Ind. Eng. Chem. Res. 1998, 37, 221-227
221
GENERAL RESEARCH Comparison between Different Explicit Expressions of the Effective Hard Sphere Diameter of Lennard-Jones Fluid: Application to Self-Diffusion Coefficients Carlos M. Silva,† Hongqin Liu,‡ and Euge´ nia A. Macedo*,† LSRE Laboratory of Separation and Reaction Engineering, Departamento de Engenharia Quı´mica, Faculdade de Engenharia da Universidade do Porto, Rua dos Bragas, 4099 Porto Codex, Portugal, Chemical Engineering Department, Beijing University of Chemical Technology, 100029 Beijing, People’s Republic of China, and Instituto Politec´ nico de Braganc¸a, Braganc¸a, Portugal
The effective hard sphere diameter (EHSD) method has been widely used to calculate both equilibrium and transport properties. Various EHSD explicit equations have been proposed in the literature according to different criteria. In this work a comparison between different expressions has been carried out and suggestions are given following the comparison. As an example of the applications, the EHSD method has been applied to the prediction of the selfdiffusion coefficients of the Lennard-Jones fluid, by use of the hard sphere model proposed by Erpenbeck and Wood. It is found that the temperature-dependent Boltzmann EHSD and both temperature- and density-dependent Lado modified Weeks-Chandler-Andersen EHSD give the best results. Introduction The researches on Lennard-Jones (6-12) (LJ) fluid attached much attention owing to its simplicity combined with realism. Its potential possesses both repulsive and the inverse sixth power London forces:
[( ) ( ) ]
φLJ(r) ) 4LJ
σLJ r
12
-
σLJ r
6
(1)
Here LJ is the depth of the potential well, which occurs at r ) 21/6σLJ, and σLJ is the collision diameter for the low energy collisions. The LJ model predicts thermodynamic properties of some simple substances quite well (e.g., noble gases), especially quantities derived from the internal energy: the agreement found for argon is surprising, in light of the fact that its properties can be generated numerically on a computer. Kirkwood and Monroe-Boggs (1942) have noticed that the form of the radial distribution function is primarily determined by repulsive forces, while attractive interactions play a secondary role. Such results promoted significantly the development of the perturbation approaches for dense fluids, which usually combine hardspheres (HSs) as an appealing and tractable first approximation, for the major excluded-volume and packing effects, with an effective diameter dependent on temperature and possibly on density, to account for the softness of the repulsive potential. Barker and Henderson (1967) (BH) presented the first successful analysis for the LJ fluid, adopting the HS and * To whom all correspondence should be addressed. † Universidade do Porto. ‡ Beijing University of Chemical Technology. § Instituto Polite ´ cnico de Braganc¸a.
LJ systems as the unperturbed and perturbing potentials. Their mathematical procedure gave rise to the so-called BH criterion for the effective hard-sphere diameter (EHSD). Another milestone in the perturbation theory is the work of Weeks, Chandler, and Andersen (Weeks et al., 1971; Andersen et al., 1971), who split up the LJ potential into a reference part containing all repulsive forces and a perturbing part containing all forces of attraction. Accordingly, the repulsive LJ fluid, hereafter called WCA LJ, was defined as
φWCA(r) )
{
φLJ(r) + LJ
r e 21/6σLJ
0
r > 21/6σLJ
(2)
The WCA theory asserts that the response given by the Fourier transform of the reference correlation function can be approximated by the response of a HS system with an as yet undefined diameter. Imposing the long wavelength responses to be exactly the same, a new physically reasonable criterion for the EHSD (henceforth denoted by WCA EHSD) has arisen. The WCA theory converges faster than BH theory, although its success is based upon the fact that errors from some rather poor approximations finally almost cancel each other in the end result. Verlet and Weis (1972) have developed corrections and analytical expressions for the thermodynamic functions with better results at high densities, which render possible a new formulation to estimate the WCA EHSD. However, their expression is implicit and a trial and error method is needed for the calculations. Lado (1984) proposed another modification to improve consistency with thermodynamic criteria, yielding a new EHSD (written as LWCA in this paper).
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222 Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998
The EHSD method has been widely used for the estimation of equilibrium properties, but particularly welcome is its extension to the field of transport phenomena accomplished by many authors. Heyes (1987, 1988) calculated the self-diffusion coefficients of the LJ fluid, employing an empirical EHSD expression and parameters obtained from the molecular dynamics (MD) data available for that fluid. Speedy et al. (1989) presented a prediction model for the self-diffusivities of the LJ fluid. Simultaneously, the LJ models have been observed to be applicable to simple nonpolar real substances (Ben-Amotz and Herschbach, 1990; Sun et al., 1994). Dymond (1985), Erkey et al. (1990), and Harris (1992) have studied real fluids using empirical correlations for the EHSDs. In our recent work (Liu et al., 1997), the EHSD method, coupled with an energetic term, yielded satisfactory results for the self-diffusivities of real substances, except hydrogen-bonding molecules, over the whole ranges of liquid and gas. Its scope has been found to comprehend molecules as polar as acetonitrile (AAD ) 3.49% for 60 data points) and nonspherical as n-hexadecane (AAD ) 5.36% for 25 data points). Since the 80s many authors have proposed explicit equations for the EHSD according to the above mentioned criteria: BH, WCA, and LWCA, respectively. Simplest is a criterion suggested by Boltzmann (B), which specifies the diameter as the distance of closest approach, averaged over collisions that reach the repulsive wall of the potential. This criterion has been used by some researchers working mainly on transport properties (Andrews, 1976; Speedy et al., 1989; BenAmotz and Herschbach, 1990). All four theoretical criteria predict a significant temperature dependence of the diameter, but only the WCA and LWCA predict a density dependence. Since many explicit expressions for the EHSD have been proposed in the literature, it seems fundamental to carry out a systematic comparison in order to facilitate and legitimate its selection. This paper presents such a study performed from the point of view of diffusive phenomena. Accordingly, we start presenting the EHSD equations (compiled in Appendix A) and compare their behavior and trends; following that, the self-diffusion coefficients of the LJ fluid, obtained in the literature by the MD simulation method, are predicted with these diameters; in the last section, a comparison follows the results obtained and recommendations are given as well. It is hoped that further insights into EHSD models and diffusion will be gained. Comparison between Different EHSD Expressions As mentioned in the Introduction, the EHSDs can be split into two distinct groups according to their temperature and density dependencies. The EHSD expressions belonging to the first group are only temperaturedependent, while the second group comprises both temperature and density dependencies. The comparison that will be carried out in this work embraces only explicit equations. All of them are listed in Appendix A. Temperature-Dependent Expressions. (1) Empirical Expressions. Heyes (1987, 1988) used an empirical polynomial function with five parameters to represent the EHSD. This correlation will not be considered for comparison, as the model parameters were optimized by fitting MD diffusivity data for the
LJ fluid. Hammonds and Heyes (1988) followed the same procedure and suggested an expression with three parameters. Sun et al. (1994) utilized the same expression, but the parameters were obtained from the PVT data of some real fluids by using the Ross variational approach. (2) Boltzmann EHSD Expression. The Boltzmann criterion for the EHSDs has been used by several authors working mainly on transport properties (Andrews, 1976; Speedy et al., 1989; Ben-Amotz and Herschbach, 1990). It approximates the EHSD by the distance of closest approach (σB) of a colliding pair of molecules with the average relative kinetic energy, subjected to soft repulsive interactions. For a central force potential φ(σB), this corresponds to the criterion:
φ(σB) ) f kT
(3)
where f is a constant determined by a suitable thermal average of the kinetic energy and k is the Boltzmann constant. Although not rigorous, this appealing criterion has been found to give good results for diffusion coefficients over a wide range of density and temperature (Speedy et al., 1989; Liu et al., 1997). Expressions for σB can be obtained by combining eq 1 or 2 with eq 3. For the WCA LJ fluid, the general form for the Boltzmann diameter is (Ben-Amotz and Herschbach, 1990):
[ ( )]
σ* B(T*) ) R* 0 1 +
T* T*0
1/2 -1/6
(4)
where T* ) T/(LJ/k) is the reduced temperature, 1/6 σB(T*) ) σB(T*)/σLJ is the reduced EHSD, R* 0 ) 2 , and T* ) 1/f is a constant. 0 For a dense fluid it is not apparent what the value of f should be. Andrews (1976) used f ) 1.5, while Speedy et al. (1989) used f ) 2. Ben-Amotz and Herschbach (1990) have taken both R*0 and T*0 as adjustable parameters in fitting the Carnahan-Starling-van-derWaals (CS-vdW) formula to equation-of-state (EoS) data evaluated by Monte Carlo (MC) simulations for the LJ fluid: R* 0 ) 1.1532 and T* 0 ) 0.527. In a previous work (Liu et al., 1997), we used eq 4 with a self-diffusion hard sphere model to reproduce the MD simulation results available for the WCA LJ fluid and found that the optimized value for T*0 is 0.7559 (f ) 1/6 1.3229) with R* 0 ) 2 . (3) BH EHSD Expression. This effective diameter criterion was introduced by Barker and Henderson (1967) in their perturbation theory and involves integrating over the repulsive portion of the potential:
σBH )
∫0r [1 - exp(-βφLJ)] dr 0
(5)
Here β ) 1/kT and r0 is the value of r for which φLJ ) 0. According to this definition, the EHSD is also only temperature-dependent. Several analytical expressions appeared from eq 5, based on the numerical calculation results (e.g., Verlet and Weis, 1972; Miyano and Masuoka, 1984; Ben-Amotz and Herschbach, 1990; Nezbeda, 1993; de Souza and Ben-Amotz, 1993). Figure 1 illustrates the temperature dependence of ten existent EHSD expressions. For the BH EHSD, the equations suggested by Miyano and Masuoka (1984), Ben-Amotz and Herschbach (1990), and Nezbeda (1993) give very similar results, while those of Verlet and Weis
Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998 223
Figure 1. Comparison between ten temperature-dependent EHSD expressions: (1) BH EHSD by Verlet and Weis (1972); (2) BH EHSD by Miyano and Masuoka (1984); (3) BH EHSD by de Souza and Ben-Amotz (1993); (4) BH EHSD by Nezbeda (1993); (5) BH EHSD by Ben-Amotz and Herschbach (1990); (6) B EHSD by Liu et al. (1997); (7) B EHSD by Speedy et al. (1989); (8) correlation by Sun et al. (1994); (9) B EHSD by Ben-Amotz and Herschbach (1990); (10) correlation by Hammonds and Heyes (1988).
(1972) and De Souza and Ben-Amotz (1993) overestimate and underestimate the values, respectively. Compared with our fitting results (Liu et al., 1997) and the Boltzmann diameter by Ben-Amotz and Herschbach (1990), it can be found that the BH rule overestimates the EHSD, while the expressions due to Hammonds and Heyes (1988) and Speedy et al. (1989) underestimate it. Particularly interesting is the fact that the B EHSD (eq 4) with R*0 ) 1.1532 and T* 0 ) 0.527, obtained from equilibrium data by Ben-Amotz and Herschbach (1990), gives results very near to those exhibited by the same expression with the parameters fitted to MD diffusivity data for the WCA LJ fluid (Liu et al. 1997). This fact points out a rough thermodynamic consistency, since diameters optimized from equilibrium and nonequilibrium properties come close together. The EHSDs determined from the expressions proposed by Speedy et al. (1989) and Hammonds and Heyes (1988) seem too low compared with the others. These low values result from the data base used to extract the parameters (Liu et al., 1997). It is curious to notice that the EHSD values given by Sun et al. (1994) for real fluids lie between those from the BH and B EHSD. Temperature- and Density-Dependent Expressions. (1) WCA EHSD. The well-known WCA perturbation theory (Weeks et al., 1971; Andersen et al., 1971) yields an EHSD dependent on both temperature and density:
∫0∞yHS(r;σWCA)[exp(-βφWCA) - exp(-βφHS)]r2 dr ) 0
(6)
where yHS is the background correlation function of the 3 HS fluid with diameter σWCA(T*,F*); F* ) FσLJ is the reduced density. The analytical expressions for σWCA(T*,F*) have been proposed by some authors (Miyano and Masuoka, 1984;
Figure 2. Comparison between five temperature- and densitydependent EHSD expressions, illustration of the density dependence: (1) WCA EHSD by Ben-Amotz and Herschbach (1990); (2) WCA EHSD by Miyano and Masuoka (1984); (3) WCA EHSD by de Souza and Ben-Amotz (1993); (4) WCA EHSD by Nezbeda (1993); (5) LWCA EHSD by Ben-Amotz and Herschbach (1990).
Figure 3. Comparison between five temperature- and densitydependent EHSD expressions, illustration of the temperature dependence (also shown are B and BH EHSDs): (1-5) same as in Figure 2; (6) BH EHSD by Ben-Amotz and Herschbach (1990); (7) B EHSD by Ben-Amotz and Herschbach (1990).
Ben-Amotz and Herschbach, 1990; Nezbeda, 1993; de Souza and Ben-Amotz, 1993). Ben-Amotz and Herschbach (1990) suggested once more eq 4 but treated T*0 as density-dependent. The Verlet and Weis method (Verlet and Weis, 1972) was rejected in this study, since it implies a trial and error procedure for the calculation: after starting with a value for the BH diameter, it requires iterating to converge to the WCA diameter. Figures 2 and 3 show a comparison between these expressions and reveal the following: (1) the temperature dependency is clearly more pronounced and thus significant; (2) only at high densities the density dependency becomes very appreciable, which has already
224 Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998
been pointed out by Ben-Amotz and Herschbach (1990); (3) very similar results can be obtained in the temperature range 0.5 e T* e 11.0. (2) LWCA EHSD. As was pointed out by Lado (1984), there is a fundamental shortcoming in the WCA theory: the lack of thermodynamic consistency. In fact, the pressure calculated from the virial equation does not confirm the value obtained from the WCA free energy. To overcome this problem, Lado (1984) recommended an alternative temperature- and density-dependent criterion for the EHSD:
∫0∞ [exp(-βφWCA) - exp(-βφHS)]
∂yHS 2 r dr ) 0 ∂σ
DHS V0 + ) 1.0 + 0.054034 DE V 6.3656
Application to Self-Diffusion Coefficients of the LJ Fluid In principle, a good EHSD model should work equally well for both thermodynamic and transport properties, which demands the expressions obtained from different properties to appear congruent. Particularly opportune and fortunate is the consistency found in Figure 1 between the B EHSD given by Ben-Amotz and Herschbach (1990) (PVT data; MC simulations) and that by Liu et al. (1997) (self-diffusivity data: MD simulations). In this paper we apply the EHSD method to predict the self-diffusion coefficients of the LJ fluid, which makes necessary an equation describing the HS system. The aim of such study is to select the best expressions from the point of view of the diffusive phenomena. On account of the nonexistence of good theoretical analytical expressions for the HS fluid transport properties, the empirical and semiempirical correlations based on the MD simulation results still hold their importance. However, this problem has not yet been solved satisfactorily, owing to the inconsistencies found between some groups of data, which led to the appearance of several different equations, whose correct choice becomes confusing. Following a complete critical comparison of the existent HS representations (Liu et al., 1997), here we have adopted the most recent work on simulation and correlation for this model fluid (Erpenbeck and Wood, 1991). The equation of Erpenbeck and Wood (1991) is a polynomial function according to
V0 V
2
-10.9425
( ) V0 V
3
(8)
where the three coefficients (0.054034, 6.3656, -10.9425) are optimized parameters, V is the molar volume, V0 ) Nσ3/x2 is the closed packing volume, N is the Avogadro constant and DE is the Enskog self-diffusivity given by
DE ) D0[g(σ)]-1
(7)
Ben-Amotz and Herschbach (1990) have derived an expression for σLWCA(T*,F*) following this criterion. It is also given by eq 4 with T* 0 treated as densitydependent, in much the same way as for the preceding case. In this work, five equations for temperature- and density-dependent diameters derived from eqs 6 or 7 are compared with each other and represented in Figures 2 and 3. Figure 2 shows that for the WCA theory, the four expressions give quite similar results, except in the region of high density. The LWCA values are lower than those of WCA. Figure 3 illustrates the temperature dependence of these expressions, along with the results due to Ben-Amotz and Herschbach (1990) for the BH and B theories. All the WCA and LWCA EHSD models, together with this BH equation, almost overlap. The predictions of de Souza and BenAmotz (1993) are lower at high temperatures. The B EHSD of Ben-Amotz and Herschbach (1990) performs differently, since its values are clearly lower in all temperature range.
( )
(9)
Here, D0 is the self-diffusivity from the kinetic theory of dilute gases:
D0 )
3 1 kT 1/2 8 Fσ2 πm
( )
(10)
m is the mass of the molecule, and g(σ) is the radial distribution function at contact, which for spheres of diameter σ is expressed in terms of η ) πF*/6 by the Carnahan-Starling model (1969):
g(σ) )
1 - 0.5η (1 - η)3
(11)
Taking account of the essential assumption of the EHSD method (i.e., the properties of a fluid can be expressed by the HS model, if the molecular diameter is replaced by an EHSD, σeff), the general expression for the LJ diffusivities can be written as
DLJ(T*,F*) ) DHS[σeff(T*,F*)]
(12)
where the reduced density is now defined as F* ) F(σeff)3. Based on the comparisons given above, the EHSD equations to be used along with eq l2 are those derived by Ben-Amotz and Herschbach (1990), eq 4, for BH, WCA, LWCA, and B criteria. The B EHSD expression of Speedy et al. (1989) has been also adopted for comparison. In all the calculations we have used the * ) F*DLJ/σLJ(LJ/ reduced self-diffusivity defined as DLJ 1/2 m) . There are many works in the literature concerning the MD simulations for the self-diffusion coefficients of the LJ fluid. In Table 1, the data sources used in this work are listed. They cover the whole ranges of temperatures and densities found in practical problems. Table 1. Self-Diffusivity Data Sources for the LJ Fluid authors
T*
Carelli et al. (1976) Michels and Trappeniers (1978) Lucas and Moser (1979) Heyes (1983)a Hammonds and Heyes (1988)a Heyes (1987)a Heyes (1988) Heyes and Powles (1990) Borgelt et al. (1990) Heyes et al. (1993) Kincaid et al. (1994)
0.687-1.90 1.30-1.50
0.30-0.84 0.00-0.3024
25 50
0.788-4.02 0.71-4.58 0.72-10.0
0.30-0.85 0.20-1.113 0.40-1.06
10 54 67
1.4562 0.72-10.0 0.722-6.00 0.673-2.54 0.707-6.02 2.00
0.10-1.0017 0.30-1.22 0.50-1.40 0.7803-0.8839 0.50-1.00 0.05-0.50
a
F*
Some data are included in Heyes (1988).
NDP
16 211 26 46 4 6
Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998 225 Table 2. EHSD Prediction Results (AAD) for the LJ Fluid (G* ) 0.0-1.22, T* ) 0.72-10.0, NDP ) 435)a T*-dependent EHSD eq 12
T* and F*-dependent EHSD
σBH(T*)
σB(T*)
σBS(T*)
σWCA(T*,F*)
σLWCA(T*,F*)
19.47
10.69
42.37
14.27
11.58
a
The expressions for all EHSDs are from Ben-Amotz and Herschbach (1990), except the Boltzmann model of Speedy et al. (1989), σBS(T*).
Table 2 contains the calculation results. It is found that the B EHSD (eq 4) with parameters optimized by Ben-Amotz and Herschbach (1990) (by fitting the CSvdW formula to EoS data from MC simulations) gives rise to the best predictions (average absolute deviation AADB ) 10.69%). The temperature- and densitydependent LWCA and WCA EHSDs provide acceptable results, although the former works better. The large error achieved with Speedy’s equation (Speedy et al., 1989) (AADBS ) 42.37%) comes from the questionable database they have used in its derivation (see Liu et al. 1997). The performance exhibited by the EHSD equations based on the BH, WCA, and LWCA criterias AADLWCA < AADWCA < AADBHsseems to confirm the trends observed in thermodynamics and already pointed out in the Introduction. Figure 4 shows the deviations for the EHSD expressions being compared. All methods give larger discrepancies at low temperatures, especially near the triple point, which emphasizes the rising role that the attractive forces play in that region. These deficiencies have persuaded Liu et al. (1997) to couple the repulsive contribution offered by the EHSD method with an energetic term allowing for the attractive forces. Figure 4 also lets us know that the BH, WCA, and LWCA EHSDs predict lower diffusivities, while the B EHSD works in the opposite way. Because the selfdiffusion coefficients increase with decreasing diameter, the relative magnitudes of the various EHSD expressions illustrated in Figure 3 warrant those results.
bach (1990), and Nezbeda (1993) are recommended. For the WCA rule, all four expressions (Ben-Amotz and Herschbach, 1990; Miyano and Masuoka, 1984; de Souza and Ben-Amotz, 1993; Nezbeda, 1993) give quite similar results. The LWCA EHSD due to Ben-Amotz and Herschbach (1990) is very near to the WCA equations. For all cases the expressions derived by BenAmotz and Herschbach (1990) are recommended. (2) The EHSD method can be used as a predictive tool for the self-diffusion coefficients of the LJ fluid, if a proper HS model is combined with a suitable EHSD expression. Among the three EHSD criteria supplied by the perturbation theories, the LWCA provides best estimates, corroborating the good performance already observed in thermodynamics. The B EHSD with parameters optimized from PVT data (Ben-Amotz and Herschbach, 1990) achieves the best results, leading us to conclude that a temperature dependence is sufficient for reasonable EHSD representations. (3) Taking into account the prediction results (Table 2) and the relative magnitudes of the EHSDs used (Figures 1-3), it can be asserted that the self-diffusivities are strongly dependent on the molecular diameter. For instance, AADLWCA ) 11.58% and AADWCA ) 14.27%, although the two effective diameters almost overlap (curves 1 and 5 in Figures 2 and 3). (4) An interesting and valuable finding in this work is the consistency of the B EHSD derived by Ben-Amotz and Herschbach (1990) from equilibrium data with that obtained from self-diffusivity data (Liu et al., 1997) (curves 9 and 6 in Figure 1, respectively), which explains the good results it furnishes. It should be emphasized that the merits of the B EHSD are due principally to its convincing theoretical basis, once its fundamentals rest on the transport phenomena domain. Acknowledgment H.L. acknowledges with thanks the financial support from Fundac¸ a˜o Oriente (Lisboa, Portugal), and C.M.S. acknowledges Junta Nacional de Investigac¸ a˜o Cientı´fica e Tecnolo´gica (Lisboa, Portugal) for financial support. List of Symbols
Figure 4. Prediction deviations vs reduced temperature for the LJ fluid, from different EHSD expressions combined with the HS model, eq 8: (squares) BH EHSD; (filled circles) B EHSD; (circles) WCA EHSD; (triangles) LWCA EHSD.
Conclusions and Discussion According to the comparison accomplished for the available EHSD expressions, we come to the following conclusions. (1) For the BH criterion, the equations proposed by Miyano and Masuoka (1984), Ben-Amotz and Hersch-
NDP AAD ) average absolute deviations: (100/NDP)∑i)1 cal MD MD |D*i - D* |/D* i i B ) Boltzmann BH ) Barker-Henderson BS ) Boltzmann-Speedy EHSD CS-vdW ) Carnahan-Starling-van-der-Waals D ) self-diffusion coefficient, cm2/s D* ) reduced diffusivity: F*D/σ(/m)1/2 D0 ) diffusivity of dilute gas given by eq 10, cm2/s EHSD ) effective hard sphere diameter EoS ) equation of state f ) parameter in the Boltzmann EHSD expression g(r) ) radial distribution function HS ) hard sphere k ) Boltzmann constant, 1.38048 × 10-23 J/K LWCA ) Lado modified WCA LJ ) Lennard-Jones m ) mass of the molecule, kg MC ) Monte Carlo MD ) molecular dynamic N ) Avogadro constant NDP ) number of data points r ) radial coordinate T ) temperature, K
226 Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998 T* ) reduced temperature: kT/ T*0 ) parameter in eq 4; equals 1/f for the B EHSD V ) molar volume, cm3 V0 ) closed packing volume: Nσ3/x2, cm3 WCA ) Weeks-Chandler-Andersen y ) background correlation function: gHS(r) exp(-βφHS) Greek Letters
5
aiT*0.093(i-1)]} ∑ i)1
1/6 σ* {1 - exp[ BHMM(T*) ) 2
(A.7)
where a1 ) -13.95687, a2 ) 24.13697, a3 ) -18.01784, a4 ) 6.355364, and a5 ) -0.869877. Ben-Amotz and Herschbach (1990):
σ*BHAH(T*) ) 1.1154[1 + (0.5685T*)1/2]-1/6
R* 0 ) parameter in Boltzmann EHSD, eq 4 β ) 1/kT η ) πF*/6 ) energy parameter, J/mol φ0 ) reference potential, J/mol φ(r) ) potential energy, J/mol σ ) molecular diameter, cm σ* ) reduced diameter: σ/σLJ σeff ) effective hard sphere diameter, cm F ) number density, cm-3 F* ) reduced density, Fσ3
(A.8)
de Souza and Ben-Amotz (1993):
σ*BHSA(T*) )
{ [
21/6 1 + 1 +
]}
T* + a2T*2 + a3T*4 a1
1/2 -1/6
(A.9)
where a1 )1.1287, a2 ) -0.05536, and a3 ) 0.0007278. Nezbeda (1993):
Subscripts
σ*BHN(T*) ) σtP exp[b1 ln(t) + b2 ln2(t) + b3 ln3(t)]
C ) critical point t ) triple point
(A.10)
Appendix A. Various Expressions for the Effective Hard-Sphere Diameter
where t ) T*/Tt, σt ) 1.02851, Tt ) 0.69, b1 ) -0.03208, b2 ) -0.00520, and b3 ) 0.00015.
Temperature-Dependent Expressions. (1) Empirical Correlations. Hammond and Heyes (1988):
(
σ*HH(T*) ) 1.0217 1 -
)
0.0178 1 1.256 T* T*1/12
(A.1)
(2) Ross Variational Perturbation Theory. Sun et al. (1994):
(
σ*SBT(T*) ) 1.2566 1 -
)
0.2075 1 0.177 T* T*1/12
(A.2)
WCA EHSD. The parameters of all models are derived by fitting the numerical integration results for the WCA criterion, eq 6. The first work was presented by Verlet and Weis (1972), but trial and error method is needed when their expression is used. In this work, only explicit expressions are considered. Miyano and Masuoka (1984):
σ*WCAMM(T*,F*) ) σ* BHMM(T*)[1 + S(η) D(T*)] (A.11a)
The parameters are obtained from the PVT data by using the Ross variational approach. (3) Boltzmann EHSD. Speedy et al. (1989): 1/6 1/2 -1/6 σ* BS(T*) ) 2 [1 + (2T*) ]
Temperature and Density-Dependent Expressions
(A3)
where σ* BHMM(T*) is given by eq A.7 and
S(η) ) (1 - 5.224η + 5.72η2 - 2.623η3 + 1.693η4)/ (1 - η)3 (A.11b) ln D(T*) ) a6 + a7 ln(1/T* + a8)
Ben-Amotz and Herschbach (1990):
σ*BAH(T*) ) 1.1532[1 + (1.8975T*)1/2]-1/6 (A.4) The parameters are obtained by fitting the CS-vdW formula to EoS data from MC simulation results. Liu et al. (1997):
(A.11c)
Here a6 ) -6.411461, a7 ) -0.551805, and a8 ) 0.05. Ben-Amotz and Herschbach (1990):
[ ( )]
σ* WCAAH(T*,F*) ) 1.1137 1 +
T* T*0
1/2 -1/6
(A.12a)
with
σ*B(T*) ) 21/6[1 + (1.3229T*)1/2]-1/6
(A.5)
The parameter in eq A.5 was obtained from fitting the MD data for the WCA fluid at F* e 0.95. (4) Barker-Henderson EHSD. The parameters in all models of this group are obtained by fitting the numerical integration results for the BH criterion, eq 5. Verlet and Weis (1972):
0.3837 + 1.068/T* σ* BHVW(T*) ) 0.4293 + 1/T* Miyano and Masuoka (1984):
(A.6)
(T*0)-1/2 ) 0.72157 + 0.04561F* - 0.07468F*2 + 0.12344F*3 (A.12b) de Souza and Ben-Amotz (1993):
σ*WCASA(T*,F*) )
{ [
21/6 1 +
T* + a2T*2 + a3T*4 a1(1 + a4F* + a5F*2 + a6F*3)
]}
1/2 -1/6
(A.13)
where a1 ) 1.5001, a2 ) -0.03367, a3 ) 0.0003935, a4 ) -0.09835, a5 ) 0.04937 and a6 ) -0.1415.
Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998 227
Nezbeda (1993):
σ*WCAN(T*,F*) )
[
σ*BHN(T*) d1 + d2td7 +
d3 + d4td8F0
]
1 + d5F0 + d6F02
(A.14)
where σ*BHN(T*) is given by eq A.10, F0 ) F*σ3t , d1 ) 0.975101, d2 ) 0.040766, d3 ) -0.014391, d4 ) -0.002316, d5 ) 0.075294, d6 ) -0.313557, d7 ) 0.018699, and d8 ) 0.115357. Lado WCA EHSD. Ben-Amotz and Herschbach (1990)
[ ( )]
σ* LWCAAH(T*,F*) ) 1.1152 1 +
T* T*0
1/2 -1/6
(A.15a)
where
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Received for review April 14, 1997 Revised manuscript received October 7, 1997 Accepted October 21, 1997X IE970281S Abstract published in Advance ACS Abstracts, December 15, 1997. X
