Phase instability and local dynamics in directional solidification

Recently a general phase equation has been derived from the boundary integral equation, and preliminary results on the Eckhaus instability were given [K. Brattkus and C. Misbah, Phys. Rev. Lett. 64, 1935 (1990)]. The first focus of the present study is devoted to an extensive analysis of both the derivation of the phase equation and the computation of the Eckhaus boundaries from the low-velocity regime until the planar-front restabilization. We pay a special attention to the experiments on liquid crystals [J. M. Flesselles, A. J. Simon, and A. J. Libchaber, Adv. Phys. 40, 1 (1991)]. The special shape of the Eckhaus boundaries in the present situation provides a simple hint for experimental investigations. The second line of this paper is motivated by a strong wish to go further towards the understanding of the diverse variety of dynamical manifestations observed in experiments, such as oscillatory modes and ‘‘chaotic’’ motions. The study of these phenomena is greatly facilitated by focusing on the large-velocity regime where the front dynamics turns out to be described by a local equation. We outline here the derivation of that equation appropriate for liquid-crystal experiments. A full study on this equation, going from order to chaos, is presented elsewhere [K. Kassner, C. Misbah, and H. Müller-Krumbhaar, Phys. Rev. Lett. 67, 1551 (1991)]. Among other results presented here we show that the wavelength of the pattern λ scales with the growth velocity V and the thermal gradient as λ∼${\mathit{V}}^{\mathrm{-}1}$f(G/V). At the fold singularity for steady symmetric solutions, we find that λ∼${\mathit{V}}^{\mathrm{-}1}$, which is in agreement with experiments on liquid crystals. This scaling is to be contrasted to the one obtained in the small-Péclet-number limit λ∼${\mathit{V}}^{\mathrm{-}1/2}$f(G/V) [K. Kassner and C. Misbah, Phys. Rev. Lett. 66, 445 (1991)].