^{1,a)}, Mariano López de Haro

^{1,b)}and Santos B. Yuste

^{1,c)}

### Abstract

Different theoretical approaches for the thermodynamic properties and the equation of state for multicomponent mixtures of nonadditive hard spheres in dimensions are presented in a unified way. These include the theory by Hamad, our previous formulation, the original MIX1 theory, a recently proposed modified MIX1 theory, as well as a nonlinear extension of the MIX1 theory proposed in this paper. Explicit expressions for the compressibility factor, Helmholtz free energy, and second, third, and fourth virial coefficients are provided. A comparison is carried out with recent Monte Carlo data for the virial coefficients of asymmetric mixtures and with available simulation data for the compressibility factor, the critical consolute point, and the liquid-liquid coexistence curves. The merits and limitations of each theory are pointed out.

This work has been supported by the Ministerio de Educación y Ciencia (Spain) through Grant No. FIS2007-60977 (partially financed by FEDER funds).

I. INTRODUCTION

II. GENERAL BACKGROUND

III. SOME APPROXIMATE THEORETICAL DEVELOPMENTS

A. MIX1 approximation

B. Paricaud’s modified MIX1 theory (mMIX1)

C. Hamad’s proposal

D. The Santos–López de Haro–Yuste (SHY) proposal

E. A nonlinear MIX1 theory

IV. RESULTS

A. Virial coefficients

B. Compressibility factor

C. Demixing

V. CONCLUDING REMARKS

### Key Topics

- Equations of state
- 11.0
- Helmholtz free energy
- 10.0
- Cumulative distribution functions
- 5.0
- Photon density
- 4.0
- Data analysis
- 3.0

## Figures

Plot of the composition-independent fourth virial coefficients , , and vs the size ratio for a nonadditivity parameter . The dotted lines correspond to the original MIX1 theory, Eq. (3.7), the short-dashed lines correspond to the mMIX1 theory, Eq. (3.7) with , the thin solid lines correspond to the nlMIX1 theory, Eq. (3.34), the long-dashed lines correspond to Hamad’s proposal, Eq. (3.20), and the thick solid lines correspond to the SHY proposal, Eq. (3.27). The symbols are Monte Carlo data from Ref. 17.

Plot of the composition-independent fourth virial coefficients , , and vs the size ratio for a nonadditivity parameter . The dotted lines correspond to the original MIX1 theory, Eq. (3.7), the short-dashed lines correspond to the mMIX1 theory, Eq. (3.7) with , the thin solid lines correspond to the nlMIX1 theory, Eq. (3.34), the long-dashed lines correspond to Hamad’s proposal, Eq. (3.20), and the thick solid lines correspond to the SHY proposal, Eq. (3.27). The symbols are Monte Carlo data from Ref. 17.

Same as in Fig. 1, but for .

Same as in Fig. 1, but for .

Same as in Fig. 1, but for .

Same as in Fig. 1, but for .

Same as in Fig. 1, but for .

Same as in Fig. 1, but for .

Same as in Fig. 1, but for .

Same as in Fig. 1, but for .

Same as in Fig. 1, but for .

Same as in Fig. 1, but for .

Plot of the compressibility factor vs the nonadditivity parameter for a symmetric binary mixture of nonadditive hard spheres at and two different compositions. The dotted lines correspond to the original MIX1 theory, Eq. (3.3), the short-dashed lines correspond to the mMIX1 theory, Eq. (3.3) with , the thin solid lines correspond to the nlMIX1 theory, Eq. (3.31), the long-dashed lines correspond to Hamad’s proposal, Eq. (3.18), and the thick solid lines correspond to the SHY proposal, Eq. (3.24). The symbols are results from Monte Carlo simulations (Refs. 10 and 11).

Plot of the compressibility factor vs the nonadditivity parameter for a symmetric binary mixture of nonadditive hard spheres at and two different compositions. The dotted lines correspond to the original MIX1 theory, Eq. (3.3), the short-dashed lines correspond to the mMIX1 theory, Eq. (3.3) with , the thin solid lines correspond to the nlMIX1 theory, Eq. (3.31), the long-dashed lines correspond to Hamad’s proposal, Eq. (3.18), and the thick solid lines correspond to the SHY proposal, Eq. (3.24). The symbols are results from Monte Carlo simulations (Refs. 10 and 11).

Plot of the compressibility factor vs the nonadditivity parameter for an equimolar asymmetric binary mixture of nonadditive hard spheres with size ratio at . The dotted line corresponds to the original MIX1 theory, Eq. (3.3), the short-dashed line corresponds to the mMIX1 theory, Eq. (3.3) with , the thin solid line corresponds to the nlMIX1 theory, Eq. (3.31), the long-dashed line corresponds to Hamad’s proposal, Eq. (3.18), and the thick solid line corresponds to the SHY proposal, Eq. (3.24). The symbols are results from Monte Carlo simulations (Ref. 13).

Plot of the compressibility factor vs the nonadditivity parameter for an equimolar asymmetric binary mixture of nonadditive hard spheres with size ratio at . The dotted line corresponds to the original MIX1 theory, Eq. (3.3), the short-dashed line corresponds to the mMIX1 theory, Eq. (3.3) with , the thin solid line corresponds to the nlMIX1 theory, Eq. (3.31), the long-dashed line corresponds to Hamad’s proposal, Eq. (3.18), and the thick solid line corresponds to the SHY proposal, Eq. (3.24). The symbols are results from Monte Carlo simulations (Ref. 13).

Plot of the compressibility factor vs the size ratio for binary mixtures of nonadditive hard spheres with and (upper panel) and (lower panel). The dotted lines correspond to the original MIX1 theory, Eq. (3.3), the short-dashed lines correspond to the mMIX1 theory, Eq. (3.3) with , the thin solid lines correspond to the nlMIX1 theory, Eq. (3.31), the long-dashed lines correspond to Hamad’s proposal, Eq. (3.18), and the thick solid lines correspond to the SHY proposal, Eq. (3.24). The symbols are results from Monte Carlo simulations (Ref. 13).

Plot of the compressibility factor vs the size ratio for binary mixtures of nonadditive hard spheres with and (upper panel) and (lower panel). The dotted lines correspond to the original MIX1 theory, Eq. (3.3), the short-dashed lines correspond to the mMIX1 theory, Eq. (3.3) with , the thin solid lines correspond to the nlMIX1 theory, Eq. (3.31), the long-dashed lines correspond to Hamad’s proposal, Eq. (3.18), and the thick solid lines correspond to the SHY proposal, Eq. (3.24). The symbols are results from Monte Carlo simulations (Ref. 13).

Plot of the reduced critical density vs the nonadditivity parameter for symmetric binary mixtures of nonadditive hard spheres. The short-dashed line corresponds to the mMIX1 theory, the thin solid line corresponds to the nlMIX1 theory, the long-dashed line corresponds to Hamad’s proposal, and the thick solid line corresponds to the SHY proposal. The symbols are results from Monte Carlo simulations (Refs. 25–27).

Plot of the reduced critical density vs the nonadditivity parameter for symmetric binary mixtures of nonadditive hard spheres. The short-dashed line corresponds to the mMIX1 theory, the thin solid line corresponds to the nlMIX1 theory, the long-dashed line corresponds to Hamad’s proposal, and the thick solid line corresponds to the SHY proposal. The symbols are results from Monte Carlo simulations (Refs. 25–27).

Liquid-liquid coexistence curves for several binary mixtures of nonadditive hard spheres in the reduced density vs composition plane (top panel) and in the reduced pressure vs composition plane (bottom panel). From top to bottom, the set of curves correspond to (absent in the bottom panel), (1,0.2), and (5/6,0.1818). The short-dashed lines correspond to the mMIX1 theory, the thin solid lines correspond to the nlMIX1 theory, and the thick solid lines correspond to the SHY proposal. The diamonds indicate the locations of the respective critical consolute points. The other symbols are results from Monte Carlo simulations: Ref. 25 (filled circles), Ref. 28 (open circles), and Ref. 9 (filled squares).

Liquid-liquid coexistence curves for several binary mixtures of nonadditive hard spheres in the reduced density vs composition plane (top panel) and in the reduced pressure vs composition plane (bottom panel). From top to bottom, the set of curves correspond to (absent in the bottom panel), (1,0.2), and (5/6,0.1818). The short-dashed lines correspond to the mMIX1 theory, the thin solid lines correspond to the nlMIX1 theory, and the thick solid lines correspond to the SHY proposal. The diamonds indicate the locations of the respective critical consolute points. The other symbols are results from Monte Carlo simulations: Ref. 25 (filled circles), Ref. 28 (open circles), and Ref. 9 (filled squares).

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