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Lennard-Jones Fluid and Diffusivity: Validity of the Hard-Sphere Model for Diffusion in Simple Fluids and Application to CO2 R. Ravi* and V. Guruprasad Department Of Chemical Engineering, Indian Institute Of Technology Madras, Chennai-600036, India
The validity of the hard-sphere (HS) model in determining the self-diffusion coefficient of simple fluids is studied using the Lennard-Jones (LJ) fluid as a test fluid. The latest simulation results on the LJ fluid are used along with the various perturbation theories to assess the HS model separately in the liquid, gas, and supercritical regions as a function of both density and temperature. Apart from the commonly used methods of estimating the effective hard-sphere diameter (EHSD), the Gibbs-Bogoliubov criterion, which provides an EHSD that is more representative of the intermolecular interactions, is used. This criterion gives a vastly different assessment of the HS model compared to other methods in the supercritical and liquid regions. All methods result in considerable errors in the low-density dilute gas region at lower temperatures. Next, the diffusivity of carbon dioxide is studied. Nine different sets of LJ potential parameters and three non-LJ potentials are considered. The LJ potential based on the critical constants of carbon dioxide yields diffusivities closest to the experiment. Introduction The Lennard-Jones (LJ) fluid has remained a workhorse for modeling real systems ranging from simple fluids to complex materials such as proteins and colloids.1,2 The fluid is governed by an interaction potential uLJ containing a characteristic distance parameter σLJ and a characteristic energy parameter LJ. The potential has a harshly repulsive short-range part and a more slowly varying attractive part. Apart from its utility in modeling the properties of real substances, the LJ fluid provides an ideal testing ground for statistical mechanical theories on account of extensive simulation results available for both its equilibrium and nonequilibrium properties. In this work, both of these aspects of the LJ fluid are exploited within the context of a specific property, namely, the diffusivity. In the first part, the most recent molecular dynamics (MD) simulation results3 on the diffusivity of the LJ fluid are used to test the hard-sphere (HS) model for diffusivity. In the second part, the applicability of the LJ model to the diffusivity of a real fluid, namely, carbon dioxide, is investigated. The HS model continues to be widely studied for predicting the properties of real fluids.4 This is because the properties of a real fluid are believed to be “largely determined by the repulsive part of the intermolecular potential for which the hardsphere interaction is the simplest representation”.5 Further, the HS model has a single parameter, namely, the hard-sphere diameter, and analytical expressions are available for the properties of a system of hard spheres. In order to apply these equations for a real substance, the effective hard-sphere diameter (EHSD) of the substance at the specified temperature and density must be determined. Thus, the real substance is to be viewed as equivalent to a system of hard spheres characterized by the EHSD. Three of the most commonly used methods of estimating the EHSD are the perturbation theories labeled after their contributors, namely, Barker-Henderson (BH),6 WeeksChandler-Andersen (WCA),7 and Lado.8 The hard-sphere model can be a particularly good approximation for simple fluids, which may be roughly defined9 as those * To whom correspondence should be addressed. E-mail: rravi@ iitm.ac.in.
whose properties are largely determined by harshly repulsive short-range forces with the more slowly varying attractive forces playing only a secondary role. This excludes ionic liquids, liquid metals, etc., which exhibit longer-range attractive forces and/ or a softer core. However, Ben-Amotz and Stell,4 who have suggested a quantitative measure of the softness of the “core” of a potential, have developed modified perturbation methods that predict the equilibrium properties of even systems with softcore potentials with “unprecedented accuracy”. These developments are bound to broaden the applicability of the HS model. Although reference fluids other than hard spheres can be accommodated within the framework of a perturbation theory, Ben-Amotz and Stell point out that the “cost-benefit quotient” of computational cost to predictive accuracy is particularly low when a system of hard spheres is used as the reference fluid. Given the importance of the HS model as outlined above, it is imperative to assess the accuracy of the model when applied to real systems. Once the intermolecular potential governing a real system is specified, then the EHSD corresponding to the potential can be calculated using standard methods such as the perturbation methods mentioned earlier. Then any property of the real system such as diffusivity can be calculated by evaluating the corresponding property of the equivalent system of hard spheres characterized by the EHSD. This may be compared with experimental data. However, this approach does not yield a true test of the HS model for predicting the property of interest because one cannot determine whether any discrepancy between the predictions of the theory and the experiment is due to the defects in the theory or in the model potential used to represent the real system.2 In this context, the role of computer simulations cannot be overemphasized. If one has access to accurate computer simulations of a property for a particular model potential, then one may apply the theory for the same model potential. Then, any discrepancy between the predictions of the theory and the simulation results may be attributed to the defects of the theory alone and we get a true assessment of the validity of the theory in predicting a property of interest. The interrelation between theory, experiment, and simulations and the role of each have been discussed elsewhere.10
10.1021/ie071073v CCC: $40.75 © 2008 American Chemical Society Published on Web 01/23/2008
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In the context of the present work, the availability of extensive simulation results for the self-diffusion coefficient of the LJ fluid, which may be regarded as a prototype of a simple fluid, makes the test of the HS model possible. In fact, Silva et al.11 addressed this question earlier in a detailed manner. They obtained the average absolute deviation (AAD)PT-Sim expressed in percent and defined as
(AAD)PT-Sim )
100 NP
NP
∑ i)1
(
)
|DiPT - DiSim| DiSim
(1)
by studying more than 400 states of differing densities and temperature. In eq 1, NP refers to the number of points or states at which the calculations are carried out and the subscripts PT and Sim refer to perturbation theory and simulation, respectively. Apart from the perturbation theories mentioned above, they also used the modified Boltzmann (B) criterion12 to determine the EHSD. They found that this criterion performed the best followed by the theories of Lado, WCA, and BH, in that order. The present work differs from that of the work of Silva et al.11 in the following aspects: (a) The simulation results employed in the work of Silva et al. were from ∼10 different sources to cover the range of densities and temperatures required. In this work, we use the recent simulation results of Meier et al.,3 who used a higher number of particles (1372) than ever before in their MD simulation. They also studied the largely unexplored low-density gas region. They found that simulations performed with 256 particles yielded systematically lower diffusion coefficients than those with 1372 particles. Further, for supercritical isotherms, they found that, at higher densities, the density dependence of the diffusion coefficient was a strong function of the number of particles used in the simulation. Zabaloy et al.1 have obtained analytical expressions for the diffusivity of the LJ fluid based on the simulation results of Meier et al. Recently, the “comprehensive” simulation results of Meier et al. have been used to test various kinetic theories of transport.2 (b) While the AAD calculated by Silva et al. is no doubt very useful, it is instructive to study the validity of the HS model in each phase region of the LJ fluid. This is accomplished here by studying the validity for supercritical isotherms as well as for subcritical isotherms and in the latter for both the gas and liquid regions. Furthermore, the effect of density, as well as that of temperature, is explicitly studied. The deviation between the predictions of the theory and the simulation is obtained so that the nature of the discrepancy, i.e., whether the theory overestimates or underestimates, may be clearly seen. (c) In addition to the above stated methods of determining the EHSD, the Gibbs- Bogoliubov (GB) variational scheme as implemented elsewhere13 is used. This scheme, in its general form, incorporates explicitly the attractive part of the potential in the defining equation for determining the EHSD and, thus, can be regarded as providing an EHSD that is more representative of the potential. Young14 has pointed out that the EHSD calculated by the GB method (σGB) is typically lower than that predicted by the WCA criterion (σWCA). The value of the potential at an intermolecular distance of σGB is roughly 3kBT/ 2, while that at σWCA is only kBT. Thus, Young argues that the GB method provides an EHSD that is more representative of the collisions between fluid particles while the WCA method effectively describes soft (low-energy) collisions alone. Thus, the GB criterion can be expected to give a more realistic assessment of the effectiveness of the hard-sphere model. Further, the recent success of such variational methods in
Table 1. Average Absolute Deviation (eq 1) for the Modified Boltzmann (B), Barker-Henderson (BH), Weeks-Chandler-Andersen (WCA), Lado, and Gibbs-Bogoliubov (GB) Criteria in Different Phase Regions; The Number in Brackets Indicates the Number of Points Used in the Calculation theory
gas (56)
liquid (42)
supercritical (60)
B BH WCA Lado GB
17.66 16.88 16.89 16.78 21.98
5.05 18.41 13.93 10.67 41.64
3.82 23.4 19.68 16.98 9.05
predicting the equilibrium properties of fluids4 described by softcore potentials provides further motivation for investigating their effectiveness in describing nonequilibrium properties such as the diffusion coefficient. A comment is in order regarding the utility of the results obtained here as well as in the earlier work of Silva et al. These results are an estimate of the errors one would incur when applying the HS model for diffusivity. Although they have been obtained using the simulation results on LJ fluid, they are an indication of the errors involved for simple fluids in general. For complex fluids, they may be regarded as providing a lower bound for the errors. Although the LJ fluid has a wide range of applicability1,2 in modeling real substances, here it plays the role of a test fluid on account of extensive simulation results available for it. Thus, the attempt here is not to find a substitute for the LJ fluid through the HS fluid. This would be a futile exercise, because simulation results are available for the LJ fluid and there is no need to resort to a more approximate model, namely, the HS model. For fluids where such simulation results are not available, one may try to model the fluid as an effective LJ fluid. If such an option is not feasible, then the HS-based perturbation theory provides a route for the calculation of diffusivity, and the results of this work provide an estimate of the error one would incur in such a case. In the second part of this work, we use the simulation results of Meier et al.3 to assess the various LJ parameters that have been proposed in the literature for CO2, specifically with a view to predicting its diffusivity. We also examine some non-LJ potentials that have been proposed in the literature for CO2. Applicability of Hard-Sphere Theory for Self-Diffusion in Simple Fluids Methodology. For the perturbation methods of Boltzmann, Barker-Henderson, Weeks-Chandler-Andersen, and Lado, the analytical expressions proposed by Ben-Amotz and Herschbach12 for determining the EHSD of the LJ fluid are used. Although many equations are available for the self-diffusion coefficient of the LJ fluid,15-17 we use the equations of Zabaloy et al.1 because they represent a fit to the latest and most comprehensive simulation results.3 For calculating the selfdiffusion coefficient of the HS fluid, we use the expression obtained by Erpenbeck and Wood.18 Here again, other expressions are available.19 Very recently, Heyes et al.5 have performed MD simulations to calculate the diffusivity of the HS fluid, and they found that their results were very much in agreement with the expression of Erpenbeck and Wood. The phase diagram of the LJ fluid is adapted from the work of Tang.20 In Figures 1-9, the deviation (%) is defined as
Deviation (%) )
DPT - DSim × 100 DSim
(2)
Table 1 gives the values of the AAD (eq 1) in the various regions.
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Figure 1. Plot of deviation (eq 2) versus reduced density (FσLJ3) for the supercritical region (Lado criterion); T* ) kBT/LJ.
Figure 4. Plot of deviation (eq 2) versus reduced density (FσLJ3) for the liquid region (Lado criterion); T* ) kBT/LJ.
Figure 2. Plot of deviation (eq 2) versus reduced density (FσLJ3) for the supercritical region (Boltzmann criterion); T* ) kBT/LJ.
Figure 5. Plot of deviation (eq 2) versus reduced density (FσLJ3) for the liquid region (Boltzmann criterion); T* ) kBT/LJ.
Figure 3. Plot of deviation (eq 2) versus reduced density (FσLJ3) for the supercritical region (GB criterion); T* ) kBT/LJ.
Results Figures 1-3 concern the supercritical region. Here, the Lado criterion consistently underestimates the diffusion coefficient (Figure 1), with the degree of underestimation increasing sharply at higher densities. The behaviors for the BH and WCA theories are very similar and, hence, not shown. In the case of the Boltzmann criterion (Figure 2), the density dependence for a given isotherm is much weaker than that for the Lado criterion, except at the higher densities. In contrast to the above theories, the GB criterion overestimates at all temperatures and densities studied (Figure 3). The degree of overestimation increases with an increase in density and with a decrease in temperature.
Figure 6. Plot of deviation (eq 2) versus reduced density (FσLJ3) for the liquid region (GB criterion); T* ) kBT/LJ.
In the liquid region, the predictions of the Lado, BH, and WCA theories are qualitatively similar to those in the supercritical region (Figure 4). The Boltzmann criterion does the best with a slight degree of overestimation at most conditions studied (Figure 5), while the GB criterion significantly overestimates the diffusion coefficients (Figure 6). In the case of the GB criterion, we adopted the scheme outlined in Osman and Ali,13 which uses the triangulation approximation to calculate one of the integrals. In order to check the effect of this approximation, we used the results of Mansoori and Canfield,21 who implemented the algorithm without any approximation. However, the degree of overestimation only increased slightly at higher densities.
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Figure 7. Plot of deviation (eq 2) versus reduced density (FσLJ3) for the gas region (GB criterion); T* ) kBT/LJ.
Figure 8. Plot of deviation (eq 2) versus reduced density (FσLJ3) for the gas region (Boltzmann criterion); T* ) kBT/LJ.
Figure 9. Plot of deviation (eq 2) versus reduced density (FσLJ3) for the gas region (Lado criterion); T* ) kBT/LJ.
In the gas region, the GB criterion again consistently overestimates the diffusion coefficient (Figure 7). Similar results were found for the Boltzmann criterion (Figure 8), although the degree of overestimation was, in general, lower than that of the GB criterion. The other three criteria overestimate at the lower temperatures but underestimate at higher temperatures (for example, Figure 9 for the Lado criterion). Discussion Figures 1-9, along with Table 1, reinforce the need for studying the validity of the HS model in each region separately and as a function of density as well as temperature. For instance,
in both the liquid and supercritical regions, the deviation of the predictions of the hard-sphere theories from the simulation results increases sharply with an increase in density, especially at higher densities. This behavior points to the inadequacy of the hard-sphere expression for diffusion coefficient in these regions where attractive forces can play an important role in determining the diffusivity. Also, on comparing Figures 1 and 4 with Figure 9, one can observe the qualitatively different behavior exhibited by the Lado criterion (the behavior of the BH and WCA criteria are very similar) in the liquid and supercritical regions as opposed to the gas region. While Figures 1 and 4 indicate a consistent underestimation of the diffusion coefficient, Figure 9 indicates significant overestimation at the lower temperatures studied. Although the GB criterion overestimates the diffusion coefficient in all regions, the degree of overestimation is far higher in the liquid region (Table 1). Long-range correlations are known to be important in the supercritical region as well, but yet the GB criterion does well here. This can be related to the consistent degradation in its performance with the decrease in temperature in general. This may be seen from Figure 3 (supercritical region) also, where the predictions arising from the GB criterion improve considerably at the higher temperatures, even at high densities. These results could be attributed to the hightemperature approximation inherent in the GB method.9,14 In general, the GB criterion predicts a lower EHSD than the BH, WCA, and Lado criteria. As pointed out by Young,14 the WCA criterion effectively accounts for only low-energy collisions, resulting in a larger diameter, while the GB method yields a EHSD that is a weighted average of diameters representing collisions of a wider energy range. The net result is a consistent overprediction of the diffusion coefficient using the EHSD of the GB criterion. Thus, it is clear that the EHSD that would give diffusion coefficients in agreement with the simulation results in the liquid and supercritical regions lies between the prediction of the GB criterion and those of the other perturbation theories. In the low-density gas region at lower reduced temperatures, all theories predicted an EHSD that was lower than the molecular diameter obtained by Meier et al.3 to fit their simulation results to the dilute gas Boltzmann expression for the diffusion coefficient. The deviation was sometimes as high as 30%. This results in a significant overestimation at these conditions for all methods (Figures 7-9). Even the modified Boltzmann criterion, which performs very well in other regions, fails in the low-density gas region, especially at lower temperatures (Figure 8). Attractive forces are known to play a greater role in determining the structure of a fluid at low densities.9 This is because packing forces become increasingly predominant at higher densities. It appears that this feature has an influence on the diffusion coefficient as well. From Table 1, it is seen that the Boltzmann criterion performs the best overall, followed by GB in the supercritical region and Lado in the liquid region. These results compare well with those of Silva et al.,11 who report an AAD that is an average over a range of states covering the various regions in the phase diagram of the LJ fluid. There again, the Boltzmann criterion performed best followed by those of Lado, WCA, and BH, in that order; the GB criterion was not used in that study. Among the criteria due to Lado, WCA, and BH, the Lado criterion performs best for equilibrium properties as well.4 While the WCA criterion is generally regarded as superior to the BH criterion in that its reference potential includes all the repulsive forces in the original potential, it suffers from a thermodynamic consistency problem.
Ind. Eng. Chem. Res., Vol. 47, No. 4, 2008 1301 Table 2. (AAD)Sim-Exp (Eq 3) for Different Sets of LJ Parameters; The Number in Brackets Indicates the Number of Points Used in the Calculation temperature (°C)
LJ/kB ) 231.7 K σLJ ) 3.644 Å17
LJ/kB ) 195.2 K σLJ ) 3.628 Å22
LJ/kB ) 241.5 K σLJ ) 3.655 Å23
LJ/kB ) 241.6 K σLJ ) 3.448 Å15
LJ/kB ) 236.1 K σLJ ) 3.72 Å27
35 (6) 50 (7) 75 (7) 100 (5)
5.37 9.56 4.22 2.44
6.14 4.74 4.85 4.34
7.25 11.91 5.94 4.24
8.26 5.74 7.59 8.82
11.04 15.85 9.01 7.24
This was rectified by Lado. Thus, it is interesting to note the fundamental superiority of the Lado criterion as compared to the BH and WCA criteria, reflected in its superior performance in describing equilibrium as well as nonequilibrium properties. It must be noted that the Boltzmann criterion used here is not the original criterion attributed to Boltzmann but a modified one. This modified criterion involves a factor f that has been determined using simulation results on equation of state for LJ fluids.12 This might explain its better performance. In fact, as noted by Silva et al.,11 the original Boltzmann criterion results in significant errors. In the case of non-LJ potentials, where such simulation results (to determine the appropriate value of f) may not be available, our results show that it would be better to use the GB criterion in the supercritical region and the Lado criterion in the liquid region. Apart from the above limitation, the numerical superiority of the modified Boltzmann criterion for predicting the diffusivity of the LJ fluid might actually mask the limitations of the HS model. If one adopts the viewpoint that the GB criterion provides a more realistic assessment of the HS model, as argued for earlier, then one would be forced to arrive at a much more modest assessment of the HS model than the modified Boltzmann criterion provides, as Figures 3, 6, and 7 and Table 1 indicate. Another point worth noting is that, in the case of equilibrium properties such as the free energy, the perturbation theory provides a correction to the free energy apart from determining the EHSD. In this work as well as in previous related work,11 only the hard-sphere expression for the diffusivity is used. Although corrections to the hard-sphere expression have been accounted for semiempirically,15 it would be worthwhile to investigate corrections to the diffusivity within a perturbation-theory framework. It would then become clear if the success of the hard-sphere model in predicting equilibrium properties can be carried over to nonequilibrium properties as well. Application to a Real FluidsCarbon Dioxide We now undertake the application of the above results for a real substance. To study the properties of a real substance, one must first choose a model intermolecular potential and then specify the parameters that appear in the model. Carbon dioxide, a well-studied substance, is chosen for the case study, partly in view of its widespread applications as a solvent under supercritical conditions. Both LJ as well as non-LJ potentials for CO2 are considered. Lennard-Jones Potentials for CO2. We found at least nine different sets of LJ parameters for CO2 in the literature. Some were obtained from diffusivity data itself,15,16,22,23 some from critical constants,17,24 a few from viscosity data,25,26 and one from PVT (pressure-volume-temperature) data.27 Two sets of parameters obtained by the same method, such as from diffusivity data, could still differ because of the different equations used and /or different sources of data employed. In view of the availability of recent simulation results3 over a wide range of conditions and using a much larger number of particles in the simulation than before, it seems worthwhile to reassess
Table 3. Effective Lennard-Jones Parameters for Non-LJ Fluids potential
LJ/kB (K)
σLJ (Å)
R2
IMP32
260.87 282.61 333.33
3.66 3.697 3.636
0.9995 0.9745 0.999
m-6-833 exp-634
the performance of these potential parameters in predicting the diffusivity of CO2. The results are presented in Table 2 at four different temperatures in terms of the average absolute deviation (AAD)Sim-Exp between simulation and experimental results, defined as
(AAD)Sim-Exp )
100 NP
NP
∑ i)1
(
)
|DiSim - DiExp| DiExp
(3)
where DExp is the experimental value of the diffusion coefficients28-30 and DSim is the value of the diffusion coefficient calculated using the equations of Zabaloy et al.1 While all temperatures studied are above the critical temperature of CO2, not all states are in the supercritical range. For temperatures 35 and 50 °C, ∼50% of the states correspond to the supercritical region, while at 75 and 100 °C, ∼25% of the states are in the supercritical regime. The results are shown for only five of the nine potentials studied, since the performances of the other four potentials were much worse than those reported in Table 2. From the table, it may be seen that the AAD is lowest for potential parameters17 using critical constants (Tc ) 304.2 K, Fc ) 6.41 × 1027 molecules/m3) followed by the potential parameters proposed by Dariva et al.,22 which were obtained from diffusivity data. The latter set of parameters was found to show better agreement with the experiment at higher densities, while the former did so at lower densities. These two sets are denoted as LJ1 and LJ2 for convenience. Non-LJ Parameters for CO2. It has been argued that, given the importance of long-range correlations in determining the properties of a supercritical fluid, the LJ model may not be adequate from a fundamental viewpoint.31 Hence, alternate potentials were looked into. One is the m-6-8 potential of Hanley and Klein,32 in which the parameters are obtained from viscosity data. The isotropic multipolar (IMP) potential33 has, in addition to an LJ potential, a term involving the quadrupole moment. In addition, the effective spherical (exp-6) potential of Johnson and Shaw34 was also studied. It must be pointed out that simulation results for diffusivities are not available for these potentials. Two options are available for calculating the diffusion coefficients. One is to use the best perturbation theory in the appropriate region and account for the deviation in an approximate manner based on the results obtained for the LJ fluid. An alternative is to fit the non-LJ potentials to effective LJ potentials. If the fit is sufficiently good, then the simulation results1,3 may be used directly to calculate the diffusion coefficients. From Table 3, it is clear that the fit is quite good (R2 values are close to unity). Table 4 displays the results of the predictions of the diffusion coefficients. The predictions of the two best LJ potentials are also given for comparison. It is clear that the IMP potential is the best among the non-LJ
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Table 4. (AAD)Sim-Exp (Eq 3) for Different Potentials; The Number in Brackets Indicates the Number of Points Used in the Calculation LJ temperature (°C)
117
222
IMP32
m-6-833
exp-634
35 (6) 50 (7) 75 (7) 100 (5)
5.37 9.56 4.22 2.44
6.14 4.74 4.85 4.34
10.81 15.16 9.35 7.44
16.80 21.88 14.89 12.64
21.52 24.34 19.70 17.08
potentials. However, all the three non-LJ potentials give predictions that are inferior to the best of the two LJ potentials. Thus, the LJ model appears to be adequate for predicting the diffusion coefficient of CO2 even in the supercritical region over the range of conditions studied in this work. Conclusions The validity of the hard-sphere model in predicting the selfdiffusion coefficient of simple fluids is studied using the Lennard-Jones fluid as a test fluid. The most recent molecular dynamics simulation results have been used to arrive at the conclusions. The importance of obtaining estimates of the validity separately in the various regions in the phase diagram of the LJ fluid as a function of both density and temperature is demonstrated. Five different methods of estimating the EHSD have been considered. This includes the GB criterion, which is expected to yield a more realistic assessment of the validity of the HS model compared to the other criteria on account of it being more representative of the collisions between the fluid particles. On the basis of this criterion, we arrive at a more modest estimate of the success of the HS model in predicting diffusivities when compared with the predictions of the modified Boltzmann criterion. The limitations inherent in using the EHSD predicted by the modified Boltzmann criterion for estimating diffusivity are pointed out. In the second part of the work, the diffusivity of carbon dioxide is considered. Nine different sets of LJ potential parameters proposed in the literature for CO2 are evaluated in terms of their abilities to predict the diffusivity of CO2 by comparing the simulation results with experimental values. In addition, three non-LJ potentials are also considered. It is found that the LJ parameters based on the critical temperature and density of carbon dioxide yield results closest to the experiment. Nomenclature D ) self-diffusion coefficient/diffusivity kB ) Boltzmann constant T ) temperature T* ) reduced temperature (kBT/LJ) uLJ ) Lennard-Jones potential LJ ) energy parameter in the Lennard-Jones potential σLJ ) distance parameter in the Lennard-Jones potential F ) particle number density F* ) reduced density (FσLJ3) Subscripts/Superscripts Exp ) experimental PT ) perturbation theory Sim ) simulation AbbreViations B ) Boltzmann BH ) Barker-Henderson EHSD ) effective hard-sphere diameter GB ) Gibbs-Bogoliubov
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ReceiVed for reView August 7, 2007 ReVised manuscript receiVed October 22, 2007 Accepted November 16, 2007 IE071073V
