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ANISOTROPIC PHASE FIELD EQUATIONS OF ARBITRARY ORDER

机译：任意阶的各向异性相场方程

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摘要

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+δΣ~N-n=1{a-n cos(2nθ)+b-n sin(2nθ)} where θ is the angle with respect to a fixed axis, and δ 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 2Nth degree phase field equation.We derive the classical result (T-T-E) [s]-E=-κ{σ (θ)+ a~n (θ)} 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, σ is the surface tension and κ is the curvature. If there is only one mode in the anisotropy [i.e., the sum contains only one term: A-n cos (2nθ)] then the anisotropy can be obtained without full solutions of the equations if the surface tension is interpreted as the sharp interface limit of excess free energy obtained by the solution of the 2Nth 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.
机译：我们使用具有详细各向异性的微观相互作用哈密顿量以a-0 +δΣ〜Nn = 1 {an cos（2nθ）+ bn sin（2nθ）}的形式相互作用，导出了一组高阶相场方程。哈密​​顿量是通过复数傅里叶级数展开的，并产生一个自由能和相位场方程，在空间变量中具有任意高阶导数。对这些相场方程根据界面厚度进行形式渐近分析，以获得界面条件。通过保留至少2N度的相位场方程，可以捕获2N倍各向异性。我们推导出经典结果（TTE）[s] -E =-κ{σ（θ）+ a〜n（θ）}，其中TTE为界面处的温度与相之间的平衡温度之差，[s] -e是相之间的熵差，σ是表面张力，κ是曲率。如果各向异性中只有一种模式[即，总和仅包含一个项：cos（2nθ）]，那么如果将表面张力解释为过量的尖锐的界面极限，则可以在不完全求解方程式的情况下获得各向异性通过2N次微分方程的解获得的自由能。该技术依靠使用复数变量和组合恒等式重写导数和，并对任意阶微分方程执行形式渐近分析。

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