Toward a simple density functional theory of nonuniform solids
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I. INTRODUCTION It has become popular recently to approach the problem of phase transitions from the viewpoint of liquid state physics. An early example of this is the work of Onsager concerning the effect of particle shape on the properties of colloids.' In it he examines the isotropicnematic liquid crystal phase transition using the general method of Mayer and Mayer to evaluate the free energy of the system in terms of the second virial coefficient, generalized from the uniform fluid to include orientation by the introduction of an angle-dependent distribution function. Unable to evaluate higher-order virial coefficients, Onsager approximates the third virial coefficient in terms of the second and ends the expansion at third order. This was inspired by Boltzmann's evaluation of the third virial coefficient for hard spheres of equal diameter in which he derives the relation between the two and three body cluster integrals. In pursuit of a density functional description of the liquid-amorphous solid phase transition, and of the thermodynamic properties of the high-density solid, we have developed a simple real space free-energy density functional expression, which describes quite accurately the pressure behavior of the solid phase hard sphere system. The high-density solid pressures are in remarkable agreement with the exact compressibilities given by Salsburg and Wood,2 both for crystalline and amorphous packings. In retrospect, our present theory is very reminiscent of Onsager's theory of the isotropic-nematic liquid crystal system mentioned earlier. Our theory, at the lowest level, resembles Onsager's if one associates his orientational distribution function with our single-particle density,/^). In Sec. II we present the derivation of the free-energy functional. In Sec. Ill we consider the hard-sphere system. We conclude in Sec. IV. II. THEORY We begin by considering N spherical atoms in a volume Fat temperature T, with mean density p, interact274
J. Mater. Res. 3 (2), Mar/Apr 1988
http://journals.cambridge.org
ing via pair potential (x). We assume the existence of a single-particle distribution function, p(x), which describes the local density of particles at point x in a nonuniform system^ and leave the explicit specification of p(x) until later. In the absence of all external fields, our density functional expression for the ideal gas Helmholtz free energy for the nonuniform system is3 O
= j dxp{x){ln[p(x)] - 1 } ,
where /? is the inverse temperature scaled by Boltzmann's constant. In the presence of interactions we seek the interaction Helmholtz free energy as a single-particle density functional. The usual way to proceed is to functionally Taylor expand the interaction Helmholtz free energy of the nonuniform system about the uniform system.3 The basic point is that the direct correlation function for the nonuniform phase acts like the expansion coefficient. This object is not known, so one approximates the direct correlation function of the nonuniform phase with that of the uniform fluid. In a differen
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