Phase Field Modelling of Morphologies Driven by Tetragonal Interfacial Energy Anisotropy

A wide range of morphologies, such as dipyramids, rods, plates, and their truncated variants with more than one type of facet, are experimentally observed in tetragonal systems. In many cases, the origin of these morphologies is the strong anisotropy in the interfacial free energy. Phase field models that are restricted to the second-rank tensor term for the gradient energy coefficient can give rise to neither square cross sections nor such faceted and truncated morphologies. In this paper, using phase field simulations based on the implementation of a family of extended Cahn–Hilliard (ECH) models, we show that we can obtain all these experimentally observed morphologies. In addition to describing the formulation briefly, we (i) list the sufficient and necessary nonzero and independent parameters needed to describe the anisotropic interfacial free energy and (ii) enumerate the constraints on these parameters based on the demand that the interfacial free energy is positive definite. We analyze the precipitate morphologies and show that they indeed are equilibrium morphologies consistent with the Wulff construction. Our study is useful in understanding the origin of multiple facets that are formed as a result of interfacial free energy anisotropy; it can also serve as a guide for choosing parameters for obtaining specific morphologies consistent with tetragonal anisotropy.