Interfacial instabilities in directional solidification of dilute binary alloys: the Kuramoto-Sivashinsky equation (Q579713)

Directional solidification processes in the presence of an impurity are studied in the limit in which the dimensionless parameter \(\overline {W}=GD/VC_ 0(g+f_ c)\) is small. Here \(G\) is the imposed temperature gradient, \(D\) is the diffusion coefficient of the impurity, \(V\) is the imposed transport velocity, \(g\) is equal to minus the slope of the liquidus line, \(C_ 0\) is the impurity concentration at the liquid side of a planar interface, and \(f_ c\) is a coefficient reflecting deviations from local thermal equilibrium. The dynamics of interfacial kinetics becomes important in this limit and the phenomenological model of \textit{S. R. Coriell} and \textit{R. F. Sekerka} [J. Cryst. Growth 61, 499--508 (1983)] is used to model these processes. In this limit, the Kuramoto-Sivashinsky equation is shown to be an asymptotically valid description of the interfacial dynamics. The Kuramoto-Sivashinsky equation is known to exhibit intermittancy superimposed on a relatively stable array of cusps or wrinkles \textit{[D. M. Michelson} and \textit{G. I. Sivashinsky}, Acta Astronaut. 4, 1207--1221 (1977; Zbl 0427.76048)] and thus may give a reasonable limiting description of the solidification interface just before coherency is lost. These cusps may also be important in the initiation of dendritic growth by serving as defect points.