^{1}and David A. Kofke

^{1,a)}

### Abstract

We report results of calculations of the second through sixth virial coefficients for four prototype Lennard-Jones (LJ) mixtures that have been the subject of previous studies in the literature. Values are reported for temperatures ranging from to , where here the temperature is given units of the LJ energy parameter of one of the components. Thermodynamic stability of the mixtures is studied using the virial equation of state (VEOS) with the calculated coefficients, with particular focus on characterizing the vapor-liquid critical behavior of the mixtures. For three of the mixtures, vapor-liquid coexistence and critical data are available for comparison at only one temperature, while for the fourth we can compare to a critical line. We find that the VEOS provides a useful indication of the presence and location of critical behavior, although in some situations we find need to consider “near-miss” critical behavior, where the classical conditions of criticality are nearly but not exactly satisfied.

Funding for this research was provided by Grant Nos. CTS-0414439 and CHE-0626305 from the U.S. National Science Foundation and the University at Buffalo School of Engineering and Applied Sciences. Computational support was provided by the University at Buffalo Center for Computational Research. Additional resources were provided by the Open Science Grid, which is supported by the National Science Foundation and the U.S. Department of Energy’s Office of Science. We are grateful to Dr. Javier Pérez-Pellitero for providing data from Ref. 23 in tabular form.

I. INTRODUCTION

II. MODELS AND METHODS

A. Models

B. Thermodynamic stability

C. Evaluation of virial coefficients

D. Simulation details

III. RESULTS AND DISCUSSION

IV. CONCLUDING REMARKS

### Key Topics

- Critical point phenomena
- 34.0
- Equations of state
- 6.0
- Monte Carlo methods
- 6.0
- Vapor pressure
- 5.0
- Chemical potential
- 2.0

## Figures

Virial coefficients for mixture I for . Values are made dimensionless by the critical density of the pure LJ fluid (Ref. 23) . Mixture I is symmetric with respect to exchange of the species, so some of the coefficients are identical, as indicated in the legends. Data are presented on a scale. Calculated coefficients are plotted with their 67% confidence limits (which are often smaller than the line thickness) and lines are drawn to join them using an interpolation scheme presented elsewhere (Ref. 29).

Virial coefficients for mixture I for . Values are made dimensionless by the critical density of the pure LJ fluid (Ref. 23) . Mixture I is symmetric with respect to exchange of the species, so some of the coefficients are identical, as indicated in the legends. Data are presented on a scale. Calculated coefficients are plotted with their 67% confidence limits (which are often smaller than the line thickness) and lines are drawn to join them using an interpolation scheme presented elsewhere (Ref. 29).

Mixture I spinodals in the composition-density plane. Dotted lines are calculated from VEOS4, solid lines from VEOS5, and dot-dashed lines from VEOS6. Each group of lines from left to right in the figure corresponds to the indicated temperature (in units of ). Spinodal lines terminate where the critical criterion [Eq. (7)] is satisfied (even though spinodal condition continues to be met at higher density). Black dot-dashed line traversing figure from to 1 at the far right is the line of critical points according to VEOS6.

Mixture I spinodals in the composition-density plane. Dotted lines are calculated from VEOS4, solid lines from VEOS5, and dot-dashed lines from VEOS6. Each group of lines from left to right in the figure corresponds to the indicated temperature (in units of ). Spinodal lines terminate where the critical criterion [Eq. (7)] is satisfied (even though spinodal condition continues to be met at higher density). Black dot-dashed line traversing figure from to 1 at the far right is the line of critical points according to VEOS6.

Critical line for mixture I in the pressure-temperature plane. Dotted line (circles) is computed from VEOS4, solid line (squares) VEOS5, and dot-dashed line (triangles) VEOS6. Open symbols at left correspond to equimolar mixture and critical lines for each pure substance begin at the right and trace identical curves (owing to the symmetry of the mixture).

Critical line for mixture I in the pressure-temperature plane. Dotted line (circles) is computed from VEOS4, solid line (squares) VEOS5, and dot-dashed line (triangles) VEOS6. Open symbols at left correspond to equimolar mixture and critical lines for each pure substance begin at the right and trace identical curves (owing to the symmetry of the mixture).

Comparison of mixture I spinodal at to established binodals (Ref. 22) at the same temperature. Dotted and solid lines are computed from VEOS4 and VEOS5, respectively, and end where the VEOS indicates a critical point; the dot-dashed line is computed from VEOS6 and does not exhibit a critical point.

Comparison of mixture I spinodal at to established binodals (Ref. 22) at the same temperature. Dotted and solid lines are computed from VEOS4 and VEOS5, respectively, and end where the VEOS indicates a critical point; the dot-dashed line is computed from VEOS6 and does not exhibit a critical point.

Same as Fig. 2, but for mixture II.

Same as Fig. 2, but for mixture II.

Critical line for mixture II in the pressure-temperature plane. Dotted line (circles) is computed from VEOS4, solid line (squares) VEOS5, and dot-dashed line, VEOS6.

Critical line for mixture II in the pressure-temperature plane. Dotted line (circles) is computed from VEOS4, solid line (squares) VEOS5, and dot-dashed line, VEOS6.

Critical line for mixture II in pressure-composition (top) and temperature-composition (bottom) planes. Dotted line (circles) is computed from VEOS4, solid line (squares) VEOS5, and dot-dashed line VEOS6.

Critical line for mixture II in pressure-composition (top) and temperature-composition (bottom) planes. Dotted line (circles) is computed from VEOS4, solid line (squares) VEOS5, and dot-dashed line VEOS6.

Schematic of behavior exhibited by critical-point indicators in mixtures III and IV. Lines are the indicated second derivatives (in arbitrary units) evaluated along the spinodals. The abscissa is a coordinate such as density or mole fraction that measures movement along the spinodal line. Along the spinodal, the two second derivatives are necessarily of opposite sign. The “near-miss” condition occurs when lines turn away from zero before crossing it. Two critical points emerge as temperature is varied. Further changes in temperature may move the second critical point to very high density or see it eliminated entirely.

Schematic of behavior exhibited by critical-point indicators in mixtures III and IV. Lines are the indicated second derivatives (in arbitrary units) evaluated along the spinodals. The abscissa is a coordinate such as density or mole fraction that measures movement along the spinodal line. Along the spinodal, the two second derivatives are necessarily of opposite sign. The “near-miss” condition occurs when lines turn away from zero before crossing it. Two critical points emerge as temperature is varied. Further changes in temperature may move the second critical point to very high density or see it eliminated entirely.

Mixture III spinodals in the composition-density plane. Dotted lines are calculated from VEOS4, solid lines from VEOS5, and dot-dashed lines from VEOS6. Each group of lines in the figure corresponds to the indicated temperature (in units of ). Spinodal lines terminate where the critical criterion [Eq. (7)] is satisfied (even though spinodal condition continues to be met at higher density). The × symbol indicates the location of a near-miss critical point for VEOS4 and VEOS5, and spinodal lines are drawn past them. Black dot-dashed line at far right is the line of critical points according to VEOS6; the line is drawn thicker at conditions where critical point is given by near-miss criterion (roughly between and 0.6).

Mixture III spinodals in the composition-density plane. Dotted lines are calculated from VEOS4, solid lines from VEOS5, and dot-dashed lines from VEOS6. Each group of lines in the figure corresponds to the indicated temperature (in units of ). Spinodal lines terminate where the critical criterion [Eq. (7)] is satisfied (even though spinodal condition continues to be met at higher density). The × symbol indicates the location of a near-miss critical point for VEOS4 and VEOS5, and spinodal lines are drawn past them. Black dot-dashed line at far right is the line of critical points according to VEOS6; the line is drawn thicker at conditions where critical point is given by near-miss criterion (roughly between and 0.6).

Spinodals for mixture III in the pressure-composition plane. Top: dotted lines are calculated from VEOS4, solid lines from VEOS5, and dot-dashed lines from VEOS6. Each group of lines in the figure corresponds to the indicated temperature (in units of ). Termination of lines and use of × symbol is as in Fig. 9. Bottom: comparison of mixture III spinodal at to established binodals (Ref. 22) at the same temperature, with × symbol indicating the near-miss critical point. Line types are as in top figure.

Spinodals for mixture III in the pressure-composition plane. Top: dotted lines are calculated from VEOS4, solid lines from VEOS5, and dot-dashed lines from VEOS6. Each group of lines in the figure corresponds to the indicated temperature (in units of ). Termination of lines and use of × symbol is as in Fig. 9. Bottom: comparison of mixture III spinodal at to established binodals (Ref. 22) at the same temperature, with × symbol indicating the near-miss critical point. Line types are as in top figure.

Critical line for mixture IV in the pressure-composition plane. Dotted line (circles) is computed from VEOS4; solid line (squares), VEOS5; dot-dashed line (triangles), VEOS6. Line is made thicker and open symbols are used where classical criterion [Eq. (7)] is not satisfied, and critical line is instead eliminated using near-miss critical condition. Symbols indicate temperatures where the virial coefficients were given directly by the Mayer-sampling calculations, and lines connecting them were computed using virial coefficients given by interpolation of the measured coefficients (Ref. 29). In addition, the × symbols indicate MC simulation data (Ref. 23).

Critical line for mixture IV in the pressure-composition plane. Dotted line (circles) is computed from VEOS4; solid line (squares), VEOS5; dot-dashed line (triangles), VEOS6. Line is made thicker and open symbols are used where classical criterion [Eq. (7)] is not satisfied, and critical line is instead eliminated using near-miss critical condition. Symbols indicate temperatures where the virial coefficients were given directly by the Mayer-sampling calculations, and lines connecting them were computed using virial coefficients given by interpolation of the measured coefficients (Ref. 29). In addition, the × symbols indicate MC simulation data (Ref. 23).

## Tables

Parameters for the LJ mixtures examined in this work. For all mixtures, the species-1 size and energy parameters and are unity and provide the length and energy scales for the system. The table lists the unlike and species-2 size and energy parameters for the four mixtures in units of the species-1 parameters.

Parameters for the LJ mixtures examined in this work. For all mixtures, the species-1 size and energy parameters and are unity and provide the length and energy scales for the system. The table lists the unlike and species-2 size and energy parameters for the four mixtures in units of the species-1 parameters.

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