Phase field simulations of faceted growth for strong anisotropy of kinetic coefficient

Facet formation during crystal growth is simulated by using the phase field model in two dimensions. Instead of moderate anisotropy of the often-used form 1 þ d cos 4y; several functions having strong anisotropy are explored. For simplicity, the interfacial energy is assumed to be isotropic, so only the anisotropy in the kinetic coefficient is considered. This results in the formation of a nearly flat face when the anisotropy function has a narrow minimum at a certain direction, for example 45 � for four-fold symmetry. Two types of functions are studied in this paper; Type 1: q1ðy Þ¼ 1 þ d � 2dð1 � cos4yÞ n =2 n ; and Type 2: q2ðy Þ¼ 1 � d þ 2dtanhðk=jtan2yjÞ: A ‘‘facet’’ is formed at the 45 � direction for each case. This ‘‘facet’’ is not completely flat for q1; but a real facet is obtained for q2: The crystal shapes depend on the parameters d; n and k in the anisotropy functions. A wider facet is formed for larger d for both q1 and q2; whereas, larger values of n in q1 and k in q2 lead to more pronounced facets. Results obtained by using the phase field model are in good agreement with Wulff shapes for the kinetic coefficient. Finally, corner formation is simulated by using similar anisotropy functions with maxima at 45 � :

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