In a solidifying system of finite extent, the solute redistribution which takes place at the advancing interface, combined with diffusion in the liquid and possibly in the solid, leads to a final segregation profile. This might range from the solute being highly concentrated in the region which solidified last to being uniformly distributed over the whole region. The region being considered might be as small as a thickening dendrite arm (l~10-100 mm) or as large as a macroscopic bar (l~100mm).

However, in practice conditions during solidification are often such that the distribution of solute before and after solidification conforms to simple analytical equations. For, example, a dendrite arm (l~50mm) typically thickens at such a slow rate (v ~ 1mm/s) that the boundary layer, (D_L/v) is >>l. This means that diffusion will maintain C_L approximatelly uniform throughout the liquid during solidification. In fact, this condition is in practice also commonly maintained in much larger systems by the action of convection in the liquid.

Diffusion in the solid can often also be treated in a simple way. Sometimes (usually with interstitial solutes only) diffusion is so fast that the composition is mantained uniform throughout the solid. The situation then conforms to the Lever Rule assumptions. Although solute partitions during solidification, it ends up being uniformly distributed. With substitutional solutes, on the other hand, while the above condition does not hold, diffusion is much slower and can often be neglected. The solid, once formed, retains the same composition. The distribution of solute conforms to the Scheil equation. This type of solidification tends to cause solute to segregate to the later-solidifying regions (for k<1).

The behaviour can be explored using this simulation. This is based on finite difference calculations which are quite complex to carry out.

Derive Lever Rule
Derive Scheil Equation