^{1,a)}and Jeffrey R. Errington

^{2,b)}

### Abstract

We present a novel computational methodology for determining fluid-phase equilibria in binary mixtures. The method is based on a combination of highly efficient transition-matrix Monte Carlo and histogram reweighting. In particular, a directed grand-canonical transition-matrix Monte Carlo scheme is used to calculate the particle-number probability distribution, after which histogram reweighting is used as a postprocessing procedure to determine the conditions of phase equilibria. To validate the methodology, we have applied it to a number of model binary Lennard-Jones systems known to exhibit nontrivial fluid-phase behavior. Although we have focused on monatomic fluids in this work, the method presented here is general and can be easily extended to more complex molecular fluids. Finally, an important feature of this method is the capability to predict the entire fluid-phase diagram of a binary mixture at fixed temperature in a single simulation.

V.K.S. gratefully acknowledges financial support from a National Research Council postdoctoral research associateship at the National Institute of Standards and Technology. J.R.E. acknowledges the financial support provided by startup funds from the University at Buffalo.

I. INTRODUCTION

II. GRAND-CANONICAL TRANSITION-MATRIX MONTE CARLO FOR BINARY MIXTURES

A. General overview

B. Directed sampling of particle-number space

III. MIXTURE PROPERTIES FROM THE MACROSTATE DISTRIBUTION

A. Thermodynamic properties

B. Phase equilibria

IV. COMPUTATIONAL DETAILS

V. RESULTS AND DISCUSSION

A. Mixture I

B. Mixture II

C. Mixture III

D. Mixture IV

E. Mixture V

F. Discussion

VI. CONCLUSIONS

### Key Topics

- Probability theory
- 29.0
- Chemical potential
- 23.0
- Monte Carlo methods
- 23.0
- Thermodynamic properties
- 15.0
- Phase equilibria
- 14.0

## Figures

Region in the plane over which the particle-number probability distribution must be calculated. In general, the region of interest for the fluid-phase behavior of a binary mixture is bounded by a right, scalene triangle.

Region in the plane over which the particle-number probability distribution must be calculated. In general, the region of interest for the fluid-phase behavior of a binary mixture is bounded by a right, scalene triangle.

Illustration of the directed sampling scheme of particle-number space in the plane. The system is forced to sample the entire composition range at fixed total particle number using isochoric semigrand Monte Carlo simulation. When , sampling is restricted to the composition range defined by pure component 1 and the point of intersection of the hypotenuse of the bounding right triangle and the line . The statistics gathered along each value are stitched together by performing phantom insertion and deletion moves (dashed lines) that are never accepted but whose acceptance probabilities are used to update the overall collection matrix.

Illustration of the directed sampling scheme of particle-number space in the plane. The system is forced to sample the entire composition range at fixed total particle number using isochoric semigrand Monte Carlo simulation. When , sampling is restricted to the composition range defined by pure component 1 and the point of intersection of the hypotenuse of the bounding right triangle and the line . The statistics gathered along each value are stitched together by performing phantom insertion and deletion moves (dashed lines) that are never accepted but whose acceptance probabilities are used to update the overall collection matrix.

Normalized particle-number probability distribution for a binary Lennard-Jones mixture designed to mimic an argon-krypton mixture. The distribution presented here was produced by taking a raw PNPD obtained from mixture transition-matrix Monte Carlo and reweighting it with respect to the chemical potentials of both species. This particular PNPD corresponds to liquid-vapor phase coexistence at and .

Normalized particle-number probability distribution for a binary Lennard-Jones mixture designed to mimic an argon-krypton mixture. The distribution presented here was produced by taking a raw PNPD obtained from mixture transition-matrix Monte Carlo and reweighting it with respect to the chemical potentials of both species. This particular PNPD corresponds to liquid-vapor phase coexistence at and .

Liquid-vapor phase diagram for an argon-krypton mixture, mixture I, in the pressure-composition plane at and . The solid lines are the predictions of the mixture TMMC simulation algorithm and are composed of several thousand points. Circles correspond to the experimental data points of Schouten *et al.* (Ref. 45), and crosses correspond to Gibbs-ensemble data generated by Panagiotopoulos (Ref. 41). and are mole fractions of argon in the liquid and vapor phases, respectively.

Liquid-vapor phase diagram for an argon-krypton mixture, mixture I, in the pressure-composition plane at and . The solid lines are the predictions of the mixture TMMC simulation algorithm and are composed of several thousand points. Circles correspond to the experimental data points of Schouten *et al.* (Ref. 45), and crosses correspond to Gibbs-ensemble data generated by Panagiotopoulos (Ref. 41). and are mole fractions of argon in the liquid and vapor phases, respectively.

Pressure-composition diagram for the size-asymmetric Lennard-Jones mixture, mixture II, at reduced temperatures and 1.00. The solid lines are the predictions of the mixture TMMC simulation algorithm and are composed of several thousand points. Crosses correspond to the Gibbs-ensemble calculations of Harismiadis *et al.* (Ref. 42). and are the mole fractions of species 1, the smaller of the two components, in the liquid and vapor phases, respectively.

Pressure-composition diagram for the size-asymmetric Lennard-Jones mixture, mixture II, at reduced temperatures and 1.00. The solid lines are the predictions of the mixture TMMC simulation algorithm and are composed of several thousand points. Crosses correspond to the Gibbs-ensemble calculations of Harismiadis *et al.* (Ref. 42). and are the mole fractions of species 1, the smaller of the two components, in the liquid and vapor phases, respectively.

Liquid-vapor phase diagram in the pressure-composition plane for a symmetric Lennard-Jones mixture, mixture III, at and 1.15. The solid lines are the predictions of the mixture TMMC simulation algorithm and are composed of several thousand points. Gibbs-ensemble data (×) generated by Panagiotopoulos *et al.* (Ref. 6) and Gibbs–Duhem integration data (+) generated by Mehta and Kofke (Ref. 9) are also plotted. and are the mole fractions of species 1 in the liquid and vapor phases, respectively.

Liquid-vapor phase diagram in the pressure-composition plane for a symmetric Lennard-Jones mixture, mixture III, at and 1.15. The solid lines are the predictions of the mixture TMMC simulation algorithm and are composed of several thousand points. Gibbs-ensemble data (×) generated by Panagiotopoulos *et al.* (Ref. 6) and Gibbs–Duhem integration data (+) generated by Mehta and Kofke (Ref. 9) are also plotted. and are the mole fractions of species 1 in the liquid and vapor phases, respectively.

Fluid-phase diagram in the pressure-composition plane for mixture III at and 0.85. , , and represent a vapor phase, a liquid rich in component 1, and a liquid rich in component 2, respectively. and are the mole fractions of species 1 in the liquid and vapor phases, respectively. Horizontal tie lines are drawn to denote the location of three-phase coexistence.

Fluid-phase diagram in the pressure-composition plane for mixture III at and 0.85. , , and represent a vapor phase, a liquid rich in component 1, and a liquid rich in component 2, respectively. and are the mole fractions of species 1 in the liquid and vapor phases, respectively. Horizontal tie lines are drawn to denote the location of three-phase coexistence.

Particle-number probability distribution for mixture III at the triple point at a temperature .

Particle-number probability distribution for mixture III at the triple point at a temperature .

Liquid-vapor phase diagram in the pressure-composition plane for mixture IV at five temperatures, , 0.80, 0.90, 1.00, and 1.10. The solid lines are the predictions of the mixture TMMC simulation algorithm and are composed of several thousand points. Crosses correspond to the Gibbs–Duhem integration data of Pandit and Kofke (Ref. 43). and are the mole fractions of species 1 in the liquid and vapor phases, respectively.

Liquid-vapor phase diagram in the pressure-composition plane for mixture IV at five temperatures, , 0.80, 0.90, 1.00, and 1.10. The solid lines are the predictions of the mixture TMMC simulation algorithm and are composed of several thousand points. Crosses correspond to the Gibbs–Duhem integration data of Pandit and Kofke (Ref. 43). and are the mole fractions of species 1 in the liquid and vapor phases, respectively.

Liquid-vapor phase diagram in the pressure-composition plane for mixture V at five temperatures , 0.90, 1.00, 1.10, and 1.20. , , and represent a vapor phase, a liquid rich in component 1, and a liquid rich in component 2, respectively. and are the mole fractions of species 1 in the liquid and vapor phases, respectively. Horizontal tie lines are drawn to denote the location of three-phase coexistence.

Liquid-vapor phase diagram in the pressure-composition plane for mixture V at five temperatures , 0.90, 1.00, 1.10, and 1.20. , , and represent a vapor phase, a liquid rich in component 1, and a liquid rich in component 2, respectively. and are the mole fractions of species 1 in the liquid and vapor phases, respectively. Horizontal tie lines are drawn to denote the location of three-phase coexistence.

## Tables

Binary Lennard-Jones parameters.

Binary Lennard-Jones parameters.

Comparison of azeotrope pressure and composition for mixture IV between Pandit and Kofke’s (PK) Gibbs–Duhem integration results (Ref. 43) and mixture TMMC (M-TMMC) predictions.

Comparison of azeotrope pressure and composition for mixture IV between Pandit and Kofke’s (PK) Gibbs–Duhem integration results (Ref. 43) and mixture TMMC (M-TMMC) predictions.

Comparison of triple point predictions for mixture V between Lopes’ Gibbs-ensemble predictions (Ref. 44) and mixture TMMC (M-TMMC).

Comparison of triple point predictions for mixture V between Lopes’ Gibbs-ensemble predictions (Ref. 44) and mixture TMMC (M-TMMC).

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