Grain Growth Controlled by Diffusion

For diffusion-controlled growth, consider a material flux (moles/s) passing through an imaginary spherical surface in the liquid matrix a distance of R from a grain with a radius of a. This material flux must be equal to the rate of the mass change of the grain. If the a phase is pure A and the molar volumes of a and are the same. 

However, as the annealing time increases, the particles that have been growing can begin to dissolve and theoretically only the largest grain present at the 

The change in average grain size with annealing time was predicted by the LSW theory. However, since Eq. (15.15) holds for each grain, we may just assume that da/dt Then, the integration of Eq. (15.15) gives 

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