Nonlinear theory of self-similar crystal growth
by
S. Li, J.S. Lowengrub, P.H. Leo and V. Cristini
in
Journal of Crystal Growth, 267, pp. 703 - 713, 2004.
Category: Journal Article
Keywords: A1. Diffusion; A1. Morphological stability; A1. Mullins–Sekerka instability; A1. Quasi–Newton method; A2. Compact
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Abstract:
In this paper, we demonstrate the existence of noncircular shape-invariant (self-similar) growing and melting twodimensional 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 selfsimilar 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 Jk 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.
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