Elastic crystals with a triple point

The peculiar behavior of active crystals is due to the presence of evolving phase mixtures the variety of which depends on the number of coexisting phases and the multiplicity of symmetry-related variants. According to Gibbs’ phase rule, the number of phases in a single-component crystal is maximal at a triple point in the p-T phase diagram. In the vicinity of this special point the number of metastable twinned microstructures will also be the highest—a desired effect for improving performance of smart materials. To illustrate the complexity of the energy landscape in the neighborhood of a triple point, and to produce a workable example for numerical simulations, in this paper we construct a generic Landau strain-energy function for a crystal with the coexisting tetragonal (t), orthorhombic (o), and monoclinic (m) phases. As a guideline, we utilize the experimental observations and crystallographic data on the t-o-m transformations of zirconia (ZrO₂), a major toughening agent for ceramics. After studying the kinematics of the t-o-m phase transformations, we re-evaluate the available experimental data on zirconia polymorphs, and propose a new mechanism for the technologically important t-m transition. In particular, our proposal entails the softening of a different tetragonal modulus from the one previously considered in the literature. We derive the simplest expression for the energy function for a t-o-m crystal with a triple point as the lowest-order polynomial in the relevant strain components, exhibiting the complete set of wells associated with the t-o-m phases and their symmetry-related variants. By adding the potential of a hydrostatic loading, we study the p-T phase diagram and the energy landscape of our crystal in the vicinity of the t-o-m triple point. We show that the simplest assumptions concerning the order-parameter coupling and the temperature dependence of the Landau coefficients produce a phase diagram that is in good qualitative agreement with the experimental diagram of ZrO₂.