Density functional theory of freezing of classical liquids, and packings of binary hard disc mixtures

This thesis consists of two parts: the first part deals with the theory of nonuniform classical fluids and, in particular, with the problem of freezing of classical liquids into crystalline solids. We propose an extended formulation of the Modified Weighted Density Approximation (MWDA) in order to treat the problem of freezing of the classical One Component Plasma (OCP). We call the new theory Extended Modified Weighted Density Approximation, and designate it simply by EMA. The EMA retains the non-perturbative character of the MWDA, and the use of low-order functions of the uniform system in order to obtain information about its nonuniform counterpart, but is now exact up to third order in the functional expansion of the excess free energy of the nonuniform system around a uniform liquid. It succeeds to predict the freezing parameters of the OCP in very good agreement with simulation data. The EMA is subsequently applied to the well-known problem of freezing of the classical Hard Sphere liquid, giving results that are sensitive to the choice for the third-order direct correlation function of the uniform liquid. Certain models for this function lead to a significant improvement over the already quite satisfactory results of the MWDA; a different model leads to no freezing at all. As a related but separate problem in the realm of nonuniform classical fluids, we also develop a set of exact integral relations for classical uniform and nonuniform liquids and liquid mixtures, discovering a host of interesting relations and sum rules.; The second part deals with the problem of packing of binary hard disc mixtures. We construct the phase diagram of such a system, varying both the size and number ratios, based on the principle of minimization of the specific volume. We examine a large variety of candidate structures and choose the ones that optimize the packing fraction. Over ten distinct pure alloy phases are discovered at different stoichiometries, as well as large regions of the phase diagram that are covered by "lattice-gas" and "random tiling" phases. We discuss the relevance of this work to quasicrystals, and its possible extensions to three dimensions.