Mathematical Formulation of Free Dendrite Growth from a Pure Melt

Abstract

We first study the problem of free dendrite growth from a pure substance. Dendrite growth from a binary mixture will be studied in Chap. 8. Typical free dendrite growth was shown in Fig. 1.6. In general, the dendrite growth, starting from a tiny seed, proceeds via a very complex dynamic process until, at a later stage of evolution, a permanent pattern is displayed. The details of pattern formation during the entire evolution will obviously be affected by many factors, in particular by the details of the initial conditions. To describe and predict the whole history of growth is very difficult and not meaningful. From a theoretical point of view the most interesting thing, which is also the objective of the present monograph, is to study the behavior of the system during the later stages of evolution, when the dendrite is fully developed, with a sufficiently long stem, so that the effects of the initial growth conditions and the situation at the root will diminish to a minimum. Thus we assume that at the stage t ≫ 1 , a free dendrite grows with a characteristic velocity U into an undercooled pure melt with the undercooling temperature T _∞ < T _M0, where T _M0 is the melting temperature of a flat interface. The characteristic velocity U may be chosen as the average velocity of the dendrite tip, which may depend on the time variable t and may finally become a constant at t = ∞.