Simulation of Vapor− Liquid Equilibria for Alkane Mixtures

agreement with experiment is satisfactory. Two commonly used equations of state are also used to describe experimental coexistence curves. In the abse...
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Ind. Eng. Chem. Res. 1998, 37, 3195-3202

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Simulation of Vapor-Liquid Equilibria for Alkane Mixtures Shyamal K. Nath, Fernando A. Escobedo, Juan J. de Pablo,* and Issarin Patramai Department of Chemical Engineering, University of Wisconsin, 1415 Johnson Drive, Madison, Wisconsin 53706

A newly proposed united-atom force field is used to predict the phase behavior of binary mixtures of long and short alkanes over wide ranges of temperature and pressure. Given the chemical similarity of the components, no adjustable binary parameters are introduced. In general, agreement with experiment is satisfactory. Two commonly used equations of state are also used to describe experimental coexistence curves. In the absence of binary data, it is found that molecular simulations provide a fairly reliable means of estimating thermodynamic properties for the highly asymmetric mixtures of short and long alkanes studied in this work. I. Introduction

Table 1. Potential Energy Functions

Molecular simulations can now be used to generate phase diagrams for pure, relatively complex fluids. From simple Lennard-Jones1 fluids to fairly complex polyatomic molecules,2-4 coexistence (e.g., vapor-liquid equilibria, VLE) properties can now be simulated with good accuracy. Simulation of phase equilibria for long, polymeric molecules has also become possible with the development of configurational-bias and expandedensemble techniques.3,5-8 Several years ago Laso et al.3 presented what appears to be the first simulation study of phase equilibria for long alkanes. Since that time, much progress has been achieved in the area of simulation of polymeric molecules. It is now possible to use molecular simulations to generate phase diagrams for mixtures of moderately long molecules with relative ease; further, the prospects for these types of calculations are likely to continue improving over the years to come. In this study, we present results for the phase behavior of binary mixtures of short and long alkanes (e.g., ethane and tetracontane). These mixtures are encountered in a variety of industrial applications, and for years they have provided a challenging prototype system on which to test the predictive capabilities of equations of state. In this work, the results of simulations are compared to experimental data and to predictions of the SAFT9,10 and Peng-Robinson11 equations of state. The paper is organized as follows. We begin with a description of the model and the simulation methods employed to generate phase diagrams, followed by a brief account of the equations of state considered in this work. We then present vapor-liquid equilibrium (VLE) results for pure alkanes, followed by Henry’s law constants and coexistence curves for binary systems. We compare the predictions of two equations of state and the results of simulations to experimental data. We conclude with a few remarks concerning the future of molecular simulations for study of realistic mixtures of industrial importance and the reliability of equations of state for oligomeric and polymeric systems.

Kr V(r)/kB ) (r - beq)2 2

* To whom correspondence is addressed. Phone: (608) 2627727. Fax: (608) 262-0832.

Bond Stretching Potential

Kr ) 96500 K/Å2

beq ) 1.54 Å Bond Bending Potential24,4

Kθ V(θ)/kB ) (θ - θeq)2 2 Kθ ) 62500 K/rad2

θeq ) 114.0°

Torsional Potential25 V(φ)/kB ) V0 + V1(1 + cos φ) + V2(1 - cos 2φ) + V3(1 + cos 3φ) V0 ) 0 V1 ) 355.04 K V2 ) -68.19 K V3 ) 701.32 K Nonbonded Interaction Potential

r V(r)/kB ) 4 σ

[( ) - (σr) ]

σCH2 ) 3.79 Å σCH2 ) 3.93 Å

12

6

Ethylene CH2 ) 84.7 K Alkanes

σCH3 ) 3.825 Å

Ethane CH3 ) 100.6 K

σCH3 ) 3.857 Å

Propane CH3 ) 102.6 K CH2 ) 45.8 K

σCH3 ) 3.91 Å

Butane and Longer CH3 ) 104.0 K CH2 ) 45.8 K

II. Models and Simulation Details A united-atom representation of the alkanes is adopted throughout this work (e.g., CH3 and CH2 groups are described by single interaction sites). For simple nalkanes, we use the recently proposed NERD force field.8 Table 1 gives intra- and intermolecular NERD parameters. As described in the original reference, this force field was arrived at by simultaneous regression of second-virial-coefficient and phase-coexistence data for pure alkanes.8 In all calculations, a cutoff radius of 13.8 Å was employed for (site-site) Lennard-Jones interactions and standard tail corrections were implemented.12 As explained in Table 1, a Lennard-Jones potential energy function is adopted to describe site-site interactions for both sites located more than three bonds apart on the same molecule and sites located on different molecules. For mixtures, some freedom can be exercised in the choice of unlike pair interactions. In this work

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we use Lorentz-Berthelot combining rules:

1 σij ) (σii + σjj) 2

(1)

ij ) xiijj

(2)

Given the similar chemical nature of the components studied here, these simple rules are expected to be sufficiently accurate. All coexistence curve simulations were conducted in the Gibbs ensemble.1,3,5 For mixtures, all simulations were conducted at constant pressure and temperature. The transfer of long alkane molecules between coexisting phases was possible through the use of an expanded Gibbs method.7 For detailed accounts of these techniques, readers are referred to the original publications.1,3,7 Molecules were rearranged by a combination of reptation, translation, and configurational-bias Monte Carlo moves.3,4 To estimate the Henry’s constants of alkanes in polyethylene, constant pressure simulations were performed for a pure system of chains of C70. A configurational-bias method was subsequently used to determine the excess chemical potential of alkanes in that system, according to procedures outlined in the literature.13 The sizes of the systems investigated in this work ranged from 300 molecules, for mixtures of methane and pentane, to 600 molecules for mixtures of ethane (or ethylene) and tetracontane. Equilibrium averages were collected for about 2 × 107 simulation steps, of which about 5% were volume moves. Twenty to fourty percent were transfer moves, and the rest of them were equilibration moves. Second virial coefficients were determined from the average interaction between two isolated molecules according to expressions provided by Harismiadis and Szleifer.16 Critical temperatures were estimated by means of a simple Ising scaling-law analysis of simulation data using a critical exponent β ) 0.32; no finitesize or cross-over corrections were implemented in this work. III. Equations of State (EoS) Cubic equations of state with a one-fluid type of mixing rule were among the first phenomenological models capable of providing a satisfactory description of the phase behavior of alkane mixtures; they have been extensively used in industry for a variety of applications (e.g., for petroleum reservoir fluids and gasprocessing plants). Agreement with experimental data, however, often relies on empirically determined binary interaction parameters. A known shortcoming of such models is that the quality of their predictions tends to deteriorate as the asymmetry between individual components increases. For systems that exhibit large-size asymmetries (e.g., solvent-plus-polymer systems), a number of equations based on statistical mechanical theories have been developed in recent years. In this context, binary mixtures of a small alkane (e.g., ethane) and successively longer alkanes (e.g., from pentane to tetracontane) provide good prototype systems to test the predictive capabilities and range of applicability of such equations (i.e., without use of any ad-hoc binary parameter).

In this work we examine the binary-alkane phase diagrams predicted by two equations of state: the Peng-Robinson equation,11 which is an example of a phenomenological equation, and the statistical associating fluid theory (SAFT),9,10 which is representative of statistical-mechanical-based equations of state. Both of these equations are widely used in practice to correlate or predict the phase behavior of alkane mixtures, and it is therefore relevant to address how these models compare with molecular simulation results. A. Peng-Robinson (PR). This equation provides an expression for the pressure of the form:

P)

ai RT V - bi V2 + 2Vb - b 2 i i

(3)

where ai and bi are substance-specific parameters that depend only on temperature. A mixture is assumed to obey the same equation with mixture parameters, am and bm, calculated from the one-fluid mixing rules:

am )

∑∑xixjaij

bm )

∑∑xixjbij

where aij ) (aiaj)0 5 and bij ) (bi + bj)/2, and where xi is the molar fraction of component i. B. SAFT. The SAFT compressibility factor is assumed to consist of the sum of several contributions, that is,

Z ) Z(reference) + Z(chain) + Z(association)

(4)

where the term Z(reference) corresponds to a reference system of spherical sites (e.g., hard sphere plus a dispersion term). The term Z(chain) describes the effect of bonding of spherical sites to form chains, and the term Z(association) is introduced to account for association between nonbonded sites. Detailed expressions for each of the terms in eq 4 can be found in the original references.10 For mixtures, rigorous statistical-mechanical expressions are used to describe the hardsphere, chain, and association terms: an empirical mixing rule is needed, however, for the dispersion term (e.g., a volume-fraction approximation). For both the PR and SAFT EoS, substance-specific parameters related to attraction energies and molecular volumes are generally determined by regression of experimental liquid densities and saturation pressures. In this work, however, we also follow a second, alternative approach to adjust the parameters of the PR EoS. In the “conventional” approach, parameters are calculated from the original correlations proposed by Peng and Robinson.11 In the second approach, ai and bi for the subcritical component (i.e., the long alkane) are found by fitting experimental liquid densities and saturation pressures at each temperature of interest;22 these “optimized” parameters are employed to partially correct for the fact that the original Peng-Robinson correlation was developed for short alkanes only. It is emphasized that, given the chemical similarity between the components of the mixtures studied in this work, all binary interactions parameters are set to zero for both simulations and the PR and SAFT EoS’s. IV. Results and Discussion Figure 1 shows results for the second virial coefficient of ethylene, ethane, butane, hexane, and octane. The lines correspond to experimental data; the symbols

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Figure 1. Comparison of experimental and simulated second virial coefficient for small alkanes. Experimental data (lines) are taken from the compilation of Dymond and Smith;15 two sets of experimental results are reported for hexane (the second set of data is shown by diamonds). Simulation data (triangles for ethylene and squares for the alkanes) were obtained employing the method described in ref 16.

correspond to results of Monte Carlo simulations. For lower alkanes, agreement between calculated and experimental second virial coefficients is highly satisfactory over a wide range of temperature. As chain length increases, however, distinct deviations arise between experiment and theory, particularly at low reduced temperatures. Such deviations between theory and experiment can be attributed to a number of shortcomings of our simple force field, including the neglect of three-body interactions. Figure 2 shows experimental and simulated orthobaric densities for several pure alkanes, including ethylene, ethane, and dodecane. In general, agreement between simulation and experiment is highly satisfactory. For alkanes longer than C12 there is a lack of experimental data at high temperatures: we have used the NERD force field to predict the critical properties of long alkanes with reasonable success.8 These properties are required by many correlations for prediction of thermophysical properties. Figure 3 shows simulated and experimental weight fraction Henry’s constants for ethylene, butane, and hexane dissolved in polyethylene at 1 bar and at several temperatures. Note that, in these calculations, polyethylene melts were represented by a collection of C70 chains; beyond this molecular weight the density of long alkane oligomers does not change significantly and we do not expect alkane Henry’s constants to exhibit appreciable differences in systems of longer chains. In general, agreement between simulation and experiment is satisfactory, particularly when we consider that the NERD force field was derived from pure component data and that no binary parameters were employed in these calculations. Figure 4a shows a pressure vs composition diagram for a binary methane-pentane mixture at T ) 104.4 °C. The symbols correspond to the results of our

Figure 2. Orthobaric densities for small-to-intermediate alkanes. The symbols are simulation results for the NERD force field (see Table 1): triangles for ethylene and squares for the alkanes. The full lines correspond to experimental data from ref 17. The broken line corresponds to an extrapolation from experimental data using an ising fit. The error bars (e.g., (the standard deviation of the average values) are smaller than 3 times a symbol size near the critical points and less than twice the symbol size at lower temperatures.

Figure 3. Experimental (diamonds) and simulation (squares) data for Henry’s constant of small hydrocarbons in molten polyethylene.

simulations (squares) and to experimental data (diamonds).21 The solid lines correspond to the predictions of the PR EoS with optimized parameters and the dotted lines correspond to the predictions from the SAFT equation of state. Figure 4b shows the corresponding pressure vs density diagram for the same methanepentane mixture. The simulations were conducted without the use of binary parameters; agreement with

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Figure 5. Pressure vs composition diagram for a binary mixture of ethane-heptane at T ) 92.84 °C. The symbols correspond to the results of our simulations (squares) and to experimental data (diamonds).21 The lines correspond to the predictions of SAFT (dashed) and the PR EoS (full). Error bars are as indicated in Figure 4.

Figure 4. (a) Pressure vs composition diagram for a binary methane-pentane mixture at T ) 104.4 °C. The symbols correspond to the results of our simulations (squares) and to experimental data (diamonds).21 The lines correspond to the predictions of SAFT (dashed) and the PR EoS (full). In (b) the pressure vs density diagram is shown for the same methanepentane mixture. The solid line corresponds to experimental data and the squares to simulation results. Error bars of the simulation data are about twice the size of the symbols; deviations tend to be somewhat larger, however, for higher pressures (near the critical value).

experiment is satisfactory, particularly at low-tointermediate pressures. As pressure increases, however, simulations tend to slightly overpredict both the vapor and liquid mole fractions of methane. The deviation between simulated and experimental compositions is consistent with that of the simulated Henry’s constants, where the NERD model slightly underpredicts the Henry’s constants of small alkanes in polyethylene.

From Figure 4a, it is also apparent that simulations predict an onset of miscibility at a slightly lower pressure than observed experimentally. The coexistence curve predicted by SAFT shows a deviation from experimental trends that is akin to that of our simulation data, but slightly more pronounced; SAFT predicts the onset of full miscibility at a much higher pressure than experimentally observed. In contrast, the PR EoS generates a phase diagram that is in quantitative agreement with experimental data. For this system and other similar mixtures of small alkanes, conventional and optimized PR EoS parameters provide almost indistinguishable predictions. For clarity, we show results only for the optimized parameters; differences between both approaches become noticeable only for mixtures containing alkanes longer than decane. Figure 5 shows simulated and experimental phase equilibrium data along with EoS predictions for the binary ethane-heptane system at constant temperature (T ) 92.84 °C). Agreement between simulation and experiment is again satisfactory. As for the methanepentane system, however, deviations between simulation and theory become more pronounced as pressure is increased. These results indicate that the NERD force field does not capture accurately the effect of pressure on the fluid density. The reason for such a shortcoming can be traced to the fact that NERD unitedatom parameters were developed to describe experimental data of orthobaric densities and second virial coefficients; such properties are inherently associated with low-to-intermediate pressure conditions. For the ethane-heptane system, the PR EoS provides excellent agreement with experiment, while the SAFT predictions exhibit somewhat larger deviations. As for the methanepentane system, SAFT predicts a larger miscibility gap along the pressure axis. Figure 6a,b show simulated and experimental coexistence curves and EoS predictions for the ethane-

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Figure 7. Temperature vs composition diagram for a mixture of ethane-eicosane at P ) 96.526 bar. The symbols correspond to the results of our simulations (squares) and to experimental data (diamonds).23 The lines correspond to the predictions of SAFT (dashed) and the PR EoS with conventional parameters (dotted line) and optimized parameters (full line). Error bars of the simulation data are about 3 times the size of the symbols; deviations tend to be somewhat larger, however, near the LCST.

Figure 6. Phase diagrams for a mixture of ethane-decane. (a) Pressure vs composition diagram at T ) 137.8 °C. (b) Temperature vs composition diagram at P ) 96.526 bar. The symbols correspond to the results of our simulations (squares) and to experimental data (diamonds).21 The lines correspond to the predictions of SAFT (dashed) and the PR EoS (full). Error bars of the simulation data are about 3 times the size of the symbols.

decane binary system at T ) 137.77 °C and at P ) 96.526 bar, respectively. The phase diagram at constant temperature (Figure 6a) shows similar agreement between simulated, experimental, and EoS predictions as observed for the previous two mixtures. The results of simulations compare slightly more favorably with experiment than SAFT. The results shown in Figure 6b are interesting in that at this pressure the system exhibits a closed-loop miscibility diagram, with a lower critical solution temperature (LCST) at about T ) 100 °C and an upper critical solution temperature (UCST) at about T ) 260 °C. The agreement between experi-

Figure 8. Temperature vs composition diagram for a mixture of ethane-tetracontane at P ) 96.526 bar. The symbols correspond to the results of our simulations (squares). The lines correspond to the predictions of SAFT (dashed) and the PR EoS with conventional parameters (full line). Error bars of the simulation data are as indicated in Figure 7.

ment and simulations is fair. Both PR and SAFT overpredict the temperature range of the miscibility gap. Figures 7 and 8 show results for the ethane-eicosane and ethane-tetracontane systems at a pressure of P ) 96.526 bar, respectively. These systems also show

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Figure 9. Temperature vs composition diagram for a mixture of ethylene-eicosane at P ) 96.526 bar. The symbols correspond to the results of our simulations (squares) and to experimental data (diamonds).26 The lines correspond to the predictions of SAFT (dashed) and the PR EoS with conventional parameters (dotted line) and optimized parameters (full line). Error bars of the simulation data are as indicated in Figure 7.

Figure 10. Temperature vs composition diagram for a mixture of ethylene-tetracontane at P ) 96.526 bar. The symbols correspond to the results of our simulations (squares) and to experimental data (diamonds).26 The lines correspond to the predictions of SAFT (dashed) and the PR EoS with conventional parameters (full line). Error bars of the simulation data are as indicated in Figure 7.

evidence of an LCST (at 60.5 °C for the ethane-eicosane mixture). The EoS’s predict upper critical solution temperatures above 400 °C, which is above the point at which the components become thermally unstable. For the ethane-eicosane system (Figure 7), simulation results provide the best agreement with experiment at higher temperatures (around 430 K) but they tend to overestimate the ethane composition in the eicosanerich phase, thereby resulting in a curve that lies slightly inside the experimental curve and predicting a LCST that lies slightly above the experimental value. The curve predicted by SAFT in Figure 7 lies well-inside the experimental curve (ethane concentrations in the dense phase are too high), but the LCST is predicted accurately. For this system, conventional and optimized PR EoS parameters yield significantly different results; the former generate a curve that lies outside the experimental curve, slightly underestimating the LCST, while the latter yields results that lie inside the experimental curve, thereby overpredicting the LCST. These PR EoS results should be contrasted with the high-quality predictions obtained for mixtures of smaller alkanes. The pronounced decline of predicting capability illustrates the sensitivity of the results to the values of pure component parameters and, more importantly, the fact that a one-fluid mixing rule tends to deteriorate as the asymmetry between components increases. For the ethane-tetracontane system, unfortunately, experimental data to perform comparisons are not available. Figures 9 and 10 show results for the ethyleneeicosane and ethylene-tetracontane systems, respectively. The pressure is again P ) 96.526 bar. For these systems agreement with experiment is remarkable and clearly superior to that provided by SAFT and PR. The improved performance of the simulation approach for the ethylene-containing systems over the ethanecontaining mixtures is probably due to a better descrip-

tion of ethylene by the force field. In agreement with the trends observed in ethane-long alkane systems, SAFT systematically overpredicts the concentration of the long alkane in the dense phase; the deviations associated with the PR EoS are less consistent. Further, deviations between experiment and the predictions of these equations become larger as the length of the long alkane increases. Surprisingly, as can be seen in Figures 7-10, even for systems containing eicosane or tetracontane, SAFT does not seem to offer a significant advantage over the PR EoS (at least at the conditions studied here). These results lend credence to the idea that, in the absence of experimental data, simulations can indeed be used to provide reasonable estimates of the phase behavior of highly asymmetric alkane mixtures. For long alkanes, or branched alkanes, the experimental data necessary to determine pure component EoS parameters are generally not available. For eicosane and tetracontane, for example, conventional PR EoS parameters require Tc, Pc, and the acentric factor; critical properties need to be estimated by extrapolation of experimental data for shorter alkanes. The predicted phase diagrams for mixtures containing eicosane and tetracontane (Figures 7-10) turn out to be highly sensitive to, for example, Pc. For the “optimized” PR EoS parameters, experimental saturation pressures do not cover the entire temperature range of interest and therefore some extrapolation is necessary. The situation worsens significantly for alkanes longer than eicosane, where extrapolations of that sort are likely to be more unreliable. These problems are common to the SAFT and other widely used equations of state. In contrast, the simulation approach advocated here has the advantage that it is always applicable, regardless of alkane chain length: in this case, the extrapolations are made

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at the level of the molecular model and can therefore be carried out in a consistent and unambiguous manner. V. Conclusions Monte Carlo simulations in an expanded Gibbs ensemble have been conducted to study the ability of a united-atom force field (NERD) to describe 1ow- and high-pressure vapor-liquid equilibria of pure alkanes and their mixtures. The simulation results for mixtures have been compared to experimental data and the predictions of two well-known equations of state, the Peng-Robinson and the SAFT equation. The systems studied included pure short and long alkanes, and mixtures of methane, ethane, or ethylene with several longer alkanes (from pentane to tetracontane). It has been found that simulation results typically provide quantitative agreement with experimental data for the VLE of pure alkanes. For mixtures, the simulation approach gives satisfactory agreement, although it is never completely quantitative at elevated pressures. For mixtures of short alkanes, the agreement between simulation and experiment is not superior to that achieved by common equations of state. In general, however, the performance of the PR and the SAFT EoS’s deteriorates significantly for mixtures containing long alkane molecules. In this regard, the simulation approach is more consistent in that it provides the same level of agreement with experimental data for both, systems with short and with long alkanes. A systematic trend is found between simulation and experimental results; the deviations tend to be larger at higher pressures, which is a reflection of the approach employed to obtain the parameters of the NERD force field. It is anticipated that if a similar united-atom force field is developed on the basis of low- and high-pressure properties of alkanes, a significant improvement of performance would be observed for high-pressure VLE calculations. In this regard, it would be advantageous to use a model that takes into account the anisotropy introduced by the hydrogen atoms of an alkane; such anisotropy is likely to become important when subtle packing effects become significant (i.e., at elevated densities). The anisotropic united-atom model of Toxvaerd,19 for example, has been shown to provide good agreement with experimental liquid densities of alkanes over a wide range of pressures. The ability of this model to describe phase equilibria for alkane mixtures will be examined in a forthcoming publication. For the systems studied here, it is expected that the simple Lorentz-Berthelot mixing rules (without binary parameters) should not be a source of significant deviations between simulation and experimental results. Nonetheless, the issue of the accuracy of such mixing rules to describe high-pressure properties of alkane mixtures should probably be studied in more detail. Contrary to our original expectations, it has been shown that, in the absence of any adjustable binary interaction parameters, high-pressure phase diagrams for systems ranging from methane-pentane to ethanetetracontane and ethylene-tetracontane, the PR EoS (with either conventional or optimized pure component parameters) outperforms SAFT. Note, however, that while the purely predictive capability of SAFT appears to be only semiquantitative, we still expect it (on the basis of literature studies) to be advantageous for systems containing polymer-like molecules and associating components (also note that binary SAFT

interaction parameters, if employed, could be less temperature-sensitive than those used with cubic EoS). The use of a cubic EoS with a one-fluid mixing rule has been found suitable for the prediction of phase diagrams of the simple nonpolar systems studied here; as expected, however, the quality of the predictions begins to deteriorate as the size asymmetry of the components becomes larger. Acknowledgment The authors wish to dedicate this manuscript to John Prausnitz on the occasion of his 70th birthday. We are grateful to Prof. W.G. Chapman for making his SAFT code available to us and to Prof. Theodorou for providing results from ref 27 prior to publication. Financial support from a PECASE award of the National Science Foundation (CTS-9629135) is gratefully acknowledged. One of the authors (J.J.dP.) is also grateful to the Camille and Henry Dreyfus Foundation for a TeacherScholar Award. Note added in proof: A similar study of simulation of phase equilibria for alkane mixtures has been recently presented by Spyriouni et al.27 These authors employed different force fields and different simulation schemes from those adopted by us; their simulation results also show good agreement with experimental data. Literature Cited (1) Panagiotopoulos, A. Z. Direct Determination of Phase Coexistence Properties of Fluids by Monte-Carlo Simulation in a New Ensemble. Mol. Phys. 1987, 61, 813. (2) de Pablo, J. J.; Prausnitz, J. M.; Strauch, H.; Cummings, P. T. Molecular Simulation of Water Along the Liquid Vapor Coexistence Curve from 25 °C to the Critical-Point J. Chem. Phys. 1990, 93, 7355. (3) Laso, M.; de Pablo, J. J.; Suter, U. W. Simulation of PhaseEquilibria for Chain Molecules. J. Chem. Phys. 1992, 97, 2817. (4) Smit, B.; Karaborni, S.; Siepmann, J. I. Computer Simulations of Vapor-Liquid Phase Equilibria of n-Alkanes J. Chem. Phys. 1995, 102, 2126. Siepmann, J. I.; Karaborni, S.; Smit, B. VaporLiquid-Equilibria of Model Alkanes J. Am. Chem. Soc. 1993, 115, 6454. Siepmann, J. I.; Karaborni, S.; Smit, B. Simulating the Critical-Behavior of Complex Fluids. Nature 1993, 365, 330. (5) Mooij, G. C. M. A.; Frenkel, D.; Smit, B. Direct Simulation of Phase Equilibria of Chain Molecules. J. Phys. Condensed Matters 1992, 4, L255. (6) Escobedo, F. A.; de Pablo, J. J. Monte-Carlo Simulation of the Chemical-Potential of Polymers in an Expanded Ensemble. J. Chem. Phys. 1996, 103, 2703. (7) Escobedo, F. A.; de Pablo, J. J. Expanded Grand Canonical and Gibbs Ensemble Monte Carlo Simulation of Polymers. J. Chem. Phys. 1996, 105, 4391. (8) Nath, S. K.; Escobedo, F. A.; de Pablo, J. J. On the Simulation of Vapor-Liquid Equilibria for Alkanes. J. Chem. Phys. 1998, 102, 9905. (9) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. SAFT - Equation of State Solution Model for Associating Fluids. Fluid Phase Equilib. 1989, 52, 31. Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New Reference Equation of State for Associating Fluids. Ind. Eng. Chem. Res. 1990, 29, 1709. (10) Huang, S.; Radosz, M. Equation of State for Large, Polydisperse and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284. Huang, S.; Radosz, M. Equation of State for Small, Large, Polydisperse and Associating Molecules: Extension to Fluid Mixtures. Ind. Eng. Chem. Res. 1991, 30, 1994. (11) Peng, D.; Robinson, D. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59. (12) Allen, M. P.; Tildesley, D. J. Computer Simu1ation of Liquids; Oxford University Press: New York, 1987.

3202 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 (13) de Pablo, J. J.; Laso, M.; Suter, U. W. Simulation of Solubility of Alkanes in Polyethylene Macromolecules 1993, 26, 6181. (14) TRC Thermodynamic Tables, Hydrocarbons, Thermodynamic Research Center, Texas A&M, University System, College Station, Texas, 1984 and 1995. (15) Dymond, J. H.; Smith, E. B. The Virial Coefficients of Gases; Clarendon Press: Oxford, 1969. (16) Harismiadis, V. I.; Szleifer, I. 2nd Virial-Coefficients of Chain MoleculessA Monte-Carlo Study. Mol. Phys. 1994, 81, 851. (17) Canjar, L. W.; Menning, F. S. Thermodynamic Properties and Reduced Correlations for Gases; Gulf Publishing Co.: Houston, TX, 1967. (18) Perry’s Chemical Engineers’ Handbook; Perry, R. H., Green, D. W., Eds.; McGraw-Hill Book Company: New York, 1984. (19) Toxvaerd, S. J. Equation of State of Alkanes II. J. Chem. Phys. 1997, 107, 5197. (20) Dee, G. T.; Ougizawa, T.; Walsh, D. J. The PressureVolume-Temperature Properties of Polyethylene, Poly(dimethyl siloxane), Poly(ethylene glycol) as a Function of Molecular Weight. Polymer 1992, 33, 3462. (21) Gmehling, J.; Onken, U. Vapor-Liquid Equilibrium Data Collection; Chemistry Data Series; DECHEMA: Frankfurt/Main, 1977-1990.

(22) Joke, J.; Zudkevitch, D. Prediction of Liquid-Phase Enthalpies with the Redlich-Kwong Equation of State. Ind. Eng. Chem. Fundam. 1970, 9, 545. (23) Peters, C. J.; Roo, J. L.; Lichtenthaler, R. N. Measurements and Calculations of Phase-Equilibria of Binary-Mixtures of Ethane+Eicosane. 1. Vapor + Liquid Equilibria. Fluid Phase Equil. 1987, 34, 287. (24) Van der Ploeg, P.; Berendsen, H. J. C. Molecular Dynamics Simulation of a Bilayer Membrane. J. Chem. Phys. 1991, 94, 5650. (25) Jorgensen, W. L.; Madura., J. D.; Swenson, C. J. Optimized Intermolecular Potential Functions for Liquid Hydrocarbons. J. Am. Chem. Soc. 1984, 106, 6638. (26) de Loos, Th. W.; Poot, W.; Lichtenthaler, R. N. Fluid Phase Equilibria in Binary Ethylene + n-Alkane Systems Ber. BunsenGes. Phys. Chem. 1984, 88, 855. (27) Spyriouni, T.; Economov, I. G.; Theodorou, D. N. Phase Equilibrium of Chain Molecules Predicted through a Novel Simulation Scheme. Phys. Rev. Lett. 1998, 80, 4466.

Received for review January 12, 1998 Revised manuscript received April 8, 1998 Accepted April 9, 1998 IE980021Q