Directional solidification at high speed. I. Secondary instabilities

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Abstract
We present an extensive analytical and numerical analysis of secondary instabilities in directional solidification in the limit of high speed, which is by now accessible in real experiments. The important feature in this regime is that front dynamics are quasilocal. From symmetry and scaling arguments, we write down the general form of the nonlinear equation for the interface, an equation to which the present study pertains. In order to determine the values of the coefficients, a derivation from the fully nonlocal model was performed. We consider the general case where mass diffusion is allowed in both phases, and its special restrictions to the one-sided model (appropriate for regular materials), and the symmetric one (appropriate for liquid crystals). We first focus on the appearance of cellular structures (primary instability). In the symmetric case the structures are rather shallow, in accord with experiments. In the one-sided model, the front generically develops, in a certain region of parameter space, cusp singularities. These can be avoided by allowing a small amount of diffusion in the growing phase; the front then reaches a stationary state. Stationary states are in turn subject to instabilities (secondary instabilities). Besides the Eckhaus instability, we find parity-breaking (PB), vacillating-breathing (VB), and period-halving (PH) bifurcations, regardless of the details of the model, a fact which points to their genericity. Another line of research developed in this paper is the analytical analysis of these bifurcations.