Disequilibrium fractionation during solidification

Let us consider the influence of a solid-liquid interface advancing at a constant velocity on the solid-liquid fractionation of an element i. In the case of unidirectional solidification, it is convenient to consider that liquid crosses the immobile interface with an absolute constant velocity v, while a solid-liquid fractionation coefficient K is applied to the fractionation of element i. Let us assume that the interface is at x=0, the medium being solid for x 0. Liquid fills the half-space 0 x oo and 

After some manipulation, concentration of the liquid at the interface = 0 becomes 

Let us investigate the long-term asymptotic properties of this solution. Given that 

Using the property of the error function listed in Appendix 8B 

The steady-state profile in the liquid is calculated in Section 9.6. 

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