Effect of gravity on coupled convective and interfacial instabilities during directional solidification

Abstract

We investigate the role of gravity in a linear stability analysis of the onset of coupled convective and morphological instabilities during directional solidification at constant velocity of a dilute alloy of tin in lead. For solidification vertically upwards, the temperature gradient alone would cause a negative density gradient and the solute gradient alone would cause a positive density gradient. Two types of instability are found, a convective type that occurs for long wavelengths and a morphological type that occurs for short wavelengths. In general, these are coupled but the morphological instabilities are practically independent of gravity and thus correspond to the predictions of previous morphological stability theory in which density changes and convection are neglected. The convective instabilities depend strongly on gravity; for a growth velocity of V = 1 μm/s and a temperature gradient in the liquid of G L = 200 K/cm, the critical concentrations for convective instabilities are 3.1 × 10 −4, 3.1 × 10 −2 and 2.39 wt. % for g e = 980 cm/s 2, 10 −4 g e and 10 −6 g e, respectively. For low velocities, the convective instabilities occur at much lower solute concentrations than the morphological instabilities whereas at high velocities, the reverse is true. At intermediate velocities where the changeover takes place, there are oscillitory instabilities of mixed character whose periods range from 60 s at g e to 6 × 10 4 s at 10 −6 g e.