Pattern selection with anisotropy during directional solidification

The effects of surface-tension anisotropy on interface morphology during the directional solidification of a binary alloy are studied. The long-wave evolution equation derived by Brattkus & Davis to describe growth near the absolute stability limit is generalized to include the effects of a surface tension with cubic anisotropy. The special cases of growth in the [001], [011] and [111] directions are considered. The resulting evolution equations are derived, and amplitude equations governing roll/rectangle and roll/hexagon competition are obtained. The coefficients of the amplitude equations depend on the surface-tension anisotropy, and determine how pattern selection is influenced by the presence of geometrically preferred directions. Anisotropy leads to changes in the existence and stability criteria for each pattern, to imperfect bifurcations, and to loss of degeneracy in bifurcations.