The square gradient model in a two-phase mixture I. \Equilibrium properties

In order to describe a nonuniform equilibrium mixture with an interface between two coexisting phases it is necessary to consider contributions to the Helmholtz energy which depend on the gradients of for instance the density. Van der Waals \cite{vdW/sg, vdW/translation} was the first to introduce such a term, which is very important in the interfacial region for a one-component system. Cahn & Hilliard \cite{cahnhilliard/fens/I} extended this analysis to a binary mixture by introducing gradient terms of the mol fraction. We give an systematic extension of the gradient theory to three-dimensional multi-component systems.