At an advancing solidification front, rejected solute tends to build up in the liquid ahead of the front. The nature of this build-up depends on the growth velocity, V, the partition coefficient, k, and the diffusivity of the solute in the liquid, D_L.

The value of D_L is similar for most systems (since most liquids have similar atomic structure) and k is fixed for a given system.

For given values of D_L, k, and V, the changing solute profile as the interface advances is dictated by the nature of the diffusion in the liquid. In the simulation shown here, a finite difference method is applied to predict this profile. (Diffusion in the solid is considered neglible.)

Once a steady state is established, the solute profile is described by a simple exponential decay expression. The characteristic distance for the exponential decay is given by D_L/V, which is often refered as a boundary layer thickness and given the symbol d.

Explore the effects of different  k, and V on the solute profile. A point of interest is the time it takes for a steady state to become established. This can be predicted as the time for which the characteristic diffusion distance is approximately equal to the boundary layer thickness. Using this condition, derive an expression for the time required and check it by running the simulation with different values of V. 

[ Derivation of equation ]