Radial Distribution Functions Of Mixtures Research Articles

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.

Calculation of binary hard-sphere mixture radial distribution functions at contact from an equation of state

A new method for calculation of the radial distribution functions (RDFs) at contact from a specified hard-sphere mixture equation of state (EOS) is proposed. The method is applied to two available accurate EOS expressions and new analytical formulas for the RDFs are obtained for binary additive hard-sphere mixtures. The results of the new formulas are compared with available computer simulation data and with those of other RDF expressions from the literature. The evidence to date suggests that the new formula is more accurate than alternative formulas currently available.

Analytical representations for the radial distribution functions of mixtures of adhesive spheres

A study has been made of the pole topology of the Laplace transforms of the pair distribution functions (PDFs) of a binary mixture of adhesive hard spheres (AHS) both for the Percus-Yevick equation and the mean spherical model (MSM). Expressions are given that describe how the distribution of the poles in the left half of the complex plane varies with the system parameters for the special case of the MSM for symmetric binary AHS mixtures. The locations of the poles closest to the imaginary axis are known to determine the asymptotic form of the PDFs, i.e. either exponentially monotonic or exponentially oscillatory decaying. As a byproduct of this inquiry analytical r space representations of the PDFs are derived that allow their accurate and efficient determination over the entire r range.

Deriving integral equations for radial distribution functions of multicomponent mixtures on the basis of scale transformations in the phase space

It is shown that scale transformations of the coordinate part of the phase space for one of the mixture components correspond to virtual variations in the local density for the same component in a thermodynamic system. The investigation results are used to construct different variants of a generating functional with the goal of deriving a system of integral equations for the radial distribution functions of mixtures. An equation of state, which is a modification of the Tate equation, is obtained. Systems of integral equations that imply, in the limit, the Perkus-Yevick equations and systems of equations for hypernetted chains are derived for radial distribution functions.

Radial distribution functions and their role in modeling of mixtures behavior

Radial distribution (Pair correlation) functions are the primary linkage between macroscopic thermodynamics properties and intermolecular interactions of fluids and fluid mixtures. Numerous theories of mixtures exist based on the definition of the radial distribution functions. Utilization of such theories require the knowledge about mixture radial distribution functions and their relations. Generally the exact radial distribution functions of real fluid mixtures are not available. In this paper a number of approximations for the mixture radial distribution functions are presented. Application of these approximation can result in analytic mixture theories and mixing rules. A uniform statistical mechanical technique is presented through which one can derive equation of state mixing rules based on different RDF approximations. The relative merits of different sets of mixing rules derived by this technique and their applications for modeling of the behavior of mixtures are discussed.

Application of approximations to mixture radial distribution functions to the development of mixing rules

Abstract Li M.-H. and Wang Y.-L., 1992. Application of approximations to mixture radial distribution functions to the development of mixing rules. Fluid Phase Equilibria, 72: 89-110. The mixing rules utilizing the approximations for mixture radial distribution functions have been presented for extension of an equation of state of polar fluids to mixtures. The van der Waals one-fluid approximation and the mean density approximation (MDA) for radial distribution functions in a mixture have been applied to derive mixing rule models. To investigate further the composition dependence of the MDA, the composition-dependent cross-interaction energy parameter is also introduced; the resulting model is referred to as the modified mean density approximation (MMDA). In calculations of the vapor-liquid equilibria for non-ideal fluid mixtures, satisfactory results were obtained by the MMDA. When performing the vapor-liquid equilibrium calculations for systems with data over a wide range of temperatures, the MMDA also yielded reasonable results. Furthermore, the proposed method can be used to extend any existing accurate corresponding-states-type pure-fluid equation of state to mixtures.

A study of the Buckingham exponential-six fluid mixtures

Extensive Monte Carlo computer simulation results for the configurational energy, compressibility and the radial distribution functions of mixtures of exponential-six molecules at 352 state points and presented. These data are used to test mixture models that have been proposed, and to develop a model which describes the exp-6 fluid quite well. Using this model, explicit expressions are obtained for the partition function for pure fluids and mixtures of exp-6 molecules.It is also shown how, by redefinition of the free volume and mean potential terms, simple models can be used to describe the behavior of exp-6 fluids and that with this definition the behavior of the exp-6 and square-well fluids are very similar.

Mixture radial distribution functions: Are they all independent?

New constraints of radial distribution functions in the canonical ensemble of multicomponent mixtures are derived. These constraints indicate that mixture radial distribution functions are not all independent from each other. One of the important applications of such constraints is to test the approximation theories of mixture distribution functions. A number of examples are reported for the application of the proposed constraints.

Conformality in the Kirkwood–Buff solution theory of statistical mechanics

The Kirkwood–Buff solution theory of statistical mechanics is examined in the light of the conformal solution approximation of the mixture radial distribution functions. By joining the mixture compressibility equation of the Kirkwood–Buff solution theory with the mixture energy and virial equations of statistical mechanics, a set of density and temperature dependent mixing rules has been developed which are used here to calculate properties of molecular fluids with varying size and interaction energy differences. It is demonstrated that the conformality approximation in the compressibility equation produces mixture results with a deviation, from the exact mixture data, on the opposite side of the predictions of the van der Waals theory of mixtures. The Kirkwood–Buff relation for the composition derivative of the chemical potential is also integrated by combining it with the conformal solution approximation and compared with the simulation data.

Mean density approximation and hard sphere expansion theory: A review

This review surveys research dealing with the Mean Density Approximation (MDA) and the Hard Sphere Expansion (HSE) theory developed by Leland and coworkers. MDA and its modifications provide a simple way to predict radial distribution functions of mixtures from the pure fluid information. Comparisons with computer simulation data show the MDA to be superior to the van der Waals approximation for mixture radial distribution functions. Derivations of the HSE theory and the HSE Conformal Solution Theory (HSE-CST) are also described. For Lennard-Jones mixtures, the HSE theory is proven to be superior to the van der Waals theory by using a proper method to determine the hard-sphere diameter. The major problem associated with the development of a consistent method to determine the hard-sphere diameter of the HSE-CST and the requirements regarding the extension of the HSE-CST to polar mixtures are discussed. It is concluded that the HSE-CST is capable of predicting, satisfactorily, the vapor-liquid equilibrium properties of mixtures containing hydrocarbons and/or slightly quadrupolar or polar compounds such as CO 2 and H 2S.

Improved conformal solution theory for mixtures with large size ratios

Previous studies on model Lennard-Jones systems have shown that conformal solution theories tend to fail when mixtures contain molecules with large size differences. This failure can be attributed to deficiencies in the underlying mean density approximation (MDA) for the mixture's radial distribution functions which was originally proposed by Leland and coworkers. In this communication, we present an improved mean density approximation which is obtained from a straightforward application of statistical mechanical perturbation theory. Comparisons of the new theory with simulation PVT data for systems with large size ratios show it to be far superior to van der Waals one-fluid and MDA n-fluid theories, and an improvement upon hard sphere expansion and perturbation theories as well. Comparisons are also made with chemical potential simulation data, and for these data the new theory is competitive with perturbation theory.

A test of the mean density approximation for Lennard-Jones mixtures with large size ratios

The mean density approximation for mixture radial distribution functions plays a central role in modern corresponding-states theories. This approximation is reasonably accurate for systems that do not differ widely in size and energy ratios and which are nearly equimolar. As the size ratio increases, however, or if one approaches an infinite dilution of one of the components, the approximation becomes progressively worse, especially for the small molecule pair. In an attempt to better understand and improve this approximation, isothermal molecular dynamics simulations have been performed on a series of Lennard-Jones mixtures. Thermodynamic properties, including the mixture radial distribution functions, have been obtained at seven compositions ranging from 5 to 95 mol%. In all cases the size ratio was fixed at two and three energy ratios were investigated, ɛ22/ɛ11=0.5, 1.0, and 1.5. The results of the simulations are compared with the mean density approximation and a modification to integrals evaluated with the mean density approximation is proposed.

Fluids of hard spheres with nonadditive diameters

Monte Carlo results are reported for the equation of state and radial distribution functions of mixtures of hard spheres with nonadditive diameters. Comparison is made with the predictions of perturbation theories of various types and methods based on a one-component reference fluid are found to give the most satisfactory results.

Correlation functions of classical fluids. I. The radial distribution functions of mixtures of Kihara molecules in the Percus-Yevick approximation and their thermodynamic functions

The Percus-Yevick integral equation is solved for systems of molecules interacting with the Kihara potential. Radial distribution functions for the pure fluid systems of argon, krypton, and xenon, and for the binary mixture argon-nitrogen are obtained. For the one-component system, the use of the Kihara potential yields equivalent results in comparison with previous theoretical and experimental studies. For the case of argon, a numerical potential proposed by Dymond and Alder is also employed and the results show superior agreement with the x-ray diffraction data. The solutions of the two-component case cover a wide range of temperatures, densities, and compositions, including those of the liquid state. The variations of the radial distribution functions with density, temperature, and composition are ascertained. It is found that the main features of the radial distribution functions, such as the peak heights, their relative magnitudes, and the zeros of the total correlation functions, vary under quite complex rules, yet in most part are verifiable by experiments. The macroscopic properties of Ar–N2 mixtures, such as the internal energy and the Helmholtz free energy, are also calculated. Hiroike's formula for the calculation of the free energy is adopted which avoids the usual procedure of integrating a pressure term that has two values, one from the virial theorem and the other from the fluctuation-compressibility derivation. Agreements with values derived from the Strobridge equation of state are close for the internal energy and less satisfactory for the Helmholtz free energy. Their variations with temperature, density, and composition are also demonstrated.