A new integral equation for the radial distribution function of a hard sphere fluid

Based on a proposal by Shinomoto, a new integral equation is derived for the radial distribution function of a hard-sphere fluid using mainly geometric arguments. This integral equation is solved by a perturbation expansion in the density of the fluid, and the results obtained are compared with those from molecular dynamics simulations and from the Born-Green-Yvon (BGY) and Percus-Yevick (PY) theories. The present theory provides results for the radial distribution function which are intermediate in accuracy between those obtained from the BGY and from the PY theories.     

Equation of state and correlation function contact values of a hard sphere mixture

Recently, there has been considerable activity in the study of the equation of state and correlation functions of a hard sphere mixture when the concentration of the large spheres is exceedingly small. This system is of considerable interest both as a simple prototype of an asymmetric mixture and as a simple but useful model of a colloidal suspension. We review our ad hoc formulae for the correlation function contact values and discuss the evidence, both pro and con, from exact and nearly exact theorems, computer simulations and experiment. Our conclusion is that our formulae are, on the whole, correct but the large sphere–large sphere correlation function contact value seems to need revision when the concentration of the large spheres is small but not exceedingly small. We propose a modest revision that preserves the seemingly correct aspects of our formula and this is in good agreement with the most recent simulations.     

An accurate equation of state of a hard convex body fluid mixture

A method enabling to calculate the contact-point values of the pair correlation function of convex body fluids from a semi-empirical equation of state is presented and the accurate Nezbeda equation of the pure hard convex body fluid is extended to mixtures. Comparison of results for one- and two-component systems with Monte Carlo simulation data shows excellent agreement.     

Integral equations for the radial distribution functions of binary mixtures. Equation of state for a binary mixture

A system of integral equations obtained earlier for the radial distribution functions is used for the investigation of the equation of state of binary mixtures in the gaseous, critical, and intermediate regions in the direction isomorphic to the single-component liquid. The question of the universality of critical phenomena in mixtures is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 92–96, January, 1986.     

An approximate equation for the radial distribution function in the hard sphere model

We investigate the spatially homogeneous solution of a new approximate equation for the radial distribution function of classical hard-core systems. Its bifurcation may be connected with a phase transition.     

Density expansion of the radial distribution and bridge functions of the hard sphere fluid

The bridge function and the background correlation function (and consequently the radial distribution function) of the pure hard sphere fluid are expanded up to the sixth power in density. The calculations are based on the Ree–Hoover representation of the diagrams and Monte Carlo integration. The coefficients as functions of the particle–particle separation are fitted to splines taking into account discontinuities in higher derivatives up to the term of the order of (r- const)⁵.     

Molecular dynamics results for the radial distribution functions of highly asymmetric hard sphere mixtures

Molecular dynamics (MD) results for the radial distribution functions of mixtures of large and small hard spheres are reported for size ratios whose (large/small) values are 1, 2.5, 5, 7.5, and 10 in the region where the concentration of the large spheres is very small. The MD contact values of these functions are compared with formulae due to Boublik, Mansoori, Carnahan, Starling, Leland, Grundke, and Henderson, Viduna and Smith, Henderson, Trokhymchuk, Woodcock, and Chan, as well as new formulae that are considered here. The new formulae give good agreement for the large–small contact values and reasonably good agreement for the large–large contact values. The Viduna–Smith formula is satisfactory for the small–small contact value and quite reasonable for the small–large contact value. Undoubtedly, further improvements are possible. These results give insight into what may be called the colloidal limit, where the size ratio is exceedingly large while the concentration of the large spheres is exceedingly small, and into the passage to this limit.     

Universal cubic equation of state and contact values of the radial distribution functions for multi-component additive hard-sphere mixtures

A universal cubic equation of state (UC EOS) is proposed based on a modification of the virial Percus-Yevick (PY) integral equation EOS for hard-sphere fluid. The UC EOS is extended to multi-component hard-sphere mixtures based on a modification of Lebowitz solution of PY equation for hard-sphere mixtures. And expressions of the radial distribution functions at contact (RDFC) are improved with the form as simple as the original one. The numerical results for the compressibility factor and RDFC are in good agreement with the simulation results. The average errors of the compressibility factor relative to MC data are 3.40%, 1.84% and 0.92% for CP3P, BMCSL equations and UC EOS, respectively. The UC EOS is a unique cubic one with satisfactory precision among many EOSs in the literature both for pure and mixture fluids of hard spheres.     

Elementary theory of the equation of state and the pair distribution function for the hard sphere system

We present an intuitive method to calculate the pair distribution function and the equation of state for the hard sphere system by an elementary kinetic theory. The resulting equation of state not only reproduces exactly the second and the third virial coefficients, but also exhibits a remarkable agreement with molecular-dynamics results.     

On the compressibility equation of state for multi-component adhesive hard sphere fluids

The compressibility equation of state (EOS) for a multi-component sticky hard sphere model alternative to Baxter's one is investigated within the mean spherical approximation (MSA). For this model and this closure, as well as for a more general class of models and closures leading to Baxter functions q_ij(r) with density-independent stickiness coefficients, no compressibility EOS can exist for mixtures, unlike the one-component case (in view of this, an EOS recently reported in the literature turns out to be incorrect). The reason is the failure of the Euler reciprocity relation for the mixed second-order partial derivatives of the pressure with respect to the partial densities. This is in turn related to the inadequacy of the approximate closure (in particular, the MSA). A way out to overcome this drawback is presented in a particular example, leading to a consistent compressibility pressure, and a possible generalization of this result is discussed.     

Approximate solution of the Percus-Yevick equation for a binary mixture of hard spheres with non-additive diameters

We present an approximate solution of the Percus-Yevick integral equation for a binary mixture of hard spheres with non-additive diameters. Defining R_ij the distance of closest approach between particles of species i and j by R ₁₂ = ½(R ₁₁ + R ₂₂) + α, we obtain a closed set of equations for the direct correlation functions c_ij (r) when 0 < α ? min [½(R ₂₂ - R ₁₁), ½R ₁₁]. Our expressions for c_ii (r), and for c ₁₂(r) in the range 0 < r ? ½[R ₂₂ - R ₁₁] - α, agree with those previously obtained by Lebowitz and Zomick.     

On some empirical expressions of the contact values of the pair distribution functions and fluid-fluid phase separation in hard sphere mixtures

(Molec. Phys., 2001, 99, 2055–2060)     

True mixture versus effective one component fluid models of asymmetric binary hard sphere mixtures: a comparison by simulation

The pair distribution function of solute particles determined by simulation of true mixtures with diameter ratios of 5, 10 and 20 is compared with that obtained in their representation by effective one-component fluids with pairwise additive interactions. The pair distribution function near contact is found to be overestimated by the effective fluid approach, for all size ratios. In the domain of the phase diagram accessible to the simulation algorithm, the deviation is found to be moderate. Its consequences for the coexistence lines are discussed.     

The orientational pair correlation functions in a dense hard sphere fluid at long times

The two-particle contribution to the potential part of the stress tensor autocorrelation function of a dense hard sphere fluid is studied. It is shown that the long-time decay is given as the solution of a diffusion equation for the relative particle in a potential of mean force. The diffusion constant needed in order to accurately reproduce molecular dynamics results is found to be somewhat lower than the self-diffusion constant.     

A globally accurate theory for a class of binary mixture models

The self-consistent Ornstein-Zernike approximation results for the 3D Ising model are used to obtain phase diagrams for binary mixtures described by decorated models, yielding the plait point, binodals, and closed-loop coexistence curves for the models proposed by Widom, Clark, Neece, and Wheeler. The results are in good agreement with series expansions and experiments.     

On the nonequilibrium statistical mechanics of a binary mixture. I. The distribution functions

In this paper, we obtain the generalization of the BBGKY hierarchy for a binary mixture of chemically neutral particles. Using modified boundary conditions different from the ones proposed by Bogoliubov, we solve the hierarchy, and obtain explicitly the set of two-particle distribution functions for the several species of the mixture, up to first order in the density.     

Modified cell theory: Free volume distributions and the equation of state for D-dimensional hard sphere systems

A generalized cell model, using cells of different sizes, is applied to hard rods, disks and spheres. Structures is discussed in terms of free volumes. The derived equation of state is exact for rods. For disks and spheres it provides a good approximation in the dense fluid and solid state.     

Perturbation theory for equilibrium properties of a binary mixture of hard discs

The radial distribution function (RDF) and thermodynamic properties of a two-dimensional hard-disc mixture are calculated by using the perturbation theory. Numerical results are given for theRDF, pressure and excess-free energy of the binary mixture of both additive and non-additive hard discs. It is found that the thermodynamic properties of the binary mixture of non-additive hard discs increase with Δ, the non-additive parameter.     

混合物状态方程的计算

多介质流体动力学过程的数值模拟往往涉及混合物状态方程的计算. 做图法和Newton 法是混合物状态方程计算常采用的方法, 前者虽直观精度却差, 后者计算效率高却只具有局部收敛性, 当解与其初始猜测值相差较远时Newton法不一定能够获得收敛解. 为此, 本文给出一种具有大范围收敛性的嵌入算法(imbedding method)求解混合物状态方程, 其基本思想是通过引入嵌入参数, 将待解的混合物状态方程和易解的混合物状态方程线性组合, 构成嵌入方程组, 当嵌入参数从0连续地变化到1 时, 嵌入方程组的解由易解的混合物状态方程的解连续地变化为待解的混合物状态方程的解. 嵌入方程组可由Newton法迭代求解, 也可转化为以嵌入参数为自变量的常微分方程组, 从而易于由成熟的计算方法如梯形法等进行求解. 进一步利用热力学基本关系, Maxwell形式的微分方程描述了压力和温度随嵌入参数的演化速率与应变速率和组分质量分数演化速率的关系. 对铅锡混合物热力学量的计算表明了本文算法的有效性.