A free-energy density functional for the inhomogeneous hard-sphere Quid mixture is derived from general basic considerations and yields explicit analytic expressions for the high-order direct correlation functions of the uniform Auid. 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 scaledparticle 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. PACS numbers: 61.20. -p, 05.20. -y Density-functional theories of inhomogeneous classical fluids' and of freezing have received increasing attention in recent years. Expansions of inhomogeneous fluidor crystal properties around a uniform reference state are generated by the m-particle direct correlation functions (DCF's), c,whi ch are functional density derivatives of the excess (relative to the ideal gas) Helmholtz free energy F,". The expansions are truncated after second order because very little is known about highorder DCF's, c " for n~3. Model free-energy functionals, formally equivalent to an infinite-order expansion, always employ weighted (coarse-grained) densities which are tailored to reproduce available properties of the homogeneous fluid, notably c (r) and sum rules. Applications to hard spheres (the reference system for classical fluids) almost invariably use the analytic solution of the Percus-Yevick (PY) equation' ' for c Only recently significant simulation results for the triplet function c~( r, r') of the uniform one-component softsphere fluid near freezing were obtained which compare favorably with a factorized ad hock form and with the weighted density approximation (WDA) for hard spheres,both gauged by a given c . The desirable more comprehensive approach to inhomogeneous fluids should be able to derive the uniform fluid properties. In this Letter, I make a step in that direction, and find that (a) general basic constraints on the nature of the freeenergy functional fully dictate its complete form, to yield a comprehensive analytic description of the hard-sphere fluid mixture that contains both the PY and scaledparticle theories in a unified way. (b) Applications of this derived free energy raise questions about the convergence of the density-functional theory of freezing.The lowest-order graph in the diagrammatic (virial) expansion of the excess (relative to the ideal gas) chemical potential corresponds to pair exclusion. For the inhomogeneous fluid mixture of hard spheres characterized by the set of one-particle densities jp;(r )j, it reads &F.,(fp;(r)1)i'k g T g"d r'p~(r)8(~r -r'~-(R;+R~)) =gp~(r), bp(r) p-0 jw ith the unit step function, 8(x )0)=0, 8(x~0) = I, providing an obvious possible weight function for obtaining the coarse-grained densities p~(r). Instead, on the basis of previous work on uniform fluids, I seek a description in terms of characteristic functions...
A geometrically based fundamental-measure free-energy density functional unified the scaled-particle and Percus-Yevick theories for the hard-sphere fluid mixture. It has been successfully applied to the description of simple ͑''atomic''͒ three-dimensional ͑3D͒ fluids in the bulk and in slitlike pores, and has been extended to molecular fluids. However, this functional was unsuitable for fluids in narrow cylindrical pores, and was inadequate for describing the solid. In this work we analyze the reason for these deficiencies, and show that, in fact, the fundamental-measure geometrically based theory provides a free-energy functional for 3D hard spheres with the correct properties of dimensional crossover and freezing. After a simple modification of the functional, as we propose, it retains all the favorable Dϭ3 properties of the original functional, yet gives reliable results even for situations of extreme confinements that reduce the effective dimensionality D drastically. The modified functional is accurate for hard spheres between narrow plates (Dϭ2), and inside narrow cylindrical pores (Dϭ1), and it gives the exact excess free energy in the Dϭ0 limit ͑a cavity that cannot hold more than one particle͒. It predicts the ͑vanishingly small͒ vacancy concentration of the solid, provides the fcc hard-sphere solid equation of state from closest packing to melting, and predicts the hard-sphere fluid-solid transition, all in excellent agreement with the simulations.
A semi-empirical `universal' corresponding-states relationship, for the dimensionless transport coefficients of dense fluids as functions of the reduced configurational entropy, was proposed more than twenty years ago and established by many simulations. Here it is shown analytically, by appealing to Enskog's original results for the inverse-power potentials, that the quasi-universal entropy scaling can be extended also to dilute gases. The analytic form and the possible origin for the entropy scaling for dense fluids are discussed in view of this unexpected result. On the basis of the entropy scaling we predict a minimum in the shear viscosity as a function of temperature for all soft inverse-power potentials, in quantitative agreement with the available simulations.
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.
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