A composition density functional theory for mixtures based upon an infinitely polydisperse reference. I. Formalism and theory

A general, statistical mechanical theory which relates the properties of mixtures of different compositions is presented. It is developed within a semigrand canonical framework, and thus the mixtures are formally described by species chemical potential differences, rather than directly by composition. The introduction of a set of n-particle composition distribution functions leads to a composition-space superposition approximation (CSSA), which forms the only approximate part of the treatment. A functional expansion of the canonical partition function in terms of the composition density is used to develop systematic corrections to the CSSA. Infinitely polydisperse mixtures [D. A. Kofke and E. D. Glandt, J. Chem. Phys. 90, 439 (1989)] are shown to be the composition-space analogs of homogeneous pure fluids, and the scaling properties of these mixtures make them ideal as a reference in the theory. The required input is the density-invariant composition of the infinitely polydisperse reference. The validity of the method is demonstrated on hard-particle fluids using accurate equations of state from the literature. Although based on a polydisperse reference, the treatment is equally applicable to discrete, i.e., conventional mixtures. In its most stringent test - the prediction of pure-fluid properties - the theory based on an infinitely polydisperse reference displays quantitative agreement with known behavior.