Pattern selection in solidification

Abstract

Directional solidification of alloys produces a wide variety of cellular or lamellar structures which, depending upon growth conditions, may be reproducibly regular or may behave chaotically. It is not well understood how these patterns are selected and controlled or even whether there ever exist sharp selection mechanisms. A related phenomenon is the spatial propagation of a pattern into a system which has been caused to become unstable against pattern-forming deformations. This phenomenon has some features in common with the propagation of sidebranching modes in dendritic solidification. In a class of one-dimensional models, the nonlinear system can be shown to select the propagating mode in which the leading edge of the pattern is just marginally stable. This stability principle, when applicable, predicts both the speed of propagation and the geometrical characteristics of the pattern which forms behind the moving front. A boundary-layer model for fully two or three dimensional solidification problems appears to exhibit similar mathematical behavior.