Equilibrium Multi-precipitate Configurations

In this paper, we formulate a phase-field model for the computation of equilibrium configurations of multiple phases that arise out of a solid-state precipitate reaction, in the presence of coherency stresses. Here, we utilize the phase-field framework to minimize the sum of the elastic and the interfacial energies for a given volume of the precipitates using an extension of the volume-preserved Allen-Cahn algorithm (Garcke et al. in Math Models Methods Appl Sci 18(08):1347-1381, 2008; Bhadak et al. in Metall Mater Trans A 49A(11):5705-5726, 2018). Using this technique, we investigate the precipitate organization for three solid-state reactions. The first is the classical two-phase precipitate reaction that leads to the formation of core-shell microstructures, where we clarify the influence of elasticity on the formation of such clusters. Following this, we investigate two symmetry-breaking transitions (cubic to tetragonal) and (hexagonal to orthorhombic), that lead to the formation of multi-variant clusters where we study the organization of the precipitates as a function of the elastic properties of the precipitate and the matrix.