A star-function based density functional study of the adsorption of Lennard-Jones fluid near its supercritical states

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Abstract

The goal of this paper is to show that the wetting behavior of simple fluids on a repulsive solid surface – especially the “drying” phenomenon – is closely related to the proximity of the supercritical state of the bulk fluids to their vapor–liquid coexistence region. We propose here a new DFT (the star-function based density functional theory, s-DFT) that is based on the functional Taylor expansion of the intrinsic free energy F[ρ] and the singlet direct correlation Cw(1)(r) to arrive at closed-form expressions for both quantities without truncations or approximations. The two formulas are mutually consistent because of the introduction of a star function Sw that has been shown to be the functional primitive of the bridge function Bw, i.e. δSw*/δρ=Bw (L.L. Lee, J. Chem. Phys. 97 (1992) 8606 [34]). The new formulation is applied to the Lennard-Jones molecules adsorbed on a planar hard wall (LJ/HW). We carried out new Monte Carlo simulations for this system. Since the s-DFT uses a bridge function Bw, we demonstrate (i) the existence of a set of data (inverted from the MC information) that can perform as the bridge function and reproduce accurately the density profiles ρw(1)(r); (ii) this set of data can be “fitted” by a function-form with acronym ZSEP; finally (iii) ZSEP expresses the bridge function Bw in terms of a new renormalized basis function γH, i.e. Bw(γH). The existence of a bridge function dispels some of the misconceptions that the bridge-function based formulations did not describe the “drying” behavior. We also show that for the high density case ZSEP equation can qualify as a “closure relation”, but seems to deteriorate for the two low-density supercritical states that are close to the bulk saturated liquid phase boundaries.

Introduction

More than a hundred years ago, Johannes Diderik van der Waals proposed the first significant theory [1], [2], [3], [4], [5] for inhomogeneous fluids experiencing phase transition. He was interested in the vapor–liquid transition of fluids with nonuniformity, just as in their behavior in uniform systems. This theory is known as the square-gradient theory (SGT) of classical fluids.

For a planar vapor–liquid interface, the Helmholtz free energy F per area A was given asFA=dzf0+c2dρw(1)(z)dz2where f0 is the bulk (uniform fluid) Helmholtz free energy per volume; ρw(1)(z) is the nonuniform singlet density profile as a function of the distance z from the interface; and c in the Rayleigh–van der Waals approximation can be written as an integral involving the attractive potential. The derivative dρw(1)(z)/dz is evaluated at distance z from the interface (note that no “average density” is taken over the distances). Given this local character of the SGT density profile it is not surprising that in applications, it was found that SGT is suitable for spatially slowly varying density profiles and does not perform well for steep density profiles. There are also other approaches to the inhomogeneous fluids sharing the same local character of the SGT. We can count, among others, the molecular gradient theories: such as the WLMB (Wertheim–Lovett–Mou–Buff) [6], [7], Zwangzig–Triezenberg [8], and the YBG (Yvon–Born–Green) [9], [10] theories, the wall Ornstein–Zernike (WOZ) theory [11], [12] developed in the 1970s, and recently the bridge function-based DFTs (B-DFT) [13], [14], [15], [16], [17].

The modern origin of DFT is usually attributed to Hohenberg and Kohn [18] who in 1964 formulated the grand potential approach for quantum electron gas. Later, in the 1980s the density functional theory evolved into a weighted density formulation (WDA) in search of a coarse-grained density ρ¯(r) (for example, ρ¯(r)drρw(1)(r)ω(r,r), where ω(r,r′) is some type of weighting function) to be used in the free energy expression [19], [20], [21], [22], [23], [24], [25], [26]. This development culminated in the fundamental measure theory (FMT) of Rosenfeld [27] which is based on the hard-sphere geometric measures within a scaled-particle theory (SPT) [28] framework. These DFTs are called the non-local density approximations (NLDA) because of the use of the weighted densities ρ¯(r). The NLDAs have met success in wetting, phase transition, freezing, and recently in electric double layer studies. This approach constitutes a major trend in current DFT applications.

We note that in 1990s, a number of attempts have been made which applied the successful WDA theories developed for nonuniform systems to uniform fluids. This is based on the insight of Percus [29], [30] who proposed the concept of “test particles” (he called them source particles). These test particles in a homogeneous fluid are treated by Percus as sources of an external potential, w(r). This equivalence puts the nonuniform systems and uniform systems on equal footing. For example, a planar wall can be considered as the result of a spherical test particle whose diameter has grown to infinity. On the other hand, a member molecule in a bath of a homogeneous fluid can be thought of as the cause of nonuniformity while its pair potential (interacting with other molecules) is viewed as an “external” force (this picture is particularly fitting for solvation studies). Notable among these crossovers were the application of Curtin and Ashcroft's [21], [22] WDA theory to the homogeneous hard sphere fluids (Denton and Ashcroft 1991 [31]), and the application of Tarazona's [32] WDA to the same kind of fluids (Brenan and Evans 1991 [33]).

On the other hand the liquid theories for uniform fluids have vigorously developed in the past decades, and highly accurate results can be obtained for many pair potentials, not just limited to the hard spheres. It is worthwhile to ask if we can “transplant” the successful liquid theories to nonuniform systems via the Percus intuition (namely going the opposite direction of Denton and Ashcroft: from uniform to nonuniform fluids). In this work, we propose a star-function-based density functional theory (s-DFT) that shall produce a pair of expressions for the Helmholtz free energy functional (HFEF) F[ρ] and the singlet direct correlation function (1-DCF) C(1)(r;[w]) that are consistent with each other. The notation C(1)(r;[w]) is to exhibit that C(1) is a function of the vector distance r, while at same time it is a functional of the external potential w(r). The square brackets [.] indicate functional dependence on the argument. These two quantities are linked by a star function Sw that was proposed [34] earlier in 1992. The subscript w indicates nonuniform quantities (under influence of the external potential w). Functional expansions are made for the excess HFEF Fex and C(1)(r;[w]) (Section 4). The 1-DCF expansion is terminated by a bridge functional, Bw(r;[w]); while the Fex expansion is terminated by the star series Sw*. The notation Bw(r;[w]) is similar to the one adopted for C(1)(r;[w]) and Sw. The star series can be shown to be the functional primitive of the bridge functional [34]. This theory is formally exact (with no truncations) and uses only local densities.

As a second development, we shall show that the Euler–Lagrange equation in DFT is equivalent to the wall Ornstein–Zernike equations [11], [12] proposed in the 1970s (Section 3). As the solution of WOZ required a bridge function Bw, the Euler–Lagrange equation can also use a bridge function for its solution. Through this Bw, the WOZ can be linked to the recent bridge function-based DFTs [13], [14], [15], [16], [17]. Previous research [35] has shown that with the usual closures, the WOZ has not been satisfactory. We shall examine however if the successful uniform liquid theory on the bridge functions can be transferred to nonuniform fluid studies.

We divide the study and search of the bridge functions into at least two categories: (i) the existence of the bridge function (the existence question); and (ii) the relation of this bridge function to other correlation functions found in the liquid theory (i.e. the “closure relation”). Subscript 0 shall indicate uniform fluid properties (i.e. at w=0). The first question whether the bridge function B0 (or B) exists or not can be formally answered by its definition in cluster expansion [36] or its definition via functional expansion [37]. Mathematically, B is a well defined quantity, and it exists in this sense! However, the expansions are infinite series – consequently numerical determination of B is a separate issue and is a problem in practice: it may not be accessible to numerical computation. The second category, this being the gist of the liquid state theory, is to relate the bridge function, in closed form, to other known correlation functions. These relations are called the closure equations. In the literature, there are quite a number of closure equations, such as the Percus–Yevick (PY) [38] equation and the hypernetted chain (HNC) [39], [40] equation. Note that all existing closures are approximate relations. Some may perform better than others, depending on the pair potential in question. For example, the PY closure is better for hard spheres [41], while the HNC closure is better for the Coulomb potential [42]. They express the bridge function B(f) in terms of another correlation function, f. Some chose the correlation function to be the indirect correlation function [43] (icf), γ (≡ h  C(2) = total correlation  pair direct correlation). Others [44], [45] chose f = the thermal potential θ (≡ ln y, y = the cavity function). We shall give an analysis (Section 5) on the issue of the chosen correlation function from resummation of the functional expansion. The difficulty in obtaining a closure relation is the fact that the bridge functions are functionals, not functions. To render a functional B[f(r)] into a function B(f), one has to eliminate all dependence on the domain values r. This in general is a daunting task even in mathematics. In the literature, many researchers [46], [47], [48], [49], [50] have attempted to “renormalize” f(r), i.e. transforming f(r) into f*(r) by adjusting the functional form of f in the hope that ultimately the equality holds: B[f(r)]=Bˆ(f*), where Bˆ is some optimized function form. This is called the unique functionality principle. We shall propose a new renormalization procedure, to be performed on the indirect correlation, γ (Section 5).

We shall apply this new theory to a simple nonuniform system: Lennard-Jones fluid adsorbed on a planar hard wall (LJ/HW). There are interesting phenomena [19], [20] in wetting of surfaces, such as prewetting or drying behavior. Drying happens when the liquid layers of a bulk liquid, coming into contact with a wall, begin to form vapor-like densities, or “cavitation” as we call it. The LJ/HW system exhibits drying behavior at certain conditions. This system has been investigated a number of times in the past and the WDA theory [51], [52], [53] is able to describe this behavior reasonably well. We consider this system as a first test of the new theory.

We shall examine the supercritical states of Lennard-Jones fluid at T* = kT/ɛ = 1.35, and densities ρ* = ρσ3 = 0.50, 0.65, and 0.82. ɛ and σ are the parameters of the Lennard-Jones potential (ɛ is the energy parameter and σ the size parameter)u(r)=4εσr12σr6Fig. 1 shows the state points examined relative to the pure Lennard-Jones vapor–liquid equilibrium (VLE) phase boundaries as determined from computer simulations [54], [55]. All three densities (ρ* = 0.50, 0.65, and 0.82) are in the supercritical state (T* = 1.35 is about 2.5% above the LJ critical temperature Tc*=1.317 as computed by simulations [54], [55]). The state ρ* = 0.50 is situated closest to the critical point and the vapor–liquid phase boundary. The state at ρ* = 0.65 is a bit farther but still subject to the effects of proximity to the saturated liquid line; while the state ρ* = 0.82 is in the “dense” fluid state due to its high density. Thus we have a good collection of three supercritical states: being near, not far away, and far removed from the liquid-side bubble-point line.

We carried out independent Monte Carlo (MC) simulations for the LJ/HW system. In all cases (all three densities), we are able to determine the bridge functions Bw(z) from the MC data. Namely, the existence question is answered positively. It is possible to characterize the bridge function within our approach. On the second hand, determination of the closure relation meets partial success. For the high density case (ρ* = 0.82), the ZSEP closure [56] developed earlier performs well (the closure relation works); while for cases with drying (ρ* = 0.50 and 0.65), no closure relation has been found to be adequate (Section 6).

In Section 2, we review the conventional DFT from Hohenberg–Kohn to establish notation. In Section 3, we show the connection of the Euler–Lagrange equation to the wall Ornstein–Zernike integral equation by hinging the 1-DCF in one theory to the bridge function in the other. Next in Section 4, we make Taylor's functional expansions of the singlet direct correlation and the HFEF. The star series Sw* is introduced into the s-DFT. These will be the basic formulas in the star-function based DFT. In Section 5 we examine several aspects of the present approach: the inversion of MC data to obtain the bridge functions, the resummation of the bridge function expansion, and the renormalization of the base function f* = γH in the bridge function. The results of calculations for the LJ/HW system are given in Section 6. Conclusions are drawn in Section 7.

Section snippets

Review of the density functional theory

In order to establish the notation and the theoretical framework without ambiguity, we recapitulate the Hohenberg and Kohn [18] formulation briefly. The grand potential Ω is defined for a fluid in a nonuniform system, where the nonuniformity is induced by an external potential w(r)ΩF[ρw]+drρw(1)(r;[w]){w(r)μ0}Ω thus defined is a functional of the singlet density (one-body density) ρ(1)(r;[w]) of a fluid in general, not necessarily at equilibrium. The singlet density ρ(1)(r;[w]) is a

Connection to the wall Ornstein–Zernike equation

In the 1970s, a wall-particle Ornstein–Zernike theory [11], [12] was proposed that is based on the uniform-fluid Ornstein–Zernike (OZ) equations for a binary mixture of two species: “0”  tag of fluid particles with mole fraction x0, and “1”  tag of so-called “wall” particles of mole fraction x1. In view of Percus’ [29], [30] “test particle” picture, the wall particles are considered initially as a species of large spheres and eventually their diameters grow to infinity (into a flat wall of zero

Expansions of the singlet DCF and HFEF and the star function

In this section, we shall give the functional Taylor expansions of the singlet direct correlation and the HFEF. Since the developments are well-known [19], [20] and have been summarized in Ref. [56], we give only the essential results below. First, from Eq. (2.10) the functional derivative of the HFEF is the singlet direct correlation. From Lebowitz and Percus [57], higher order derivatives of 1-DCF areC(n)(1,2,3,,n)δn1C(1)(1)δρ(2)δρ(n)for all n > 1. Note that the arguments 1, 2, 3, etc. are

The existence of the bridge function and renormalization

To implement the s-DFT, we need the Euler–Lagrange equation (4.6) for the singlet density ρw(z) and the definition (4.10) for the icf γw. In cylindrical coordinates (with z = direction perpendicular from the wall)ρw(z)=ρ0exp[βw(z)+γw(z)+Bw(z)]where the icf γw(z) is obtained from a convolutionγw(z)=ρ002πdθ0drrC0(2)(r)rrdzhw(z+z)where hw(z)=δρw(z)/ρ0. The uniform pair-DCF C0(2)(r) needed in (5.2) can be obtained from separate calculations with the pure LJ fluid at the desired bulk

Application to the Lennard-Jones/hard wall system

To test the present local density approach, we shall examine a classical simple fluid system: the Lennard-Jones fluid adsorbed on a planar hard wall. This system is of interest, because the fluid contains molecules with attractive forces while the wall is purely repulsive. The wall, being hard, will not hide any attractive effects from the fluid (in other words, there is no wall attraction to interfere with the fluid attraction). In addition, it would be of interest to examine the “drying”

Conclusions

We employ the functional expansion methods to formulate a new density functional theory, the star-function based density functional theory. This theory gives consistent formulas for the singlet DCF and the free energy functional F[ρw(z)]. The new theory belongs to the class of local density approximations in DFT.

In order to test its performance, we made calculations on the Lennard-Jones Fluid/Hard Wall system. For the three densities (ρ0*=0.50, 0.65, and 0.82, T* = 1.35) studied, we show (i) the

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