Asymptotic analysis of interfacial stability with surface tension anisotropy for directional solidification of alloys

An asymptotic analysis of interfacial stability with the surface tension anisotropy is derived for directional solidification of alloys. The zero and first order solution for multi-scale and the zero order solution for single-scale are obtained. The dispersion relation adapted to all wave-length is presented by combining the short and the long wave-length solutions. It was concluded that the zero order asymptotic solution is equivalent to constitutional undercooling theory and the first order solution is consistent with M-S theory. The instability zone of control parameter enlarges with the surface tension anisotropy, especially for the high temperature gradient and high pulling velocity cases.