On the incorporation of cubic and hexagonal interfacial energy anisotropy in phase field models using higher order tensor terms

Abstract

In this paper, we show how to incorporate cubic and hexagonal anisotropies in interfacial energies in phase field models; this incorporation is achieved by including upto sixth rank tensor terms in the free energy expansion, assuming that the free energy is only a function of coarse grained composition, its gradient, curvature and aberration. We derive the number of non-zero and independent components of these tensors. Further, by demanding that the resultant interfacial energy is positive definite for inclusion of each of the tensor terms individually, we identify the constraints imposed on the independent components of these tensors. The existing results in the invariant group theory literature can be used to simplify the process of construction of some (but not all) of the higher order tensors. Finally, we derive the relevant phase field evolution equations.

Theoretical and computational comparison of models for dislocation dissociation and stacking fault/core formation in fcc crystals

“…Further, attention is restricted here for simplicity to isotropic gradient 1 energy. 1 As discussed recently by Nani and Gururajan (2014), in general, an explicit dependence of ψ on second-order ∇∇φ and third-order ∇∇∇φ gradients, and the corresponding fourth-and sixorder tensor moduli, are required for cubic or hexagonal (i.e., anisotropic) gradient energy. For isotropic systems as well as symmetries such as tetragonal, second-rank tensor gradient moduli suffice.…”

Simulation of Intermetallic Solidification Using Phase-Field Techniques

“…However, although minimisation of the complex free energies associated with sublattice phases is straightforward the solute trapping problems arising can be severe. For solidification of AlNi, with c = 0.3 (30% Ni) we measured the extent of solute trapping using the metric given in Equation (9):…”

“…An alternative approach which has shown considerable promise in the simulation of faceted morphologies is the so called Extended Cahn-Hilliard Model (ECHM), in which, rather than introducing the crystallographic anisotropy via an orientation depended coefficient, , to the gradient energy, higher order tensorial gradient energy terms are introduced [7]. The approach has a number of advantages, including that the stationary crystal shape does not exhibit sharp corners [8,9] even when the interfacial energy is high enough for the corresponding Wulff shape to do so. However, here we have adopted a pragmatic view and used a regularised conventional phase-field model, due to the extensive literature on extending such models to multi-phase systems.…”

Phase field modelling of precipitate morphologies in systems with tetragonal interfacial free energy anisotropy

“…In this paper, we briefly describe the ECH model [24,25,30,31] to study the morphological evolution of precipitates in systems with tetragonal interfacial free energy anisotropy; the detailed formulation can be found elsewhere [26]. Our description is based on a scalar, compositional order parameter (c); however, the extension to nonconserved order parameters and to combinations of conserved and non-conserved order parameters is straightforward.…”

“…Our aim in this paper is to study the morphology of precipitates in systems with tetragonal interfacial free energy anisotropy -using phase field models. Phase field models are ideal for the study of morphology of precipitates and crystallites; recently, we have used the Extended Cahn-Hilliard (ECH) model to study the precipitate morphologies in systems with cubic and hexagonal interfacial free energy anisotropy [24][25][26]. Some aspects of the tetragonal symmetry (distinction between c and a,b axes) can be introduced using the classical Cahn-Hilliard equation with second rank gradient free energy coefficient; and ECH models are not necessary [27][28][29].…”