At equilibrium, the shape of a strongly anisotropic crystal exhibits corners when for some orientations the surface stiffness is negative. In the sharp-interface problem, the surface free energy is traditionally augmented with a curvature-dependent term in order to round the corners and regularize the dynamic equations that describe the motion of such interfaces. In this paper, we adopt a diffuse interface description and present a phase-field model for strongly anisotropic crystals that is regularized using an approximation of the Willmore energy. The Allen–Cahn equation is employed to model kinetically controlled crystal growth. Using the method of matched asymptotic expansions, it is shown that the model converges to the sharp-interface theory proposed by Herring. Then, the stress tensor is used to derive the force acting on the diffuse interface and to examine the properties of a corner at equilibrium. Finally, the coarsening dynamics of the faceting instability during growth is investigated. Phase-field simulations reveal the existence of a parabolic regime, with the mean facet length evolving in t, with t the time, as predicted by the sharp-interface theory. A specific coarsening mechanism is observed: a hill disappears as the two neighbouring valleys merge.

中文翻译：

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动力学控制晶体生长中刻面的正则化相场模型

在平衡状态下，当某些取向的表面刚度为负值时，强各向异性晶体的形状会显示角。在尖锐界面问题中，表面自由能传统上会增加一个曲率相关项，以便圆角和规范描述此类界面运动的动态方程。在本文中，我们采用漫反射界面描述并提出了强各向异性晶体的相场模型，该模型使用 Willmore 能量的近似值进行正则化。Allen-Cahn 方程用于模拟动力学控制的晶体生长。使用匹配渐近展开的方法，表明该模型收敛于Herring提出的尖锐界面理论。然后，应力张量用于推导出作用在扩散界面上的力并检查平衡角的特性。最后，研究了生长过程中刻面不稳定性的粗化动态。相场模拟揭示了抛物线状态的存在，平均小平面长度以 t 为单位，以 t 为单位，正如锐界面理论所预测的那样。观察到一种特定的粗化机制：当两个相邻的山谷合并时，一座小山消失了。