^{1,2,a)}, P. A. Hlushak

^{2}and A. Trokhymchuk

^{2}

### Abstract

The theory, which utilizes an exponential enhancement of the first-order mean spherical approximation (FMSA) for the radial distribution functions of the hard-core plus square-well fluid, is adopted to study the properties of the simplest model of the core-softened fluids, i.e., the hard spheres with a square-shoulder interaction. The results for structure and thermodynamic properties are reported and compared against both the Monte Carlo simulation data as well as with those obtained within the conventional FMSA theory. We found that in the region of low densities and low temperatures, where the conventional FMSA theory fails, the exponential-based FMSA theory besides being qualitatively correct also provides with a notable quantitative improvement of the theoretical description.

S.H. was supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Geoscience Research Program, through Grant No. ERKCC72 to Oak Ridge National Laboratory, which is managed for DOE by UT Battelle, LLC under Contract No. DE-AC05-00OR22725.

I. INTRODUCTION

II. GENERAL CONSIDERATION

A. Towards a model and its theoretical treatment

B. Computer simulation details

C. FMSA theory for the radial distribution function of the HCSS fluid

D. SEXP/FMSA theory for the thermodynamics of the HCSS fluid

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Cumulative distribution functions
- 23.0
- Monte Carlo methods
- 17.0
- Particle distribution functions
- 15.0
- Density functional theory
- 9.0
- Perturbation theory
- 9.0

## Figures

Radial distribution functions of the HCSS fluid with square-shoulder width parameter λ = 1.1 at the reduced temperature *T** = *k* _{ B } *T*/ɛ = 0.5 and for several densities ρ* = ρσ^{3} = 0.1, 0.2, 0.4, 0.6, and 0.8 as it is specified in the figure. The dashed (black) lines denote the results of the conventional FMSA theory, while the solid (red) lines mark the results of the SEXP/FMSA theory. Symbols correspond to the MC simulations data. Insets show radial distribution functions of the MC simulation for larger distances from the centre.

Radial distribution functions of the HCSS fluid with square-shoulder width parameter λ = 1.1 at the reduced temperature *T** = *k* _{ B } *T*/ɛ = 0.5 and for several densities ρ* = ρσ^{3} = 0.1, 0.2, 0.4, 0.6, and 0.8 as it is specified in the figure. The dashed (black) lines denote the results of the conventional FMSA theory, while the solid (red) lines mark the results of the SEXP/FMSA theory. Symbols correspond to the MC simulations data. Insets show radial distribution functions of the MC simulation for larger distances from the centre.

Radial distribution functions of the HCSS fluid with square-shoulder width parameter λ = 1.2 at the reduced temperature *T** = *k* _{ B } *T*/ɛ = 0.5 and for several densities ρ* = ρσ^{3} = 0.1, 0.2, 0.4, 0.6, 0.8, and 0.7639 (η = 0.4) as it is specified in the figure. On the subplots for ρ* = ρσ^{3} = 0.1, 0.2, 0.4, 0.6, 0.8, the dashed (black) lines denote the results of the conventional FMSA theory, while the solid (red) lines mark the results of the SEXP/FMSA theory. Symbols correspond to the MC simulations data. Insets show radial distribution functions of the MC simulation for larger distances from the centre. The bottom-right subplot for (ρ* = 0.7639 or η = 0.4) compares the radial distribution functions predicted by the FMSA and SEXP/FMSA theories with the results of the optimized random phase approximation (ORPA) and optimized cluster theory (OCT). ^{ 4 } Thick solid lines of the bottom-right subplot denote the results of the Monte Carlo simulation. ORPA, OCT, and MC results for this subplot are from Ref. ^{ 4 } .

Radial distribution functions of the HCSS fluid with square-shoulder width parameter λ = 1.2 at the reduced temperature *T** = *k* _{ B } *T*/ɛ = 0.5 and for several densities ρ* = ρσ^{3} = 0.1, 0.2, 0.4, 0.6, 0.8, and 0.7639 (η = 0.4) as it is specified in the figure. On the subplots for ρ* = ρσ^{3} = 0.1, 0.2, 0.4, 0.6, 0.8, the dashed (black) lines denote the results of the conventional FMSA theory, while the solid (red) lines mark the results of the SEXP/FMSA theory. Symbols correspond to the MC simulations data. Insets show radial distribution functions of the MC simulation for larger distances from the centre. The bottom-right subplot for (ρ* = 0.7639 or η = 0.4) compares the radial distribution functions predicted by the FMSA and SEXP/FMSA theories with the results of the optimized random phase approximation (ORPA) and optimized cluster theory (OCT). ^{ 4 } Thick solid lines of the bottom-right subplot denote the results of the Monte Carlo simulation. ORPA, OCT, and MC results for this subplot are from Ref. ^{ 4 } .

Radial distribution functions of the HCSS fluid with square-shoulder width parameter λ = 1.5. The layout and notations are the same as in Fig. 2 .

Radial distribution functions of the HCSS fluid with square-shoulder width parameter λ = 1.5. The layout and notations are the same as in Fig. 2 .

Radial distribution functions of the HCSS fluid with square-shoulder width parameter λ = 1.8. The layout and notations are the same as in Fig. 1 .

Radial distribution functions of the HCSS fluid with square-shoulder width parameter λ = 1.8. The layout and notations are the same as in Fig. 1 .

Internal energy β*E*, compressibility factor *Z* = β*P*/ρ, and chemical potential βμ of the HCSS fluids with the square-shoulder width parameter λ = 1.1 (left column) and λ = 1.2 (right column) at the reduced temperature *T** = 0.5. The dashed (black) lines denote the results of the conventional FMSA theory, while the solid (red) lines mark the results of the SEXP/FMSA theory. Symbols correspond to the MC simulations data and insets show low density regions of the plots.

Internal energy β*E*, compressibility factor *Z* = β*P*/ρ, and chemical potential βμ of the HCSS fluids with the square-shoulder width parameter λ = 1.1 (left column) and λ = 1.2 (right column) at the reduced temperature *T** = 0.5. The dashed (black) lines denote the results of the conventional FMSA theory, while the solid (red) lines mark the results of the SEXP/FMSA theory. Symbols correspond to the MC simulations data and insets show low density regions of the plots.

Internal energy β*E*, compressibility factor *Z* = β*P*/ρ, and chemical potential βμ of the HCSS fluids with the square-shoulder width parameter λ = 1.5 (left column) and λ = 1.8 (right column) at the reduced temperature *T** = 0.5. The dashed (black) lines denote the results of the conventional FMSA theory, while the solid (red) lines mark the results of the SEXP/FMSA theory. Symbols correspond to the MC simulations data. The filled triangles correspond to the MC simulations data of Zhou and Solana. ^{ 38 } Insets show low density regions of the plots.

Internal energy β*E*, compressibility factor *Z* = β*P*/ρ, and chemical potential βμ of the HCSS fluids with the square-shoulder width parameter λ = 1.5 (left column) and λ = 1.8 (right column) at the reduced temperature *T** = 0.5. The dashed (black) lines denote the results of the conventional FMSA theory, while the solid (red) lines mark the results of the SEXP/FMSA theory. Symbols correspond to the MC simulations data. The filled triangles correspond to the MC simulations data of Zhou and Solana. ^{ 38 } Insets show low density regions of the plots.

## Tables

MC simulation data for the values of the radial distribution function at the distances of discontinuities, i.e., *r* = σ and *r* = λσ of the HCSS fluid with different values of the width of repulsive square shoulder. Several densities ρσ^{3} are given in the table and temperature is fixed at *T** = 0.5.

MC simulation data for the values of the radial distribution function at the distances of discontinuities, i.e., *r* = σ and *r* = λσ of the HCSS fluid with different values of the width of repulsive square shoulder. Several densities ρσ^{3} are given in the table and temperature is fixed at *T** = 0.5.

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