Interfacial wave theory of solidification: Dendritic pattern formation and selection of growth velocity

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Abstract
This work deals with the global instability mechanism of solidification from melt. It leads to a wave theory for solving long-standing problems–the pattern formation of dendritic growth and the selection of the tip velocity. One of the most important results drawn from the present work is that the selection condition of the dendrite’s tip velocity can be found even in the absence of the anisotropy of surface tension. Two distinct sets of unstable global modes for the system are obtained: (1) the global trapped wave (GTW) modes, which describe the characteristics of waves trapped in the region between the tip point and the turning point; (2) modes that display a mechanism involving wave emission at the turning point and signal reflections between the turning point and the leading edge of the tip, abbreviated as WEASR. Uniformly valid asymptotic expansions for the GTW modes and the quantum conditions of corresponding eigenvalues are derived. The requirement that the total perturbed interfacial energy must be finite eventually rules out all the WEASR modes. The presence of the self-sustaining GTW mode in the system, however, very well explains the origin and persistence of the pattern formation in dendrite growth. A unique global neutral stable state is found, which gives the tip velocity at the later stage of growth. The present theory shows good agreement with the available experimental data for a nearly isotropic material.