Hard-sphere Reference Fluid Research Articles

Excess Entropy Scaling Law for Diffusivity in Liquid Metals.

Understanding how dynamic properties depend on the structure and thermodynamics in liquids is a long-standing open problem in condensed matter physics. A very simple approach is based on the Dzugutov contribution developed on model fluids in which a universal (i.e. species-independent) connection relates the pair excess entropy of a liquid to its reduced diffusion coefficient. However its application to “real” liquids still remains uncertain due to the ability of a hard sphere (HS) reference fluid used in reducing parameters to describe complex interactions that occur in these liquids. Here we use ab initio molecular dynamics simulations to calculate both structural and dynamic properties at different temperatures for a wide series of liquid metals including Al, Au, Cu, Li, Ni, Ta, Ti, Zn as well as liquid Si and B. From this analysis, we demonstrate that the Dzugutov scheme can be applied successfully if a self-consistent method to determine the packing fraction of the hard sphere reference fluid is used as well as the Carnahan-Starling approach to express the excess entropy.

The theory of non-electrolyte solutions: an historical review

Starting with the work of van der Waals in 1873, a historical review is given of theories of non-electrolyte mixtures that are based on a well-defined Hamiltonian, and thus can be tested against molecular simulation, as well as experiment. Most of the first 100 years covered here were devoted to attempts to find a successful theory of simple mixtures, culminating in the van der Waals 1-fluid theory of conformal solutions, and perturbation theories based on a hard-sphere reference fluid. The last 40 years has seen the more rapid development of theories for fluids of more complex molecules, including strongly polar liquids, chain molecules and liquids in which molecular association is important. 1 Based on a Molecular Physics Lecture, ‘The Theory of Non-Electrolyte Solutions: A Recent History’, delivered at Imperial College London, 25 September 2009.

Relationship between local molecular field theory and density functional theory for non-uniform liquids

The local molecular field theory (LMF) developed by Weeks and co-workers has proved successful for treating the structure and thermodynamics of a variety of non-uniform liquids. By reformulating LMF in terms of one-body direct correlation functions we recast the theory in the framework of classical density functional theory (DFT). We show that the general LMF equation for the effective reference potential φ(R)(r) follows directly from the standard mean-field DFT treatment of attractive interatomic forces. Using an accurate (fundamental measures) DFT for the non-uniform hard-sphere reference fluid we determine φ(R)(r) for a hard-core Yukawa liquid adsorbed at a planar hard wall. In the approach to bulk liquid-gas coexistence we find the effective potentials exhibit rich structure that can include damped oscillations at large distances from the wall as well as the repulsive hump near the wall required to generate the low density "gas" layer characteristic of complete drying. We argue that it would be difficult to obtain the same level of detail from other (non-DFT based) implementations of LMF. LMF emphasizes the importance of making an intelligent division of the interatomic pair potential of the full system into a reference part and a remainder that can be treated in mean-field approximation. We investigate different divisions for an exactly solvable one-dimensional model where the pair potential has a hard-core plus a linear attractive tail. Results for the structure factor and the equation of state of the uniform fluid show that including a significant portion of the attraction in the reference system can be much more accurate than treating the full attractive tail in mean-field approximation. We discuss further aspects of the relationship between LMF and DFT.

Understanding liquids: reflections about Melbourne, 1966–67

The progress in the development of the theory of liquids from early lattice or cell theories, which were fashionable until the 1960s, the formulation of perturbation theories that employ a hard sphere reference fluid, and a simple method of estimating the second-order term is reviewed. With the present availability of several useful simulation methods and theories, it may be difficult for modern scientists to realize that this work with John Barker during my stay in Melbourne in 1966–1967 was the first successful theory of the liquid state. Ironically, perturbation theory is similar to the van der Waals theory, which until recently was regarded, incorrectly, as only of pedagogical interest.

Influence of the attractive pair-potential in density functional models of nucleation

The influence of different types of attractive pair-potentials in density functional models of homogeneous nucleation is analyzed. The models considered here are based on the local-density approximation of the free-energy functional and make use of the same hard-sphere reference fluid (the Carnahan-Starling approximation). The analyzed magnitude is the ratio between the obtained density functional theory nucleation barrier height and the corresponding classical result. Some recent studies suggest that such an energy ratio follows a universal scaling relation which is independent of the form of the pair-potential. The present analysis confirms that the results are weakly dependent on the form of the interaction potential, provided that its asymptotic decay is strong enough as for those that are usually considered in practice: Yukawa, Lennard-Jones, Square-Well, etc. However, when the asymptotic decay at infinity of the attractive pair-potential is weak enough, the conjectured scaling behavior ceases to be applicable and the nucleation barrier ratio can be reduced significantly.

Comprehensive investigation about the second order term of thermodynamic perturbation expansion

Monte Carlo simulations are carried out for the second order term in the thermodynamic perturbation expansion around a hard sphere reference fluid. The sample potentials considered cover a wide spectrum: From two frequently employed, namely hard sphere plus square well potential and hard core attractive Yukawa potential, to two kinds of repulsive potentials, namely hard sphere plus square shoulder potential and hard sphere plus triangle shoulder potential; the investigated potential range also extends from extremely short range to rather long range. The obtained simulation data are used to evaluate performance of two theoretical approaches, i.e., a traditional macroscopic compressibility approximation (MCA) and a recent coupling parameter expansion. Extensive comparison shows that the coupling parameter expansion provides a reliable method for accurately calculating the second order term of the high temperature series expansion, while the widely accepted MCA fails quantitatively or even qualitatively for most of the situations investigated.

Free Volumes and the Anomalous Self-Diffusivity of Attractive Colloids

Free volume theories for the dynamics of dense fluids commonly assume (i) that diffusivity increases with average free volume per particle and (ii) that the size distribution of free volumes can be approximated by that of an equivalent hard-sphere reference system. We use molecular simulations to demonstrate that these assumptions break down when one considers concentrated suspensions of particles with short-range attractions. In these systems, self-diffusivity shows nonmonotonic dependencies on both average free volume and the strength of the interparticle attraction. Moreover, when interparticle attractions are strong, the shape of the free volume distribution is qualitatively different than that of the corresponding hard-sphere reference fluid. We propose a conceptual revision to the traditional free volume perspective that takes into account both the size distribution and the persistence time of the free volumes, and we demonstrate that it can qualitatively capture the disparate behaviors of a model fluid with short-range attractions and its hard-sphere reference fluid.

Two-body integrals for hard sphere fluid based on Tang–Lu RDF expression

Two-body integral with hard sphere reference fluid is of significance in thermodynamic modelling with perturbation theory. In this paper, the two-body integral is recalculated by using Tang–Lu radial distribution function (RDF) expression, which is superior to P–Y RDF in representing the structure of hard sphere fluid in whole density region and especially in contact-distance region. It is found that the two-body integral values calculated with Tang–Lu RDF are between those of Larsen et al.’s and Henderson et al.’s, and their difference is negligible. In order to investigate the accuracy of two-body integrals, properties of Lennard–Jones fluid are calculated with B–H perturbation theory under local-compressible approximation and compared with computer simulation results. It is shown that the accuracy of internal energy and compressibility calculated with two-body integrals of this paper and with Larsen et al.’s are 6.78, 8.60, 6.85 and 9.08%, respectively for 170 computer simulation data. As a whole, the accuracy of the present two-body integral with Tang–Lu RDF is a little superior to that of Larsen et al.’s.

A SAFT–DFT approach for the vapour–liquid interface of associating fluids

We present a density functional theory (DFT) based on the statistical associating fluid theory (SAFT) bulk free energy to describe the behaviour of inhomogeneous associating molecular fluids, with a special emphasis on the vapour–liquid interface. The molecules are described in terms of hard-sphere (HS) segments which interact through an arbitrary intermolecular potential. Off-centre square-well bonding sites are incorporated to mediate the association between molecules. The approach presented here is based on a perturbation theory about a HS reference fluid under the local density approximation (LDA); the contributions due to chain formation and association are also considered at the local density level. The Helmholtz free energy functional due to the dispersive attractive interactions is treated at the mean-field level by ignoring the correlations. This description is essentially a ‘van der Waals’ theory of non-uniform fluids. The incorporation of a SAFT free energy of the bulk associating fluid represents the simplest extension of the theory to deal with inhomogeneous associating systems (SAFT–DFT). The resulting functional is used to investigate the effect of the range of the attractive interactions on the interfacial properties, such as the density profile and surface tension, for two different intermolecular potential models, namely the Yukawa and square-well. This simple SAFT–DFT approach is also used to make quantitative comparisons with the experimental surface tension for water.

On the thermodynamic properties of the Lennard-Jones fluid

An analytical scheme for the evaluation of the compressibility factor of a simple fluid is presented. It is based on the clasical perturbation theory and on a recent (analytical) approximation for the radial distribution function of the hard-sphere fluid. The scheme, which provides, in the case of the Lennard-Jones fluid, explicit expressions for both structural and thermodynamic quantities in a closed form, requires the hard-sphere fluid equation of state as the sole input. Using the Carnahan–Starling equation of state for the hard-sphere reference fluid, the prediction of the critical point as well as the behavior of the pressure of the Lennard-Jones fluid are an improvement in relation to previous work and compare rather well with recent numerical simulations. The same applies to the radial distribution function computed with the present scheme and the Weeks–Chandler–Andersen approach. From a knowledge of the freezing and melting transitions in the hard-sphere system, the liquid and solid branches of the reduced temperature vs. reduced density curve at coexistence for the Lennard-Jones fluid are simply obtained.

Thermodynamic properties and phase equilibria of branched chain fluids using first- and second-order Wertheim’s thermodynamic perturbation theory

We present an extension of the statistical associating fluid theory (SAFT) for branched chain molecules using Wertheim’s first- and second-order thermodynamic perturbation theory with a hard-sphere reference fluid (SAFT-B). Molecules are formed by hard spherical sites which are tangentially bonded. Linear chains are described as freely jointed monomeric units, whereas branched molecules are modeled as chains with a different number of articulation points, each of them formed by three arms. In order to calculate the vapor–liquid equilibria of the system, we have considered attractive interactions between the segments forming the chain at the mean-field level of van der Waals. The Helmholtz free energy due to the formation of the chain is explicitly separated into two contributions, one accounting for the formation of the articulation tetramer, and a second one due to the formation of the chain arms. The first term is described by the second-order perturbation theory of Phan et al. [J. Chem. Phys. 99, 5326 (1993)], which has been proven to predict the thermodynamic properties of linear chain fluids in a similar manner to Wertheim’s approach. The formation of the chain arms is calculated at Wertheim’s first-order perturbation level. The theory is used to study the effect of the chain architecture on the thermodynamic properties and phase equilibria of chain molecules. The equation predicts the general trends of the compressibility factor and vapor–liquid coexistence curve of the system with the branching degree, in qualitative agreement with molecular simulation results for similar models. Finally, SAFT-B is applied to predict the critical properties of selected light alkanes in order to assess the accuracy of the theory. Experimental trends of the critical temperature of branched alkanes are qualitatively captured by this simple theory.

Mapping of theoretical equations of state for molecular fluids on a biquadratic equation

A biquadratic equation of state for the attractive hard sphere chain fluid with dipolar interactions is developed. The background of the development is the mapping of molecular model equations onto a simplified mathematical form. In course of this mapping procedure known properties such as virial coefficients and physical limits are accounted for. For the hard sphere reference model a simple theoretical equation is developed, which is almost as accurate as the Carnahan–Starling (CS) equation and includes the nearly correct high density limit. The perturbation of this reference model due to chain formation and dipolar forces is accounted for by variations of the coefficients in the reference model. These coefficients are constant values for the hard sphere reference fluid and become dependent on the chain length parameter or the reduced dipole moment for the chain fluid or the dipolar fluid, respectively. By simple mathematical manipulations one can recover the usual equation of state form which consists of a sum of additive terms for each molecular property.

Equations of state from generalized perturbation theory: Part II. The Lennard-Jones fluid

Sowers, G.M. and Sandier, S.I., 1991. Equations of state from generalized perturbation theory. Part II. The Lennard-Jones Fluid. Fluid Phase Equilibria, 67: 127-150. We use perturbation theory to derive extensions of our equations of state used previously for the hard-core Lennard-Jones fluid to the more realistic soft-core Lennard-Jones fluid. We find that the use of a “hybrid” Barker-Henderson diameter (see Aim and Nezbeda, 1987) for the hard sphere reference fluid leads to perturbation free energies for the soft-core Lennard- Jones fluid that are similar to the perturbation free energies for a hard-core Lennard-Jones fluid. For the range of temperatures and densities considered here, one of our equations of state is more accurate than a 33-parameter equation (see Adachi et al., 1988) even though it has only four adjustable parameters. The other equation of state is a straightforward, theoretically based extension of a square-well equation of state (see Lee et al., 1985) to the Lennard-Jones fluid. Both equations of state can be used to correlate the phase diagrams and supercritical PVT data of noble gases and methane, and perform especially well in the correlation of saturated liquid volumes.

Thermodynamics and structure of liquid metals from the charged hard-sphere reference fluid

Perturbative variational calculations of thermodynamic and structural properties of liquid metals, based on the use of ab initio and highly reliable nonlocal pseudopotentials for the electron-ion interactions and of the fluid of charged hard spheres as a reference system, have been reported recently for the liquid alkali metals from Na to Cs near the freezing point. We extend in this work the above-mentioned calculations in two directions. Firstly, we discuss the predicted temperature dependence of the liquid structure factor for the same alkali metals over a limited temperature range above the freezing point. Secondly, we examine the usefulness of the approach for metals with relatively strong electron-ion interactions, namely Li and several polyvalent metals (Mg, Cd, Al, In, Tl and Pb). The charged hard-sphere reference system leads to lower values of the Helmholtz free energy and to slightly improved values of the excess entropy for all the liquid metals that we evaluate, even though polyvalent ones overall appear to be relatively close to fluids of neutral hard spheres. For the liquid alkali metals at elevated temperatures, the calculated structure factors are of similar quality as in our previous work, that is, they show a systematic shift in the positions of peaks and valleys to slightly larger wave numbers and peak heights that are somewhat underestimated with increasing temperature. However, for liquid polyvalent metals, our approach yields quite good agreement with experiment for the positions of maxima and minima in the liquid structure factor, while it tends to overemphasize somewhat these structures.

Influence of thermodynamic variables and molecular parameters in the solubility of gases in liquids

Abstract

Clustering and some other physical effects of van der Waals potentials

Molecular Dynamics and Monte Carlo computations for the original van der Waals model potentials, consisting of a hard core and inverse power attractive tails, are reported. The results show that the cusp at the minima of these potentials gives rise to anomalous phenomena in the low and intermediate density regions which derive from clustering effects. A new improved analytic equation-of-state for the van der Waals model is presented. This is based on a simple but accurate semi-empirical expression for the hard sphere reference fluid and Andrews' development for the attractive perturbation. A comparison of this first order perturbation equation with the ‘exact’ machine data highlights the limitations arising from clustering effects. The pair radial distribution function in the low density regime is not well-represented by using an expansion about the hard sphere distribution function and the attractive potential's second and third virial coefficients, used in Modified Enskog Theory (MET).

Small-angle scattering from colloidal dispersions

A theory of the small-q behaviour of the static structure factor S(q) for monodisperse colloidal dispersions is presented in which the long-range forces between the colloid particles are treated in the random phase approximation. By using a hard-sphere reference fluid, explicit results for the coefficients to (q4) in a small-q expansion of c(q), the Fourier transform of the direct correlation function, are obtained.

Blip function calculation of the radial distribution functions for a binary liquid mixture

It is shown that the use of the high temperature approximation to blip function theory together with a two-component hard sphere reference fluid leads to reasonably accurate predictions for the radial distribution functions in a moderate temperature, moderate density, binary Lennard-Jones mixture.

Perturbation theory for a mixture of hard spheres and square-well molecules

Perturbation theory is applied to a mixture of hard spheres and square-well molecules. If the higher order perturbation terms are adequately accounted for, excellent agreement with machine simulation results is obtained. In particular, good values for the heat capacity are obtained. This is in contradiction with Alder's conjecture that the hard-sphere reference fluid is unable to deal with concentration fluctuations. In addition, an approximation to the higher order terms in perturbation theory which appears useful at low densities is suggested. Finally, the theory of random mixtures (RM) and the van der Waals (vdW) theory of mixtures, which are identical for this system, are considered. The estimates of the higher order perturbation terms given by the RM and vdW theories are seriously in error for this system.

First-Order Perturbation Theory for the Distribution Functions of a Dense Fluid

A first-order expansion of the Percus-Yevick functional is derived for the properties of a real fluid as a perturbation about a hard-sphere reference fluid. The resulting expression is an integral equa ion relating the real fluid distribution functions and intermolecular potential to those of the reference fluid. Use of a realistic approximation in calculating the integral in the Ornstein-Zernike equation leads to real-fluid distribution functions that are in closer agreement with molecular dynamics results than those of the usual first-order Percus-Yevick equation.