3764
J. Phys. Chem. B 2006, 110, 3764-3772
Nanoscopic Liquid Bridges between Chemically Patterned Atomistic Walls† C. J. Hemming and G. N. Patey* Department of Chemistry, UniVersity of British Columbia, VancouVer, British Columbia V6T 1Z1, Canada ReceiVed: NoVember 2, 2005
A binary liquid mixture, containing the Lennard-Jones molecules A and B, in equilibrium with a bulk liquid reservoir near the point of phase separation, confined between atomistic chemically patterned walls, is studied by grand canonical Monte Carlo simulations. In the bulk, the B-rich phase is stable and the A-rich phase is metastable. The walls bear patches attractive to A; when the walls are close, A-rich liquid bridges condense between the patches. The normal and lateral forces on the walls are measured as a function of the wall separation and of the lateral displacement between the patches on opposite walls. When there are one or two molecular layers in the bridge and the wall lattice constant is close to that of crystalline A, the normal and lateral forces depend strongly on the registry of the wall lattices, varying in an oscillatory manner.
I. Introduction When a fluid near a phase transition is confined in a thin layer in a pore, heterogeneities on the pore wall can induce the phase transition locally, resulting in a spatially inhomogeneous “bridge phase”, in which different phases of the fluid occupy regions in the pore with interfaces separating them. In this paper we are concerned with a liquid mixture with two components, A and B, in a pore made from two parallel planar solid walls. The bulk liquid exhibits phase separation into an A-rich phase and a B-rich phase. The walls are chemically patterned, bearing patches which interact attractively with component A. If they are immersed in a B-rich bulk phase which is nearly saturated with A and hence near coexistence, an A-rich phase can condense between the patches, forming A-rich liquid “bridges” that connect patches on opposite walls. The behavior of highly confined fluids between inhomogeneous surfaces has been a subject of increasing interest. Many new techniques for preparing surfaces with well-defined nanoscale patterning have emerged, and the development of the surface forces apparatus means that shear and normal forces on surfaces separated by layers of liquid a few molecules thick can be measured directly.1-3 Additionally, there has been increasing general interest in all aspects of the behavior of matter when nanoscopic length scales are involved, motivated by the prospect of technological application in new devices or materials with designed and controlled nanoscale structure. The formation of A-rich bridges between the walls in our system is an example of a confinement-induced phase transition.4,5 In highly confined fluids the interaction with the walls has an increasingly strong influence as the wall-to-wall separation decreases. In our system of study the A-rich phase is a metastable state that is stabilized by the attractive patches on the walls. A similar phenomenon, except with chemically uniform walls, is observed in surface forces apparatus (SFA) experiments with the nonpolar fluid octamethylcyclotetrasiloxane (OMCTS) containing traces of water.6 The mica walls of the SFA are hydrophilic, and at a critical separation a water†
Part of the special issue “Michael L. Klein Festschrift”. * Author to whom correspondence should be addressed. E-mail:
[email protected].
rich phase condenses between them. This condensation exerts large attractive net forces on the walls. The critical separation depends on the water concentration and has been observed to be as large as 14 nm. Greberg and Patey found a similar wallinduced demixing transition in simulations of a two-component liquid mixture that is essentially the same as the A-B model that we study in this paper.7 Overduin and Patey studied liquid bridges in the same liquid system with walls bearing stripes attractive to A.8 As in the experiments, the confinement-induced phase transitions are associated with strong net forces acting on the walls, attracting them to each other. The phenomenon of a confinement-induced phase transition is a very general one, and for a given example one can construct a system that may form bridge phases simply by varying in space the wall-fluid interaction between two forms, each of which gives rise to a different confined phase. Liquid-solid bridges between physically corrugated walls in a one-component confined fluid were studied in simulations by Curry et al.9 In this case, it is the wall-wall separation h itself that varies in space. Ro¨cken, Tarazona, and co-workers studied liquid-vapor bridges for a one-component fluid between chemically patterned walls in a lattice gas model and with density functional theory.10,11 Bock, Schoen, Diestler, and co-workers used a lattice gas model and simulations to study similar one-component bridges, with particular emphasis on the effects of imposing shear strain on the bridges by misaligning parallel stripes on opposite walls.12-20 Sacquin-Mora et al. studied one-component bridges between elliptical patches subjected to torsional strain via rotation of the patches.21-23 Bridges made up of a liquid polymer were studied by Yaneva et al.24 For the two-component liquid mixture that we study here, in addition to the work done by Overduin and Patey, we have investigated the behavior of the bridges under shear strain in a previous paper.8,25 The results that we obtained in ref 25 regarding the static properties of liquid bridges under shear strain are very similar to those reported by Bock, Schoen, and Diestler for onecomponent liquid-vapor bridges.17,20 At sufficiently general levels of description that ignore microscopic details of the system and do not address dynamics, such as the mean-field lattice gas models studied by Bock et al. and by Ro¨cken and Tarazona, the two systems are essentially identical. In the cases
10.1021/jp056331l CCC: $33.50 © 2006 American Chemical Society Published on Web 01/21/2006
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studied, the walls bear parallel, regularly spaced attractive stripes. With the walls in alignment, meaning stripes on opposite walls directly face each other, there is no shear force on the walls. As the walls are displaced laterally relative to each other, misaligning the stripes, a lateral restoring force acting to bring the stripes back into alignment appears. This is linear in the lateral displacement for small displacements, but the strength of the restoring force falls below the linear value at larger displacements. Eventually the force reaches a maximum value; for further displacements the force decreases and eventually approaches zero. The maximum value corresponds to the shear force that must be applied to the walls to initiate sliding motion. Beyond the yield point where the maximum occurs, bridges disintegrate, losing material. The results of Bock and Schoen suggest that this yielding is a first-order transition.18 In our earlier paper, we studied liquid bridges between walls that bore chemical patterning but possessed no atomistic structure. (We will refer to walls bearing no atomistic structure as “smooth”, and walls with no chemical patterning will be termed “chemically uniform” or “chemically unpatterned”.) The wall-wall separations h considered in that work were in the range 4σ e h e 9σ, where σ is a “molecular” diameter. In general it is found for confined liquids that at layer thicknesses less than 10σ confinement effects due the discrete molecular nature of the fluid begin to appear; quantities such as normal force on the walls and the density show signs of being “quantized”, exhibiting oscillations as a function of wall separation. This is associated with the layering of fluid molecules parallel to the walls. There are a variety of results from simulations of different systems that suggest that, as a general rule, for still smaller wall separations than those considered in ref 25, atomistic wall structure can have a strong effect on the properties of confined fluids. This paper investigates the regime of small wall separations, 1.8σ e h e 6σ, with atomistic walls, with the aim of determining the conditions and separations at which atomistic wall structure becomes relevant, and the relative magnitude of effects due to wall structure compared to those arising from bridge formation. Note that h in this paper is not exactly comparable with h in ref 25 because of the different wall potentials used. This paper is organized as follows. In section 2, details of the model and the Monte Carlo simulations are presented. The results are discussed in section 3, and the conclusions of the paper appear in section 4. II. Model and Simulations Our grand canonical Monte Carlo (GCMC) simulations use the method of Adams, in which the configurational chemical potentials are held constant.26 The configurational chemical potential for species R is
µ′R ) µR - µideal + kT ln FjRσ3 R σ3 ) µR + kT ln 3 Λ
(1)
where V is the volume of the simulation cell, FjR is the average density of species R, and Λ ≡ (2πp2/mkT)1/2 is the thermal de Broglie wavelength. The A-B binary liquid model is defined by the cut-andshifted Lennard-Jones (LJ) potential
uij(rij) )
{
4
[( ) σ rij
12
- λR(i)R(j)
() () σ rij
6
-
σ rc
12
+ λR(i)R(j)
( )] σ rc
0
6
for rij e rc for rij > rc (2)
where uij is the pairwise potential between fluid atoms i and j, rij ) |ri - rj| is the distance between them, R(i) ) A or B is the type of fluid atom i, rc ) 5σ is the cutoff radius, and λAA ) 1 and λAB ) λBA ) λBB ) 0.5. The cut-and-shifted LJ potential was used so that molecular dynamics studies can be performed using the identical potential. Properties of the fluid in the bulk are given in Table 1. These were measured in grand canonical Monte Carlo simulations in a simulation cell measuring 10σ × 10σ × 10σ with periodic boundary conditions. Monte Carlo moves employed consisted of displacements, insertions, deletions, and changes of molecule type. An initial period of 107-108 moves was allowed for equilibration, and data were collected over 2 × 108 to 4 × 108 moves. Properties of metastable phases were measured by imposing a minimum or maximum on the value of the mean density F. The pressure was determined from the virial expression27
P)
〈N〉kT
1
-
V
riju′ij(rij)〉 ∑ i>j
〈
3V
(3)
where the broken brackets represent the grand canonical ensemble average computed over the configurations of the Monte Carlo simulation. Keeping µ′A fixed such that βµ′A ) -2.61, with β-1 ) kT ) 1.15, coexistence between an A-rich phase and a B-rich phase occurs at βµ′* B = -0.743 (on the basis of interpolation from Table 1). We perform our GCMC simulations in the confined system fixing βµ′A ) -2.61, βµ′B ) -0.174, and kT ) β-1 ) 1.15. At these values the B-rich bulk phase is stable, and the A-rich bulk phase is metastable. The confined system contains two parallel planar walls of atoms, with fluid molecules between the walls. The bottom wall lies in the z ) 0 plane, and the top wall is in the z ) h plane. The system is periodic in the x- and y-directions. Each wall consists of one layer containing Nw atoms arranged in a twodimensional square lattice. Wall atoms are fixed at their lattice sites; we anticipate that this use of fixed-atom walls rather than more realistic “live” walls in which atoms may move will not affect results significantly. In addition, fixed-atom walls are much more tractable theoretically and simpler to implement in simulations.28 Two kinds of atoms are used in the walls: “attractive” (a), which interact with A fluid atoms via a cut-and-shifted LJ potential and with B atoms via a Weeks-Chandler-Anderson (WCA) potential, and “neutral” (n), which interact with both A and B fluid atoms via the same WCA potential. The pair potential between wall atom i and fluid atom j is
{
Vij(rij) ) σw 4w rij
[( ) ( ) ( ) ( ) ]
0
12
-
σw rij
6
-
σw
rR(i)R(j) c
12
+
σw
rR(i)R(j) c
6
for rij e rR(i)R(j) c for rij > rR(i)R(j) c (4)
where rij ) |ri - rj| is the distance between the atoms, w and σw are, respectively, the energy and length parameters for the interaction, R(i) ) a or n is the type of wall atom i, R(j) ) A
3766 J. Phys. Chem. B, Vol. 110, No. 8, 2006
Hemming and Patey
TABLE 1: Bulk Properties of the Liquid Defined by the Potential in Eq 2 at β ) 1.15E and βµ′A ) -2.61a B-rich phase
A-rich phase
βµ′B
βPσ3
Fσ3
xB
F′
βPσ3
Fσ3
xB
F′
-0.174 -0.348 -0.522 -0.696 -0.730 -0.870
0.531 0.483 0.440 0.400 0.393 0.364
0.321 0.306 0.292 0.279 0.277 0.268
0.883 0.856 0.822 0.780 0.771 0.727
0.438 0.414 0.391 0.369 0.365 0.349
0.397 0.393 0.393 0.393 0.390 0.389
0.649 0.661 0.666 0.670 0.671 0.672
0.028 0.018 0.013 0.010 0.010 0.008
0.657 0.666 0.670 0.673 0.673 0.675
LJ 3 B 3 A The quantity F′ ≡ (σLJ A ) F + (σB ) F is a “packing density”, LJ LJ 1/6 where σA ) σ, σB ) 2 σ, and FA and FB are the mean number densities of A and B. Italics indicate metastable states. a
TABLE 2: Parameters for Atomistic Walls system
orientation
dimensions (Lxa-1 × Lya-1)
aσ-1
w-1
Na
C D E
I II II
12x2 × 6x2 18 × 8 20 × 10
1.2 1.2 1
1 1 0.8
48 48 60
or B is the type of fluid atom j, rR(i)R(j) is the cutoff radius, with c aB nA nB 1/6 raA c ) rc ) 5σ and rc ) rc ) rc ) 2 σw. The top wall is a duplicate of the bottom wall but the lattice sites are translated in the x-direction by a distance ∆. Specifically, whenever there is a bottom wall atom at (xi, yi, 0), there is a top wall atom of the same type at (xi + ∆, yi, h). The full potential for the system is
Φ({ri}) )
Vij(rij) + ∑ ∑ Vij(rij) + ∑ Vhw(ri) ∑ uij(rij) + i∈W ∑∑ j∈F i∈W j∈F i∈F
i,j∈F i>j
t
b
(5) where F is the set of N fluid atoms, Wt and Wb are respectively the sets of top and bottom wall atoms, each containing Nw atoms. Terms have been added corresponding to hard walls at z ) 0 and z ) h
Vhw(ri) )
{
0 0 e zi e h ∞ otherwise
(6)
to reflect that fluid molecules may not be found outside the walls. There is no interaction between wall atoms; since the wall atoms are fixed such an interaction would make a constant contribution to all quantities. It would be straightforward to calculate these contributions if desired. We investigate systems with several different kinds of walls. Table 2 gives the parameter values defining the systems used, which we designate C through E (skipping A and B to avoid confusion with the fluid molecule types). For all systems, σw ) σ. The parameter a is the lattice constant, and Na is the number of attractive atoms in a single wall. The attractive wall atoms occupy adjacent rows parallel to the y-axis. The lattice orientation refers to whether the square unit cells have their sides at an angle of 45° to the x- and y-axes (I) or parallel to the axes (II). Figure 1 shows single walls for systems C and D. In one-component fluids confined between atomistic walls, confinement-induced crystallization of the fluid can occur. Regarding the possibility of wall-induced crystallization, we are primarily concerned with the potential formation of facecentered cubic (fcc) crystals, as this is the stable crystalline structure for an LJ potential. The square-lattice walls are (100) planes of an fcc crystal, when considered individually. The
Figure 1. Single walls for systems C (top panel) and D (bottom panel) (Table 2). Wall atom positions are represented by filled circles for attractive wall atoms and empty circles for neutral.
direction of relative wall displacement as ∆ is varied is the [010] direction for orientation I and the [011] direction for orientation II. For orientation I, when the walls are in-registry (meaning that the top wall atoms are directly over the bottom wall atoms, i.e., ∆ ) ma for integer m), an fcc lattice can be completed by packing an odd number of layers of spheres between the walls. For out-of-registry walls (∆ ) (m + 1/2)a, integer m) with orientation I, fcc lattices are completed with an even number of layers between the walls. For orientation II, fcc lattices can be completed only with the walls in registry with an odd number of layers between them. In our GCMC simulations of the confined system, the (h, ∆) parameter space was surveyed systematically, where h is the wall separation and ∆ is the lateral relative wall displacement. The same types of Monte Carlo moves were employed as for the simulations of the bulk system described above. An initial period of 108 moves was allowed for equilibration, and data were collected over 4 × 108 moves. The quantities measured included the outward normal force on the walls and the lateral restoring force on the walls. These may be determined either from direct expressions or from expressions obtained via a virial route, as described by Henderson.29 For the normal pressure, the direct expression is
P)
1
〈(
zij
∑ ∑ V′ij(rij)r 2A i∈F j∈W
-
ij
t
)〉
zij
∑ V′ij(rij)r j∈W
ij
b
(7)
The direct expression for the restoring force on the walls is
FR )
〈(
xij
1
∑ - ∑ V′ij(rij)r 2 i∈F j∈W t
ij
-
)〉
xij
∑ V′ij(rij)r j∈W b
ij
(8)
Expressions derived from Henderson’s virial route make use of the expression for the differential
(dΩ)µA,µB,T ) -P dV + γ dA + γL dLx - FR d∆
(9)
where Ω ) Ω(µA, µB, T, V, A, Lx, ∆) is the grand potential, which is the thermodynamic potential applicable for the grand canonical ensemble in which we perform our simulations, A ) LxLy, V ) Ah ) LxLyh, γ is the surface tension, γL is the line tension, and FR is the lateral restoring force. Note from eq 1 that holding µA, µB, and T constant is equivalent to holding µ′A, µ′B, and T constant since µ′R - µR depends only on T and no other thermodynamic variables. In the virial method one
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considers the change in the grand potential under an infinitesimal displacement in which the independent thermodynamic state variables V, h, Lx, or ∆ change. This is done by applying an infinitesimal displacement field e(r) to the fluid molecules, where one chooses a displacement field that changes the system geometry in the desired manner, typically by changing one of the independent variables while leaving the others constant. Since the wall potential depends explicitly on the independent thermodynamic variables, through h and ∆, we must apply the appropriate change to it as well. To the first order, the average change in the grand potential is
〈
δΩ ) -kT
∇·ei + ei·∇iΦ({ri}; h, ∆) + ∑ i∈F
δh
∂
Φ({ri}; h, ∆) + δ∆
∂h
∂ ∂∆
〉
Φ({ri};h, ∆) (10)
where ei ≡ e(ri). The latter two terms arise from the change in the wall potential via its explicit dependence on h and ∆. Although such terms do not appear explicitly in the formula as written by Henderson, it is made clear in ref 29 that their presence is required. To calculate the pressure normal to the wall, P, using eq 10, an appropriate displacement field that changes V but leaves A, Lx, and ∆ unchanged is e(r) ) ′(0, -y, z), where ′ is a small parameter. Then δh ) A-1δV, δ∆ ) 0, and δV ) ′V. We obtain
P) 1 V
kT〈N〉
〈(
-
V zij2 zij2 zij2 u′ij(rij) + V′ij(rij) + V′ij(rij) j∈F rij j∈Wb rij j∈Wt rij
∑ ∑ i∈F
∑
∑
i>j
)〉
(11)
For FR, we use the displacement field e(r) ) ′(zi - h/2, 0, 0). Then δh ) 0 and δ∆ ) ′h, and we obtain
FR ) -
〈(
1 h
xijzij (zij - h/2)xij u′ij(rij) + V′ij(rij) + rij j∈F rij j∈Wb
∑ ∑ i∈F
∑
i>j
∑
j∈Wt
(zij + h/2)xij rij
V′ij(rij)
)〉
(12)
Use of the virial theorem30
〈
qj
〉
∂Φ({qi}) ) δjkkT ∂qk
(13)
where the qj are the generalized coordinates, which in our case are (q1, q2, ...) ) (x1, y1, z1, ...), shows that eqs 11 and 12 reduce, respectively, to eqs 7 and 8 when the ensemble average is taken. The reduction also uses the identities z2ij ) zizij for j ∈ W b, z2ij ) (zi - h)zij for j ∈ Wt, and 〈∑i∈F ∑j∈W t zijV′ij(rij)/rij〉 ) -〈∑i∈F ∑j∈W b zijV′ij(rij)/rij〉. III. Results and Discussion Studies have established that the influence of atomistic wall structure on in-plane ordering depends strongly on the commensurability between the wall lattice and the structure of the crystal lattice formed by the fluid upon solidification.31,32 Consequently we have investigated walls with structures of
Figure 2. Normal pressure, system D. The dotted curves at the bottom are the projection of the level curve βPσ3 ) 0.525 onto the (hσ-1, ∆σ-1) plane. The shaded area indicates the bridge-forming region.
varying commensurability, from system C, which is highly commensurate with the fcc structure of solid A, to system E in which the lattice constant is significantly different. We will first discuss the intermediate case, system D, in which the lattice constant, a ) 1.2σ is close to the expected lattice constant for solid A (at T ) 0 and P ) 0, a ) 21/6σ = 1.122σ), but the relative wall displacement in the [011] direction only allows fcc crystal formation when the walls are in-registry. Following this, we will discuss systems C and E with respect to their similarities and differences with system D. A. System D. For system D the walls each bear six rows of eight attractive atoms, and the relative wall displacement with varying ∆ is in the [011] direction. Figure 2 shows the normal pressure βPσ3 as a function of the independent variables (h, ∆) for system D. At low h, there are oscillations in the h-direction, forming a “trough-and-ridge” structure. These are due to the layering of the fluid molecules in the direction perpendicular to the walls (i.e., the layers form parallel to the walls), which is a consequence of the discrete molecular nature of the fluid. This is the same well-known layering phenomenon observed whenever a liquid is near a wall.1,33,34 In Figure 2 one can distinguish by inspection the region in which bridge formation occurs. The round-shaped region surrounds the low-h, low-∆ corner of the (h, ∆) space, extending to ∆ ≈ 6σ at low h and to h ≈ 6σ at ∆ ) 0. For h greater than about 3σ, the bridgeforming region is associated with a depression in the βPσ3 surface; to assist in visualization of the region, a contour corresponding to βPσ3 ) 0.525, a value lying slightly below the bulk value βPbulkσ3 ) 0.531, is shown. With these contours as a guide, shading has been added to indicate the bridgeforming region. The depression in the surface indicates the presence of a net attractive force between the walls due to bridge formation. Oscillations in the h-direction extend out to larger h inside the bridge-forming region because the bridges are composed of the more densely packed A-rich phase. For ∆ > 6σ, there is no overlap between the regions of attractive atoms on opposite walls (within the 0 e ∆ e Lx/2 range studied); hence the boundary of the bridge-forming region occurs there. Outside the bridge-forming region, the first layer to fill the pore is A-rich, and the second layer and all further fluid to enter the pore is B-rich; this is established from FA(x, z) and FB(x, z) data, where the (x, z)-density profile for species R is
FR(x, z) )
1 Ly
∫0L dy FR(x, y, z) y
(14)
where FR(x, y, z) is the local density of species R. Consistent with the low packing density of the B-rich liquid, layering effects are visible only for the formation of the first two layers. The second ridge in the bridge-forming region (at h ≈ 2.6-2.7σ
3768 J. Phys. Chem. B, Vol. 110, No. 8, 2006
Figure 3. Lateral restoring force, system D. The level curve FRσ-1 ) -1 is projected onto the (hσ-1, ∆σ-1) plane at bottom.
Figure 4. Restoring force versus ∆ for system D. Top panel (lower to upper): h ) 2.2σ, 3.8σ, 4.4σ, 5σ. Bottom panel: FR(∆) for h ) 1.8σ (solid curve), smooth curves drawn through the maxima and minima (dashed, upper and lower), and the average of the upper and lower smooth curves (heavy dotted).
does not line up with the ridge outside the region, at h ≈ 2.93.0σ, because of the larger diameter of the B molecules. In addition to the oscillations in the h-direction caused by layering, there are oscillations in the ∆-direction, which are stronger at small h. These are due to the atomistic wall structure and do not appear with smooth walls.25 Their wavelength is 1.2σ, equal to the lattice period in the direction of wall displacement. Figure 3 shows the lateral restoring force on the walls, σ-1FR. Oscillations in the h-direction due to layering are present. Oscillations in the ∆-direction arising from atomistic wall structure are present and become extremely large at the smallest h values. As in the βPσ3 plot, both h-direction and ∆-direction oscillations extend to larger h within the bridgeforming region. Figure 4 (top panel) shows some selected data from Figure 3 plotted as σ-1FR(∆) curves. The curve for the largest h value, h ) 5σ, has a structure free from oscillations, with a general form very similar to the FR(∆) curves obtained for systems with smooth walls in ref 25. The ∆ ) 0 position, at which the wall lattices are in-registry and the stripes on opposite walls are aligned facing each other, is a point of stable mechanical equilibrium since FR ) 0 and (∂FR(0)/∂∆)h < 0, where our notation suppresses other constant state variables. (Note that for all systems studied in this paper, by symmetry FR(0) ) FR(Lx/2) ) 0, necessarily.) For small displacements from the ∆ ) 0 equilibrium point, there is a restoring force
Hemming and Patey linear in ∆ acting to return the walls to alignment. As ∆ increases, the restoring force strength becomes increasingly less than the linearly extrapolated value; eventually it passes through a maximum and decreases to a small value. The minimum of FR(∆) is the yield strength of the bridge; it is the minimum shear stress that must be applied to initiate sliding of the walls. At the portion of the curve corresponding to bridge yielding, where |FR| decreases after its maximum, the bridges undergo rupture, which occurs via a decrease in the density of A in the middle of the bridge. The FR(∆) curves for h ) 4.4σ and 3.8σ in Figure 4 (top panel) have a shape similar to the h ) 5σ curve, but with small oscillations present. The yield strength of the bridge increases with decreasing h, as seen for bridges between smooth walls. For the h ) 2.2σ curve, the oscillations are much larger; however, one can view this curve as consisting of oscillations occurring around a curve of the same basic shape as the h ) 5σ curve. That is, we may imagine there being an “underlying” smooth curve passing through the midpoint of the oscillations, and it is reasonable to think of the restoring force FR(∆) as arising from the addition of two contributions: the contribution associated with the “underlying” curve due to the presence of the bridge and the oscillatory contribution arising from wall granularity. The length scales associated with these two contributions are, respectively, the attractive stripe width and the lattice constant of the walls. The curves in Figure 4 (top panel) were selected for clarity in illustration; they give the impression that the amplitude of the oscillatory contribution decreases monotonically as h increases, but this is in fact not the case. On the h ) 2.2σ curve of Figure 4 (top panel), the amplitude of the ∆-direction oscillations is smaller at larger ∆, where there is no overlap between the attractive patches on opposing walls. When patches overlap there is a region of A-rich liquid with attractive wall atoms on each side. A region of attractive wall atoms presents a much more highly corrugated potential to the A molecules than does a region of neutral atoms, due to the difference between the respective LJ and WCA interactions. Consequently the system is less sensitive to variations in ∆ when there is no overlap between the patches of attractive atoms. Note that for the h ) 2.2σ curve, ∆ ) 0 is not a point of stable mechanical equilibrium, because (∂FR(0)/∂∆)h > 0. Rather than having the two wall lattices in-registry, the preferred configuration is out-of-registry; there is a nearby mechanically stable point at ∆ = a/2 (and another at ∆ = -a/2 by symmetry). If a shear stress close to but less than the yield stress is applied, then there may be several points of mechanical equilibrium. Similarly, with zero applied shear stress and the attractive patches near antialignment (∆ ≈ Lx/2), there are several mechanically stable points. One can consider the thermodynamics of systems with a given applied shear stress using methods developed by Diestler and Schoen et al.35-37 With their methods, working in the grand isostress isostrain ensemble (fixed shear stress and normal strain), one could determine which of the mechanically stable points is the thermodynamically stable state at fixed applied shear stress; the other points will be thermodynamically metastable. Figure 4 (bottom panel) shows the FR(∆) curve for the smallest h value studied, h ) 1.8σ. The oscillations due to atomistic wall structure are extremely large. The figure shows a decomposition of the FR(∆) curve into oscillations occurring symmetrically about an “underlying” curve and the “underlying” curve itself, which is the midpoint of the upper and lower envelopes of the oscillations. This illustrates that our view of
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Figure 5. Grand isostress potential ∆φ(h, ∆) ) Ω(h, ∆) - Ω(1.8σ, 0) + PbulkA(h - 1.8σ), system D.
Figure 6. A atoms per attractive wall atom, system D.
the FR(∆) curve as being made up of an oscillatory component and a contribution from the effect of the bridge remains applicable. The shape of the midpoint curve that we have obtained is somewhat different from the shape of the h ) 5σ curve in Figure 4 (top panel); however in light of the qualitative and informal nature of the decomposition concept we do not attach significance to this. Since P ) -A-1(∂Ω/∂h)∆ and FR ) (∂Ω/∂∆)h, starting from a reference point (h0, ∆0) we may integrate the FR(h, ∆) and P(h, ∆) data to obtain the relative grand potential ∆Ω(h, ∆) ) Ω(h, ∆) - Ω(h0, ∆0). In many applications of interest, however, the walls will experience an applied normal stress per unit area Pbulk due to their being immersed in a bulk reservoir. The net force per unit area pushing the walls apart will then be Pnet ) P - Pbulk. For such a case, a more useful quantity than ∆Ω(h, ∆) is ∆φ(h, ∆) ≡ PbulkA(h - h0) + ∆Ω(h, ∆). The grand isostress potential, φ ) PextAh + Ω, introduced by Schoen et al., is the appropriate thermodynamic potential for a system in the presence of an applied normal stress PextA.37,38 Minima of this quantity represent metastable (or stable) configurations if the walls are allowed to move freely in the presence of the external pressure Pext. Figure 5 shows the ∆φ(h, ∆) surface for system D. The funnel-like shape sloping downward toward the low-h, low-∆ corner indicates the bridge-forming region. The ∆φ(h, ∆) surface clarifies the origin of the relative phases of the oscillations in the P(∆) and FR(∆) curves. Figure 6 shows 〈NANa-1〉 for system D. This is the number of A molecules in the pore for each attractive wall atom (on one wall). It is an estimate of the coverage of the attractive patches by A in the bridge; there is as well a small contribution from A atoms in the B-rich phase outside the bridge. The bridgeforming region stands out distinctly. Within it, oscillations in the ∆-direction are apparent, and there is also evident structure associated with fluid layering. This latter structure takes the form of plateaus where the steepness |(∂〈NANa-1〉/∂h)∆| is near a minimum in the h-direction. The first two plateaus occur at h ≈ 2.2σ and h ≈ 3.1σ. There are corresponding troughs in the ∆φ(h, ∆) surface (Figure 5), with one located at h ≈ 3.1σ and the other at h ≈ 2.2σ outside the bridge-forming region, varying down to h ≈ 1.8σ within the bridge-forming region. On the basis of the well-known layering phenomena in systems with chemically uniform walls,1,33,34 we expect these plateaus cor-
Figure 7. Snapshots from system D simulations with h ) 2.2σ, ∆ ) 0, looking in (a) the y-direction and (b) the z-direction, with h ) 2.9σ, ∆ ) 0.5a in the (c) y- and (d) z-directions. A molecules are white, B molecules are gray, and attractive and neutral wall atoms are, respectively, light and dark crosses.
Figure 8. Density profile F(x, z) for system D, h ) 2.2σ, ∆ ) 0.
respond to configurations at which integral numbers of layers of liquid molecules fit into the pore well. Thus the first plateau at h ≈ 1.8σ-2.2σ should correspond to the presence of a single layer of A in the bridge, and the h ≈ 3.1σ plateau should correspond to a bridge with two molecular layers. These assignments are confirmed by configuration snapshots from the simulations, shown in Figure 7 for h ) 2.2σ, ∆ ) 0 (parts a and b) and h ) 2.9σ, ∆ ) 0.5a (parts c and d), for which there are present, respectively, one and two confined layers. The density profile F(x, z) ) FA(x, z) + FB(x, z) for h ) 2.2σ, ∆ ) 0 is shown in Figure 8. Although there is only one confined layer in the bridge, the profile in Figure 8 shows two peaks, which reflects that in the z-direction the wall potential is doublewelled. The ∆φ(h, ∆) surface (Figure 5) establishes that with one confined layer, thermodynamically favored values of ∆ are ∆ = ka where k is an integer, where the wall lattices are in-registry. For two confined layers, out-of-registry values ∆ = (k + 1/2)a are favored. This is consistent with behavior found for a onecomponent fluid between chemically uniform atomistic walls, where crystallization may occur if the walls are in-registry and an odd number of fluid layers or present or if they are out-ofregistry and an even number of layers are present, and a restoring force appears for displacement of the walls from these alignments.34,38,39 We believe a similar explanation holds in our system even though crystallization does not occur. Values of h
3770 J. Phys. Chem. B, Vol. 110, No. 8, 2006
Hemming and Patey
Figure 9. A lattice with two confined layers with walls out of registry. Black circles indicate molecules with y-coordinates ka, k an integer, while white circles indicate molecules at y ) (k + 1/2)a.
and ∆ consistent with the formation of an fcc lattice containing the fluid molecules and wall atoms are favored, regardless of whether crystallization of the confined fluid occurs. If the confined fluid remains liquid, then density profiles show that the molecules are predominantly found near lattice sites, and long-range order is present in the liquid. It has similarly been observed for one-component liquids that atomistic walls induce long-range order in the contact layers of the confined fluid.31,33 Presumably, energetically favorable local structure in the liquid resembles the solid structure; by placing the molecules near lattice sites they may occupy attractive wells of the potentials of wall and fluid neighbors, and the number of fluid neighbors can be maximized. Consequently, in-registry wall lattices are favorable for an odd number of confined fluid layers, and outof-registry lattices are favored with an even number. For our system, the density profile for h ) 2.2σ, ∆ ) 0 (Figure 8) shows localization of liquid molecules between wall atoms in the x-direction, consistent with their favoring fcc lattice sites. The configuration snapshot at these same (h, ∆) values (Figure 7a) indicates some tendency to occupy sites between wall atoms. Geometrically, formation of a perfect fcc lattice with one confined layer with lattice constant a ) 1.2σ occurs with walls in registry at a separation h ) ax2 = 1.70σ, consistent with the downward-sloping ∆φ(h, ∆) surface at h ) 1.8σ in Figure 5. For two confined layers, formation of an fcc lattice is not possible geometrically at any value of ∆ due to the relative displacement in the [011] direction. Still, it is clear that packing can be improved by displacing the wall lattices from the inregistry position. Figure 9 suggests one possible packing with site spacing a ) 1.2σ and two confined layers with walls outof-registry. Although this is not an fcc lattice, any regular and relatively densely packed lattice should still provide a basis for favorable liquid structure by the same arguments advanced for the fcc case. For the Figure 9 structure h ) a(x2 + x3/2) = 2.74σ, agreeing well with the observed location of the trough in the ∆φ(h, ∆) surface at h ≈ 2.8-2.9σ, particularly if one allows for some expansion due to disorder in the liquid. B. Systems C and E. For system E the walls have a smaller lattice constant, a ) σ; hence the commensurability with the solid A structure is lower than that for system D. The relative wall displacement is still in the [010] direction. The P(h, ∆), FR(h, ∆), and 〈NANa-1〉 results are shown in Figure 10. The plots are similar to those for system D in that there is a bridgeforming region, and features due to layering of liquid molecules are visible, specifically: ridge-and-trough structure in the P(h, ∆) surface, rows of bumps and indentations in the FR(h, ∆) surface, and a tiered structure in the 〈NANa-1〉 surface. The oscillations in the ∆-direction are much smaller than those for
Figure 10. System E properties. Top panel: normal pressure, with projection of level curve βPσ3 ) 0.525. Middle panel: restoring force, with projection of level curve FRσ-1 ) -1. Bottom panel: A atoms per attractive wall atom.
Figure 11. Total density profile F(x, z) ) FA(x, z) + FB(x, z) for system E with h ) 3.6σ and ∆ ) σ.
system D, except in the FR(h, ∆) surface where they are of comparable magnitude but do not extend out as far in the h-direction. The reduced oscillation amplitude is a consequence of the lower commensurability between wall and solid A lattices, which reduces the templating influence of the walls on the liquid. This is consistent with the finding in a number of systems that commensurability plays a large role in determining how strongly atomistic wall structure influences liquid structure. For system C, the wall lattice constant is a ) 1.2σ, as in system D. The relative wall displacement is in the [011] direction, meaning that with two confined layers formation of an fcc lattice is possible when the walls are out-of-registry. The P(h, ∆), FR(h, ∆), and 〈NANa-1〉 surfaces for system C are shown in Figure 12. The noteworthy differences compared to the other systems are: first, the oscillations in the ∆-direction have period ax2, which is the period of the wall lattice in the ∆-direction, and second, the amplitude of the ∆-direction oscillations is much greater than that for system D (and greater than that for system E too). Figure 13 compares some βσ3P(∆) curves for systems C, D, and E. Note that the lower envelopes of the oscillations are essentially the same for systems C and D; the amplitude difference is due to the larger P(∆) maxima in system C. A
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J. Phys. Chem. B, Vol. 110, No. 8, 2006 3771
Figure 14. Restoring force for h ) 1.8σ (top panel) and 2.4σ (bottom panel) for systems C (solid), D (long dashes), and E (short dashes).
Figure 12. System C properties. Top panel: normal pressure, with projection of level curve βPσ3 ) 0.525. Middle panel: restoring force, with projection of level curve FRσ-1 ) -1. Bottom panel: A atoms per attractive wall atom.
Figure 15. Grand isostress potential ∆φ(h, ∆) ) Ω(h, ∆) - Ω(1.8σ, 0) + PbulkA(h - 1.8σ), system C.
Figure 16. Snapshot for system A with h ) 1.8σ, ∆ ) 0. See Figure 7 caption for key.
Figure 13. Pressure for h ) 1.8σ (top panel) and 2.4σ (bottom panel) for systems C (solid), D (long dashes), and E (short dashes).
comparison of σ-1FR(∆) curves is shown in Figure 14; again, the system C curves show the largest oscillation amplitude. The free energy ∆φ(h, ∆) ) Ω(h, ∆) - Ω(1.8σ, 0) + PbulkA(h - 1.8σ) for system C, obtained by integrating FR(h, ∆) and P(h, ∆), is shown in Figure 15. Comparison with the ∆φ(h, ∆) surface for system D (Figure 5) reveals that at h ) 1.8σ, the out-of-registry configurations are more unfavorable in system C than those in system D. With one confined layer and the wall lattices out of registry, the atoms of one wall are directly over the attractive wells of the other wall. If a confined molecule is to occupy an attractive well, then it will experience strong
repulsion from the atom of the other wall which has the same x- and y-coordinates. This does not happen with system D, where the confined liquid molecules can always lie at y-coordinates between the rows of wall atoms, where the attractive wells due to the walls are located. That is, system C frustrates favorable liquid packing more strongly when the wall lattices are at the most unfavorable registry than does system D. For systems D and E, the structure of the A-rich bridge was found to be substantially disordered (cf. Figure 7), suggesting that the bridges are liquid. The best candidate for a system to allow bridge solidification is system C at the smallest wall separation with the walls in registry. Figure 16 shows a typical configuration for this system; the bridge molecules are typically found near lattice sites in the middle of cages formed by four molecules in each wall; however there are also some disordered regions where the bridge molecules are not near the centers of the cages. A series of canonical MC simulations with only diffusive moves, i.e., in which the only moves were displace-
3772 J. Phys. Chem. B, Vol. 110, No. 8, 2006 ments distributed uniformly within the cube ((0.15σ, (0.15σ, (0.15σ) with sides 0.30σ (where the displaced z-coordinate is taken modulo h) were performed. For these, the populations NA and NB were chosen to be equal to the average populations 〈NA〉, 〈NB〉 determined from the GCMC simulations. The A molecules in the bridge spent most of their time near the centers of cages at positions corresponding to fcc lattice (stretched in the z-direction) sites but were able to move between sites. Over long times, their motion was diffusive in the x- and y-directions. Hence we regard the bridges, even at this small separation, as being liquid with ordering imposed by the atomistic wall structure. IV. Conclusions Two key effects associated with bridge formation at larger wall separations are the presence of net attractive normal force on the walls and a lateral restoring force appearing when the patches on opposite walls are misaligned, exhibiting a maximum strength where yield occurs, with the bridge disintegrating for displacements beyond this point. These effects remain present at the smaller wall separations we have investigated here; however, other effects due to fluid layering and atomistic wall structure appear as well. The normal pressure and lateral restoring force exhibit oscillations with the wall separation h, with wavelength equal to the diameter of the fluid molecules. These quantities also exhibit oscillations with the lateral wall offset ∆, which have wavelength equal to the length over which the wall lattices go through one registry cycle. Similar effects have been observed in one-component Lennard-Jones liquid confined between atomistic walls. Oscillations in the h-direction are due to layering of fluid molecules, an effect that does not depend on atomistic wall structure. Oscillations in the ∆-direction are due to whether the registry of the wall lattices facilitates or frustrates favorable packing of fluid molecules. Favorable registries and wall separations are those which an fcc crystal could form between the walls; although the confined fluid remains liquid in all cases considered here, presumably favorable local structures in the liquid resemble the solid structure. The atomistic walls induce long-range order in the confined liquid. The A-rich phase of which the bridges are formed is denser than the B-rich phase comprising the surrounding fluid. Hence, when bridges are present, oscillations in the h-direction extend out to larger h, to a value corresponding to three confined layers, whereas when no bridge is present they extend out for only one confined layer. Similarly, oscillations in the ∆-direction are stronger when bridges are present, an effect that is also in part due to the fact that the walls appear less strongly corrugated to the B-rich phase. As a qualitative approximation, we may view the normal pressure and restoring force results obtained as being a simple superposition of contributions from three different effects: a contribution from bridge formation, a contribution from fluid layering, and a contribution from atomistic wall structure. The strength of the oscillations in the ∆-direction due to atomistic wall structure depends strongly on the commensurability of the wall lattices with the lattice constant of crystalline A. For one confined fluid layer, the effect of atomistic walls dominates the effect of bridge formation; the oscillation amplitude is much larger than the magnitude of forces due to bridge formation. With two confined layers and high commensurability between wall lattices and solid A, the oscillation amplitude is comparable to the magnitude of bridge formation forces. With lower commensurability, the effects of atomistic wall structure are significant only at one confined layer.
Hemming and Patey Acknowledgment. The financial support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. This work has been enabled by the use of WestGrid computing resources, which are funded in part by the Canada Foundation for Innovation, Alberta Innovation and Science, BC Advanced Education, and the participating research institutions. WestGrid equipment is provided by IBM, Hewlett-Packard, and SGI. References and Notes (1) Klein, J.; Kumacheva, E. Science 1995, 269, 816-819. (2) Klein, J.; Kumacheva, E. J. Chem. Phys. 1998, 108, 6996-7009. (3) Kumacheva, E.; Klein, J. J. Chem. Phys. 1998, 108, 7010-7022. (4) Evans, R.; Marconi, U. M. B. J. Chem. Phys. 1987, 86, 71387148. (5) Nakanishi, H.; Fisher M. E. J. Chem. Phys. 1983, 78, 3279-3293. (6) Christenson, H. K.; Fang, J.; Israelachvili, J. N. Phys. ReV. B 1989, 39, 11750. (7) Greberg, H.; Patey, G. N. J. Chem. Phys. 2001, 114, 7182-7188. (8) Overduin, S. D.; Patey, G. N. J. Chem. Phys. 2002, 117, 33913397. (9) Curry, J. E.; Zhang, F.; Cushman, J. H.; Schoen, M.; Diestler, D. J. J. Chem. Phys. 1984, 101, 10824-10832. (10) Ro¨cken, P.; Tarazona, P. J. Chem. Phys. 1996, 105, 2034-2043. (11) Ro¨cken, P.; Somoza, A.; Tarazona, P.; Findenegg, G. J. Chem. Phys. 1998, 108, 8689-8697. (12) Diestler, D. J.; Schoen, M.; Curry, J. E.; Cushman, J. H. J. Chem. Phys. 1994, 100, 9140-9146. (13) Schoen, M.; Diestler, D. J. Chem. Phys. Lett. 1997, 270, 339344. (14) Schoen, M.; Diestler, D. J. Phys. ReV. E 1997, 56, 4427-4440. (15) Bock, H.; Schoen, M. Phys. ReV. E 1999, 59, 4122-4136. (16) Schoen, M.; Bock, H. J. Phys.: Condens. Matter 2000, 12, A333A338. (17) Bock, H.; Schoen, M. J. Phys.: Condens. Matter 2000, 12, 15451568. (18) Bock, H.; Schoen, M. J. Phys.: Condens. Matter 2000, 12, 15691594. (19) Bock, H.; Diestler, D. J.; Schoen, M. J. Phys.: Condens. Matter 2001, 13, 4697-4714. (20) Bock, H.; Diestler, D. J.; Schoen, M. Phys. ReV. E 2001, 64, 046124. (21) Sacquin, S.; Schoen, M.; Fuchs, A. H. Mol. Phys. 2002, 100, 29712982. (22) Sacquin-Mora, S.; Fuchs, A. H.; Schoen, M. Phys. ReV. E 2003, 68, 066103. (23) Sacquin-Mora, S.; Fuchs, A. H.; Schoen, M. J. Chem. Phys. 2004, 121, 9077-9086. (24) Yaneva, J.; Milchev, A.; Binder, K. J. Chem. Phys. 2004, 121, 12632-12639. (25) Hemming, C. J.; Patey, G. N. J. Chem. Phys. 2004, 121, 65086517. (26) Adams, D. J. Mol. Phys. 1975, 29, 307-311. (27) Allen, M. P.; Tildesley, D. J. Computer Simulations of Liquids; Oxford University Press: New York, 1987; p 47. (28) Nijmeijer, M. J. P.; van Leeuwen, J. M. J. J. Phys. A: Math. Gen. 1990, 23, 4211-4235. (29) Henderson, J. R. J. Phys.: Condens. Matter 1999, 11, 629-643. (30) Huang, K. Statistical Mechanics, 2nd ed.; Wiley: New York, 1987; p 137. (31) Schoen, M.; Hess, S.; Diestler, D. J. Phys. ReV. E 1995, 52, 25872602. (32) Thompson, P. A.; Robbins, M. O. Phys. ReV. A 1990, 41, 68306837. (33) Toxvaerd, S. J. Chem. Phys. 1981, 74, 1998-2005. (34) Rhykerd, C. L., Jr.; Schoen, M.; Diestler, D. J.; Cushman, J. H. Nature 1987, 330, 461-463. (35) Diestler, D. J.; Schoen, M.; Cushman, J. H. Science 1993, 262, 545-547. (36) Schoen, M.; Diestler, D. J.; Cushman, J. H. Phys. ReV. B 1993, 47, 5603-5613. (37) Bordarier, P.; Schoen, M.; Fuchs, A. H. Phys. ReV. E 1998, 57, 1621-1635. (38) Schoen, M.; Diestler, D. J.; Cushman, J. H. J. Chem. Phys. 1994, 100, 7707-7717. (39) Schoen, M.; Rhykerd, C. L., Jr.; Diestler, D. J.; Cushman, J. H. Science 1996, 245, 1223-1225.
