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.

We use extended Cahn-Hilliard (ECH) equations to study faceted precipitate morphologies; specifically, we obtain four sided precipitates (in 2-D) and dodecahedron (in 3-D) in a system with cubic anisotropy, and, six-sided precipitates (in 2-D, in the basal plane), hexagonal dipyramids and hexagonal prisms (in 3-D) in systems with hexagonal anisotropy. Our listing of these ECH equations is fairly comprehensive and complete (upto sixth rank tensor terms of the Taylor expansion of the free energy). We also show how the parameters that enter the model are to be obtained if either the interfacial energy anisotropy or the equilibrium morphology of the precipitate is known. IntroductionProperties of crystalline materials are anisotropic due to the anistropy of the underlying continuum. In particular, the interfacial energy in crystalline systems is anisotropic and can have a strong influence on the formation and evolution of microstructures. Phase field models, which are best suited for the study of the formation and evolution of microstructures (see [1,2,3,4,5] for some recent reviews), have been used quite successfully to study the effect of interfacial energy anisotropy on microstructures and their evolution: see ] for some representative examples.The phase field models that incorporate interfacial energy anisotropy do so in one of two ways: (A) replace the gradient energy coefficient by an anisotropic function or polynomial, and (B) include higher order terms in the Taylor series expansion of the free energy functional. In this paper, we take the second approach -which is an extension of the original Cahn-Hilliard equation along the lines shown by Abinandanan and Haider [6] (hereafter referred to as ECHAH) and Torabi and Lowengrub [7] (hereafter referred to as ECHTL); this approach is argued to be advantageous in terms of the levels of anisotropy one can incorporate [6] and the kinetics remaining diffusionlimited [13]; in addition, the first method requires regularisation [22] for large anisotropies while ours does not.During solid-solid phase transformations interfacial energy anisotropy is known to lead to faceted precipitates: see for example, PbS precipitates in Na-doped PbTe system [31], Al 3 Sc precipitates in Al(Sc) alloys [32], Pt precipitates in sapphire [33], several metallic precipitates in internally reduced oxides [34], and Al 3 Ti precipitates in Al [35]. Our objective in this paper is to obtain, using phase field modelling, faceted precipitates in cubic systems that distinguish between 100 and 111 (for which one has to necessarily include fourth rank tensor terms [6]) and those that prefer 110 over both 100 and 111 (for which one has to necessarily include sixth rank tensor terms [8]); additionally, the sixth rank tensor terms can also be used to study

Grand-potential based multiphase-field model is extended to include surface diffusion. Diffusion is elevated in the interface through a scalar degenerate term. In contrast to the classical Cahn-Hilliard-based formulations, the present model circumvents the related difficulties in restricting diffusion solely to the interface by combining two second-order equations, an Allen-Cahn-type equation for the phase field supplemented with an obstacletype potential and a conservative diffusion equation for the chemical potential or composition evolution. The sharp interface limiting behavior of the model is deduced by means of asymptotic analysis. A combination of surface diffusion and finite attachment kinetics is retrieved as the governing law. Infinite attachment kinetics can be achieved through a minor modification of the model, and with a slight change in the interpretation, the same model handles the cases of pure substances and alloys. Relations between model parameters and physical properties are obtained which allow one to quantitatively interpret simulation results. An extensive study of thermal grooving is conducted to validate the model based on existing theories. The results show good agreement with the theoretical sharp-interface solutions. The obviation of fourth-order derivatives and the usage of the obstacle potential make the model computationally cost-effective.