The coarsening kinetics of two misfitting particles in an anisotropic crystal

We examine the influence of elastic stress on the Ostwald ripening kinetics of two elastically and diffusionally interacting, misfitting spherical particles in an anisotropic crystal. The coupled equations of elasticity and diffusion are solved analytically in series form in a bispherical coordinate system when the matrix supersaturation is small, local equilibrium obtains at the interface, particle and matrix possess the same cubic elastic constants, and the stress engenered by compositional inhomogeneity is negligible. Expressions are obtained for the matrix composition field, the local normal interfacial velocities of the particles, the isotropic particle growth rates, and the velocity of the particles' centers of mass. Inverse coarsening, or the growth of a smaller particle at the expense of a larger particle, is predicted for particle alignments along the elasticity soft 〈100〉 directions in nickel and for the 〈110〉 and 〈111〉 directions in molybdenum. Coarsening rates are often significantly enhanced for other particle orientations with respect to those of the stress-free case. The elastic stresses also change the functional dependence of the particle growth rate on particle size suggesting that the temporal exponents observed during classical ripening may not obtain in stressed systems. These predictions indicate that elastically-induced preferential coarsening strongly influences microstructural development in two-phase coherent alloys.