Document [original]

Fluids confined by nanopatterned substrates

vorgelegt von

Diplom Chemiker

Henry Bock

aus Eisenh¨uttenstadt

Der Fakult¨at II

Mathematik und Naturwissenschaften

der Technischen Universit¨at Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

Dr. rer. nat.

genehmigte Dissertation

Promotionsausschuß:

Berichter: Prof. Dr. M. Schoen

Berichter: Prof. Dr. S. Hess

Tag der m¨undlichen Pr¨ufung: 15. August 2001

Berlin 2001

D83

Zusammenfassung

Der Einﬂuß chemischer Heterogenit¨at der Substratoberﬂ¨achen auf Phasenver-

halten und mechanische Eigenschaften von Fluiden in begrenzender Geome-

trie wird untersucht. In der vorliegenden Arbeit sind Fluide durch chemisch

strukturierte, planparallele Substrate auf schlitzf¨ormige Bereiche nanoskopis-

cher Dicke eingeschr¨ankt. Die chemisch strukturierten Substrate bestehen aus

alternierenden stark und schwach adsorbierenden Streifen.

Zur Untersuchung des Phasenverhaltens werden Phasendiagramme eines (ein-

fach–kubischen) Gittergases berechnet. Eine modulares Verfahren erlaubt die

analytische Berechnung von Phasendiagrammen bei verschwindender Temper-

atur (T=0). F¨ur T>0 wird das Gittergas innerhalb einer Molekular-

feldn¨aherung behandelt. Das komplexe Phasenverhalten ist unter anderem

dadurch gekennzeichnet, daß die Porenkondensation in zwei aufeinander fol-

gende Phasen¨uberg¨ange erster Ordnung aufspaltet. W¨ahrend des ersten ¨

Uber-

ganges wird eine Fl¨ussigkeitsbr¨ucke zwischen den sich gen¨uberliegenden stark

adsorbierenden Substratteilen gebildet. Br¨uckenphasen unterscheiden sich von

allen anderen Phasen, da sie bei allen Abst¨anden zwischen den W¨anden laterale

Inhomogenit¨aten aufweisen. Um den Einﬂuß der Vereinfachungen dieses Modells

zu kontrollieren, werden wesentliche Ergebnisse mit Hilfe eines kontinuierlichen

Modells ¨uberpr¨uft.

Durch relative Verschiebung der W¨ande gegeneinander, k¨onnen Fl¨ussigkeits-

br¨ucken einer Scherdeformation αsxausgesetzt werden, die zu einer nicht ver-

schwindenden Scherspannung Tzx f¨uhrt. Die entsprechenden Scherspannungskur-

ven Tzx (αsx) stimmen qualitativ mit denen ¨uberein, die f¨ur feste Filme zwischen

atomar strukturierten W¨anden gefunden werden. Bei kleinen αsxverh¨alt sich

eine Fl¨ussigkeitsbr¨ucke wie eine Hooke‘sche Feder, gefolgt von einem Bereich

zunehmender Nichtlinearit¨at, bis schließlich ein ein Maximum, der sogenan-

nte Haltepunkt erreicht ist. Variationen der Breite der stark und schwach ad-

sorbierenden Streifen f¨uhren zu einer Verschiebung des Haltepunktes, ¨andern

jedoch die generelle Form der Scherspannungskurven nicht. Mit Hilfe einer

Theorie korrespondierender Zust¨ande k¨onnen die Scherpannungskurven auf die

Koordinaten des Haltepunktes normiert und durch eine systemparameterfreie

universelle Kurve repr¨asentiert werden.

Publikationen von Teilen dieser Dissertation

1. Phase behavior of a simple fluid confined between chemically corrugated

substrates, Henry Bock and Martin Schoen, Phys. Rev. E 59, 4122 (1999)

2. Shear-induced phase transitions in confined lattice gases, Martin Schoen

and Henry Bock, J. Phys.: Condens Matter 12, A333 (2000)

3. Thermophysical properties of confined fluids exposed to a shear strain,

Henry Bock and Martin Schoen, J. Phys.: Condens Matter 12, 1545 (2000)

4. Shear-induced phase transitions in fluids confined between chemically dec-

orated substrates, Henry Bock and Martin Schoen, J. Phys.: Condens

Matter 12, 1569 (2000)

5. Phase behaviour of fluids confined between chemically decorated substrates,

Henry Bock, Dennis J. Diestler and Martin Schoen, J. Phys.: Condens.

Matter, 13, 4697 (2001)

Abstract

The impact of chemical heterogeneity of solid surfaces on phase behavior and

mechanical properties of conﬁned simple ﬂuids is investigated. In the present

work ﬂuids are conﬁned to a slit of nanoscopic width by chemically decorated,

plane–parallel substrates consisting of alternating slabs of weakly and strongly

adsorbing solid.

The phase behavior is explored by calculating phase diagrams using a simple–

cubic lattice–gas model. A modular approach is developed which allows us to

calculate the phase diagram analytically at vanishing temperature (T=0).

At higher temperatures T>0 the lattice gas is treated within the mean–ﬁeld

approximation. A rich phase behavior is observed which depends on the large set

of parameters needed to describe the heterogeneous conﬁnement. Caused by the

chemical heterogeneity of the substrates the capillary condensation is split into

two successive ﬁrst–order phase transitions. During the ﬁrst transition the gap

between opposing strongly adsorbing wall parts is ﬁlled with liquid surrounded

by vapor, that is a bridge phase forms. Bridge phases are distinguished from all

other phases in that they are laterally inhomogeneous at all planes z=const.

The mean–ﬁeld approximation to the lattice–gas model is used since it provides

phase diagrams at moderate computational expense. To control the inﬂuence of

the simpliﬁcations inherent in the lattice model, the main ﬁndings are veriﬁed

qualitatively employing a parallel continuous model, which is treated by Monte

Carlo simulations.

By misaligning the the opposite substrates, bridge phases can be subjected

to a shear strain αsx. Because of their unique internal structure bridge phases

support a nonvanishing shear stress Tzx. The shear stress curve Tzx(αsx)is

qualitatively similar to the one characteristic of solidlike ﬁlms conﬁned be-

tween atomically structured substrates, in that the response to small strains

is Hookean, followed by an increasingly nonlinear regime up to the yield point

where Tzx(αsx) assumes its maximum. Variation of the width of strongly and

weakly adsorbing slabs causes the yield point to shift, but does not alter the

general form of Tzx(αsx). With the aid of a theory of corresponding states,

Tzx (αsx) is renormalized by yield stress and strain such that the results can be

represented uniquely by a master curve independent of any system parameters.

Contents

1 Introduction 1

2 The mean-ﬁeld lattice-gas model 12

2.1 Thelatticegasmodel ........................ 12

2.1.1 Grandpotential ....................... 12

2.1.2 ThePrototype ........................ 14

2.1.3 Nearest–neighbor lattice gas in magnetic language . . . . 16

2.2 Exactsolutions............................ 17

2.2.1 ThegrandpotentialatT=0................. 17

2.2.2 MorphologiesatT=0 .................... 20

2.2.3 Thermodynamically stable morphologies at T=0 . . . . . 26

2.3 Mean-FieldTheory.......................... 32

2.3.1 Variational treatment of the mean ﬁeld model . . . . . . . 33

2.3.2 Bulk phase diagram of the mean-ﬁeld lattice gas . . . . . 36

2.3.3 ThelimitT=0 ........................ 40

2.4 Phase behavior for T >0 ...................... 42

2.4.1 Grandpotentialcurves.................... 42

2.4.2 Phasediagrams........................ 46

3 The continuous model 56

3.1 Thecontinuousmodel ........................ 57

3.2 Thermodynamics........................... 60

3.3 StatisticalThermodynamics..................... 63

3.3.1 Partition function of the grand mixed isostress-isostrain

ensemble ........................... 64

CONTENTS

3.3.2 Theclassicallimit ...................... 66

3.3.3 Molecular expressions for stress tensor components . . . . 68

3.4 MonteCarlosimulation ....................... 72

3.4.1 Thegeneralmethod ..................... 72

3.4.2 Application and Implementation . . . . . . . . . . . . . . 74

4Results 78

4.1 Phasebehavior ............................ 79

4.1.1 Variationofwalldistance .................. 79

4.1.2 Variation of chemical corrugation . . . . . . . . . . . . . . 90

4.1.3 Theimpactofshearstrain.................. 93

4.2 Thermo-mechanicalproperties....................103

4.2.1 Shear stress of conﬁned ﬂuids . . . . . . . . . . . . . . . . 103

4.2.2 Thermodynamic stability . . . . . . . . . . . . . . . . . . 110

5 Discussion and conclusions 118

A Jacobi–Newton iteration 125

B Fluid–substrate forces 131

C Implementation of the ﬂuid–substrate potential 134

Chapter 1

Introduction

Among the three classical states of (single–component bulk) matter, namely

gas, liquid and solid there is a clearcut distinction as far as gas and solid are

concerned. While the latter is characterized by a highly symmetric and periodic

microscopic structure, the former lacks any inherent structural features, that is

gases are completely disordered from a molecular perspective. From a structural

point of view liquids are somewhat intermediate to both gas and solid, that is

oﬀ the gas–liquid near–critical regime they exhibit short–range positional order

vanishing on a lengthscale set by the range of intermolecular forces.

The three states of matter may be transformed into one another by chang-

ing the thermodynamic conditions, that is temperature T, pressure p,ordensity

ρ=N/V (Nnumber of molecules, Vvolume). The associated phase transitions

are accompanied by signiﬁcant changes in certain system properties. For exam-

ple, the liquid–solid phase transition is characterized by a qualitative change in

the degree of molecular order. Therefore liquid–solid phase transitions appear

even at very high pressures where properties such as the mean intermolecular

distances are much the same in both phases [1]. Moreover, a liquid–solid crit-

ical point is unknown so far. In other words, liquid–solid phase transitions in

inﬁnitely large bulk systems are always discontinuous (i.e., ﬁrst–order accord-

ing to Ehrenfest‘s classiﬁcation [2]) and accompanied by release of latent heat.

On the contrary, a key feature of liquid–gas phase transitions is a change in

the mean density. If the temperature Tis increased coexisting gas and liquid

2CHAPTER 1. INTRODUCTION

phases become more and more alike the higher Tis. At the critical point they

become indistinguishable and the gas–liquid phase transition disappears. This

oﬀers the interesting possibility to pass from any gas state to any liquid state

continuously along a partially supercritical path in thermodynamic state space.

Because one can gradually transform gases into liquids and vice versa the term

“ﬂuid” is often used to refer to both nonsolid states of matter if a more detailed

classiﬁcation is not required.

The ﬁrst quantitative picture of gas–liquid phase transitions in the bulk

emerged from the thesis of van der Waals published in 1873 [3] (for an English

translation see [4]). In his thesis van der Waals proposed an equation of state

capable of describing real ﬂuids whose properties are determined by intermolec-

ular forces, attraction and repulsion that is. Because of attractive interactions

the van der Waals equation predicts gas–liquid phase transitions for suﬃciently

low temperatures and existence of a critical point. This is diﬀerent from liquid–

solid transitions which are driven by repulsive interactions, as solidiﬁcation of

hard sphere ﬂuids clearly indicates [5]. In this latter system solidiﬁcation occurs

solely on account of entropic eﬀects.

As far as ﬂuids are concerned it is almost commonplace that a container of

some sort is required to keep them. In view of this it seems surprising that only

after more than a hundred years after van der Waals physicists became aware of

the impact of the container walls on the phase behavior of ﬂuids. The interaction

of ﬂuids with solid substrates and its consequences for the phase behavior of ﬂu-

ids near solid surfaces was ﬁrst realized by Cahn [6] and Ebner and Saam [7, 8].

These works were concerned with ways a ﬂuid wets a solid surface. Since then

these wetting phenomena are nowadays perceived as substrate–induced phase

transitions in the classical thermodynamic sense. They can be investigated ex-

perimentally by measuring the amount of ﬂuid adsorbed on a solid surface or by

determining the thickness of the adsorbed ﬁlm. Experiments are usually carried

out under isothermal conditions and (relative) pressures P/Psat ≤1wherePsat

is the saturated–gas pressure for the given temperature T.IfTis below the

so–called wetting temperature (Tw<T

c,Tccrititcal temperature), the amount

of adsorbed ﬂuid Γ remains ﬁnite (partial wetting) even for P/Psat =1,that

is at (bulk) gas–liquid coexistence; for Tw<T<T

c, Γ becomes inﬁnitely large

(complete wetting), that is, the thickness of the wetting layer attains macro-

scopic dimensions for P/Psat .1. In addition, Γ may diverge continuously

(critical wetting) or discontinuously (ﬁrst–order wetting) if T−→ Twsuch that

P/Psat = 1 is maintained. For suﬃciently low temperatures, Γ may change dis-

continuously but remains ﬁnite for states oﬀ gas–liquid coexistence, that is for

P/Psat <1 [9]. During this prewetting transition the thickness of the wetting

layer remains ﬁnite, since the inﬁnitely thick ﬁlm (i.e., the liquid) is still ther-

modynamically unstable. Points {(T,P)|P/Psat <1}where prewetting takes

place form the prewetting line which ends at the prewetting critical point. More-

over, the formation of the wetting layer may appear in a sequence of ﬁrst–order

phase transitions layer by layer (layering transitions) such that the thickness

of the adsorbed liquid ﬁlm eventually may become inﬁnitely large (roughening)

[10, 11].

Wetting of solid substrates plays also an important rˆoleincommontechnical

applications. Consider, for instance, painting solid surfaces. To obtain particu-

larly nice–looking, smooth surfaces the paint should form a homogeneous layer

on the surface. Thus, wetting of the surface by the paint should be optimum.

As another example consider special coatings designed to enhancing the wet-

ting of glass surfaces by water. These coatings prevent water from forming

small droplets that would otherwise blur the view through the glass. On the

contrary, in the automobile industry, where one is concerned with safety aspects,

water–repellent glass is utilized to optimize the driver‘s vision in rain. Instead

of perfect wetting one aims at optimizing the “drying” characteristics of the

glass so that water can easily form droplets rolling oﬀ the glass surface without

diﬃculty [12].

In nature this latter problem is already solved. Leaves of many plants are not

wetted by water. The most prominent representative is the so–called “sacred

lotus” (Nelumbo nucifera) [13]. This is particularly surprising since the leaf

consists mostly of water. The key to understanding this so–called lotus–leaf

phenomenon is surface roughness, that is the surface of the leaf is covered with

asperities characterized by a large aspect ratio. In addition, the surface is coated

with water–repellent material.

Wetting of homogeneous planar substrates has been intensively studied over

4CHAPTER 1. INTRODUCTION

the last twenty years [10, 14, 15, 16]. To explore wetting of heterogeneous sur-

faces and eventually design and fabricate surfaces with speciﬁc and controllable

local wetting characteristics, an improved understanding of molecular aspects

of wetting of heterogeneous surfaces is required. At the nanometer lengthscale,

heterogeneity can be achieved by endowing surfaces with geometrical or chemical

patterns. Various techniques have been developed to decorate smooth homoge-

neous surfaces with such structures. For example, lithography frequently used

in the fabrication of microelectronic chips can be used to produce structures

with typical sizes of the order of a few tenths of a micrometer. Using syn-

chrotron radiation (X-Ray) lithography is capable of generating structures on

the micrometer lengthscale, with a remarkably high aspect ratio, that is with

a height of up to two millimeters [17]. On the other hand, very small struc-

tures from 10 µm to 10 nm are available by using masks in the lithographic

process which are formed by a (self–assembled) monolayer of colloidal particles

located directly on the surface [18, 19]. Even smaller structures can be real-

ized in thin (submonolayer) ﬁlms which relieve surface stress by formation of

thermodynamically stable, periodically ordered domains on the nanometer scale

[20, 21, 22]. Moreover, using scanning probe techniques single atoms can be ma-

nipulated [23, 24]. Vapor deposition through grids and microcontact printing

are alternative methods.

The manufacturing of micro– and nanostructures using microcontact print-

ing is another well established technique [25, 26, 27, 28, 29]. The core of this

method is a polymer stamp which is set up lithographically with subsequent

wet chemical etching. In this case the “ink” for the stamp consists of a chemical

substance which after printing forms a self assembled monolayer anchored to

the surface. The result of this treatment is a chemically heterogeneous surface

nearly ﬂat on an atomic lengthscale. Microcontact printing is a relatively simple

technique. Stamps can be used several times and a wide range of pattern sizes

from tens of centimeters to tens of nanometers is accessible, so that this method

may in principle be used in industrial applications. It is furthermore noteworthy

that microcontact printing can also be used to decorate curved surfaces [30].

If ﬂuids are exposed to patterned surfaces they are no longer spatially uni-

form (i.e., homogeneous) as in the bulk. Take as an example a hard sphere ﬂuid

exposed to a periodic array of opposite and parallel hard wedges characterized

by a dihedral angle Θ and a distance sxbetween the “tips” of this sawtooth–

shaped substrate [31]. In Ref. [31], Schoen and Dietrich demonstrated by means

of grand canonical ensemble Monte Carlo (GCEMC) simulations that in general

the ﬂuid in the corner of such a wedge is more ordered than in the vicinity of

the tips. For the special case Θ = π/2 a substrate–induced solidlike structure

of fourfold in–plane symmetry is observed which cannot exist in the bulk.

Moreover, ﬂuids adsorbed on patterned surfaces comprise a rich variety of

morphologies and transitions between them [32]. Consider, for instance, a ﬂuid

in single (macroscopic) wedge. Using an eﬀective interface Hamiltonian, Rejmer

et al [33] showed that wedge ﬁlling may compete with wetting of the sloping

substrates forming the surface of the wedge. That is, depending on the opening

angle of the wedge a so–called preﬁlling line appears in the phase diagram.

It represents a line of ﬁrst–order phase transitions where the ﬁlling height of

the liquid in that wedge jumps from a microscopic to a macroscopically large

value, while the thickness of the wetting layer suﬃciently far away from the

corner of the wedge does not vary discontinuously. The inﬂuence of the speciﬁc

geometry of the structured surface has also been investigated by comparing

ﬁlling transitions in wedges and cones [34].

Studies of ﬂuids wetting chemically structured surfaces indicate a strong in-

ﬂuence of the substrate structure on the ﬁlm morphology. Both experimental

and theoretical investigations of ﬂuids exposed to chemically striped surfaces

show formation of liquid channels along the chemical stripes [35]. A large

number of channel morphologies have been observed, that is, depending on

the thermodynamic conditions channels on single stripes comprise diﬀerent vol-

umes [36]. Moreover, liquid can spill over to neighboring channels to form big-

ger channels which cover more then one chemical stripe. Transitions between

these morphologies, which can be either continuous or discontinuous, are also

of particular practical interest [37, 38, 39, 40, 41](see below). Experimentally

it has been observed that liquid channels may become unstable and undergo a

transformation where a bulge occurs along the channels. If the striped surface

domains exhibit corners, bulges will be located preferentially at these corners

[35]. In Ref. [35] the size of the structured domains is of the order of a few

6CHAPTER 1. INTRODUCTION

µm so that optical microscopy can be employed to analyze the morphologies of

wetting ﬁlms. If on the other hand, these structures “live” on the nanometer

lengthscale they cannot be resolved by optical microscopy. In this case scan-

ning force microscopy (SFM) in tapping mode can be utilized as an alternative

technique [42, 43, 44, 45].

The results summarized above have contributed to a novel ﬁeld of techni-

cal applications referred to as “microﬂuidics” where one is concerned with the

controlled transportation of tiny amounts of valuable liquids. By locally mod-

ifying the wetting characteristics of an underlying solid substrate “lanes” for

the transportation of liquids can be created such that liquid does not spill over

neighboring substrate parts. It is therefore necessary to understand the inter-

play between ﬂuid–substrate forces and the morphology of the ﬂuid on such a

patterned substrate [46]. An example where these principles are invoked already

is a continuous ﬂow mixer created on a patterned silicon wafer. This device is

capable of mixing nanoliters of ﬂuids on timescales of less then 10 µs. Since an

investigation of kinetics of chemical reactions is limited by the time needed to

mix the reactants, this device enables one to study very fast reaction kinetics

unaccessible by conventional mixing technology on account of their much larger

mixing times in the range of milliseconds. At the same time only very small

amounts of the chemical substances are needed in many applications. Thus,

microﬂuidics provides useful tools to treat valuable ﬂuids such as human DNA

[47, 48, 49, 50, 51].

Another interesting application has been reported by Wang et al who ob-

served that the wettability of glass coated with TiO2can be changed by ultra-

violet irradiation [52]. Upon illumination the coated glass is wetted by water,

whereas the original, unilluminated coating is water–repellent. Light–induced

variation of wetting characteristics may also be used to imprint chemical pat-

terns onto a monolayer ﬁlm of a polymeric material immobilized on a silicon wa-

ver [53]. The chemical heterogeneity was imprinted by illuminating the coating

with ultraviolet light (350nm) through a mask, whereupon the illuminated poly-

mer undergoes a transition from the trans– to the cis–isomer. Thus, it results

in a surface with cis and trans domains, which are either wetted (cis–isomer) or

nonwetted (trans–isomer) by water, respectively. This system exhibits another

interesting property: by illuminaton of the surface with blue light (455nm) the

structures can be erased completely (transformation to the trans–isomer) and

new patterns can be created by subsequent illumination with ultraviolet light.

These examples may suﬃce to demonstrate the richness and practical im-

portance of wetting of a single solid surface. However, the behavior of ﬂuids

becomes even more fascinating if such a ﬂuid is conﬁned by two (or more) solid

substrates to spaces of nanoscopic dimension(s), that is spaces comparable in

size with the range of the ﬂuid–solid interaction potential. Conﬁnement adds

a new lengthscale with profound consequences for the properties of ﬂuids. The

most obvious consequence is that conﬁnement precludes roughening and com-

plete wetting because formation of ﬁlms of macroscopic thickness is impossible

(see Ref. [54]). Prewetting and layering, on the other hand, compete with

capillary condensation which is the analogue of the bulk gas–liquid phase tran-

sition. In general, conﬁnement shifts the coexistence curve of a conﬁned ﬂuid

with respect to the bulk. This shift has been observed in sorption experiments,

where a ﬂuid vapor is conﬁned to a (random) porous material (VYCOR, CPG)

[55, 56, 57]. In particular, a depression of the pore critical temperature and

an increase of the the pore critical density with respect to the bulk have been

reported [55, 56, 57]. Consequently, the gas–liquid two–phase region (in the T–ρ

representation of the phase diagram) has been shifted to higher densities and

lower temperatures. The two–phase region is much narrower compared with the

bulk. The gas branch of a conﬁned ﬂuid‘s phase diagram is more aﬀected than

the liquid branch. The shift of the coexistence lines and of the critical point is

the stronger the more severe conﬁnement is (see ﬁgure 7 in Ref. [55]).

Several theoretical methods are available to study ﬂuids in disordered porous

materials [58]. Rosinberg and co–workers investigated the phase behavior of a

lattice gas exposed to a disordered (solid) matrix, the latter being realized by

placing “wall” particles at random positions on the lattice [59, 60, 61]. Their

main ﬁndings agree with experimental results discussed above, that is depres-

sion of the critical temperature, increase of the critical density, and a nar-

rower gas–liquid two–phase–region. Moreover, at suﬃciently low temperatures

and densities additional phases appear. In parallel Monte Carlo studies of a

Lennard–Jones LJ(12,6) ﬂuid exposed to a random porous matrix Page and

8CHAPTER 1. INTRODUCTION

Monson observed that one of these transitions is associated with partial con-

densation of the ﬂuid in regions where the spherical particles forming the solid

matrix are more densely packed [62, 63]. R¨ocken and co–workers used a mean–

ﬁeld lattice gas model [64] and a density–functional approach [65] to investigate

ﬂuids conﬁned by planar, chemically heterogeneous substrates. In their model

the chemical heterogeneity of the walls is represented by a wall potential vary-

ing sinusoidally in one lateral direction (x) while it is uniform in the other (y).

The walls are arranged such that chemically identical parts of the substrates are

exactly opposite, that is the substrates are in registry. For this system R¨ocken

and co–workers observed a split capillary condensation occurring as two discon-

tinuous phase transitions. The ﬁrst transition is related to partial condensation

of the ﬂuid in regions where the walls are strongly attractive, that is liquid ﬁlls

the gap between the opposite strongly attractive substrate parts. In a second

step the ﬂuid condenses in the remainder of the system. Thus, eventually the

entire pore is ﬁlled with liquid. The occurrence of this two–stage capillary con-

densation was found to depend on the period of the substrate potential, that

is it appears only if this period is large compared with the “diameter” of a

ﬂuid molecule. Fluids conﬁned by geometrically inhomogeneous substrates may

form similar liquid bridges in the narrower regions of the pore depending on the

thermodynamic conditions [66, 67]. Recently Schoen studied a Lennard–Jones

LJ(12,6) ﬂuid conﬁned between solid substrates endowed with wedge–shaped

furrows using GCEMC [67]. The furrows are arranged such that they are pe-

riodic in one lateral direction and translationally invariant in the other one. A

comprehensive overview of actual experiments and theoretical progress is given

in reference [68].

Besides phase behavior mechanical properties of conﬁned ﬂuids are of inter-

est. An appropriate device to carry out measurements of mechanical properties

of conﬁned phases is the surface forces apparatus (SFA) [69]. The core of an

SFA consists of two macroscopically curved cylinders (radius of curvature ca.

1cm), arranged such that their axes are at right angles. This conﬁguration

minimizes the contact area on the cylinder surfaces. Due to the macroscopic

curvature of the cylinders the surfaces can be taken as parallel on a molecular

lengthscale around the point of minimum distance. In most cases the surfaces

of the two cylinders are coated with mica, which can be prepared with atomic

smoothness over molecularly large areas. The whole setup is immersed in a

reservoir of the ﬂuid of interest. Between the cylinder surfaces a thin ﬁlm forms

in thermodynamic equilibrium with the bulk reservoir. The distance between

the two cylinders, which is a measure of ﬁlm thickness, is determined by optical

interferometry. In one particular setup of the SFA the force exerted in direction

normal to the ﬂuid–substrate interface is maintained such that the ﬁlm thick-

ness may ﬂuctuate thermally. This is done by attaching springs to the upper

cylinder whereas the lower cylinder remains stationary. In addition a conﬁned

ﬁlm can be exposed to a shear strain by attaching a movable stage to the upper

substrate via another spring device and moving it at some constant velocity,

in a 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 elon-

gation of the spring, the shear stress sustained by the ﬁlm can be determined.

Beyond a critical shear strain (i.e., at the so-called “yield point” corresponding

to the maximum shear stress sustained by the ﬁlm) the shear stress declines

abruptly and 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

the critical shear stress is once again attained so that the stick-slip cycle repeats

itself periodically.

A key issue still under discussion is whether or not the rheological behavior

of conﬁned phases reﬂects conﬁnement–induced solidiﬁcation or not (see [70, 71]

and references therein). For instance, Klein and Kumacheva carried out SFA

experiments in which an octamethylcyclotetrasiloxane (OMCTS) ﬁlm conﬁned

between mica surfaces is exposed to a shear strain [70, 71]. In SFA experi-

ments OMCTS plays a prominent rˆole because of its approximately spherically

symmetric molecular structure so that models based upon “simple” ﬂuids (i.e.,

ﬂuids composed of molecules having only translational degrees of freedom) can

be employed theoretically to understand many important aspects of SFA exper-

iments [72]. In their work Klein and Kumacheva ﬁnd that for large substrate

separations of 1160 ˚

A “conﬁned” OMCTS behaves essentially like bulk liquid.

In this case a characteristic relative lateral displacement of the upper substrate

10 CHAPTER 1. INTRODUCTION

is observed on account of thermal noise (see Fig. 6(a) in [70]). This motion

remains unaltered if the distance between substrate surfaces is reduced down

to approximately 62 ˚

A. However, for a slightly smaller substrate separation of

about 54 ˚

A the lateral motion of the upper substrate suddenly disappears as if

the ﬁlm would be capable of “glueing” the substrate to some ﬁxed position in

space (see Figs. 6(b) and 6(c) of [70]). Klein and Kumacheva take the abrupt

disappearance of lateral substrate motion as evidence of conﬁnement–induced

solidiﬁcation of OMCTS in the narrow gaps between the mica surfaces. If the

above ﬁlms are exposed to oscillatory shear forces, only the thinnest one is

capable of sustaining a shear stress which Klein and Kumacheva take as fur-

ther evidence for a liquid–solid phase transition in OMCTS ﬁlms triggered by

conﬁnement.

Theoretically, most previous studies support the notion of solidiﬁcation of

simple ﬂuids conﬁned by commensurately structured substrate surfaces [73].

However, in this case the ﬂuid and the substrates are composed of the same sort

of particles. Thus, under favorable geometrical conditions one expects a strong

template eﬀect triggering solidiﬁcation.

To shed more light on the rˆole of solidiﬁcation as far as mechanical properties

of conﬁned ﬂuids are concerned we investigate a model system in which the

substrates are perfectly smooth on an atomic lengthscale but decorated with

chemical structures such that the conﬁned phase is prevented from solidifying.

As will be shown below the partially condensed ﬂuid bridges are capable of

sustaining a shear strain. A comprehensive understanding of their rˆole can only

be achieved if one understands their phase behavior as well. This can be done

conveniently by a combination of two complementary treatments. The ﬁrst is

a lattice gas, where we invoke a mean–ﬁeld approximation for the intrinsic free

energy. The second model employs a Lennard-Jones (LJ)(12,6) ﬂuid treated in

Monte Carlo simulations.

In chapter 2 we introduce the lattice model. Applying a modular approach,

developed in this work we identify possible morphologies of the conﬁned lattice

gas and derive exact expressions for the respective grand potentials are derived

in the limit of vanishing temperature (T= 0). The phase diagram can then be

obtained analytically at T= 0. For higher temperatures (T>0) the model is

treated at mean–ﬁeld level which becomes exact in the limit T= 0. Within the

mean–ﬁeld approximation the equilibrium phase diagram is obtained numeri-

cally at higher temperatures (T>0). A continuous analogue of this model is

introduced in chapter 3. It can only be treated by the Monte Carlo method

also introduced in that chapter. Chapter 4 is devoted to a presentation of the

results. Phase diagrams for various sets of model parameters are presented.

Our main ﬁndings are veriﬁed with the aid of the continuous model. Further-

more, the continuous model is utilized to study bridge phases exposed to shear

strains. A theory of corresponding states is employed to derive a master–curve

description for the shear stress curves free of any model–dependent parameters.

A discussion of the results and the conclusions drawn from them are represented

in chapter 5.

Chapter 2

The mean-field lattice-gas

model

2.1 The lattice gas model

2.1.1 Grand potential

We consider a ﬂuid made up of structureless molecules, that is molecules without

internal degrees of freedom. Their positions are constrained to the nx×ny×nz

nodes of a simple cubic lattice, where the lattice constant is inﬁnitesimally

larger than the molecular diameter. Any site can be occupied by at most a

single molecule. This restriction is caused by the inﬁnite (hard–core) repulsion

between molecules occupying the same site. Attractive interactions between

molecules are limited to nearest neighbours. They are described by a square–

well potential, where depth and width of the attractive well are ﬀ(coupling

constant) and , respectively, We also assume that the ﬂuid is conﬁned in the

z–direction between two plane–parallel substrates. A molecule located at site i

is subjected to an (external) ﬁeld Φi. Thus, the Hamiltonian of the lattice gas

can be written

HLG(s)=−ﬀ



i=1

νi



sisj+



i=1

Φisi−µ



i=1

si(2.1)

2.1. THE LATTICE GAS MODEL 13

where µis the chemical potential and sistands for the occupation number of

site i,namely

si=









0,empty site

1,occupied site ,

(2.2)

in a given conﬁguration s={s1,s

2,...,s

N},N=nxnynzis the number of

lattice sites, and νiis the number of nearest neighbour sites of site i.Forthe

present conﬁned simple–cubic lattice

νi=









5,if iis located next to the substrate

6,otherwise .

(2.3)

Since Φiis arbitrary and because the second and the third terms on the right side

of (2.1) are both linear in the occupation numbers it is convenient to introduce

an “intrinsic” chemical potential via

µLG

i:= µ−Φi(2.4)

so that (2.1) simpliﬁes to

HLG(s)=−ﬀ



i=1

νi



sisj−



i=1

µLG

isi.(2.5)

The grand partition function of the conﬁned lattice gas can be cast as

ΞLG =

exp −βHLG(s)(2.6)

=

exp 



−β

−ﬀ



i=1

νi



sisj−



i=1

µLG

isi







where the sums are taken over all sets sand β=(kBT)−1(Tis the temperature

and kBis Boltzmann’s constant). To make contact with thermodynamics we

invoke the customary statistical–physical expression for the grand potential

ΩLG =−β−1ln ΞLG (2.7)

=−β−1ln 

exp 



−β

−ﬀ



i=1

νi



sisj−



i=1

µLG

isi







where the second line follows directly from (2.6). The grand potential in (2.7)

is the quantity of prime interest. Consequently, the subsequent discussion will

focus on it.

14 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL



Figure 2.1: Schematic of the prototypical model: cubic lattice gas conﬁned between

substrates consisting of strongly attractive stripes alternating periodically with weakly

attractive ones. Sites at which a molecule is subject to the strongly attractive substrate

(Φi=−fs) are indicated by dark gray squares; those at which a molecule is subject

to the weakly attractive substrate (Φi=−fw) are denoted by light gray squares. A

molecule in central region (black circle) interacts with its six nearest neighbours. The

four in the x–z–plane are depicted as gray circles; the two in the y–direction are not

shown.

2.1.2 The Prototype

So far the external potential Φiis completely arbitrary. However, henceforth it

will be associated with the conﬁning substrates. Various conﬁnement scenarios

are realized through diﬀerent choices for Φi. The prototype consists of two par-

allel substrates in the x–y–plane conﬁning the lattice gas in the ±z–direction.

Each substrate comprises stripes composed of diﬀerent chemical species, whose

interaction with the lattice gas is strongly (coupling constant fs)orweakly

(coupling constant fw) attractive, respectively (see ﬁgure 2.1). The stripes are

locatedintherange1≤x≤ns,−∞ <y<+∞(weak) and ns<x≤nx,

2.1. THE LATTICE GAS MODEL 15

−∞ <y<+∞(strong), such that they are parallel with the y-axis and alter-

nate periodically (period nx)inthex–direction. Periodicity in the x–direction

is realized by applying periodic boundary conditions. Since the external poten-

tial is translationally invariant in the y–direction, system properties are trans-

lationally invariant in that direction. The attraction with the substrates is

short–range: only molecules located at sites across the x–y–planes at z=1,n

interact with the substrates. In addition, the substrates may be misaligned in

the x–direction by shifting the strongly attractive portion of the upper substrate

by ∆nxlattice sites in the +x–direction. For this purpose it is convenient to in-

troduce a parameter α:= ∆nx/nxto specify the misalignment of the substrates

quantitatively where α0≤α≤αmax =min1

2,(nx−1)/2nx.Ifα=0the

substrates are “in registry”, i.e. strongly and weakly attractive portions of both

substrates are exactly opposite each other; α=αmax if the misalignment is

maximum (i.e., substrates “out of registry”). Notice, αvaries discontinuously

because of the discrete nature of the lattice. Thus, αis a measure of shear strain

imposed on the conﬁned lattice gas.

The external potential Φiof the prototype is given by

Φi=Φ

[1]

i+Φ

[2]

i(2.8)

where

Φ[2]

i≡Φ[2] (x, z)=











∞,z>n

−fs,1+αnx≤x≤ns+αnx

−fw,1≤x<1+αnx

−fw,n

s+αnx<x≤nx











z=nz

0,z<n

(2.9)

speciﬁes the interaction of the lattice gas with the upper substrate. Likewise

Φ[1]

i≡Φ[1] (x, z)=











∞,z<1

−fs,1≤x≤ns

−fw,n

s<x≤nx



z=1

0,z>1

(2.10)

represents the interaction with the lower substrate.

16 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

2.1.3 Nearest–neighbor lattice gas in magnetic language

By applying “magnetic” language the prototype can be transformed into an

Ising magnet in a local magnetic ﬁeld hi. This can be demonstrated by starting

from the Ising Hamiltonian [74]

HI=−J

i,j

σiσj−



i=1

hiσi(2.11)

where σi=±1 is a double–valued spin-variable ( +1 and -1, corresponding to

“spin up” and “spin down”, respectively). Coupling constant Jis a measure of

the strength of the interaction between neighboring spins and hiis the (local)

external ﬁeld acting on lattice site i. In (2.11) 

i,j

stands for summation over

all nearest–neighbour pairs. Notice that (2.11) is valid regardless of the speciﬁc

lattice considered. It is therefore convenient to rewrite (2.5) in a similar fashion

HLG =−ﬀ

i,j

sisj−



i=1

µLG

isi.(2.12)

Spin variables can be translated into occupation numbers by means of the trans-

formation

σi=2si−1.(2.13)

Thus, replacing σiin (2.11) according to (2.13) we obtain after some rearrange-

ments

HI=−4J

i,j

sisj+2

i,j

(Jνi−hi)si+



i1

2Jνi+1

.(2.14)

Comparison with (2.12) shows that

HI=HLG +



i1

2Jνi+1

(2.15)

based upon the transformation rules

ﬀ=4JJ=ﬀ

4(2.16)

µLG

i=2(Jνi−hi)hi=Jνi−νi

2(2.17)

converting lattice–gas into magnetic language.

The close correspondence between lattice–gas and magnetic language as re-

ﬂected by (2.15) is caused by the restriction to nearest–neighbor interactions

2.2. EXACT SOLUTIONS 17

and the fact that both σiand siare double–valued. However, there are dif-

ferences between the languages. Their origin are the diﬀerent symmetries of

the intermolecular interactions and the interaction of a lattice molecule with

the external ﬁelds (hi,µ

i). First, in both languages the interaction with the

external ﬁeld has the same symmetry. For example, in lattice–gas language a

particular site icontributes −µLG

ito the Hamiltonian if it is occupied and 0

otherwise. In magnetic language, on the other hand, this contribution is −hior

+hiif the spin at site iis “up” (+1) or “down” (−1), respectively. Shifting the

magnetic ﬁeld by −hiat each lattice site it is simple to realize that sites with

spin “up” contribute −2hito the Hamiltonian whereas the contribution of sites

with spin down vanishes. Comparison with lattice–gas language reveals the like

symmetry. The term 2hion the right side of (2.17) is caused by this shift of the

external ﬁeld.

Consider now the symmetry of the interparticle interaction. Imagine two

neighbouring spins are parallel (“up” or “down”) contributing −Jto the Hamil-

tonian if they are parallel (both “up” or both “down”) and +Jif they are an-

tiparallel. In lattice–gas language we observe a nonvanishing contribution to

HLG only if two neighbouring sites are occupied. Inspecting the translation

rules for the ﬁelds (2.17), one realizes that an Ising model without an external

ﬁeld hi= 0 is equivalent to a lattice–gas with a ﬁeld µLG

i=2Jνi.

2.2 Exact solutions

2.2.1 The grand potential at T=0

To understand the phase behavior of the prototype introduced in section 2.1.2

we seek (global) minima of its grand potential given in (2.7) for ﬁxed µand

T. In the limit of vanishing temperature the phase diagram can be determined

analytically. To demonstrate this we begin by calculating the grand potential

in that limit. Starting from (2.6) we assume that a conﬁguration s0exists such

that it corresponds to the maximum term in the sum on sin (2.6). It is then

convenient to separate the maximum term from the remainder according to

Ξ=exp−βHLG (s0)+

s=s0

exp −βHLG (s)(2.18)

18 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

where HLG (s) is given by (2.5). The largest term in (2.18), exp −βHLG (s0)is

determined by the set s0for which HLG (s0) is smallest, i.e. the one minimizing

the Hamiltonian. From (2.18)

Ξ=exp−βHLG (s0)

1+ 

s=s0

exp −βHLG (s)−HLG (s0)

(2.19)

and

Ω=−β−1ln Ξ (2.20)

=HLG (s0)−β−1ln 



1+ 

s=s0

exp −βHLG (s)−HLG (s0)





obtain without further ado. In the limit T−→ 0thesumin{...}in (2.20)

vanishes rapidly since β−→ ∞. Hence the logarithmic term in (2.20) vanishes

much more rapidly than linearly. Thus, for T=0

Ω=HLG (s0)=−ﬀ



i=1

νi



si0sj0+



i=1

µLG

isi0.(2.21)

According to (2.21) the partition function reduces to a single term in the limit

of vanishing temperature. In other words the maximum term method is exact

in that limit, which has some important and useful implications. Consider, for

example, the mean value Mof any thermodynamic observable M.AtT=0

it is given by

M=M(s0) (2.22)

according to the above rationale. Obviously, this implies there are no correla-

tions, that is

sisj−sisj=si0sj0−si0sj0=0 ∀i, j (2.23)

and consequently

(M−M)2=M2−M2=M2(s0)−M2(s0)=0.(2.24)

Notice also that (2.21) is free of any entropic contributions in accordance

with the third law. At thermodynamic equilibrium and for T=0thesetof

s0minimizes the grand potential, and therefore it minimizes the total energy

of the lattice gas. Obviously, energetically equivalent sites (i.e., sites at which

2.2. EXACT SOLUTIONS 19

lattice–gas molecules are subjected to identical interactions with their neighbors

and with the external potential) must have the same value of the occupation

numbers. This permits one to deduce several important conclusions concern-

ing the prototype (see ﬁgure 2.1). For example, along the y–direction all sites

are exposed to the same µLG

isince Φi[see (2.9),(2.10)] is translationally invari-

ant in this direction. We may therefore restrict the discussion to an eﬀectively

two–dimensional problem. Because of the spatial variation of Φi, regions char-

acterized by identical occupation numbers can be identiﬁed. The reason for that

is not immediately obvious but will become clear shortly when we delineate a

strategy to identify such regions by applying a modular approach. This allows

us to construct a hierarchy of increasingly complex modules sequentially from

simpler ones, starting from the bulk. Any module, which gives rise to a set of

so–called morphologies {M}, consists of a juxtaposition of one or more of the

previous (simpler) modules. We introduce the term “morphology” to refer to

the set of energetically homogeneous regions in which the occupation numbers

are identical at all sites pertaining to such a region according to the above dis-

cussion. However, occupation numbers will generally diﬀer between diﬀerent

such regions. Thus,

M:= {si}.(2.25)

The modular approach to construct more complex modules from simpler

ones consists of two steps. In the ﬁrst one, auxiliary surfaces are introduced in

the simpler module by breaking a certain number of bonds. This reduces νifrom

6 to 5 for all sites located at this newly created surface according to (2.3). In

the second step two simpler modules are juxtaposed and the auxiliary surfaces

between them are removed. Thereby new bonds are created now connecting

the original simpler modules across the interface. The grand potential of a

given morphology within the more complex module can therfore be expressed

as a sum of the grand potentials of the simpler ones, plus corrections which

account for the breaking of bonds between nearest neighbors in the simpler

modules and the making of new bonds across the interfaces between modules

that make up the new composite (more complex) module. Since the system

consists of a certain number of regions ˆnwith equal occupation numbers, we

replace individual occupation numbers by occupation numbers for the entire

20 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

region by introducing the notion of block occupation numbers ˆsi. Therefore the

deﬁnition of the morphologies (2.25) simpliﬁes to

M:= {ˆsi}(2.26)

Since ˆsi[like sisee (2.2)] is double-valued, the number of morphologies conceiv-

able in principle is given by

number of morphologies = 2



n.(2.27)

As we will see shortly for the system of interest, nis always small so that only a

few morphologies need to be considered. This is because the structure of Φiis

still quite simple. For more complex external ﬁelds such as, for instance, the one

characterizing random porous media [58, 61], 2



nmay become overwhelmingly

large. Because of the small number of possible morphologies in the present case

one can construct the phase diagram at T= 0 in a straightforward fashion

since Ω is an analytic function of the block occupation numbers based upon the

modular approach described above. This shall be demonstrated in the following

section.

2.2.2 Morphologies at T=0

Bulk lattice gas.



Figure 2.2: Bulk module: All sites are

identical. Only one (double–valued) block

occupation number ˆs0accounts for all pos-

sible morphologies.

In the simplest case (Φ ≡0,ν

i= 6) (i.e., the bulk lattice gas) all sites are

equivalent, so that only one block occupation number ˆs0is required. Thus,

2.2. EXACT SOLUTIONS 21

(2.21) reduces to

Ωb=−N ν

2ﬀˆs2

0+µˆs0=: Nω0.(2.28)

In (2.28), ω0is the grand–potential density (per site) of the bulk lattice gas.

Because ˆs0is double–valued, (2.28) gives two possible morphologies [see also

(2.27)], namely a “gas” characterized by Mg={0}having grand potential

Ωg

b= 0 and corresponding to an entirely empty lattice (ˆs0= 0). In addition a

“liquid” exists characterized by Ml={1}and Ωl

b=−N ν

2ﬀ+µwhere all

sites are occupied (ˆs0= 1). Gas and liquid phases may coexist at µgl

xdeﬁned

through

Ωg

b(µgl

x)=Ω

b(µgl

x)=:Ωgl

b(µgl

x)=0=Nν

2ﬀ+µgl

x(2.29)

from which µgl

x−1

ﬀ=−ν/2=−3 is easily deduced (see section 2.3.2 and [74]).

Thus, for µ<µ

x, gas is the thermodynamically stable phase, whereas for

µ>µ

xliquid is the stable phase.

Hard substrates.



Figure 2.3: Hard-wall module: Sites of

groups A and B have diﬀerent coordina-

tion numbers 5 and 6, respectively. Never-

theless, all sites have the same occupation

number ˆs0(see text).

The next slightly more complicated situation is one in which a lattice gas is

conﬁned in the z–direction by two planar hard substrates represented by

Φi≡Φhs (z)=



∞z<1,z >n

01≤z≤nz

,(2.30)

where Φhs (z) serves to introduce “surfaces” in the spirit of section 2.2.1. From

an inspection of (2.30) it is obvious that the system is translationally invariant

in directions parallel to the walls, but comprises two types of sites. All sites

22 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

which are located in lattice planes next to the walls, i.e. the surface planes, have

a lower coordination number (νi= 5) than “core” sites (νi=6). However,all

sites are still occupied equivalently and possible morphologies can be described

by a single occupation number ˆs0as for the bulk. To see this, suppose two types

of sites exist labeled “A” and “B” with associated block occupation numbers

ˆsAand ˆsB, respectively (see ﬁgure 2.3). The set of B sites can be subdivided by

distinguishing sites labeled Bswhich are connected to A sites and sites labeled

Biwhich are connected to other B sites only. Imagine now ˆsB=1andˆsA=0

then sites Bshave only νi=5occupied nearest–neighbor sites. If it is favorable

to occupy the Bssites then it must be even more favorable to occupy sites

A, because in that case A sites have νi= 5 occupied nearest–neighbor sites

(as Bsbefore) and Bssites capture one additional occupied nearest–neighbor

site (νi= 6). Thus, the latter occupation scenario is energetically preferential.

In principle, this rationale can be applied to any other distribution of A and

B sites, hence one concludes occupation numbers of all sites to be identical

ˆsA=ˆsB=ˆs0. It is important to realize that by introducing a hard substrate

which corresponds only to the breaking of certain bonds no new morphologies

arise. This observation validates and motivates the modular approach. It holds

regardless of the complexity of the original modules.

According to our modular approach the present conﬁned lattice gas may

be viewed as a bulk system, plus “surfaces”. We can then express the grand

potential of the conﬁned phase as

Ωhs =Ω

b+ ∆Ω (2.31)

where Ωbpertains to the bulk module and the correction ∆Ω accounts for

the interactions that are missing for molecules in the surface planes z=1

and z=nz. Since each nearest–neighbor interaction contributes −ﬀs2/2per

particle to the conﬁgurational energy of the original bulk module, and since

there are nxnymolecules in each surface and two surfaces, the total correction

is nxnyﬀs2. We can therefore rewrite (2.31) as

Ωhs =nxnynzω0+nxnyﬀs2

0=nxnynzω0+ﬀs2

0(2.32)

where ω0is deﬁned by (2.28). The only eﬀect of conﬁnement is an upward shift

in the chemical potential at gas–liquid coexistence. By solving the analogue of

2.2. EXACT SOLUTIONS 23

(2.29), Ωg

hs(µgl

x)=Ω

hs(µgl

x), we obtain µgl

x−1

ﬀ=−ν/2+1/nz=−3+1/nz.As

expected, this shift vanishes in the limit of large substrate separations (i.e., as

nz→∞).

Chemically homogeneous substrates.



Figure 2.4: Homogeneous-wall module:

Two occupation numbers ˆs0,ˆs1account

for the occupation of the two types of sites.

Dotted lines demarcate the position of the

auxiliary surfaces. Gray squares repre-

sent sites at which lattice–gas molecules

are subject to attractive interaction with

the substrates.

The situation discussed above becomes slightly more complicated if one re-

places (2.30) by

Φi≡Φhom (zi)=









∞z<1,z >n

−fs,z=1,z =nz

02≤z≤nz−1

(2.33)

that is by chemically homogeneous substrates capable of attracting the lattice

gas in addition to merely conﬁning it. Caused by the external potential Φi

(2.33) the system now comprises two types of sites (subscripts 0 and 1), since

the energy contribution from the external potential might dominate all other

contributions. Sites of type 0 having the intrinsic chemical potential µLG

0=µ

[see (2.4)] are located in the region {z|2≤z≤nz−1}whereas sites of type

1 with µLG

1=fs +µare located at {z|z=1,z=nz}. The grand potential

of the lattice gas conﬁned between homogeneous attractive substrates can thus

be determined by sandwiching an nx×ny×(nz−2) hard–substrate module

[which consists of a slab of uniformly occupied (ˆs1= 0 or 1) sites] between two

nx×ny×1 hard–substrate modules (i.e., identical thin slabs of nxnyuniformly

24 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

occupied sites). Using the modular principle, we can express the grand potential

of the composite “homogeneous” module as

Ωhom =Ω

[0]

hs +2Ω

[1]

hs + ∆Ω (2.34)

where

Ω[0]

hs := nxny(nz−2) ω0+ﬀˆs2

0

Ω[1]

hs := nxnyω1+ﬀˆs2

1(2.35)

and Ωhs stands for the grand potential of the previous member of the hierarchy

of modules, namely a slab between hard substrates. The index 0 denotes the

central module at µLG

0≡µ, while index 1 pertains to the other two (identical)

modules at µLG

1=µ+fs. Since Ωhs already accounts for the breaking of bonds

to create auxiliary surfaces (see discussion of the previous simpler module), the

correction ∆Ω in (2.34) is due solely to the formation of bonds across the two

interfaces and is given by −2nxnyﬀˆs0ˆs1. Therefore, we can rewrite (2.34) as

Ωhom =nxnyψ(2.36)

where

ψ:= 2 ω1+ﬀˆs2

1+(nz−2) ω0+ﬀˆs2

0−2ﬀˆs0ˆs1.(2.37)

Since the present module consists of two types of sites, four diﬀerent morpholo-

gies arise from the homogeneous module [see (2.27)]. These can be identiﬁed

by sets of occupation numbers M={ˆs0,ˆs1},whereˆs0and ˆs1are the block

occupation numbers of the central and outer slabs, respectively.

Chemically heterogeneous substrates.

Consider now the prototype: a lattice gas between substrates decorated with

strongly attractive stripes (fs) that alternate periodically with weakly attractive

stripes (fw)inthex–direction (see section 2.1.2). We restrict our consideration

to the case of perfectly aligned substrates, i.e. ∆nx= 0. Thus, within one

period the potential can be represented as

Φi≡Φhet (x, z)=











∞z<1,z >n





−fs 1≤x≤ns

−fw ns<x≤nx



z=1,z =nz

02≤z≤nz−1.

(2.38)

2.2. EXACT SOLUTIONS 25



Figure 2.5: Heterogeneous-wall module:

Four occupation numbers ˆsw

0,ˆsw

1,ˆss

0,ˆss

1ac-

count for the occupation of the four groups

of sites. Dotted lines demarcate the

boundaries of the groups. Gray squares

symbolize attractive interaction with the

wall. Dark gray squares indicate “strong”

interaction (fs) and light gray squares in-

dicate “weak” interaction (fw). (See sec-

tion 2.1.2 for details.)

Because of the diﬀerent attraction acting on sites of the surface planes, these

planes have to be subdivided into two groups. Since the central region is

bounded by these two groups, it subdivides into two groups as well (see ﬁg-

ure 2.5). Again following the modular principle, we can determine the grand

potential by juxtaposing (in the x–direction) two modules corresponding to

the previous, simpler one: the lattice gas between homogeneous attractive sub-

strates. Thus, we can write the grand potential as

Ωhet =Ω

[w]

hom +Ω

[s]

hom + ∆Ω (2.39)

where from (2.36)

Ω[u]

hom =nunyψu,u=s,w (2.40)

and from (2.37)

ψu=2(ωu

1+ﬀˆsu

1ˆsu

1)+(nz−2) ωu

0+ﬀˆsu

0ˆsu

0−2ﬀˆsu

0ˆsu

1,u=s,w.(2.41)

Ω[s]

hom and Ω[w]

hom are the grand potentials of the lattice gas between strongly

attractive substrates of width nsand weakly attractive substrates of width nw=

nx−ns, respectively. Note that the regions of the composite module now carry

two indices, one denoting the strength of the attraction (w or s) and the other

denoting the particular slab of the “homogeneous” module (0 referring to the

central slab and 1 to the outer slabs).

The correction in (2.39) can be derived as follows. We must ﬁrst create

surfaces by breaking bonds between nearest neighbors across a plane (paral-

lel with the y–z–plane) in the “homogeneous” module. This process increases

26 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

Ω by the amounts nyﬀ[2ˆsu

1ˆsu

1+(nz−2) ˆsu

0ˆsu

0] for weak (u = w) and strong

(u = s) substrates. We must then join the strong and weak “homogeneous”

modules by forming bonds across the interfaces. This joining decreases HLG by

nyﬀ[2ˆsw

1ˆss

1+(nz−2) ˆsw

0ˆss

0]. Thus, the total grand potential for the “heteroge-

neous” module can be expressed

Ωhet =ny[nsψs+nwψw+χss +χww −2χsw] (2.42)

where

χuu =ﬀ[2ˆsu

1ˆsu

1+(nz−2) ˆsu

0ˆsu

0],u=s,w

χsw =ﬀ[2ˆsw

1ˆss

1+(nz−2) ˆsw

0ˆss

0] (2.43)

A consequence of the lower symmetry of the prototype is a larger number of

possible morphologies. Inspection of (2.41)–(2.43) reveals that the grand poten-

tial is determined by the set M:= {ˆsw

0,ˆss

0,ˆsw

1,ˆss

1}, where each block occupation

number can independently assume the value 0 or 1. Thus, 16 diﬀerent mor-

phologies are possible in principle according to (2.27). This fairly large number

can be reduced substantially on physical grounds (i.e. by taking into account

the relative magnitudes of fs,fw,andﬀ). For example, if both fs and fw

are small compared to ﬀ, the morphology characterized by M={0,0,1,1}is

physically not sensible because it refers to a situation where sites at which the

lattice gas is exposed to a reduced total attraction (i.e., in the immediate vicin-

ity of the substrate) are occupied whereas energetically more favorable (nz−2)

bulk sites remain empty. By similar considerations most of the remaining mor-

phologies can be ruled out, without the necessity of actually calculating their

grand potentials.

2.2.3 Thermodynamically stable morphologies at T=0

The analysis of potentially possible morphologies of the prototype in section

2.2.2 can now be employed to construct the phase diagram at T= 0. Henceforth

we employ the customary dimensionless units (distance in units of ,energyin

units of ﬀ, temperature in units of ﬀ/kB). As an example we consider the

case ns=nw=10(nx= 20), nz= 10, fw =0.0, and 0.0≤fs ≤2.0.

With the aid of ﬁgures 2.5 and 2.6 it can be seen that the only physically

2.2. EXACT SOLUTIONS 27

(a)

(b)

(c)

Figure 2.6: Subset of morphologies of the prototype M:= {ˆsw

0,ˆss

0,ˆsw

1,ˆss

1}.Black

dots indicate occupied sites. a) gas Mg={0,0,0,0},b)dropletMd={0,0,0,1},c)

bridge Mb={0,1,0,1}, d) vesicle Mv={1,1,0,1}, e) liquid Ml={1,1,1,1}

28 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

(d)

(e)

Figure 2.6: (Continued)

sensible morphologies are characterized by Mg={0,0,0,0}[empty lattice, i.e.

“gas” morphology, ﬁgure 2.6(a)], Ml={1,1,1,1}[full lattice, i.e. “liquid”

morphology, ﬁgure 2.6(e)], Md={0,0,0,1}[liquid–ﬁlled lanes stabilized by

the strongly adsorbing stripes, i.e. “droplet” morphology, ﬁgure 2.6(b)], Mb=

{0,1,0,1}[ﬂuid “bridge” morphology connecting strongly adsorbing stripes,

ﬁgure 2.6(c)], and Mv={1,1,0,1}[gas–ﬁlled lanes immersed in high–density

ﬂuid, i.e. “vesicle” morphology, ﬁgure 2.6(d)].

Using (2.41), (2.42), and (2.43), we derive expressions for the grand potential

of these morphologies. The trivial one is the “gas”, that is the empty lattice

Mg={0,0,0,0}, for which

Ωg(µ)≡0.(2.44)

The simplest nontrivial morphology is the “droplet” Md={0,0,0,1}. Its grand

2.2. EXACT SOLUTIONS 29

potential is given by

Ωd(µ)=ny −2nsν−2

2+µ+2−2nsfs!,(2.45)

where ν(= 6) is the number of nearest neighbors (of the bulk). Eventually, a

“bridge” morphology Mb={1,0,1,0}characterized by

Ωb(µ)=ny"−nsnzν

2+µ+nz+ns−2nsfs#(2.46)

may form connecting the strongly attractive stripes of the substrates along the

z–direction. It is also conceivable that under favorable conditions a “vesicle”

Mv={1,1,1,0}may exist. Its grand potential is given by

Ωv(µ)=ny"(2nw−nxnz)ν

2+µ+2+nx−2nsfs#.(2.47)

Eventually, all lattice sites may be occupied to yield a morphology to which we

refer as “liquid” Ml={1,1,1,1}. The grand potential of this liquid is given by

Ωl(µ)=ny"−nxnzν

2+µ+nx−2nsfs −2nwfw#.(2.48)

To construct the phase diagram we must identify the morphology having the

lowest value of the grand potential for a speciﬁc chemical potential. Therefore

we consider the grand–potential curve Ωα(µ). Its slope is given by the partial

derivative of the grand potential with respect to the chemical potential.

∂Ω

∂µT=0

=−



i=1

si0=N≤N (2.49)

where Nis the number of occupied sites. Evidently, Nis independent of µat

T= 0. In other words, because each morphology is in its ground state, there

are no density ﬂuctuations (see also section 2.2.1) implying

∂2Ω

∂µ2T=0

= 0 (2.50)

which is also obtained directly from (2.49). Thus, for T=0,Ω

α(µ)isa

straight line with negative slope. Of course, because of (2.49) all grand po-

tential curves may have diﬀerent slopes and intersect the ordinate at diﬀerent

values Ωα(0),α=g,d,b,v,l (see ﬁgure 2.7).

The phase having the lowest value of Ω for a given µis thermodynamically

stable (all other parameters ﬁxed). Two morphologies are coexisting phases at

30 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL



-8

-4

-3.05 -3 -2.95



Figure 2.7: Line density of the grand potential Ωα(µ)/nyversus chemical potential µ

for various morphologies α= g (gas), d (droplet), b (bridge), v (vesicle), and l (liquid)

indicated in the ﬁgure. For all systems fs =1.0, fw =0.0, T=0.

a chemical potential µαβ

xobtained as a solution of

Ωαµαβ

x=Ω

βµαβ

x,(2.51)

where Ωα,β µαβ

xis the absolute minimum of the grand potential for the given

temperature. Figure 2.7 shows Ωα(µ)/nyobtained from (2.44)-(2.48). If µis

suﬃciently low the gas morphology is thermodynamically stable. At µgl =−3,

Ωgµglintersects with Ωlµglwhere µgl =µgl

xis a solution of (2.51). Beyond

that intersection the liquid is thermodynamically stable. Moreover, at µgl

x,Ω

bis

equal to both Ωgand Ωl. Thus, we have three-phase-coexistence at µgl

x≡µgbl

tr ,

deﬁned by

Ωgµgbl

tr =Ω

bµgbl

tr =Ω

lµgbl

tr (2.52)

and therfore a triple point {Tgbl

tr ,µ

gbl

tr }. The “width” of the one–phase region

of bridge morphologies vanishes for T= 0, that is it consists of the triple point

only.

Based upon (2.49) and (2.50) and taking into account the diﬀerent values of

2.2. EXACT SOLUTIONS 31

Ωα(0) several characteristics of the phase behavior are readily deduced. From

(2.44)-(2.48) it is obvious that Ωg(µ) has the largest slope of all grand potential

curves. Thus, if one decreases the chemical potential suﬃciently the gas will

eventually become thermodynamically stable, i.e. the gas phase is stable over

the region −∞ <µ≤µgα

x. Similarly the stability region of the liquid phase is

given by µβl

x≤µ<+∞.Inotherwords,forT= 0 the phase diagram always

consists of stable gas and liquid regimes, with associated mean densities ρg=

Ng/N=0andρl=Nl/N= 1, respectively. Any other stable phase βmust

have a mean density ρβsuch that ρg< ρβ< ρl. The stability region of phase

βis bounded by coexistence “points” (µgβ

x,µ

βl

x) on both “sides”. Extending

this argument to more than 3 phases one realizes that phases associated with

lower densities are stable at lower values of the chemical potential than higher–

density phases. This behavior at T= 0 is universal and extendable to other

lattice geometries.

From the explicit expressions for Ωα(µ) given in (2.44)-(2.48) one can cal-

culate µαβ analytically for all morphologies. Comparing the associated grand

potentials at µαβ one identiﬁes points of phase coexistence µαβ

xand therefore

the range over which a given phase αis thermodynamically stable. Since each

pair of grand–potential lines has exactly one intersection, 10 such intersections

are possible for the 5 grand–potential curves given in (2.44)-(2.48). A subset is

given by the following equations

µgd =−2+ 1

ns−fs (2.53)

µgb =−3+ 1

nz−2

fs (2.54)

µgl =−3+ 1

−2nsfs−2nwfw

nxnz

(2.55)

µdb =−3+ 1

ns−1

nz−2(2.56)

µdl =−3+(nw−ns)−2(1 + nwfw)

nwnz+ns(nz−2) (2.57)

µbl =−3−1

nz−2

fw.(2.58)

From (2.55) – (2.58) one notices that some of the expressions are independent of

certain system parameters. For example, µgb in (2.54) does not depend on fw

and nw. This is an important and useful result to understand the dependence of

32 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

the phase diagram on certain system parameters at higher temperatures (T>0,

see section 2.4).

2.3 Mean-Field Theory

In the preceding section the phase behavior of the prototype was analyzed based

upon an analytical solution of the grand canonical ensemble partition function

at T= 0. Unfortunately, for nonzero temperatures, such an exact solution

is not known. Thus, there is no way to calculate the phase diagram of the

prototype analytically for T>0. However, several numerical techniques have

been developed which can be employed in principle.

A direct (and formally exact) method is based upon Monte Carlo simulation.

In Monte Carlo properties of the system of interest are calculated as ensemble

averages over a suﬃciently long Markov chain of conﬁgurations such that in the

limit of an inﬁnite number of conﬁgurations, these are distributed according to

probability densities characteristic of the speciﬁc statistical physical ensemble

in question. Calculating such quantities as the mean density or internal energy

as functions of thermodynamic state variables, one may employ thermodynamic

integration techniques to obtain the variation of, say, the grand potential along

a suitably chosen path in thermodynamic state space. If these paths are chosen

such that the absolute value of Ω can be calculated at the starting point, this

procedure allows one to determine the phase behavior numerically as shown by

Binder for the bulk Ising magnet in three dimensions [75].

However, there are essentially two drawbacks of this method. First, it is

rather time consuming since several Monte Carlo simulations have to be carried

out to determine one point on the phase diagram. This is particularly cum-

bersome for the prototype if one wishes to explore its phase behavior in the

multidimensional space of model parameters. Second, and more importantly,

thermodynamic integration works only if the path chosen in state space does

not accidentally cross a line of ﬁrst–order phase transitions at which ensemble

averages change discontinuously by an a priori undetermined amount. This, in

turn, implies that one needs to have at least a rough idea of the phase diagram

prior to determining it by Monte Carlo. Since the discussion in section 2.2

2.3. MEAN-FIELD THEORY 33

indicated that a large number of phases may exist for the prototype even for

nonzero temperatures, choosing a suitable path in state space beforehand seems

hopeless.

An alternative to Monte Carlo simulations is to approximate ΩLG in (2.7).

To obtain such an approximation we utilize Bogoliubov’s theorem [76] which

provides the possibility of ﬁnding the “best” approximation within a number

of constraints imposed on the system in a controlled and transparent manner.

This approach will be discussed in the following section.

2.3.1 Variational treatment of the mean field model

We begin again with the Hamiltonian of the nearest neighbor lattice gas given

in equation (2.5). Let us also introduce the Hamiltonian of a system having no

intermolecular attractions [nonattractive (NA) lattice gas, ﬀ= 0, see (2.5)]

HNA =−



i=1

µNA

isi(2.59)

where µNA

iis the associated intrinsic chemical potential. Intermolecular repul-

sion is still accounted for because of the double–valued occupation numbers si

[see (2.2)]. The purpose of (2.59) is to determine µNA

isuch that the system of

nonattracting lattice–gas molecules, which can be treated analytically at all tem-

peratures, becomes the best approximation to the lattice gas with intermolecular

interactions governed by HLG in (2.5). To obtain this best approximation we

employ Bogoliubov‘s theorem.

Suppose a system with Hamiltonian Hwhich can be split according to

H=H0+∆H(2.60)

where H0is the Hamiltonian of a reference system and ∆His diﬀerence between

the latter and H. In the present context we identify Hwith HLG and take

the nonattractive lattice gas as reference system (H0=HNA). Therefore ∆H

accounts for attractive interactions between lattice–gas molecules. Let us deﬁne

H(λ)=HNA +λ∆H,0≤λ≤1 (2.61)

where H(λ=0)≡HNA and H(λ=1)≡HLG. Thus, by increasing the

value of the dimensionless coupling parameter λfrom 0 to 1 (“turning on” the

34 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

perturbation ∆H)onecanvaryH(λ) smoothly between HNA and HLG.The

grand potential of a system governed by H(λ)isgivenby

−βΩ(λ) = ln Ξ = ln 

exp [−βH(s,λ)] .(2.62)

Let us now diﬀerentiate (2.62) to obtain

dΩ(λ)

dλ =Ξ

−1

∆Hexp [−βH(s,λ)] (2.63)

=∆Hλ

and

d2Ω(λ)

dλ2=−β$(∆H−∆Hλ)2%λ≤0∀λ(2.64)

which is negative semideﬁnite for all λ.Thus,Ω(λ)isconcavefor0≤λ≤1.

Since the perturbation ∆His assumed to be small, we approximate Ω(λ)bya

Taylor series around λ=0,thatis

ΩLG ≃ΩNA +dΩ(λ)

dλ λ=0

λ+··· (2.65)

Because Ω(λ) is concave [see (2.64)], it follows that Ω(λ) lies below its tangent

at λ= 0, that is the right side of (2.65) must always be larger or equal to ΩLG.

Thus, from (2.63) and (2.65) one arrives at the Bogoliubov inequality (λ=1)

ΩLG ≤ΩNA +∆Hλ=0 .(2.66)

This last expression can be interpreted the following way. The exact grand

potential of the system of interest, i.e. HLG (perturbed system) is always lower

or equal to the grand potential of the unperturbed system plus the perturbation

(averaged over the states of the unperturbed system). Thus, ΩNA +∆Hλ=0

is an upper bound of ΩLG. For the present system we may therefore write

ΩLG ≤ΩNA +HLG −HNAλ=0 .(2.67)

Now we are in a position to solve our initial problem, that is to determine the

best choice of µNA

iin (2.59). Since the right side of (2.67) is an upper bound

of ΩLG we minimize ΩNA +HLG −HNAλ=0 with respect to µNA

i. To derive

2.3. MEAN-FIELD THEORY 35

expressions for ΩNA and HLG −HNAλ=0 as functions of µNA

iwe start from

ΞNA =

exp &β



i=1

µNA

isi'(2.68)

(

i=1



si=0

exp βµNA

isi=

(

i=1 1+expβµNA

i

where the second line follows from absence of attractive interactions between

molecules located at diﬀerent sites. The grand potential is then given by [see

(2.62)]

ΩNA =−β−1ln ΞNA =−β−1



i=1

ln 1+expβµNA

i (2.69)

From (2.5) and (2.59) we calculate

HLG −HNAλ=0 =)−ﬀ



i=1

νi



sisj−



i=1 µLG

i−µNA

isi*λ=0

(2.70)

=−ﬀ



i=1

νi



jsisjλ=0 −



i=1 µLG

i−µNA

isiλ=0

=−ﬀ



i=1

νi



jsiλ=0 sjλ=0 −



i=1 µLG

i−µNA

isiλ=0

=−ﬀ



i=1

νi



ρiρj−



i=1 µLG

i−µNA

iρi

where the local densities are deﬁned by ρi≡siλ=0. The third line of (2.70)

follows because in the nonattracting system there are no intermolecular corre-

lations

sisjλ=0 −siλ=0 sjλ=0 =0 .(2.71)

Since ΩNA +HLG −HNAλ=0 is a functional of µNA

iwe seek solutions of the

variational expression

δΩNA +HLG −HNAλ=0

δµNA

=0 .(2.72)

Equation (2.72) determines the local density in terms of the optimum intrinsic

chemical potential µNA

iof the corresponding nonattractive system, namely

ρi=1

1+exp(−βµNA

i)(2.73)

which can be rearranged to give

exp (−βµNA

i)= ρi

1−ρi

.(2.74)

36 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

Inserting now (2.74) into (2.70) together with (2.67) gives us the optimized

grand potential

ΩLG ≤−β−1



i=1

[ρiln ρi+(1−ρi)ln(1−ρi)] (2.75)

−ﬀ



i=1

νi



ρiρj−



i=1

µLG

iρi

=: ΩMF

to which we refer as the grand potential in mean–ﬁeld approximation, since it

explicitly neglects intermolecular correlations [see (2.70)].

From (2.75) it is furthermore clear that ΩMF is a functional of the local

density. The discussion in [77] makes it clear that for given intrinsic chemical

potential µ:= {µi}(we shall drop the superscript “MF” henceforth to simplyfy

notation) and ﬁxed T, Ω is minimum if the set ρ:= {ρi}corresponds to the

local density at thermodynamic equilibrium. The latter can be determined as a

solution of the variational expression

δΩ[ρ]

δρi

= 0 (2.76)

from which we obtain a set of coupled transcendental equations (i.e., Euler–

Lagrange equations), namely

β−1ln ρi

1−ρi−ﬀ

νi



ρj−µi=0 i=1,...,N.(2.77)

The set of equations (2.77) can be solved numerically by applying an iterative

procedure discussed in detail in appendix A.

2.3.2 Bulk phase diagram of the mean-field lattice gas

As an illustration and as a useful reference system we apply the above consid-

erations to the bulk lattice gas where it is convenient to introduce the grand

potential density

ω=Ω

N(2.78)

where Nis the “volume” (number of lattice sites). The bulk is characterized by

Φi=0,i=1,...,N. We employ a simple cubic lattice so that νi= 6 regardless

2.3. MEAN-FIELD THEORY 37

of the lattice site considered. Therefore, the bulk lattice gas is uniform, that is

ρi=ρ, i =1,...,N. In that case (2.75) simpliﬁes to

ω(T,ρ)=T[ρln ρ+(1−ρ)ln(1−ρ)] −ν

2ρ2−µρ (2.79)

where we have employed the same dimensionless units already introduced in

section 2.2.3. Since ωis a function of the uniform density, (2.76) simpliﬁes to

∂ω

∂ρT,µ

= 0 (2.80)

from which

Tln ρ

1−ρ−νρ −µ= 0 (2.81)

follows with the aid of (2.77). From stability considerations one knows, that

any two systems are in thermodynamic equilibrium if their respective sets of

intensive variables have the same values [78]. These equilibrium constraints can

be utilized to determine coexisting states as follows. Let “α”and“β”denote

diﬀerent thermodynamic states whose intensive variables satisfy the equations

Tα=Tβ(2.82)

µα=µβ(2.83)

Pα=Pβ(2.84)

where Pstands for the pressure. Since we are exclusively concerned with isother-

mal conditions T=Tα=Tβis satisﬁed aprioriand does no longer need to

be listed explicitly. From (2.81) it is clear that bulk phases are characterized

by their densities ρ. Thus, to identify coexisting phases (say, αand β), we are

seeking densities ραand ρβsatisfying the constraints (2.83) and (2.84). From

(2.81) we have

µ(ρ)=Tln ρ

1−ρ−νρ. (2.85)

or equivalently

µ(x)=Tln (1

2+x)

2−x)−ν1

2+x.(2.86)

where the transformation ρ=x+1/2 is introduced to make symmetry properties

of µtransparent. Moreover, we introduce

µ(x):=µ(x)+ν

2=T ln 1

2+x−ln 1

2−x!−νx (2.87)

38 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

which is an odd function of x. Suppose xαand xβsolve (2.83). Then −xαand

−xβare solutions as well. In particular, xα

∗and xβ

∗may be solutions where

xα

∗=−xβ

∗. In this special case the two solutions are identical such that the far

right side of (2.87) must be zero for symmetry reasons. Thus,

µ(xα

∗)=µ(−xα

∗)=−ν

2.(2.88)

To reduce the manifold of solutions of (2.83) we employ the pressure

P(ρ)=−ω(ρ)=−Tln(1 −ρ)−ν

2ρ2(2.89)

where we have used (2.79) and (2.85) to derive the far right side. Replacing ρ

by xas before we obtain after some straightforward algebra

P(x)=−T ln (1

2−x)

2+x)+ln1

2+x!−ν

2x2+ν

2x+ν

8(2.90)

=µ(x)+P(−x)+ν

where we have used (2.85) and (2.89) to arrive at the last line, which may be

recast as

P(x)−µ(x)=P(−x)+ν

2.(2.91)

This last expression satisﬁes the constraint (2.84) only if µ(x) is given by (2.88)

so that only the single solution xα

∗=−xβ

∗is compatible with both constraints

(2.83) and (2.84). Thus, at coexistence

µαβ

x=−ν

2(2.92)

and the phase diagram

µx(T)=−ν

2,0≤T≤Tc(2.93)

is a straight line parallel to the T-axis starting at T= 0 and ending at the

critical temperature Tcto be speciﬁed shortly [see ﬁgure 2.8(b)]. From (2.85)

and (2.92)

µ(ρ)=Tln ρ

1−ρ−νρ =−ν

2(2.94)

can be obtained immediately. Equation (2.94) has a trivial solution ρ=1/2as

one can easily verify. For this solution to be thermodynamically stable

∂2ω

∂ρ2T

ρ(1 −ρ)−ν≥0 (2.95)

2.3. MEAN-FIELD THEORY 39

0.0

0.5

1.0

1.5

0.0 0.25 0.5 0.75 1.0



(a)

-3.1

-3.0

-2.9

0.0 0.5 1.0 1.5



(b)

liquid

gas

Figure 2.8: Bulk phase diagram: (a) T-ρrepresentation (b) µ-Trepresentation. The

critical point (Tc=3/2, µc=−ν/2=−3) is indicated by the black dot.

40 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

has to be satisﬁed. Rearranging (2.95) gives (ν=6)

T≥|νρ(1 −ρ)|ρ=1/2=3

2(2.96)

where the equality holds for the critical temperature T=Tc=3/2. Thus,

ρ=1/2 is a thermodynamically stable solution only if T≥Tc. Since for

T=Tc,ρ=ρc=1/2 satisﬁes (2.80) and the equality in (2.95) simultaneously

we associate with it the density at the bulk critical point. Hence, from (2.92),

µc=−3 is readily obtained. For T<T

c,ρ=1/2 does not satisfy the inequality

in (2.95) and therefore corresponds to a maximum of ω. Hence, the two other

solutions xα

∗and xβ

∗must correspond to minima of ωand therefore represent

metastable morphologies or thermodynaically stable phases of the bulk lattice

gas (see ﬁgure 2.8).

2.3.3 The limit T=0

An interesting limiting situation of the mean–ﬁeld treatment is the limit of

vanishing temperature (T= 0) where the lattice gas may be treated analytically

according to the discussion in section 2.2.3. Let us begin by examining the bulk

system (Φi=0,ν

i=ν= 6) whose equation of state is given by [see (2.81)]

Tln ρ

1−ρ−νρ −µ=0 .(2.97)

Setting µto µx(T) [see (2.93)], (2.97) is an equation for the densities of coexisting

phases. Unfortunately, (2.97) cannot be solved in closed form for ρ.Insteadwe

apply a graphical method recasting (2.97) as

ρ=1

ν(T+x−µx) (2.98)

where we have introduced the deﬁnition

ρ=: 1

1+exp(−+x)(2.99)

Plotting (2.98) and (2.99) versus +xfor T<T

cwe ﬁnd that (2.97) has three

real roots −+x0,0,and+x0corresponding to the densities 1 −ρ0,1/2andρ0,

respectively. This result is in accordance with the one obtained in section 2.3.2,

where we have also shown that ρ=1/2 is an unstable solution of (2.97) below Tc.

The remaining solutions must therefore be minima associated with coexisting

2.3. MEAN-FIELD THEORY 41

gas (1−ρ0) and liquid (ρ0) phases. To determine the densities of the coexisting

phases in the limit T→0 we replace µx(T)by−ν/2 [see (2.93)]. After some

straightforward algebraic manipulations this leads to

+x=νρ−1

21

T(2.100)

which shows that for 0 ≤ρg<1/2 (gas branch) +xgoes to negative inﬁnity if

Tgoes to zero. If, on the other hand, +xis in the range 1/2<ρ

l≤1 (liquid

branch) then +xdiverges to positive inﬁnity. With the help of (2.99), which is

temperature independent, one obtains

lim

T→0ρ0=









(2.101)

If the chemical potential is not equal to the chemical potential at liquid gas

coexistence, (2.100) must be written

+x=(νρ −µ)1

T.(2.102)

In that case

lim

T→0+x=









−∞ ,0≤ρ≤µ/ν

+∞,µ/ν ≤ρ≤1

(2.103)

such that (2.101) is again recovered. Thus, in the limit of vanishing temperature

the grand–potential density (2.79) simpliﬁes to

ω0≡ω(T=0)=−ν

3ρ2

0+µρ0.(2.104)

Because of (2.101), (2.104) is equivalent to

ω0=−ν

3s2

0+µs0.(2.105)

where the (mean) density has been replaced by the occupation number s0.We

thus arrive at the gratifying result that, for the bulk system, the mean-ﬁeld

approximation of the lattice gas agrees with the exact result at T=0givenin

(2.28).

The above reasoning can be extended to the situation of primary interest,

namely the prototype (see section 2.1.2) where the analogue of (2.97) is given

by the Euler–Lagrange equations (2.77). Deﬁning a parameter ηi

ηi:= µi+

νi



ρj(2.106)

42 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

(2.77) simpliﬁes to

Tln ρi

1−ρi−2ρi−ηi=0.(2.107)

Since (2.107) has exactly the same form as (2.97) for the bulk system, the same

reasoning can be applied to conclude

lim

T→0ρi=ρ0,i =









i=1,...,N(2.108)

meaning that all lattice sites are either ﬁlled or empty irrespective of the value

of ηi.Thus,atT= 0 densities in (2.75) can be replaced by occupation numbers

Ω0=−



i=1

νi



s0,is0,j −



i=1

µis0,i .(2.109)

Again, this result is in accordance with the exact grand potential at vanishing

temperature [see (2.21)]. The set of stable solutions of (2.109) or, equivalently

(2.21), constitute the “morphologies” at T= 0. Thus, we have demonstrated

that in the limit T= 0, the mean–ﬁeld treatment becomes exact. This is also

reﬂected by (2.23) because ﬂuctuations vanish in that limit.

2.4 Phase behavior for T >0

The mean–ﬁeld approximation of the grand canonical potential introduced in the

preceding section is now used to investigate the phase diagram of the conﬁned

lattice gas for T>0.

2.4.1 Grand potential curves

By solving (2.77) numerically one obtains a set of equilibrium densities

ρα:= {ρα

1,ρ

2,...,ρ

N}(2.110)

where the superscript αis introduced to indicate that for a given Tand µseveral

solutions of (2.77) may exist. Using (2.75) we calculate values Ωα=Ω[ρα,T,µ]

under isothermal conditions. Similar to (2.49)

∂Ωα(µ)

∂µ T>0

=−Nα(µ)≤0 (2.111)

2.4. PHASE BEHAVIOR FOR T >043

where, unlike the case T=0,Nαis no longer independent of µ.Thus,forT>0

the grand–potential curves in ﬁgure 2.9 are characterized by a nonvanishing

curvature. A measure of this curvature is the isothermal compressibility κα

associated with ρα,thatis

∂2Ωα(µ)

∂µ2=−(Nα)2

Nκα

T<0 (2.112)

where the inequality is a consequence of thermodynamic stability [78], disregard-

ing explicitly the “vacuum” (i.e., N=0,κ

T= 0). From the above considera-

tion one realizes that the grand potential is a monotonically decaying function

with slope −Nα(µ) and a curvature determined by the isothermal compressibil-

ity. Coexisting phases are identiﬁed by intersections of grand–potential curves

having the lowest value of Ω as detailed in section 2.2.3. Unfortunately, ﬁnding

these intersections is complicated even in relatively simple systems since one has

to make sure that one takes into account all solutions of (2.77). Appendix A

presents a strategy to overcome this diﬃculty.

To investigate the eﬀects of varying Twe consider the case fw =0.0and

fs =1.0, where the interaction of a molecule with the “weak” stripe is purely

repulsive (i.e., hard substrate) and the interaction with the “strong” stripe is

characterized by fs ≡ﬀ. Thus, for molecules located at sites in the planes

z=1andz=nz, the interaction with the “strong” stripes exactly compensates

the interaction with the nearest neighbor that has been lost on account of the

creation of the “surfaces” of the hard–substrate module (section 2.2.2). Fig-

ure 2.9 presents grand potential curves at diﬀerent temperatures. The ﬁgures

show that {Ωα(µ)}are only slightly bent. Curvature increases with increas-

ingtemperature,indicatingalargerκTat higher T[see (2.112)]. Moreover,

with increasing temperature one observes a pronounced shift to lower values

of the grand potential. In addition, the number of stable phases and the to-

tal number of solutions of (2.77) varies with Tas well. Figure 2.9(a) shows

that for T=0atriplepointµgbl

tr =−3 exists at which gas, liquid and bridge

phases coexist. Following the evolution of Ωα(µ) one notices from the plot

in ﬁgure 2.9(b) that for T=0.6 the triple point has given way to a narrow

one–phase region −3.004 <µ<−2.998 in which bridge phases are thermo-

dynamically stable. Hence, for {(T,µ)|T=0.6,µ<−3.004}gas phases are

thermodynamically stable whereas this is the case for liquid phases over the

44 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL



-8

-4

-3.05 -3 -2.95

(a)



-4

-3.05 -3 -2.95

(b)



Figure 2.9: The grand potential Ωα(µ)/nyversus chemical potential µfor various

morphologies α= g (gas), d (droplet), b (bridge), v (vesicle), and l (liquid) indicated

in the ﬁgure. In all cases, nw=ns=10(nx= 20), nz= 10, fs =1.0andfw =0.0.

(a) T= 0 (for comparison), (b) T=0.6, (c) T=0.9, (d) T=1.2.

2.4. PHASE BEHAVIOR FOR T >045



-20

-16

-12

-8

-3.05 -3 -2.95

(c)



-36

-32

-28

-3.05 -3 -2.95

(d)



Figure 2.9: (Continued)

46 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

range {(T,µ)|T=0.6,µ>−2.998}.

This picture changes substantially for T=0.9 [ﬁgure 2.9(c) ]. Now the

gas phases are stable for thermodynamic states {(T,µ)|T=0.9,µ<−3.044}.

Over the range {(T,µ)|T=0.9,−3.044 <µ<−3.010}droplet morphologies

(Md={0,0,0,1}for T= 0) represent thermodynamically stable phases. At

µgd

x=−3.044 gas and droplet phases coexist. The region of bridge phases,

{(T,µ)|T=0.9,−3.010 <µ<−2.990}has considerably widened compared with

T=0.6 [ﬁgure 2.9(b)]. Bridge and droplet phases coexist at µdb

x=−3.010

whereas bridge and vesicle phases (Mv={1,1,0,1}for T=0)coexistat

µbv

x=−2.990. Vesicle phases are thermodynamically stable over the range

{(T,µ)|T=0.9,−2.990 <µ<−2.956}, eventually coexisting with liquid at

µvl

x=−2.956, which is then stable for all larger chemical potentials.

For the highest temperature T=1.2 one deduces from ﬁgure 2.9(d) that only

gas, bridge and liquid phases are thermodynamically stable over the respective

ranges {(T,µ)|T=1.2,µ<−3.013},{(T,µ)|T=1.2,−3.013 <µ<−2.988},

and {(T,µ)|T=1.2,µ>−2.988}where µgb

x=−3.013 and µbl

x=−2.988.

2.4.2 Phase diagrams

From the consideration of grand–potential curves in section 2.4.1 we are now in

a position to determine lines of discontinuous phase transitions (i.e., coexistence

lines) through the analogue of (2.51), that is

Ωαµαβ

x(T)=Ω

βµαβ

x(T)(2.113)

where µαβ

x(T) stands for the coexistence line, that is the set of values of the

chemical potential at which phases αand βcoexist at a given temperature T.

The phase diagram can then be represented by

µx(T)=,

α,β

µαβ

x(T),∀α=β(2.114)

that is, the union of all coexistence lines between all pairs of phases. Thus, one

can perceive µx(T) as a “web” of coexistence lines, whose structure depends

implicitly on system parameters fw,fs,nw,ns,andnz. As these parameters

vary, the web evolves. The next two paragraphs give an impression of the

mean-ﬁeld phase diagrams of a conﬁned lattice gas. We especially focus on the

2.4. PHASE BEHAVIOR FOR T >047

impact of the external potential of the prototype, which is characterized by the

strength of attraction of the two diﬀerent substrate parts denoted by fw and

fs, respectively. We split the discussion into two parts. The ﬁrst one focuses

on the inﬂuence of fs on the phase diagram while holding fw ﬁxed; the second

one discusses the variation of fw at ﬁxed fs.

The impact of fs Figure 2.10 illustrates the impact of increasing attraction

between the lattice gas and the “strong” stripe. For fs =0.5, ﬁgure 2.10(a)

reveals a tiny one–phase region for bridge phases indicated by the bifurcation

in µx(T)atT≃1.375. The coexistence lines involving bridge phases terminate

at the respective critical temperatures Tgb

c≃1.461 and Tbl

c≃1.440. One

also notes a bifurcation in µx(T)atT≃0.980, indicating the existence of a

vesicle phase. The vesicle–liquid coexistence line ends at its critical temperature

Tvl

c≃1.005.

Increasing the ﬂuid–substrate interaction to fs =1.0 [see ﬁgure 2.10(b)]

causes µx(T) to move down to lower chemical potentials compared with the plot

in ﬁgure 2.10(a). The gas–bridge–liquid triple point has also shifted all the way

to T= 0 [see also ﬁgure 2.9(a) ] so that the one–phase region of bridge phases

is now much wider compared with the case fs =0.5. At the same time Tvl

has not changed at all but the coexistence line µvl

x(T) is longer now. However,

a new bifurcation appears at T≃0.815, corresponding to the appearance of

a droplet phase that can coexist with gas or bridge phases. We note that the

phase diagram is symmetric with respect to µc=−3, as it must be on account

of the symmetry of the grand potential.

For even larger fs =1.1 one sees from ﬁgure 2.10(c) that the gas–bridge–

liquid triple point vanishes, that is even for T= 0 a range of chemical potentials

exists over which bridges are the thermodynamically stable phases. This eﬀect

results from a lowering of the temperature at which the “droplet” bifurcation

occurs, along with a shift of µgb

x(T)andµgd

x(T) toward lower values of µfor

T<T

c. As before, however, all four critical temperatures remain unaltered.

A slight further increase of the strength of the ﬂuid–substrate attraction to

fs =1.2 eventually causes µgd

x(T) to become detached from the other coex-

istence lines as the plot in ﬁgure 2.10(d) clearly shows. The remainder of the

phase diagram appears to be unaﬀected by the increase of fs.Consequently,

48 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

one ﬁnds three chemical potentials for T= 0 at which pairs of phases (i.e., gas–

droplet, droplet–bridge, bridge–liquid) coexist. The one–phase region of droplet

phases is already quite large. It increases further if the interaction of the lattice

gas with the “strong” stripe is increased to fs =1.5 [see ﬁgure 2.10(e)]. For this

value of fs one notices the appearance of a very short additional coexistence line

beginning at Tbd2

tr ≃1.001 and µ≃−3.027 with negative slope. An inspection

of the local densities indicates that this coexistence line reﬂects layering tran-

sitions between droplet phases and new bilayer droplet phases (2) localized at

the “strong” stripe. The layering transitions disappear at a critical temperature

Td2

c≃1.035.

The impact of fw Plots in ﬁgure 2.11 illustrate variations of µx(T) with

increasing interaction between lattice gas and the “weak” stripe. Comparing

ﬁgure 2.11(a) with ﬁgure 2.11(b) one notices that µgd

x(T) starting at µ=−3.400

for T= 0 remains unaﬀected. However, the vesicle and layered phases, both

visible in ﬁgure 2.11(a), disappear. At the same time the triple–point temper-

ature corresponding to droplet–bridge–liquid coexistence is signiﬁcantly raised

to Tdbl

tr ≃1.310.

As fw increases further to 1.0, plots in ﬁgure 2.11(c) show that µgd

x(T)is

still unaﬀected. On the other hand, the bifurcation appearing in ﬁgure 2.11(b)

apparently shifts to a temperature of about 0.817. However, an inspection of the

phase diagrams in the equivalent T–ρrepresentation in ﬁgure 2.12(a) and ﬁgure

2.12(b) shows that the bifurcation temperature is actually not associated with

bridge phases, which have already become metastable for this fw [see ﬁgure

2.12(b)]. Instead the coexistence line branching oﬀ at Tdml

tr ≃0.817 corresponds

to a line of discontinuous transitions between droplet phases and monolayer

(m) phases adsorbed on the entire substrate [see ﬁgure 2.13(a), ﬁgure 2.13(b)]

and may thus be regarded as a diﬀerent type of layering transition triggered

predominantly by the “weak” part of the substrate.

If fw =1.5 the decorated substrate of the prototype degenerates to a chem-

ically homogeneous one wetted by the lattice gas. In this case µx(T) consists

of µgm

x(T) ending at Tgm

c≃1.018 and µml

x(T) terminating at its respective

critical temperature Tml

c≃1.452 <Tbulk

c=3

2on account of conﬁnement where

we use superscript “m” to indicate that the droplet phase has been replaced by

2.4. PHASE BEHAVIOR FOR T >049

the monolayer as indicated by the representative plot of local density in ﬁgure

2.13(b). However, in the present case the local density in this monolayer no

longer depends on x.

The phase diagrams presented in ﬁgure 2.11 exhibit yet another interesting

characteristic. From that ﬁgure one immediately realizes that, for example,

µgd

x(T) is almost independent of the special choice of fw.Thisisalsothecase

at T= 0. From the equation for the intersection point of the grand–potential

curves of the gas and the droplet morphologies [see (2.53)]

µgd(T=0)=−2

fs +1

nz−ν

2(2.115)

it is obvious that µgd(T= 0) does not depend on fw. The origin of this

independence is, that lattice sites exposed to the weak parts of the ﬂuid–wall

potential i.e., (fw) are not involved in the gas–droplet phase transition [see

ﬁgure 2.6(a),(b)].

50 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

-3.6

-3.4

-3.2

-3

0 0.5 1 1.5



(a)

gl gv

vl gb

-3.6

-3.4

-3.2

-3

0 0.5 1 1.5



(b)

gd db

bvvl

-3.6

-3.4

-3.2

-3

0 0.5 1 1.5



(c)

bl vl bv

Figure 2.10: Phase diagrams in µ–Trepresentation for nw=ns=10(nx= 20),

nz=10andfw =0.0(a)fs =0.5, (b) fs =1.0, (c) fs =1.1, (d) fs =1.2, (e)

fs =1.5. (



) analytical solution for T=0.

2.4. PHASE BEHAVIOR FOR T >051

-3.6

-3.4

-3.2

-3

0 0.5 1 1.5



(d)

bl vl bv

-3.6

-3.4

-3.2

-3

0 0.5 1 1.5



(e)

bl vl bv

Figure 2.10: (Continued)

52 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

-3.6

-3.4

-3.2

-3

00.5 11.5



(a)

bl vl bv

-3.6

-3.4

-3.2

-3

00.5 11.5



(b)

Figure 2.11: As ﬁgure 2.10, but for fs =1.5. (a) fw =0.0, (b) fw =0.5, (c) fw =1.0,

(d) fw =1.5. Open and ﬁlled circles in ﬁgure 2.11(c) signify thermodynamic states

for which local densities are plotted in ﬁgure 2.13(a) and ﬁgure 2.13(b), respectively.

(



) analytical solution for T=0.

2.4. PHASE BEHAVIOR FOR T >053

-3.6

-3.4

-3.2

-3

00.5 11.5



(c)

-3.6

-3.4

-3.2

-3

00.5 11.5



(d)

Figure 2.11: (Continued)

54 CHAPTER 2. THE MEAN-FIELD LATTICE-GAS MODEL

0.5

1.5

0 0.5 1



(a)

0.5

1.5

0 0.5 1



(b)

Figure 2.12: Phase diagrams in T–ρrepresentation for fs =1.5. (a) fw =0.0, (b)

fw =1.0 corresponding to ﬁgure 2.11(a) and ﬁgure 2.11(c), respectively. (



)analytical

solution for T= 0. Note that in the immediate vicinity of the critical points the phase

diagram could not be determined because of failure of numerical method to converge

(see appendix A).

2.4. PHASE BEHAVIOR FOR T >055

10 20 30 40 0

0.5



(a)

10 20 30 40 0

0.5



(b)

Figure 2.13: The local density ρ(x, z) of lattice gases between prototypal chemically

decorated substrates (see ﬁgure 2.1) where fw =1.0, fs =1.5, and T=0.9[seeﬁgure

2.11(c)]. (a) µ=−3.30, (b) µ=−3.11.

Chapter 3

The continuous model

In the preceding chapter we introduced the lattice gas as a convenient model

to determine phase diagrams of ﬂuids conﬁned between chemically decorated

surfaces at limited computational expense. However, the lattice–gas has sev-

eral shortcomings. Its main disadvantage is perhaps the restriction of ﬂuid

molecules to discrete lattice sites. Thus, the maximum number of neighbors

of a molecule is constant and the distance between nearest neighbors is ﬁxed

(and determined by the lattice structure). This is fairly unrealistic as far as real

ﬂuids are concerned, in which molecules move continuously in space. Moreover,

we neglect correlations altogether within the mean–ﬁeld approach employed in

section 2.3.1. However, the latter may be abandoned in favor of more sophisti-

cated treatments culminating eventually in more complex free–energy function-

als than the one given in (2.75). On the other hand, the apparent simplicity

of the present lattice–gas model oﬀers the possibility of calculating phase di-

agrams with little computational eﬀort. This is particularly useful because of

the multidimensional parameter space {T,µ,fs,

fw,n

s,n

w,n

z}governing the

present model. However, to gain some insight into the impact of discreteness

and the mean–ﬁeld approximation it seems desirable to compare the lattice–gas

results with those obtained for a continuous model incorporating intermolecular

correlations.

3.1. THE CONTINUOUS MODEL 57



«×

Figure 3.1: A schematic diagram of a simple ﬂuid conﬁned by a chemically hetero-

geneous model pore. Fluid molecules (gray spheres) are spherically symmetric. Each

substrate consists of a sequence of crystallographic planes separated by a distance δ

along the z-axis and comprises slabs of wall atoms interacting diﬀerently with ﬂuid

molecules. “Strong” and “weak” interactions are indicated by gray and white surface

areas, respectively. The surface planes of the two opposite substrates are separated by

adistanceszand are misaligned by αsx.

3.1 The continuous model

The continuous analogue of the lattice gas model (see section 2.1.2) consists of

a ﬂuid ﬁlm composed of spherically symmetric molecules sandwiched between

the surfaces of two solid substrates [79] (see ﬁgure 3.1). The substrate surfaces

are planar, parallel to the x–y–plane and separated by a distance szalong the

z–axis of the coordinate system. We assume all interparticle interactions (ﬂuid-

ﬂuid and ﬂuid-substrate) to exhibit a distance dependence described by the

Lennard-Jones LJ(12,6) potential

u(r)=4 σ

r12 −σ

r6!(3.1)

where is the well depth, σthe molecular “diameter”, and rthe distance be-

tween the centers of a pair of particles (ﬂuid molecules or substrate atoms).

58 CHAPTER 3. THE CONTINUOUS MODEL

Since the substrates in our model are chemically heterogenous they are made

of atoms of two chemical species diﬀering in their respective interaction strength

with the ﬁlm molecules. Thus, in (3.1) ≡fs (u≡ufs) to indicate that the

interaction of ﬁlm molecules with substrate atoms is “strong” and ≡fw

(u≡ufw) if the interaction is “weak”, respectively. We assume substrate atoms

and ﬁlm molecules to be of equal size, that is σ=σﬀ=σfs =σfw.

Each substrate comprises alternating slabs of atoms of the two species (see

ﬁgure 3.1). The “strong” and “weak” slabs have widths dsand dw, respectively,

in the x–direction, are semiinﬁnite in the z–direction, and inﬁnite in the y–

direction. The semiinﬁnite character in the z–direction is accounted for by an

inﬁnite number of crystallographic planes in the half space sz/2≤|z|<∞

separated by a distance δ(see ﬁgure 3.1). The substrates are thus periodic in

x–direction of period sx=ds+dwand homogeneous in the y–direction along

lines x=const,z= const. Substrate atoms are assumed to occupy the sites of

the (100) plane of the face-centered-cubic (fcc) lattice. The lattice constant lis

taken to be the same for both chemical species.

Since we are concerned with the eﬀect of chemical heterogeneity at the

nanoscale on the behavior of conﬁned fluids, we expect details of the atomic

structure not to matter greatly. Therefore, we adopt a mean–ﬁeld representa-

tion of the ﬂuid–substrate interaction [79], which we obtain by averaging ufs and

ufw over positions of substrate atoms in the x–y–plane. The resulting coarse–

grained potential can be expressed as

Φ[k](x, z)=nA

∞



m=−∞

∞



m=0

∞

−∞

dy









−ds/2+msx

−sx/2+msx

dxufw (|r−r|) (3.2)

ds/2+msx

−ds/2+msx

dxufs (|r−r|)+

sx/2+msx

ds/2+msx

dxufw (|r−r|)









where nA=2/l2is the areal density of the (100) plane of the fcc lattice and r

denotes the position of a ﬁlm molecule. The position of a wall atom is given by

r0=[x,y,z+(−1)k(sz/2+mδ)], where k=1,2 refers to lower and upper

substrate, respectively. Thus, the sum over maccounts for contributions to

Φ from successive crystallographic planes, while the sum over mrepresents the

inﬁnite (periodic) extent of the substrates in ±x–direction.

3.1. THE CONTINUOUS MODEL 59

Since the substrates comprise periodic heterogeneities in the x–direction we

introduce αsxas a quantitative measure of displacement of the upper substrate

relative to the lower. The dimensionless registry parameter 0 ≤α≤1/2isde-

ﬁned such that for α= 0 the substrates are perfectly aligned, that is chemically

identical parts of both substrates are exactly opposite each other; for α=1/2

the misalignment is maximum. Introducing the transformations

x−→ x =x−(k−1)αsx−x(3.3)

y−→ y =y−y

z−→ z =z−z+(−1)ksz

2+mδ

and interchanging the order of integration, we can evaluate the integrals on

the right side of (3.2) analytically [79]. After lengthy algebraic manipulations

detailed in [79], one gets

-a

dx

∞

−∞

dyu(|r−r0|)=21σ

32 I3(x,z;ds,s

x,s

z)|x =x−b

x =x−a(3.4)

−σI4(x,z;ds,s

x,s

z)|x =x−b

x =x−a

where aand brepresent the integration limits in (3.2). In (3.4)

I3:= xσ10

9(z)2√R9 1+8

7S+48

35S2+64

35S3+128

35 S4!,(3.5)

I4:= xσ4

3(z)2√R3[1 + 2S],(3.6)

R:= (x)2+(z)2,(3.7)

and

S:= R

(z)2.(3.8)

Deﬁning the function

∆(x,z):=21

32I3(x,z)−I4(x,z) (3.9)

we can cast the ﬂuid–substrate potential in ﬁnal form as

Φ[k](x, z)= −3π

2nAσ

∞



m=−∞

∞



m=0

(3.10)

(fw −fs){∆[x

u(x, ds),z]−∆[x

l(x, ds),z]}

+fw {∆[x

u(x, sx),z]−∆[x

l(x, sx),z]}

60 CHAPTER 3. THE CONTINUOUS MODEL



Figure 3.2: A schematic diagram of the ﬂuid ﬁlm conﬁned by chemically striped

substrates. The system of interest, i.e., the lamella is bounded by black lines.



zis

also shown.(See also ﬁgure 3.1.)

where we simpliﬁed notation by introducing

x

l(x, d):=x−(k−1)αsx−1

2d+msx(3.11)

x

u(x, d):=x−(k−1)αsx+1

2d+msx(3.12)

and d=ds,s

x, respectively.

3.2 Thermodynamics

Since equilibrium properties of conﬁned ﬂuids are the focal point of this the-

sis, (classical) thermodynamics provides the theoretical framework. To apply

thermodynamics we have to distinguish between the thermodynamical system

of interest and its surroundings (see ﬁgure 3.2). In addition we need to specify

their interactions. Here we regard the system to be a ﬁnite piece (lamella) of the

(inﬁnite) ﬁlm having dimensions sx×sy×sz. The environment thus comprises

the remainder of the inﬁnite ﬁlm plus the substrates. The system is bounded

in the z–direction by two solid surfaces located at z=±sz/2 and in lateral

directions by planes at x=±sx/2andy=±sy/2, respectively, since we place

the origin of the coordinate system at the center of the lamella (see ﬁgure 3.1).

By moving the surface planes, which may viewed as imaginary pistons, the

ﬂuid lamella can do work on its surroundings and vice versa. In general me-

3.2. THERMODYNAMICS 61

chanical work dWmech done by the system on its surroundings is given by

−dWmech =

j·T·d∆j.(3.13)

where the sum jin (3.13) extends over planes jhaving area Aj=|Aj|and a

normal pointing in the j–direction (see ﬁgure 3.2). In 3.13, Tis the stress tensor

(see chapter 13 in [80]), and d∆jaccounts for the (inﬁnitesimal) displacement of

plane j. In the absence of shear strains, Tis diagonal and can be representd by

a3×3 matrix. If on the other hand, a shear strain is applied in the x–direction,

say,

−dWmech =TxxAxdsx+TyyAydsy+TzzAzdsz+TzxAzdαsx.(3.14)

where we assume the solid substrates to be rigid, so that they cannot be com-

pressed or sheared. In the y–direction a shear strain cannot be applied, because

the ﬂuid–substrate potential is translationally invariant in that direction [see

(3.10)].

Therefore reversible transformations of the lamella are govened by Gibbs’

fundamental equation

dU =TdS+µdN −dWmech .(3.15)

All variables in (3.15) have their usual meaning: Uis the internal energy, T

is temperature, Sentropy, µchemical potential, and Nthe number of ﬂuid

molecules. Substituting dWmech in (3.15) by (3.14) we rewrite Gibbs’ funda-

mental equation (3.15) as

dU =TdS+µdN +TxxAxdsx+TyyAydsy+TzzAzdsz+TzxAzdαsx(3.16)

where Udepends on the set {S, N, sx,s

y,s

z,αs

x}of natural variables.

However, other thermodynamic potentials will prove to be convenient for our

purposes, because they depend on diﬀerent sets of natural variables. A straight-

forward way of deriving these potentials is to start from a closed expression for

Uand replace its natural variables by successive Legendre transformations [81].

A closed form of Uis always available if the system in question is homogeneous

in at least in one spatial dimension. Take as an example a bulk ﬂuid which is

homogeneous in all three spatial dimensions. Therefore,

U(λS, λN, λsx,λs

y,λs

z)=λU (S, N, sx,s

y,s

z)λ∈R,(3.17)

62 CHAPTER 3. THE CONTINUOUS MODEL

that is Uis a homogeneous function of degree one of all its extensive variables.

For the system of interest (3.17) is no longer valid because the conﬁned ﬂuid is

subject to an external potential whose dependence on xand zcauses the ﬂuid

to be inhomogeneous in general in these two directions [see (3.10) and ﬁgure

3.1]. Therefore, we replace (3.17) by

U(λS, λN, λsy)=λU (S, N, sy,)λ∈R(3.18)

where we have dropped sx,szand αsxkeeping in mind that these strains are

supposed to remain ﬁxed. Equation (3.18) expresses the homogeneity of the

ﬂuid (i.e., the translational invariance of system properties) in the y–direction.

Because Uin (3.18) is a homogeneous function of degree one in its remaining

extensive variables, we apply Euler’s theorem [81] to obtain a Gibbs-Duhem

equation

U(S, N, sy)=∂U

∂S S+∂U

∂N N+∂U

∂sy

sy=TS+µN +TyyAysy(3.19)

where we have identiﬁed the partial diﬀerentials with the help of (3.16). The

diﬀerential form of the Gibbs-Duhem equation is obtained by inserting (3.16)

into the exact diﬀerential of (3.19), that is

0=−SdT −Ndµ+AysydTyy +TxxAxdsx+TzzAzdsz+TzxAzdαsx.(3.20)

Equation (3.20) expresses the fact that of the six variables {T,µ,Tyy,s

x,s

z,αs

only ﬁve may be varied independently in a reversible transformation of the

conﬁned ﬂuid.

From (3.19) two other useful potentials can be derived. For example, the

Legendre Transformation

Ω=U−TS−µN =TyyAysy(3.21)

deﬁnes the grand potential Ω, whose exact diﬀerential is given by

dΩ(T,µ,sz,s

y,s

z,αs

x) = (3.22)

−SdT −Ndµ+TxxAxdsx+TyyAydsy+TzzAzdsz+TzxAzdαsx

where (3.16) has also been used. From (3.21)

Ψ=Ω−TzzAzsz=U−TS−µN −TzzAzsz=(Tyy −Tzz)Aysy.(3.23)

3.3. STATISTICAL THERMODYNAMICS 63

is introduced as the grand mixed isostress-isostrain potential, where from (3.23)

and (3.22)

dΨ(T,µ,sx,s

y,T

zz,αs

x) = (3.24)

−SdT −Ndµ+TxxAxdsx+TyyAydsy+AzszdTzz +TzxAzdαsx

follows. This latter potential will turn out to be useful in section 3.3 where

conﬁned ﬂuids are exposed to shear strains under ﬁxed normal load, i.e. ﬁxed

Tzz.

3.3 Statistical Thermodynamics

Thermodynamic considerations detailed in the preceding paragraph led to closed

expressions for both grand (3.21) and grand mixed isostress–isostrain potentials

(3.23), that is expressions for Ω and Ψ in terms of stresses and conjugate strains.

However, in order to use (3.21) or (3.23) molecular expressions for Tyy and Tzz

are needed. That is, we are seeking expressions for stress tensor components in

terms of the interaction potentials given in (3.1) and (3.10). This can be done

within the framework of statistical thermodynamics which combines the notion

of molecules and their properties with the principles of thermodynamics [76].

Consider the ﬂuid lamella, introduced in section 3.2, at ﬁxed Tand µ.We

assume the “shape” of the lamella to be ﬁxed, that is sx,sy,andαsx(see

ﬁgure 3.1) remain constant. Let us furthermore consider the lamella to be

under ﬁxed normal load, i.e. we maintain Tzz in addition to the other vari-

ables. The thermodynamic state of the lamella can thus be speciﬁed by the set

{T,µ,sx,s

y,T

zz,αs

x}. In other words, the lamella is thermally and materially

coupled to its environment and exchanges work due to compression (expansion)

in the normal z–direction against a constant (external) load Tzz. This implies

that the number of molecules Nas well as the distance szbetween the conﬁning

walls may vary during the course of evolution of the lamella.

64 CHAPTER 3. THE CONTINUOUS MODEL

3.3.1 Partition function of the grand mixed isostress-iso-

strain ensemble

From a molecular perspective the lamella is assumed to be in a microstate spec-

iﬁed by variables {sz,N,j}where jaccounts for a collection of suitable quan-

tum numbers necessary to specify the discrete microstate of energy E(sz,N,j)

of a system containing Nmolecules conﬁned by substrates separated by dis-

tance sz. Since we are exclusively concerned with thermodynamic equilib-

rium, E(sz,N,j) can be obtained (at least in principle) by solving the (time-

independent) Schr¨odinger equation [78], such that |jis a stationary eigenstate

of the N–molecule system. For suﬃciently large None can envisage a large

number of microstates {sz,N,j}having energy E(sz,N,j) consistent with the

macrostate of the lamella determined by {T,µ,sx,s

y,T

zz,αs

x}.Considernow

a large number of N(virtual) replicas of the lamella each of which may be in

one of the available microstates. The ensemble of replicas is perfectly isolated

such that its total energy

Et=n(sz,N,j)E(sz,N,j) (3.25)

is ﬁxed. In (3.25), stands for szNjand n(sz,N,j)isthenumberof

replicas in microstate jcomprising Nmolecules between substrates separated

by sz, that is the occupation number of this microstate. Perfect isolation of the

super-system of replicas (i.e., the ensemble) furthermore implies

N=n(sz,N,j) (3.26)

Nt=n(sz,N,j)N(3.27)

Azsz,t=n(sz,N,j)Azsz(3.28)

where Ntand Azsz,tare the total number of molecules and total volume of

the ensemble, respectively. From the above it is clear that there are many sets

of occupation numbers (i.e., distributions) consistent with the four constraints

(3.25)-(3.28). A particular distribution {n}can be realized

W(n)= N!

sz.

n(sz,N,j)! (3.29)

times. On account of perfect isolation of the ensemble the principle of equal a

prioi probabilities applies to all microstates [82]. Thus, we may deﬁne a mean

3.3. STATISTICAL THERMODYNAMICS 65

occupation number of a particular microstate as

nsz,N,j =

{n}

n(sz,N,j)W(n)



{n}

W(n)(3.30)

where the sums run over all distributions {n}. Therefore, the probability of

ﬁnding an arbitrarily selected replica in a speciﬁc microstate is given by

Psz,N,j =nsz,N,j

N=1

N

{n}

n(sz,N,j)W(n)



{n}

W(n).(3.31)

To proceed beyond (3.31) we invoke as a key assumption the existence of a most

probable distribution n∗overwhelming all others such that the maximum-term

method [83] applies and (3.31) can be recast as

Psz,N,j =n∗(sz,N,j)

N.(3.32)

The assumption that no term except n∗contributes to the sum in (3.31) is not

per se justiﬁed. It may, however, be shown a posteriori that the distribution of

naround n∗is essentially given by a δ-function provided N−→∞[83]. This

δ-function-like character of the distribution of ncan, on the other hand, be

demonstrated independently invoking function-theoretical arguments put for-

ward by Darwin and Fowler [84]. In this latter case no assumption concerning

existence of n∗is required.

Within the framework of the present approach, n∗is the distribution of mi-

crostates maximizing W(n), or equivalently ln W(n). By introducing a set of

Lagrangian multipliers {λi}we account for the constraints (3.25)-(3.28). Max-

imizing ln W(n) subject to the constraints (3.25)-(3.28) we eventually obtain

n∗

sz,N,j =exp(−λ0)exp(−λ1Azsz)exp(−λ2N)exp[−λ3E(sz,N,j)].(3.33)

where the set {λi}remains yet to be determined. Strictly speaking n∗is the

distribution making Wextreme. However, on physical grounds it can be demon-

strated that this extremum is indeed a maximum. Summing (3.33) over all

microstates and utilizing (3.26) we can immediately replace λ0so that (3.32)

becomes

Psz,N,j =1

χexp[−λ1Azsz−λ2N−λ3E(sz,N,j)].(3.34)

66 CHAPTER 3. THE CONTINUOUS MODEL

where

χ:= exp[−λ1Azsz−λ2N−λ3E(sz,N,j)] (3.35)

is the partition function of the grand mixed isostress-isostrain ensemble. Fol-

lowing Schoen [72] one may calculate the exact diﬀerential

dE(sz,N,j)=[E(sz,N,j)dPsz,N,j +Psz,N,jdE(sz,N,j)].(3.36)

with aid of (3.34) and (3.35). After a sequence of tedious but straightforward

manipulations detailed in [72] one may eventually compare (3.36) with Gibbs’

fundamental equation (3.16) to arrive at

dU ≡dE(sz,N,j)=1

λ3

dS +λ1

λ3

Azdsz+λ2

λ3

dN+dW (3.37)

where dW is the work required to alter the shape of the lamella (see above).

Comparison with (3.16) also permits one to deduce (see [83])

λ3≡β=1

kBT(3.38)

where kBis Boltzmann’s constant,

λ2=−βµ (3.39)

and

λ1=−βTzz (3.40)

With the help of these identiﬁcation we obtain from (3.35) [72]

−kBTln χ=U−TS−TzzAzsz−µN(3.41)

so that by comparison with the thermodynamic equation (3.23)

−kBTln χ= Ψ (3.42)

is derived as the desired expression relating the thermodynamic potential Ψ of

the lamella to the properties of its individual molecules via χ.

3.3.2 The classical limit

So far the treatment of the grand mixed isostress-isostrain ensemble is quantum–

mechanically exact. However, here we are concerned with classical ﬂuids, that

3.3. STATISTICAL THERMODYNAMICS 67

is ﬂuids in a temperature range such that typical intermolecular separations rij

are large compared to the thermal de Broglie wavelength Λ = h2/2πmkBT1/2

where his Planck’s constant and mis the mass of a ﬂuid molecule. Under these

conditions one may employ Kirkwood’s method [78] to show that in (3.35)



exp (−βE(sz,N,j)) Λ/rij →0

−−−−−−→1

h3NN!--exp −βH(pN,qN)dpNdqN

(3.43)

where



i=1

2m+UrN,N,s

z(3.44)

is the classical Hamiltonian of an N–particle ﬂuid and piis the momentum of

the i-th molecule. Because of (3.44), integration over momentum subspace in

(3.43) can be carried out analytically to give [see (3.35)]

χ=

sz

exp (βTzzAzsz)exp(βµN)Q(sz,N) (3.45)

where

Q(sz,N)=Z(sz,N)

N!Λ3N(3.46)

and

Z(sz,N)=-

drNexp −βU rN,N,s

z (3.47)

is the conﬁguration integral. The total potential energy UrN,N,s

zof the

system of interest can be expressed as

UrN,N,s

z=UFF +UFS .(3.48)

In (3.48), UFF represents the ﬂuid–ﬂuid contribution of the potential energy

given by

UFF =1



i=1



j=1=i

uﬀ(rij ) (3.49)

where

rij =|rij|=/(xi−xj)2+(yi−yj)2+(zi−zj)2(3.50)

is the distance between molecules iand j. The ﬂuid–substrate contribution can

be expressed as [see (3.10)]

UFS =U[1]

FS +U[2]

FS =



k=1



i=1

Φ[k](xi,z

i).(3.51)

68 CHAPTER 3. THE CONTINUOUS MODEL

A macroscopic (thermodynamic) quantity Oof the system of interest can

now be identiﬁed with the ensemble average Oof its microscopic analogue

OrN,N,s

z. In the grand mixed isostress-isostrain ensemble Ois given by

[see (3.34),(3.35)]

O=O=1

χ

sz

exp (βTzzAzsz)exp(βµN) (3.52)

×1

N!Λ3N-

drNOrN,N,s

zexp −βU rN,N,s

z

=: 

sz

N-

drNOrN,N,s

zf0rN,N,s

z

where f0rN,N,s

zis the probability density of the grand mixed isostress-

isostrain ensemble in the classical limit. Consider, for example, the simple case

OrN,N,s

z=N. From (3.52) O=Nfollows then immediately. For other

quantities of interest, for example, the stress tensor components O=Tzx and

O=Tyy, the functional form of OrN,N,s

zis more complicated as we shall

demonstrate in the following section.

3.3.3 Molecular expressions for stress tensor components

In this section we derive microscopic expressions for the three stress tensor

components Tzx,Tyy and Tzz and for the shear modulus c44 (assoziated with

Tzx) which are key quantities here. From (3.24) it is evident that the shear

stress Tzx is given by

Tzx =1

Az∂Ψ

∂(αsx)T,µ,sx,sy,Tzz

.(3.53)

Using (3.42)we can rewrite (3.53) as

Tzx =−1

βAzχ

∂χ

∂(αsx)(3.54)

=−1

βAzχ

exp(βTzzAzsz)

exp(βµN)1

N!Λ3N

∂Z(sz,N)

∂(αsx).

where we have also used (3.45)-(3.47). The partial derivative on the second line

of (3.54) can be rewritten more explicitly as

∂Z(sz,N)

∂(αsx)=∂

∂(αsx)

(

i=1

sy/2

−sy/2

dyi

sz/2

−sz/2

dzi

sx/2+(αsx)(zi/sz+1/2)

−sx/2+(αsx)(zi/sz+1/2)

dxi

(3.55)

×exp{−β[UFF {r}N+UFS {r}N]}

3.3. STATISTICAL THERMODYNAMICS 69

where we have assumed that the lamella is sheared homogeneously. To solve

(3.55) it is convenient to interchange the order of integration and diﬀerentia-

tion. Therefore both operations must be independent, which can be achieved

by introducing dimensionless variables [85]

x−→ +x=x

sx−α+z+1

2(3.56)

y−→ +y=y

(3.57)

z−→ +z=z

(3.58)

following Hill [83] and McQuarrie [82]. The distance between molecules iand j

in dimensionless variables is therefore given by [85]

rij =/s2

x(+xi−+xj+α(+zi−+zj))2+s2

y(+yi−+yj)2+s2

z(+zi−+zj)2.(3.59)

Using (3.56) to (3.58), (3.55) may be recast as

∂Z

∂(αsx)=−β(sxsysz)N

(

i=1

1/2

−1/2

d+xi

1/2

−1/2

d+yi

1/2

−1/2

d+zi(3.60)

×exp{−β[U{r}N]}∂

∂(αsx)UFF {r}N+UFS {r}N .

Replacing in (3.54), ∂Z/∂(αsx) by (3.60) and comparing the result with (3.52)

we ﬁnd that

Tzx =TFF

zx +TFS

zx (3.61)

where

TFF

zx := 1

Az0∂

∂(αsx)UFF {r}N1(3.62)

and

TFS

zx := 1

Az0∂

∂(αsx)UFS {r}N1.(3.63)

Using (3.49)

TFF

zx =1

Az)−1

2sz



i=1



j=1=i

u

FF(rij )xij zij

rij *(3.64)

=: 1

Az0−1

2sz

Wzx1

where the last equality deﬁnes Clausius’ virial Wzx [83]. Similarly, From (3.63)

and (3.51) one has (see [85] for details)

TFS

zx =−1

Az)1



k=1



i=1

f[k]

x(xi,z

i)zi+sz

2*(3.65)

70 CHAPTER 3. THE CONTINUOUS MODEL

where f[k]

x(xi,z

i)isthex–component of the force exerted by wall kon a ﬂuid

molecule ilocated at (xi,z

i). This force is given by [see (3.10) and appendix B]

f[k]

x(xi,z

i)=−∂Φ(xi,z

∂x .(3.66)

Thus, equations (3.64) and (3.65) provide the “virial” route to Tzx.

Alternatively, it is possible to diﬀerentiate Z(sz,N) in (3.54) directly [85],

by applying Leibniz’s rule for the diﬀerentiation of an integral [81]. Since the

integrations over xiand ziare independent of αsxone may rewrite (3.55)

∂Z

∂(αsx)=

(

i=1

sy/2

−sy/2

dyi

sz/2

−sz/2

dzi

∂

∂(αsx)g(3.67)

where

g:=

sx/2+(αsx)(zi/sz+1/2)

−sx/2+(αsx)(zi/sz+1/2)

dx1g1(3.68)

and

g1:=

(

i=2

sx/2+(αsx)(zi/sz+1/2)

−sx/2+(αsx)(zi/sz+1/2)

dxiexp[−βU {r}N].(3.69)

With aid of Leibniz’s rule one obtains

∂

∂(αsx)g=

sx/2+(αsx)(zi/sz+1/2)

−sx/2+(αsx)(zi/sz+1/2)

dx1

∂

∂(αsx)g1(3.70)

+g1 x1=+

2+αsxzi

2!zi

2

−g1 x1=−sx

2+αsxzi

2!zi

2.

Because the system is periodic in the x–direction of period sxit follows from

(3.69) that the last two terms in the integrand of (3.70) are identical and thus

cancel. Applying Leibniz’s rule Ntimes we eventually obtain

∂Z

∂(αsx)=

(

i=1

−sy

dyi

−sz

dzi

2+αsx(zi

sz+1

−sx

2+αsx(zi

sz+1

dxi(3.71)

×∂

∂(αsx)exp −βU rN .

Since the ﬂuid–ﬂuid interaction is independent of αsx,∂UFF/∂(αsx)≡0[see

(3.1), (3.50)]. From (3.3) it is furthermore clear that the interaction of ﬂuid

3.3. STATISTICAL THERMODYNAMICS 71

molecules with the lower wall [k=1 in (3.51)] is also independent of αsx,thatis

∂U[1]

FS/∂(αsx)≡0. Thus we arrive at

Tzx =−1

Az)N



i=1

f[2]

x*=: −1

Az$F[2]

x%.(3.72)

The last line of (3.72) gives Tzx as the x–component of the total force exerted

by the ﬂuid lamella on the walls. Consequently, we refer to it as the “force”

expression. “Force”[see (3.72)] and “virial” [see (3.61), (3.64),(3.65)] routes to

Tzx are useful since they provide a check on internal consistency of the simulation

results to be presented below.

By analogy with the derivation of the shear stress molecular expressions for

Tyy can also be derived. Since the ﬂuid–substrate potential Φ[k](x, z) (3.10) is

translationally invariant in y–direction the partial derivative of Φ[k](x, z) with

respect to yvanishes. Thus, Tyy is given by

Tyy =−1

βAz0N

sz1+1

Az)1

2sz



i=1



j=1=i

u

FF(rij )y2

rij *.(3.73)

A force expression for Tyy similar to (3.72) does not exist since the system

does not contain a solid substrate in the y–direction. Consequently, arguments

similar to the one between (3.69) and (3.70) do not exist. Similarly, one may

derive a molecular expression for Tyy in the grand canonical ensemble, that is

Tyy =−1

βV N+1

V)1



i=1



j=1=i

u

FF(rij )y2

rij *.(3.74)

In the grand canonical ensemble the wall distance szis a natural variable of Ω

[see (3.21)]. Therefore, a virial expression for Tzz can be derived

Tzz =−N

βV +1

V)1



i=1



j=1=i

u

FF(rij )z2

rij *(3.75)

−1

V)2



k=1



i=1

f[k]

z(xi,z

i)(zi+(−1)k(sz

2)*.

By arguments parallel to the ones above a force expression[see (3.69), (3.70)](see

Ref. [79])

Tzz =−1

2Az"$F[1]

z%−$F[2]

z%# (3.76)

also exists.

72 CHAPTER 3. THE CONTINUOUS MODEL

Another quantity of interest in the context of this work is the shear modulus

c44 := &∂2Ψ

∂(αsx)2'T,µ,sx,sy,Tzz

=∂Tzx

∂(αsx)T,µ,sx,sy,Tzz

(3.77)

in Voigt’s notation [80]. By a tedious but straightforward calculation parallel

to the one detailed in [86] one can show from (3.45),(3.54) and (3.77) that

Ac44 =−β $F[2]

2%−$F[2]

x%2!+)∂2U[2]

∂(αsx)2*.(3.78)

From (3.72), and (3.78) it is clear that

∂2U[2]

∂(αsx)2=−∂F[2]

∂(αsx)=−



i=1

∂f[2]

x(+xi,z

∂(αsx).(3.79)

The next section introduces Monte Carlo simulations as a method to calculate

the required ensemble averages.

3.4 Monte Carlo simulation

As we have seen in the preceding sections, thermodynamic properties of the

conﬁned ﬂuid lamella can be calculated from molecular expressions as ensemble

averages. This program requires knowledge of the probability density f0deﬁned

in (3.52). However, f0is accessible only if the conﬁguration integral (Z)isknown

apriori. To calculate it one could in principle introduce additional simplifying

assumptions or resort to numerical techniques instead. Unfortunately, a deeper

analysis reveals that even numerical approaches have great diﬃculty to obtain an

estimate of Z(see, for example, Chap.2 of [87]). To circumvent these problems

one therefore seeks a numerical method capable of computing ensemble averages

without requiring knowledge of the conﬁguration integral. This is accomplished

by the Monte Carlo (MC) method.

3.4.1 The general method

To calculate ensemble averages in Monte Carlo simulations we need to discretize

the space of microstates Γ={rN,s

z,N}and replace (3.52) by

O= lim

M→∞



k=1

O(Γk)f0(Γk)



k=1

f0(Γk)

(3.80)

3.4. MONTE CARLO SIMULATION 73

Within the concept of importance sampling [88], where microstates {Γk}are

generated according to f0(i.e., their “importance”), (3.80) can be simpliﬁed

further to

O= lim

M→∞



m=1

O(Γm) (3.81)

where the prime attached to the summation sign to emphasizes generation of

microstates with a probability proportional to f0. Importance sampling can be

realized conveniently if microstate generation proceeds as a (stationary) Markov

process [89] Since the implementation of Markov processes in the context of

Monte Carlo simulations is well explained in the literature [79, 87, 90], we sum-

marize only brieﬂy the main concepts.

Let p(n) be the vector of probabilities of all microstates at time t+1. Then

the immediately preceding vector p(m) is related to p(n) by

p(n) = Pp(m) (3.82)

where Pis the transition matrix. Matrix elements Pnm (transition probabilities)

represent the probability of a transition from state “m” to state “n”. If we

take Pto be time–independent and apply it repeatedly [i.e., P···Pp(m)] the

resulting probability vector will eventually become stationary, that is, it satisﬁes

the equation

π=Pπ(3.83)

where πis the limiting (stationary) probability eigenvector of P.Itcanbe

shown that a sequence of microstates (i.e., a Markov chain) will indeed become

stationary and that πis unique if Pis constructed such that in principle any

state n can be reached in a ﬁnite number of steps (i.e., repeated application of P)

from any other state [90]. In practice, we therefore need to construct Psuch that

πis proportional to the probability density of the ensemble in question. This

task becomes much simpler if we invoke the principle of microscopic reversibility

Pnmπm=Pmnπn(3.84)

which automatically satisﬁes (3.83). However, it is extremely diﬃcult (if not

impossible) to realize transitions between microstates with the correct proba-

bility. Instead we perform “trial” transitions between microstates governed by

a certain probability α(the “underlying matrix”[91] of the Markov chain) and

74 CHAPTER 3. THE CONTINUOUS MODEL

then decide whether the trial state is taken as a new state “n” or not; if not,

the system remains in its original state “m”. Hence, (3.84) is now given by

αnmPnmπm=αmnPmnπn.(3.85)

Assuming a symmetric underlying matrix (i.e., αmn =αnm), we can rearrange

(3.85) to give

Pnm

Pmn

=πn

πm

.(3.86)

In essence, αmn =αnm expresses the assumption that trial state “n” is gen-

erated from “m” with the same probability with which trial state “m” would

be generated from “n” as the initial state. Equation (3.86) is a formulation of

the principle of detailed balance. Equation (3.86) can be implemented following

Metropolis et al. [92] by choosing

Pnm =min1,πn

πm.(3.87)

The advantage of (3.87) is that Pnm depends only on the ratio πn/πm,sothat

the unknown (and in most cases inaccessible) partition function cancels out, as

we shall demonstrate in the following section.

3.4.2 Application and Implementation

According to (3.87) a trail state “n” is immediately accepted if πn≥πm.Ifon

the other hand, πn<π

m, (3.87) can be written more explicitely as

Pnm =πn

πm

=f0(Γn)

f0(Γm)(3.88)

where it is clear from (3.52) that the conﬁguration integral cancels between

denominator and numerator so that on the basis of a Markov process the

importance–sampling concept [see (3.81)] can indeed be implemented.

Because of the functional form of f0in (3.52) it seems sensible to gener-

ate a numerical representation of a Markov chain through a sequence of three

consecutive and independent processes. In the ﬁrst of these Nand szremain

constant and a (randomly or sequentially) selected ﬂuid molecule iis displaced

at random according to

r(n)

i=r(m)

i+δr(1−2ξ) (3.89)

3.4. MONTE CARLO SIMULATION 75

where 1:= (1,1,1) and ξis a vector whose three components are (pseudo–)

random numbers distributed uniformly on the interval [0,1]. In (3.89), δris the

side length of a small cube centered on r(m)

i.SinceNand szremain constant

between initial (m) and new trial conﬁgurations (n) it is easy to verify from

(3.52) and (3.88) that

nm = min [1,exp(−β∆Unm)] (3.90)

where

∆Unm =



j=i"uﬀ(r(n)

ij )−uﬀ(r(m)

ij )#+



k=1 "Φ[k](x(n)

i,z(n)

i)−Φ[k](x(m)

i,z(m)

i)#

(3.91)

is the change in conﬁgurational energy associated with the displacement of

molecule i. The acceptance ratio for displacement steps generally depends on δr.

Following standard practice [72] we adjust δrduring a Monte Carlo simulation to

maintain an overall acceptance of roughly 50% of all attempted displacements.

Based upon this criterion, δrdepends on the actual thermodynamic state, that

is on Tand the average density of the ﬂuid.

In the next process a molecule is either created at a randomly selected po-

sition rior a randomly selected, already existing molecule is removed from its

present position ri. Both processes are attempted with equal probability. Since

szremains ﬁxed and the spatial positions of all other molecules are unaltered it

is a straightforward matter to demonstrate from (3.52) and (3.88) that

PII

nm =min(1,exp(r±)) (3.92)

where

r±=±B∓ln N∓βU±(3.93)

B=βµ −ln Λ3

V(3.94)

U±=



j=i

uﬀ(rij )+



k=1

Φ[k](xi,z

i) (3.95)

and Nis the number of ﬂuid molecules after addition (r+)orprior to removal

(r−) of molecule i. Notice that in contrast to the displacement step, where the

acceptance ratio can be adjusted through variations of δr, no such adjustment

76 CHAPTER 3. THE CONTINUOUS MODEL

is possible here. The acceptance ratio is solely determined by the physical

situation, that is by temperature and density of the ﬂuid.

The third and ﬁnal process involves compression/expansion of the conﬁned

ﬂuid eﬀected by random displacements of the substrates according to

s(n)

z=s(m)

z+δs(1 −2ξ) (3.96)

where ξis a random number uniformly distributed on [0,1] and δsis a small

displacement increment again adjusted during the course of the simulation such

that roughly 50% of all compression/expansion attempts are accepted. Because

of (3.96)

z(n)

i=z(m)

s(n)

s(m)

z∀i=1,...,N . (3.97)

Since during compression/expansion attempts Nremains ﬁxed this process is

realized with probability

PIII

nm = min [1,exp(rs)] (3.98)

where

rs=β"TzzAzs(n)

z−s(m)

z−∆Unm#+Nln &s(n)

s(m)

z'(3.99)

as one can verify from (3.52) and (3.98). In (3.99)

∆Unm =1



i=1



j=i"uﬀ(r(n)

ij )−uﬀ(r(m)

ij )#(3.100)



k=1



i=1 "Φ[k](x(n)

i,z(n)

i)−Φ[k](x(m)

i,z(m)

i)#

The last term in (3.99) (as well as the factor Vin (3.94)) arise for reasons

detailed in [87].

The three processes are carried out sequentially. Suppose the system con-

tains Ninit molecules at the beginning of a sequence. Then Ninit displacements,

Ninit creation/destruction events and one compression/expansion attempt con-

stitute a Monte Carlo cycle in the grand mixed isostress-isostrain ensemble.

This ratio is convenient because the last step displaces all Ninit molecules at

once on account of (3.97). Thus, (3.100) involves in principle numerical opera-

tions of order N2whereas a single displacement or creation/destruction attempt

requires numerical operation of order N. Monte Carlo results to be presented in

3.4. MONTE CARLO SIMULATION 77

chapter 4 are typically based on 37500 cycles with systems containing roughly

200 to 3000 ﬂuid molecules (depending on the thermodynamic state).

Chapter 4

Phase behavior and

thermomechanical

properties

In the two preceding chapters mean-ﬁeld theory and Monte Carlo simulations

were introduced, which are now applied to investigate the phase behavior and

thermomechanical properties of conﬁned ﬂuids. We already demonstrated (see

section 2.4) that ﬂuids conﬁned by chemically corrugated walls can be expected

to exhibit a complex phase behavior. In order to unravel the dependence of

the phase diagram on the various model parameters {nz,n

s,n

w,

fs,

fw,αs

x},

we will now employ the mean-ﬁeld lattice model discussed in chapter 2 to study

the inﬂuence of variations of individual model parameters in depth. Since the

mean-ﬁeld treatment is based on a number of simplifying assumptions, it seems

sensible to verify its predictions independently within the framework of the more

realistic continuous model of a ﬂuid conﬁned between nanopatterned substrates

which we introduced in chapter 3. As pointed out in section 3.4, this latter

model can be treated only by means of (Monte Carlo) computer simulations

which are computationally much more demanding than solving the mean-ﬁeld

model. Therefore, we restrict Monte Carlo simulations to a veriﬁcation of key

predictions of the mean-ﬁeld treatment.

4.1. PHASE BEHAVIOR 79

In parallel experiments conﬁned ﬂuids are explored from various points of

view. For example, gas adsorption in porous materials [55] demonstrated that

the phase behavior is strongly aﬀected by the pore width (wall distance sz).

These experiments provide clear evidence for the impact of conﬁnement on the

phase diagram of a ﬂuid adsorbed in a porous medium which diﬀers from that of

a corresponding bulk ﬂuid the more the narrower the pores are. Adsorption ex-

periments on chemically nanopatterned substrates showed selective adsorption

on substrate regions composed of the chemical compound preferred energetically

by the adsorbent [22]. Properties of the adsorbed ﬂuid such as, for instance its

evaporation rate [22] depend signiﬁcantly on the pattern size, i.e. the chemical

corrugation (cr). Inﬂuence of pore size and chemical corrugation on the phase

behavior of conﬁned ﬂuids will be delineated in this chapter.

With the help of the surface forces apparatus (SFA) [69] mechanical prop-

erties of thin ﬂuid ﬁlms can be measured directly on the nanoscale. In an SFA

experiment the ﬂuid is conﬁned between two solid substrates and maintained

under constant pressure or load applied in the direction normal to the ﬂuid-

substrate interface. Under these conditions a conﬁned ﬂuid of a typical thick-

ness of one to ten molecular diameters can be exposed to a shear strain. The

impact of shear strain on both phase behavior and thermomechanical properties

like the conjugate shear stress and modulus will be discussed below.

4.1 Phase behavior

4.1.1 Variation of wall distance

The thermodynamic state of a ﬂuid conﬁned between chemically heterogeneous

walls depends on various system parameters {nz,n

s,n

w,

fs,

fw,αs

x}.Inthis

multidimensional parameter space the degree of conﬁnement, that is nz,ispar-

ticularly important as far as the phase behavior of the conﬁned ﬂuid is con-

cerned. This has been demonstrated by Thommes and Findenegg [55] who

determined the coexistence curve of pure SF6in controlled–pore glasses (CPG)

of nominal pore widths of 31 and 24 nm, respectively. Their results show that

the pore critical point shifts to temperatures below the bulk critical tempera-

ture Tcand to densities above the bulk critical density (see ﬁgure 7 in [55]).

80 CHAPTER 4. RESULTS

Consequently, the entire phase diagram of the pore ﬂuid is displaced to higher

densities and lower temperatures. This shift was found to be stronger for the

narrower pores. Thus, it seems sensible to begin a discussion of conﬁnement

eﬀects by investigating the impact of nzon the phase diagram µx(T) within the

framework of the lattice–gas model of a ﬂuid conﬁned by chemically decorated

substrate surfaces (see ﬁgure 2.1) which we introduced already in section 2.1.

The lattice–gas model is particularly convenient for this purpose, because it

is numerically not too demanding. To compute phase diagrams for T>0we

utilize the method detailed in section 2.2.3.

Figure 4.1(a) shows plots of phase diagrams in µ–Tprojection for various

degrees of conﬁnement (i.e., nz). The horizontal line represents the bulk phase

diagram which we include for comparison. It consists of a single coexistence

curve µgl

x(T)=µc=−3 [see (2.92) and ﬁgure 2.8]. Along µgl

x(T) gas and liquid

(bulk) phases coexist. Thus, µgl

x(T) is a line of ﬁrst–order phase transitions

terminating, of course, at T=Tc=3/2 (2.96). We employ dimensionless units

introduced already in section section 2.1.1.

More subtle eﬀects are observed if the lattice gas is conﬁned by solid sub-

strates as plots in ﬁgure 4.1(a) show. For suﬃciently large nz[see, for example,

nz= 15 in ﬁgure 4.1(a)] the coexistence line shifts to lower chemical potentials

compared with the bulk and is no longer parallel to the temperature axis. Fur-

thermore, the critical point moved to lower Tand µin qualitative agreement

with experimental ﬁndings [55]. The diﬀerence between the bulk gas–liquid co-

existence line and that shown for nz= 15 is not solely due to a conﬁnement

eﬀect, as we will see shortly.

Figure 4.2 clearly shows that the full phase diagram for the conﬁned ﬂuid

consists of more than one coexistence line. This has been overlooked in the

past [64, 65, 79, 93, 94, 95, 96]. Detailed investigations reveal that the upper

coexistence line (i.e., the one at higher chemical potentials) µdl

x(T) in ﬁgure 4.2

refers to ﬁrst–order phase transitions between droplet and liquid phases (see

section 2.4.2). The lower one, on the other hand, is identiﬁed as the gas–droplet

coexistence line µgd

x(T). The local density ρ(xi,z

i)=ρi(see section 2.3.1) of

a typical droplet phase is shown in ﬁgure 4.3(a). The droplet phase consists of

ﬂuid–ﬁlled columns (in the y–direction) adsorbed along the strongly attractive

4.1. PHASE BEHAVIOR 81

-3.00

-3.02

-3.04

1.2 1.3 1.4 1.5

-3.00

-3.02

-3.04

1.2 1.3 1.4 1.5

-3.00

-3.02

-3.04

1.2 1.3 1.4 1.5

-3.00

-3.02

-3.04

1.2 1.3 1.4 1.5

-3.00

-3.02

-3.04

1.2 1.3 1.4 1.5



bulk

(a)

-3.03

-3.025

-3.02

-3.015

1.3 1.35 1.4



(b)

Figure 4.1: (a) Part of phase diagrams in T–µprojection for various conﬁned lattice

gases as a function of substrate separation nzindicatedinﬁgure(α=0,nx= 14, ns=

8, fs =1.4, fw =0.3); (—)µdl

x(T),(−−−)µdb

x(T), (−·−)µbl

x(T). Corresponding

bulk coexistence curve is also shown. (b) as (a) but on an enhanced scale showing

only coexistence–curve branches in grey box of (a); (•): denotes ﬁxed thermodynamic

state of conﬁned T=1.325, µ=−3.0235 see text.

82 CHAPTER 4. RESULTS

-3.4

-3.3

-3.2

-3.1

-3

0 0.5 1 1.5







 



 



 

Figure 4.2: As ﬁgure 4.1 but here the full phase diagrams for two substrate separations

nz=15andnz= 9 are shown. Notice, the gas–droplet coexistence line µgd

x(T)(·· ·· ··)

is independent of nz(within computational accuracy). Analytic solutions (2.53) for

T= 0 are indicated by (



substrate parts. During a ﬁrst–order phase transition from the gas (where the

system is almost empty) to the droplet phase [that is upon crossing µgd

x(T)] the

columns ﬁll. Filling is obviously induced by the presence of a single surface and

controlled by the strongly attractive substrate parts. Thus, µgd

x(T) is expected

to be independent of the wall separation nz. This notion is supported by plots

in ﬁgure 4.2 where µgd

x(T)isthesamefornz= 9 and 15. This holds all the

way down to T= 0 where the analytic solution [see (2.53)] is given by

µgd

x(T=0)=−2−fs −1

(4.1)

which is independend of nzas expected. On the contrary [see (2.58)]

µdl

x(T=0)=−3+(nw−ns)+2(1−nwfs)

nwnz+ns(nz−2) (4.2)

does depend on nzexplicitly. Hence, one expects this dependence to persist

even at nonzero temperatures. Plots of µdl

x(T) in ﬁgure 4.2 conﬁrm this notion.

4.1. PHASE BEHAVIOR 83

-15

30 45

246810

0.5



(a)

-15

30 45

246810

0.5



(b)

-15

30 45

246810

0.5



(c)

Figure 4.3: Typical microscopic structures of the lattice gas conﬁned between chemi-

cally striped substrates (see ﬁgure 2.1). Plots show local density ρ(xi,z

i) as a function

of position in the x–z–plane. The thermodynamic state is speciﬁed by T=1.1, and (a)

droplet phase (µ=−3.06), (b) bridge phase (µ=−3.03), (c) liquid phase (µ=−3.01)

(see text). Substrate parameters are ns= 15, nw= 15, nz=8.0 with fs =1.6and

fw =0.4.

84 CHAPTER 4. RESULTS

Moreover, if nzdecreases, a bifurcation appears at T=Tdbl

tr .Aninspectionof

the local densities reveals that only (inhomogeneous) liquid [ﬁgure 4.3(c)] and

droplet phases [ﬁgure 4.3(a)] coexist along the line µdl

xT<T

dbl

tr .AtT=Tdbl

the latter two are in thermodynamic equilibrium with a bridge phase [see ﬁgure

4.3(b)]. As can be seen from the plot of local density in ﬁgure 4.3(b) the bridge

phase consists of a high-density regime spanning the gap between the strongly

attractive wall parts surrounded by a low density ﬂuid controlled by the weakly

attractive portions of the substrates. As the plot in ﬁgure 4.3(b) illustrates,

bridge morphologies are inhomogeneous in one lateral direction (x) as reﬂected

by their alternating high– and low–density regimes. Intuitively one would expect

the stability of the bridges to depend strongly on the relative position of the

substrates and particularly on nz(we defer a detailed discussion of the impact

of misaligning the walls in the x–direction to section 4.1.3 where the eﬀect of

exposing ﬂuid bridges to shear deformations is analyzed in depth).

For T>T

dbl

tr , the phase diagram µx(T) consists of two branches. The

upper one, µbl

x(T), is a line of ﬁrst–order phase transitions involving liquid and

bridge phases whereas the lower one, µdb

x(T), corresponds to bridge and droplet

phases, respectively. Both branches terminate at their respective critical points

µbl

c,Tbl

cand µdb

c,Tdb

c. According to (2.92) the phase diagram µx(T)of

the lattice gas in the given temperature region is formed by µgd

x(T), µdl

x(T),

µdb

x(T), µbl

x(T), and the point µdbl

tr ,Tdbl

tr .

Comparing in ﬁgure 4.1(a) coexistence curves for nz= 8 and 9, we see that

the triple point is lowered the more the more severe the conﬁnement is, that is

the smaller nzis. Simultaneously, µbl

cincreases whereas µdb

cdecreases such that

the one–phase region of the bridges widens. Because of these rather complex

variations of µx(T) with nzthe following sequence of discontinuous phase tran-

sitions may be envisioned. Suppose {µ, T }is chosen such that the conﬁned ﬂuid

in the limit nz−→ ∞ is a “droplet”. This is illustrated in ﬁgure 4.1(b) where

for a speciﬁc thermodynamic state determined by T=1.325 and µ=−3.0235

and nz= 10 this state point falls below all branches of µx(T). However, as

the substrate separation decreases, one notices from the plot corresponding to

nz= 9 that the same thermodynamic state now pertains to the one–phase

regime of liquid phases, that is it falls above all branches of µx(T). Thus, in

4.1. PHASE BEHAVIOR 85

going from nz=10tonz= 9 the conﬁned lattice gas undergoes a ﬁrst–order

phase transition from a droplet to a liquid phase. For an even smaller substrate

separation nz= 8 one sees from ﬁgure 4.1(a) that the triple point has shifted to

rather small values µdbl

tr ,Tdbl

tr and that the one–phase region of bridge phases

has widened considerably. Thus, as can be seen from the parallel plot in ﬁgure

4.1(b), the thermodynamic state eventually belongs to the one–phase region

of bridge phases where it remains for all smaller nz. Hence, as one decreases

the substrate separation from nz=9tonz= 8 an originally liquid phase is

transformed into a bridge phase during a ﬁrst–order phase transition.

To conﬁrm this successive appearance of phase transitions with decreased

substrate separation, parallel results for the continuous model were obtained in

a sequence of Monte Carlo simulations (see section 3.4) in the grand canonical

ensemble where T=1.0, µ=−11.50, sx= 12.0, ds=4.0, dw=8.0 with fs =

1.25 and fw =0.001 [94] (we deviate from the the lattice–gas notation ns−→

ds,n

w−→ dw,n

x−→ sxto emphasize that these dimensions can be varied

continuously). Under these conditions the thermodynamically stable phase of a

corresponding bulk system is a gas phase of mean density ¯ρ=N/V ≃0.036

[97]. As before for the lattice gas we express all quantities in the customary

dimensionless (i.e., “reduced”) units. However, here we again deviate from the

lattice–gas by expressing length in units of σ[see (3.1)].

Depending on the degree of conﬁnement (sz) a ﬂuid in the continuous model

may form a droplet, liquid or bridge phase similar to the conﬁned lattice gas

(see ﬁgure 4.3). Typical structures of these three phases are illustrated by plots

of the local density in ﬁgure 4.4. Within the framework of the continuous model

the local density is deﬁned as

ρ(x, z):=N(x, z)

∆x∆zsy

.(4.3)

In (4.3), N(x, z) is the number of ﬂuid molecules in a given conﬁguration that

are located in a square prism of dimensions ∆x×sy×∆zcentered on a point

(x, z). As in ﬁgure 4.3(a), ρ(x, z) in ﬁgure 4.4(a) is representative of a typical

bridge morphology. In the absence of a shear strain (α= 0, see ﬁgure 3.1),

ρ(x, z) is symmetric with respect to x=0andz= 0 as it must be. As in

the lattice–gas model, a bridge phase may condense or evaporate upon varying

thermodynamic conditions. The microscopic structure of liquid and droplet

86 CHAPTER 4. RESULTS

-6

-3.5 03.5



(a)

-6

-3.5 03.5



(b)

-6

-4 04



(c)

Figure 4.4: Typical microscopic structures of ﬂuids conﬁned between chemically

striped substrates (see ﬁgure 3.1). Plots show local density ρ(x,z) as a function

of position in the x–z–plane. (a) bridge phase (sz=7.2), (b) liquid phase (sz=7.5),

T=1.0andµ=−11.50; substrate parameters are sx=12.0, ds=4.0, dw=8.0 with

fs =1.25 and fw =0.001.

4.1. PHASE BEHAVIOR 87

-3

-2

-1

-6 -4 -2 0 2 4 6

Figure 4.5: Contour lines ρ(x, z)=0.25 (−·−), 0.70 (—) corresponding to the plot

in ﬁgure 4.4(a).

phases for α= 0 is illustrated by the plots in ﬁgure 4.4(b) and ﬁgure 4.4(c).

From the plot in ﬁgure 4.4(a) one notices that ρ(x, z) is a nonmonotonic

function of zalong any line x= const in the high–density regime. Nonmono-

tonicity of the local density is also clearly visible in plots of contour lines of

ρ(x, z) (i.e., lines along which ρ(x, z) = const). The plot in ﬁgure 4.5 shows

a sequence of “islands” along the z–axis surrounded by a closed line of lower

density. Apparently, the islands are resolved and well separated by a distance of

approximately ∆z≃1 between centers of neighboring islands. Thus, it seems

plausible to associate these islands with molecular strata parallel with the con-

ﬁning substrates. Stratiﬁcation reﬂects substrate–mediated intermolecular cor-

relations. With increasing distance from a substrate, stratiﬁcation diminishes in

the continuous model [see ﬁgure 4.4(a)] due to the decay of the ﬂuid–substrate

potential. This is reﬂected by a declining amplitude of oscillations in ρ(x, z)

with increasing distance from a substrate which can also be seen in ﬁgure 4.5

where the islands shrink in transverse (i.e., x–) direction as |z|→0. Within the

lattice–gas model, stratiﬁcation cannot be resolved in the local–density plots

88 CHAPTER 4. RESULTS

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14



Figure 4.6: Average density ¯ρas function of substrate separation szfor continuous

model. Grand canonical ensemble Monte Carlo simulations were carried out for T=

1.0, µ=−11.5, sx=12.0, ds=4.0, αsx=0.0. Solid lines are intended to guide the

eye.

(ﬁgure 4.3). In the lattice model positions of ﬂuid molecules are restricted to

sites of the simple cubic lattice (see section 2.1.2), where the lattice constant 

is equivalent to the diameter of a ﬂuid molecule in the continuous model. Thus,

we observe the local density only at discrete positions whose distances from

the substrate surfaces are integer multiples of . Thus, applying this logic to

ρ(x, z) in the continuous model one would have to calculate it at points whose

distance from the substrate surfaces is given in integer multiples of σ(provided

sx/σ ∈N). With such a coarse resolution in the z–direction one would, however,

capture only maxima of ρ(x, z) in the stratiﬁed region of the bridges therfore in-

evitably missing the nonmonotonic variation of ρ(x, z) present only on a smaller

lengthscale (i.e., a ﬁner grid) as plots in ﬁgures 4.4(a) and 4.5 show. Since a

lengthscale smaller than does not exist for the lattice model nothing can be

said about stratiﬁcation by deﬁnition.

Morphological changes with decreasing sz, as illustrated by the sequence

of plots in ﬁgures 4.4(a) – (c), may be cast in a more compact manner by

4.1. PHASE BEHAVIOR 89

calculating the average density

¯ρ:= 1

sxsz

sx/2

−sx/2

sz/2

−sz/2

dzρ(x, z)=N

V(4.4)

for various substrate separations sz. A plot of ¯ρ(sz) in ﬁgure 4.6 exhibits two

discontinuities. An analysis of ρ(x, z)atsz≃8.2 permits one to conclude that

at this substrate separation a ﬁrst–order phase transition involving droplet and

liquid phases occurs whereas the one at sz≃7.5 refers to a transition between

a liquid and a bridge phase. Therefore, the sequence of phase transitions in

ﬁgure 4.6 resembles precisely the scenario observed for the lattice gas shown

in ﬁgure 4.1(b) and discussed above. Oscillations of ¯ρin ﬁgure 4.6 over the

range 2 .sz.6 reﬂect stratiﬁcation of the conﬁned ﬂuid which becomes more

pronounced the smaller szis.

However, investigations of phase transitions by Monte Carlo simulations in

the spirit of ﬁgure 4.6 are frequently plagued by metastability, that is existence

of a sequence of conﬁgurations rN

kk=1,...,M corresponding only to a local min-

imum of Ω where Mcan be quite substantial. In other words, the “lifetime” of

a metastable thermodynamic state can be large compared with the time over

which the microscopic evolution of the system can be pursued on account of

limited computational speed. The origin of metastability is lack of ergodicity in

the immediate vicinity of a ﬁrst–order phase transition which arises on account

of the microscopically small systems employed in computer simulations [98].

Metastability is manifest as hysteresis in a sorption isotherm (like the one plot-

ted in ﬁgure 4.6), that is a range of ﬁnite width ∆szaround the true transition

point over which for the same Tand µ,¯ρ(sz) is a double–valued function. To

distinguish metastable from the thermodynamically (i.e. globally) stable phase

one needs to compare Ω for the two states pertaining to diﬀerent branches of

the sorption isotherm at the same µand sz. The one having lowest Ω is the

globally stable phase; the other one is only metastable. In ﬁgure 4.6 we plot only

data for thermodynamically stable phases identiﬁed according to this rationale,

where Ω was calculated from (3.21) during the Monte Carlo simulation.

90 CHAPTER 4. RESULTS

4.1.2 Variation of chemical corrugation

Sorption experiments have clearly demonstrated that the phase behavior of

conﬁned ﬂuids depends strongly on the degree of conﬁnement (i.e, pore width)

as far as nanoporous media are concerned [55]. It is thus conceivable that the

stability of such a conﬁned phase may also depend on the degree of chemical

corrugation of the substrate. To investigate its impact on the phase diagram we

deﬁne

cr=ns

ns+nw

(4.5)

as a quantitative measure of chemical corrugation.

Figure 4.7 shows phase diagrams for four diﬀerent substrates of cr=6/8,

6/10, 6/12, 6/14. In all cases the width of the strongly attractive substrate

part ns= 6 is maintained. Diﬀerent dergrees of corrugation, are therefore

eﬀected by varying the width of the weakly attractive strip nw.Themost

surprising observation from ﬁgure 4.7 is that the droplet–bridge coexistence line

µdb

x(T)doesnotshiftwhithcr. Likewise the associated critical point µdb

c,Tdb

c

remains nearly unaﬀected by a change of the width of the weakly attractive strip.

On the contrary, the triple point µdbl

tr ,Tdbl

tr exhibits a signiﬁcant dependence

on crleaving, however, the chemical potential at the triple point µdbl

tr roughly

unchanged since µdb

x(T) is nearly parallel to the temperature axis. The variation

of the triple point temperature Tdbl

tr is related to a shift of the droplet-liquid

and the bridge-liquid coexistence lines µdl

x(T)andµbl

x(T), respectively, to lower

chemical potentials as nwdecreases. For cr=6/8, µdl

x(T) eventually shifts to

lower chemical potentials compared with the chemical potential at the droplet-

bridge critical point, that is µdl

x(T)<µ

c. Therefore, this system does not have

a thermodynamically stable bridge phase.

The degree of corrugation also aﬀects the succession of phase transitions

if the substrate separation is varied at constant crfollowing the discussion in

section 4.1.1. Consider, for example, the case cr=6/8. Here the one–phase

region of the bridge phase disappeared completely. Suppose Tand µare chosen

such that the point {T,µ}is located slightly below µdl

x(T). Decreasing nzcauses

µdl

x(T) to move down in chemical potential (see ﬁgure 4.1 in section 4.1.1). Thus,

the system undergoes a ﬁrst order phase transition from the droplet to the liquid

phase. If nwis suﬃciently small (i.e. cris suﬃciently large) no stable bridge

4.1. PHASE BEHAVIOR 91

-3.00

-3.05

-3.10

-3.15

1.0 1.1 1.2 1.3 1.4 1.5

-3.00

-3.05

-3.10

-3.15

1.0 1.1 1.2 1.3 1.4 1.5

-3.00

-3.05

-3.10

-3.15

1.0 1.1 1.2 1.3 1.4 1.5

-3.00

-3.05

-3.10

-3.15

1.0 1.1 1.2 1.3 1.4 1.5

-3.00

-3.05

-3.10

-3.15

1.0 1.1 1.2 1.3 1.4 1.5



 







 







 







 





Figure 4.7: As ﬁgure 4.1, but for various corrugation crindicated in ﬁgure (α=0,

nx= 14, nz=8,fs =1.4andfw =0.3) (—)µdl

x(T),(−−−)µdb

x(T), (−·−)

µbl

x(T). Corresponding bulk coexistence curve is also shown.

phase appears in the phase diagram even for smaller nz. Therefore, the system

remains in its liquid state for all smaller values of nz.Forcrbelow a certain

threshold (cr=6/8 in ﬁgure 4.7) a one–phase region of bridges appears and

increases in size with decreasing cr. Consider now a system where cr=6/14

and a point {T,µ}below µdb

x(T). Decreasing nzthen causes the one-phase

region of the bridges to widen, while the entire phase diagram moves downward

(see ﬁgure 4.1). Thus the initial droplet phase is transformed into a bridge

phase as the state point {T,µ}crosses the line of ﬁrst order phase transition,

that is µdb

x(T). Decreasing nzfurther results in an even wider one–phase region

of bridges. Thus, for all smaller nzthe state point remains in that one–phase

region.

To conﬁrm these conjectures we turn to the continuous model where we

calculate isotherms ¯ρ(sz) for several systems conﬁned by diﬀerently corrugated

substrates (see ﬁgure 4.8). As before for the lattice model we ﬁx the width of

the strongly attractive stripe, ds= 4, and vary the corrugation over the range

92 CHAPTER 4. RESULTS

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14



Figure 4.8: Mean density ¯ρas a function of substrate separation szand various

degrees of chemical corrugation of substrate cr=4/7(



), cr=4/10 (•), cr=4/12

(



), and cr=4/14 (



). Solid lines are intended to guide the eye.

4/14 ≤cr≤4/7. At large wall separations sz&11 all systems exhibit similar

(low) mean densities ¯ρindicating that the thermodynamically stable state of all

systems in that region is the droplet phase. However, as szdecreases diﬀerent

eﬀects are detected. The system having the highest value cr=4/7, undergoes

a discontinuous phase transition from the droplet to the liquid phase at a wall

separation sz≃11, as the plot in ﬁgure 4.8 shows. (That the higher density

state at the discontinuous change in ¯ρ(sz) is, in fact, liquidlike was conﬁrmed by

an inspection of the local density of the conﬁned ﬂuid, see above.) Because of the

reduced net attraction as crdecreases, all other systems exhibit this transition

(known as capillary condensation) but at smaller sz. For the system having the

lowest value of the corrugation cr=4/14, only partial condensation is observed.

That is at sz≃8.3 the initial droplet phase undergoes a discontinuous phase

transition to form a bridge phase, thereby conﬁrming the predictions of the

parallel lattice gas calculations.

For cr=4/10 a steep but continuous decay of ¯ρaround sz= 6, is observed.

4.1. PHASE BEHAVIOR 93

As shown in [94] the isothermal compressibility exhibits a cusp–like maximum in

the vicinity of sz≃6 indicating large density ﬂuctuations. Applying ﬁnite–size

scaling arguments [79] it was concluded in [94] that over the range 5.5≤sz≤6.5

the thermodynamic state of the conﬁned ﬂuid may be in the vicinity of the

bridge–liquid critical point.

4.1.3 The impact of shear strain

The preceding sections clearly illustrate the complex dependence of the sta-

bility of bridge phases on both conﬁnement and chemical corrugation. If the

substrates are perfectly aligned (i.e. αsx=0) a bridge phase is characterized by

a high-density regime spanning the gap between the strongly attractive parts

of the opposing substrates [see ﬁgure 4.9(a)]. Intuitively one expects these

structures to be capable of resisting shear deformations to a certain degree. A

shear strain may be applied to a bridge phase by misaligning the corrugated

substrates. The plots in ﬁgure 4.9(a) and ﬁgure 4.9(b) illustrate the eﬀect of

applying a shear strain αsx>0 to a typical bridge phase. Depending on the

thermodynamic state a bridge phase will sustain a maximum shear strain but

will eventually be either “torn apart” and undergo a ﬁrst–order phase transition

to a droplet phase [see ﬁgure 4.9(c)] or condense and form a liquid phase [see

ﬁgure 4.9(d)]. Corresponding phase diagrams µx(T) plotted in ﬁgure 4.10 show

that upon increasing αfrom its initial value of zero causes the triple point to

shift to higher Tdbl

tr and µdbl

tr . Simultaneously, the one–phase region of bridge

phases shrinks. The one–phase regime of bridge phases may, however, vanish

completely for some α<α

max depending on substrate separation (i.e., nz),

chemical corrugation (i.e., cr), or strength of interaction with the chemically

diﬀerent parts of the substrate (i.e., fw,fs). Notice that for the special case

αmax =1/2 (i.e., nxeven) the one–phase region of bridge phases must vanish

in the limit α=αmax for symmetry reasons. In addition, ﬁgure 4.10 shows

that critical temperatures Tbl

cand Tdb

cdepend only weakly on the shear strain

unlike µbl

cand µdb

csuch that the critical points are essentially shifted upwards

and downwards, respectively, as αincreases.

Consider now a speciﬁc isotherm T={(µ, TT)|TT=1.25}in ﬁgure 4.10,

intersecting with diﬀerent branches of the (same) phase diagram µx(T) at dif-

94 CHAPTER 4. RESULTS

1234567



(a)

1234567



(b)

Figure 4.9: Local density ρ(x,z) for conﬁned lattice at T=1.25, µ=−3.03658.

Substrates are characterized by nx= 14, nz=7,nw=8,ns=6,fw =0.4, and

fs =1.6. (a) bridge phase (α= 0), (b) bridge phase (α=5/14), (c) droplet phase

(α=1/2), (d) liquid phase (α=1/2). Plots in (c) and (d) correspond to coexisting

phases (see ﬁgure 4.10). Two periods of ρ(x, z)inthex–direction are shown because

of periodicity of the lattice model (see section 2.1.2).

4.1. PHASE BEHAVIOR 95

1234567



(c)

1234567



(d)

Figure 4.9: (Continued)

96 CHAPTER 4. RESULTS

-3.06

-3.04

-3.02

-3.00

1.0 1.1 1.2 1.3 1.4 1.5

-3.06

-3.04

-3.02

-3.00

1.0 1.1 1.2 1.3 1.4 1.5



Figure 4.10: As ﬁgure 4.1, but for various shear strains αindicated in ﬁgure (nx= 14,

ns=6,nz=7,fs =1.6, fw =0.4). Intersections between isotherm

(vertical

solid line, see text) and coexistence–curve branches represent coexisting phases. (



)

µgb

x(T

), (



)µbl

x(T

), α=0;(•)µgb

x(T

), (◦)µbl

x(T

), α=1/7; (



)µgl

x(T

), α=2/7.

ferent chemical potentials. According to the deﬁnition of µx(T), (2.114) each

intersection corresponds to a pair of (separately) coexisting phases. For ex-

ample, at µdb

x(TT)≃−3.053 and α= 0 a droplet phase coexists with a (more

dilute) bridge phase whereas a (denser) bridge phase coexists with a liquid phase

for µbl

x(TT)≃−3.029. Because the one–phase region of bridge phases shrinks

with α(see ﬁgure 4.10), the “distance” ∆µx(TT):=µgb

x(TT)−µbl

x(TT)→0

the larger αbecomes, that is with increasing shear strain. From the plot in

ﬁgure 4.10 it is clear that a shear strain exists such that ∆µx=0,thatis

TT≤Tdbl

tr (αnx). For this and larger shear strains only a single intersection

remains, corresponding to coexisting droplet and liquid phases (see ﬁgure 4.10).

If the conﬁned ﬂuid in the continuous model is exposed to a shear strain, it

undergoes structural transformations similar to the ones just discussed for the

4.1. PHASE BEHAVIOR 97

-4

-2

;



(a)

-4

-2

;



(b)

Figure 4.11: As ﬁgure 4.9, but for continuous model. Plots show two periods of

ρ(x,z)inthex–direction because of periodic boundary conditions. (a) unsheared

bridge phase, αsx=0.0; (b) sheared bridge phase, αsx=7.5; (c) droplet phase,

αsx=10.0; (d) liquid phase, αsx=10.0. Plots in (c) and (d) correspond to coexisting

phases. In all cases T=0.7, µ=−8.15, sx=20.0, ds=10.0, and sz=8.0.

98 CHAPTER 4. RESULTS

-4

-2

;



(c)

-4

-2

;



(d)

Figure 4.11: (Continued)

4.1. PHASE BEHAVIOR 99

-4

-2

-10 0 10 20 30

(a)

-4

-2

-10 0 10 20 30

(b)

Figure 4.12: Contour lines ρ(x, z)=0.10 (−·−), 0.75 (—) corresponding to plots

in ﬁgure 4.11.

100 CHAPTER 4. RESULTS

-4

-2

-10 0 10 20 30

(c)

-4

-2

-10 0 10 20 30

(d)

Figure 4.12: (Continued)

4.1. PHASE BEHAVIOR 101

lattice gas. For example, a bridge phase can sustain a shear strain [see ﬁgure

4.11(a), ﬁgure 4.11(b)]. Comparing the corresponding contour plots in ﬁgure

4.12(a) and ﬁgure 4.12(b) one sees that as a result of the applied deformation,

centers of molecular strata are displaced in the +x–direction. If the shear strain

exceeds a certain threshold one expects from the lattice–gas results (see ﬁgure

4.10) that the bridge phase undergoes a ﬁrst–order phase transition. Depending

on the “position” of the thermodynamic state with respect to µgb

x(T)andµbl

x(T)

either a droplet or a liquid phase may form as a result. Both situations are

realized as plots in ﬁgure 4.11(c) and ﬁgure 4.11(d) reveal.

Because of the similarity between the lattice–gas calculations and the Monte

Carlo simulations for the continuous model, it seems instructive to study the

phase behavior in the latter if the conﬁned ﬂuid is exposed to a shear strain.

This may be done conveniently by calculating ¯ρ[see (4.4)] as a function of µ

and αsx. Because of the microscopic size of the simulation cell, results are

again aﬀected by metastability in the immediate vicinity of a phase transition.

To identify coexisting phases along a sorption isotherm we adopt the procedure

described above in section 4.1.1.

For suﬃciently low µ<µ

dbl

x(TT) one expects a droplet phase to exist along

a subcritical isotherm T=(µ, T )Ttr <T

T<min Tgb

c,Tbl

c,T

T=const

(see

ﬁgure 4.10). At an intersection between Tand µdb

x(T) the droplet phase will

undergo a spontaneous transformation to a bridge phase. In a corresponding

plot of ¯ρversus µone should see a discontinuous jump to a higher density.

Eventually, another intersection between Tand µbl

x(T) exists and a second

discontinuous jump to an even higher value of ¯ρshould be visible in that plot.

Both of these transitions are indeed observed in ﬁgure 4.13 for αsx=0,µ≃

−8.40, and µ≃−7.98, respectively. Notice that in ﬁgure 4.13, µbl

x(TT)forαsx=

0.0 exceeds its bulk counterpart µbulk

x(TT), that is for µbl

x(TT) the corresponding

bulk phase is liquid. This can be rationalized by noting that the low(er)–density

part of a bridge phase is predominantly involved in this second transition. Recall

also that this part of a bridge phase is stabilized by the weak portions of both

(perfectly aligned) substrates characterized by fw ﬀ. Hence, the second

ﬁrst–order transition is inhibited rather than supported by the substrates (with

respect to the bulk) because of the dominating repulsive interaction of a ﬂuid

102 CHAPTER 4. RESULTS

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-8.5 -8.4 -8.3 -8.2 -8.1 -8 -7.9



Figure 4.13: Sorption isotherms ¯ρ(µ) from grand canonical ensemble Monte Carlo

simulations (continuous model); (◦), (−−−)αsx=0.0; (•), (−−−): αsx=2.5;

(



), (····): αsx=5.0; (



), (−·−): αsx=7.5; (



), (−··−): αsx=10.0. Also shown

are corresponding bulk data (



), (—). Results were obtained for T=0.7, sx=20.0,

ds=10.0, and sz=8.0.

molecule with the weak part of the substrate.

If a shear strain is applied, the region of overlap of the weak substrate

parts in the x–direction shrinks [see ﬁgure 3.1 and (3.10)] such that a ﬂuid

molecule located at {x|ds/2≤|x|≤sx/2,αs

x=0.0}is exposed to a stronger

net ﬂuid–substrate attraction. Consequently, one expects an associated shift

of µbl

x(TT) to lower values. The plot in ﬁgure 4.13 conﬁrms the expectation.

In addition, ﬁgure 4.10 shows that the one–phase region shrinks because Tdbl

shifts to higher temperatures and because the slope of the coexistence lines

does not change much. The plot in ﬁgure 4.10 therefore suggests that for α>0

the two discontinuities in ¯ρ(µ) approach each other so that the branch of ¯ρ(µ)

pertaining to bridge phases becomes narrower with increasing αsx. This eﬀect is

indeed visible in ﬁgure 4.13 where the width of the intermediate–density branch

of ¯ρ(µ) (corresponding to thermodynamically stable bridge phases) diminishes

4.2. THERMO-MECHANICAL PROPERTIES 103

from |∆µ|≃0.42 (αsx=0.0) to |∆µ|≃0.14 (αsx=7.5). Finally, if the shear

strain is large enough, the lattice–gas results in ﬁgure 4.10 suggest that for a

given temperature TT,Tdbl

tr (αsx)>T

Tfor suﬃciently large shear strains (see the

curve for α=2/7 in ﬁgure 4.10). Hence, under these circumstances one would

expect ¯ρ(µ) to exhibit just a single discontinuity referring to a phase transition

between droplet and liquid phases. The plot in ﬁgure 4.13 for αsx= 10 conﬁrms

this notion.

4.2 Thermo-mechanical properties

4.2.1 Shear stress of confined fluids

Thermodynamic stability of bridge phases turned out to depend on system

parameters (cr,nz,fs,fw) determining the inﬂuence of substrate heterogeneity

on the ﬂuid ﬁlm. Now we focus on mechanical stability and thermomechanical

properties of the conﬁned phases. From ﬁgure 4.12(a) the local density of bridge

phases appears to be inhomogeneous in a direction (x) perpendicular to the

substrate heterogeneities for all distances between the walls (i.e., along lines z=

const). Comparing ﬁgure 4.12(a) with ﬁgures 4.12(c) and (d) one realizes that

ﬂuid and droplet phases diﬀer from bridge phases signiﬁcantly in that they both

exhibit a central region where the respective local densities are approximately

independent of x, that is, gas and liquid phases are approximately homogeneous

in this central region. Thus, in the latter two phases the walls can slide more

or less freely over one another without substantial resistance, because their

homogeneous portions cannot sustain shear deformations. If, on the other hand,

the conﬁned ﬂuid is capable of connecting only certain parts of the substrates by

ﬂuid, as the ﬂuid bridges do (see ﬁgure 4.12(a) and (b)), the conﬁned phases may

sustain shear deformations, that is exhibit a nonvanishing shear stress (Tzx).

To extract thermomechanical properties within the continuous model, we

apply Monte Carlo simulations in the grand mixed isostress-isostrain ensemble

(see section 3.4). Under the present thermodynamic conditions, T=1.0, µ=

−11.455, Tzz =0.0, and 2.0≤ds≤8.0 the conﬁned ﬂuid forms a bridge phase

for various values of αsx. The key quantity calculated in the present Monte

Carlo simulations is the stress curve Tzx (αsx) accessible via (3.64), (3.65) and

104 CHAPTER 4. RESULTS

0.05

0.1

0.15

012345



Figure 4.14: Typical stress curve Tzx (αsx) for a monolayer bridge phase and cr=

5/10. Solid line is a least-squares ﬁt of a polynomial to the (discrete) Monte Carlo data

points (



) intended to guide the eye. The dashed line has been included to emphasize

the Hookean behavior at small strains.

the alternative expressions in (3.72). Regardless of the thermodynamic state

and the thickness (i.e., sz) of a bridge phase, a typical stress curve plotted in

ﬁgure 4.14 exhibits the following features:

1. For vanishing shear strain (i.e., α=0),Tzx (0) ≡0 for symmetry reasons.

2. Tzx (αsx) depends linearly on the shear strain αsxin the limit α→0, that

is the response of the bridge phase to small shear strains follows Hooke’s

law.

3. For larger shear strains, negative deviations from Hooke’s law are ob-

served, eventually leading to a yield point αyd,Tyd

zx deﬁned by the con-

stitutive equation

∂Tzx

∂(αsx)T,µ,sx,sy,Tzz α=αyd

= 0 (4.6)

or, alternatively using the shear modulus [see (3.78)],

c44 αydsx=0,ﬁxed T,µ,sx,s

y,T

zz.(4.7)

4.2. THERMO-MECHANICAL PROPERTIES 105

4. For symmetry reasons, Tzx (sx/2) ≡0 (i.e., for α=1/2).

These general characteristics of stress curves have also been observed previously

in simulations of “simple”–ﬂuid ﬁlms conﬁned between chemically homogeneous

but atomically structured (i.e., discrete) substrates [86, 98, 99, 100, 101, 102,

103, 104, 105, 106, 107, 108]. The substrates were composed of a single layer of

Lennard–Jones atoms arranged according to a plane of the face–centered cubic

lattice. In the earlier studies the unstrained phase was solidlike on account of a

template eﬀect imposed on the conﬁned ﬂuid by the discrete nature of the sub-

strate material. No solidiﬁcation occurs here under the present thermodynamic

conditions.

The impact of substrate corrugation.

However, as far as the present model is concerned, the degree of chemical corru-

gation of the substrate cr[see (4.5)] has signiﬁcant consequences for the yield–

point location αyd,Tyd

zx . Plots of stress curves for various values of crare

shown in ﬁgure 4.15(a). For monolayer bridge phases and ﬁxed sx=10one

can see from ﬁgure 4.15(a) that both Tyd

zx and αyd are smallest for the smallest

cr=2/10. For cr<2/10 only droplet phases are thermodynamically stable

because the strongly attractive portion of the substrate is too narrow to sup-

port formation of denser (bridge) phases. As crincreases both Tyd

zx and αyd

increase until they reach their maximum values αydsx,Tyd

zx ≈(2.740,0.169)

for cr=5/10. For larger cr>5/10 the plots in ﬁgure 4.15(a) show that both

Tyd

zx and αyd decrease again until αydsx,Tyd

zx ≈(1.550,0.069) for cr=8/10

which is the largest substrate corrugation for which bridge phases were ob-

served. For cr>8/10 only thermodynamically stable liquid phases formed in

the simulations, incapable of sustaining a shear strain.

One also notices from ﬁgure 4.15(a) that stress curves for cr=2/10, 3/10,

and 4/10 apparently do not cover the entire range of shear strains. In these cases

the bridge phase undergoes a shear–induced phase transition at some threshold

αcsxto form a droplet phase (see discussion in section 4.1.3). This droplet

phase, by virtue of its microscopic structure [see ﬁgure 4.12(c)], is incapable of

sustaining a shear stress. Thus, at αcsx,Tzx drops to zero discontinuously such

that Tzx ≡0 for all {α|αc≤α≤1/2}. For the sake of clarity we do not plot

106 CHAPTER 4. RESULTS

0.05

0.1

0.15

0 1 2 3 4 5

0.05

0.1

0.15

0 1 2 3 4 5

0.05

0.1

0.15

0 1 2 3 4 5

0.05

0.1

0.15

0 1 2 3 4 5

0.05

0.1

0.15

0 1 2 3 4 5

0.05

0.1

0.15

0 1 2 3 4 5

0.05

0.1

0.15

0 1 2 3 4 5



(a)

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1



(b)

Figure 4.15: (a) Stress curve Tzx (αsx;cr) for various chemical corrugations cr=2/10

(+), 3/10 (×), 4/10 (◦), 5/10 (



), 6/10 (



), 7/10 (∗), 8/10 (



). Solid lines are intended

to guide the eye. (b) Reduced stress curve



Tzx (



α;cr) [see (4.8)] where symbols are

referring to data plotted in (a). The solid line is a representation of (4.13).

4.2. THERMO-MECHANICAL PROPERTIES 107

this part of the stress curves in ﬁgure 4.15(a).

Despite this nonmonotonic variation of the yield–point location with crit

turns out that within the theory of corresponding states (see section 12-7 in [82])

it is feasible to renormalize stress curves such that all data points fall onto a

unique master curve. Renormalization is eﬀected by introducing dimensionless

variables

αsx→+α:= αsx

αydsx

(4.8)

Tzx →+

Tzx (+α):=Tzx (+α)

Tyd

Normalization by αyd and Tyd

zx is consistent with the theory of corresponding

states because it was pointed out in [86] that the yield point may be perceived as

a shear critical point analogous to the liquid–gas critical point in one–component

homogeneous ﬂuids. If the simulation data plotted in ﬁgure 4.15(a) are renor-

malized according to this recipe, they can indeed be represented by a master

curve as the plot in ﬁgure 4.15(b) shows.

Universality of stress curves.

The remarkable insensitivity of +

Tzx (+α) in ﬁgure 4.15(b) to variations of cr(3.78)

can be rationalized as follows. Because of the Hookean regime in the limit

αsx→0, c44 should be approximately constant and positive in this limit. A

typical plot in ﬁgure 4.16 conﬁrms this notion. However, because of (4.7) one

expects c44 to decline from its Hookean value as αsx→αydsxalso in agreement

with ﬁgure 4.16. Furthermore, since ﬁgure 4.16 shows that the variation of c44

with αsxis not too strong over the range α0≤α≤αyd , it seems sensible

to expand c44 in a power series according to

c44 (αsx)=

∞



k=0

d(k)c44

d(αsx)kα=0

(αsx)k=

∞



k=0

ak(αsx)k≃a0+a2(αsx)2(4.9)

where we refer to the far right side as the small–strain approximation. Notice

that the set of coeﬃcients {ak}refer to the unstrained bridge phase (i.e., α=

0). A molecular expression for a0≡c44 (0) is given in (3.78). In the small–

strain approximation a2accounts for deviations from Hookean behavior and

may therefore be interpreted as a measure of plasticity of the unshered conﬁned

ﬁlm.

108 CHAPTER 4. RESULTS

-0.2

-0.1

0.1

0.2

0 1 2 3



Figure 4.16: Shear modulus c44 as function of shear strain αsx.(◦): Monte Carlo

simulations in grand mixed stress–strain ensemble; (—): representation of small–

strain approximation c44 (αsx)=a0+a2(αsx)2[see (4.9), (4.10)]. Note: systems

having negative values of c44 are mechanically unstable.

The vanishing of the coeﬃcients a2k−1(k=1,...,∞) in (4.9) can be

rationalized by symmetry considerations. As detailed in appendix B the x–

component of the force exerted by the walls on a ﬂuid molecule is an odd

function of x. For the two highly symmetric cases α=0andα=1/2, respec-

tively, the probability density of the grand mixed isostress-isostrain ensemble

f0sz,N,rN[see (3.52)] is symmetric with respect to the plane x=0. There-

fore $F[k]

x%≡0, k=1,2. Thus from the deﬁnition of {ak}in (4.9) and (3.77)

it is furthermore clear that for α=0,a2k−1≡0(k=1,...,∞). However, we

note in passing that these coeﬃcients do not vanish apriorifor α= 0 since the

f0sz,N,rNis asymmetric in this case. From (3.77) and (4.9) we obtain the

(shear stress) equation of state

Tzx (αsx)=

αsx

-0

d(αsx)c44 (αsx)≃a0αsx+1

3a2(αsx)3(4.10)

4.2. THERMO-MECHANICAL PROPERTIES 109

Table 4.1: Comparison of shear modulus c44 from molecular expression and

yield–point location.

From (4.11) From (3.78)

crszαydsxTyd

zx c44 (0) c44 (0)

2/10 2.113 1.350 0.075 0.084 0.079

4/10 2.075 2.499 0.161 0.096 0.088

4/10 3.057 2.588 0.101 0.058 0.060

5/10 2.069 2.743 0.169 0.092 0.101

6/10 3.044 2.412 0.095 0.059 0.066

based upon the small–strain approximation. In principle, a0and a2are deter-

mined by ordinate and initial curvature of the function c44 (αsx)(α→0) (see

ﬁgure 4.16). The latter is extremely diﬃcult to extract from a molecular expres-

sion given the typical accuracy with which the shear modulus can be calculated

in our Monte Carlo simulations (see ﬁgure 4.16). However, an accurate estimate

is possible based upon (4.6) which, together with (4.10) leads to

a0≡c44 (0) = 3

Tyd

αydsx

(4.11)

and

a2≡1

d2c44 (αsx)

d(αsx)2α=0

=−3

Tyd

(αydsx)3(4.12)

in terms of yield stress and strain. These latter quantities can be determined

with high precision from (3.64), (3.65), (3.72), and plots similar to the ones

shown in ﬁgure 4.14, ﬁgure 4.15(a), and ﬁgure 4.21(a). Validity of (4.11) is

illustrated by table 4.1 where we compare it with the shear modulus obtained

directly from the molecular expression (3.78) for a selection of unsheared bridge

phases. Normalizing the (shear) equation of state (4.10) by using (4.8) and

subsequently inserting (4.11) and (4.12) permits one to recast (4.10) as

Tzx =+α3−+α2

2(4.13)

where up to the yield point 0 ≤+α≤1and0≤+

Tzx ≤1 are dimensionless quan-

tities so that (4.13) may be viewed as a master (stress) equation in agreement

with the plot in ﬁgure 4.15(b). We emphasize that the master equation is a

direct consequence of the small–strain approximation. A unique representation

110 CHAPTER 4. RESULTS

of stress curves is precluded if, on the other hand, one includes higher–order

terms proportional to a2k(k≥2) in the expansion (4.9) because then expres-

sions for αydsxand Tyd

zx [determined via (4.7), (4.10)] depend on the expansion

coeﬃcients in a complex way. Thus, contrary to Ref [86] there is no hope to

obtain a unique expression like (4.13) free of any materials constants {a2k}.

4.2.2 Thermodynamic stability

From a fundamental point of view, bridge phases comprising diﬀerent numbers

of molecular strata may be viewed as diﬀerent thermodynamic phases. This

interpretation is evident from (3.23) indicating that these diﬀerent bridge mor-

phologies (generally corresponding to diﬀerent values of szand Tyy) will exhibit

diﬀerent values of the grand mixed isostress-isostrain potential Ψ [see (3.23)].

A multiplicity of morphologies exists despite the fact that the thermodynamic

state is uniquely speciﬁed by the set {T,µ,sx,s

y,T

zz,αs

x}of natural variables

of Ψ. However, from an equilibrium perspective only the morphology corre-

sponding to the global minimum of Ψ is a thermodynamically stable phase; the

others must be metastable.

Fortunately, only a small, ﬁnite number of possible morphologies can ex-

ist under the present thermodynamic constraints. This can be understood by

considering the (normal) compressional stress Tzz (see (3.75), (3.76)) plotted as

a function of substrate separation szin ﬁgure 4.17(a). Data plotted in ﬁgure

4.17 were obtained in Monte Carlo simulations in the grand canonical ensemble

which has the advantage that the wall distance szbelongs to the set of natural

variables instead of Tzz. The plot in ﬁgure 4.17(a) shows that Tzz is a damped

oscillatory function of sz. These oscillations are ﬁngerprints of stratiﬁcation

(see section 4.1.1), that is the formation of new ﬂuid layers as the substrate

separation increases at constant Tand µ[79]. Damping can be ascribed to the

decreasing inﬂuence of the wall potential Φ(x, z)ifszincreases. Eventually the

system undergoes a phase transition to a droplet phase (see ﬁgure 4.6). Droplet

phases exhibit a local density which is low but approximately uniform in the core

region [see ﬁgure 4.11(c)]. This region increases with increasing sz.Thusthe

local density in the core region equals more and more that of the corresponding

4.2. THERMO-MECHANICAL PROPERTIES 111

-1

-0.5

0.5

2 3 4 5 6

-0.8

-0.6

-0.4

2 3 4 5 6

(a)

(b)



Figure 4.17: (a) Normal compressional stress Tzz (a) (see (3.75), (3.76)) as function

of substrate separation from Monte Carlo simulations in grand canonical ensemble (◦)

(αsx=0.0). Solid lines are intended to guide the eye. (b) as (a) but for line density of

the grand potential ω[see (4.16)]. Intersections between the latter and vertical lines

demarcate (meta– or thermodynamically) stable states in the grand mixed stress–

strain ensemble for Tzz =0.0 (see text).

bulk phase (i.e. bulk phase at the same Tand µ). As a result

lim

sz→∞ Tzz (sz)=−Pbulk (4.14)

where Pbulk (µ, T )≃0.03 is the bulk pressure. In other words, because strati-

ﬁcation diminishes with increasing sz, oscillations in Tzz (sz) vanish eventually,

too [109]. Therefore, the plot in ﬁgure 4.17(a) shows that under the present

conditions and for sz≥6.0 stratiﬁcation becomes unimportant.

In the grand mixed stress–strain ensemble morphologies consistent with the

set {T,µ,sx,s

y,T

zz,αs

x}of state variables can now be identiﬁed with intersec-

tions between the oscillatory curve Tzz (sz) and the isobar Tzz =const≤0.

However, only intersections for which dTzz/dsz≥0 correspond to (thermo-

dynamically or meta–) stable states as pointed out in [110]; intersections for

which dTzz/dsz<0 pertain to mechanically unstable states which cannot be

112 CHAPTER 4. RESULTS

-0.5

0.5

2 3 4 5 6

-0.6

-0.5

-0.4

2 3 4 5 6

(a)

(b)



Figure 4.18: As ﬁgure 4.17, but for αsx=2.25.

-0.5

0.5

2 3 4 5 6

-0.6

-0.5

-0.4

2 3 4 5 6

(a)

(b)



Figure 4.19: As ﬁgure 4.17, but for αsx=2.50.

4.2. THERMO-MECHANICAL PROPERTIES 113

realized in the grand mixed stress–strain ensemble. The thermodynamically

stable phase corresponds to the intersection having the smallest (areal) grand

potential density (3.21)

ω=Ω/Az=Tyysz(4.15)

at Tzz = 0. Based upon this rationale, an inspection of ﬁgure 4.17 shows that

the thermodynamically stable, unstrained morphology (α=0.0) is a monolayer

ﬁlm with sz≃2.1(Tzz =0.0). If conﬁned ﬁlms are progressively sheared a

parallel analysis of plots in ﬁgure 4.18 and ﬁgure 4.19 shows that the minimum

of ωfor sz≃2.1 becomes shallower while another minimum around sz≃3.1,

corresponding to a bilayer ﬁlm, becomes deeper with increasing shear strain.

Eventually the depth of the latter minimum exceeds that pertaining to the

monolayer ﬁlm so that a bilayer ﬁlm becomes the thermodynamically stable

phase. Thus, a shear strain exists such that ωis the same for mono– and

bilayer ﬁlms, meaning that a monolayer phase coexists with a bilayer phase.

To obtain a more concise picture of thermodynamic stability of diﬀerent

ﬁlm morphologies we plot the (areal) grand mixed isostress-isostrain potential

density (3.23)

ζ=Ψ/Az=(Tyy −Tzz)sz(4.16)

as a function of αsxin ﬁgure 4.20 for the same system analyzed in ﬁgures 4.17-

4.19. In a sequence of Monte Carlo simulations in the grand mixed isostress–

isostrain ensemble we calculate ζdirectly from (4.16), (Tzz = 0) using the molec-

ular expression for Tyy given in (3.73). An alternative expression for ζ(αsx)can

be obtained by integrating (3.24)

ζ(αsx)=ζ(0) +

αsx

-0

d(αsx)Tzx (αsx),ﬁxed T,µ,sx,s

y,T

zz (4.17)

≃ζ(0) + a0

2(αsx)2+a2

12 (αsx)4

where the second line is based upon the small–strain approximation (4.10). Full

lines in ﬁgure 4.20 are representations of (4.17) where the constants a0and

a2were determined as in section 4.2.1. Solid lines plotted in ﬁgure 4.20(a)

are therefore obtained without further adjusting a0and a2;ζ(0) is taken from

Monte Carlo simulations for unstrained bridge phases. The excellent agreement

114 CHAPTER 4. RESULTS

-1.4

-1.3

-1.2

-1.1

-1

-0.9

012345





(a)

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

0 1 2 3 4





(b)

Figure 4.20: (a) Areal free–energy density ζas function of shear strain αsxfor mono–

(◦), bi– (



), and trilayer (+) morphologies calculated in grand mixed stress–strain

ensemble Monte Carlo simulations [see (4.16), (3.73)] for cr=6/10. Solid lines are

calculated from (4.17). (b) As (a), but for cr=4/10.

4.2. THERMO-MECHANICAL PROPERTIES 115

between ζ(αsx) from the Monte Carlo simulations in the grand mixed stress–

strain ensemble and the small–strain–approximation in (4.17) highlights once

more the validity of the latter for all α≤αyd. However, the plot in ﬁgure 4.20(a)

also shows that the small–strain assumption is doomed to fail for suﬃciently

large shear strains in accord with one’s expectation.

From the plots in ﬁgure 4.20(a) one also notices that ζ(and therefore Ψ,

Az= const) is lowest for a monolayer bridge phase over the range 0.0≤αsx.

2.3 indicating that the monolayer is the thermodynamically stable phase in this

regime. Figure 4.20(a) also shows that intersections α∗sxexistatwhichΨfor

a pair of diﬀerent morphologies assumes the same value. Thus, at α=α∗these

diﬀerent phases coexist so that the points α=α∗correspond to ﬁrst–order phase

transitions between bridge phases comprising diﬀerent numbers of molecular

strata. While there is no obvious relationship between α∗for the coexistence

of mono– and bilayer morphologies and αyd, we notice that for all the cases

investigated a monolayer ﬁlm is the thermodynamically stable morphology for

all α≤αyd so that up to the yield point, plots in ﬁgure 4.15 apparently pertain

to thermodynamically stable phases.

Thicker ﬁlms are therfore thermodynamically stable only if the shear strain

exceeds the yield strain. For example, plots in ﬁgure 4.20(a) for cr=6/10 show

that ζfor a bilayer bridge phase is lower than for the corresponding monolayer

bridge phase over the range 2.3.αsx≤5.0 where the bilayer bridge phase

is the thermodynamically stable phase according to the above discussion. An

additional trilayer bridge phase was investigated for cr=4/10 as plots in ﬁgure

4.20(b) show. For cr=4/10 the bilayer is thermodynamically stable over the

range 2.4.αsx.3.3 whereas the trilayer ﬁlm seems to be thermodynamically

stable over the range 3.3.αsx.4.0 where all three curves end. However,

for the trilayer morphology the statistical error of ζ(αsx) is already quite large

because Tyy is small (see ﬁgures 4.17-4.19). For α≃4.0 bridge phases become

unstable and the system undergoes a ﬁrst–order phase transition to a gas phase

(see section 4.1.3).

It is furthermore noteworthy that universality of stress curves, in the sense of

section 4.2.1, is not restricted to monolayer ﬂuids. Plots of +

Tzx versus +αin ﬁgure

4.21(b) show that simulation data for mono–, bi–, and trilayer bridge phases can

116 CHAPTER 4. RESULTS

0.05

0.1

0.15

012345

0.05

0.1

0.15

012345

0.05

0.1

0.15

012345



(a)

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1



(b)

Figure 4.21: (a) As ﬁgure 4.15(a), but for mono– (◦), bi– (



), and trilayer (+)

morphologies and cr=4/10. (b) as ﬁgure 4.15(b) but for data points plotted in (a).

4.2. THERMO-MECHANICAL PROPERTIES 117

also be mapped onto the master curve (4.13) according to the treatment detailed

in the previous section. Again, the stress curves in ﬁgure 4.21(a) end at some

αcsxbecause the bridge phases evaporate (see section 4.1.3).

Chapter 5

Discussion and conclusions

Phase behavior and mechanical properties of simple ﬂuids conﬁned by chemically

corrugated substrates were investigated theoretically in Monte Carlo simulations

in the grand canonical and a grand mixed isostress-isostrain ensemble. These

computer simulations supplement density–functional calculations of a parallel

lattice model based upon a mean–ﬁeld treatment of the intrinsic free energy. In

general, conﬁnement introduces a new lengthscale with profound consequences

for phase behavior and materials properties of soft condensed matter (see chap-

ter 4). In this work we are exclusively concerned with conﬁning substrates en-

dowed with stripes of chemically diﬀerent materials. The widths of these stripes

add yet another relevant lengthscale to the system. Hence, a relatively large set

of parameters is needed to describe a model system with nanopatterned sub-

strates. To investigate the eﬀect of variations of these parameters on the phase

behavior of the conﬁned ﬂuid requires an approach which, on one hand, is ca-

pable of representing the heterogeneity of the conﬁning substrates and, on the

other hand, permits one to obtain phase diagrams at moderate computational

expense. The mean–ﬁeld treatment (see section 2.3.1) of a conﬁned lattice gas

(see ﬁgure 2.1) serves this purpose in an almost ideal manner.

However, the mean–ﬁeld lattice–gas model is based upon two key assump-

tions. First, positions of ﬂuid molecules are restricted to sites of a simple (cubic)

lattice and, second, intermolecular correlations are explicitly neglected. Based

upon these two assumptions the grand potential becomes a functional of the

118

119

local density Ω[ρ] [see (2.75)]. From a variational treatment of Ω[ρ] one obtains

a set of transcendental (Euler–Lagrange) equations [see (2.77)] which must be

solved to identify thermodynamically stable phases and metastable morpholo-

gies associated with global and local minima of the grand potential, respectively.

In the limit of vanishing temperature (T= 0) the lattice–gas model may

be treated analytically (see section 2.2.3) because the grand potential is given

by the internal energy only, while entropic contributions according to the Third

Law of Thermodynamics vanish. The identiﬁcation of thermodynamically stable

phases and transformations between them requires knowledge of all possible

morphologies. A systematic investigation of these morphologies at T= 0 in the

case of a single homogeneous surface was ﬁrst presented by Pandit et al [11]. The

chemical heterogeneity of the present substrates, on the other hand, induces a

variety of new morphologies diﬀering from those reported in Ref. [11] in both

number and structure. However, a systematic classiﬁcation of morphologies

is still possible utilizing the modular approach detailed in section 2.2.2. The

modular approach developed in this thesis enables one to identify the complete

set of morphologies and subsequently to determine the full phase diagram at

T=0.

For higher temperatures (T>0) an analytic solution of the lattice–gas model

is no longer possible. Instead, the full phase diagram is obtained by solving the

mean-ﬁeld equations (2.77) iteratively. Since the mean–ﬁeld treatment is exact

in the limit of vanishing temperature (see section 2.3.3), exact solutions derived

at T= 0 serve as suitable initial guesses for the iteration procedure. This is

particularly convenient because it can be rationalized that the iteration proce-

dure utilized here provides the full equilibrium phase diagram [see appendix A].

Therefore, the present approach prevents one from accidentally missing parts

of the phase diagram, as in previous work where less sophisticated techniques

were used [79, 64, 65, 93, 94, 95, 96].

The exact phase diagrams µx(T) obtained here turn out to be ramiﬁed webs

of coexistence lines µαβ

x(T), that is, lines of ﬁrst order phase transitions between

phases αand β. The structure of the web depends distinctly on the system

parameters. As these parameters vary, the web evolves, where some of the

coexistence lines are (almost) independent of the variation of certain model

120 CHAPTER 5. DISCUSSION AND CONCLUSIONS

parameters (see ﬁgure 2.11 and ﬁgure 4.2). This is already evident from the

analytic solutions in (2.53) at T= 0, where some of the expressions for µαβ

xare

independent of certain subsets of system parameters (see section 2.2.3). Thus,

the solutions at T= 0 are also useful to predict variations of the phase diagram

due to changes of single system parameters.

Unfortunately, the two key assumptions of the mean–ﬁeld lattice gas, namely

discretization of space and lack of correlations are quite unrealistic as far as

ﬂuids are concerned. Moreover, the impact of both assumptions is apriori

uncontrollable. Therefore, we attempt to verify predictions of the lattice–gas

model with the aid of a more realistic continuous model in which the treatment

of intermolecular correlations is exact.

Equilibrium properties of the continuous model were obtained by Monte

Carlo simulations in the grand canonical ensemble, which treats the conﬁned

phase as an open system in the thermodynamic sense. The price paid for the

more realistic modeling of conﬁned ﬂuids is a much greater demand for computer

time by Monte Carlo simulations compared with the lattice model. This, in

turn, limits the scope of Monte Carlo methods so that only key predictions

of the lattice–gas treatment could be checked. These concern the sequence of

phase transitions expected upon variations of substrate separation in section

4.1.1 where the excellent qualitative agreement between the two approaches

is illustrated. Therefore, we conclude that the mean–ﬁeld lattice gas model

is capable of representing the phase behavior of conﬁned ﬂuids qualitatively

correctly as far as ﬁrst–order phase transitions are concerned.

Phase transitions of ﬂuids conﬁned by chemically decorated substrates can be

divided into two groups: surface–induced and conﬁnement–induced transitions.

Surface–induced transitions controlled by a single chemically heterogeneous sur-

faces have been studied extensively by Dietrich and co-workers [37, 111, 112,

113]. The simplest conﬁnement–induced phase transition is the analogue of the

bulk liquid–gas transition known as “capillary condensation”. If the conﬁn-

ing substrates are decorated with chemical patterns composed of strongly and

weakly adsorbing materials, the strongly attractive parts of the substrate may

induce partial condensation of the ﬂuid whereupon a so–called bridge phase may

form. Consequently, bridge phases consist of a high–density regime ﬁlling the

121

gap between the strongly attractive substrate parts, surrounded by low–density

gas [see ﬁgure 4.3(b)]. Bridge phases are distinguished by their inhomogeneity

in the direction perpendicular to the alternating chemical stripes regardless of

the distances between the conﬁning walls. The present work is largely devoted

to a systematic investigation of thermodynamic and mechanical properties of

bridge phases (see section 4.1) reported ﬁrst by R¨ocken and co-workers [64, 65].

This unique local structure causes a bridge phase to sustain a nonvanishing

shear strain (to which they “respond” with a nonvanishing shear stress). Ap-

plication of a shear strain is possible since the substrates are inhomogeneous

in one lateral direction (x), so that their misalignment in that direction (which

is ameasure of the applied shear strain) can be speciﬁed quantitatively. In the

limit of small deformations the shear stress increases linearly with the conjugate

strain (see ﬁgure 4.14). This can be interpreted as elastic deformation of the

bridges, that is Hookean behavior. Negative deviations from this Hookean re-

sponse at higher shear strains indicate plastic deformation. Increasing plasticity

eventually causes the shear stress to reach a maximum (yield point) and to de-

cay for strains exceeding the yield strain. Thus, shear-stress curves Tzx(αsx)of

ﬂuid bridges exhibit a qualitatively similar shape as those obtained for conﬁned

solidlike ﬁlms by Schoen et al [100]. However, these latter shear–stress curves

diﬀer in both lengthscale of the deformation and height of the maximum (i.e.

yield stress) from the ones calculated here [see ﬁgure 5.1(a)]. For bridge phases,

Tzx(αsx) varies nonmonotonously with the degree of substrate corrugation (see

ﬁgure 4.15).

In view of the Hookean behavior in the limit of small shear strains, the shear

modulus c44 can be expanded in powers of the shear strain [see (4.9)]. This

small–strain approximation eventually permits one to deduce the dependence

of Tzx on αsxin the vicinity of the unstrained bridge, that is for αsx=0

[see (4.10)]. Within the framework of a theory of corresponding states the

expressions for Tzx(αsx) obtained from the small strain–approximation can be

renormalized so that a master curve is obtained which no longer depends on

any system parameters [see (4.13) and ﬁgure 4.14].

The theory of corresponding states is not limited to the present bridge phases

but may also be applied to conﬁned solidlike ﬁlms as ﬁgure 5.1 clearly indicates.

122 CHAPTER 5. DISCUSSION AND CONCLUSIONS

0.5

1.5

0 0.1 0.2 0.3 0.4 0.5



(a)

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1



(b)

Figure 5.1: a)StresscurveTzx (α) for a solidlike ﬁlm forming between discrete sub-

strates (from [100]). Solid line is a least-squares ﬁt of a polynomial to the (discrete)

Monte Carlo data points (



) intended to guide the eye. b) Reduced stress curve



Tzx (



α)

[see (4.8)] calculated from the data plotted in (a). The solid line is a representation of

the master curve (4.13)

123

Consequently, a conﬁned fluid bridge exhibits qualitatively the same shear stress

curves as a conﬁned solid. In their SFA experiments (see chapter 1) Klein and

Kumacheva interpreted the appearence of a nonvanishing shear stress as clear

evidence for solidiﬁcation of the conﬁned ﬁlm [70, 71]. In the light of the present

results this is highly questionable.

This work is exclusively concerned with one–component (i.e., pure) conﬁned

phases. However, in future investigations it would be very intriguing to replace

the present single–component ﬂuid by binary (A–B) ﬂuid mixtures. Conse-

quently, three types of ﬂuid–ﬂuid interactions arise, namely A–A, B–B, and

A–B between molecules of the two mixture components. In addition, there

are two types of interactions between A molecules and the substrate and B

molecules and the substrate, respectively (assuming the substrate to be chemi-

cally homogeneous). In other words, the phase behavior of the conﬁned binary

mixture depends on ﬁve independent types of interaction potentials rather than

two as in a pure ﬂuid. It is therefore conceivable that additional phase transi-

tions may occur including liquid–liquid transitions. Wilding et al demonstrated

that even a symmetric binary bulk mixture (that is, A–A and B–B interactions

are identical and only the A–B interaction is diﬀerent) exhibits quite a complex

phase diagram [114]. In another study, Gelb et al recently studied a symmet-

ric binary mixture conﬁned to a cylindrical pore with homogeneous walls [115].

In their model none of the mixture components is preferentially adsorbed by

the pore wall. They observed a demixed ﬂuid phase consisting of alternating

“plugs” of ﬁnite size that are composed predominantly of one or the other molec-

ular species, respectively. This interesting morphology is very similar to bridge

phases analyzed here in that high– and low–density regimes of the bridge are

replaced by A–rich and B–rich regions (i.e., the “plugs”). However, within the

framework of a mean–ﬁeld lattice gas model of a conﬁned binary mixture for-

mation of such a plug morphology seems highly unlikely since nothing supports

internal interfaces energetically as long as the wall potential is translationally

invariant. Thus, the question arises whether the mean–ﬁeld lattice gas is inca-

pable of describing “plug–phases” (with ﬁnite plug size) in principle or whether

these plugs may form as metastable morphologies in the simulations of Gelb et

al [115].

124 CHAPTER 5. DISCUSSION AND CONCLUSIONS

So far we were concerned with a unique chemical structure of the substrate,

namely alternating chemical stripes of variable width causing the symmetry to

be broken in directions normal to the walls and perpendicular to the stripes.

Thus, system properties are translationally invariant only in the direction par-

allel to the stripes, that is, the ﬂuid is homogeneous in that direction. Therefore,

in the continuous model a Gibbs–Duhem equation exists, oﬀering the possibil-

ity to calculate thermodynamic potentials directly from ensemble averages of

“mechanical” properties [see (3.23) and (3.21)]. However, this “mechanical” ap-

proach cannot be extended to substrates coated with chemically patterns such

that properties of the conﬁned ﬂuid are not translationally invariant in any di-

rection. An example of such a substrate structure was recently employed by

Lenz et al [40] who studied the wetting of a single surface endowed with ring–

shaped surface domains of macroscopic extent. The lack of a closed analytic

expression for Ω in terms of stresses and strains is a nontrivial obstacle if one

wishes to investigate the phase behavior of ﬂuids interacting with such a low–

symmetry substrate in the spirit of section 3.2 of this work. The problem may,

however, be overcome by thermodynamic integration [75]. The main diﬃculty

then is that the integration path must not intersect with any coexistence line.

This, however, is problematic again since it requires knowledge of the apriory

unknown phase diagram. Depending on the details of the system in question the

latter may be rather complex as we have shown above. Applying the lattice–gas

model to substrates decorated with circular domains is also nontrivial since these

domains can be represented only very roughly by a sensible lattice structure.

Nevertheless, substrates endowed with circular chemical pattern seem quite

intriguing. In the spirit of the work of Lenz et al [40] one may envision circular

patterns consisting of alternating rings of weakly and strongly adsorbing solid.

If a ﬂuid is conﬁned by such decorated substrates partial condensation triggered

by the rings may occur in very much the same way in which bridges arise in

this study. However, in the former system this phase transition may also be

controlled by the curvature of the circular pattern with which the substrate is

decorated.

Appendix A

Jacobi–Newton iteration

The numerical procedure used to determine phase diagrams of the conﬁned

lattice gas (see ﬁgure 2.1) is presented schematically in ﬁgure A.1. We begin

with the analytic phase diagram at vanishing temperature (T= 0) (see section

2.2). For T>0 the phase diagram is obtained by computing grand potential

curves Ωα(µ) (of morphologies α) to identify thermodynamically stable phases

and phase coexistence [see section 2.4]. Since it is aprioriunknown which

morphologies will become coexisting phases at the actual temperature Tk>0,

we assume these to be the same as for the preceding temperature Tk−1(see

ﬁgure A.1). This is possible since for the initial temperature T0=0theexact

phase diagram is available.

However, one may accidentally miss triple points where a new coexisting

phase arises. In principle, this problem cannot be avoided aprioribut can be

circumvented a posteriori because the triple point can be determined accurately

as detailed below. To compute grand potential curves Ωα(µ) for morphologies

αat a given temperature Tk, equilibrium densities ρα={ρα

i}(which depend

on µ) must be determined [see (2.75)].

For a particular morphology α(henceforth, we drop the superscript αto

simplify notation) given Tkand µthis can be done by solving the variational

expression [see (2.77)] using the Jacobi–Newton iteration technique [116] pro-

ceeding in alternating sequences of “local” and “global” minimization steps. For

125

126 APPENDIX A. JACOBI–NEWTON ITERATION

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Figure A.1: Scheme of the iterative technique used to calculate phase diagrams of

the conﬁned lattice gas.

127

a particular site ione can rewrite (2.77) using (2.106)

Tln ρi

1−ρi−2ρi−ηi= 0 (A.1)

where ηi=ηi(µ, Φi,ρ

j) is a function of the chemical potential µ,theexternal

potential at lattice site i,Φ

i, the number of nearest neighbors in the x–z–plane,

and their associated local densities ρj={g

lρj},whereg

lρiis the local density at

lattice site iin the l-th local and g-th global minimization step. An estimate of

the density g

l+1 ρiis obtained via

l+1ρi=g

lρi−f(g

lρi)

f(g

lρi),i=0,1,2,... (A.2)

where from (A.1)

f(g

lρi)=Tln

lρi

1−g

lρi−2g

lρi−ηi(µ, Φi,g

lρj)(A.3)

and

f(g

lρi)= df(ρi)

dρ

i!g

lρi

lρi(1 −g

lρi).(A.4)

It is important to realize that throughout each local minimization cycle η={ηi}

is maintained at its initial value assigned at the beginning of that particular

cycle. The iterative solution of (A.2) is halted if max

ig

l∗ρi−g

l∗−1ρi≤10−7.

Local minimization is performed by visiting each lattice site consecutively; the

local cycle ends once all sites have been considered.

Global minimization then involves updating the local density of the entire

lattice according to g+1

0ρi=g

l∗

ρithus providing new initial values for the next

local minimization cycle [see (A.2) and ﬁgure A.1]. Global minimization is

carried out until max

ig+1

0ρi−g

0ρi≤10−7.

After determining phase coexistence and identifying thermodynamically sta-

ble phases at temperature Tk, one proceeds by increasing temperature Tk+1 =

Tk+∆Tand returning to step 5 of the scheme ﬁgure A.1).

There are some technicalities of the technique presented above worth men-

tioning:

•Inspecting (A.4) and (A.3) one realizes, that in the two limiting cases

ρi=0orρi= 1 the second term in (A.2) is zero. Therefore the iteration

fails if it is initialized using either ρi= 0 or 1. Instead, values like 0.0001

or 0.9999 should be employed in practice.

128 APPENDIX A. JACOBI–NEWTON ITERATION

0.0

0.5

1.0

1.5

0.0 0.5 1.0



(a)

βγ

-3.3

-3.2

-3.1

-3.0

0.0 0.5 1.0 1.5



(b)

Figure A.2: Phase diagram of a mean-ﬁeld lattice gas conﬁned by homogeneous

walls (nx= 10, ewall =1.1). Full lines represent the phase diagram µx(T), dotted

lines indicate the ﬁrst accidental calculation path. (a) T-ρrepresentation (b) µ-T

representation

129

•If fin (A.3) has a zero at g

lρithis zero is usually close to an extremum

of f. Both zero an d extremum get closer the lower the temperature is.

Therefore, at low temperatures initial densities are required which are

very close to the sought (equilibrium) solutions. For that reason Newton’s

method usually fails at low temperatures (T.0.4). A possibility to tackle

this problem is to introduce a damping coeﬃcient c(0 <c≤1) such that

(A.2) is replaced by

l+1ρi=g

lρi−cf(g

lρi)

f(g

lρi),i=0,1,2,... . (A.5)

However, in practice the accessible (low) temperature region remains lim-

ited on account of the sensitivity of initialization (usually T&0.25).

•Very low temperatures (T<0.1) can be reached by starting at a suﬃ-

ciently high temperature (Tw0.4) where Newton’s method succeeds and

which is still low enough to employ the analytical (T= 0) solutions to

initialize the iterative procedure. The calculation then proceeds as above

(see ﬁgure A.1) but decreasing Tin a stepwise fashion.

•The most substantial problem is the appearance of one or more triple

points in the phase diagram. At a triple point (Tαγβ

tr ,µ

αγβ

tr ) a coexistence

line µαβ

x(T) bifurcates into two branches µαγ

x(T)andµγβ

x(T), because a

new phase γbecomes thermodynamically stable. The diﬃculty is that a

priory one does not know where the triple point will appear and what the

structure of the new phase is. Thus, one may accidentally miss the new

phase altogether. Suppose, for instance, the phase diagram shown in ﬁgure

A.2 is calculated. Below the triple point temperature Tαγβ

tr we proceed

as detailed above. If the calculation is continued without care one may

stay on the branch µαβ(T) which is now metastable [see ﬁgure A.2(b)].

However, the metastable path µαβ(T) ends at the critical temperature

Tαγ

csince Ωα(µ, T ) vanishes at this temperature. Thus, for T&Tαγ

the solution “jumps” to the one belonging to coexistence line µγβ

x(T)[see

ﬁgure A.2(b)]. Now this artiﬁcial “jump” enables us to determine the

triple point (Tαγβ

tr ,µ

αγβ

tr ), since it is clear now that such a triple point

exists and what the new phase γis. Todosooneneedstogoback

to a temperature below the triple point T≤Tαγβ

tr,whichis,ofcourse

130 APPENDIX A. JACOBI–NEWTON ITERATION

still unknown. To locate Tαγβ

trone has to go back to successively lower

temperatures until Ωγµαβ

x=Ω

αµαβ

x≡Ωβµαβ

x. This procedure

permits us to eventually calculate the complete phase diagram.

•In practice, the critical point cannot be reached. This is because the

respective minima of the grand potential representing the two coexisting

phases are very close. Therefore they are hardly distinguishable causing

the iteration technique to fail numerically. Therefore, all phase diagrams

in the T–ρrepresentation are not closed in the vicinity of the critical point.

Appendix B

Fluid–substrate forces in

the continuous model

This appendix provides explicit expressions for the components f[k]

x(xi,z

i)and

f[k]

z(xi,z

i) of the force exerted by the substrates on a ﬁlm molecule ilocated at

apoint(xi,z

i) (see [79]).The z–component of that force is deﬁned by

f[k]

z(xi,z

i):=−∂Φ[k](xi,z

∂z (B.1)

where Φ[k](xi,z

i) is the mean–ﬁeld representation of the external potential [see

(3.10)]. From (B.1) one obtains after lengthy but straightforward algebra (see

[79])

f[k]

z(xi,z

i)=3π

2nAσ2

∞



m=−∞

∞



m=0

(B.2)

×(fw −fs){Θ[x

u(x, ds),z]−Θ[x

l(x, ds),z]}

+fw {Θ[x

u(x, sx),z]−Θ[x

l(x, sx),z]}

where x

uand x

lare deﬁned by (3.11) and (3.12), respectively,

Θ(x,z):=21

32K3(x,z)−K4(x,z)(B.3)

K3(x,z):=−2

xσ11

(z)3√R9 1+8

7S+48

35S2+64

35S3+128

35 S4!(B.4)

−xσ11

z√R11 1+8

7S+48

35S2+64

35S3+128

35 S4!

−2

(x)3σ11

(z)5√R9 8

7+96

35S+192

35 S2+512

35 S3!

131

132 APPENDIX B. FLUID–SUBSTRATE FORCES

and

K4(x,z):=−22

xσ5

(z)3√R3+xσ5

z√R53(1 + 2S)(B.5)

−4

(x)3σ5

(z)5√R3.

The quantities Rand Sare deﬁned by (3.7) and (3.8), respectively.

Similarly, one obtains

f[k]

x(xi,z

i):=−∂Φ[k](xi,z

∂x (B.6)

=3π

2nAσ2

∞



m=−∞

∞



m=0

×(fw −fs){θ[x

u(x, ds),z]−θ[x

l(x, ds),z]}

+fw {θ[x

u(x, sx),z]−θ[x

l(x, sx),z]}

where

θ(x,z)=I1(x,z)−I2(x,z)(B.7)

I1(x,z):=21

324σ2

R11

(B.8)

and

I2(x,z):=

4σ2

R5

.(B.9)

Because I1and I2are even functions of x [see also (3.7)], θis an even

function of x as well. For the special case αsx= 0, that is absence of a shear

strain, one easily obtains [see (3.11) and (3.12)]

−x

u(x, d)=x

l(−x, d, )and −x

l(x, d)=x

u(−x, d) (B.10)

since the transformation m−→ −mis inconsequential for the summations in

(B.6) ∞



m=−∞ ≡

∞



−m=−∞

.(B.11)

Thus,

θ[x

u(−x, d),z]−θ[x

l(−x, d),z] (B.12)

=θ[−x

l(x, d),z]−θ[−x

u(x, d),z]

=−5θ[x

u(x, d),z]−θ[x

l(x, d),z]6

133

and therefore f[k]

xis an odd function of x(provided αsx=0).

Appendix C

Implementation of the

fluid–substrate potential

From an inspection of (3.10) it is immediately clear that the ﬂuid-wall poten-

tial Φ[k]cannot be computed directly, since inﬁnite summation is numerically

impossible. However, following Schoen et al. [79] it turns out that in practice

a suﬃciently accurate numerical representation of Φ[k]is obtained by replacing

the double sum in (3.10) according to

∞



m=−∞

∞



m=0 −→



m=−2



m=0

.(C.1)

As shown in the same reference it is too time consuming to calculate the ﬂuid-

wall interaction in every simulation step, even in the case of truncated summa-

tions (C.1). Instead of computing Φ[k]during each simulation step, the trun-

cated version of Φ[k]is calculated prior to the simulation on a ﬁne grid xh,z

spanning the entire simulation cell. The potential at an arbitrary point in the

x–z–plane (Φ[k]is translationally invariant in y–direction) is then calculated by

bilinear interpolation [117].

The computational eﬀort can be reduced further by regarding the symmetry

of Φ[k][see (3.10)]. From their deﬁnitions [see (3.5), (3.6)] one identiﬁes I3and

I4to be odd functions of x (3.3) and therefore, ∆ [see (3.9)] to be an odd

134

135

function of x likewise. Thus,

∆x+1

2d, z−∆x−1

2d, z(C.2)

=−∆−x−1

2d, z+∆−x+1

2d, z

=∆

−x+1

2d, z−∆−x−1

2d, z

where the second equality reveals that Φ[1] is an even function of x. Therefore

it is suﬃcient to calculate Φ[1] in the region 0 ≤x≤sx/2 and to apply the

transformation

[1] : x−→ +x=|x|:Φ

[1](x, z)=Φ

[1](+x, z).(C.3)

Since the upper wall “2” is identical with the lower wall “1” except the spatial

position, Φ[2] is an even function of (x−αsx). Thus, the interaction contribution

from the upper wall can be obtained by introducing a transformation similar to

(C.3)

[2] : 5x−→ +x=|x−αsx|

z−→ −z6Φ[2](x, z)=Φ

[1](+x, −z).(C.4)

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Curriculum Vitae

Dipl. Chem. Henry Bock

Date of birth: 15/07/1970

Place of birth: Eisenh¨uttenstadt/Germany

Nationality: German

Marital Status: married, 2 children

Education

09/1985 - 09/1989 Carl-Friedrich-Gauß-Gymnasium Frankfurt (Oder)

11/1989 - 11/1990 Military service

04/1991 - 01/1997 Undergraduate, Chemistry Department, Technische Uni-

versit¨at Berlin

01/1997 Diploma in Chemistry (roughly equivalent to Masters de-

gree)

07/1997 - present Ph.D. student, thesis title “Fluids conﬁned by nanopat-

terned substrates”

Professional Experience

01/1998 - present Member of scientiﬁc staﬀ Sonderforschungsbereich 448

“Mesoscopically structured composite systems” (Chairman

G. H. Findenegg), TU Berlin

04/1998 - present Teaching Assistant in courses “Statistical Thermodynam-

ics” and “Mathematics for Chemists”