The freezing transition of a hard sphere fluid subject to the Percus-Yevick approximation

A classical density functional theory is applied to the calculation of the fluid-solid transition for hard spheres, using the Percus-Yevick (PY) direct correlation function. Three algebraic conditions are established for the coexistence densities and the Lindemann parameter. The terms neglected in the present analysis are small and the present theory, in our eyes, is essentially an exact solution given the PY approximation. No fluid-solid transition is found for the bcc lattice, whereas for expanded fcc lattices, the agreement with previous density functional theory-based theories is good.     

A fundamental measure theory for the sticky hard sphere fluid

We construct a density functional theory (DFT) for the sticky hard sphere (SHS) fluid which, like Rosenfeld's fundamental measure theory (FMT) for the hard sphere fluid [Y. Rosenfeld, Phys. Rev. Lett. 63, 980 (1989)], is based on a set of weighted densities and an exact result from scaled particle theory (SPT). It is demonstrated that the excess free energy density of the inhomogeneous SHS fluid Φ(SHS) is uniquely defined when (a) it is solely a function of the weighted densities from Kierlik and Rosinberg's version of FMT [E. Kierlik and M. L. Rosinberg, Phys. Rev. A 42, 3382 (1990)], (b) it satisfies the SPT differential equation, and (c) it yields any given direct correlation function (DCF) from the class of generalized Percus-Yevick closures introduced by Gazzillo and Giacometti [J. Chem. Phys. 120, 4742 (2004)]. The resulting DFT is shown to be in very good agreement with simulation data. In particular, this FMT yields the correct contact value of the density profiles with no adjustable parameters. Rather than requiring higher order DCFs, such as perturbative DFTs, our SHS FMT produces them. Interestingly, although equivalent to Kierlik and Rosinberg's FMT in the case of hard spheres, the set of weighted densities used for Rosenfeld's original FMT is insufficient for constructing a DFT which yields the SHS DCF.     

Phase behavior of weakly polydisperse sticky hard spheres: perturbation theory for the Percus-Yevick solution

We study the effects of size polydispersity on the gas-liquid phase behavior of mixtures of sticky hard spheres. To achieve this, the system of coupled quadratic equations for the contact values of the partial cavity functions of the Percus-Yevick solution [R. J. Baxter, J. Chem. Phys. 49, 2770 (1968)] is solved within a perturbation expansion in the polydispersity, i.e., the normalized width of the size distribution. This allows us to make predictions for various thermodynamic quantities which can be tested against numerical simulations and experiments. In particular, we determine the leading order effects of size polydispersity on the cloud curve delimiting the region of two-phase coexistence and on the associated shadow curve; we also study the extent of size fractionation between the coexisting phases. Different choices for the size dependence of the adhesion strengths are examined carefully; the Asakura-Oosawa model [J. Chem. Phys. 22, 1255 (1954)] of a mixture of polydisperse colloids and small polymers is studied as a specific example.     

Kirkwood factor for dipolar hard sphere fluids. hindered rotation model

Possibilities of using various physicochemical models of polar liquids for describing the dielectric properties of model systems (one-, two-, or threedimensional dipolar hard spheres) are analyzed. Simple analytical formulas for the Kirkwood factor of the model systems are derived using the generalized hindered rotation model in a nearest neighbor interaction approximation. In the onedimensional case, an exact formula is obtained. For two- and three-dimensional spheres, the formulas adequately reproduce the available data of computer simulations over a wide range of thermodynamic parameters. In the lowtemperature limit (highly polar fluid), the expressions for the Kirkwood factor coincide with those in Pople’s bent hydrogen bond model. The associated equilibrium model also adequately describes the available experimental data in this limit, but leads to nonphysical results at high temperatures. The worst results are obtained in Wertheim’s mediumsphere approximation. Translated fromZhumal Struktumoi Khimii, Vol. 39, No. 5, pp. 843–850, September–October, 1998.     

Solution of the Percus-Yevick equation for hard disks

The authors solve the Percus-Yevick equation in two dimensions by reducing it to a set of simple integral equations. They numerically obtain both the pair correlation function and the equation of state for a hard disk fluid and find good agreement with available Monte Carlo results. The present method of resolution may be generalized to any even dimension.     

Structure of nonuniform hard sphere fluids from shifted linear truncations of functional expansions

Percus showed that approximate theories for the structure of nonuniform hard sphere fluids can be generated by linear truncations of functional expansions of the nonuniform density rho(r) about that of an appropriately chosen uniform system. We consider the most general such truncation, which we refer to as the shifted linear response (SLR) equation, where the density response rho(r) to an external field phi(r) is expanded to linear order at each r about a different uniform system with a locally shifted chemical potential. Special cases include the Percus-Yevick (PY) approximation for nonuniform fluids, with no shift of the chemical potential, and the hydrostatic linear response (HLR) equation, where the chemical potential is shifted by the local value of phi(r). The HLR equation gives exact results for very slowly varying phi(r) and reduces to the PY approximation for hard core phi(r), where generally accurate results are found. We show that a truncated expansion about the bulk density (the PY approximation) also gives exact results for localized fields that are nonzero only in a "tiny" region whose volume V(phi) can accommodate at most one particle. The SLR equation can also exactly describe a limit where the fluid is confined by hard walls to a very narrow slit. This limit can be related to the localized field limit by a simple shift of the chemical potential, leading to an expansion about the ideal gas. We then try to develop a systematic way of choosing an optimal local shift in the SLR equation for general phi(r) by requiring that the predicted rho(r) is insensitive to small variations about the appropriate local shift, a property that the exact expansion to all orders would obey. The resulting insensitivity criterion (IC) gives a theory that reduces to the HLR equation for slowly varying phi(r) and is much more accurate than HLR both for very narrow slits, where the IC agrees with exact results, and for fields confined to "tiny" regions, where the IC gives very accurate (but not exact) results. However, the IC is significantly less accurate than the PY and HLR equations for single hard core fields. Only a small change in the predicted reference density is needed to correct this remaining limit.     

Geometry and energy effects on the structure of hard sphere fluids

The density profile of molecules is an interesting property in the theoretical study of inhomogeneous fluids. In this work, the density profile of a hard sphere fluid around various hard and soft spheres is studied using a well-established version of the density functional theory called “modified fundamental measure theory”. The results obtained from this study show that an increase in the size of the central molecule leads to both an enhanced density profile at the contact point and an amplified layering structure. Similar effects can be observed when attraction between original and bulk molecules increases. Another finding of the present study is that increasing the size of the central molecule has a quasi-attractive role of an entropic origin called ‘depletion potential’. Since the molecule is not very large, the depletion potential can be mapped in a hard core with an attractive Yukawa tail. Increasing molecule softness, however, has an opposite effect and causes the height of the first peak of the density profile to diminish and the inhomogeneity of the structure to lighten.     

Computer simulation study of the closure relations in hard sphere fluids

We study, using Monte Carlo simulations, the cavity and the bridge functions of various hard sphere fluids: one component system, equimolar additive, and nonadditive binary mixtures. In particular, we numerically check the assumption of local dependency of the bridge functions from the indirect correlation functions, on which most of the existing integral equation theories hinge. We find that this condition can be violated either in the region around the first and second neighbors shell, or inside the hard core, for the systems here considered. The violations manifest themselves clearly in the so-called Duh-Haymet plots of the bridge functions versus the indirect correlation functions and become amplified as the coupling of the system increases.     

A practical integral equation for the structure and thermodynamics of hard sphere Coulomb fluids

A closure for the Ornstein-Zernike equation is presented, applicable for fluids of charged, hard spheres. From an exact, but intractable closure, we derive the radial distribution function of nonlinearized Debye-Hückel theory by subsequent approximations, and use the information to formulate a new closure by an extension of the mean spherical approximation. The radial distribution functions of the new closure, coined Debye-Hückel-extended mean spherical approximation, are in excellent agreement with those resulting from the hyper-netted chain approximation and molecular dynamics simulations, in the regime where the latter are applicable, except for moderately dilute systems at low temperatures where the structure agrees at most qualitatively. The method is numerically more efficient, and more important, convergent in the entire temperature-density plane. We demonstrate that the method is accurate under many conditions for the determination of the structural and thermodynamic properties of homogeneous, symmetric hard-sphere Coulomb systems, and estimate it to be a valuable basis for the formulation of density functional theories for inhomogeneous or highly asymmetric systems.     

The complete T-->V,R energy conversion in three-body collisions within the hard sphere model

It is shown that in hard sphere (impulsive) collisions of atoms with diatomic molecules, complete conversion of the collision energy into the internal energy of the diatomic partner is possible for any number of impacts between the elastic balls representing the particles. The corresponding collision geometries and relations between the masses of the particles are described in detail.     

The electric double layer capacitance for a hard sphere ion-dipole electrolyte in the mean field approximation

The electric double layer capacitance for a hard sphere ion-dipole system in the neighbourhood of a plane charged wall is calculated in the mean field approximation. To order ka the capacitance predicts the same structural features as the MSA capacitance.     

An analytical approximation for the orientation-dependent excluded volume of tangent hard sphere chains of arbitrary chain length and flexibility

Onsager-like theories are commonly used to describe the phase behavior of nematic (only orientationally ordered) liquid crystals. A key ingredient in such theories is the orientation-dependent excluded volume of two molecules. Although for hard convex molecular models this is generally known in analytical form, for more realistic molecular models that incorporate intramolecular flexibility, one has to rely on approximations or on computationally expensive Monte Carlo techniques. In this work, we provide a general correlation for the excluded volume of tangent hard-sphere chains of arbitrary chain length and flexibility. The flexibility is introduced by means of the rod-coil model. The resulting correlation is of simple analytical form and accurately covers a wide range of pure component excluded volume data obtained from Monte Carlo simulations of two-chain molecules. The extension to mixtures follows naturally by applying simple combining rules for the parameters involved. The results for mixtures are also in good agreement with data from Monte Carlo simulations. We have expressed the excluded volume as a second order power series in sin?(γ), where γ is the angle between the molecular axes. Such a representation is appealing since the solution of the Onsager Helmholtz energy functional usually involves an expansion of the excluded volume in Legendre coefficients. Both for pure components and mixtures, the correlation reduces to an exact expression in the limit of completely linear chains. The expression for mixtures, as derived in this work, is thereby an exact extension of the pure component result of Williamson and Jackson [Mol. Phys. 86, 819-836 (1995)].     

Evaluating the accuracy of a density functional theory of polymer solutions with additive hard sphere diameters

We assess the accuracy of a density functional theory for athermal polymer solutions, consisting of solvent particles with a smaller radius than that of the monomers. The monomer and solvent density profiles in a slit bound by hard, flat, and inert surfaces are compared with those obtained by a Metropolis Monte Carlo simulation. At the relatively high density at which the comparison is performed, there are considerable packing effects at the walls. The density functional theory introduces a simple weight function to describe nonlocal correlations in the fluid. A recent study of surface forces in polymer solutions used a different weighting scheme to that proposed in this article, leading to less accurate results. The implications of the conclusions of that study are discussed.     

Collisions, caging, thermodynamics, and jamming in the barrier hopping theory of glassy hard sphere fluids

An ultralocal limit of the microscopic single particle barrier hopping theory of glassy dynamics is proposed which allows explicit analytic expressions for the characteristic length scales, energy scales, and nonequilibrium free energy to be derived. All properties are shown to be controlled by a single coupling constant determined by the fluid density and contact value of the radial distribution function. This parameter quantifies an effective mean square force exerted on a tagged particle due to collisions with its surroundings. The analysis suggests a conceptual basis for previous surprising findings of multiple inter-relationships between characteristics of the transient localized state, the early stages of cage escape, non-Gaussian or dynamic heterogeneity effects, and the barrier hopping process that defines the alpha relaxation event. The underlying physical picture is also relevant to fluids of nonspherical molecules and sticky colloidal suspensions. The possibility of a unified view of liquid dynamics is suggested spanning the range from dense gases to the zero mobility jammed state.     

Virial coefficients of the binary distribution function for hard sphere systems

Analytical representations have been obtained for the first time for several Mayer diagrams of virial decomposition of the binary distribution function for a system of hard spheres. The contributions of these diagrams to the binary function itself and to the pressure factor are estimated.     

Lanthanide salts solutions: representation of osmotic coefficients within the binding mean spherical approximation

Osmotic coefficients of aqueous solutions of lanthanide salts are described using the binding mean spherical approximation (BIMSA) model based on the Wertheim formalism for association. The lanthanide(III) cation and the co-ion are allowed to form a 1-1 ion pair. Hydration is taken into account by introducing concentration-dependent cation size and solution permittivity. An expression for the osmotic coefficient, derived within the BIMSA, is used to fit data for a wide variety of lanthanide pure salt aqueous solutions at 25 degrees C. A total of 38 lanthanide salts have been treated, including perchlorates, nitrates, and chlorides. For most solutions, good fits could be obtained up to high ionic strengths. The relevance of the fitted parameters has been discussed, and a comparison with literature values has been made (especially the association constants) when available.     

Analytic solution of the mean spherical approximation for a multi-component plasma

The mean spherical approximation for a mixture of charged hard spheres in a uniform neutralizing background is solved analytically. The factor correlation functions and the excess thermodynamic properties are explicitly expressed through a single parameter, which can be obtained by solving an algebraic equation.     

Thermodynamic perturbation theory of simple liquids: Monte Carlo simulation of a hard sphere system and the Helmholtz free energy of SW fluids

The Monte Carlo method is used to simulate similar sized hard sphere systems in a wide range of densities (from η 0.005 to 0.530 with a step of η = 0.005). The models are used to calculate the coefficients of the thermodynamic perturbation theory series for SW fluids up to the fourth order. The width of the attraction zone of the SW potential λ is varied from 1 to 2.5 sphere diameters. The analytical expressions approximating the obtained coefficients by polynomials with respect to the variables η and λ are determined. The absolute accuracy of the approximation is estimated to be better than ±0.001. All the necessary data for the calculation of the Helmholtz free energy of SW fluids up to the fourth-order perturbation theory are given.