A New Formulation for Dendritic Crystal Growth in two Dimensions
The objective of this paper is to study the growth of dendritic crystals in two spatial dimensions plus time. The paper makes three contributions, (i) We propose a new dynamic criterion to test physical mechanisms that might produce the velocity selection that is observed experimentally. We have not yet determined how the results of this criterion compare with those obtained by other criteria, such as microscopic solvability, (ii) The (known) equations of motion are restated in terms of orthogonal parabolic coordinates, a natural coordinate system in which to study perturbations of a parabolic (Ivantsov) interface. Among its other advantages, this formulation permits a larger class of behaviours far from the tip of the crystal than is allowed in the usual representation, (iii) On an initially parabolic interface, the analogue of the Mullins-Sekerka instability is more delicate than previously had been assumed. In particular, we find numerically that the range of unstable “wavenumbers” is bounded away from zero; i.e., sufficiently low wavenumbers are stable. Moreover, our preliminary calculations show parabolae, characterised by Peclet numbers of order 1, for which the linear instability is completely suppressed by enough surface tension. Such suppression is impossible on a flat interface.
KeywordsSurface Tension Short Time Scale Peclet Number Flat Interface Dendritic Crystal
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