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A generalized mixing rule for hard-sphere equations of state of Percus–Yevick type
Fluid Phase Equilibria 142 Ž1998. 131–147
A generalized mixing rule for hard-sphere equations of state of Percus–Yevick type Mohammad K. Khoshkbarchi, Juan H. Vera
)
Department of Chemical Engineering, McGill UniÕersity, Montreal, Quebec, Canada H3A 2B2 Received 25 May 1996; accepted 5 September 1997
Abstract An analogical approach has been used to extend to mixtures the radial distribution function of Percus–Yevick type of equations of state for hard spheres. The proposed approach follows the same formalism employed by Mansoori et al. to extend to mixtures the Carnahan–Starling equation of state. The generality of the proposed method permits to extend to mixtures equations of state with different singularities. In this work, it is applied to an equation of state which meets the correct low density and high density limits, previously proposed by Khoshkbarchi and Vera. The results show that both the radial distribution function and the resulting equation of state for a mixture of hard spheres, developed in this study, can accurately represent the computer simulation data while satisfying the correct limit at close-packing. q 1998 Elsevier Science B.V. Keywords: Hard spheres; Equations of state; Mixtures
1. Introduction Modelling of the properties of the systems of hard spheres is an essential element of most modern theories of liquids and liquid mixtures. Systems of hard sphere mixtures are the simplest nontrivial models which can mimic the properties of the real fluids. They also serve as the reference system in statistical mechanical theories of liquids such as the perturbation theory for liquids. Although, there are many studies reported in literature for the equations of state of hard spheres for pure liquids w1–3x, few studies were conducted to develop equations of state for the systems of mixture of hard spheres w4,5x. All of the equations of state for hard spheres reported in literature suffer from a major deficiency as they fail to meet the correct limit at close-packing, where the compressibility factor must tend infinity. These equations, although they may represent well the computer simulation data at low to moderate densities, predict a finite value at close-packed density and their singularity )
Corresponding author.
0378-3812r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 8 - 3 8 1 2 Ž 9 7 . 0 0 2 3 0 - 6
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
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corresponds to a physically unrealistic condition. This shortcoming, common to all equations of state derived from scaled particle theory w6x, is due to the use of spherical cells, which ignores the fact that when spherical cells are close-packed, there is a volume which is not available for occupation by the centre of the hard spheres. This unaccessible volume, although not important at low densities, has an important effect on the properties of hard sphere systems in the high density region. Recently Wang et al. w7x proposed a new form of volume dependence for the equation of state of hard spheres which has a pole at the close-packed limit while representing accurately the compressibility factor simulation data over a wide range of density. The new volume dependence is based on the definition of cells with tetrakaidecahedron geometry, with centres placed on the corners of a rhombohedral, and the probability of finding the centre of each of these cells occupied by the centre of a hard sphere. Since an array of cells with this geometry fills up all the space, the compressibility factor tends to infinity when the hard spheres are in the close-packed state. Based on this new volume dependence, Wang et al. w7x proposed three equations of state for hard spheres which accurately correlated the compressibility factor and reproduced well the virial coefficients of a hard sphere system, while fulfilling the close-packed limit condition. These equations were rather complicated for practical use, thus, we recently proposed a simplified equation of state for hard spheres, which meets the correct limit at close-packing w8x. In this study we propose a generalized mixing rule for the radial distribution function, from which the equation of state and other thermodynamic properties of a mixture of hard spheres can be obtained.
2. Thermodynamic framework In this study, one of the forms of the equation of state for pure compound hard sphere systems proposed by Wang et al. w7x is selected, and extended to a mixture of hard spheres. Among the three general forms of equations proposed by Wang et al. w7x, the following general Percus–Yevick type form is chosen:
Ý bi j i Zs
is0
Ž1 y j .
Ž1.
3
where Z is the compressibility factor, bi is a numerical parameter, with b 0 s 1. The dimensionless variable j is defined as:
js
V0
h s
V
h0
1 s
'2
rs 3
Ž2.
with
p hs
6
rs 3
Ž3.
and
h0 s
p'2 6
Ž4.
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where V is the total volume, V0 s Ns 3r62 is the volume of the system at the close-packing state of N spheres, r s N r V is the number density, and s is the diameter of the hard spheres. The term Ž1 y j . in the denominator of Eq. Ž 1. guarantees the correct limit, i.e., that at the close-packed state, where j s 1, the compressibility factor tends towards infinity. This form of equation has been chosen because it can represent the computer simulation data and the virial coefficients with less uncertainty than the other forms proposed by Wang et al. w7x. This form of equation of state of hard spheres is a generalization of both forms of the Percus–Yevick equation of state for hard spheres w9,10x and of the Carnahan–Starling equation w1x. The virial coefficients, for a series in terms of powers of h , generated by this general form of the equation of state of hard spheres, bi , can be calculated, for i G 1, according to the following relation: bi s Ž biy1 q 3 biy1h 0iy2 y 3 biy2 h 0iy3 q biy3h 0iy4 . rh 0iy1 Ž5. Ž . with b k s 0 for k F 0. The radial distribution function RDF , at contact value, can be calculated using the virial theorem. The equation of the virial theorem, at contact value, for hard spheres has the form w11x: Z s 1 q 4h 0 j g Ž s . Ž6. ( ) where g s is the RDF at contact value of the distance of separation between the centres of two hard spheres. By combining Eq. Ž6. with Eq. Ž 1. , the contact value radial distribution function associated with Eq. Ž1. is obtained as: 1 1q Ž b 2 y 3 . j q Ž b 3 q 1 . j 2 q Ý biq1 j i 4h 0 is3 gŽs . s Ž7. 3 Ž1 y j . In the derivation of Eq. Ž7. we have incorporated the known Žexact. values of the first two virial coefficients for hard spheres, i.e., b 1 s 1 and b 2 s 4. With these values, Eq. Ž5. gives b 1 s 4h 0 y 3 which generates the first term of the numerator of Eq. Ž 7. . Eq. Ž 7. is the general form of the RDF for equations of state of the Percus–Yevick form, i.e., Eq. Ž1.. For the purpose of extending its applicability to mixtures, it is convenient to re-write Eq. Ž 7. as: 1 G1 G2 gŽs . s q Ž8. 2 q 3 Ž1 y j . Ž1 y j . Ž1 y j . with 1 G1 s 8h 0 q b 2 y 3 j Ž9. 4h 0 and 1 G2 s Ž 4h0 q b 2 q b 3 y 2 . q Ý bi j iy3 j 2 Ž 10. 4h 0 is4 We have recently proposed a modified form of the Eq. Ž 1. for a pure compound hard sphere system w8x, which meets the correct limits and reproduces well the first six virial coefficients. It has the form: 1 2 5 9 71 1y j y j 2 y j 3 q j 5 q j 12 25 5 4 50 50 Zs Ž 11. 3 Ž1 y j .
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According to Eq. Ž7. , the contact value radial distribution function for Eq. Ž11. takes the form: 1q gŽs . s
1
17 y
4h 0
1 9 71 j y j 2 q j 4 q j 11 5 4 50 50
Ž1 y j .
Ž 12.
3
where h 0 is a numerical constant given by Eq. Ž 4. . Eq. Ž12., can be rewritten as: gŽs . s
y0.232 q 0.061 j 2 q 0.479j 9 j 2
0.852 j
1
Ž1 y j .
q
Ž1 y j .
2
q
Ž1 y j .
3
Ž 13.
The expansion for the RDF given by Eq. Ž13. is used below to extend Eq. Ž11. for a mixture of hard spheres.
3. Generalized mixing rule An analogical approach is adopted here to extend the pure compound hard sphere RDF, discussed in Section 2, to the case of a multicomponent mixture of hard spheres. The pairwise RDF, g ab , of two hard spheres a and b with different diameters, in a multicomponent hard sphere system, can be written as a function of the hard spheres diameter, si , and their number densities, r i , as: g ab s g Ž sa , sb , . . . , ra , rb , . . . .
Ž 14 .
From a comparison of the expressions for the pure compound RDF of the Percus–Yevick Ž P–Y. and the Carnahan–Starling Ž CS. equations of state and their extension to the multicomponent RDF for hard sphere systems, the analogy is directly obtained. For generality, we consider first the form proposed by Mansoori et al. w5x ŽMCSL. , to extend to mixtures the Carnahan–Starling ŽCS. equation of state. For a pure compound, the RDF of the CS equation can be written as: CS
g Žs . s
Ž 3hr2.
1 1yh
q
Ž1 y h .
2
q
Ž h 2r2. Ž1 y h .
Ž 15.
3
The extension of the RDF of the C–S equation to a multicomponent system, proposed by Mansoori et al. w5x Ž MCSL., can be written as w12x: MCSL g ab s
1 1 y h3
q
ž
sa sb h 2 sa b h 3
/
Ž 3h3r2. Ž1 y h3 .
2
q
ž
sa sb h 2 sa b h 3
2
/
Ž h32r2 . Ž1 y h3 .
3
Ž 16.
where sab s Ž sa q sb .r2, and h is the generalized packing factor of hard spheres in spherical cells defined as: p hk s Ý r j sj k Ž k s 0,1,2,3 . Ž 17 . 6 js1 with the summation running over all components j in the mixture. Comparison of Eqs. Ž3. and Ž17. indicates that for a pure compound system h 3 s h. Notably the same factor appears in a form of a power series in Eq. Ž16. modifying all terms of Eq. Ž15.. Based on the comparison of the pure
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compound and multicomponent RDF of the CS equation shown in Eqs. Ž 15. and Ž 16., the following generalized relation is proposed for extending the pure compound RDF to the multicomponent RDF of a general P–Y type equation of state for hard spheres: g ab s
Ý is0
ž
sa sb h 2 sab h 3
i
/
Gi Ž h 3 .
Ž1 y h3 .
iq1
Ž 18.
with G 0 s 1. The maximum value of i is one unit less than the exponent of the denominator of the expression for Z for a pure compound. Eqs. Ž 16. and Ž 18. are extensions the form obtained by Lebowitz w13x for the P–Y equation, using the method of functional Taylor expansion. For the P–Y equation the summation includes only the term with i s 1 and G 1 s 3h 3r2. The term G 2 and higher terms are equal to zero. Thus, for P–Y equation, the extension to mixtures reduces to the first two terms of the right hand side of Eq. Ž16. w6,12x. The multi-component hard sphere RDF for the Percus–Yevick equation of state can also be derived from the solution of the general virial theorem with direct correlation function of the hard spheres as shown by Lebowitz and Robinson w14x. For other equations of state for hard spheres, Eq. Ž18. has G 0 s 1, and the values of G 1, G 2 , etc., are those arising from the pure compound form of g Ž s .. In fact, the MCSL equation of state used an analogical extension of the Percus–Yevick result to obtain a multicomponent RDF from the pure compound CS equation of state w5,15x. For MCSL, the summation in Eq. Ž18. extends to i s 2 with G 1 s 3h 3r2 and G 2 s h 32r2 w12x. For a system of hard spheres in tetrakaidecahedron cells, as proposed by Wang et al. w7x, the generalized packing factor of hard spheres, j i , is defined as:
ji s
1
Ý r j sj i Ž i s 0,1,2,3 . '2 js1
Ž 19 .
Comparison of Eq. Ž19. with Eq. Ž2. indicates that for a pure component system j 3 s j . Based on the similarity of the pure compound RDF’s of the Percus–Yevick type of equations of state and of the equation proposed by Khoshkbarchi and Vera w8x, the following generalized mixing rule is proposed to extend the pure compound RDF to a multicomponent RDF for a hard sphere system with tetrakaidecahedron cells: g ab s
Ý is0
ž
sa sb j 2 sab j 3
i
/
Gi Ž j 3 .
Ž1 y j 3 .
iq1
Ž 20.
where G iŽ j 3 . is a generalized function of j 3 , arising from the functional form of g Ž s . when expanded in terms of powers of Ž1 y j 3 .y1. As for P–Y and MCSL forms, G 0 s 1 in Eq. Ž20.. It is interesting to observe that, based on the definitions of the packing factors in both the spherical cells h2 and the tetrakaidecahedron cells, the value of the ratio in Eq. Ž 18. is identical to the value of the h3 j2 ratio in Eq. Ž 20.. Thus, the factors modifying the corresponding terms of the series in Eqs. Ž 18. j3 and Ž20., are the same. The only difference in Eqs. Ž18. and Ž20. is that, Eq. Ž20., in contrast to Eq. Ž18., has a singularity at the close-packed limit. The use of other limiting packing array generating different singularities is also possible w8,16x.
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Similarly to the case for spherical cells w6x, the application of Eq. Ž 20. to the pure compound RDF in tetrakaidecahedron cells gives a parametric expression for the RDF of the mixture of hard spheres. The expressions for G iŽ j 3 . for the terms of the numerator in Eq. Ž20. can be obtained from the direct comparison with the pure compound contact value RDF, Eq. Ž 13.. The validity of Eq. Ž20. can be tested by considering the properties of the hard sphere system under certain conditions. The first condition dictates that the multicomponent hard sphere RDF should reduce to single component hard sphere RDF for systems composed of the same diameter hard spheres, i.e., sa s sb . This condition is obviously satisfied by the definition of the G iŽ j 3 . terms. The second condition is that for a binary system for which one of the hard spheres becomes a point particle, i.e., sa ™ 0, and the hard sphere becomes a point particle, the like-molecule pairwise RDF of the other component, g bb , should reduce to the pure compound hard sphere RDF, b . This, in turn, suggests that the pairwise RDF of the point particle component, g aa , should reduce to a form that allows the point particles to have only access to the portion of the total volume of the system which is not occupied by the cells containing hard spheres of type b . This leads to the following form for g aa as: 1 g aa s Ž 21. 1yj3 The consideration of Eq. Ž21. is similar to the consideration of the co-volume term in the van der Waals type equations of state. Finally, the multicomponent hard sphere RDF should approach to unity as the behavior of the hard sphere system tends to the ideal gas behavior, i.e., the diameters of all components tend to zero. Eq. Ž 20. clearly satisfies all these conditions. Thus, the hard sphere RDF obtained from the simplified pure compound hard sphere RDF, Eq. Ž 7., has the form: g ab s
1
Ž1 y j 3 .
q
ž
sa sb j 2 sa b j 3
/
0.851 j 3
Ž1 y j 3 .
2
q
ž
sa sb j 2 sa b j 3
2
/
y0.232 q 0.061 j 32 q 0.479j 39 j 32
Ž1 y j 3 .
3
Ž 22. where j i is the packing factor of hard spheres in tetrakaidecahedron cells defined by Eq. Ž19.. Once the RDF of the system is known all other thermodynamic functions can be calculated. The compressibility factor of a mixture is related to its RDF through the statistical mechanical virial equation as w12x: d uab 2p P s kTr y ra rb H0` g Ž r . r 3d r Ž 23. Ý Ý 3 a b d r ab For the case of a mixture of hard spheres, for which the assumption of pairwise additivity is exact, Eq. Ž23. simplifies to the following form with the RDF at the contact value w12x: P s kTr q
2p kT 3
Ý Ý ra rb g Ž sa b . sa3b a
Ž 24.
b
Combining the RDF of a mixture, Eq. Ž 22., with Eq. Ž24., gives the equation of state for the system fora mixture of hard spheres, as: Zs1q
0.740 j 3 Ž 1 q 3Y1 .
Ž1 y j 3 .
q
1.26 j 32 Ž Y1 q Y2 .
Ž1 y j 3 .
2
q
j 33 Y2 Ž y0.690 q 0.180 j 32 q 1.42 j 39 .
Ž1 y j 3 .
3
Ž 25.
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
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In Eq. Ž25., Z is the compressibility factor, and the dimensionless groups Y1 and Y2 are given by: Y1 s
j1 j2 j0j3
and Y2 s
j 23
Ž 26.
j 0 j 32
The expressions for the residual Helmholtz free energy A R, and the residual chemical potential m R, can be calculated from Eq. Ž25.. It should be noted that a residual property m R, is usually defined with respect to an ideal gas system at the same temperature T, pressure P, and number of moles N, of the system w17x as: X R T ,P,N sX T ,P,N yX ig T ,P,N
Ž 27.
where superscript ig refers to the ideal gas state. In some statistical thermodynamic texts, however, the residual properties have been defined with respect to an ideal gas system at the same temperature T, volume V, and number of moles N, of the system w12x as: X r T ,V , N s X T ,V , N y X i g T ,V , N
Ž 28.
For the sake of clarity, the residual properties at the same temperature, volume, and number of moles of the system are shown by the superscript r. The corresponding residual Helmholtz free energy can be calculated, using either of these definitions, from an equation of state of the form Z s Z(T, V, N), and one of the following relations: AR T , P , N
r
s NkT A r T ,V , N
r
s NkT
H0
H0
Zy1
r Zy1
r
d r y ln Z
Ž 29.
dr
Ž 30.
The variables in square brackets on the left hand side of Eqs. Ž29. and Ž30., indicate the variables held constant for comparison with the ideal gas. The independent variables of the equation of state are T, V, and N, for both cases. The residual chemical potential can be calculated from Eq. Ž 29. or Eq. Ž 30. , depending on the case, and the following relation:
m ir kT
Ar s NkT
qr
ž
E Ž A rrNkT . Er i
/
Ž 31. T ,V , r jqi
The expressions for the residual Helmholtz free energy, the residual Gibbs free energy and the residual chemical potential are presented in Appendix A.
4. Results and discussion The generalized mixing rule proposed here can be tested by judging the results obtained for a mixture of hard spheres using the equation of state which satisfies the correct limit for compressibility factor. This equation has been employed to calculate the compressibility factor for systems composed of hard spheres with various diameters and packing factors and the results have been compared with computer simulation data. For comparison, we have included the results obtained with the MCSL equation of state.
138
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
Fig. 1. Prediction of compressibility factor simulation data vs. h for mixture of hard spheres with the ratio of diameters of s2 s 3 and x 2 s 0.5. v: Computer simulation data w18x; : Results using Eq. Ž25.; - - -: Results using MCSL. s1
Fig. 1 shows the results of the prediction of compressibility factor as a function of h for an equi-molar fraction mixture of two different hard spheres with the ratio of diameters s 2rs 1 s 3. The molecular dynamic ŽMD. data of Fig. 1 were reported by Alder w18x. The Monte Carlo Ž MC. data in Figs. 2–7 were reported by Jackson et al. w19x. Figs. 2 and 3 show the results of the prediction of compressibility factor as a function of h at two different mole fractions for mixtures of two different hard spheres with the ratio of diameters s 2rs 1 s 5r3. The mole fraction of the component with bigger diameter in the systems shown in Figs. 2 and 3 are 0.1019 and 0.5, respectively. Figs. 4 and 5 show the results of the prediction of compressibility factor as a function of h at two different mole fractions for mixtures of two different hard spheres with the ratio of diameters s 2rs 1 s 5. The mole fraction of the component with bigger diameter in the systems shown in Figs. 4 and 5 are 0.1019 and 0.8981, respectively. Figs. 6 and 7 show the results of the prediction of compressibility factor as a function of h at two different mole fractions for mixtures of two different hard spheres with the ratio of diameters s 2rs 1 s 20. The mole fraction of the component with larger diameter in the systems shown in Figs. 6 and 7 are 0.1019 and 0.5, respectively. Fig. 8 shows the results of the prediction of compressibility factor as a function of mole fraction of the hard spheres at three different values of h for a system of mixtures of two hard spheres with the ratio of diameters of s 2rs 1 s 2. The molecular dynamic data are those reported by Fries and Hansen w20x. As shown in Figs. 1–8, the equation of state for mixture of hard spheres proposed in this work can accurately represent the compressibility factor computer simulation data over a wide range of density and mole fraction and for different ratios of diameters of the hard spheres. Figs. 1–7 also show the comparison of the results obtained from the equation of state for hard spheres proposed here with those obtained from MCSL.
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139
Fig. 2. Prediction of compressibility factor simulation data vs. h for a mixture of hard spheres with the ratio of diameters of s2 s 5r3 and x 2 s 0.1019. v: Computer simulation data w19x; : Results using Eq. Ž25.; - - -: Results using s1 MCSL.
Fig. 3. Prediction of compressibility factor simulation data vs. h for a mixture of hard spheres with the ratio of diameters of s2 s 5r3 and x 2 s 0.5. v: Computer simulation data w19x; : Results using Eq. Ž25.; - - -: Results using MCSL. s1
140
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
Fig. 4. Prediction of compressibility factor simulation data vs. h for a mixture of hard spheres with the ratio of diameters of s2 s 5 and x 2 s 0.1019. v: Computer simulation data w19x; : Results using Eq. Ž25.; - - -: Results using MCSL. s1
Fig. 5. Prediction of compressibility factor simulation data vs. h for a mixture of hard spheres with the ratio of diameters of s2 s 5 and x 2 s 0.8981. v: Computer simulation data w19x; : Results using Eq. Ž25.; - - -: Results using MCSL. s1
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141
Fig. 6. Prediction of compressibility factor simulation data vs. h for a mixture of hard spheres with the ratio of diameters of s2 s 20 and x 2 s 0.1019. v: Computer simulation data w19x; : Results using Eq. Ž25.; - - -: Results using MCSL. s1
Fig. 7. Prediction of compressibility factor simulation data vs. h for a mixture of hard spheres with the ratio of diameters of s2 s 20 and x 2 s 0.5. v: Computer simulation data w19x; : Results using Eq. Ž25.; - - -: Results using MCSL. s1
142
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Fig. 8. Prediction of compressibility factor simulation data vs. mole fraction for a mixture of hard spheres with the ratio of s2 diameters of s 2. v: Computer simulation data w20x; : Results using Eq. Ž25.. s1
Figs. 1–3 show that the results obtained with the new form, which meets the correct limits at close packing and at low density, are perfectly comparable to the results obtained with the MCSL equation. This proves that the generalized mixing rule can be used with confidence as more sophisticated forms of the pure compound hard sphere equation of state become available. Figs. 2, 3 and 6 clearly indicate the advantage gained in the compressed region when using an equation of state with the correct behaviour at close packing. For high ratios of the diameters of the hard spheres, such as 20, the results obtained in this work, and shown in Fig. 6, are in better agreement with computer simulation data than those obtained with MCSL. It must be clearly stated that the use of j 3 s 1 as the pole for the equation of state for mixtures, for the case of tetrakaidecahedron cells, can be as problematic as the use of h 3 s 1 as the pole for the case of spherical cells. In fact the value of the pole for a mixture of hard spheres with different diameters is not known. The only basis for this selection is that the latter is the rigorous result which was obtained by Lebowitz w13x from the closure of the integral Percus–Yevick equation and it was tacitly incorporated in the MCSL equation. However, due to the rhombohedral array of centres in the case of tetrakaidecahedral cells, the problem, if any, is expected to appear at larger diameter ratios than in the case of spherical cells. Some of the figures presented here compare the results of our equation of state for densely packed systems. At high densities, the solidification of hard spheres or a meta-stable equilibrium phenomenom may occur. Thus, it seems important to make sure about the quality of simulation data obtained from the literature. Our survey in the literature showed that the simulation data for hard sphere systems at high density presented by Jackson et al. w19x suit the best our purposes. This is mainly due to the fact that for the Monte Carlo Ž MC. simulation they have used the isobaric
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Table 1 Comparison of the RDF computer simulation data ŽMD. reported by Fries and Hansen w20x with values obtained from Eq. Ž25., from MCSL equation and from P–Y equation for the RDF at the contact value of the unlike hard spheres with the ratio s2 of diameters of s2 s1 h x MD P–Y MCSL Eq. Ž25. 0.45 0.50 0.55 0.45 0.50 0.55 0.59 0.45 0.50 0.55
0.05000 0.05000 0.05000 0.10156 0.10156 0.10156 0.10156 0.19922 0.19922 0.19922
5.20 6.10 9.00 4.60 5.70 7.40 11.00 4.30 5.10 6.70
4.35 5.41 6.85 4.09 5.05 6.36 7.79 3.80 4.67 5.85
5.14 6.70 8.99 4.71 6.08 8.08 10.40 4.28 5.46 7.16
4.75 5.98 7.77 4.67 5.98 8.02 11.26 4.54 5.91 8.11
ensemble, MCŽ T, P, N ., rather than the canonical ensemble, MCŽ T, V, N .. This is important because it is generally accepted that MCŽ T, P, N . simulations generate more stable and physically meaningful configurations than MCŽ T, V, N . simulations. Jackson et al. w19x have also performed molecular dynamic simulations, MD, from one of the stable MC configurations to check their results, and good agreement was found between the results obtained from both methods. Although for hard sphere
Fig. 9. Prediction of residual Gibbs free energy simulation data vs. h for a mixture of hard spheres with the ratio of s2 diameters of s 3 and x 2 s 0.5. v: Computer simulation data w18x; : Results using Eq. ŽA.5.. s1
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systems at high densities, the danger of solidification always exists, special care was taken by Jackson et al. w19x to significantly reduce this possibility. Table 1 compares the computer simulation data for the RDF at the contact value for the unlike hard spheres with the results obtained from Eq. Ž 22. developed in this study, with the MCSL equation of state w5x and with the P–Y equation w14x, for the systems shown in Fig. 7. As it can be seen from this comparison, in general, Eq. Ž 22. can represent the contact value computer simulation RDF data better than the other two models. Fig. 9 compares computer simulation data for the residual Gibbs free energy for a mixture of hard spheres w18x, with the results obtained from Eq. ŽA.5. of Appendix A. As it can be seen from this comparison, the residual model presented in this work can represent the residual Gibbs free energy computer simulation data with good accuracy. This can be construed as a proof of the reliability of the proposed mixing rule, which can be directly extended to more complex models.
5. Conclusions A generalized analogical approach has been proposed to extend the pure compound RDF to the multicomponent RDF of hard sphere systems. The approach is based on the similarity of the pure compound and multicomponent RDF for the Percus–Yevick and the Carnahan–Starling equations of state of hard spheres. The generalized approach was applied to an equation of state for pure compound hard spheres proposed by Khoshkbarchi and Vera w8x, which meets the correct limit at the close-packed state of hard spheres. The equation of state for a mixture of hard spheres thus obtained was employed to predict the computer simulation compressibility factor and contact value RDF. No adjustable parameter has been employed. The results show that the equation of state developed here can accurately predict the simulation data while satisfying the correct limit at close-packed. Expressions for the residual Helmholtz free energy and the residual chemical potential, based on the new equation of state, have been derived.
6. List of symbols
A bi Gi G gŽ s . gi k N P r T
Helmholtz free energy Parameters of Eq. Ž 1. Functions of Eq. Ž18. Gibbs free energy Radial distribution function at contact value Function of Eq. Ž14. Boltzman constant Number of spheres Pressure Radial distance Absolute temperature
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
ui j V V0 x Z
145
Pair potential between hard spheres Total volume Ns 3 Total volume of hard spheress '2 Mole fraction Compressibility factor
Greek letters
bi h h0
Coefficients of virial expansion p rs 3 6 p'2
m
6 Chemical potential
j
Packing factor in tetrakaidecahedron cells s
r s
Number density Diameter of a hard sphere
V0 V
Superscripts CS MCSL R r
Carnahan–Starling Mansoori–Carnahan–Starling-Leland Residual, w T, P, N x Residual, w T, V, N x
Subscripts
a b
Hard sphere of type a Hard sphere of type b
Acknowledgements The authors are grateful to the Natural Sciences and Engineering Research Council of Canada for financial support. Appendix A Combining Eqs. Ž25. and Ž 30. provides with an expression for the residual Helmholtz free energy as: A r T ,V , N 1.260Y1 y 13.70Y2 0.4550Y2 s q y F1Y2 q 13.25Y2 y 1.260Y1 y F2 ln D Ž A.1 . NkT D D2
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
146
where
Ds1yj3
Ž A.2 .
F1 s 64.44j 3 q 25.65j 32 q 13.25j 33 q 7.455j 34 q 4.260 j 35 q 2.366 j 36 q 1.217j 37 q 0.5325j 38 q 0.1578 j 39
Ž A.3 . Ž A.4 .
F2 s 0.7400 q 0.9600Y1 q 77.23Y2 The residual Gibbs free energy, G r, can be calculated as: GR T,P,N
A r T ,V , N
q Z y 1 y ln Z Ž A.5 . NkT NkT Combining Eqs. Ž31. and ŽA.1. provides with an expression for the residual chemical potential, m r w T, V, N x, as: s
m r T ,V , N
A r T ,V , N
y F3
s kT q
NkT
F2 j 3 R3 D
q
j2 R3 D
j 3 Ž 1.260Y1 y 13.70Y2 .
y 0.9600Y1
R3 D2 1
1
R1
y
y
R2
R3
y
D
q F1 j 3Y2
ln D
1
q
30.11 2
R0
y 140.6 q
1
3 y
R3
0.9100Y2 j 3 3
D R3
y 1.260 F4
1yD
D
1 q
R2
R0
Ž A.6 .
where Rk s
r i si k
Ž A.7 .
Ý r j sj k j
F3 s Y2 F4 s Y1
0.4550
1.365 y
R0 1
R2 1
y R0
0.9100 q
1 y
R1
R3 1 q
R2
R3
Ž A.8 . Ž A.9 .
References w1x N.F. Carnahan, K.E. Starling, Equation of state of nonattracting rigid spheres, J. Chem. Phys. 51 Ž1969. 635–636. w2x B. Barboy, W.M. Gelbart, Series representation of the equation of state for hard particle fluids, J. Chem. Phys. 71 Ž1979. 3053–3062. w3x T. Boublik, I. Nezbeda, P-V-T behaviour of hard body fluids: Theory and experiment, Collection Czechoslovak Chem. Commun. 51 Ž1986. 2301–2432. w4x Boublik, 1971. w5x G.A. Mansoori, N.F. Carnahan, K.E. Starling, T.W. Leland Jr., Equilibrium thermodynamic properties of the mixture of hard spheres, J. Chem. Phys. 54 Ž1971. 1523–1525.
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w6x H. Reiss, Scaled Particle Methods in the Statistical Thermodynamics of Fluids, Advances in Physical Chemistry, Vol. IX, pp. 1–84, in: I. Prigogine ŽEd.., Interscience Publishers, New York, 1965. w7x W. Wang, M.K. Khoshkbarchi, J.H. Vera, A new volume dependence for the equation of state of hard spheres, Fluid Phase Equilib. 115 Ž1996. 25–38. w8x M.K. Khoshkbarchi, J.H. Vera, A simplified hard sphere equation of state meeting the high and low density limits, Fluid Phase Equilib. 130 Ž1997. 189–194. w9x M.S. Wertheim, Exact solution of the Percus–Yevick integral equation for hard spheres, Phys. Rev. Letters 10 Ž1963. 321–323. w10x M.S. Wertheim, Analytic solution of the Percus–Yevick equation, J. Math. Phys. 5 Ž1964. 643–651. w11x T.L. Hill, Statistical Mechanics, Principles and Selected Applications, Dover Publ., New York, 1956. w12x T.M. Reed, K.E. Gubbins, Applied Statistical Mechanics, McGraw Hill, New York, 1973. w13x J.L. Lebowitz, Exact solution of generalized Percus–Yevick equation for mixture of hard spheres, Phys. Rev. 133 Ž1964. A895–A899. w14x J.L. Lebowitz, J.S. Rowlinson, Thermodynamic properties of mixtures of hard spheres, J. Chem. Phys. 41 Ž1964. 133–138. w15x T. Boublik, Hard-sphere equation of state, J. Phys. Chem. 53 Ž1970. 471–472. w16x Iglesias-Silva, Hall, 1994. w17x J.M. Smith, H.C. Van Ness, M.M. Abbott, Introduction to Chemical Engineering Thermodynamics, McGraw-Hill, New York, 1996. w18x B.J. Alder, Studies in molecular dynamics: III. A mixture of hard spheres, J. Chem. Phys. 40 Ž1964. 2724–2730. w19x G. Jackson, J.S. Rowlinson, F. van Swol, Computer simulation of mixture of hard spheres, J. Phys. Chem. 91 Ž1987. 4907–4912. w20x P.H. Fries, J.P. Hansen, A Monte Carlo study of semi-dilute hard sphere mixtures, Mol. Phys. 48 Ž1983. 891–901.
A generalized mixing rule for hard-sphere equations of state of Percus–Yevick type Mohammad K. Khoshkbarchi, Juan H. Vera
)
Department of Chemical Engineering, McGill UniÕersity, Montreal, Quebec, Canada H3A 2B2 Received 25 May 1996; accepted 5 September 1997
Abstract An analogical approach has been used to extend to mixtures the radial distribution function of Percus–Yevick type of equations of state for hard spheres. The proposed approach follows the same formalism employed by Mansoori et al. to extend to mixtures the Carnahan–Starling equation of state. The generality of the proposed method permits to extend to mixtures equations of state with different singularities. In this work, it is applied to an equation of state which meets the correct low density and high density limits, previously proposed by Khoshkbarchi and Vera. The results show that both the radial distribution function and the resulting equation of state for a mixture of hard spheres, developed in this study, can accurately represent the computer simulation data while satisfying the correct limit at close-packing. q 1998 Elsevier Science B.V. Keywords: Hard spheres; Equations of state; Mixtures
1. Introduction Modelling of the properties of the systems of hard spheres is an essential element of most modern theories of liquids and liquid mixtures. Systems of hard sphere mixtures are the simplest nontrivial models which can mimic the properties of the real fluids. They also serve as the reference system in statistical mechanical theories of liquids such as the perturbation theory for liquids. Although, there are many studies reported in literature for the equations of state of hard spheres for pure liquids w1–3x, few studies were conducted to develop equations of state for the systems of mixture of hard spheres w4,5x. All of the equations of state for hard spheres reported in literature suffer from a major deficiency as they fail to meet the correct limit at close-packing, where the compressibility factor must tend infinity. These equations, although they may represent well the computer simulation data at low to moderate densities, predict a finite value at close-packed density and their singularity )
Corresponding author.
0378-3812r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 8 - 3 8 1 2 Ž 9 7 . 0 0 2 3 0 - 6
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
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corresponds to a physically unrealistic condition. This shortcoming, common to all equations of state derived from scaled particle theory w6x, is due to the use of spherical cells, which ignores the fact that when spherical cells are close-packed, there is a volume which is not available for occupation by the centre of the hard spheres. This unaccessible volume, although not important at low densities, has an important effect on the properties of hard sphere systems in the high density region. Recently Wang et al. w7x proposed a new form of volume dependence for the equation of state of hard spheres which has a pole at the close-packed limit while representing accurately the compressibility factor simulation data over a wide range of density. The new volume dependence is based on the definition of cells with tetrakaidecahedron geometry, with centres placed on the corners of a rhombohedral, and the probability of finding the centre of each of these cells occupied by the centre of a hard sphere. Since an array of cells with this geometry fills up all the space, the compressibility factor tends to infinity when the hard spheres are in the close-packed state. Based on this new volume dependence, Wang et al. w7x proposed three equations of state for hard spheres which accurately correlated the compressibility factor and reproduced well the virial coefficients of a hard sphere system, while fulfilling the close-packed limit condition. These equations were rather complicated for practical use, thus, we recently proposed a simplified equation of state for hard spheres, which meets the correct limit at close-packing w8x. In this study we propose a generalized mixing rule for the radial distribution function, from which the equation of state and other thermodynamic properties of a mixture of hard spheres can be obtained.
2. Thermodynamic framework In this study, one of the forms of the equation of state for pure compound hard sphere systems proposed by Wang et al. w7x is selected, and extended to a mixture of hard spheres. Among the three general forms of equations proposed by Wang et al. w7x, the following general Percus–Yevick type form is chosen:
Ý bi j i Zs
is0
Ž1 y j .
Ž1.
3
where Z is the compressibility factor, bi is a numerical parameter, with b 0 s 1. The dimensionless variable j is defined as:
js
V0
h s
V
h0
1 s
'2
rs 3
Ž2.
with
p hs
6
rs 3
Ž3.
and
h0 s
p'2 6
Ž4.
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
133
where V is the total volume, V0 s Ns 3r62 is the volume of the system at the close-packing state of N spheres, r s N r V is the number density, and s is the diameter of the hard spheres. The term Ž1 y j . in the denominator of Eq. Ž 1. guarantees the correct limit, i.e., that at the close-packed state, where j s 1, the compressibility factor tends towards infinity. This form of equation has been chosen because it can represent the computer simulation data and the virial coefficients with less uncertainty than the other forms proposed by Wang et al. w7x. This form of equation of state of hard spheres is a generalization of both forms of the Percus–Yevick equation of state for hard spheres w9,10x and of the Carnahan–Starling equation w1x. The virial coefficients, for a series in terms of powers of h , generated by this general form of the equation of state of hard spheres, bi , can be calculated, for i G 1, according to the following relation: bi s Ž biy1 q 3 biy1h 0iy2 y 3 biy2 h 0iy3 q biy3h 0iy4 . rh 0iy1 Ž5. Ž . with b k s 0 for k F 0. The radial distribution function RDF , at contact value, can be calculated using the virial theorem. The equation of the virial theorem, at contact value, for hard spheres has the form w11x: Z s 1 q 4h 0 j g Ž s . Ž6. ( ) where g s is the RDF at contact value of the distance of separation between the centres of two hard spheres. By combining Eq. Ž6. with Eq. Ž 1. , the contact value radial distribution function associated with Eq. Ž1. is obtained as: 1 1q Ž b 2 y 3 . j q Ž b 3 q 1 . j 2 q Ý biq1 j i 4h 0 is3 gŽs . s Ž7. 3 Ž1 y j . In the derivation of Eq. Ž7. we have incorporated the known Žexact. values of the first two virial coefficients for hard spheres, i.e., b 1 s 1 and b 2 s 4. With these values, Eq. Ž5. gives b 1 s 4h 0 y 3 which generates the first term of the numerator of Eq. Ž 7. . Eq. Ž 7. is the general form of the RDF for equations of state of the Percus–Yevick form, i.e., Eq. Ž1.. For the purpose of extending its applicability to mixtures, it is convenient to re-write Eq. Ž 7. as: 1 G1 G2 gŽs . s q Ž8. 2 q 3 Ž1 y j . Ž1 y j . Ž1 y j . with 1 G1 s 8h 0 q b 2 y 3 j Ž9. 4h 0 and 1 G2 s Ž 4h0 q b 2 q b 3 y 2 . q Ý bi j iy3 j 2 Ž 10. 4h 0 is4 We have recently proposed a modified form of the Eq. Ž 1. for a pure compound hard sphere system w8x, which meets the correct limits and reproduces well the first six virial coefficients. It has the form: 1 2 5 9 71 1y j y j 2 y j 3 q j 5 q j 12 25 5 4 50 50 Zs Ž 11. 3 Ž1 y j .
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
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According to Eq. Ž7. , the contact value radial distribution function for Eq. Ž11. takes the form: 1q gŽs . s
1
17 y
4h 0
1 9 71 j y j 2 q j 4 q j 11 5 4 50 50
Ž1 y j .
Ž 12.
3
where h 0 is a numerical constant given by Eq. Ž 4. . Eq. Ž12., can be rewritten as: gŽs . s
y0.232 q 0.061 j 2 q 0.479j 9 j 2
0.852 j
1
Ž1 y j .
q
Ž1 y j .
2
q
Ž1 y j .
3
Ž 13.
The expansion for the RDF given by Eq. Ž13. is used below to extend Eq. Ž11. for a mixture of hard spheres.
3. Generalized mixing rule An analogical approach is adopted here to extend the pure compound hard sphere RDF, discussed in Section 2, to the case of a multicomponent mixture of hard spheres. The pairwise RDF, g ab , of two hard spheres a and b with different diameters, in a multicomponent hard sphere system, can be written as a function of the hard spheres diameter, si , and their number densities, r i , as: g ab s g Ž sa , sb , . . . , ra , rb , . . . .
Ž 14 .
From a comparison of the expressions for the pure compound RDF of the Percus–Yevick Ž P–Y. and the Carnahan–Starling Ž CS. equations of state and their extension to the multicomponent RDF for hard sphere systems, the analogy is directly obtained. For generality, we consider first the form proposed by Mansoori et al. w5x ŽMCSL. , to extend to mixtures the Carnahan–Starling ŽCS. equation of state. For a pure compound, the RDF of the CS equation can be written as: CS
g Žs . s
Ž 3hr2.
1 1yh
q
Ž1 y h .
2
q
Ž h 2r2. Ž1 y h .
Ž 15.
3
The extension of the RDF of the C–S equation to a multicomponent system, proposed by Mansoori et al. w5x Ž MCSL., can be written as w12x: MCSL g ab s
1 1 y h3
q
ž
sa sb h 2 sa b h 3
/
Ž 3h3r2. Ž1 y h3 .
2
q
ž
sa sb h 2 sa b h 3
2
/
Ž h32r2 . Ž1 y h3 .
3
Ž 16.
where sab s Ž sa q sb .r2, and h is the generalized packing factor of hard spheres in spherical cells defined as: p hk s Ý r j sj k Ž k s 0,1,2,3 . Ž 17 . 6 js1 with the summation running over all components j in the mixture. Comparison of Eqs. Ž3. and Ž17. indicates that for a pure compound system h 3 s h. Notably the same factor appears in a form of a power series in Eq. Ž16. modifying all terms of Eq. Ž15.. Based on the comparison of the pure
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
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compound and multicomponent RDF of the CS equation shown in Eqs. Ž 15. and Ž 16., the following generalized relation is proposed for extending the pure compound RDF to the multicomponent RDF of a general P–Y type equation of state for hard spheres: g ab s
Ý is0
ž
sa sb h 2 sab h 3
i
/
Gi Ž h 3 .
Ž1 y h3 .
iq1
Ž 18.
with G 0 s 1. The maximum value of i is one unit less than the exponent of the denominator of the expression for Z for a pure compound. Eqs. Ž 16. and Ž 18. are extensions the form obtained by Lebowitz w13x for the P–Y equation, using the method of functional Taylor expansion. For the P–Y equation the summation includes only the term with i s 1 and G 1 s 3h 3r2. The term G 2 and higher terms are equal to zero. Thus, for P–Y equation, the extension to mixtures reduces to the first two terms of the right hand side of Eq. Ž16. w6,12x. The multi-component hard sphere RDF for the Percus–Yevick equation of state can also be derived from the solution of the general virial theorem with direct correlation function of the hard spheres as shown by Lebowitz and Robinson w14x. For other equations of state for hard spheres, Eq. Ž18. has G 0 s 1, and the values of G 1, G 2 , etc., are those arising from the pure compound form of g Ž s .. In fact, the MCSL equation of state used an analogical extension of the Percus–Yevick result to obtain a multicomponent RDF from the pure compound CS equation of state w5,15x. For MCSL, the summation in Eq. Ž18. extends to i s 2 with G 1 s 3h 3r2 and G 2 s h 32r2 w12x. For a system of hard spheres in tetrakaidecahedron cells, as proposed by Wang et al. w7x, the generalized packing factor of hard spheres, j i , is defined as:
ji s
1
Ý r j sj i Ž i s 0,1,2,3 . '2 js1
Ž 19 .
Comparison of Eq. Ž19. with Eq. Ž2. indicates that for a pure component system j 3 s j . Based on the similarity of the pure compound RDF’s of the Percus–Yevick type of equations of state and of the equation proposed by Khoshkbarchi and Vera w8x, the following generalized mixing rule is proposed to extend the pure compound RDF to a multicomponent RDF for a hard sphere system with tetrakaidecahedron cells: g ab s
Ý is0
ž
sa sb j 2 sab j 3
i
/
Gi Ž j 3 .
Ž1 y j 3 .
iq1
Ž 20.
where G iŽ j 3 . is a generalized function of j 3 , arising from the functional form of g Ž s . when expanded in terms of powers of Ž1 y j 3 .y1. As for P–Y and MCSL forms, G 0 s 1 in Eq. Ž20.. It is interesting to observe that, based on the definitions of the packing factors in both the spherical cells h2 and the tetrakaidecahedron cells, the value of the ratio in Eq. Ž 18. is identical to the value of the h3 j2 ratio in Eq. Ž 20.. Thus, the factors modifying the corresponding terms of the series in Eqs. Ž 18. j3 and Ž20., are the same. The only difference in Eqs. Ž18. and Ž20. is that, Eq. Ž20., in contrast to Eq. Ž18., has a singularity at the close-packed limit. The use of other limiting packing array generating different singularities is also possible w8,16x.
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
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Similarly to the case for spherical cells w6x, the application of Eq. Ž 20. to the pure compound RDF in tetrakaidecahedron cells gives a parametric expression for the RDF of the mixture of hard spheres. The expressions for G iŽ j 3 . for the terms of the numerator in Eq. Ž20. can be obtained from the direct comparison with the pure compound contact value RDF, Eq. Ž 13.. The validity of Eq. Ž20. can be tested by considering the properties of the hard sphere system under certain conditions. The first condition dictates that the multicomponent hard sphere RDF should reduce to single component hard sphere RDF for systems composed of the same diameter hard spheres, i.e., sa s sb . This condition is obviously satisfied by the definition of the G iŽ j 3 . terms. The second condition is that for a binary system for which one of the hard spheres becomes a point particle, i.e., sa ™ 0, and the hard sphere becomes a point particle, the like-molecule pairwise RDF of the other component, g bb , should reduce to the pure compound hard sphere RDF, b . This, in turn, suggests that the pairwise RDF of the point particle component, g aa , should reduce to a form that allows the point particles to have only access to the portion of the total volume of the system which is not occupied by the cells containing hard spheres of type b . This leads to the following form for g aa as: 1 g aa s Ž 21. 1yj3 The consideration of Eq. Ž21. is similar to the consideration of the co-volume term in the van der Waals type equations of state. Finally, the multicomponent hard sphere RDF should approach to unity as the behavior of the hard sphere system tends to the ideal gas behavior, i.e., the diameters of all components tend to zero. Eq. Ž 20. clearly satisfies all these conditions. Thus, the hard sphere RDF obtained from the simplified pure compound hard sphere RDF, Eq. Ž 7., has the form: g ab s
1
Ž1 y j 3 .
q
ž
sa sb j 2 sa b j 3
/
0.851 j 3
Ž1 y j 3 .
2
q
ž
sa sb j 2 sa b j 3
2
/
y0.232 q 0.061 j 32 q 0.479j 39 j 32
Ž1 y j 3 .
3
Ž 22. where j i is the packing factor of hard spheres in tetrakaidecahedron cells defined by Eq. Ž19.. Once the RDF of the system is known all other thermodynamic functions can be calculated. The compressibility factor of a mixture is related to its RDF through the statistical mechanical virial equation as w12x: d uab 2p P s kTr y ra rb H0` g Ž r . r 3d r Ž 23. Ý Ý 3 a b d r ab For the case of a mixture of hard spheres, for which the assumption of pairwise additivity is exact, Eq. Ž23. simplifies to the following form with the RDF at the contact value w12x: P s kTr q
2p kT 3
Ý Ý ra rb g Ž sa b . sa3b a
Ž 24.
b
Combining the RDF of a mixture, Eq. Ž 22., with Eq. Ž24., gives the equation of state for the system fora mixture of hard spheres, as: Zs1q
0.740 j 3 Ž 1 q 3Y1 .
Ž1 y j 3 .
q
1.26 j 32 Ž Y1 q Y2 .
Ž1 y j 3 .
2
q
j 33 Y2 Ž y0.690 q 0.180 j 32 q 1.42 j 39 .
Ž1 y j 3 .
3
Ž 25.
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
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In Eq. Ž25., Z is the compressibility factor, and the dimensionless groups Y1 and Y2 are given by: Y1 s
j1 j2 j0j3
and Y2 s
j 23
Ž 26.
j 0 j 32
The expressions for the residual Helmholtz free energy A R, and the residual chemical potential m R, can be calculated from Eq. Ž25.. It should be noted that a residual property m R, is usually defined with respect to an ideal gas system at the same temperature T, pressure P, and number of moles N, of the system w17x as: X R T ,P,N sX T ,P,N yX ig T ,P,N
Ž 27.
where superscript ig refers to the ideal gas state. In some statistical thermodynamic texts, however, the residual properties have been defined with respect to an ideal gas system at the same temperature T, volume V, and number of moles N, of the system w12x as: X r T ,V , N s X T ,V , N y X i g T ,V , N
Ž 28.
For the sake of clarity, the residual properties at the same temperature, volume, and number of moles of the system are shown by the superscript r. The corresponding residual Helmholtz free energy can be calculated, using either of these definitions, from an equation of state of the form Z s Z(T, V, N), and one of the following relations: AR T , P , N
r
s NkT A r T ,V , N
r
s NkT
H0
H0
Zy1
r Zy1
r
d r y ln Z
Ž 29.
dr
Ž 30.
The variables in square brackets on the left hand side of Eqs. Ž29. and Ž30., indicate the variables held constant for comparison with the ideal gas. The independent variables of the equation of state are T, V, and N, for both cases. The residual chemical potential can be calculated from Eq. Ž 29. or Eq. Ž 30. , depending on the case, and the following relation:
m ir kT
Ar s NkT
qr
ž
E Ž A rrNkT . Er i
/
Ž 31. T ,V , r jqi
The expressions for the residual Helmholtz free energy, the residual Gibbs free energy and the residual chemical potential are presented in Appendix A.
4. Results and discussion The generalized mixing rule proposed here can be tested by judging the results obtained for a mixture of hard spheres using the equation of state which satisfies the correct limit for compressibility factor. This equation has been employed to calculate the compressibility factor for systems composed of hard spheres with various diameters and packing factors and the results have been compared with computer simulation data. For comparison, we have included the results obtained with the MCSL equation of state.
138
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
Fig. 1. Prediction of compressibility factor simulation data vs. h for mixture of hard spheres with the ratio of diameters of s2 s 3 and x 2 s 0.5. v: Computer simulation data w18x; : Results using Eq. Ž25.; - - -: Results using MCSL. s1
Fig. 1 shows the results of the prediction of compressibility factor as a function of h for an equi-molar fraction mixture of two different hard spheres with the ratio of diameters s 2rs 1 s 3. The molecular dynamic ŽMD. data of Fig. 1 were reported by Alder w18x. The Monte Carlo Ž MC. data in Figs. 2–7 were reported by Jackson et al. w19x. Figs. 2 and 3 show the results of the prediction of compressibility factor as a function of h at two different mole fractions for mixtures of two different hard spheres with the ratio of diameters s 2rs 1 s 5r3. The mole fraction of the component with bigger diameter in the systems shown in Figs. 2 and 3 are 0.1019 and 0.5, respectively. Figs. 4 and 5 show the results of the prediction of compressibility factor as a function of h at two different mole fractions for mixtures of two different hard spheres with the ratio of diameters s 2rs 1 s 5. The mole fraction of the component with bigger diameter in the systems shown in Figs. 4 and 5 are 0.1019 and 0.8981, respectively. Figs. 6 and 7 show the results of the prediction of compressibility factor as a function of h at two different mole fractions for mixtures of two different hard spheres with the ratio of diameters s 2rs 1 s 20. The mole fraction of the component with larger diameter in the systems shown in Figs. 6 and 7 are 0.1019 and 0.5, respectively. Fig. 8 shows the results of the prediction of compressibility factor as a function of mole fraction of the hard spheres at three different values of h for a system of mixtures of two hard spheres with the ratio of diameters of s 2rs 1 s 2. The molecular dynamic data are those reported by Fries and Hansen w20x. As shown in Figs. 1–8, the equation of state for mixture of hard spheres proposed in this work can accurately represent the compressibility factor computer simulation data over a wide range of density and mole fraction and for different ratios of diameters of the hard spheres. Figs. 1–7 also show the comparison of the results obtained from the equation of state for hard spheres proposed here with those obtained from MCSL.
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
139
Fig. 2. Prediction of compressibility factor simulation data vs. h for a mixture of hard spheres with the ratio of diameters of s2 s 5r3 and x 2 s 0.1019. v: Computer simulation data w19x; : Results using Eq. Ž25.; - - -: Results using s1 MCSL.
Fig. 3. Prediction of compressibility factor simulation data vs. h for a mixture of hard spheres with the ratio of diameters of s2 s 5r3 and x 2 s 0.5. v: Computer simulation data w19x; : Results using Eq. Ž25.; - - -: Results using MCSL. s1
140
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
Fig. 4. Prediction of compressibility factor simulation data vs. h for a mixture of hard spheres with the ratio of diameters of s2 s 5 and x 2 s 0.1019. v: Computer simulation data w19x; : Results using Eq. Ž25.; - - -: Results using MCSL. s1
Fig. 5. Prediction of compressibility factor simulation data vs. h for a mixture of hard spheres with the ratio of diameters of s2 s 5 and x 2 s 0.8981. v: Computer simulation data w19x; : Results using Eq. Ž25.; - - -: Results using MCSL. s1
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
141
Fig. 6. Prediction of compressibility factor simulation data vs. h for a mixture of hard spheres with the ratio of diameters of s2 s 20 and x 2 s 0.1019. v: Computer simulation data w19x; : Results using Eq. Ž25.; - - -: Results using MCSL. s1
Fig. 7. Prediction of compressibility factor simulation data vs. h for a mixture of hard spheres with the ratio of diameters of s2 s 20 and x 2 s 0.5. v: Computer simulation data w19x; : Results using Eq. Ž25.; - - -: Results using MCSL. s1
142
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
Fig. 8. Prediction of compressibility factor simulation data vs. mole fraction for a mixture of hard spheres with the ratio of s2 diameters of s 2. v: Computer simulation data w20x; : Results using Eq. Ž25.. s1
Figs. 1–3 show that the results obtained with the new form, which meets the correct limits at close packing and at low density, are perfectly comparable to the results obtained with the MCSL equation. This proves that the generalized mixing rule can be used with confidence as more sophisticated forms of the pure compound hard sphere equation of state become available. Figs. 2, 3 and 6 clearly indicate the advantage gained in the compressed region when using an equation of state with the correct behaviour at close packing. For high ratios of the diameters of the hard spheres, such as 20, the results obtained in this work, and shown in Fig. 6, are in better agreement with computer simulation data than those obtained with MCSL. It must be clearly stated that the use of j 3 s 1 as the pole for the equation of state for mixtures, for the case of tetrakaidecahedron cells, can be as problematic as the use of h 3 s 1 as the pole for the case of spherical cells. In fact the value of the pole for a mixture of hard spheres with different diameters is not known. The only basis for this selection is that the latter is the rigorous result which was obtained by Lebowitz w13x from the closure of the integral Percus–Yevick equation and it was tacitly incorporated in the MCSL equation. However, due to the rhombohedral array of centres in the case of tetrakaidecahedral cells, the problem, if any, is expected to appear at larger diameter ratios than in the case of spherical cells. Some of the figures presented here compare the results of our equation of state for densely packed systems. At high densities, the solidification of hard spheres or a meta-stable equilibrium phenomenom may occur. Thus, it seems important to make sure about the quality of simulation data obtained from the literature. Our survey in the literature showed that the simulation data for hard sphere systems at high density presented by Jackson et al. w19x suit the best our purposes. This is mainly due to the fact that for the Monte Carlo Ž MC. simulation they have used the isobaric
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
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Table 1 Comparison of the RDF computer simulation data ŽMD. reported by Fries and Hansen w20x with values obtained from Eq. Ž25., from MCSL equation and from P–Y equation for the RDF at the contact value of the unlike hard spheres with the ratio s2 of diameters of s2 s1 h x MD P–Y MCSL Eq. Ž25. 0.45 0.50 0.55 0.45 0.50 0.55 0.59 0.45 0.50 0.55
0.05000 0.05000 0.05000 0.10156 0.10156 0.10156 0.10156 0.19922 0.19922 0.19922
5.20 6.10 9.00 4.60 5.70 7.40 11.00 4.30 5.10 6.70
4.35 5.41 6.85 4.09 5.05 6.36 7.79 3.80 4.67 5.85
5.14 6.70 8.99 4.71 6.08 8.08 10.40 4.28 5.46 7.16
4.75 5.98 7.77 4.67 5.98 8.02 11.26 4.54 5.91 8.11
ensemble, MCŽ T, P, N ., rather than the canonical ensemble, MCŽ T, V, N .. This is important because it is generally accepted that MCŽ T, P, N . simulations generate more stable and physically meaningful configurations than MCŽ T, V, N . simulations. Jackson et al. w19x have also performed molecular dynamic simulations, MD, from one of the stable MC configurations to check their results, and good agreement was found between the results obtained from both methods. Although for hard sphere
Fig. 9. Prediction of residual Gibbs free energy simulation data vs. h for a mixture of hard spheres with the ratio of s2 diameters of s 3 and x 2 s 0.5. v: Computer simulation data w18x; : Results using Eq. ŽA.5.. s1
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systems at high densities, the danger of solidification always exists, special care was taken by Jackson et al. w19x to significantly reduce this possibility. Table 1 compares the computer simulation data for the RDF at the contact value for the unlike hard spheres with the results obtained from Eq. Ž 22. developed in this study, with the MCSL equation of state w5x and with the P–Y equation w14x, for the systems shown in Fig. 7. As it can be seen from this comparison, in general, Eq. Ž 22. can represent the contact value computer simulation RDF data better than the other two models. Fig. 9 compares computer simulation data for the residual Gibbs free energy for a mixture of hard spheres w18x, with the results obtained from Eq. ŽA.5. of Appendix A. As it can be seen from this comparison, the residual model presented in this work can represent the residual Gibbs free energy computer simulation data with good accuracy. This can be construed as a proof of the reliability of the proposed mixing rule, which can be directly extended to more complex models.
5. Conclusions A generalized analogical approach has been proposed to extend the pure compound RDF to the multicomponent RDF of hard sphere systems. The approach is based on the similarity of the pure compound and multicomponent RDF for the Percus–Yevick and the Carnahan–Starling equations of state of hard spheres. The generalized approach was applied to an equation of state for pure compound hard spheres proposed by Khoshkbarchi and Vera w8x, which meets the correct limit at the close-packed state of hard spheres. The equation of state for a mixture of hard spheres thus obtained was employed to predict the computer simulation compressibility factor and contact value RDF. No adjustable parameter has been employed. The results show that the equation of state developed here can accurately predict the simulation data while satisfying the correct limit at close-packed. Expressions for the residual Helmholtz free energy and the residual chemical potential, based on the new equation of state, have been derived.
6. List of symbols
A bi Gi G gŽ s . gi k N P r T
Helmholtz free energy Parameters of Eq. Ž 1. Functions of Eq. Ž18. Gibbs free energy Radial distribution function at contact value Function of Eq. Ž14. Boltzman constant Number of spheres Pressure Radial distance Absolute temperature
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
ui j V V0 x Z
145
Pair potential between hard spheres Total volume Ns 3 Total volume of hard spheress '2 Mole fraction Compressibility factor
Greek letters
bi h h0
Coefficients of virial expansion p rs 3 6 p'2
m
6 Chemical potential
j
Packing factor in tetrakaidecahedron cells s
r s
Number density Diameter of a hard sphere
V0 V
Superscripts CS MCSL R r
Carnahan–Starling Mansoori–Carnahan–Starling-Leland Residual, w T, P, N x Residual, w T, V, N x
Subscripts
a b
Hard sphere of type a Hard sphere of type b
Acknowledgements The authors are grateful to the Natural Sciences and Engineering Research Council of Canada for financial support. Appendix A Combining Eqs. Ž25. and Ž 30. provides with an expression for the residual Helmholtz free energy as: A r T ,V , N 1.260Y1 y 13.70Y2 0.4550Y2 s q y F1Y2 q 13.25Y2 y 1.260Y1 y F2 ln D Ž A.1 . NkT D D2
M.K. Khoshkbarchi, J. H. Vera r Fluid Phase Equilibria 142 (1998) 131–147
146
where
Ds1yj3
Ž A.2 .
F1 s 64.44j 3 q 25.65j 32 q 13.25j 33 q 7.455j 34 q 4.260 j 35 q 2.366 j 36 q 1.217j 37 q 0.5325j 38 q 0.1578 j 39
Ž A.3 . Ž A.4 .
F2 s 0.7400 q 0.9600Y1 q 77.23Y2 The residual Gibbs free energy, G r, can be calculated as: GR T,P,N
A r T ,V , N
q Z y 1 y ln Z Ž A.5 . NkT NkT Combining Eqs. Ž31. and ŽA.1. provides with an expression for the residual chemical potential, m r w T, V, N x, as: s
m r T ,V , N
A r T ,V , N
y F3
s kT q
NkT
F2 j 3 R3 D
q
j2 R3 D
j 3 Ž 1.260Y1 y 13.70Y2 .
y 0.9600Y1
R3 D2 1
1
R1
y
y
R2
R3
y
D
q F1 j 3Y2
ln D
1
q
30.11 2
R0
y 140.6 q
1
3 y
R3
0.9100Y2 j 3 3
D R3
y 1.260 F4
1yD
D
1 q
R2
R0
Ž A.6 .
where Rk s
r i si k
Ž A.7 .
Ý r j sj k j
F3 s Y2 F4 s Y1
0.4550
1.365 y
R0 1
R2 1
y R0
0.9100 q
1 y
R1
R3 1 q
R2
R3
Ž A.8 . Ž A.9 .
References w1x N.F. Carnahan, K.E. Starling, Equation of state of nonattracting rigid spheres, J. Chem. Phys. 51 Ž1969. 635–636. w2x B. Barboy, W.M. Gelbart, Series representation of the equation of state for hard particle fluids, J. Chem. Phys. 71 Ž1979. 3053–3062. w3x T. Boublik, I. Nezbeda, P-V-T behaviour of hard body fluids: Theory and experiment, Collection Czechoslovak Chem. Commun. 51 Ž1986. 2301–2432. w4x Boublik, 1971. w5x G.A. Mansoori, N.F. Carnahan, K.E. Starling, T.W. Leland Jr., Equilibrium thermodynamic properties of the mixture of hard spheres, J. Chem. Phys. 54 Ž1971. 1523–1525.
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w6x H. Reiss, Scaled Particle Methods in the Statistical Thermodynamics of Fluids, Advances in Physical Chemistry, Vol. IX, pp. 1–84, in: I. Prigogine ŽEd.., Interscience Publishers, New York, 1965. w7x W. Wang, M.K. Khoshkbarchi, J.H. Vera, A new volume dependence for the equation of state of hard spheres, Fluid Phase Equilib. 115 Ž1996. 25–38. w8x M.K. Khoshkbarchi, J.H. Vera, A simplified hard sphere equation of state meeting the high and low density limits, Fluid Phase Equilib. 130 Ž1997. 189–194. w9x M.S. Wertheim, Exact solution of the Percus–Yevick integral equation for hard spheres, Phys. Rev. Letters 10 Ž1963. 321–323. w10x M.S. Wertheim, Analytic solution of the Percus–Yevick equation, J. Math. Phys. 5 Ž1964. 643–651. w11x T.L. Hill, Statistical Mechanics, Principles and Selected Applications, Dover Publ., New York, 1956. w12x T.M. Reed, K.E. Gubbins, Applied Statistical Mechanics, McGraw Hill, New York, 1973. w13x J.L. Lebowitz, Exact solution of generalized Percus–Yevick equation for mixture of hard spheres, Phys. Rev. 133 Ž1964. A895–A899. w14x J.L. Lebowitz, J.S. Rowlinson, Thermodynamic properties of mixtures of hard spheres, J. Chem. Phys. 41 Ž1964. 133–138. w15x T. Boublik, Hard-sphere equation of state, J. Phys. Chem. 53 Ž1970. 471–472. w16x Iglesias-Silva, Hall, 1994. w17x J.M. Smith, H.C. Van Ness, M.M. Abbott, Introduction to Chemical Engineering Thermodynamics, McGraw-Hill, New York, 1996. w18x B.J. Alder, Studies in molecular dynamics: III. A mixture of hard spheres, J. Chem. Phys. 40 Ž1964. 2724–2730. w19x G. Jackson, J.S. Rowlinson, F. van Swol, Computer simulation of mixture of hard spheres, J. Phys. Chem. 91 Ž1987. 4907–4912. w20x P.H. Fries, J.P. Hansen, A Monte Carlo study of semi-dilute hard sphere mixtures, Mol. Phys. 48 Ž1983. 891–901.