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In this talk, we demonstrate the existence of non-circular 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 and dynamical numerical simulations. 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. This is joint work with Shuwang Li, Perry Leo and Vittorio Cristini.

This is joint work with Shuwang Li, Perry Leo and Vittorio Cristini.