Coupling-parameter expansion in thermodynamic perturbation theory

Ramana, A. Sai Venkata; Menon, Suresh

doi:10.1103/physreve.87.022101

Cited by 21 publications

(8 citation statements)

References 18 publications

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“…One eventual misguided view is that the numerical differentiation method may be not powerful in the higher-order derivatives, and this may induce attention to a semi-numerical and semi-analytical method recently suggested in literature. 20 The present study proves that the worries are unnecessary. Moreover, two added points greatly enhance the serviceability of the numerical differentiation method in comparison with the method.…”

Section: Discussionsupporting

confidence: 61%

“…Moreover, two added points greatly enhance the serviceability of the numerical differentiation method in comparison with the method. 20 First, in the present Fortran code for the CPSE, we use double precision real number; due to unavoidable loss of number of significant digits in the calculations, the actual accuracy is reduced greatly. By using double double precision real number, the actual accuracy expects to be improved and even higher-order derivatives may become tractable.…”

Section: Discussionmentioning

confidence: 99%

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Excellence of numerical differentiation method in calculating the coefficients of high temperature series expansion of the free energy and convergence problem of the expansion

Zhou

Solana

2014

The Journal of Chemical Physics

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In this paper, it is shown that the numerical differentiation method in performing the coupling parameter series expansion [S. Zhou, J. Chem. Phys. 125, 144518 (2006); AIP Adv. 1, 040703 (2011)] excels at calculating the coefficients ai of hard sphere high temperature series expansion (HS-HTSE) of the free energy. Both canonical ensemble and isothermal-isobaric ensemble Monte Carlo simulations for fluid interacting through a hard sphere attractive Yukawa (HSAY) potential with extremely short ranges and at very low temperatures are performed, and the resulting two sets of data of thermodynamic properties are in excellent agreement with each other, and well qualified to be used for assessing convergence of the HS-HTSE for the HSAY fluid. Results of valuation are that (i) by referring to the results of a hard sphere square well fluid [S. Zhou, J. Chem. Phys. 139, 124111 (2013)], it is found that existence of partial sum limit of the high temperature series expansion series and consistency between the limit value and the true solution depend on both the potential shapes and temperatures considered. (ii) For the extremely short range HSAY potential, the HS-HTSE coefficients ai falls rapidly with the order i, and the HS-HTSE converges from fourth order; however, it does not converge exactly to the true solution at reduced temperatures lower than 0.5, wherein difference between the partial sum limit of the HS-HTSE series and the simulation result tends to become more evident. Something worth mentioning is that before the convergence order is reached, the preceding truncation is always improved by the succeeding one, and the fourth- and higher-order truncations give the most dependable and qualitatively always correct thermodynamic results for the HSAY fluid even at low reduced temperatures to 0.25.

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Section: Discussionsupporting

confidence: 61%

Section: Discussionmentioning

confidence: 99%

Excellence of numerical differentiation method in calculating the coefficients of high temperature series expansion of the free energy and convergence problem of the expansion

Zhou

Solana

2014

The Journal of Chemical Physics

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“…The second sum terminates at p, due to the monotonic property of the elements in {B p }. Now, Equation (11) reduces to an identity if {I p } are chosen, so that the second sum is 1 for p = 1 and 0 for p ≥ 2. A recursion formula for I p , for p ≥ 2, then follows on separating out the last term in this sum:…”

Section: Chen-m öBius Formulamentioning

confidence: 99%

“…First of all, effective pair interaction potentials are derived from the SCE formula by employing lattice inversion techniques, and split into repulsive and attractive components while using the Weeks-Chandler-Anderson (WCA) prescription [10]. These components are used in an accurate thermodynamic perturbation theory, called couplingparameter expansion (CPE) [11], for solving the Ornstein-Zernike equation (OZE) and an appropriate closure relation. The coupling parameter (0 ≤ λ ≤ 1) tunes the strength of the attractive component of the potential, and all correlation functions are expressed as Taylor's series in λ around λ = 0.…”

Section: Introductionmentioning

confidence: 99%

Scaling of Phase Diagram and Critical Point Parameters in Liquid-Vapour Phase Transition of Metallic Fluids

Menon¹

2021

Condensed Matter

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The first objective of this paper is to investigate the scaling behavior of liquid-vapor phase transition in FCC and BCCmetals starting from the zero-temperature four-parameter formula for cohesive energy. The effective potentials between the atoms in the solid are determined while using lattice inversion techniques as a function of scaling variables in the four-parameter formula. These potentials are split into repulsive and attractive parts, as per the Weeks–Chandler–Anderson prescription, and used in the coupling-parameter expansion for solving the Ornstein–Zernike equation that was supplemented with an accurate closure. Thermodynamic quantities obtained via the correlation functions are used in order to obtain critical point parameters and liquid-vapor phase diagrams. Their dependence on the scaling variables in the cohesive energy formula are also determined. An equally important second objective of the paper is to revisit coupling parameter expansion for solving the Ornstein–Zernike equation. The Newton–Armijo non-linear solver and Krylov-space based linear solvers are employed in this regard. These methods generate a robust algorithm that can be used to span the entire fluid region, except very low temperatures. The accuracy of the method is established by comparing the phase diagrams with those that were obtained via computer simulation. The avoidance of the ’no-solution-region’ of the Ornstein-Zernike equation in coupling-parameter expansion is also discussed. Details of the method and complete algorithm provided here would make this technique more accessible to researchers investigating the thermodynamic properties of one component fluids.

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“…The Taylor series of RDF truncated up to second order gives third order CPE for f(ρ). The numerical procedure we used is as explained in our previous paper 11 and is not repeated here. Equation (19) has been first derived by Zhou 5 in a different way.…”

Section: Theorymentioning

confidence: 99%

Generalized coupling parameter expansion: Application to square well and Lennard-Jones fluids

Ramana

2013

The Journal of Chemical Physics

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The coupling parameter expansion in thermodynamic perturbation theory of simple fluids is generalized to include the derivatives of bridge function with respect to coupling parameter. We applied seventh order version of the theory to Square-Well (SW) and Lennard-Jones (LJ) fluids using Sarkisov Bridge function. In both cases, the theory reproduced the radial distribution functions obtained from integral equation theory (IET) and simulations with good accuracy. Also, the method worked inside the liquid-vapor coexistence region where the IETs are known to fail. In the case of SW fluids, the use of Carnahan-Starling expression for Helmholtz free energy density of Hard-Sphere reference system has improved the liquid-vapor phase diagram (LVPD) over that obtained from IET with the same bridge function. The derivatives of the bridge function are seen to have significant effect on the liquid part of the LVPD. For extremely narrow SW fluids, we found that the third order theory is more accurate than the higher order versions. However, considering the convergence of the perturbation series, we concluded that the accuracy of the third order version is a spurious result. We also obtained the surface tension for SW fluids of various ranges. Results of present theory and simulations are in good agreement. In the case of LJ fluids, the equation of state obtained from the present method matched with that obtained from IET with negligible deviation. We also obtained LVPD of LJ fluid from virial and energy routes and found that there is slight inconsistency between the two routes. The applications lead to the following conclusions. In cases where reference system properties are known accurately, the present method gives results which are very much improved over those obtained from the IET with the same bridge function. In cases where reference system data is not available, the method serves as an alternative way of solving the Ornstein-Zernike equation with a given closure relation with the advantage that solution can be obtained throughout the phase diagram with a proper choice of the reference system.

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Coupling-parameter expansion in thermodynamic perturbation theory

Cited by 21 publications

References 18 publications

Excellence of numerical differentiation method in calculating the coefficients of high temperature series expansion of the free energy and convergence problem of the expansion

Excellence of numerical differentiation method in calculating the coefficients of high temperature series expansion of the free energy and convergence problem of the expansion

Scaling of Phase Diagram and Critical Point Parameters in Liquid-Vapour Phase Transition of Metallic Fluids

Generalized coupling parameter expansion: Application to square well and Lennard-Jones fluids

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