Diffusion fields associated with prolate spheroids in size and shape coarsening

Abstract

The diffusion field or solute concentration distributed around a prolate spheroidal particle simulating a rod- or needle-shaped precipitate has been solved for varying aspect ratios of the precipitate and varying concentrations along the precipitate surface owing to the curvature effect. With the prolate spheroidal coordinates, the principal curvatures of the prolate spheroidal surface are derived as functions of the angular variable, and the Laplace field equation is separated into two Legendre equations either on the radial variable or on the angular variable. The analytical solution fitting the present boundary conditions is secured as the sum of a Legendre function of the second kind of order zero and a Legendre series. The Legendre function of the second kind gives the concentration distribution when the curvature effect can be ignored, whereas the Legendre series represents the concentration contributed from the curvature effect. Numerical results of normalized concentrations are presented as functions of the radial and angular variables for selected aspect ratios. The tangent component of the concentration gradients, contributed from the curvature effect, may cause complicated mass transfer, which is responsible for shape coarsening.