Calculations of free energies in liquid and solid phases: Fundamental measure density-functional approach

Transcription

1 PHYSICAL REVIEW E 69, (2004) Calculations of fee enegies in liquid solid phases: Fundamental measue density-functional appoach Vadim B. Washavsky Xueyu Song Ames Laboatoy Depatment of Chemisty, Iowa State Univesity, Ames, Iowa 50011, USA (Received 6 Febuay 2004; published 28 June 2004) In this pape, a theoetical desciption of the fee enegies coelation functions of had-sphee (HS) liquid solid phases is developed using fundamental measue density-functional theoy. Within the famewok of Weeks-Chle-Andesen petubation theoy, fee enegies of liquid solid phases with many inteaction potentials can be obtained fom these chaacteistics of the HS system within a single theoetical desciption. An application to the Lennad-Jones system yields liquid-solid coexistence esults in good ageement with the ones fom simulations. DOI: /PhysRevE PACS numbe(s): h I. INTRODUCTION Fee enegy is an impotant themodynamical chaacteistic of condensed matte systems. If the dependence of feeenegy on tempeatue the bulk density is known many othe themodynamic quantities of the system, such as chemical potential, pessue, compessibility, can be calculated. It also becomes possible to study phase behavios, citical tempeatues, intefacial popeties, so on. Calculations of fee enegies using numeical expeiments: Monte Calo (MC) o molecula dynamics (MD) simulations yield accuate esults, but still ae timeconsuming pocesses [1]. Petubation theoetical appoaches to the calculations of fee enegies in fluids [2 8] solids [9,10] pesent an attactive altenative. One of the most eliable petubation theoies fo fee-enegy calculations was developed by Weeks, Chle, Andesen (WCA) [2,] fo Lennad-Jones (LJ) fluid was late genealized to fluids nea the feezing line [8] also to solids [9,10]. In the famewok of WCA theoy the fee enegy is sepaated into two contibutions: one of them is the fee enegy of an appopiate efeence system wheeas the second one is the petubative pat. In geneal, a had-sphee (HS) system with an appopiate HS diamete is chosen as the efeence system, the coelation function of that HS system is also utilized in the petubation calculations. A ecent vesion of WCA theoy [10] allows to calculate the fee enegies both of liquid of solid phase within a unified famewok. It has been successfully applied to the fee-enegy calculations of liquid solid phases with diffeent types of intemolecula potentials [10 12]. In those studies popeties of the HS efeence system wee obtained fom MC o MD simulation esults. Fo example, HS adial distibution functions of liquid solid phases ae paametized using simulation data [9,11,1,14], so ae the HS fee enegies of liquid solid phases [15,16]. Such a stategy is quite successful fo a single component system, but the paametization of multicomponent systems will become extemely deming theoetical appoaches, such as integal equations [17], do not yield accuate esults as the simulations yet. In the pesent wok we developed a unified theoetical appoach of calculating the fee enegies of liquid solid phases using a fundamental measue density-functional theoy. Namely, we can obtain the popeties of liquid solid phases fo a HS efeence system within a single theoetical famewok. At the same time, ou appoach can be genealized to multicomponent systems since the fundamental measue densityfunctional theoy is known to povide accuate esults fo mixtues [18,19]. This pape is oganized as following. The modified WCA petubation theoy fo the fee enegies of liquid solid phases is pesented in Sec. II to set up the stage fo ou theoetical development. The fundamental measue densityfunctional theoy is descibed in Sec. III applied in Sec. IV to calculate the fee enegies coelation functions of HS liquid solid phases. The esults of Sec. IV ae used in Sec. V to compute the fee enegies of Lennad-Jones liquid solid phases. Some conclusions ae given in Sec. VI. II. PERTURBATION THEORY FOR THE FREE ENERGIES OF LIQUID AND SOLID PHASES In the famewok of WCA petubation theoy, the intemolecula isotopic pai potential V is divided into two pats: V = V 0 + V 1, whee V 0 is the shot-anged steeply epulsive pat 0, V 1 the weake long-anged attactive pat = V 1 B, V,. =V B, V 0 The fist one is teated as a efeence potential second one as a petubation. To teat liquid solid phases on the same footing, the paamete is chosen as [10] = * + sa c *. 4 Hee * is the distance whee the potential V has a minimum, a c is the neaest-neighbos distance of the solid lattice /2004/69(6)/06111(10)/$ The Ameican Physical Society

2 V. B. WARSHAVSKY AND X. SONG PHYSICAL REVIEW E 69, (2004) 1 1 s =0, , 1 2 1, 2. 5 In this equation 1 =0.97 c, 2 =1.01 c, c is the density whee * =a c.if 2 (the solids) then Eqs. (4) (5) give =a c.if 1 (the liquid case) this choice educes to the WCA sepaation = *. At high densities nea the feezing line an equilibium neaest-neighbo sepaation will lie close to =a c athe than = * due to the stong epulsion of othe molecules. Such a choice makes the ange of V 0 shink with density. It educes the coesponding HS diamete allows to avoid difficulty associated with the metastable HS fluid. The function B is equal to B = V dv. d = Moe details of the potential sepaation can be found in Ref. [10]. The fee enegy can be exactly expessed as [20] F = F d d 1 d 2 V , 2, F 0 is the fee enegy of the efeence system =0, is the coupling paamete fo the inteaction potential V;=V 0 +V , 2 ; is the pai distibution function when the potential is V 12 ; but the density is. To lowest ode appoximation, Eq. (7) yields F = F d 1 d 2 V , 2, which can be also witten as [21] F/N = F 0 /N dv 1 g 0, whee N is the numbe of paticles g 0 12 the angleaveaged coelation function in the efeence system 1 g 0 12 = d 4V 2 d , In the WCA famewok, the efeence system is futhe appoximated by a had-sphee system with the intemolecula pai potential = V HS +, d 11 0, d, whee d is a tempeatue dependent had-sphee diamete since a lot of infomation about the HS system was collected fom theoetical teatments numeical expeiments [17]. To appoximate the efeence system with the intemolecula potential V 0 by a HS system, the had-sphee diamete d is chosen to satisfy the equation (2) de V 0 e V HS y HS /d =0, 12 whee =1/k B T (k B is the Boltzmann constant, T tempeatue), y HS is the cavity function [11,22] of the HS system. With such a choice of HS diamete, the efeence system F 0 g 0 ae elated to the ones of the HS system as: F 0 =F HS +O 2 g 0=g HS +O. The small paamete is equal to d d B 1 2 d d e V 0, 1 =0 d B =0 d1 e V 0 14 is the Bake-Hendeson diamete [4]. Instead of Eq. (12) we will use a fast accuate but appoximate way of solving fo d with help of the fomula (1), whee d = d B , 0 = y HS = d, =2 0 + dy HS /dx x=/d=1 17 [Eq. (15) is obtained fom Eq. (12) as an expansion on the small paamete [1] by neglecting the extemely small high-ode tem of 0 d/d B 1 de V0 /d; fo example, fo Lennad-Jones system with tempeatues T * =0.75,2.74 this tem is espectively]. Using the above equations, we finally have F/N = F HS /N dv 1 g HS /d

3 CALCULATIONS OF FREE ENERGIES IN LIQUID AND PHYSICAL REVIEW E 69, (2004) To yield accuate esults of the fee enegy in Eq. (18), the HS fee enegy F HS coelation function g HS fo liquid solid phases ae obtained fom simulations. In this wok we will get the popeties of the HS efeence system by using a theoetical appoach athe than numeical simulations. Namely, we will use fundamental measue density functional to calculate the fee enegies coelation functions of both liquid solid phases within a single famewok. 2 y = d 2 y, y = d 2 y, v2 y = e y d 2 y, III. FUNDAMENTAL MEASURE DENSITY-FUNCTIONAL THEORY The density-functional theoy (DFT) is a poweful tool fo studying the popeties of fluids solids (see eview of Evans [20]). In the famewok of DFT, the had-sphee fee enegy F HS is consideed as a functional of numbe density pofile. It consists of two pats: an ideal-gas contibution F id the excess fee enegy F ex ove the ideal-gas pat whee F HS = F id + F ex, F id = k B T dv ln [In this wok, the k B TN ln k B T ln contibution to the fee enegy chemical potential espectively ae neglected since they do not affect ou esults besides a tempeatue dependent constant. is the themal de Boglie wavelength.] The g canonical potential functional HS can also be constucted HS = F HS d U, 21 whee is chemical potential, U is the extenal potential. The vaiational pinciple HS =0 22 detemines the equation fo the equilibium density pofile in an extenal potential. Vaious appoximations to the density functional have been poposed [2 27] to constuct the excess fee enegy F ex fo the HS system. In ou study we will apply the fundamental measue (FM) density functional poposed by Rosenfeld [28] extended by many othes [19,29 1]. Below we list the main equations of FMT fo one-component HS system used in this wok. Fist of all a set of weighted densities is defined n = d 2 with the density-independent weight functions given by ˆ y = e y e y d 2 y. 27 Hee, e y is a unit vecto along the y diection, the Heaviside step function, the Diac delta function, v2 ˆ ae the vecto tenso weighted densities, espectively. The excess pat of the fee enegy can be witten in the following fom: whee F ex = d i n, i=1 1 = n 2 d 2ln1 n, 2 = n n v2 2d1 n, = f n Vaious appoximations of FMT employed diffeent functions f in Eq. (1). The fist appoximation of f was given by Rosenfeld [28]: f V1 = n 2 2 n 2 n v PY 1 = 1 n 2, we will denote this vesion of the theoy as V1PY. In the homogeneous limit the functional Eqs. (28) () ecove the Pecus-Yevick excess fee-enegy density. Functional deivative of the F ex in the homogeneous limit yields the PY pai distibution function. In the 1D limit (had ods) it epoduced the exact density functional of 1D system. Although the functional is successful fo the desciption of liquids it completely failed to descibe liquid-solid coexistence. The poblem was ovecome patially in Ref. [29] with the intoduction

4 V. B. WARSHAVSKY AND X. SONG PHYSICAL REVIEW E 69, (2004) n 2 f V2 2 n 2 v2 = 4 24n 2 (denoted as V2PY). A systematic way to impove the Rosenfeld functional lies in the fact that in the 0D limit (a spheical cavity holds no moe than one paticle) the excess feeenegy functional should satisfy exactly known solution, which is cucial to descibe solids. Using this dimensional cossove agument, a new vesion (denoted as T1PY) [0] using f T1 = 9 16 detnˆ, 5 a vaiation (denoted as T2PY) [1] with f T2 = 16 n v 2 nˆn v2 n 2 n 2 v2 tnˆ + n 2 tnˆ 2 6 can be used to descibe many popeties of HS solids. Fo example, the vesion with = f T2 PY (T2PY) yields coect asymptotics fo the HS fcc lattice nea close packing density even can be used to descibe the metastable HS bcc lattice [1]. The contibution can also be impoved to descibe the fluid phase bette, namely, in the limit of bulk density the Canahan-Staling (CS) equation of fluid state should be epoduced athe than the PY equation of state. Recently [19,2,] this has been achieved by the intoduction of n CS = 2 1 n 2 1 n 2 +ln1 n. 7 T2 A combination of this function with f yields a new vesion of functional, = f T2 CS (denoted as T2CS) was discussed in detail [19,]. Fo instance T2CS vesion of FMT gave the best values of the coexisting HS liquid solid densities. This is the vesion that we will use to compute popeties of HS liquid solid phases. Some details of calculations of the weighted densities n can be found in Appendix A. IV. PROPERTIES OF LIQUID AND SOLID PHASES OF THE HARD-SPHERE SYSTEM A. Fee enegies coelation functions of HS solids In a solid the density pofile can be appoximated as a sum of identical Gaussian peaks centeed at the lattice sites R i, = R i = i /2 i e R i 2. 8 Given the weighted functions in Sec. III the weighted density can be witten as n = n R i. 9 i Moeove, the scala, vecto, tenso weighted densities n can be found in analytical foms (see Appendix B). FIG. 1. Contibutions of diffeent tems in the fee-enegy functional to the fee-enegy density F HS /V at d =1.05 using T2CS. Fo the fcc solid phase of a HS system, using the symmety of the cystal the volume of integation can be educed to a simplex coesponding to 1/48 of the unit cell [4]. The contibutions fom all 1 sites of the unit cell (fo fcc lattice) to the total weighted densities can be calculated. Fo a given bulk density the functional dependence of the solid fee enegy on the Gaussian paamete of density distibution [Eq. (8)] was plotted in Fig. 1. The minimum of this function gives the equilibium values of fee enegy paamete. Figue 1 illustates the diffeent contibutions fom the dependence of F HS /V on fo d =1.05 (whee F i =d i, i=1,2,). Fo all vesions of FMT except V1PY, Fig.1is qualitatively the same. The dependence of equilibium fee enegy F HS /N on the densities d is shown in Fig. 2. The calculated esults ae in excellent ageement with MC simulations [16]. The coelation function can be calculated using the appoach of Rascon et al. [5,6]. In contast to liquid phase, the stuctual popeties of a solid phase is mainly detemined by one paticle density. Let 2 be the density pobability of finding a paticle at 2 without fixing any othe paticles. The pobability density of finding a second paticle at 2 povided the fist paticle is aleady fixed at 1 is equal to 2 1, 2 / 1. Thus, two angula aveage pobability densities can be defined FIG. 2. Dependence of the fee-enegy density F HS /N on d. Results of the pesent theoy T2CS ae plotted as a solid line, MC simulation esults [16] as diamonds

5 CALCULATIONS OF FREE ENERGIES IN LIQUID AND PHYSICAL REVIEW E 69, (2004) FIG.. Details of the fist peak of the pai distibution function g fo the solid at d =1.04 fo MC simulation esults [11] diffeent vesions of the fundamental measue density-functional theoy (DFT). g 0 1 = 4V 2 d 1 d 1 1 +, 40 1 g = 4V 2 d 1 d 2 1, 1 +, 41 whee g 0 is the coelation function when the paticle at the oigin is not fixed g is the coelation function, which descibes the stuctue of solid in the extenal field of the paticle fixed at the oigin. It can be shown that the function g should be used fo the petubation calculations of the fee enegy in Eq. (18) [21]. Putting Eq. (8) into Eq. (40), the coelation function g 0 0 can be educed to the sum of contibutions g i fom the successive peaks of lattice sites [5] g 0 i = n i 4R i 2 1/2 e R i 2 /2 + e + R i 2 /2 i 0, 42 whee n i is the coodination numbe of shell i R i being the coesponding adius. Rascon et al. [5,6] showed that fo small value of compessibility T =/P T (which is the case fo HS solids) the successive peaks of g diffe negligibly fom the ones of g 0 except the fist peak with i=1, thus, g = g 1 + g 0 i. 4 i2 The fist peak of g (denoted as g 1) is due to the neaest neighbos which bea the most impotant pat of the shotange coelations can be paametized as [5] g 1 = A e /2, 44 whee the paametes A, 1, 1 can be found fom the sum ules conditions given below. FIG. 4. Pai distibution function g fo the solid at d =1.05. The solid line coesponds to the esult of the pesent theoy. The dashed line is the Monte Calo esults [11]. (1) Viial theoem: P/ =1+4g d HS. 45 (2) The nomalization of g 1 to the neaest-neighbos numbe: 1 n 1 dg 1 =1. () The mean location of the neaest neighbos : 1 n 1 dg 1 = 1 n 1 dg Using the aleady found dependence of pessue P on the density in the equations above all the successive peaks of g can be obtained. We have calculated the coelation functions using the diffeent vesions of the fundamental measue densityfunctional theoy. It is found (see Fig. ) that the esults of the T2 vesions of the theoy gave the best ageement between the contact value of the coelation functions fom ou calculation MC simulations [11]. Thus, we ae mostly inteested in the T2CS vesion of the theoy since it gives the best liquid solid popeties. Fo this vesion the esulting coelation functions agee vey well with the MC esults [11] fo the vaious densities (Figs 4 5). (Fo the low densities the compessibility T gets highe, thus, the sum ule [Eq. (47)] gets less accuate [6]; as a esult nea melting density the ageements get a bit wose. Thee ae FIG. 5. The same as Fig. 4 except d =

6 V. B. WARSHAVSKY AND X. SONG PHYSICAL REVIEW E 69, (2004) also no significant diffeences also if any othe DFT appoaches ae used.) Thus, all of ou late calculations will be based upon the T2CS vesion of the DFT theoy. B. Coelation functions in HS liquid In the homogeneous limit the T2CS vesion of the FM functional educes to the CS fee enegy [15], F HS /N =lnd , 48 whee =/6d is the packing faction. To descibe the HS liquid stuctue we note that in a liquid phase the density pofile 2 seen fom the oigin is just the bulk density. Fixing a test paticle at the oigin leads to the shot-anged density pofile [7] 2 1, 1 +/=g. It can be seen fom Eqs. (40) (41) that in the homogeneous limit the coelation function g 0 g in solid educe to 1 g, espectively. Hence the pai distibution function in a liquid g is equal to the density distibution of paticles in the extenal potential ceated by the test paticle fixed at the oigin divided by the bulk density [7], g = /. 49 The density distibution can be calculated fom the solution of the integal equation obtained fom the vaiational pinciple fo the g canonical potential HS, Eqs. (19) (22) (28) with = exp F ex / + U 50 =lnd U=V HS. Note that simila equation as Eq. (50) was used in Ref. [2], but fo a diffeent vesion of the Rosenfeld functional, namely fo one with = f V1 CS (vesion V1CS). The details fo the calculation of F ex / in Eq. (50) can be found in Appendix C. This integal equation (50) can be solved by the iteation method, i.e., at its i-step i+1 in = i out =  i in, 52 whee  is the integal opeato on the ight-h side of Eq. (50). Thee is a poblem fo the convegence of the method, namely the weighted function n at the small is highe than 1, a signatue that 1 tem is an appoximation in thee dimension. To avoid this we used a mixing scheme, i+1 in = i out + 1 i in, 5 with the small mixing paamete, which depends on the bulk density. To speed up dastically the convegency of iteations Eqs. (50), (51), (5), we developed the special optimization method. We have found that the coelation functions g fom the V1CS T2CS vesions of FMT fo FIG. 6. Pai distibution function g of liquid at densities d =0.7,0.8,0.9,1.0. The solid line is the esult of ou theoy, the dashed line is the esult of MC simulations [1]. Fo claity, cuves fo d =0.8,0.9,1.0 ae shifted upwads by 1,2,, accodingly. low densities d ae almost the same, with the some diffeence aisen fo the high densities. It is also seen fom Fig. 6 that fo all the densities that the DFT coelation functions ae in excellent ageement with the MC esults of Velet- Weis [1] (fo the high density d =1.0 the ageements get a bit wose). The pai diect coelation function c 2 in liquids can also be obtained as the second functional deivative of the excess fee enegy in the homogeneous limit. Since the pesent functional employs the weight functions of FMT based on one cente convolutions it cannot accounts fo the decaying tail of c 2 in the egion d. As a esult the function g obtained fom c 2 via the Onstein-Zenike equation does not yield esults which ae in good agement with the simulations. Possible impovement of the theoy equies the inclusion of the two-centes convolutions [19,]. V. LENNARD-JONES LIQUID AND SOLID COEXISTENCE In this section we will apply ou method to a simple model system with the Lennad-Jones (LJ) intemolecula potential, whose liquid-solid coexistence behavios ae well known fom simulations, V LJ = , 54 whee ae the enegy length paametes of the LJ potential. Using the coelation functions of the efeence HS solid liquid fom the DFT calculations the iteative solution of the equation fo the HS diamete d [Eq. (15)] can be found. Fo some given tempeatues T * =k B T/=0.75,2.74 vaious liquid solid densities * = the fee enegies F ex /N=F/N lnd +1 both in liquid solid phases wee obtained with the help of Eq. (18). Some of the esults ae given in Table I they ae in excellent ageements with the esults of MC simulations [8,10]. At the fixed tempeatue T the coexistence of solid liquid phases equied the equality of the chemical potentials pessues P in the both phases. The coexisting liquid * l solid * s densities can be calculated using the Maxwell double-tangent constuction, i.e., the condition when

7 CALCULATIONS OF FREE ENERGIES IN LIQUID AND PHYSICAL REVIEW E 69, (2004) TABLE I. Excess (with espect to an ideal gas at the same tempeatue density) fee enegy pe paticle F ex /N fo the Lennad-Jones system obtained fom MC simulations [8,10] the pesent theoy. Values of the HS packing faction =/6d obtained in this wok ae also given. F ex /N F ex /N MC simulations This wok T * =0.75 Liquid Solid T * =2.74 Liquid Solid F solid /N F liquid /N plotted vesus the invese densities have a common tangent. Fo T * =0.75 we found that the coexisting densities the Lindemann atio ae s * =0.959, l * = ; fo T * =2.74, s * =1.214, l * = , which ae vey close to the esults obtained by MC simulations [8] s * =0.97, l * =0.875, 0.15 s * =1.179, l * =1.11, 0.14 espectively. If diffeent theoies would be applied to compute F liquid /N F solid /N the diffeences in the models could affect significantly on the esults of coexisting densities. In ou pesent appoach the solid liquid fee enegies ae obtained within a single theoetical famewok, thus, such eos ae avoided. VI. CONCLUSIONS In the pesent study we have developed a theoy to calculate the fee enegies of liquid solid phases within a single theoetical famewok. To this end the fundamental measue DFT was applied to the theoetical calculations of the fee enegies coelation functions in HS liquid solid phases. These popeties of the HS systems wee used in the WCA petubation theoy to calculate the fee enegies both in solid liquid phases of the Lennad-Jones system. The obtained esults ae in a good ageement with simulations. The pesent study can be extended along seveal diections. Fo example, such a appoach can be used to compute phase behavios of simple metallic systems [12] also of the multicomponent mixtue (alloy) systems [9]. Anothe possible application of this appoach is the theoetical desciption of the solid-melting inteface [40,41] of the diffeent substances. ACKNOWLEDGMENTS This eseach was sponsoed in pat by the Division of Mateials Sciences Engineeing, Office of Basic Enegy Sciences, U. S. Depatment of Enegy, unde Contact No. W-7405-ENG-82 with Iowa State Univesity (V.B.W. X.S.) by NSF Gant No. CHE00758 (X.S.). APPENDIX A: WEIGHTED DENSITIES IN THE LIQUID At Appendixes A B we summaize some popeties of the weighted densities (pat of them was povided aleady somewhee [29,4]) also we will give some new details of calculations. To tansfom the expession fo the weighted densities [Eq. (2)] the vecto y = can be intoduced y 2 = 2 = cos ( is the angle between the vectos ). With the fixed = = it follows that y+ sin d=1/ydyy=y. Using this the thee-dimensional integal educes to two onedimensional integals n = d + yydy. A1 Using the Rosenfeld expessions fo the scala, vecto, tenso weight functions [Eqs. (24) (27)] all the inteested weighted densities can be found. Such, the scala weighted densities n 2 n educe to one-dimensional integals n 2 = d +d/2 d/2 d A2 n = +d/2 d d2 2 d/2 4 + d d/2 2 4 d 0 2. A To tansfom the expessions fo vecto tenso densities we wite

8 V. B. WARSHAVSKY AND X. SONG PHYSICAL REVIEW E 69, (2004) e y = cos e + sin e A4 ( is the angle between the vectos y; e e the unit vectos to be paallel pependicula to the diection). Using 2 = y 2 = 2 +y 2 2y cos we find that cos = y , 2y sin = 42 2 y /2. A5 2y Finally, putting Eqs. (26), (27), (A5) into Eq. (A1) the vecto n v2 tenso nˆ weighted densities can be found as nˆ = i=1 n v2 = n v2 e, nˆ ii = n ii e i e i, i=1 A6 whee e 1, e 2, e ae the unit othogonal vectos e =e +d/2 n v2 = 2 d/2 n 11 = n 22 = n = +d/2 d d/2 d d2 4, A7 +d/2 2 d d/2 d2 Fom Eqs. (2) (27) the elations, d , A8 d 2 d n v2 = n, tnˆ = n 2, A9 A10 A11 also follow, which ae useful to check the esults. In the bulk with = the expessions Eqs. (A2), (A), (A7) (A9) educe just to n 2 = d 2, n = 6 d, n v2 =0, n ij = d2 ij. A12 n 2 = d 2 e d/2 2 e d/2 + 2, n = 1 2ef d ef d 2 B1 2 e d/2 e d/ , B2 n v 2 = 1+e 2d 1 e 2d 1 dn 2, n 11 = n 22 = n v 2, d B n = n 2 2n v 2. B4 d The total scala, vecto, tenso weighted densities ae the sum of the contibutions fom the diffeent lattice sites [Eq. (9)], moeove, the vecto tenso contibutions must be tansfomed to a common efeence fame. To this end we intoduce the laboatoy-fixed fame e x,e y,e z elated to the cystal lattice planes. As a esult the thee position () dependent vectos e 1, e 2, e in Eq. (A6) can be tansfomed as e i = a i, e i = 1,2,; = x,y,z B5 with the tansfomation matix [42] y A = a i, = xz x x yz y 0 z, B6 whee =x 2 +y 2 1/2. Finally, in the efeence fame the contibutions fom one lattice site to vecto tenso densities with help of Eqs. (A6) (B) (B6) ae given by the expessions v n 2 v = n 2 v e = n 2 a, e, B7 APPENDIX B: WEIGHTED DENSITIES IN SOLID When the density is given by the Gaussian distibution aound the zeo =/ /2 exp 2 the coesponding weighted densities can be found analytically fom Eqs. (A2), (A), (A6) (A11), nˆ = n ii e i e i i=1 = n ii a i, a i, e e. i=1, B8

9 CALCULATIONS OF FREE ENERGIES IN LIQUID AND PHYSICAL REVIEW E 69, (2004) APPENDIX C: CALCULATION OF F ex / In this Appendix C we calculate F ex / which aises in Eq. (50). It can be ewitten as whee F ex / = F, F = d n. C1 C2 So, the poblem educes to the calculation of evey F.To do this, fist of all the expession + + F = 2 d 0 n y ydy C can be obtained fom Eq. (C2) by the same way as Eq. (2) was tansfomed to Eq. (A1). Fo the scala weight functions with =2, Eqs. (A2) (A) can be utilized fo F 2, F [in this equations the functions /n 2 o /n should be substituted instead ]. To calculate the est of the functions F we wite n v2 = n v2 e, C4 Next, as it was done at Eqs. (A4) (A5) the vecto e y can be witten as e y = cos e + sin e C6 ( is angle between the vectos y ), the expessions cos ˆ = y , 2y sin ˆ = 42 2 y /2 2y C7 can be found. Finally, putting Eqs. (26), (27), (C4) (C7) into Eq. (C) we have F v2 = d/2 F ii = +d/2 +d/2 2d d/2 n ii d d n v2, d d i =1,2 C8 C9 nˆ ii = n ii e ie i i = 1,2,, C5 whee e 1, e 2, e ae the unit othogonal vectos (e is diected along the vecto ). +d/2 F = d d/2 d 2 1 d n. C10 [1] D. Fenkel B. Smit, Undesting Molecula Simulation: Fom Algoithms to Applications, 1st ed. (Academic Pess, New Yok, 1996). [2] J. D. Weeks, D. Chle, H. C. Andesen, J. Chem. Phys. 54, 527 (1971). [] D. Chle, J. D. Weeks, H. C. Andesen, Science 220, 786 (198). [4] J. A. Bake D. Hendeson, Phys. Rev. A 1, 1266 (1970). [5] G. A. Mansooi F. B. Canfield, J. Chem. Phys. 51, 4958 (1969). [6] J. C. Rasaiah G. Stell, Mol. Phys. 18, 249 (1970). [7] M. Ross, J. Chem. Phys. 71, 1567 (1979). [8] H. S. Kang, C. S. Lee, T. Ree, F. H. Ree, J. Chem. Phys. 82, 414 (1986). [9] J.-J. Weis, Mol. Phys. 2, 296 (1976). [10] H. S. Kang, T. Ree, F. H. Ree, J. Chem. Phys. 84, 4547 (1986). [11] Y. Choi, T. Ree, F. Ree, J. Chem. Phys. 95, 7548 (1991). [12] X. Song J. R. Mois, Phys. Rev. B 67, (200). [1] L. Velet J. Weis, Phys. Rev. A 5, 99 (1972). [14] J. M. Kincaid J.-J. Weis, Mol. Phys. 4, 91 (1977). [15] N. F. Canahan K. E. Staling, J. Chem. Phys. 51, 65 (1969). [16] K. R. Hall, J. Chem. Phys. 57, 2252 (1972). [17] J. P. Hansen I. R. McDonald, Theoy of Symple Liquids (Academic, New Yok, 1986). [18] J. A. Cuesta, Y. Matinez-Raton, P. Taazona, J. Phys.: Condens. Matte 14, (2002). [19] R. Roth, R. Evans, A. Lang, G. Kahl, J. Phys.: Condens. Matte 14, 1206 (2002). [20] R. Evans, in Fundamentals of Inhomogeneous Fluid, edited by D. Hendeson (Wiley, New Yok, 1992). [21] C. Rascon, L. Medeos, G. Navascues, Phys. Rev. Lett. 77, 2249 (1996). [22] D. Hendeson E. W. Gundke, J. Chem. Phys. 6, 601 (1975). [2] T. V. Ramakishnan M. Yussouff, Phys. Rev. B 19, 2775 (1979). [24] P. Taazona, Phys. Rev. A 1, 2672 (1985). [25] W. A. Cutin N. W. Ashcoft, Phys. Rev. A 2,

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