Pattern formation in directional solidification

The one-sided model for a nearly planar solidification front advancing at steady velocity is studied. The model neglects impurity diffusion in the solid; the interface is stabilized by the imposition of a thermal gradient. The front, located at $z_s=z_s\mathrm{}(x,y)\mathrm{}$, is described by the coefficients ${\epsilon }_{\overrightarrow{\mathrm{k}}}$ of the Fourier expansion for $z_s$, and equations of motion ${\epsilon }_{\overrightarrow{\mathrm{k}}}=f_{\overrightarrow{\mathrm{k}}}\left(\right\{\epsilon \left\}\right)$ are derived in the approximation where the velocity $\overrightarrow{\mathrm{v}}$ of the interface is small. The functions $f_{\overrightarrow{\mathrm{k}}}$ are expressed as infinite polynomials in the $\left\{\epsilon \right\}$. Stationary solutions $f_{\overrightarrow{\mathrm{k}}}=0\mathrm{}$ are sought with the help of a consistent truncation scheme. Truncations which involve keeping terms of up to fifth order in ${\epsilon }_{\overrightarrow{\mathrm{k}}}$ are used. Stationary profiles for both one- and two-dimensional fronts are obtained numerically, whose features are in reasonable agreement with experimental observations. In particular, the one-dimensional solutions exhibit a relatively well-developed cellular structure; this is in contrast with what happens in more conventional analyses, where higher-order nonlinearities are not accounted for. The two-dimensional stationary interfaces are of various types, displaying twofold or sixfold symmetry. It appears to be the first time that calculations of two-dimensional structures are reported.