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Nonlinear theory of self-similar crystal growth and melting

机译：自相似晶体生长和熔化的非线性理论

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摘要

In this paper, we demonstrate the existence of noncircular shape-invariant (self-similar) growing and melting two-dimensional crystals. This work is motivated by the recent three-dimensional studies of Cristini and Lowengrub in which the existence of self-similar shapes was suggested using linear analysis (J. Crystal Growth, 240 (2002) 267) and dynamical numerical simulations (J. Crystal Growth 240 (2003) in press). Here, we develop a nonlinear theory of self-similar crystal growth and melting. Because the analysis is qualitatively independent of the number of dimensions, we focus on a perturbed two-dimensional circular crystal growing or melting in a liquid ambient. Using a spectrally accurate quasi-Newton method, we demonstrate that there exist nonlinear self-similar shapes with k-fold dominated symmetries. A critical heat flux J_k is associated with each shape. In the isotropic case, k is arbitrary and only growing solutions exist. When the surface tension is anisotropic, k is determined by the form of the anisotropy and both growing and melting solutions exist. We discuss how these results can be used to control crystal morphologies during growth.
机译：在本文中，我们证明了非圆形形状不变（自相似）生长和熔化的二维晶体的存在。这项工作是受最近Cristini和Lowengrub的三维研究的启发而进行的，其中使用线性分析（J. Crystal Growth，240（2002）267）和动态数值模拟（J. Crystal Growth）提出了自相似形状的存在。 240（2003）。在这里，我们建立了自相似晶体生长和熔化的非线性理论。由于分析在质量上与维数无关，因此我们专注于在液态环境中生长或熔化的二维二维圆形晶体。使用光谱精确的拟牛顿法，我们证明存在具有以k折叠为主的对称性的非线性自相似形状。临界热通量J_k与每种形状相关联。在各向同性的情况下，k是任意的，并且仅存在增长的解。当表面张力为各向异性时，k由各向异性的形式决定，并且存在生长和熔化溶液。我们讨论了如何将这些结果用于控制生长过程中的晶体形态。

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