Effect of concentration-dependent diffusion coefficients on precipitate growth kinetics

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Abstract
A theoretical analysis is presented for the growth kinetics of a phase precipitated from supersaturated solid solution in a system with concentration‐dependent diffusion coefficients. It is shown that at high supersaturations and for all precipitate shapes the graphs relating growth rate to supersaturation for constant diffusion coefficients can be used, provided the diffusion coefficient is taken as D̄, the weighted average coefficient. For low supersaturations, the same graphs can be used provided the diffusion coefficient is taken as D̄ for spherical precipitates and D̿ for planar precipitates, where D̿ is very close to the first moment of the diffusion coefficient about the interface composition. At other supersaturations the diffusion coefficient to be used lies intermediate between D̄ and D̿ for all precipitate shapes, the actual value depending on the shape of the D‐vs‐concentration curve. For cylindrical precipitates, the growth rate always lies exactly halfway between the growth rates for planar precipitates and spherical precipitates. The procedure outlined here should give growth rates accurate to within 1.5% of the difference between growth rates calculated for diffusion coefficients at the phase‐boundary composition and well away from the phase boundary. For many purposes it is sufficient to use one diffusion coefficient covering all precipitate shapes and all supersaturations. This is best taken as D̄ and will give maximum errors of 6% for spherical precipitates and 15% for planar precipitates.