A simple model for coherent equilibrium

Abstract

The recent discoveries by Williams of unusual phase diagram features for coherent phase equilibrium are explored in detail for a model that leads to a simple formulation involving only the minimization of a polynomial free energy function that includes an elastic energy term. The elastic energy term in general makes the use of the common tangent construction invalid. It requires one to distinguish between the phase diagram concepts of “field boundaries” where new phases make their appearance and “tie line ends” marking compositions of coexisting phases. The elastic energy term invalidates the phase rule and the theorem that tie line ends lie on field boundaries, and several other theorems about metastable equilibrium. Two-phase coexistence can terminate at a special point, which we call the Williams point, where a ratio of elastic to chemical energy reaches a definite value. Beyond the Williams point a first-order coherent phase transition occurs without phase coexistence or composition difference, and the T 0 curve becomes an equilibrium feature of the coherent phase diagram.