Plane Front Solidification, Constitutional Undercooling

In the growth of single crystals (or other application) where uniform composition is required, it is also necessary to maintain a near planar solidification front. In the above discussion of directional solidification, it was shown that the segregation coefficient causes a buildup of the rejected phase in front of the advancing solidification front. 

Final composition profile of a directionally solidified ingot under diffusion-controlled conditions (no convective mixing). 

Local freezing temperature along the growing sample. The temperature ahead of the solidification interface must increase at least at the mte shown by Gi to stabilize the plane front growth interface. If the thermal gradient is insufficient, as illustrated by slope Gz, the growth interface breaks down into a dendritic structure with a mushy zone (mixture of solid dendrites and interdendritic fluid) that extends to the point where the temperature is above the local freezing point. 

A simple constitutional supercooling (CS) criterion was developed by Rutter and Chalmers (1954) that predicts the ratio of the gradient required to stabilize the interface to the growth velocity for a given solidification system. Taking the derivative of Equation 13.4 at x = 0, 

Mullins and Sekerka (1964) developed a more rigorous theory based on a stability analysis that included the liquid-solid interfacial energy, which can provide a stabilizing effect on the interface. However, the difference between the two theories is so small that, for the most part, the more conservative CS criteria can be used for experiment design. 

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