Directional solidification in the presence of an impurity may be described by a set of impurity-concentration and thermal-diffusion equations coupled at a free boundary. Small deviations of the interface from planarity can be described by a single fourth-order equation. This equation is derived by a long-wavelength, small-amplitude expansion in the limit of a small distribution coefficient. We present an alternative asymptotic approach that isolates and preserves the crucially important nonlinearities in their original form, and thus preserves the proper behavior at large amplitudes during pattern formation. The resulting evolution equation is in better agreement with the physical phenomena of front destabilization and droplet creation than are previously presented models. The formation of different solidification patterns is numerically elucidated.

• Received 19 June 1987

DOI:https://doi.org/10.1103/PhysRevB.37.7603