Effect of surface tension anisotropy on cellular morphologies

Abstract

Morphological stability theory predicts the conditions for which a planar crystal-melt interface is unstable for directional solidification of a binary alloy at constant velocity. For conditions near the onset of instability, we carry out a three-dimensional weakly nonlinear analysis to second order in the interface deformation, taking into account the effects of latent heat generation and anisotropy of the crystal-melt surface tension. We consider the growth of a cubic crystal in the [001], [011], and [111] directions. Linear stability theory predicts that, for growth in the [011] direction (two-dimensional bands are preferred. For growth in the [011] direction (four-fold axis), weakly-nonlinear theory predicts steady-state solutions having six-fold symmetry. For growth in the [111] direction (three-fold axis), steady-state solutions with six-fold symmetry are only possible for a particular alignment of the hexagonal array with respect to the crystal axes; otherwise, the solutions only have three-fold symmetry. In the second-order theory developed here, all small-amplitude solutions are unstable. If the subcritical solution branch regains stability at larger amplitudes, the small-amplitude solutions would indicate whether this branch would correspond to stable cells or nodes. For an alloy with distribution coefficient less than unity, and with the thermal conductivity of the crystal greater than that of the melt, the theory predicts solute patterns with nodes near the instability, except possibly at very high growth velocities.