Thermodynamic perturbation theory for polydisperse colloidal suspensions using orthogonal polynomial expansions

We present a method for calculating the thermodynamic and structural properties of a polydisperse liquid by means of a thermodynamic perturbation theory: the optimized random phase approximation (ORPA). The approach is an extension of a method proposed recently by one of us for an integral equation application [Phys. Rev. E 54, 4411 (1996)]. The method is based on expansions of all $\sigma$-dependent functions in the orthogonal polynomials $p_i\left(\sigma \right)$ associated with the weight function $f_{\Sigma }\left(\sigma \right)$, where $\sigma$ is a random variable (in our case the size of the particles) with distribution $f_{\Sigma }\left(\sigma \right)$. As in the one-component or general $\mathcal{N}$-component case, one can show that the solution of the ORPA is equivalent to the minimization of a suitably chosen functional with respect to variations of the direct correlation functions. To illustrate the method, we study a polydisperse system of square-well particles; extension to other hard-core or soft-core systems is straightforward.