Enhanced KR-Fundamental Measure Functional for Inhomogeneous Binary and Ternary Hard Sphere Mixtures

Transcription

1 Commun. Theor. Phys. 55 (2011) Vol. 55, No. 1, January 15, 2011 Enhanced KR-Fundamental Measure Functional for Inhomogeneous Binary and Ternary Hard Sphere Mixtures ZHOU Shi-Qi ( Ð) State Key Laboratory of Powder Metallurgy, Central South University, Changsha , China School of Physics Science and Technology, Central South University, Changsha , China (Received February 23, 2010; revised manuscript received March 24, 2010) Abstract An enhanced KR-fundamental measure functional (FMF) is elaborated and employed to investigate binary and ternary hard sphere fluids near a planar hard wall or confined within two planar hard walls separated by certain interval. The present enhanced KR-FMF incorporates respectively, for aim of comparison, a recent 3rd-order expansion equation of state (EOS) and a Boublík s extension of Kolafa s EOS for HS mixtures. It is indicated that the two versions of the EOS lead to, in the framework of the enhanced KR-FMF, similar density profiles, but the 3rd-order EOS is more consistent with an exact scaled particle theory (SPT) relation than the BK EOS. Extensive comparison between the enhanced KR-FMF-3rd-order EOS predictions and corresponding density profiles produced in different periods indicates the excellent performance of the present enhanced KR-FMF-3rd-order EOS in comparison with other available density functional approximations (DFAs). There are two anomalous situations from whose density profiles all DFAs studied deviate significantly; however, subsequent new computer simulation results for state conditions similar to the two anomalous situations are in very excellent agreement with the present enhanced KR-FMF-3rd-order EOS. The present paper indicates that (i) the validity of the naive substitution elaborated in the present paper and peculiar to the original KR-FMF is still in operation even if inhomogeneous mixtures are being dealt with; (ii) the high accuracy and self-consistency of the third order EOS seem to allow for application of the KR-FMF-third order EOS to more severe state conditions; and (iii) the naive substitution enables very easy the combination of the original KR-FMF with future s more accurate but potentially more complicated EOS of hard sphere mixtures. PACS numbers: Hh, g, a Key words: classical density functional theory, hard sphere fluid, hard sphere colloids 1 Introduction Fundamental measure functional (FMF) had gained much attention of researchers in liquid theory community [1] since its introduction in the late eighties and early nineties of the last century, [2 3] its popularity originates from two features peculiar to the FMF. In first place the FMF does not use Ornstein Zernike (OZ) integral equation (IE) theory as input, and constitutes an independent theoretical development; consequently the FMF provides a closure [4] to the OZ IE theory through recourse to a test particle trick. [5] In the next place, the FMF is intrinsically a formulation suitable for mixtures, its use in the case of mixtures only involves simple sum of relevant integrals over component index, and does not touch upon any new approximations or self-consistency problem. It also should be pointed out that the FMF is not without its theoretical origins, it heavily depends on the idea of scaled particle theory (SPT) [6] and weighted density approximation (WDA) concept. [7] Particularly, the WDA still find its opportunity in field of inhomogeneous non-hard sphere fluids, which is however beyond the power of the FMF at least in the current situation, therefore it is reasonable to say that the WDA concept is of more general theoretical significance. What is embedded in the original Rosenfeld FMF (thereafter marked as R-FMF) and its equivalent version of Kierlik Rosinberg (KR) FMF is the SPT equation of state (EOS) or a Percus Yevick (PY) compressibility EOS. It is well-known that the PY compressibility EOS overestimates the bulk pressure p at high packing fraction and the FMF exactly accords with the wall contact theorem, which declares that the contact value of the density profile of the hard sphere fluid at a planar wall is equal to βp underlying the density functional approximation (DFA) in question, consequently both the R-FMF and KR-FMF also overestimate the densities at the contact point and its vicinity of the hard sphere fluid near a hard wall at high coexistence bulk densities. This undesirable property directly motivates revision of the R-FMF by Roth, Evans, Lang, and Kahl who reported in Ref. [8] systematic descriptions about the modification way (MW) in their White Bear version. The White Bear version of the R-FMF (thereafter abbreviated as White Bear FMF) uses a Mansoori Carnahan Starling Leland (MCSL) EOS, [9] Supported by the National Natural Science Foundation of China under Grant No Corresponding author, chixiayzsq@yahoo.com c 2011 Chinese Physical Society and IOP Publishing Ltd

2 No. 1 Communications in Theoretical Physics 47 which is a generalization to the υ-component hard sphere fluid of an accurate, one-component Carnahan Starling EOS, [10] as an input, and achieves the desired results; subsequently, Malijevský [11] incorporated Boublík s multicomponent extension [12] of Kolafa s EOS [13] for pure HS fluid by following closely to the same MW resulting in the White Bear FMF. It was discovered that the Malijevský FMF based on the BK EOS exhibits minor improvements over the White Bear FMF based on the MCSL EOS for the region near the wall. It is known that the KR-FMF is actually equivalent to the R-FMF, and profiting from the absence of the vector weighted densities, the KR-FMF is computationally simpler than the R-FMF, there should be enhanced version corresponding to the KR-FMF. Such enhanced KR-FMF was recently attempted by the present author [14] in which the FMF was further extended in a unified approach to non-hard sphere fluids in inhomogeneous state. The MW leading to the enhanced KR-FMF is completely different from that causing the White Bear FMF, it seems necessary to test the validity of the MW- KR in situation of multicompoent hard sphere mixtures. Furthermore a new EOS for hard sphere mixtures was suggested [15] recently by expanding the EOS in powers of the total packing fraction with coefficients depending on the first three of the SPT variables; it was shown that the new EOS is more accurate than the MCSL EOS and an extended Carnahan Starling EOS introduced by Santos et al. [16] However, an enhanced R-FMF version corresponding to the new EOS is not advanced due to the complexity of the new EOS; consequently it makes sense to incorporate the new EOS into the KR-FMF to advance an enhanced KR-FMF. Another aim of the present paper is to make clear the difference between the two MWs for the R-FMF and the KR-FMF, and test the validity of the MW-KR in situation of binary and ternary hard sphere mixtures. The paper is arranged as follows. In Sec. 2 both the MW-R [8] and the MW-KR are presented, and a discussion is given to the distinction between the two ways. In Sec. 3 the new third order EOS [15] and the BK EOS [12 13] are respectively incorporated into the KR-FMF, the resultant enhanced KR-FMFs are tested by application to binary sphere mixtures near a planar hard wall at selected state points; one of the two enhanced KR-FMFs with better quality will be tested extensively for binary and ternary hard sphere mixtures near a planar hard wall or confined within two planar hard walls. Finally conclusions are collected in Sec Modification Ways for R-FMF and KR- FMF In order to help in expanding our discussion, we first briefly recount the R-FMF and its White Bear version together with relevant symbols and terminologies also used in the present enhanced KR-FMF. Making use of an exact low density limit for the excess Helmholtz free energy functional F ex [{ρ i }] and dimensional analysis, Rosenfeld [2] wrote the F ex [{ρ i }] in the form βf ex [{ρ i }] = drφ({n α (r)}), (1) where β = 1/kT is inverse thermal energy with k a Boltzmann constant and T an absolute temperature, the free energy density Φ divided by kt is a general function of the weighted densities n α (r) with α labeling four scalar and two vector weights Φ = f 1 (n 3 )n 0 + f 2 (n 3 )n 1 n 2 + f 3 (n 3 ) n 1 n 2 + f 4 (n 3 )n f 5(n 3 )n 2 n 2 n 2, (2) n n α (r) = dr ρ i (r )ω (α) i (r r ), (3) i=1 the weighting functions ω (α) i (r), characterizing the geometry of the particle i and being density independent, are required to satisfy normalization conditions ω (α) i (k = 0) = R (α) i, α = 0, 1, 2, 3, (4) for the first four scalar weighting functions, and ω (α) i (k = 0) = 0, (5) for the two vector ones. Throughout the text, R (α) i are the fundamental measures characterizing particle i: R (0) i = 1, R (1) i = R i, R (2) i = 4πRi 2, R(3) i = (4/3)πRi 3, the relevant R i denotes the radius of particle i, and is related to the relevant diameter σ i by R i = σ i /2. It is noted that n α with α = 0, 1, 2, 3 will reduce to scaled particle variables ξ α = n i=1 ρi b R(α) i (α = 0, 1, 2, 3) in the bulk limit; for convenience s sake, thereafter in the text, the ξ α, when needed, will be written as n α with the implication that n α is actually ξ α in the bulk limit. The function forms for (r) were reported in Ref. [2] and not repeated here. The functions f 1, f 2, f 3, f 4, f 5 can be determined by requiring that the resulting functional satisfies an exact SPT relation, which given the function form for Φ marked by Eq. (2), turns into βp = Φ, (6) n 3 ω (α) i Eq. (6) is an exact relation in the framework of the FMF. On the other hand, the FMF form for Φ marked by Eq. (2) also can give an expression for the bulk pressure denoted by βp(φ). By substituting the left side of Eq. (6) by βp(φ), Rosenfeld obtained the function forms for f 1, f 2, f 3, f 4, f 5 ; furthermore, it is noted that f 3, f 5, which fix the dependence of the FMF on the vector weighted densities, are related to f 2, f 4 by simple equalities, namely f 3 = f 2 and f 5 = 3f 4. Because of imposition of the Eq. (6), R-FMF exactly meets the exact relation. In the MW-R, all of the above discussion, particularly the two equalities of f 3 = f 2 and f 5 = 3f 4, are assumed still valid, but Eq. (6) is not used; instead, one

3 48 Communications in Theoretical Physics Vol. 55 equals βp(φ) to an externally imported bulk EoS to determine f 1, f 2, f 4. Two examples using the MW-R are reported respectively in Refs. [8] and [11], where the MCSL EOS and the BK EOS for HS mixtures are respectively employed as input. It was shown that the two enhanced versions of the R-FMF outperform the original R-FMF at region near the wall and the BK enhanced version is slightly better than the MCSL enhanced one, obviously this subtle difference derives from the distinction between the two underlying EOSs. It is noted that since the Eq. (6) is not imposed in the MW-R, the resultant enhanced FMF-R might not meet the exact relation, i.e. Φ/ n 3 might be not equal to the imported βp, this is termed as pressure inconsistency. [8,15] As a comment, one notes that in the MW-R the function forms for f 1, f 2, f 4 are obtained by solving the differential equation resulting from equaling βp(φ) to an externally imported EOS; then the MW-R will become less tractable if the underlying EOS tends to be more complicated. Before we will describe the MW-KR, we will first briefly recount the derivation of the original KR-FMF, which is a little different from that of the R-FMF and from which one will intuitively arrive at the MW-KR through recourse to the experience of the WDA applications. By tailoring Φ({n α (r)}) on its uniform Percus Yevick (PY) or scaled particle limit, and choosing the same normalization conditions as Eqs. (4) and (5), Kierlik and Rosinberg [3] found that the first four scalar weighting functions ω (α) i (α = 0, 1, 2, 3) are sufficient to recover the hard sphere PY bulk second order direct correlation function (DCF) C (2) 0 hs PY ij (r) by functional differentiation of βf ex [{ρ i }], i.e. Kierlik and Rosinberg simply chose the function form of the scalar part of Φ({n α (r)}) as that of the PY compressibility excess Helmholtz free energy density (divided by kt) expressed in terms of scaled particle variables; consequently, Φ entering the KR-FMF is free of vector-like contribution and only comprises a single scalar term. The concrete function forms of ω (α) i (α = 0, 1, 2, 3) for recovering C (2) 0 hs PY ij (r) in the uniform limit of the KR-FMF are different from those of the R-FMF and repeated as follows ω (3) i (r) = Θ(R i r), ω (2) i (r) = δ(r i r), ω (1) i (r) = (1/8π)δ (R i r), ω (0) i (r) = (1/8π)δ (R i r) + (1/2πr)δ (R i r). (7) It should be pointed out that the differential of δ should be done with respect to the argument of δ. It is noted that any FMF should be guaranteed to recover the exact low density limit; the KR-FMF provides a deconvolution of the Mayer function different from that of the Rosenfeld FMF, but of course obey the exact low density result for the βf ex [{ρ i }]. By above brief recapitulation of the KR-FMF derivation, one realizes that the derivation of the KR-FMF is closer to that of the WDA, and somewhat deviates from that of the original R-FMF. The basic procedure is choosing the appropriate free energy expression, and specifying the weighting functions to recover the bulk 2nd-order DCF corresponding to the chosen free energy expression. The only distinction is that the free energy is expressed in terms of SPT variables, and accordingly four weighting functions are needed to produce four weighted densities. Then, it is reasonable to inspect and improve the KR-FMF based on experience of the WDA applications. Experiences of the WDA tell us that improvement on any single part of an approximate scheme will, broadly speaking, help in accuracy provided that all parts of the approximate scheme are reasonable. One of examples is a so-called BDFA [17] whose performance for inhomogeneous hard sphere fluid improves once the PY bridge functional is substituted by a Verlet-modified (VM) bridge functional, even if one uses the C (2) 0 hs PY ij (r) to calculate the inhomogeneous indirect correlation function. The current situation is similar with that of the BDFA: the weighting functions Eq. (7) are of course reasonable as they lead to the reasonable C (2) 0 hs PY ij (r) in the uniform limit and the KR-FMF is actually equivalent to the long-tested R-FMF; then replacing the reasonable Φ PY by any other more accurate Φ should help in improving the performance of the approximation considered. Based on the above understanding, the present author [14] attempted replacing Φ PY by a more accurate one Φ MCSL derived from the MCSL EOS, [9] and at the same time the original four scalar weighting functions ω (α) i (r) (α = 0, 1, 2, 3) is preserved intact. Such a naive substitution indeed induces an undesirable nonself-consistency between the PY type weighting function and the MCSL type excess free energy density, but it has no bad effect on the performance [14] of the resultant enhanced KR-FMF, which on the one hand, preserves the exact low-density limit of the original KR-FMF only if the externally imported free energy density reduces to the Φ PY in the zero density limit, on the other hand, improves significantly, as expected, the predictions of density profile of a single component hard sphere fluid at single hard wall contact location and its vicinity, and of the bulk hard sphere 2nd-order DCF, obtained from functional differentiation. There are also such problem as pressure inconsistency derived from the MW-KR, but the degree of pressure inconsistency obviously and uniquely depends on the imported EOS itself; however, even if one takes into consideration that the MW-R employs the same structure marked by Eq. (2) for the free energy density Φ, and the two vector weighted densities vanish in bulk limit, existence or not of such dependence is not obvious for the MW-R since the relevant Φ is obtained by solving the differential equation.

4 No. 1 Communications in Theoretical Physics 49 As a comment, one notes that in the MW-KR one simply substitutes Φ PY by chosen more accurate expression for the free energy density and keeps all others constant, this is in contrast with the practice of MW-R in which more accurate expression for the bulk pressure is employed as input, and the new expression for the free energy density is obtained by solving the differential equation. Since the MW-KR gets around solving the probably very complicated differential equation, the naive substitution makes easy application of any complicated EOS to formulate enhanced version of the KR-FMF. To summarize, the MW-KR is given by Eqs. (1), (3), (7) with the subscript α confined to 0, 1, 2, 3, and the free energy density Φ being specified by any reliable EOS. For a single component HS fluid, the validity of the MW-KR is confirmed [14] in the case of the MCSL EOS, now the validity in case of binary and ternary HS mixtures will be checked. The verification procedure is indispensible since the derivation of the MW-KR is based on experience, it belongs to reasonable ratiocination in mathematics. 3 Enhanced KR-FMF With the aid of the just discussed MW-KR, we input respectively a recent EOS stemming from a third order expansion [15] and a previous EOS coming from a Boublík s extension [12] of Kolafa s EOS [13] for HS mixtures into the KR-FMF. The 3rd-order EOS provides an expression for the free energy density Φ as follows [15] here ( Φ 3rd = Φ SPT + Φ 4 (n 3 ) 1 18π n3 2 n n 1n 2 n Φ 5 (n 3 ) 1 36π (n3 2n πn 1 n 2 n 3 3), ) (8a) Φ SPT = n 0 ln(1 n 3 ) + n 1n 2 (1 n 3 ) + n π(1 n 3 ) 2, (8b) Φ 4 (n 3 ) = (3/2)n2 3 n 3 (1 n 3 ) 2 ln(1 n 3 ) n 3 3 (1 n 3) 2, (8c) Φ 5 (n 3 ) = (9/2)n2 3 n3 3 3n 3 3(1 n 3 ) 2 ln(1 n 3 ) n 4 3 (1 n 3) 2 whereas the BK EOS specifies the Φ as below, (8d) Φ BK = n 0 ln(1 n 3 )+ n 1n 2 (1 n 3 ) + n 3 2Φ 3 (n 3 ) 108πn 2 3 (1 n 3) 2, (9a) here Φ 3 (n 3 ) = 8(1 n 3 ) 2 ln(1 n 3 )+8n n2 3 +2n3 3. (9b) Then, the two enhanced KR-FMF versions result when one substitutes respectively Φ 3rd or Φ BK into Eq. (1) and employs the Eqs. (3) and (7) for calculation of n α (r) occurring in Eq. (1). We will check the validity of the MW-KR in the case of binary and ternary hard sphere mixtures. According to routine, one will calculate the component density profiles and compare them with the corresponding computer simulation data available in literature. Given the functional approximation for F ex [{ρ i }], one will make use of a variational principle for the grand potential to acquire the density profile equation. The grand potential Ω[{ρ i }] is related to the Helmholtz free energy functional F[{ρ i }] via a Legendre transform: n Ω[{ρ i }] = F[{ρ i }] + drρ i (r)(ϕ i (r) µ i ), (10) i=1 where µ i is chemical potential of component i in the coexistence bulk fluid, ϕ i (r) is the external potential imposed on the component i. F[{ρ i }] is de-coupled into an idea part F id [{ρ i }] and an excess part F ex [{ρ i }]: F[{ρ i }] = F id [{ρ i }] + F ex [{ρ i }]. (11) The ideal part is exactly given by n βf id [{ρ i }] = drρ i (r)(ln(λ 3 i ρ i(r)) 1). (12) i=1 F ex [{ρ i }] represents all of many body correlation effects originating from the inter-particle interactions, and is not known exactly; for the case of hard sphere mixtures, F ex [{ρ i }] is given by Eqs. (1), (3), and (7) combined with Eq. (8) or Eq. (9). Minimizing the grand potential Ω[ρ] gives the density profile equation ρ i (r)=ρ i b exp{ βϕ i(r)+c (1) i (r; [{ρ i }]) C (1) 0i ({ρi b })}, (13) where ρ i b is number density of the coexistence bulk fluid, the non-uniform first order DCF C (1) i (r; [{ρ i }]) is mathematically 1st-order functional derivative of F ex [{ρ i }] w.r.t ρ i (r) C (1) i (r; [{ρ i }]) = δβf ex[{ρ i }] δρ i (r) = 3 α=0 dr Φ3rd or BK ({n α (r )}) n α (r ) ω (α) i ( r r ). (14) Taking uniform limit {ρ i } {ρ i b }, C(1) i (r; [{ρ i }]) reduces to the uniform first order DCF C (1) 0i ({ρi b }) 3 C (1) Φ 3rd or BK ({n α }) 0i ({ρi b})= n α {nα} {ξ R(α) i.(15) α} α=0 According to routine, we will mainly employ a single planar hard wall as external field for aim of verification, this external field with the wall situated at z = 0 is given as below { ϕi (z) = 0, R i < z, (16), otherwise. It is shown that the present two enhanced KR-FMF versions accurately satisfy the contact theorem ρ(r i ) = βp, (17) i here, βp entering the Eq. (17) is exactly that one stemming from the EOS underlying the functional in question.

5 50 Communications in Theoretical Physics Vol. 55 In the present context, the BK EOS for HS mixtures is given by βp BK = n 0 1 n 3 + n 1n 2 (1 n 3 ) 2 + n3 2(9 2n 3 2n 2 3) 108π(1 n 3 ) 3, (18) whereas Φ 3rd ({n α }) corresponds to an EOS as below βp 3rd = n 0 + n 1n 2 (1 + (1/3)n 2 3 ) 1 n 3 (1 n 3 ) 2 + n3 2 (1 (2/3)n 3 + (1/3)n 2 3 ) 12π(1 n 3 ) 3. (19) Fig. 2 The same as in Fig. 1 but different parameter values for the coexistence bulk fluid. Fig. 1 Normalized density profiles for binary hard sphere mixtures near a planar hard wall obtained from computer simulation [18] (symbols) and the present enhanced KR-FMF-3rd-order EOS (solid lines) and enhanced KR-FMF-BK EOS (dashed lines); the parameter values for the coexistence bulk fluid are indicated in the figure. The density profiles due to the two enhanced KR-FMF are presented in Figs. 1 and 2 for binary hard sphere mixtures near a single planar hard wall, the calculation are performed for four particular state points. The binary hard sphere mixtures are completely characterized by the hard sphere diameter ratio α = σ 1 /σ 2, the concentration x = x 2, and a total packing fraction η, related to the bulk density ρ b according to the relation η = π 6 [x + (1 x)α3 ]ρ b σ 3 2, (20) here, we follow the convention σ 1 < σ 2 for binary mixtures. It is shown that the two enhanced KR-FMF predict very similar density profiles, which are almost indistinguishable on the figure scale and very accurately reproduce the computer simulation results [18] except for one state point for which the deviation between predicted density profiles and exact computer simulation results are clearly obvious particularly for the region near the wall; checking Fig. 7 in Ref. [18] one observes that another two liquid other theories termed as PY1 and PY2 theories and the original KR-FMF [3] also exhibit large deviation from the computer simulation results for this particular state point. Considering that our predictions very accurately satisfy the contact theorem and the PY compressibility EOS underlying the original KR-FMF should not induce so large pressure errors for such small packing fraction, probably the reported simulation results for this particular state point contain obvious error; the unexpected anomaly motivates publication [11] of new and more accurate computer simulation results for similar state points, we will come back to this issue a little later. Although we have not displayed the results of PY1, PY2 and the original KR-FMF in the present figures for aim of clarity, the coordinate axis of the present figures and those in Ref. [18] 18 have the same scale, one can clearly discover that the two enhanced KR-FMF versions outperform the original KR-FMF and the PY1 and PY2 theories. Now what we face with is to choose one of the two enhanced KR-FMFs

6 No. 1 Communications in Theoretical Physics 51 for further testing and future s application; obviously one should choose the version with lower pressure inconsistency on the assumption that the two enhanced versions are actually indiscernible at least for the four state points examined. In Fig. 3 the absolute value of percent relative error between βp BK or βp 3rd and the corresponding βp from relation Eq. (7) are presented for three sets of parameters specifying the ternary hard sphere mixtures whose definitions will be given a little later; it is easily found that generally the pressure inconsistency relevant to the 3rd-order EOS is lower than that inherent in the BK EOS. Consequently, we will employ the 3rd-order EOS for use in the later part of this paper. hard sphere mixtures is only concerned with simple sum over the component index. Fig. 4 (a) Density profiles of binary hard sphere mixture near a planar hard wall, the parameter values for the coexistence bulk fluid are indicated in the figure; the symbols are for computer simulation [19] and solid lines the present enhanced KR-FMF-3rd-order EOS. (b) Concentration profile of the relevant small size component. Fig. 3 Absolute value of percent relative error (PRE) for deviation from the Eq. (7); solid lines are for the 3rdorder EOS and dashed lines the BK EOS, the parameter values for the bulk fluid are indicated in the figures. In Figs. 4 and 5, the predictions from the enhanced KR-FMF-3rd-order-EOS are compared with the simulation results [19] for binary hard sphere mixtures at two different state points, the calculated quantities include density profiles of each component and concentration profiles of the smaller spheres; these situations had previously been calculated with a hybrid WDA. [20] By comparison with Figs. 1 and 2 in Ref. [20] one finds that the advantage in accuracy of the enhanced KR-FMF-3rd-order-EOS over the hybrid WDA is evident. As for the computational intensity, although the hybrid WDA uses a globally averaged density as argument of the involved weighting functions, it is obviously still more time-consuming than the FMF; consideration of computational complexity is completely in favorable of the FMF whose extension to υ-component Fig. 5 The same as in Fig. 4 but different parameter values for the coexistence bulk fluid.

7 52 Communications in Theoretical Physics Vol. 55 Contents in Figs. 6 8 are similar to those in Figs. 4 and 5 except that the simulation results are supplied earlier in Ref. [21] for three different state points, and the previous theoretical results are with a third order perturbation DFT approach. [22] Comparing the present Figs. 6 8 with Figs. 1 3 in Ref. [22] one finds that the advantage in accuracy of the present approach over the previous approach is displayed even more obviously than that in the case of the present Figs. 4 and 5 and the hybrid WDA. [20] An anomaly occurs for the state point in Fig. 7; as one observes, this anomaly is very similar to that in the present Fig. 2(a), we will employ recent and more accurate simulation results for similar bulk state points to continue our verification procedure. Fig. 6 (a) Density profiles of binary hard sphere mixture near a planar hard wall, the parameter values for the coexistence bulk fluid are indicated in the figure; the symbols are for computer simulation [21] and solid lines the present enhanced KR-FMF-3rd-order EOS. (b) Concentration profile of the relevant small size component. Fig. 8 The same as in Fig. 6 but different parameter values for the coexistence bulk fluid. Fig. 7 The same as in Fig. 6 but different parameter values for the coexistence bulk fluid. Figures 9 11 are also for binary hard sphere mixtures near a planar hard wall, but the diameter ratio α = 1/3 is fixed, and the large size component is in very dilute state such that both components occupy a similar volume, this is somewhat similar to the situation in Fig. 2(a) except that the concentration of large size component is more diluted in the present case. Figures 9 11 successively describe increasing values of the total bulk packing fraction η, the simulation results are reproduced from Ref. [11], the predictions from the present approach are displayed for comparison aim. Although we have not displayed the results of the enhanced R-FMF-BK-EOS in the present Figs. 9 11, the coordinate axis of the present figures and

8 No. 1 Communications in Theoretical Physics 53 those in Ref. [11] have the same scale, one can clearly discover that the two enhanced version, i.e. the present one based on the MW-KR and that one in Ref. [11] based on the MW-R, are, in predicting density profiles, almost indistinguishable on the figure scale and both very satisfactorily reproduce the computer simulation results; however, it was confirmed in Ref. [11] that the R-FMF-BK-EOS is faintly more accurate than the R-FMF-MCSL-EOS in the region near the hard wall for a binary mixture at a less harsh coexistence condition. It was shown [11] that the enhanced R-FMF-BK EOS is obviously more accurate than the PY theory for the cases of Figs. 9 11, whereas a rational function approximation, although overall accurate, may predict unphysical negative values in the vicinity of the first minimum. Fig. 9 Normalized density profiles of binary hard sphere mixture near a planar hard wall, the symbols are for computer simulation [11] and solid lines the present enhanced KR-FMF-3rd-order EOS. The coexistence bulk fluids are for a fixed diameter ration of α = 1/33, the total packing fraction is close to η = 0.2. Top panels, η = and x 2 = 0.063; middle panels, η = and x 2 = 0.042; bottom panels, η = and x 2 = The insets provide a wider range and more details of the structures.

9 54 Communications in Theoretical Physics Vol. 55 Fig. 10 The same as in Fig. 9 but the total packing fraction is close to η = 0.3. Top panels, η = and x 2 = 0.062; middle panels, η = and x 2 = 0.037; bottom panels, η = and x 2 = Now we investigate the ternary hard sphere mixture confined within two hard walls with an inter-wall separation H, the corresponding ϕ i (z) is given as { ϕi (z) = 0, R i < z < H R i, (21), otherwise. The ternary hard sphere mixture is completely characterized by a diameter ratio α i = σ i /σ 2, a component concentration x i, and the total bulk packing fraction η, which is related to the bulk density ρ b according to the relation η = π 6 ρ bσ x i α 3 i, (22) i=1 here, the component diameter σ i follows the convention σ 1 < σ 2 < σ 3. In Ref. [23] three systems are simulated, of the same total bulk packing fraction η = 0.337, and other parameters given by (system 1) x 1 = x 2 = x 3 = 1/3, and σ 1 = 0.4σ, σ 2 = σ, σ 3 = 1.6σ, (system 2) x 1 = x 2 = x 3 = 1/3, and σ 1 = 0.5σ, σ 2 = σ, σ 3 = 1.5σ, and (system 3) x 1 = 0.6, x 2 = 0.2, x 3 = 0.2, and σ 1 = 0.5σ, σ 2 = σ, σ 3 = 1.5σ. For all three systems H is fixed at

10 No. 1 Communications in Theoretical Physics 55 12σ. In Figs the present predictions are presented together with corresponding simulation results reported in Ref. [23], which also reported the corresponding predictions based on a self-consistent DFA approach. Although the previous predictions are not displayed in the present figures, the coordinate axis of the present figures and those in Ref. [23] have the same scale, one can easily have a comparison between the two theoretical routes. One can find that the enhanced KR-FMF-3rd-order-EOS is a little inferior to the self-consistent DFA approach at contact point with the wall. Since the wall separation H is so large that the present density profiles are almost identical with the density profiles due to a single hard wall, one can think that the inferiority of the enhanced KR-FMF-3rd-order- EOS stems from an inaccuracy of the underlying EOS. It should be pointed out that the self-consistent DFA approach heavily depends on the OZ integral equation theory as input; for highly asymmetrical hard sphere mixtures, the OZ theory will become less satisfactory. On the contrary, there are publications, [24] which indicated that the FMF can be used as bridge function approximation to the OZ IE theory for highly asymmetrical hard sphere mixtures, and the results are satisfactory. Fig. 11 The same as in Fig. 9 but the total packing fraction is close to η = 0.4. Top panels, η = and x 2 = 0.062; middle panels, η = and x 2 = 0.036; bottom panels, η = and x 2 =

11 56 Communications in Theoretical Physics Vol. 55 Fig. 12 Density profiles of a ternary hard sphere mixture confined between two planar hard walls separated by an interval H = 12σ. The coexistence bulk fluid is characterized by x 1 = x 2 = x 3 = 1/3, σ 1 = 0.4σ, σ 2 = σ, σ 3 = 1.6σ, and η = The symbols refer to the computer simulation results [23] and solid lines the present enhanced KR-FMF-3rd-order EOS. Fig. 14 Density profiles of a ternary hard sphere mixture confined between two planar hard walls separated by an interval H = 12σ. The coexistence bulk fluid is characterized by x 1 = x 2 = x 3 = 1/3, and σ 1 = 0.5σ, σ 2 = σ, σ 3 = 1.5σ, and η = The symbols refer to the computer simulation results [23] and solid lines the present enhanced KR-FMF-3rd-order EOS. Fig. 13 Concentration profiles of the small- and medium-sized hard spheres for a ternary hard sphere mixture confined between two planar hard walls separated by an interval H = 12σ. The key is the same as in Fig. 12. Fig. 15 Concentration profiles of the small- and medium-sized hard spheres for a ternary hard sphere mixture confined between two planar hard walls separated by an interval H = 12σ. The key is the same as in Fig. 14.

12 No. 1 Communications in Theoretical Physics 57 Fig. 16 Density profiles of a ternary hard sphere mixture confined between two planar hard walls separated by an interval H = 12σ. The coexistence bulk fluid is characterized by x 1 = 0.6, x 2 = 0.2, x 3 = 0.2, and σ 1 = 0.5σ, σ 2 = σ, σ 3 = 1.5σ, and η = The symbols refer to the computer simulation results [23] and solid lines the present enhanced KR-FMF-3rd-order EOS. Fig. 17 Concentration profiles of the small- and medium-sized hard spheres for a ternary hard sphere mixture confined between two planar hard walls separated by an interval H = 12σ. The key is the same as in Fig Conclusion The present investigation indicates that the naive substitution for the enhanced KR-FMF is valid for binary and ternary hard sphere mixtures, the pressure inconsistency in the enhanced KR-FMF only depends on the inputted EOS for the hard sphere mixtures. An inevitable problem associated with the previous MW-R is that a differential equation has to be solved to specify the unknown functions f 1 f 2 f 4 occurring in the expression for Φ to be used in the enhanced R-FMF, the solving of differential equation will become less tractable when more complicated HS EOS is involved. It was indicated [15] that lowering the pressure inconsistency is a viable route to increase the accuracy of the HS EOS, but this undoubtedly will lead to complicated expression for the HS EOS, and consequently possibly causes difficulty in their application in the MW-R. Fortunately, the present MW-KR is not plagued by this undesirable property, the naive substitution allows for any complicated HS EOS to be incorporated into the enhanced KR-FMF since no any differential equation needs to be solved in the MW-KR; therefore, the MW-KR is more applicable to potentially complicated hard sphere EOS than the MW-R. Excluding the abnormal situations displayed in Figs. 2(a) and 7(a) whose existence is not indisputable according to point of view of Roth and Dietrich, [25] in normal circumstances observable small deviation between the present KR-FMF-3rd-order- EOS and computer simulation at contact point with the wall indicates that there still needs for improvement of the underlying HS 3rd-EOS. For high enough coexistence bulk density in the case of single component hard sphere fluid, Ref. [11] indicated that the R-FMF-BK practically reproduces the original R-FMF results as well as the R- FMF-MCSL results except for a narrow region adjacent to the wall where the R-FMF-BK shows a minor improvement over the R-FMF-MCSL, which however significantly improves upon the original R-FMF; although the relevant results of the present KR-FMF-3rd-order-EOS are not presented in the present paper, we note that it almost reproduces the results of the R-FMF-BK over whole interfacial region as occurs in the present Figs It should be pointed out that as observed in Ref. [11], for this extreme situation, significant deviations occur at the first minimum and second maximum between all the considered FMFs in Ref. [11] and the corresponding simulation data, we note that this is also common to our enhanced KR-FMF-3rd-order-EOS. Existence of such evident deviations possibly displays the inherent defect of the FMF. In fact, the FMF is also one kind of WDA, in bulk limit the original R-FMF reduces to the SPT in thermodynamics and PY structure functions, both of which are approximate. As a result, the original R-FMF and its various approximate revised versions are also approximate. The success achieved with the FMF might stem from the appropriate use of the scale conception; the obvious defects

13 58 Communications in Theoretical Physics Vol. 55 displayed in the case of highly oscillatory density profile of fluid state and use of tensor weighted densities in the case of solid state, [26] often regarded as an extreme example of fluid state, imply that there is a necessity to introduce new concept for an essential improvement! Acknowledgments It is the author s great pleasure to thank the anonymous referee for valuable comments. References [1] V.B. Warshavsky and X.Y. Song, Phys. Rev. E 77 (2008) ; R. Roth, M. Rauscher, and A.J. Archer, Phys. Rev. E 80 (2009) ; J.F. Lutsko, J. Chem. Phys. 129 (2008) ; P. Bryk and L.G. MacDowell, J. Chem. Phys. 129 (2008) ; J.A. Capitan, Y. Martinez-Raton, and J.A. Cuesta, J. Chem. Phys. 128 (2008) ; V. Botan, F. Pesth, T. Schilling, and M. Oettel, Phys. Rev. E 79 (2009) [2] Y. Rosenfeld, Phys. Rev. Lett. 63 (1989) 980. [3] E. Kierlik and M.L. Rosinberg, Phys. Rev. A 42 (1990)3382; S. Phan, E. Kierlik, M.L. Rosinberg, B. Bildstein, and G. Kahl, Phys. Rev. E 48 (1993) 618. [4] Y. Rosenfeld, M. Schmidt, M. Watzlawek, and H. Lowen, Phys. Rev. E 62 (2000) [5] J.K. Percus, In: H. L. Frisch, A.L. Lebowitz, (eds) The Equilibrium Theory of Classical Fluids, Benjamin, New York (1964) p [6] H. Reiss, H.L. Frisch, and J.L. Lebowitz, J. Chem. Phys. 31 (1959) 369. [7] S. Nordholm, M. Johnson, and B.C. Freasier, Aust. J. Chem. 33 (1980) 2139; M. Johnson and S. Nordholm, J. Chem. Phys. 75 (1981) 1953; P. Tarazona, Phys. Rev. A 31 (985) 2672; W.A. Curtin and N.W. Ashcroft, Phys. Rev. A 32 (985) [8] R. Roth, R. Evans, A. Lang, and G. Kahl, J. Phys.: Condens. Matter 14 (2002) [9] G.A. Mansoori, N.F. Carnahan, K.E. Starling, and T.W. Leland, J. Chem. Phys. 54 (1971) [10] N.F. Carnahan and K.E. Starling, J. Chem. Phys. 51 (1969) 635. [11] A. Malijevský, J. Chem. Phys. 125 (2006) ; A. Malijevský, S.B. Yuste, A. Santos, and M. López de Haro, Phys. Rev. E 75 (2007) [12] T. Boublík, Mol. Phys. 59 (1986) 371. [13] T. Boublík and I. Nezbeda, Collect. Czech. Chem. Commun. 51 (1986) [14] S.Q. Zhou, Submitted to Commun. Theor. Phys. 54 (2010) [15] H. Hansen-Goos and R. Roth, J. Chem. Phys. 124 (2006) [16] A. Santos, S.B. Yuste, and M. López de Haro, Mol. Phys. 96 (1999) 1. [17] S.Q. Zhou and E. Ruckenstein, J. Chem. Phys. 112 (2000) [18] J. Noworyta, D. Henderson, S. Sokolowski, and K.Y. Chan, Mol. Phys. 95 (1998) 415. [19] S. Sokolowski and J. Fischer, Mol. Phys. 70 (1990) [20] S.C. Kim, C.H. Lee, and B.S. Seong, Phys. Rev. E 60 (1999) [21] Z. Tan, U. Marini Bettolo Marconi, F. van Swol, and K.E. Gubbins, J. Chem. Phys. 90 (1989) [22] N. Choudhury and S.K. Ghosh, J. Chem. Phys. 110 (1999) [23] C.N. Patra and S.K. Ghosh, J. Chem. Phys. 118 (2003) [24] A. Ayadim, J.G. Malherbe, and S. Amokrane, J. Chem. Phys. 122 (2005) [25] R. Roth and S. Dietrich, Phys. Rev. E 62 (2000) [26] P. Tarazona, Phys. Rev. Lett. 84 (2000) 694.