Since the properties of an alloy can depend strongly on the concentration of solutes that it contains, being able to quantitatively predict the concentrations in the solid is desirable, but a mathematical description of solidification is very difficult in the general case. Solutions can be obtained when certain assumptions are made. For example, we can assume:
- There is no diffusion the solid phase, DS = 0. This will hold if the characteristic diffusion distance is much smaller than the length of the sample, << L
- There is complete mixing in the liquid phase, giving a uniform concentration in the liquid. This may occur because of convection, or can be aided by mechanical mixing.
Using these assumptions we can derive the Scheil equation, which describes the composition of the solid and liquid during solidification, as a function of the fraction solidified. Work through the animation to see how the conditions required to derive the Scheil equation are obtained:
Note: This animation requires Adobe Flash Player 8 and later, which can be downloaded here.
Equating the amount of solute in the shaded areas gives:
( CL - Cs ) df = ( 1 - f ) dC
Solving this gives us the Scheil equation, for the profile of solute in the completely solidified bar:
Cs = k C0 ( 1 - fs )k-1
where CS is the concentration of solute in the solid, at a fractional distance along the bar, fS, C0 is the initial concentration of the liquid, and k is the partition coefficient.
To see a full derivation of the equation, click here.
The Scheil equation predicts a profile with a concentration that tends towards infinity at the end of the bar, but clearly there will be an upper limit on the concentration of any solid forming. In the case where A and B form a complete solid solution across the composition range, the liquid will, at some point, reach a concentration of 100 %B and hence solidify as pure B. An alternative case arises in a binary eutectic system of A and B, where a solid eutectic structure will form when the concentration of the liquid reaches the eutectic concentration.
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