Growth Limited by Heat Conduction and Mass Diffusion Simultaneously

3 Growth Limited by Heat Conduction and Mass Diffusion Simultaneously 

During further solidification, the solute concentration in the liquid in front of the advancing interface will then quickly build up to form a concentration spike  

In this example, the equilibrium concentrations maintained at the interface are functions of the interface temperature, which in turn is a function of time. In addition, the velocity of the interface, v (i.e., rate of solidification), depends simultaneously upon the mass diffusion rates and the rates of heat conduction in the two phases, as may be seen by examining the two Stefan conditions that apply at the interface. For the mass flow the condition is 

A complete general solution therefore requires solving for temperature and concentration fields in both phases which satisfy all boundary conditions as well as the two coupled Stefan conditions. Solving this problem is a challenging task [4, 5] however, an analysis of the concentration spike under certain simplifying conditions when v is known is given in Section 22.1.1. 

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