Nonlinear evolution of directional solidification patterns

We have studied directional solidification theoretically for high velocities, where a time-dependent description is essential. The model equations can be simplified into an asymptotically valid strongly nonlinear equation. This quasilocal approximation preserves the essential dynamic features of the system. Analyzing the dynamic evolution of an isotrope-nematic interface, we find, besides the usual parity-breaking solution (PB), a vacillating-breathing (VB) mode, which is associated with spatial period-doubling. Both the PB and VB modes can be understood analytically, the former as the consequence of a $q$–$2q$ interaction, the latter on the basis of a perturbative approach. As the relevant system parameter, a renormalized thermal gradient, is decreased, the VB mode goes unstable with respect to parity breaking and acquires a lateral drift velocity. The interface motion is then quasiperiodic. Further decrease of the thermal gradient drives the interface into a chaotic state. We suggest that the quasiperiodicity scenario is generic for systems in which both an oscillatory and a parity-breaking instability exist. This expectation is supported by a study of amplitude equations for the same system, in which only the two most important modes have been retained.