Anisotropic phase field equations of arbitrary order

Abstract

We derive a set of higher order phase field equations using a microscopic interaction Hamiltonian with detailed anisotropy in the interactions of the form $a_{0}+\delta\sum_{n=1}^{N}{a_{n}\cos( 2n\theta) + b_{n}\sin( 2n\theta) }$ where $\theta$ is the angle with respect to a fixed axis, and $\delta$ is a parameter. The Hamiltonian is expanded using complex Fourier series, and leads to a free energy and phase field equation with arbitrarily high order derivatives in the spatial variable. Formal asymptotic analysis is performed on these phase field equation in terms of the interface thickness in order to obtain the interfacial conditions. One can capture $2N$-fold anisotropy by retaining at least $2N^{th}$ degree phase field equation. We derive, in the limit of small $\delta,$ the classical result $( T-T_{E} ) [s]_{E}=-\kappa {\sigma( \theta ) + \sigma^{''}( \theta) }$ where $T-T_{E}$ is the difference between the temperature at the interface and the equilibrium temperature between phases, $[s]_{E}$ is the entropy difference between phases, $\sigma$ is the surface tension and $\kappa$ is the curvature. If there is only one mode in the anisotropy [i.e., the sum contains only one term: $A_{n}\cos( 2n\theta) $] then this identity is exact (valid for any magnitude of $\delta$) if the surface tension is interpreted as the sharp interface limit of excess free energy obtained by the solution of the $2N^{th}$ degree differential equation. The techniques rely on rewriting the sums of derivatives using complex variables and combinatorial identities, and performing formal asymptotic analyses for differential equations of arbitrary order.