Routes to Cell Formation and Hidden Ramps in Directional Solidification

Abstract

Precise measurements of interface position as a crystal grows in a temperature gradient (directional solidification) show that while cellular interfaces are usually steady, stationary planar interfaces are more difficult to achieve. Since the interface’s position is directly related to its temperature that is in turn related to impurity concentration at the interface, we found directional melting provided a sensitive tool to probe the quality of the crystal grown and obtain a measure of the partition coefficient, k, that determines the nature of the planar-cellular bifurcation.

In directional solidification, the allowed band of linearly unstable wavelengths for cellular patterns without boundary effects extends over 3 orders of magnitude, i.e. is much broader than “canonical” hydrodynamic pattern forming systems. In stark contrast, the observed band is much less than one order. We present experimental observations suggesting that soft boundary conditions intrinsic to the experiment are responsible for the dramatic discrepancy between theoretical expectations and experimental observations. In particular, we found that at onset, the interface first develops structure in the vertical, shorter dimension because of curvature at the liquid-solid-glass contact. We use this information to install a slowly varying “ramp”, where the control parameter varies spatially from above to below threshold, by inducing a concentration gradient parallel to the solid-liquid interface in the longer direction and find that this leads to further collapse of the allowed band.