On the incorporation of cubic and hexagonal interfacial energy   anisotropy in phase field models using higher order tensor terms
  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. 
E. S. Nani and M. P. Gururajan
cond-mat.mtrl-sci math-ph math.MP