Formulation of phase-field energies for microstructure in complex crystal structures

The unusual properties of many multifunctional materials originate from a structural phase transformation and consequent martensitic microstructure. Phase-field models are typically used to predict the formation of microstructural patterns and subsequent evolution under applied loads. However, formulating a phase-field energy with the correct equilibrium crystal structures and that also respects the crystallographic symmetry is a formidable task in complex materials. This paper presents a simple method to construct such energy density functions for phase-field modeling. The method can handle complex equilibrium structures and crystallographic symmetry with ease. We demonstrate it on a shape memory alloy with 12 monoclinic variants. The unusual behavior of shape-memory alloys, ferroelectrics, and other multifunctional materials is driven by a structural phase transformation. Below the critical phase-transformation temperature, the crystal structure changes and this leads to multiple, symmetry-related variants. The different variants can form microstructural mixtures to satisfy applied boundary conditions. Changes in the applied loads can cause microstructural rearrangements rather than the lattice distortions that are characteristic of typical materials. This ability to rearrange microstructure leads to unusual and technologically important behavior as seen in shape-memory alloys, ferroelectrics, and other multifunctional materials [1, 2, 3, 4]. To enable design with multifunctional materials, it is important to be able to predict the microstructural patterns. Phase field models are widely used for this purpose [5, 6, 7, 8, 9]. They obtain microstructural patterns by minimizing the free energy E of a specimen Ω. In the case of shape-memory alloys, that are the focus of this paper, E consists of Winter, the surface energy of the interfaces between different variants, and Waniso, the anisotropy energy that penalizes deviation of the strain (x) from the crystallographically preferred states.