Thermodynamic properties of molecular fluid mixtures of hard ellipsoids

Thermodynamic properties of molecular fluid mixtures of hard ellipsoids are calculated. Numerical results are given for equation of state and excess-free energy of the binary mixture of both additive and non-additive hard ellipsoids. It is found that the equation of state and free energy of mixtures increase with increase of anisotropy parameterx ₀.     

On the critical non-additivity driving segregation of asymmetric binary hard sphere fluids

A previously proposed version of thermodynamic perturbation theory, appropriate for singular pair interactions between particles, is applied to binary mixtures of hard spheres with non-additive diameters. The critical non-additivity Δ_C required to drive fluid–fluid phase separation is determined as a function of the ratio ξ ≤ 1 of the diameters of the two species. Δ_C(ξ) is found to decrease with ξ and to go through a minimum for ξ ? 0.015 before increasing sharply as ξ → 0, irrespective of the total packing-fraction η of the mixture. These results are the basis of an estimate of the range of size ratios for which a binary mixture of additive hard spheres exhibit a fluid–fluid miscibility gap. This range is conjectured to be 0.01 ? ξ ? 0.1.     

Thermodynamics of pseudo-hard body mixtures

Thermodynamic properties of binary mixtures of hard spheres of various size and pseudo-hard bodies, mimicking the short-range non-additive repulsive interactions in realistic models of water, have been determined over the entire concentration range using standard NVT Monte Carlo simulations. Virial coefficients of the mixture have also been computed. Having no other theoretical tool currently available, a perturbed virial expansion is examined with respect to its potential to estimate/predict the properties of the mixture without resorting to any fitting of simulation data. The perturbed virial expansion is found to perform quite accurately for the mixtures containing larger spheres, whereas for small spheres dissolved in water the result is only qualitatively correct.     

Quantum corrections to thermodynamics of polar hard sphere fluids

The quantum corrections to the thermodynamic properties of polar hard sphere fluids and fluid mixtures are estimated taking into account the influence of dipole and quadrupole moments. Expressions are given for the second virial coefficient, free energy and pressure and results are given for different values ofμ* andϑ*. The first order quantum correction arises due to the translational contribution only. The quantum effect increases with density,μ* andϑ*. Numerical results are also estimated for binary mixtures of (i) hard spheres and dipole hard spheres and (ii) hard spheres and quadrupole hard spheres. The ‘excess’ free energy for dipole hard sphere binary mixture is also reported. It is found that the ‘excess’ quantum effect depends on the concentration and the particle diameter ratio and increases with increase ofμ* andϑ*.     

Perturbation expansions and series acceleration procedures. Part I. ε-convergence and critical parameters

A simple acceleration of convergence technique known as the ‘ε-convergence algorithm’ (ea) is applied to determine the critical temperatures and exponents. Several illustrations involving well-known series expansions appropriate to two- and three-dimensional Ising models, three-dimensional Heisenberg models, etc., are given. Apart from this, a few recently studied ferrimagnetic systems have also been analysed to emphasise the generality of the approach. Where exact solutions are available, our estimates obtained from this procedure are in excellent agreement. In the case of other models, the critical parameters we have obtained are consistent with other estimates such as those of the Padé approximants and group theoretic methods. The same procedure is applied to the partial virial series for hard spheres and hard discs and it is demonstrated that the divergence of pressure occurs when the close-packing density is reached. The asymptotic form for the virial equation of state is found to beP/ρkT ∼ (1 −ρ/ρ _c ⁻¹ for hard spheres and hard discs. Apart from the estimation of ‘critical parameters’, we have applied theea and the parametrised Euler transformation to sum the partial, truncated virial series for hard spheres and hard discs. The resulting values of pressure so obtained, compare favourably with the molecular dynamics results.     

Pertubation theory for systems with strong short-ranged interactions

We propose a variant of thermodynamic perturbation theory based on the Mayer f-function which is applicable to strongly repulsive, and even singular interactions. The expansion of the free energy is successfully tested against known ‘exact results’ for hard-sphere fluids, and then applied to binary mixtures of particles with non-additive hard cores or shouldered potentials. The resulting phase diagrams agree well with existing simulation data and theoretical predictions.     

Perturbation expansions and series acceleration procedures: Part-II. Extrapolation techniques

Three new procedures for the extrapolation of series coefficients from a given power series expansion are proposed. They are based on (i) a novel resummation identity, (ii) parametrised Euler transformation (pet) and (iii) a modifiedpet. Several examples taken from the Ising model series expansions, ferrimagnetic systems, etc., are illustrated. Apart from these applications, the higher order virial coefficients for hard spheres and hard discs have also been evaluated using the new techniques and these are compared with the estimates obtained by other methods. A satisfactory agreement is revealed between the two.     

Equation of state of mixtures of simple fluids at high pressures

Recently Lee and Levesque have extended the Weeks-Chandler-Andersen pure fluid perturbation theory to mixtures and have compared the results to the Leonard-Henderson-Barker mixture theory. The results seem to favour the Leonard-Henderson-Barker theory. This and other previous comparisons of mixture theories have been mostly confined to the study of ‘zero pressure’ thermodynamic properties. In this paper we compare the Lee-Levesque, Leonard-Henderson-Barker and the van der Waals one-fluid theories to high pressure equation of state data for helium-xenon mixtures. This system is modelled by a binary mixture of Lennard-Jones fluids and the hard sphere reference system is characterized by the Grundke-Henderson hard sphere mixture radial distribution function. The Lee-Levesque theory compares favourably with experimental equation of state data up to pressures of 2000 atmospheres. The Leonard-Henderson-Barker and van der Waals one theories are satisfactory. Although the van der Waals one theory yields the poorest results, it does offer the advantage of having the greatest ease of computation.

Previous theoretical and Monte Carlo calculations of mixture properties have assumed the geometric mean rule for the mixed interaction energy parameter, ε ₁₂, with consequent disagreement with experimental results. We point out that ε ₁₂ can be determined from mixed second virial coefficient data and use such improved determinations of ε ₁₂. We show that this method yields significantly improved theoretical predictions.     

Thermodynamic Curvature of the Binary van der Waals Fluid

The thermodynamic Ricci curvature scalar R has been applied in a number of contexts, mostly for systems characterized by 2D thermodynamic geometries. Calculations of R in thermodynamic geometries of dimension three or greater have been very few, especially in the fluid regime. In this paper, we calculate R for two examples involving binary fluid mixtures: a binary mixture of a van der Waals (vdW) fluid with only repulsive interactions, and a binary vdW mixture with attractive interactions added. In both of these examples, we evaluate R for full 3D thermodynamic geometries. Our finding is that basic physical patterns found for R in the pure fluid are reproduced to a large extent for the binary fluid.     

Two disks in a box

Exact analytical expressions are derived for the thermodynamic properties of two discs in square and rhomboidal boxes with hard walls, and in a square cell with periodic boundaries. With periodic boundaries the equation of state has a van der Waals loop and a second order cusp. In a box with hard walls there is a third order “glass transition” which seems to capture the essence of the glass transition observed in systems of a few hundred hard spheres.     

Perturbation theory of polar hard Gaussian overlap fluid mixtures

A perturbation theory of polar hard Gaussian overlap fluid mixture is discussed. Explicit analytic expressions for the second and third varial coefficients are given. Numerical results are estimated for the thermodynamic properties of quadrupolar hard Gaussian overlap fluid and fluid mixture. It is found that the excess free energy and internal energy depend on concentrationsc ₁,c ₂, molecular diameter ratioR, shape parameterK and the quadrupole momentsQ*₁,Q*₂.     

Analytical approach to the thermodynamics and density distribution of crystalline phases of hard spheres

We extend the well-known free-volume approach to describe the thermodynamic properties and density distribution of crystalline phases of hard hypersphere systems. Despite its extreme simplicity the approach yields results which are in quantitative agreement with simulation data. The theory can, in particular, describe the properties of the body-centred cubic phase of hard spheres, for which density-functional approaches provide unphysical results, allowing for the application of perturbation theory to situations where, as is the case in some colloidal systems, the body-centred cubic is one of the most interesting phases. The theory is also tested by applying it to systems of hard discs.     

Mixtures of polar and nonpolar molecules

The site-site Ornstein-Zernike (SSOZ) equation with mean spherical approximation closure is solved analytically for a mixture of hard dumbbells and polar hard dumbbells. The solution reduces to that of the pure polar hard dumbbell fluid at the polar species density rather than the total density. The thermodynamic properties of the mixture are obtained using the zero-pole approximation (ZPA) to the free energy. The mixture is shown to separate into two mixed phases, one rich in the nonpolar species and the other rich in the polar species. This phase separation terminates in an upper critical solution temperature. The excess thermodynamic functions are presented and the mixture exhibits both positive and negative values of the excess volume. The negative values of the excess volume occur in mixtures rich in the polar component.     

Perturbation theory for a mixture of polarizable liquids

Thermodynamic perturbation theory developed for pure polar liquids with significant polarizability is applied to a mixture with polar and nonpolar polarizable components. Expressions for the average and unperturbed dipole moments of the components of the mixture are presented and the dipole and induced contributions to the thermodynamic functions of the mixture are calculated. The equation of state is obtained and used to calculate the excess properties of a binary mixture of particles interacting with a stockmeyer potential. It is shown that the induced dipole moments contribute significantly to the thermodynamic properties of this model solution and that their effects must be taken into account in predicting the properties of real liquid mixtures containing polar components.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 69–73, March, 1985.     

Structure factors in a two-dimensional binary colloidal hard sphere system

ABSTRACT

Hard disks are one of the simplest interacting many-body model system in two dimensions (2D). Here, we present a comprehensive set of measurements of the static structure factors for quasi-2D monodisperse fluids and two different binary colloidal hard sphere mixtures: a small size ratio (SSR) system with a negligibly small negative non-additivity and a large size ratio system with a significantly larger non-additivity. We compare the experimental results for the monodisperse and SSR systems to those calculated using density functional theory (DFT) for additive mixtures. Furthermore, we determine the zero-wavevector limits of the static structure factors for the monodisperse and binary hard sphere fluids directly from an analysis of number and concentration fluctuations. For the monodisperse case, this leads to the isothermal compressibility, which agrees very well with DFT, and is consistent with the scaled particle theory equation of state for hard disks. For the binary fluids, the partial static structure factors are used to calculate the Bhatia–Thornton structure factors, and we find qualitative agreement with DFT for the SSR mixture. Finally, the zero-wavevector limits of the Bhatia–Thornton structure factors are determined and directly related to the thermodynamic factor, the dilatation factor and the isothermal compressibility.     

Thermodynamic properties of some gallium-based binary alloys

We have studied the concentration dependence of the free energy of mixing, concentration-concentration fluctuations in the long-wavelength limit, the chemical short-range order parameter, the enthalpy and entropy of mixing of Ga-Zn, Ga-Mg and Al-Ga binary alloys at different temperatures using a quasi-chemical approximation for compound forming binary alloys and that for simple regular alloys. From the study of the thermodynamic quantities, we observed that thermodynamic properties of Ga-Zn and Al-Ga exhibit positive deviations from Raoultian behaviour, while Ga-Mg exhibits negative deviation. Hence, this study reveals that both Ga-Zn and Al-Ga are segregating systems, while chemical order exists in Ga-Mg alloy in the whole concentration range. Furthermore, our investigation indicate that Al-Ga binary alloy have a tendency to exhibit ideal mixture behaviour in the concentration range 0?c_Al?0.30 and 0.7?c_Al?1.     

Fourth virial coefficients for globular molecules

A binary mixture of hard rods with non-additive distances of contact, in one dimension, is studied. Special attention is paid to the case that two particles (hard rods) of the same species can come into contact in the presence of a particle of the other species between them. The system is treated by means of the ensemble which is specified by the total number of particles of one species, the chemical potential of the particles of the other species, the temperature and the pressure. The free energy in this ensemble is shown to be expressed by the eigenvalue with the smallest absolute value of an integral equation. The equation of state is obtained by numerical differentiation of the free energy.     

Far-infra-red absorption of non-polar liquids

A new conformal solution theory using a single pure fluid as a reference substance for the calculation of thermodynamic properties of fluid mixtures is developed. The perturbation theory developed by Weeks, Chandler and Andersen (WCA) and by Verlet and Weis (VW) is used to calculate the reference properties. The mean density approximation and corresponding state principle are used to eliminate the higher order terms in the mixture system and to derive the pseudo-parameters for the reference system. The mixture properties are obtained from the reference properties and their corresponding hard sphere excess functions defined as the properties of the mixture less the value of the properties for the hard sphere mixture.

The excess functions of mixing for several liquid mixtures of Lennard-Jones fluids, obeying the Lorentz-Berthelet rule, are calculated by the new method (VW-HSE). Comparison with the results of other theories and Monte Carlo data shows definite improvement. Since only the properties of a pure reference fluid are directly calculated, the method can be applied to more complicated multicomponent systems without additional computational effort as required by other theories.     

Is it possible to infer the equation of state of a mixture of hard discs from that of the one-component system?

Based on exact asymptotic properties of the composition-independent virial coefficients of a binary mixture of hard discs in the limits α = σ₂/σ₁ → 0, α → 1 and α → ∞, R. J. Wheatley (1998, Molec. Phys., 93, 965) has recently proposed an approximate interpolation equation for these coefficients. In this note, the equation of state equivalent to this interpolation is obtained, expressing the compressibility factor of the mixture in terms of that of the pure system. An extension to an arbitrary number of components is also given. The equation of state derived here is compared with another one recently proposed by following a different route (Santos, A., Yuste, S. B., and López de Haro, M., 1999, Molec. Phys., 96, 1) and with Monte Carlo simulation results. It is shown that the latter equation is more accurate than the former one, at least for not too disparate mixtures (0.7 < α < 1).     

Density functional for ternary non-additive hard sphere mixtures

Based on fundamental measure theory, a Helmholtz free energy density functional for three-component mixtures of hard spheres with general, non-additive interaction distances is constructed. The functional constitutes a generalization of the previously given theory for binary non-additive mixtures. The diagrammatic structure of the spatial integrals in both functionals is of star-like (or tree-like) topology. The ternary diagrams possess a higher degree of complexity than the binary diagrams. Results for partial pair correlation functions, obtained via the Ornstein-Zernike route from the second functional derivatives of the excess free energy functional, agree well with Monte Carlo simulation data.