1122
Ind. Eng. Chem. Res. 2002, 41, 1122-1128
A General Expression for the Ordered-Packed Volume Fraction of Hard Spheres of Different Diameters Cyrus Ghotbi†,‡ and Juan H. Vera*,† Department of Chemical Engineering, McGill University, Montreal, Quebec, Canada H3A 2B2, and Chemical Engineering Department, Sharif University of Technology, Tehran, Iran
An expression is proposed to evaluate the ordered-packed volume fraction of hard spheres of unequal diameters in multicomponent mixtures. This value can be used with different mixing rules and with any one-component hard-sphere equation of state (EOS) meeting the correct close-packed limit. It is shown that the new method improves the results produced by conventional mixing rules for the individual radial distribution functions (RDFs) at contact value of hard-sphere mixtures. Particular attention is given to the case of the large diameter ratio especially when the larger spheres are at low concentration. Based on the observation that, in a binary mixture, the value of the RDF at contact value of the larger spheres approaches the value of the RDF at contact value of a one-component hard-sphere fluid, the proposed expression is used to correct the RDF at contact value of a pure hard-sphere fluid in the Santos et al. mixing rule. The EOS obtained by combining the Ghotbi-Vera one-component EOS with the corrected Santos et al. mixing rule is used to predict different thermodynamic properties, and the results are compared with the computer-simulated data. Introduction Close-ordered or random packing of spheres is of interest in science, engineering, and medicine.1-6 The classical problem of sphere packing, a matter of discussion even today despite its long history, is to find out how densely a large number of spheres of different diameters can be packed closely together.1 For random close packing of identical spheres into a three-dimensional space, a rate-dependent densification algorithm achieves a packing fraction between 0.642 and 0.649 while a Monte Carlo scheme gives a packing fraction of 0.68.6 It has been argued that close packing of identical spheres into a three-dimensional space can occupy 0.78 of the total space.3,4 As Rogers2 remarked, “many mathematicians believe and all physicists know” that the densest possible packing fraction for identical spheres is
η0p )
Π Πx2 ) ) 0.7404 6 x18
(1)
In eq 1 we have used the subscript p to indicate that this result is only valid for the close-packing of hard spheres of equal diameter. In the well-known Kelvin’s conjencture,7,8 the value of the close-packing volume fraction for identical spheres is also η0p. Weaire and Phelan9 proposed a new geometrical structure with a surface exergy which is approximately 0.3% less than that of Kelvin’s solution. The nature of the empty space between two or more layers of close-packed spheres has been the subject of detailed study.1 In the case of mixtures of spheres of * To whom correspondence should be addressed. E-mail:
[email protected]. † McGill University. ‡ Sharif University of Technology. E-mail: ghotbi@ sina.sharif.ac.ir.
very unequal diameters, the small ones can fill the spaces created by the closed packing of the large spheres. This arrangement can produce a close-packed volume fraction even larger than 0.9.5 In thermodynamics, the study of the behavior of the hard-sphere fluid is usually the first step for the development of perturbation theories used to represent the behavior of real fluids. In the particular case of the study of colloidal systems and surface phenomena, the ratio of the diameters of different spheres is very large while the composition of the larger spheres is usually very small. This is a particularly difficult case to treat from the modeling point of view. The equation of state (EOS) of Carnahan-Starling,10 which is commonly used for one-component hard-sphere fluids, has a simple form and good accuracy in the low to moderate density region. However, this equation fails to satisfy the close-packed limit, which is important for the dense fluid region. Iglesias-Silva and Hall11 proposed two EOSs for onecomponent hard-sphere fluids based on the random close packing of spheres with a packing fraction limit value of 2/π. Some hard-sphere EOSs which satisfy the closepacked limit fraction value of η0p have been proposed in the literature.12-16 Extension of a one-component hard-sphere EOS to a multicomponent mixture is normally done using mixing rules.17-20 The accuracy of these mixing rules in predicting the radial distribution function (RDF) at contact value, the compressibility factor, and the excess chemical potential has been reported15,16 for a wide range of packing fraction, composition, and diameter ratio values for binary and ternary hard-sphere mixtures. For mixtures of hard spheres with large diameter ratios, in the high-density region, all of the aforementioned mixing rules fail to represent accurately the compressibility factor and especially the individual RDF at contact value.15,16,21 Henderson et al.22 proposed expressions for the RDF at contact value for a binary
10.1021/ie010793d CCC: $22.00 © 2002 American Chemical Society Published on Web 02/09/2002
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1123
hard-sphere fluid in order to give better results when the diameter ratio of the spheres is very large and the composition of the larger spheres is very small. In this work, we propose an expression to correct the close volume packing fraction for multicomponent hardsphere fluids. This expression is used in combination with the one-component hard-sphere EOS recently proposed by Ghotbi and Vera16 and with the generalized mixing rule proposed by Santos et al.20 for mixtures of hard spheres. Special attention is given to mixtures with large diameter ratios in the high-density region. Correction for the Ordered-Packed Volume Ratio for a Mixture of Hard Spheres Ghotbi and Vera16 recently proposed two new closedvirial hard-sphere EOSs for a one-component fluid meeting the correct Kelvin limit at the close-packed condition. These equations are expressed in terms of the ratio of the volume V0 occupied by the spheres at the close-packed limit to the total volume V of the system:
ξ ) V0/V ) η/η0p
(2)
For the purposes of this study, we have selected the simplest of these equations which has the form
ZP ) 1 + 2.96ξ + 5.48ξ2 + 7.46ξ3 + 8.49ξ4 + 8.9ξ5 - 2.8ξ8 (3) 1-ξ The pure-compound RDF at contact value is obtained as
gP(σ) )
ZP - 1
(4)
4ηP0ξ
where ZP and gP(σ) are the compressibility factor and the RDF at contact value for a one-component hardsphere fluid and η0p ) (π/x18) is the pure closed-packed volume fraction, which is independent of the diameter of the hard spheres. Extension of the above equations to multicomponent hard-sphere fluids can be performed using the method proposed by Santos et al.20 According to this method, the compressibility factor and the RDF at contact value for a mixture are related to those for a pure-compound hard-sphere fluid by the following relations:
(
Zmix ) 1 + (ZP - 1)
gij(σij) )
)
Y1 + Y 2 η (1 + Y1 - 2Y2) + 2 1-η (5)
1 1 η2 σ iσj + gP(σ) 1-η 1 - η η σij
[
]
(6)
where
Y1 ) Y2 )
η1η2 ξ1ξ2 ) η0 η ξ0ξ η23 η0η2
)
ξ23 ξ0ξ2
(7)
(8)
σij ) ηk )
ξk )
ηk η0p
σi + σj 2
Π
Fjσjk ∑ 6 j
)
1
x2
∑j Fjσjk
(9) (10)
(11)
In the above equations, σi is the hard-sphere diameter of compound i, Fj ) Nj/V is the number density of compound j, and η3 is the volume packing fraction. For simplicity, we use here the abbreviations ξ ) ξ3 and η ) η3. It has been shown15,16 that, for a binary hard-sphere mixture with a large diameter ratio in the high-density region, the combination of eq 4 with eq 6 produces results with large deviations from computer-simulated data20 for the individual RDF at contact value. This is especially noticeable for a binary system of hard spheres with large diameter ratios at low large-sphere concentrations. The deviation of the compressibility factor from the simulated data21 is also considerable in this case. The same results were also obtained,15,16,21 using the Boublik-Mansoori-Carnahan-Starling-Leland (BMCSL) EOS for hard spheres.17,18 Detailed consideration of previous results15,16 shows that, for a binary hard-sphere fluid, if σ2 . σ1, then the values obtained with eq 5, and with eq 6 for the spheres with larger diameter, are quite close to those obtained for a pure compound, i.e., g22 = gP and also Zmix = ZP. This fact suggests that the values of the RDF at contact value and the compressibility factor can be predicted accurately by eqs 5 and 6 if the pure-compound properties used in these equations are replaced with modified values which correct for mixture effects. The modification of the pure-compound property values in eqs 5 and 6, ZP and gP, can be performed by correcting the value of the pure-compound close-packed volume fraction (η0p) for the effect due to the packing of unequal hard spheres. As stated above, the value of the η0p for identical spheres is π/x18 = 0.7404. In the case of mixtures of spheres of unequal diameters, the space created by close-packing spheres with a large diameter in Kelvin tetrakaidecahedron cells23 can be filled by spheres with a smaller diameter. If the second kind of spheres is small enough to completely fill all of the holes, to the maximum packing limit, the value of the close packing volume fraction can be increased up to 0.7404 + (1 0.7404) × 0.7404 ) 0.9326. If the process is continued by adding a third component of small enough spheres, the value of the close-packing volume fraction raises to 0.9825. The assumption that the third kind of spheres will be small enough to fill 0.7404 of the total free volume created by the Kelvin structure of the largediameter spheres will seldom be valid in practical applications. Random packing, in addition, tends to decrease the occupied volume. Thus, for the case of packing of a multicomponent system of spheres of different diameters, it seems logical to consider a corrected close-packed volume fraction of the form
η0m ) η0p + F(C, xi, σi)
(12)
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Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002
where C is the number of components and xi is the mole fraction of compound i with diameter σi in the mixture. The function F should become zero in the case of a onecomponent hard-sphere fluid, C ) 1, or at any composition for the case of the mixture of hard spheres with equal diameters, σj ) σi. When η0p is replaced by η0m in eq 11, the values of gij and Zmix in eqs 5 and 6 are corrected as follows:
ξm ) η/η0m
(13)
(
)
Y1 + Y2 Ar K1 - (1 + Y1 - 2Y2) ln(1 - η) (19) ) NkT 2 and
(
)
µri M1,i + M2,i Ar ) + K1 + kT NkT 2 Y1 + Y2 ∂K1 - (M1,i - 2M2,i) ln(1 - η) + 2 ∂Fi T,V,Fj×b9i
(
)( )
(1 + Y1 - 2Y2)ηR3,i (20) 1-η
Zm ) 1 + 2.96ξm + 5.48ξm2 + 7.46ξm3 + 8.94ξm4 + gm(σ) )
8.9ξm5 - 2.8ξm8 (14) 1 - ξm
Zm - 1 4η
(15)
In the above equations, k is the Boltzmann constant, and T is the absolute temperature. In addition,
K1 ) -3.14ξm - 0.31ξm2 + 0.45ξm3 + 0.6ξm4 + 0.56ξm5 + 0.47ξm6 + 0.4ξm7 - 6.1 ln(1 - ξm) (21)
and the Santos et al.20 mixing rule then takes the form
(
Zmix ) 1 + (Zm - 1)
)
Y1 + Y2 η + (1 + Y1 - 2Y2) 2 1-η (16)
M1,i ) Y1(R2,i + R1,i - R3,i - R0,i)
(22)
M2,i ) Y2(3R2,i - R0,i - 2R3,i)
(23)
with
and
gij(σij) )
1 1 η2 σ iσj + gm(σ) 1-η 1 - η η σij
[
]
F(C, xi, σi) )
100
(
∑i xiσi2
∑i ∑j
xixjσiσj
σik
(24)
∑j xjσj
k
Comparison of eqs 5 and 16, and also of eqs 3 and 14, shows that the single difference between the present treatment and the Santos et al.20 mixing rule is that ξm, as defined by eq 13, replaces ξ ) ξ3. Similarly, comparison of eqs 6 and 17, and also of eqs 4 and 15, shows that the function gm, defined by eq 15, is the corrected value of the radial distribution function gp, for the case of packing of a multicomponent system of hard spheres of different diameters. To find a suitable expression for F(C, xi, σi) in eq 12 to correct η0p, different functions were studied. The deviations of the values of the compressibility factor, the pair RDF at contact value, and the excess chemical potentials from the simulated data21,23,24 for a wide range of packing fractions, compositions, and diameter ratios were calculated. Based on this study, the following empirical expression to correct η0p is proposed.
(C - 1)0.5
Rk,i )
(17)
)
-1
and
( ) ∂K1 ∂Fi
) (Zm - 1)[R3,i - 0.019(C - 1)0.5E]
where 2 S(2) j + σi -
E)
∑j xjσj
(27)
∑j ∑l xjxlσjσl
(28)
∑j xjσj2
(29)
Sj )
(18)
2S(2) j Sjσi Sjl
[(x2 - 0.019(C - 1)0.5)Sjl + 0.019(C - 1)0.5S(2) j ] (26)
Sjl )
In the case of a one-component hard-sphere fluid of any diameter or for a mixture of hard spheres of the same diameter at any composition, η0m, in eq 12, reduces to η0p. Equation 18 is, certainly, not unique. However, of the many expressions tested, it consistently gave the best results. Although no physical interpretation is associated with eq 18, the effect of the correction it introduces increases with the number of species present and also as the difference in diameters between species becomes larger. The expressions for the residual Helmholtz free energy, Ar, and the residual chemical potential, µri , generated using eqs 10-16 are
(25)
T,V,Fj*i
and
S(2) j ) Results and Discussion
The close-packing volume fraction of identical hard spheres, η0p, has been replaced by the modified closepacking volume fractions, η0m, because of the packing of unequal hard spheres in the definition of ξ ) η/η0. The combination of the corrected Ghotbi-Vera16 EOS with the Santos et al.20 mixing rule has been used to calculate the pair RDF at contact value, the compressibility factor, and the chemical potentials. For comparison with the computer-simulated values reported in the litera-
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1125
Figure 1. Pair radial distribution function at contact value, g22, as a function of the mole fraction of the larger spheres, x2, for a binary hard-sphere mixture with η ) 0.5899 and σ2/σ1 ) 5/3: 0, simulated datum;21 - - -, BMCSL EOS;17,18 - - -, Henderson’s approach;22 - - -, eqs 3 and 5; s, this work.
Figure 2. Pair radial distribution function at contact value, g11, as a function of the mole fraction of the larger spheres, x2, for a binary hard-sphere mixture with η ) 0.5569 and σ2/σ1 ) 20: 0, simulated data;21 - - -, BMCSL EOS17,18 and Henderson’s approach;22 - - -, eqs 3 and 5; s, this work.
ture,21,24,25 we have also included the results obtained with the BMCSL17,18 EOS, the results reported by Henderson et al.22 using an ad hoc function to correct for mixing rule, and the results obtained by the combination of the original Ghotbi-Vera16 EOS with the Santos et al.20 mixing rule. Figures 1-4 show the results of the prediction of the pair RDF at contact value, using different EOSs for the binary hard-sphere mixtures, as a function of the mole fraction of the larger spheres, x2, for the value of η ) 0.5899 with a diameter ratio of σ2/σ1 ) 5/3 and for the value of η ) 0.5569 with a diameter ratio of σ2/σ1 ) 20. In Figures 1-4, the computer-simulated values21 are also shown. As expected, for relatively low diameter ratios of binary hard spheres such as 5/3, the results obtained in this work are similar to those obtained using eqs 4 and 6 with the constant value of η0P ) π/x18. For high diameter ratios, such as 20, in binary hard-sphere mixtures the pair RDFs at contact values calculated in
Figure 3. Pair radial distribution function at contact value, g12, as a function of the mole fraction of the larger spheres, x2, for a binary hard-sphere mixture with η ) 0.5569 and σ2/σ1 ) 20: 0, simulated data;21 - - -, BMCSL EOS;17,18 - - -, Hendreson’s approach;22 - - -, eqs 3 and 5; s, this work.
Figure 4. Pair radial distribution function at contact value, g22, as a function of the mole fraction of the larger spheres, x2, for a binary hard-sphere mixture with η ) 0.5569 and σ2/σ1 ) 20: 0, simulated data;21 - - -, BMCSL EOS;17,18 - - -, Henderson’s approach;22 - - -, eqs 3 and 5; s, this work.
this work are different from those obtained from any of the other equations. This difference becomes more important in the region of low concentration of larger spheres (spheres 2) for g22. As shown in Figures 2-4, the predicted values of the pair RDF at contact value are in good agreement with the computer-simulated values21 for high diameter ratios and in low concentration regions of the larger spheres. In Figures 2 and 3, it can be observed that the values of g11 and g12 obtained in this work for high diameter ratios vary more sensibly with composition in comparison with the other equations. These values are similar to the values obtained using eqs 4 and 6 with the constant close-packing volume fraction in the region of high concentration of the larger spheres, and they
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Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002
Table 1. Comparison the Pair RDF at Contact Value, gij, the Compressibility Factor, Z, and the Pure RDF at Contact Value, gp, with Computer-Simulated Values21 no.
R
x2
η
1
20
0.1019
0.5178
2
3
4
5
6
7
8
9
10
11
20
20
20
5
3
3
5/3
5/3
5/3
5/3
0.1019
0.5
0.037
0.5
0.0648
0.0648
0.5
0.5
0.5
0.5
0.5569
0.3004
0.5569
0.3016
0.2
0.4515
0.1047
0.4982
0.5271
0.5899
MC BMCSL Henderson eqs 3 and 5 this work MC BMCSL Henderson eqs 3 and 5 this work MC BMCSL Henderson eqs 3 and 5 this work MC BMCSL Henderson eqs 3 and 5 this work MC BMCSL Henderson eqs 3 and 5 this work MC BMCSL Henderson eqs 3 and 5 this work MC BMCSL Henderson eqs 3 and 5 this work MC BMCSL Henderson eqs 3 and 5 this work MC BMCSL Henderson eqs 3 and 5 this work MC BMCSL Henderson eqs 3 and 5 this work MC BMCSL Henderson eqs 3 and 5 this work
approach the values obtained using BMCSL EOS in the region of low concentration of the larger spheres. The values calculated with Henderson’s approach22 deviate considerably from those obtained with all other models for the case presented in Figure 1. For the case presented in Figure 2, Henderson’s results correspond exactly with those of the BMSCL EOS and both of these models correspond closely for the cases presented in Figures 3 and 4. The values of g22, obtained in this work for the high diameter ratio, as shown in Figure 4, deviate considerably from the values calculated using any of the other equations, as the concentration of the larger spheres decreases, and they come closer to simulated results. Table 1 shows the comparison of the results obtained by using different EOSs with the computer simulation
g11(σ11)
g12(σ12)
g22(σ22)
Z
2.25 2.247 2.247 2.307 2.248 2.54 2.479 2.479 2.57 2.486 1.44 1.476 1.476 1.483 1.48 2.69 2.488 2.488 2.583 2.484 1.6 1.629 1.629 1.653 1.646 1.52 1.528 1.528 1.539 1.535 3.35 3.294 3.294 3.451 3.43 1.19 1.252 1.252 1.254 1.253 4.63 4.427 4.427 4.654 4.642 4.48 5.079 5.079 5.393 5.377 6.03 7.105 7.105 7.866 7.833
2.41 2.41 2.415 2.517 2.406 2.7 2.687 2.696 2.853 2.694 1.56 1.518 1.519 1.531 1.525 2.74 2.705 2.714 2.877 2.69 1.8 1.767 1.774 1.8 1.789 1.68 1.676 1.687 1.684 1.678 4.03 4.177 4.334 4.266 4.233 1.28 1.287 1.288 1.289 1.287 5.05 5.173 5.338 5.32 5.304 6.24 6.004 6.22 6.213 6.193 9.09 8.627 9.022 9.222 9.282
5.56 6.73 6.682 6.733 5.563 6.75 8.458 8.387 8.522 6.843 2.48 2.485 2.484 2.493 2.436 5.33 8.783 8.594 8.772 6.805 2.59 2.531 2.523 2.536 2.503 2.01 2.158 2.144 2.118 2.106 8.74 7.42 8.489 6.708 6.644 1.33 1.346 1.344 1.346 1.344 6.64 6.54 6.364 6.429 6.408 7.82 7.712 7.484 7.579 7.552 11.52 11.48 11.09 11.48 11.43
3.472 3.719 3.712 3.778 3.471 4.095 4.476 4.466 4.588 4.115 2.759 2.757 2.756 2.764 2.729 3.15 3.413 3.403 3.519 3.241 2.975 2.979 2.976 2.99 2.968 1.951 1.955 1.958 1.959 1.956 6.266 6.228 6.412 6.324 6.286 1.506 1.505 1.505 1.506 1.505 11.5 11.48 11.48 11.55 11.52 14.03 13.97 13.96 14.09 14.05 22.46 22.2 22.21 22.95 22.85
gp/gm 6.61 6.64 5.49 8.29 8.39 6.75 2.48 2.49 2.43 8.29 8.39 6.54 2.49 2.5 2.47 1.76 1.76 1.75 4.69 4.7 4.66 1.32 1.32 1.32 5.94 5.96 5.94 6.96 7 6.98 10.22 10.53 10.48
data for the pair RDF at contact value and the compressibility factor21 for a binary mixture. In comparison with the other equations, the use of the modified closepacking volume fraction, eq 18, in the calculation of ξ improves considerably the prediction of g22 and of the compressibility factor for high diameter ratios and low concentration of the larger spheres. As expected, for high diameter ratios of binary hard-sphere mixtures, the predicted values of g22 are very close to the predicted values of the one-fluid RDF at contact value, gp, in the case of the BMCSL EOS or the Santos et al. mixing rule. For this work, the predicted values of g22 are very close to the corrected values of gm, which are in good agreement with the computer-simulated values. It should be noted that the results obtained in this study for low
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1127 Table 2. Averaged Absolute Deviations (% AADa) and Averaged Deviations (% ADb) of the Pair RDF at Contact, gij, and the Compressibility Factor, Z, from the Simulated Values21 (x1 ) 0.50-0.96, σ2/σ1 ) 1.67-20, and η ) 0.10-0.56) % AADa (% AD)b mixing rule
EOS
g11
g12
g22
Z
Boublik-Mansoori Santos et al.
Carnahan-Starling eq 3 this work (eq 14)
3.93 (+1.86) 5.65 (+5.18) 5.54 (+4.38)
1.78 (-1.01) 2.23 (+1.61) 1.4 (+0.31)
8.76 (+6.14) 9.41 (+5.22) 4.78 (-0.12)
1.85 (+1.17) 2.35 (+2.19) 0.62 (+0.17)
a
% AAD (Property) )
100
N
∑ABS N i)1
b
% AD (Property) )
100
N
∑ N i)1
[
[
]
(Property)cal. - (Property)sim. (Property)sim.
]
i
(Property)cal. - (Property)sim. (Property)sim.
i
Table 3. Averaged Absolute Deviations (% AAD) of the Pair RDF at Contact, gij, the Compressibility Factor, Z, and the Excess Chemical Potentials, βµiex, from the Simulated Values23 (x1 ) 0.06-0.75, σ2/σ1 ) 0.3-0.9, η ) 0.30-0.49) mixing rule
EOS
% g11
% g12
% g22
%Z
% βµ1ex
% βµ2ex
Boublik-Mansoori Santos et al.
Carnahan-Starling eq 3 this work (eq 14)
2.32 4.01 4.53
0.59 2.44 2.05
0.51 3.26 2.92
0.44 0.20 0.20
0.48 0.26 0.25
0.39 0.28 0.33
Table 4. Averaged Absolute Deviations (% AAD) of the Pair RDF at Contact, gij, and the Compressibility Factor, Z, for Ternary Mixtures from the Simulated Values24 (x1 ) x2 ) x3 ) 1/3 or x1 ) 1/6, x2 ) 1/3, x3 ) 1/2, σ1 ) 1, σ2 ) 0.6, σ3 ) 0.3, η ) 0.35, 0.40, 0.45) mixing rule
EOS
% g11
% g22
% g33
% g12
% g13
% g23
%Z
Boublik-Mansoori Santos et al.
Carnahan-Starling eq 3 this work (eq 14)
0.89 3.16 4.40
1.70 4.30 3.24
1.95 6.02 5.30
0.70 1.07 1.24
1.48 5.14 4.21
0.99 4.91 4.06
0.21 0.98 0.14
diameter ratios are also perfectly comparable to the results obtained with the other equations. Tables 2-4 show the percent average absolute deviations (% AAD) of the calculated properties from the computer-simulated data. To show the distribution of the deviations, the percent average deviations (% AD) of the calculated properties from the simulated data are also shown in Table 2. The simulated data used for comparison in Tables 2-4 are those given by Yau et al.21 and by Barosova et al.24 for binary mixtures and by Sindelka and Boublik25 for ternary mixtures. As shown in Table 2, there is a considerable improvement in the prediction of the individual pair RDF at contact value and the compressibility factor for high diameter ratio, which results in an important decrease of the % AAD and produces much better distribution of the deviations of these properties. Tables 3 and 4 show that employing a corrected closepacking volume fraction, eq 16, in the combination of the Ghotbi-Vera16 EOS with the Santos et al.20 mixing rule produces better results, in most of the cases, for a wide range of packing fractions, compositions, and diameter ratios in comparison with the values obtained using the unmodified close-packing volume fraction for identical spheres. While the pair RDF values in the low to moderate density region for binary and ternary mixtures calculated by the BMCSL EOS better represent the simulated values, the compressibility factor and the chemical potentials are better predicted by the method proposed in this work. This phenomenon can be explained by the better compensation of the deviations of the pair RDF values obtained in this work in comparison with those obtained in the BMCSL EOS. Conclusions The prediction of the individual RDF at contact value using BMCSL EOS or the combination of eq 4 with eq
6 produced important deviations from the simulated data for binary hard-sphere fluids with large diameter ratios. While, for these mixtures, the individual RDF at contact value of the larger sphere, g22, in the Santos et al. mixing rule approaches to gp, an expression was proposed to replace the pair RDF at contact value of a pure-compound hard sphere in eq 4. In this expression the close-packing volume fraction of unequal hardsphere mixtures is a function of diameter, composition, and number of different spheres participating in the mixture. Using the modified values of the close-packing volume fraction obtained in this study, eqs 12 and 18, in the definition of ξm ) η/η0m, the prediction of the values of g22 and the compressibility factor for the mixtures of hard spheres with large diameter ratios improved considerably. The combination of eq 11, Ghotbi-Vera EOS,16 and Santos et al.20 mixing rule gave also an accurate reproduction of the compressibility factor and the excess chemical potentials for a wide range of packing fraction, composition, and diameter ratio for binary and ternary hard-sphere fluids. The proposed expression can be combined with any other accurate EOS for a one-component hard-sphere fluid which meets the ordered close-packed limit. Acknowledgment The authors are grateful to the Natural Sciences Research Council of Canada (NSERC) for financial support. Nomenclature A ) Helmholtz free energy C ) number of components E ) function of eq 23 F ) correction term defined in eq 10 g ) radial distribution function at contact value
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Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002
k ) Boltzmann constant K1 ) function of eq 17 M1,i ) (∂Y1/∂Fi)T,V,Fj×b9i M2,i ) (∂Y2/∂Fi)T,V,Fj×b9i N ) number of spheres P ) pressure RK,i ) σik/∑jxjσkj S ) summation on σ T ) absolute temperature V ) total volume V0 ) total volume of hard spheres at ordered closed-packing limit x ) mole fraction Y1 ) volume-independent function Y2 ) volume-independent function Z ) compressibility factor Greek Letters η ) packing fraction η0 ) ordered closed-packing limit for identical spheres µi ) chemical potential of component i ξ ) V0/V F ) number density σ ) hard-sphere diameter Superscript r ) residual [T, V, N] Subscripts i ) hard sphere of type i j ) hard sphere of type j P ) pure m ) modified values mix ) mixture
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Received for review September 25, 2001 Revised manuscript received December 11, 2001 Accepted December 12, 2001 IE010793D
