Equilibrium thermodynamic properties of binary hard-sphere mixtures from integral equation theory

A local self-consistent Ornstein-Zernike (OZ) integral equation theory (IET) is proposed to provide a rapid route for obtaining thermodynamic and structural information for any thermodynamically stable or metastable state points in the bulk phase diagram without recourse to traditional thermodynamic integration, and extensive NVT-Monte Carlo simulations are performed on a recently proposed honeycomb potential in three dimensions to test the theory's reliability. The simulated quantities include radial distribution function (rdf) and excess internal energy, pressure, excess chemical potential, and excess Helmholtz free energy. It is demonstrated that (i) the theory reproduces the rdf very satisfactorily only if the bulk state does not enter deep into a two phases coexistence region; (ii) the excess internal energy is the only one of the four thermodynamic quantities investigated amenable to the most accurate prediction by the present theory, and the simulated pressure is somewhat overestimated by the theoretical calculations, but the deviation tends to vanish along with rising of the temperature; (iii) using the structural functions from the present local self-consistent OZ IET, a previously derived local expression, due to the present author, achieves even a higher accuracy in calculating for the excess chemical potential than the exact virial pressure formula for the pressure, and the resulting excess Helmholtz free energy is in surprisingly same with the simulation results due to offset of the errors. Based on the above observations, it is suggested that it may be a good procedure to integrate the theoretical excess internal energy along the isochors to get the excess Helmholtz free energy, which is then fitted to a polynomial to be used for calculation of all of other thermodynamic quantities in the framework of the OZ IET.

It has been shown that an improved prediction of the thermodynamic quantity for the hard homonuclear diatomics can be performed with an interpolation scheme that relates the thermodynamic property of the hard sphere to that of tangent hard spheres at the same density. Using the analytic expressions based on the solution of an integral equation an excess Helmholtz free energy per particle and chemical potential for hard homonuclear diatomic fluid have been computed. Calculations are carried out for hard homonuclear diatomics with reduced internuclear separations of 0.1 to 1 at reduced densities of 0.2 to 0.9. Our findings for the excess Helmholtz free energy from the Percus–Yevickand Martynov–Sarkisov approximations presents good agreement with available accurate data, having maximum deviations of 15% from it over the separation and density ranges of the calculations. The excess chemical potential from the Martynov–Sarkisov approximation shows better agreement with accurate available data than those from the Percus–Yevick approximation. For the excess chemical potential, a maximum deviation of 9.5% over the range of the calculations has been shown up for the Percus–Yevick approximation.

A non-hard sphere (HS) perturbation scheme, recently advanced by the present author, is elaborated for several technical matters, which are key mathematical details for implementation of the non-HS perturbation scheme in a coupling parameter expansion (CPE) thermodynamic perturbation framework. NVT-Monte Carlo simulation is carried out for a generalized Lennard-Jones (LJ) 2n-n potential to obtain routine thermodynamic quantities such as excess internal energy, pressure, excess chemical potential, excess Helmholtz free energy, and excess constant volume heat capacity. Then, these new simulation data, and available simulation data in literatures about a hard core attractive Yukawa fluid and a Sutherland fluid, are used to test the non-HS CPE 3rd-order thermodynamic perturbation theory (TPT) and give a comparison between the non-HS CPE 3rd-order TPT and other theoretical approaches. It is indicated that the non-HS CPE 3rd-order TPT is superior to other traditional TPT such as van der Waals/HS (vdW/HS), perturbation theory 2 (PT2)/HS, and vdW/Yukawa (vdW/Y) theory or analytical equation of state such as mean spherical approximation (MSA)-equation of state and is at least comparable to several currently the most accurate Ornstein-Zernike integral equation theories. It is discovered that three technical issues, i.e., opening up new bridge function approximation for the reference potential, choosing proper reference potential, and/or using proper thermodynamic route for calculation of f(ex-ref), chiefly decide the quality of the non-HS CPE TPT. Considering that the non-HS perturbation scheme applies for a wide variety of model fluids, and its implementation in the CPE thermodynamic perturbation framework is amenable to high-order truncation, the non-HS CPE 3rd-order or higher order TPT will be more promising once the above-mentioned three technological advances are established.

A recently proposed non hard sphere (HS) coupling parameter expansion (CPE)thermodynamic perturbation theory (TPT) is evaluated for its performance over a widerange of temperatures, particularly in tackling a so-called low temperature problem,i.e. the traditional high temperature series expansion TPT performance becomesprogressively less satisfactory as the temperature drops, and finally becomes qualitativelyincorrect. Accordingly, we have performed Monte Carlo simulations in the canonicalensemble for a honeycomb model potential, and the pressure, excess internalenergy, excess Helmholtz free energy, excess chemical potential, and excess enthalpyhave been obtained over wide density and temperature ranges for the fluid phase.These new simulation data, together with published simulation data for an HS + square well fluid at very low temperatures, have been used to test the performanceof the non HS CPE third-order TPT. A general scheme for assimilating partof the tail term of the potential function considered into a reference system isproposed, which ensures running normality of the non HS perturbation codeat subcritical temperatures and smooth transition into a commonly used HSperturbation scheme when the temperatures considered rise infinitely. It is found thatthe non HS CPE third-order TPT is very satisfactory or even very accurate forthese extremely low temperatures for which traditional Zwanzig type TPT isqualitatively incorrect and even an HS CPE third-order TPT also seriously fails insome cases. Among the thermodynamic quantities considered the most difficultone to predict theoretically is the constant volume excess heat capacity, and forthis quantity the non HS CPE third-order TPT is also obviously superior to itscompetitors.

Monte Carlo simulations have been carried out for a hard sphere square well model fluid with well widths of lambda = 1.01, 1.02 and 1.04 to obtain thermodynamic properties, such as pressure, excess Helmholtz free energy and internal energy, constant volume excess heat capacity, excess chemical potential, and excess enthalpy. The extremely narrow well widths considered allowed us to perform simulations at extremely low temperatures without reaching the liquid-vapour transition, which is metastable for these values of lambda. These simulation data have been used to explore the low temperature behavior of two thermodynamic perturbation theories (TPT), namely a high temperature series expansion truncated at second order within the local compressibility approximation (thereafter denoted as 2nd-order LCA-TPT) and a recently proposed coupling parameter series expansion truncated at 3rd-, 4th- and 5th-order (thereafter denoted as 3rd-order, 4th-order and 5th-order TPT, respectively). It has been found that the 2nd-order LCA-TPT is qualitatively incorrect in most of the cases analyzed, whereas the 3rd-order TPT is quantitatively correct in most of them. With increasing the well width, and consequently the temperatures considered, the 3rd-order TPT quickly becomes more accurate. Among the six thermodynamic quantities analyzed, the one most difficult to predict accurately is the constant volume excess heat capacity, for which the TPT based on the coupling parameter expansion provides satisfactory results only in the case of lambda = 1.04. For the cases studied, the performance of the 3rd-order, 4th-order, and 5th-order TPT are essentially equal; the change of the order of the truncation sometimes results in an improvement of some thermodynamic quantities within certain regions of the thermodynamic surface of states, while worsening in others.

The Ornstein-Zernike (OZ) integral equation theory is a powerful approach to simple liquids due to its low computational cost and the fact that, when combined with an appropriate closure equation, the theory is thermodynamically complete. However, approximate closures proposed to date exhibit pressure or free energy inconsistencies that produce inaccurate or ambiguous results, limiting the usefulness of the Ornstein-Zernike approach. To address this problem, we combine methods to enforce both pressure and free energy consistency to create a new closure approximation and test it for a single-component Lennard-Jones fluid. The closure is a simple power series in the direct and total correlation functions for which we have derived analytical formulas for the excess Helmholtz free energy and chemical potential. These expressions contain a partial molar volumelike term, similar to excess chemical potential correction terms recently developed. Using our bridge approximation, we have calculated the pressure, Helmholtz free energy, and chemical potential for the Lennard-Jones fluid using the Kirkwood charging, thermodynamic integration techniques, and analytic expressions. These results are compared with those from the hypernetted chain equation and the Verlet-modified closure against Monte Carlo and equations-of-state data for reduced densities of ρ^{*}<1 and temperatures of T^{*}=1.5, 2.74, and 5. Our closure shows consistency among all thermodynamic paths, except for one expression of the Gibbs-Duhem relation, whereas the hypernetted chain equation and the Verlet-modified closure exhibit consistency between only a few relations. Accuracy of the closure is comparable to the Verlet-modified closure and a significant improvement to results obtained from the hypernetted chain equation.

A comprehensive comparison between thermodynamic perturbation theory and first-order mean spherical approximation: Based on discrete potentials with hard core

The structure of a binary mixture of nonadditive hard spheres confined in a slit pore is studied by the integral equations method in which the confining medium acts as a giant particle at infinite dilution. The adsorption/desorption curves are studied as a function of the composition and density, when the homogeneous bulk mixture is near the demixing instability. The Ornstein-Zernike integral equations are solved with the reference functional approximation closure in which the bridge functions are derived from Rosenfeld's hard sphere functional for additive hard sphere. To study the high composition asymmetry regime in which a population inversion occurs, we developed an approximate closure that overcomes the no solution problem of the integral equation. By comparison with simulation data, this method is shown to be sufficiently accurate for predicting the threshold density for the population inversion. The predictions of simpler closure relations are briefly examined.

A comparison is made of the capabilities of various equation of state (EOS) + excess free energy (Gex) models in representing the activity‐coefficient models they incorporate. Such EOS models need to reproduce the excess Gibbs free energy behavior of the activity‐coefficient models as closely as possible in order to represent low‐pressure vapor–liquid equilibrium behavior of complex mixtures accurately, and also to make accurate the vapor–liquid predictions at higher temperatures and pressures using only low‐pressure information. It is demonstrated that the EOS models that use the excess Helmholtz free energy instead of the excess Gibbs free energy, and that combine the EOS with activity‐coefficient models at the infinite pressure limit, are successful for calculations of the type just described.

We apply the potential distribution theorems for the cavity distribution functions to the development of thermodynamic formulas for fused-sphere chain molecules. Alternative forms of the potential distribution theorems are derived: in terms of the cavity functions, and in terms of the singlet direct correlation functions. We point out the connections to integral equation theories. To determine the behavior of fused dispheres, we examine the successful Wertheim thermodynamic perturbation theory (TPT) at different bond lengths l in light of the cavity functions. For ternary mixtures of spheres S and B, and fused dispheres (SB), we discover a confluence point where all cavity functions at different mixture compositions converge. This takes place at the tangent disphere limit l=d (l being the bond length, and d, the hard sphere diameter). This point is also in common with the excess Helmholtz free energy from the TPT theory for tangent dumbbells. The cavity functions are obtained from the accurate equation of state of Boublík. To verify the chemical potentials calculated, we compare with new Monte Carlo simulations for mixtures of hard spheres and dumbbells. TPT does not hold for l&amp;lt;d. In order to have a quantitative expression for fused disphere properties, we propose an interpolation formula that performs well for both symmetric dispheres and asymmetric dispheres. This formula, though empirical, performs better than similar interpolative schemes proposed by Phan–Kierlik–Rosinberg. We have also derived purely thermodynamic formulas based on the TPT theory. These formulas can be exploited if one uses many of the existing thermodynamic properties correlations for mixtures.

Monte Carlo (MC) simulations have been performed for the restricted primitive electrolyte model at a Bjerrum parameter B= 1.546 and three extremely small concentrations. The precision of the obtained thermodynamic properties (the excess energy, the heat capacity and the osmotic coefficient) is unprecedented in the literature. The objective is to study the first deviations from the Debye-Huckel limiting law (DHLL) without any assumptions. The values of the mentioned thermodynamic properties are significantly different from the values obtained from the extended Debye-Huckel expression (DH). MC values are situated between DH and DHLL values and are close to the values of the DHX theory (which uses a DH screened potential as the effective potential connected with the radial distribution functions). However, the MC values are also significantly different from the DHX values. The MC values of the radial distribution functions at contact are statistically identical to the values given by the DHX theory, but the contact values are quite uncertain because of the rare close encounters at extreme dilution. Nevertheless, the osmotic coefficients are well determined, since the contact term is vanishingly small in comparison to the third of the excess energy. The excess Helmholtz free energy has been calculated in the same Metropolis sampling by means of the Salsburg–Chesnut method. For an infinite Metropolis Markov chain, this method is shown to lead to the correct result for the excess electrostatic free energy. However, the method has not quite converged even with 106 configurations, and the electrostatic free energies are found to be ca. 5% higher than the DHLL values. The DH theory predicts up to 5% lower values. Because of the present systematic deviation, the electrostatic entropies calculated by the difference between the excess energy and the electrostatic free energy are 15–20% lower than the DHLL values.

We have derived an analytic expression for the contact value of the local density of binary hard-sphere systems under gravity. We have obtained the crossover conditions for the Brazil-nut type segregation of binary hard-sphere mixtures and binary hard-sphere chain mixtures from the segregation criterion, where the segregation occurs when the density (or the pressure) of the small spheres at the bottom is higher than that of the large spheres, or vice versa. For the binary hard-sphere chain mixtures, the crossover condition for the segregation depends on the number of monomers composed of hard-sphere chains as well as the mass and the diameter of each species. The fundamental-measure theories (FMTs) and local density approximation (LDA) are employed to examine the crossover condition for the segregation of the gravity-induced hard-sphere mixtures. The calculated results show that the LDA does not explain the density oscillation near the bottom, whereas the modified fundamental-measure theory (MFMT) compares with molecular dynamics simulations.

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Binary mixtures of hard spheres: how far can one go with the virial equation of state?

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