10360
J. Phys. Chem. B 2001, 105, 10360-10366
Density Functional Theory Based on the Universality Principle and Third-Order Expansion Approximation for Adhesive Hard-Sphere Fluid near Surfaces Shiqi Zhou* Research Institute of Modern Statistical Mechanics, Zhuzhou Institute of Technology, Wenhua Road, Zhuzhou City 412008, P. R. China ReceiVed: April 12, 2001; In Final Form: August 9, 2001
The high-order perturbative density functional theory (Phys. ReV. E 2000, 61, 2704) was reformulated to be adapted for a nonuniform Lennard-Jones fluid with an attractive tail included in the interaction potential as a whole. The second-order direct correlation function of a uniform adhesive hard-sphere fluid was divided into a hard-sphere-like part and an attractive-tail part, and then the density functional formalism based on the universality principle and bridge function concept was employed to treat the nonuniform first-order direct correlation function for the hard-sphere-like part. The reformulated high-order perturbative density functional theory was employed to treat the attractive-tail part for the nonuniform case. The two parts were added together to construct the nonuniform first-order direct correlation function of the adhesive hard-sphere fluid. Then, the ensued result was substituted into the density profile equation in the density functional theory to predict the nonuniform density distribution profile and the contact density for an adhesive hard-sphere fluid subject to an external field. Good agreement with the corresponding computer simulation data was obtained.
I. Introduction Considerable attention has been paid to theoretical and experimental research on inhomogeneous fluids in the past 20 years, particularly with regard to their importance in the understanding of many problems, such as wetting transition, capillary condensation, liquid crystals, catalysis, the process of nucleation and crystal growth, the separation of mixtures, and many other interfacial phenomena.1-5 Extensive studies for these inhomogeneous fluids have been carried out with various versions of density functional theory (DFT), which were devised originally for an inhomogeneous hard-sphere fluid.6-9 However, a hard-sphere fluid cannot reproduce many of the interesting properties of real fluids that result from the attractive forces among molecules. Thus, it is important to study a suitable model with an attractive interaction as well as a repulsive one. One such example is the adhesive hard-sphere model, which consists of a hard core repulsion as well as a short-range attraction and was originally introduced by Baxter,10 who obtained an analytical solution for the Percus-Yevick (PY) approximation to the Ornsterin-Zernike (OZ) equation. Because of its tractability, this potential has found widespread use in many fields. This potential represents the extreme case of the hard-sphere system, with the square well or Yukawa attractive tail in the limit of an infinitely strong and infinitesimally short range attraction. It has already been proven to be useful in displaying the phase behavior in both the pure fluid and the mixtures,11 in modeling the interfacial equilibrium12 and the solvation interactions in colloidal solutions,13 and in describing the association phenomena in the ionic fluids.14 The inhomogeneous density distribution is a very important quantity from which the distance dependence of the adsorption isotherm, solvation force, and so forth (in the case of parallel plates immersed in the fluid) could be obtained. The inhomogeneous density distribution for the adhesive hardsphere fluid subject to an external potential had been studied by some versions of DFT.15-17 In ref 15, the hard-sphere-like (as short-range reference system) second-order direct correlation * Corresponding author. E-mail:
[email protected].
function (DCF) is separated from the second-order DCF of a uniform adhesive hard-sphere fluid; the difference is regarded as the attractive-tail part. Then, the hard-sphere-like part is treated by the weighted density approximation (WDA)18 directly to the nonuniform first-order DCF; the attractive-tail part is treated by the second-order perturbative expansion approximation. In refs 16 and 17, the hard-sphere-like part is treated by WDA19 to the excess Helmholtz free energy of the nonuniform fluid; the attractive-tail part is treated by the third-order perturbative expansion approximation.20 The predictions from refs 16 and 17 are superior to those of ref 15. A completely new DFT formalism9 was proposed recently which resulted from the use of the universality concept of the free-energy density functional and collected all of the orders beyond the second order of the functional perturbative expansion in the form of a bridge functional. Universality means the independence of the free-energy density functional or its firstorder functional derivative (the first-order DCF) on the external potential responsible for the formation of the density distribution. Thus, one can determine the universal functional form of the free-energy density functional or its first functional derivative from one special external potential that results from the particle chosen from the bulk and situated at the origin.21 The predictions from the new DFT formalism compare very well with the computer simulation results for the confined hard-sphere fluid. Also, the new DFT formalism was extended to the LennardJones (LJ) fluid and mixture case of a hard-sphere fluid and an LJ fluid.22,23 The suggested methodology in ref 9 is computationally much more modest than WDA: it has strict mathematical rigor, it embodies rich physical content, and its approximation comes only from the approximation included in the approximate bridge functional. At the same time, on the basis of the simple WDA (SWDA),24 the high-order perturbative DFT6e was also proposed for a nonuniform hard-sphere fluid. In ref 20, the kernel function of the employed third-order DCF was proposed from the symmetrical and intuitive considerations, not from strict mathematical derivation. The kernel function is short-range, but it is employed to approximate the long-range
10.1021/jp011399w CCC: $20.00 © 2001 American Chemical Society Published on Web 09/19/2001
DFT for Adhesive Hard-Sphere Fluid
J. Phys. Chem. B, Vol. 105, No. 42, 2001 10361
part of the interaction potential in refs 20 and 25. In ref 9, the new methodology requires that the bulk bridge functional be expressed as a functional of the indirect correlation function; such types of bridge functionals include the PY bridge functional, Verlet-modified (VM) bridge functional, and MartynovSarkisov (MS) bridge functional. However, all of these bridge functional forms are only suitable for a nonuniform hard-sphere fluid; among these bridge functional forms, the VM bridge functional produces the most accurate predictions for a nonuniform hard-sphere fluid density distribution profile.9 When the methodology in ref 9 was employed to study nonuniform adhesive hard-sphere fluids, even the VM bridge functional did not produce good predictions, although we did not plot the results in the present paper. The numerical version of the methodology was proposed recently,26 in which the RogersYoung (RY) approximation was combined with the OZ equation to extract the bridge functional as a functional of the indirect correlation function numerically, and then the interpolation procedure was used to specify the bridge functional from these numerically obtained points. When we used the numerical version to study the nonuniform adhesive hard-sphere fluid, again poor predictions were made because the RY approximation could not provide an accurate radial distribution function for the bulk adhesive hard-sphere fluid. It is the aim of the present paper to reformulate the high-order perturbative DFT6e for long-range interaction potential fluids (with attractiveness) in section II. Then in section III, the reformulated high-order perturbative DFT is employed to treat the attractive-tail part of the adhesive hard-sphere fluid, and the new methodology introduced in ref 9 is employed to handle the hard-sphere-like part to predict the density distribution profile and the contact density of the adhesive hard-sphere fluid subject to an external potential. Finally, some concluding remarks are made in section IV. II. High-Order Perturbative DFT for a Nonuniform Fluid with an Attractiveness Tail Included in the Interaction Potential In ref 6e, the high-order DCF of a uniform fluid of bulk density, Fb, was derived as follows:
C0(n)(r,r1,...,rn-1;Fb) ) n-1
C0(1) (Fb) [C0(1)′(Fb)]n
∫C0(2)(r0,r;Fb) C0(2)(r0,r1;Fb) ...
(2)
(2)
C0 (r0,rn-2;Fb) C0 (r0,rn-1;Fb) dr0
n g 3 (1)
∫dr1 (F(r1) ∝
Fb)C0(2)(|r-r1|;Fb) +
{
∫dr1 (F(r1) -
F(r) ) Fb exp -βφext(r) +
Fb)C0 (|r-r1|;Fb) + (2)
n-1
∝
C0(1) (Fb)
n)3
(n - 1)![C0(1)′(Fb)]n
∑
∑ (n - 1)!∫dr1 ∫dr2 ...
n-1
(2)
×
∫C0(2)(r,r′′;Fb)[∫C0(2)(r′,r′′;Fb) (F(r′) - Fb) dr′]n-1 dr′′
(4)
In ref 6e, eq 4 was truncated at the second order, third order, and fifth order to be carried out numerically for a nonuniform hard-sphere fluid, and a good agreement with simulation data was obtained. Now, to adapt eq 4 for a long-range potential fluid whose C0(1)n-1(Fb) could not be obtained accurately, we truncated the series in eq 4 at the third order, and then eq 4 became
{
F(r) ) Fb exp -βφext(r) +
∫dr1 (F(r1) -
Fb) C0 (|r-r1|;Fb) +
C0(1)(Fb)
(2)
2[C0(1)′(Fb)]3
×
∫C0(2)(r,r′′;Fb)[∫C0(2)(r′,r′′;Fb) (F(r′) - Fb) dr′]2 dr′′
(5)
For a long-range potential fluid such as an LJ fluid considered in the present paper, from the integral equation theory (IET), we can only obtain the second-order DCF accurately. It is difficult, although not impossible, to obtain accurate values for C0(1)′′(Fb) and C0(1)′(Fb), so we are using a constant, F, to substitute the coefficient C0(1)′′(Fb)/2[C0(1)′(Fb)]3 in eq 5. Then, the value of F is determined by the sum rule, which specifes the bulk pressure by the hard-wall contact density
(6)
where Fw is the hard-wall contact density which can be obtained from F(0) in eq 5 when the external potential has the following form:
φext(z) ) ∝
z/σ < 0
)0
0 < z/σ
(7)
P is the bulk pressure which can be obtained from the equation of state for LJ28 8
n)3
m)1
where φext(r) is the external potential responsible for the generation of the density distribution, F(r), and β ) 1/kT with k as the Boltzmann constant and T as the absolute temperature. A combination of eqs 1-3 leads to the density profile equation of the following form:
P* ) F*T* +
1
∫drn-1 ∏[F(rm) - Fb]C0(n)(r,r1,...,rn-1;Fb)
F(r) ) Fb exp{-βφext(r) + C(1)(r;[F]) - C0(1)(Fb)} (3)
P ) FwkT
where C0(n) is the nth-order DCF of the uniform fluid and C0(1)n-1(Fb) is the (n - 1)th-order derivative with respective to Fb. Equation 1 was derived from SWDA,24 and although SWDA is the most crude approximation, both eq 1 and SWDA satisfy some well-known sum rules. The expansion of C(1)(r;[F(r)]) around the uniform system of bulk density, Fb, is as follows:
C(1)(r;[F(r)]) ) C0(1)(Fb) +
In the formalism of DFT, the density profile equation of a nonuniform single-component fluid is
6
aiF*(i+1) + F∑biF*(2i+1) ∑ i)1 i)1
(8)
where P* ) Pσ3/, T* ) kT/, F* ) Fbσ3, and σ and are the potential parameters characterizing the interaction range and strength, respectively. The parameters ai, bi, and F can be obtained from ref 28.
10362 J. Phys. Chem. B, Vol. 105, No. 42, 2001
Zhou
Figure 1. Density profiles of a Lennard-Jones fluid (Fbσ3 ) 0.75 and kT/ ) 1.304) confined between two hard walls with H/σ ) 5. The dots represent the corresponding MC results.25 Only half of the slit is shown.
Because the constant F is only related to the bulk parameter and is not connected to the external potential parameters, F in this special case can be used for an arbitrary external potential case. To proceed computationally, the quantity C0(2)(r;Fb) is needed. In the present paper, C0(2)(r;Fb) is obtained by numerically solving the OZ equation
∫
h(r) ) C0(2)(r;Fb) + Fb dr′ h(r′) C0(2)(r,r′;Fb)
(9)
combined with the closure equation
h(r) + 1 ) exp[-βu(r) + h(r) - C0(2)(r;Fb) + b(r)] (10) In eq 10, b(r) is the sum of diagrams that are free of nodal circles and is conventionally called a bridge function. The following expression for b(r), proposed by Duh and Henderson,29 for a uniform LJ fluid is employed,
b(r) ) -s2/2[1 + s(5s + 11)/(7s + 9)]
(11A)
where s ) h(r) - C0(2)(r;Fb) - βu2(r) and u2(r) ) -4(σ/r)6 exp[(-1/Fx)(σ/r)6Fx], with Fx ) Fbσ3. For an LJ fluid, the interaction potential has the following form:
u(r) ) 4[(σ/r)12 - (σ/r)6]
(11B)
Now, eq 5 will be applied to an LJ fluid confined between two hard walls at a separation, H. For this case, the external potential has the following form:
φext(z) ) ∝ )0
z/σ < 0.5 or z/σ > H - 0.5 0.5 < z/σ < H - 0.5
(12)
The density profile equation (eq 5) is plotted and compared with the corresponding computer simulation results25 in Figures 1 and 2 for H/σ ) 5 and 13, respectively. Also, the predictions from the second-order DFT were plotted in the same figures. From Figures 1 and 2, one can see that the predictions of the present method are far more superior to those from the secondorder DFT and are in very good agreement with the computer simulation results. This fact indicates the importance of the incorporation of the third-order term and concludes that the present reformulated version of the original high-order perturbative DFT is suitable for even the long-range potential fluid as a whole. For a long-range potential fluid, for example, the
Figure 2. Same as in Figure 1 but with H/σ ) 13.
LJ fluid, all of the previous studies employed exclusively perturbative methods,30 contrary to the present way of treating the pair interaction potential as a whole. The form of the present third-order DCF of a uniform fluid is similar to that proposed in ref 20, which is of the following form:
∫
C0(3)(r1,r2,r3) ) B dr4 a(r4-r1) a(r4-r2) a(r4-r3) (13) where the kernel function is
a(r) )
σ 6 Heaviside - r 3 2 πσ
(
)
(14A)
and, obviously, the kernel function a(r) is short-range and a different interaction potential fluid enjoys the same kernel function; however, this is not reasonable. The present kernel function is exactly the second-order DCF itself, and it is derived from SWDA.24 The combination of the second-order DCF and radial distribution function determines all of the thermodynamics and structural properties of the fluid, so it necessary and reasonable for one to derive the high-order DCF from the second-order DCF, not from the intuitive and simple a(r). In fact, for the calculated cases in Figures 1 and 2, the predictions from the present third-order perturbative DFT, with the secondorder DCF of a uniform fluid as the kernel function of thirdorder DCF, are superior to those based on the previous thirdorder perturbative DFT,25 in which the kernel function of thirdorder DCF is of the following simple form:
a(r) ) (1 - x2/σ2)Heaviside(|σ - x|)
(14B)
For clarity, we did not plot the predictions from ref 25 in Figures 1 and 2. One can expect that the reformulated third-order DFT will break down in some regions of phase space (especially in the regions near the critical points); this is common for all of the perturbative DFTs6 based on IET, but this does not matter, as we will explain. The prime application of the reformulated thirdorder DFT lies in the treatment of the long-range part or attractive-tail part of the real fluid interaction potential; the shortrange part is treated by a nonperturbative approximation, for example, the WDA, the fundamental measure theory by Rosenfeld,8 or the new formalism in ref 9. The separate treatment of the interaction potential is standard and is exactly what van der Waals and liquid-state perturbative theory are based on. All of the DFT is actually a device by which the input parameters from IET can be employed to predict the properties of a nonuniform fluid, so whether the separate
DFT for Adhesive Hard-Sphere Fluid
J. Phys. Chem. B, Vol. 105, No. 42, 2001 10363
treatment is applied does not determine the phase space for which the DFT is suitable; what is deterministic is the accuracy of the input parameters from IET. III. Mixed-Order DFT for a Nonuniform Adhesive Hard-Sphere Fluid
(
(15)
where τ is a dimensionless parameter which increases with increasing temperature and τ-1 is a parameter representing a measure of the adhesiveness of the potential. When we separate the hard-sphere-like (as a short-range reference system) secondorder DCF C0ref(2)(r,Fb,τ) from the second-order DCF C0adh(2)(r,Fb,τ) of a uniform adhesive hard-sphere fluid, the difference is denoted by C0attr(2)(r,Fb,τ):
C0attr(2)(r,Fb,τ) ) C0adh(2)(r,Fb,τ) - C0ref(2)(r,Fb,τ) (16) and the corresponding first-order DCFs are denoted by C0attr(1)(Fb,τ), C0adh(1)(Fb,τ), and C0ref(1)(Fb,τ) for a uniform fluid and Cattr(1)(r,Fb,τ), Cadh(1)(r,Fb,τ), and Cref(1)(r,Fb,τ) for the corresponding nonuniform fluid. According to eq 16, we have
Cattr(1)(r,Fb,τ) ) Cadh(1)(r,Fb,τ) - Cref(1)(r,Fb,τ)
(17)
C0attr(1)(Fb,τ) ) C0adh(1)(Fb,τ) - C0ref(1)(Fb,τ)
(18)
Then, we treat the hard-sphere-like part with the methodology proposed in ref 9
Cref(1)(r;[F],τ)
Substituting eqs 19 and 20 into eq 3 for the basic density profile equation in the DFT, we obtain
F(r) ) Fb exp{-βφext(r) +
∫(F(r′) - Fb) C0ref(2)(|r-r′|;Fb,τ) dr′ + B[∫(F(r′) - Fb) C0ref(2)(|r-r′|;Fb,τ) dr′]} + ∫dr1 (F(r1) - Fb) C0attr(2)(|r-r1|;Fb,τ) + F∫C0attr(2)(|r-r′′|;Fb,τ)[∫C0attr(2)(|r′-r′′|;Fb,τ) (F(r′) Fb) dr′]2 dr′′ (21) With regards to the expression for C0adh(2)(r,Fb,τ), we employed the result from the PY approximation to the OZ equation
1 C0adh(2)(r,Fb,τ) ) -a - br/R - ηa(r/R)3 2 1 2 1 ηλ R/r Θ(R - r) + λδ(r - R-) (22) 12 12
(
)
R , R′ < r e R ) -In 12τ(R - R′) ) 0, r > R
∫dr1 (F(r1) - Fb) C0attr(2)(|r-r1|;Fb,τ) + F∫C0attr(2)(|r-r′′|;Fb,τ)[∫C0attr(2)(|r′-r′′|;Fb,τ) (F(r′) -
C0attr(1)(Fb,τ) +
Fb) dr′]2 dr′′ (20)
As explained in the Introduction, in some DFTs for real fluids, such as adhesive hard-sphere fluids and electrolyte fluids, the DCF is usually divided into a hard-sphere-like part and a tail part. Then, the hard-sphere-like part is treated by the DFT suitable for a nonuniform hard-sphere fluid, and the tail part is treated by the second- or third-order perturbative DFT. So, the wide application of the present reformulated third-order perturbative DFT also lies in the case of the previously mentioned mixed-type DFT in which the different parts of the interaction potential were treated by different DFT approximations. We will report an application of the mixed-type DFT based on the methodology in ref 9 and the present reformulated third-order perturbative DFT for an inhomogeneous adhesive hard-sphere fluid. The interaction potential of an adhesive hard-sphere fluid has the following form:
βφ(r) ) ∝, 0 < r < R′
Cattr(1)(r;[F],τ) )
)
where a, b, and λ are a function of η ) Fbσ3π/6 and τ.10 We chose the following form for C0ref(2)(r,Fb,τ):
1 C0ref(2)(r,Fb,τ) ) -a - br/R - ηa(r/R)3 Θ(R - r) (23) 2
(
)
Because the hard-sphere-like part is treated by the new DFT formalism which includes all of the orders beyond the secondorder of the functional perturbative expansion in the form of a bridge functional, we can denote the present formalism as the mixed-order DFT. No coupling term results from the above separation of correlation functions because the correlation functions are the functional derivative of the excess Helmholtz free energy and the excess Helmholtz free energy is additive. According to the definition of the functional derivative, the functional derivative of the sum of two quantities is equal to the sum of the functional derivative of each quantity. Now, we apply the present formalism to the case where the adhesive hard-sphere fluid is confined between two hard walls situated at z ) -0.5R and z ) H + 0.5R, respectively. For this case, the external potential is given by
βφext(z) )
)
∫dr1 (F(r1) - Fb)C0ref(2)(r,r1;Fb,τ) + B[∫dr1 (F(r1) - Fb)C0ref(2)(r,r1;Fb,τ)] (19)
C0ref(1)(Fb,τ) +
where B stands for the bridge functional for a hard-sphere fluid, such as the PY and VM approximations in which the bridge functional is expressed as a function of the indirect correlation function. The difference (i.e., the attractive-tail part) is expressed by the present third-order DFT.
{
0 0
