A Perturbation Theory approach to Equation of State and Excess energy of Size-Asymmetric Square-Well Fluid Mixtures

 

Roquia Perween¹, Binay Prakash Akhouri²*

¹University Department of Physics, Ranchi University, Ranchi - 834008, Jharkhand (India).

^1,2Department of Physics, Suraj Singh Memorial College, Kanke - 834008, Ranchi, Jharkhand (India).

*Corresponding Author E-mail: binayakhouri@yahoo.in

 

 

INTRODUCTION:

Simple model fluids play a central role in understanding the thermodynamics behavior of real of real molecular systems. Among these, the square-well potential has long been recognized as one of the most useful intermolecular models, as it combines a hard-core repulsion with a finite-range attraction while remaining analytically and computationally tractable^1-15. For simple liquids, the hard-core component of the potential is the primary determinant of their structure and the non-hard-core component mainly serves to generate a uniform background potential within which the molecules can move. In many practical systems, however, the constituent species differ significantly in molecular size. Such size asymmetry is encountered in colloid-polymer mixtures, surfactant solutions, electrolyte models, and complex fluid formulations.

 

Size disparity strongly influences packing effects, excluded volume interactions, and the relative importance of attractive forces, thereby leading to thermodynamic behavior that cannot be captured adequately by symmetric or single-component models. Consequently, the development of reliable equations of state and free-energy expressions for size-asymmetric square-well mixtures remains an important problem in statistical thermodynamics. Recent years have seen a surge in interest in using   computer simulation data to predict thermodynamic properties of fluids^16-43. The results of these simulations have been utilized in the formulation of an equation of state, the prediction of excess properties, and the investigation of mixing rules for fluid mixtures. The effectiveness of a novel mixing rule grounded in statistical mechanical perturbation theory in predicting the compressibility factor and excess energy of binary mixtures has been tested. In this work, we present a perturbation theory based equation of state and thermodynamic description of size-asymmetric square-well mixtures. The computer simulations¹ data were used to predict compressibility factors and excess energy for binary mixture for diameter ratios. The paper has been organized as follows. In section I, the theoretical framework of perturbation theory-square-well fluid mixtures. Section II extends this formalism to size-asymmetric square-well mixtures, where explicit expressions for the excess free energy, equation of state, and related thermodynamic quantities are derived. Section III contains the results and discussion with emphasis on equation of state excess free energy, whereas Section IV contains the conclusion of the present study. Size-asymmetry strongly influences exclude-volume (repulsive) interactions. Large particles restrict the free volume available to small particles and packing fraction and local structure change significantly. This directly affects the excess Helmholtz free energy and the equation of state.

 

1.      Perturbation Theory of Square-Well Fluid Mixture: In the theory of perturbation, the well-known sum has the form, , whereby  is the reference potential, and  is the perturbation of the former, the former being the reference potential. Consider a binary mixture .

 

Here, the potential of the hard sphere serves as the reference potential.

 

Helmholtz free energy: The Helmholtz free energy in high temperature expansion (HTE) has the form¹ as given in equation (4).

For SW mixtures:

 

Hence excess free energy

 

Equation (4) provides the basis for calculating the equation of state and other thermodynamic properties of size-asymmetric SW mixture. The Boltzmann constant, denoted by, is the potential's energy parameter and is considered as the free energy's perturbation term. In this expansion, zero term is acquired by integrating an appropriate equation of state, and computer simulation is used to derive the higher order terms. The compressibility factor and excess energy  can be written as follows:

 

Compressibility factor: It is defined as the ratio of the actual pressure of a fluid to the pressure predicted by the ideal gas law at the same temperature and density:

                                (5)

                (6)

 

Excess energy: It is defined as the difference between the internal energy of a real interacting system and that of an ideal system at the same temperature, volume, and composition:

 

                                                   (7)

and

 

Pavlyukhin Perturbation Theories:¹⁵

The details of Pavlykhuin perturbation theory are given in Reference15. The expression for free energy term has the form.

 

                                                                  (9)

 

One can approximate by different polynomials of in the range of   values. For a fixed value of , the dependence on is actually given as a polynomial, i.e it is smoothed enough. For a system of N particles, we introduce the total number of pair of particles whose interatomic distance distances satisfy the condition

 

. Evidently,

2. Square Well Fluid Binary Mixture:

To understand the perturbation theory for mixtures, we have, the expression,

 

                (10)

where is the energy parameter and  is the distance parameter, for a particle of types interacting with a particle of types. The size asymmetry is given by  and cross diameter using Lorentz rule is . The potential for interaction between  and  particles in the SW mixes under consideration is

                         (11)

 

Comparison of a new mixing rule with the existing mixing rule: The equation (9) provides the expression for the Helmholtz energy of square well (SW) mixtures having the HS reference fluid as. BMCSL equation of state provides an accurate expression for the compressibility factor and excess Helmholtz free energy of size asymmetric hard-sphere mixtures^15-27 and is widely used as the reference system in perturbation theories such as Pavlyukhin and square-well perturbation theories. The HS reference fluid is produced by integrating the Boublik-Mansoori-Carnhan-Starling-Leeland (BMCSL) model EOS^27,28. 

 

                                                                                             (12)

 

For a hard-sphere mixture, the excess Helmholtz free energy per particle in BMCSL form is

 

                                                                               (13)

 

in which

 

where is the component 's diameter in the HS mixture with a mole fraction of.

 

The BMCSL equation works effectively at higher temperatures; however it should be noted that the perturbation theory loses accuracy at very low temperatures due to the non-convergence of inverse power of expansion of temperature.  At low densities the van der Waals one fluid mixing rules work reasonably well. However, when wide ranges of densities and compositions are involved, they are no longer satisfactory and also Dieters’ proposition does not work satisfactory. Size asymmetry is characterized by the ratio  . A fixed value of is chosen to systematically study the effect of size disparity on compressibility factor and the excess Helmholtz energy. In this case, the BMCSL equation's tested ratio, produces better results.

 

Mixing rule: Below are the standard mixing rules used for size-asymmetric square-well mixtures within the Pavlyukhin-BBMSCL perturbative frame-work.

 

                                     (15)

                         (16)

 

In size-asymmetric square-well mixtures, unlike-particle interactions are described using standard Lorentz-Bertholet mixing rules. The effective hard-sphere is given by, while the square well depth follows.

 

These mixing rules allow consistent incorporation of size disparity into the BMCSL reference system and the perturbative attractive contribution in Pavlyukhin theory. Here,

 

The one fluid model of van der Waals is verified within the framework of perturbation theory pertaining to fluid mixtures. In addition to the van der Waals mixing rules the other considered mixing rules taken here for comparison with simulation data are that of mixing rules of Deters’⁸, mixing rules of Ely⁹ and mixing rules of Jonha¹⁰ and the new mixing rules of Binay and Solana¹. The new mixing rules are:

 

                                          (17)

      (18)

 

The newly implemented mixing rule has influence of temperature, density and composition. Note that the Jonah mixing rules function on both composition and temperature, while the van der Waals mixing rule requires solely on composition. Dieter's mixing rules, on the other hand, depend on both composition and density.

 

RESULT AND DISCUSSION:

Monte Carlo simulations provide accurate reference data for the excess Helmholtz free energy and equation of state of size asymmetric mixtures^1-43. Comparison of Monte Carlo results with BMCSL-based and Pavlyukhin perturbation-theory equation of state enables validation of the reference hard-sphere description and assessment of perturbative contributions, thereby establishing the reliability of the theory for realistic asymmetric fluids. Monte Carlo NVT simulations results¹ were used to justify the EOS and the excess energy for  square well binary mixtures with = 1.5, diameter ratio,potential depths ratio= 1 and 0.5, and the mole fractions = 0.25, 0.50 and 0.75, reduced temperatures  = 1.25, 1.50, 2.0 and 2.5 wherewith  and reduced densities= 0.1 to 0.9 with  step 0.1. For the ratio, the comparison of simulation results and the two approximations of Ely and Jonah for  are found to be excellent for any value of. But in case of  a little deviation is observed for other values of i.e., etc.

 

Figure 1: Attractive contribution to the excess energy and compressibility factor, for fluid mixture in square wells with diameter ratios /of 2/3 and energy ratios/ of 0.5.

 

Figure 2: Attractive contribution to the excess energy  and compressibility factor, for fluid mixture in square wells with diameter ratios / of 2/3 and energy ratios/ of 1.

 

In above Figs. 1and 2, the simulation results for  = 0.25, 0.50, and 0.75 are represented by circles (●), squares (■), and triangles (▲), respectively.  The outcomes of the van der Waals one fluid theory, Ely, Jonah, and present mixing rules are dotted (….), dashed (- - -), dot-dashed (−∙ −∙ −∙), and continuous (─) curves, respectively. The reduced temperatures are taken as=1.25,1.50,2.0 and 2.5 for both and  in Figs. (1) and (2) from top to down. The points and curves of Zatt for each mole fraction have been moved down by 0.5 with respect to those immediately above.

 

CONCLUSION:

The new mixing rules proposed by Solana and Akhouri¹ yields improved performance compared to the mixing rules proposed by Dieters’⁸, Ely⁹, and Jonha¹⁰. The success of the new mixing rule emphasizes the importance of incorporating physically meaningful diameter and energy mixing schemes and hence this proposed mixing rule offers a more realistic description of intermolecular interactions. Hence, Binay-Solana mixing rule can be attributed to its better treatment of molecular size asymmetry and interaction parameter.

 

CONFLITS OF INTEREST:

The authors declare that there are no conflicts of interest.

 

REFERENCES:

1.      Akhouri BP, Solana JR. Monte Carlo simulation of the thermodynamic properties of square-well fluid mixtures by a one-component effective fluid model. Molecular Simulation. 2020; 46:102-110. https://doi.org/10.1080/08927022.2019.1677903.

2.      Akhouri BP, Solana JR. Monte Carlo simulation and theoretical calculation of the thermodynamic properties of binary hard-core Lennard-Jones fluid mixtures. Molecular Simulation. 2020; 46(14): 1117–1124. https://doi.org/ 10.1080/08927022.2020.1806264

3.      Akhouri BP, Solana JR. Thermodynamic properties of hard-core attractive Yukawa fluids: single-component monomers, binary mixtures and chains. Journal of Molecular Liquids. 2021; 338: 116493. https://doi.org/10.1016/j.molliq.2021.116493

4.      Akhouri BP, Solana JR. On the choice of the effective diameter in the high-temperature expansion for the Lennard–Jones fluid. Molecular Physics. 2022; 120(7): e2028918. https://doi.org/10.1080/00268976.2022.2028918

5.      Akhouri BP, Solana JR. Thermodynamic properties of Ar, Kr and Xe from a Monte Carlo-based perturbation theory with an effective two-body Lennard-Jones potential. Physica A.  2022; 608: 128280/1-14. https://doi.org/1 10.1016/j.physica.2022.128280

6.      Akhouri BP, Perween R, Solana JR. Thermodynamic properties of square-well fluid mixtures by a one-component effective fluid model Molecular Simulation. 2024; 50: 238-247. https://doi.org/10.1080/08927022.2023.2292558

7.      Akhouri, BP, Solana JR. The role of higher-order terms in perturbation approaches to the monomer and bonding contribution in a SAFT-type equation of state for square-well chain fluids. Mol. Phys. 2018;116: 1706–1721.https://doi.org/10.1080/00268976.2018.144802

8.      Deiters UK. Density-dependent mixing rules for the calculation of fluid phase equilibria at high pressures. Fluid Phase Equilib. 1987;33: 267–293.https://doi.org/10.1016/0378-3812(87)85024-8

9.      Ely JF. The extended corresponding states method applied to the nitrogen–methane system. In Equations of State—Theories and Applications; ACS Symposium Series 300; American Chemical Society: Washington, DC, 1986; 331–350.https://doi.org/10.1021/bk-1986-0300.ch020

10.   Jonah DA. Comment on “Systematic investigation of the phase behavior in binary fluid mixtures. II. Calculations based on the Carnahan–Starling–Redlich–Kwong equation of state”. J. Chem. Phys. 1992;97: 3867–3868.https://doi.org/10.1063/1.463933

11.   Solana JR. Perturbation Theories for the Thermodynamic Properties of Fluids and Solids; CRC Press: Boca Raton, FL, 2013.https://doi.org/10.1201/b15485

12.   Zwanzig RW. High-temperature equation of state by a perturbation method. I. Nonpolar gases. J. Chem. Phys. 1954; 22: 1420–1426.https://doi.org/10.1063/1.1740409

13.   Paricaud P. Local composition model for the calculation of the surface tension of mixtures. J. Chem. Phys. 2006; 124: 054508.https://doi.org/10.1063/1.2165187

14.   Barker JA., Henderson D. What is “liquid”? Understanding the states of matter. Rev. Mod. Phys. 1976; 48: 587–671.https://doi.org/10.1103/RevModPhys.48.587

15.   Espíndola-Heredia R, Malijevský A. Optimized equation of state of the square-well fluid of variable range, based on a fourth-order free-energy expansion. J. Chem. Phys. 2009; 130: 024509.https://doi.org/10.1063/1.3054380

16.   Carnahan NF, Starling K.E. Equation of state for non-attracting rigid spheres. J. Chem. Phys. 1969; 51: 635–636.https://doi.org/10.1063/1.1672048

17.   Wertheim MS. Exact solution of the Percus–Yevick integral equation for hard spheres. Phys. Rev. Lett. 1963; 10: 321–323.https://doi.org/10.1103/PhysRevLett.10.321

18.   Verlet L. Weis, J. J. Perturbation theory for the thermodynamic properties of simple liquids. Phys. Rev. A 1972; 5: 939–952.https://doi.org/10.1103/PhysRevA.5.939

19.   Akhouri BP, Roquia P. Equation of state and excess energy for size asymmetric square well mixture. AIP Conf.Proc.3398,080005, 2026; 080005-1-10. http://doi.org/10.1063/5.0318166

20.   Bravo Yuste S, Santos A. Radial distribution function for hard spheres. Phys. Rev. A 1991; 43: 5418–5423.https://doi.org/10.1103/PhysRevA.43.5418

21.   Tang Y, Lu BC. -Y. Thermodynamic perturbation theory for square-well fluids. J. Chem. Phys. 1995; 103,:7463–7470.https://doi.org/10.1063/1.470437

22.   Barker JA., Henderson D. Perturbation theory and equation of state for fluids. J. Chem. Phys. 1967; 47: 2856–2861.https://doi.org/10.1063/1.1712308

23.   Smith WR, Henderson D, Barker JA. Perturbation theory of fluids. J. Chem. Phys. 1970; 53: 508–515.https://doi.org/10.1063/1.1673822

24.   Largo J, Solana JR. Square-well fluids: perturbation approach. Mol. Simul. 2003; 29: 363–371.https://doi.org/10.1080/0892702031000109262

25.   Zhang BJ, Equation of state for square-well fluids. Fluid Phase Equilib. 1999; 154: 1–10. https://doi.org/10.1016/S0378-3812(98)00362-1

26.   Pavlyukhin YT. Thermodynamic properties of square-well fluids. J. Struct. Chem. 2012; 53: 476–486.https://doi.org/10.1134/S0022476612030113

27.   Zhou S, Solana JR. Higher-order perturbation expansion for square-well fluids. J. Phys. Chem. B 2013; 117: 9305–9313.https://doi.org/10.1021/jp404328f

28.   Largo L, Solana JR. Extended perturbation theory for square-well fluids. Mol. Phys. 2016; 114: 2391–2399.https://doi.org/10.1080/00268976.2016.1176034

29.   Zhou S, Solana JR. Fourth-order perturbation theory for square-well fluids. J. Chem. Phys. 2013; 138: 244115.https://doi.org/10.1063/1.4811176

30.   Sastre F, Moreno-Hilario E, Sotelo-Serna MG. et al. Equation of state for square-well fluids. Mol. Phys. 2017; 116: 351–360.https://doi.org/10.1080/00268976.2017.1357786

31.   Gil-Villegas A, del Río F, Benavides AL. SAFT-type equation of state for square-well fluids. Fluid Phase Equilib. 1996; 119: 97–112.https://doi.org/10.1016/0378-3812(96)03061-5

32.   Leland TW, Rowlinson JS, Sather GA. Thermodynamic properties of square-well fluids. Trans. Faraday Soc. 1968; 64: 1447–1460.

33.   Mansoori GA, Leland TW. Statistical mechanics of hard-sphere mixtures. J. Chem. Soc., Faraday Trans. II 1972; 68: 320–344.https://doi.org/10.1039/F29726800320

34.   Vidales AL, Benavides A, Gil-Villegas A. Square-well chain fluids. Mol. Phys. 2001; 99: 703–710.https://doi.org/10.1080/00268970110034328

35.   Longuet-Higgins HC. Statistical mechanics of liquids. Proc. R. Soc. London A 1951; 205: 247–269.https://doi.org/10.1098/rspa.1951.0043

36.   Smith WR. Perturbation theory of liquids. Can. J. Chem. Eng. 1972; 50: 271–274. https://doi.org/10.1002/cjce.5450500303

37.   Cook D, Rowlinson JS. Thermodynamics of square-well fluids. Proc. R. Soc. London A 1953; 219: 405–418.https://doi.org/10.1098/rspa.1953.0157

38.   Boublík T. Hard-sphere equation of state. J. Chem. Phys. 1970; 53: 471–472. https://doi.org/10.1063/1.1673824

39.   Mansoori GA, Carnahan NF, Starling KE, Leland TW. Equation of state for hard-sphere mixtures. J. Chem. Phys. 1971; 54: 1523–1525.https://doi.org/10.1063/1.1675048

40.   Akhouri, BP. Shape factors in hard convex body equations of state. Asian journal of Research in Chemistry. 2017;10(1): 36-41. http:/doi.org/10.5958/0974-4150.2017.00007.4

41.   Akhouri BP; Kaur S. Viscosity for isotopes of helium with HCB model. Asian journal of Research in Chemistry.2016; 9(12): 623-632.http:/doi.org/10.5958/0974-4150.2016.00084.5

42.   Akhouri BP, Akhtar K, Akhouri TP. Helmholtz energies from equations of state and their relative deviations. Asian journal of Research in Chemistry.2015; 8(9) :545-551.http:/doi.org/10.5958/0974-4150.2015.00092.9

43.   Akhouri BP, Roquia P. An improved SAFT-type equation of state for variable-Range square-well chain fluids. AIP Conf.Proc.3398,080006, 2026; 080006-1-15. http:/doi.org/10.1063/5.0318167

 

Received on 23.01.2026      Revised on 17.02.2026 Accepted on 12.03.2026      Published on 27.05.2026 Available online from May 30, 2026 Asian J. Research Chem.2026; 19(3):263-269. DOI: 10.52711/0974-4150.2026.00041 ©A and V Publications All Right Reserved
This work is licensed under a Creative Commons Attribution-Non Commercial-Share Alike 4.0 International License. Creative Commons License.