 10−5 m a3 = ⋅ 6π ⋅ 2 ⋅ 0.05 N∕m ⋅ 10−5 m 10 4 2.8 × 10 Pa (5.75) = 5.05 × 10−21 m3 ⇒ a = 1.71 × 10−7 m The neck height is [ ]1∕3 0.052 N2 ∕m2 ⋅ 10−5 m hn ≈ = 3.2 × 10−10 m (2.8 × 1010 Pa)2 The neck height is about as large as an atomic diameter. Therefore, the DMT model is suitable, and the adhesion is Fadh = 4π ⋅ 0.05 N∕m ⋅ 10−5 m = 6.3 𝜇N Experimentally, the predictions of the theories for the behavior of spheres in contact can be measured in different ways. Adhesion forces have, for instance, been measured by a centrifuge for more than 40 years (Figure 5.22). A significant part of the knowledge about the behavior of powders stems from such experiments [296]. The centrifugal force that is required to detach particles from a planar surface is measured [297–300]. Usually, the detachment force of many particles is measured in a single experiment, allowing statistical evaluation of the data. This is especially useful

Figure 5.22 Schematic of centrifuge method to determine adhesion forces of particles on surfaces. Friction forces can also be analyzed when particles are placed on a horizontal surface.

5.9 Summary

in the case of irregularly shaped particles where the contact area and adhesion force will depend on the random orientation of the particles relative to the surface. Thus, the centrifuge technique has been used to characterize the behavior of powders for the pharmaceutical or food industry. When tilting the surfaces to which the particles are attached, the centrifuge technique can also be used to study friction forces. There are, however, also disadvantages to this technique. One limit is that the rotational speed of the available ultracentrifuges is limited due to the material stability of the rotor. This restricts adhesion measurements using the centrifuge method to particles larger than a few microns. Otherwise, the centrifugal force is not strong enough to detach the adhering particles from the surface. In addition, the contact time and load are difficult to vary. Some of the disadvantages of the technique were overcome by the use of the colloidal probe technique to measure adhesion forces (for a review see [213]). The colloidal probe technique offers the advantage that the same particle can be used for a series of experiments and its surface can be examined subsequently. The accessible range of particle size is typically limited to a range between 1 and 50 μm. The tedious sample preparation limits the number of different particles used within one study, for practical reasons. Therefore, the colloidal probe and centrifugal methods complement each other. The JKR theory predicts correct contact radii for relative soft surfaces with effective radii larger than 100 μm. This was shown in direct force measurements by the SFA [301, 302] or specifically designed systems. For smaller spheres, it was verified using the colloidal probe technique [303].

5.9 Summary ●

●

The van der Waals force includes three kinds of dipole–dipole interactions: Keesom, Debye, and London dispersion components. Usually the dispersion interaction dominates. Between molecules, the van der Waals energy decreases with 1∕D6 . For macroscopic bodies, the decay is less steep and depends on the specific shape of the interacting bodies. For example, for two infinitely extended bodies separated by a gap of thickness x the van der Waals energy per unit area is AH 12πx2 The Hamaker constant is determined by the dielectric permittivities and the optical properties of the interacting media. Derjaguin’s approximation allows us to calculate the force (or energy) between bodies of arbitrary shape from the force per unit area (or energy per unit area) between two planar surfaces, provided the radius of curvature of the bodies is large compared to the typical decay lengths of the forces involved. Experimentally, surface forces between solid surfaces can be determined using an AFM or SFA. w=−

●

●

●

131

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5 Surface Forces ●

●

●

●

In an aqueous medium, the electrostatic double-layer force is present. For distances x larger than the Debye length λD , it decays roughly exponentially: F ∝ exp(−x∕λD ). The stability of dispersions in aqueous media can often be described by the DLVO theory, which contains the double-layer repulsion and the van der Waals attraction. In some applications, other effects are important that are not considered in DLVO theory. At short ranges and for hydrophilic particles, the hydration repulsion prevents aggregation. Hydrophobic particles, in contrast, tend to aggregate due to the hydrophobic force. At separations of a few molecular diameters, the solvation force due to the specific structure of the confined liquid can be substantial. Polymers at surfaces can be used to stabilize dispersions by steric interaction. The interaction between fine particles is often dominated by their mechanical properties such as Young’s modulus. This was first considered by Hertz theory. Adhesion between spherical particles increases with the radius of the particles and is described by JKR and DMT theories.

5.10 Exercises 5.1

What is the van der Waals force per unit area for two planar, parallel layers of molecules (one molecule thick) having surface densities (molecules per unit area) 𝜌𝜎A and 𝜌𝜎B ?

5.2

Verify approximation Eq. (5.26).

5.3

In some applications, clays are combined with polymers, which leads to composite materials with improved thermoelastic properties. The polymer not only surrounds the clay particles but also intercalates into the layers of the clay. In that way, the polymer changes its properties because it is confined between neighboring clay layers. In one particular case, we have polystyrene confined between two planar parallel layers of mica separated by 1 nm, and we are interested in the effect of the van der Waals force. To estimate this effect, calculate the van der Waals energy per unit area w. Then calculate the area for which this van der Waals energy is equal to the thermal energy kB T. The temperature we are interested in is 110 ∘ C, roughly the glass transition temperature of polystyrene. Do the same for silicon oxide. For simplicity, take 𝜀, n, and 𝜈e to be the same as at 20 ∘ C.

5.4

In coating technology, surfaces are coated to protect them from corrosion. This is usually done by a layer containing, for example polymer and pigment. As a model system, thin polystyrene films on silicon oxide have been studied [304, 305]. Does polystyrene form a stable layer, or would you expect it to dewet the surfaces assuming that van der Waals forces dominate?

5.10 Exercises

5.5

In atomic force microscopy, the tip shape is often approximated by a parabolic shape with a certain radius of curvature R at the end. Calculate the van der Waals force for a parabolic tip versus distance. We only consider nonretarded contributions. Assume that the Hamaker constant AH is known.

5.6

Calculate the force between a conical and a parabolic object and a planar surface for an exponential force law f = f0 e−κx .

5.7

Derive Eq. (5.64). To do so, consider that a shell of thickness R0 around each particle is not available to the dissolved molecules because they cannot get closer than R0 . This inaccessible volume is reduced when two particles approach each other. Calculate this reduced inaccessible volume and multiply it with the osmotic pressure. Then assume that Rp ≫ R0 .

5.8

Plot the contact radius versus load (up to a load of 5 μN) for two spherical powder particles of silicon oxide and a radius of R = 2 μm using the Hertz and the JKR model (𝛾S = 0.05 N/m).

5.9

In an aqueous electrolyte, we have spherical silicon oxide particles. The dispersion is assumed to be monodisperse with a particle radius of 1 μm. Please estimate the concentration of monovalent salt at which aggregation sets in. Use the DLVO theory and assume that aggregation starts when the energy barrier decreases below 10 kB T. The surface potential is assumed to be independent of the salt concentration at −20 mV. Use a Hamaker constant of 0.4 × 10−20 J.

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6 Contact Angle Phenomena and Wetting Wetting in general includes all phenomena involving contacts between three phases of which one is liquid and one is fluid (liquid or gaseous). A typical situation is a liquid wetting a solid surface in a gaseous environment. Instead of the gaseous environment, we may have another immiscible liquid. Wetting phenomena can be observed every day around us. Examples are raindrops on a window, dispersing a powder such as cocoa in milk, or the penetration of a liquid into porous soil. Applications include printing, fogging of glasses, condensation and evaporation, wetting of filters, flotation, oil recovery, soldering and lubrication. In many applications, we would like to achieve complete wetting. Examples are the application of coatings and paints, the distribution of herbicides on the surface of leaves, or the insecticides on the epidermis of insects. In other cases, we want to avoid wetting. For example, water should not wet rainproof clothes. Road pavements should not be wetted easily by water to prevent it from penetrating into small cracks and fissures and then – particularly in winter, upon freezing – destroy the pavement. Introductions to the subject include [306, 307]. A historic perspective is given in [308].

6.1 Young’s Equation 6.1.1

Equilibrium Contact Angle

Young’s equation is the basis for a quantitative description of wetting phenomena. If a drop of a liquid is placed on a solid surface, there are two possibilities: the liquid spreads on the surface completely (contact angle Θ = 0∘ ) or a finite contact angle is established. In the first case, we talk about perfect, complete, or total wetting. In the second case, a three-phase contact line – briefly called contact line – is formed. At this line, three phases are in contact: the solid, the liquid, and the vapor (Figure 6.1). It is convenient to distinguish two regimes: ●

For contact angles less than 90∘ , wetting is favorable. Adhesive forces between the liquid and the solid cause the liquid to spread. We talk about partial wetting and high wettability, and a liquid is said to be mostly wetting.

Physics and Chemistry of Interfaces, Fourth Edition. Hans-Jürgen Butt, Karlheinz Graf, and Michael Kappl. © 2023 WILEY-VCH GmbH. Published 2023 by WILEY-VCH

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6 Contact Angle Phenomena and Wetting

Gas

h

Figure 6.1 Liquid drop with circular contact area on a planar solid surface.

h′

Liquid

a a′ Solid

●

R

R′

Contact angles above 90∘ indicate that wetting of the surface is unfavorable. Cohesive forces within the liquid cause the drop to ball up and minimize its contact with the solid surface. This regime of low wettability and high contact angles is described as nonwetting.

With respect to water sometimes the terms hydrophilic, water loving, and hydrophobic, water hating, are used. Here, we use the term hydrophobic for surfaces that form a contact angle above 90∘ with water. Hydrophilic is applied to describe a surface that forms a contact well below 90∘ . Until now, we have used one symbol for the surface tension 𝛾. It usually referred to the surface tension of a liquid, and it is equal to the surface Gibbs energy (Eq. (3.36)). In other cases, it denoted an interfacial tension. In any case, it was clear which interfacial energy was meant. In this chapter, we must distinguish between the surface energy of the liquid–vapor, 𝛾L , the interfacial energy of the solid–vapor, 𝛾S , and the interfacial energy of the solid–liquid interface, 𝛾SL . Young’s equation relates the contact angle to the interfacial energies [15, 70, 71, 309]: 𝛾L cos Θ = 𝛾S − 𝛾SL

(6.1)

If the interfacial energy of the bare solid surface is higher than that of the solid–liquid interface (𝛾S > 𝛾SL ), the right-hand side of Young’s equation is positive. Then cos Θ must be positive and the contact angle is lower than 90∘ . If the solid–liquid interface is energetically less favorable than the bare solid surface (𝛾S < 𝛾SL ), the contact angle will exceed 90∘ because cos Θ must be negative.

6.1.2

Derivation

We derive Young’s equation for a typical example: a small sessile drop on a planar solid surface. Other geometries can be treated in an analogous way. The solid surface is assumed to be smooth, homogeneous, rigid, insoluble, and nonreactive. The liquid is not volatile or the surrounding atmosphere is assumed to be saturated with its vapor. Then we can neglect evaporation and the volume of the liquid is constant. To derive Young’s equation, we consider the change in the Gibbs free energy dG when the drop spreads an infinitesimal amount. While the drop spreads, the radius of the circular contact zone increases from a to a′ = a + da (Figure 6.1). Its height decreases from h to h′ = h + dh (dh is negative). If the change in the Gibbs free

6.1 Young’s Equation

energy is negative, the process will occur spontaneously. For positive dG, the drop will contract. In equilibrium, which is the energetically most favorable situation, we have dG = 0. For simplicity, we assume that the drop is small enough so that gravitation is negligible. As a consequence, its shape is that of a spherical cap. The result is also valid for large drops, but for negligible gravitation, the mathematical treatment is simpler. When the drop spreads, some free solid surface is changed to a solid–liquid interface. The change in area dASL = 2ada leads to a change in surface energy (𝛾SL − 𝛾S )dASL . In addition, the surface area of the liquid–gas interface changes. Elementary geometry tells us that the surface of a spherical cap is AL = 𝜋(a2 + h2 )

(6.2)

A small change in the contact radius leads to a change in the liquid surface area of 𝜕AL 𝜕AL da + dh = 2𝜋ada + 2𝜋hdh (6.3) 𝜕a 𝜕h Unfortunately, the change in surface area depends on two variables: a and h (we need not consider a change in Θ; it is of second order). However, these two variables are not independent because the volume of the drop is constant. The volume of a spherical cap is 𝜋 V = (3a2 h + h3 ) (6.4) 6 In general, a small change in this volume is dAL =

𝜕V 𝜋 𝜕V da + dh = 𝜋ahda + (a2 + h2 )dh 𝜕a 2 𝜕h Since the volume is assumed to be constant (dV = 0), we have dV =

−𝜋ahda =

2ah 𝜋 2 dh (a + h2 )dh ⇒ =− 2 2 da a + h2

(6.5)

(6.6)

Using Pythagoras’ law, R2 = a2 + (R − h)2 ⇒ a2 = 2Rh − h2 , we get 2ah dh a =− =− da R 2Rh − h2 + h2 and

(6.7)

( ) h R−h a dAL = 2𝜋ada − 2𝜋h da = 2𝜋a 1 − da = 2𝜋a da = 2𝜋a cos Θda R R R (6.8)

Now, we can write the total change in the Gibbs free energy as follows: dG = (𝛾SL − 𝛾S )dASL + 𝛾L dAL = 2𝜋a(𝛾SL − 𝛾S )da + 2𝜋a𝛾L cos Θda

(6.9)

Setting dG = 0 immediately leads to Young’s equation. In a more general derivation, the Helmholtz or Gibbs energy of the system is minimized, taking gravitation and a possible adsorption of dissolved substances into account [70, 71, 310–312].

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6 Contact Angle Phenomena and Wetting

γL cos Liquid

γL γSL

Figure 6.2 Forces acting on a segment of the three-phase contact line. The three-phase contact line is supposed to be oriented normal to this page.

Vapor γS Solid

Young’s equation is also valid if we replace the gas phase by a second, immiscible liquid. The derivation would be the same; we only need to replace 𝛾L and 𝛾SL by the corresponding interfacial energies. For example, we could determine the contact angle of a water drop on a solid surface under oil. Instead of having a gas saturated with the vapor, we must have a second liquid saturated with dissolved molecules of the first liquid. Young’s equation can alternatively be derived by a force balance. Therefore, we consider a segment of the contact line of length dl (Figure 6.2). In equilibrium, the contact line does not move, which implies that all forces in horizontal direction need to balance. In the horizontal direction, the surface energy of the solid draws the segment with a force 𝛾S dl to the right. The interfacial energy of the solid–liquid interface draws the segment in the opposite direction with a force 𝛾SL dl. The horizontal component of the liquid surface tension is 𝛾L dl cos Θ. In equilibrium, the horizontal components must cancel each other out, leading to 𝛾S dl − 𝛾SL dl − 𝛾L dl cos Θ = 0

(6.10)

Division by dl and rearranging leads to Eq. (6.1). Here, we derived Young’s equation by balancing horizontal forces. What about the vertical force component of the liquid’s surface tension? This vertical force component is balanced by a small deformation of the solid surface and the resulting vertical internal stress. Usually, the deformation of the solid surface is atomistically small. Only for very soft materials can it reach length scales that are observable by optical microscopy [313]. Example 6.1 An ionic liquid is a salt in the liquid state. In contrast to a molten salt, the term is restricted to salts whose melting temperature is below 100 ∘ C. Ionic liquids are practically nonvolatile but still have a low viscosity. The ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate is fluid at room temperature. When a drop of this ionic liquid is placed on the surface of a soft, partially cross-linked polydimethylsiloxane (PDMS), the polymer is elastically deformed. The polymer is pulled upward at the three-phase contact line due to the vertical component of the interfacial tension 𝛾L sin Θ (Figure 6.3a). This deformation can be imaged with a confocal microscope in reflection mode (thick black lines to the left and right of the drop in Figure 6.3b). Due to the curvature inside the drop, the capillary pressure ΔP is higher than in the surrounding atmosphere. This pressure compresses the PDMS underneath the drop. To make this effect visible, a fluorescent marker was added to the ionic liquid, so the liquid can be seen as a gray

γL

2 µm

6.1 Young’s Equation

Liquid P

Soft solid (a)

Surface of PDMS

Air Ionic liquid

50 µm

(b)

Figure 6.3 (a) Schematic of sessile liquid drop on elastic surface. The surface tension of the liquid pulls the surface up at the periphery, while the capillary pressure in the liquid compresses the soft material. (b) Experimental results obtained with a drop of 1-butyl-3-methylimidazolium hexafluorophosphate on PDMS. The plot is a cross section through the drop recorded with a laser scanning confocal microscope [314]. Please note that the horizontal and vertical scales are different. The dotted lines are results of calculations with elastic theory. The PDMS surface was not perfectly horizontal so that the left edge appears slightly lower than the calculated result.

area in Figure 6.3b. In the equilibrium configuration, the vertical interfacial forces are just balanced by the elastic stresses in the polymer. Young’s equation has been verified by Monte Carlo [315] and molecular dynamics simulations [316–318]. Experimental verification is, however, limited. The problem lies in the solid–gas and solid–liquid interfacial energies. They cannot be measured independently. Only changes in 𝛾S and 𝛾SL can be measured [70, 319, 320]. The underlying difficulty is that solid surfaces can be formed plastically or elastically (Chapter 8). Depending on the method of formation, the interfacial energy can be different. One of the assumptions for equilibrium is that the contact line can freely move on the surface. In reality, this is usually not the case. Defects in the surface or roughness tend to pin the contact line. As a result, in measurements a whole range of contact angles is measured rather than one single value, depending on how the measurement is carried out. This range is limited by the receding (lower limit) and the advancing contact (upper limit) angles. They give a range for the equilibrium contact angle as given by Young’s equation. Another limit of Young’s equation is the fact that liquids with a curved surface evaporate or condense. Let us consider, for example, a sessile drop of a volatile liquid. The drop’s surface is curved. This curvature leads to an increased vapor pressure according to the Kelvin equation. Eventually, the drop will evaporate even if it is exposed to saturated vapor (saturated with respect to a planar liquid surface) [321, 322]. Thus, a drop of a volatile liquid will hardly be in real equilibrium. Example 6.2 A drop of water is placed on a surface and forms a certain contact angle. The humidity in the laboratory is most likely not 100%. Thus, in a strict sense, the system is not in equilibrium. Can we still apply Young’s equation? With good approximation, yes. The reason is that in the vicinity of the drop, the local vapor pressure is close to saturation. For water at 25 ∘ C, the saturation vapor

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6 Contact Angle Phenomena and Wetting

pressure is 3169 Pa. Directly at the drop’s surface. The diffusion coefficient of water molecules in the gas phase in air at normal pressure is D = 2.4 × 10−5 m2 ∕s. To reach local equilibrium within a range Δr of say 10 μm takes a time 𝜏 ≈ Δr 2 ∕6D = 0.69 μs. Thus, 10 μm away from the drop we are close to 100% humidity, unless the drop is moving very fast.

6.1.3

Complete Wetting, Surface Forces, and the Core Region

As mentioned at the beginning of this chapter, when a liquid contacts a solid surface, it either forms a defined contact angle or completely wets the surface. To describe the tendency of a liquid to spread spontaneously, the spreading coefficient is defined [323, 324]: S = 𝛾S − 𝛾SL − 𝛾L

(6.11)

For S < 0, a finite contact angle will form. For S = 0, a liquid spreads. What about the case that S > 0? Practically, one can bring any high-energy solid surface in contact with a liquid. The liquid spreads and forms a film. For a volatile liquid, molecules would also condense from the vapor onto the solid until a film is formed. As a result, the free solid surface is replaced by a solid–liquid interface plus a liquid surface. Thus, in equilibrium, the free energy per unit area will be 𝛾S = 𝛾SL + 𝛾L and S will not exceed zero. Thus, the case S = 0, which may appear very particular, is no exception at all. One of the advantages of Young’s equation is that its derivation is independent of the atomistic structure of the liquid and solid surfaces at the contact line. For this reason, we only need to consider macroscopic quantities and need not to take care about the details in the core region, i.e. the microscopic region directly around the contact line. As long as the structure in the core region is preserved upon a translation of the contact line, the macroscopically observed contact angle outside the core region Θ is determined by the far-field interfacial energies 𝛾S , 𝛾L , and 𝛾SL by Young’s equation [113, 306]. For a more detailed understanding of wetting and spreading, we now turn toward the mesoscopic regime. This leads to an extended definition of the spreading coefficient. Let us consider a thin liquid film of thickness h on a flat, rigid, inert, homogeneous solid substrate (Figure 6.4a). As described in Chapter 5, in general, surface forces will be acting between the solid–liquid and the liquid–gas interfaces. Per unit area these surface forces are expressed by Π(h) (Eq. 5.38), where Π is called disjoining pressure. To describe this change in energy per unit area, the interface potential g(h) is introduced; g is also called effective interface potential, binding potential or wetting potential [325]: g(h) = 𝛾S (h) − 𝛾SL − 𝛾L

(6.12)

Here, 𝛾S (h) is the free energy per unit area of the solid surface covered with a film of thickness h. g(h) is related to the disjoining pressure by Π = −𝜕g∕𝜕h. For finite film thickness g(h) describes the contribution of the interaction between the two interfaces to the free energy. This interpretation is more obvious when writing

6.1 Young’s Equation

P0=

Vapor

(h0)

Liquid (a)

h

Core region

P0=2γS/R

h0

Solid

R

(b)

γS(h) γSL+γL

h0 h h0

(c)

(d)

(e)

Figure 6.4 Wetting on the mesoscopic length scale: (a) Schematic of a thin liquid film of thickness h on a flat solid surface. (b) A drop in coexistence with a thin liquid film on a solid surface. Please note, the film and drop are not to scale. The film, if existing, is of the order of 1 nm thick, while the drop is macroscopic. (c–e) Three schematic examples of changes in free solid surface energy with film thickness and the corresponding core regions of a drop.

𝛾S (h) = 𝛾SL + 𝛾L + g(h). For large film thickness, when their distance is much larger than the range over which surfaces forces act, the interface potential is zero, g(h → ∞) = 0. Usually, a continuum description by the interfacial potential is applied to films which are several molecules thick. With the effective film thickness h = ΓM∕(𝜌L − 𝜌g ), we can extend the definition to molecular films [325]; here, M is the molar mass of the liquid. The limit h → 0 corresponds to the solid surface without any liquid present. In many situations, the interface potential has a minimum at a defined film thickness h0 . In equilibrium and in the presence of a liquid, the latter will form a film of thickness h0 . Here, we need to be precise with the notation. When we write 𝛾S without brackets, we refer to the equilibrium surface energy per unit area, thus 𝛾S = 𝛾S (h0 ). The spreading coefficient and the interfacial potential are related by g(h0 ) = S. Now, we can relate the equilibrium contact angle to the interfacial potential at h = h0 . With Young’s equation, we obtain cos Θ = 1 +

g(h0 ) 𝛾L

(6.13)

Thus, a drop with a certain contact angle usually coexists with a thin liquid film covering the whole solid surface (Figure 6.4b). Such a coexistence raises the question: How does the region between the drop and the film look like? The shape of the transition region is determined by a modified Laplace equation [326–328]: ( ) 1 1 ΔP0 = 𝛾L + + 𝜌L gh + Π(h) (6.14) R1 R2

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6 Contact Angle Phenomena and Wetting

To derive Eq. (6.14), it is assumed that in equilibrium no pressure difference should exist in the liquid phase. Any pressure difference would lead to a flow, and a reshaping of the liquid until the pressure is balanced. Thus, the pressure difference between the liquid and the surrounding fluid phase, ΔP0 , is constant. It is balanced by pressure caused by the curvature of the liquid and the disjoining pressure. Here, we assumed that the disjoining pressure only depends on the local thickness of the film, irrespective of its slope. In most practical cases, we can further assume that the curvature of the contact line is low compared to the curvature of the liquid surface in a direction normal to its surface. For the shape of the liquid close to the contact line, we can also neglect gravity and simplify Eq. (6.14): ΔP0 = 𝛾L [

𝜕 2 h∕dx2 1 + (𝜕h∕𝜕x)2

]3∕2 + Π(h)

(6.15)

Let us discuss three typical examples [328]. In Figure 6.4c, the free solid surface energy 𝛾S (h) and thus the interface potential shows a minimum. A mesoscopic film coexists with a drop of defined macroscopic contact angle. Figure 6.4d shows an example, in which the liquid is attracted by van der Waals forces. The disjoining pressure for large distances is given by Eq. (5.17): Π = −AH ∕6𝜋h3 . For a solid–liquid interface interacting with a liquid–air interface, the Hamaker constant AH is negative (Table 5.3). As a result, the disjoining pressure is positive; it tends to increase the thickness of the liquid layer. In Figure 6.4d, we further assumed that at the intermediate range, we have an attractive disjoining pressure, as would, for example be the case for water on a hydrophobic surface [329]. For a very short range, we assumed that a molecular layer is attracted by the solid, leading to an increase of 𝛾S (h). The drop coexists with an adsorbed layer of molecules. Figure 6.4d would be typical for glass or silicon dioxide coated with a hydrophobic silane wetted by water in humid atmosphere. In case the attraction is so weak, that at a given temperature molecules do not adsorb, the drop coexists with a completely bare solid surface (Figure 6.4e). In summary: In the vicinity of the contact line, in the so-called core region, surface forces such as van der Waals or electric double-layer forces may deform the liquid surface. Considering the range of surface forces, the typical size of the core region is of the order of 10–100 nm. The contact angle is measured outside the core region. It is the angle the liquid surface forms with the solid surface when the interface is extrapolated to the contact line by using the Laplace equation. It is usually measured by a camera or optical microscopes that cannot resolve the core region.

6.1.4 Line Tension, Wetting Transitions, Estimation of Interfacial Energies Line Tension

Spreading is usually accompanied by a change in the length of the contact line. For example, if a drop with a circular contact area spreads, the length of the contact line increases by 2𝜋da. Just as with the formation of a new surface area, the creation of a contact line also requires energy. This energy per unit length is called

6.1 Young’s Equation

the line tension 𝜅l . For small droplets, the line tension needs to be taken into account (review [330]) and a term must be added to Young’s equation [331–333]: 𝜅 𝛾L cos Θ = 𝛾S − 𝛾SL − l (6.16) a Here, a is the radius of curvature of the three-phase contact line. For a drop with a circular contact area or radius, a is the contact radius. Example 6.3 Estimate the line tension for cyclohexane liquid from its heat of vaporization Δvap U = 30.5 kJ/mol at 25 ∘ C. Then the density is 𝜌 = 773 kg∕m3 , and its molecular weight is M = 84.2 g/mol. As in Example 2.4, we picture the liquid as being composed of cubic molecules. The size of each cube was calculated from the density of cyclohexane to be aM = 0.565 nm. In the bulk, each molecule is supposed to directly interact with six neighbors. The energy per bond is thus Δvap U∕6NA . At the rim two fewer bonds can be formed and the energy loss per molecule is 2Δvap U∕6NA . Thus, the energy difference per unit length is Δvap U 30 500 J∕mol 𝜅l = = = 2.98 × 10−11 J∕m 3NA aM 3 ⋅ 6.02 × 1023 mol−1 ⋅ 0.565 × 10−9 m The result leads to the right order of magnitude. For water, the same calculation results in an estimated line tension of 7 × 10−11 J∕m. Typical line tensions are on the order of few times 10−11 N [334]. Thus, macroscopic wetting phenomena will not be influenced. Line tension may, however, have a influence on the shape of nanodrops as they occur transiently in heterogeneous nucleation [335]. Surface tensions are always positive. It requires work to create more interface. A negative surface tension would imply that the system could spontaneously reduce its free energy by forming more interface. Then the interface would tend to become infinitely large. In contrast, line tensions can also be negative [333]. It can be favorable to form an additional contact line. Wetting Transitions

What happens with the contact angle when we increase the temperature? Let us consider a sessile drop of a partially wetting liquid (0∘ < Θ < 90∘ ) on a solid surface in equilibrium with its vapor. In general, 𝛾L decreases much faster with increasing temperature than 𝛾S − 𝛾SL . Hence, with increasing temperature, the contact angle decreases [336]. For a sufficiently high temperature, a wetting temperature Tw must exist where S = 0 and, therefore, Θ = 0, provided the liquid does not reach its boiling point first. Below the wetting temperature, a finite contact angle is observed. Above Tw , the liquid forms a continuous film on the solid. The process is called wetting transition [337]. Example 6.4 Figure 6.5 shows the contact angle of n-octane and 1-octene on a silanized silicon wafer versus temperature. To determine the wetting temperature, it is more appropriate to plot cos Θ versus temperature and extrapolate to

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6 Contact Angle Phenomena and Wetting

10 1.000 (°)

8 0.996 cos

6 Contact angle

144

4 n-octane 1-octane

2 0 25

0.992 0.988 0.984

30

35

40

45

Temperature (°C)

(a)

50

55

25 (b)

30

35

40

45

50

55

Temperature (°C)

Figure 6.5 Variation of contact angle Θ (a) and of cos Θ (b) with temperature for n-octane and 1-octene drops situated on a silicon wafer coated with a monolayer of hexadecyltrichlorosilane. Source: Adapted from Law [338].

cos Θ = 1 (Figure 6.5b). The wetting transitions occur at Tw = 45.4 ∘ C for octane and Tw = 51.2 ∘ C for octene. Estimation of Interfacial Energy

Contact angle measurements can be used to determine the surface energy of a solid, 𝛾S . In particular, for polymers, it is a common method to measure the contact angles with different liquids and calculate the surface energy. Then, however, one needs to know 𝛾SL . Therefore, it would be helpful to express 𝛾SL through 𝛾L and 𝛾S . Since 𝛾SL , 𝛾L , and 𝛾S are independent parameters, we can only hope to find an approximate expression, and we must use additional information. Girifalco, Good, and Fowkes considered solids and liquids in which the molecules are held together by van der Waals forces [336, 339, 340]. Then, in a thought experiment, they separated two materials at the interface (Figure 6.6). The required difference in energy per cross-sectional area, also called the work of adhesion, is w = 𝛾1 + 𝛾2 − 𝛾12 . Two new surfaces are formed while the interfacial area disappears. Rearrangement leads to 𝛾12 = 𝛾1 + 𝛾2 − w

(6.17)

Now, we recall Section 5.2.4. The work of separating two solids to an infinite distance against the van der Waals attraction is w=

A12 12𝜋 D20

(6.18)

Here, D0 is a typical interatomic spacing and A12 is the Hamaker constant for the interaction of material 1 with material 2 across a vacuum or gas. Here, we implicitly assumed that the surfaces molecules do not rearrange after separating the two materials. According to Eq. (5.27), the “mixed” Hamaker constant can be expressed by the geometric rule √ A12 ≈ A11 A22 (6.19)

6.2 Wetting of Real Surfaces

Figure 6.6 Two materials are separated at their interface.

2 2 1 1

The Hamaker constants of the single materials are related to the surface tensions Eq. (5.29) 𝛾1 =

A11 24𝜋 D20

and 𝛾2 =

A22 24𝜋 D20

Substituting all these results into Eq. (6.17) finally leads to √ 𝛾12 = 𝛾1 + 𝛾2 − 2 𝛾1 𝛾2

(6.20)

(6.21)

The model of Girifalco, Good, and Fowkes has been extended to other interactions. For example, if we assume that the surface energies are the sum of van der Waals (dispersive) and polar interactions, one often uses [341, 342] √ √ p p 𝛾SL ≈ 𝛾S + 𝛾L − 2 𝛾Sd 𝛾Ld − 2 𝛾S 𝛾L (6.22) p

p

with 𝛾S = 𝛾Sd + 𝛾S and 𝛾L = 𝛾Ld + 𝛾L . The superscripts “d” and “p” indicate dispersive and polar interactions. For a discussion, see [343].

6.2 Wetting of Real Surfaces 6.2.1

Advancing and Receding Contact Angles

Early experiments showed that contact angles not only depend on the interfacial energies but also on the surface structure, the pretreatment of the surfaces and on contamination [344]. Agnes Pockels1 noticed that the contact angle of an advancing liquid which had just stopped advancing is larger than the contact angle of a drop which was forced to recede [345]. She also noticed that drops, where the difference between advancing and receding contact angles is low, slide easily while drops with a large difference do not slide at all when the substrate is being tilted. In the 1920s and 1930s, it was established that the measurable quantities which characterize a specific surface are the advancing and receding contact angles, Θa and Θr , respectively. They limit the range of accessible contact angles. When a contact angle exceeds Θa , the contact line will start to advance. When the actual contact angle falls below Θr , it will recede. The difference between the two is called contact angle hysteresis, ΔΘ = Θa − Θr [346]. The situation is a bit unsatisfactory because the equilibrium contact angle, which appears in Young’s equation, cannot be measured. Contact angle hysteresis is essential for our daily life because it provides friction to drops. Without contact angle hysteresis, drops would slide off any surface, even at 1 Agnes L.W. Pockels, 1862–1935; German autodidactic physicochemist, housewife in Braunschweig, Lower Saxony.

145

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6 Contact Angle Phenomena and Wetting

Figure 6.7 Schematic of a two-dimensional drop sliding down a tilted plate.

r a

α w

very small inclination. This sounds good news to anybody wearing glasses, because even in the rain, drops would not blur the view. However, many applications, such as coating, painting, flotation, would not be possible without contact angle hysteresis. To better understand drop sliding, we consider a drop on an inclined plate and calculate the tilt angle 𝛼, where the drop starts moving off a surface. The gravitational force along the plate is mg sin 𝛼, where m is the mass of the drop. Just before the drops start to move, this gravitational force is equal to the capillary force. To simplify the calculation of the capillary force, we take the drop to be two-dimensional (Figure 6.7). Its width, w, corresponds to the width of the contact area of the drop. At the rear of the drop, the lateral force is 𝛾L w cos Θr . At the front, the force in lateral direction is −𝛾L w cos Θa . Thus, the sum of lateral capillary forces is [336, 347, 348]: ( ) Fc = w𝛾L k cos Θr − cos Θa (6.23) Here, k is a geometrical factor. For the two-dimensional drop, k = 1. In reality, the actual contact angle gradually changes from Θa at the front of the drop to Θr at the rear. As a result, k can be slightly different from unity, depending on the specific geometry of the drop [349, 350].

6.2.2

Measurement of Contact Angles

Several methods have been established to measure advancing and receding contact angles (reviews: [336, 351]). The most common method of measuring contact angles is to image a sessile drop by a video camera while the drop is inflated or deflated via, e.g. a micropipette (Figure 6.8a). As described in Section 2.4, a light source is positioned behind the drop, which then appears dark. The contact angle is determined directly with a goniometer or the contour is fitted by a computer with a line, a segment of a circle or an ellipse, or the Laplace equation. The advancing contact angle Capillary Liquid Solid

a

Plate, tape or fiber

r

rr

(a) a r

(b)

Gas

a

Gas

r aa

Liquid

(c)

(d)

Figure 6.8 Methods to measure advancing Θa and receding contact angles Θr : (a) sessile drop, (b) captive bubble, (c) Wilhelmy plate, and (d) inclined plane.

6.2 Wetting of Real Surfaces

is recorded just before the contact line starts to advance when increasing the volume of the drop. The receding contact angle is measured just before the contact line starts to recede when decreasing the volume of the drop. Alternatively, we can measure the contact angle at the edge of a bubble (Figure 6.8b). This method is called a captive or sessile bubble method. In this case, a bubble is positioned usually at the top of a cell, which is otherwise filled with liquid. The volume of the bubble is increased and decreased. Θa is measured when the volume decreases and Θr when the volume increases. The method is less sensitive to contamination of the interface. In addition, the vapor phase is automatically saturated. A widely used technique is the Wilhelmy plate method introduced in Section 2.4 (Figure 6.8c). If the contact angle is larger than zero, then the force with which the thin plate is pulled into the liquid is 2l𝛾L cos Θa∕r . Here, l is the width of the plate. The contact angles are either determined from the contour images or from the force. When the plate is lowered into the liquid, Θa is obtained. Θr is measured when the plate is slowly moved out of the liquid, just before the contact line starts to move. A common method to measure advancing and receding contact angles is on a tilted plate (Figure 6.8d). By inclining the supporting surface, gravity can gradually be increased until at some tilt angle, the drop starts moving. While tilting, the shape of the drop is observed by a camera. Afterward, the contact angles are determined from the images taken just before the drop starts sliding. One word of caution. Often, static contact angles are reported, sometimes even with a precision better than 1∘ . This does usually not make sense, in particular not for water as a liquid. Static contact angles are measured by placing a drop of water onto a horizontal surface and detecting the angle optically. The observed contact angle will be somewhere between Θr and Θa , depending on how precisely the drop is pipetted onto the surface. It would be a coincidence if it is close to the global equilibrium contact angle. If a single person does the pipetting for a series of drops, the observed contact angle may be quite similar. It does, however, not characterize the surface. To characterize a specific liquid–surface combination, advancing and receding contact angles should be reported.

6.2.3

Causes of Contact Angle Hysteresis

Various causes for contact angle hysteresis have been discussed. The contribution of each factor depends on the particular situation. Possible causes of hysteresis are the following: ●

Surface roughness [352–355]. Although many surfaces appear flat and homogeneous to the naked eye, at the submicroscopic scale they are usually rough. This can lead to hysteresis, as illustrated for a drop with a contact angle of 90∘ in Figure 6.9. Here, the wetting line must move over a microscopically small cylindrical protrusion. If the three-phase contact line advances from left to right (position i), it will at some point come into contact with the protrusion. There, it jumps immediately to a position, where it again assumes a contact angle of 90∘

147

148

6 Contact Angle Phenomena and Wetting

i, ii

i

iii

ii

iii

Liquid app

(b)

Solid (a)

Figure 6.9 A drop advancing over a solid surface with a microscopic cylindrical protrusion: (a) optically visible situation; (b) schematic illustration of microscopic details.

●

●

●

(ii). In our case, this is at the top of the protrusion. Then the drop is hindered from spreading further in order to keep locally a contact angle of 90∘ . Macroscopically, the three-phase contact line sticks at the protrusion. The apparent, macroscopic contact angle is now much larger than 90∘ , even though the microscopic contact angle remains constant at 90∘ (iii). Only after overcoming the protrusion will it continue to spread until it reaches the next protrusion. A rough surface can be pictured as a collection of densely packed protrusions. They might not be as steep as in our example, but the effect is qualitatively similar. When the liquid recedes, the same effect occurs, and we observe a hysteresis. Most surfaces are not perfectly homogeneous but expose chemically or structurally different regions. This becomes obvious when looking at the spreading of a liquid on surfaces with a micropatterned heterogeneity [336, 356–358]. Such heterogeneity of the solid surface leads to line pinning and, thus, to hysteresis [359–362], just as described for roughness. Upon advancing, the three-phase contact line is pinned by relatively lyophobic (“liquid hating”) regions; when receding, the three-phase contact line is pinned by lyophilic (“liquid loving”) regions. Early on scientists adopted the concept that the solid somehow adapts to the presence of the liquid or its vapor [336, 363]. Many polymers reconstruct due to a reorientation of side groups, a selective exposure of specific segments, and the diffusion of liquid into the polymer or even swelling of the polymer [342, 364, 365]. Surfaces have been deliberately coated with mixed polymer brushes which change their structure depending on the type of fluid they are exposed to [366, 367]. One may also consider the replacement of a contamination/adsorption layer as an adaptive process. As soon as a surface is exposed to air, water and airborne hydrocarbons adsorb. When the surface is wetted, the adsorbed layer changes its structure, is replaced, or dissolves. At the three-phase contact line, the surface tension exerts strong forces on the surface. For instance, if we consider a water drop on a polymer surface, typical contact angles are 90∘ . The surface tension pulls upward on the solid surface. Based on molecular dynamics simulations [368], we estimate the wetting line to have a width of 𝛿 ≈ 10 nm, the force F per unit length l can be related to the effective

6.2 Wetting of Real Surfaces

stress exerted on the solid surface: 𝛾L = F∕l = Pl𝛿∕l = P𝛿. With 𝛾L = 0.072 N∕m, and 𝛿 = 10 nm, the effective stress is on the order of P = 72 × 105 Pa. On soft surfaces, such as weakly cross-linked PDMS, this stress can change the surface structure, cause mechanical deformation at the moving wetting line [70, 314, 369], and lead to contact angle hysteresis [313].

6.2.4

Surface Roughness and Heterogeneity

Roughness and heterogeneity of surfaces can lead to contact angle hysteresis. In addition, it influences the macroscopic equilibrium contact angle (for a review see [370]). Due to surface structures or inhomogeneity, a macroscopic system may not be in its global free energy minimum. To help it reach the global energy minimum, the sample can be vibrated [371]. Vibrations can help to overcome line pinning so that the contact line moves out of a local energy minimum and finds a position close to the lowest global free energy. It is, however, unclear if the contact line really comes to a halt in the global energy minimum or some other metastable position [372]. To describe the effect of surface roughness on the equilibrium contact angle, Wenzel proposed the following equation [370, 373]: cos Θapp = Rrough cos Θ

(6.24)

Here, Θapp is the apparent contact angle that we observe on a length scale much larger than the scale of roughness. Rrough is the ratio between the actual and projected surface area. Since Rrough is always greater than or equal to one (Rrough ≥ 1), surface roughness decreases the apparent contact angle for Θ < 90∘ , while for poorly wetted surfaces (Θ > 90∘ ) the apparent contact angle increases. If a molecularly hydrophobic surface is rough, the appearance is that of an even more hydrophobic surface. If a hydrophilic surface is roughened, it becomes more hydrophilic. Equation (6.24) is, however, of limited use, because on rough surfaces, contact angle hysteresis is often large and a change in the equilibrium contact angle may not even be noticeable. In addition, the exact value of Rrough will be hard to determine for most surfaces, especially for micro- and nanoscale roughness. Cassie and Baxter considered a smooth but chemically heterogeneous surface. If there are two different kinds of regions with contact angles Θ1 and Θ2 that occupy the surface ratios f1 and f2 , then the apparent average contact angle is [374] cos Θapp = f1 cos Θ1 + f2 cos Θ2

(6.25)

Again, the apparent contact angle is the contact angle observed on a length scale much larger than the variation in surface energy. Example 6.5 Drelich et al. [357] measured the contact angles of water drops on patterned surfaces. They created parallel stripes of hexadecanethiol, HS(CH2 )15 CH3 , and diundecane disulfide carboxylic acid, [S(CH2 )11 COOH]2 , on gold, which were 2.5 and 3.0 μm wide, respectively. Pure hexadecanthiol monolayers (Section 9.3.1) are hydrophobic and showed an advancing contact angle of 107.8∘ at pH 7.0.

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6 Contact Angle Phenomena and Wetting

The disulfides were more hydrophilic (Θa = 50.1∘ ) because the carboxylic acid is exposed on the surface. For the micropatterned surface, a contact angle of 3.0 2.5 cos 107.8∘ + cos 50.1∘ = 0.211 ⇒ 2.5 + 3.0 2.5 + 3.0 = 77.8∘

cos Θapp = Θapp

is calculated using the Cassie equation Eq. (6.25). Experimentally, Drelich et al. observed an advancing contact angle of 77∘ ± 3∘ parallel to the stripes, which is in good agreement with the calculated value. Perpendicular to the stripes, the experimental contact angle deviated significantly from that calculated using the Cassie–Baxter equation. In that direction, no macroscopic equilibrium was reached, and the wetting line was pinned. An important factor is the length scale of surface structures and their shape. If, for example, a surface is covered with cylindrical protrusion as in Figure 6.9, and they are 0.1 μm in size, they may escape optical detection. Macroscopically, one would observe a hysteresis. A microscopic look would reveal the wetting line as sliding continuously over the surface, always forming the contact angle given by Young’s equation. The fact that macroscopically we observe a contact angle hysteresis does not necessarily violate Young’s equation. It may only reflect our inability to look more precisely. In general, Young’s equation can be applied if a system is in global equilibrium on a length scale one is looking at. The relevance of length scales becomes obvious when analyzing the wetting of textured, porous, or micropatterned surfaces. In particular, arrays of micropillars have been intensely studied in order to better understand and control wetting [370, 375, 376]. In some cases, a large hysteresis of more than 100∘ has been observed. Macroscopic wetting is then dominated by line pinning rather than interfacial energies

6.2.5

Superhydrophobic Surfaces

Micropillar arrays are also used to understand superhydrophobicity; surfaces are called superhydrophobic, when they form a very high contact angle with drops of water [351]. When placing a water drop on top of a micropillar array, the water will usually wet the whole surface and penetrate between the pillars. This state is called Wenzel state [373] (Figure 6.10a). If the surface is made of a hydrophobic material such as a hydrocarbon or fluorinated hydrocarbon and the drop is placed carefully, the water will not slide down the pillar walls and stay on top of the micropillars. An air layer is formed underneath the drop. For vertical walls of the micropillars, this is the case if the advancing contact angle is above 90∘ . This state is called Cassie state (Figure 6.10b) [378]. To achieve superhydrophobicity, the water needs to remain in the Cassie state. As long as the bottom side of a liquid drop is for a large part in contact with air, the apparent contact angle will be high. Such a high apparent contact angle follows from the Cassie–Baxter equation (6.25). If we take material 2 to be air (with Θ2 = 180∘ ) and the surface fraction f2 ≫ f1 , Θapp can be very high even for Θ1

6.2 Wetting of Real Surfaces

Water app Water

Oil

app

app

a

app

app

app

a

a

φ (a)

Solid

(b)

Solid

(c)

Solid

Figure 6.10 Schematic of a water drop on an array of micropillars with vertical walls in the Wenzel state (a). For Θa < 90∘ , water penetrates between the pillars. For Θa > 90∘ , the water can remain on top of the micropillars leading to the Cassie state (b). To reach a Cassie state with nonpolar liquids, the micropillars need to have an overhanging structure (c). Figure 6.11 Scanning electron microscope image of a leaf of the sacred lotus Nelumbo nucifera. Source: Barthlott and Neinhuis [380], Springer Nature.

20 µm being only slightly above 90∘ . A surface is called “superhydrophobic” when water has a high apparent receding contact angle, typically higher than 150∘ [379]. Drops easily roll off such surfaces (see Exercise 6.6). Some plant leaves, the wings of many insects, and the feathers of many birds are superhydrophobic. It not only keeps them dry. When water drops roll off, they pick up dirt on their way and clean the surface. The effect is called the lotus effect because it was first described for the surface of lotus leaves [380]. The surfaces of these leaves are hydrophobic and extremely rough (Figure 6.11). They can be viewed more like a surface with many hydrophobic spikes. A water drop placed on top only comes into contact with the ends of the spikes. Air is trapped below the drop so that the contact angle increases and hysteresis is low. In addition, dust particles will also rest only on the spikes, forming very weak contacts. When water drops slide over the leaf surface, capillary forces between particles and drops are higher than adhesion forces to the leaf surface, leading to particle removal. Though many ways are known to make superhydrophobic surfaces [351, 370, 373, 381, 382], it is still a challenge to fabricate mechanically robust, transparent, and UV-stable superhydrophobic layers in a cost-efficient process. An even greater challenge is to make surfaces that repel not only water but also oil. Such surfaces are called superoleophobic or, since they repel polar as well as nonpolar liquids, superamphiphobic [383–386]. To fabricate robust superamphiphobic surfaces, two conditions need to be fulfilled: first, the material needs to have a low surface

151

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6 Contact Angle Phenomena and Wetting

energy such as perfluoroalkanes, hydrocarbons, or methylated siloxanes. Second, the surface needs to be coated by nano- and microscopic protrusions with steep or even overhanging sidewalls. The slope of the sidewalls has to be lower than the advancing contact angle on a flat, smooth surface of the same material, 𝜑 < Θa (Figure 6.10e) [336, 387]. Thus, for superhydrophobic surfaces, usually a slope of 90∘ is sufficient because many materials have Θa > 90∘ . However, for oil-repellent surfaces, overhanging structures with 𝜑 < 90∘ are required [383, 388] because oils do not form such high contact angles, not even on fluorinated surfaces.

6.2.6

Surfaces with Low Sliding Angle

For many applications such as in heat conversion or for fog harvesting drops should have a low lateral adhesion so that they slide off easily. For this purpose, they do not necessarily need to have a high apparent contact angle. It is sufficient that the apparent contact angle hysteresis is low so that according to Eq. (6.23), the lateral adhesion is low. One strategy to reduce lateral adhesion is to take a rough or porous surface and impregnate it with a lubricant [389]. The lubricant is a liquid, which wets the porous surface but is immiscible with the liquid of the drop; to distinguish the two we call it “lubricant.” Such surfaces are called lubricant-infused surfaces (LIS), also called slippery lubricant-infused surfaces (SLIPS) (Figure 6.12a). A deposited drop partially rests on the top faces of the underlying microstructure and partially on lubricant. The lubricant layer reduces contact line pinning [390, 391]. Compared to super liquid-repellent surfaces, the drop is primarily in contact with the lubricant rather than air. As a result, this lubricated state is more stable against external perturbation or pressure. A disadvantage is the depletion of lubricant by evaporation, by gravitational drainage, or by being dragged away by sliding drops [392, 393]. An alternative strategy is to graft a flexible polymer such as PDMS, polyethylene glycol, or perfluorinated polyether to the surface (Figure 6.12b). Such liquid-like polymers serve as a lubricant and reduce the lateral adhesion [394–396]. The lateral adhesion is, however, not as strongly reduced as for LIS. To further reduce lateral adhesion, a combination of LIS and polymer brushes can be made by adding a lubricant to the polymer brush [397, 398] (Figure 6.12c). A lubricant can be any second liquid which mixes with the brush but not with the liquid in the drop. In fact, usually after the synthesis of, e.g. PDMS brushes, some oligomers are still remaining and act as lubricant. However, with time, the lubricant is again washed out, and the contact angles gradually approach that of the pure brush.

Lubricant

(a)

Liquid

Rough or porous substrate

Polymer brush

Air

(b)

Substrate

Lubricant

(c)

Figure 6.12 Schematic of a liquid drop on lubricant-infused surfaces (a), a surface grafted with a flexible polymer brush (b), and a polymer brush surface with dissolved lubricant (c).

6.3 Important Wetting Geometries

6.3 Important Wetting Geometries 6.3.1

Capillary Rise

An important wetting phenomenon, and a procedure for measuring the contact angle at the same time, is the rise of a liquid in a capillary tube. If a capillary is lowered into a liquid, the liquid often rises in the capillary until a certain height is reached (Figure 6.13a). For a capillary with a circular cross section of radius rc , this height is given by [15, 16, 399] hmin =

2𝛾L cos Θa rc g𝜌

(6.26)

Here, g is the acceleration of free fall and 𝜌 is the density of the liquid. For details of capillary rise, see [1]. Historically, already Jurin2 observed in 1718 that the height of a liquid rising in a capillary tube is inversely proportional to the diameter of the capillary. For a long time, capillary rise was the most commonly used technique to measure the surface tension of liquids [400–402]. Rather than measuring the height of the liquid after spontaneous imbibition, one may also start with a filled capillary and let the liquid flow out until a stable configuration is reached. Due to contact angle hysteresis, the observed height will be higher because in this case, the receding contact angle is relevant. To calculate this maximal height, hmax , the receding contact angle, Θr , has to be inserted in Eq. (6.26). As long as the actual height is between hmin and hmax , the system will be stable. Example 6.6 Water in trees rises in capillaries that are called xylem. They are 5–200 μm in radius and are completely wetted (Θ = 0). What is the maximum height water can rise in a capillary with a radius of 5 μm? Using Eq. (6.26), we calculate h=

2 ⋅ 0.072 N∕m 5 × 10

−6

m ⋅ 9.81 m∕s2 ⋅ 997 kg∕m3

= 2.94 m

Air/ vapor

h

Liquid

(6.27)

Capillary

h

2rc (a)

(b)

Figure 6.13 Rise of a liquid in a partially wetted capillary (a) and capillary depression in a nonwetted capillary (b). 2 James Jurin, 1684–1750. English scientist and physician.

153

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6 Contact Angle Phenomena and Wetting

Obviously, trees can grow to larger heights. The tallest trees, the coastal redwood in California (Sequoia sempervirens), reach 115 m. To reach such heights, capillary pressure in the xylem is not sufficient. The driving force to create the necessary pressure is evaporation of water through small pores at the surface of leaves. These pores have diameters < 50 nm. They generate a sufficiently high Laplace pressure. As long as no bubbles form and the water column in the xylem remains intact, water can rise even to 115 m. Intermolecular forces between the water molecules are strong enough to hold such a column of water. For a review, see [403]. To derive Eq. (6.26), we neglect contact angle hysteresis and assume that an equilibrium contact angle exists. We consider the change in the Gibbs free energy upon an infinitesimal rise dh of the liquid. Since the shape of the liquid–vapor interface does not change upon an infinitesimal rise, the change in the Gibbs free energy is dG = −2𝜋rc ⋅ dh ⋅ (𝛾S − 𝛾SL ) + 𝜋rc2 𝜌gh ⋅ dh

(6.28)

The first term represents the surface work. This is actually driving the process: the liquid rises because a high-energy solid surface is exchanged for a low-energy solid–liquid interface. The second term corresponds to the work required to lift the liquid with a weight 𝜋rc2 h𝜌 by a step dh against gravity. In equilibrium, the gravitation term is equal to the contribution of the interfacial tensions, and we have dG = −2𝜋rc (𝛾S − 𝛾SL ) + 𝜋rc2 𝜌gh = 0 ⇒ 2(𝛾S − 𝛾SL ) = rc 𝜌gh (6.29) dh Replacing 𝛾S − 𝛾SL by 𝛾L cos Θ, we immediately get Eq. (6.26). A liquid rises for partially wetted surfaces (Θa < 90∘ ). If the liquid does not wet the inner surface of the capillary and the advancing contact angle is greater than 90∘ , the liquid is pressed out of the capillary. Work must be carried out to fill the capillary with liquid. For this reason, it is difficult to get water into hydrophobic polymeric capillaries. In order to inject water into hydrophobic capillaries, one has to apply an external pressure: ΔP =

2𝛾L cos Θa rc

(6.30)

How fast does a liquid imbibe into a capillary? Neglecting gravitation, that is, for a horizontal capillary, the distance l penetrated by a liquid into a capillary after a time t is given by the Lucas–Washburn equation [404, 405]: √ rc 𝛾L t cos Θa l= (6.31) 2𝜂 In many applications, e.g. in pharmaceutical or food industry and mining, powders come into contact with a liquid, and we would like to quantify their wetting behavior [406, 407]. For mostly wetting powders, the usual way to do this is by the capillary rise method also called Washburn method [408, 409]. In a capillary rise measurement, the powder is pressed into a tube typically 1 cm in diameter (Figure 6.14). This porous material is then treated as a bundle of thin capillaries with a certain effective radius rceff [404, 405, 410]. To measure this effective radius, first a completely

6.3 Important Wetting Geometries

Figure 6.14 Capillary rise method to quantify wetting properties of powders or porous materials.

Capillary Powder Filter

Liquid

wetting reference liquid with a low surface tension 𝛾Lref is used. Either the speed of the liquid rise is measured (this technique is sometimes referred to as the capillary penetration technique [411]) or the pressure required to keep the liquid out of the porous material is determined. This backpressure is equal to the Laplace pressure of the liquid, given by ΔPref =

2𝛾Lref rceff

(6.32)

Knowing 𝛾Lref , we obtain rceff . Then we measure the Laplace pressure with the liquid of interest. It is 2𝛾 cos Θ (6.33) ΔP = L eff a rc Comparing the two pressures directly leads to the contact angle. One limitation of the capillary rise method is that it averages over many particles and the actual size distribution remains unknown. In addition, it relies on the assumption that a powder can be treated as a bundle of capillaries and depends on the specific model applied. It is not applicable to lyophobic liquids because they do not penetrate. In this case, the sessile drop method is often applied [407]. Capillary rise is particularly pronounced in the confined geometry of a thin capillary. It also occurs on single walls (Figure 6.15). For example, if a vertical plate is partially immersed in liquid, the liquid will form a meniscus. The height of the meniscus is given by [16, 307, 399] √ 2𝛾L hmin ∕ max = (1 − sin Θa∕r ) (6.34) 𝜌g Here, hmin is obtained with Θa when the plate is lowered into the pool or when the plate is brought into contact with the liquid and wets spontaneously. The maximal height hmax is measured when withdrawing a previously immersed plate out of the liquid pool, and then, Θr is the relevant contact angle.

6.3.2

Particles at Interfaces

Small particles bind to liquid–gas interfaces if the contact angle is not zero. As we will see later, this is of fundamental importance in applications such as flotation

155

156

6 Contact Angle Phenomena and Wetting

Figure 6.15 Schematic of a liquid meniscus rising on a vertically immersed wall.

h

r

h

R Large particle

Small particle Liquid (a)

(b)

Figure 6.16 Small spherical particle in a liquid–gas interface for which gravitation is negligible (a). The shape of the interface remains unchanged by the presence of the particle. For larger particles (b), the liquid interface is deformed, and a net capillary force stabilizes the particle and prevents it from sinking.

or the stabilization of emulsions. For this reason, particles at interfaces have been studied extensively [412, 413]. For simplicity, we start by considering small, spherical particles. Small refers to particles for which we can neglect gravitational effects and buoyancy. For Θr > 0∘ , a particle is stable at the liquid surface (Figure 6.16). Its position in the interface is simply determined by the fact that, in equilibrium, the liquid surface is unperturbed. A planar liquid surface will also be planar with the adsorbed particle. This is easy to understand. If the liquid surface is curved, then the capillary force due to surface tension will result in a force in the normal direction. In the absence of external forces, this capillary force will draw the particle to a new position until the curvature is zero again. If the particle is significantly larger than ≈ 10 μm, then gravitation pulls it down. The liquid surface becomes curved so that the capillary force compensates for the gravitational force. For many applications, the work required to remove a particle from the gas–liquid interface is important. For example, in flotation (Section 6.5.1), mineral particles adhere to gas bubbles, and we would like them to remain in the interface rather than go back into the bulk liquid. To calculate the change in the Gibbs free energy upon moving a small particle from its equilibrium position in the interface into a bulk liquid, where it is completely immersed, we again neglect contact angle hysteresis. We first need to know the surface area of the particle that is exposed to the gas phase. It is (see also Eq. (6.2)) (r 2 + h2 ) = R2 (sin2 Θ + 1 − 2 cos Θ + cos2 Θ) = 2R2 (1 − cos Θ)

(6.35)

with r = R sin Θ and h = R − R cos Θ. When moving the particle from the interface into liquid, the solid–liquid and liquid–vapor areas increase while the solid–gas

6.3 Important Wetting Geometries

interfacial area decreases. This results in a change in the Gibbs free energy of [332] ΔG = 2R2 (1 − cos Θ)(𝛾SL − 𝛾S ) + R2 𝛾L sin2 Θ

(6.36)

With Young’s equation, we get ΔG = −2R2 (1 − cos Θ) 𝛾L cos Θ + R2 𝛾L sin2 Θ = R2 𝛾L (sin2 Θ − 2 cos Θ + 2cos2 Θ) = R2 𝛾L (cos2 Θ − 2 cos Θ + 1) = R2 𝛾L (cos Θ − 1)2

(6.37)

Without derivation, we report the force required to pull a spherical particle out of an interface of interfacial tension 𝛾L [332, 414, 415]: F = 2R𝛾L sin2

Θ 2

(6.38)

Example 6.7 To remove a hydrophobic glass microsphere (Θ = 90∘ ) of 5 μm radius from a water surface (𝛾L = 0.072 J/m2 ), work of 5.6 × 10−12 J is required. This is roughly 109 kB T. The adhesive force is 1.13 μN. We conclude that under normal conditions, the particle will remain in the interface. For comparison, the gravitational force on the particle is only the order 5 pN.

6.3.3

Network of Fibers

For the water repellency of feathers or applications such as water repellent clothing or textile filters, a network of fibers is important [416, 417]. As a simple model, we consider a bundle of parallel, horizontal, cylindrical fibers separated by a certain spacing. This spacing is assumed to be much smaller than the capillary constant, so that the shape of the liquid surface is assumed to be determined only by the Laplace equation (2.6). For a low external pressure such a network will not allow a liquid to pass, unless the contact angle is zero (Figure 6.17) [374, 416]. The liquid forms a contact angle with the solid, which determines how far it will penetrate into the interfibrous spacing. If an external hydrostatic pressure is applied, water moves further into the interfibrous spacing. It penetrates the network only if the pressure is so high that, for a given spacing, the liquid surface becomes unstable and a drop is formed. Example 6.8 Effect of oil on aquatic birds. Most aquatic birds keep their feathers hydrophobic with wax. For them, it is essential that the feathers repel water. The air entrapped in and under the feathers provides a good heat insulation. It keeps the birds afloat and light so that they can fly. Due to low surface tension, oil has a strong tendency to wet all kinds of solid surfaces. It also wets the surfaces of feathers, which then deprives the birds of all the essentials described. In addition, heavy oil is sticky, destroys the feather structure, and prevents the birds from moving freely.

157

158

6 Contact Angle Phenomena and Wetting

Liquid

(a)

(b)

Figure 6.17 Schematic of parallel cylindrical fibers (oriented normal to the paper) with a liquid on top. In (a, b) no external hydrostatic pressure is applied. For this reason, the liquid surface is planar (in equilibrium). The contact angle is larger than 90∘ in (a). In (b), it is lower than 90∘ but higher than 0∘ . In (c), a hydrostatic pressure is applied. (a) Θ > 90∘ . (b) Θ < 90∘ . (c) Θ > 90∘ with pressure.

(c)

6.4

Dynamics of Wetting and Dewetting

In this section, we deal with moving contact lines. Reviews on the subject are [418, 419]. We distinguish spontaneous and forced wetting. In forced wetting, externally imposed hydrodynamical or mechanical forces cause the solid–liquid interfacial area to increase beyond static equilibrium. Forced wetting plays an essential role in industrial coating processes where a thin layer of liquid is deposited continuously onto a moving solid surface. Failure of the liquid to displace sufficient air from the substrate limits the speed at which a coating can be applied, leading to air entrainment otherwise. Forced wetting is also important in polymer processing and enhanced oil recovery. Spontaneous spreading, on the other hand, is the advancing of a liquid on a solid surface toward thermodynamic equilibrium. The driving force is not imposed externally, but stems from liquid–solid interactions. Spontaneous spreading is of practical relevance in the application of paints, adhesives lubricants, in detergency, flotation, for the migration of inks, and so on. Dynamical aspects of wetting are far from being fully understood, which is certainly due to their inherent complexity.

6.4.1

Spontaneous Spreading

When we place a liquid drop with Θ ≪ 90∘ on a horizontal hard, inert, and smooth surface, the interfacial forces cause the drop to spread (Section 2.4). The gradient of curvature of the liquid surface and, thus, the capillary (Laplace) pressure between the center and the periphery of the drop causes the drop to expand. The contact radius increases with time t as [420] a = a0 t𝛼 . Here, 𝛼 is the spreading exponent, and a0 is a coefficient depending on the initial conditions and the material’s properties. The spreading exponent varies depending on the effect limiting the speed of wetting. Typically, in the first few milliseconds, the inertia of the liquid mass transported limits the spreading speed. In this initial regime, 𝛼 = 0.5 [421, 422].

6.4 Dynamics of Wetting and Dewetting

In the second regime, wetting is limited by viscous hydrodynamic effects. Spreading slows down and 𝛼 = 0.10 − 0.13 [423–426], depending also on drop size. For a liquid with a viscosity 𝜂 above 0.05 Pa s the viscous regime dominates right from the beginning [421]. When the drop has reached a contact radius of typically 1 cm or larger, gravitation becomes the dominant force driving the expansion of the liquid. Then 𝛼 ≈ 1∕8 [425]. Spontaneously spreading drops are known to form a thin (< 0.1 μm) primary or precursor film [426, 427]. The first indication of a precursor film was described by Hardy [428]. He placed a small drop of acetic acid onto a dry glass plate near a corner. The liquid formed a flat lens with a defined contact angle. He noticed, however, that even in the opposite corner friction was reduced. He concluded that the liquid had formed an invisibly thin film over the whole glass plate. Its thickness and extension are determined by surface forces [429]. In the precursor film, energy is dissipated by viscous friction. The liquid transport in the precursor film is driven by the disjoining pressure in the precursor film, which sucks liquid from the wedge of the drop. Example 6.9 Cazabat and coworkers [430] observed the spreading of PDMS on cleaned silicon wafer. PDMS is a nonvolatile polymer that is liquid at room temperature. The PDMS used had a mean molecular weight of 9.7 kg/mol and a viscosity of 0.2 Pa s. Spreading was observed by an ellipsometer that had a lateral resolution of 30 μm. In Figure 6.18 we can see that, in front of the drop, there is a precursor film of ≈ 0.7 nm thickness. When a drop of water is placed on a hydrophobic surface such as polypropylene, it does not spread but forms a high contact angle. Adding surfactant can change it, but typically the contact angle remains high and the water does not spread. In 1990, however, Ananthapadmanabhan et al. [431] observed that the addition of a specific silicone surfactant led to complete and fast spreading. The phenomenon was termed superspreading. It turned out that this surfactant belongs to a whole class of silicone surfactants with a specific structure, all showing a phase behavior, which facilitates spreading. For a review see [432].

10 Thickness (nm)

Figure 6.18 Spreading of a drop of PDMS on a silicon wafer observed with an ellipsometer 19 hours after deposition. Source: Redrawn after De Coninck et al. [430].

Drop

8 6 Prewetting layer

4 2 0

–6

–4

–2 0 2 Radius (mm)

4

6

159

160

6 Contact Angle Phenomena and Wetting

Figure 6.19 Schematic velocity dependence of dynamic contact angles ΘD .

D

180°

Rupture of film

Entrainment of air

a r

0° R e c e d in g

6.4.2

0 Velocity

A d v a n c in g

Dynamic Contact Angles

A contact line advances when the actual contact angle exceeds the advancing contact angle and it recedes when it falls below the receding contact angle. The force per unit length of contact line in the horizontal direction is 𝛾L (cos Θ0 − cos ΘD ), where Θ0 is the equilibrium contact angle. As a result, the dynamic contact angle increases monotonically with the wetting advancing velocity, and the receding contact angle decreases with the receding velocity (Figure 6.19). When increasing the wetting velocity more and more, at some point bubbles are entrained. For fast coating processes bubble entrainment can be a severe problem [433, 434]. On the receding side, either a film is formed [433] or tiny droplets are left behind once the speed exceeds a certain threshold [435]. A typical example of water drops sliding down a tilted hydrophobic plate is shown in Figure 6.20. The dynamic advancing and receding contact angles are not symmetric and show different velocity dependency. While the advancing side only shows a weak increase with velocity, the receding contact angles changes drastically within the explored velocity range. The two open circles at zero velocity were determined by gradually tilting the plate and taking the front (Θa ) and rear contact angles (Θr ) just before a drop starts sliding. Several experimental arrangements are used to measure and analyze the dynamic wetting of liquids on solid surfaces (Figure 6.20). Typical geometries are drops sliding down a tilted plate, a sessile drop on a solid surface, liquid–fluid displacement through a capillary tube, steady immersion or withdrawal of fibers, plates, or tapes from a pool of liquid, and the rotation of a horizontal cylinder in a liquid (Figure 6.21). Experiments on forced wetting show that, in general, the apparent contact angle depends not only on the speed v but also on the viscosity 𝜂 and the surface tension 𝛾L of the liquid. Often, these are correlated, and the influence on the contact angle can be described by one parameter only, the so-called capillary number: v𝜂 Ca ≡ (6.39) 𝛾L The fundamental reason for the fact that velocity, surface tension, and viscosity influence dynamic wetting not independently but as one parameter, Ca, is revealed

6.4 Dynamics of Wetting and Dewetting

Contact angle (°)

120 r

100

a

80 60 40 –0,50

–0,25

–0,00 Velocity (m/s)

0,25

0,50

Figure 6.20 Dynamic advancing and receding contact angles measured with 33 μL water drops sliding down a tilted, hydrophobized silicon wafer [436]. Results were obtained with different samples and at different tilt angles.

(a)

(b)

(c)

(d)

Figure 6.21 Some geometries used to study the dynamics of contact angle phenomena: (a) spreading drop; (b) liquid advancing in a capillary; (c) plate, tape, or fiber immersed in a liquid; and (d) rotating cylinder in a pool of liquid.

by hydrodynamic theory [423, 424, 437–440]. The flow of liquid near the wetting line is determined by two antagonistic forces: a pressure gradient caused by the curvature of the liquid surface. According to the Laplace equation (2.6), the pressure gradient is proportional to the surface tension. Flow is hindered by viscous forces, which are proportional to the viscosity. Thus, the flow velocity at a given dynamic contact angle should be proportional to 𝛾L and inversely proportional to 𝜂. To be able to compare dynamic contact angles obtained with different liquids, the capillary number, rather than the velocity, is used as the abscissa. Example 6.10 To study the contact angle at different velocities, Blake and Shikhmurzaev [441] drew a film of poly(ethylene terephthalate) (PET) vertically into a glass tank filled with a water/glycerol mixture. This is the commonly used plunging-tape experiment. Dynamic contact angles ΘD were determined either directly, with an optical goniometer, or from high-speed video images. Figure 6.22 shows results obtained for two mixtures: 16% glycerol (𝜂 = 0.0015 Pa s, 𝛾L = 0.0697 N/m, Θa = 72.5∘ ) and 59% glycerol (𝜂 = 0.010 Pa s, 𝛾L = 0.0653 N/m, Θa = 64.5∘ ). At high wetting speeds, both graphs increase monotonically up to a contact angle of 180∘ . At such high capillary numbers, a layer of air is entrained between the

161

6 Contact Angle Phenomena and Wetting

Figure 6.22 Dynamic advancing contact angles versus capillary number for two glycerol/water mixtures on a film of PET. Source: Redrawn after [441].

180 Dynamic contact angle (˚)

162

160

ƞ = 1.5 mPa s ƞ = 10 mPa s

140 120 100 80 60

1E-5

1E-4

1E-3

0.01

0.1

Ca = vƞ/γL

liquid and the solid. This intermediate layer of air is unstable, and at some point, the liquid eventually comes into direct contact with the solid. Despite the fact that the viscosities differ by a factor of 7, the curves agree reasonably well up to capillary numbers of 0.1. The difference at low capillary numbers is due to the fact that the advancing contact angles differ by 8∘ . For high advancing velocity, wetting is usually limited by the entrainment of air [434, 442]. The entrainment of air is an important limit, for example when coating paper, metals, or textile fibers. In experiments with plunging tapes of fibers, air entrainment usually occurs around Ca ≈ 0.1 − 1. For water, typical maximal wetting velocities of 5–10 m/s are observed. Despite much research, our ability to quantitatively predict ΘD versus Ca is fairly limited. One reason for the poor understanding is that wetting involves length scales from the molecular to the macroscopic. Observations are, however, usually limited to several microns and larger. Second, depending on the materials and temperature, spontaneous or forced wetting, the capillary numbers easily span several orders of magnitude and different physical effects limit the wetting velocity. There are two main approaches to describing dynamic wetting theoretically (reviews are [419, 443]). They differ in how energy is being dissipated. Since spontaneous spreading and forced wetting involve a movement of the wetting line, both theories are applicable to both situations. In one approach, the contact line has to overcome local or atomic energy barriers. On the molecular scale and driven by thermal fluctuations, liquid molecules near the contact line are jumping from the liquid phase to binding sites on the solid surface or they are jumping from one binding site to the next overcoming energy barriers of the order of kB T. Thus, the movement of the contact line is a collective thermal motion of liquid molecules under the influence of the capillary driving force. The capillary driving force is 𝛾L (cos Θ0 − cos ΘD ). If the actual dynamic contact angle ΘD is larger than Θ0 , more liquid molecules move forward and the liquid front advances. For ΘD < Θ0 , more liquid molecules jump from the surface region into the liquid and

6.4 Dynamics of Wetting and Dewetting

the liquid front recedes. This idea has led to the molecular kinetic theory, which describes dynamic contact angles as a result of molecular adsorption and desorption processes at the moving contact line [444]. In the second approach, known as the hydrodynamic theory, continuum hydrodynamic theory is applied [423, 438–440, 445]. Energy dissipation due to hydrodynamic flow near the wetting line limits the wetting velocity. The microscopic contact angle is assumed to be constant and governed by short-range intermolecular forces. The change in dynamic contact angle is attributed to a bending of the liquid surface on the mesoscopic scale by viscous flow. There are two problems when applying the hydrodynamic theory: (i) Mathematically, an infinite stress occurs at the wetting line. This must be removed, for example by introducing local slip [438, 439]. (ii) In the theory, a parameter describing the transition between the microscopic and macroscopic scales (described by classical continuum theory) must be introduced. Still, molecular-kinetic and hydrodynamic theories or combinations of both [441] have been successfully applied to describe dynamic contact angles versus velocity [446]. Often, however, the parameters required to fit experimental curves are somewhat arbitrary or unrealistic. In particular for low velocities, it is doubtful that the underlying physical processes are really determining the contact angles. In such cases, the adaptation of surfaces to the presence of the liquid [363] or small elastic deformations [313] may influence dynamic contact angles. In addition, the kinetics of wetting can be influenced by evaporation and condensation at the contact line [447, 448].

6.4.3

Coating and Dewetting

In many applications, solid substrates are coated to protect them against corrosion, scratching, or wear, to improve their appearance, or to obtain specific adhesion and wetting properties. Coatings are usually applied as liquids [449]. The liquid contains the substance to be coated either dissolved or in the form of dispersed particles. After evaporation of the liquid, the dissolved substance or dispersed particles remain on the substrate. Good coating is achieved when the material forms a continuous film. Many coatings contain substantial amounts of polymers to provide the desired homogeneity and mechanical stability. To produce polymer films, the polymer is first dissolved or dispersed in a suitable, volatile solvent. Three methods are widely used to coat surfaces with polymer films [450] (Figure 6.23): ●

●

Dip coating [433]: The plate or fiber to be coated is immersed in a pool containing the solution. Then it is pulled out at a constant speed. The speed determines the layer thickness; a fast withdrawal leads to thick films. Excess solution is allowed to drain into the pool. The remaining solvent evaporates, leaving the polymer film behind on the plate. Thicknesses between 20 nm and 50 μm can be made by dip coating. Spin coating, also called spin casting [451–453]: The solid in the form of a flat disc is placed horizontally on a rotor. The liquid melt or solution is dropped onto the

163

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6 Contact Angle Phenomena and Wetting

Plate fiber Nozzle Liquid Plate

Liquid Dip coating

Spin coating

Spray coating

Figure 6.23 Dip, spin, and spray coating are common techniques to form polymer films on substrates.

●

rotating substrate. Due to the centrifugal force, the liquid wets the whole substrate uniformly. After the solvent has evaporated, a polymer film is left on the surface. For academic purposes, care should be taken to get rid of residual solvent by annealing in a vacuum. Using spin coating, very thin, homogeneous films can be made and the thickness can be adjusted with good precision. Spin coating is used, for instance to cover silicon wafers with photoresist in microcircuit fabrication. Disadvantages of spin coating are that a lot of material is wasted, it is a batch process, and only relatively small, symmetric plates can be coated. Spray coating: The solution is sprayed via a nozzle onto the substrate. After evaporation of the liquid, the dissolved or dispersed coating material is left on the substrate. Spray coating is a flexible and inexpensive coating technique in particular for large areas. The surface of the coating is, however, less homogeneous and smooth than for layers made by dip coating and in particular by spin coating.

A relevant question with respect to coating is: How stable are the polymer films? When the polymer wets the solid (Θ = 0∘ ), polymer films are thermodynamically stable. For Θ > 0∘ , the films are only metastable. When a thin, metastable film is heated above the glass transition temperature, dewetting may take place. Holes start to form spontaneously, usually at small defects. The holes increase in size until only a network of polymer lines is formed that eventually breaks up into individual droplets (Figure 6.24) [304]. The stability of films with a thickness of 1–100 nm is determined by long-range surface forces, mainly van der Waals forces [305, 450, 454].

6.5 Applications 6.5.1

Flotation

Flotation is a method to separate various kinds of solid particles from each other. It is of enormous importance to the mining industry, where it is used for large-scale processing of crushed ores [193, 455]. The desired mineral is separated from the gangue

6.5 Applications

200 μm

26 min

9h

33 h

Figure 6.24 Dewetting of a polystyrene film of 28 nm thickness from a silicon wafer at 121 ∘ C. At certain nucleation sites, which can be any kind of irregularities on the silicon surface, the film ruptures and the holes grow in size. Source: Courtesy of C. Lorenz.

Figure 6.25 Schematic of a flotation system. The bubbles are usually much larger than the particles so that the water–air interface is almost planar with respect to the particles.

Air

Concentrate

Froth

Air bubble

Propeller

Mineral particles

Pulp

or nonmineral-containing material. Originally, the procedure was applied only to some sulfides and oxides (iron oxides, rutile, and quartz). Meanwhile, many minerals such as gold, borax, pyrite, phosphate minerals, fluorite, calcite, and apatite are separated by flotation. Other important applications of flotation are the purification of coal, deinking of paper for recycling [456], and the removal of unwanted material in water purification [457]. In flotation of ore, the material is first crushed to particles with a size under 0.1 mm. The particles are mixed with water and form a sol. This sol is called pulp. The pulp flows into a container and air bubbles are passed through (Figure 6.25). The mineral-rich particles bind to the air bubbles by hydrophobic forces and are carried to the surface of the container. A stable foam, also called froth, is formed. With the froth, the mineral-rich particles can be skimmed off and removed. The wetting properties of the particles play a crucial role in flotation. We have already discussed the equilibrium position of a particle in a water–air interface (Section 6.3.2). The higher the contact angle, the more stably a particle is attached to a bubble Eq. (6.37), and the more likely it will be incorporated into the froth. Some minerals naturally have a hydrophobic surface and thus a high flotation efficiency. For other minerals, surfactants are used to improve the separation [455]. They are called collectors, which adsorb selectively on the mineral and render its surface hydrophobic. Activators support collectors. Depressants reduce a collector’s effect. Frothing agents increase the stability of the foam.

165

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6 Contact Angle Phenomena and Wetting

6.5.2

Detergency

Washing and cleaning in aqueous solutions is a complex process involving the cooperative interaction of different physical and chemical influences [458]. Washing laundry, household cleaning, and dishwashing all require the removal of dirt by a combined action of solubilization of water-soluble dirt, mechanical effort, thermal energy, and surfactants. Here we concentrate on the action of surfactants and take laundry as an example. Detergency is about the theory and practice of the removal of foreign material from solids by surface-active substances. In textiles, oily substances usually attach to the fibers (e.g. animal fats, fatty acids, and hydrocarbons). Also, dust, soot, and other solid particles must be removed in a washing process. To test the effectiveness of a surfactant, textiles are often polluted with standard dirt mixtures and cleaned with a standard washing procedure (Launderometer). Cleanliness can be measured on the basis of the optical reflectivity of white textiles. Dirt particles spontaneously leave a solid surface if it is energetically favorable to replace the dirt–solid interface (SD) by two interfaces: the dirt–aqueous solution interface (DW) and the solid–aqueous solution interface (SW). Here, the solid is a textile fiber or any other material that we want to clean (Figure 6.26). The change in the Gibbs free energy must be negative: ΔG = A(𝛾DW + 𝛾SW − 𝛾SD ) ≤ 0

(6.40)

Here, A is the contact area. This condition can be simplified to 𝛾SD ≥ 𝛾DW + 𝛾SW

(6.41)

In this estimation, we have assumed that the shape of the dirt does not change upon removal. For liquid dirt, like grease or oil, the contact angle, given by cos Θ = (𝛾SW − 𝛾SD )∕𝛾DW , should be as high as possible. We can formulate a general requirement for a good surfactant: an effective surfactant should reduce 𝛾SW and 𝛾DW , without γDW γSW

Dirt

γSD

γDW γSW

Dirt

γSD

Textile fiber

(a)

(b)

Figure 6.26 Removal of solid (a) and liquid (b) dirt from a textile fiber.

6.5 Applications

reducing 𝛾SD too strongly. A decrease in the surface tension of water – visible by bubble formation – is not a proof of an effective surfactant for detergency. An important characteristic of a surfactant is its ability to keep dirt particles in solution (suspending power). Without this ability, a washing procedure would only lead to a uniform distribution of the dirt. Obviously, the surfactant is bound to the surface of the dirt particles and keeps them dispersed. Aggregation and flocculation are prevented, for instance, by electrostatic repulsion. With respect to liquid dirt, it was originally assumed that the cleaning effect of surfactants was caused by their ability to accommodate hydrophobic substances inside micelles (surfactant aggregates; see Section 11.2). However, this does not seem to be the dominating factor because the cleaning ability already increases with surfactant concentrations below the CMC, thus before micelles are formed.

6.5.3

Microfluidics

Microfluidics is about the flow of tiny amounts of liquids. The prefix micro indicates that, at least in two dimensions, the liquid should be confined in micrometer dimensions. If we are dealing, for instance, with a channel, its width or diameter should be much below 1 mm to earn the title microchannel. A microliter is already a relatively large volume in microfluidics since it is equal to the volume of 1 mm3 . Microfluidics has become an important field of research and development mainly driven by biological and analytical applications [459, 460]. It allows us to analyze small amounts of substances. For this reason, it is widely used in DNA and protein analysis. Miniaturization is also attractive in chemical synthesis, and significant effort is put into making chemical reactions on a microchip (lab-on-chip). Advantages are that high concentrations can be reached even with tiny amounts of substances. Heat is dissipated fast so that reactions occur practically at constant temperature. This reduces the danger of overheating and explosions. The flow is always laminar because the Reynolds3 number Re =

𝜌vd 𝜂

(6.42)

is much lower than one. Here, v is its average velocity and d is a typical dimension of fluid confinement. When trying to manipulate fluids on the micron scale, formidable problems must be overcome. Miniature pumps, valves, switches, and new analytical tools must be developed. To illustrate this, we discuss the fundamental problem of transporting a liquid through a cylindrical capillary tube. In the macroscopic world, we would apply a pressure ΔP between the two ends. According to the law of Hagen–Poiseuille, the volume of liquid V transported per time t is (assuming laminar flow) 𝜋r 4 ΔP V = C t 8𝜂L

(6.43)

3 Osborne Reynolds, 1842–1912, professor of engineering at Owens college, now University of Manchester.

167

168

6 Contact Angle Phenomena and Wetting

B

3 C

A

Sample pulling

Absorption at 210 nm

D

2 5

1

Sample injection

Detector

4

5

10 Time (min)

15

Figure 6.27 Simple electrophoretic microfluidic system (Example 6.11).

Here, rC is the radius and L the length of the capillary tube. We see that, due to the strong dependence on the radius, we would have to apply enormous pressure to maintain a significant flow. For this reason, sometimes electro-osmosis (Section 4.4.2) is used to drive a flow of aqueous solutions in microfluidics. Example 6.11 A simple microfluidic electrophoretic device is shown in Figure 6.27. The device allows us to separate different platinum complexes used in anticancer therapy [461]. In our case, five platinum complexes were separated. The electrophoretic device consists of reservoirs for the sample (A), buffer (B), and waste (C) and (D). The channels in quartz capillaries were 70 μm in diameter. The aqueous solution contained 50 mM SDS, 25 mM Na2 B4 O7 , 50 mM NaH2 PO4 at pH 7.0 in addition to the analyte. Applying a positive voltage at A and a negative voltage at C leads to an electro-osmotic flow from A to C because quartz (SiO2 ) is negatively charged at neutral pH (Table 4.1). A small plug of sample solution is drawn into the separation channel (BD). Then the voltage between A and C is switched off, and a voltage between B and D is applied. B is positively charged, D is negative. The platinum complexes are carried with the osmotic flow toward D. Superimposed is the electrophoretic drift, which separates the platinum complexes. Unfortunately, platinum complexes are usually not charged in water and no electrophoresis occurs. For this reason, the author used a trick and added SDS. The platinum complexes are enclosed in negatively charged SDS micelles. Depending on their precise structure, the micelles have a different size and charge. In this way, a mixture can be separated into its components. Molecules were detected by optical adsorption at a wavelength of 210 nm.

6.5.4

Electrowetting

For many applications, it is desirable to be able to adjust the wetting properties of a solid surface for aqueous solutions. One method to achieve this is called electrowetting [462–464]. In electrowetting, an electric potential is applied between

6.6 Thick Films: Spreading of One Liquid on Another

0V

U Oil

Insulating Drop layer

h 0

Contact angle (˚)

160 140

43 mN/m

120 100

20 mN/m

80 60

Conducting material (a)

0 (b)

20

40

60

80

Voltage (V)

Figure 6.28 (a) Electrowetting experiment of a liquid drop on a metal coated with a thin insulating layer. (b) Two measurements (symbols, [465]) and fits with Eq. (6.44) described in Example 6.12.

a metal surface and a liquid via an electrode (Figure 6.28). The metal is coated with an insulating layer of thickness h. Fluoropolymer coatings turned out to be suitable materials. During the last decade, their thickness was reduced to a few tens of nanometers. The change in contact angle is described by cos Θ(U) = cos Θ0 +

𝜀𝜀0 U 2 2h𝛾L

(6.44)

Here, Θ0 is the contact angle without applied voltage, 𝜀 is the dielectric permittivity of the insulating polymer film, and U is the applied voltage. The sign of the applied potential is not relevant. The second term is always positive, and thus the contact angle decreases when a potential is applied. Usually, the aqueous drop is submerged in an oil to reduce contact angle hysteresis. Then, the interfacial tension of the water–oil interface, 𝛾WO , has to be inserted rather than 𝛾L . Example 6.12 To see the influence of interfacial tension on electrowetting, Chevalliot et al. [465] coated a glass plate with a conducting layer of ITO and a 1.3 μm thick insulating layer of fluorinated Parylene. The cell was immersed in a mixture of silicone oils. About 1 μL aqueous drops containing 0.02 wt% NaCl and the neutral surfactant Triton X102 were placed on the Parylene. The interfacial tension was varied by adjusting the surfactant concentration. A DC voltage was applied via a Pt electrode to the drop; ITO was grounded. The contact angle was measured with a video camera in side view. The contact angle decreased with increasing applied voltage as predicted by Eq. (6.44). When reducing 𝛾WO , the slope of the graph increases. It is not yet clear, why at some contact angle (≈ 57∘ ), the graphs saturate.

6.6 Thick Films: Spreading of One Liquid on Another What happens when we replace the solid surface by a second liquid? Let us put a drop of liquid B on the surface of liquid A in their vapor (Fluid C). The two liquids

169

170

6 Contact Angle Phenomena and Wetting

are assumed not to mix and the density of B should be lower than the density of A, that is, the case for many oils on water. There are two possibilities: either liquid B spreads or it forms a lens with a defined edge. Just as for solid substrates, liquid B spreads spontaneously on liquid A, if the spreading coefficient, also called spreading pressure SA∕B = 𝛾A − 𝛾B − 𝛾AB

(6.45)

is bigger than zero (see also Eq. (6.11)). Here, 𝛾A and 𝛾B are the surface tensions of the pure liquids and 𝛾AB is the interfacial tension between the two liquids. The condition results from a simple energy balance: for SA∕B > 0, it is energetically favorable for liquid B to spread. More precisely, the Gibbs energy of the system decreases if liquid B spreads on A and forms a homogeneous film. A complication now arises. 𝛾A and 𝛾B are values for the pure liquids. In a real situation, at least a tiny amount of molecule A is dissolved in liquid B, and vice versa. Therefore, we should actually use 𝛾A(B) and 𝛾B(A) instead of 𝛾A and 𝛾B . Here, 𝛾A(B) is the surface tension of liquid A, in which molecules B are dissolved up to saturation. For simplicity, we continue using the first notation. If the spreading coefficient is lower than zero, then a well-defined drop of liquid B forms on the surface of A. The shapes of the surface of liquid A, the liquid B surface, and the A–B interface are determined by the Laplace equation. The angles Θ1 , Θ2 , and Θ3 (Figure 6.29a) are related by 𝛾A(B) cos Θ3 = 𝛾B(A) cos Θ1 + 𝛾AB cos Θ2

(6.46)

Like Young’s equation, Eq. (6.46) can be derived from a force balance. At equilibrium, the contact line does not move. Hence, the sum of the forces acting in the horizontal direction must be zero. The surface of liquid A pulls the contact line outward with a force (per unit length) of 𝛾A(B) cos Θ3 . The surface of liquid B pulls the contact line inward with a force 𝛾B(A) cos Θ1 , as does the liquid A–liquid B interface 𝛾AB cos Θ2 . Balancing these three forces directly leads to Eq. (6.46). Θ1 , Θ2 , and Θ3 are measured with respect to the horizontal. Setting Θ2 = Θ3 = 0∘ leads to Young’s equation. Often, one is more interested in the angles between the Fluid C 3 1

Liquid B 2

Liquid A

(a)

γAC

γCB γAC

C B A

(b)

γAB

γAB (c)

γCB

Figure 6.29 A drop of liquid B (e.g. oil) on the surface of an immiscible liquid A (e.g. water) in a fluid C. Fluid C is typically the vapor phase.

6.6 Thick Films: Spreading of One Liquid on Another

Figure 6.30 Drop volume (a) and spinning drop tensiometers (b) to measure the interfacial tension between two immiscible liquids.

Liquid B 2r Liquid A (b) (a)

2rc

interfaces, ΘA , ΘB , and ΘC (Figure 6.29b). They can be derived by the so-called Neumann triangle. In the Neumann triangle, the three interfaces are represented by vectors with a length proportional to their interfacial tensions and a direction parallel to the respective interface (Figure 6.29c). In equilibrium, the sum of the three vectors is zero since the net force acting on the contact line needs to cancel. Using the sine rule, we obtain [21, 399]: 𝛾AC 𝛾 𝛾 = AB = BC (6.47) sin ΘB sin ΘC sin ΘA To measure interfacial tension between two immiscible liquids 𝛾AB , most of the methods described in Section 2.4 for liquid–vapor interfaces can be adapted. The vapor phase is replaced by the second liquid. For example, the equivalent of the drop weight method is a drop volume tensiometer (Figure 6.30a) [466, 467]. Liquid of one density is pumped through a capillary of radius rC into a second liquid of a different density and the time between drops produced is measured. If the interfacial tension is high, drops are large, while for a low interfacial tension, the drops are small. From the flow rate and the number of drops, the mean volume of a drop Vd can be determined. The interfacial tension is then given by 𝛾AB =

Vd (𝜌A − 𝜌B )g 2𝜋rC

(6.48)

The drop volume tensiometer is often used to analyze dynamic effects like adsorption kinetics. One method that is able to measure very low interfacial tensions is the spinning drop method (Figure 6.30b). The liquid with the higher density – let it be liquid A – is filled into a horizontal capillary. Through a septum, which closes the capillary, a drop of liquid B is injected with a syringe into the center of the capillary. When the capillary is rotated rapidly, liquid B will go to the center, forming a drop around the axis of revolution. With increasing speed of revolution, the drop elongates since the centrifugal force increasingly opposes the surface tensional drive toward minimum interfacial area. At sufficiently high speed of revolution, the drop is shaped like an elongated cylinder. From the radius of the cylinder r and the angular rotation frequency 𝜔, we can calculate the interfacial tension [468]: 𝛾AB = 0.25(𝜌A − 𝜌B )𝜔2 r 3 . The spinning drop method has a high sensitivity of up to 10−6 mN/m. It is frequently

171

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Table 6.1 Interfacial tensions 𝛾AB of some liquids with water at 25 ∘ C [470, 471]. 𝜸 AB

𝜸 AB

(mN/m)

(mN/m)

n-Heptane

50.2

Cyclohexane

50.2

n-Octane

50.8

Toluene

36.1

n-Decane

51.2

Hexanol

n-Dodecane

52.8

n-Perfluorohexane

57.2

Dichloromethane

52.8

n-Perfluoroheptane

50.6

6.8

used in emulsion technology to measure the effect of surfactants on interfacial tension [469]. Examples of interfacial tensions of some organic liquids with water are listed in Table 6.1.

6.7 Summary ●

For smooth, homogeneous, inert surfaces, Young’s equation relates the contact angle to the interfacial tensions 𝛾S , 𝛾L , and 𝛾SL : 𝛾L cos Θ = 𝛾S − 𝛾SL

●

●

●

●

●

●

The work required to create a new three-phase contact line per unit length is called line tension. It is typically in the order of 0.1 nN. For tiny liquid drops, the line tension can significantly influence the wetting behavior. A liquid rises spontaneously in a lyophilic (Θ < 90∘ ) capillary. The height increases with decreasing capillary radius. From lyophobic (Θ > 90∘ ) capillaries, a liquid is expelled. Contact angles are commonly measured by the sessile drop, the captive bubble, and the Wilhelmy plate method. To characterize the wetting properties of powders, the capillary rise method is used. In many applications, the advancing and receding contact angles are different. Contact angle hysteresis can be caused by surface roughness, heterogeneity, dissolved substances, and structural changes of the solid at the three-phase contact line. For large contact angle hysteresis, contact line pinning, and thus the history of the system determines the macroscopic contact angle rather than Young’s equation. In the vicinity of the contact line, in the so-called core region, surface forces such as van der Waals or electric double-layer forces may deform the liquid surface. Considering the range of surface forces the typical size of the core region is of the order of 10–100 nm. The macroscopic contact angle is measured outside the core region and is accessibly by optical microscopy.

6.8 Exercises ●

When setting an immiscible liquid on top of another liquid, a macroscopic thick film or a drop forms, depending on the interfacial tensions. This is quantified using the spreading coefficient.

6.8 Exercises 6.1

A specific water-based paint (𝛾 = 62 mN/m, 𝜌 = 1050 kg/m3 ) has a contact angle of 60∘ on a certain steel. How large an area can you coat on a horizontal plate with 100 mL? In general, adding surfactant decreases the contact angle. A specific surfactant is supposed to reduce the contact angle to 20∘ and the surface tension to 𝛾 = 38 mN/m. How much does the coated area increase? Recall Eq. (6.34). For simplicity, assume that the rim is an edge rather than being curved.

6.2

A small drop is placed on a planar solid surface. Verify that the upward component of the surface tensional force along the three-phase contact line is just compensated by the Laplace pressure acting downward on the contact area.

6.3

A hydrophilic capillary (Θ = 0∘ ) with an inner radius of 1 mm is vertically lowered into a reservoir of water. How high does the water rise? We repeat the experiment with the same capillary filled with a powder of hydrophilic, spherical particles of 5 μm radius. Estimate the height of the liquid column once the system has reached equilibrium. For simplicity, we assume that the particles are densely packed.

6.4

A small drop is placed on a solid surface. Its contact angle is 90∘ for a contact radius of 0.5 mm. Its line tension is κ l = 0.5 nN and its surface tension 50 mN/m. How does the contact angle change depending on the contact radius a? How does the contact angle change for a negative line tension of −0.5 nN? Plot Θ–vs–a for both cases.

6.5

In a classic paper, Owens and Wendt estimated the surface energy of polymers by measuring the contact angle with water and methylene iodide [341]. They measured:

𝚯 with water (∘ )

𝚯 with methylene iodide (∘ )

Polyethylene

94

52

Polyvinylchloride

87

36

Polystyrene

91

35

Poly(methyl methacrylate)

80

41

173

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6 Contact Angle Phenomena and Wetting

The dispersive and polar components of the surface tensions of the liquids p were estimated to be 𝛾Ld = 21.8 mN∕m and 𝛾L = 51.0 mN∕m for water and p d 𝛾L = 49.5 mN∕m and 𝛾L = 1.3 mN∕m for methylene iodide. This estimation was done by measuring contact angles with various hydrocarbons and assuming that there were only nonpolar interactions. What are the surface energies, 𝛾S , of the polymers? 6.6

Water drops of 2, 5, and 10 μL volume are placed on a superhydrophobic plate with Θr = 150∘ . To make sure the drops roll off, how much would one need to incline the plate?

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7 Solid Surfaces 7.1

Introduction

Molecules at a surface are arranged in a different way from molecules in the bulk. This statement is, in general, true for solids and liquids. There is, however, a major difference: when a liquid is deformed, no barrier prevents molecules from entering or leaving the surface. In the new equilibrium state, each molecule covers the same area as in the original undistorted state. The number of molecules in the surface has changed, but the area per molecule remains the same. Such a deformation is called plastic. If a solid is deformed by small external forces, it will react elastically. A new surface area of a solid can be created by stretching. Since the molecules are not mobile, the number of molecules usually remains constant, but the area occupied by one molecule increases. The shape of the solid was generated in the past, and it is usually not determined by the surface tension. Individual atoms and molecules are only able to vibrate around their mean position but are otherwise fixed to a certain site. This elastic increase in surface area applies, however, only within limits. Many solids are somewhat mobile and can flow very slowly. In that case, methods and models of capillarity can be applied. One case where capillarity plays an important role is sintering. In sintering a powder is heated. At a temperature of roughly two-thirds of the melting point, the surface molecules become mobile and can diffuse laterally. In that way the contact areas of neighboring particles melt and menisci are formed. When cooling, the material solidifies in this new shape and forms a continuous solid. In this chapter, we mainly deal with the microscopic structure of crystalline solid surfaces (introduction: [472]) and with the methods used to analyze their structure and the chemical composition (introductions are [473, 474]). Most solids are not crystalline on their surface. This is certainly true for amorphous solids. It is also true for many crystalline or polycrystalline solids because for many materials, the molecular structure at the surface is different from the bulk structure. Many surfaces are, for example, oxidized under ambient conditions. A prominent example is aluminum, which forms a hard oxide layer as soon as it is exposed to air. Even in an inert atmosphere or in an ultrahigh vacuum (UHV), the surface molecules might form an amorphous layer on the crystalline bulk solid. Physics and Chemistry of Interfaces, Fourth Edition. Hans-Jürgen Butt, Karlheinz Graf, and Michael Kappl. © 2023 WILEY-VCH GmbH. Published 2023 by WILEY-VCH

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7 Solid Surfaces

Since many natural surfaces are amorphous, it may seem somewhat academic to study crystalline surfaces. But there are good reasons to do so: ●

●

●

The well-defined structure of crystalline surfaces allows for a comparison of experiments on different samples of the same material. The periodic structure of crystalline surfaces facilitates theoretical description and allows us to use powerful diffraction methods to analyze them. Crystalline surfaces are important in the semiconductor industry. Many modern semiconductor devices depend on the defined production of crystalline surfaces.

For the investigation of clean crystalline surfaces, UHV must be used. To keep a surface clean for time periods of hours, a gas pressure below 10−7 Pa is required (Exercise 2.1). Providing such conditions is quite demanding. A textbook on vacuum technology is [475]. A review on the history and current developments in vacuum technology can be found in [476].

7.2

Description of Crystalline Surfaces

7.2.1

Substrate Structure

Let us start with the simple case of an ideal crystal with one atom per unit cell. A surface is obtained by cutting this crystal. We are interested in cuts, which lead to crystal planes because they are usually more stable than completely random cuts. We further assume that the relative arrangement of atoms does not change. The resulting surface structure can be described by specifying the bulk crystal structure and the relative orientation of the cutting plane. This ideal surface structure is called the substrate structure. The orientation of the cutting plane and, thus, of the surface is commonly notated by use of the so-called Miller1 indices. Miller indices are determined in the following way (Figure 7.1):2 The intersections of the cutting plane with the three crystal axes are expressed in units of the lattice constants. Then the inverse values of these three numbers are taken. This usually leads to noninteger numbers. All numbers are multiplied by the same multiplicator to obtain the smallest possible triple of integer numbers. The triple of these three numbers h, k, and l is written as (hkl) to indicate the orientation of this plane and all parallel planes. Negative numbers are written as n̄ instead of −n. The notation {hkl} is used to specify the (hkl) planes and all symmetrical equivalent planes. In a cubic crystal, for example (100), (010), and (001) are all equivalent and summarized as {100}. For crystals with a hexagonal structure, we have three equivalent a axes under relative angles of 60∘ and one c axis perpendicular to them, corresponding to four lattice vectors. Correspondingly, four Miller indices (h, k, i, l) are commonly used 1 William H. Miller, 1801–1880; British crystallographer. 2 A more general definition of the Miller indices is given in the appendix on diffraction in the context of the reciprocal lattice.

7.2 Description of Crystalline Surfaces

Figure 7.1 Notation of a cutting plane by Miller indices. The three-dimensional crystal is described by the three-dimensional unit cell vectors a1 , a2 , and a3 . The indicated plane intersects the crystal axes at the coordinates (3,1, 2). The inverse is (1∕3,1∕1,1∕2). The smallest possible multiplicator to obtain integers is 6. This leads to the Miller indices (263).

a3 a2 a1

(100)

(110)

(111)

Figure 7.2 Low-index surfaces for a face-centered cubic crystal.

for the hexagonal system. They are then called Miller–Bravais indices. Note that the fourth index i is related to the first two indices by i = −(h + k). In surface science, often low-index surfaces, that is crystal surfaces, with low Miller indices, are of special interest. Figure 7.2 shows the three most important low-index surfaces of a face-centered cubic lattice. The (100) is equivalent to the (010) and the (001) surfaces, whereas the (110) is equivalent to the (011) and the (101) surfaces. Crystalline surfaces can be divided into five Bravais lattices (Figure 7.3) according to their symmetry. They are characterized by the lattice angle 𝛼 and the lengths of the lattice vectors a1 and a2 . The position vectors of all individual surface atoms can be indicated by r = na1 + ma2

(7.1)

Here, n and m are integers.

7.2.2

Surface Relaxation and Reconstruction

Atoms at solid surfaces have missing neighbors on one side. Driven by this asymmetry, the topmost atoms often assume a structure different from the bulk. They might form dimers or more complex structures to saturate dangling bonds. In the case of a surface relaxation, the lateral or in-plane spacing of the surface atoms remains

177

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Figure 7.3 The five two-dimensional Bravais lattices. Square a1 = a2, α = 90°

Rectangular a1 ≠ a2, α = 90°

Centered rectangular a1 ≠ a2, α = 90°

α Hexagonal a1 = a2, α = 120°

Unreconstructed fcc(110) surface

Figure 7.4

Oblique a1 ≠ a2, α ≠ 90°

Missing-row (MR) reconstructed

Pairing-row (PR) reconstructed

Typical reconstructions of face-centered cubic (110) surfaces.

unchanged, but the distance between the topmost atomic layers is altered. In metals, for example, first layer is typically 1–10% closer to the bulk than expected from the lattice constant [477]. The reason is the presence of a dipole layer at the metal surface that results from the distortion of the electron wave functions at the surface. When the distances of the atoms are changed, this is called surface reconstruction. Reconstruction is, for instance, observed with the (100) faces of Au, Ir, Pt, and W. Figure 7.4 shows two types of surface reconstructions that lead to a doubling of the lattice spacing in one direction. Semiconductor surfaces tend to exhibit surface reconstruction due to the directional character of the dangling bonds at the surface. For reconstructed crystalline surfaces, the position vectors of the atoms must be described by new unit cell vectors b1 and b2 : r = n′ b1 + m′ b2

(7.2)

Often, there is a relationship between a1 , a2 and b1 , b2 of the form b1 = pa1

and b2 = qa2

with p and q being integers. The surface structure is denoted in the form: A(hkl)(p × q) where A is the chemical symbol of the substrate.

(7.3)

7.2 Description of Crystalline Surfaces

Top layer (adatoms) Second layer Third layer

10 nm

Figure 7.5 (a) Schematic of 7 × 7 reconstruction of Si(111) surface. (b) Image of a Si 7 × 7 surface obtained with a combination of STM and a special noncontact AFM mode [480] (Section 7.7.3). Only the adatoms are visible under the chosen imaging conditions. Single defects of the surface structure are resolved (circles). Source: Picture courtesy of E. Meyer. Basis

Basis

Surface

Surface

Bulk solid

Bulk solid Substrate structure

Reconstructed surface

Figure 7.6 Example of necessary extension of basis in the case of surface atoms positioned at different heights.

Example 7.1 Gold (111) is an example of a surface without reconstruction. It is denoted by Au(111)(1 × 1). The (100) surface of silicon exhibits a (2 × 1) reconstruction and is denoted by Si(100)(2 × 1). The 7 × 7 surface reconstruction of Si(111) (Figure 7.5) is another example of a complex surface reconstruction [478, 479]. In general, the surface structure can be more complex as every lattice site may be occupied by more than one atom. This can be described by the so-called basis that indicates the fixed relative orientation of the atoms or molecules. In this case, the lattice type and the basis must be given for a complete description of the crystal structure. In addition, it is possible that the surface atoms will lie at different heights (Figure 7.6), and then the basis must be extended to more than just the surface layer of atoms.

7.2.3

Description of Adsorbate Structures

If molecules adsorb to a crystalline surface, they will often form a crystalline overlayer called a superlattice. This occurs when the adsorbates bind preferentially

179

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7 Solid Surfaces

Au

Rh

(a)

(b)

Pd

(c)

√ √ Figure 7.7 Adsorbate superlattices:√(a) Rh(110)(1 × 1)-2H; (b) Au(111)( 3 × 3) √ R 30∘ –CH3 (CH2 )n SH; (c) Pd(100)c(2 2 × 2) R45∘ –CO.

to specific sites. Then the positions of the adsorbed molecules can be written as follows: rad = n′′ c1 + m′′ c2

(7.4)

Again, n′′ and m′′ are integers. Usually, there are fewer adsorbed molecules than there are atoms in the underlying crystal surface. Therefore, the unit cell formed by the vectors c1 and c2 is often larger than the unit cell of the underlying crystal lattice described by b1 and b2 . The structure of the adsorbate lattice is expressed by the ratios of the lengths of the unit cell vectors: p′ = c1 ∕b1 and q′ = c2 ∕b2 . When the adsorbate lattice is rotated with respect to the underlying substrate lattice by an angle 𝛽, the value of 𝛽 preceded by an “R” is indicated. All this together leads to the so-called Wood’s notation A(hkl)c(p′ × q′ )R𝛽 − B Here, A and B are the chemical symbols of substrate and adsorbate, respectively. The letter “c” is added in case of a centered unit cell of√ the adsorbate. As examples, √ the (1 × 1)-2H structure of hydrogen on Rh(110), the ( 3 × 3) R 30∘ structure √ √of chemisorbed alkanethiols on Au(111) (like Xe on graphite), and the c(2 2 × 2) R 45∘ adsorption structure of CO on Pd(100) (the latter occurring at a surface coverage of 𝜃 = 0.5) are shown in Figure 7.7 [481–484].

7.3

Preparation of Clean Surfaces

To prepare crystalline surfaces, usually the starting material is a suitable, pure, three-dimensional single crystal. From this crystal, a slice of the desired orientation is cut. Therefore, the crystal must be oriented. Orientation is measured by X-ray diffraction. Hard materials are then grounded and polished. Soft materials may be cleaned chemically or electrochemically. The surfaces are still mechanically stressed, contaminated, or chemically changed, for example oxidized. In principle, electrochemical processes in liquid can be used to generate clean crystalline surfaces. The problem is that an electrochemical setup is not compatible with a UHV

7.3 Preparation of Clean Surfaces

environment. There are, however, combined UHV/electrochemistry (UHV-EC) instruments, where the transfer of electrochemically treated samples into the UHV chamber without contact with air is possible [485]. Still, the sample surfaces may undergo structural or compositional changes during the transfer. Usually, in situ methods of surface preparation are preferred that can be applied within the UHV chamber. In principle, there are two possible strategies: one can either try to clean an existing crystalline surface or create a fresh surface. Methods for both approaches are described in the following.

7.3.1

Thermal Treatment

Heating of a material may cause desorption of weakly bound species from the surface and can therefore be used to clean surfaces. A positive side effect is that annealing reduces the number of surface defects since it increases the diffusion rates of surface and bulk atoms. There can also be some unwanted side effects: surface melting and other types of phase transitions may occur well below the bulk melting point, leading to other than the desired surface structure. Example 7.2 To obtain a clean tungsten W(110) surface, the crystal is heated to 1600 ∘ C for about one hour in the presence of 10−6 mbar of oxygen. This oxidizes surface contaminations. A consecutive flash heating to 2000 ∘ C leads to decomposition and desorption of the oxide layer.

7.3.2

Plasma or Sputter Cleaning

In plasma cleaning, a plasma is used to remove contaminants or surface layers. A plasma is an overall electrically neutral gas of electrons and ions. Depending on operating conditions and applications, this type of cleaning process is also called glow discharge cleaning, plasma etching, (plasma) sputtering, or plasma ashing (introduction: [486]). For plasma treatment, the sample is placed inside a chamber and the pressure of the process gas used for the plasma is lowered to about 0.1–1 mbar (10–100 Pa). In any gas and at any given moment, a certain number of atoms are already ionized (although very few), for example by cosmic background radiation. The preexisting free electrons are accelerated by an applied external DC or AC electric field. If the pressure within the vacuum chamber is sufficiently low to allow the electrons to gain enough kinetic energy before they hit the next atom, they will excite and ionize additional gas atoms. In this way, more and more electrons are liberated, and these in turn further ionize other atoms. Eventually, a plasma is created within the chamber. Excited atoms or ions within the plasma will lose energy by emission of photons. Due to this light emission from the plasma, this process is sometimes also called glow discharge cleaning. DC voltages can be used for plasma treatment of conductive surfaces. For insulators, AC voltages must be used to avoid sample charging, which would repel ions and disrupt the sputtering process. The impact of electrons and ions onto the sample surface can lead to sample heating, removal of atoms from the surface by ion

181

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7 Solid Surfaces

impact, and emission of secondary electrons, which help to maintain the plasma. When noble gases are used as process gases to create the plasma, sputtering, that is, removal of surface atoms by ion impact, will be the dominant process. Reactive species like oxygen will lead not only to additional chemical etching of the surface and much faster removal rates but also to changes of the surface chemistry, for example, an increased density of OH groups at the sample surface. Organic contaminants may be removed very efficiently by oxygen or air plasma. Here, oxygen radicals created in the plasma react with organic contaminants, forming H2 O, CO, CO2 , or low-molecular-weight hydrocarbons, which are volatile and easily removed by vacuum pump. One application of oxygen plasma cleaning is the removal of photoresist layers, where the oxygen reacts with the organic photoresist layer, reducing it to an ash layer that can be rinsed off. This process is therefore known as plasma ashing. After sputtering, annealing is often necessary to remove adsorbed or embedded noble gas atoms and to heal defects of the crystal surface structure created by the bombardment. Sputtering is an excellent and versatile surface-cleaning technique for elemental materials. Care must be taken when applying it to composite materials like alloys. Sputtering rates depend on the component and lead to changes in surface stoichiometry. In these cases, cleavage may be the better choice. The necessary annealing process can also be critical and even induce surface contamination. A prominent example is iron, which will usually contain some sulfur that segregates at the surface during annealing. With such materials, consecutive cycles of annealing and sputtering may be required to obtain a clean crystalline surface. An in-depth discussion of sputtering can be found in [487, 488]. As described previously, plasma treatment of surfaces is usually carried out within closed vacuum chambers. Sample loading and vacuum pumping allow only batchwise processing. However, for industrial applications, inline processes with high throughput are preferred. This has led to the development of atmospheric pressure plasma jets [489] for surface treatment.

7.3.3

Cleavage

Cleavage of bulk crystals to expose clean, defined lattice planes is possible for brittle substances. Some materials, like mica or highly oriented pyrolytic graphite (HOPG), that exhibit a layered structure are readily cleaved by just peeling off some layers or using a razor blade. For other materials, the so-called doublewedge technique can be used (Figure 7.8). Cleavage is a quick way to produce fresh surfaces from brittle or layered materials. There are, however, limitations: cleavage may produce metastable surface configurations that are different from the equilibrium structure. Depending on the material, only certain cleavage planes can be realized. Crystals usually cleave along nonpolar faces so that positive and negative charges compensate within the surface. GaAs, for example, can only be cleaved along the nonpolar {110} planes, whereas the polar {100} and {111} faces cannot be obtained.

7.4 Thermodynamics of Solid Surfaces

Figure 7.8 Double-wedge technique for cleavage of brittle materials. To ensure proper cleavage, the crystal is precut and positioned between the wedges according to its crystallographic orientation.

7.3.4

Deposition of Thin Films

An alternative approach to creating fresh clean surfaces is the deposition of thin films on top of an existing substrate. There exists a broad variety of methods to create high-quality surface by thin-film deposition such as physical vapor or chemical vapor deposition or molecular beam epitaxy. These methods are described in more detail in Chapter 9.

7.4

Thermodynamics of Solid Surfaces

7.4.1

Surface Energy, Surface Tension, and Surface Stress

In our description of liquid surfaces, surface tension was of fundamental importance. When we try to extend the definition of surface tension to solids, a major problem arises [490]. When the surface of a liquid increases, the number of surface atoms increases in proportion since the relaxation processes are very quick without showing a yield value. Hence, the area per molecule ΣA in the liquid surface remains constant, but the number of surface molecules N increases. For a solid surface, this plastic increase of the surface area is not the only possible process since for solids the relaxation usually occurs much slower. For a solid, we also have an elastic increase in the surface area. When the solid surface is increased by stretching it elastically, the distance between neighboring surface molecules changes, and thus the area per molecule changes while the number of surface atoms remains constant. The parameters for characterizing changes in surface energetics are therefore ΣA and N.

183

184

7 Solid Surfaces

The change in surface area is commonly described in terms of the surface strain 𝜀tot . Its change is defined by the change in surface area divided by the whole surface area A: dA d𝜀tot = = d𝜀p + d𝜀e (7.5) A The surface strain is divided into a plastic strain d𝜀p and an elastic strain d𝜀e . Let us consider this a bit more in depth. An excess energy ES is associated with each surface molecule. By dividing this excess energy by the molecular surface area, we get the surface energy ES ∕ΣA . For any reversible process, the work done to create a new surface area that can be written as the product of a generalized force and a generalized displacement. We call the former generalized surface-intensive parameter or surface energy 𝛾 S . The latter is taken as the increase in surface area dA. The work done is dW = 𝛾 S dA. It is equal to the increase in surface energy d(ES N). We thus have 𝜕ES 𝜕N dW = 𝛾 S dA = d(ES N) = ES dA + N dA (7.6) 𝜕A 𝜕A or 𝜕ES 𝜕N 𝛾 S = ES +N (7.7) 𝜕A 𝜕A To interpret 𝛾 S , we first consider the case of plastic and elastic changes in surface area separately before we turn to the general case. It must be emphasized that by plastic we mean a reversible change in surface area by bringing new atoms to the surface without changing their relative distances. This is different from the meaning of plastic deformation in the sense of solid mechanics, where it usually denotes an irreversible change of the material. For a plastic change in surface area, ΣA is constant since the surface conformation does not change. The change in surface area is given by dA = ΣA dN, and we can write ( ) 𝜕ES 𝜕ES | 𝜕ES | = 1 = =0 (7.8) 𝜕A pla 𝜕(NΣA ) ||ΣA ΣA 𝜕N This expression must equal zero since the energy per surface molecule does not depend on the total number of molecules at the surface. The index “pla” indicates that we are considering a purely plastic process. For the surface-intensive parameter, we obtain ( ) E 𝜕A∕ΣA || 𝜕N S 𝛾pla = ES = ES = S ≡𝛾 (7.9) | 𝜕A pla 𝜕A |ΣA ΣA In the case of a plastic change in surface area, the change in the surface-intensive parameter is like that of a liquid. We therefore call it surface tension and denote it by the symbol 𝛾. For an elastic deformation, N remains constant. A change in surface area is given by dA = NdΣA . Since the number of molecules does not change with a change in surface area, 𝜕N∕𝜕A = 0, and we write Eq. (7.7) as follows: ( ) 𝜕ES 𝜕ES N 𝜕ES || S 𝛾ela = N = = (7.10) 𝜕A ela N 𝜕ΣA ||N 𝜕ΣA

7.4 Thermodynamics of Solid Surfaces

Inserting the result of Eq. (7.9) into Eq. (7.10) leads to S 𝛾ela =

𝜕(𝛾ΣA ) 𝜕𝛾 = 𝛾 + ΣA 𝜕ΣA 𝜕ΣA

(7.11)

For an elastic process d𝜀e ≡ dA∕A = dΣA ∕ΣA , and we get S 𝛾ela =𝛾 +A

𝜕𝛾 𝜕𝛾 =𝛾+ 𝜕A 𝜕𝜀e

(7.12)

The change in surface-intensive parameter equals the sum of the surface tension 𝛾 and its derivative with respect to the elastic surface strain 𝜕𝛾∕𝜕𝜀e or S 𝛾ela ≡Υ=𝛾+

𝜕𝛾 𝜕𝜀e

(7.13)

The quantity Υ is also called surface stress. Equation (7.13) was derived in a classic paper by Shuttleworth [491, 492]. The Shuttleworth equation indicates that, to know the surface stress, we need to know not only the surface tension but also the dependence of 𝛾 on the elastic strain 𝜀e . Let us now consider the general case. We substitute ES in Eq. (7.7) by 𝛾ΣA : 𝜕(𝛾ΣA ) 𝜕ΣA 𝜕𝛾 𝜕N 𝜕N +N = 𝛾ΣA + N𝛾 + NΣA 𝜕A 𝜕A 𝜕A 𝜕A 𝜕A With d(NΣA ) = NdΣA + ΣA dN and A = NΣA we get 𝛾 S = 𝛾ΣA

𝛾S = 𝛾

𝜕(NΣA ) 𝜕𝛾 𝜕𝛾 + NΣA =𝛾+ 𝜕A 𝜕A 𝜕𝜀tot

(7.14)

(7.15)

To highlight plastic and elastic contributions, we rewrite Eq. (7.15) using 𝜕A∕𝜕A = 𝜕(Aela + Apla )∕𝜕A and d𝜀tot = dA∕A: 𝜕(Aela + Apla )

𝜕𝛾 𝜕Aela 𝜕Aela 𝜕A ( ) 𝜕Apla 𝜕Aela 𝜕𝛾 =𝛾 + 𝛾 +A 𝜕A 𝜕Aela 𝜕A d𝜀p d𝜀e =𝛾 +Υ d𝜀tot d𝜀tot

𝛾S = 𝛾

𝜕A

+A

(7.16)

The change in the Gibbs energy is given by the reversible work 𝛾 S dA required to expand the surface against the surface tension 𝛾 and the surface stress Υ. When the increase in surface area is purely plastic, only the first term on the right-hand side of Eq. (7.16) contributes, and we get the familiar 𝛾dA. If the stretching is purely elastic, only the second term will contribute. In the general case, there is a contribution from both the elastic and the plastic increase in surface area. 𝛾 S is therefore not a thermodynamic state variable since it depends on the history of the solid. It depends on the way new surface area is created, whereas the surface tension 𝛾 and the surface stress Υ are quantities that are independent of the specific process. There is another fundamental difference between solid and liquid surfaces. Crystals can respond differently in different directions when the surface area is increased. As a result, the number of equations increases by a factor of two as we must consider contributions for the two in-plane coordinates separately.

185

186

7 Solid Surfaces (10) small γ

Shape at equilibrium

(01) medium γ (11) large γ

Figure 7.9

Wulff construction for determining the equilibrium configuration of a crystal.

For many applications, it is useful to know which shape a crystal would assume in equilibrium. Equilibrium means when plastic deformation is allowed, for instance, during annealing. The surface tensions of a crystal are in general different for the different crystal faces. What shape does the crystal assume for a given volume when its entire free surface energy is minimal? A general, quasigeometrical solution of this problem was suggested by Wulff [493] as indicated in Figure 7.9: 1. Draw a group of vectors with common starting points. The length of the vectors should be proportional to 𝛾 of the crystal face, the direction is perpendicular (normal) to it. 2. Draw at the end of each vector one plane perpendicular to the vector direction. The body included by these planes is the equilibrium shape of the solid.

7.4.2

Determining Surface Energy

For the case of liquids, determining the surface tension is usually a straightforward procedure and well-established experimental methods exist. However, for solid surfaces, the situation is much more complex. In experimental approaches to determining surface energy parameters such as the surface tension, the surface stress, and the internal surface energy of solids, it is hard to separate the different contributions. Creation of new surface of a solid can hardly be done purely elastically or purely plastically, and the structure of the freshly created surface may not necessarily be the real equilibrium structure. This is one of the reasons why different methods may lead to different parameters and results cannot be compared directly. We start by discussing how surface energy parameters can be calculated. Bonds in covalent solids are dominated by short-range interactions. The internal surface energy is simply calculated as half of the energy that is necessary to split the bonds that pass through a certain cross-sectional area. This is called the nearest neighbor broken bond model. The Gibbs free surface energy is not much different from this value because, at room temperature, the entropic contribution is usually negligible. Example 7.3 The energy per bond between two carbon atoms in diamond is 376 kJ∕mol. If diamond is split at the (111) face, then 1.83 × 1019 bonds per m2 break.

7.4 Thermodynamics of Solid Surfaces

Figure 7.10 Splitting of a crystal: thought experiment for the calculation of the surface energy.

1. Step

2. Step

Thus, uΣ ≈

376 kJ∕mol × 1.83 × 1019 m−2 2 × 6.02 × 1023 mol−1

= 5.7 N∕m

This simple calculation demonstrates that the surface energies of solids can be much higher than the corresponding values of liquids. Noble gas crystals are held together by van der Waals forces. To calculate their surface energy, we proceed as indicated in Section 5.2.4. In a thought experiment, we split the crystal and calculate the required work (Figure 7.10). This splitting at fixed atomic positions is, however, only the first step. In a second step, we allow the molecular positions to rearrange according to the new situation. This is done by computer simulations. A certain interatomic potential must be assumed. Often, the Lennard–Jones potential, V(rij ) = C1 rij−12 − C2 rij−6 , is used. Here, rij is the distance between atoms i and j. The constants C1 and C2 depend on the material. The first term is due to the short-range repulsion of overlapping electron orbitals. It is called Pauli repulsion. The second term represents the van der Waals dipole attraction. As a result, the atoms close to the surface usually increase their distance because the attraction of second and third neighbors is missing. The surface energy is reduced. A similar calculation can be done for ionic crystals (Table 7.1). In this case, the Coulomb interaction is taken into account, in addition to the van der Waals attraction and the Pauli repulsion. Although the van der Waals attraction contributes little to the three-dimensional lattice energy, its contribution to the surface energy is significant and typically 20–30%. The calculated surface energy depends sensitively on the particular choice of the interatomic potential. Two methods are used to calculate the surface energy of metals: (i) As in the case of noble gases and ionic crystals, the surface energy is calculated from the interaction potential between atoms. (ii) Alternatively, we can use the model of free electrons in a box whose walls are the surfaces of the metal. This quantum mechanical view is independent of the specific material. The wave functions of the electrons have nodes at the walls. When the crystal is split, some previous states of the electrons are no longer available due to the additional boundary condition. The electrons are thereby forced to occupy states of higher energy. This additional energy is the surface energy.

187

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7 Solid Surfaces

Table 7.1 Calculated surface tensions 𝛾 and surface stresses Υ of ionic crystals [494, 495] and metals [477] for different surface orientations compared to experimental results. 𝜸 calc

Crystal

Orientation

LiF

(100)

480

(110)

1047

𝜸 exp

𝚼calc

340a)

1530

𝚼exp

407 450a)

CaF2

(111)

NaCl

(100) (110)

425

KCl

(100)

170

(110)

350

Na

(110)

212

300a), 190b)

415

375

256 110a), 173b)

295

210

250

120

540

320

401

Pb

(111)

270

Fe

(100)

2880

720

(110)

2450

2500

Ag

(111)

760

1200

790

1400

Au

(111)

710

1500

1770

1180

1390 1560

a) From cleavage experiments. b) Extrapolated from liquid. All values are given in mN/m (10−3 N/m).

There are only a few and often indirect methods of measuring the surface energy parameters of solids. The problems are that surfaces quickly contaminate unless placed in UHV and that they are usually inhomogeneous. For a review, see [495]: ●

●

●

●

● ●

For low-energy solids, for example many polymers, the surface tension can be obtained from contact angle measurements. This was described in Section 6.4 and the exercise “Surface energy of polymers.” For higher-energy solids, complete wetting occurs, and thus the contact angle is zero. From the value of the liquid, we measure the surface tension of the melt and rely on the fact that, close to the melting point, the free surface enthalpy of the solid is 10–20% larger. From a decrease in the lattice constant in small crystals caused by the compression due to surface stress, the lattice spacings can be measured with the help of X-ray diffraction or by LEED experiments (Section 7.8.2). From the work that is necessary for cleavage, we measure the work required to split a solid. The problem is that often mechanical deformations consume most of the energy and the surfaces can reconstruct after cleaving [490]. From adsorption studies with the help of inverse gas chromatography (IGC). From mechanical measurement of the surface tension, when the surface tension of a solid changes, the surface tends to shrink (if Υ increases) or expand (if Υ decreases). This leads, for instance, to the deflection of bimetallic cantilevers or a contraction of ribbons.

7.4 Thermodynamics of Solid Surfaces ●

From the heat generated upon immersion. Material in the form of a powder with a known overall surface area is immersed in a liquid. The free surface enthalpy of the solid is set free as heat and can be measured with precise calorimeters.

7.4.3

Surface Steps and Defects

When a crystal is cut at a small angle 𝜗 relative to a low-index surface, the surface exhibits steps or ledges that separate low-index terraces. The average distance of steps with height h is given by d = h∕ sin 𝜗, where 𝜗 is the angle between the low index surface and the actual surface. Such a surface is called vicinal. Figure 7.11 shows an example of a vicinal surface on a simple cubic crystal. If there is an additional tilt of the cutting plane, the ledges will have kinks (Figure 7.11b). Real surfaces will always exhibit a certain number of defects at temperatures above 0 K (Figure 7.12). This is true despite the fact that defects have a positive energy of formation compared to an ideal crystalline surface. What stabilizes these defects is the change in entropy connected with the induced disorder. For this reason, a certain average number of defects – which increases with temperature – will be present. As a consequence, real surfaces will not exhibit such evenly sized terraces or evenly spaced kinks as suggested by Figure 7.11. The terrace-ledge-kink (TLK) model [496] can provide a more realistic description of vicinal surfaces. Figure 7.11 (a) Vicinal surface of a cubic crystal that has been cut at a small angle relative to {100} plane exhibiting monoatomic steps; (b) an additional tilt leads to kinks in the ledges.

d

(a)

(b)

Figure 7.12 Schematic view of a vicinal surface with surface defects.

Terrace

Vacancy Ledge adatom Ledge Surface adatom

Kink Kink adatom

189

190

7 Solid Surfaces

(a)

(b)

Figure 7.13 Examples of (a) an edge and (b) a screw dislocation.

The distribution of the terrace widths is calculated taking into account the entropic repulsion between ledges. The confinement of a ledge between two neighboring steps (which cannot be crossed) leads to a reduction of the number of possible kink configurations and therefore a decrease in entropy. This entropic repulsion follows a d−2 law, where d is the mean distance between the ledges [497]. Elastic strain and dipole interaction of steps may lead to an additional repulsion, which also follows a d−2 dependence [498]. Defects lead to a roughening of crystal surfaces with increasing temperature, as already predicted by Burton et al. [499]. However, calculations for low-index surfaces yield roughening transition temperatures well above the melting point. The reason is the high formation energy of a ledge on a low-index surface. For vicinal surfaces, where ledges are already present, roughening can occur by the formation of kinks, and roughening transition temperatures on the order of half the melting point are observed [500]. Hoogeman et al. [501] were able to directly image the roughening transition using high-speed STM (Section 7.7.3). From a statistical analysis of the image data, they were able to extract the kink creation energy and step interaction energy. A review of the topic of steps on surfaces is given in [502]. An important class of defects are dislocations. Dislocations are not thermodynamically stable but kinetically trapped. The two primary types of dislocations are edge and screw dislocations (Figure 7.13). An edge dislocation corresponds to an additional half-plane of atoms in the bulk crystal. Screw dislocations create a step on the surface that starts from the emergence of the dislocation at the surface. Dislocations are often characterized by their so-called Burgers vector. The Burgers3 vector of a dislocation is found in the following way. Draw a closed path consisting of an arbitrary sequence of lattice vectors within the perfect crystal lattice in the absence of any dislocation. Then follow the same sequence of lattice vectors while encircling a dislocation. This second path will not be closed due to the dislocation. The vector from the starting point to the endpoint of the second path is the Burgers vector. 3 Johannes Martinus Burgers, 1895—1981; Dutch physicist.

7.5 Surface Diffusion

Example 7.4 The Burgers vector for the edge dislocation in Figure 7.13 would be an in-plane vector perpendicular to the additional lattice plane, with a length of one lattice constant. For the screw dislocation, one gets a Burgers vector perpendicular to the surface. Its length is again identical to the lattice constant. Surface defects always involve local variations in electronic states and binding energies. Therefore, surface defects are crucial in processes such as adsorption, nucleation, and surface reactions or catalysis. For example the step of a screw dislocation can eliminate the nucleation barrier for crystal growth.

7.5

Surface Diffusion

The lateral motion of adsorbed species due to thermal activation is an important phenomenon in crystal growth or for chemical reactions in catalysis; for an introduction see [503]. Also, in thin-film deposition (Section ), adsorption of atoms or molecules is typically followed by surface diffusion, which then allows nucleation and structure formation. Lateral mobility of adsorbates is usually a thermally activated process, that is, a certain energy barrier Ed must be overcome for migration of one adsorption site to the next. Two regimes can be distinguished: ●

●

Ed ≤ kB T: when the activation energy is smaller than the thermal energy of the adsorbate, adsorbed molecules can move relatively freely on the surface without being restricted to specific sites. Surface migration can be described as 2D Brownian motion. This situation corresponds to the presence of physisorbed species with low surface coverage (𝜃 ≪ 1, much less than a monolayer.) Thermal desorption is likely to occur in parallel to lateral migration. Ed ≫ kB T: when the activation energy is much larger than the thermal energy, the adsorbates will mostly reside at the adsorption sites, which in general reflect the symmetry of the crystal surface. Thermal vibrations will eventually lead to stochastic jump processes to neighboring binding sites. This situation is typical for chemisorbed species. From theoretical considerations, the condition Ed ≫ kB T appears to be fulfilled already for Ed > 5kB T [504].

Most experimental studies of surface diffusion have been carried out under conditions where Ed ≫ kB T. Therefore, we will focus in what follows on the treatment of this situation. For detailed discussion of both regimes and the transition between them, we refer to [504, 505]. If the density of adsorbates is much lower than single monolayer coverage so that there is no lateral interaction between single adsorbates, we talk about tracer diffusion. If there is a high concentration of adsorbates, collective diffusion will occur. For such a situation, lateral interaction between the adsorbates must be taken into account. In the presence of concentration gradients, collective diffusion leads to mass transport along the gradient, which is called mass transfer diffusion. In the absence of concentration gradients, there will still be intrinsic diffusion, that is random motion of adsorbates without a net flux of material.

191

192

7 Solid Surfaces

7.5.1

Theoretical Description of Surface Diffusion

The fundamental equations describing mass transfer diffusion phenomena are Fick’s first and second laws [506].4 Fick’s first law relates the material flux J due to diffusion with the concentration gradient ∇c that acts as driving force: J = −D∇c

(7.17)

Here, c is in units of molecules or moles per square meter. The flux describes the number of molecules or moles crossing a unit line per second. The diffusion coefficient D is a material parameter and given in square meters per second. D will usually not be a constant but it depends on the surface coverage 𝜃 since lateral interactions between adsorbates will influence their diffusion rates. Fick’s second law describes how a given concentration profile decays with time due to diffusion: 𝜕c = D∇2 c (7.18) 𝜕t Fick’s second law makes it possible to obtain diffusion coefficients by producing defined concentration profiles and following their decay over time. Data analysis involves the solution of the differential equation for the corresponding boundary conditions. For example, one could assume the deposition of molecules along a line with a certain number of molecules per unit length u. The evolution of the concentration profile c(x, t) due to spreading of the molecules in the x-direction perpendicular to the initial line orientation is then given by a Gaussian distribution: ( ) u x2 c(x, t) = √ exp − (7.19) 4Dt 4𝜋Dt Another possible starting configuration would be covering one half-space of the surface. The initial step concentration profile with c(x, 0) =

c0 ∶ x ≤ 0 0 ∶ x>0

decays according to [ ( )] c0 x c(x, t) = 1 + erf √ 2 4Dt

(7.20)

(7.21)

One often has the impression that diffusion is highly effective at short time scales while for longer times it is slow. For example in Figure 7.14, the equilibration after 0.25 seconds looks quite different to the initial concentration profile. When looking at the graphs calculated for later times, the changes seem less and less substantial. The reason for this impression is that we are usually accustomed to advective transport, where the distances covered increases linearly with time. In contrast, for diffusion, the characteristic distance over which concentration profiles √ on surfaces are equilibrated, Δr, increase with the square root of time: Δr = 4Dt. 4 Adolf Eugen Fick, 1829–1901; German physician and physiologist.

7.5 Surface Diffusion

–600 –400 –200

–600 –400 –200

(a)

(b)

Figure 7.14 Spreading of a step profile (a) and a line profile (b) with time by diffusion. The graphs were calculated with a diffusion coefficient of 10−7 m2 /s with Eqs. (7.21) and (7.19), respectively.

This square root dependence is even clearer when considering tracer diffusion. Rather than looking at concentration distributions, we ask the question: How far does a particle move by Brownian motion in a given time? For tracer diffusion of a single adsorbate or adatom at a surface, the diffusion coefficient can be calculated from the mean square displacement along the x- and y-coordinates using the Einstein–Smoluchowski equation: ⟨Δx2 ⟩ = 2Dt, ⟨Δy2 ⟩ = 2Dt

and

⟨Δr 2 ⟩ = 4Dt

(7.22) √

Thus, the mean distance covered by a molecule diffusing on a surface is ⟨Δr 2 ⟩ = √ 4Dt. The diffusion coefficient for tracer diffusion may deviate from that defined previously for mass transfer diffusion. Both will coincide only in the limit of noninteracting adsorbates, for example in the limit 𝜃 → 0. For tracer diffusion on an ideal crystal surface, we can come up with a microscopic model. We picture the diffusion of an adsorbate by random jumps from one adsorption site to the next driven by thermal fluctuations. The mean square displacement in such a situation is ⟨Δr 2 ⟩ = 𝜈l2 t

(7.23)

where 𝜈 is the jump frequency and l the jump distance. Note that the use of an average jump frequency for such a stochastic process implies the average over many hopping events. Therefore, Eq. (7.23) is strictly only valid for the limit t → ∞. In the simplest case of a square lattice and assuming nearest neighbor jumps, l is equal to the lattice constant a. The diffusion coefficient is given by (Eq. 7.22): D=

⟨Δr 2 ⟩ l2 𝜈 = 4t 4

(7.24)

To get from one adsorption site to the next, the adsorbate will have to overcome an energy barrier Ed . The hopping frequency can be calculated from transition state

193

194

7 Solid Surfaces

theory [507]: 𝜈 = 𝜈0 exp

(

ΔS kB

)

( ) E exp − d kB T

(7.25)

Here, 𝜈0 denotes the attempt frequency to overcome the barrier and ΔS is the change in entropy between adsorption site and transition state. This leads to the fundamental equation for tracer diffusion ) ( ( ) Ed l2 𝜈0 ΔS D = D0 exp − with D0 = exp (7.26) 4 kB T kB All temperature-independent terms have been incorporated into the preexponential factor D0 . Equation (7.26) makes it possible to determine the activation energy Ed and the preexponential factor D0 by an Arrhenius plot, that is, by plotting ln D versus 1∕T. Since ΔS is usually close to zero, a simple estimate of D0 can be obtained by assuming ΔS = 0. Taking l equal to a typical lattice constant of a ≈ 0.3 nm and setting the attempt frequency equal to a typical vibrational frequency of an adsorbate of 1013 s−1 [508] leads to a universal estimate of D0 = 10−7 m2 ∕s for the preexponential factor. Indeed, values of that order are often observed experimentally. Activation energies for surface diffusion are typically on the order of 5–20% of the respective adsorption energies. This implies significant lateral mobility of adsorbates due to surface diffusion will occur at temperatures well below those temperatures at which significant thermal desorption of adsorbates is observed. Since surface diffusion is a thermally activated process, one can define an onset temperature by the condition that 𝜈 = 1 Hz, that is, that on average one diffusive jump of a single atom occurs per second. Typical values of the onset temperature for the self-diffusion of metal atoms are in the order of 10% of the melting temperature of the metal. Tables of diffusion coefficients for many different substrates and adsorbates can be found in [509]. Example 7.5 For the diffusion of copper atoms on a Cu(100) surface, a diffusion barrier of Ed = 0.36 eV was found [510]. This corresponds to an onset temperature of 139 K, which is close to 10% of the melting temperature of copper Tm = 1356 K. For the more closely packed Cu(111), the energy barrier is about one order of magnitude lower, with Ed = 40 meV [511], and the onset of diffusion occurs already at 15 K, corresponding to approximately 1% of Tm . A special case is the diffusion of hydrogen, where at low temperatures, tunneling may become the dominant process for surface diffusion [512]. Lauhon and Ho observed that for hydrogen the diffusion rate on Cu(100) becomes temperature independent for T < 60 K [513], whereas for deuterium it follows an Arrhenius behavior for all temperatures tested. The exact crystal structure of the surface will have a strong influence on the diffusion coefficient. For more closely packed crystal faces, diffusion barriers are usually lower. The (110) surface of face-centered cubic metals exhibits a row structure with surface channels (Figure 7.2). On such surfaces, one-dimensional diffusion along the rows is commonly observed. However, for Pt(110), diffusion across channels was found to have almost the same activation energy as migration along the channels

7.5 Surface Diffusion

[514]. It was soon after shown that this is due to an atom exchange process, where the adatom in the channel takes the position of one of the lattice atoms forming a row and the former lattice atom ending up in the next channel as a new adatom [515]. Since then a number of different jump processes have been identified, where a single atomistic event leads to translocations over distances larger than a single lattice constant [516, 517]. However, even in the presence of such longer jumps, surface diffusion rates are dominated by the normal single-unit hopping processes.

7.5.2

Measurement of Surface Diffusion

The existence of surface diffusion was first noted by Hamburger [518] in 1918. He concluded from the distinct mosaic structure of vacuum-deposited silver films that silver atoms do not stick at the point they strike a surface but must have a significant mobility that allows them to collide with other silver atoms to form crystallites. The first quantitative measurements of surface diffusion appeared in the 1930s. For a review of the early history of surface diffusion measurements see [519]. Critical requirements for quantitative measurements of surface diffusion are a sufficiently high quality of the vacuum to preserve defined conditions during experiments and sensitive enough detection methods for the surface concentration of diffusing species. Two main approaches to measuring surface diffusion have evolved: ● ●

Direct observation of single atom or molecule motion undergoing diffusion. Creation of defined concentration profiles to observe their evolution with time due to mass transport diffusion.

For the direct observation of tracer diffusion, atomic resolution imaging is needed with a time resolution that matches the hopping frequencies of the diffusing species. This either limits such methods to systems with high diffusion barriers or requires sufficiently low sample temperatures to slow down the diffusion process. The observation of tracer diffusion became possible with the invention of the field ion microscope (FIM) by Erwin Müller in 1951 [520]. The FIM was the first instrument that allowed imaging of single atoms. It quickly became the most prominent method for fundamental studies of surface diffusion [521]. In the FIM, a sharp metal tip with a radius less than 50 nm is placed in a UHV chamber with a low level of background gas (typically He or Ne) and cooled to temperatures below 100 K. A high positive voltage of ≈ 10 kV is applied to the tip, creating an electric field of some 10 V∕nm at the tip surface. Gas atoms close to the tip get polarized by this high electric field and are attracted to the surface, where they adsorb and equilibrate to the low temperature of the tip, losing their kinetic energy. The high local field leads to tunneling of electrons from the gas molecule into the tip, and the ionized atoms are repelled from the tip in a perpendicular direction from the tip surface. They then hit a multichannel plate detector (replacing nowadays the phosphorous screen of the original design), creating a projected image of the FIM tip with a magnification in the order of 106 . Since the local field strength at the tip surface varies with the position of the tip atoms, the ion image resembles the atomic structure of the tip surface.

195

196

7 Solid Surfaces

For surface diffusion experiments, the species of interest is first adsorbed at cryogenic temperatures onto the tip surface and the structure is imaged. The electric field is then switched off in order not to perturb the diffusion process, which occurs at an elevated temperature. The tip is then cooled again to freeze the state after a defined diffusion time and an image is taken of the frozen state. By comparing series of subsequent images, the mean square displacement of atoms can be obtained and the diffusion coefficient is calculated from Eq. (7.22). Due to the extremely high fields applied in the imaging process, FIM is limited to conductive samples with high binding energy such as refractory metals. A further development of the FIM was the atom probe FIM or short atom probe, where a mass spectrometer with single-ion sensitivity was added [522]. Using field evaporation, single atoms from the FIM tip can be ionized and sampled by the mass spectrometer that resides behind a small central hole in the multichannel plate. This allows an atom-by-atom analysis of tip composition. With the introduction of the scanning tunneling microscope (STM, Section 7.7.3) an alternative instrument for direct imaging of single atom movement became available. With the STM, many more material combinations can be analyzed. Using sufficiently low temperatures, diffusion of adatoms can be kept slow enough to ensure that only single-jump events will occur between each image scan. Example 7.6 The diffusion of nitrogen atoms on a Ru(0001) crystal face was monitored by STM imaging (Figure 7.15). The nitrogen atoms were adsorbed by exposure of the surface to NO, which then dissociated at a monoatomic step of the Ru(0001) surface. The diffusion of the nitrogen atoms away from the step was imaged with an STM at a temperature of T = 300 K. The oxygen atoms diffuse so much faster that they quickly disappear out of the imaging region. A linear increase of the mean square displacement with the time of the single nitrogen atoms was observed (Figure 7.15c), as expected for tracer diffusion (Eq. 7.22). The tracer diffusion coefficient derived from the linear fit in Figure 7.15c is D = 3.4 × 10−22 m2 ∕s. The inset of Figure 7.15c shows a Gaussian fit of the concentration profile using Eq. (7.19) at t = 7080 seconds with the same value of the diffusion coefficient. The excellent fit without any adjustable parameters demonstrates that under these experimental conditions, the tracer diffusion coefficient and mass transfer diffusion coefficient were identical. The average distance of the nitrogen atoms have moved during the ≈ 2 hours time window of the experiment is slightly more than 20 Å. For measuring mass transport diffusion on surfaces, first a defined concentration gradient is established. This can be achieved simply by evaporation or gas exposure through a mask that shields part of the sample. The decay of that gradient due to mass transport diffusion can then be monitored with any detection method that is sensitive to the surface concentration of adsorbed species [523]. In early studies, spatially resolved photoelectric current measurements were used. Today, photoemission electron microscopy (PEEM) is more common. In PEEM, the local work function change due to absorption is exploited. In other studies, scanning Auger electron spectroscopy or scanning HREELS (Sections 7.9.2 and 7.9.3) was employed.

7.5 Surface Diffusion

800 n(x, 7080 s)

30 20

(a)

< x2 > (Å2)

600

10 0

400

0 20 40 60 80 x (Å)

200 0 0 (b)

(c)

2000

4000 6000 Time (s)

8000

Figure 7.15 STM topography images of nitrogen atoms on a Ru(0001) surface taken six minutes (a) and two hours (b) after NO exposure. Image size is 18 nm × 20 nm. Nitrogen atoms adsorb by dissociation of NO at the monoatomic step of Ru. (c) The mean square displacement of nitrogen increases linearly with time, as described by Eq. (7.22). The inset shows a Gaussian concentration profile. Source: Zambelli [524], American Physical Society.

In laser-induced thermal desorption [525] (LITD), a complete desorption of molecules from the surface within the focal spot of a laser beam is achieved by a sufficiently intense laser pulse. A second laser pulse is applied to the same spot after a certain delay time Δt, typically a few seconds, to desorb the molecules that have diffused back into the depleted zone. For each pulse, the amount of molecules desorbed is determined by a mass spectrometer. Since the amplitude of the mass spectrometer signal should be proportional to the number of molecules desorbed, one can obtain the diffusion coefficient from a plot of the normalized signal amplitude A(Δt)∕A0 versus lag time Δt, where A0 is the amplitude for the first desorption pulse at t = 0. The diffusion into a circular spot of radius r can be calculated from Fick’s second law as follows: √ 2 DΔt A(Δt) = (7.27) r 2𝜋 Since the decay of the concentration profiles is commonly monitored with methods that have a much more limited spatial resolution compared to the methods applied in tracer diffusion experiments, the diffusion coefficients in such experiments need to be much higher in order to lead to measurable changes in surface concentration on the relevant length scales (μm to mm) within a reasonable amount of time. A common feature of all methods described so far is the use of UHV to maintain clean and well-defined surfaces. Applying UHV, however, creates a “pressure gap” between these fundamental studies and typical industrial applications where catalytic reactions at surfaces often occur at ambient or even higher pressures. Furthermore, industrial catalysts will not consist of well-defined single crystalline surfaces. To obtain information about surface diffusion in porous media, several indirect methods have been developed. Typically, the kinetics of uptake into or permeation through a pellet of a sample is recorded and the observed behavior is analyzed by

197

198

7 Solid Surfaces

a model that combines assumptions about the adsorption isotherm and about the relative contributions of bulk and surface diffusion. For a review of such methods, see [526].

7.6

Solid–Solid Interfaces

Until now, we have concentrated on solid surfaces. This section addresses solid–solid interfaces. Such interfaces play an essential role in the stability of materials. In particular, the mechanical properties of metals are strongly influenced by grain boundaries. The stability of materials is a complex subject in itself, and we will not deal with it within the scope of this book. We will only provide some basic concepts and classifications. In the semiconductor industry, the production of solid–solid interfaces with a well-defined microscopic structure is of strong commercial interest. Another important area of research is the interface between polymers and inorganic materials since this interface is of vital importance in composite materials and for polymeric adhesives. In connection with composite materials often the term interphase is used. It describes a polymer region of typically 1–100 nm thickness that is in direct contact with the inorganic surface. There are different criteria of how to classify solid–solid interfaces. One is the sharpness of the boundary. It could be abrupt on an atomic scale such as in III–IV semiconductor heterostructures prepared by molecular beam epitaxy. In contrast, interdiffusion can create broad transitions. Surface reactions can lead to the formation of a thin layer of a new compound. The interfacial structure and composition will therefore depend on the temperature, diffusion coefficient, miscibility, and reactivity of the components. Another criterion is the crystallinity of the interface. The interface may be crystalline–crystalline, crystalline–amorphous, or completely amorphous. Even when both solids are crystalline, the interface may be disturbed and exhibit a high density of defects. Thermodynamic considerations may help to find the most likely equilibrium shapes, but these may not be realized due to kinetic limitations. As a rule of thumb, minimizing the interfacial energy usually means maximizing atomic matching to reduce the number of broken bonds. For crystalline–crystalline interfaces, we further discriminate between homophase and heterophase interfaces. At a homophase interface, composition and lattice types are identical on both sides; only the relative orientations of the lattices differ. At a heterophase interface, two phases with different compositions or Bravais lattice structures meet. Heterophase interfaces are further classified according to the degree of atomic matching. If the atomic lattice is continuous across the interface, we talk about a fully coherent interface. At a semicoherent interface, the lattices only partially fit. This is compensated for by periodic dislocations. At an incoherent interface,there is no matching of lattice structure across the interface. The most important example of homophase interfaces are grain boundaries. Figure 7.16 shows a model for a tilt grain boundary for a simple cubic crystal. The two grains are tilted relative to each other and joined to form the grain

7.6 Solid–Solid Interfaces

Figure 7.16 Symmetric small-angle tilt boundary. Source: Drawn after [528]).

ϑ

D

boundary. The former surface ledges become edge dislocations. The number of edge dislocations per unit length is given by 2 sin(𝜗∕2) 1 = (7.28) D b for small angles 𝜗. Here, b is the absolute value of the Burgers vector characterizing the dislocations (in this simple case, it is identical to the step height h). Each dislocation has an elastic strain energy that depends on the spacing of the dislocations. For small angles (𝜗 ≤ 15∘ ), the grain boundary energy per unit area 𝛾GB is calculated by summing up the contributions of all edge dislocations [527]. This results in the Read–Shockley equation ( ) b 𝜏b − ln 𝜗 with E0 = (7.29) 𝛾GB = E0 𝜗 1 + ln 2𝜋r0 4𝜋(1 − 𝜈) Here, 𝜏 and 𝜈 are the shear modulus and Poisson ratio of the bulk material, and r0 is the so-called core radius of the dislocation, which is a measure of the energy of a single dislocation. 𝜗 is in radians so as to be dimensionless. It is instructive to have a closer look at the dependence of 𝛾GB on 𝜗 for small angles: d𝛾GB ∕d𝜗 goes to infinity for 𝜗 → 0. This implies that the grain boundary energy rises steeply with the introduction of the first ledge. The reason is the long-range stress field associated with an isolated dislocation. This is quite different from the solid–vapor interface, where the energy of a ledge is localized and no long-range stresses occur. With increasing 𝜗, the slope becomes more shallow, as the dislocations get closer and their stress fields start to cancel. Another simple type of grain boundary is the twist boundary, where the lattice planes of the grains are rotated relative to each other. In this case, the interface consists of a cross grid of screw dislocations. In the more general case, combinations of these two simple types of dislocation will occur. For higher-angle grain boundaries, the cores of the dislocations start to overlap and the simple summation of elastic strains no longer holds. To describe high-angle grain boundaries, the coincident site lattice (CSL) model [529, 530] can be used. It starts by considering that if two contacting lattices are rotated relative to each other around a common lattice point, there will be a coincidence of other lattice points at

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certain angles. The coincident points form a lattice themselves, which is termed a CSL. The degree of coincidence can be characterized by the parameter Σ=

area of coincidence lattice cell area of original lattice cell

(7.30)

The grain boundary energy 𝛾GB should be proportional to Σ. For small values of Σ, high coincidence occurs and the number of broken bonds can be minimized. Σ = 1 corresponds to complete coincidence of the ideal crystal. Experimentally, it was found that the correlation between 𝛾GB and Σ is not that simple due to volume expansions or translations at the grain boundaries. A principal problem of the CSL model is that even arbitrarily small variations of the lattice orientation lead mathematically to a complete loss of coincidence. This is physically not reasonable because an arbitrarily small deviation should have a small effect. This problem was solved by the O-lattice theory [531]. For a comprehensive treatment of solid–solid interfaces and grain boundaries, see [532, 533].

7.7

Microscopy

In the following sections, we introduce some of the most important experimental techniques of surface characterization. For the interested reader, a broad range of books on this topic is available (e.g. [534, 535]). We start by discussing microscopy, continue with diffraction, and finally focus on spectroscopic methods.

7.7.1

Optical Microscopy

Optical microscopy is often the first step in surface analysis since it is fast and easy to perform. It can be an aid in selecting the area of interest on a sample for further analysis with more complex methods. The application of classical optical microscopy to surface science is, however, limited because the maximum lateral resolution is on the order of the optical wavelength (≈ 500 nm). With opaque solids, the light penetrates into the material, giving optical microscopy a poor surface sensitivity. In addition, the depth of field is limited, which calls for flat, polished surfaces or allows only plane sections of the sample to be viewed. Classical microscopy techniques are based on the incoherent scattering of visible light by the sample surface. Using coherent scattering of light, new methods of phase contrast microscopy became possible, and it can be used to detect local changes in refractive index of transparent samples. Another method is interference microscopy, where the light that is scattered from the surface is made to interfere with light that is reflected from an “ideally flat” surface within the instrument. In this way subnanometer resolution, normal to the surface, is achieved. The lateral resolution is still limited by the wavelength of light. In confocal microscopy, the resolution of the optical microscope is increased, and the background light is minimized by introducing a pin hole. Rather than illuminating the whole sample, only a tiny spot is illuminated and only the light

7.7 Microscopy

coming from this spot is viewed. Focusing the illuminating light and viewing the scattered light is done via the same objective lens. The sample is raster-scanned in three dimensions, and the light intensity is displayed versus the position on the sample. Three-dimensional images with a resolution of 200–400 nm laterally and 1–2 μm normal to the surface can be obtained.

7.7.2

Electron Microscopy

With the development of quantum mechanics in the early twentieth century, it became clear that microscopic particles such as atoms, electrons, and neutrons behave in some cases like waves. Both views, the classical picture of a particle characterized by a momentum p and that of a wave with a wavelength 𝜆, are only two different, complementary, viewpoints of the same physical object. Both quantities are related by the de Broglie5 relation, 𝜆 = h∕p, where h is Planck’s constant. Some experiments are better understood in terms of the particle approach, whereas others are better described in the wave picture. To understand the resolution of electron microscopes and later diffraction techniques, the wave approach is more instructive. The resolving power of electron microscopes is substantially better than that of a light microscope since the wavelength of the electrons, given by the de Broglie relation 𝜆=

h h = √ p 2me Ekin

(7.31)

is smaller. Here, me is the electron mass and Ekin is the electrons’ kinetic energy. In an electron microscope, the electrons get their kinetic energy by an applied electric potential of 1–400kV. This leads to wavelengths of 0.4–0.02 Å. Electron beams can be deflected by electromagnetic fields. This allows the construction of electron optics using electromagnetic lenses. A broad overview on electron microscopy can be found in [536]. We distinguish two basic types of electron microscope: the transmission electron microscope (TEM) and the scanning electron microscope (SEM). In transmission electron microscopy, an electron beam is transmitted through a sample. All electrons have the same energy and, thus, the same wavelength because they are accelerated by the same voltage. The scattered electrons are projected onto a phosphorous screen or an imaging plate to obtain an image. Since the electrons must pass through the whole sample, the TEM is not a surface-sensitive method, and we will not discuss it further here. SEMs are routine instruments for obtaining a view of the surface structure with typical resolutions of 1–20 nm. In addition, SEMs have a high depth of field. They make it possible to obtain a three-dimensional impression of the sample topography. In an SEM (Figure 7.17), the electron beam is emitted from a heated filament or a field emission tip. The electrons are accelerated by an electric potential on the order of 1–400 kV. A condenser lens projects the image of the source onto the condenser 5 Louis Victor de Broglie, 1892–1987; French physicist who proposed Eq. (7.31) in his PhD thesis; Nobel Prize in Physics, 1929.

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Figure 7.17 Schematic of a scanning electron microscope (SEM).

Electron gun

Primary electron beam

Condensor

Aperture Scanning coils

Objective

Secondary electrons e−

e−

Detector for secondary electrons

Sample

aperture. The beam is focused by an objective lens and raster-scanned by scanning coils over the sample. It is important to minimize the spot size of the primary electron beam because it determines the resolution of the instrument. When the primary electrons hit the sample surface, they pass part of their energy to electrons in the sample, resulting in the emission of secondary electrons. These secondary electrons have low energies (≈ 20 eV; therefore, only those secondary electrons escape from the sample that were close (≈ 1 nm) to the surface of the sample. This makes scanning electron microscopy highly surface-sensitive. The secondary electrons are collected by a detector, and their intensity is displayed versus the position of the primary beam on the sample. To avoid electric sample charging by the electron beam, the objects must either be conducting or they must be covered with a thin conductive layer (e.g. by sputtering of metal or carbon). Otherwise, the electron beam gets deflected by the surface charges and imaging is no longer possible. Another way to circumvent the charging of the sample surface is through the use of low acceleration voltages for the primary electrons (≈ 1 kV) so that charge deposited by the primary electrons gets balanced by the charge removed by the secondary electrons. In standard SEMs, the sample is placed in a vacuum of ≈ 10−7 mbar, so that the electrons are not scattered by gas molecules. This limits the applicability of SEM for biological samples that are usually denatured during the drying in vacuum unless some special fixation procedures or cryo-SEM is applied. In environmental scanning electron microscopes (ESEM) [537], it is also possible to obtain an image at gas pressures of up to 30 mbar. The trick with this type of instrument is to separate the sample chamber with a detector from the rest of the instrument by pressure-limiting apertures. These allow the electron beam to pass

7.7 Microscopy

from the high-vacuum part with the electron gun and most of the electron pathway into the sample chamber while reducing the number of gas molecules that can enter the high-vacuum area. The ESEM uses a different type of detector that makes use of the presence of the higher residual gas pressure in the sample chamber. The secondary electrons emitted from the sample are accelerated toward the positively biased detector. Collisions between the electrons and the residual gas atoms set free additional electrons by ionization of the atoms. On the one hand, this leads to an amplification of the electron signal. On the other hand, the remaining positively charged gas ions can neutralize the excess negative charge that would otherwise build up at the surface. This allows for the investigation of nonconducting samples without the need for metalization. Depending on the type of SEM, it is possible to obtain more information than just the sample topography. Besides the secondary electrons that are detected for the topographic imaging, there are also elastically backscattered primary electrons and X-rays generated by the interaction of the primary electrons with the sample atoms. The backscattering probability depends on the mean atomic number of the material. By detecting only the high-energy backscattered electrons, one can obtain a material contrast image. However, this imaging mode has a lower signal-to-noise ratio due to the smaller number of backscattered electrons. In addition, it is less surface-sensitive since the penetration depth of the backscattered electrons is in the range of several hundred nanometers. The X-ray emission can be used for X-ray spectroscopy of the sample, which allows for quantitative analysis of the sample composition (see description of EDX below).

7.7.3

Scanning Probe Microscopy

Scanning probe microscopes (SPMs) have dramatically increased our possibilities for analyzing surfaces. The two most important representatives of the family of SPMs are the STM and the atomic force microscope (AFM), also called the scanning force microscope (SFM). In the STM [538], a fine metal tip is brought into close proximity of the sample. The sample must be electrically conductive. Between the sample and tip, a voltage V of typically 0.1–1 V is applied. At a distance of approximately 1 nm, a tunnel current begins to flow. The current intensity depends strongly on the distance. The sample is scanned underneath the metal tip by a scanner that consists of piezoelectric crystals. Usually, the tunneling current is kept constant and an electronic feedback system regulates the z (height) position of the sample while scanning. Practically, in this way, the distance D is kept constant. This height information of the sample is finally plotted in the xy-plane. In this way a picture of the sample surface is obtained. The tunneling current I is proportional to √ V Φ −KD√Φ I∝ e (7.32) D Here, Φ is the mean work function of tip and sample, Φ = (Φtip + Φsample )∕2, and K is a constant. In addition, the current is approximately proportional to the density of

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I

U

z y (a)

x

Scanner

(b)

Figure 7.18 (a) Schematic of a scanning tunneling microscope (STM). (b) STM image (2.7 nm × 2.7 nm) of atomic structure of a copper(111) surface imaged in an aqueous medium after electrochemical cleaning [540]. Source: Image courtesy of P. Broekmann and K. Wandelt.

electronic states near the Fermi level [539]. With the STM, the density of the electron states is thus scanned. Atomic resolution up to the imaging of atomic defects in crystal surfaces is possible with the STM (Figure 7.18). Surfaces can be scanned in a vacuum, air, and, with a little more effort and experience, in liquid. With the STM we are, however, restricted to electrically conductive samples such as metals or semiconductors. In addition, it is possible to scan thin nonconducting layers (≈ 1 nm thick) on conducting substrates. Thus, it is possible to take images of certain organic monolayers or adsorbates. One might think that it is difficult to produce tips that allow atomic resolution imaging. This is fortunately not the case. Often, a wire made of Pt/Ir that is cut with simple scissors produces atomically resolved images. Some researchers further etch the tip to obtain more reproducible results. The restriction to conducting samples was overcome with the invention of the AFM [212]. In this microscope, a tip attached to a cantilever is scanned over the sample surface (Figure 7.19). The end of the tip is usually in mechanical contact with the sample. The sample is moved by a piezoelectric scanner in the xy-plane, as with the STM. The critical element in an AFM is the cantilever with the tip. Both are usually made using microfabrication techniques. To measure the up and down movements of the tip, a laser beam is focused on the back of the cantilever. From there, the laser beam is reflected toward a photodetector. When the cantilever bends, the direction of the reflected beam changes. With the help of the photodetector, this change is converted into an electrical signal. As in the STM, an electronic feedback control keeps the cantilever deflection constant by adjusting the z position of the sample. This height information of the sample is finally plotted in the xy-plane and a topographical picture of the sample surface is obtained. For an introduction to AFM, see [541]. The resolution of the AFM depends on the radius of curvature of the tip and its chemical condition as well as the operating conditions of the instrument. Solid crystal surfaces can often be imaged with atomic resolution. At this point, however, we need to specify what atomic resolution is. Periodicities of atomic spacing are, in fact,

7.7 Microscopy

Laser light Photo detector Lens 100 μm

(b) 1 μm

Tip Sample Piezoelectric scanner

(a)

(c)

Figure 7.19 Schematic of an atomic force microscope (AFM). In (a, b), scanning electron micrographs show a cantilever (b) and a tip (c) in more detail. The tip, which in operation points downward to the sample, is pointing toward the observer (b) and upward (c).

easily reproduced. To resolve atomic defects is much more difficult and usually is not achieved with standard atomic force microscopy. When it comes to steps and defects, the STM has a higher resolution. On soft, deformable samples, for example on many biological materials, the resolution is reduced due to mechanical deformation. Practically, a real resolution of a few nanometers is achieved routinely. One danger in atomic force microscopy is the destruction of fragile objects on the sample. Although the force applied by the tip to the sample is only in the order of 1 nN, the pressure can easily reach 1000 bar because the contact area is so small. One way to avoid deformation of fragile objects is to use the so-called tapping mode. In tapping mode, the cantilever is vibrated at its resonance frequency. At the end of the cantilever, where the tip is, the vibration amplitude is typically 1–10 nm. When we approach the surface with such a vibrating cantilever, at some point, the amplitude will decrease, simply because the tip starts to hit the surface. Instead of scanning the surface at constant deflection or constant height, we scan at constant reduction of the vibration amplitude. As a result, most of the time the tip is not in actual contact with the surface. It just hits the surface for a short time at intervals given by the resonance frequency. The tapping mode is often less destructive than the contact mode because the contact time is very short and shear is prevented. In addition, we obtain information about the local mechanical properties such as the elasticity [542, 543]. One disadvantage of the tapping mode is the slightly lower resolution. It is usually impossible to image crystals with atomic resolution. An important improvement for high-resolution imaging was the introduction of frequency modulation atomic force microscopy (FM-AFM) [544], where the shift in resonance frequency of the AFM probe due to interactions with the sample surface is used as feedback to control tip-sample distance. This allows a noncontact operation of AFM since a frequency shift sets already in several nanometers above the surfaces within the range of surface forces. With FM-AFM, true atomic resolution of surfaces in UHV [545] and

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liquid is achieved [242, 546, 547]. A detailed discussion on high resolution by AFM is given in [548, 549]. With an AFM surfaces can be scanned in UHV, air, and, most favorably, in liquids. Liquids have the advantage that the force between tip and sample, which might lead to a possible deformation of fragile sample structures, is smaller than in air or UHV. During the last two decades, a wealth of different scanning probe microscope techniques based on scanning tunneling and atomic force microscopy have evolved that make it possible to obtain more information than just topography. Examples are friction, adhesion, elasticity, conductance, electron densities, magnetization, and surface charges. For more information see [550, 551].

7.8

Diffraction Methods

For solid surfaces with a crystalline structure, we can apply diffraction techniques to analyse their molecular structure. In a diffraction experiment, the sample surface is irradiated with electrons, neutrons, atoms, or X-rays, and the angular distribution of the outgoing intensity is detected. The analysis of diffraction patterns is a formidable task, and in the first subsection, we only introduce a simple case, which nevertheless contains the main features. A more general formalism for the interested reader is described in the appendix.

7.8.1

Diffraction Patterns of Two-Dimensional Periodic Structures

To analyze diffraction patterns, we use the wave picture. The incident beam is treated as a coherent wave with a large extension perpendicular to its direction of propagation. Large means that it extends over a much larger distance than a typical atomic spacing. We start our discussion by asking: What is the diffraction pattern of a row of atoms of equal spacing d, irradiated perpendicular to its axis (Figure 7.20)? The atoms are supposed to partially scatter the incoming beam. They act as new sources of spherical waves all of which are in phase. We observe the resulting diffraction pattern far from the sample as compared with the size of the sample. This implies that the rays at a certain point of observation are almost parallel, although they emanate from different atoms of the sample. Rays coming from two neighboring atoms interfere constructively if the difference in path length is just an integer number of wavelengths. This is the case for Δ = n𝜆 = d sin 𝜗

(7.33)

where n is an integer called the order of the diffraction peak and 𝜗 is the angle of observation. This is fulfilled for all outgoing waves that are on a cone around the axis along the one-dimensional structure, with a cone angle of 2 × (90∘ − 𝜗). If we insert a detector plate, the observed diffraction pattern will consist of hyperbolic curves. A real two-dimensional crystal consists of a periodic structure in not only one but in two directions. This second periodic structure in another direction also leads to a condition for constructive interference. The conditions for both directions must be

7.8 Diffraction Methods

Figure 7.20 Condition for constructive interference for a one-dimensional array of scattering centers (a). Constructive interference occurs along the cones that reflect the rotational symmetry of the one-dimensional arrangement (b). For a two-dimensional crystal, constructive interference is obtained along lines (c). The incoming rays are directed from the top or bottom.

ϑ Δ

90°– ϑ d

(a) Incident radiation

Detector plate

(b)

(c)

met at the same time. Both conditions are met only where the cones of both orientations intersect. As a result, constructive interference occurs only on lines starting from the point of incidence. On a detector plate, we would observe spots where these lines cross the plate.

7.8.2

Diffraction with Electrons, X-Rays, and Atoms

Surface diffraction methods should meet three conditions: 1. The resolution must lie in the range of some atomic layers since only over this depth does the surface differ from the volume. This criterion, together with the angle of incidence, sets an upper limit for the wavelength. 2. The method must be surface-sensitive, that is, the observation depth must be defined largely by the first atomic layer. For this reason, the penetration depth of the beam should be small. 3. The method must be nondestructive, that is, no irreversible changes in the surface structure should occur. Surface diffraction experiments need to be carried out in UHV. Otherwise, the surfaces are covered with a monolayer of adsorbed molecules. At this point, the reader might ask: why do we not have to use UHV in the STM or AFM? In both techniques, the tip penetrates through the surface contamination layer. In the STM, it is often just invisible because contamination layers are usually not good conductors. In atomic force microscopy, it might disturb the imaging process, but often the tip pushes through mechanically.

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Grid for post amplification Sample –

e

Screen, +5 kV

(a)

(b)

Figure 7.21 Schematic of a LEED setup (a) and a LEED diffraction pattern obtained from a Pt(111) surface with electrons of 350 eV energy. (b) Image of a diffraction pattern courtesy of M. Zharnikov and M. Grunze.

The most common diffraction method to look at surfaces is low-energy electron diffraction (LEED). With typical electron energies of 20–500 eV, the de Broglie wavelength is at 3–0.5 Å. The penetration depth is relatively independent of the material and typically 4–10 Å. In a LEED experiment, a collimated, monoenergetic beam of electrons is directed toward the sample (Figure 7.21). The elastically backscattered electrons interfere and form a pattern of maxima and minima that is visualized on a fluorescent screen after amplification. Since the wavelength of the electrons lies in the range of atomic spacings, only a few reflexes appear on the screen. When we increase the electron energy, the angle between the reflexes becomes smaller and the reflexes move closer on the screen. Example √ 7.7 With an electron energy Ekin of 50 eV, the wavelength is 𝜆 = h∕ 2me Ekin = 1.73 Å (Eq. 7.31). With a lattice constant of 3.0 Å, the first reflex appears under an angle of 35.2∘ . When we increase the electron energy to 100 or 200 eV, then the wavelengths are 𝜆 = 1.23 and 0.87 Å. The corresponding first intensity maxima appear under angles of 24.2∘ and 16.9∘ , respectively. The incident electrons might change the sample structure. For low-energy electrons, these might be electronic excitations, which quickly disappear in metals and semiconductors, which can, however, be severe in ionic or covalent crystals. The more insulating a solid is, the greater the danger of changes, such as desorption, place change, dissociation, and other secondary processes, which are caused by charge transfer in atoms. High-energy electrons with energies in the range of 1–5 keV can penetrate deeper into a solid. Therefore, in high-energy electron diffraction, also called reflective high-energy electron diffraction, RHEED, HEED, or RED, we usually work at a grazing incidence angle to achieve a high surface sensitivity with small penetration depth. The technology can usually not compete with LEED and is used if there is not enough space in the UHV chamber to accommodate LEED. The interaction of X-rays with matter is weak compared to that of electrons. This leads to a large penetration depth of several microns and allows for surface

7.9 Spectroscopy

Figure 7.22 Schematic of an atomic beam diffractometer. Pressure-limiting apertures separate the zones with different vacuum qualities.

Detector Monochromator

Gas reservoire ≈106 Pa

10−1 Pa

10−5 Pa 10−8 Pa

Pump 1

Sample

Pump 2 Pump 3

inspection only at grazing incidence. The technique is called grazing incidence X-ray diffraction. At the same time, the diffraction intensity from a single-surface monolayer is small; therefore, particularly high intensity X-ray sources are required such as synchrotrons. X-ray diffraction has a big advantage: since interactions with the sample are scarce, multiple scattering effects can often be neglected, which makes a comparison with calculations much simpler. Atomic beams are also suitable for diffraction experiments. The repulsive interaction with the surface is so strong that low-energy atoms are already reflected by the topmost atomic layer. Due to its pronounced sensitivity to the topmost surface layer, the method of atomic beam diffraction is especially useful for the study of adsorbates and superlattices. Typical energies of the incident atoms are under 0.1 eV. At such low energies, no radiation damage occurs. To generate a monochromatic atomic beam, the atoms are allowed to escape from a gas reservoir through a nozzle. Apertures and a chopper collimate and monochromatize the beam (Figure 7.22).

7.9

Spectroscopy

In Chapter 6, we introduced methods to analyze the structure of surfaces. Now, we discuss methods to analyze the chemical composition of surfaces and probe the electronic states of molecules at the surface.

7.9.1

Optical Spectroscopy of Surfaces

Optical spectroscopy is a well-established technique for characterizing bulk materials. For the identification of molecules, infrared (IR) spectroscopy has proven especially useful. IR spectroscopy probes the resonant absorption of IR radiation for wavelengths of 16–2.5 μm due to vibrational transitions of molecules. Absorption bands in the range of 1600–4000 cm−1 yield information on the presence of specific functional groups. The so-called fingerprint region in the range of 600–1600 cm−1 gives unique information on the exact structure of the compound.

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Today, most IR spectrometers are not simple monochromatic absorption spectrometers but so-called Fourier transform (FT) spectrometers. They essentially consist of a Michelson interferometer. In an FTIR instrument, light from a broadband IR source is split into two beams by a semitransparent mirror. One beam is reflected off a fixed mirror, and the other one off a movable mirror. After passing through the sample, the beam intensity I(𝛿), depending on the variable path difference 𝛿 between the two beams, is recorded. Since I(𝛿) is essentially the Fourier cosine transform of the spectrum I(𝜆), the latter can be calculated from I(𝛿) by an inverse Fourier cosine transform. In an FTIR spectrometer, all wavelengths are measured simultaneously, while in a traditional dispersive instrument, consecutive measurements in small steps for each wavelength must be done. For the same signal-to-noise ratio, measurement times are therefore cut down in FTIR spectroscopy from tens of minutes to a few seconds (multiplex or Fellget advantage). Since no narrow slits are necessary to obtain high spectral resolution, a much higher optical throughput is possible for FTIR spectrometers (Jacquinot advantage). As the only part in an FTIR spectrometer that needs to move is a mirror, constructions can be simple and robust. In addition, FTIR spectrometers are self-calibrating. For calibration, the interference fringes of a He-Ne laser that is also coupled into the Michelson interferometer serve as a wavelength calibration standard (Connes advantage). A common problem in IR spectroscopy is the influence of water vapor and CO2 . Both have strong absorption bands in the IR range. Background spectra must be taken without the sample itself and IR spectrometers are often purged with dry N2 during measurements or even placed in a vacuum. For a comprehensive book on FTIR, see [552]. As was already mentioned in Section 7.7.1, optical methods are usually not really surface-sensitive since the penetration depth, and thus the surface layer probed, cannot be smaller than the wavelength of light. In particular, for IR with its long wavelengths this can be a problem. Nevertheless, there have been several approaches to utilizing IR spectroscopy for surface science (for an older but comprehensive review see [553]). One possible approach to minimizing the penetration depth is the use of total internal reflection of light. Light coming from an optically dense medium 1 is totally reflected from an interface with an optically less dense medium 2 if the angle relative to the surface normal is larger than a certain critical angle 𝜃c . This critical angle for the so-called total internal reflection is given by 𝜃c = arcsin

n2 n1

(7.34)

Here, n1 and n2 are the refractive indices of the respective media. No net energy is transferred into medium 2 under total internal reflection. Still there is an exponentially decaying electromagnetic field reaching into medium 2. The penetration depth dp of this so-called evanescent field, that is, the distance at which the amplitude of the electric field has decayed to 1∕e of its value at the interface, is given by dp =

𝜆 √ 4𝜋 (n1 sin 𝜃)2 − n22

Here, 𝜆 is the wavelength, and 𝜃 is the angle of incidence with 𝜃 > 𝜃c .

(7.35)

7.9 Spectroscopy

Example 7.8 Total internal reflection of light with a wavelength of 𝜆 = 500 nm at the interface between fused silica with a refractive index of n1 = 1.46 and water with n2 = 1.33 leads to a critical angle of 𝜃c = 65.6∘ . For the same configuration and an angle of incidence of 75∘ , one obtains a penetration depth of dp = 85 nm. When a material 3 with a refractive index n3 > n2 is brought within the range of the penetration depth of an evanescent wave, part of the light is transmitted. The intensity of the reflected beam decreases. This attenuation of the reflected intensity is called attenuated total reflection (ATR). ATR can be used to detect the presence of particles close to an interface. It can also be used for spectroscopy since absorption bands of a material close to the interface will lead to a stronger ATR signal. It is routinely used for IR spectroscopy of opaque samples, which are not accessible to transmission IR spectroscopy. For this so-called ATR-IR spectroscopy, the sample is mounted on top of a prism or waveguide and the IR absorption of the sample is probed by the evanescent field that penetrates from the prism into the sample [554–556]. Typical penetration depths for ATR-IR spectroscopy are on the order of 1 μm. Instead of detecting the reduction in the intensity of the reflected beam, one can also use the evanescent wave to excite the fluorescence of molecules close to the interface. With this total internal reflection fluorescence (TIRF) spectroscopy, detection of single molecules within a range few hundred nanometers above the interface is feasible [557]. For the investigation of monolayers or ultrathin films on surfaces, infrared reflection absorption spectroscopy (IRRAS, sometimes also denoted as IRAS) has proven to be effective. The sample is illuminated under gracing incidence and the absorption of IR radiation in the reflected beam is detected. Of special interest is IRRAS for molecules adsorbed to metal surfaces. The dynamic dipole moment induced in an adsorbed molecule by the incident IR radiation leads to a mirror dipole in the metal. For an orientation of the dipole parallel to the surface, it gets compensated by its image, while for a dipole perpendicular to the substrate, amplification occurs. Any component of the electric field parallel to the metal surface is suppressed. This leads to a special selection rule on metal substrates: only vibrations having a component of the transition dipole moment aligned perpendicularly to the surface can contribute to the IRRAS spectrum [558] and only p-polarized light will excite this transition. The sensitivity of IRRAS was further improved by phase modulation (PM-IRRAS), where a high photoelastic modulator is applied to switch between s- and p-polarization. The s- and p-polarized components Ip and Is of the reflected beam are recorded with lock-in detection and the ratio (Ip − Is )∕(Ip + Is ) is formed. In this ratio, the contributions of all randomly oriented molecules are canceled out [559]. A special type of FTIR spectroscopy is diffuse reflectance infrared FT spectroscopy or DRIFT spectroscopy (for a review see [560, 561]). Here, diffuse reflections from a sample surface, typically a powder, are collected and analyzed by an FTIR spectrometer. Originally, the technique was applied to analyze the powder material itself [562]. It soon became apparent that DRIFT can also be used to detect adsorbates in

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powder samples [563]. Quantitative information can be extracted from the spectra using appropriate theoretical models for diffuse reflectance [564, 565]. A breakthrough for the application of IR spectroscopy to surfaces was the introduction of nonlinear optical effects such as second harmonic generation (SHG) and sum frequency generation (SFG). Nonlinear effects come into play when the electric fields of the electromagnetic waves become comparable to the intrinsic electric field felt by the electrons in a molecule. Such strong fields can only be achieved by pulsed lasers. The theoretical foundation for understanding nonlinear optical processes at surfaces was laid already by Bloembergen and Pershan in 1962 [566]. Only in 1987 did the first sufficiently strong lasers become available to actually carry out SFG experiments and record IR spectra of single monolayers [567] and probe their molecular orientation [568, 569]. As a noninvasive optical method, SFG is not limited to solid surfaces in vacuum but can be applied to almost any kind of interface in ambient conditions. Even buried interfaces can be probed, as long as the absorption of upper layers is not too strong [570]. In a sum frequency experiment, two pulsed laser beams are overlapped spatially and temporally. The first one has a fixed frequency 𝜈1 in the optical range. The second one is a tunable IR laser with frequency 𝜈2 . The nonlinear polarization leads to emission of photons at the angular frequency 𝜔3 = 𝜔1 + 𝜔2 with an intensity approximately given by: ISFG (𝜈) ∝ |𝜒 (2) |2 Ivis (𝜈1 )IIR (𝜈2 )

(7.36)

Here, 𝜒 (2) is the second-order susceptibility, and Ivis (𝜈1 ) and IIR (𝜈2 ) are the intensities of the two input lasers. The surface sensitivity of SFG results from the fact that 𝜒 (2) is zero for any medium with inversion symmetry. Bulk materials, where the molecules are randomly oriented, do not contribute. Only at the interface, where the symmetry is broken, can SFG occur (for a detailed theoretical discussion of SFG see [571]). This selection rule makes SFG an effective technique to probe molecules at even submonolayer coverage [572, 573]. SFG allows vibrational spectroscopy of surface molecules since 𝜒 (2) will be greatly enhanced when the IR laser frequency matches the frequency of a molecular vibration mode. By tuning the wavelength of the IR laser over a broad range, vibrational spectra of the surface species can be recorded. Wavelengths from the IR fingerprint region and the visible laser wavelength result in a sum of frequency wavelength in the visible range, which allows for the use of optical detectors with single-photon sensitivity compared to much less sensitive IR detectors in classical IR spectroscopy. Therefore, detection of submonolayer coverage is achieved. By varying the polarization of the two incoming lasers and analyzing the polarization of the outgoing SFG signal, information about the conformation and orientation of the surface molecules can be obtained [571, 574, 575]. While the single pulses used in SFG instruments are extremely short (picosecond to femtosecond range), the acquisition time for a spectrum takes typically on the order of seconds since many pulses need to be summed up to get a sufficient signal to noise. To make use of the ultrafast laser pulses for dynamic studies, one can perform pump-probe-type experiments in which an intense pump laser pulse is used

7.9 Spectroscopy

to induce a chemical reaction or molecular reorientation. The evolution of the molecular system is then probed by SFG pulses with a variable delay time after the pump pulse. Such ultrafast pump-probe experiments allow one to study chemical reactions or relaxation processes with femtosecond time resolution (for a review see [576]).

7.9.2

Spectroscopy Using Inner Electrons

Spectroscopy of inner electrons allows us to identify the elemental species of surface atoms because the energy levels of the inner electrons are hardly affected by chemical bonds. In X-ray photoemission spectroscopy (XPS or ESCA for electron spectroscopy for chemical analysis) a sample is exposed to monochromatic photons of high energy (> 1 keV) [577, 578]. These photons excite the inner electrons of the sample atoms, and the kinetic energy of the emitted photoelectrons is measured. Their kinetic energy equals h𝜈 − Eb , where h𝜈 is the energy of the incoming photon and Eb is the binding energy of the electron. Example 7.9 Figure 7.23 shows two X-ray photoemission spectra. In Figure 7.23a, a survey spectrum recorded on palladium radiated by Mg K𝛼 X-rays (1253.6 eV) is shown. The horizontal axis is inverted because, practically, the kinetic energy of the emitted electrons is plotted. The spectrum shows a small peak at 51 eV, which corresponds to the binding energy of a 4p electron. A peak at 86 eV (4s) is not visible because the absorption cross section is too low. At 335 eV, a peak of the 3d electron is visible, followed by two peaks corresponding to two 3p orbitals (at 531 and 559 eV). The highest binding energy is observed for the 3s electrons (670 eV). Figure 7.23b shows a detail of a nitrogen spectrum. It was recorded on a mixed monolayer of dipalmitoyl phospatidylcholine (DPPC) and the lipopeptide surfactin (at 1: 1 molar ratio) on mica deposited by Langmuir–Blodgett transfer at a film pressure of 20 mN/m (Chapter 12). For XPS, the film was irradiated with monochromatized Al K𝛼 X-ray radiation (energy 1486.6 eV). The energy of electrons emitted from the nitrogen 1s orbital (Eb = 399 eV) was detected. 3D

Counts per second

Pd

N

3p 3s 4P 800

(a)

600

400

200

Binding energy (eV)

0

410 (b)

405

400

395

Binding energy (eV)

Figure 7.23 (a) XPS survey spectrum recorded on palladium. (b) Nitrogen 1s spectrum of a surfactin/DPPC monolayer on mica. The vertical scale shows the number of counts per second. Source: Adapted from Deleu et al. [579].

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Why is this technique surface-sensitive, although the incoming X-rays penetrate, typically, microns into the sample? It is surface-sensitive because only those electrons that are emitted by atoms close to the surface have the chance to leave the sample and be detected. With the availability of synchrotrons as high-intensity tunable X-ray sources, another technique experienced a breakthrough. This technique is called extended X-ray absorption fine structure (EXAFS). In EXAFS, the precise shape of an X-ray absorption band is measured. The photoelectron ejected from the absorbing atom can be described as an outgoing wave that will be scattered by the surrounding atoms. The interference between the outgoing wave and the scattered waves leads to a fine structure of the absorption band. The periodicity of the fine structure is related to the distances between the absorbing atom and its neighbors. The amplitudes of the periodic peaks contain information about the number of neighboring atoms. Therefore, EXAFS probes the local structure around the absorbing atoms. The advantage is that not only can crystals be analyzed, but so too can polycrystalline material without long-range order. This was essential, for instance, in the study of catalysis. When an electron of an inner shell is removed by X-rays or primary electrons, an electron from an outer shell fills the hole. During this process either a photon with an energy that corresponds to the difference between the two involved energy levels is emitted or this energy is transferred to another outer electron. This outer electron is then emitted; it is called an Auger electron. We can analyze either the energy of the emitted X-rays, which is called energy dispersive X-ray analysis (EDX), or the energy of the emitted Auger electrons, which is called Auger electron spectroscopy (AES). The emission of an Auger electron is thus characterized by three energy levels. Like XPS, AES and EDX are suitable for an element analysis. AES is more suitable for elements of low atomic numbers, while with EDX heavier elements can be sensitively detected. EDX is often used for elemental analysis in electron microscopy, where the electron beam is used to eject the inner electrons. It should be noted, however, that the penetration depth of the primary electrons is in the order of several hundred nanometers, so the method has only limited surface specificity. The three methods are schematically summarized in Figure 7.24. Why is there no method in which we excite with photons and detect the emitted photons? The reason is simply that such a method would not be surface-sensitive. The incoming photons penetrate deeply into the solid and the emitted photons are still detected from large depths.

7.9.3

Spectroscopy with Outer Electrons

To obtain information about molecular structure and the bonds between atoms, the energies of the exciting photons or electrons must be reduced to match the energy levels of outer electrons. In contrast to the inner electrons, which occupy isolated atomic orbitals, the energies of the outer electronic structure are determined by valence or conduction bands.

7.9 Spectroscopy XPS

AES e–

Photon

e–,

Sample

Ekin EVac

e–

EDX Auger e

Photon

–

e–

Sample

e–

e–

Photon

Sample e–

e–

hν

hν

EF

Valence band hν

Valence band hν, e

−

Valence band −

hν, e e−

LII/III LI e−

Eb K

Figure 7.24 Term diagram for XPS, AES, and EDX. The vacuum energy Evac defines the zero point of the energy scale. The binding energy of electrons Eb , the Fermi energy EF , and the kinetic energy of free electrons Ekin are indicated.

An important technique is UV photoemission spectroscopy (UPS), which is based on the outer photoelectric effect (in contrast to XPS, where we use the inner photoelectric effect). Photons with energies of 10–100 eV are used to ionize atoms and molecules at the surface. The energy of emitted electrons is detected. To study the adsorption of molecules to surfaces, often difference spectra are analyzed that are measured before and after the adsorption. These difference spectra are compared to the spectrum of the molecules in the gaseous phase. A versatile tool to analyze vibrations of surface atoms and adsorbed molecules is high-resolution electron energy loss spectroscopy (HREELS) [580]. Monoenergetic low energy electrons (1–10 eV) are directed to the surface. Most of them are backscattered elastically. Some of them, however, excite vibrations of surface atoms or molecules. Their energy is thus reduced by the energy of the vibration. This allows one to take energy spectra of surface excitations.

7.9.4

Secondary Ion Mass Spectrometry

In secondary ion mass spectrometry (SIMS), the sample surface is sputtered by an ion beam and the emitted secondary ions are analyzed by a mass spectrometer (for a review see [581, 582]). Due to the sputtering process, SIMS is a destructive method. Depending on the sputtering rate, we discriminate static and dynamic SIMS. In static SIMS, the primary ion dose is kept below 1012 ions∕cm2 to ensure that, on average, every ion hits a “fresh” surface that has not yet been damaged by the impact of another ion. In dynamic SIMS, multiple layers of molecules are removed at typical sputter rates of 0.5–5 nm/s. This implies a fast removal of the topmost layers of material but allows for quantitative analysis of the elemental composition. We start with a discussion of dynamic SIMS. In dynamic SIMS, the most common primary ion sources are oxygen duoplasmatrons, which produce O− or O+2 , or

215

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7 Solid Surfaces

caesium guns, which provide Cs+ ions. These two elements are preferred for their enhancement effects. Oxygen implantation leads to the formation of oxides in the surface layer. During the secondary ion emission process, the bonds of the oxides break, mainly leaving the oxygen as a negative ion and the original surface material as a positive ion. Caesium acts as an efficient electron donor and mainly negative secondary ions are generated. Only a small fraction of the sputtered material will actually be ionized. The ionization probability depends on the element species and on the matrix material. Most SIMS instruments use magnetic sector mass analyzers, where the different mass-dependent deflection of the ions in a magnetic field is used to separate the elements. The other types are quadrupole mass analyzers. There the ions move in the center of a quadratic arrangement of four parallel cylindrical electrodes. A combination of static and alternating electric fields provide stable oscillations only for ions with a specific charge-to-mass ratio. Only these ions are able to exit the aperture at the end of the analyzer. After mass separation, the ions must be collected by a detector. An electron multiplier can be used to count single ions. This allows for high sensitivity but limits detectable count rates to 106 ions per second. For higher ion currents, a Faraday cup is used to collect the ions and measure the accumulated charge. It is also possible to use locally resolved SIMS. SIMS imaging allows one to visualize the lateral variation of secondary ion intensities. This can be achieved by two different methods: ●

●

A focused primary ion beam is raster-scanned over the surface and the intensity of the secondary ion current is stored as a function of beam position. The result is visualized on a computer screen. The lateral resolution is limited by the diameter of the primary ion beam that can be as small as 20 nm. In an ion microscope, a sample is “illuminated” with a broad ion beam (25–250 μm). The secondary ions are filtered by mass spectrometers that conserve their spatial distribution. Ions are then visualized by microchannel plates as image detectors. The maximum lateral resolution of this method is on the order of 1 μm.

The continuous sputtering of the surface during analysis allows for the measurement of depth profiles. Therefore, the secondary ion intensity of the element of interest is recorded versus sputtering time. The time axis is then converted to a length scale by measuring the depth of the sputtering crater at the end of the experiment, for example, using a profilometer. For static SIMS, the time-of-flight SIMS (TOF-SIMS) technique is most suitable. For TOF-SIMS, a pulsed primary ion beam is used. The secondary ions emitted from the surface after one pulse are accelerated by an electric potential U. These ions gain the velocity v that depends on the mass m and charge Q of the ion according to mv2 ∕2 = QU or √ 2QU v= (7.37) m

7.10 Summary

The ions will hit the detector at a distance L after a time span Δt =

L L = √ v 2QU∕m

(7.38)

In this way, the mass of the ions can be deduced from their arrival time at the detector and a complete mass spectrum is acquired for every pulse. Since after each pulse, we must wait until all ions have arrived at the detector, TOF-SIMS is limited to low sputter rates. This does not allow depth profiling but makes TOF-SIMS virtually nondestructive compared to dynamic SIMS.

7.10 Summary ●

●

●

●

●

Crystalline surfaces can be classified using the five two-dimensional Bravais lattices and a basis. Depending on the surface structure, the basis may include more than just the first surface layer. The substrate structure of a surface is given by the bulk structure of the material and the cutting plane. The surface structure may differ from the substrate structure due to surface relaxation or surface reconstruction. Adsorbates often form superlattices on top of the surface lattice. For the study of crystalline surfaces, UHV is required. The preparation of clean crystalline surfaces is usually carried out within the UHV system by cleavage, sputtering, evaporation, thermal treatment, or molecular beam epitaxy. The creation of a new surface on a solid can be done either plastically – described by the surface tension – or elastically – described by the surface stress. The equilibrium shape of a crystal can be determined from the Wulff construction. Surface energies can be calculated using a nearest-neighbor broken bond model: in a first approximation, the surface energy corresponds to the energy required to break all bonds of the bulk crystal along the cutting plane. In practice, surfaces may lower their energy by surface relaxation or surface reconstruction. Real surfaces have defects. On vicinal surfaces, terraces, ledges, and kinks are present. Other important types of defects are screw and edge dislocations. Grain boundaries are the most important type of homophase solid–solid interfaces. An example of crystalline heterophase interfaces with high relevance for technological applications are semiconductor heterostructures. Optical microscopy may be used for surface examination, but in many cases, it is not really surface-sensitive. SEM allows one to study surfaces with high resolution in a vacuum. The STM can achieve atomic resolution on conducting substrates. The AFM can be used on most solid surfaces for high-resolution imaging. Diffraction techniques using X-rays, electrons, or atomic beams can reveal the surface structure of crystalline materials. While standard optical spectroscopy has little surface specificity, special IR spectroscopic modes such as IRRAS, DRIFT, or SFG allow for the identification of molecular species at surfaces. Spectroscopic methods probing the energy levels of inner or outer electrons such as XPS, EDX, AES, UPS, or SIMS can give information on the surface composition and the electronic energies at a surface.

217

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7.11 Exercises 7.1

Draw√schematically a hexagonal fcc(111) lattice and indicate the (1 × 2) and √ the ( 7 × 7)R 19.1∘ overlayer structures.

7.2

In the chapter about the AFM, it is mentioned that the force between tip and sample is usually lower in liquids than in air or UHV. Why?

7.3

Calculate the structure factor of the diamond lattice (appendix).

7.4

In a TIRM setup, a zinc crown glass with a planar surface and refractive index of n = 1.52 is brought into contact with hexadecane (n = 1.44). You want to create an evanescent light field at 𝜆 = 500 nm with an intensity decaying with a decay length of 100 nm. What angle of incidence would you need to choose?

7.5

In Example 7.6, the tracer diffusion of N on Ru(0001) was observed with STM. What would be the minimum frame rate for STM imaging in that system that would have on average no more than a single jump per frame for a diffusing atom? The lattice constant of Ru(0001) is 2.7 Å.

7.6

Calculate the diffusion coefficient for Cu on Cu(100) (Example 7.5) for (a) the respective onset temperature of 139 K and (b) room temperature. How far would a Cu atom diffuse under these conditions on average within one hour? The lattice constant of a Cu(100) surface is 2.5 Å.

219

8 Adsorption 8.1 Introduction Adsorption is the enrichment of a substance at an interface. In the first four sections, we concentrate on the adsorption of small, uncharged gas molecules to a solid surface. When a solid surface is exposed to a gas, gas molecules will adsorb to its surface due to attractive intermolecular forces such as the omnipresent van der Waals forces. The amount adsorbed is determined by several parameters. The most important one is the partial pressure P. At the surface, the rotational and vibrational freedom of adsorbed molecules is usually reduced. Even their electrical properties may change. Some molecules diffuse laterally or the molecules might react on the surface; these processes are highly important in the understanding of catalysis [503]. Finally, molecules might desorb again into the gaseous phase. Adsorption and desorption rates determine the equilibrium amount on the surface. We extend our description to adsorption at solid–liquid interfaces. For some systems, we can use the same models as for gas adsorption on a solid surface, we only have to replace the partial pressure P by the concentration c. Instructive reviews about adsorption are Refs. [5, 583–585]. A good overview on the adsorption of water to solid surfaces is Ref. [586].

8.1.1

Definitions

Let us first introduce the most important definitions [587]. The material in the adsorbed state is called adsorbate. The substance to be adsorbed (before it is on the surface) is called the adsorpt or adsorptive. The substance, onto which adsorption takes place, is the adsorbent (Figure 8.1). An important question is how much of a material is adsorbed to an interface. This is described by the adsorption function Γ = f (P, T), which is determined experimentally. It indicates the number of adsorbed moles per unit area. In general, it depends on the temperature. A graph of Γ versus P at constant temperature is called an adsorption isotherm. For a better understanding of adsorption and to predict the amount adsorbed, adsorption isotherm equations are derived. They depend on the specific theoretical model used. For some complicated models, the equation might not even be an analytical expression. Physics and Chemistry of Interfaces, Fourth Edition. Hans-Jürgen Butt, Karlheinz Graf, and Michael Kappl. © 2023 WILEY-VCH GmbH. Published 2023 by WILEY-VCH

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8 Adsorption

Adsorptive

Figure 8.1 Definitions of adsorbent, adsorpt (= adsorptive), and adsorbate.

Adsorbate

Adsorbent

In this chapter, the symbol P is used for the partial pressure of the gas or vapor species considered. It is lower than the total pressure if other gases are present. P0 is the saturation vapor pressure at which the gas is in equilibrium with a reservoir of the liquid with a planar surface. Therefore, P0 is also called equilibrium vapor pressure. For example, we may consider the adsorption of water vapor in air. At 25∘ C and under ambient conditions, the total pressure is 105 Pa, P0 = 3169 Pa, and P can be in the range of 0–3169 Pa. If P exceeds P0 condensation sets in. All gases adsorb to solid surfaces below a critical temperature because of the van der Waals attraction. In general, when adsorption is dominated by physical interaction rather than chemical bonding, we talk about physisorption. Physisorption is characterized by several features [588]: ● ● ●

●

The sublimation energy is in the order of 20–40 kJ/mol. The adsorbate is still relatively free to diffuse on the surface and to rotate. The molecular structure of the solid does not change with physisorption except for some molecular solids (e.g. ice, paraffin, polymers). An adsorption equilibrium is quickly established. When lowering the pressure, the gas desorbs reversibly (except in porous solids).

If the adsorption energy is in the order of chemical binding energies, we talk about chemisorption. Characteristic properties are the following: ● ●

● ●

Typical sublimation energies of 100–400 kJ/mol. Often there are specific binding sites. The adsorbate is relatively immobile and usually does not diffuse on the surface. Even on covalent or metallic solids, there is often a surface reconstruction. Due to the strong binding, experiments in UHV are possible because the molecules practically do not desorb.

When the molecules penetrate the surface layer and diffuse into the structure of the bulk solid, we use the term absorption. For example, when polydimethylsiloxane is exposed to toluene vapor, toluene molecules are absorbed and diffuse into the bulk of the material. They are not enriched at its surface. Sometimes it is difficult or even impossible to distinguish between adsorption and absorption. Then it is convenient to use the more general term sorption. Rather adsorbent, adsorbate, and adsorptive, we talk about sorbent, sorbate, and sorptive. Oxidation can be viewed as the chemisorption of oxygen. For example, nickel and silicon are oxidized at ambient conditions. The resulting oxide layer is thermodynamically more stable and passivates the pure material below it. Another important

8.1 Introduction

example is the oxidation of aluminum which provides the metal with a very hard roughly 100 nm thick aluminum oxide (Al2 O3 ) layer. To stabilize the aluminum surface even more and to passivate it against reactive chemicals, the thickness of the oxide layer can be increased electrochemically. This procedure is called the eloxal process (electrolytical oxidation of aluminum).

8.1.2

The Adsorption Time

A useful parameter to characterize adsorption is the adsorption time. Let us first assume that no forces act between the surface and a gas molecule. Then, if a molecule hits the surface, it is reflected elastically with the same energy. An energy transfer between the surface and gas molecule does not take place. As a consequence “hot” molecules do not cool down even when they hit a cold surface. The residence time in proximity to the surface can be estimated using two results of the kinetic theory of gases, namely (1∕2) mv2 = (3∕2) kB T and v2x = v2 ∕3, by 𝜏≈

2Δx 2Δx = √ vx kB T∕m

(8.1)

Here, Δx is the thickness of the surface region which has to be passed twice and vx is the mean velocity normal to the surface. Example: N2 at 25 ∘ C, Δx = 1 Å, vx ≈ 300 m/s, 𝜏 ≈ 7 × 10−13 seconds. This is in the order of typical molecular vibration periods of 10−13 seconds. An attractive force between the gas molecule and the surface increases the average residence time of the molecule at the surface to 𝜏 = 𝜏0 eQ∕RT

(8.2)

with 𝜏0 ≈ 10−13 …10−12 seconds. Q is the heat of adsorption. To be more precise, we have to identify the inverse of 𝜏0 with a surface bond vibration frequency. Values for various atoms and molecules adsorbed to well-defined surfaces are given in Table 8.1. Heats of adsorption up to 10 kJ/mol refer to practically no adsorption and residence times are below 10 ps. Q = 20 − 40 kJ/mol is characteristic for physisorption. Residence times become significantly longer and can assume quite different values, depending on the precise value of Q. Chemically adsorbed molecules (Q ≥ 100 kJ/mol) practically do not leave the surface again. Table 8.1 Heats of adsorption Q, surface bond vibration frequencies 𝜏0−1 , and adsorption times 𝜏 at 27 ∘ C. H/W(100)

Hg/Ni(100)

CO/Ni(111)

N2 /Ru(100)

Xe/W(111)

Q (kJ/mol)

268

115

125

31

40

𝜏0−1 (Hz)

3 × 1013

1012

8 × 1015

1013

𝜏 (s)

10

33

Source: Results from Ref. [591].

8

10

7 × 10

5

3 × 10

1015 −8

9 × 10−9

221

222

8 Adsorption

Another useful parameter is the accommodation coefficient 𝛼. The accommodation coefficient is defined by the temperature of the molecules before the impact T1 , the surface temperature T2 , and the temperature of the reflected molecules T3 [590]: T − T1 𝛼= 3 (8.3) T2 − T1 For an elastic reflection, the mean velocity of the molecules before and after hitting the surface are identical and so are the temperatures: T1 = T3 . Then 𝛼 = 0. If the molecules reside a long time on the surface they have the same temperature, after desorption, as the surface: T2 = T3 and 𝛼 = 1. Thus, the accommodation coefficient is a measure of how much energy is exchanged before a molecule leaves the adsorbent again.

8.1.3

Classification of Adsorption Isotherms

Depending on the physicochemical conditions, a great variety of adsorption isotherms are experimentally observed. In most cases at sufficiently low pressure, the adsorbed amount increases linearly with the pressure. It is described by Henry’s law1 : Γ = KH P

(8.4)

KH is a constant in units of mol/m2 /Pa. It is the ideal limiting law for low P. Henry’s law may also be viewed as the first term in a series because any adsorption isotherm can be expressed as a series Γ = KH P + K2 P2 + K3 P3 + .... Here, K1 , K2 , etc. are positive or negative coefficients describing the specific combination of adsorptive and adsorbent. The first systematic attempt to classify adsorption isotherms goes back to 1940 and was reported by Brunauer et al. [591, 592]. They proposed five different types of isotherms. With three additional types, these five isotherms constitute the modern IUPAC classification shown in Figure 8.2 [587]. These types are not restricted to perfectly flat, uniform, smooth solid surfaces. Often, adsorption experiments are conducted with porous materials or powders. Type I is concave with respect to the abscissa and it approaches a limiting value when the gas pressure P approaches the saturation pressure P0 . Type I isotherms are observed by microporous solids which have only a relatively small external surface. Once all pores have been filled, the isotherm saturates. The limiting amount taken up is determined by the accessible microporous volume rather than the internal surface area. A steep increase as in type I(a) isotherms is observed for materials with narrow micropores of a diameter below 1 nm. Type I(b) isotherms are found for materials having wider pore size distributions. Type II is the most common type found with nonporous or macroporous adsorbents; examples are shown in Figures 8.5 and 8.10. Usually the first concave part is attributed to the adsorption of a monolayer. Roughly, at point B, a monolayer covers the surface. For higher pressures, more layers adsorb on top of the first one. Eventually, if the pressure reaches the saturation vapor pressure, condensation leads to macroscopically thick layers. 1 William Henry, 1775–1836; English Chemist.

8.1 Introduction I(a)

I(b)

II

III

P0 B P

P

IV(a)

P

IV(b)

V

P

VI

P0 B

B P

P

P

P

Figure 8.2 Schematic plot of eight types of physisorption isotherms commonly observed for the physisorption of gases to solids. The abscissa is the partial pressure P given in Pa or the relative partial pressure P∕P0 . Gibbs dividing interface c

Solid

Vapor

cb x

Figure 8.3 Schematic of the gas concentration along a coordinate normal to a solid surface. In general, the molecules are concentrated at the solid surface. All molecules in the hatched area constitute the surface excess Γ. The gas concentration in the bulk vapor phase is cb .

Type III is convex with respect to the P axis. It is not very common. Convex isotherms indicate cooperative effects. The presence of already adsorbed molecules favors the adsorption of more molecules. This may be due to attractive lateral interactions between adsorbed molecules. It also occurs if a gas molecule prefers to bind on an already adsorbed molecules rather than onto the bare solid surface. Type IV(a) is characterized by a hysteresis loop and a saturation plateau at P∕P0 ≈ 0.6–0.95 (example Figure 8.19). Such hysteresis was first observed by van Bemmelen and Zsigmondy2 et al. [593, 594]. The hysteresis is associated with capillary condensation taking place in mesopores. At low pressure, a single layer of molecules adsorbs to the surface. At intermediate pressures, multilayers start to form and at a certain point, capillary condensation sets in until the pores are filled. The saturation at high pressures is caused by the reduction of effective surface area once all pores have been filled. For cylindrical or conical pores with a width below ≈ 4 nm reversible isotherm as in Type IV(b) are observed. 2 Richard Adolf Zsigmondy, 1865–1929; Austrian Chemist of Hungarian origin. 1925 Nobel prize for chemistry. Prof. in Göttingen

223

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8 Adsorption

Type V is uncommon. It represents a cooperative adsorption (Type III) on porous adsorbents. The porosity causes capillary condensation, which leads to the hysteresis loop. Stepwise adsorption characterizes Type VI isotherms (example Figure 8.13). Subsequent layers are adsorbed to a uniform, nonporous surface. The step height corresponds to the amount adsorbed in each layer. In the simplest case, it remains constant for two or three layers.

8.1.4

Presentation of Adsorption Isotherms

An adsorption isotherm is a graph of the amount adsorbed versus the pressure of the vapor phase (or concentration in the case of adsorption from solution). The amounts adsorbed can be described by different variables. The first one is the surface excess Γ in mol/m2 . In Chapter 3, we stressed the fact that surface excess depends on the specific choice of Gibbs dividing plane. For the adsorption of impenetrable solid surfaces, Gibbs dividing plane is conveniently chosen to coincide with the physical boundary between solid and gas (Figure 8.3). All the volume accessible to the vapor molecules counts as vapor phase. The thermodynamic surface excesses occurring in Gibb’ s adsorption isotherm reduce to the corresponding analytical concentrations. It is also the reason why in the practical literature on adsorption, we find little about the Gibbs dividing plane. Using the Gibbs approach, we can convert the number of moles adsorbed N 𝜎 to the surface excess by N𝜎 (8.5) A where A is the total surface area. Adsorption is often studied using powders or porous materials because of the high surface-to-volume ratio. In a typical experiment, the volume (V) or the mass (m = V∕𝜌) adsorbed per gram of adsorbent, is measured. Theoretical models, on the other hand, describe an adsorption per surface area, i.e. m∕A or N 𝜎 ∕A. Adsorbed mass and amount adsorbed can be converted by N 𝜎 = m∕M, where M is the molar mass. In order to compare theoretical isotherms to experimentally determined adsorption results, the specific surface area needs to be known. The specific surface area Σ (in m2 /kg) is the surface area per kg of adsorbent. Once the specific surface area is known, the area can be calculated by A = mad Σ, where mad is the mass of the adsorbent. Γ=

8.2 Thermodynamics of Adsorption 8.2.1

Heats of Adsorption

Heats of adsorption are important characteristics of adsorption because they provide information regarding the driving forces for adsorption. Several heats, energies, enthalpies, and other quantities of adsorption have been defined. For a detailed discussion see Refs. [595, 596].

8.2 Thermodynamics of Adsorption

We first introduce the integral molar energy of adsorption: int 𝜎 Δad Um = Um − Um g

(8.6)

Sorry about all the sub- and superscripts, but we need to precisely define the therint is the energy difference between one mole of gas modynamic quantities. Δad Um g 𝜎 adsorbed Um (per mol) and the same amount free in the gas phase Um . The next important quantities are the integral molar enthalpy of adsorption int 𝜎 Δad Hm = Hm − Hm g

(8.7)

and the integral molar entropy of adsorption int 𝜎 = Sm − Sm Δad Sm g

(8.8)

They are defined in complete analogy to the integral molar energy. The difference between the energy and the enthalpy of adsorption is usually small. If we treat the free gas as being ideal and neglecting the influence of other background gases int = Δ H int + RT. At 25 ∘ C RT is only 2.4 kJ/mol. present, the difference is Δad Um ad m For this reason for an estimation, we do not need to worry too much about whether a heat of adsorption is the adsorption enthalpy or the internal adsorption energy. Let us consider how these quantities are related to experimentally determined heats of adsorption. An essential factor is the condition under which the calorimetric int is equal to the experiment is carried out. Under constant volume conditions, Δad Um total heat of adsorption. In such an experiment, a gas reservoir of constant volume is connected to a constant volume adsorbent reservoir (Figure 8.4). Both are immersed in the same calorimetric cell. The total volume remains constant, and there is no volume work. The heat exchanged equals the integral molar energy times the amount of gas adsorbed: int Q = −N 𝜎 Δad Um

(8.9)

int is negative (otherwise, the substance would not adsorb) and heat In general, Δad Um is released upon adsorption.

Figure 8.4 Schematics of calorimeters for measuring heats of adsorption under constant volume and constant pressure conditions. The active volume is filled with the adsorbent usually in the form of a powder.

Gas reservoir

Adsorbent

Constant volume

Constant pressure

225

226

8 Adsorption

In practice, most calorimetric experiments are carried out under constant pressure. In this case, the heat exchanged is equal to the integral enthalpy of adsorption: int Q = −N 𝜎 Δad Hm

(8.10)

int is negative. Again, heat is released upon adsorption and Δad Hm The integral molar entropy of adsorption is obtained from a well-known thermodynamic relation: for a reversible, isothermal process, the heat is equal to the change in entropy multiplied by the temperature. This directly leads to int Δad Sm =

int Δad Hm T

(8.11)

int is negative because adsorbed molecules have less freedom to move around Δad Sm and their order is higher than in the gas phase. One word about the Gibbs energies of adsorption. In equilibrium, the molar 𝜎 g Gibbs energy of adsorption is zero: Δad Gint m = 𝜇 − 𝜇 = 0. The reason is simple. In equilibrium and for constant P and T, the chemical potential of the molecules in the gas phase 𝜇 g is equal to the chemical potential of adsorbed molecules 𝜇𝜎 . What is not zero is the standard Gibbs energy of adsorption

Δad G0m = 𝜇 0𝜎 − 𝜇 0g

(8.12)

It is a molar quantity, but since the name is so long, the “molar” is not explicitly stated. The problem with Δad G0m is that we cannot measure it directly, but its value depends on the specific model of adsorption used. We return to it when introducing the Langmuir model.

8.2.2

Differential Quantities of Adsorption

We have introduced integral molar quantities, which indicate that there are corresponding differential quantities. Integral refers to the fact that the total amount of adsorbed gas is involved. In contrast, the differential molar energy of adsorption is determined only by the last infinitesimal amount adsorbed. It is defined as follows: dU 𝜎 || dU g || diff Δad Um = − (8.13) | 𝜎 dN |T,A dN 𝜎 ||T,A U g is the total internal energy of the free gas. Since usually the amount adsorbed is small compared to the total amount of gas in the reservoir, the properties of the free g gas do not change much during adsorption. Thus, dU g ∕dN 𝜎 = Um which leads to diff Δad Um =

dU 𝜎 || g − Um dN 𝜎 ||T,A

(8.14)

It involves the change of the internal surface energy upon adsorption of an infinitesimal amount of gas at constant temperature and total surface area. We have to distinguish between integral and differential quantities because the energy changes with the amount adsorbed. This can have at least three causes: first, most surfaces are energetically heterogeneous and the binding sites with a high binding energy are occupied first. Second, in general, the first monolayer has a different

8.2 Thermodynamics of Adsorption

binding energy from the next layer because its adsorption is dominated by the interaction of the solid adsorbent with the gas molecule. For the second layer, the interaction between adsorbed gas molecules with gas molecules is important. Third, if molecules interact laterally with neighboring molecules repulsively on the surface it is energetically more favorable for molecules to adsorb to a partially covered surface. By analogy, the differential molar enthalpy of adsorption and the differential molar entropy of adsorption are defined as follows: dH 𝜎 || g − Hm dN 𝜎 ||T,A dS𝜎 || g = − Sm dN 𝜎 ||T,A

diff Δad Hm =

(8.15)

diff Δad Sm

(8.16)

Physical adsorption of gases on solids is virtually always enthalpically driven diff (Δad Hm < 0). Entropically driven adsorption can exist but usually the entropy of molecules on a surface is much lower than in the gas phase. Vibrational, rotational, and also translational degrees of freedom are restricted on surfaces. Example 8.1 As an example, a typical adsorption isotherm for benzene adsorbing to graphitized carbon black is shown in Figure 8.5 [597]. Graphitized carbon blacks are produced by heating carbon in the absence of air to 3000∘ C. The initially spherical carbon particles become polyhedral with faces consisting mainly of homogeneous basal planes of single crystal graphite. In the adsorption isotherm, three regimes can be distinguished:

(a)

20 18 6 16 4 14 12 2 10 0 8 0.0 6 4 2 0 0.0

0.1

0.2

0.2

0.3

0.4

Diff. heat of adsorption (kJ/mol)

At very low pressure (P∕P0 < 0.1), the adsorption isotherm rises steeply. Adsorbing molecules find many free binding sites. These very few molecules on the surface have a chance to bind to strong binding sites at grain boundaries. This

Γ (μmol/m2)

●

0.4

0.6 P/P0

0.8

1.0

46 44 42 40 38 36 34 0

(b)

2

4 6 Γ (μmol/m2)

8

10

Figure 8.5 (a) Adsorption isotherm for benzene (C6 H6 ) adsorbing to graphitized carbon black at 20 ∘ C. The insert shows the adsorption isotherm for low coverage in more detail. Dotted lines indicate mono- or multilayer coverage at multiples of 4.12 μmol/m2 . The equilibrium vapor pressure of benzene at 20 ∘ C is P0 = 10.2 kPa. (b) Differential heat of adsorption versus adsorbed amount. The dashed line corresponds to the heat of condensation of bulk benzene. Source: Redrawn after Ref. [597].

227

228

8 Adsorption

●

●

can also be seen from the differential heats of adsorption: at a coverage below 0.3 μmol/m2 , the heat of adsorption is maximal. Monolayer coverage is reached at a pressure of P∕P0 ≈ 0.1. At this point, the steep slope of the adsorption isotherm levels off. For the first monolayer, a roughly constant heat of adsorption of 43 kJ/mol is observed. This is about 9 kJ/mol higher than the heat of condensation of benzene. (At higher vapor pressures P∕P0 > 0.1) multilayers are formed. In the multilayer region, the slope becomes steeper again with increasing pressure. For P → P0 , the adsorbed layer gets very thick because macroscopic condensation sets in. The differential heat of adsorption is slightly above the heat of condensation, but substantially lower than the value for the first monolayer.

8.3 Adsorption Models 8.3.1

The Langmuir Adsorption Isotherm

A simple model to describe adsorption was presented by Langmuir3 [598]. Langmuir assumed that on the surface there are a certain number of binding sites per unit area S (Figure 8.6). S is in units of mol/m2 (or m−2 ). Of these binding sites S1 are occupied with adsorbate and S0 = S − S1 are vacant. The adsorption rate in moles per second and per unit area is proportional to the number of vacant binding sites S0 and to the pressure: kad PS0 . Here, kad in s−1 Pa−1 is a constant characterizing the rate of adsorption. The desorption rate is proportional to the number of adsorbed molecules S1 and equal to kde S1 , where kde in s−1 is a constant. In equilibrium, the adsorption rate must be equal to the desorption rate; otherwise, the number of adsorbed molecules would change. This leads to ( ) kde S1 = kad PS0 = kad P ⋅ S − S1 kad P S ⇒ kde S1 + kad PS1 = kad PS ⇔ 1 = S kde + kad P S1 ∕S is the coverage 𝜃. With KL = kad ∕kde , we get the Langmuir equation: KL P 𝜃= 1 + KL P

Figure 8.6 Schematic drawing of the Langmuir adsorption model.

Gas molecule

Occupied binding site

(8.17)

Vacant binding site

Solid surface 3 Irving Langmuir, 1881–1957; American physicist and chemist spent most of his time at the General Electric Company. Nobel Prize for chemistry in 1932.

8.3 Adsorption Models

1.0 0.8 θ

Figure 8.7 Langmuir adsorption isotherms plotted as coverage 𝜃 versus relative vapor pressure for three different Langmuir constants. The Langmuir constants are given in units of P0 .

KL = 20 KL = 10

0.6

KL = 5

0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

P/P0

The Langmuir constant KL is given in Pa−1 . For low partial pressure (KL P ≪ 1), the Langmuir isotherm increases linearly and Henry’s law (Eq. 8.4) is obtained. Typical Langmuir adsorption isotherms are plotted in Figure 8.7 for different values of the Langmuir constant. If adsorption from solution is considered, the partial pressure P has to be replaced by the concentration c and the Langmuir constant is given in units of l/mol instead of Pa−1 . Alternatively, the Langmuir adsorption isotherm equation can be expressed by the number of adsorbed moles per gram or per unit area of substrate: Γ=

Γmon KL P 1 + KL P

(8.18)

Here, Γmon corresponds to a situation, where all binding sites are occupied and a monolayer of molecules is bound. Γmon is related to the surface area occupied by one ( ) adsorbed molecule 𝜎A by Γmon = Σ∕ N0 𝜎A (for Γ in mol/g) or Γmon = 1∕(N0 𝜎A ) (for Γ in mol/m2 ). What is the significance of the constants kad and kde ? kde is the inverse of the adsorption time: kde =

1 −Q∕kB T e 𝜏0

(8.19)

It is low for strongly adsorbing molecules with a high heat of adsorption. In order to calculate kad , we remember the kinetic theory of ideal gases. Equation (2.1) tells us how many gas molecules of mass m hit a certain area A per second. If we take the area to be the active area of one binding site 𝜎A , the number of molecules hitting one binding site per second is (eq. 2.1) √

𝜎A P

(8.20)

2𝜋mkB T

If we assume that each molecule that hits the surface sticks to it, expression (8.20) is equal to kad P, and we get kad = √

𝜎A 2𝜋mkB T

(8.21)

229

230

8 Adsorption

In reality, the sticking probability can be lower than one. Thus, kad given in Eq. (8.21) must be considered as an upper limit. For the Langmuir constant, we obtain 𝜎 𝜏 KL = KL0 eQ∕kB T with KL0 = √ A 0 (8.22) 2𝜋mkB T Example 8.2 Let us estimate the Langmuir constant for the physisorption of a gas to a solid surface at a temperature of 120 K. We take 𝜏0 to be 10−13 seconds, use a typical molecular cross-section of 10 Å2 , and assume a heat of adsorption of 20 kJ/mol. As a gas, we consider nitrogen (M = 0.028 kg/mol). It is more convenient to use Eq. (8.22) in molar rather than molecular units: KL = KL0 eQ∕RT

with

N 𝜎 𝜏 KL0 = √ A A 0 2𝜋MRT

(8.23)

Inserting the values leads to KL0 = 4.55 × 10−10 Pa−1 and KL = 0.23 Pa−1 . The sticking probability can also be used for the calculation of the bulk condensation rate of a liquid. After all, if the partial pressure of a vapor P exceeds its equilibrium vapor pressure P0 , the vapor condenses. We define a condensation coefficient or sticking probability as the ratio between the actual condensation and the upper limit in Eq. (8.21). The sticking probability can be determined from molecular beam experiments. Examples: For N2 on tungsten, the sticking probability at 27∘ C is 0.61 [599]. For H2 O on ice at 200 K, it is close to one [600]. The sticking probability can also be significantly below one. Then the Langmuir constant is reduced accordingly. The kinetic derivation has the disadvantage that it refers to a certain model. The Langmuir adsorption isotherm, however, applies under more general conditions, and it is possible to derive it with the help of statistical thermodynamics [5, 601]. Necessary and sufficient conditions for the validity of the Langmuir equation (8.18) are the following: The molecules bind to well-determined binding sites on the adsorbent; each binding site can bind only one molecule; the binding energy is independent of the presence of other bound molecules.

8.3.2

The Langmuir Constant and Gibbs Energy of Adsorption

The Gibbs energy of adsorption depends on the specific model used (Section 8.2.1). Here, we demonstrate this with the Langmuir model and derive a relation between the standard Gibbs energy of adsorption Δad G0m and the Langmuir constant. Therefore, we treat the binding and desorption of gas molecules to surface binding sites like a chemical reaction. Chemical equilibria are commonly characterized by an equilibrium constant K. For the dissociation reaction AB ⇌ A + B, this constant is given by K=

[A] [B] [AB]

(8.24)

It is related to the standard Gibbs energy of the reaction Δr G0m by Δr G0m = −RT ln K

(8.25)

8.3 Adsorption Models

Here, [AB], [A], and [B] are the concentrations (for gas reactions, the pressures) of the bound (educt) and dissociated molecules (products), respectively. Let us apply this formalism to the adsorption of gas molecules to free sites on a surface. The adsorption equilibrium constant in units of Pa is Kad =

S0 P S1

(8.26)

It is related to the standard Gibbs energy of adsorption by Δad G0m = −RT ln Kad

(8.27)

Inserting S − S1 = S0 leads to ( ) S − S1 P SP Kad = ⇒ S1 = S1 Kad + P

(8.28)

Since 𝜃 = S1 ∕S, we immediately get 𝜃=

P Kad + P

with

( ) Δ G0 Kad = exp − ad m RT

(8.29)

Comparison with Eq. (8.18) shows that KL =

1 Kad

(8.30)

When inserting real numbers in Eqs. (8.27) and (8.29), it is important to remember that pressures are given in units of normal pressure, that is 105 Pa.

8.3.3

Langmuir Adsorption with Lateral Interactions

One assumption of the Langmuir model is that the adsorbed molecules do not interact with each other. It is possible to modify the theory and take such an interaction into account. Therefore, we assume that each binding site has n neighboring binding sites. The average number of neighbors of an adsorbed molecule is n𝜃. When we denote the additional binding energy related to the interaction between a pair of neighboring molecules by EP , we can consider lateral interactions by modifying the Langmuir equation. Therefore, it is convenient to write the Langmuir equation as follows: 𝜃 = KL′ P 1−𝜃 and modify the Langmuir constant: ( ) ( ) Q + nEP 𝜃 nEP 𝜃 KL′ = KL0 exp = KL exp RT RT

(8.31)

This equation is sometimes called the Frumkin–Fowler–Guggenheim (FFG) isotherm [602–604]. For 𝛽 ≡ nEP ∕RT < 4 lateral interactions cause a steeper increase of the adsorption isotherm in the intermediate pressure range. Characteristic of all Langmuir isotherms is a saturation at high partial pressures P∕P0 → 1.

231

8 Adsorption

Amount adsorbed (μmol)

0.4 0.8 0.6

β=6

β=4

θ

232

β=2 β=0

0.4 0.2 0.0 0.01 (a)

0.1

1

81.8 K 0.3 0.2 0.1 0.0 0.0

10

KLP

79.2 K

(b)

0.1 0.2

0.3 0.4

0.5 0.6 0.7

Pressure (KPa)

Figure 8.8 (a) Frumkin–Fowler–Guggenheim (FFG) adsorption isotherms (coverage 𝜃 versus the pressure in units of KL−1 ). The curves were calculated using Eq. (8.31) with 𝛽 = 0, 2, 4, 6. For 𝛽 = 6, the physically correct adsorption curve is plotted as a continuous curve while the one calculated with Eq. (8.31) is plotted as a dotted curve. (b) Adsorption isotherms for krypton adsorbing to the (0001) plane of graphite at two different temperatures. The dotted curves were fitted using Eq. (8.31) with 𝛽 = 4.5. Experimental results were taken from Ref. [605].

A remarkable shape is calculated with Eq. (8.31) for 𝛽 > 4. A region is obtained where the 𝜃-versus-P curve has a negative slope (dotted curve in Figure 8.8). This is physically meaningless: the coverage would decrease with increasing pressure and for one pressure, there would be three possible values of 𝜃. In reality, this is a region of two-phase equilibrium. Single adsorbed molecules and clusters of adsorbed molecules coexist on the surface. The situation is reminiscent of the three-dimensional van der Waals equation of state which can be used to describe condensation. As an example, Figure 8.8 shows adsorption isotherms of krypton on the (0001) face of graphite. The dashed lines were fitted using Eq. (8.31) with 𝛽 = 4.5. In reality, the coverage increases steeply and the two-phase region can be identified. Figure 8.8 shows another typical feature of adsorption: The amount adsorbed decreases with increasing temperature.

8.3.4

The BET Adsorption Isotherm

The Langmuir model is well suited to describe adsorption at submonolayer coverage. The maximal adsorption is that of a monolayer. Langmuir adsorption isotherms all saturate at high vapor pressures. This is unrealistic for many cases. In order to consider the adsorption of multilayers, Brunauer, Emmett, and Teller extended the Langmuir theory and derived the so-called BET adsorption isotherm [606] (Figure 8.9). It is assumed that the adsorption heat for the first layer Q1 has a particular value. For all further layers, the heat of adsorption Qc corresponds to the heat of condensation of the liquid. Another condition is that desorption and adsorption take place only directly between vapor and surface. Adsorbed molecules are not

8.3 Adsorption Models

S2

S1

S0

S1

S2

S3

S1 S0 S2 S0

Figure 8.9 BET model of adsorption.

allowed to move from one layer directly to another. In equilibrium, the desorption rate for each layer must be equal to the adsorption rate. We have the following adsorption and desorption rates: 1 Adsorption to vacant surface sites kad PS0 Desorption from first layer = a1 S1 e−Q1 ∕RT c Adsorption to the ith layer kad PSi−1 Desorption from the ith layer = ac Si e−Qc ∕RT

(8.32)

Here, a1 and ac are frequency factors like 1∕𝜏0 . As a result, we get Γ = ( Γmon 1−

P P0

C ) [ 1+

P P0

P ] P 0 (C − 1)

(8.33)

Here, Γmon is the number of adsorbed moles in one full monolayer per unit area (each binding site is occupied exactly once), P0 is the saturation vapor pressure, and C=

1 ac kad c a1 kad

e(Q1 −Qc )∕RT ≈ e(Q1 −Qc )∕RT

(8.34)

Equation (8.33) shows that, Γ∕Γmon becomes infinite for P∕P0 → 1. This is what we expect because condensation sets in. For the second part of Eq. (8.34) see Exercise 8.3. Since the BET adsorption isotherm is so widely used, we describe a simple derivation [1]. It is convenient to define two parameters, 𝛼 and 𝛽, according to 𝛼=

1 P kad

a1

eQ1 ∕RT

and

𝛽=

c kad P

ac

eQc ∕RT

(8.35)

Using these parameters, we can write S1 = 𝛼S0

and

S2 = 𝛽S1

(8.36)

In general Si = 𝛽 i−1 S1 = 𝛼𝛽 i−1 S0 = C𝛽 i S0

(8.37)

The number of moles adsorbed per unit area is given by Γ = S1 + 2S2 + 3S3 + · · · =

∞ ∑ i=1

iSi = CS0

∞ ∑ CS0 𝛽 i𝛽 i = − 𝛽)2 (1 i=1

(8.38)

The series converges because 𝛽 < 1 as can be seen from Eq. (8.36); the ith layer has less molecules than the underlying layer. Monolayer coverage can be written as follows: ∞ ∑ CS0 𝛽 𝛽 i = S0 + (8.39) Γmon = S0 + S1 + S2 + S3 + … = S0 + CS0 1−𝛽 i=1

233

8 Adsorption

Now, we divide Eq. (8.38) by Eq. (8.39): Γ Γmon

=

CS0 𝛽∕(1 − 𝛽)2 C𝛽 = S0 + CS0 𝛽∕ (1 − 𝛽) (1 − 𝛽) [1 − 𝛽 + C𝛽]

(8.40)

An essential step is to realize that the factor 𝛽 is equal to P∕P0 . To see this, let us consider the adsorption of a vapor to its own liquid at equilibrium and hence at P0 . We take this situation to be similar to the situation of the vapor adsorbing to the ith (not the first) layer. Taking the rate of binding equal to the rate of desorption, we write c P0 Γmon = ac Γmon e−Qc ∕RT kad

(8.41)

On both sides, we insert the total number of binding sites per unit area Γmon because all sites can serve as binding and desorption sites. This immediately leads to a P0 = cc e−Qc ∕RT (8.42) kad By inserting Eq. (8.42) into Eq. (8.35), we get 𝛽 = P∕P0 and with Eq. (8.40) obtain the BET isotherm Eq. (8.33). Figure 8.10 shows how BET isotherms depend on the parameter C. For high values of C, the binding of vapor molecules directly to the surface is strong compared to the intermolecular interaction. Therefore, at least for low pressures, a Langmuir type of adsorption is obtained. Only at high pressures, do the molecules start to form multilayers. For low values of C, the molecules prefer binding to themselves while the binding energy to the surface is low. For this reason, the first monolayer only forms at relatively high pressures. Once it has formed, it is easier for the next molecules to adsorb. 5 0.8

3

Thickness (nm)

4 Γ/Γmom

234

C = 0.3 C=1

2 C=3

1 0 0.0 (a)

0.4 Alumina 0.2

C = 50 C = 10

0.2

0.6

0.4

0.6 P/P0

0.8

0.0 0.0

1.0 (b)

Silica 0.2

0.4

0.6

0.8

1.0

P/P0

Figure 8.10 (a) BET adsorption isotherms plotted as total number of moles adsorbed, Γ, divided by the number of moles in a complete monolayer, Γmon , versus the partial pressure, P, divided by the equilibrium vapor pressure, P0 . Isotherms were calculated for different values of the parameter C. (b) Adsorption isotherms of water on a sample of alumina (Baikowski CR 1) and silica (Aerosil 200) at 20 ∘ C (P0 = 2.7 kPa, redrawn from Ref. [607]). The BET curves were plotted using Eq. (8.33) with C = 28 (alumina) and C = 11 (silica). To convert from Γ∕Γmon to thickness, the factors 0.194 and 0.104 nm were used, which correspond to Γmon = 6.5 and 3.6 water molecules per nm2 , respectively.

8.3 Adsorption Models

BET isotherms are widely used to fit experimental adsorption isotherms. Two examples, the adsorption of water to alumina and silica, are shown in Figure 8.10b. The adsorption of alumina can be fitted with the BET equation up to a relative pressure of 0.4, which is quite typical. For silica, the fit is acceptable even up to P∕P0 = 0.8.

8.3.5

Adsorption on Heterogeneous Surfaces

Surfaces are usually not perfectly homogeneous. Different crystal faces are exposed to the vapor, defects and other deviations from the perfect lattice are present. Often, there are different types of molecules as in steel (e.g. Fe, C, Ni, Co) or in glass (e.g. SiO2 , B, Na, K) and their concentrations on the surface might vary locally. On heterogeneous surfaces, the binding energy of an adsorbate will generally not be a fixed value, but there is a distribution of binding energies. The probability of finding a binding site in the energy interval Q…Q + dQ is described by a distribution function f (Q)dQ. The experimentally observed adsorption is the sum of all adsorption events on all different kinds of binding sites. At a fixed temperature, the coverage is [1, 608] ∞

𝜃(P) =

∫0

𝜃 H (Q, P) f (Q) dQ

(8.43)

The distribution function is normalized: ∞

∫0

f (Q)dQ = 1

(8.44)

For the adsorption isotherm 𝜃 H (Q, P) on a well-determined homogeneous part of the surface, the Langmuir equation is often used. A relatively well-known adsorption isotherm, the Freundlich4 isotherm [609] ( ) kB∗T P Q 𝜃= (8.45) P0 is obtained with an exponentially decaying distribution of adsorption sites accord∗ ing to f (Q) ∝ e−Q∕Q and assuming a Langmuir behavior for 𝜃 H [610]. Here, Q∗ is a constant which characterizes the distribution of adsorption energies. A requirement in the derivation is that Q∗ > kB T (Exercise 8.6). This requirement, however, is not a severe restriction because for Q∗ < kB T adsorption is negligible. The decreasing slope of the adsorption isotherm is a result of the heterogeneity. There are adsorption sites, which have a high affinity, and regions, which have a low affinity. The high affinity sites are occupied first, which accounts for the steep increase at low pressure. Another reason is sometimes a lateral repulsion between adsorbed molecules. Often, one would like to conclude something about the distribution of the binding energies using measured adsorption isotherms. This is difficult. Usually, certain assumptions concerning 𝜃 H have to be made. For a detailed discussion of adsorption of gases on heterogeneous surfaces, see Ref. [611]. 4 Herbert Max Finlay Freundlich, 1880–1941; German physicochemist. Professor in Berlin and Minneapolis.

235

236

8 Adsorption

8.4 Experimental Aspects of Adsorption from the Gas Phase 8.4.1

Measuring Adsorption to Planar Surfaces

Historically, adsorption has first been measured for powders and textured materials. One reason was that many industrial and natural materials are textured and porous: textiles, paper, bricks, sand, porous rocks, food products, zeolites, etc. Another simple reason was that only for materials with a high-specific surface area the amount adsorbed could be detected. More recently, methods have also been developed to measure the adsorbed amount on single, flat surfaces, and not only onto powders. Conceptually, it is more instructive to start with planar surfaces and then proceed to powders and porous materials. Adsorption to isolated surfaces can be measured with a quartz crystal microbalance (QCM) [612]. The QCM consists of a thin quartz crystal that is plated with electrodes on the top and bottom (Figure 8.11). Since quartz is a piezoelectric material, the crystal can be deformed by an external voltage. By applying an AC voltage across the electrodes, the crystal can be excited to oscillate in a transverse shear mode at its resonance frequency. This resonance frequency is highly sensitive to the total oscillating mass. For an adsorption measurement, the surface is mounted on such a QCM. Upon adsorption, the mass increases, which lowers the resonance frequency. This reduction of the resonance frequency is measured and the mass increase is calculated [612–614]. QCMs can also be operated in aqueous medium and are thus suitable to detect the adsorption of biomolecules [615]. Another technique applicable to single flat surfaces is ellipsometry. In ellipsometry, the change in polarization of a light beam upon reflection from a surface is detected. Ellipsometry is not only used in adsorption studies, but it can in general be used to measure the thickness of thin layers. To understand ellipsometry, we first have to remember that the reflection of light from a surface depends on the direction of polarization. Light polarized parallel and perpendicular to the surface is reflected differently. Amplitude and phase change in a different way depending on polarization. In ellipsometry, we direct monochromatic, polarized light onto a surface. At the interface, the polarization changes. Two parameters, the so-called ellipsometric angles Δ and Ψ, are measured. The first parameter is Δ = 𝛿in − 𝛿out , where 𝛿in is the phase difference between the parallel and perpendicular component of the incoming wave. 𝛿out is the phase difference between the parallel and perpendicular component of the outgoing wave. The second parameter is given | | by tan Ψ = |Rp | ∕ ||Rs ||, where Rp and Rs are the reflection coefficients for the light | | Shear oscillation

Electrodes

Quartz crystal

Adsorbed layer

AC

Figure 8.11 Working principle of a quartz crystal microbalance.

8.4 Experimental Aspects of Adsorption from the Gas Phase

Laser

Detector

Polarizer

Polarizing unit

Compensator

Polarizer

(a)

Sample

Polarizing unit

Polarizer Polarizer Compensator

(b)

Figure 8.12 (a) Schematic of an ellipsometer. At panel (b), a so-called null ellipsometer is shown. In a null ellipsometer, polarizer and analyzer are rotated alternately until the null is found.

polarized parallel and perpendicular. From the amplitude ratios and the phase shift between the incident and reflected light, we can determine the thickness of a thin film, assuming that the refractive index of the material is known. This is, however, far from being trivial. The equations are not simple and usually rely on severe assumptions [616, 617]. Different experimental configurations are used to determine Δ and Ψ (Figure 8.12). For a practical introduction, see Ref. [616]. A laser provides linearly polarized light. It is converted to circularly polarized light by a quarter-wave plate. A quarter-wave plate is an optical element that introduces a relative phase shift of 90∘ between the constituent orthogonal components of the electric field vector. As a result, we get circularly polarized light which has equal amplitudes of perpendicular and parallel components and a phase shift of 90∘ . A first polarizer is used to select any linear polarization without changing the amplitude. After reflection from the sample surface, the light will be elliptically polarized. A compensator, which introduces an adjustable phase shift, is used to convert the elliptic back to a linear polarization. This phase shift is the value Δ. The orientation of the resulting linearly polarized light is measured with a second polarizer called analyzer. It is adjusted until the intensity at the detector is zero. This provides Ψ. The angle of incidence is chosen close to the Brewster angle of the sample to ensure a maximal difference in Rp and Rs . In common practice, however, the compensator is often placed before the reflection (Figure 8.12b). This arrangement is called a null ellipsometer or polarizer-compensator-sample-analyzer (PCSA) ellipsometer.

237

8 Adsorption

Figure 8.13 Adsorption of nitrogen to a single basal plane of graphite at a temperature of 46.2 K as determined by ellipsometry. Plotted is the change in the ellipsometric angle △ versus pressure. The subsequent adsorption of at least four layers at defined pressures can be discriminated. Source: Redrawn from Ref. [618].

1.0

Δ (°)

238

0.5

0.0

0

100

200

300

400

Pressure (Pa)

As one example, the adsorption isotherm for nitrogen to the basal plane of graphite is shown in Figure 8.13.

8.4.2

Measuring Adsorption to Powders and Textured Materials

Several experimental methods are applied to measure adsorption isotherms to materials with a high specific surface area. The main problem is to determine the amount adsorbed. One method is a gravimetric measurement. In a gravimetric measurement the weight increase as a function of the pressure is determined. The adsorbent, usually in the form of a powder, is placed into a bulb and kept at the desired temperature. The bulb is mounted on a sensitive balance. Before the experiment, the bulb is evacuated. Then the gas of interest is admitted into the bulb at a certain pressure. The increase in weight is measured. Dividing by the total surface area of the adsorbent, we get the amount of adsorbed gas. The pressure is increased, and the weight measurement is repeated. In this way, a whole isotherm is recorded. In volumetric measurements, the volume of an adsorbed gas at constant pressure and temperature is determined. Therefore, we first determine the “dead space” or volume of the bulb by admitting some nonadsorbing (or weakly adsorbing) gas such as helium. Then, after evacuating the bulb, the gas of interest is admitted into the bulb. This is done at constant pressure and temperature. The volume admitted into the bulb minus the dead space is the volume adsorbed. For all experiments with porous materials or powders, we need to know the specific surface area. The most widely used procedure for the determination of the surface area of finely divided and porous materials is the BET gas adsorption method [583, 587]. The principle of the method is rather simple. We measure an adsorption isotherm and fit the result with the BET isotherm Eq. (8.33). Then we assume a reasonable value for the cross-section area of a gas molecule 𝜎A . The specific surface ∑ area is = nmon 𝜎A NA . Examples are given in Table 8.2. Some cross-sectional areas for suitable gases in Å2 are the following: N2 : 16.2; O2 : 14.1; Ar: 13.8; n-C4 H10 : 18.1. How do we practically determine the specific surface area? We measure the adsorption isotherm of a defined mass of adsorbent mad and fit it with the BET

8.4 Experimental Aspects of Adsorption from the Gas Phase

Table 8.2 Calculated specific surface areas Σ in m2 /g for a batch of anatase, porous glass, a silica gel, and a special sample of the protein albumin measured with different gases [619]. N2

Ar

Anatase, TiO2

13.8

11.6

Porous glass

232

217

Silica gel

560

477

Albumin

11.9

10.5

O2

CO

CH4

14.3

CO2

9.9

NH3

159

207

9.6 164

464

C2 H2

550

455 10.3

equation. Usually, the BET model describes adsorption for 0.05 < P∕P0 < 0.35 reasonably well, and we can restrict the measurement to that pressure range. Most commonly, the volumetric technique is applied. As a result of our measurements, we get the volume of gas adsorbed, V ad . To be meaningful, the conditions (pressure, temperature) at which the volume of the gas is reported, have to be given. Usually, standard conditions are chosen, although the actual experiment is often done at lower temperature. For the analysis, it is convenient to transform the BET adsorption equation. First, we express the number of moles adsorbed ad , where V ad is the volume of gas required by the volumes: n∕nmon = V ad ∕Vmon mon to get one complete monolayer. Inserting into Eq. (8.33) and rearrangement leads to P∕P0 P∕P0 (C − 1) 1 + (8.46) ( ) = ad ad ad CVmon CVmon V 1 − P∕P0 P∕P

0 As a result, we plot V ad 1−P∕P versus P∕P0 . This should give a straight line with a 0) ( ad ) ( ad )( slope (C − 1) ∕ CVmon and an intersection with the ordinate at 1∕ CVmon . From ad can be calculated. The specific surface area the slope and intersection C and Vmon ( ) ad N 𝜎 ∕ V m is given by Σ = Vmon A A m ad , where Vm is the molar volume of the gas phase.

8.4.3

Adsorption to Porous Materials

Definitions and Classification

Many solids of high surface area are to some extent porous. Then not only the specific surface area is of interest, but we also want to know the texture. For a powder of porous particles, the texture is defined by the detailed geometry of the void and pore space (Figure 8.14). We distinguish between open pores, which are connected by a channel to the surface of the particles as opposed to closed pores. Closed pores are not accessible by the vapor. They only reduce the density. Void is the interstice between particles. It is instructive to distinguish between powder porosity and particle porosity. Powder porosity is the ratio of the volume of the voids plus the volume of open pores and ink-bottle pores to the total volume occupied by the powder. Particle porosity is the ratio of open pore volume to the volume of the whole particle.

239

240

8 Adsorption

Open pore

Ink-bottle pore

Void Interconnected pore

Closed pore

Figure 8.14 Schematic of a powder made of porous particles. The different types of pores are indicated.

IUPAC recommends to classify pores according to their size [587, 620]: ●

●

●

Macropores have a diameter larger than 50 nm. Macropores are so wide that gases adsorb virtually to flat surfaces. Mesopores are in the range of 2–50 nm. Capillary condensation often dominates the filling of mesopores. Below the critical temperature, multilayers arise. The pore size limits the number of layers. Micropores are smaller than 2 nm. In micropores, the structure of the adsorbed fluid is significantly different from its macroscopic bulk structure. Confined liquids are a highly active area of research because of their unique properties. Important examples are zeolites, which are used for catalysis or to soften water in laundry.

This classification is certainly not perfect because the filling of pores is also determined by their shape (cylinders, slits, cones, irregular) and pores may be separate or connected. To characterize the size of pores, the hydraulic radius ah was introduced. Its definition is similar to the characteristic length scale introduced in chapter 1. The hydraulic radius is the ratio of pore and void volume to surface area. For a long cylindrical pore of length l and radius r (r ≪ l) the hydraulic radius is for instance r 𝜋r 2 l = (8.47) 2𝜋rl 2 Above a certain critical temperature, no multilayers should adsorb. Hence, even porous solids should behave like flat surfaces, if no pores of molecular size are present. Capillary condensation plays a role only below the critical temperature. Measuring pore size distributions is often not an easy task, once we deal with meos- or even micropores. Electron microscopy can help, but typically special sample preparation is required. In addition, the images have to be carefully analyzed to get quantitative information of pore sizes. ah =

8.4 Experimental Aspects of Adsorption from the Gas Phase

Adsorption and Capillary Condensation

To illustrate adsorption into porous materials, we first consider as a model the adsorption of vapor into a cylindrical pore of radius rc . The pore is closed at one end (Figure 8.15) [621, 622]. At low pressure, a thin layer of molecules adsorbs to the surface. Capillary condensation only plays a minor role at the edges at the bottom of the pore. As the pressure increases, the thickness of the adsorbed layer increases. This leads to a reduced radius of curvature for the liquid cylinder. Once a critical radius is reached, capillary condensation sets in and the whole pore fills with liquid. According to the Kelvin equation (Eq. 2.26), this critical radius is reached when rc − h = −

2𝛾L Vm RT ln(P∕P0 )

(8.48)

Here, h is the thickness of the adsorbed layer. The amount adsorbed in the pore resembles a Type IV isotherm (Figure 8.2), except that the adsorption isotherm is reversible; adsorption and desorption isotherms coincide. Capillary condensation can be used to determine the pore size distribution of a mesoporous material [46, 620]. The Kelvin equation provides an absolute size standard. The plateau of a Type IV isotherm is usually taken as the starting point. We assume that all pores are filled. In the first step of the desorption process, e.g. from P∕P0 = 0.95 → 0.90, only capillary condensate is removed. In the following steps, both condensate from the cores of pores is removed and gas desorbes from multilayers in the larger pores (i.e. those pores already emptied of condensate). Assuming a reasonable adsorption isotherm for multilayer adsorption, the pore size distribution can be calculated in a stepwise way from a measurement of gas pressure and the amount adsorbed. Usually, nitrogen at −195.8 ∘ C or argon at −185.7 ∘ C are used as a gas because the thickness of their respective multilayers are largely insensitive to the specific adsorbent and the same equipment can be used for determining the surface area and mesoporous size distribution; −195.8 and −185.7 ∘ C are the respective boiling temperatures at normal pressure. One example is shown in Figure 8.16. Different mathematical procedures have been proposed to derive the pore size distribution from nitrogen adsorption isotherms [583, 623, 624]. They rely on several assumptions: (i) the pores are rigid and have a well-defined shape and fixed

h

2rc

V

0

P/P0

1

Figure 8.15 Filling and emptying of a cylindrical solid pore with liquid from its vapor and the schematic adsorption isotherm. The dashed circles indicate the curvature given by the Kelvin equation at a certain pressure.

241

8 Adsorption

0.8 ΔV/Δr (ml/nm)

0.5

VN2 (ml)

242

0.4 0.3

0.6 0.4 0.2

0.2 0.0 (a)

0.2

0.4 P/P0

0.6

0.8

0.0

1.0 (b)

0

1

2

3

4

5

Pore radius (nm)

Figure 8.16 Dollimore and Hill [624] analyzed the pore structure of a silica gel by nitrogen adsorption at −196∘ C. (a) The amount adsorbed of liquid N2 per gram of adsorbent measured in the desorption branch. (b) From these values, the amounts of nitrogen removed in each step are calculated. Using a semiempirical adsorption isotherm [ ]1∕3 h = 0.43 −5∕ ln(P∕P0 ) nm for the thickness of adsorbed multilayers, the pore size distribution could be calculated step by step. It is reported in pore volume per gram of adsorbent and per interval of pore radius. One assumption in all calculations was that the pores are cylindrical. This specific silica gel is dominated by pores of 2 nm radius. Isotherms saturate at high P∕P0 , because filling up of pores decreases the available surface area.

contact angle. Usually, the contact angle is taken to be zero and a cylindrical shape is assumed. (ii) Filling and emptying of pores does not depend on the location within the sample. Effectively, this implies that small pores are always connected by larger pores to the outside. Otherwise, a large pore might be disconnected to the vapor phase by a liquid slug. (iii) Adsorption on the pore walls proceeds in the same way as on the corresponding open surface. (iv) The Kelvin equation is valid over the whole range of pore sizes with the same surface tension of the liquid. In reality, interfacial forces [621, 625] and the curvature of the liquid surface [626] might change the effective surface tension. Due to slow diffusion and specific adsorption, cryogenic adsorption usually cannot be used to analyze micropores around 1 nm or below. In recent years, carbon dioxide has turned out to be a suitable probe molecule to characterize microporous materials. It is adsorbed even at ambient temperature, allowing for faster diffusion rates and penetration into the narrowest micropores. Typical operation temperatures are 273 K, at which it condenses at 35 bar. Adsorption/desorption isotherms are then analyzed based on computer simulations [627]. It is evident that the use of the adsorption method to determine mesopore size distributions is subject to a number of uncertainties. Uncertainties arise from the assumptions made and the complexity of real pore structures. When a hysteresis is observed, it is not evident, which branch should be used for the analysis. A fundamental problem is that an adsorption isotherm is not unambiguously related to a specific pore size distribution. Rather, many different pore size distributions lead to the same adsorption isotherm. Still, physisorption is one of the few nondestructive methods to characterize mesoporosity.

8.4 Experimental Aspects of Adsorption from the Gas Phase

Hysteresis and Ink-Bottle Effect

Adsorption to porous materials is often characterized by hysteresis in the adsorption behavior. Such a hysteresis is observed when, after the adsorption process, a desorption experiment is done in which the pressure is progressively reduced from its maximum value and the desorption isotherm is measured. During the desorption process, the liquid phase vaporizes from the pores. The desorption isotherm does not precisely track the adsorption isotherm, but lies above it. The topology of the pore network plays a role [628]. Interconnected pores show a different adsorption and desorption isotherm than pores closed at one end. For a cylindrical pore of radius rc , this is illustrated in Figure 8.17a [622, 623, 629, 630]. The adsorption cycle starts at a low pressure. A thin layer of vapor adsorbed onto the walls of the pore (i). With increasing pressure, the thickness of the adsorbed layer increases. This leads to a reduced radius of curvature for the liquid cylinder a. Once a critical radius ac is reached (ii), capillary condensation sets in and the whole pore fills with liquid (iii). When decreasing the pressure again, at some point the liquid evaporates. This point corresponds to a radius am which is larger than ac . Accordingly, the pressure is lower. 3

4

rc

Γ

2am

3 Liquid 4

1

2

2 a

dc

1

Vapor 0

1

P/P0

(a) 3

4 Γ

Liquid

3 4 2

1 2 1

Vapor Solid

0

P/P0

1

(b)

Figure 8.17 Filling and emptying of an interconnected cylindrical pore (a) and an inkbottle pore (b) with liquid from its vapor and the schematic adsorption/desorption isotherms.

243

244

8 Adsorption

During the desorption process, vaporization can occur only from pores that have access to the vapor phase, and not from pores that are surrounded by other liquid-filled pores. There is a “pore blocking” effect in which a metastable liquid phase is preserved below the condensation pressure until vaporization occurs in a neighboring pore. Therefore, the relative pressure at which vaporization occurs depends on the size of the pore, the connectivity of the network, and the state of neighboring pores. For a single “ink bottle” pore this is illustrated in (Figure 8.17b). The adsorption process is dominated by the radius of the large inner cavity while the desorption process is limited by the smaller neck. Porous Materials

Our understanding of the adsorption from vapor and liquid phases into porous materials has greatly progressed in the last two decades. It is driven by many applications such as gas exchange and storage, catalysis, decontamination, soil treatment, or purification of air or water. Materials with different pore size distributions, chemical composition, and porosity have been developed (Figure 8.18) [631]. Activated carbons are the oldest and still widely used porous materials [632]. Activated carbon is made from different carbon sources such as coconut shells, wood, or coal. The first two materials are dried and charred at temperatures up to 600 ∘ C. Afterwards it is “activated” either chemically or physically. During activation, the carbon becomes porous and is partially oxidized. In chemical activation, the carbon materials are heated to 250–600 ∘ C in a bath of acid, base, or other chemicals. In physical activation, it is exposed to an oxidizing atmosphere at typically 600–1100 ∘ C. Activated carbon can be viewed as an assembly of graphene crystallites which are chemically modified. Activated carbons have a wide pore distribution, which varies depending on how they were fabricated. Typical specific surface areas are 500–3000 m2 /g. Mesoporous silicates are synthesized by a sol–gel process from functional silanes, e.g. tetraethoxysilane (Figure 8.18b), using supramolecular assemblies as a template [633]. Usually, cylindrical surfactant micelles (chapter 11) or micelles from block copolymers are used as a template. Once the silicon oxide has been formed, the material is heated to burn the organic templates away. Typical pore diameters are 2–10 nm, depending on the template molecule. An example of the adsorption to such a material, called MCM-41, is shown in Figure 8.19. With increasing pressure, first a layer is adsorbed to the surface. Up until a pressure of P∕P0 ≈ 0.45 is reached, adsorption could be described by a BET adsorption isotherm equation [45, 634]. Then capillary condensation sets in. At a pressure of P∕P0 ≈ 0.75, all pores are filled. This leads to a very much reduced accessible surface and practically to saturation. When reducing the pressure, the pores remain filled until the pressure is reduced to P∕P0 ≈ 0.6. At P∕P0 ≈ 0.45, all pores are empty and are only coated with roughly a monolayer. Adsorption and desorption isotherms are indistinguishable again below P∕P0 ≈ 0.45. Zeolites are microporous aluminosilicate minerals with typical pore diameters of 0.3–1 nm [635]. The term “zeolite” comes from the Greek “zeo” (to boil) and “lithos”

8.4 Experimental Aspects of Adsorption from the Gas Phase

Carbon matrix (a)

Functional groups, e.g. –OH, COOH

O O Si O O

O2

Si

Polycondensation

Calcination

Micelles (b)

+ Organic linker

Metal center (c)

MOF

(d)

Figure 8.18 Schematic of different porous materials. (a) The hierarchical porosity shown for activated carbon. (b) Mesoporous silica made in this example by templating an aqueous solution containing tetraethoxysilane and surfactants, which form cylindrical micelles. (c) Example of the structure of zeolite A. (d) Formation of metal-organic frameworks.

Γ (mol/kg)

30

20

10

0 0.0

0.2

0.4

0.6

0.8

1.0

P/P0

Figure 8.19 Adsorption isotherms of argon at 87 K in siliceous material MCM-41 with cylindrical pores of 6.5 nm diameter. Source: Redrawn after Ref. [634].

245

246

8 Adsorption

(stone). It was coined in 1756, by the Swedish mineralogist Axel Fredrik Cronstedt. He observed that when heating zeolites, a lot of steam is produced. The steam comes from water that had been adsorbed in the porous material. Zeolites occur naturally as a mineral and are synthesized on an industrial scale [636]. Metal–organic frameworks (MOF). MOFs are composed of two components: a metal ion or metal cluster and an organic linker molecule (Figure 8.18d) [637]. Pore diameters range from 0.3 to 1.3 nm and specific surface areas reach up to 4500 m2 /g. MOFs are less rigid than the other materials and in some cases, change their structure when they adsorb molecules. Each new compound gets a new number. One example is MOF-5. It consists of a cubic structure with an oxygen-centered Zn4 O tetrahedron bridged by 1,4-benzenedicarboxylate (HOOC-C6 H4 -COOH) as linkers. MOF-5 has been suggested for hydrogen storage [638]. Clays. In aqueous medium, clays are an important porous material with a high specific surface area. Clays are fine-grained natural soil materials containing clay minerals. Clay minerals are characterized by two-dimensional sheets of aluminosilicates.

Mercury Intrusion Porosimetry

A widely used technique to measure macro- and mesopore distributions is mercury intrusion porosimeter, or briefly mercury porosimetry [620, 639]. It is for example applied to analyze cement-based materials or soils. In a mercury porosimeter, the porous sample of known mass is placed into the sample chamber. To remove residual gas and adsorbed liquid the sample chamber is evacuated. Then the chamber is filled with mercury. Since mercury does not wet most materials spontaneously, it does not intrude into empty pores unless a pressure is applied. Progressively increasing pressure P is applied to the mercury while monitoring the volume of mercury in the sample chamber. After each step, the system is allowed to equilibrate. The intruded volume is plotted as a function of the applied pressure. Then the pressure is stepwise reduced, and the extruding volume is recorded. To analyze volume-versus-pressure curves, it is usually assumed that the pores have a cylindrical shape and are all accessible to the outer surface of the specimen. The pressure required to fill a cylindrical pore of radius r is according to Eq. (6.30) [405] P=

2𝛾Hg |cos Θ| r

(8.49)

Here, 𝛾Hg is the surface tension of mercury. The contact angle of mercury depends on the solid and lies typically between Θ = 125∘ and 150∘ . The fraction of pores having radii in the interval r…r + dr is obtained from the volume of mercury that intrudes or extrudes in the pressure range P…P + dP. The volume pore-size distribution function DV (r), defined by DV (r) = −

dV dr

(8.50)

8.4 Experimental Aspects of Adsorption from the Gas Phase

is the volume of pores with a radius r per unit interval of pore radius [640]. With Eq. (8.49), we get dP∕dr = −P∕r. We can modify Eq. (8.50) dV dP P dV dV =− = dr dP dr r dP Often, the volume log radius distribution DV (log r) is used:

(8.51)

DV (r) = −

DV (log r) =

r dV dV P dV = D (r) = = d log r log e V log e dP d log P

(8.52)

Commercial mercury porosimeters can typically measure a pressure range between 0.004 and 400 MPa leading to a minimal pore diameter of 3 nm. Practically, the interpretation of volume-versus-pressure curves suffers from a number of problems. Like in adsorption experiments, a fundamental problem is that measured intrusion/extrusion curves are not uniquely linked to a specific pore size distribution. Many distribution functions can reproduce a measured curve. Additional information, for example from X-ray tomography [640] or scanning electron microscopy [641], is required to get a realistic picture. The first intrusion curve typically leads to a different volume distribution than the extrusion curve because mercury is entrapped; one example is shown in Figure 8.20. This has been attributed to interconnected pores, to ink-bottle pores, or to the fact that the advancing and receding contact angles of mercury on the specific material can be different. Often, a sudden increase in volume of voids filled with only slight increase in pressure is observed. It was soon recognized that Eq. (8.49) fails to explain such a behavior in most cases. For a powder, the observation of a breakthrough pressure is natural. Once the pressure is high enough for mercury to penetrate the narrowest voids between particles, it will penetrate the whole powder. If we, for example, consider a powder consisting of spherical particles of equal radius Rp which are hexagonally closed-packed, the pressure required for mercury to penetrate is

Extrusion

15 10 5 0

(a)

10

20

Rel. pore volume (%)

Cumulative pore volume (%)

25

Intrusion

Intrusion

8 Extrusion 6 4 2 0

0.01

0.1

1 Pressure (MPa)

10

100

0.01

(b)

0.1

1

10

Pore diameter (μm)

Figure 8.20 Using mercury porosimetry, Krakowiak et al. [641] analyzed a common building brick. (a) Experimental cumulative change in volume versus applied pressure. The material had a porosity of 23%, which was intruded by mercury when gradually increasing the pressure to 100 MPa. When reducing the pressure, mercury came out of the material. Some mercury was trapped so that even at zero pressure 10% of the brick was still filled with mercury. (b) Relative pore volume distribution DV (log r) versus pore diameter. The pore diameter was calculated from the pressure with Eq. (8.49) assuming that Θ = 130∘ .

247

248

8 Adsorption

P = 11.3 ⋅ 0.72 ⋅ 𝛾Hg ∕Rp [642]. For a cubic packing, the breakthrough pressure is lower: P = 4.49 ⋅ 0.73 ⋅ 𝛾Hg ∕Rp . A random packing is somewhere in between. Both equations are valid for Θ = 130∘ .

8.4.4

Chemisorption and Temperature-programmed Desorption

Chemisorption is usually connected with a chemical reaction. It is not surprising that the molecules must overcome an activation energy EA (Figure 8.21). Often, the molecules first physisorb to the surface and in a second, much slower, step the bond is established. For desorption both the adsorption energy Q and the activation energy must be overcome. The desorption energy Edes is thus larger than the adsorption energy. Experimentally, information about the adsorption and desorption rates is obtained with the help of temperature-programmed desorption (TPD) [643]. In a TPD experiment, molecules are deposited onto a cold sample kept in ultrahigh vacuum. Then, the sample is heated with a defined heating rate. Desorbing molecules are detected using a mass spectrometer. Usually, distinct maximums are observed which correspond to the breaking of specific bonds. For well-defined crystalline samples, the method is also called thermal desorption spectroscopy (TDS). A modified desorption method is flash desorption. In flash desorption, a surface is instantaneously heated up (normally in vacuum), and we measure the temporal desorption of material, for instance with a mass spectrometer. Heating is usually done with a laser pulse (PLID, pulsed laser-induced thermal desorption). Despite the fact that TPD is conceptually rather simple, the resulting desorption spectra are often challenging to analyze. Dissociation of molecules or interactions with neighboring adsorbed molecules influence desorption spectra [644]. For the simple case of a first-order desorption process, we briefly describe the analysis; a first-order desorption is described by d𝜃∕dt = −kde 𝜃. Desorption is assumed to be an activated process. The desorption rate, that is the decrease in coverage or the number of molecules coming off the surface per unit time, is ∗ kde =−

d𝜃 = a𝜃 e−Edes ∕RT dt

(8.53)

Here, a ≈ 10−13 s−1 is a frequency factor. In the experiment, we increase the temperature of the surface linearly with a rate 𝛽 (in K/s): T = T0 + 𝛽t. A desorption Typical for chemisorption

Typical for physisorption

Potential energy EA Edes Q

Q

(a)

Distance

(b)

Distance

Figure 8.21 Potential energy profile versus distance for chemisorption and physisorption.

8.5 Adsorption from Solution ∗ maximum occurs at a temperature Tm . At this maximum, we have dkde ∕dt = 0. Differentiation of Eq. (8.53) leads to ∗ dkde

E 𝛽 d𝜃 −Edes ∕RTm e + a𝜃 e−Edes ∕RTm des2 = 0 dt dt RTm E 𝛽 ⇒ a e−Edes ∕RTm = des2 RTm =a

(8.54)

Thus, the desorption energy can be calculated from Tm and a. An example, the desorption of thiols from gold is described in Ref. [645].

8.5 Adsorption from Solution In this section, we consider the adsorption of molecules to surfaces in liquids (reviews: [5, 583, 646, 647]). Adsorption from a solution is important, e.g. to remove pollutants [648], in chromatography and for catalysis. Adsorption from solution is a diverse subject. Here, we can only introduce some basic, general features. We restrict ourselves to small, uncharged species and dilute solutions. The important subject of polymer adsorption is described in Ref. [649]. Protein adsorption is reviewed in [650] and the adsorption of surfactants is discussed in Refs. [651, 652]. Adsorption of ions and formation of surface charges was treated in Section 4.1. In dilute solutions, there is no problem in positioning the Gibbs dividing plane, and the analytical surface excess is equal to the thermodynamic one, as it occurs in the Gibbs equation. For liquid mixtures, one needs to be more precise. Then either the so-called relative surface excess or the reduced surface excess is commonly presented as a function of the volume or mass fraction of a certain component. Some basic concepts developed for vapor adsorption can also be applied to the adsorption from dilute solution. However, there are also fundamental differences. In solution, adsorption is always an exchange process. It is an exchange process in two ways: First, a molecule adsorbing to a surface has to replace solvent molecules. Second, the adsorbing molecule gives up part of its solvent environment. This has several practical consequences: ●

●

Molecules adsorbing from the gas phase are small. Otherwise, they would not be in the vapor phase. In contrast, even macromolecules dissolve in a suitable liquid. Examples are water-soluble proteins. In some cases, adsorption to a surface leads to a conformational change of the macromolecules which often make the process irreversible. A prominent example are proteins adsorbing to hydrophobic surfaces [650]. Some proteins, called soft proteins, unfold and expose their hydrophobic parts to the surface. In water (or other liquids with high dielectric permittivity) surfaces and many dissolved molecules carry a charge. Molecules with a charge opposite to that of the surface tend to adsorb while molecules with the same charge are repelled.

249

8 Adsorption ●

●

Molecules do not only adsorb because they are attracted by the surfaces but also because the solution might reject them. An example are hydrophobic substances in water. They readily adsorb to many surfaces because of their dislike for water, rather than a strong interaction with the adsorbent. A thermodynamic treatment has to take into account the exchange character of adsorption. If one adsorbing molecule replaces ν solvent molecules at the surface, the whole reaction is AL + νS𝜎 ⇌ A𝜎 + νSL

●

(8.55)

Here, AL is the adsorptive in solution, A𝜎 is the adsorbate at the surface, SL represents a solvent molecule surrounded by other solvent molecules, while S𝜎 is a solvent molecule at the surface. Multilayer formation is less common in solutions than in the gas phase because the interaction with the adsorbent is screened by the solvent, and the adsorbing molecules have alternative partners. As a result, many adsorption isotherms can be fitted with the Langmuir or Freundlich isotherms. The Freundlich isotherm is usually applied in the empirical form: Γ = KF c1∕n

(8.56)

Γ is typically reported in g/m2 . KF and n are the Freundlich constants used to characterize the system. As an example of adsorption from solution Figure 8.22 shows the isotherm of n-docosane (C22 H46 ) and n-octacosane (C28 H58 ) adsorbing to graphite in n-heptane (C7 H16 ). The longer-chain alkanes are strongly preferred to heptane, indicating that they adsorb in a flat position. A sigmoidal shape is observed with C22 H46 , a hint that the molecules also interact laterally with neighbors. For C28 H58 , adsorption is so strong that even trace amounts all adsorb to the surface. The conclusions – parallel 80 n-C22H46 60 Γ (μmol/g)

250

n-C28H58

40

20

0 0.000 (a)

2 nm 0.002

0.004 0.006 Mole fraction

0.008 (b)

Figure 8.22 (a) Adsorption isotherms of long-chain n-alkanes from n-heptane to graphite. Source: Redrawn from Ref. [653]. (b) Scanning tunneling microscope image of n-heptacosane (C27 H56 ) adsorbed to the basal plane of graphite form an organic solution. The alkanes lay flat on the graphite surface and are highly oriented. Source: Rabe and Buchholz [654], American Association for the Advancement of Science - AAAS.

8.6 Summary

orientation and cooperativity between the adsorbed molecules – are supported by structural studies with the STM. Measuring adsorption from solution is almost a new world compared to the adsorption of gases. When analyzing adsorption to powders or porous material, the traditional method is the immersion method. A known mass of solid with known specific surface area is added in a convenient container filled with solution. The container is sealed and kept at constant temperature. After equilibration, a sample of the supernatant is withdrawn and analyzed to measure the change in concentration. Concentrations can, for example be monitored by measuring the refractive index or by UV or IR spectroscopy. Knowing the volume of the solution and the total surface area of the adsorbent, the surface excess can be calculated. For example see Exercise 8.8. One problem of the immersion method is that for each concentration a new sample has to be prepared. This is tedious and not very accurate. To always use the same sample in flow-through methods, the adsorbent is successively brought into contact with solutions of increasing concentration. After passing the porous adsorbent, the concentration of the solution is measured with the same methods as in the immersion method. In the open-flow method, at each step a fresh solution of known concentration continuously flows through the sample. The concentration at the outlet is lower than the concentration at the inlet as long as equilibrium is not reached. The amount adsorbed is obtained by integrating the concentration with respect to volume or mass of solution which has passed the adsorbent. In circulation methods, the same solution passes continuously and repeatedly through the sample until equilibrium is reached. To measure adsorption to planar surfaces ellipsometry is again applicable [655, 656]. A quantitative measurement is, however, much more demanding. The main reason is that the refractive indices of the adsorbed layer is usually close to that of the bulk liquid. This makes it difficult to separately determine the refractive index and the thickness of the layer.

8.6 Summary ●

●

●

Adsorption isotherms show the amount of a substance adsorbed versus the partial pressure or the concentration. Physisorption is characterized by a relatively weak binding with adsorption energies of a few 10 kJ/mol. In chemisorption, a chemical bond is established between the adsorbate and the adsorbent; desorption energies are in the range of 100–400 kJ/mol. To describe adsorption, Langmuir assumed that independent binding sites exist on the adsorbent. This leads to the adsorption equation: KL P 𝜃= 1 + KL P The Langmuir adsorption isotherm saturates at high partial pressure and the maximal amount adsorbed is that of a monolayer. It is suited to describe submonolayer coverage.

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8 Adsorption ●

●

●

●

In the monolayer coverage region, usually adsorption is more realistically described by the BET model. The BET model considers multilayer adsorption. The adsorption isotherm goes to infinity at relative partial pressures close to one, which corresponds to condensation. Experimentally, adsorption isotherms are determined by gravimetric or volumetric measurements for powders or porous adsorbents. For isolated flat surface, a quartz microbalance or an ellipsometer can be applied to measure adsorption isotherms. For porous adsorbents, capillary condensation is a common phenomenon. It often leads to hysteresis in the adsorption curve. In liquids, adsorption is an exchange process in which adsorbed molecules replace liquid molecules.

8.7 Exercises 8.1

Discuss the significance of the Langmuir constant KL . At which pressure is the adsorbent half covered?

8.2

Estimate the Langmuir parameter KL for nitrogen at 79 K assuming 𝜎A = 16 A2 , 𝜏0 = 10−12 seconds, M = 28 g/mol and Q = 2 kcal/mol.

8.3

BET adsorption isotherm: Verify

8.4

What is the hydraulic radius for a powder of close-packed spherical particles of radius Rp ? A total of 74% of the volume is filled by the particles.

8.5

Kern and Findenegg measured the adsorption of n-docosane (C22 H46 ) in heptane solution to graphite [653]. They used a porous graphite with a specific surface area of 68 m2 /g as determined from BET adsorption isotherms with N2 . Γmax , which is assumed to correspond to monolayer coverage, is found to be 88.9 μmol/g. Can you conclude something about the structure of the adsorbed molecules? What is the area occupied by one molecule compared to its size?

8.6

Verify Eq. (8.45). That is, derive the Freundlich adsorption isotherm assuming an exponential distribution of adsorption energies. You find help in Ref. [657].

8.7

Behm et al. studied the adsorption of CO to Pd(100) [484]. Therefore, they did a series of TDS experiments at a coverage of 𝜃 = 0.15. Using the following heating rates 𝛽, they observed desorption peaks.

1 ac kad c a1 kad

e(Q1 −Qc )∕RT ≈ e(Q1 −Qc )∕RT in Eq. (8.34).

𝛽 (K/s)

0.5

1.2

2.5

4.9

8.6

15.4

25

Tm (K)

449

457

465

473

483

489

492

8.7 Exercises

Calculate the desorption energy Edes and the frequency factor a assuming first-order kinetics. Therefore, rewrite Eq. (8.54) so that on one side you have ( 2 ) ln Tm ∕𝛽 . 8.8

To enrich iron ore and separate the ore from its tailings, one way is to add a polymeric flocculant which helps the particles in the dispersion to aggregate and then to separate by gravity assisted sedimentation. To improve this process, Dash et al. [658] analyzed the adsorption of anionic polyacrylamide to a dispersion of iron ore. They filled mio = 80 g iron ore (mean particle radius Rp = 5 μm) into an cylinder with a volume of V0 = 1 l. In six measurements, the cylinder was completely filled with an aqueous polymer solution of initial concentration ci = 0.1, 1, 6, 19, 62, and 86 mg/l. The cylinder was closed and shaken for 24 hours in order to achieve equilibrium adsorption. Then the particles were centrifuged down. The final concentration of polymer in the supernatant was measured to be cf = 0.0083, 0.079, 5.66, 18.57, 61.19, and 85.06 mg/l. The density of the iron ore is ρ = 3500 kg/m3 . Calculate the polymer adsorbed in mg/m2 and plot it versus the final concentration. Fit the results with the Langmuir (Eq. (8.18)) and Freundlich isotherms (Eq. (8.56)).

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9 Surface Modification 9.1 Introduction In this chapter, we discuss how solid surfaces can be modified. Surface modification is essential for many applications, for example to reduce or enhance friction, wear, or adhesion, to make implants biocompatible, to coat sensors [659, 660], or in microfabrication [661]. Solid surfaces can be changed by a variety of processes such as adsorption, thin-film deposition, chemical reactions, or etching. The technique used depends on the type of surface modification, the material to be deposited, and the intended thickness of a surface layer. For example, for paints and coatings film thicknesses from 1 μm to several tens of microns are common. Such films are applied by brush or via spraying. Processes during film formation are complex, and a discussion would exceed the scope of this book. An introduction can be found in [662]. Some techniques to deposit thin layers of organic material have already been described (Section 6.4.3). Here, we focus on surface modifications employing physical deposition and chemical reactions to deposit inorganic or organic material on a surface. Additionally, we describe the removal of material from a surface by physical or chemical methods. Inorganic films are usually deposited by physical or chemical vapor deposition. They require a vacuum for reasons of purity and minimization of secondary reactions with gas molecules. Such reactions occur if the molecules to be deposited have the chance to collide at an energy high enough to overcome the activation barrier. The probability of a collision in the gas phase is a function of the mean free path λ, which depends on the pressure. The lower the pressure, the higher the mean free path. Another measure for vacuum quality is the covering time 𝜏 necessary to cover a surface with a monomolecular layer, that is, a monolayer of gas molecules (Exercise 2.1). Different types of vacuum are defined in Table 9.1, together with 𝜏 and the mean free path. For high and ultrahigh vacuums, λ becomes larger than the typical size of the chamber.

Physics and Chemistry of Interfaces, Fourth Edition. Hans-Jürgen Butt, Karlheinz Graf, and Michael Kappl. © 2023 WILEY-VCH GmbH. Published 2023 by WILEY-VCH

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Table 9.1 Different categories of vacuum together with the typical mean free path λ and time for monolayer coverage 𝜏 (assuming a sticking coefficient of one) for nitrogen at 20 ∘ C. Pressure (mbar)

Mean free path 𝝀

Covering time 𝝉

Rough vacuum

1000–1

60 nm–60 μm

3 ns–3 μs

Fine vacuum

1–10−3

60 μm–6 cm

3 μs–3 ms

High vacuum

−3

10 –10

6 cm–600 m

3 ms–30 s

Ultrahigh vacuum

< 10−7

> 600 m

> 30 s

−7

9.2 Physical and Chemical Vapor Deposition 9.2.1

Physical Vapor Deposition

In physical vapor deposition (PVD), films are deposited onto a substrate by condensation of a vaporized material without involving chemical reactions (monographs [663, 664]; review [665]). For PVD, the material to be deposited must first be brought into the vapor phase. This is achieved by high-temperature evaporation, plasma sputtering, ablation with laser pulses, or electric arc discharge. Evaporation and sputtering are the most common methods. PVD is used for the production of thin films, most often metallic, of several nanometers to micrometers. For thicker layers, internal stresses of the films may become too high, leading to delamination. Evaporation is a simple and well-understood vacuum technique for thin-film deposition (for the basics, see [666]). The material to be deposited is heated until it starts to evaporate. Since evaporation is done in a vacuum, the mean free path of the molecules is long, and they move practically in straight lines to condense onto the substrate that is placed at an appropriate position. Mainly two methods are used for evaporation: ●

●

For materials with a low melting point, such as gold, the material is placed into a small heat-resistant container (evaporation boat) usually made of tungsten or tantalum (Figure 9.1a). The evaporation boat is heated by an electrical current. Only small amounts of material can be evaporated, and if it is an alloy, the composition might change due to the different partial pressure of the single components in the alloy. Materials with high melting points that are difficult to evaporate can be deposited using electron beam evaporation. An electron beam is guided onto the material by an electromagnetic field to melt it locally or sublime atoms.

Typically, amorphous or polycrystalline layers of some 10 nm thickness are produced by evaporation. To obtain a crystalline surface, the sample can be annealed during or after evaporation. The film thickness is usually monitored by a quartz crystal microbalance (Section 8.4.1). Sputter deposition. Sputtering can be used not only to remove surface layers (Section 7.3.2) but also to deposit material on surfaces (for the basics see [666]).

9.2 Physical and Chemical Vapor Deposition

Sample

Vacuum chamber filled with inert gas at 1 Pa

Vacuum chamber P < 10–3 Pa

HV

Shutter Material to be deposited

(a)

Target

Sample

(b)

Figure 9.1 Schematic of an evaporation (a) and a sputter chamber (b). HV: high voltage.

For this purpose, we include a target composed of the material to be deposited into a vacuum chamber (Figure 9.1b). In addition, a nonreacting gas such as argon is introduced into the chamber at a typical pressure of 1 Pa. The gas is ionized by applying an electric field. The target material is bombarded by Ar+ ions, which knock out atoms from the target. The atoms that are ejected from the target partially condense onto the sample surface and form a thin layer. An important parameter is the kinetic energy of the Ar+ ions. Low-impact energies are not sufficient to break the bonds of the target atoms and high-energy ions would penetrate deeply into the target without disrupting the surface. Typically, energies of several hundred electron volts to several kiloelectron volts are used. To increase the efficiency of the sputtering process, magnetron sputtering was introduced [667]. The main difference is the presence of a magnetic field close to the target, which forces the electrons to follow helical paths around the magnetic field lines. This leads to increased probability of ionizing collisions with neutral gas molecules near the target. The enhanced ionization of the plasma near the target leads to a higher sputter rate. In addition, the plasma can be sustained at a lower gas pressure. While sputtering requires a more complex setup than evaporation, it offers several advantages: ●

●

●

Better step coverage and more homogeneous coverage of large areas. This is mainly due to the larger area of the sputter target. It acts like a planar emitter compared to the pointlike emission for evaporation. Sputtering of alloys can be done without much change in the stoichiometry of the resulting film. Sputtering of metals leads to films with larger grain sizes that exhibit higher electrical conductivity.

Sometimes, different techniques are combined. For example, a combination of evaporation and sputtering is ion plating (for a review see [668]). In ion plating, the evaporated metal atoms like titanium are ionized by electrons or plasma ions. A negative applied potential accelerates the metal ions toward the substrate. By adding a reaction gas like nitrogen, a titanium nitride layer is formed on the solid

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support. This layer is harder and chemically more stable compared to other plating methods. It is an ideal support for adsorbing a new layer with a different material such as gold. This is used, for example, for gold plating in the jewelery industry. In molecular beam epitaxy (MBE) [669], molecular beams are used to deposit epitaxial layers onto the surface of a heated crystalline substrate (typically at 500–600 ∘ C). Epitaxial means that the crystalline structure of the grown layer matches the crystalline structure of the substrate. This is possible only if the two materials are the same (homoepitaxy) or if the crystalline structure of the two materials is very similar (heteroepitaxy). In MBE, a high purity of the substrates and the ion beams must be ensured. Effusion cells are used as beam sources (Example 2.1). MBE is of high technical importance in the production of III–V semiconductor compounds for electronic and optoelectronic devices. For overviews, see [670, 671]. The growth of epitaxial layers (epilayers) on top of a crystalline substrate follows one of three primary modes. The prevailing mode depends on the interaction of the adatoms with either the substrate or the other adatoms in the epilayers [672]. In a simple model that assumes equilibrium between film and vapor phase and neglects irregularities of the substrate like steps and edges [673], the growth mode is determined by the quantity ΔE = γSf + γSs − γSi . Here, γSf and γSs are the surface-free energies of film and substrate, respectively. γSi is the interfacial free energy between film and substrate. If ΔE ≤ 0, the substrate will be wetted by the film and layer-by-layer growth, called Frank–van der Merwe growth [674], will occur (Figure 9.2, left). However, the value of γSi depends not only on the chemical interaction but also on the mechanical strain due to lattice mismatch between film and substrate. Since the total strain increases linearly with the number of atomic layers, stable Frank–van der Merwe growth will occur only for zero lattice mismatch or if the strain is sufficiently released by dislocations or film buckling. Otherwise, the lattice mismatch leads to an increasing strain energy with increasing layer thickness. Above a certain critical layer thickness, typically on the order of a few monolayers, this induces reorganization (Stranski–Krastanow transition) in the epilayers to form three-dimensional islands on top of a thin two-dimensional Frank-van der Merwe Solid substrate

Stranski-Krastanov

Volmer-Weber 0 1 are more likely to form W/O emulsions. Two more detailed guiding principles that are used for practical emulsion formulation are Bancroft’s rule of thumb and the more quantitative concept of the hydrophile–lipophile balance (HLB) scale: ●

●

Surfactants are enriched at the interface, but they are also dissolved in the aqueous and oil phases. Some surfactants are more soluble in water, others in oil. In essence, Bancroft’s rule states that the continuous phase of an emulsion will be the phase in which the emulsifier is preferentially soluble [883, 884]. Griffin suggested an empirical quantitative HLB scale that characterizes the tendency of a surfactant to form W/O or O/W emulsions [885]. The HLB is a direct measure of the hydrophilic character of a surfactant: the larger the HLB, the more hydrophilic the compound. For most surfactants, the scale runs from 3–20. Surfactants with low HLB values (3–6) tend to stabilize W/O emulsions, while those with high HLB values (8–18) tend to form O/W emulsions. How HLB values are determined is described in [886, 887].

Later we introduce another parameter, the phase inversion temperature (PIT), which helps us to predict the structure of emulsions stabilized by nonionic surfactants. The PIT concept is based on the idea that the type of an emulsion is determined by the preferred curvature of the surfactant film. For an introduction to the HLB and PIT concepts, see [888]. Up to this point, we have concentrated on static effects. Often, dynamic effects are equally important in stabilizing emulsions. An emulsifier that can dissolve only slowly from an interface has a stabilizing effect because during the merging of two drops the total surface diminishes, that is, emulsifier must desorb from the interface. In several processes, for instance, in the splitting-off of a droplet from a larger one or directly after the merging of two drops, the surfactant is not distributed homogeneously at the interface; then the Marangoni effect (Section 3.5.4) comes into play. Due to the nonuniform surfactant distribution, those regions with a low surfactant concentration have a high surface tension, and vice versa. Surfactant will flow toward the area of low surfactant concentration and high surface tension. This Marangoni effect quickly reestablishes a uniform distribution of surfactant. It is usually faster than diffusion equilibration.

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In 1903, Ramsden noticed that some proteins and colloidal particles enrich at water–air or water–oil interfaces [889]. Four years later, Pickering published a systematic study of emulsions that are stabilized by colloidal particles situated at the oil–water interface [890]. These are referred to as Pickering emulsions or solid-stabilized emulsions (for a review see [891]). To understand the responsible effect, please remember that a particle assumes a stable position at the liquid–liquid interface if the contact angle is not zero (Section 6.3.2). Upon coalescence of two drops, the solid particles would have to desorb from the interface. This is energetically unfavorable. Common examples of the stabilizing contribution of solid particles are margarine and butter. Both are W/O emulsions. The water droplets are stabilized by small fat crystals. Whether, upon stabilizing with powders, an O/W or a W/O emulsion is formed depends largely on the contact angle [892]. Also, the effect of additional surfactants can often be explained by their influence on the contact angle. In recent years, Pickering emulsions have attracted renewed interest due to the availability of nanoscale powders. Example 11.5 The stabilizing effect of powders was impressively demonstrated by making liquid marbles in air [893, 894]. Liquid marbles (Figure 11.12) are obtained by making a small amount of water (typically 1 mm3 ) roll on a hydrophobic powder. The powder particles go into the interface and completely coat it so that, after spontaneous formation of the spherical drop, only the solid caps of powder particles come into contact with the solid support. Some inorganic electrolytes stabilize O/W emulsions. One example is potassium thiocyanide (KSCN), which dissociates in the aqueous phase. The anion SCN− adsorbs at the interface, which becomes negatively charged. As a result, the oil droplets repel each other electrostatically.

11.3.4 Evolution and Aging Freshly prepared macroemulsions change their properties with time. The time scale can vary from seconds (then it might not even be appropriate to talk about an emulsion) to many years. To understand the evolution of emulsions, we must Hydrophobic powder particle Water

Solid surface (a)

(b)

Figure 11.12 Liquid marble on a planar solid surface stabilized by a hydrophobic powder. A schematic (a) and a video image (b) are shown. The marble and its mirror image can be seen in (b). Figure courtesy of David Quéré.

11.3 Macroemulsions

Creaming Flocculation

Coalescence and phase separation

Figure 11.13 Typical steps in the evolution of a macroemulsion.

take different effects into account. First, any reduction of the surface tension reduces the driving force of coalescence and stabilizes emulsions. Second, repulsive interfacial film and interdroplet forces can prevent droplet coalescence and delay demulsification. Here, all those forces discussed in Section 5.5.2 are relevant. Third, dynamic effects such as the diffusion of surfactants into and out of an interface can have a dramatic effect. Evolution and, eventually, demulsification of macroemulsions proceed through a series of steps (for a review see [895]). For different macroemulsions, the individual steps might be different. Rather typical is that the dispersed drops first form loose clusters without losing their identity. This process is called flocculation4 (Figure 11.13). It is caused by the secondary energy minimum, discussed in Section 5.5.2. In fact, for O/W emulsions, the same interactions are present: attractive van der Waals forces between oil droplets destabilize the emulsion, the electrostatic double-layer repulsion stabilizes it. For W/O emulsions, the interactions are different. Since oil has a low dielectric permittivity, only a few ions are dissolved (see Eq. 11.8), and there is no significant double-layer repulsion. Still, a secondary energy minimum might exist. In the secondary energy minimum, the surface films are not in direct molecular contact because the repulsive energy maximum prevents such contact. Flocculation kinetics can be described in different ways. Here we introduce a treatment first suggested by Smoluchowski [896] and described in [877](p. 417) and [897]. The formalism can also be used to treat the aggregation of sols. A prerequisite for coalescence is that droplets encounter each other and collide. Smoluchowski calculated the rate of diffusional encounters between spherical droplets of radius R. The rate of diffusion-limited encounters is 8𝜋DRc2 , where c is the concentration of droplets (number of droplets per unit volume). For the diffusion coefficient D, we use the Stokes–Einstein relation D = kB T∕(6𝜋𝜂R). The rate of diffusion-limited encounters is, at the same time, the upper limit for the decrease in droplet concentration. Both 4 In chemical engineering and mineral processing, the term flocculation is used for aggregation induced by the addition of polymer while coagulation indicates aggregation induced by the addition of electrolyte.

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11 Surfactants, Micelles, Emulsions, and Foams

rates are equal when each encounter leads to coalescence, then the rate of encounters is given by 8𝜋RkB T 2 4k T dc = −8𝜋DRc2 = − c = − B c2 dt 6𝜋𝜂R 3𝜂 or dc = −kf c2 dt

with kf =

4kB T 3𝜂

(11.16)

The rate constant of flocculation kf depends only on the viscosity of the solution and is independent of the size of the droplets. Equation (11.16) is only valid for droplets of identical size. Fortunately, since it does not depend on the droplet size, the flocculation rate does not change dramatically if we consider that aggregates are formed and not all droplets have the same size. Example 11.6 For water at 25 ∘ C (𝜂 = 8.91 × 10−4 Pa s), the rate of flocculation is kf = 6.19 × 10−18 m3 s−1 . This rate constant is valid if we give the concentration in the number of droplets (or particles) per cubic meter. If we report the concentration in moles of droplets per liter (M), the rate constant is kf = 3.71 × 109 M−1 s−1 . If an energy barrier E∗ for coalescence exists, the rate of effective encounters is reduced. We can take this into account by adding a Boltzmann factor: ∗ dc = −kf c2 e−E ∕(kB T) dt

(11.17)

This differential equation is solved by 1 1 = + kf∗ t c c0

with kf∗ = kf e−E

∗ ∕(k T) B

(11.18)

Here, c0 is the original concentration of droplets at the beginning. The quantity d(1∕c)∕dt, which should be equal to kf or kf∗ , is used as a measure of the initial flocculation rate. Whether flocculated or not, drops can migrate in the gravitational field and increase the concentration either at the bottom or at the top of the vessel. The latter effect is known as creaming. Once drops have been brought into close proximity they can coagulate into the primary energy minimum. There the surface films come into direct molecular contact, but they might still keep their identity as separate droplets. Eventually, the surface film ruptures and the two droplets merge into a bigger one. This leads to coalescence and demulsification.

11.3.5 Coalescence and Demulsification Let us turn to the final step, which is coalescence. We can identify three crucial steps in the process of droplet coalescence (Figure 11.14):

11.4 Microemulsions

Oil

Oil

Water Figure 11.14 Three steps of droplet coalescence after flocculation or creaming for an O/W emulsion. ●

●

●

For two drops to coalesce, the two surfactant films must first come into molecular contact. Due to the ever-present van der Waals attraction, two neighboring droplets deform, giving a flat contact area. Depending on the repulsive forces between the two surfactant films, the lamella is more or less stable. The two surfactant films must fuse, forming a neck with direct contact between the dispersed liquid in the two droplets. If the two surfactant films are poorly developed, the lamella will rupture easily, while a fully saturated film could resist the fusion process. Fluctuations of the surfactant surface density can trigger the fusion process. Temporarily, bare spots attract each other and can break the lamella locally. The neck must grow in size so that the two drops eventually merge completely. In this process, the surfactant film can remain intact, but its area and curvature changes.

Any of the three steps can be rate determining depending on the specific nature of the system [898]. In many applications, our goal is to destabilize an emulsion and obtain two separate phases. Mechanical methods are most common for demulsification. Centrifugation, for instance, leads to creaming. Gentle stirring often accelerates coalescence. In some cases, the addition of a cosurfactant or salt might destabilize an emulsion. As we will see in the next section, emulsions are stable only if the spontaneous curvature of the surfactant agrees with the actual radius of the droplets. A cosurfactant with a different spontaneous curvature from the main surfactant might lead to an unstable situation.

11.4 Microemulsions Thermodynamically, stable emulsions were discovered some 70 years ago [898–901] and were treated as a very special case for a long time. Thermodynamic stability not only refers to the fact that they do not change with time; it also implies that microemulsions react reversibly to changes in temperature or composition. Microemulsions tend to form spontaneously. No powerful stirring or strong agitation is required. Also, the size of the structures formed does not depend on

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the energy input as with macroemulsions (Eq. 11.15). In a microemulsion, the volume fraction of surfactant is usually significant. Reviews on microemulsions are [902, 903].

11.4.1 Size of Droplets We start our description of microemulsions by asking: what is the most likely radius of drops R for given volume fractions of continuous phase, dispersed phase 𝜙d , and surfactant 𝜙s ? This radius can be estimated using two equations. First, the total volume of the dispersed phase is given by 4 V𝜙d = n ⋅ 𝜋R3 3

(11.19)

Here, n is the number of droplets in the total volume V. For the second equation, we assume that all surfactant goes into the interface and that only a negligible amount is dissolved in the continuous and dispersed phases. In addition, we assume that the surfactant is oriented normal to the interface so that the thickness of the surfactant film is equal to the total length of a surfactant molecule Ls ; then we get the volume occupied by surfactant: V𝜙s = n ⋅ 4𝜋R2 ⋅ Ls

(11.20)

Dividing Eq. (11.19) by (11.20) leads to R=

3Ls 𝜙d 𝜙s

(11.21)

This is the radius of droplets of the dispersed phase. It is a necessary condition for an emulsion to be thermodynamically stable. It turned out that the parameter Ls should be considered as an effective length of a surfactant molecule that can differ from the geometric length by as much as a factor of two. Example 11.7 Consider an O/W emulsion with a volume fraction of surfactant of 10% and a volume fraction of oil of 30% (𝜙s = 0.1, 𝜙d = 0.3). For a surfactant 2 nm in length, the radius of oil droplets is R = 18 nm. For a W/O emulsion with the same parameters, we have 𝜙d = 0.6 and R = 36 nm. Droplets in microemulsions are usually quite small so that the length of the surfactant is often not negligible. A complication arises as to what precisely is the radius. It can be significant whether we choose the radius to include the whole surfactant rather than, for instance, only the tail. The conventional choice is to position the radius at the so-called neutral surface that is, the plane whose area is constant upon bending. This neutral surface is usually located close to the plane where the head groups meet the hydrocarbon chains. For this reason, it is chosen to be at the core radius RC (Figure 11.15).

11.4 Microemulsions

Figure 11.15 Schematic of an O/W droplet stabilized by surfactant.

Water

Oil Rc

11.4.2 Elastic Properties of Surfactant Films Surfactants form semiflexible elastic films at interfaces. In general, the Gibbs free energy of a surfactant film depends on its curvature. Here we are not talking about the indirect effect of the Laplace pressure but a real mechanical effect. In fact, the interfacial tension of most microemulsions is very small so that the Laplace pressure is low. Since the curvature plays such an important role, it is useful to introduce two parameters, the principal curvatures C1 =

1 R1

and C2 =

1 R2

(11.22)

which are the inverse of the two principal radii of curvature (Section 2.3.1). The curvature can be positive or negative. We count it as positive if the interface is curved toward the oil phase. Following Helfrich, we express the Gibbs free energy of curvature by an integral over the area considered [904]: [ ] 1 ̄ C dA G= k(C1 + C2 − C0 )2 + kC (11.23) 1 2 ∫ 2 C0 is called the spontaneous curvature, which is a more general parameter than the surfactant parameter NS , defined by Eq. (11.4). It makes it easier to discuss the phase behavior of microemulsions because we get away from the simple geometric picture. The parameters k and k̄ have the dimensions of energy and denote the bending rigidity (also called bending elastic modulus) and the saddle-splay modulus (also called the modulus of Gaussian curvature), respectively. This equation has played a pivotal role in our understanding of mono- and bilayers [905]. The film is characterized by a stable equilibrium curvature, corresponding to a minimum in the Gibbs free energy, ̄ at C = C = C k∕(2k + k). 1

2

0

Example 11.8 We consider a film with zero spontaneous curvature (C0 = 0). What is the elastic energy for bending such a film to a sphere of radius R? With C1 = C2 = 1∕R, we obtain [ ( )2 ] 1 2 k̄ G= k + 2 dA ∫ 2 R R [ ] ̄ 2k k ̄ = 4𝜋R2 2 + 2 = 4𝜋(2k + k) R R

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336

11 Surfactants, Micelles, Emulsions, and Foams

It is independent of the radius. It does not depend on the radius because with decreasing radius the bending energy per unit area increases but the surface area decreases. Both effects just compensate each other. See also Exercise 11.6. ̄ The bending rigidity of surfacWhat are typical values for the parameters k and k? tant films and the saddle-splay modulus are typically in the range of −2· · ·2kB T at room temperature [906]. Factors that reduce k are short alkyl chains, cosurfactants, double-chain surfactants with unequal chains, and cis-unsaturated bonds. For the saddle-splay modulus, only a few measurements have been done. It tends to be negative with an amount much smaller than the bending rigidity for the same system. The spontaneous curvature can vary in a range of −0.5 to +0.5 nm−1 , depending on the polar head group, the length and number of apolar chains, and the nature of the oil. Most microemulsions that have been studied contain four or even more components. In addition to water, oil, and surfactant, usually a cosurfactant is added. Typical cosurfactants are alcohols. With their small head group and relatively large hydrophobic tail, they tend to decrease the spontaneous curvature of surfactants. The spontaneous curvature of a surfactant film determines, to a high degree, the structure of a microemulsion. Let us illustrate this for an O/W microemulsion and let us neglect the influence of the saddle-splay modulus. When the spontaneous curvature of the film is equal to the curvature of the oil droplets, C0 = 2∕R, we expect a stable situation [907]. When the spontaneous curvature is larger than 2∕R, the film can relax toward a lower Gibbs energy by decreasing the drop size and expelling emulsified oil to the bulk phase. When C0 is much lower than 2∕R, the film relaxes by forming larger droplets.

11.4.3 Factors Influencing the Structure of Microemulsions What are the most important factors influencing the type of a microemulsion? Here again, we must distinguish between nonionic and ionic surfactants. For nonionic surfactants, often alkylethylene glycols, temperature is the dominating parameter for the structure of a microemulsion. For ionic surfactants, mostly SDS or CTAB, the salt concentration dominates the phase behavior. Alkyl polyglycosides take an intermediate position, although they are nonionic. Here we only discuss nonionic surfactants to show some principles. Microemulsions stabilized with nonionic surfactants tend to form an O/W type of microemulsion at low temperature, while at high temperature, W/O microemulsions are more common. A phase inversion temperature (PIT) exists at which the microemulsion changes from an O/W to a W/O type. This PIT is an important parameter characterizing a microemulsion [908]. The reason for this phase inversion becomes clear when we recall that the tendency of the head group to bind water decreases with temperature (see end of Section 11.2.2). As a result, the size of the head group area decreases. In contrast, hydrocarbon chains tend to occupy a larger volume since their thermal fluctuations become more and more prominent. As a result, the spontaneous curvature changes from favoring an O/W emulsion to favoring a W/O microemulsion.

11.4 Microemulsions

45 L2

T (°C)

40 35

3ϕ Lα

30 25

L1

20 0.0

0.1

ϕS

0.2

0.3

Oil

Water n1 > 1 and that there is an optical phase shift of 𝜋 if light is reflected from a material with higher optical density and a phase shift of 0 if the reflecting material has a lower optical density. What is the condition for destructive interference in the cases n1 > 1 and n1 > n2 ?

371

13 Solutions to Exercises Chapter 2: Liquid Surfaces 1

Surface contamination: For the estimation, we use Eq. (2.1) and take nitrogen as a gas. With a molar mass of 28 g/mol, the molecular mass is m=

0.028 kg∕mol 6.02 × 1023 ∕mol

= 4.7 × 10−26 kg 2

The area occupied by one adsorbed molecule is in the order of 𝜎A ≈ 10 Å = 0.1 nm2 . To get an average of 0.1 hits in 𝜎A (this leads to 10% coverage) in a time span Δt, we solve P𝜎 Δt 0.1 = √ A ⇒ 2πmkB T √ 0.1 2 ⋅ 4.7 × 10−26 kg ⋅ 1.38 × 10−23 ⋅ 298 J = 9.7 × 10−9 Pa P= 10−19 m2 ⋅ 3600 s We assumed that the coverage was low so that any approaching molecule finds a free surface. Otherwise, we would have had to solve a differential equation and obtain an exponential dependence on Δt. 2

Molecules at the surface: The total number of water molecules in a drop of 1 nm radius is N=

4πR3d NA 3M

=

4π(10−9 m)3 ⋅ 1000 kg∕m3 ⋅ 6.02 × 1023 ∕mol = 140 3 ⋅ 0.018 kg∕mol

Drops with 10 and 100 nm radii contain 1.4 × 105 and 1.4 × 108 molecules, respectively. A monolayer has a thickness of ( )1 ( )1 3 0.018 kg∕mol M 3 d= = = 0.31 nm 23 3 𝜌NA 1000 kg∕m ⋅ 6.02 × 10 ∕mol

Physics and Chemistry of Interfaces, Fourth Edition. Hans-Jürgen Butt, Karlheinz Graf, and Michael Kappl. © 2023 WILEY-VCH GmbH. Published 2023 by WILEY-VCH

372

13 Solutions to Exercises

The number of molecules at the surface divided by the total number of molecules is 4πR2d d 4∕3 ⋅ πR3d

=

3d Rd

For Rd = 1 nm 90% of the molecules are at the surface. For 10 and 100 nm radius, this ratio decreases to 9% and 0.9%. 3

Water running out of a box: Since the plastic is not wetted by water, the maximum radius of a water drops coming out of the box is 100 μm. To push a water drop of 100 μm radius out of a hole, a pressure of 2 ⋅ 0.072 N∕m 2𝛾 = = 1440 Pa R 10−4 m is required. This is equal to the hydrostatic pressure P = 𝜌𝛾h at a height ΔP =

ΔP = h=

2𝛾 = 𝜌gh ⇒ R

2 ⋅ 0.072 N∕m

100 × 10−6 m ⋅ 9.81 m∕s2 ⋅ 1000 kg∕m3

= 0.147 m

Thus, the water runs out of the box until a height of 14.7 cm is reached. 4

Layer of water: From Figure 2.9c, we can estimate the height and contact angle of the drop: h = 3.1 mm and Θ = 70∘ . With Eq. (2.16), we get [ ]2 [ ]2 h 3.1 × 10−3 m 𝛾= g𝜌 = 9.81 m∕s2 ⋅ 997 kg∕m3 2 sin(Θ∕2) 1.15 = 0.071 kg∕s2 = 0.071 N∕m

5

Wilhelmy plate method: The force on the plate is 2l𝛾 cos Θ = 2 ⋅ 0.01 m ⋅ 0.072 N∕m ⋅ 0.707 = 1.02 × 10−3 N

6

Drop-weight method: With a mass of m = 2.2 × 10−3 kg∕100 = 2.2 × 10−5 kg, the volume of one drop is V=

m 2.2 × 10−5 kg = = 2.8 × 10−8 m3 𝜌 773 kg∕m3

Since hexadecane wets the capillary, we choose the outer diameter of the capillary: 𝜙=

rc 2 × 10−3 m = = 0.659 V 1∕3 (2.8 × 10−8 m3 )1∕3

The correction factor is f = 0.615. Using this correction factor, we get 2.2 × 10−5 kg ⋅ 9.81 m∕s2 mg N f rc = = 0.0279 2π m 2π ⋅ 0.615 ⋅ 2 × 10−3 m The real surface tension of hexadecane is 27.1 mN/m at 25∘ C. 𝛾=

Chapter 3: Thermodynamics of Surfaces

400

x (nm)

300 RP

100

r

2x (a)

200

0 0.0

(b)

0.2 0.4 0.6 0.8 Relative vapor pressure

Figure 13.1 (a) Condensing liquid which forms a pendular meniscus between a perfect sphere and a flat surface. (b) Radius of pendular meniscus versus relative vapor pressure.

7

Capillary condensation: We start with Kelvin’s equation: ( ) ( ) PK 𝛾V 1 1 1 1 RT ln 0 = 𝛾Vm + = 𝛾Vm − ≈− m ⇒ P0 R1 R2 x r r 𝛾Vm r=− RT ln(P0K ∕P0 ) using the same notation as in Section 2.6. Roughly, we have (Figure 13.1a) R2P + (x + r)2 = (RP + 2r)2 ⇒ R2P + x2 + 2rx + r 2 = R2P + 4rRP + 4r 2 ⇒ x2 + 2rx = 4rRP + 3r 2 ⇒ x2 ≈ 4rRP √ √ √ √ 4 ⋅ 0.072 N ⋅ 18 × 10−6 m3 ⋅ 5 × 10−6 m √ 4𝛾Vm RP m mol x= − = √− K RT ln(P0 ∕P0 ) 8.31 molJ K ⋅ 298 K ⋅ ln(P0K ∕P0 ) √ 1 = 0.102 μm − K ln(P0 ∕P0 )

8

Force on a ring: As long as the ring is submerged, the force is glπrc2 (𝜌Pt − 𝜌Bu ). The maximal force acting on the ring when it is pulled out of the pool is 2l𝛾 + glπrc2 𝜌Pt . The difference is ΔF π 2 ΔF = 2l𝛾 + glπrc2 𝜌Bu ⇒ 𝛾 = − rc g𝜌Bu 2 2l kg 2.08 × 10−3 N π m N −3 𝛾= − (0.3 × 10 m)2 ⋅ 9.81 2 ⋅ 810 3 = 0.0249 2 ⋅ 0.04 m 2 m s m

Chapter 3: Thermodynamics of Surfaces 1

Alternative definition of interfacial tension: From Eq. (3.42), we get 𝜕F 𝜎 || =𝛾 𝜕A ||T,Ni𝜎

(13.1)

373

374

13 Solutions to Exercises

2

Surface entropy and energy of ethyl acetate: 𝜕𝛾 Δ𝛾 s𝜎 = − ≈− 𝜕T ΔT ( ) 1 0.02339 N − 0.02513 N 0.02049 N − 0.02339 N =− + 2 15 K m 25 K m −5 N = 11.6 × 10 mK 𝜕𝛾 N N N 𝜎 u =𝛾 −T = 0.02339 + 298 K ⋅ 11.6 × 10−5 = 0.05796 𝜕T m mK m

3

Orientational surface entropy: The entropy deficit of one mole of surface molecules as compared to bulk molecules is given by R times the logarithm of the ratio of numbers of configurations in the surface to that of the bulk: ΔSm = −R ln 2 = 5.8 J∕K∕mol. To convert this to a surface entropy, we have to estimate the surface area per mole of molecules Am . Assuming a cubic shape of 2∕3 1∕3 the molecules, Am = (Vm ∕NA )2∕3 NA = Vm NA . Thus, s𝜎 ≈ −

R ln 2 2∕3

1∕3

Vm N A ≈−

(18 × 10−6

8.31 J K−1 mol−1 0.693 J = 9.9 × 10−5 2 −1 2∕3 −1 1∕3 23 3 m K m mol ) (6.02 × 10 mol )

For ethanol, we get 4.6 × 10−5 J∕m2 ∕K and for toluene 3.0 × 10−5 J∕m2 ∕K. This is of the same order of magnitude as the experimental values of 15.6 × 10−5 (water), 8.3 × 10−5 (ethanol), and 11.8 × 10−5 J∕m2 ∕K (toluene). Thus, orientational entropy plays a significant role. 4

Surface excess: For low mole fractions, the mole fraction is proportional to the activity and concentration. For a mole fraction of 0.002, we get Γ≈−

(0.0619 − 0.0674)N∕m 0.002 mol ⋅ = 2.22 × 10−6 8.31 ⋅ 2.98 J∕mol 0.003 − 0.001 m2

or 1.34 propanol molecules per nm2 . For 0.004 and 0.006, the surface excess is 3.34 × 10−6 and 4.36 × 10−6 mol∕m2 . The surface excess does not increase linearly with the mole fraction. 5

Pressure in bubble: Without surfactant, the pressure inside the bubble is 2 ⋅ 0.072 N∕m 2𝛾 = = 14.4 Pa R 0.01 m With surfactant, the surface tension is reduced according to the Gibbs adsorption isotherm Eq. (3.64). To apply Eq. (3.64), we need to know the surface excess: 1 mol Γ= = 1.43 × 1018 m−2 = 2.38 × 10−6 −18 2 m2 0.7 × 10 m a Δ𝛾 ΓRTΔa a 𝜕𝛾 Γ=− ≈− ⇒ Δ𝛾 = − RT 𝜕a RT Δa a −6 2.38 × 10 mol∕m2 ⋅ 2480 J∕mol ⋅ 0.002 mol∕L J Δ𝛾 = − = 5.90 × 10−3 2 0.002 mol∕L m ΔP =

Chapter 4: Charged Interfaces and the Electric Double Layer

The surface tension is reduced to 0.0720 N∕m − 0.0059 N∕m = 0.0661 N∕m, which leads to a pressure of 13.2 Pa. 6

Adsorption from gas phase: The concentration (activity) a of gas molecules is proportional to the partial pressure P: a = KP, where K is a constant. Inserting into Eq. (3.64): P 𝜕𝛾 1 𝜕𝛾 KP 𝜕𝛾 Γ=− =− =− RT 𝜕(KP) RT 𝜕P RT 𝜕 ln P

Chapter 4: Charged Interfaces and the Electric Double Layer 1

Debye length: Inserting 𝜀 = 25.3 for ethanol and a concentration of 6.02 × 1022 salt molecules per cubic meter into Eq. (4.9), and with λD = 𝜅 −1 , we get √ 25.3 ⋅ 8.85 × 10−12 A s ∕V∕m ⋅ 1.38 × 10−23 J∕K ⋅ 298 K λD = = 17.3 nm 2 ⋅ 6.02 × 1022 ∕m3 ⋅ (1.60 × 10−19 A s)2 √ For water, the decay length is larger according to λD = 0.304 nm∕ c0 = 30.4 nm.

2

Potential–versus–distance at high surface potentials: With a Debye length of 6.80 nm and using Eqs. (4.10) and (4.23), we get figure 13.2.

3

Capacitance of an electric double layer: From the concentration, we calculate the Debye length and with Eq. (4.36) the capacitance per unit area: c = 0.001 M ⇒ λD = 9.6 nm ⇒ CdA = 0.072 F∕m2 and c = 0.1 M ⇒ λD = 0.96 nm ⇒ CdA = 0.72 F∕m2 . To obtain the total capacities, we multiply with the surface area, which is π ⋅ 0.4 × 10−3 m ⋅ 0.05 m = 6.3 × 10−5 m2 (neglecting the end cap). The total capacities are 4.5 and 45 μF, respectively.

4

Cation concentration at a silicon oxide surface: The concentration of cations is −19 −21 increased by a factor ee𝜑0 ∕kB T = e(1.60×10 As⋅0.07V)∕(4.12×10 J) = 15.3. Despite the Linear Full

100 Potential ψ (mV)

Figure 13.2 Potential–versus–distance for surface potentials of 60, 100, and 140 mV using the solution of the linearized and the full Poisson–Boltzmann equation for an aqueous solution with 2 mM KCl.

10 0

5 10 Distance x (nm)

15

375

376

13 Solutions to Exercises

low bulk concentration, we therefore expect a concentration of 0.76 M. This corresponds to 4.6 × 1026 counterions per cubic meter. The average distance is only (4.6 × 1026 ∕m3 )−1∕3 ≈ 1.3 nm. With a bulk concentration of H+ of [H+ ]0 = 10−9 M, the local pH at the surface is − log[H+ ] = − log(15.3 × 10−9 ) = 7.8. 5

Charge of a sphere: To relate the surface potential to the surface charge of the particle, we proceed as before for planar surfaces (Chapter 4.2.5): ∞

∞

d Q = 4π𝜎R = −4π 𝜌e r dr = 4π𝜀𝜀0 ∫ ∫ dr R R [ ]∞ d𝜑 = 4π𝜀𝜀0 r 2 dr R 2

2

( ) 2 d𝜑 r dr dr

With d𝜑∕dr = −𝜑0 R(𝜅r + 1)e−𝜅(r−R) ∕r 2 , we get [ ]∞ Q = −4π𝜀𝜀0 𝜑0 R(𝜅r + 1)e−𝜅(r−R) R = 4π𝜀𝜀0 𝜑0 R(𝜅R + 1) 6

Capacitance of mercury: For 𝜑0 = 0 and considering that cosh 0 = 1, Eq. (4.35) simply reads CdA = 𝜀𝜀0 ∕λD . With 𝜀 = 78.4, we get c0 c0 c0 c0

= 0.001 M = 0.01 M = 0.1 M =1M

⇒ ⇒ ⇒ ⇒

λD λD λD λD

= 9.61 nm = 3.04 nm = 0.961 nm = 0.304 nm

⇒ ⇒ ⇒ ⇒

CdA CdA CdA CdA

= 7.2 μF∕cm2 = 22.9 μF∕cm2 = 72.3 μF∕cm2 = 229 μF∕cm2

Compared to the experimental results, this shows that Gouy–Chapman theory fails to describe the capacitance of the double layer, especially at high salt concentration. 7

Gibbs free energy of double layer around a spherical particle: In analogy to Eq. (4.39), we need to replace the surface charge density by the total charge of the particle. For the total charge of the particle see exercise 4.5. 𝜑0

) 𝜑0 R G=− = −4π𝜀𝜀0 R 1 + 𝜑′ d𝜑′ ∫ λD ∫ 0 0 0 0 ( ) [ ′2 ]𝜑0 𝜑0 Q2 R = −4π𝜀𝜀0 R 1 + =− ( λD 2 8π𝜀𝜀 R 1 + (

Qd𝜑′0

0

8

0

R λD

)

Electro-osmosis: The flow velocity is (Eq. 4.73): |v0 | = 𝜀𝜀0

ζEx A s 0.03 V ⋅ 100 Vm−1 m = 78.5 ⋅ 8.85 × 10−12 = 2.08 × 10−6 −1 0.001 kgs−1 m−1 𝜂 s Vm

The flow is directed from plus to minus.

Chapter 5: Surface Forces

9

Electrophoresis: The drift velocity of the particles with respect to the liquid is given by Eq. (4.81). With λD = 3.04 nm ≪ R, we have |v| =

78.5 ⋅ 8.85 × 10−12 A s∕V∕m ⋅ 0.02 V ⋅ 100V∕m μm = 1.39 0.001 kg∕s∕m s

(13.2)

The particle drifts toward the negative electrode with the liquid flow at a velocity of 2.08 − 1.39 = 0.7 μm/s. It is not a great idea to use them as markers.

Chapter 5: Surface Forces 1

Van der Waals force of layer of molecules: The energy between a single molecule A and a planar monolayer of surface density 𝜌𝜎B is ∞

Wmol∕pl

∞

2πrdr dr 2 = = −πCAB 𝜌𝜎B 2 2 3 2 ∫ (D + r 2 )3 (D + r ) 0 0 [ ]∞ πCAB 𝜌𝜎B π𝜌𝜎B CAB 1 = =− 2 (D2 + r 2 )2 0 2D4 −CAB 𝜌𝜎B ∫

The energy and force per unit area for a whole planar layer of molecules A at surface density 𝜌𝜎A are, respectively, wpl∕pl = − 2

π𝜌𝜎A 𝜌𝜎B CAB 2D4

,

fpl∕pl = −

2π𝜌𝜎A 𝜌𝜎B CAB D5

Hamaker constant between metals across vacuum/gas: Inserting expression (5.25) for 𝜀1 and 𝜀2 and 𝜀3 = 1 into the integral of Eq. (5.22) leads to the integral )2 1 dν ∫ 2 + ν2e ∕ν2 ν2 + ν2e ∕2 ν1 ν1 [ ( √ )]∞ √ 4 νe ν 2 2 ν = + 3 arctan 2 2 2 4 νe (ν + νe ∕2) νe νe ν1 [√ √ ( √ )] 4 ν ν 2π 2 ν1 2 ≈ e − 2 2 1 2 − 3 3 4 νe νe (ν1 + νe ∕2) 2νe νe ∞

(

)2

ν2e ∕ν2

∞

ν4 dν = e 4∫

(

Here, we wrote arctan x = x − x3 ∕3 ± … and because ν1 ≪ νe used only the linear term. This expression can be further simplified: ( ) ν4e πν 2ν1 2ν1 π − 4 − 4 ≈ √e ≈ √ 3 4 ν ν e e 2ν 4 2 e

Multiplication by 3h∕(4π) leads to Eq.√ (5.26). The first term in Eq. (5.22) can be neglected because 3∕4 ⋅ kB T ≪ (3∕(16 2))hνe .

377

378

13 Solutions to Exercises

3

Van der Waals energy across a polymer film: To obtain the energy per unit area (Eq. 5.16), we first need to calculate the Hamaker constant. For identical materials 1 and 2 Eq. (5.24) is ( ) (n21 − n23 )2 𝜀1 − 𝜀3 2 3hν 3 + √e ( AH ≈ kB T )√ 2 4 𝜀1 + 𝜀3 16 2 n2 + n2 n1 + n23 1 3 With 𝜀1 = 𝜀2 = 5.4, 𝜀3 = 2.55, n1 = n2 = 1.58, n3 = 1.59 (Table 5.2) and with νe = 2.7 × 1015 Hz, we get ( ) 3 5.4 − 2.55 2 AH ≈ ⋅ 5.28 × 10−21 J ⋅ 4 5.4 + 2.55 (2.496 − 2.528)2 3 ⋅ 6.63 × 10−34 J s ⋅ 2.7 × 1015 + ⋅ √ √ (2.50 + 2.53) ⋅ 2.50 + 2.53 16 2 s = 0.533 × 10−21 J AH A 12πD2 kB T = kB T ⇒ A = 2 AH 12πD 12π ⋅ 1 nm2 ⋅ 5.28 × 10−21 J 12π ⋅ 1 nm2 ⋅ 5.28 A= = = 376 nm2 0.53 0.53 × 10−21 J For silicon oxide, we get AH ≈ 0.97 × 10−21 J ⇒ A = 205 nm2 . wA =

4

Polystyrene films on silicon oxide: Inserting the appropriate values (n2 = 𝜀2 = 1) into Eq. (5.24): ( )( ) 1 − 2.55 3 4.7 − 2.55 AH ≈ ⋅ 4.12 × 10−21 J ⋅ 4 4.7 + 2.55 1 + 2.55 4.83 × 10−19 J ⋅ (2.372 − 2.528) (1 − 2.528) +√ (√ ) √ √ 2.372 + 2.528 ⋅ 1 + 2.528 ⋅ 2.372 + 2.528 + 1 + 2.528 = −4.0 × 10−22 J + 7.4 × 10−21 J = 7.0 × 10−21 J A positive Hamaker constant corresponds to an attractive force between the silicon oxide–polystyrene and the polystyrene–air interfaces. This implies that the film is not stable. If it is thin enough and has the chance, for instance when annealing, the film ruptures and holes are formed.

5

Van der Waals force on atomic force microscope tip: For a parabolically shaped tip, we have r 2 ∕(2R) = x − D. With a cross-sectional area at height x of A = πr 2 = 2πR(x − D) for x ≥ D, we get dA∕dx = 2πR and ∞

∞

∞

2πRAH A R dx A R dA dx = − H = − H2 F(D) = f (x) dx = − ∫ ∫ 6πx3 dx 3 ∫ x3 6D D

6

D

D

Exponential force law: Parabolic shape: ∞

∞

2πRf0 −𝜅D dA F(D) = f (x) dx = f0 e−𝜅x 2πR dx = e ∫ ∫ dx 𝜅 D

D

Chapter 5: Surface Forces

Figure 13.3 Excluded volume for gap between two spheres when dissolved molecules have a radius R0 .

R0 Rp

r h

D

Conical shape: ∞

F(D) =

∫

f0 e−𝜅x 2π tan2 𝛼 ⋅ (x − D) dx

D ∞ ⎛∞ ⎞ −𝜅x = 2πf0 tan 𝛼 ⎜ xe dx − D e−𝜅x dx⎟ ∫ ⎜∫ ⎟ ⎝D ⎠ D [ −𝜅x ]∞ 2πf tan2 𝛼 e D 0 = 2πf0 tan2 𝛼 − 2 (𝜅x + 1) + e−𝜅x = e−𝜅D 𝜅 𝜅 𝜅2 D 2

7

Depletion attraction: When two spheres approach each other closer than 2R0 , the excluded volume is reduced by the volume indicated in gray in Figure 13.3. This volume is twice the end cap volume: π V = 2 h(3h2 + r 2 ) 6 With h = R0 − x∕2, r 2 = (Rp + R0 )2 − (Rp + x∕2)2 , and the osmotic pressure ckB T, we get ) ( Rp x 2 x 3 4 3 2 2 W = ckB Tπ 2Rp R0 + R0 − 2RP R0 x − R0 x + + 3 2 12 For R0 ≪ Rp and x ≪ RP the terms R30 , R20 x, and x3 ∕12 can be neglected and we get Eq. (5.64).

8

Contact radius between silica spheres: We use E = 5.4 × 1010 Pa, ν = 0.17, R∗ = R∕2 = 10−6 m, W = 2𝛾S = 0.1 N/m. The contact area is πa2 . The reduced Young’s modulus is 2 ⋅ 0.971 1 1 − ν2 = 2 = = 3.60 × 10−11 Pa−1 ⇒ E∗ = 2.8 × 1010 Pa E∗ E 5.4 × 1010 Pa Hertz model: √ √ √ ∗ 3 × 10−6 m3 m3 3 3 3 3R F = F = 2.67 × 10−17 F a= 11 ∗ 4E N 1.12 × 10 N JKR model: In contrast to the Hertz model where the minimal load is zero, we can apply negative loads, that is, we can even pull on the particles. The greatest negative load is equal to the adhesion force 3π𝛾S R∗ = 0.471 μN. Results calculated with Eq. (5.69) are plotted in figure 13.4.

379

13 Solutions to Exercises

Contact radius (nm)

380

50

JKR

40

Hertz

30 20 10 0

9

Figure 13.4 Contact radius versus load for the two spherical SiO2 particles.

60

0

1

2 3 Load (μN)

4

5

Dispersion in aqueous electrolyte: According to Derjaguin’s approximation (5.35), the force between two particles is ( e𝜑 ∕(2k T) )2 D A R − e 0 B −1 F = πRw(D) = 64πRc0 kB TλD e𝜑 ∕(2k T) e λD − H 2 12D e 0 B +1 The energy is ( W(D) = 64πRc0 kB TλD

∫

e𝜑0

( = 64πRc0 kB Tλ2D

)2 ∞

e𝜑0

e 2kB T − 1 e 2kB T + 1 e

e𝜑0 2kB T

e

e𝜑0 2kB T

−1 +1

)2

∞ ′

e

− λD

D

A R dD′ dD − H 12 ∫ D′2 ′

D

e

D

− λD

D

−

AH R 12D

When plotting the energy versus distance for different salt concentrations, we find that at 0.28 M, we have an energy barrier of roughly 10kB T.

Chapter 6: Contact Angle Phenomena and Wetting 1

Sessile layer of liquid: The thickness of the film after letting the liquid spread is (Eq. 2.16) √ √ 𝛾L 0.062 N∕m Θ 60∘ h=2 sin = 2 sin = 1.23 mm 2 3 g𝜌 2 2 9.81 m∕s ⋅ 1050 kg∕m With A = V∕h, the area is 0.081 m2 . Addition of surfactant leads to a reduced thickness of 0.334 mm and a covered area of 0.299 m2 .

2

Small sessile drop: The force due to the capillary pressure is πa2 ΔP. Assuming that the drop is shaped like a spherical cap of radius R, the capillary pressure is ΔP = 2𝛾L ∕R. The contact radius a is given by a = R sin Θ. Inserting leads to 2𝛾L = 2πa𝛾L sin Θ R This expression agrees with the integrated vertical component of the surface tension acting at the contact line. F = πa2 ΔP = πaR sin Θ ⋅

Chapter 6: Contact Angle Phenomena and Wetting

3

Capillary rise: With Eq. (6.26): h=

2𝛾L cos Θ 2 ⋅ 0.072 N∕m = −3 = 1.47 cm rc g𝜌 10 m ⋅ 9.81 m∕s2 ⋅ 997 kg∕m3

When the capillary is filled with close-packed spherical particles, the rise of the liquid is determined by the size of the voids. As an approximation, we characterize the voids by an inscribed sphere of radius 0.16 Rp . Taking this value as the effective radius, the height can be estimated as Eq. (6.33) h=

2𝛾L cos Θ rceff g𝜌

=

2 ⋅ 0.072 N∕m 0.16 ⋅ 5 × 10−6 m ⋅ 9.81 m∕s2 ⋅ 997 kg∕m3

= 19 m

For a more detailed discussion, see [987]. 4

Line tension effects (Figure 13.5): Rearranging Eq. (6.16), we obtain cos Θ =

𝛾S − 𝛾SL − 𝜅l ∕a 𝛾L

To determine 𝛾S − 𝛾SL , we apply 𝛾L cos Θ + 𝜅l ∕a = 𝛾S − 𝛾SL . Thus, at a = 0.5 mm, we have 0.05

5

N 0.5 × 10−9 N N ⋅ cos 90∘ + = 10−6 = 𝛾S − 𝛾SL −4 m m 5 × 10 m

Surface energy of polymers: With Eq. (6.22), we obtain √ √ p p 𝛾SL = 𝛾S + 𝛾L − 2 𝛾Sd 𝛾Ld − 2 𝛾S 𝛾L √ √ p p ⇒ 𝛾L = 𝛾SL − 𝛾S + 2 𝛾Sd 𝛾Ld + 2 𝛾S 𝛾L Inserting Young’s equation 𝛾L cos Θ = 𝛾S − 𝛾SL in the previous equation leads to √ √ p p 𝛾L = −𝛾L cos Θ + 2 𝛾Sd 𝛾Ld + 2 𝛾S 𝛾L √ √ 𝛾 p p ⇒ L (cos Θ + 1) = 𝛾Sd 𝛾Ld + 𝛾S 𝛾L 2

160 Contact angle (deg.)

Figure 13.5 Contact angle versus contact radius for a drop with 𝛾L = 0.05 N∕m and Θ = 90∘ (at a = 0.5 mm) and a line tensions of 𝜅l = ±0.5nN.

140 120

+0.5 nN

100 80 60

−0.5 nN

40 20 0 1E−8

1E−7

1E−6

Contact radius (m)

381

382

13 Solutions to Exercises

We can solve this equation, we have at least two sets of results (with different liquids). For polyethylene, we use (without units): √ √ 72.8 p Water ∶ (1 + cos 94∘ ) = 33.86 = 𝛾Sd 21.8 + 𝛾S 51.0 2 √ √ 50.8 p MI ∶ (1 + cos 52∘ ) = 41.04 = 𝛾Sd 49.5 + 𝛾S 1.3 2 √ Multiplying the last equation by 21.8∕49.5 = 0.6636 and subtracting the second equation from the first yields √ √ √ p p p 33.86 − 41.04 ⋅ 0.6636 = 6.63 = 𝛾S 51.0 − 0.6636 𝛾S 1.3 = 6.38 𝛾S ) ( 6.63 2 mN p ⇒ 𝛾S = = 1.08 6.38 m ( )2 √ √ 33.86 − 1.08 ⋅ 51.0 √ mN 33.86 = 𝛾Sd 21.8 + 1.08 ⋅ 51.0 ⇒ 𝛾Sd = = 32.06 21.8 m mN mN p d 𝛾S = 𝛾S + 𝛾S = (32.06 + 1.08) = 33.1 m m For poly(vinyl chloride): √ √ 72.8 p Water ∶ (1 + cos 87∘ ) = 38.3 = 𝛾Sd 2.18 + 𝛾S 51.0 2 √ √ 50.8 p MI ∶ (1 + cos 36∘ ) = 45.9 = 𝛾Sd 49.5 + 𝛾S 1.3 2 √ ( ) 7.84 2 mN p p 38.3 − 45.9 ⋅ 0.6636 = 7.84 = 6.38 ⋅ 𝛾S ⇒ 𝛾S = = 1.51 6.38 m ( )2 √ √ 38.3 − 1.51 ⋅ 51.0 √ mN 38.3 = 𝛾Sd 21.8 + 1.51 ⋅ 5.10 ⇒ 𝛾Sd = = 40.0 21.8 m mN mN p d 𝛾S = 𝛾S + 𝛾S = (40.0 + 1.5) = 41.5 m m For polystyrene, we obtain 𝛾S = 42.1 mN∕m. For PMMA 𝛾S = 40.2 mN∕m. 6

Roll-off angle: We apply Eq. (6.23) with k = 1: ( ) 𝜌Vg sin 𝛼 = w𝛾L cos Θr − cos Θa To be on the safe side, we set cos Θa = cos 180∘ = −1. To calculate w, we assume that the contact angle is 140∘ and that the drop is shaped like a spherical cap. [ ]1 3 3⋅2 × 10−9 m3 For geometric reasons, the width is w = 2 sin 150∘ = π(1 − cos 150∘ )2 [2 + cos 150∘ ]

0.78 mm. w = 1.07 and 1.34 mm for V = 5 and 10 μl, respectively. The tilt angles, where the drops will safely roll off the plate, are 6.4, 10.2, and 19.0∘ for 2, 5, and 10 μL drops, respectively.

Chapter 7: Solid Surfaces 1

Overlayer structures on a fcc(111) surface: See Figure 13.6.

2

Force in atomic force microscopy: It is lower because the van der Waals attraction is reduced, and the meniscus force is absent.

Chapter 7: Solid Surfaces

Figure 13.6 Hexagonal fcc(111) lattice with (1 × 2) (a) and √ √ ( 7 × 7) R19.1∘ (b) overlayer structures.

(a)

3

(b)

Structure factor of diamond: The diamond lattice can be viewed as a face-centered cubic (fcc) lattice with a basis of c1 = 0, c2 = (a∕4)(ex + ey + ez ), where a is the lattice constant of the cubic lattice and ex , ey , ez are unit vectors along the cubic axes (Figure 13.7). The primitive vectors of the fcc lattice are indicated in the figure. a a a a1 = (ey + ez ), a2 = (ex + ez ), a3 = (ex + ey ) 2 2 2 The lattice vectors of the reciprocal lattice are given by a2 × a3 2π b1 = 2π = (e + ez − ex ) a1 ⋅ a2 × a3 a y a3 × a1 2π b2 = 2π = (e + ez − ey ) a1 ⋅ a2 × a3 a x a1 × a2 2π b3 = 2π = (e + ey − ez ) a1 ⋅ a2 × a3 a x Then we get SG = 1 + exp(−iq ⋅ d2 ) with q = n1 b1 + n2 b2 + n3 b3 a q ⋅ d2 = (n1 b1 + n2 b2 + n3 b3 ) ⋅ (ex + ey + ez ) 4 π = (n1 + n1 − n1 + n2 − n2 + n2 + n3 + n3 − n3 ) 2 π = (n1 + n2 + n3 ) 2 [ ] 1 π SG = 1 + exp −i (n1 + n2 + n3 ) = 1 + (−1) 2 (n1 +n2 +n3 ) 2 SG = 2 ± i for n1 + n2 + n3 twice an even number SG = 1 ± i

for n1 + n2 + n3

odd

SG = 0 ± i

for n1 + n2 + n3

twice an odd number

Figure 13.7 Crystal structure of diamond.

a

383

384

13 Solutions to Exercises

4

TIRM: Since the intensity is field strength squared, the penetration depth of the intensity is only half of that for the field strength. With Eq. (7.35), we get ( )2 ( ) ( )2 ( ) n2 2 λ 1.44 2 5 × 10−7 m 2 sin 𝜃 = + = + −7 n1 1.52 4πn1 dp 4π ⋅ 1.52 ⋅ 10 m = 0.966 The required angle of incidence is 79.4∘ .

5

Tracer diffusion: From Eq. (7.26), we obtain the hopping rate ν=

4D 4 ⋅ 3.6 × 10−22 m2 ∕s = = 0.02∕s l2 (2.7 × 10−10 m)2

To have no more than a single hopping event per frame, one should have less than 50 s scanning time per frame, which is a reasonable time scale for STM imaging. 6

Temperature dependence of diffusion: (a) The onset temperature was defined as the temperature at which ν = 1 Hz. With Eq. (7.26) (2.5)2 1 l2 ν= = 1.56 × 10−22 m2 s 4 4 s (b) From the Arrhenius behavior of the diffusion, we conclude [ ( )] exp(−Ed ∕(kB T2 )) Ed 1 1 D(T2 ) = D(T1 ) = D(T1 ) exp − exp(−Ed ∕(kB T1 )) kB T1 T2 D=

With T1 = 139 K and T2 = 298 K, we obtain [ ( )] 2 1 0.36 ⋅ 1.6 × 10−19 J 1 −22 m D(298 K) = 1.56 × 10 ⋅ exp − s 1.38 × 10−23 J∕K 139 K 298 K = 1.42 × 10−13 m2 s The average distance a Cu atom moves within 1 h is given by Eq. (7.22): √ √ 15 nm at T = 139 K ⟨Δr 2 ⟩ = 4Dt = 45 μm at T = 298 K

Chapter 8: Adsorption 1

Significance of Langmuir parameter: Half coverage is reached at a pressure KL P1∕2 1 1 KL P1∕2 = ⇒ + = KL P1∕2 ⇒ 2 1 + KL P1∕2 2 2 1 KL P1∕2 1 = ⇒ P1∕2 = 2 2 KL

Chapter 8: Adsorption

2

Langmuir parameter: With Eq. (8.22), we get 6.02 × 1023 ∕mol ⋅ 16 × 10−20 m2 ⋅ 10−12 s KL0 = √ 2π ⋅ 0.028 kg∕mol ⋅ 8.31 J∕K∕mol ⋅ 79 K = 8.96 × 10−9

m s2 = 8.96 × 10−9 Pa−1 kg

KL = KL0 eQ∕(RT) = 8.96 × 10−9 Pa−1 ⋅ e 3

2000⋅4.184 8.31⋅79

= 3.10 × 10−3 Pa−1

BET adsorption isotherm equation: From Eq. (8.32), we get 1 kad

a1

=

S1 −Q ∕(RT) e 1 S0 P

In thermodynamic equilibrium, this is equal to 1∕P because the Boltzmann factor e−Q1 ∕(RT) determines the occupancy of the two states, and we have i S1 ∕S0 = eQ1 ∕(RT) . With the same argument, we get kad ∕ai ≈ 1∕P. Therefore, the prefactor is ≈ 1. 4

Hydraulic number for close-packed spheres: We consider a volume V that contains n spheres. Here, n is assumed to be a large number so that it is practically continuous. The volume filled by particles is Vf = n ⋅ 4∕3 ⋅ πR3p . The total surface area of all particles is A = n ⋅ 4πR2p . The free (“void”) volume is Vv = 0.26V = 0.26 ⋅ n ⋅ 4∕3 ⋅ πR3p ∕0.74. Thus, the hydraulic number is ah =

5

0.26 ⋅ n ⋅ 4∕3 ⋅ πR3p Vv 0.26 = = R = 0.117 Rp A 0.74 ⋅ 3 p 0.74 ⋅ n ⋅ 4πR2p

Docosane on graphite: The number of molecules per unit area is 88.9 × 10−6 mol mol NA = 1.31 × 10−6 2 ⋅ 6.02 × 1023 mol−1 68 m2 m = 7.89 × 1017 m−2 The area per molecule is 1∕7.89 × 1017 m−2 = 1.27 × 10−18 m2 = 1.27 nm2 . The cross-sectional area of an alkane chain is roughly 0.45 nm × 0.45 nm = 0.2 nm2 . The area occupied by a molecule lying flat on the surface is roughly 0.45 nm × 0.12 nm × 22 = 1.2 nm2 . Conclusion: the molecules are lying flat on the surface.

6

Freundlich adsorption isotherm: The exponential distribution of adsorp∗ tion energies is described by f (Q) = e−Q∕Q ∕Q∗ ; division by Q∗ ensures that

385

386

13 Solutions to Exercises ∞

∫0 f (Q)d Q = 1. Inserting Eq. (8.18) 𝜃 = KL P∕(1 + KL P) with KL = KL0 eQ∕(kB T) Eq. (8.23) and f (Q) into Eq. (8.43) we get ∞

𝜃(P) =

∫ 0

Q

P KL0 e kB T 1 + P KL0 e ∞

1 = ∗ Q ∫ 0

Q

1 + P KL0 e

(P KL0 ) Q∗ Q∗

1 − QQ∗ e dQ Q∗

P KL0 e kB T kB T

=

Q kB T

∞

Q kB T

e

− k QT B

kB T Q∗

Q

P KL0 e kB T

∫

Q

1 + P KL0 e kB T

0

dQ

kB T ( Q )− ∗ Q P KL0 e kB T dQ

With the dimensionless numbers m = kB T∕Q∗ and p = PKL0 and by substituting k Tdy dy y = eQ∕(kB T) , we get dQ = k yT ⇒ dQ = B y leading to B

∞

∞

1

1

k T pm py p y−m 𝜃(p) = ∗ (p y)−m B dy = dy ∫ ∫ Q 1 + py y m 1+p y Usually, Q∗ > kB T and m < 1. Otherwise, no significant adsorption would occur. For m < 1, the integral can be solved. Integration from 1 to ∞ leads to [657] ) ( mπ pm p2 p3 p 𝜃(p) = − + ∓ … for p < 1 (13.3) −m sin (mπ) 1−m 2−m 3−m ( ) 1 1 𝜃(p) = 1 − m − 2 ±… for p > 1 (13.4) p(1 + m) p (2 + m) The solution is simplified if the integration is carried out from y = 0 to ∞ rather than 1 to ∞. This implies taking into account also negative binding energies, that is, taking the distribution from Q = −∞ to −∞ rather than Q = 0 to −∞. With this approximation, one obtains ∞

p y−m mπ 𝜃(p) = dy = pm m ∫ 1 + py sin(mπ)

(13.5)

0

Eq. (13.5) has the form of the Freundlich isotherm Eq. (8.45) ( )m P mπ 𝜃= = pm P0 sin(mπ) if we identify P0 with ( )1∕m sin(mπ) 1 P0 = mπ KL0 The approximate solution, however, only agrees for low pressure with the exact solution (13.3) and (13.4). With increasing m, the range where approximation and exact solutions agree becomes larger (Figure 13.8).

Chapter 8: Adsorption

5

1.0

=

0 .2

m=

0.5

m = 0.5 m = 0.25

m

0.8 0.6

θ

Figure 13.8 Adsorption isotherms calculated with an exponential distribution of ∗ adsorption energies f (Q) = e−Q∕Q ∕Q∗ for Langmuir type adsorption. Two distributions with m = kB T∕Q∗ = 0.5 and m = 0.25 are plotted. Continuous lines were calculated with Eqs. (13.3) and (13.4). Dashed lines are Freundlich isotherms Eq. (13.5).

0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

PKL0

Figure 13.9 ln(Tm2 ∕𝛽) versus 1∕Tm for CO on Pd(100). 1n(Tm2/β)

13 12 11 10 9 0.0020

0.0021

0.0022

1/Tm

7

Thermal desorption spectroscopy: Edes 𝛽

𝛽 aR −Edes ∕(RTm ) = e ⇒ 2 Edes Tm ( 2) ( ) E E Edes Tm 𝛽 aR ln 2 = ln − des ⇒ ln = des + ln Edes RTm 𝛽 RTm aR Tm

ae−Edes ∕(RTm ) =

2 RTm

⇒

2 ∕𝛽) versus 1∕T , we should get a straight line with a slope E ∕R If we plot ln(Tm m des and an intercept at ln(Edes ∕(aR)) (Figure 13.9). From a linear fit of the graph, we get

Edes kJ = 17 900 ⇒ Edes = 149 R ( mol ) Edes E ln = −27.1 ⇒ a = des e27.1 = 1.1 × 1016 Hz aR R 8

Adsorption from solution: With Δc = ci − cf , the total mass of the polymer adsorbed is mad = VL Δc. Here, VL is the total volume of the liquid. VL = V0 − VP , where VP is the total volume occupied by the particles. VP = mio ∕𝜌. To calculate: Γ = mad ∕AP , where AP is the total surface of all particles. With AP = N4πR2P and the total number of particles N=

mio 𝜌 4πR3P ∕3

387

13 Solutions to Exercises

Figure 13.10 Polymer adsorbed versus concentration after equilibrium is established.

Langmuir

0.06 Γ (mg/m2 )

388

0.04 Freundlich 0.02 0.00

0

40 60 80 20 Concentration cf (mg/l)

we get AP = 3mio ∕(𝜌Rp ). This leads to ( ) m m 𝜌RP Δc Γ = ad = V0 − io AP 3mio 𝜌 The graphs are shown in Figure 13.10. The Langmuir isotherm was fitted with Γmon = 0.067 mg∕m2 and KL = 0.105 l∕mg. The Freundlich isotherm Eq. (8.56) mg l1∕3 was plotted with KF = 0.015 m2 mg and n = 3. 1∕3

Chapter 9: Surface Modification 1

Hydrophobization/hydrophilization: (a) To hydrophobize a silicon wafer, we first create a high density of hydroxy groups for the consecutive coupling reaction with the silanes. This can be done in an oxygen-containing plasma cleaner or by immersion in a mixture of hydrogen peroxide, ammonia, and water (e.g. H2 O2 ∶ NH4 OH ∶ water∕1 ∶ 1 ∶ 5 vol. ratio). Under basic conditions, the hydrogen peroxide oxidizes organic material on the silicon surface. The ammonia dissolves ions by complexing them. Then the wafer is rinsed with pure water, which is removed by subsequent rinsing with different organic solvents with decreasing polarity. The silane, for example, octadecyltrichlorsilane (OTS), is dissolved in a water-free organic solvent. The silicon substrate is immersed for several hours in the silane solution. Then it is washed with organic solvents with increasing polarity to remove physisorbed silanes. (b) To form a hydrophilic thiol monolayer on gold, a thiol having a hydrophilic rest group such as 11-mercaptoundecanol or 11-mercaptoundecanoic acid are dissolved, for example, in ethanol to 1 mM. Then gold is immersed for typically 1–12 h. (c) Contact angle measurements with water are applied to determine hydrophobicity or hydrophilicity (Chapter 6). If Θ > 90∘ , then the surface is hydrophobic. For hydrophilic surfaces, Θ should be well below 90∘ . For comparison, the contact angle on the substrate before surface modification should be measured. Film thickness may be measured by grazing incidence X-ray diffraction (Section 12.3.3) provided the substrate is flat and smooth. AFM or STM provides information about the homogeneity or roughness of the surface layer up to a molecular resolution (Section 7.7.3).

Chapter 10: Friction, Lubrication, and Wear

2

Polymer plasma film: The polymer coating is transparent and protects the aluminum layer from oxidation in a humid environment. Otherwise, the headlights would lose their reflectivity. The film thickness must be smaller than ≈ 50 nm to avoid interference of light even at small angles. Otherwise, this would lead to colored headlights (see end of Section 12.4).

3

Neurochip: We measure electric signal and we chose a conductive coating on the substrate. Let us take gold. Gold can be deposited on the polystyrene through masks by sputtering. Neurons like soft, hydrophilic surfaces such as polyelectrolytes that contain a lot of water. To adsorb polyelectrolytes on gold without the risk of washing the polymers away during cultivation, we hydrophilize the gold, for example, by thiols containing hydrophilic end groups. Usually, the end groups must be capable of covalently binding the polyelectrolyte on top.

4

Microfabrication. First, the silicon surface is oxidized by exposing it to an oxygen-containing atmosphere at a high temperature to get a thickness of some 100 nm (thermal oxidation). The native silicon oxide layer is only about 1–2 nm thick. Later, the oxide layer serves as an insulator. Second, chromium and then gold are sputtered on the oxidized silicon. Chromium increases the adhesion of gold to the silicon substrate. Then a photoresist is spin-coated on top. It is sensitive to light. The radiation through a mask polymerizes the negative photoresist locally. After radiation, the unpolymerized photoresist is rinsed away (the hole in this case). Afterward, the metals in the unprotected regions are etched away. Metals, especially gold, can be removed by wet isotropic etching with potassium iodine/iodine 1:1 aqueous solution, not with aqua regia for reasons of selectivity. The polymerized photoresist is removed with an organic solvent. Then the walls of the channel are formed. A new, special polymer is spin-coated on the substrate to the desired thickness. This polymer differs from the inexpensive photoresist because later it comes into contact with the liquid. Therefore, it should have a long-term stability, should not form cracks, should be stable against different chemicals, and should be hydrophilic (or easy to hydrophilize) because otherwise water will not run through the channel. This time the positive photoresist SU(8) is used. The light-exposed part can be washed away afterward. In the last step, we put a glass cover on top. This is achieved by hydrophilizing the glass substrate on one side, for example, with oxygen plasma, and then pressing it onto the polymer. After contacting the polymer surface, the glass substrate can no longer be removed because it adheres to the polymer.

Chapter 10: Friction, Lubrication, and Wear 1

Sliding on an inclined plane: The block starts sliding when the gravitational force component parallel to the surface, F∥ = mg sin 𝛼, exceeds the static friction, FF = 𝜇s F⟂ (Fig. 13.11). Here, F⟂ = mg cos 𝛼 is the force component normal to

389

390

13 Solutions to Exercises

FF

Figure 13.11 Schematic of a block sliding down an inclined plane.

F mg α

α F⊥ F

the surface (the load). Thus, sliding starts at an angle given by mg sin 𝛼 = 𝜇s mg cos 𝛼 ⇒ 𝜇s = tan 𝛼 ⇒ 𝜇s = arctan 0.5 = 26.6∘ When the block slides, the force parallel to the inclined plane is F∥ = mg sin 𝛼 − 𝜇k mg cos 𝛼 = mg(sin 𝛼 − 𝜇k cos 𝛼) It is accelerated with constant acceleration a = F∥ ∕m. The speed after a distance L = 1 m is √ √ v = 2aL = 2gL(sin 𝛼 − μk cos 𝛼) √ m m = 2 ⋅ 9.81 2 ⋅ 1 m ⋅ (sin 26.6∘ − 0.4 cos 26.6∘ ) = 1.32 s s 2

True contact area: The contact surface will deform until the pressure is equal to the yield stress 𝜎y . If Fg is the weight force, A the true contact area, 𝜌 the density, and V the volume of the cube, we can write 𝜎y = Fg ∕A = 𝜌Vg∕A ⇒ A = 𝜌Vg∕𝜎y . Inserting numbers: A=

7.85 kg∕cm3 ⋅ (10 cm)3 ⋅ 9.81 m∕s2 109 N∕m2

= 7.7 × 10−8 m2 = 7.7 × 104 μm2

This corresponds to 7700 microcontacts of 10 μm2 each. 3

Hydrodynamic lubrication: The sliding speed is v = 2π ⋅ 0.005 m ⋅ 80 Hz = 1.26 m∕s. The friction force is FF =

0.1 m ⋅ 2π ⋅ 0.005 m ⋅ 1.26 m∕s ⋅ 2.49 × 10−3 Pa s

3 × 10−6 m The torque is M = 3.29 N ⋅ 0.005 m = 0.0164 Nm.

= 3.29 N

Chapter 11: Surfactants, Micelles, Emulsions, and Foams 1

Hydrophobic effect drives micellization: Since C8 E6 and C12 E6 are nonionic, we apply Eq. (11.3). The Gibbs energies of micellization are ΔGmic m = RT ln 0.0098 = −11.5 and −23.4 kJ/mol, respectively. Taking the difference and division by 4 leads to a value of 3.0 kJ/mol; the difference between C8 E6 and C12 E6 are exactly four methylene groups.

2

Volume of hydrocarbon chain: We assume that the volume VC of n-alkanes is the sum of the volumes of all methylene (∼ CH2 ∼) units plus twice the volume of a

Chapter 12: Thin Films on Surfaces of Liquids

methyl group (∼ CH3 ): 𝜌=

M∕NA M ⇒ (nC − 2)VCH2 + 2VMe = (nC − 2)VCH2 + 2VMe 𝜌NA

Using the values for hexane and dodecane, we get two equations: 4VCH2 + 2VMe = 10VCH2 + 2VMe =

86.2 × 10−3 kg∕mol 655 kg∕m3 ⋅ 6.02 × 1023 ∕mol 170.3 × 10−3 kg∕mol 749 kg∕m3 ⋅ 6.02 × 1023 ∕mol

= 0.219 nm3 = 0.378 nm3

Subtracting the two equations yields 6VCH2 = 0.159 nm3 ⇒ VCH2 = 0.0265 nm3 . Inserting this value into one of the two equations, we get VMe = 0.0565 nm3 . Using the total number of carbon atoms nC as a parameter, we get for a single, saturated alkyl chain VC = (0.0265 nC + 0.060) nm3 . 3

Gibbs free energy of emulsion formation: The Gibbs free energy is ΔGem = 𝛾A, where A is the total surface area of all drops. This area is A = n ⋅ 4πR2 , with n being the number of drops. With a total volume of V = n ⋅ 4πR3 ∕3 ⇒ n = 3V∕(4πR3 ) we get ΔGem = 3𝛾V∕R.

4

3 Two merging drops: For constant volume, we have 2 43 πr 3 = 43 πrnew ⇒ rnew = √ 2 Aold 3 2⋅4πr 2 r leading to A = 4πr2 = √ 2 = 1.26. 3 2 = new

new

√ 3

2⋅

2

5

Rate of flocculation: The concentration is c0 = 1015 droplets/m3 . With kf = 6.2 × 10−18 m3 ∕s, the initial decrease in concentration is −dc∕dt = kf c2 = 6.2 × 10−18 m3 ∕s ⋅ (1015 ∕m3)2 = 6.2 × 1012 droplets/(m3 s). So we can estimate that within the first second, the relative decrease in concentration is only 0.6%.

6

Bending energy: For the bending energy, we get ( ) 20kB T 1 2 2 2k = 2 = g𝜎 = k 2 R R R2 R = 5 nm ⇒ g𝜎 = 3.3 mN∕m, R = 20 nm ⇒ g𝜎 = 0.21 mN∕m, R = 100 nm ⇒ g𝜎 = 0.0082 mN∕m.

Chapter 12: Thin Films on Surfaces of Liquids 1

Pressure in monolayer: From P = π∕d, we can calculate the corresponding 0.03 N∕m three-dimensional pressure as P = 25×10−10 m = 12 × 106 N∕m2 ≈ 120 atm.

2

Transition enthalpy and entropy: To obtain the two-dimensional Clapeyron equation, we transformed pressure P to lateral pressure π and volume V to area

391

392

13 Solutions to Exercises

per molecule 𝜎A . Resolving with respect to ΔHc : ΔHc = −

𝜕πc TΔ𝜎c 𝜕T

(13.6)

We determine the area change in the plateau of the phase transition as indicated in Figure 12.4 for 30 ∘ C. The corresponding temperatures must be transformed into Kelvin. To determine 𝜕πc ∕𝜕T, you can plot πc –versus–T, make a linear regression analysis, and take the slope. It turns out to be 𝜕πc ∕𝜕T ≈ 1.1 × 10−3 N∕m∕K. We extract the transition pressures and molecular areas from Figure 12.4, insert the numbers into Eq. (13.6), and get the enthalpies. Division by temperature leads to the entropies.

T(K)

𝛑c (mN∕m)

𝚫𝝈A (Å )

2

𝚫Hc (kJ∕mol)

293.2

5

52 − 77 = −25

49

0.17

298.2

9

50 − 68 = −18

36

0.12

303.2

16

50 − 64.5 = −14.5

29

0.096

𝚫Sc (kJ∕(K mol))

Two remarks: (i) The area change is negative because the area after the phase transition is lower than before. (ii) The enthalpy of condensation is on the order of the standard enthalpies of vaporization of hydrocarbons. 3

X-ray reflection: (a) To calculate the thickness from the angle we use Bragg’s equation (12.9). Curve i: 𝛼1 = 1.75∘ ⇒ d1 = 25 Å. Curve ii: 𝛼2 = 1.25∘ ⇒ d2 = 35 Å. The thickness increases with increasing surface pressure. At higher pressure, the thickness is equal to the molecular length as calculated from the structure formula (d ≈ 9 Å + 10 ⋅ 2.5 Å = 34 Å). The molecules are oriented normal to the water surface. (b) The effective volume is the product of molecular area times thickness: 3 3 3 V = 𝜎A d. Curve i: V1 = 25 × 110 Å = 2750 Å . Curve ii: V2 = 35 × 35 Å = 3 1225 Å . The effective volume decreases with pressure because the lipid alkyl chains change from a disordered to nearly fully stretched (alltrans) conformation. In the disordered phase, there is still a high fraction of free volume within the monolayer. Additionally, water molecules bound to the sugar headgroup are expelled with compression (dehydration). 3

(c) From the electron density profile, we get an electron density of 0.33 e∕Å . The reciprocal is the volume of one electron in a water molecule. Water has 3 10 electrons. Thus, the volume of one water molecule is 30 Å . The theoretical volume of a water molecule can simply be calculated from the density of water 3 (𝜌 = 1.0 g∕cm3 ) by V = M∕(𝜌NA ) = 30 Å . Here, M = 18 g/mol is the molar mass. Thus, the experimental value is reasonable. 3

3

(d) d1 = 14 Å; d2 = 9 Å; 𝜌1 = 0.29 e∕Å ; 𝜌2 = 0.37 e∕Å . Film 1 can be interpreted as the a polar alkyl chain part of the monolayer. Film 2 corresponds to

Chapter 12: Thin Films on Surfaces of Liquids

the layer of polar headgroup. The value of 𝜌1 is typical for a liquid-expanded alkyl chain layer. For a liquid-condensed alkyl layer, it would be close to that of water. The headgroup contains relatively heavy oxygen atoms and its electron density is higher. 4

Optical interference: We assume that the light is coupled perpendicular to the thin film (𝛼 = 90∘ ). Case 1: 1 < n1 < n2 . The path difference between the beams reflected at the two interfaces 1 and 2 is n1 2d. The condition for the first minimum is (destructive interference) that the total path difference between 1 and 2 should be λ∕2. Thus, we get 2n1 d = λ∕2 or d = λ∕(4n1 ). Case 2: 1 < n1 > n2 . The optical density of the thin film is higher than that of the surrounding media. We need to consider a phase shift of π at the interface between n1 and n2 . The total path difference is π or 2n1 d − λ∕2. For the first minimum, we have a path difference of λ∕2, which leads to 2n1 d − λ∕2 = λ∕2 or d = λ∕(2n1 ). If d is small enough, the path difference is always λ∕2 for all wavelengths. The thin film appears black (see Section 11.5.3).

393

395

14 Analysis of Diffraction Patterns Here we introduce the more general theory of diffraction at crystal surfaces. First, we analyze for which directions of the outgoing radiation we get constructive interference and observe diffraction peaks. In the second part, we discuss what determines the intensities of these maxima.

14.1

Diffraction at Three-Dimensional Crystals

14.1.1 Bragg Condition A simple framework to understand the diffraction of electromagnetic radiation or particles by periodic structures was introduced by Bragg. In this model, we look for the interference of rays that are scattered from different parallel crystal planes with a spacing d (Figure 14.1). We first treat the case of specular reflection. In specular reflection, the angle between incident rays and the lattice plane is the same as the angle between outgoing rays and the lattice plane. The condition for constructive interference is given by the Bragg equation: nλ = 2d sin 𝜗

(14.1)

Here, n is an integer that is called the order of the diffraction peak. If we measure the angles under which diffraction peaks occur, then we can determine the distances between the lattice planes. In the case of a two-dimensional array (crystal surface) or a three-dimensional periodic arrangement (bulk crystal), we must satisfy a Bragg equation for each dimension simultaneously. Constructive interference occurs only at angles were the diffraction patterns of different dimensions overlap with each other. This strongly limits the number of observed diffraction peaks.

14.1.2 Laue Condition To discuss diffraction for the more general case, where the condition of specular reflection is given up, it is convenient to introduce the so-called wave vector. The direction of the wave vector k is that of the propagating wave. Its length is given by 2π (14.2) |k| = λ Physics and Chemistry of Interfaces, Fourth Edition. Hans-Jürgen Butt, Karlheinz Graf, and Michael Kappl. © 2023 WILEY-VCH GmbH. Published 2023 by WILEY-VCH

396

14 Analysis of Diffraction Patterns

Instead of the scattering angle 𝜗, we use the so-called scattering vector q. The scattering vector is the difference between the incoming wave vector ki and the outgoing wave vector kf :

𝜗

d sin𝜗

(14.3)

q = kf − ki

𝜗

d

Figure 14.1 Scattering of

Since we assume elastic scattering (no change in energy radiation at parallel lattice and wavelength), ki and kf have the same length, planes. |ki | = |kf |. The maximum length of the scattering vector is limited to 2|ki |, which corresponds to the case of backscattering. Let us consider two lattice atoms, one at the origin and the other one at the position ∑ R = mi ai from the origin. The ai are the primitive vectors of the lattice and mi are integers (Figure 14.2a). What is the difference in path length between a wave scattered at the origin and a wave that is scattered at position R? Compared to the atom at the origin, the incoming wave front must travel the additional path R cos α = Rki ∕ki to reach the second atom. On the other hand, the distance between the second atom and the detector is shorter by Rkf ∕kf = Rkf ∕ki . The total path difference for the two waves scattered in the direction of kf by the two atoms is given by Δ = (R∕ki )(ki − kf ) = Rq∕ki . Constructive interference occurs for Δ=

1 2π qR = nλ = n ki ki

(14.4)

This means the scattering vector must satisfy the condition: qR = 2πn

(14.5)

Equation (14.5) is known as the Laue1 condition for constructive interference. How are the Laue and Bragg conditions connected? In Figure 14.2b, the wave vectors of the incident and outgoing radiation and the scattering vector are drawn for the Bragg reflection of Figure 14.1. We can conclude that for specular reflection, the scattering vector q is always perpendicular to the lattice plane. Its length is given by |q| = 2|ki | sin 𝜗 =

α

R

4π sin 𝜗 λ

–ki

kf q

ki 𝜗

ki (a)

(14.6)

d

kf 𝜗

Figure 14.2 (a) Path difference between scattered waves from two atoms with a relative distance R. (b) Incident wave vector ki , outgoing wave vector kf , and scattering vector q for Bragg reflection (Figure 14.1). The length of the scattering vector is |q| = 2|ki | sin 𝜃 = (π∕λ) sin 𝜃.

(b)

1 Max von Laue, 1879–1960; German professor for physics; Nobel Prize in Physics in 1914 for the discovery of X-ray diffraction in crystals.

14.1 Diffraction at Three-Dimensional Crystals

To get constructive interference, we need to satisfy the Bragg condition. Inserting Eq. (14.1) into Eq. (14.6) yields |q| = 2π∕d. In other words, we observe a diffraction peak if the scattering vector is perpendicular to any of the lattice planes and its length is equal to 2π d Here, d being the lattice spacing between this set of lattice planes. |q| = n

(14.7)

14.1.3 Reciprocal Lattice Let us return to our task of finding all possible diffraction peaks for a given crystal lattice. What are the possible scattering vectors that lead to constructive interference? This question can be answered in an elegant way by defining the so-called reciprocal lattice: let a1 , a2 , and a3 be the primitive vectors of the crystal lattice. We choose a new set of vectors according to b1 = 2π

a2 × a3 , a1 ⋅ (a2 × a3 )

b2 = 2π

a3 × a1 , a1 ⋅ (a2 × a3 )

b3 = 2π

a1 × a2 a1 ⋅ (a2 × a3 ) (14.8)

The dots between the vectors denote the scalar (inner) product and the crosses denote the cross-product (outer product) of the vectors. These vectors bi are in units of m−1 , which is proportional to the inverse of the lattice constants of the real space crystal lattice. This is why one calls the three-dimensional space spanned by these vectors the reciprocal space. The lattice defined by these primitive vectors is the reciprocal lattice. Due to the cross product in the denominator, the primitive reciprocal vectors have the following properties: ●

●

b1 is perpendicular to a2 and a3 and, therefore, normal to the lattice plane spanned by a2 and a3 . Accordingly, b2 is perpendicular to a1 and a3 , and b3 is perpendicular to a1 and a2 . The length of each vector bj is 2π∕dj , where dj is the distance between the lattice planes perpendicular to bj . This is ensured by the numerators a1 ⋅ a2 × a3 in Eq. (14.8). For this reason, each of these primitive reciprocal lattice vectors satisfies the Laue condition Eq. (14.7) and is a possible scattering vector for constructive interference.

These two properties can be summarized into { 0 for j ≠ k aj ⋅ bk = 2π δjk = 2π for j = k

(14.9)

Another property is that for every set of parallel lattice planes, there are reciprocal lattice vectors that are normal to these planes. The shortest one of these reciprocal lattice vectors is used to characterize the planar orientation. The components (h, k, l) of this vector are called Miller indices, and the direction of the plane is denoted by (hkl) for the single plane or {hkl} for a set of planes (Section 7.2.1). Why is it useful to introduce such a complicated set of vectors? This becomes obvious when we look at

397

398

14 Analysis of Diffraction Patterns

the scalar product between a real space lattice vector R and a reciprocal lattice vector q. Expressing these vectors by the corresponding primitive vectors, we can write R ⋅ q = (l1 a1 + l2 a2 + l3 a3 ) ⋅ (m1 b1 + m2 b2 + m3 b3 ) = l1 m1 π + l2 m2 π + l3 m3 π = nπ

(14.10)

with n = l1 m1 + l2 m2 + l3 m3 being an integer. This means that any reciprocal lattice vector automatically satisfies the Laue condition Eq. (14.5). For this reason, all reciprocal lattice vectors are possible scattering vectors that lead to constructive interference. A more detailed analysis shows that every scattering vector that leads to constructive interference is also a reciprocal lattice vector, that is, a reciprocal lattice contains all possible scattering vectors. More precisely, the vectors from the origin to every point of the reciprocal lattice give the set of possible scattering vectors with constructive interference for the corresponding real space lattice. For an infinite crystal, this would lead to an infinite number of vectors. How many diffraction peaks we can expect to observe in a diffraction experiment? To answer this question, we recall that for elastic scattering, the length of the scattering vector cannot become larger than 2|ki |. Taking into account this additional limit, |q| ≤ 2|ki |

(14.11)

we end up with a small number of diffraction peaks.

14.1.4 Ewald Construction The so-called Ewald2 construction (Figure 14.3) allows us to find all possible scattering vectors q for b3 b1 a given incident wave vector ki and a given crystal lattice. It is a simple geometric construction that autoki kf matically includes the additional boundary condition 0 Eq. (14.11). As the first step, we construct the recipq rocal lattice as described previously. Take one of the lattice points as the origin. Draw the wave vector ki Figure 14.3 Ewald conof the incident wave starting from this origin. Then struction. Given an incident draw a sphere with radius |ki | around the end point wave vector ki , a sphere of of ki (so that it passes through the origin). If we then radius |ki | is drawn around draw the outgoing wave vector kf ending in the cen- the end point of ki . Diffraction peaks are observed only if the ter of the sphere, then its starting point will always scattering vector q ends on be on the surface of the sphere. The scattering vector this sphere. q = kf − ki must then connect this starting point with the origin. For constructive interference to occur, the scattering vector q must be a lattice vector of the reciprocal lattice (that is it must connect two lattice points). 2 Paul P. Ewald, 1888–1985; physicist and crystallographer, professor at the universities of Munich, Stuttgart, Queen’s University at Belfast, and the Polytechnic Institute of Brooklyn.

14.2 Diffraction at Surfaces

This is possible only if some reciprocal lattice points (in addition to the origin) lie on the surface of this sphere. Therefore, we get diffraction peaks only for very special values of ki and q. Usually we need to vary either the wavelength (this corresponds to varying the radius of the Ewald sphere) or the direction of the incident beam (this corresponds to a rotation of the sphere around the origin) to observe a reasonable number of possible diffraction peaks.

14.2 Diffraction at Surfaces The formalism of the reciprocal lattice and the Ewald construction can be applied to the diffraction at surfaces. As an example, we consider how the diffraction pattern of a LEED experiment (Figure 7.21) results from the surface structure. The most simple case is an experiment where the electron beam hits a crystal surface perpendicularly, as shown in Figure 14.4. Since we do not have a Laue condition to satisfy in the direction normal to the surface, we get rods vertical to the surface instead of single points. All intersecting points between these rods and the Ewald sphere will lead to diffraction peaks. For this reason, we always observe diffraction peaks in LEED if the wave vector of the electrons is longer than the shortest reciprocal lattice vector, that is, if the wavelength of the electrons is short enough. This is in contrast to the situation for three-dimensional diffraction, where diffraction peaks are observed only for special combinations of wave and scattering vectors. From Figure 14.4, we can see that the intersections between the rods and the Ewald sphere occur at the points where the component q|| of the scattering Incident electron beam Vacuum

q|| ki

Bulk solid

q –ki kf

b2 b1

(a)

(b)

Figure 14.4 Ewald construction for surface diffraction: (a) side view of reciprocal lattice at surface. Constructive interference occurs for all intersection points of the vertical rods with the Ewald sphere. This is equivalent to the condition where the component q|| of the scattering vector parallel to the surface is identical to a reciprocal lattice vector of the surface lattice; (b) top view of reciprocal surface lattice. The circle is the projection of the Ewald sphere. If we disregard the radiation scattered into the crystal, the number of lattice points within the circle (corresponding to the intersections of the rods with the Ewald sphere) is identical to the maximum number of observed diffraction peaks.

399

400

14 Analysis of Diffraction Patterns

Real space lattice (top view)

Reciprocal lattice (top view) [110]

[110]

2.55 Å

a2

3.61 Å

a1

2.46 Å–1

b2 b1

1.74 Å–1 [001]

[001]

Figure 14.5 Ewald construction for surface diffraction at the Cu(110) surface.

vector parallel to the surface is identical to a reciprocal lattice vector of the surface lattice. Therefore, the Laue condition for surface diffraction is given by q|| ⋅ R = 2π n

(14.12)

Here, R is a lattice vector of the real space surface lattice and n is an integer. The number of diffraction peaks observed in a real experiment will, however, be limited by the fact that only backscattered radiation (i.e. with kf directed away from the surface) will reach the detector. Example 14.1 LEED pattern of the Cu(110) surface at an electron energy of 70 eV. Copper has an fcc lattice with a lattice constant of 3.61 Å (Figure 14.5). For this reason, the (110) surface has a rectangular lattice with lattice constants a1 = 3.61 Å √ and a2 = 3.61 Å. 2∕2 = 2.55 Å (Figure 7.2). The reciprocal lattice is a rectangular −1 lattice with the lattice constants b1 = 2π∕3.61 Å = 1.74 Å and b2 = 2π∕2.55 Å = √ −1 2.46 Å . The wavelength of the electrons is (Eq. 7.31) λ = h∕ 2me eU = 1.47 Å. The −1 radius of the Ewald sphere is given by |ki | = 2π∕1.47 Å = 4.27 Å . If the electrons approach normal to the surface, the resulting pattern is therefore a rectangular lattice with relative lengths of the sides of 1.74: 2.46 and consisting of 15 diffraction spots.

14.3

Intensity of Diffraction Peaks

The Laue and Bragg conditions give us information about the angular distribution of diffraction peaks. To calculate the peak intensities, we need to know more about the scattering properties of the atoms or molecules in the crystal. In the case of X-rays and electrons, the scattering probability is proportional to the electron density ne (r) within the crystal. Since ne (r) must have the same periodicity as the crystal lattice, we can write it as a three-dimensional Fourier series (using the notation eikx = cos kx + i sin kx): +∞ ∑

ne (x, y, z) =

nklm ei(kx+ly+mz) =

k,l,m=−∞

=

∑ g

ng eig⋅r

+∞ ∑

nklm ei(k,l,m)⋅(x,y,z)

k,l,m=−∞

(14.13)

14.3 Intensity of Diffraction Peaks

Here, ng are the Fourier coefficients and g is the vector with the components (k, l, m). From the requirement of lattice periodicity, we can conclude ne (r) = ne (r + R) for any lattice vector R = ha1 + ka2 + la3 . As a Fourier series, it is written ∑ ∑ ne (r) = ng eig⋅r+R = ng eig⋅r eig⋅R (14.14) g

g

ne (r) = ne (r)eig⋅R ⇒ 1 = eig⋅R = cos(g ⋅ R) + i sin(g ⋅ R) ⇒ g ⋅ R = 2πn

(14.15)

The last condition is identical to Eq. (14.10). Therefore, g must be a reciprocal lattice vector. This shows us another important property of the reciprocal lattice: it is the Fourier transform of the real space lattice. The procedure of obtaining a crystal structure from a diffraction experiment should be straightforward. From the diffraction pattern, we get the reciprocal lattice vectors and can construct the reciprocal lattice. The atomic arrangement should then be easily calculated by an inverse Fourier transform. Unfortunately, this is not possible for several reasons. The observed diffraction pattern contains only the intensity, but not the phase, of the diffracted waves. The reconstruction is thus not unique. In addition, multiple diffraction events, which were not accounted for in the above calculations, might change the diffraction pattern. Therefore, the analysis is done going the opposite way. First, a plausible atomic arrangement is assumed. From this, we calculate the diffraction pattern and compare it with the experimental results. The initial atomic arrangement is then adjusted until calculated and experimental diffraction patterns agree. Up to this point, we have not made any specific assumptions about the real space lattice. It could contain more than one atom per lattice point and more than one type of atoms. In such a case, the lattice would be described using a Bravais lattice plus a basis (Section 7.2.2). To obtain the intensity of the diffracted wave for crystals with a basis, we must simply sum up the contributions from all scattering points within the unit cell. The scattering probability for a crystal of N unit cells with an electron density ne (r) is proportional to N

∫unit cell

ne (r)e−iqr dV = NSG

(14.16)

where the integration runs over the volume of one unit cell. SG is the so-called geometrical structure factor. It contains information about the positions of the atoms within the unit cell and takes into account the interference between the rays scattered from the different basis sites within the unit cell. Since the intensity of a diffracted wave is proportional to the square of the amplitude, the measured intensities of the diffraction peaks are proportional to |SG |2 . If the basis contains atoms of different elements, it is useful to separate the phase shifts due to the geometric arrangement from the scattering characteristics of the atoms. If rj is the vector from the origin of the unit cell to the center of atom j (denoting the position of atom j within the

401

402

14 Analysis of Diffraction Patterns

contribution of atom j to the total electron density at r, the structure factor can be written as follows: ∑ SG = nj (r − rj )e−iqr dV ∫ j =

∑ e−iqrj j

∫

nj (𝝆)e−iq𝝆 dV

∑ = Fj e−iqrj

(14.17)

j

with 𝜌 = r − rj and the atomic form factor Fj =

∫

nj (𝝆)e−iq⋅𝝆 dV

(14.18)

which is determined by the scattering characteristics of atom j. If we write rj = xj a1 + yj a2 + zj a3 , then we get q ⋅ rj = (xj a1 + yj a2 + zj a3 ) ⋅ (h b1 + k b2 + l b3 ) = 2π(xj h + yj k + zj l) (14.19) and SG (hkl) =

∑

Fj e−i2π(xj h+yj k+zj l)

(14.20)

j

as the structure factor of the (hkl) plane. The atomic form factor accounts for the internal structure of the different atoms or molecules. It will also be different for X-rays and neutrons since the former probe the electron distribution of the target, while the latter interact with the nuclei of the atoms. Therefore, the analysis of the positions of the reflexes indicates mainly the lattice constants and angles. The intensity of the reflexes contains mainly information about the atomic configuration within a unit cell (structure factor) and the scattering behavior of the single atoms (form factor). The description of a given crystal by a lattice and a basis is not unique. One might, for example, choose to double the size of the unit cell by including more atoms in the basis. This would also lead to a different reciprocal lattice. This seems to lead to a contradiction since the diffraction pattern should only depend on the crystal and not how we choose our description. As we will see in Example 14.2, the choice of a different basis leads to a change in the structure factor, so that the combination of reciprocal lattice and structure factor always leads to the same diffraction pattern. Example 14.2 A body-centered cubic lattice can also be regarded as a simple cubic lattice with primitive lattice vectors (a, 0, 0), (0, a, 0), and (0, 0, a) and a basis r1 = 0 and r2 = a∕2(1, 1, 1). The reciprocal lattice will then be a simple cubic lattice with primitive lattice vectors 2π (1, 0, 0), a

2π (0, 1, 0), a

2π (0, 0, 1) a

(14.21)

14.3 Intensity of Diffraction Peaks

and with a structure factor SG = F(1 + e−i(a∕2)q⋅(1,1,1) )

(14.22)

Substituting 2π ⋅ [n1 ⋅ (1, 0, 0) + n2 ⋅ (0, 1, 0) + n3 ⋅ (0, 0, 1)] a into Eq. (14.22), we get q=

SG = F(1 + e−iπ(n1 +n2 +n3 ) ) = F(1 + (−1)n1 +n2 +n3 ) { 2F for n1 + n2 + n3 even = 0 for n1 + n2 + n3 odd

(14.23)

(14.24)

with n1 , n2 , and n3 being integers. Points that have an odd sum of coordinates in the simple cubic reciprocal lattice do not lead to diffraction peaks. This makes it equivalent to a reciprocal fcc structure with side length 4π∕a, which is precisely the reciprocal lattice we would have obtained if we had treated the body-centered cubic lattice as a Bravais lattice rather than as a lattice with a basis.

403

405

Appendix A Symbols and Abbreviations Many symbols are not unique for a certain physical quantity but are used two or even three times. We use the symbols as they are usually used in the relevant literature. Since the scope of this book includes many disciplines and thus different scientific communities, multiple usage of symbols is unavoidable. In molecular chemistry and physics, for instance, 𝜇 is the dipole moment, while in engineering, 𝜇 symbolizes the friction coefficient. A

Area (m2 )

AH

Hamaker constant (J)

a

Activity (mol∕m3 ), contact radius (m), frequency factor (Hz)

a1 , a2

Unit vectors of a two-dimensional unit cell of lengths a1 and a2 (m)

ah

Hydraulic radius (m) used to characterize pore sizes

b1 , b2

Unit vectors of a reconstructed unit cell of lengths b1 and b2 (m) or reciprocal lattice vectors

C

Capacitance (C∕V), van der Waals coefficient (J m6 ), BET constant

C

A

Differential capacitance per unit area (C∕V∕m2 )

C1 , C2

Two principal curvatures (m−1 )

C0

Spontaneous curvature (m−1 )

CMC

Critical micelle concentration (mol∕L)

c

Number concentration (number of molecules m−3 ) or amount concentration (mol∕m3 or mol∕L = M)

c1 , c2

Unit vectors of a two-dimensional overlayer of lengths c1 and c2 (m)

D

Distance (m), diffusion coefficient (m2 ∕s)

D0

Interatomic spacing used to calculate adhesion (m), typically 1.7 Å

E

Electric field strength (V∕m), Young’s modulus (Pa)

E

∗

EF E

𝜎

Reduced Young’s modulus (Pa) Fermi energy (J) Surface dilational elastic modulus (N∕m)

F

Helmholtz free energy (J), force (N)

Fadh

Adhesion force (N)

Physics and Chemistry of Interfaces, Fourth Edition. Hans-Jürgen Butt, Karlheinz Graf, and Michael Kappl. © 2023 WILEY-VCH GmbH. Published 2023 by WILEY-VCH

406

Appendix A Symbols and Abbreviations

FF

Friction force (N)

FL

Load (N)

F𝜎 , f 𝜎

Interfacial Helmholtz energy in Gibbs convention (J) and interfacial Helmholtz energy per unit area (J∕m2 ), respectively

f

Force per unit area (N∕m2 )

G

Gibbs energy (J)

Gm , G0m 𝜎

G ,g

𝜎

Molar Gibbs energy and standard molar Gibbs energy (J∕mol) Interfacial Gibbs energy (J) in Gibbs convention or surface elastic modulus (N∕m) and interfacial Gibbs energy per unit area (J∕m2 ), respectively

H

Enthalpy (J)

h

Height of a liquid with respect to a reference level (m), Planck’s constant, layer thickness (m)

I

Electric current (A)

J

Nucleation rate (s−1 m−3 )

K

Spring constant (N∕m), equilibrium constant (e.g. mol∕L)

k

Bending rigidity (J)

k

Saddle-splay modulus (J)

l

Length of one chain link (m) in a polymer chain

L0

Thickness of a polymer brush (m)

Lc

Length of alkyl chain (m)

Ls

Effective length of surfactant (m)

M

Molar mass (kg∕mol), torque in Chapter 10 (N m)

m

Mass (kg), molecular mass (kg/molecule)

mad

Mass of adsorbent (kg)

N

Number of molecules (dimensionless or mol), number of segments in a linear polymer chain

Nagg

Aggregation number of surfactant micelles

Ni

Number of molecules of a certain species i (dimensionless or mol)

NS

Surfactant parameter

n

Refractive index, integer number

nC

Number of carbon atoms in an alkyl chain

P

Total pressure or partial pressure (Pa), probability

P0

Equilibrium vapor pressure of a vapor in contact with a liquid having a planar surface (Pa)

P0K

Equilibrium vapor pressure of a vapor in contact with a liquid having a curved surface (Pa)

p

Momentum (kg m∕s), integer coefficient, exponent in Freundlich adsorption isotherm

Q

Electric charge (A s), heat (J), quality factor of a resonator

q

Heat per unit area (J∕m2 ), wave vector momentum (m−1 ), integer coefficient

R

Radius of a (usually) spherical object (m), gas constant

R1 , R2

Two principal radii of curvature (m)

Appendix A Symbols and Abbreviations

Rb

Radius of a spherical bubble (m)

Rc

Core radius of droplet (m)

Rd

Radius of a spherical drop (m)

Re

Electric resistance (Ω)

Rg

Radius of gyration of a polymer (m)

Rp

Radius of a spherical particle (m)

R0

Size of a polymer chain (m)

r

Radius (m), radial coordinate in cylindrical or spherical coordinates

rc

Radius of a capillary (m)

S

Entropy (J∕K), number of adsorption binding sites per unit area (mol∕m2 ), spreading coefficient (N∕m)

S0 , S1 𝜎

𝜎

Number of vacant and occupied adsorption binding sites per unit area (mol∕m2 )

S ,s

Interfacial entropy in Gibbs convention (J∕K) and interfacial entropy per unit area (J∕K∕m2 ), respectively

T

Temperature (K)

TΘ

Theta temperature (K)

t

Time (s)

U

Internal energy (J), applied or measured electric potential (V)

U 𝜎 , u𝜎

Interfacial internal energy (J) and interfacial internal energy per unit area (J∕m2 ), respectively

V

Volume (m3 )

Vm

Molar volume (m3 ∕mol)

v

Velocity (m∕s)

W

Helmholtz free energy (J), work (J)

w

Helmholtz free energy for the interaction between two surfaces per unit area (J∕m2 )

x, y, z

Cartesian coordinates (m), y is also the reduced electric potential

Z

Valency of an ion

𝛼

Polarizability (C m2 ∕V), angle, accommodation coefficient

𝛾

Surface tension or interfacial energy (N∕m). Specifically, 𝛾L is the surface tension of a liquid and 𝛾SL is the interfacial energy of a solid–liquid interface

𝛾s

Surface strain (no unit)

Γ 𝛿

Interfacial excess (mol∕m2 ), grafting density of polymer (mol∕m2 or m−2 ) Thickness of the hydration layer (m), indentation (m), lag angle (∘ )

Δ

In connection with another symbol it is a difference, ellipsometric parameter

𝜀

Dielectric permittivity

𝜁

Zeta potential (V)

𝜂

Viscosity (Pa s)

𝜂d𝜎 𝜂s𝜎

Surface dilational viscosity (N s∕m)

Θ

Contact angle (∘ )

Surface shear viscosity (N s∕m)

407

408

Appendix A Symbols and Abbreviations

𝜃 𝜅

Coverage of a surface by adsorbed molecules √ Capillary constant 𝛾∕g𝜌 (m), inverse of the Debye length (m−1 ), line tension (N)

𝜅e

Electric conductivity (A∕V∕m)

𝜅l 𝜅d𝜎

Line tension (J∕m) = 1∕E

𝜎

Surface dilatation compressibility (m∕N)

𝜆

Wavelength (m)

𝜆D

Debye length (m)

𝜇

Chemical potential (J∕mol), dipole moment (C m), friction coefficient

𝜇k , 𝜇s

Coefficient of kinetic and static friction, respectively

𝜇r

Coefficient of rolling friction (m)

ν

Frequency (Hz)

π

Film pressure (N∕m)

Π

Disjoining pressure (Pa)

𝜌

Mass density (kg/m3 ), molecular density of a pure phase (molecules/m3 ); in contrast, the density of dissolved molecules is denoted by c

𝜌e

Electric charge density (C∕m3 )

𝜎

Surface charge density (C∕m2 )

𝜎A

Area occupied by a single molecule on a surface (m2 ), head group area of a surfactant (m2 )

Σ

Specific surface area of a powder or porous material (m2 ∕kg). Parameter characterizing the degree of coincidence at grain boundaries

𝜏

Stress, that is, force per unit line (N∕m)

𝜏c

Yield stress (N∕m2 )

𝜏s

Slip time

Υ

Surface stress (J∕m2 )

𝜙

Volume fraction

𝜑

Galvani potential (V)

Φ

Thermionic work function (J)

𝜒

Surface potential (V)

𝜓

Electric Volta potential (V)

𝛹

Ellipsometric parameter

𝜔

Angular frequency (Hz)

Fundamental Constants Atomic mass unit

u

1.660 54 × 10−27 kg

Avogadro constant

NA

6.022 14 × 1023 mol−1

Boltzmann constant

kB

1.380 66 × 10−23 J∕K

Electron mass

me

9.109 39 × 10−31 kg

Elementary charge

e

1.602 18 × 10−19 C

Appendix A Symbols and Abbreviations

Faraday constant

FA = eNA

96 485.3 C∕mol

Gas constant

R = k B NA

8.314 51 J∕K∕mol

Planck constant

h

6.626 08 × 10−34 J s

Speed of light in vacuum

c

2.997 92 × 108 m∕s

Standard acceleration of free fall

g

9.806 65 m∕s2

Vacuum permittivity

𝜀0

8.854 19 × 10−12 A s∕V∕m

Conversion Factors 1 eV = 1.602 18 × 10−19 J 1 dyne = 10−5 N 1 erg = 10−7 J 1 kcal = 4.184 kJ 1 torr = 133.322 Pa = 1.333 mbar 1 bar = 105 Pa 1 poise (P) = 0.1 Pa s 1 Debye (D) = 3.336 × 10−30 C m 1 V = 1 J/A/s 0 ∘ C = 273.15 K kB T∕e = 25.69 mV at 25 ∘ C

409

411

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441

Index a abrasion 305 accommodation coefficient 222 activated carbons 244 activator 165 adhesion force(s) 24, 129–131, 151, 289, 379 rolling friction 288–289 adhesive 130, 135, 157, 158, 198, 295, 305, 306 adhesive contact 295, 306 adsorbate 191–194, 204, 209, 212, 217, 219, 220, 228, 235, 250, 251, 284, 292 adsorbate structures 179–180 adsorpt 219, 220 adsorption BET adsorption isotherm 232–235 chemisorption and temperature-programmed desorption 248–249 classification of adsorption isotherms 222–224 and desorption rates 219 differential quantities of 226–228 energy 248 heats of 224–226 Langmuir adsorption isotherm 228–230 Langmuir adsorption with lateral interactions 231–232

Langmuir constant and Gibbs energy 230–231 on heterogeneous surfaces 235 planar surfaces, measuring to 236–238 to porous materials capillary condensation 241–242 definitions and classification 239–241 hysteresis and ink-bottle effect 243–244 mercury intrusion porosimetry 246–248 powders and textured materials 238–239 from solution 249–251 time 221–222 adsorption isotherm 219, 222, 224, 227, 238 for pentanol 48 presentation of 224 adsorption time(s) 124, 221–222, 229 adsorptive 219, 220, 222 advancing contact angle 139, 146, 149, 150, 152, 154, 160, 162 aerosol 2, 21 AgCl 75, 82, 361 agglomerate of silicon oxide particles 3 aggregation number 315, 316, 318, 320 AgI 75, 76, 81, 114 aging behavior 304

Physics and Chemistry of Interfaces, Fourth Edition. Hans-Jürgen Butt, Karlheinz Graf, and Michael Kappl. © 2023 WILEY-VCH GmbH. Published 2023 by WILEY-VCH

442

Index

alcohol 49, 120, 347, 350 alkyl glycosides 313 alkyl polyglycosides 313, 336 alkylbenzenesulfonate 310 alkylcarboxylate 310 alkyldimethylpropanesultaine 312 alkylethylene glycol 47, 311, 313, 336, 344 alkylethylene oxides 313 alkylglucoside 312 alkylsulfate 309, 310, 318, 319 Al2 O3 76, 77, 221, 262, 264 Amontons’ law 284–286, 288, 289, 295 of friction 284 amphiphilic molecules 48, 304, 305, 309, 347–351, 367, 368 amphoteric surfactants 313 anionic surfactants 309, 313 annealing 164, 181, 182, 186, 378 antifoam agents 304 antifoamers 304, 343 antioxidants 304 antiwear additive 305 AOT 311 apparent contact angle 149, 150, 152, 160 aquaplaning effect 297 aquatic birds 157 Archard’s wear equation 305 asperities 285–287, 295–297, 299, 305, 306 association colloids 314 atomic beam diffraction 209 atomic beams 209, 217 atomic force microscope (AFM) 110, 111, 120, 123, 126, 131, 133, 179, 203–207, 217, 263, 264, 291, 283, 293, 295, 321, 367, 378, 388 atomic form factor 402 atomic layer deposition (ALD) 261, 262, 280 atomic stick-slip friction 293 attenuated total reflection (ATR) 211, 355

attenuated total reflection infrared spectroscopy (ATR-IR) 211, 355 Auger electron spectroscopy (AES) 196, 214, 215, 217

b Bénard–Marangoni convection 50 Bancroft’s rule 329, 344 Barrel etching 275, 276 bending elastic modulus 335 bending rigidity 335, 336 BET adsorption isotherm 232–235, 238, 252, 385 bilayers 105, 112, 121, 314–323, 335 binding potential 140 biological membranes 322–323 black films 342 block copolymer 244, 314 bolaform surfactant 313 Boltzmann equation 56, 57, 59–61, 89, 114, 115, 117, 375 boundary lubrication 284, 285, 296, 299–300, 303, 306, 307 boundary slip 300 Boussinesq number 366 Bragg condition 359, 357, 395–397, 400 Bravais lattices 177, 178, 217 Brewster angle microscopy 354, 355, 367, 368 bridging forces 125 Brownian motion 2, 111, 193 bubble in a liquid 15, 22 bulk systems, thermodynamic functions for 33–34 Burgers vector 190, 191, 199 butter 3, 330

c calorimeter 189, 225 canal surface viscosimetry 365 capacitances 63, 82–83 capacity 81–83 diffuse layer 63, 67, 72 Helmholtz layer 67

Index

Stern layer 67–68 capillary condensation 23–26, 31, 223, 224, 240–244, 252, 373 capillary constant 15, 24, 157 capillary force 24–26, 146, 151, 156 capillary length 15, 365 capillary number 160–162 capillary penetration technique 155 capillary rise 17, 50, 153–155, 381 capillary waves 357, 365, 366 captive bubble method 147 Cassie state 150, 151 Cassie–Baxter equation 149, 150 cationic surfactants 313 cavitation 122, 306 centrifuge 90, 130 centrifuge method 130, 131 ceramic foams 338 cetyltrimethylammonium bromide (CTAB) 122, 216, 313, 314, 320, 321, 336, 351 Champagne 30 characteristic length scale 3, 15, 324 charged surfactants 49, 318, 320 chemical binding energies 220 chemical vapor deposition (CVD) 259–262 diamond films 260–262 index 259 diamond films 260 chemisorbed polymers 268, 269 chemisorption 220, 248–249, 251, 261, 266 chronoamperometry 82, 83 clay 3, 78, 119, 132, 246 cleavage 106, 182, 183, 188, 217 closed system 38, 39 cloud point 316 critical micellization concentration (CMC) 48, 167, 310, 314–319, 322, 324 of alkylsulfates 318–319

CMC see critical micellization concentration (CMC) coagulation 112, 117, 119, 331 coalescence 326, 330–333 coarsening process 343 coefficient of friction 284, 285, 288, 292, 297 coherent interface 198 coincident site lattice (CSL) model 199, 200 cold welding 286, 306 collector 165 colloid(s) 1, 2 dispersion 1 probe technique 111, 131, 289, 295, 296 systems 1, 2 competitive ablation and polymerization scheme (CAP) 272 composite material 3, 132, 198 concentration profile, of solute 37, 47, 48, 192, 195–197 condensation coefficient 230 conductivity 55, 69, 79, 81, 88, 89, 257, 260, 325, 326, 338, 361 conductometric titration 81, 82 confined liquid 119–121, 300 confocal microscopy 200 constant charge 79, 116 constant potential 116, 117 contact angle 15, 135–137, 141, 142, 264 advancing 145–146 hysteresis 145–148, 169 receding 145–146, 151 contact line, phase 135 continuity equation 86 continuous phase 321, 324–326, 328, 329, 334 conversion factors 409 core region 140–142 corrosion inhibitors 305 cosurfactants 333, 336 Coulomb force 93, 112 Coulomb’s law of friction 284, 286, 293 creaming 331–333

443

444

Index

creaming effect 332, 333 critical micellar concentration (CMC) 48, 167, 310, 314–318, 322, 324 critical radius 28, 241, 243 critical temperature 220, 240, 350, 352 crystalline solid surfaces adsorbate structures 179–180 clean surfaces, preparation of 181 cleavage 182–183 thermal treatment 181 diffraction methods 206–209 electron microscopy 201–203 optical microscopy 200–201 optical spectroscopy of surfaces 209–213 periodic structure of 176 scanning probe microscopes (SPMs) 203–206 secondary ion mass spectrometry 215–217 spectroscopy of inner electrons 213–214 spectroscopy with outer electrons 214–215 substrate structure 176–178 surface diffusion 191–198 thermodynamics of 183–191 crystalline surfaces 176–180, 197, 217, 301, 302 curvatures 14, 15, 39, 335 of sphere, cylinder, and drop 12 curved liquid surface 10–13, 20, 21 cyclic voltammetry 83 cylindrical micelles 245, 320, 321

d DDAB 311 de-inking of paper 165 Debye interaction 96 Debye length 5, 57, 58, 60–63, 79, 87–90, 121, 132, 328, 375 Debye–Hückel approximation 57 Debye–Hückel model 61 deep reactive ion etching (DRIE) 275, 280

defoamers 343, 344 demulsification 324, 326, 328, 331–333 demulsifiers 304 density, liquid surface 5 depletion forces 126–127 depressant 165, 304 Derjaguin approximation 107–109, 114, 117 desorption energy 248, 249, 253 detergency 3, 158, 166–167, 309 detergents 45, 309, 313 dewetting 158–165 dialkyldimethylammonium bromide 311 dialkylglycerophosphatidyl choline, nomenclature of 348 diamond CVD film 260–262 diblock copolymer 313, 314 diffraction at surfaces 399–400 diffraction at three-dimensional crystals Bragg condition 395, 396 Ewald construction 398–399 Laue condition 395–397 reciprocal lattice 397–398 diffraction peaks intensity 400–403 diffuse double layer 54, 57, 58, 63, 64, 91 diffuse electric double layer 58, 65 Diffuse Reflectance Infrared Fourier Transform Spectroscopy 211 diffusion 191, 193–195, 197 coefficient 140, 192–194, 196–198, 331 surface 191–198 dilational elastic modulus 362, 363 dilational surface viscosity 362 dilational viscosity 362, 365, 366 dimeric surfactants 313 dipalmitoyl phospatidylcholine (DPPC) 213, 348, 352, 354 dip coating 163, 164 dipole moment 53, 80, 91, 94–97, 101, 105, 211, 262, 352, 354, 359, 360, 369 dipole–dipole interactions 95–97, 131, 354

Index

dipoles, in a monolayer 353, 356, 360, 362 disjoining pressure 109–110, 114, 115, 125, 140, 142, , 159, 341 dislocation 190 edge 190 screw 190 dislufide 263 dispersants 304 dispersing agent 324, 326 dispersion 2 force 96 interaction 96 DLVO theory 116–119, 121–122, 132 DMT model 129, 130 dodecyl trimethylammonium bromide 313 doping 79, 277 double-layer force 112–119, 132, 142, 331 double-wedge technique 182, 183 drainage 152, 340, 343, 344 DRIFT spectroscopy 211 drop in vapor 21 drop shape analysis 16 drop volume method 171 drop-weight method 17, 18, 32, 372 droplet coalescence 331–333 droplets in microemulsions 334 dropping mercury electrode 73 dry etching 274, 275, 277 dry foam 340, 341 dry friction 284–286, 295, 299 Du Noüy ring tensiometer 18, 19 dynamic contact angles 160–163 dynamic friction 284, 286, 287, 307 dynamic methods 19 dynamic superlubrictiy 302 dynamic viscosity 84, 303

e Einstein–Smoluchowski equation 193 elastic deformation 128, 163, 184, 285

elastic modulus 335, 362–364 elastic properties of surfactant films 335–336 elasto-hydrodynamic lubrication 298 electric conductivity, of dilute emulsions 326 electric double layer 53, 62–64, 91, 112, 328 in aqueous electrolyte 54 capacitance of 81 mathematical description 55–68 around a sphere 61 Gibbs energy 64–65 Gouy–Chapman model, for planar surfaces 57–59 Grahame equation 62–63 one–dimensional case 59–61 Poisson–Boltzmann equation 56 Stern model 67–68 electric field strengths 360 electric surface charge density 57 electrified interfaces 68 capacitances 82–83 charged surfaces 73–80 mercury 74 mica 78–79 oxides 76–78 semiconductors 79–80 silver iodide 75–76 electrocapillarity 70–73 electrokinetic phenomena 83–91 potentiometric colloid titration 80–82 types of potentials 68–70 zeta potential 84 electro-osmosis and streaming potential 86–88 electrophoresis and sedimentation potential 88–91 Navier–Stokes equation 84–86 electro-osmosis 84, 86–88, 90, 91, 168, 376 electrocapillarity 55, 70–74 electrocapillary curves 72–74 electrocapillary measurements 55 electrochemical cell 69, 70

445

446

Index

electrochemical potential 69, 71 electrodeposition 280 electron beam evaporation 256 electron beam lithography 280 electron microscopy 196, 201–203, 240, 247, 325 electroneutrality 58, 62, 67, 71, 113 electron spectroscopy for chemical analysis (ESCA) 213 electrophoresis 84, 88–91, 168, 377 electroplating 280 electrostatic double-layer force 112 DLVO theory 116–119 electrostatic interaction between two identical surfaces 112–116 electrostatic double-layer repulsion 53, 331, 341 electrostatic interaction between two identical surfaces 112–116 electrowetting 168–169 ellipsometry 159, 236–238, 251 Eloxal process 221 empirical law 6, 284, 286, 307 emulsifier 328, 329 emulsion(s) 1, 156, 172, 324, 326, 328–331, 333, 336, 337, 338, 391 energy dispersive X-ray analysis 214, 215 energy of adsorption differential 226 integral 225 enthalpy 38, 41–43, 188, 189, 225–227, 391, 392 enthalpy of adsorption 226 differential 227 integral 225 entrainment of air 162 entropy of adsorption differential 227 integral 225 environmental scanning electron microscopes (ESEM) 202, 203 erosion 305–306 etching 274 barrel 275, 276

chemical 274 deep reactive ion 275, 276, 280 dry 275 ion beam 275 physical 274 plasma 275, 276 reactive ion 275, 276 reactive ion beam 275 sputter 275 etching techniques 274–278 Euler’s theorem 41, 42 evanescent field 210, 211 evaporation 17, 19, 22, 40, 50, 135, 136, 152, 154, 164, 196, 256, 257, 348, 366 Ewald construction 398 for surface diffraction 399 at Cu(110) surface 400 extended X-ray absorption fine structure (EXAFS) 214 extensive parameters 33, 34 external phase 1, 324 extreme pressure additive 305

f far-field interfacial energies 140 fatty acids 166, 305, 309, 314, 347, 351, 352 Fe3 O4 77 Fiber 157–158 Fick’s law 192, 197 field ion microscope 195 film balance 341, 342, 348, 365, 366, 368, 369 film pressure 213, 348–354, 358, 359, 366, 368, 369 vs. surface area per molecule 351 flocculation 126, 331, 332 kinetics 331 rate constant 332 flotation 3, 104, 146, 155, 156, 158, 164–166, 309, 338 fluid lubrication 297 fluid volume element, in flow field 85

Index

fluorescence microscopy 353, 354, 367, 368 foams 338–341, 343, 344 applications 338–339 soap films 341–343 structure 339–341 focused ion beam 280 forced wetting 158, 160, 162 Frank–van der Merwe growth 258 free solid surface energy 141, 142 fretting wear 306 Freundlich adsorption isotherm 235, 252, 385 Freundlich isotherms 250, 253, 387 friction 283, 302 Amontons’ law 284–286 and adhesion 289–290 and lubrication see lubrication Coulomb’s law 284–286 macroscopic 292 measurement techniques 290–292 microscopic 293–296 rolling 288–289 static 286–288 friction coefficients 284, 292, 295–297, 299, 303 on ice 292 kinetic 286–287, 290, 295, 297, 299, 300 rolling 288–289 static 286–287, 290 friction force microscopy (FFM) 291, 293–295, 301, 302, 307 pictures, of graphite surface 294 friction forces 130, 131, 284, 290, 291, 303 friction modifiers 305 Froth 165, 338, 340 Frothing agent 165 Frumkin–Fowler–Guggenheim (FFG) adsorption isotherms 231, 232 Frumkin–Fowler–Guggenheim (FFG) isotherm 231, 232 FT-IR spectrometers 210 fundamental constants 408–409

g galactocerebroside structure 358 Galvani potential 68, 71 gas–gas interfaces 1 Gecko 100 Gemini surfactants 313 geometrical structure factor 401 germanium 79 Gibbs adsorption isotherm 45–47, 72, 328 derivation 45–46 experimental aspects 48–49 ideal interface in 47 system of two components 46–47 Gibbs dividing plane 34, 36, 37, 40, 43, 51, 223, 224, 249 Gibbs energy 33, 41, 64, 230, 231 adsorption 226–228, 230–231 of electric double layer 64–65 interfacial 43 micellization 317–318 Gibbs free energy 9, 27–29, 95, 121, 126, 136–137, 154, 156–157, 166, 322, 326, 335, 343, 376, 391 of micellization 317 Gibbs free interaction energy 116, 118 Gibbs model, of ideal interface 35 Gibbs monolayers 347, 349, 363 glow discharge 181 Gouy–Chapman model 55, 57–59, 63 Gouy–Chapman theory 54, 62, 63, 66, 67, 83, 91, 376 Gouy–Chapman–Stern model 54, 67 grafting-from 269–270, 273 grafting-to 124, 269 Grahame equation 62–63, 68, 75, 90 grain boundary 198 tilt 198–199 twist 199 granular matter 2 granular system 2 grazing incidence X-ray diffraction 209, 356, 357, 388 Guggenheim, Edward Armand 34

447

448

Index

h Hagen–Poiseuille law 167 Hamaker constant 99, 101, 102, 104–106, 118, 131, 142, 144, 145, 299, 377, 378 heat of adsorption 221, 224, 225, 227–230, 232 HEED 208 helium 106, 238 surface energy of 106 Helmholtz energy 33, 38, 39, 41, 95 interfacial 42 Helmholtz free energy 95, 96 Helmholtz layer 54, 67, 91 Henry adsorption isotherm 222–224 Hertz model 127–129, 133, 285, 379 heteroepitaxy 258, 260 heterogeneous nucleation 26, 27, 143 heterogeneous surfaces 235 heterophase interface 198, 217 hexadecyl trimethylammonium bromide 313 high energy electron diffraction 208 high internal phase emulsions (HIPE) 325 higher-angle grain boundaries 199 highly oriented pyrolytic graphite (HOPG) 182, 293, 301 Hofmeister series 66 homoepitaxy 258, 259 homogeneous nucleation 26, 27 homophase interfaces 198 hydration force 112, 121, 328 hydration repulsion 121, 132, 318, 328 hydraulic radius 240, 252 hydrodynamic lubrication 296–300, 307, 390 hydrodynamic theory 161, 163 hydrophile–lipophile balance (HLB) 329, 344 scale 329 hydrophilic surface 121, 136, 149, 281, 318, 388, 389 hydrophobic effect 318, 390 hydrophobic force 122, 132, 165

hydrophobic interaction 122, 268 hydrophobic surface 136, 142, 149, 159, 165 hysteresis, isotherm 243

i ice 220, 230, 292 ideal freely jointed chain 122 ideal interface 34, 35, 40, 41, 46, 47 immersion method 251 impedance spectroscopy 83 inclined plane tribometers 290 incoherent interface 198 infrared reflection absorption spectroscopy (IRRAS/IRAS) 211, 217, 355 infrared (IR) spectroscopy 209–217, 355–356 Iniferter 271 ink 123, 239, 244, 247 ink bottle pore 239, 244, 247 inner Helmholtz plane (IHP) 54 inner phase 1, 68, 324–326 inner phase volume fraction 325 inner potential 56, 68–70 inorganic films 255 insoluble monolayers 347, 348, 368 intensity of diffraction peaks 400–403 interface 1 Gibbs adsorption isotherm 45–51 derivation 45–46 experimental aspects 48–49 Marangoni effect 49–51 system of two components 46–47 pure liquids 43–45 thermodynamic relations for systems with 38–43 equilibrium conditions 39–40 Gibbs energy 41 Helmholtz energy 38–39 interface location 40 interfacial excess energies 41–43 internal energy 38–39 interfaces and colloids 1, 3 interface potential 140–142

Index

interface thickness 5 interfacial enthalpy 42, 43 interfacial excess 35, 41, 43, 47, 71, 72 Helmholtz energy 41 interfacial Gibbs energy 43, 51 interfacial Helmholtz energy 41–43 interfacial potential 141 interfacial tension 3, 10, 40, 41, 70, 72, 136, 138, 154, 157, 169–173, 327–329, 335, 373 interference microscopy 200 internal energy 33, 38 densities 35 interfacial 35, 41–43 internal surface energy 41, 43–45, 186, 226 interphase 34, 198 intersegment force 125 inverse gas chromatography 188 inverted micelles 319, 321, 322 ion beam etching 275 ion etching 275, 376 ion mass spectrometry 215–217 ion plating 257, 280 ionic liquids 138 ionic surfactants 316–319, 328, 336, 344 ionizing electrode method 360, 361 isoelectric point (iep) 84 isolated system 38

j JKR model 128–131, 133, 289, 290, 295, 379

k Keesom interaction 95–97, 106 Kelvin equation 20–23, 28, 31, 139, 241–242, 373 Kelvin length 20–21, 26 Kelvin probe 360 Kelvin question 23 kinematic viscosity 84, 300, 303 kinetic friction coefficient 286–287, 290 kinetic theory of ideal gases 5, 221, 229

Kirchhoffs law 70 Krafft temperature 316 Kugelschaum 339–340

l lab-on-chip 167 lamellar phase 321, 338 Langmuir adsorption isotherm 228–230, 232, 251 Langmuir constant 229–231, 252 Langmuir equation 228, 230, 231, 235 Langmuir trough 348–349, 368 Langmuir–Blodgett transfer technique 366–368 Laplace equation 11–15, 40, 141, 142 application 14–15 derivation of 13–14 fundamental implications 12 Laplace pressure 11, 17, 20–21, 24–25, 154–155, 158, 173, 327, 335, 343, 344 laser-induced thermal desorption (LITD) 197, 248 lateral force microscopy (LFM) 291 Laue condition 395–400 Launderometer 166 LB film 367–368 LB transfer 213, 366–368 Lifshitz theory 100–105 LIGA 280 line tension 142–145, 172, 354, 369, 381 linear polymer 122–123 lipid 3, 320, 366, 392 lipid bilayers 105, 112, 121, 321–323, 348, 354–355 Lippmann equation 72, 74 liquid-condensed phase 351–352 liquid crystals 1, 10, 316, 350 liquid-expanded phase 351–352 liquid foams 110, 338, 339, 343, 344 liquid–liquid interfaces 1, 3, 10, 326, 328, 330, 364 liquid marble 330

449

450

Index

liquid phase 6, 27–28, 35, 56, 71, 142, 162, 243–244, 278, 315, 317, 343, 349, 351, 359–360 liquid superlubricity 302–303 liquid surface 5, 183 see also Young–Laplace equation capillary condensation 23–26 curved 10–13 Kelvin equation 20–22 microscopic structure of 5–6 nucleation theory 26–31 surface tension 6–10 lithography 276, 278–281 London interaction 96, 131 Lotus effect 151 low pressure chemical vapor deposition (LPCVD) 259 low-energy electron diffraction (LEED) 208, 399 diffraction pattern 208 experiment 399 low-energy electrons 208, 215, 306 lower critical solution temperature (LCST) 270–271 lubricant 158, 303 infused surfaces 152 solid 302 lubrication 283, 296–298 boundary 296, 299–300 elasto-hydrodynamic 298 fluid 297 hydrodynamic 296–298 lubricants 303–305 mixed 296, 299 superlubricity 301–303 thin film 300–301

m macroemulsion 323–328 coalescence and demulsification 332–333 evolution and aging 330–332 stabilization 328–330 macropore 240

macroscopic friction 283–284, 292, 296, 302, 307 macroscopic quantities 140 macroscopic wetting 143, 150 magnetic needles 365 Marangoni effect 49–51, 329, 343 margarine 330 maximum bubble pressure method 17, 18 Maxwell relations 42 MCM-41 244 mean free path 255–256, 259 mean micellar aggregation number 315 membrane proteins 322–323 membrane resistance, specific 322 Meniscus force 24, 122, 382 mercury 10, 55, 70, 73–74, 76, 376 mercury porosimetry 246–248 mesophases 350 mesopores 223, 240 mesoporous silica 244, 245 mesoscopic film 142 metal–electrolyte interface, with applied potential 70 metal foams 338 metal organic chemical vapor deposition 260 metal–organic frameworks (MOF) 245–246 methylene diphenyl diisocyanate (MDI) 339 mica 73, 78–79, 110, 121, 132, 213, 296, 366 micelles 48, 121–122, 167–168, 244–245, 304, 309–344, 390 micellization 317, 318 for ionic surfactants 316 thermodynamics of 317–319 microcontacts 285–286, 289, 290, 292, 293, 295, 299, 390 microelectromechanical systems (MEMS) 280, 300–301 microemulsion 324, 334 droplet size 334–335 factors influencing structure 336–338

Index

microfluidics 88, 167, 168 microlithography 278 micropillar arrays 150 micropores 222, 240 microscopic friction 293–296 microtribology 293 milk 3, 135, 324–325 Miller–Bravais indices 177 Miller indices 176–177, 397 miniemulsion 324 mixed lubrication 296, 299–300, 305, 307 MnO2 77 modified Laplace equation 141 modulus of Gaussian curvature 335 molar Gibbs energy 21, 27–28, 226, 317, 327 molecular beam epitaxy (MBE) 183, 198, 217, 258–259, 280 molecular kinetic theory 163 molecular structure of liquid–vapor interface 9 of water 6 monolayers 347, 350–358, 360, 361, 366, 367 with an applied shear 362 coverage 256 experimental techniques 353 IR spectroscopy 355 optical microscopy 353 rheologic properties of liquid surfaces 361 sum frequency generation spectroscopy 355 surface potential 359 X-ray reflection and diffraction 356 Gibbs 347, 349 insoluble 347–348 Langmuir–Blodgett transfer 366–368 mesophases 350 phase behavior of 352 surface potential measurements 360, 361 monomolecular amphiphilic films, phases of 350 multicomponent liquids 35

n nanocontacts 295 nanolithography 280 nanoparticles 3, 269, 322 nanotribology 292, 293, 296, 307 Navier–Stokes equation 84–87, 297 Nearest neighbor broken bond model 186, 217 Nernst equation 75, 76 Neumann triangle 171 neutron reflectivity 48 Newtonian fluid 84, 297, 298 no-slip boundary condition 300 nodoids 12, 13 non-DLVO forces, in aqueous medium 121–122 non-polar liquids 55, 151, 323 nonionic surfactants 313, 316–318, 329, 336, 344 nonlinear optical effects 212 nucleation 3, 22, 26–31, 143, 165, 191, 259, 260 nucleation theory 26–31 null ellipsometer 237

o 4-octyl-4′ -cyanobiphenyl (8CB) 10 oil-in-water emulsion 325, 328–330 O-lattice theory 200 oligomeric surfactants 313 optical microscopy 138, 172, 200–201, 217, 353–366 optical tweezers 111, 112 organic monomolecular layer, LB transfer technique 367 oscillating drop 19, 20 osmotic stress method 112 Ostwald ripening 21, 324, 343 outer Helmholtz plane (OHP) 54, 80 outer phase 324 O/W droplet 335 O/W microemulsion 336, 337 oxidation 220, 221, 371, 277, 292, 304, 306, 389

451

452

Index

oxide

76, 78, 80, 175, 181, 220, 221, 269, 278, 292, 313, 389

p packing ratio 319 paint 2, 3, 50, 135, 158, 255 partial wetting 135 particle 3, 25–26, 61, 64, 69, 155, 165 quartz 99 spherical 24 pendant bubbles 17 pendant drops 16 phase diagram, of a water-in-octane-C12 E5 emulsion 337 phase inversion temperature (PIT) 329, 336–338 phase modulation IRRAS 211 phopholipids 348, 352 phosphatidyl cholines (PC) 347 phosphatidyl ethanolamines (PE) 347 phosphatidyl glycerol (PG) 347 phosphatidyl serines (PSs) 347 phospholipids 66, 321, 347, 348, 351, 352, 354, 368 chemical structure of 348 photoemission electron microscopy (PEEM) 196 photolithography 278–281 photoresist 164, 182, 276, 278–281, 389 physical etching 274, 275 physical vapor deposition (PVD) 256–259 Physisorption 220, 223, 230, 242, 248, 251, 266 Physisorption isotherms 223 Pickering emulsions 330 pin-on-disk tribometer 290 plasma ashing 181, 182 plasma cleaning 181, 182, 266 plasma-enhanced chemical vapor deposition (PECVD) 259 plasma etching 181, 275, 276 plasma polymerization 272–274

plastic deformation 184, 186, 284–286, 289, 292, 306 Plateau border 340, 343, 344 PLAWM trough 349 plunging-tape experiment 161 Pluronics 314 PNIPAM 270, 271 point of zero charge (pzc) 72, 74, 75, 77, 81, 84 Poisson equation 56, 62, 87 Poisson–Boltzmann equation 56, 57, 59–62, 65–67, 87, 89, 91, 114, 115, 117, 375 polarizability 66, 95, 100, 101 polarizable electrode 66, 76, 95, 96, 100, 101 polarizer-compensator-sample-analyzer (PCSA) ellipsometer 237 poly(dimethylsiloxane) 8 poly(ethyleneimine hydroxid) (PEI) 267 polyallylamine 267, 273 polydimethylsiloxane (PDMS) 8, 138, 139, 149, 152, 159, 220, 269, 270, 296, 313 polyederschaum 340 polyelectrolyte 64, 122, 267–268, 303, 389 polyhedral foam 340 polylysine 267 polymer(s) 76, 122–125, 266, 268, 328 brush 124 film 163–165 properties of 122–123 structure 124 polymer-coated surfaces, force between 123–126 polymeric surfactant 313, 314, 344 polymerization 268, 270, 271 atom transfer radical polymerization (ATRP) 271, 274 grafting-from 270, 273 plasma 272–274 pulsed plasma 273

Index

reversible addition fragment chain transfer (RAFT) 271 surface-initiated (SIP) 269, 271 polymethyl methacrylate (PMMA) 280, 382 polypropylene 159 polystyrene 92, 103, 104, 126, 132, 165, 173, 267, 277, 281, 340, 378, 382, 389 polystyrene sulfonate 267 polyurethane (PUR) foams 338, 339 pore-size distribution 246 pore space 239 porous material 31, 154, 155, 222, 224, 236, 238–246, 251 porous solid 2, 23, 220, 222, 240, 339 potential determining ions 75, 76, 80, 81, 84 potentiometric colloid titration 80–82 potentiometric titration 80–82 of latex particles 82 pour point depressants 304 powder 50, 130–131, 135, 154–155, 175, 189, 211, 224–225, 236, 238–240, 247, 251, 330 Prandtl–Tomlinson model 301, 302 precursor film 159 primary energy minimum 118, 332 principal curvatures 14, 15, 335 propanol 8, 45, 103, 120, 374 protein 53, 58, 66, 76, 89, 112, 123, 167, 239, 249, 272, 322, 323, 328, 330, 353 pulp 165 pulsed plasma polymerization 273 pulsed plasma technique 273 pump-probe-type experiments 212 pure liquids 35, 43–45, 51, 170, 341

r radiation scattering at parallel lattice planes 396 radius of curvature 11, 13, 23–25, 120, 143, 204, 241, 243, 296, 337 radius of gyration 123, 124 RCA method 266 reactive ion beam etching 275 reactive ion etching 275, 276 Read–Shockley equation 199 real vs. ideal isotherms 352 receding contact angle 139, 145–147, 153, 160, 161, 247, 264 reciprocal lattice 383, 397–405 RED 58, 208 refined base oils 303 refractive index 100, 102, 103, 200, 211, 237, 251, 260, 368 relative adsorption 36 relative equilibrium vapor pressure 22 retardation 101, 106 retarded Van der Waals forces 105–106 reversible electrode 76, 82, 83 Reynolds number 167, 297 RHEED 208 rheologic properties of liquid surfaces 361–366 rheology 361 rheometer 364, 365 ring tensiometer 18, 19, 31 robust superamphiphobic surfaces 151 rodlike micelles 320 rolling friction 288–289, 307 roughness 26, 83, 100, 110, 111, 139, 148–150, 172, 266, 268, 285, 286, 296, 307, 356, 357, 388

s q quartz crystal microbalance (QCM) 236, 256, 291, 293 quartz particle 99, 103, 168, 236, 252, 256, 283, 291, 294

saddle-splay modulus 335, 336 saturated alkyl chains 347 scanning electron microscope/microscopy (SEM) 111, 201–206, 217 environmental 202

453

454

Index

scanning force microscope (SFM) 111, 203 scanning tunneling microscope (STM) 179, 190, 196, 197, 203–205, 207, 217, 251, 384, 388 Schulze–Hardy rule 116 second harmonic generation (SHG) 212 secondary energy minimum 118, 119, 331 secondary ion mass spectrometry (SIMS) 215 dynamic 215 static 216 sedimentation potential 84, 88–91 self-assembled monolayer (SAM) 262–266, 270, 280 self-energy 55 semicoherent interface 198 semiconductor 79–80, 176, 178, 198, 208, 217, 258, 260–262, 278–280 semiconductor surfaces 178, 262 semiconductor–electrolyte interface potential 79, 80 sessile bubbles 17 sessile drop 15, 16, 31, 136, 139, 143, 146, 155, 160, 380 shape, of water drop resting on flat surface 16 shear rate 84, 85, 297, 298, 364 shear thinning 297, 298 shear thinning effect 297, 298 shear viscosity 362–365 Shuttleworth equation 185 silanes 244, 262–266, 269, 280, 388 silanization 265, 266, 289 silicon, semiconductor 79 silicone surfactants 159, 313 silver chloride 75 silver iodide 75 simple liquid structure 120 single “ink bottle” pore 244 single-chain surfactants 320 sintering 175 SiO2 2, 76, 77, 99, 102–104, 114, 132, 168, 259, 262, 264, 277, 289, 380

size of a polymer 123 slip phase 300 slippery lubricant-infused surfaces (SLIPS) 152 small-angle neutron scattering (SANS) 315, 318 small-angle X-ray scattering (SAXS) 315 soap(s) 303, 309, 314, 326, 341–344, 368 soap films 341–344, 368 sodium dodecanoate 309 sodium dodecyl sulfate (SDS) 48, 49, 168, 309, 310, 314, 316–318, 320, 336, 344, 355, 356 solid foams 338, 339 solid friction 284 solid–liquid interfaces 219 solid mechanics 184 solid–solid interfaces 1, 198–200, 217 solid-stabilized emulsions 330 solubility and CMC 317 soluble amphiphiles 309, 347 solvation forces 119, 120 specific membrane resistance 322 specific surface area 2, 80, 224, 236, 238, 239, 244, 246, 251, 252 spherical particles 3, 24, 61, 90, 112, 118, 126–132, 156, 247, 326, 381 in contact 128 spin coating 163, 164, 278, 279 spinning drop method 171 spontaneous curvature 333, 335–337 of surfactant film 336 spontaneous spreading 158–160, 162 spray coating 164 spreading coefficient 140, 141, 170, 173 sputter deposition 256 sputter etching 275 sputtering 181, 182, 215–217, 256, 257, 280, 389 static contact angle 147 static friction coefficient 286, 290 steric and depletion interaction 122–127 steric force 122, 124, 126, 328 steric repulsion 121, 328

Index

Stern layer 54, 57, 67–68, 83 Stern model 67–68 stick-slip friction 286–289, 294, 301, 302 atomic 293 stick-slip motion 286–287, 300–302, 305 sticking probability 230 Stokes equation 84–87, 297 Stokes law 89 Stranski–Krastanov growth 258, 259 streaming potential 84, 86–88, 91 Stribeck diagram 299, 300 structural superlubricity 301 structure factor 383, 401–403 Styrofoam 338 SU-8 280 substrate structure 176–177, 179, 217, 279 sum frequency generation (SFG) 212, 355, 356 spectroscopy 212, 355, 356 superamphiphobic surfaces 151 superhydrophobic surfaces 150–152 superhydrophobicity 150 superlattice 179–180, 209, 217 superlubricity 301 dynamic 302 liquid 302–303 structural 301 thermal 302 superoleophobic surfaces 151 superposition principle 94 superspreading 159 surface 1 dilational viscosity 362 elastic modulus 362 shear viscosity 362 surface active agents 46, 309 surface active molecules, types of 314 surface charge 53–58, 62–63, 66–78, 80–81, 84, 86–87, 89, 91, 112–114, 116–117, 202, 206, 249, 376 surface diffusion 191–198 surface dilational viscosity 362, 365–366 surface elastic modulus 362–364, 366 surface energies 129, 217

interfacial 44 internal 44 solid 183–188 surface enthalpy 43, 188–189 surface entropy 7, 42–45, 374 surface excess 34–37, 43, 47–48, 51, 223–224, 249, 251, 347, 349–350, 363, 374 surface fatigue 306 surface force apparatus (SFA) 110, 111, 119, 122, 123, 131, 283, 291, 293, 295–296, 300, 303, 307 surface forces 93 Derjaguin approximation 107–109 disjoining pressure 109–110 measurement 110–112 surface intensive parameter 184–185 surface melting 181, 292, 306 surface modification chemical vapor deposition 259–262 etching techniques 274–278 lithography 278–280 physical vapor deposition 256–259 soft matter deposition physisorption of polymers 266–268 plasma polymerization 272–274 polymerization 268–271 self-assembled monolayers 262–266 surface potential 57, 62, 69, 76, 113 vs. charge 63 monolayer 359–361 jump 69 measurements 360, 361 surface reconstruction 178–179, 217, 220 surface relaxation 177–179, 217 surface rotational rheometry 364, 365 surface roughness and heterogeneity 149–150 surface shear viscosity 362–363, 365 surface strain 184–185, 362–353 surface stress 183–186, 188, 217 surface structure 145, 149–150, 176, 178–179, 181–182, 201, 207, 217, 261–262, 276, 280, 399

455

456

Index

surface tension 6–8, 15, 39–41, 44, 45, 143, 183–186, 188 of charged surfactants 49 gradient in 50 measurement technique 15–20 solid 185 unit of 7 surface viscosity 362–363, 366 surfactant(s) 46, 165, 167, 309, 327, 329 aggregates distribution 316 aggregates formed by 321 aggregate structure 319–322 amphoteric 313 anionic 309 bolaform 313 cationic 313 characteristic property of 314 dimeric 313 gemini 313 ionic 316 nonionic 313, 316 oligomeric 313 polymeric 313 structure of 310–312 temperature influence 316, 317 tetrameric 313 trimeric 313 zwitterionic 313, 319 surfactant films, elastic properties of 335–336 surfactant numbers 320 surfactant parameter 319–321, 335, 344 suspending power 167 suspension 2, 53, 304 symbols 42, 59, 114, 120, 136, 169, 178, 180, 184, 220, 260, 264, 290, 350, 405–409

t tears of wine 50 Teflon 106 surface energy of 106 temperature-programmed desorption 248–249 TEMPO 271

tensides 309 tensiometer bubble pressure 17 drop volume 171 du Noüy ring 18 pendant drop 16 spinning drop 171 Wilhelmy plate 19 terrace-ledge-kink (TLK) 189 terraces 189, 217 tetrameric surfactants 313 thermal decomposition 304 thermal desorption spectroscopy (TDS) 248, 252, 387 thermal superlubricity 302 thermal treatment 181, 182, 217 thermodynamic considerations 198 thermodynamic functions, for bulk systems 33–34 thermodynamics of micellization 317–319 theta solvent 123 theta temperature 123 thick films 163, 169–171, 173, 280 thin film 104, 140 balance 341, 342 lubrication 300–301 thiols 180, 249, 263–264, 266, 280, 389 three-phase contact line 19, 135, 138, 143, 147–148, 172 tilted plate method 146–147 TiO2 76, 77, 264, 266 TOF-SIMS 216–217 toluene 51, 126, 220, 322–323, 339, 374 toluene diisocyanate (TDI) 339 Tomlinson’s model 293, 294 total internal reflection 210, 211 total internal reflection fluorescence (TIRF) 211 total internal reflection microscopy (TIRM) 111, 384 transmission electron microscope (TEM) 201, 296 triblock copolymer 314 tribochemical reactions 306

Index

tribology 4, 283 tribometers 290 tribosystem 284 trifunctional silane 265 trimeric surfactant 313 twist boundary 199

u ultrafast laser pulses 212 ultrafast pump-probe experiments 213 ultrahigh vacuum (UHV) 31, 175–176, 180–181, 188, 197, 205–208, 217, 220, 248, 255, 299, 301, 303 ultraviolet photon spectroscopy 70 unduloids 12–13 unit of surface tension 7 unsaturated alkyl chains 347 UV photoemission spectroscopy (UPS) 215, 217

v vacuum, types 255, 256 van der Waals attraction 99, 187, 220 between macroscopic solids 97–106 Lifshitz theory 100–105 microscopic approach 97–100 retarded Van der Waals forces 105–106 surface energy 106 between molecules 93–97 energy 98, 99 equation of state 97, 350 force 98, 99, 101, 102, 104–107, 110, 117, 120, 219, 295, 299, 331 interaction 96–98, 101, 164, 187, 220, 331, 333, 352, 354 type equation 350 vapor condenses 22, 26, 230 vapor pressure, of drop 20, 22 vertical force component 13, 138 vibrating electrode method 360 vicinal surface 189–190, 217 viscoelastic hysteresis 289 viscosity 84, 298, 303, 326

dynamic 84, 303 kinematic 84, 300, 303 of lubricants 298 viscosity index (VI) 303–304 viscosity modifiers 304 void 239 volatile liquid 139, 140 volatility 304 Volmer–Weber growth 258–259 volta potential 68

w water purification 165 water-in-oil emulsion 304, 325, 329 wave vector 358, 395–396, 398–399 wear 283, 305, 306 abrasion 305 defined 305 erosion 305–306 fretting 306 wearless friction 293, 307 Wenzel equation 149 Wenzel state 150–151 wet and dry etching 274 wet foam 339–340 wetting applications 135 detergency 166–167 and dewetting coating 163–164 dynamic contact angles 160–163 spontaneous spreading 158–160 electrowetting 168–169 flotation 164–165 microfluidics 167–168 of real surfaces advancing and receding contact angles 145–146 contact angle hysteresis, causes of 147–149 measurement of contact angles 146–147 superhydrophobic surfaces 150–152 surface roughness and heterogeneity 149–150

457

458

Index

wetting (contd.)

y

surfaces with low sliding angle 152 road pavements 135 thick films 169–172 wetting geometries capillary rise 153–155 network of fibers 157 particles at interfaces 155–157 wetting line 135, 143, 147–150, 162–163 wetting potential 140 wetting transition 143–144 Wilhelmy plate method 19, 31, 32, 146–147, 349, 365–366, 372 Wood’s notation 180 work function 70, 196, 203 wormlike micelles 320 Wulff construction 186, 217

yield stress 285–286, 289, 390 Young’s equation 135–140 derivation 136–140 equilibrium contact angle 135–136 interfacial energy, estimation of 144–145 line tension 142–143 surface forces 140–142 wetting transitions 143–144 Young–Laplace equation 11, 17, 40 application 14–15 derivation of 13–14 fundamental implications 12

x X-ray diffraction 356, 357, 359, 400 grazing incidence diffraction 357, 359 reflection 356–359, 369 X-ray photoemission spectroscopy (XPS) 213–215, 217

z zeolites 236, 240, 244–246 zeta potential 83–84 electro-osmosis and streaming potential 86–88 electrophoresis and sedimentation potential 88–91 Navier–Stokes equation 84–86 zwitterionic amphiphiles 347 zwitterionic surfactants 313, 319

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