Morphological stability analysis of directional solidification in thin samples with lateral heat transfer

Recent experimental evidence indicates that the onset of instability of a planar interface during the directional solidification of a binary system depends on the thickness of the sample. We have investigated the effect that finite lateral heat transfer between a thin sample and the containing cell can have on the morphological stability problem. We have found that such an effect contributes significantly to the stability of the planar interface when D_T≈√d/h'. D_T is the thermal diffusivity, υ the pulling speed and d the thickness of the sample. h' is a heat transfer coefficient in a phenomenological law according to which the heat flux between the sample and the cell is given by the product of h' with the temperature difference between the sample and the cell. Since it appears that D_T/υ≫√d/h' in the above mentioned experiment, there are probably no appreciable effects arising from a finite lateral heat transfer in the experimentally relevant range of velocities and thicknesses, given a reasonable estimate of the heat transfer coefficient h'. However, h' might be smaller than we have estimated and it would be interesting to determine it experimentally in order to conclusively rule out finite lateral heat transfer alone as the origin for the thickness dependence of the critical velocity observed in the experiments.