Hard-sphere Fluid Mixture Research Articles

Half widths of neutron spectra in dense helium-neon gas mixtures.

Results are presented of seven inelastic-neutron-scattering experiments on dense gas mixtures of helium and neon, performed on the Interfaculty Reactor Institute time-of-flight spectrometer RKS2. The concentration ratio of neon in the mixture is varied from 0 to 1 at 47 K and 285 bar. The neutron spectra S(k,\ensuremath{\omega}), obtained for wave numbers 2.5k30 ${\mathrm{nm}}^{\mathrm{\ensuremath{-}}1}$, are centered around the frequency \ensuremath{\omega}=0 with half widths ${\mathrm{\ensuremath{\omega}}}_{\mathit{H}}$(k). The experimental ${\mathrm{\ensuremath{\omega}}}_{\mathit{H}}$(k) agree well with the revised Enskog theory for equivalent binary hard-sphere fluid mixtures and show de Gennes minima similar to those in monoatomic fluids.

Free energy model for inhomogeneous fluid mixtures: Yukawa-charged hard spheres, general interactions, and plasmas

A free energy model for the inhomogeneous hard-sphere fluid mixture was derived recently [Phys. Rev. Lett. 63, 980 (1989)], which is based on the fundamental geometric measures of the particles. Along with an updated assessment of its accuracy, this model is first generalized for charged hard-sphere fluid mixtures, in which every particle carries a central Yukawa charge, and it is then extended to general fluid mixtures in external fields. The Yukawa-charged hard-sphere mixture provides a quite general reference system for many interesting physical systems including plasmas, molten salts, and colloidal dispersions, the screening parameter enabling to interpolate between the long range Coulomb forces and the short range hard cores. A special renormalization property of the Yukawa potential provides the means to derive the exact Onsager-type lower bound for the potential energy of the mixture, and its related asymptotic strong-coupling limit of the liquid pair correlation functions. These results are obtained analytically for the general homogeneous mixture with Yukawa interactions. They enable to extend the fundamental measure free energy model to inhomogeneous charged Yukawa mixtures, with the charge contributions given by a truncated second order expansion from the uniform (bulk) fluid limit. The resulting free energy model, which interpolates between the ideal-gas and ‘‘ideal-liquid’’ limits, then leads to a self-consistent method for calculating the density profiles for general fluid mixtures in external fields. This method is equivalent to an ansatz of ‘‘universality of the bridge functional.’’ The ‘‘bridge functional’’ consists of all the terms beyond the second order, in the expansion of the excess free energy functional around a reference uniform fluid. The self-consistency is imposed by applying the general method in the special case when the external potential is generated by a ‘‘test particle’’ at the origin of coordinates. In this limit, our general method for nonuniform fluids corresponds to an established and successful theory for the bulk uniform fluid pair structure, namely the thermodynamically consistent modified-hypernetted-chain theory, with the bridge functions now generated by an explicit and demonstratively accurate, ‘‘universal,’’ hard-sphere bridge functional. As a stringent test for the general model, the strongly coupled one-component plasma, in the bulk and near a hard wall, is considered in some detail.

Phase stability of dense multicomponent charged and uncharged hard-sphere fluid mixtures

We report calculations of the phase diagram and spinodal decomposition line of a charged hard-sphere fluid mixture constituted of two different cations, one anion and one neutral particle species. In the high-density regime the model is suitable to mimic molten silicate and metal halides mixtures, as those forming a magmatic fluid; in the low-density regime it can conveniently describe the properties of charged colloidal suspensions or micellar solutions. In this work the attention is focused mainly on the high-density regime of the mixture, and phase stability conditions are determined through the knowledge of the Gibbs free energy of mixing, Gmix at constant pressure, as calculated in the mean spherical approximation. It turns out that the solubility of the neutral component in the ionic mixture depends substantially on the potential-to-kinetic-energy ratio, that is, on the so-called coupling strength or plasma parameter, and increases when this decreases. This result is rationalized by considering the different terms contributing to Gmix and, specifically, the confi-gurational Coulombic internal energy term and entropic term. The fine balance between these two quantities is detailed and the onset of phase segregation discussed on these same bases. Such a representation also allows us to interpret why mixtures like those envisaged here are characterized by a phase diagram with an upper consolution point and why the critical temperature turns out to be an increasing function of the pressure. The implications of this last result seem interesting since, e.g., molten salt-silicate mixtures and, therefore, presumably fused rocks or magmas, could segregate not only under cooling, but also at constant temperature as the result of pressure variations.

A mean spherical approximation study of phase stability in charged colloidal solutions

The phase stability of charged hard sphere fluid mixtures modelling colloidal solutions is studied through the Mean Spherical Approximation (MSA). The isothermal compressibility and the Gibbs free energy of mixing are calculated, and from them, the liquid-vapor and the liquid-liquid decomposition spinodal line are determined, respectively. The MSA results favorably compare with those available from other theories as far as the liquid-vapor instabiity is concerned. For liquid-liquid decomposition the theory allows one to construct the full temperature concentration spinodal which shows an upper consolution point in temperature. Discussion and interpretation of this and related features of the phase stability line are reported

Calculation of the entropy of binary hard sphere mixtures from pair correlation functions

We evaluate the entropy of several binary hard sphere fluid mixtures using two approximate expressions that require as input only the pair correlation functions, gαβ(r). An approximation based on the incompressible limit is found to be accurate for medium to high density fluids.

Free-energy model for the inhomogeneous hard-sphere fluid in D dimensions: Structure factors for the hard-disk (D=2) mixtures in simple explicit form.

A free-energy model for the inhomogeneous hard-sphere fluid mixture was derived recently [Phys. Rev. Lett. 63, 980 (1989)], which is based on the fundamental geometric measures of the particles. It yields explicit analytic predictions for the direct correlations of the uniform fluid in three dimensions, which are in overall good agreement with extensive Monte Carlo simulation results for the pair and triplet functions. In the present paper, general features of the model are further explicated, with special emphasis on those features enabling a straightforward application to the inhomogeneous hard-disk fluid mixture. In particular, an analytic theory for the structure factors of the hard-disk mixture is derived. The present results should be useful as reference for the statistical-thermodynamic analysis of two-dimensional systems.

Free-energy model for the inhomogeneous hard-sphere fluid: ‘‘Closure’’ relation between generating functionals for ‘‘direct’’ and ‘‘cavity’’ distribution functions

A free-energy model for the inhomogeneous hard sphere fluid mixture was derived recently [Phys. Rev. Lett. 63, 980 (1989)] which is based on the fundamental geometric measures of the particles. It features convolution forms for the direct correlation functions as obtained by functional differentiation from the free energy. General features of the model are further explicated in this paper. In particular, fused-spheres overlap forms for the cavity distribution functions are obtained by recursive application of the model to the free energy of composite particles. It is shown how the scaled-particle theory provides an explicit ‘‘closure’’ relation between these two alternative description of the correlations.

Free-energy density functional for the inhomogeneous hard-sphere fluid: Application to interfacial adsorption

We propose a simplified version of the free-energy density functional for the inhomogeneous hard-sphere fluid mixture recently derived by Rosenfeld [Phys. Rev. Lett. 63, 980 (1989)]. This new functional, which requires four distinct weight functions, generates a triplet-direct-correlation function for the one-component fluid, which is in good agreement with Monte Carlo simulation results. It also performs well in describing the density profile of the hard-sphere fluid in contact with hard and soft walls. Yet, it is not suitable for studying the freezing transition.

The thermal conductivity of dense fluids and their mixtures using the effective diameter revised enskog theory

A method for calculating the thermal conductivity of dense fluids and their mixtures is described. This method relies on the results of the revised Enskog theory for hard-sphere fluid mixtures, and the use of the temperature and density dependent diameters given by Mansoori and Canfield, and Rasaiah and Stell. In addition, a correction in order to take into account the internal degrees of freedom contribution was introduced. A comparison with the conformal solution model, and with accurate experimental data was made. A very good agreement was found.

Phase stability of multicomponent highly asymmetric electrolytes: A charged hard sphere model study

The phase diagrams and the spinodal lines of different types of charged hard sphere fluid mixtures (CHSM) are determined through the use of the mean spherical approximation (MSA). The attention in the paper is particularly addressed to the low density regime of these systems, where the physical parameters are chosen in such a way so to mimic polyelectrolyte solutions with an added salt, or dilute electrolyte solution mixtures of simple salts. The phase diagrams at densities corresponding to the molten salt regime, obtained in a previous paper, are also reported, in order to get a complete picture of the phase stability behavior of CHSM. In all cases investigated the MSA predicts the existence of a closed solubility loop in the low density domain of the phase diagram. The location and the extension of this feature turns out to be crucially affected by the pressure, by the diameter ratios of the ionic components, and by the asymmetry of the ionic charges. A comparison of the MSA predictions in the low density regime with more refined theoretical approaches, or with computer simulation data, has not been possible for reasons which are explained in the text. However, the theory is expected to work at the same level of accuracy as in other physical contexts, where its predictions for the phase stability conditions of various systems have been found to be qualitatively reliable; moreover, for the case in which the model mimics a polyelectrolyte solution with an added salt, the form of the theoretical spinodal closely resembles the cloud points curve of an ionic micellar solution experimentally investigated by other authors.

Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing.

A free-energy density functional for the inhomogeneous hard-sphere fluid mixture is derived from general basic considerations and yields explicit analytic expressions for the high-order direct correlation functions of the uniform fluid. It provides the first unified derivation of the most comprehensive available analytic description of the hard-sphere thermodynamics and pair structure as given by the scaled-particle and Percus-Yevick theories. The infinite-order expansion around a uniform reference state does not lead, however, to a stable solid, thus questioning the convergence of the density-functional theory of freezing.

Phase stability of dense charged hard sphere fluid mixtures

The mean spherical approximation (MSA), the hypernetted-chain (HNC) approximation, and Monte Carlo (MC) calculations are used in order to investigate the phase stability of dense charged hard sphere fluid mixtures. According to the MSA, spinodal decomposition occurs in these systems in a certain range of the physical parameters. The HNC results indicate the existence of a spinodal line in qualitative agreement with the MSA prediction. Local compositions obtained by MC calculations show the expected behavior in a mixture approaching segregation. The concentrations of the coexisting phases are also calculated within the MSA, and the characteristics of the phase diagrams of different types of mixture are discussed. A detailed analysis is then made of the thermodynamic quantities which define the Gibbs free energy of mixing; this allows one to understand in more detail the segregation process. Finally, the applicability of the present approach for studying the phase stability conditions of molten salt mixtures and ionic micellar solutions is discussed.

The structure of a hard sphere fluid mixture confined to a slit

The structure of a binary fluid mixture of hard spheres of different diameters is studied using both computer simulation by the Monte Carlo method and density functional theory. The density functional used contains a term third order in the densities, chosen to ensure that sum rules previously obtained are satisfied. The density functional also uses the two-particle direct correlation function of the homogeneous fluid. For this work, this function is approximated by the PY result obtained by Lebowitz and this limits the accuracy of the theory. Within this limit, the agreement between the theoretical results and the simulated results for the density profiles of the two species in the fluid is good.

Correlation Method for the Shear Viscosity of Fluid Mixtures at Moderate Densities

A method to predict the shear viscosity of fluid mixtures is described. The method relies on the results of the Revised Enskog Theory for hard-sphere fluid mixtures and the use of temperature and density dependent diameters to model each species of the real mixture. The predictions are compared against the empirical Dean and Stiel correlation scheme for twenty six binary mixtures mainly of hydrocarbons. While the overall qualitative agreement is satisfactory, quantitative results may differ substantially. Possible sources of this discrepancy are briefly discussed.

Scaled field particle theory of the structure and the thermodynamics of isotropic hard particle fluids

The pair structure and the thermodynamics of isotropic fluids composed of hard particles of different shapes and concentrations are considered by a new approach which (i) unifies the diagramatic Percus–Yevick theory and the geometric scaled particle theory, and (ii) leads to analytic accurate approximations for the direct correlation functions and cavity distribution functions. The scaled-particle interpolation idea of Reiss et al. is achieved diagramatically, approximating the full expansion of the direct correlation functions by appropriately renormalized low order graphs, in which the size of the field particle (‘‘black circle’’) is scaled. It is shown that the lowest order scaled field particle approximation, involving only pair excluded volumes as variables, and pair overlap volumes as functions, contains the exact solution of the Percus–Yevick integral equation for the general hard sphere mixture in D dimensions. An exact geometric relation obeyed by the excluded volume (covolume) of two fused convex bodies and another single convex body, is used to show that the scaled field particle theory is equivalent to the scaled particle expansion of the chemical potential in terms of the fundamental measures of the hard particles. This is in contrast to the conventional scaled particle Taylor series expansion in powers of the linear scale of the solute particle. The first order scaled field particle approximation and the ‘‘fundamental measure’’ scaled particle theory become identical when the fluid mixture contains only convex particles. For a D-dimensional (D=odd) hard spheres fluid mixture our new analysis enables to obtain the complete exact solution of the Percus–Yevick integral equation for the structure directly from its compressibility equation of state. This leads to the derivation of simple, analytic, geometric approximations for the direct correlation functions and cavity distribution functions of the general isotropic hard particle fluid, conformal to those for the hard sphere fluid mixture.

The bulk viscosity in dense fluids

A method for calculating bulk viscosity is described. This method relies on the results of the revised Enskog theory for hard-sphere fluid mixtures and the use of the temperature- and density-dependent diameter of Mansoori-Canfield and Rasaiah-Stell to model each species of the real mixtures. Using this method the predicted values of the bulk viscosity of several mixtures of hydrocarbons and noble gases were calculated. These results are mainly predictive.

Fluids of dimerizing hard spheres, and fluid mixtures of hard spheres and dispheres

We continue the study of a fluid of hard spheres which contain an off-center attraction site, with the range of attraction restricted so that dimers, but no higher s-mers can be formed. Two approximations, an integral equation and a version of thermodynamic perturbation theory, were previously found to perform well when tested against Monte Carlo simulations. Here, the limit in which the attraction becomes an infinitely strong glue spot of infinitesimal extent is evaluated analytically for the two theories. The resulting equation of state may be identified with the equation of state of a mixture of hard spheres and hard dispheres. In the limit of complete dimerization the system is equivalent to a one-component fluid of hard dispheres. The equation of state of hard dispheres, known from Monte Carlo simulations, is accurately represented by the Tildesley–Streett (TS) fitting function. In the pure disphere limit, our version of thermodynamic perturbation theory predicts an equation of state which is numerically almost indistinguishable from the TS fitting function. For the integral equation, the disphere limit of the compressibility equation of state is also very good, showing a maximum deviation from the TS fitting function of 2% over the density range of the simulations. The virial equation of state is much less satisfactory, with the deviation rising to 15% at the highest density.

Transport coefficients of fluid mixtures

On the basis of the successful description of the equilibrium properties of simple fluids and fluid mixtures using perturbation theory, the consequences of including density- and temperature-dependent diameters in the formulas for the transport coefficients of dense hard-sphere fluid mixtures are investigated. The advantages and limitations of this approach for the correlation of the experimental data of real mixtures, together with numerical estimates for particular mixtures, are discussed. On the other hand, recent mean field kinetic theories which include the effect of the attractive tail in the intermolecular potential are employed to derive transport coefficients for mixtures. Numerical results are presented and comparison with other theories is also made.

The Enskog theory for multicomponent mixtures. II. Mutual diffusion

We present a detailed description of the mutual diffusion coefficients of binary and ternary dense fluid mixtures of hard spheres, as given by the revised Enskog theory (RET) of van Beijeren and Ernst and the standard Enskog theory (SET) of Tham and Gubbins. The formulas for the diffusion coefficients [see part I of this series, J. Chem. Phys. 78, 2746 (1983)] involve the contact values of the equilibrium pair distribution functions and the chemical potentials, for which the Carnahan–Starling approximation is used. The formulas, which were obtained by making an expansion in Sonine polynomials, are evaluated up to the third order and the convergence of the Sonine polynomial expansion is discussed. Except at low densities, the SET cannot be used to describe diffusion in hard-sphere mixtures. We find, using the CS approximation, that the SET gives values for the direct mutual diffusion coefficients that become negative at very high densities. A study is made of the dependence of the RET mutual diffusion coefficients on a variety of the parameters that determine the mixture (density, composition, molecular masses, and diameters) and a summary of the trends found is given. In particular, the influence of the addition of a third component on the mutual diffusion of two other components is discussed.

Equilibrium properties of hard-sphere fluid mixture in the semiclassical limit

High-temperature expansions for the density independent part of the to it are obtained for a hard-sphere binary mixture. The ‘‘excess’’ quantum corrections to the excess properties, such as excess second and third virial coefficients and excess free energy, are also reported. It is found that the excess quantum effect depends on the concentration and the particle diameter ratio.