Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

# Derivation of the Scheil equation

Equating the amount of solute in the shaded areas gives:

CL - C>S ) df = ( 1 - f ) dCL

Substituting for CS, using:

CS = k C0

where k is the partition coefficient gives:

CL ( 1- k ) df = ( 1 - f ) dCL

We then integrate, from zero to fS, the fraction solidified, and from C0 (since CL = C0 when fS = 0) to CL:

$\int_0^{{f_S}} {\frac{{df}}{{\left( {1 - f} \right)}} = \int_{{C_0}}^{{C_{\rm{L}}}} {\frac{{d{C_{\rm{L}}}}}{{{C_{\rm{L}}}\left( {1 - k} \right)}}} }$

$- \ln \left( {1 - {f_{\rm{s}}}} \right) = \frac{1}{{1 - k}}\ln \frac{{{C_{\rm{L}}}}}{{{C_0}}}$

Taking exponentials of both sides, and rearranging gives:

CL = C0 ( 1 - fS )k-1

We are more interested in the concentration of the solid, so this is usually written as:

CS = k C0 ( 1 - fS )k-1

This is known as the Scheil equation.