capillary condensation between mesocopically rough surfaces

Colloids and Surfaces

A: Physicochemical and Engineering Aspects 206 (2002) 253–266

Capillary condensation between mesocopically roughsurfaces

Martin Schoen *Stranski-Laboratorium fur Physikalische und Theoretische Chemie, Sekr. TC 7,

Fakultat II Mathematik und Naturwissenschaften Technische Uni�ersitat Berlin, Straße des 17. Juni 124, D-10623 Berlin, Germany

Abstract

Grand canonical ensemble Monte Carlo (GCEMC) simulations are employed to study a Lennard–Jones (12, 6)fluid confined between furrowed walls. The walls are constructed by gouging triangular grooves in planar substratesthat are structureless on the molecular scale. The furrows are infinitely long in one transverse direction (y) andcharacterized by a period (sx) and amplitude (D) of corrugation. For a fixed separation sz=10� (� is the ‘diameter’of a fluid molecule) between the substrates the phase behavior of the confined fluid is investigated as a function ofsx, D, and temperature T. It depends crucially on substrate corrugation. For example, capillary condensation (i.e.confinement-induced condensation of gas) is observed for sx�30� as � increases under isothermal conditions. In thelimit D=0 (planar substrate) capillary condensation is a discontinuous phase transition. For a nonplanar substrate(D=5�) it occurs as a two-stage process where first liquid continuously fills the furrows until a nearly planargas–liquid interface has formed separating liquid in the furrows from gas in the ‘inner’ subvolume of the system. Inthe second stage the gas condenses spontaneously. For sx�30� an initial gas condenses partially thereby formingfluid bridges. Bridges are characterized by high(er)-density fluid stabilized by the ‘tips’ of the opposite furrows. Thefluid bridges enclose gas-filled ‘holes’ which shrink in size if � increases further. Eventually, the ‘holes’ vanishspontaneously in favor of a liquid phase. © 2002 Elsevier Science B.V. All rights reserved.

Keywords: Confined fluids; Phase transitions; Rough surfaces; Monte Carlo simulation

PACS numbers: 64.70.Nd; 64.60.− i; 61.46.+w; 61.20.Ja

www.elsevier.com/locate/colsurfa

1. Introduction

Modern techniques permit experimentalists tostudy the phase behavior of fluids in spaces ofdimensions comparable to the range of inter-molecular interactions. Realizations of such

severely confined systems include fluids in zeolites[1], where the solid matrix can be synthesized witha very regular though complex chemical structure.Other examples include Vycor or controlled-poreglasses (CPG) consisting of a disordered andstrongly interconnected mesoporous solid matrix[2]. However, in recent years it became also possi-ble to synthesize such mesoporous media with asimple, yet very regular structure. An example is

* URL: http://www.tu-berlin.de/� insi/ag–schoen.E-mail address: [email protected] (M.

Schoen).

0927-7757/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved.

PII: S0 927 -7757 (02 )00080 -8

http://www.tu-berlin.de/~insi/ag_schoen

mailto:[email protected]

M. Schoen / Colloids and Surfaces A: Physicochem. Eng. Aspects 206 (2002) 253–266254

MCM-41 [3] consisting of hexagonal arrays ofsingle, presumably disconnected cylindrical poreswith a narrow width distribution (see, for exam-ple, Fig. 1(c) of [4]).

Use of these highly regular materials to studythe phase behavior of confined fluids is especiallyappealing from a theoretical perspective, becausethey permit the solid matrix to be modeled assingle idealized cylindrical or slit-shaped pores toa good approximation. The status of gas– liquidequilibria in disordered porous matrices, on theother hand, is far less clear at present [5]. If suchan idealized pore is sufficiently wide condensation(evaporation) of gas (liquid) can approximately bedescribed by Kelvin’s equation [6] or a somewhatmore sophisticated mean-field treatment of theconfined fluid [7]. Unlike the former, the latter iscapable of predicting a confinement-induced de-pression of the critical temperature (relative to thebulk value) in qualitative agreement with experi-mental data [8]. Unfortunately, both approachesfail to capture more subtle features of gas– liquidequilibria in confinement like the shift of thecritical point to higher (mean) densities (com-pared with the bulk). Shifts of the critical point ofconfined fluids to both lower temperature andhigher density have been reported experimentally[8] and theoretically on the basis of more sophisti-

cated lattice models [9] or density functional ap-proaches [10].

However, all these considerations are valid forrather idealized situations in which the fluid isconfined to slits or cylinders with infinitesimallysmooth surfaces. In nature, on the other hand,most surfaces are rough on nanoscopic lengthscales. One of many examples are minerals of thesepiolite or palygorskite group consisting of alter-nating stacks of tetrahedral and octahedral silicatesheets [12]. Moreover, recent advances in technol-ogy permit one to deliberately create highly regu-lar but laterally nonflat structures. For instance,Rauscher et al. investigated the confinement oforganic molecules to one-dimensional trenchesconstructed by vapor deposition of CaF2 onstepped Si(111) surfaces [11]. Other examples arisein the context of structured semiconductor sur-faces [13–15], templates for the self-assembly ofsmall particles [16–18], and microfluidics [19–23].

Numerous experimental [24–27] and theoretical[28–37] evidence exists, indicating that surfaceroughness has a significant impact on wettingphenomena. However, these studies are limited tocoarse-grained approaches which average laterallyover the height of the local microscopic structureof fluids filling the grooves and covering the as-perities of such disordered substrates. More de-tailed microscopic investigations of wetting ofnonplanar substrates are just beginning to emerge[38,39]. These latter works are based upon effec-tive interface-hamiltonian models that are solvednumerically.

Even less is known about the properties offluids confined to nanoscopic volumes by roughsubstrates. For example, Chmiel et al. applied amean-field density-functional theory to the Len-nard–Jones (LJ) (12,6) fluid confined betweenplanar walls having a superimposed square-wavebarrier [40]. They found that capillary condensa-tion occurs in a two-stage process: liquidlikebridges form first in the narrow gaps between thetops of the opposing barriers, followed by con-densation in the wider gaps, so that eventually thewhole volume is filled with liquid. This two-stageprocess has also been reported for fluids confinedbetween planar but chemically decorated substratesurfaces by several authors [41–46]. A model sim-

Fig. 1. Side view of slit-pore with furrowed walls. Shadowedboxes display equations of planes of sides of ‘central’ furrows.Subscripts 1 and 2 refer, respectively, to lower and uppersubstrates.

M. Schoen / Colloids and Surfaces A: Physicochem. Eng. Aspects 206 (2002) 253–266 255

ilar to the one employed by Chmiel et al. [40] wasstudied by Curry et al. with particular emphasison shear-strain induced solid– liquid phase transi-tions and diffusive transport [47].

In [40,47] the fluid is confined between sub-strates whose separation varies discontinuously be-tween two characteristic values. This work, on thecontrary, is concerned with substrates whoseroughness is characterized by a certain period sx

and amplitude D such that the separation betweenthe substrate surfaces varies continuously as afunction of lateral (x) position. The substrate maybe imagined to be constructed by gouging trian-gular grooves (furrows) in an infinitesimallysmooth planar surface (see Fig. 1), a model whichhas almost assumed the status of a paradigm of amesoscopically rough substrate surface [38,39,48–50]. For example, Schoen and Dietrich carried outgrand canonical ensemble Monte Carlo(GCEMC) simulations to examine the entropy-driven packing of hard spheres in the groove of afurrowed substrate [48]. It was shown that underfavorable conditions the furrow can induce solid-like order of fourfold in-plane symmetry. Thisobservation was verified independently by Hen-derson et al. in a density-functional calculationbased upon the Evans–Tarazona functional [51].More recently, Diestler and Schoen [50] extendedthe work of [48] to ‘simple’ LJ (12, 6) fluids inwhich fluid molecules interact with the furrowedsubstrate �ia continuously varying potential func-tions. They observed that slight deviations fromthe limiting case of a planar substrate suffice tosuppress nearly completely the oscillatory solva-tion forces frequently observed both experimen-tally [52] and theoretically [53].

In contrast to [50] this work is solely concernedwith the phase behavior of confined fluids as afunction of substrate roughness controlled by theparameters sx and D. As in the earlier studies[48,50] results are obtained in GCEMC simula-tions. The remainder of the manuscript is orga-nized as follows. The model is summarized inSection 2. Section 3 is devoted to a discussion ofcentral theoretical aspects. Results are presentedin Section 4 and the paper concludes in Section 5with a summary and discussion of the main find-ings and their potential implications.

2. Model system

Consider a fluid composed of spherically sym-metric molecules (i.e. a ‘simple’ fluid wheremolecules possess only translational degrees offreedom). The fluid is confined by two solid sub-strates that occupy the half spaces sz/2�z��and −��z� −sz/2. Engraved into the sub-strates are triangular furrows extending from −��y��. In the x-direction the furrows arearranged as a periodic sequence where

�=2 arctan� sx

2D�

(1)

is the dihedral angle formed by the sloping sidesof a furrow of depth D and period sx (see Fig. 1).Fluid molecules interact with substrate atoms �iathe LJ (12, 6) potential

uLJ(r)=4��

r�12

−��

r�6n

(2)

where � is the depth of the attractice well and r isthe distance between a fluid molecule and a sub-strate atom. Assuming fluid molecules and sub-strate atoms to be of equal ‘size’, � denotes their‘diameter’. To avoid complications that wouldattend specification of the detailed molecularstructure of the substrate, a mean-field approxi-mation is employed for the fluid–substrate inter-action potential. It is derived by averaging Eq. (2)over the positions of the substrate atoms. Theresulting mean-field expression describing the in-teraction of a fluid molecule with the upper sub-strate (designated by the superscript 2) is given by

� [2](x, z)=�0[2](z)+�1

[2](x, z) (3)

where �1[2] and �0

[2] correspond to averages overthe ‘hills’ between the furrows and the remainderof the substrate, respectively. As detailed in [50],�0

[2] can be expressed in closed form as

�0[2]

(z)

=2��s�

3

3� 2

15� �

sz/2+D−z�9

−� �

sz/2+D−z�3n

(4)

Similarly,


�1

[2](x, z)

=3��s�sxD

4��

m= −�

� 1

0

dzz�� 1

0

dx(x�, z�)

+� 1

0

dx(x�, z�)n

(5)

where �s is the solid density,

x�=x−sx

2(xz+2m−1)

x=x−sx

2(xz− z+2m+1)

z�=z−Dz−sz

2(6)

and

(x, z)=2132� �

R(x, z)�11

−� �

R(x, z)�9

(7)

In Eq. (7), R(x, z)=�x2+z2. The summationon m in Eq. (1) takes into account contributionsfrom the infinite periodic sequence of triangularfurrows in the x-direction. Similar expressionsobtain for the interaction of a fluid molecule withthe lower substrate, � [1](x, z). On account ofsymmetry considerations detailed in [50], the totalmean–field interaction potential (referring to in-teractions with both substrates) at any point (x, z)in the ‘central’ furrow (m=0) can be cast as

�(x, z)=� [2](x �,�z �)+� [2](x �,− �z �) (8)

This symmetry is exploited to simplify the com-putation of the fluid–substrate interaction poten-tial (see [50]).

3. Theoretical background

3.1. Thermomechanical properties

Since the principal object of this work is todelineate phase transitions that can occur in thesimple fluid confined by furrowed substrate sur-faces, it is convenient to treat the system (i.e. theconfined fluid) in the grand canonical ensemble,whose characteristic function is the grand poten-tial (�), which is a function of the thermody-

namic state represented by the set �={T, �, sx,sy, sz}, where T is the (absolute) temperature, � isthe chemical potential of the fluid and sx, sy, andsz are the (compressional) strains (see Fig. 1) Sincethe fluid–substrate potential depends on the x-and z-coordinates of fluid molecules (see Eq. (8)),system properties are translationally invariantonly in the y-direction. Thus, if the subset �� =�/{sy} is kept constant, is a homogeneous func-tion of degree one in sy. Hence, Euler’s theoremapplies and one obtains a mechanical expressionfor , namely

�(�)=Tyy(�)Aysy=��(�)V (9)

where V�Aysy=sxsy(sz+D) is the volume of thesystem and �� Tyy is the grand-potential den-sity. A molecular expression

�� = −�N�kBT

V+

12V� �

N

i=1

�N

j=1�1

rij u �(rij)y ij2

(10)

can be derived through manipulations detailed inthe Appendix of [50]. In Eq. (10), yij is the y-com-ponent of the unit distance vector rij=rij/�rij �,u �(r)=du/dr (r� �r�), and �…� signifies an aver-age in the grand canonical ensemble. In the bulklimit (i.e. �=�, sz�), �� b in Eq. (9)where

�b= −�N�kBT

V+

16V

�N

i=1

�N

j=1� i

rij u �(rij)�=PV

(11)

on account of the higher symmetry of the bulkfluid. In Eq. (11), P is the bulk pressure.

3.2. Phase coexistence

From the exact differential of the grand poten-tial and Eq. (9) one has

−��

��

�T,sx,sy,sz

= −1V��

��

�T,sx,sy,sz

=�N�

V

=1V�

V

dx�(r)=� (12)

indicating that the grand potential (density) mustdecrease with increasing � since the (number)density � is positive definite (disregarding explic-


itly the vacuum). In Eq. (12) the local density �

(r) is introduced as an obvious generalization of �

in an inhomogeneous system.On account of the external field represented by

� (x, z) different morphologies characterized bydifferent �(r)’s are generally expected. Because ofEq. (12) this implies different ‘rates’ at which thegrand-potential density increases with �. Supposenow two morphologies and � exist with associ-ated grand-potential densities ��

a and �� . Under

isothermal conditions (i.e. with T, sx, sy, and sz

fixed) and according to the above rationale theequation

��a (� �)=��

� (� �) (13)

may have a solution �xa�� at which the mor-

phologies and � correspond to coexisting phases.If a third morphology � exists, one may havethree solutions � �, ��, and �� from equationsanalogous to Eq. (13) involving pairs of thesemorphologies. Suppose the mean densities associ-ated with the morphologies satisfy the inequality

� �� (14)

irrespective of �. Because of Eq. (12) this implies��

��

�T,sx,sy,sz

��

�

��

�T,sx,sy,sz

��

�

��

�T,sx,sy,sz

(15)

since the grand-potential density is a monotonousfunction of �. Denoting by ��

� the value of thegrand-potential density at � � etc., three differentscenarios are discernible as one can verifygeometrically:

1. ��

��

��

In this case only morphologies and � com-port with thermodynamically coexisting phasesat � ��x

�; at �� and � � morphologies �, �

and , � are only metastable.

2. ��=��

�=��

��=� �=� �

The three intersections coincide at the giventemperature thereby defining a triple point

{Ttr, � tr ��} at which all three morphologies are

thermodynamically stable.

3. ��

��

��

This describes a situation in which two pairsof separately coexisting morphologies arethermodynamically stable phases, namely �

and � at ��x�� and and � at � ��x

�;at � � morphologies and � are onlymetastable.

4. Results

4.1. Technical aspects

Based upon the above considerations, GCEMCsimulations are employed to investigate the phasebehavior of ‘simple’ fluids confined by nonplanarsubstrate surfaces. Since the focal point is theimpact of surface roughness, this work is predom-inantly concerned with variations of the ampli-tude of this roughness (D) and its lateral period(sx) (see Eq. (1)). Therefore, the distance sz be-tween the ‘tips’ of the opposing furrows located atxtip= �sx/2 (see Fig. 1) is maintained at 10�

which is large enough for capillary condensationto occur in the limit of planar substrates (i.e. for�=� ; see [54] and Section 4.3 below). Hence-forth, all quantities will be expressed in the cus-tomary dimensionless (i.e. ‘reduced’) units, that islength is given in units of �, energy in units of �,and temperature in units of �/kB.

As explained in detail in [50], the generation ofa Markov chain of configurations rN={r1, r2, ···,rN} in GCEMC proceeds in pairs of events: trialdisplacements and attempts to create or destroyfluid molecules. Both events are realized accord-ing to the probability density governing the grandcanonical ensemble. Suppose a particular configu-ration k contains Nk fluid molecules, the sequenceof Nk displacements followed by Nk creation–de-struction attempts constitutes a ‘Monte Carlo(MC) cycle’. Results presented below are based


upon runs of 5×104–105 MC cycles with50�N 4000 depending on thermodynamic stateand specific substrate geometry (i.e. sx and D). InGCEMC it is furthermore advantageous (see de-tailed discussion in [50]) to model the fluid–fluidinteraction according to the shifted-force versionof the LJ (12, 6) potential given by

uLJ(r)usf(r)=

�uLJ(r)−uLJ(rc)+duLJ/dr �r=r c(rc−r), r�rc

0 rrc

(16)

where rc=2.5.Since the goal of this work is to illustrate the

impact of surface roughness for a representativesubcritical thermodynamic state, the temperaturewas fixed in most cases at T=0.65. However, tolocate the triple point of gas–bridge– liquid coex-istence T=0.70 and T=0.75 were also used. Aspointed out by Fan and Monson, who based theirstudy of prewetting transitions at planar sub-strates on usf with the same value of rc, T=0.65 isstill above the bulk triple point [55]. To employ atemperature as low as possible is advantageousbecause the difference in mean density between allpossible morphologies increases with decreasingT. One, therefore, expects from Eq. (12), � and�� to have maximally different slopes so that Eq.(13) can be solved numerically more accurately.

4.2. Bulk system

For later comparison it seems sensible to beginby locating the liquid–gas transition in the bulk.Fig. 2 shows a plot of �b �ersus � (see Eq. (11)).Two curves are discernible both having negativeslopes in accordance with Eq. (12). The one char-acterized by the smaller slope corresponds to gas(g) whereas the other one refers to liquid (l) states.Both curves intersect [see Eq. (13)] at �x

gl −7.666. The corresponding value of the saturationpressure Px

gl 6.8×10−3 is in good agreementwith the value Px

gl 6.5×10−3 given in Table Vof [55]. This holds also for the densities �x

g 0.0116 and �x

l 0.764 of the coexisting gas andliquid phases. The corresponding values reported

Fig. 2. Grand-potential densities �bg (�) and �b

l (�) for bulkgas (g) and liquid (l) (T=0.65). Solid lines are intended toguide the eye.

by Fan and Monson, who employed an isother-mal– isobaric–ensemble Monte Carlo method, are�x

g =0.0109 and �xl =0.763 [56]. Since the phase

characterized by the absolute minimum of (or� since V=const) is thermodynamically stable,��x

gl is the one-phase gas region whereas ��x

gl is the one-phase region of liquid (note that forsufficiently large ��x

gl solid(s) may become ther-modynamically stable, a situation which we disre-gard here explicitly).

From the plots in Fig. 2one notices that datapoints pertaining to liquid states exhibit a sub-stantially larger statistical error than those be-longing to gas states. This is not surprising inview of the typical densities of liquid states rang-ing from � l 0.793 for �= −7.300 to � l 0.722for �= −8.000. These densities are close to thosefor which the present GCEMC algorithm be-comes prohibitively inefficient according to asser-tions by Allen who pointed out that below anacceptance probability threshold of approximately10−4 for the creation/destruction event in thealgorithm, reliability of (conventional) GCEMCbreaks down [57]. For liquid states acceptanceprobabilities varying between about 1.5×10−4

(�= −7.300) and 4.0×10−3 (�= −8.200) wereobserved here which are still significantly largerthan the threshold value. However, these valuesare already small enough to substantially reducethe accuracy with which thermal averages can becomputed in GCEMC runs based upon the


present number of cycles. On the other hand, asthe plot of �b

l in Fig. 2 shows, these runs are stillsufficiently long to permit an accurate determina-tion of the liquid–gas phase transition. Creation/destruction acceptance probabilities aresubstantially larger for states pertaining to �g

b.For example over the same range −8.200��−7.300 acceptance probabilities for gas statesvary between 4.4×10−1 and 3.5×10−1. Thus, itis not surprising that �b

g can be determined inGCEMC with greater accuracy than �b

l .

4.3. Furrowed substrate

As far as confined fluids are concerned we takethe planar limit [i.e. D=0, �=�, see Eq. (1)] as areference. On account of confinement �x

gl isshifted by an increment ��0 relative to its bulkvalue �x

gl −7.666 (see Fig. 2). Plots in Fig.3show that �� increases with increasing corruga-tion amplitude D (i.e. with decreasing dihedralangle �, sx fixed). This is the chemical potential ofgas– liquid coexistence of the confined fluid shiftsback initially towards the bulk value. Eventually,�� reaches a plateau, that is it becomes approxi-mately independent of D in the limit of suffi-ciently large corrugation amplitude. Tounderstand this effect, it is instructive to investi-gate the dependence of the set of points

Fig. 4. (a) The set L(�) [see Eq. (17)] for states in theone-phase gas region identified in the figure (sx=10, D=18.66). Solid lines represent z1(x) and z2(x) (see Fig. 1). (b) as(a) but L�(�) [see Eq. (19)] and sx=40 (D=5).

Fig. 3. Shift of chemical potential at gas– liquid coexistence ��

relative to bulk (��0) (− ) as a function of amplitude ofcorrugation D for sx=10, T=0.65. Solid line is intended toguide the eye.

L(�)

=�(x, z)��x ��sx

2,�z ��sz

2−1,�(x, z ; �)=0.03

(17)

separating regions of higher density in the vicinityof the substrates (i.e. �z ��sz/2−1) from those oflower density located towards the center of thesystem (i.e. �z ��sz/2−1). Plots of L (�) forD 18.66 (�=�/6) in Fig. 4(a) illustrate thegrowth of a liquidlike film on the substrates priorto capillary condensation. As � increases thethickness of this film increases mainly in the cor-ner of the furrow (i.e. around x�0) whereas inthe vicinity of its asperities (i.e. ‘tips’) located at


�x �=5, �z �=5, L (�) is nearly independent of �.For the largest �= −7.750 at the verge of con-densation, L (�) depends only weakly on z,thereby representing an approximately planar liq-uid–gas interface. In other words, as � ap-proaches �x

gl from below the interior of the furrow(�z ��sz/2) fills gradually with liquid until a nearlyplanar interface is established. Capillary conden-sation then occurs in the remainder of the system,that is in the subvolume V� ={(x, y, z)��x ��sx/2,�y ��sy �z ��sz/2}. It is driven by the interactionwith the solid substrate which is mediated by thefluid occupying the interior of the furrow. Thus,as D increases the interaction between fluidmolecules located in V� and the solid substrateweakens such that for sufficiently large D conden-sation in V� is dominated by the interaction with

the fluid inside the furrow and not by that withthe solid substrate. In the limit of large D onemay therefore view the gas– liquid transition in V�as capillary condensation of a fluid confined by(approximately) planar fluid interfaces as plots ofL (�) in Fig. 4(a) suggest. The form of this fluidinterface is very similar to the one observed forpolystyrene films wetting furrowed silicon sub-strates prior to annealing [58].

Let us now fix D=5 but increase the period ofcorrugation over the range 10�sx�40 in anothersequence of GCEMC simulations. For sx=10and from the plots in Figs. 2 and 3 one deter-mines �x

gl = −7.794 whereas plots of ��g and

��l in Fig. 5(a) indicate that for sx=20 gas and

liquid coexist at a slightly more negative chemicalpotential �x

gl −7.817. Thus, increasing the pe-

Fig. 5. Grand potential density �� as function of chemical potential � for T=0.65. (a) ��g (�), ��

l (�) for sx=20, D=5; (b)��

g (�), ��b (�), and ��

l (�) for sx=30, D=5; (c) as (b) but for sx=40; (d) as (b) but for D=7.


Fig. 6. Local density �(x, z) for points in the upper half of the x–z plane (see Fig. 1), sx=30, and D=5; (a) gas morphology, (b)bridge morphology at phase coexistence (�x

gb= −7.762, see Fig. 5(b)].

riod of corrugation shifts the phase coexistence tolower chemical potentials compared with bothbulk and the case sx=10.

This picture changes for sx=30. Plots in Fig.5(b) show that besides ��

g and ��l a third mor-

phology characterized by ��b may form. Before

analyzing its structure it is immediately apparentfrom Eq. (12) that the mean density of this newmorphology must be intermediate to that of bothgas and liquid morphologies. Moreover, Eq. (13)now has two solutions �x

gb −7.844 and �xbl −

7.762 at which pairs of morphologies must coexistindependently [see Fig. 5(b)]. This situation corre-sponds to scenario 3 of Section 3.2.

What is the morphology of the coexisting phasesat �x

gb −7.844 and �xbl −7.762? To answer

this question an inspection of local densities isuseful. Plots in Fig. 6(a) and (b) show �(x, z) [59]for the two coexisting phases at �x

gb −7.844.The plot in Fig. 6(a) shows a gas morphology

characterized by a thin film of fluid adsorbed onthe surfaces of the furrow whereas the so-called‘bridge’ morphology in Fig. 6(b) consists of twoparts: high(er)-density fluid spanning the gaps be-tween the opposite substrates in the vicinity of thetips and low(er)-density fluid in the ‘inner’ volumebetween the furrowed substrates. The two regionsare separated by an interface. The plot in Fig. 6(b)also shows that the high(er)-density portion offluid is stratified in the vicinity of the substrate asindicated by the nonmonotonic dependence of �(x,z) on distance from the substrate along the surfacenormal of the furrow. On account of the decreasingstrength of fluid–solid attraction, stratification be-comes less pronounced as one moves away fromthe substrate. It is noteworthy that the high(er)-density portion of the bridge phase at �x

gb −7.844 exhibits a remarkable width in thex-direction (z=0) of about ten molecular diame-ters (periodic boundary conditions!).


At �xbl −7.762 [see Fig. 5(b)] a bridge phase

coexists with a liquid phase as plots of the localdensity in Fig. 6(c) and (d) illustrate. However,the high(er)-density part of the bridge morphol-ogy has grown in the x-direction, that is the fluidbridge is significantly wider compared with theone at �x

bl −7.844 [see Fig. 6(b)] and, conse-quently, the low(er)-density region has shrunk insize. Liquid morphologies, like gas morphologies,do not exhibit any inner interfaces between re-gions of higher and lower density as the plot of �

(x, z) in Fig. 6(d) shows.The plot in Fig. 5(b) shows that for sx=30

bridge morphologies are thermodynamically sta-ble over the range −7.844�� −7.762 since��

b is lower than either ��g and ��

l . This one-phase region widens if the period of corrugationincreases from sx=30 to 40 as a comparison ofFig. 5(b) and (c) shows. For example, for sx=30the ‘width’ of the one-phase region of bridgemorphologies is ��1� � 0.082 whereas ��1� � 0.167 for sx=40.

The dependence of ��1� on sx suggests that thetriple-point temperature T tr

gbl also depends on sx ina way such that T tr

gbl decreases with increasing sx

(D fixed). From Fig. 5(b) and (c) one expects Ttr

(sx=30)Ttr (sx=40) both of which are, how-ever, lower than the actual temperature T=0.65.To verify these conjectures ��1� � was determined

Fig. 8. Excess coverage per unit area �(�, T)/Ay as a functionof P/P sat; (�) sx=20, (�) sx=30, (�) sx=40 (D=5). Linesare intented to guide the eye.

for sx=30 and T=0.70, 0.75. Plots in Fig. 7show that ��1� � increases with T. At T tr

gbl,��1� ��0 by definition. Extrapolating therefore��1� � linearly as indicated in Fig. 7, T tr

gbl 0.6 isestimated for sx=30. The same linear extrapola-tion yields T tr

gbl 0.43 for sx=40 thereby confirm-ing the notion of triple-point depression withincreasing corrugation period (see Fig. 7). Thereader should appreciate that a determination ofT tr

gbl for these large corrugation periods is verytime consuming since it is based on plots likethose shown in Fig. 5(b) and (d) so that thenumber of points used in the extrapolation in Fig.7 had to be small.

Experimentally, one is frequently concernedwith measuring the excess coverage, that is theexcess amount adsorbed by a porous medium inthermodynamic equilibrium with a bulk reservoirmaintained at the same T and P/P sat�1 whereP sat denotes the pressure of saturated bulk gas.The excess coverage is defined as

�(�, T):

=� sx /2

−sx/2

dx� z2(x)

z1(x)

dz [�(x, z ; T, �)−�b(�, T)]

=Ay [�(�, T)−�b(�, T)] (18)

where �(�, T)=�N(�, T)�/V and �b(T, �) arethe average density of the confined fluid and ofthe bulk reservoir, respectively. In Fig. 8, �(�, T)

Fig. 7. ‘Width’ of one-phase region of fluid bridges ��1� � asfunction of temperature T for sx=30 (�) and sx=40 (�)(D=5). Straight solid lines are intended to guide the eye andextrapolate to ��1� �=0 (see text).


is plotted as a function of P/P sat for sx=20, 30,and 40 (D=5) where P is calculated from Eq.(11) and P sat�Px

gl 6.8×10−3 (see Section 4.2).Data points plotted in Fig. 8 represent thermody-namically stable phases determined from parallelplots of �� (�) in Fig. 5(a) and (c). Thus, wedeliberately ignore hysteresis frequently observedin experimental sorption isotherms.

As one expects from Fig. 5(a), �(�, T) forsx=20 exhibits a single discontinuity correspond-ing to a phase transition between gas and liquid atP/P sat 0.7802. According to the discussionabove capillary condensation occurs as a two-stage process involving bridge phases for sx20[see Fig. 5(b) and (c), Fig. 6(b) and (c)]. From theplot in Fig. 8 one sees that the gas–bridge transi-tion is shifted to lower P/P sat whereas the bridge–liquid transition is shifted to higher P/P sat

compared with the case sx=20 where no interme-diate bridge phase is observed. The location of the(thermodynamic) phase transition also dependson sx. For example, the gas–bridge transitionoccurs at P/P sat 0.7392 (sx=30), 0.6817 (sx=40), whereas the bridge– liquid transition is ob-served at P/P sat 0.8574 (sx=30), 0.9093(sx=40), indicating that the one-phase region ofbridges widens with increasing sx in accordancewith plots of �� (�) in Fig. 5(b) and (c).

It is interesting along each of the three sorp-tion-isotherm branches plotted in Fig. 8 the slopeof �(�, T) is almost independent of sx. Since(��/��)T is a measure of the isothermal com-pressibility of the confined fluid [relative to thebulk, see Eq. (12), we conclude that this quantityas well as the mean density of gas, bridge, andliquid phases is not sensitive to variations of theperiod of substrate corrugation.

It is furthermore instructive to investigate iso-chores defined by

L�(�)=�

(x, z)��x ��sx

2,z1(x)�z�z2(x),�(x, z ; �)

=0.03

(19)

for the furrow of sx=30. Plots in Fig. 5(b) showthat up to �x

gb −7.844 gas morphologies arethermodynamically stable. For ��x

gb bridge

morphologies are thermodynamically stable andL�(�) is represented by a closed line locatedbetween the surfaces of the furrows (see Fig. 9).In this case L�(�) encloses gas filled ‘holes’ alsovisible in the plots of �(x, z) in Fig. 6(b) and (c).As � increases towards �x

bl the area enclosed byL(�) shrinks, indicating that the high(er)-densityportion of the bridge morphology grows into itslow(er)-density part. In other words, the size ofthe gas filled ‘holes’ shrinks as phase coexistencebetween bridge and liquid is approached. At �x

bl

the ‘holes’ vanish spontaneously.Finally, if one maintains the period of corruga-

tion but increases its amplitude from D=5 toD=7 (sx=30), a comparison of Fig. 5(c) and (d)shows that in the latter case ��1� � 0.120 indi-cating that an increasing corrugation amplitudestabilizes bridge morphologies. In other words,increasing D lowers Ttr at fixed sx. Note that avariation of D affects only ��

l leaving �g and �b

nearly unchanged.

5. Summary and discussion

This work is devoted to a study of confinement-controlled phase transitions in a classical fluidwhere the confining substrate surfaces are nonpla-nar. The furrowed substrates are characterized byperiod sx and amplitude D of corrugation both ofwhich are related through the dihedral angle �

(see Fig. 1). GCEMC simulations are employed toinvestigate the phase behavior of a ‘simple’ fluidfor different sx and D.

Two scenarios can be distinguished. For suffi-ciently small sx�30 only gas and liquid mor-phologies participate in discontinuous phasetransitions. For a given temperature the chemicalpotential at gas– liquid coexistence is generallylower than its bulk counterpart. Thus, the phasetransition seems similar to conventional capillarycondensation of fluids between planar substratesurfaces (see Fig. 3). However, a more detailedanalysis reveals two separate stages in the gas– liq-uid transformation as � approaches �x

gl from be-low [see Fig. 4(a)]. During the initial stage thefurrow is continuously filled with liquid until anearly planar interface separates this liquid from a


gas occupying the subvolume V� . At �xgl the gas in

V� undergoes a discontinuous phase transition andcondenses. Thus, only part of the thermodynamicsystem (residing in the total volume V) is involvedin the phase transition unlike in conventionalcapillary condensation in ideal slits or cylinders.

Condensation is triggered by an attractive fieldof superimposed contributions from gas– liquidinteractions across the interface formed betweenthem in stage one and interactions between thegas molecules and the solid substrate. The latterbecome weaker with increasing D since the homo-geneous solid background represented by �0

[2](z)[see Eq. (4)] recedes from the position of the innergas– liquid interface. Formation of the inner gas–liquid interface, on the other hand, depends onlynegligibly on D. Thus, as D increases, gasmolecules in V� are exposed to an attractive fieldwhose total strength levels out but does not van-ish on account of the liquid filling the interior ofthe furrow. This causes �x

gl to approach a con-stant value with increasing D which remainssmaller than �x

gl in the bulk.

Condensation in furrows with a relatively smallcorrugation period is distinctly different from thatobserved in larger ones. Here a thin film is ad-sorbed initially (i.e. at sufficiently low chemicalpotentials) on the surface of the solid substrate.Eventually this film becomes metastable and un-dergoes a discontinuous phase transition which is‘partial’ in nature, that is it occurs only in asubvolume in which the thermodynamic systemresides. During this partial condensation a bridgemorphology arises which forms at the tips of thefurrow. In such a bridge morphology portions ofhigh(er)-density fluid surround gas filled ‘ holes’whose centers coincide with the origin of thecoordinate system (see Fig. 9). As � increasesfurther these ‘holes’ shrink in size until phasecoexistence between bridge and liquid morpholo-gies is reached. At �x

bl the fluid in the ‘holes’condenses spontaneously and liquid morphologiesbecome thermodynamically stable.

In view of the present results and those ob-tained earlier for fluid phases confined by chemi-cally corrugated substrates, where similar bridge

Fig. 9. The set L�(�) for sx=30 (D=5) in the one-phase region of bridge morphologies [see Fig. 5(b)]. Solid lines represent z1(x),z2(x) (see Fig. 1).


morphologies were also observed, it seemsjustified to view the bridge morphology as repre-sentative of a generic thermodynamic phasepresent in fluids confined by (chemically or geo-metrically) structured or disordered matrices.Hence, these bridges should contribute to theoverall phase behavior of fluids adsorbed in meso-porous media of irregular shape, like, for exam-ple, aerogels, Vycor, or CPG. Because fluidsadsorbed in these matrices are strongly confined,an experimental determination or their morphol-ogy under different thermodynamic conditions isdemanding. In this context positron/positroniumannihilation techniques seem very promising [60].In these experiments one measures the ratio of �

photons emitted during the annihilation of ortho-and para-positronium. This ratio is affected bywhether or not a porous medium is filled withmaterial or not. Hence, the experiments revealinformation about density changes within theporous medium and therefore about the phasebehavior of confined fluids. Positron/positroniumexperiments have been employed to investigategas– liquid transitions of CO2 in nanometer pores[60], the structure of solid CO2 in Vycor [61], andcapillary condensation of N2–Ar binary mixtures[62].

Another interesting application of this type ofexperiments was reported in [63,64] where thelocal free volume in amorphous polymers wasinvestigated. Here positron-lifetime measurementswere utilized to determine the sizes of microscopic‘holes’ in the amorphous material. Even thoughthese ‘holes’ with typical radii of 0.1 nm are muchsmaller than the ones characteristic of bridge mor-phologies reported here, it might be feasible toestablish fluid bridges as thermodynamic phasesby studying the phase behavior of fluids in irregu-lar-shaped mesoporous materials using theseexperiments.

Acknowledgements

The author is grateful to Professor D.J. Diestler(University of Nebraska-Lincoln) for criticallyreading the manuscript. Financial support fromthe Sonderforschungsbereich 448 ‘Mesoskopisch

strukturierte Verbundsysteme’ is gratefullyacknowledged.

References

[1] O. Talu, Adv. Colloid Interf. Sci. 76 (1998) 227.[2] L.D. Gelb, K.E. Gubbins, R. Radhakrishnan, M. Sliwin-

ska-Bartkowiak, Rep. Prog. Phys. 62 (1999) 1573.[3] J.S. Beck, J.C. Vartuli, W.J. Roth, M.E. Leonowicz, C.T.

Kresge, K.D. Schmidt, C.T.-W. Chu, D.H. Olson, E.W.Shepard, S.B. McCullen, J.B. Higgins, J.L. Schlenker, J.Am. Chem. Soc. 114 (1992) 10834.

[4] A. Schreiber, I. Ketelsen, G.H. Findenegg, Phys. Chem.Chem. Phys. 3 (2001) 1185.

[5] E. Kierlik, M.L. Rosinberg, G. Tarjus, P. Viot, Phys.Chem. Chem. Phys. 3 (2001) 1201.

[6] J.S. Rowlinson, B. Widom, Molecular Theory of Capil-larity, Clarendon Press, Oxford, 1982.

[7] M. Schoen, D.J. Diestler, J. Chem. Phys. 109 (1998) 5596.[8] M. Thommes, G.H. Findenegg, Langmuir 10 (1994) 4277.[9] H. Nakanishi, M.E. Fisher, J. Chem. Phys. 78 (1983)

3279.[10] R. Evans, U. Marini Bettolo Marconi, P. Tarazona, J.

Chem. Phys. 84 (1986) 2376.[11] H. Rauscher, T.A. Jung, J.-L. Lin, A. Kirakosian, F.J.

Himpsel, U. Rohr, K. Mullen, Chem. Phys. Lett. 303(1999) 363.

[12] B.F. Jones, F. Galan, Reviews in Mineralogy HydrousPhyllosilicates, vol. 19, Bookcrafters, Chelsea, 1988.

[13] W. Hansen, J.P. Kotthaus, U. Merkt, in: M. Reed (Ed.),Semiconductors and Semimetals, vol. 35, Academic Press,London, 1992, p. 279.

[14] P. Bonsch, D. Wullner, T. Schrimpf, A. Schlachletzki, R.Lacmann, J. Electrochem. Soc. 145 (1998) 1273.

[15] D.W.L. Tolfree, Rep. Prog. Phys. 61 (1998) 313.[16] M. Trau, N. Yao, E. Kim, Y. Xia, G.M. Whitesides, I.A.

Aksay, Nature (London) 390 (1997) 674.[17] T.A. Winningham, H.P. Gillis, D.A. Choutov, K.P. Mar-

tin, J.T. Moore, K. Douglas, Surf. Sci. 406 (1998) 221.[18] F. Burmeister, C. Schafle, B. Keilhofer, C. Bechinger, J.

Boneberg, P. Leiderer, Adv. Mater. 10 (1998) 495.[19] J.A. Mann Jr, L. Romero, R.R. Rye, F.G. Yost, Phys.

Rev. E 52 (1995) 3967.[20] R.R. Rye, F.G. Yost, J.A. Mann, Langmuir 12 (1996)

4625.[21] R.R. Rye, F.G. Yost, J.A. Mann, Langmuir 12 (1996)

555.[22] J.B. Knight, A. Vishwanath, J.P. Brody, R.H. Austin,

Phys. Rev. Lett. 80 (1998) 3863.[23] M. Grunze, Science 283 (1999) 41.[24] D. Beaglehole, J. Phys. Chem. 93 (1989) 893.[25] S. Garoff, E.B. Sirota, S.K. Sinha, H.B. Stanley, J. Chem.

Phys. 90 (1989) 7505.[26] P.S. Pershan, Ber. Bunsenges. Phys. Chem. 98 (1994) 372.[27] V. Panella, J. Krim, Phys. Rev. E 49 (1994) 4179.


[28] J.L. Harden, D. Andelman, Langmuir 8 (1992) 2547.[29] P. Pfeifer, M.W. Cole, J. Krim, Phys. Rev. Lett. 65 (1990)

663.[30] J. Krim, J.O. Indekeu, Phys. Rev. E 48 (1993) 1576.[31] E. Cheng, M.W. Cole, P. Pfeifer, Phys. Rev. B 39 (1989)

12962.[32] G. Giugliarelli, A.L. Stella, Physica A 239 (1997) 467.[33] G. Palasantzas, Phys. Rev. B 51 (1995) 14612.[34] K. Topolski, D. Urban, S. Brandon, J. De Coninck, Phys.

Rev. E 56 (1997) 3353.[35] J.Z. Tang, J.G. Harris, J. Chem. Phys. 103 (1995) 8201.[36] R. Netz, D. Andelman, Phys. Rev. E 55 (1997) 687.[37] P.S. Swain, A.O. Parry, J. Phys. A 30 (1997) 4597.[38] K. Rejmer, S. Dietrich, M. Napiorkowski, Phys. Rev. E

60 (1999) 4027.[39] A.O. Parry, A.J. Wood, C. Rascon, J. Phys.: Condens.

Matter 13 (2001) 4591.[40] C. Chmiel, K. Karykowski, A. Patrykiejew, W. Ryzsko,

S. Sokolowski, Mol. Phys. 81 (1994) 691.[41] P. Rocken, P. Tarazona, J. Chem. Phys. 105 (1996) 2034.[42] P. Rocken, A. Somoza, P. Tarazona, G.H. Findenegg, J.

Chem. Phys. 108 (1998) 8689.[43] H. Bock, M. Schoen, Phys. Rev. E 59 (1999) 4122.[44] H. Bock, M. Schoen, J. Phys. C: Condens. Matter 12

(2000) 1569.[45] A. Vishnyakov, E.M. Piotrovskaya, E.N. Brodskaya, Ad-

sorption 4 (1998) 207.[46] W. Gac, A. Patrykiejew, S. Sokolowski, Surf. Sci. 306

(1994) 434.[47] J.E. Curry, F. Zhang, J.H. Cushman, M. Schoen, D.J.

Diestler, J. Chem. Phys. 101 (1994) 10824.

[48] M. Schoen, S. Dietrich, Phys. Rev. E 56 (1997) 499.[49] C. Rascon, A.O. Parry, Nature (London) 407 (2000) 986.[50] D.J. Diestler, M. Schoen, Phys. Rev. E 62 (2000) 6615.[51] D. Henderson, S. Sokolowski, D. Wasan, Phys. Rev. E 57

(1998) 5539.[52] M.L. Gee, P.M. McGuiggan, J.N. Israelachvili, A.M.

Homola, J. Chem. Phys. 93 (1895) 1990.[53] M. Schoen, in: M. Borowko (Ed.), Computational Meth-

ods in Surface and Colloid Science, Marcel Dekker, NewYork, 2000, p. 1.

[54] R. Evans, J. Phys.: Condens. Matter 2 (1990) 8989.[55] Y. Fan, P.A. Monson, J. Chem. Phys. 99 (1993) 6897.[56] J.E. Finn, P.A. Monson, Mol. Phys. 65 (1988) 1345.[57] M.P. Allen, in: K. Binder, G. Ciccotti (Eds.), Monte

Carlo and Molecular Dynamics of Condensed MatterSystems, Conference Proceedings, vol. 49, SIF, Bologna,1996, p. 255.

[58] N. Rehse, C. Wang, M. Hund, M. Geoghegan, R.Magerle, G. Krausch, Eur. Phys. J. E4 (2001) 69.

[59] Because of Eq. (8), �(r))�(x, z) [see also Eq. (12)].[60] N.J. Wilkinson, J.A. Duffy, H.M. Fretwell, M.A. Alam,

Phys. Lett. A 204 (1995) 285.[61] D.W. Brown, P.E. Sokol, A.P. Clarke, M.A. Alam, W.J.

Nutall, J. Phys.: Condens. Matter 9 (1997) 7317.[62] M.A. Alam, A.P. Clarke, J.A. Duffy, Langmuir 16 (2000)

7551.[63] G. Dlubek, T. Lupke, J. Stejny, M.A. Alam, M. Arnold,

Macromolecules 33 (2000) 990.[64] A. Reiche, G. Dlubek, A. Weinkauf, B. Sandner, H.M.

Fretwell, M.A. Alam, G. Fleischer, F. Rittig, J. Karger,W. Meyer, J. Phys. Chem. B 104 (2000) 6397.

capillary condensation between mesocopically rough surfaces

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