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 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.
Bibliographical noteFunding Information:
The authors thank the generous computing resources from the Minnesota Supercomputer Institute, the Network and Academic Computing Services at University of California at Irvine (NACS), and the hospitality of Institute for Mathematics and its Application. S. Li thanks the generosity of the Department of Biomedical Engineering at UCI. S. Li also thanks Yubao Zhen for technical discussions. J. Lowengrub and V. Cristini thank the National Science Foundation (division of mathematics) for partial support.
Copyright 2008 Elsevier B.V., All rights reserved.
- A1. Diffusion
- A1. Morphological stability
- A1. Mullins-Sekerka instability
- A1. Quasi-Newton method
- A2. Compact growth
- A2. Self-similar