Fluid Density Profile Transitions and Symmetry Breaking in a Closed

2514

J. Phys. Chem. B 2007, 111, 2514-2522

Fluid Density Profile Transitions and Symmetry Breaking in a Closed Nanoslit Gersh O. Berim and Eli Ruckenstein* Department of Chemical and Biological Engineering, State UniVersity of New York at Buffalo, Buffalo, New York 14260 ReceiVed: August 11, 2006; In Final Form: January 16, 2007

The density profiles in a fluid interacting with the two identical solid walls of a closed long slit were calculated for wide ranges of the number of fluid molecules in the slit and temperature by employing density functional theory in the local density approximation. Two potentials, the van der Waals and the Lennard-Jones, were considered for the fluid-fluid and the fluid-walls interactions. It was shown that the density profile corresponding to the stable state of the fluid considerably changes its shape with increasing average density (Fav) of the fluid inside the slit, the details of changes being dependent on the selected potential. For the van der Waals potential, a single temperature-dependent critical value Fsb of Fav was identified, such that for Fav < Fsb the stable state of the system is described by a symmetric density profile, whereas for Fav g Fsb it is described by an asymmetric one. This transition constitutes a spontaneous symmetry breaking of the fluid density distribution in a closed slit with identical walls. For Fav g Fsb, a metastable state, described by a symmetric density profile, was present in addition to the stable asymmetric one. The shape of the symmetric profile changed suddenly at a value Fc-h > Fsb of the average density, the density Fc-h being almost independent of temperature. Because of the shapes of the profiles before and after the transformation, this transition was named cup-hill transformation. At the transition point, the density of the fluid near the walls decreased suddenly from a liquid-like value becoming comparable with the density of a gaseous phase, and the density in the middle of the slit increased suddenly from a gaseous-like value becoming on the order of the density of a liquid phase. For the Lennard-Jones potential, there are two temperature-dependent critical densities, Fsb1 and Fsb2, such that the stable density profile is asymmetric (symmetry breaking occurs) for Fsb1 e Fav e Fsb2 and symmetric for Fav outside of the latter interval. These critical densities occur only for temperatures lower than a certain temperature, Tsb,0. The cup-hill transition is similar to that for the van der Waals potential at low temperatures but becomes smoother with increasing temperature.

1. Introduction It is well-known that a semiinfinite fluid in contact with a solid substrate can be subjected to several kinds of transformations, such as, for example, the wetting and prewetting transitions that involve a change in the density profile of the fluid near the walls.1-12 A wetting transition occurs for temperatures higher than the wetting temperature Tw at which a liquid film of infinite thickness is formed on the solid surface. The prewetting transition occurs in the temperature range Tw < T < Tsc, where Tsc is the surface critical temperature, as a discontinuous transition from a thin liquid film to a thick liquid film of finite thickness. In all the above transitions, the presence of an infinite reservoir of molecules of bulk fluid plays an important role because in all of them a change in fluid density occurs near the solid surface and the number of molecules in that region changes via an exchange with the reservoir. Inside an open slit, the chemical potential is equal to its value in the reservoir and the number of molecules per unit area of the slit is not constant. Therefore, the grand canonical ensemble was usually employed and two versions (local and nonlocal) of density functional theory (DFT) have been used7,9,10,13 to describe the state of the fluid in the slit. If the number of molecules is constant, as happens in a closed system, the behavior of the fluid is expected to be very different * Corresponding author. E-mail: [email protected]. Telephone: (716)645-2911, ext 2214. Fax: (716)645-3822.

from that in an open system. First, in the absence of an external reservoir of molecules, neither wetting nor prewetting can occur in their conventional way. Only redistributions of molecules inside the closed volume, which nevertheless can bear some features of the phase transformations that are present in open systems, are expected to occur. The behavior of a fluid confined in a close volume should be treated in a canonical ensemble. Few papers have examined such systems.14-17 They were concerned with small spherical cavities containing a small fixed number of molecules, and suitable DFT approaches have been developed that accounted for the large fluctuations of the local density arising due to the small number of molecules. The density profile in the canonical ensemble was expressed in terms of that corresponding to a grand canonical ensemble with an additional term involving the reciprocal of the number of molecules. In a previous paper,18 we examined some aspects of the problem by considering a methanol-like fluid in a long closed slit containing a large number of molecules. For that system, the effect of the small number of molecules, which was important in the cases considered in refs 14-17, was not relevant. The DFT version employed involved a canonical ensemble and the constraint of a constant number of molecules. The fluid-walls and fluidfluid interactions were selected to be of the van der Waals type, and the cases of identical and nonidentical walls were examined. The main attention was paid to the calculation of the pressure tensor.

10.1021/jp065210y CCC: $37.00 © 2007 American Chemical Society Published on Web 02/22/2007

Fluid Transitions/Symmetry Breaking in Closed Nanoslit

J. Phys. Chem. B, Vol. 111, No. 10, 2007 2515

Figure 1. Schematic presentation of the three possible density profiles in the slit. Profile I corresponds to the state in which both walls are covered by thin films of fluid of liquid-like density, the remaining of the slit being filled by a gas-like fluid. In contrast, in the state with profile III, layers of gas-like fluid are present near the walls, the remaining of the slit being filled by a liquid-like fluid. Profile II corresponds to the state with asymmetric distribution of molecules (a mirror asymmetric profile is also possible).

In the present paper, we continue the investigation of closed slits by focusing on the profile changes that occur when the number of molecules inside the slit and the temperature are changed. In particular, the possibility of symmetry breaking of the density distribution in the case of identical walls is examined. Simple intuitive arguments based on a macroscopic theory of wetting suggested such a possibility. Let us consider the three possible density profiles, shown in Figure 1. In the first profile, both walls of the slit are covered by a liquid-like density fluid, and a gas-like density fluid is present between them. In the second, a film of liquid-like density covers one of the walls and the remaining of the slit is occupied by a gas-like fluid. Finally, in the third profile, the walls are covered by a gas-like fluid and the remaining of the slit is filled with a liquid-like fluid. Denoted by γlw, γgw, and γlg, respectively, the liquidwall, gas-wall, and liquid-gas surface tensions that satisfy the Young equation γgw - γlw ) γlg cos θ, where θ is the contact angle, one can find the differences ∆F2,1 and ∆F2,3 between the free energy of the asymmetric profile II and the free energies of the symmetric profiles I and III, respectively

∆F2,1 ) γlg(cos θ - 1) e 0

(1a)

∆F2,3 ) -γlg(cos θ + 1) e 0

(1b)

Equations 1a,b indicate that the free energy for the asymmetric profile is smaller than those for the symmetric ones, with the exception of the cases θ ) 0° or 180° when they are equal. This suggested that even for identical walls the asymmetric profile might be the stable one. Rigorously speaking, the above macroscopic considerations are not valid for nanoslits. However, they point out the possibility for an asymmetric profile to be the stable one. As another argument, supporting the possibility of symmetry breaking in a slit with identical walls, one can mention the results obtained by Merkel and Lo¨wen19 who found such a symmetry breaking in a fluid confined in two parallel identical planes (with no fluid between them). In the present paper, the van der Waals (vdW) and LennardJones (LJ) potentials were selected for the fluid-fluid and the fluid-wall interactions with the values of the parameters corresponding to argon between two parallel walls made of solid carbon dioxide. Open systems of this type were previously examined both theoretically6,8,10,11 and experimentally.20 Our calculations revealed a symmetry breaking of the density distribution in a closed slit with identical walls, which occurs in a temperature-dependent average density range specific for each potential. In addition, a new kind of transition between

Figure 2. Closed slit of width L between two, generally nonidentical, solid walls W1 and W2. σff, σfw1, and σfw2 are the hard core diameters of the fluid-fluid, fluid-wall W1, and fluid-wall W2 interactions, respectively. The y axis is perpendicular to the plane of the figure.

symmetric density profiles, which was named “cup-hill transition”, was found. 2. Model The system under consideration consists of a one-component fluid confined in a closed nanoslit between two parallel structureless identical solid walls W1 and W2 separated by a distance L. However, for reasons that will be understood later, we consider for the time being the more general case of nonidentical walls (see Figure 2). The distance between the side walls that close the slit is considered to be very large compared to L. As a result, the end effects can be considered negligible, and the density distribution can be assumed to be a function of a single coordinate h. The interactions between the fluid molecules are described as the sum between a potential energy of a reference system and an attractive perturbation. As usual, the reference system is that of hard spheres, and the repulsive forces between the hard spheres provide the following main contribution to the Helmholtz free energy

FHS )

∫V fH[F(r),T] dV

(2)

where the function fH[F(r),T] is provided by the CarnahanStarling expression21 in the form presented in ref 22

[

fH[F(r),T] ) kBTF(r) log(ηF) - 1 + ηF

]

4 - 3ηF (1 - ηF)2

(3)

In eqs 2 and 3, V is the volume, kB is the Boltzmann constant, T is the absolute temperature, F(r) is the fluid density at point r, which depends only on the distance of this point from the wall W1, ηF ) (1/6)πF(r)σ3ff is the packing fraction of the fluid molecules, and σff is their hard core diameter. The attractive part of the fluid-fluid interactions, which is considered as a perturbation, is described by a potential, φff(|r - r′|) ) 4ff[kLJ(σff/r)12 - (σff/r)6] for r g σff, r ≡ |r - r′| and φff(|r - r′|) ) 0 for r < σff, where kLJ ) 1 (LJ potential) or kLJ ) 0 (vdW potential). The interaction parameters ff and σff were selected as those of argon:10 ff/kB ) 119.76 K, σff ) 3.405 Å. Similarly, the interaction between the molecules of the fluid and the molecules of the walls are described by the potential φfwR(|r - r′|) ) 4fwR[kLJ(σfwR/r)12 - (σfwR/r)6] for r g σfwR and φfwR(|r - r′|) ) ∞ for r < σfwR, where R ) 1, 2 refers to the walls W1 and W2, respectively, with the corresponding hard core diameters σfwR and interaction energy parameters fwR. The

2516 J. Phys. Chem. B, Vol. 111, No. 10, 2007 total fluid-walls interaction is provided by potential, Φfw(h), obtained by integrating φfwR over the volume of both walls, assuming constant densities of the walls.11 The explicit forms of this potential will be specified below. The density distribution of the fluid in the slit will be calculated using the density functional theory in the local density approximation (LDA).9,10,22 The LDA does not account for the short-ranged correlations arising from excluded volume effects and cannot predict, for this reason, the density oscillations that occur near solid walls. After LDA, several more sophisticated approximations were developed, such as the weighted density approximation13,23 (WDA) and fundamental measure theory.24-26 However, the LDA provides a reasonable (at least qualitative) description of a constrained fluid as was demonstrated, in particular, in refs 13 and 23, where LDA and WDA were compared for a fluid in an open slit. The density functional Ω[F(r)] appropriate for the description of a fluid in a closed slit consists of a Helmholtz free energy of the system (all quantities are defined per unit area of one wall)

F(N,V,T) ) FHS + Uff + Ufw

(4)

and a Lagrange term - λN that accounts for the constraint of a constant number, N, of molecules of fluid in the slit, λ being a Lagrange multiplier

N)

∫V F(r) dr

(5)

{

2 1 - kLJ 5 K(z - z′) ) 2 kLJ 1 14 5 (z - z′) (z - z′)6

ψ1(z) )

1 Uff ) 2 Ufs )

∫V ∫V dr dr′F(r)F(r′)φff(|r - r′|)

∫V drF(r)[ ∫V

w1

(7)

ψ2(z) )

w2

dr′Fs2φfs2(|r - r′|)] (8)

where Vw1 and Vw2 are the total volumes of the walls, which will be considered as semiinfinite uniform bodies of densities Fs1 and Fs2, respectively. The Euler-Lagrange equation for the density profile F(h), which can be derived from the minimization of the functional eq 6, has the following dimensionless form

∂ {Fj(z)f/H[Fj(z)]} ) ∂Fj(z) ff 2πff zm dz′Fj(z′)K(z - z′) Φ (z) + λh (9) 0 kBT kBT fw

1 + log ηz +

∫

where zm ) hm/σff, hm ) L - σfw1 - σfw2 (see Figure 2), z ) h/σff, Fj(z) ) F(h)σ3ff, λh ) λ/kBT, ηz ) (π/6)Fj(z)

f/H[Fj(z)]

) -1 + ηz

4 - 3ηz (1 - ηz)2

|z - z′| g 1 (11) (12)

[

]

[

kLJ 2π 2 εfw2Fjw2s3fw2 3 15 [1 + (z - z)/s ]9 m fw2

]

1 (14) [1 + (zm - z)/sfw2]3 In eqs 13 and 14, ψR(z) (R ) 1, 2) is the interaction potential between the wall WR and the fluid, and εfwR ) fwR/ff, FjwR ) FwRσ3ff, sfwR ) σfwR/σff are the dimensionless fluid-wall interaction energy parameters, densities of the walls, and hard core diameters, respectively. Formally, eq 9 coincides with that for an open system without side walls. However, the Lagrange multiplier λ in closed systems is not known in advance, in contrast to the open systems, where λ is the chemical potential. The relation between λ and the number N of molecules is provided by the constraint eq 5, which has the following dimensionless form

∫0z

m

Fj(z) dz ) Nσ2ff

(15)

By solving eq 9 with respect to ηz and substituting the solution for Fj(z) into eq 15, one obtains for λh ) λ/kBT the expression

λh ) -log

[

6 πNσ2ff

∫0z

m

]

dzeQ(z)

(16)

where

Q(z) ) -1 -

dr′Fs1φfs1(|r - r′|) +

∫V

]

kLJ 2π 1 2 εfw1Fjw1s3fw1 3 15 (1 + z/s )9 (1 + z/s )3 fw1 fw1 (13)

(6)

In the above equations, V is the volume of the slit per unit area of one wall, and Uff and Ufw are the contributions to the total potential energy of the fluid-fluid and fluid-wall interactions, respectively. In the mean-field approximation

[

|z - z′| < 1

Φfw(z) ) ψ1(z) + ψ2(z)

Consequently

Ω[F(r)] ) F(N,V,T) - λN

Berim and Ruckenstein

2πff ∂ {Fj(z)f/H[Fj(z)]} + × kBT ∂Fj(z)

∫0z

m

dz′Fj(z′)K(z - z′) -

ff Φ (z) (17) kBT fw

The nonlinear integral equation, eq 9, was solved by numerical interations. The initial trial profile was selected as the homogeneous distribution of the fluid density for the selected number of molecules in the system. The calculations were carried out until the quantities δ1 ) |max{Fji+1} - min{Fji}|/ max{Fji} (the difference between two consecutive profiles) and δ2 ) 1 - (1/N) ∫h0m F(h) dh (the relative deviation of the real number of molecules from the selected one) became both smaller than 10-7. In the expressions of δ1 and δ2, Fji represents the density profile after the i-th iteration and max{Fji+1} (min{Fji}) is the largest (smallest) density value at the grid points of the corresponding profiles. The number of grid points was selected to be 10 per molecular diameter σff. To control the consistency of calculations, this number was increased in some cases up to 20 per molecular diameter. 3. Results

(10)

In the calculations, the specific values fwR/kB ) 153 K, σfwR ) 3.727 Å, and FwR ) 2.19 × 1028 m-3 (R ) 1, 2), which


J. Phys. Chem. B, Vol. 111, No. 10, 2007 2517 TABLE 1: Values (Per Unit Area of One Wall) of the Helmholtz Free Energies Fsym and Fasym of the Metastable (Symmetric Density Profile) and Stable (Asymmetric Density Profile) States of the Fluid in the Slit with hm ) 15σff, Respectively, for Various Average Densitiesa van der Waals potential Fjav

Fsym

Fasym

0.2319 0.3092 0.3865 0.4638

-15.10 -22.32 -28.46 -34.63

-16.62 -22.77 -28.96 -35.17

Lennard-Jones potential

a

Fjav

Fsym

Fasym

0.2319 0.3092 0.3865 0.4638

-11.66 -15.53 -19.58 -23.80

-11.90 -15.87 -19.85 -23.87

T ) 87 K. The free energies are expressed in kBT/σ2ff units.

of an asymmetric profile (such as profile II in Figure 1). The following procedure for the search of an asymmetric state of the system was employed: Let us consider two nonidentical walls (fw1 * fw2) and introduce the quantity

|

δw ) 1 -

Figure 3. Two qualitatively different sequences of profiles for the van der Waals potential and δw f 0 at T ) 87 K. (a) The sequence that has the symmetric profile (δw ) 0) as the asymptotic one. The profiles with δw e 10-2 are nondistinguishable; Fjav ) 0.03865. (b) The sequence that has the asymmetric profile (long-dashed line) as the asymptotic one. All profiles with δw e 10-4 are represented by the same, long-dashed line. The symmetric profile for δw ) 0 is represented by the solid line; Fjav ) 0.05411.

constitute the parameters for the solid carbon dioxide interacting with argon,10 were selected. The width of the slit was taken to be hm ) 15σff. First, the case of vdW potential (kLJ ) 0) was examined. The fluid density profile obtained as solution of eq 9 with fw1 ) fw2 is symmetrical about the center of the slit, and its shape depends on the number N of molecules per unit area of one of the walls (or average density Fav ) N/hm of the fluid) and temperature. (Note that the solution was symmetrical even when the initial profile guess in the iterations was selected asymmetrical.) Typical profiles for small and large Fav are sketched in Figure 1 (profiles I and III). The first of them, profile I, has a cup-like shape and the second, profile III, a hill-like shape. (As shown below, depending on the value of Fav, the simple cup- and hill-like profiles may become more complicated). The hill-like profile is characterized by a relatively high density of fluid in the middle of the slit, which is of the order of the density of a liquid phase, and by a low-density fluid in the vicinity of the walls. This profile resembles the dried state of an infinite fluid in contact with a wall. In contrast, the density of the fluid in the middle of a cup-like profile is very small, comparable to the density of a gas, whereas in the vicinity of the walls it can have a liquid-like value. This profile resembles the case of a thin liquid film on a solid surface. Because of the arguments presented in the Introduction, we checked the possibility of symmetry breaking, i.e., the existence

|

fw2 fw1

(18)

which characterizes the difference between the walls. (For identical walls, δw ) 0). Only the case fw2 < fw1 was considered. Then, a sequence of density profiles was calculated using the Euler-Lagrange equation for fixed values of the temperature T, average density Fjav, and decreasing values of δw (δw f 0, δw > 0). In the calculations, values of δw in the range 0.5 g δw g 10-8 were used. For relatively large δw (δw g 0.1), the profiles in all cases were asymmetrical due to the difference between walls. However, depending on the values of T and Fav, the sequence of profiles had two different tendencies as δw decreased to the order of 10-4, 10-8. In the first, the profiles became almost symmetrical and their difference from the profile obtained for δw ) 0 negligible. In the second, the profiles remained asymmetrical and there were no visible differences between those calculated for 10-8 e δw e 10-4. In both cases, the limiting profile (either symmetrical or asymmetrical) was considered as the asymptotic solution of the Euler-Lagrange equation for δw f 0. If it is symmetrical, then the Euler-Lagrange equation has a unique stable solution. The asymmetric profile provides an additional solution to the EulerLagrange equation, which is the stable one if its free energy is smaller than that of the symmetric profile obtained for δw ) 0. For illustration, in Figure 3a, b, two qualitatively different sequences of profiles calculated for different values of δw are presented for a slit that has one wall of solid carbon dioxide and another one of a hypothetical material for which the energy parameter fw2 can take any value smaller than fw1. The average fluid densities are Fjav ) 0.03865 for Figure 3a and Fjav ) 0.05411 for Figure 3b, and T ) 87 K. For the smaller average density (Figure 3a), the density profiles have asymmetric shapes for comparatively large δw and almost symmetrical shapes for small values of δw. The density profiles obtained for δw e 0.01 practically coincide with that calculated for identical walls (δw ) 0). For the higher average density (Figure 3b), the density profiles have highly asymmetric shapes for all values of δw, including the smallest δw ) 10-8. This behavior is qualitatively different

2518 J. Phys. Chem. B, Vol. 111, No. 10, 2007


Figure 4. Slit with the van der Waals potential. (a) Stable (solid lines) and metastable (dashed lines) symmetric profiles at T ) 87 K plotted for Fjav ) 0.00772, 0.02319, 0.04638, 0.04799, 0.06956, and 0.07129, respectively. The fluid density near the walls increases with increasing average density. (b) Stable profiles corresponding to the same average densities as the metastable ones in panel a. The density near wall W1 increases with increasing Fjav; T ) 87 K. (c) The densities at the walls W1 (solid line) and W2 (dashed line) as functions of the average density in the slit; T ) 87 K. On the segment AB, both curves coincide. (d) Temperature dependence of the critical average density Fjsb.

from that presented in Figure 3a for the smaller density. Analysis revealed that the change in the character of the density profile has occurred at an average density Fjsb ) 0.04799, which represents the critical average density above which symmetry breaking occurs. It was also found that for Fjav > Fjsb the asymmetric density profile provides a smaller Helmholtz free energy than the symmetric density profile, and hence, the former corresponds to the stable state. Several values of the free energy of the stable and metastable states are listed in Table 1. In Figure 4a, the symmetric profiles obtained as solutions of eq 9 for various Fjav and δw ) 0 are presented for T ) 87 K. The solid curves represent stable and the dashed ones metastable profiles. The stable asymmetric profiles, corresponding to the same average densities as the metastable ones of Figure 4a, are presented in Figure 4b. As a quantitative characteristic of the symmetry breaking transition, one can consider the change of the fluid densities Fj(0) and Fj(hm) at the walls of the slit for the stable state. In Figure 4c, the dependencies of Fj(0) and Fj(hm) on the average density Fjav are plotted. One can see that both densities change suddenly at Fjav ) Fjsb ) 0.04799, where symmetry breaking starts to occur. The temperature dependence of Fjsb is presented in Figure 4d. For the points (T, Fjav) under (above) the curve, the equilibrium density profile is symmetric (asymmetric). For a fixed average density, symmetry breaking occurs at temperatures smaller than a critical value Tsb, which can be estimated from Figure 4d. Thus, for Fjav ) 0.4799, Tsb ) 87 K.

In Figure 5a, the profiles calculated for T e Tsb (dashed curves) and for T > Tsb (solid curve) are presented for Fjav ) 0.4799. The first two, plotted for T ) 84 and 87 K are asymmetric, and the third for T ) 89 K is symmetric. The temperature dependence of the fluid densities at the walls is presented in Figure 5b. The rate of change of those densities with changing temperature has the largest value for T ) Tsb. It should be noted that the free energy difference between the metastable and stable states does not exceed a few percents (see Table 1), and therefore, it is not unlikely for the system to be in a metastable state. For this reason, the symmetric density profiles obtained from the solution of the Euler-Lagrange equation (eq 9) at δw ) 0 for Fjav > Fjsb, i.e., in the metastable region, will be considered below in some details. Those profiles undergo a change in shape with increasing Fjav, which can be associated with the transition between profiles I and III in Figure 1. Because of the characteristic shapes of those profiles, this transition will be named the cup-hill transition. It occurs within a very narrow range of values of Fav and has therefore an almost discontinuous character. As an example, in Figure 6a, the density profiles of argon are presented for various Fjav at T ) 87 K. In this case, the fluid-fluid interaction parameter is smaller than the fluid-walls interaction parameters (ff < fwR). The transformation of the density profile from a cup- to a hill-like shape occurs when the average density changes from Fjav,1 ) 0.25089 (dashed curve) to Fjav,2 ) 0.25120 (dotted curve). Considering the arithmetic mean value of Fjav,1 and Fjav,2 as an appropriate



Figure 5. Slit with the van der Waals potential. (a) Density profiles for Fjav ) 0.04799 and temperatures T ) 84, 87 (dashed lines), and 89 K (solid line). The first two are asymmetric, whereas the latter, calculated for T > Tsb = 87 K, is symmetric. (b) Densities at the walls W1 (solid line) and W2 (dashed line) as functions of temperature; Fjav ) 0.04799. On the segment AB, both curves coincide.

estimate of the dimensionless average density Fjc-h at which the cup-hill transition occurs, one obtains Fjc-h ) 0.25105. As in the case of symmetry breaking, one can consider the fluid density Fj(hm/2) in the middle of the slit as well as that at the walls (Fj(0) ) Fj(hm)) as quantitative characteristics of the cup-hill transition. In Figure 6b, the dependencies of Fj(hm/2) and Fj(0) on Fjav are presented. Both densities have a very rapid change near Fjc-h. The former becomes approximately 400 times larger, and the latter becomes 12 times smaller when Fjav passes through Fjc-h. The value of Fjc-h is almost independent of temperature. As shown in Figure 6c, in the range 84 K < T < 150 K it slightly decreases from Fjc-h ) 0.2516 at T ) 84 K to Fjc-h ) 0.2492 at T ) 150 K. Figure 6c can be considered as a kind of phase diagram in the Fjc-h-T plane. In the entire range of temperatures, the density profile has a cup-like shape for all points below the curve AB and a hill-like shape for all points above the curve AB. If the actual average density Fjav has a value in the range Fjav,min < Fjav < Fjav,max (Fjav,min ) 0.2492, Fjav,max ) 0.2516, see Figure 6c), then the cup-hill transition occurs at a temperature Tc-h > 84 K. In Figure 7a, the density profiles are plotted for various temperatures and Fjav ) 0.24964. The cup-hill transition occurs at Tc-h = 101.57 K. Figure 7b presents the temperature dependencies of the densities Fj(0) and Fj(hm/2). Both densities change their behavior when the temperature passes through the temperature Tc-h of the cup-hill transition. The density Fj(0) of the fluid near the

Figure 6. Slit with the van der Waals potential. (a) Density profiles for Fjav equal to 0.077294, 0.11594, 0.15459, 0.19323 (solid curves), 0.25090 (dashed curve), corresponding to the cup-like profiles, equal to 0.25120 (dotted curve), 0.30917, 0.38647 (solid curves), and corresponding to the hill-like profiles; T ) 87 K. For the cup-like profiles, the fluid density at the walls increases with increasing Fjav; for the hill-like profiles, the density in the middle of the slit increases with increasing Fjav. The cup-hill transition occurs at Fjc-h = 0.25105. (b) Dependence of the fluid density at the walls (solid curve) and in the middle of the slit (dashed curve) on the average density of the fluid for T ) 87 K. On the AB segment, both curves coincide. (c) Temperature dependence of the critical average density Fjc-h for a cuphill transition; Fjav,min ) 0.2492 and Fjav,max ) 0.2516.

walls decreases almost discontinuously at the transition point. Such a change is similar to the drying transition (or wetting by the gas) that occurs in the system solid wall-bulk fluid.27 For T > Tc-h the density Fj(0) slightly increases with increasing temperature. In contrast, the density Fj(hm/2) in the middle of the slit suddenly increases at T ) Tc-h but decreases with increasing temperature when T > Tc-h. The temperature dependence of the density profile for an average density Fjav outside the range between Fjav,min and Fjav,max differs from that observed above. The main difference consists in the absence of a transition from a cup- to hill-like profile at all temperatures. In Figure 8a, b, the density profiles and the temperature dependencies of the densities Fj(0) and Fj(hm/2) are presented for Fjav ) 0.19323, which is smaller than Fjav,min. At all temperatures, the density profile has a cup-like shape, and


Figure 7. Slit with the van der Waals potential. (a) Density profiles for Fjav ) 0.24964 and various temperatures. The cup-like profiles correspond to the temperatures 87, 96 (solid curves), and 101.55 K (dashed curve). The density at the wall for those profiles decreases with increasing temperature. The hill-like profiles correspond to the temperatures 101.6 (dotted curve), 120, 130, 140, 150, and 160 K (solid curves). The density in the middle of the slit for those profiles decreases with increasing temperature. A cup-hill transition occurs at Tc-h = 101.57 K. (b) Temperature dependence of the fluid density at the walls (solid line) and in the middle of the slit (dashed line). On the AB segment, both curves coincide.

the temperature behaviors of Fj(0) and Fj(hm/2) are monotonic with the density in the middle of the slit increasing and the density at the walls decreasing with increasing temperature. For Fjav > Fjav,max (Fjav ) 0.3865), the density profile has always a hill-like shape (Figure 9a). The temperature behavior of the densities in the middle of the slit and at the walls is in this case opposite to that for Fjav < Fjav,min, the density Fj(0) increasing and the density Fj(hm/2) decreasing with increasing temperature (see Figure 9b). One should note that for the van der Waals potential the cuphill transition always occurs between metastable states. The use of the LJ potential (kLJ ) 1) instead of the vdW one leads to the following changes in the fluid behavior. First, instead of a single critical density Fsb for the symmetry breaking transition, there are two critical densities, Fsb1 and Fsb2 > Fsb1, both being larger than Fsb at the same temperature. The stable profile is asymmetric for Fsb1 e Fav e Fsb2 and symmetric for Fav > Fsb2 and Fav < Fsb1. The temperature dependence of Fsb1 and Fsb2 is shown in Figure 10. Symmetry breaking occurs only for those (Fav, T) pairs that are located between the two curves of Figure 10. One can see that the range of Fav where symmetry breaking occurs decreases with increasing temperature. For T > Tsb,0 = 112 K, no symmetry breaking occurs and the profiles


Figure 8. Slit with the van der Waals potential. (a) Density profiles for Fjav ) 0.19323 and temperatures 87, 96, 104, 120, 130, 140, 150, 160, and 170 K. At all temperatures, the profiles have a cup-like shape. The density in the middle of the slit increases with increasing temperature. (b) Temperature dependence of the fluid density at the walls (solid curve) and in the middle of the slit (dashed curve).

are symmetrical about the middle of the slit. It should be noted that for the vdW potential the temperature Tsb,0 either does not exist or is located outside the considered temperature range (see Figure 4d). Table 1 compares the free energies of the symmetric and asymmetric profiles. The second difference between the systems with vdW and LJ potentials concerns the cup-hill transition. At low temperature, e.g., T ) 87 K, the cup-hill transition for the LJ potential remains similar to that for the vdW potential. However, at higher temperatures, e.g., T ) 104 K, the change of the profile shape during the cup-hill transition becomes smoother and the variation of the fluid densities at the walls and in the middle of the slit is not as abrupt (see Figure 11). For T e Tsb,0 the cup-hill transition occurs between metastable states, whereas for T > Tsb,0 there is no symmetry breaking and the cup-hill transition occurs between stable states. 4. Conclusion The main results of this paper can be summarized as follows: For selected values of the parameters that characterize the fluid-fluid and the fluid-solid interactions, the density profile in a closed slit with identical walls depends on the average density of molecules inside the slit and temperature. The profiles can exhibit one of the three main characteristic shapes shown in Figure 1, namely the symmetric cup- and hill-like shapes (profiles I and III, respectively) and the asymmetric shape (profile II). The cup-like profiles occur mainly in slits with low average densities and the hill-like profiles in those with large average densities of the fluid molecules. The asymmetric profiles appear in systems described by symmetric equations as a


Figure 9. Slit with van der Waals potential. (a) Density profiles for Fjav ) 0.3865 and temperatures 87, 104, 120, 130, 140, 150, 160, and 180 K. At all temperatures, the profiles have a hill-like shape. The density in the middle of the slit decreases with increasing temperature. (b) Temperature dependence of the fluid density at the walls (solid curve) and in the middle of the slit (dashed curve).

Figure 10. Temperature dependence of the critical values Fjsb1 and Fjsb2 on the average density in a closed slit with Lennard-Jones potential and hm ) 15σff for T g 87 K. Symmetry breaking occurs when Fjsb1 e Fjav e Fjsb2. For temperatures higher than Tsb,0 ) 112 K, no symmetry breaking occurs.

consequence of symmetry breaking. At a given temperature, they occur in a certain range of the average density and at a given average density at temperatures smaller than a temperature Tsb, which depends on Fav. The possible density profiles of the system as function of Fav at a fixed temperature are presented in Figure 12 for the van der Waals potential. For Fav < Fsb, the stable profile is symmetric and both walls are covered by thin liquid-like films, a gas-like


Figure 11. (a) Density profiles for Lennard-Jones potential and Fjav equal to 0.2319, 0.2628, 0.2721, 0.2783, 0.3092 (cup-like profiles), and 0.4638 (hill-like profile); T ) 104 K. The fluid density in the middle of the slit increases with increasing Fjav. (b) Dependence of the fluid density at the walls (solid curve) and in the middle of the slit (dashed curve) on the average density of the fluid for T ) 104 K.

fluid filling the remaining of the slit. No asymmetric profiles are present in this range of Fav. When the average density increases and reaches the critical value Fsb, a spontaneous symmetry breaking of the density distribution inside the slit occurs. The asymmetric profile, obtained as the asymptotic solution of the Euler-Lagrange equation for δw f 0, provides a lower Helmholtz free energy than the symmetric profile obtained as the solution of the Euler-Lagrange equation for δw ) 0. As a result, the former profile provides a stable state, whereas the second provides a metastable one. The free energy of the metastable state differs little from the free energy of the stable state. If the energy barrier between the stable and metastable states is large enough (the estimation of the height of this barrier is a separate problem that was not considered in this paper), the system can have a long lifetime in the latter state. In this state, when Fav increases further, a transition from a cup- to a hill-like profile occurs at a second critical density Fc-h > Fsb of the average fluid density. The profiles as function of Fav for the Lennard-Jones potential are presented in Figure 13 for T e Tsb,0. One should note the presence of a second critical density Fsb2 and the absence of a metastable state in the ranges Fav < Fsb1 and Fav > Fsb2. For T > Tsb,0 there is no symmetry breaking. Several issues that were not addressed in this paper should be mentioned. The first one is the dependence of the symmetry breaking on the parameters of fluid-fluid and fluid-solid interactions. In the present paper, symmetry breaking was found for a fluid-fluid energy parameter ff smaller than the fluid-



Figure 12. Possible states and the corresponding density profiles in the slit for the van der Waals potential at fixed temperature as function of the average fluid density. Note that the cup-hill transition at Fav ) Fc-h always occurs between metastable states of the system.

Figure 13. Possible states and the corresponding density profiles in the slit for Lennard-Jones potential at fixed temperature as function of the average fluid density for T < Tsb,0.

solid one. However, in ref 18, where the opposite inequality was satisfied, symmetry breaking was absent. It is likely that there is a critical value of the fluid-solid energy parameter (for a given fluid-fluid one) above which symmetry breaking occurs and it is interesting to find that value. A second issue is related to the dependence of symmetry breaking and cup-hill transition on the size of the slit and, finally, a third one to the existence of symmetry breaking in open slits.

References and Notes (1) Mu¨ller, M.; MacDowell, L. G. J. Phys.: Condens. Matter 2003, 15, R609. (2) Bonn, D.; Ross, D.; Bertrand, E.; Ragil, K.; Shahidzadeh, N.; Broseta, D.; Meunier, J. Physica A 2002, 306, 279. (3) Yochelis, A.; Pismen, L. M. Colloids Surf., A 2006, 274, 170. (4) Sweatman, M. B. Phys. ReV. E 2001, 65, 011102. (5) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; SliwinskaBartkowiak, M. Rep. Prog. Phys. 1999, 62, 1573. (6) Ancilotto, F.; Toigo, F. Phys. ReV. B 1999, 60, 9019. (7) Evans, R. J. Phys.: Condens. Matter 1990, 2, 8989. (8) Dhawan, S.; Reimel, M. E.; Scriven, L. E.; Davis, H. T. J. Chem. Phys. 1991, 94, 4479.

(9) Evans, R.; Marconi, U. M. B.; Tarazona, P. J. Chem. Phys. 1986, 84, 2376. (10) Evans, R.; Tarazona, P. Phys. ReV. A 1983, 28, 1864. (11) Ebner, C.; Saam, W. F. Phys. ReV. Lett. 1977, 38, 1486. (12) Cahn, J. W. J. Chem. Phys. 1977, 66, 3667. (13) Tarazona, P.; Marconi, U. M. B.; Evans, R. Mol. Phys. 1987, 60, 573. Tarazona, P. Phys. ReV. A 1985, 31, 2672. (14) Gonza´lez, A.; White, J. A.; Romań, F. L.; Velasco, S. Phys. ReV. Lett. 1997, 79, 2466. (15) Gonza´lez, A.; White, J. A.; Romań, F. L.; Evans, R. J. Chem. Phys. 1998, 109, 3637. (16) White, J. A.; Velasco, S. Phys. ReV. 2000, 62, 4427. (17) White, J. A.; Gonza´lez, A.; Romań, F. L.; Velasco, S. Phys. ReV. Lett. 2000, 84, 1220. (18) Berim, G. O.; Ruckenstein, E. J. Chem. Phys. 2006, 125, 164717. (19) Merkel, M.; Lo¨wen, H. Phys. ReV. E 1996, 54, 6623. (20) Mistura, G.; Ancilotto, F.; Bruschi, L.; Toigo, F. Phys. ReV. Lett. 1999, 82, 795. (21) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (22) Bieker, T.; Dietrich, S. Physica A 1998, 252, 85. (23) Maciolek, A.; Evans, R.; Wilding, N. B. J. Chem. Phys. 2003, 119, 8663. (24) Rosenfeld, Y.; Schmidt, M.; Lo¨wen, H.; Tarazona, P. J. Phys.: Condens. Matter 1996, 8, L577. (25) Tarazona, P. Phys. ReV. E 1997, 55, R4873. (26) Tarazona, P. Phys. ReV. Lett. 2000, 84, 694. (27) Tarazona, P.; Evans, R. Mol. Phys. 1984, 52, 847.

Fluid Density Profile Transitions and Symmetry Breaking in a Closed

Recommend Documents