COMPUTATIONAL METHODS HI SURFACE AND COLLOID SCIENCE docx

Nonionic Surfactants: Physical Chemistry, edited by Martin J.. Anionic Surfactants: Analytical Chemistry, Second Edition, Revised and Expanded, edited by John Cross 74.. Structure and Ph

COMPUTATIONAL METHODS HI SURFACE AND COLLOID SCIENCE

SURFACTANT SCIENCE SERIES

Department of Chemical Engineering

Massachusetts Institute of Technology

1 Nonionic Surfactants, edited by Martin J Schick (see also Volumes 19, 23,

and 60)

2 Solvent Properties of Surfactant Solutions, edited by Kozo Shinoda (see

Volume 55)

3 Surfactant Biodegradation, R D Swisher (see Volume 18)

4 Cationic Surfactants, edited by Eric Jungermann (see also Volumes 34, 37,

8 Anionic Surfactants: Chemical Analysis, edited by John Cross

9 Stabilization of Colloidal Dispersions by Polymer Adsorption, Tatsuo Sato and Richard Ruch

10 Anionic Surfactants: Biochemistry, Toxicology, Dermatology, edited by Christian Gloxhuber (see Volume 43)

11 Anionic Surfactants: Physical Chemistry of Surfactant Action, edited by E H Lucassen-Reynders

12 Amphoteric Surfactants, edited by B R Bluestein and Clifford L Hilton (see

Volume 59)

13 Demulsification: Industrial Applications, Kenneth J Lissant

14 Surfactants in Textile Processing, Arved Datyner

15 Electrical Phenomena at Interfaces: Fundamentals, Measurements, and

Ap-plications, edited by Ayao Kitahara and Akira Watanabe

16 Surfactants in Cosmetics, edited by Martin M Rieger (see Volume 68)

17 Interfacial Phenomena: Equilibrium and Dynamic Effects, Clarence A Miller and P Neogi

18 Surfactant Biodegradation: Second Edition, Revised and Expanded, R D Swisher

19 Nonionic Surfactants: Chemical Analysis, edited by John Cross

20 Detergency: Theory and Technology, edited by W Gale Cutler and Erik Kissa

21 Interfacial Phenomena in Apolar Media, edited by Hans-Friedrich Eicke and Geoffrey D Parfitt

22 Surfactant Solutions: New Methods of Investigation, edited by Raoul Zana

23 Nonionic Surfactants: Physical Chemistry, edited by Martin J Schick

24 Microemulsion Systems, edited by Henri L Rosano and Marc Clausse

25 Biosurfactants and Biotechnology, edited by Nairn Kosaric, W L Cairns, and Neil C C Gray

26 Surfactants in Emerging Technologies, edited by Milton J Rosen

27 Reagents in Mineral Technology, edited by P Somasundaran and Brij M Moudgil

28 Surfactants in Chemical/Process Engineering, edited by Darsh T Wasan, Martin E Ginn, and Dinesh O Shah

29 Thin Liquid Films, edited by I B Ivanov

30 Microemulsions and Related Systems: Formulation, Solvency, and Physical

Properties, edited by Maurice Bourrel and Robert S Schechter

31 Crystallization and Polymorphism of Fats and Fatty Acids, edited by Nissim Garti and Kiyotaka Sato

32 Interfacial Phenomena in Coal Technology, edited by Gregory D Botsaris and Yuli M Glazman

33 Surfactant-Based Separation Processes, edited by John F Scamehorn and Jeffrey H Harwell

34 Cationic Surfactants: Organic Chemistry, edited by James M Richmond

35 Alkylene Oxides and Their Polymers, F E Bailey, Jr., and Joseph V Koleske

36 Interfacial Phenomena in Petroleum Recovery, edited by Norman R Morrow

37 Cationic Surfactants: Physical Chemistry, edited by Donn N Rubingh and Paul M Holland

38 Kinetics and Catalysis in Microheterogeneous Systems, edited by M Gratzel and K Kalyanasundaram

39 Interfacial Phenomena in Biological Systems, edited by Max Bender

40 Analysis of Surfactants, Thomas M Schmitt

41 Light Scattering by Liquid Surfaces and Complementary Techniques, edited

by Dominique Langevin

42 Polymeric Surfactants, Irja Piirma

43 Anionic Surfactants: Biochemistry, Toxicology, Dermatology Second Edition,

Revised and Expanded, edited by Christian Gloxhuberand Klaus Kunstler

44 Organized Solutions: Surfactants in Science and Technology, edited by Stig

E Friberg and Bjorn Lindman

45 Defoaming: Theory and Industrial Applications, edited by P R Garrett

46 Mixed Surfactant Systems, edited by Keizo Ogino and Masahiko Abe

47 Coagulation and Flocculation: Theory and Applications, edited by Bohuslav Dobias

48 Biosurfactants: Production • Properties • Applications, edited by Nairn saric

Ko-49 Wettability, edited by John C Berg

50 Fluorinated Surfactants: Synthesis • Properties • Applications, Erik Kissa

51 Surface and Colloid Chemistry in Advanced Ceramics Processing, edited by Robert J Pugh and Lennart Bergstrom

52 Technological Applications of Dispersions, edited by Robert B McKay

53 Cationic Surfactants: Analytical and Biological Evaluation, edited by John Cross and Edward J Singer

54 Surfactants in Agrochemicals, Tharwat F Tadros

55 Solubilization in Surfactant Aggregates, edited by Sherril D Christian and John F Scamehorn

56 Anionic Surfactants: Organic Chemistry, edited by Helmut W Stache

57 Foams: Theory, Measurements, and Applications, edited by Robert K homme and SaadA Khan

Prud'-58 The Preparation of Dispersions in Liquids, H N Stein

59 Amphoteric Surfactants: Second Edition, edited by Eric G Lomax

60 Nonionic Surfactants: Polyoxyalkylene Block Copolymers, edited by Vaughn

M Nace

61 Emulsions and Emulsion Stability, edited by Johan Sjoblom

62 Vesicles, edited by Morton Rosoff

63 Applied Surface Thermodynamics, edited by A W Neumann and Jan K Spelt

64 Surfactants in Solution, edited byArun K Chattopadhyay and K L Mittal

65 Detergents in the Environment, edited by Milan Johann Schwuger

66 Industrial Applications of Microemulsions, edited by Conxita Solans and Hironobu Kunieda

67 Liquid Detergents, edited by Kuo-Yann Lai

68 Surfactants in Cosmetics: Second Edition, Revised and Expanded, edited by Martin M Rieger and Linda D Rhein

69 Enzymes in Detergency, edited by Jan H van Ee, Onno Misset, and Erik J Baas

70 Structure-Performance Relationships in Surfactants, edited by Kunio Esumi and Minoru Ueno

71 Powdered Detergents, edited by Michael S Showell

72 Nonionic Surfactants: Organic Chemistry, edited by Nico M van Os

73 Anionic Surfactants: Analytical Chemistry, Second Edition, Revised and

Expanded, edited by John Cross

74 Novel Surfactants: Preparation, Applications, and Biodegradability, edited by Krister Holmberg

75 Biopolymers at Interfaces, edited by Martin Malmsten

76 Electrical Phenomena at Interfaces: Fundamentals, Measurements, and

Ap-plications, Second Edition, Revised and Expanded, edited by Hiroyuki shima and Kunio Furusawa

Oh-77 Polymer-Surfactant Systems, edited by Jan C T Kwak

78 Surfaces of Nanoparticles and Porous Materials, edited by James A Schwarz and Cristian I Contescu

79 Surface Chemistry and Electrochemistry of Membranes, edited by Torben Smith Sorensen

80 Interfacial Phenomena in Chromatography, edited by Emile Pefferkorn

81 Solid-Liquid Dispersions, Bohuslav Dobias, Xueping Qiu, and Wolfgang von Rybinski

82 Handbook of Detergents, editor in chief: Uri Toiler

Part A: Properties, edited by Guy Broze

83 Modern Characterization Methods of Surfactant Systems, edited by Bernard

P Binks

84 Dispersions: Characterization, Testing, and Measurement, Erik Kissa

85 Interfacial Forces and Fields: Theory and Applications, edited by Jyh-Ping Hsu

86 Silicone Surfactants, edited by Randal M Hill

87 Surface Characterization Methods: Principles, Techniques, and Applications,

edited by Andrew J Milling

88 Interfacial Dynamics, edited by Nikola Kallay

89 Computational Methods in Surface and Colloid Science, edited by MaJgorzata Borowko

ADDITIONAL VOLUMES IN PREPARATION

Adsorption on Silica Surfaces, edited by Eugene Papirer

Fine Particles: Synthesis, Characterization, and Mechanisms of Growth,

edited by Tadao Sugimoto

Nonionic Surfactants: Alkyl Polyglucosides, edited by Dieter Balzer and Harald Luders

METHODS IN SURFACE AND COLLOID SCIENCE

edited by Ma+gorzata Borowko

Maria Curie-Sk-todowska University

Lublin, Poland

M A R C E L

MARCEL DEKKER, INC N E W YORK • BASEL

ISBN: 0-8247-0323-5

This book is printed on acid-free paper.

Headquarters

Marcel Dekker, Inc.

270 Madison Avenue, New York, NY 10016

Copyright © 2000 by Marcel Dekker, Inc All Rights Reserved.

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission

in writing from the publisher.

Current printing (last digit):

10 9 8 7 6 5 4 3 2 1

PRINTED IN THE UNITED STATES OF AMERICA

Interfacial systems are frequently encountered in a large variety of phenomena

in biology and industry A few examples that come to mind are adsorption,catalysis, corrosion, flotation, osmosis, and colloidal stability In particular,surface films are very interesting from a cognitive point of view Surfacescience has a long history For many years, natural philosophers werecurious about interfacial phenomena because it was quite clear that matternear surface differs in its properties from the same matter in bulk Decades

of patient analysis and laboratory experiments gave only an approximatepicture of a situation at the interface, which follows from a great complexity

of investigated systems However, much of the progress in science consists ofasking old questions in new, more penetrating, and more wide-rangingways

One of the scientific advances that shaped history during the 20th century

is the revolution in computer technology It has given a strong impetus to thedevelopment of mathematical modelling of physical processes The powerfulnew tools are vehemently accelerating the pace of interfacial research Wecan easily carry out calculations that no one had previously imagined.Computer simulations have already had quite impressive achievements insurface science, so it seems timely to write a monograph summarizing theresults

The existing books cover the simple, rather than the advanced, retical approaches to interfacial systems This volume should fill this gap

theo-in the literature It is the purpose of this volume to serve as a comprehensivereference source on theory and simulations of various interfacial systems.Furthermore, it shows the power of statistical thermodynamics that offers a

Hi

iv Preface

reliable framework for an explanation of interfacial phenomena Thisbook is intended primarily for scientists engaged in theoretical physicsand chemistry It should also be a useful guide for all researchers andgraduate students dealing with surface and colloid science

The book is divided into 18 chapters written by different experts onvarious aspects In many areas of contemporary science, one is confrontedwith the problem of theoretical descriptions of adsorption on solids Thisproblem is discussed in the first part of the volume The majority of inter-facial systems may be considered as fluids in confinement Therefore, thefirst chapter is devoted to the behavior of confined soft condensed matter.Because quantum mechanics is a paradigm for microscopic physics, quan-tum effects in adsorption at surfaces are considered (Chapter 2) The theory

of simple and chemically reacting nonuniform fluids is discussed in Chapters

3 and 4 In Chapters 5 and 6, the current state of theory of adsorption onenergetically and geometrically heterogeneous surfaces, and in randomporous media, is presented Recent molecular computer-simulation studies

of water and aqueous electrolyte solutions in confined geometries arereviewed in Chapter 7 In Chapter 8, the Monte Carlo simulation of surfacechemical reactions is discussed within a broad context of integrated studiescombining the efforts of different disciplines Theoretical approaches to thekinetic of adsorption, desorption, and reactions on surfaces are reviewed inChapter 9

Chapters 10 through 14 examine the systems containing the polymermolecules Computer simulations are natural tools in polymer science.This volume gives an overview of polymer simulations in the dense phaseand the survey of existing coarse-grained models of living polymers used incomputer experiments (Chapters 10 and 11) The properties of polymerchains adsorbed on hard surfaces are discussed in the framework of dynamicMonte Carlo simulations (Chapter 12) The systems involving surfactantsand ordering in microemulsions are described in Chapters 13 and 14.Chapters 15 through 17 are devoted to mathematical modeling ofparticular systems, namely colloidal suspensions, fluids in contact with semi-permeable membranes, and electrical double layers Finally, Chapter 18summarizes recent studies on crystal growth process

I hope that this book will be useful for everyone whose professionalactivity is connected with surface science

I would like to thank A Hubbard for the idea of a volume on computersimulations in surface science and S Sokolowski for fruitful discussions andencouragement I thank the authors who contributed the various chapters.Finally, R Zagorski is acknowledged for his constant assistance

Malgorzata Borowko

3 Integral Equations in the Theory of Simple Fluids 135

Douglas Henderson, Stefan Sokolowski, and

Malgorzata Borowko

4 Nonuniform Associating Fluids 167

Malgorzata Borowko, Stefan Sokolowski, and Orest Pizio

5 Computer Simulations and Theory of Adsorption onEnergetically and Geometrically Heterogeneous

Surfaces 245

Andrzej Patrykiejew and Malgorzata Borowko

6 Adsorption in Random Porous Media 293

Orest Pizio

7 Water and Solutions at Interfaces: Computer Simulations

on the Molecular Level 347

Eckhard Spohr

vi Contents

8 Surface Chemical Reactions 387

Ezequiel Vicente Albano

9 Theoretical Approaches to the Kinetics of Adsorption,

Desorption, and Reactions at Surfaces 439

H J Kreuzer and Stephen H Payne

10 Computer Simulations of Dense Polymers 481

Kurt Kremer and Florian Muller-Plathe

11 Computer Simulations of Living Polymers and Giant Micelles 509

Andrey Milchev

12 Conformational and Dynamic Properties of Polymer Chains

Adsorbed on Hard Surfaces 555

Andrey Milchev

13 Systems Involving Surfactants 631

Friederike Schmid

14 Ordering in Microemulsions 685

Robert Holyst, Alina Ciach, and Wojciech T Gozdz

15 Simulations of Systems with Colloidal Particles 745

Matthias Schmidt

16 Fluids in Contact with Semi-permeable Membranes 775

Sohail Murad and Jack G Powles

17 Double Layer Theory: A New Point of View 799

Janusz Stafiej and Jean Badiali

18 Crystal Growth and Solidification 851

Heiner Miiller-Krumbhaar and Yukio Saito

Index 933

Ezequiel Vicente Albano, Ph.D Instituto de Investigaciones

Fisicoquimcas Teoricas y Aplicadas, Universidad National de La Plata,

La Plata, Argentina

Jean Badiali, Ph.D Structure et Reactivite des Systemes Interfaciaux,

Universite P et M Curie, Paris, France

Matgorzata Borowko, Ph.D Department for the Modelling of

Physico-Chemical Processes, Maria Curie-Sktodowska University, Lublin,Poland

Alina Ciach, Ph.D Institute of Physical Chemistry, Polish Academy of

Sciences, Warsaw, Poland

Wojciech T Gozdz, Ph.D Institute of Physical Chemistry, Polish

Academy of Sciences, Warsaw, Poland

Douglas Henderson, Prof Department of Chemistry and Biochemistry,

Brigham Young University, Provo, Utah

Robert Hofyst, Ph.D Institute of Physical Chemistry, Polish Academy

of Sciences, Warsaw, Poland

Kurt Kremer, Ph.D Max-Planck-Institut fur Polymerforschung, Mainz,

Germany

vii

viii Contributors

H J Kreuzer, Dr.rer.nat., F.R.S.C Department of Physics, Dalhousie

University, Halifax, Nova Scotia, Canada

Andrey Milchev, Ph.D., Dr.Sci.Habil Institute for Physical Chemistry,

Bulgarian Academy of Sciences, Sofia, Bulgaria

Florian Miiller-Plathe, Ph.D Max-Planck-Institut fiir

Polymerfor-schung, Mainz, Germany

Heiner Muller-Krumbhaar, Prof Dr Institut fiir Festkorperforschung,

Forschungszentrum Jiilich, Jiilich GMBH, Germany

Sohail Murad, Ph.D Department of Chemical Engineering, University of

Illinois at Chicago, Chicago, Illinois

Peter Nielaba, Prof Dr Department of Physics, University of Konstanz,

Konstanz, Germany

Andrzej Patrykiejew, Ph.D Department for the Modelling of

Physico-Chemical Processes, Maria Curie-Sklodowska University, Lublin,Poland

Stephen H Payne Department of Physics, Dalhousie University,

Halifax, Nova Scotia, Canada

Orest Pizio, Ph.D Instituto de Quimica de la Universidad Nacional

Autonoma de Mexico, Coyoacan, Mexico

Jack G Powles, Ph.D., D.es.Sc Physics Laboratory, University of

Kent, Canterbury, Kent, England

Yukio Saito, Ph.D Department of Physics, Keio University, Yokohama,

Japan

Friederike Schmid, Dr.rer.nat Max-Planck-Institut fiir

Polymerfor-schung, Mainz, Germany

Matthias Schmidt, Dr.rer.nat Institut fiir Theoretische Physik II,

Heinrich-Heine-Universitat Dusseldorf, Diisseldorf, Germany

Martin Schoen, Dr.rer.nat Fachbereich Physik - Theoretische Physik,

Bergische Universitat Wuppertal, Wuppertal, Germany

Contributors ix

Stefan Sokotowski, Ph.D Department for the Modelling of

Physico-Chemical Processes, Maria Curie-Skiodowska University, Lublin, Poland

Eckhard Spohr, Ph.D Department of Theoretical Chemistry, University

of Ulm, Ulm, Germany

Janusz Stafiej, Ph.D Department of Electrode Processes, Institute of

Physical Chemistry, Polish Academy of Sciences, Warsaw, Poland

Structure and Phase Behavior of

Confined Soft Condensed Matter

MARTIN SCHOEN Fachbereich Physik—Theoretische Physik,

Bergische Universitat Wuppertal, Wuppertal, Germany

A Stochastic processes 22

B Implementation of stress-strain ensembles for open and

closed systems 24

C The Taylor-expansion algorithm for "simple" fluids 26

D Orientationally biased creation of molecules 28

IV Microscopic Structure 29

A Planar substrates 29

B The transverse structure of confined fluids 41

C Nonplanar substrates 45

V Phase Transitions 49

A Shear-induced phase transitions in confined fluids 49

B Liquid-gas equilibria in confined systems 56References 66

I INTRODUCTION

In many areas of contemporary science and technology one is confrontedwith the problem of miniaturizing parts of the system of interest in order tocontrol processes on very short length and time scales [1] For example, tostudy the kinetics of certain chemical reactions, reactants have to be mixed

at a sufficiently high speed By miniaturizing a continuous-flow mixer,Knight et al have recently shown that nanoliters can be mixed withinmicroseconds, thus permitting one to study fast reaction kinetics on timescales unattainable with conventional mixing technology [2] The impor-tance of designing and constructing microscopic machines gave rise to anew field in applied science and engineering known as "microfabricationtechnology" or "microengineering" [3] A central problem in the operation

of such small mechanical machines is posed by friction between movablemachine parts and wear Lubricants consisting of, say, organic fluids can beemployed to reduce these ultimately destructive phenomena Their function-ing depends to a large extent on the nature of the interaction between thefluid and the solid substrate it lubricates [4,5] In the case of micromachinesthe lubricant may become a thin confined film of a thickness of only one ortwo molecular layers The impact of such severe confinement is perhaps bestillustrated by the dramatic increase of the shear viscosity in a hexadecanefilm of a thickness of two molecular layers, which may exceed the bulk shearviscosity by four orders of magnitude [6]

Understanding the effect of confinement on the phase behavior andmaterials properties of fluids is therefore timely and important from both

a fundamental scientific and an applied technological perspective This isparticularly so because the fabrication and characterization of confiningsubstrates with prescribed chemical or geometrical structures on a nano-

to micrometer length scale can nowadays be accomplished in the laboratorywith high precision and by a variety of techniques For example, by means

of various lithographic methods [3,7] or wet chemical etching [8] the surfaces

of solid substrates can be endowed with well-defined nanoscopic lateralstructures In yet another method the substrate is chemically patterned byelastomer stamps and, in certain cases, subsequent chemical etching [9-12].The development of a host of scanning probe devices such as the atomicforce microscope (AFM) [13-17] and the surface forces apparatus (SFA)[18-22], on the other hand, enables experimentalists to study almostroutinely the behavior of soft condensed matter confined by such substrates

to spaces of molecular dimensions However, under conditions of severeconfinement a direct study of the relation between material properties andthe microscopic structure of confined phases still remains an experimentalchallenge

Structure and Phase Behavior of Soft Condensed Matter 3

Computer simulations, on the other hand, are ideally suited to addressthis particular question from a theoretical perspective Generally speaking,computer simulations permit one to pursue the motion of atoms or mole-cules in space and time Since the only significant assumption concerns thechoice of interaction potentials, the behavior of condensed matter can beinvestigated essentially in a first-principles fashion At each step of thesimulation one has instantaneous access to coordinates and momenta ofall molecules Thus, by applying the laws of statistical physics, one candetermine the thermomechanical properties of condensed matter as well asits underlying microscopic structure In many cases the insight gained bycomputer simulations was and is unattainable by any other theoreticalmeans Perhaps the most prominent and earliest example in this regardconcerns the prediction of solid-fluid phase transitions in hard-sphere fluids

at high packing fraction [23]

However, because of limitations of computer time and memory required

to treat dense many-particle systems, computer simulations are usuallyrestricted to microscopic length and time scales (with hard-sphere fluids,which may be viewed as a model for colloidal suspensions [24] (this volume,chapter by M Schmidt), and Brownian dynamics [25] as two prominentexceptions) This limitation can be particularly troublesome in investigations

of, say, critical phenomena where the correlation length may easily exceedthe microscopic size of the simulation cell In confinement, on the other hand,

a phase may be physically bound to microscopically small volumes in one ormore dimensions by the presence of solid substrates so that computer simu-lations almost become a natural theoretical tool of investigation by whichexperimental methods can be complemented It is then not surprising that thestudy of confined phases by simulational techniques is still flourishing [26],illustrated here for one particular aspect, namely the relation between micro-scopic structure and phase transitions in confined fluids In Sec II an intro-duction to equilibrium theory of confined phases will be given Sec Ill isdevoted to formal and technical aspects of computer simulations In Sec IVthe microscopic structure of confined phases will be analyzed for a number ofdifferent systems The chapter concludes in Sec V with a description of phasetransitions that are unique to phases in confined geometry

II EQUILIBRIUM THEORY OF CONFINED PHASES

A Thermodynamics

1 Experiments with the Surface Forces Apparatus

The force exerted by a thin fluid film on a solid substrate can be measuredwith nearly molecular precision in the SFA [27] In the SFA a thin film is

4 Schoen

confined between the surfaces of two cylinders arranged such that their axesare at right angles [27] In an alternative setup the fluid is confined betweenthe surface of a macroscopic sphere and a planar substrate [28] However,crossed-cylinder and sphere-plane configurations can be mapped onto eachother by differential-geometrical arguments [29] The surface of each macro-scopic object is covered by a thin mica sheet with a silver backing, which

permits one to measure the separation h between the surfaces by optical

interferometry [27] The radii are macroscopic so that the surfaces may be

taken as parallel on a molecular length scale around the point of minimum distance In addition, they are locally planar, since mica can be prepared

with atomic smoothness over molecularly large areas This setup isimmersed in a bulk reservoir of the same fluid of which the film consists

Thus, at thermodynamic equilibrium temperature T and chemical potential

li are equal in both subsystems (i.e., film and bulk reservoir) By applying an

external force in the direction normal to both substrate surfaces, the ness of the film can be altered either by expelling molecules from it or byimbibing them from the reservoir until thermodynamic equilibrium is re-established, that is, until the force exerted by the film on the surfaces equals

thick-the applied normal force Plotting this force per radius R, F/R, as a function

of h yields a damped oscillatory curve in many cases (see, for instance, Fig 1

in Ref [30])

In another mode of operation of the SFA a confined fluid can be exposed

to a shear strain by attaching a movable stage to the upper substrate (i.e.,

wall) via a spring characterized by its spring constant k [6,31,32] and moving this stage at some constant velocity in, say, the x direction parallel to the

film-wall interface Experimentally it is observed that the upper wall first

"sticks" to the film, as it were, because the upper wall remains stationary.From the known spring constant and the measured elongation of the spring,the shear stress sustained by the film can be determined Beyond a criticalshear strain (i.e., at the so-called "yield point" corresponding to the max-imum shear stress sustained by the film) the shear stress declines abruptlyand the upper wall "slips" across the surface of the film If the stage moves

at a sufficiently low speed the walls eventually come to rest again until thecritical shear stress is once again attained so that the stick-slip cycle repeatsitself periodically

This stick-slip cycle, observed for all types of film compounds rangingfrom long-chain (e.g., hexadecane) to spheroidal [e.g., octamethylcyclotetra-siloxane (OMCTS)] hydrocarbons [21], has been attributed by Gee et al [30]

to the formation of solid-like films that pin the walls together (region ofsticking) and must be made to flow plastically in order for the walls to slip.This suggests that the structure of the walls induces the formation of a solidfilm when the walls are properly registered and that this film "melts" when

Structure and Phase Behavior of Soft Condensed Matter 5

the walls are moved out of the correct registry As was first demonstrated inRef 33, such solid films may, in fact, form in "simple" fluids betweencommensurate walls on account of a template effect imposed on the film

by the discrete (i.e., atomically structured) walls However, noting that thestick-slip phenomenon is general, in that it is observed in every liquid inves-tigated, and that the yield stress may exhibit hysteresis, Granick [21] hasargued that mere confinement may so slow mechanical relaxation of the filmthat flow must be activated on a time scale comparable with that of theexperiment This more general mechanism does not necessarily involve solidfilms which can be formed only if the (solid-like) structure of the film andthat of the walls possess a minimum geometrical compatibility

2 The Fluid Lamella

For a theoretical analysis of SFA experiments it is prudent to start from asomewhat oversimplified model in which a fluid is confined by two parallel

substrates in the z direction (see Fig 1) To eliminate edge effects, the strates are assumed to extend to infinity in the x and y directions The system

sub-in the thermodynamic sense is taken to be a lamella of the fluid bounded by

the substrate surfaces and by segments of the (imaginary) planes x = 0,

x = s x , y = 0, and y = s y Since the lamella is only a virtual construct it is

convenient to associate with it the computational cell in later practical

zx

FIG 1 Schematic of two atomically structured, parallel surface planes (from Ref 134).

6 Schoen

applications (see Sees IV, V) It is assumed that the lower substrate isstationary in the laboratory coordinate frame, whose origin is at 0, andthat the substrates are identical and rigid The crystallographic structure

of the substrate is described by a rectangular unit cell having transverse

dimensions £ x x £ y In general, each substrate consists of a large number of

planes of atoms parallel with the x-y plane The plane at the film-substrate interface is called the surface plane It is taken to be contained in the x-y plane The distance between the surface planes is s z To specify the trans-

verse alignment of the substrates, registry parameters a x and a y are

intro-duced Coordinates of a given atom (2) in the upper surface plane (z — s z )

are related to its counterpart (1) in the lower surface plane (z = 0) by

(1)

Thus the extensive variables characterizing the lamellar system are entropy

S, number of fluid molecules N, s x , s y , s z , a x £ x , and a y £ y

Gibbs's fundamental relation governing an infinitesimal, reversible formation can be written

where the mechanical work can be expressed as

as a — / y / , Aa 1 a8 as B W)

The primes denote restricted summations over Cartesian components

(a, /? = *,>>, z), ds a is a displacement in the a direction, A a is the area of

the a-directed face of the lamella, and T a p is the average of the /5-component

of the stress applied to A a Note that if the force exerted by the lamella on

A Q points outward, T af3 < 0 Thus, dW mtc ^ is the mechanical work done by

the system on the surroundings Terms involving diagonal and off-diagonal

elements of the stress tensor T in Eq (3) respectively represent the work ofcompressing and shearing the lamella Note that because the substratesare rigid they cannot be compressed or sheared This is the reason for

the absence of the four off-diagonal contributions involving T xz , T vz , T xy ,

and T yx

To introduce area A = A z as an independent variable, the transformation

(4)

Structure and Phase Behavior of Soft Condensed Matter 7

is introduced In terms of these new variables Eq (2) becomes

dU = TdS + ndN + j'dA + -y" AdR + T zz Ads z

Note that the definition of R is arbitrary However, the present choice seems

simplest and has a transparent physical interpretation The work done by

the system in an infinitesimal reversible transformation at constant S, N, A,

s z , a x £ x , and a y £ y is given by

dW = T xx s y s z ds x + Ty V s x s z ds y — (T xx — T yy )s y s z ds x = 7"AdR (8)

because ds y — —s y s x] ds x It is then clear that the fourth term in Eq (5) is the

net work done by the lamella as its shape (R = s x /s y ) is changed at fixed

area

To recast the thermodynamic description in terms of independent

vari-ables that can be controlled in actual laboratory experiments (i.e., T, fi, and

the set of strains or their conjugate stresses), it is sensible to introducecertain auxiliary thermodynamic potentials via Legendre transformations.This chapter is primarily concerned with

where Eqs (5) and (10) have also been employed Other relevant potentials

can be obtained by suitable Legendre transformations of T or O with respect to, say, T zz , T zx , or T zy (see Sec VA1)

Conditions for thermodynamic equilibrium of the lamella can be derived

by considering the lamella plus its environment as an isolated supersystem.

Assuming the entropy of the supersystem to be fixed, one knows that the

8 Schoen

internal energy must be minimum in a state of thermodynamic equilibrium

In mathematical terms, an infinitesimal virtual transformation that wouldtake the system from this state must satisfy

6{U + U)>0 (12)

<5(<S + <S)>0 (13)

where 8U is given in Eqs (2), (3) and 8U by

6U = T6S + flSN + Y^' Z T A af a pSs 0 (14)

and the tilde refers to environmental variables Viewing the environment as

virtual pistons, displacements 6s a of the boundary between them and the

lamella satisfy the equation 6s a = -6s a Moreover, because the supersystem

is materially closed, 6N = -6N From these two conditions and Eqs

(12)-(14), the equilibrium conditions

(15)

are deduced Now suppose the lamella is subject to thermal, mechanical, andchemical reservoirs that maintain temperature, normal stress, and chemical

potential fixed at the values f, f zz , and // Assume also that the

"comple-mentary" strains A, R, a x £ x , and a y £ y are kept fixed Then one has, fromEqs (12) and (14)

oil + Too + \i8N + > > A a T a p6sp = o[u — To — fiN — AT zz s z \ > 0

con-ture of the substrate surfaces, T zz becomes a local quantity which varies with

the vertical distance s z = s z (x,y) between the substrate surfaces (see Fig 2).

Since the sphere-plane arrangement (see Sec II Al) is immersed in bulkfluid at pressure Pbuik? t n e t o t al force exerted on the sphere by the film in

Structure and Phase Behavior of Soft Condensed Matter

FIG 2 Side view of film confined between a sphere of macroscopic radius R and a

planar substrate surface The shortest distance between two points located on

the surface of the sphere and of the substrate, respectively, is denoted by h (from

Ref 48)

the z direction can be expressed as

F(h;^ T) = - | dx j dy[T zz (s z (x,y);n, T) + Pbulk(M, T)] (17)

which depends on the (bulk) thermodynamic state specified by T and fi This

solvation, or depletion, force plays a vital role in the context of binarymixtures of colloidal particles of different sizes [34] (this volume, chapter

by M Schmidt) Because of their practical importance for colloid-polymermixtures [35], depletion forces in binary hard-sphere mixtures have recentlyreceived a lot of attention and have been studied by a range of methods,including integral equations based upon sophisticated hypernetted chainclosure approximations [36-41], density functional theory [42,43], virialexpansion [44], and computer simulation [45-47]

To evaluate the integral in Eq (17), it is convenient to transform fromcartesian to cylindrical coordinates (see Fig 2) to obtain

which follows from Eq (11) (fixed R, a x £ x , a/ y ) and a similar expression for

the bulk reservoir

^ b u i k = -«5 buik dT - Nhulk d/i - Pb u , k dV (20)

where V is the bulk volume In Eq (19) the excess grand potential

Oex := Q, — Obulk is also introduced Assuming V = As z , the far right side

of Eq (19) obtains because the bulk phase is isotropic Furthermore, note

that f(s z (p)) vanishes in the limit s z —> oo because of [49]

s —>oo

so that f(s z ) may be interpreted as the excess normal pressure exerted on the

sphere by the fluid In Eq (19), F(h) still depends on the curvature of the substrate surfaces through R Experimentally, one is normally concerned with measuring F(h)/R rather than the solvation force itself [27], because for macroscopically curved substrate surfaces this ratio is independent of R This can be rationalized by realizing that T zz (s z ) + Pbulk vanishes on a

microscopic length scale much smaller than R The upper integration limit

in Eq (19) may then be taken to infinity to give

(OO 1 />C

A

(22)

because fiex vanishes in the limit s z —• oo according to the definition in

Eq (19) In Eq (22) we introduce uf K (h) as the excess grand potential per

unit area of a fluid confined between two planar substrate surfaces separated

by a distance h The far right side of Eq (22) is known as the Derjaguin

approximation (see Eq (6) in Ref 29) As pointed out recently byGotzelmann et al [43], the Derjaguin approximation is exact in the limit

of a macroscopic sphere (i.e., if R —• oo), which is the only case of interest

here A rigorous proof can be found in the appendix of Ref 50 A similar

"Derjaguin approximation" for shear forces exerted on curved substrateshas recently been proposed by Klein and Kumacheva [51]

Structure and Phase Behavior of Soft Condensed Matter 11

Eq (22) is a key expression because it links the quantity F(h)/R that can

be determined directly in SFA experiments to the local stress T zz availablefrom computer simulations (see Sec IV Al) It is also interesting that differ-entiating Eq (22) yields

on a nanoscopic length scale (see Sec V B 3) The reduced symmetry of theconfined phase led us to replace the usual compressional-work term

-Pbuik V in the bulk analogue of Eq (2) by individual stresses and strains.

The appearance of shear contributions also reflects the reduced symmetry ofconfined phases

1 Atomically Smooth Substrates

The simplest situation is one in which a planar substrate lacks any lographic structure Then the confined fluid is homogeneous and isotropic in

crystal-transverse (x,y) directions All off-diagonal elements of T vanish,

T xx = T yv = T\\, and Eq (5) simplifies to

By symmetry, 7 ' ^f(A) at fixed T, fi, and s z Hence, under these conditions

one can formally integrate Eq (24) to obtain

U = TS + fj,N + YA (25)

taking the zero of U to correspond to zero interfacial area From Eqs (6),

(10), and (25) one gets

12 Schoen

which is the analogue of the bulk relation O = —

straightforward to realize that

V• From Eq (9) it is

(27)

is a nontrivial quantity (because in general 7^ ^ T zz ), whereas its bulk

ana-logue vanishes trivially because T\\ = T zz = -/bulk o n account of the highersymmetry of bulk phases reflected by Eq (21) [52] From Eqs (10), (24), and(25), the Gibbs-Duhem equation

follows immediately

2 The Two-dimensional Ideal Gas in an External Potential

While the smooth substrate considered in the preceding section is sufficientlyrealistic for many applications, the crystallographic structure of the sub-strate needs to be taken into account for more realistic models The essentialcomplications due to lack of transverse symmetry can be delineated by thefollowing two-dimensional structured-wall model: an ideal gas confined in aperiodic square-well potential field (see Fig 3) The two-dimensional lamella

remains rectangular with variable dimensions s x and s y and is therefore not

subject to shear stresses The boundaries of the lamella coinciding with the x and y axes are anchored From Eqs (2) and (10) one has

Structure and Phase Behavior of Soft Condensed Matter 13

for the free energy of the ideal gas under these premises From standardtextbook considerations one also knows the statistical-physical expression[53]

where (5= l/k B T (k B is Boltzmann's constant) The canonical partition

function Q can be written more explicitly as Q — q N /N\ where the atomic

partition function is given by

ideal gas does not depend on its y coordinate (see Fig 3).

The configuration integral depends on s x in a piecewise fashion For s x in

the «th period of the potential, that is for (n — 1)1 < s x < nl (n € N), one

= Vn-\)U{2n-\)l d/2

and

J3= \{2n-\) l - + ^

Fig 4 displays plots of —T xx and —T yy versus s x From these it is clear

that both stresses are functions of the size of the lamella The most cant consequence of this is that, unlike Eq (24), Eq (29) cannot be inte-

signifi-grated at fixed T, fi, and s y in general to yield an expression analogous to

Eq (25) without additional equations of state, that is T xx = T xx (s x ),

T yy — T yy (s x ) In other words, a Gibbs-Duhem equation corresponding to

Eq (28) does not obtain for the present two-dimensional structured-wallmodel The same conclusion holds for more realistic three-dimensionalstructured-wall models [54] The lack of a Gibbs-Duhem equation forgeneral thermodynamic transformations is a direct consequence of theadditional reduction of the confined fluid's symmetry caused by the discreteatomic structure of the substrate (see Sec I I B I)

3 Coarse-grained Thermodynamics

While a Gibbs-Duhem equation does not exist for general transformations

ds -> ds' , a specialized (i.e., "coarse-grained") Gibbs-Duhem equation

Structure and Phase Behavior of Soft Condensed Matter 15

FIG 4 Plots of — 7\ x (—) and —T%, ( — ) versus sx for the ideal gas confined to the

two-dimensional periodic square-well potential depicted in Fig 3 Distance is sured in units of the period /; stress in units of the pressure of the bulk ideal gas at the

mea-given T and \i {d/l = 0.20) (from Ref 54).

may be derived for cases in which the transverse dimensions of the lamellaare changed only discretely, that is, in such a way that the surface plane at

the fluid-wall interface of the lamella always comprises an integer number n

of unit cells in both x and v directions so that

(36)

Thus, the exchange of work between the lamella and its surroundings is

effected on a coarse-grained length scale defined in units of {£ x ,d v }.

Eliminating s x and s v in Eq (11) in favor of n gives

2T\\as : n dn + T zz an 2 ds : (37)

where work contributions due to shear and deformations of the shape of the

lamella are neglected for simplicity In Eq (37), a := i x £ v is the unit-cell areaand

T vy (T,ti,n£ x ,nt y ,s z )] (38)

is the "mean" stress applied transversely on the n x n lamella If T, /i, and

s~ are fixed, T xx and T vv are periodic in s x and sv, having periods t x and

i v , respectively Thus, for the restricted class of transformations

16 Schoen

n —• n' = n±m (n, m integer), T\\ is constant provided n and n' are

suffi-ciently large for intensive properties to be independent of the (microscopic)size of the lamella Under these conditions Eq (37) can be integrated to get

Eq (39) may be differentiated subsequently to give

Equating the expressions for dfl given in Eqs (37) and (40) and rearranging

terms yields the coarse-grained Gibbs-Duhem equation

which permits one to define the (transverse) isothermal compressibility K\\

(42)

where A = n 2 a as detailed in Ref 55 Note that a similar definition is

pre-vented for general transformations ds a —> ds' a according to the discussion inSec IIB 2

C Statistical Physics

1 Stress-Strain Ensembles for Open and Closed Systems

To achieve a description of confined soft condensed matter at the molecularlevel one has to resort to the principles of statistical physics To makecontact with, say, SFA experiments it is convenient to introduce statisticalphysical ensembles depending explicitly on a suitable set of stresses andstrains For simplicity, the lamella is treated quantum mechanically, follow-ing the procedure originated by Schrodinger [56] and extended by Hill [53]and McQuarrie [57], so that its energy states are formally discrete The energy

eigenvalues Ej(N, A,R,s z , a x £ x , a y £ y ) are implicit functions of the number of

fluid molecules, extent and shape of the lamella, and the registry of thesubstrates, which control the external field acting on the fluid molecules

Index j signifies the collection of quantum numbers necessary to

deter-mine the eigenstate uniquely The ensemble comprises an astronomical

number J\f of systems each in the same macroscopic state, which, as

an example, is taken to be specified by the set {T,iJ,,A,R,a x £ x ,a y £y} of

ensemble parameters Since the ensemble is isolated, it satisfies the

Structure and Phase Behavior of Soft Condensed Matter 17

where n jNs is the number of systems having A^ molecules between substrates

separated by s z and occupying eigenstate j It is assumed that the isolated ensemble has fixed total energy E, fixed total number of molecules JV, and fixed total volume Asl The total number of ways of realizing a given dis- tribution n = {n jNs _} over the allowed "superstates" characterized by triplets (j,N,s z ) is W(n) = A/*!/ny ELv TL n jNs.- Since the number of systems is

extremely large, the most probable distribution, denoted by «*, overwhelms

all others It is found by maximizing W{ri) subject to the constraints [see

Eqs (43)] The result for the probability of a system's occupying superstate

(U) = YljNs PjNs.Ej, from which its exact differential follows as

At the molecular level one may interpret {dEj/dA) N>s ^ R e e as the

interfacial tension of the system in superstate (j,N,s z ) Similar meaning

can be attached to the other partial derivatives of Ej appearing in Eq (47).

Invoking also the principle of conservation of probability (^/7V5 dPj Ns _ = 0),

Eq (47) can be recast as

Structure and Phase Behavior of Soft Condensed Matter 19

where the far right side is obtained by comparing the statistical expression in

Eq (52) with the thermodynamic Eqs (9) and (10) By exactly the sameapproach one can also derive statistical-physical expressions for other mixedstress-strain ensembles [58,59] Finally, from Eq (45) one has for the parti-tion function

X = 2_^ exp (@T zz As z ) \ exp(/fyxJV)Q(r, N,A,R,s z , a x £ x , a/ y )

= J2 (*p{0T zz As z ) E ( 7 , » , A, R,s z ) (53)

where Q := J^.exp(/?£)) is the canonical-ensemble partition function and

E — exp(—fiti) is its counterpart in the grand canonical ensemble Since this

chapter is exclusively concerned with classical systems, Q is replaced by its

classical analogue

(54)

for the special case of spherically-symmetric molecules where Z is the configuration integral, A := (h 2 f3/2irm) is the thermal de Broglie wave-

length (h is Planck's constant and m the molecular mass) The limiting

expression Qciass can be derived from the quantum mechanical Q within

the framework of the Kirkwood and Wigner theory [53] In the classical

limit one has to replace the quantum mechanical Pj Ni _ by the analogous

probability density distribution

where U(r N ) is the configurational energy of the system and #c l a s s is the

classical counterpart of X obtained by replacing Q in Eq (53) by Qciass (s e e

also this volume, chapter by Nielaba)

2 Correlation Functions

Since we shall also be interested in analyzing the confined fluid's microscopicstructure it is worthwhile to introduce some useful structural correlationfunctions at this point The simplest of these is related to the instantaneousnumber density operator

J (*•>-') (56)

where />'1'(i*,- = r) is the probability of the center of mass of molecule / being

at ** regardless of the positions of the other molecules (and regardless of

orientation, see Sec IV A 3) Since the molecules are equivalent, P^ 1 ' is

inde-pendent of / and the summation on / in Eq (57) can be performed explicitly

to which we shall henceforth refer simply as the local density

The translational microscopic structure of the confined fluid is partiallyrevealed by correlations in the number density operator, given by

(P(r)p(r')) = £ (6(rt - r)6(r, - r')>

=/>"'(')«('-o+EE E I I

x d r k f ( r x , , r i = r , , r j = r ' , , r N ] T , i J L , T z z ) ( 6 0 ) Jv

where the "self-term" gives no new information beyond the mean density.Again invoking the equivalence of fluid molecules, we recognize the cross-term in Eq (60) as the pair distribution function

pW(r,r')=N(N-l)P [2] (r,r') ( 6 1 )

Structure and Phase Behavior of Soft Condensed Matter 21

which is related to the mean local density through the pair correlationfunction by

p W(ry)=pM( r )pM(r')gW( r y) (62)

In general, g^ is a six-dimensional function of the position of reference

(x,y,z) and observed (x',y',z') molecules However, to be consistent

with the approximation for the local density [see Eq (59)], we take g® to

be a function only of the normal coordinate (z) of the reference moleculeand the cylindrical coordinates pl 2 and zl 2 of the observed molecule (2)relative to the reference molecule (I), where the distance vector between

the two r n = Pn + z v e z and e z is the unit vector in the z direction (seeSec IV B)

III MONTE CARLO SIMULATIONS

A key problem in the equilibrium statistical-physical description of densed matter concerns the computation of macroscopic properties Omacro

con-like, for example, internal energy, pressure, or magnetization in terms of an

ensemble average (O) of a suitably defined microscopic representation

O(r N ) (see Sec I V A l and VA I for relevant examples) To perform the

ensemble average one has to realize that configurations r N := {r { ,r 2 , ,r N }

generally differ energetically so that a certain probability of occurrence isassociated with each configuration Therefore, to compute the correct value

(O), O(r) needs to be multiplied by the relevant probability density function f{r N \X), where X is a set of thermodynamic state variables (for example,

T, /i, and a combination of stresses and strains).

Analytically, the computation of ensemble averages along this route is aformidable task, even if microscopically small representations of the system

of interest are considered, because f(r N ;X) is generally a very complicated

function of the spatial arrangement of the N molecules However, with the

advent of large-scale computers some forty years ago the key problem instatistical physics became tractable, at least numerically, by means of com-puter simulations In a computer simulation the evolution of a microscopi-cally small sample of the macroscopic system is determined by computingtrajectories of each molecule for a microscopic period of observation Anadvantage of this approach is the treatment of the microscopic sample inessentially a first-principles fashion; the only significant assumption con-cerns the choice of an interaction potential [25] Because of the power ofmodern supercomputers which can literally handle hundreds of millions offloating point operations per second, computer simulations are nowadays

22 Schoen

viewed as "a third branch complementary to the two traditionalapproaches" [60]: theory and experiment

There are basically two different computer simulation techniques known

as molecular dynamics (MD) and Monte Carlo (MC) simulation In MDmolecular trajectories are computed by solving an equation of motion forequilibrium or nonequilibrium situations Since the M D time scale is aphysical one, this method permits investigations of time-dependent phenom-ena like, for example, transport processes [25,61-63] In MC, on the otherhand, trajectories are generated by a (biased) random walk in configuration

space and, therefore, do not per se permit investigations of processes on a

physical time scale (with the "dynamics" of spin lattices as an exception[64]) However, MC has the advantage that it can easily be applied tovirtually all statistical-physical ensembles, which is of particular interest inthe context of this chapter On account of limitations of space and becauseexcellent texts exist for the MD method [25,61-63,65], the present discussionwill be restricted to the MC technique with particular emphasis on mixedstress-strain ensembles

If one wishes to compute (O) numerically by means of MC one diately realizes that this requires the a priori unknown function f(r N ;X)

imme-according to which the random walk in configuration space has to be carriedout However, if the random walk is carried out in a biased way as a Markov

process it turns out that only the ratio f(fm + i]X)/f{i"^;X) is relevant to

generate a new configuration (m + 1) from a given old one (m) The sequent discussion will show that /(r^+i;X)/f{ r m'i X) is computationally

sub-accessible Because this scheme generates configurations with the correctprobability of occurrence, Omacro can be computed via the simple expression

I M

»-%5J°)%iiEW':)]' («)

where the prime is attached as a reminder that the summation is restricted

to configurations generated according to their importance (importance

sampling) and Mmax should be large enough [usually, Mmax « (9(l06-109)

is sufficient, depending on the particular physical situation and quantity ofinterest] However, before turning to practical aspects of MC, a brief intro-duction to Markov processes seems worthwhile because it rarely appears inthe literature

A Stochastic Processes

Let y(t) be a random process, that is a process incompletely determined at any given time t The random process can be described by a set of prob- ability distributions {P } where, for example, Piiyxhiyih) dy\dyi is the

Structure and Phase Behavior of Soft Condensed Matter 23

probability of finding vi in the interval [ri,Vi + d}'\] at t = t x and in the

interval \y 2 -,y 2 + dy 2 ] at another time t = t 2 Thus the set {P,,} forms a

hierarchy of probability distributions describing y(t) in greater detail the larger n is.

The simplest random process is completely stochastic so that one may

write, for example, P2{y\h-,yih) — P\{y\h)P\{}'2h)- However, here we are

concerned with a slightly more complex process known as the Markovprocess, characterized by

Piiyih^h) = Px{y x t x )K x {y x t x \y 2 t 2 ) (64)

where K x {y x t]\y 2 t 2 ) *s t n e conditional probability of finding v in the interval

L>'2- >'2 + ^ ' 2 ] a t t — h provided y = y x at an earlier time t = t x (t x < t 2 ).

Some important properties are the following:

1 Normalization, that is § K x (y x t x \y 2 t 2 ) dy 2 = 1

3 Perhaps most importantly, a Markov process has a "one-step memory";

that is, to find v in the interval [y,,, j ' , , + dy n ] at t — t n depends only on

the realization y = y n _ { at the immediately preceding time t — t n _ x but is

independent of all earlier realizations {y m t m },\ <m <n— 2.

Mathematically, this can be cast as K n _\(y\t\, ,y n -\t n -\\y n t n ) = K\{y n -\t n -\\y n t tl ).

4 Stationarity, that is P\{y\h) = P\{v\) and Pi(y\t\-, y 2 t 2 ) =

J°2(>'i V2;'2-'i)-Consider now

P2(.v n -2tn-2,y n t n ) = PiLv,,- 2 t n _ 2 , y,,-x t n _ x y n t n ) dy n _ x (65)

and assume that y(t) is a Markov process Then

^ 3 (j'n-2 hi-l, }'n-1 hi-1 •> }'n hi) = ?\ (.V'H-2 hi-2 ) ^ 1 {)'n-2 hi-2 \}'n- \hi-\)

24 Schoen

where t := t n _\ — t n _ 2 and r := t n — t n _\ Equation (68) is known as the

Chapman-Kolmogoroff equation

Suppose a small characteristic time interval rc exists such that y n _ {

changes without strongly affecting Ki(y n _ 2 \y n ;?) s 0 t n a t t n e latter may beexpanded in a Taylor series as

From Eqs (68) and (69) one gets

With Eqs (71), (72), and t = t f - t n _ 2 (dt = A') one may multiply both sides

of Eq (70) by Pi{y n -2tn-2) an<^ integrate over dy n _ 2 to obtain (seeproperties 1, 2, and 4)

(73)

For stationary situations d i ^ (y n t')/dt ! = 0 and Eq (73) is then satisfied by

where II is the probability for the transition n — 1 -» n Equation (74) reflects microscopic reversibility and is a special formulation of the principle

of detailed balance.

B Implementation of Stress-Strain Ensembles for

Open and Closed Systems

Consider now, as an illustration, a confined fluid in material and thermal

contact with a bulk reservoir and under fixed normal stress T zz For

simpli-city we assume the substrates to be in fixed registry a = a = 0 and the

Structure and Phase Behavior of Soft Condensed Matter 25

fluid to consist of "simple" molecules having only (three) translationaldegrees of freedom Under these premises one has (see Sec I I C 1] [66]

(O) = J2 f ds2 f drNf(rN; T, /i, Tzz)O{fN)

N J Jv N

= X& £ ^ j ^ " U.- apWTaA,Itf f df

where (*,•,>>„ z,) -> (x,- = ^]xn y,- = s^y^Zt = s7 l z,),i = 1, ,N so that

the integration is carried out over the unit-cube volume V The summation over s z [see Eq (53)] has been replaced by an integral, and the dimensionless

quantity B is defined by

ASz) (76)

The MC method can be implemented by a modification of the classicMetropolis scheme [25,67] The Markov chain is generated by a three-stepsequence The first step is identical to the classic Metropolis algorithm: a

randomly selected molecule i is displaced within a small cube of side length

26, centered on its original position

where 1 = (1,1,1) and £ is a vector whose three components are random numbers distributed uniformly on the interval [0,1] During the MC

pseudo-run 6 r is adjusted so that 40-60% of the attempted displacements are

accepted With the i d e n t i f i c a t i o n / ^ ; T,fi, T zz ) = P{y n ) one obtains

; 7>, T zz ) = exp(-pAU)} (78)

from Eqs (55) and (74) because N m — N m+X and s zm+l = s zim where

AU := U(fn + \;s z ) — U{f^ t \s z ) is the change in configurational energy

asso-ciated with the process f^ n —> r%+i An efficient way to compute AU is

detailed below in Sec III.C

In the second step it is decided with equal probability whether to remove(AiV = — 1) a randomly chosen molecule or to create (AJV = -fl) a new one

at a randomly chosen point in the system (see also Sec HID) FromEqs (55) and (74) the transition probabilities for addition ("+") and sub-traction ("—") are given by

n2 = m i n { l , / ( f J [ S ; T, AX, T B )/f(i%; T,n, T zz ) = exp(r±)} (79)where

(80)

26 Schoen

Since only one molecule is added to (or removed from) the system, U± is

simply the interaction of the added (or removed) molecule with the ing ones If one attempts to add a new molecule, vV is the number of mole-

remain-cules after addition, otherwise it is the number of moleremain-cules prior to

removal If a cutoff for the interaction potential is employed, long-range

corrections to U± must be taken into account because of the density change

of ±l/As z Analytic expressions for these corrections can be found in the

appendix of Ref 33

In the third and final step the substrate separation is changed accordingto

and the coordinates of fluid molecules are scaled via zw+1 = z m s z ,m+\l s

:,m-Because TV is held constant the transition probability associated with thisstep is

£'; 7 > , Tzz) = exp(r,J} (82)

where

As z := s zm+ i — s zm and the same comments concerning corrections to AU

apply as in step 2 On each pass through the three-step sequence the number

of attempts in steps 1, 2, and 3 is chosen to be N, N, and 1, respectively, in

order to realize a comparable degree of events in each of the steps Becausethe third step moves all TV molecules at once, and the first two affect onlyone molecule at a time, the sought balance is roughly achieved The algo-rithm described here can easily be amended by additional steps if, for exam-ple, one is interested in situations in which the shear stress(es) is (are) also

among the controlled parameters so that a x (and a y ) may vary too [58,59].

Applying the analysis of Wood [68] to each step of the algorithm separately,one can verify that the resulting transition probabilities indeed comply withthe requirements of a Markov process as stated in Eq (74)

C The Taylor-expansion Algorithm for "Simple"

Fluids

According to Allen and Tildesley, the standard recipe to evaluate AU in step

one of the algorithm described in Sec Ill B involves "computing the energy

of atom / with all the other atoms before and after the move (see p 159 of Ref.

25, italics by the present author) as far as "simple" fluids are concerned The

evaluation of AU can be made more efficient in this case by realizing that for short-range interactions U can be split into three contributions