Mathematics of diffusion controlled precipitates with time-dependent boundary conditions

The boundary value problem of diffusion controlled precipitation with time-dependent boundary conditions is treated by an extension of the conventional eigenfunction expansion method. In this approach, expansion coefficients and eigenvalues become functions of time and eigenfunctions become functions of space and time. Mathematical techniques to determine the relevant quantities are set forth and the general solution describing precipitate growth under time-dependent conditions is obtained. The general solution is applied to the precipitation of plates and spheres and the growth of radii as a function of time is determined. In the latter case, comparisons are made with conventional methods and the fractional error for a particle to reach half its equilibrium is estimated.