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We have studied the growth kinetics of isolated precipitates growing from asupersaturated matrix in 3-dimensions (3-D) using phase field models; we assumeisotropic interfacial energy consider both constant and variable diffusivity.We report and compare our numerical growth rates with the classic analyticalsolutions of Zener and Frank (ZF). The numerical results deviate from theanalytical ones. These deviations can be understood in terms of the generalisedGibbs-Thomson effect. Specifically, due to the higher capillary contribution in3-D (curvature is twice for a sphere compared to a circle), the precipitategrowth kinetics deviates more from ZF in 3-D as compared to 2-D. In addition,the kinetic parameter associated with the normal velocity of theprecipitate-matrix interface also modifies the deviation of the precipitatecomposition from its equilibrium value and hence its growth kinetics. In phasefield models (such as the one used by us) which use a combination of Allen-Cahnand Cahn-Hilliard type equations, we show how to choose the kinetic parameters(namely, mobility and relaxation parameter) so that the kinetic coefficient (inthe generalised Gibbs-Thomson equation) is made effectively zero. We also showthat the kinetic parameter the precipitate-matrix interface might play acrucial role in making the precipitate undergo morphological instabilities asit grows (leading to "sea-weed"-like structures).

Growth kinetics and morphological stability of precipitates in 3-D: a phase field study