Pattern selection in two-dimensional dendritic growth

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Abstract
We present an analytical treatment of the problem of pattern selection in a fully non-local symmetrical model of dendritic crystal growth. Simplifications of mathematical equations are based on the assumption that anisotropies of surface energy and kinetic effects are small. Selection rules for growth velocity and instability increments are derived at arbitrary Peclet numbers. For a dendrite growing in a channel, a double-valued velocity versus undercooling dependence is obtained. The upper branch of the solution is stable and changes into a free dendrite with increased channel width. Interplay between surface energy and kinetic effects results in morphological transition from surface energy dendrite to dense branching morphology and then to kinetic dendrite. In the framework of the boundary-layer model it is shown that at deep undercooling parabolic dendrite turns into angular dendrite and then into planar front.