Scaled field particle theory of the structure and the thermodynamics of isotropic hard particle fluids

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Abstract
The pair structure and the thermodynamics of isotropic fluids composed of hard particles of different shapes and concentrations are considered by a new approach which (i) unifies the diagramatic Percus–Yevick theory and the geometric scaled particle theory, and (ii) leads to analytic accurate approximations for the direct correlation functions and cavity distribution functions. The scaled-particle interpolation idea of Reiss et al. is achieved diagramatically, approximating the full expansion of the direct correlation functions by appropriately renormalized low order graphs, in which the size of the field particle (‘‘black circle’’) is scaled. It is shown that the lowest order scaled field particle approximation, involving only pair excluded volumes as variables, and pair overlap volumes as functions, contains the exact solution of the Percus–Yevick integral equation for the general hard sphere mixture in D dimensions. An exact geometric relation obeyed by the excluded volume (covolume) of two fused convex bodies and another single convex body, is used to show that the scaled field particle theory is equivalent to the scaled particle expansion of the chemical potential in terms of the fundamental measures of the hard particles. This is in contrast to the conventional scaled particle Taylor series expansion in powers of the linear scale of the solute particle. The first order scaled field particle approximation and the ‘‘fundamental measure’’ scaled particle theory become identical when the fluid mixture contains only convex particles. For a D-dimensional (D=odd) hard spheres fluid mixture our new analysis enables to obtain the complete exact solution of the Percus–Yevick integral equation for the structure directly from its compressibility equation of state. This leads to the derivation of simple, analytic, geometric approximations for the direct correlation functions and cavity distribution functions of the general isotropic hard particle fluid, conformal to those for the hard sphere fluid mixture.