M. López de Haro

The Enskog theory for multicomponent mixtures. IV. Thermal diffusion

Using the revised Enskog theory we derive equations for the thermal diffusion ratios kTi and thermal diffusion factors αij of multicomponent hard‐sphere mixtures for systems in mechanical equilibrium. The first ten Enskog approximations to the thermal diffusion factor αij[N] (N=1,2,...,10) are evaluated numerically for a variety of system parameter choices appropriate to binary and ternary mixtures. We find that the first Enskog approximation does not vanish; the sequence of Enskog approximations converges most rapidly when the mass of the spheres are nearly equal. The seventh Enskog approximation was estimated to lie within about 1% of the exact value for all choices of system parameters. A comprehensive numerical study of the properties of the thermal diffusion factor for binary mixtures is given, including special mixtures such as the dusty gas, the Lorentz, quasi‐Lorentz and Rayleigh gases, and isotopic mixtures. Particular emphasis is placed on showing how the properties of α12 change as the density ...

An accurate and simple equation of state for hard disks

An equation of state for a fluid of hard disks is proposed: Z=[1−2η+(2η0−1)(η/η0)2]−1. The exact fit of the second virial coefficient and the existence of a single pole singularity at the close‐packing fraction η0 are the only requirements imposed on its construction. A comparison of the prediction of virial coefficients and of the values of the compressibility factor Z with those stemming out of other known equations of state is made. The overall performance of this very simple equation of state is quite satisfactory.

On the foundations of extended irreversible thermodynamics

A theory of macroscopic systems which takes as independent variables the slow (conserved) ones plus the fast dissipative fluxes is carefully analyzed at three levels of description: macroscopic (thermodynamic), microscopic (projection operators) and mesoscopic (fluctuation theory). Such a description is compared with the memory function approach based only on the conserved variables. We find that the first theory is richer and wider than the second one, and some misunderstandings in this connection are clarified and discussed.

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 coef...

Flow of Maxwell fluids in porous media

In this paper we analyze the flow of a Maxwell fluid in a rigid porous medium using the method of volume averaging. We first present the local volume averaged momentum equation which contains Darcy-scale elastic effects and undetermined integrals of the spatial deviations of the pressure and velocity. A closure problem is developed in order to determine the spatial deviations and thus obtain a closed form of the momentum equation that contains a time-dependent permeability tensor. To gain some insight into the effects of elasticity on the dynamics of flow in porous media, the entire problem is transformed to the frequency domain through a temporal Fourier transform. This leads to a dynamic generalization of Darcys law. Analytical results are provided for the case in which the porous medium is modeled as a bundle of capillary tubes, and a scheme is presented to solve the transformed closure problem for a general microstructure.

Alternative Approaches to the Equilibrium Properties of Hard-Sphere Liquids

An overview of some analytical approaches to the computation of the structural and thermodynamic properties of single component and multicomponent hard-sphere fluids is provided. For the structural properties, they yield a thermodynamically consistent formulation, thus improving and extending the known analytical results of the Percus-Yevick theory. Approximate expressions for the contact values of the radial distribution functions and the corresponding analytical equations of state are also discussed. Extensions of this methodology to related systems, such as sticky hard spheres and square-well fluids, as well as its use in connection with the perturbation theory of fluids are briefly addressed.

Equation of state of nonadditive d-dimensional hard-sphere mixtures

An equation of state for a multicomponent mixture of nonadditive hard spheres in d dimensions is proposed. It yields a rather simple density dependence and constitutes a natural extension of the equation of state for additive hard spheres proposed by us [A. Santos, S. B. Yuste, and M. Lopez de Haro, Mol. Phys. 96, 1 (1999)]. The proposal relies on the known exact second and third virial coefficients and requires as input the compressibility factor of the one-component system. A comparison is carried out both with another recent theoretical proposal based on a similar philosophy and with the available exact results and simulation data in d=1, 2, and 3. Good general agreement with the reported values of the virial coefficients and of the compressibility factor of binary mixtures is observed, especially for high asymmetries and/or positive nonadditivities.

Is there a glass transition for dense hard-sphere systems?

The recent results for the contact value of the radial distribution function obtained from large-scale molecular dynamics and Monte Carlo simulations of systems of dense hard spheres [M. D. Rintoul and S. Torquato, J. Chem. Phys. 105, 9258 (1996)] are compared to those of the Pade equation of state for a hard-sphere fluid (derived from the knowledge of the first eight virial coefficients), including the metastable fluid region up to a packing fraction of ηg≃0.56, and above such packing fraction to the ones corresponding to an equation of state of the free-volume type which presents a simple pole at random close-packing. This latter equation involves the same value for the pressure at ηg as the Pade equation of state, and arises in a consistent way from the application of a rational-function approximation method to the computation of the radial distribution function of a hard-sphere fluid. The substantial improvement of the agreement with the simulation results over the one obtained with the equation of st...

A thermodynamic approach to non-linear viscoelasticity

We show that Extended Irreversible Thermodynamics (EIT) may be regarded as a unifying thermodynamic scheme which generates a variety of non-linear viscoelastic equations of state. In particular, we show how the so-called eight-constants Oldroyd model, with all its invariant properties, may be derived from the postulates of EIT. It is also shown that the structure of the thermodynamic based rheological equations of state is in complete agreement with that of the representation theorems formulation of constitutive relationships originally proposed by Rivlin and Ericksen. We discuss how more complex rheological models can be derived from higher order approximations of the theory. Finally, the connections between our approach and other related works are briefly discussed.

Virial coefficients and equations of state for mixtures of hard discs, hard spheres and hard hyperspheres

The composition-independent virial coefficients of a d-dimensional binary mixture of (additive) hard hyperspheres following from a recent proposal for the equation of state of the mixture (Santos, A., Yuste, S. B., and López de Haro, M., 1999, Molec. Phys., 96, 1) are examined. Good agreement between theoretical estimates and available exact or numerical results is found for d = 2, 3, 4 and 5, except for mixtures whose components are very disparate in size. A slight modification that remedies this deficiency is introduced and the resummation of the associated virial series is carried out, leading to a new proposal for the equation of state. The case of binary hard sphere mixtures (d = 3) is analysed in some detail.