Cellular dynamics during directional solidification: Interaction of multiple cells

In the directional solidification of a binary melt, the relatively small value of the surface energy at the melt/solid interface leads to a weak dependence of the dimensionless growth rate P for the onset of morphological stability on the spatial wavelength λ of the interface shape. This weak dependence, in turn, leads to a multitude of nonlinear interactions between finite-amplitude and spatially resonant cellular structures as P is increased only slightly above the critical value for the onset of instability P=$P_c$. Steadily solidifying cells with the primary wavelength ${\lambda }_c$ predicted by linear theory exist for so small a range of P that they are almost impossible to observe experimentally. These conclusions are the result of a systematic set of steady-state and transient calculations of interface morphology for a two-dimensional model of directional solidification. Extensive calculations for samples with widths of one, two, and four times ${\lambda }_c$ show an increasing level of complexity in the nonlinear dynamics as the number of cells allowed to interact is increased. Ranges of P are observed in each sample size where more than one stable state exists. Long time-scale dynamics involving the splitting of cells by a localized morphological instability are observed in the two larger samples. Complex time-periodic dynamics with periods of hundreds of diffusion time units are observed for P less than two percent above $P_c$.