Interfacial wave theory for dendritic structure of a growing needle crystal. I. Local instability mechanism

Abstract

A global interfacial wave theory is developed to demonstrate the essence and origin of the complicated dendritic structure of a growing needle crystal. In this paper, by using the multiple-variable-expansion method, the local dispersion relation for normal modes is derived in a paraboloidal coordinate system. The local instability mechanism of the interface is explored. Further, in a companion paper [J. J. Xu, following article, Phys. Rev. A 40, 1609 (1989)], a global instability mechanism is established. The existence and significance of a simple turning point in the solidification system is first identified. We obtain uniformly valid solutions in the vicinity of the turning point and investigate the interaction of the interfacial waves in that region. A remarkable mechanism of wave emission and reflection at the turning point is discovered.