Phase Field Modelling of Morphologies Driven by Tetragonal Interfacial Energy Anisotropy

Abstract

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.

Approaches to modelling the shape of nanocrystals

“…Yet it is no surprise that kinetic and thermodynamic arguments remain muddled in the current literature. Indeed, a significant number of recent papers incorrectly described kinetically controlled nanocrystal shapes as thermodynamic ones [ 65 – 67 ] with others explicitly stating that Wulff constructions can only be used to determine thermodynamic equilibrium [ 68 , 69 ]. The propagation of the incorrect idea of “one Wulff construction” has not exclusively led to confusion regarding kinetics vs thermodynamics: the Winterbottom construction has been referred to as “the Wulff construction” [ 70 ] or “modified Wulff construction” [ 71 ], and the kinetic Wulff construction has been called a “modified Wulff construction” [ 72 ].…”

Growth of Abraded Crystals Tracked in Three Dimensions

“…As an alternative approach, phase-field modeling has been used to estimate the morphology of dissolving or growing 3D crystalline systems in silico. The underlying concept is to model energetic or kinetic anisotropies of interfaces using high-order Taylor polynomials .…”