JOURNAL OF CHEMICAL PHYSICS VOLUME 118, NUMBER 3 15 JANUARY 2003

Fluid phase transitions at chemically heterogeneous, nonplanar solidsubstrates: Surface versus confinement effects

Sophie Sacquin,a) and Martin Schoenb)

Stranski-Lab. fu¨r Physikalische und Theoretische Chemie, Technische Universita¨t Berlin, Straße des 17,Juni 124, D-10623 Berlin, Germany

Alain H. Fuchsc)

Laboratoire de Chimie Physique, Universite´ de Paris-Sud, F-91405 Orsay Cedex, France

~Received 24 July 2002; accepted 22 October 2002!

The phase behavior of a ‘‘simple’’ Lennard-Jones~12,6!fluid confined between planar substrates hasbeen investigated by means of Monte Carlo simulations in the grand canonical ensemble~GCEMC!.The interaction of fluid molecules with the upper wall is purely repulsive and attractive with thelower one. A spherical cap of radiusj is embedded in the surface of the upper wall. The cap alsoattracts fluid molecules. Forj50, that is, in the case of two planar homogeneous substrates, oneobserves classical prewetting, namely a first-order phase transition from thin to thick films adsorbedon the lower substrate. WhenjÞ0, that is, when the upper substrate is decorated with athree-dimensional pattern of finite size, system properties are no longer translationally invariant inany spatial dimension. Thus, the grand potentialV is not a homogeneous function of degree one inany of its extensive variables which precludes a ‘‘mechanical’’ expression forV in terms of stressesand conjugate strains. Therefore, in order to determine the phase behavior through plots ofV versuschemical potentialm we employ a thermodynamic integration scheme under isothermal conditions(T5const). We observe a partially condensed phase, where the molecules are preferentiallyadsorbed in the gap between the spherical cap and the lower substrate, associated with either a thinor a thick film adsorbed on the lower substrate. ©2003 American Institute of Physics.@DOI: 10.1063/1.1529683#

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I. INTRODUCTION

Recent advances in microfabrication have allowedprepare heterogeneous substrate surfaces in a controlledwith domain sizes ranging from the millimeter to the nanoeter scale.1,2 The techniques employed to create these wdefined chemical structures range from photolithograp3

over local oxidation of the surface4 to atomic beams5 or theuse of diblock copolymer films.6 In the meantime many studies have drawn attention to the behavior of fluids interactwith these substrates, both from an experimental7,8 and atheoretical9–11 point of view.

In particular wetting of a solid substrate by a fluid phahas received much attention in the last decades~see Ref. 12for a recent review!, first in the case of substrates with simplanar and homogeneous surfaces,13–16 and then later in thecase of heterogeneous or structured surfaces.17–21 In particu-lar it was shown by Rasco´n and Parry that a rich phase bhavior has to be anticipated in the case of heterogenesurfaces.22 This is a consequence of the interplay betwevarious length scales like, for example, the ones determinthe surface heterogeneity and the thickness of the adsofluid film.

However, the behavior of fluids near heterogeneousfaces becomes even more fascinating and the observed

a!Electronic mail: sophie.sacquin@fluids.tu-berlin.deb!Electronic mail: martin.schoen@fluids.tu-berlin.dec!Electronic mail: alain.fuchs@lcp.u-psud.fr

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nomena more complex, when the fluid is confined to spaof nanoscopic dimensions by more than one such solid sstrate. Confinement introduces yet another length scaleprofound consequences for the phase behavior of thefined fluid.23 Fluids confined by either geometrically24–26 orchemically27–29 decorated substrates have already beenvestigated in a number of publications.

From a theoretical point of view the grand-potentialV isa key quantity in the context of the phase behavior of fluas far as thermodynamically open systems are concerneall previous studies to date~except Ref. 30!fluid propertiesare translationally invariant in at least one direction parato the substrates surfaces. This causesV to be a homoge-neous function of degree one in at least one of its exten~natural!variables. In other words, Euler’s theorem caninvoked and a closed ‘‘mechanical’’ expression forV can bederived in terms of suitably chosen stresses and strains aon the confined phase. This route is particularly usefulcomputer simulations where one can calculate these streand strains as ensemble averages from molecexpressions.31

If, on the other hand, the substrate structure is suchtranslational invariance of fluid properties is not preservedany spatial direction~a case termed ‘‘low-symmetry substrate’’ henceforth!, a ‘‘mechanical’’ expression forV doesnot exist. To computeV despite this difficulty an alternativeroute based upon a thermodynamic-integration schemebeen proposed recently.30 Although this method is computa

3 © 2003 American Institute of Physics

f radiuslso attracts

1454 J. Chem. Phys., Vol. 118, No. 3, 15 January 2003 Sacquin, Schoen, and Fuchs

FIG. 1. Schematic representation of a fluid~black spheres! confined between two planar substrates the upper one being decorated by a spherical cap oj attracting fluid molecules. The fluid–substrate interaction with the rest of the upper substrate is purely repulsive, whereas the lower substrate afluid molecules.

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tionally more demanding, it is completely general as farthe geometry of the nanopatterns decorating the subst~and therefore the symmetry of the confined fluid! is con-cerned, thus permitting studies of confined phases ofsymmetry. The purpose of this work is to apply the procdure developed in Ref. 30 to the case of a fluid confinbetween a substrate bearing a three-dimensional patterfinite size, that is to a system where both surface and cfinement effects are expected to participate and contributthe overall phase behavior of the fluid.

II. MODEL SYSTEM

As our system, sketched in Fig. 1, we consider a fluidspherically symmetric molecules confined between thefaces of two solid substrates. In principle, fluid molecuinteract with each other in a pairwise additive fashion viaLennard-Jones~LJ!~12,6!potential,

u~r !54eF S s

r D 12

2S s

r D 6G , ~1!

e being the well depth,s the ‘‘diameter’’ of a molecule, andr the distance between the centers of a pair of particles. Hever, for reasons detailed in Ref. 25 we replaceu(r ) in thesubsequent Monte Carlo simulations in the grand canonensemble~GCEMC! by

u~r !→uf f~r !

5H u~r !2u~r c!1du~r !/drur 5r c~r c2r ! r<r c

0 r .r c ,~2!

where r c is a cutoff radius whose value will be specifiebelow. From Eq.~2! it is clear that unlikeu(r ), uf f(r ) isexplicitly short-range. The substrates are separated by atancesz along thez-axis of the laboratory coordinate systemThey are semiinifinite, occupying half spaces2`,z<2sz/2 and sz/2<z,`, respectively. A spherical particlof radiusj,sz whose center is located atx50, y50, and

stes

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z5sz/2 is partly embedded in the upper substrate so that oits lower hemisphere interacts with fluid particles.

The lower substrate is composed of like atoms interaing with fluid molecules according to the LJ~12,6! potentialwhere the same values ofe and s are employed as for thefluid–fluid interaction. We employ a mean-field approximtion for the fluid–substrate potential energy achieved byeraging the~original! fluid–substrate interactions over thpositions of the substrate atoms in thex–y plane. This leadsto

F [1]~z!52perss

3

3 F 2

15S s

sz/21zD9

2S s

sz/21zD3G

5:w rep[1]~z!2watt

[1]~z!, ~3!

wherers is the ~volume!number density of wall atoms.Atoms forming the upper substrate are only repuls

with respect to their interaction with the fluid molecules,that

F [2]~z!54perss

3

45 F S s

sz/22zD9G5:w rep

[2]~z!. ~4!

The surface of the sphere embedded in the upper sstrate is also composed of Lennard-Jones particles wit~surface!number densityrs8 . Based upon a similar meanfield approximation we obtain for the potential function goerning the interaction between a fluid molecule and the enembedded sphere,

C~R;j!52pejrs8s

2

R H 2

5 F S s

R2j D 10

2S s

R1j D 10G2F S s

R2j D 4

2S s

R1j D 4G J , ~5!

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1455J. Chem. Phys., Vol. 118, No. 3, 15 January 2003 Fluid phase transitions at heterogeneous nonplanar surfaces

whereR is the distance between the centers of a fluid partand that of the sphere, that is,

R5@x21y21~sz/22z!2#1/2. ~6!

Because of the repulsive part of this interaction, we walways haveR.j ~which means fluid molecules cannot peetrate into the surface of the spherical cap!. Throughout thiswork we fix j55s. HenceR1j will always be sufficientlylarge so that terms proportional to (R1j)2n (n54,10) inEq. ~5! can safely be neglected compared with those proptional to (R2j)2n. We can then rewrite the fluid–spheinteraction as

C~R;j!→C~R;j!5j

R@c rep~R;j!2catt~R;j!#, ~7!

where

c rep~R;j!54pers8s

2

5 S s

R2j D 10

, ~8!

catt~R;j!52pers8s2S s

R2j D 4

. ~9!

Eventually, the total interaction potential between a fluid pticle and the substrates can be cast as

uf s~z,R;j!5F [1]~z!1F [2]~z!1C~R;j!. ~10!

We include also the interaction between the lower sstrate and the upper one~semi-infinite solid and sphere! lead-ing to

uss~sz ;j!54eprss

3

45 S s

szD 9

12ep2rsrs8js4

3

3H 1

30F S s

sz2j D 8

2S s

sz1j D 8G2S s

sz2j D 2

1S s

sz1j D 2J ~11!

by similar manipulations and reasoning.Thus, our model resembles situations in experiments

ploying the atomic force microscope~AFM! where one isinterested in measuring adhesion~or friction! between anAFM tip ~i.e., the spherical cap!and a polymer-coated solisubstrate under conditions of controlled relative humid~i.e., fixed chemical potentialm) ~see, for example, Ref. 32!

III. THEORY

A. Thermodynamics

The system in the thermodynamic sense is a finlamella of dimensionssx3sy3sz of the confined fluid whichwe assume to be infinite in thex- andy-directions. Hence bydisplacing the boundaries between the lamella and themainder of the confined fluid~forming the environment ofthe lamella in the thermodynamic sense! infinitesimally byds the former can exchange mechanical workdW5F•dswith the latter, whereF is the force conjugate to the~com-pressional!strain ds. In addition, the lamella is materialland thermally coupled to its environment, that is thermo

e

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-

-

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e-

-

namically it constitutes an open system. Therefore, reverstransformations of the lamella are governed by changethe grand potential whose exact differential is given by

dV~T,m,s!52SdT2Ndm1F•ds. ~12!

At this point it is convenient to introduce the stress tensoTthrough

F5 AT•T5 ~¹sa

V!T,m,$sb%bÞa, ~13!

whereAT5(Ax ,Ay ,Az)[(sysz ,sxsz ,sxsy) is the~transpose!vector whose elementsAa are areas of faces of the~rectan-gular! lamella with a normal pointing in thea-direction. InEq. ~12!, S denotes entropy,T is temperature,N is the num-ber of fluid molecules, and the vectors5(sx ,sy ,sz) such that13A•s5sxsysz5:V, whereV is the volume of the lamella.

In the absence of shear forces~a situation to which thiswork is restricted exclusively!, T is diagonal and can berepresented by a 333 matrix with ~Cartesian!componentsTxxÞTyyÞTzz, (Tab50, ;aÞb) because the fluid–substrate potential in Eq.~10! depends onR for jÞ0. On theother hand, if j50, C(R)50 @see Eq. ~7!#, such thatuf s(z,R;j)→uf s(z) @see Eq.~10!#. In this latter caseT hasonly two different diagonal components, namely,Ti5Txx

5TyyÞTzz. It is then convenient to transform variables acording to dAz5d(sxsy)5sydsx1sxdsy and rewrite Eq.~12! more explicitly as

dV~T,m,Az ,sz!52SdT2Ndm1szTidAz1AzTzzdsz .~14!

Under these conditions and for fixed values ofT, m, andsz ,V is a homogeneous function of degree one inAz . Thus,Euler’s theorem applies and we obtain (j50)

V5TiszAz5TiV, fixedT,m,sz ~15!

as a closed ‘‘mechanical’’ expression for the grand potentHowever in the general casejÞ0, V is not a homogeneoufunction of degree one inanyof its extensive variables andclosed expression parallel to Eq.~15! does not exist.

B. Statistical mechanics

In the context of this work,V is the key quantity. Froma molecular perspective,V is related to the grand partitionfunction J through33

V~j!52b21 ln J~j!

52b21 ln (N50

`exp~bmN!

L3NN!Z~j!, ~16!

whereb51/kBT (kB Boltzmann’s constant!, L is the ther-mal de Broglie wavelength and the far right side obtains athe usual integration over momentum subspace. The confiration integral is given by

Z~j!5)i 51

N E2sx/2

sx/2

dxiE2sy/2

sy/2

dyiE2sz/2

zmax( r i ;j)dzi

3exp@2bU~j!#, ~17!

where r i5u r i u, with r i5(xi ,yi)PR2 and the configurationaenergy is

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1456 J. Chem. Phys., Vol. 118, No. 3, 15 January 2003 Sacquin, Schoen, and Fuchs

U~j!51

2 (i 51

N

(j 51Þ i

N

uf f~r i j !1(i 51

N

uf s~zi ,Ri ;j!

1uss~sz ;j!. ~18!

On account of the embedded sphere the upper substrsurface is not parallel with thex–y plane. Consequently thupper limit on thez-integrations in Eq.~17! depends on thex- andy-coordinates, where~see Fig. 1!

zmax~ r ;j!5H sz/22Aj22x22y2 ~x21y2!,j

sz/2 ~x21y2!>j.~19!

From Eqs.~14! and~16!–~18! we deduce the statistical themodynamic expression for the diagonal component ofstress tensor,

Taa521

bJAa(N50

`exp~bmN!

L3NN!S ]Z

]saD

T,m,$sb%bÞa

~20!

a5x,y,z.

Details of the evaluation of]Z/]sa are presented in the Appendix.

For the limiting case of a homogeneous upper substrthat is forj50, the fluid–substrate contribution in Eq.~18!does not depend on thex- and y-coordinates of the fluidmolecules. Therefore, we obtain a simpler molecular expsion for Ti5(Txx1Tyy)/2 with the aid of Eqs.~A5!–~A11!,namely,

Ti52^N&bV

11

4V K (i 51

N

(j 51Þ i

N

uf f8 ~r i j !xi j

2 1yi j2

r i jL , j50,

~21!

where ^•••& denotes an average in the grand canonicalsemble anduf f8 (r )5duf f(r )/dr. Because of Eq.~15!, Eq.~21! permits access to the grand potential in the limiting cof a chemically homogeneous, planar upper substrate.

From Eqs.~12!, ~13!, and~16!–~18! one can also showthat

Tzz~j!5TzzFF~j!1Tzz

FS~j!1TzzSS~j!. ~22!

By manipulations parallel to those used to reach Eq.~21! wefind in the more general casejÞ0,

TzzFF~j!52

1

bAzK (

i 51

N1

Dz~ r i ;j!L

11

2AzK (

i 51

N

(j 51Þ i

N uf f8 ~r i j !zi j

r i jF zi1sz/2

Dz~ r i ;j!

2zj1sz/2

Dz~ r j ;j!G L , ~23!

e’s

e

e,

s-

n-

e

TzzFS~j!52

1

AzK (

i 51

N 9w rep[1]~zi !23watt

[1]~zi !

Dz~ r i ;j!

19w rep

[2]

Dz~ r i ;j!Fzi2zmax~ r i ;j!

sz/22ziG L

11

AzK (

i 51

N

2j

Ri2 Fc rep~Ri ;j!2catt~Ri ;j!

Ri

110c rep~Ri ;j!24catt~Ri ;j!

Ri2j G3F sz/22zi

Dz~ r i ;j!G

3@zmax~ r i ;j!2zi #L , ~24!

where we use the expression forC(R;j) given in Eq.~7!,andDz is defined in Eq.~A2!. Moreover, on account of Eq~11! we have

TzzSS~j!52

4eprss2

5 S s

szD 10

14ep2rsrs8js3

3 H 2

15F S s

sz2j D 9

2S s

sz1j D 9G2S s

sz2j D 3

1S s

sz1j D 3J ~25!

as an additional contribution. Equations~23!–~25! constitutethe so-called virial expression forTzz since the mathematicamanipulations required to derive them are inspired byroute conventionally followed to derive Clausius virial fothe pressure in the bulk phases.34

An alternative, somewhat simpler formula forTzz can bederived by directly differentiatingZ given in Eq.~17! with-out transforming variables~see Ref. 24 for details!. We obtain

Tzz~j!521

2AzK (

i 51

N 9w rep[1]~zi !23watt

[1]~zi !

sz/21zi1

9w rep[2]~zi !

sz/21ziL

21

2AzK (

i 51

Nj~sz/22zi !

RiFc rep~Ri ;j!2catt~Ri ;j!

Ri

110c rep~Ri ;j!24catt~Ri ;j!

Ri2j G L 1TzzSS~j!, ~26!

whereTzzSS(j) is given in Eq.~25!. Equation~26! is termed

‘‘force expression’’ forTzz since in its derivation one essentially expressesTzz as the z-component of the~average!force exerted by the fluid on each substrate.24 Entries inTable I indicate the good agreement between virial@see Eq.~22!# and force routes@see Eq.~26!# to Tzz which is particu-larly noteworthy because of the relatively small magnituof the numbers listed in that table.

C. Thermodynamic integration

Unfortunately, in the more general casejÞ0, that is ifthe upper substrate is decorated with a spherical cap, the

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1457J. Chem. Phys., Vol. 118, No. 3, 15 January 2003 Fluid phase transitions at heterogeneous nonplanar surfaces

no such simple ‘‘mechanical’’ route toV. In this instance weresort to a thermodynamic integration scheme. To develowe differentiate Eq.~16! with respect to the radius of thspherical capj to obtain

S ]V

]j DT,m,s

52J21 (N50

`exp~bmN!

L3NN!

3E drNF(i 51

N]C~Ri ;j!

]j1

]uss~sz ;j!

]j G3exp@2bU~rN;j!#

52K (i 51

N]C~Ri ;j!

]j1

]uss~sz ;j!

]j Lj

, ~27!

where r is a vector inR3 and we useC given in Eq. ~5!instead ofC @see Eq.~7!# to guarantee limj→0(]V/]j)50.

Equation~27! can then be formally integrated yielding

V~j!5V~0!2E0

j

dj8K (i 51

N]C~Ri ;j8!

]j81

]uss~sz ;j8!

]j8L

j8

.

~28!As outlined in Sec. III A,V(0) is given by Eq.~15! (j50) so that Eq.~28! can be recast as

v~j!5Ti21

VE0

j

dj8K (i 51

N]C~Ri ;j8!

]j8

1]uss~sz ;j8!

]j8L

j8

, T,m,sz5const, ~29!

where we introduced the grand-potential densityvªV/Vfor convenience andTi is given in Eq.~21!. In this approachwe usej as a coupling parameter in order to switch continously between the homogeneous substrate (j50) and theheterogeneous one (jÞ0). This method is similar in spirit tothe so-calledl-expansion, which has been employed in tpast as a suitable starting point in thermodynamic pertution theories35 ~see also Ref. 36!. It has also been used pviously to study the phase behavior of a fluid confinedsubstrates decorated with a two-dimensional chemicaltern of low symmetry.30

TABLE I. Normal component of the stress tensorTzz from expressionsgiven in Eqs.~22! and ~26! with j55, T50.65, andm527.825.

sz Tzz @Eqs.~23!# Tzz @Eq. ~26!#

6.0 20.0931 20.09076.5 20.0990 20.09797.0 0.0571 0.05827.5 0.0099 0.00888.0 0.0796 0.08298.5 0.0581 0.05889.0 0.0798 0.08389.5 0.0720 0.0717

10.0 0.0760 0.077910.5 0.0703 0.068711.0 0.0483 0.0480

it

-

a--

yt-

Note also that the grand-potential density in Eq.~29! stilldepends onm andsz ~for fixed T) so that, from Eq.~12!,

v~m2!5v~m1!21

VEm1

m2dm^N~m!&j ,

T,sz ,j5const ~30!

can be used to calculate the grand-potential density achemical potentialm2 provided its value atm1 is alreadyknown. Similarly, one has from Eq.~12!,

sz,2v~sz,2!5sz,1v~sz,1!1Esz,1

sz,2dszTzz~sz!,

T,m,Az ,j5const. ~31!

Equations~29!–~31! are employed below to calculateabso-lute values ofv starting from a suitably chosen referenstate where the value of grand-potential density can eaand reliably be calculated.

IV. RESULTS

A. Technical aspects

Based upon the above considerations GCEMC simutions are employed to investigate the phase behavior oLJ~12,6!-type@see Eqs.~1!–~2!# of fluid confined betweensubstrate surfaces, one of them being decorated witspherical cap of radiusj. Henceforth, we employ the customary dimensionless~i.e., ‘‘reduced’’!units, that is length isexpressed in units ofs, energy in units ofe, and temperaturein units ofe/kB ; other quantities are cast in terms of suitabcombinations of these ‘‘basic’’ quantities. We employ stadard periodic boundary conditions at the planesx56sx/2,y56sy/2, wheresx5sy520 throughout. Ifj is sufficientlysmall these latter values are large enough to mimick an initely large system with asingle isolated attractive sphericacap on the upper substrate. Hence, in the actual simulatwe associate with the computational cell the fluid lameintroduced in Sec. III A. Results reported below are excsively restricted to the casej55.

As explained in detail in Ref. 25, generation of a Markchain of m51, . . . ,M configurations$rm

N% in the GCEMCproceeds in pairs of events: trial displacements and attemto create or destroy fluid molecules. Both events are realiaccording to the probability density governing the grandnonical ensemble. If a particular configurationk containsNk

fluid molecules, the sequence ofNk displacements followedby Nk creation–destruction attempts constitutes a ‘‘GCEMcycle.’’ Results presented below are based upon runs o3104–105 MC cycles with^N& ranging approximately from50 to 3000 depending on the thermodynamic state. In allsimulations we setr c52.5 @see Eq.~2!#. Numerical integra-tions of Eqs.~29!–~31! are based upon Simpson’s rule withstep width Dj50.25 in Eq. ~29!. We also takers5rs851.0.

B. Homogeneous upper substrate

We begin the discussion considering first the special cj50, that is a fluid confined between two planar substra

1458 J. Chem. Phys., Vol. 118, No. 3, 15 January 2003 Sacquin, Schoen, and Fuchs

FIG. 2. Densityr(z) as a function of position relativeto the attractive substrate plane (z) obtained forj50and sz511; ~a! T50.55, m527.100, 27.040,27.035, and 27.000, ~b! T50.65, m527.80,27.74, 27.73, 27.71, 27.69, 27.67, 27.65, and27.60, ~c! T50.75, m528.60, 28.55, 28.50,28.48, 28.46, 28.44, 28.42, 28.40, 28.35, and28.30.

e.

1459J. Chem. Phys., Vol. 118, No. 3, 15 January 2003 Fluid phase transitions at heterogeneous nonplanar surfaces

FIG. 3. Number of fluid particlesN as a function ofchemical potentialm obtained forj50 andsz511; ~a!T50.65, the empty circles denote metastable states,~b!T50.75. Solid lines are fits intended to guide the ey

pumn-t

emr

ni

a

nd-

r ac-

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nto

ce

a

e

isatens.

the lower one being attractive and the upper one being resive. GCEMC calculations were carried out for several teperatures betweenT50.55 and 0.75. As the chemical potetial was increased three different scenarios of film growwere observed, corresponding to three characteristic tperature ranges in which prewetting may or may not occuthe attractive lower substrate.

Prewetting refers to a line of first-order phase transitioalong which a thin film adsorbed on a substrate coexists wa thick film of finite extent. Prewetting occurs for tempertures above the so-called wetting temperatureTw and van-ishes at the surface critical temperatureTsc.Tw . Prewettingis a well-established phenomenon, both theoretically13,37,38

and experimentally.15,16,39According to the above descriptioit is signaled by a discontinuity in the amount of fluid asorbed at a substrate’s surface asm→msat, wheremsat is thechemical potential at which bulk gas and liquid coexist fogiven temperatureT. Once a prewetting transition has ocurred, the film thickness grows continuously withm, even-tually diverging ~i.e., becoming macroscopic! at m5msat.

l--

h-

at

sth-

For temperaturesT,Tw , the film thickness remains microscopically small for allm,msat but changes discontinuouslto a macroscopically thick film atm5msat. Above Tsc, onthe other hand, thin films are transformed continuously imacroscopically thick ones asm→msat.

As a quantitative measure of film thickness we introduthe local density

r~z!5^N~z!&sxsydsz

, ~32!

where N(z) is the number of fluid molecules located inprism of dimensionssx3sy3dsz .

For T50.55, the thin film grows continuously until thchemical potentialm.27.035 is reached@see Fig. 2~a!#.Ideally ~i.e., for a semiinfinite system!, this chemical poten-tial should correspond tomsat at which the film becomesmacroscopically thick. In our case, however, its thicknesslimited by the presence of the upper repulsive substrwhich we need to implement for purely technical reaso

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-

1460 J. Chem. Phys., Vol. 118, No. 3, 15 January 2003 Sacquin, Schoen, and Fuchs

For all m,27.035 the density profiles are distinctly diffeent from the ones form*27.035@see Fig. 2~a!#. Keeping inmind the finite system size we conclude thatT50.55,Tw

for the present system.For T50.65, the thin film again grows continuously

first. At m.27.735 the thickness of the film changes dcontinuously. The newly formed thicker film accommodaonly four layers of fluid molecules as indicated by maximar(z) @see Fig. 2~b!#. Hence, the thickness of the new filmm.27.735 remains finite. As the chemical potential iscreased further, the thickness of the film grows rapidlycontinuously. The change in the number of fluid particles

FIG. 4. Local densityr(r ,z) as a function of position relative to the substrate plane~z! and distancer from center of the spherical cap~see Fig. 1!;~a! thin film-gas (m527.85); ~b! thin film-bridge (m527.825); ~c! thickfilm-bridge (m527.80). In all casesT50.65, j55, sz511, andsx520@see also Fig. 7~b!#.

-s

t

tt

m.27.735 is discontinuous and exhibits hysteresis@see Fig.3~a!#, suggesting that at this particular temperature oncrossing the prewetting line, that isTw,0.65,Tsc. How-ever, one should bear in mind that because of the finiteof the system in thez-direction it is rather difficult to deter-mineTw with absolute certainty. Indeed, since the fluid donot wet the repulsive~upper!substrate, the densityr(z) al-ways tends to zero whenz→sz/2, however large the chemical potential may be. Therefore, it is impossible to distguish a ‘‘very’’ thick fluid film from a macroscopic one fotemperatures close toTw unless a finite-size analysis is caried out by gradually increasingsz . However, such a study isbeside the point of the present work.

Finally for T50.75, film thickness increases continously with m @see Fig. 2~c!#. The number of fluid particlechanges continuously without any hysteresis@see Fig. 3~b!#,indicating that this particularT exceedsTsc. Closer scrutinyreveals that a discontinuity inN(m) disappears somewherin the range 0.70&Tsc&0.71.

C. Heterogeneous upper substrate

We now consider a system where the upper substradecorated by a spherical cap of radiusj.0 such that thefluid–substrate interaction is given by Eq.~10!. We restrictourselves to the temperature range 0.60<T<0.75. To distin-guish different morphologies formed by the confined fluwe use the local density

r~r ,z!5^N~r ,z!&2prdrdz

, ~33!

where N(r ,z) is the number of fluid molecules in a giveconfiguration located in an annulus of widthdr and thick-nessdz at a distancer 5(x21y2)1/2 and at a pointz from thecenter of the coordinate system. To fully characterizestructure of the fluid it will prove useful below to considealternatively the local density of the contact layer on tspherical cap~i.e., the layer of fluid molecules closest to isurface!, defined as

r~u!5^N~R56,u!&

2pR2dR cosudu, ~34!

whereN(R56,u) is now the number of particles in a ring owidth dR and thicknessRdu located at a distanceR56 andan angleu from the center of the spherical cap and the sface of the upper substrate, respectively~see inset in Fig. 5!.

For T50.65 andm527.85 @see Fig. 4~a!#a small por-tion of fluid is adsorbed on the spherical cap and onlower substrate while the remainder of the system volumoccupied by low-density gas. A thin fluid film equivalentthe one described in the previous section is adsorbed onlower substrate. Morphologies similar to the one depictedFig. 4~a! will therefore be referred to as ‘‘thin film-gas.Notice also that the statistical error increases substantiallr→0 because the area of the ring segment@and thereforeN(r ,z)] goes to zero in that limit.

For a slightly larger chemical potentialm527.80, thevolume between the spherical cap and the lower substraoccupied by fluid at a higher density compared with the p

(

1461J. Chem. Phys., Vol. 118, No. 3, 15 January 2003 Fluid phase transitions at heterogeneous nonplanar surfaces

FIG. 5. Local densityr(u) as a function of the angleuwith the upper substrate plane, on the contact layerR56), starting on the upper substrate (u50) and endingon top of the spherical cap (u5p/2); m527.9 (m),27.825 (s), 27.75 (j); in all casesT50.65, j55,sz511, andsx520.

e

-ote

l b

g-ou

so

sur-

calper-int,

cal

inncefom

forsofca-e-olids

ofob-lar,

at

the-

s.

eThis

e

vious case@see Fig. 4~c!#. Fluid now bridges the gap betwethe attractive parts of the opposite substrates. Forr .j,r(r ,z) tends to 0 whenz→sz/2, illustrating again the nonwetting character of the upper substrate. Furthermore,notices a thicker adsorbed fluid film on the lower substraBecause of these two characteristics, this morphology wilreferred to as ‘‘thickfilm-bridge.’’

For an intermediate chemical potentialm527.825, asshown in Fig. 4~b!, there is still a higher-density fluid briding the gap between lower substrate and spherical cap arthe center of the system (r→0). However along lines ofconstantz one notices a faster decay ofr(r ,z) for increasingvalues ofr. Moreover in the ‘‘outer’’ part of the gap, that ifor r .j, the lower substrate is now covered by a thin ads

FIG. 6. Integration paths used for the computation ofV; ~a! full line: thinfilm-gas@see Fig. 4~a!#;~b! dashed line: thin film-bridge@see Fig. 4~b!#;~c!dashed–dotted line: thick film-bridge@see Fig. 4~c!#. The arrows indicate thdirection in which themjsz-space is traversed by the integration path.

n

ne.e

nd

r-

ded film. Thus, we shall term this morphology ‘‘thinfilm-bridge.’’

Both plots in Figs. 4~b!and 4~c!show oscillations of thedensity in the direction perpendicular to the substratesfaces, indicating stratification of the fluid along thez-axisand in the direction normal to the surface of the sphericap. Since the layers parallel to the planar substrate andpendicular to the cap’s surface normal merge at some poplots in Figs. 4~b!and 4~c!exhibit bifurcations atr .4, z.23. The plots in Fig. 5, on the other hand, show the lodensity in the contact layer of the spherical cap~defined inthe inset of Fig. 5!as a function of the angleu, at T50.65and for three different chemical potentials. From the plotFig. 5 one notices a nonmonotonic, oscillatory dependeof r(u) on u which decays asu increases. On account orepulsive fluid–substrate interactions at small distance frthe substrate limu→0 r(u)50 ~see Fig. 5!. The oscillations inr(u) become more pronounced with increasingm. The sepa-ration between successive maxima in the plotm527.75, that is for a thick film-bridge morphology, iapproximatelyDu.0.18 which corresponds to a distanceone molecular diameter. This indicates the kind of stratifition similar to that usually expected in fluids confined btween planar, chemically homogeneous, and attractive ssubstrates~see, for example, Ref. 31!. In the present case thipacking effect is due to the three-dimensional geometrythe upper substrate, similar to what has previously beenserved for substrates decorated by two-dimensional circuattractive chemical patterns~see Ref. 30!.

Depending on the morphology considered~see Fig. 4!,v(m) is obtained via thermodynamic integration startingpoints ~a!, ~b!, or ~c! in Fig. 6 and following the differentpaths indicated by full, dashed, or dashed–dotted lines indirection of the arrows. For the thin film-gas and thick filmbridge morphologies,v(m) has been calculated in GCEMCsimulations by thermodynamic integration employing Eq~29! and ~30! for fixed T and sz511. At m1527.90 (T50.65) andj50, a thin film forms at the surface of thlower substrate and gas occupies the rest of the volume.

th

uain

in

m-e

llya-

er-

mi-

d

lly

rethe, atnonperthe

hingre-

asetial

artssed

ob-

tk-rate

lotd

ic-the

-

1462 J. Chem. Phys., Vol. 118, No. 3, 15 January 2003 Sacquin, Schoen, and Fuchs

chemical potential is sufficiently low to guarantee that, asattractive spherical cap grows in size~i.e., asj→5), theoriginal gas and thin film are not subject to any discontinous transition. Forj55 the spherical cap assumes its finsize and thermodynamic integration now proceeds by raisthe chemical potential and employing Eq.~30! along the re-mainder of the integration path~see Fig. 6!. Only pointsalong this latter path are shown in Fig. 7~b!.

For a corresponding thick film-bridge state, one beg

FIG. 7. Grand potential densityv as a function of chemical potentialm for:thin film-gas (h), thin film-bridge (m), and thick film-bridge morphologies(s); ~a! T50.60, ~b! T50.65, ~c! T50.69. Solid lines are fits to simulation data intended to guide the eye.

e

-lg

s

with j50 and a sufficiently highm57.60 such that a thickfilm, large enough in thez-direction, forms on the lowersubstrate. Oncej55 the chemical potential is nowloweredandv is again calculated by thermodynamic integration eploying Eq. ~30! ~see Fig. 6!. Again only points along thsecond integration path are shown in Fig. 7~b!.

Plots in Fig. 7 also indicate that at fixedT, v(m) is amultivalued function over certain ranges ofm. Its absoluteminimum obviously corresponds to the thermodynamicastable morphology~i.e., phase!; the others are only metstable. Ideally, metastability ends~i.e., the confined fluid be-comes unstable!if the inequality

kT[2S V

ND 2S ]2v

]m2DT,s

.0 ~35!

can no longer be satisfied. In Eq.~35!, kT is the isothermalcompressibility which is positive definite as required by thmodynamic consistency.

For T50.60 plots in Fig. 7~a!indicate that thin film-gasmorphologies are stable over the range2`,m&27.49,whereas thick film-bridge morphologies are thermodynacally stable over the range27.49&m<msol, keeping inmind the general possibility of solidification of the confinephase at a sufficiently high chemical potentialmsol, where apotential solid phase may form as a thermodynamicastable phase. At the intersectionm.27.49 thin film-gas andthick film-bridge morphologies coexist. For this temperatuthe thin film-bridge morphology has not been observed insimulations and is probably always unstable. Thereforelow temperatures we observed a prewetting phenomesimilar to the case of a slit pore with homogeneous upsubstrates. However, this time prewetting is supported bypresence of the embedded spherical cap.

For a slightly higherT50.65 plots ofv vs m in Fig. 7~b!show a third set of data points corresponding to the tfilm-bridge morphology. It is stable over a nonvanishinrange of chemical potentials, even though this one-phasegion of thin film-bridges is relatively narrow (uDmu.0.035). There are now two successive first-order phtransitions occurring in the system as the chemical potenincreases. During the first of these atm.27.850 fluid par-ticles assemble mainly in the gap between the attractive pof the two opposite substrates. This phenomenon is cauby the confinement of the fluid and has already beenserved in the case of either geometrically40 or chemically27,30

heterogeneous substrates. The second transition am.27.815 is a prewetting transition during which the thicness of the film adsorbed on the lower attractive substjumps, similar to what is observed at planar surfaces~seeSec. IV B and Refs. 13, and 14!. To our knowledge the pin Fig. 7~b! is the first illustration of a confinement-inducefirst-order phase transition succeeded by prewetting asm in-creases steadily.

Calculatingv(m) for thin film-bridge morphologies issignificantly more demanding in terms of the thermodynamintegration procedure~see Fig. 6!. In order to avoid a discontinuous phase transition during the initial stage wherespherical cap ‘‘grows’’@i.e., as j→5, see Eq.~29!#, one

-f

1463J. Chem. Phys., Vol. 118, No. 3, 15 January 2003 Fluid phase transitions at heterogeneous nonplanar surfaces

FIG. 8. Normal stressTzz as function of substrate separation sz employed in thermodynamic integration ogrand potential for thin film-bridge morphologies@seeEq. ~31! and main text#, whereT50.65, m527.825.Tzz

FF1TzzFS (m), Tzz

SS(n). Solid lines are fits intended toguide the eye.

.-lyeth

,qu

dg

eru

plesin

nerheyd toen

7thees

nusatnsi-

aellula-rac-ub-ous

ssizeate,ri-micree

anin

mo-ther-inotion-

te

needs to start from a value ofsz56 which is sufficientlysmall to prevent a discontinuous transition from occurring41

Thus, as one increasesj from 0 to 5 more and more moleculesgradually assemble in the vicinity of the increasinglarger attractive spherical cap. Once the terminal cap sizj55 is reached an additional integration is needed to carryfluid system to the desired substrate separationsz511.Along this pathv is calculated via Eq.~31!. SinceTzz de-pends nonmonotonically onsz , as the plots in Fig. 8 showthis curve needs to be known with high resolution for E~31! to provide sufficiently accurate results. Here we calclateTzz(sz) in steps ofDsz50.1. Oncesz511 the remainingintegration proceeds as discussed above for thick film-briand thin film-gas. In the plot of@Tzz

FF1TzzFS# in Fig. 8 one

should realize that forsz<6.5 the distance between the lowsubstrate and the top of the spherical cap is to small for flmolecules to form a well-defined~mono-!layer between thesubstrates. Thus, oscillations in@Tzz

FF(sz)1TzzFS(sz)#, usually

observed in the case of simple fluids confined betweennar, homogeneous substrates, only appear for larger valusz . The plot in Fig. 9 shows the density profile of a fluida severely confined system wheresz57. In this case one

FIG. 9. Local densityr(r ,z) as a function of position relative to substraplane~z! and distancer from center of the spherical cap~see Fig. 1!in thecase of a highly confined fluid;m527.825, T50.65, j55, sz57, andsx

520.

e

.-

e

id

a-of

notices again strong oscillations of the density in the corformed by the lower substrate and the spherical cap. Tare similar to those observed in the case of fluids confinea pore with walls into which V-shaped wedges have be‘‘engraved’’ ~see Refs. 25,26!.

At an even higherT50.69 @see Fig. 7~c!#the same twosuccessive phase transitions as the ones shown in Fig.~b!occur but the range of chemical potentials over whichthin film-bridge phase is thermodynamically stable becomlarger. ForT.0.70 the transitions from thin film-gas to thifilm-bridge and then to thick film-bridge are now continuo~plots of v vs m are not shown in this case!, indicating ththe critical temperatures corresponding to these phase trations are lower thanT50.70.

V. SUMMARY AND CONCLUSION

In this work we investigated the phase behavior ofLJ~12,6!-type of fluid confined between chemically as was geometrically decorated substrates via GCEMC simtions. While the lower substrate has a simple planar, atttive and chemically homogeneous surface, the upper sstrate is also planar, repulsive and chemically homogenebut decorated with an attractive spherical cap of radiujlocated at the center of its surface. Because of the finiteof this three-dimensional pattern on the upper substrproperties of the confined fluid are not translationally invaant in any direction and as a consequence thermodynapotentials are no longer homogeneous functions of degone in any of the extensive~natural!variables. As a resultEuler’s theorem does not apply and one cannot obtainanalytic closed ‘‘mechanical’’ form of the grand potentialterms of suitable sets of stresses and conjugate strains.

In order to compute the grand potentialV, whoseknowledge is indispensable to discriminate between therdynamically stable and metastable phases, we resort tomodynamic integration similar in spirit to that describedRef. 30. In this case we takej as the coupling parameter ttransform a system whose spatial properties are transla

i-

n-n

entiv

aici

e

ofmsalatth

p

y-podmis

okewifosidn-th

o

mriedes

antior-

sch

qs.

o

1464 J. Chem. Phys., Vol. 118, No. 3, 15 January 2003 Sacquin, Schoen, and Fuchs

ally invariant in bothx- and y-directions (j50) to a low-symmetry system (j55) where no such translational invarance exists in any dimension. In the former caseV(0) can becalculated via a ‘‘mechanical’’ expression involving esemble averages of certain stresses exerted by the fluid oenvironment, whereas in the latter caseV(j) can only becomputed by numerically integrating]V(j)/]j.

For j50, that is in the case of a fluid confined betwetwo planar and chemically homogeneous substrates,lower one being attractive and the upper one being repulswe observe prewetting. Over a range of temperaturesTw

<T<Tsc, the amount adsorbed at the surface of the attrtive substrate changes discontinuously from a thin to a thfilm of a finite number of molecular layers when the chemcal potential increases@see Fig. 2~b!#. This change in thamount adsorbed decreases asT is raised towardTsc. Fortemperatures exceedingTsc thin films transform continuouslyinto thick ones@see Fig. 2~c!#.

If j55 a similar prewetting transition at the surfacethe lower substrate was observed for sufficiently low teperatures. For small chemical potentials, both the surfacethe lower substrate and of the attractive spherical capcovered by a thin fluid film@see Fig. 4~a!#. For high chemicapotentials the thick film at the surface of the lower substris accompanied by a fluid bridge filling the gap betweensurface of the spherical cap and the lower substrate@see Fig.4~c!#.

For temperatures aboveT50.62 a third morphologyarises where a fluid bridge starting from the spherical caassociated with a thin film on the lower substrate@see Fig.4~b!#. Since this thin film-bridge morphology is thermodnamically stable only over a certain range of chemicaltentials, special precaution has to be taken during thermonamic integration to prevent the integration path froaccidentally crossing a line of first-order transitions. Thmay happen as one increasesj from 0 to 5 where for asufficiently large substrate separation a thin film-bridge mphology may form spontaneously from an initial gasliphase. To prevent this from happening one has to begina fairly thin confined phase where no discontinuous transmations between different phases can occur. Hence, bej and m a third integration is required along a path of icreasing substrate separation. The stability domain ofthin film-bridge morphology increases withT until onereaches its critical temperature.

Finally, if one views the spherical cap as a model tipan atomic force microscope~AFM! ~with the repulsive, pla-nar upper substrate as its model cantilever!, the plots in Fig.8 illustrate the importance of the presence of a gas atsphere in AFM experiments. Such experiments are carout to study friction and stiffness of automotive pads uncontrolled humidity~see Fig. 4 in Ref. 32!. Figure 8 showthat in the absence of any gasTzz5Tzz

SS is purely attractive(Tzz.0) and decays monotonically assz increases. Thepresence of a gas atmosphere wetting the model AFM tipthe attractive substrate adds a nonmonotonic contribuTzz

FF1TzzFS which in a real AFM experiment would be supe

imposed onTzzSS.

its

hee,

c-k-

-ofre

ee

is

-y-

r-

thr-es

is

f

o-dr

dn

ACKNOWLEDGMENTS

Two of the authors~S.S. and M.S.!are grateful for sup-port from the Sonderforschungsbereich 448 ‘‘Mesoskopistrukturierte Verbundsysteme.’’

APPENDIX: DIFFERENTIATION OF THECONFIGURATION INTEGRAL IN EQ. „20…

To calculate the partial derivative ofZ(j) in Eq. ~20!,we transform to reduced dimensionless coordinates,

xi85xi1sx/2

sx,

yi85yi1sy/2

sy, ~A1!

zi85zi1sz/2

Dz~ r i ;j!,

wherea i8P@0,1# (a85x,y,z),

Dz~ r ;j!5sz/21zmax~ r ;j!, ~A2!

andzmax( r ;j) is given in Eq.~19!.The configuration integral can then be rewritten as

Z~j!5)i 51

N E0

1

dxi8E0

1

dyi8E0

1

dzi8J~j!exp@2bU~j!#,

~A3!

where the Jacobian of the transformations defined in E~A1! is given by

J~j!5sxNsy

N)k51

N

Dz~ r k ;j!. ~A4!

From Eq.~A3! we obtain

S ]Z

]saD5)

i 51

N E0

1

dxi8E0

1

dyi8E0

1

dzi8S ]J

]sa2Jb

]U

]saD

3exp~2bU !. ~A5!

We focus first on the casesa5sx . Differentiation of theexpression forJ in Eq. ~A4! yields

]J

]sx5

NJ

sx1 (

k51

N

sxNsy

N)j Þk

Dz~ r j ;j!]Dz~ r k ;j!

]sx

5NJ

sxF11

1

N (i 5k

N

xk

Dz8~ r k ;j!

Dz~ r k ;j!G , ~A6!

whereDz85dDz/dx. From Fig. 1 it is clear that the ratiDz8/Dz depends on the domain ofx andy. Using the equa-tions of the planes of the substrate surfaces~see Fig. 1!, wehave:

Dz8~ r ;j!

Dz~ r ;j!5H x

Dz~ r ;j!@sz2Dz~ r ;j!#~x21y2!<j2

0 ~x21y2! .j2.

~A7!

From Eq.~18!, we obtain

-

as

Va

an

J.

ndepa-

ys.

e

ys.

1465J. Chem. Phys., Vol. 118, No. 3, 15 January 2003 Fluid phase transitions at heterogeneous nonplanar surfaces

]U

]sx5

1

2 (i 51

N

(j 51Þ i

N

uf f8 ~r i j !]r i j

]sx1(

i 21

N]uf s~zi ,Ri ;j!

]sx,

~A8!

whereuf f8 (r )5duf f(r )/dr. A long, painful sequence of manipulations eventually leads to

]r i j

]sx5

1

sxr i jH xi j

2 1zi j Fxi~zi1sz/2!Dz8~ r i ;j!

Dz~ r i ;j!

2xj~zj1sz/2!Dz8~ r j ;j!

Dz~ r j ;j!G J , ~A9!

wherea i j 5a i2a j (a5x,z). The partial derivative ofuf s inEq. ~A8! can be expanded as

]uf s~zi ,Ri ;j!

]sx5

]F [1]~zi !

]sx1

]F [2]~zi !

]sx1

]C~Ri ;j!

]sx

5F ]F [1]~zi !

]zi1

]F [2]~zi !

]zi1

]C~Ri ;j!

]ziG ]zi

]sx

1]C~Ri ;j!

]xi

]xi

]sx~A10!

with @see Eqs.~A1!#

]xi

]sx5

xi

sx,

~A11!]zi

]sx5

xi

sx

Dz8~ r i ;j!

Dz~ r i ;j!S sz

21zi D .

Because of the cylindrical symmetry of the system, the csa5sy gives exactly the same equations withx being re-placed byy. In the casesa5sz , we also have to take intoaccount the term]uss(sz ;j)/]sz in Eq. ~A8!.

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