ELSEVIER

23 May 1997

Chemical Physics Letters 270 (1997) 339-344

CHEMICAL PHYSICS LETTERS

Liquid-vapor coexistence in a chemically heterogeneous slit-nanopore

Martin Schoen a, Dennis J. Diestler b

a lnstitutfftr Theoretische Physik Sekr. PN 7-1, Fachbereich Physik, Technische Universitfit Berlin, H ardenbergstr. 36, D-10623 Berlin, Germany

b Department of Agronomy, University of Nebraska-Lincoln, Lincoln, NE68583-0915, USA

Received 7 February 1997

Abstract

A Lennard-Jonesium film confined between planar walls made of alternating strips of strongly and weakly adsorbing solid substrate was simulated by the grand canonical Monte Carlo method. When the walls are molecularly close, they serve as a nanoscale template, on which liquid bridging the gaps between the "strong" strips coexists with gas over the "weak" ones. The structure of the nanoscopic liquid-gas interface has the same form as that of the planar interface. When the walls become too far apart, the bridges give way to "nanodroplets" adhering to the strong strips coexisting with dilute gas.

Adsorption by porous media [1,2], swelling of clay minerals [3], formation of biocomposite materi- als [4], and friction and lubrication [5] all involve fluids confined by solid surfaces to spaces having dimensions of the order of one or a few nanometers. A fundamental understanding of these diverse phe- nomena therefore calls for a study of the molecular behavior of severely confined fluids under precisely controlled conditions. An ideal device for the experi- mental investigation of such systems is the surface forces apparatus (SFA) [6-8], the heart of which comprises a thin film sandwiched between two atom- ically smooth sheets of mica, which can be posi- tioned relative to each other with molecularly fine precision. The SFA functions as a single nanoscale slit-pore immersed in a fluid bath at constant temper- ature and pressure. Until now SFA experiments [9], and analogous computer simulations of model slit- pores [10], pertain to walls that are structurally and chemically homogeneous (in lateral dimensions) on

the nanoscale. However, since recent advances in surface probe microscopy have made feasible the construction of substrates possessing prescribed nanoscale heterogeneities [11,12], it is timely to in- vestigate the influence of nanoscale heterogeneities on the behavior of a fluid film in a slit-pore. The effects of purely structural heterogeneity were ex- plored previously by us [13] using the grand canoni- cal Monte Carlo method to simulate a Lennard-Jones (12,6) film between face-centered cubic (100) planar walls, one smooth and the other scored with rectilin- ear grooves several molecular diameters wide. If the walls are proper aligned laterally, the film consists of fluid and solid portions in thermodynamic equilib- rium (that is, fluid filled nanocapillaries (the grooves) alternating with solid columns). Epitaxial freezing of the film is promoted by the molecular-scale tem- plates of the strips between the grooves [14-16]. In the grooves, however, the template is too weak and the film consequently remains fluid there.

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340 M. Schoen, DJ. Diestler / Chemical Physics Letters 270 (1997) 339-344

Our purpose in this Letter is to present the results of a grand canonical ensemble Monte Carlo (GCEMC) study undertaken to ascertain whether strictly chemical heterogeneity on the nanoscale might also give rise to phase coexistence. To divorce chemical from structural effects, we adopt a chemi- cally patterned smooth-wall slit-pore, each wall con- sisting of alternating strips (infinite in one dimen- sion) of strongly and weakly adsorbing substrates. The concept is that film molecules should tend to congregate preferentially in the vicinity of the strong strip, leaving the film over the weak strip at much lower density. That is, we would expect the strong strip to become wet and the weak strip to remain dry. This would result in nanoscale liquid-like regions separated from gas-like regions by interfaces parallel to the strips and perpendicular to the walls.

Previous work on the character of films adsorbed on chemically patterned solid substrates has been summarized recently by Koch et al. [17]. In the current study the walls are supposed to be made of alternating slabs of monatomic solid of two types, designated " s " and " w " for strongly and weakly adsorbing [18]. The slabs are infinite in the y direc- tion and semi-infinite in the z direction, one wall occupying the half space z <~ - s z / 2 and the other the half space z >1 Sz/2 . The walls are consequently periodic in the x direction, having a period d~ + dw, where ds and dw stand for the thicknesses of the slabs. We take the film to be monatomic also and assume the potential energy of the system to be the usual pairwise sum of Lennard-Jones (LJ) (12,6) potentials tt( Fij) = 4Eij[( O'ij// t ' i j) 12 -- ( Oeij//Fij) 6 ] where rij is the distance between pair of molecules i and j, and e~/ and o'~j are respectively the depth of the attractive well and mean diameter associated with the given pair. If i and j are both film molecules, then e~j. = eFF; if i is a film molecule and j is a slab molecule, or vice versa, then e~j = Evs or evw. For simplicity we take ~r~i = ~FF for all pairs. The smooth potential energy is obtained by averaging the LJ interactions between the film and slab molecules over the positions of the latter (i.e., by smearing the molecules of each slab over the layers of that slab in which they lie [19]).The resulting potential depends both on the distance of the film molecule from the wall and on its x coordinate. Hence, the smooth walls can be pictured as chemically striped sheets

(see Fig. 1). Very far from the wall, the potential decays as the inverse cube of the distance from the wall; this long-range character has important impli- cations for the deportment of the film [20].

In the GCEMC procedure, which has been de- tailed previously [21], the film is contained in the rectangular simulation cell bordered in the z direc- tion by the two smooth walls (Fig. 1). We set s x = d s + d w and Sy = Sx. We assume the walls are aligned transversely so that strips of like type are exactly opposite one another and impose periodic boundary conditions on the cell in the x and y directions. The values of all parameters, henceforth expressed in reduced units based on the molecular diameter o- w and well depth eFF associated with the fi lm-film LJ interaction, are listed in Table 1. We set the temperature T and chemical potential /x so that the bulk phase is a dilute gas having a density n bulk= 0.036o'~ 3 and an isothermal compressibility KTbUlk 3 - I = 44.600"FFEFF, compared with that of ideal gas at the same temperature and density, (kBTnb"lk) - I = 27.78 O-3F e ~ .

TO demonstrate that the present choice of Evs and eVW corresponds to wetting and drying walls, respec- tively, we computed the mean pore density n = ( N ) / s x s y s : and the transverse isothermal com- press ib i l i ty Kyy = Sy l ( O S y / ~ T y y ) r . N ...... : = ( ( N 2 ) / ( N ) 2 - 1 ) s x s y s z / k B T for films between chemically homogeneous s and w walls as a function of s z (Fig. 2). The angular brackets in these expres- sions signify ensemble averages, N the number of film molecules and Try the diagonal component of the stress tensor. The oscillations in n and Kry for

~ d s ~ - J

j ~ " Sx - - . ~ J

Fig. 1. Schematic perspective of simulation cell. Cross-hatched region is strongly adsorbing strip of width ds; blank strips on either side of it together constitute weakly adsorbing strip of width d,,,. Rectangle (filled with vertical dotted lines) in the y = 0 plane is quadrant in which local density is plotted in Figs. 3 and 5.

M. Schoen, DJ. Diestler / Chemical Physics Letters 270 (1997)339-344 341

Table 1

Parameters of system in reduced units: distance, O'FF; energy, ~FF; temperature, EFF / k B

d s 4.0 s x 10.0 Sy 10.0 /x -11 .5 T 1.000

~ s 1.250 (FW 0.001 OYS 1.000 orv- w 1.000

the s wall in the range 1.9 < s z < 6 have a period of about one ~rrr and reflect stratification of the film, that is, on account of constraints on the packing of spherical molecules against planar walls, there is a tendency for the fluid to order itself in layers parallel with the walls in their immediate vicinity [22-26]. Rises in n coincide with maxima in Kyy and corre- spond to order-disorder transitions within existing strata [27-29]. That n is continuous through these transitions indicates they are of order higher than first. In the range 6 < s z < 14, where n rises mono- tonically while Kyy stays approximately constant, stratification has ceased; newly added molecules oc- cupy the central, essentially homogeneous region of the film. Noting that the repulsive part of the f i lm- wall interaction excludes film molecules from a thin layer next to either wall, we deduce that n ap- proaches its asymptote n,q according to the formula n ( s , ) = n ~ i q - c / s z, where c is a positive constant [19]. From a plot of n vs. Sz ~ we extrapolate to a " l imit ing" density nl~ q = 0.71. If s z were to become sufficiently large, that is of mesoscopic dimension, then a vapor phase would form in the center of the pore in equilibrium with the liquid film at the walls. This scenario is typical of a fluid that wets a solid surface [30]. In contrast, for the pore with pure w walls n rises monotonously with s z over the entire range, a s Kyy decreases monotonously. Stratification does not occur and both quantities approach their limiting bulk-phase values as expected. Extrapolating a plot of n vs. s ] i as before, we get ngas =/,/bulk =

0.036. It is interesting that the extrapolated liquid and gas film densities agree well with those knli q " bulk =

0.69, ngas-b~k = 0.036) recently determined [31] for bulk LJ liquid and gas phases coexisting at T = 1.00. We

conclude that the homogeneous w wall is drying under these conditions and that a competition be- tween wetting of the s strip and drying of the w strip is expected to take place in the pore with chemically striped walls.

Plots of n and Kyy for the film between c h e m i -

c a l l y s t r i p e d walls (Fig. 2) show that stratification dominates in the range 1.9 < se < 5, as it does in the film between pure s walls. Compared with the latter, however, n is smaller and Kyy greater on account of the contribution from the gas-like film over the w strip. Beyond s z ~ 5 the zigzag structure in n vs. s z

gives way to a broad sigmoidal rise in n coinciding with an intense cusp-like peak in Kyy. This suggests a higher-order transition of a qualitatively different nature than the order-disorder transitions at smaller

n 0,I

0.01 i00

I0

rCyy

1

0.I

+*++. ooo eoo

nbulk

J

[] o D o\ <> I

+

o

[q [3

A

<>

¢oo

I I i

<>

[3 ++ + ÷ + ++++~ 4++++ ++ + + $ * * + + + + + + +~+ ~. ++++ ~-

2 4 6 8 I0 L2 14

Sz

Fig. 2. Mean pore density n (A) and transverse isothermal com- pressibility ~yr (B) versus s z for homogeneous w pore (D), homogeneous s pore (~ ) and heterogeneous chemically striped pore ( + ) pictured in Fig. 1. Also indicated as horizontal lines are density and isothermal compressibility of bulk Lennard-Jones (12,6) fluid at T = 1.00 and /x= - 11.5.

342 M. Schoen, D.J. Diestler / Chemical Physics Letters 270 (1997) 339-344

s z. For 5.6 < s z < 8.4, Kyy decreases rapidly, ap- proaching a value less than one, typical of dense LJ liquid. Over the same range n increases monotonously, approaching the limiting value nl~q. The pore thus seems to have filled with liquid having a density approximately equal to that near the center of the pore with chemically pure s walls at the same s z. Around s z ~ 8.5 the liquid-like film evaporates suddenly. The associated discontinuity in n and Kyy

indicates that the evaporation is first order, as usual. Structural changes in the film that accompany

these various transitions can be seen nicely in con- tour maps of local density. Fig. 3 shows a sequence of maps in the range of s z below the first "spread- ing" or "fi l l ing" transition around sz = 5.6. Note that since the external field acting on the film is independent of y and has rectangular symmetry in planes y = constant, one need represent the local density in only one quadrant of a single plane (see Fig. 1). At s z = 2 (Fig. 3A) a single dense layer of molecules is indeed localized in the central region between the s strips, while a nearly uniform very low-density gas occupies the w region. For conve- nience we shall henceforth refer to the dense central portion of the film as the bridge. The sequence of plots in Fig. 3 indicates that as s, increases, addi- tional discrete dense layers abruptly appear in the bridge, which grows only slightly in the x direction. This is inferred from the shift of the intersection of

the 0.1 contour with the x axis, which changes by about one O'FF in the x direction over the range 2 < s z < 5 (compare Fig. 3A and Fig. 3D). The period between successive appearances of an addi- tional layer is also about one (TFF. This stratification is of course the same phenomenon that operates in films between c h e m i c a l l y p u r e h o m o g e n e o u s walls. In the heterogeneous pore, however, film molecules tend to pack preferentially against the attractive s strip. The ordering breaks down at the edges of the strip, giving rise to unstructured gas over the w strip, where the walls are essentially repulsive.

We can view the situation in the slit-pore as an analogue of bulk liquid-vapor equilibrium, the bridge assuming the role of the liquid. Although the contour plots of Fig. 3 reveal a somewhat irregular interface between the bridge and the gas, it is roughly parallel with the y z plane. We have fitted the local density in planes parallel to the xy plane, and hence approxi- mately normal to the interface, to the form, p ° ) ( x , z

= z0) = ~(nj + n~) - ~(n I z _ ng) tanh[2(x - x t) / D ] , normally applied to the planar liquid-vapor interface [32], where n] and ng stand for the densi- ties of the homogeneous (bulk) liquid and gas phases, x~ for the position of the interface and D for its thickness. Regarding these as unknown parameters, we used a nonlinear least-squares method [33] to fit the local density in the plane z = z0. The relative success of these fits (see Fig. 4), especially the

A ~ - . . . . O_Q___

B 0.5

0.4

0.3 Z/Sz

0.2

0.1

C 0.0 D O.O

. . . . . 0 .2

- - - - , ~ ,~ \,) S, o.o o.o o., o., o.

×/sx

Fig. 3. Contour maps of local density ( p ( I ) ( x , z ; s : ) = c) of film between walls of chemically striped slit-pore separated by distances (A) s. = 2.0; (B) s z = 3.0; (C) s~ = 4.0; (D) s: = 5.0. Numbers by contours are values c of constant density.

M. Schoen, DJ. Diestler / Chemical Physics Letters 270 (1997) 339-344 343

0 . 6

+ +

N N 0.3

o.~ \

0 l 0.I 0 . 2 0 . 3 0.4 0.5

X/Sx Fig. 4. Local density p(i) (x,z = Zo;Sz) for s: = 5.4, Zo/S: = 0 . 0

( + ) and s: = 8.6, Zo / sz = 0.2 ( ~ ) (see Figs. 5A and D). Solid

l ines represent best least-squares fits of expression in text.

interfacial thickness D--- 1.32 at s z = 5.4, which is of the order of the thickness of the typical bulk liquid-vapor interface [32], supports the analogy. However, it is worth noting that the best values of n I = 0.52 and ng = 0.072 at s z = 5.4 do not agree with the densities of the coexisting bulk phases at T = 1.00. This discrepancy demonstrates the marked effect of the walls on phase equilibria in confined films. The strong impact of confinement on the phase diagram has also been observed experimen- tally, for example by Thommes and Findenegg [34],

who measured the dependence of the critical temper- ature on pore width for SF 6 in mesoporous glasses.

As s z increases beyond about 5, the interface becomes increasingly diffuse. Note, for example, that the 0.1 contour of the map at s z = 5.4 (Fig. 5A) is no longer parallel with the z axis. At s z = 6.2 (Fig. 5B) the bridge has apparently spread over the w strip. Comparison with Fig. 2 shows that the cusp in t(yy is associated with this spreading, which has the earmarks of a higher (than first) order transition. For example, at the cusp Kyy shows a pronounced de- pendence on the size of the system [19]. For the high-density liquid-like film in the range 6.4 < s z <

8.4, the interface is absent; note that the 0.5 contour in Fig. 5C runs roughly parallel with the walls. Finally, at around s z = 8.5 the liquid film evaporates abruptly in a first-order transition (reflected in the discontinuity in n; see Fig. 2) to yield nanodroplets adhering to either s strip in equilibrium with gas whose density ( n g = 0.039) is approximately equal to that of the bulk phase (see Fig. 5D). As can be seen in Fig. 4, the interface between nanodroplet and vapor phases can again be fitted to the above expres- sion. However, the interface is more diffuse (D = 2.09) and the droplet is hardly liquid because of its very low density (n I = 0.16).

The behavior of the film in the chemically striped pore can be correlated with the behaviors of films in

A 0,0

0.0 C

i i L

0.0 0.i 0.2 0.3 0.4

X/Sx

0.5 B 0.2 0 0

0 . 5 ~

D ~ ' ' '0.0

J

0 . 4

0.3

0.2

0.i

0.4

0 . 3 N

0.2

0.1

' ' ' ' 0.0 0.I 0.2 0.3 0.4 0.5

X/Sx

Fig. 5. Contour maps of local density ( p ° ) ( x , z ; s z ) = c) of film between walls of chemical ly striped slit-pore separated by distances (A) s. = 5.4; (B) s z = 6.2; (C) s, = 8.4; (D) s~ = 8.6, chosen to illustrate changes in the molecular structure of film across " sp read ing"

transition at s z = 5.6 and evaporation at s . = 8.5.

344 M. Schoen, D.J. Diestler / Chemical Physics Letters 270 (1997) 339-344

the chemically h o m o g e n e o u s pores of strongly and weakly adsorbing types. Roughly speaking, in the range 2 < s z < 5, that part of the film over the s strip acts like the whole film in the pure s pore; the part over the w strip acts like the whole film in the pure w pore. When the heterogeneous walls get suffi- ciently far apart, however, both strips become cov- ered with liquid-like film. For a brief range of s z t h e

s strip is capable of stabilizing a liquid-like film throughout the pore. In other words, the striped walls are "temporarily" wetted by the film. Nevertheless, when s z exceeds a critical value, the long-range attraction of the s strip is so weakened by the broader w strip that the liquid-like film cannot be sustained. We surmise that if the ratio of to d w to d s were smaller, the liquid-like film would persist to larger s z. Obviously in the limit that it goes to zero, the wall becomes pure s and is wetted out to wall separations greater than s z = 14.

Acknowledgements

The authors have been generously supported by the North Atlantic Treaty Organization (Collabora- tive Research Grant CRG801927). MS is grateful to the Deutsche Forschungsgemeinschaft for a Heisen- berg fellowship (Scho 525 1-1) and DJD to the Office of Naval Research (Grant No. N00014-96-1- 0903) and the National Science Foundation (Grant No. ECS-9521288) for additional support.

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