Bounds on the Nonlinear Diffusion Controlled Growth Rate of Spherical Precipitates

Abstract

This paper considers a nonlinear diffusion problem in which there is a change of phase. The physical situation is e.g., the diffusion of solute in a crystal towards a growing precipitate. The nonlinear-diffusion coefficient is a function of the local concentration of solute. A radially symmetric precipitate which grows at a rate proportional to the square root of the time is considered in n dimensions (n=1, 2, 3); thus ρ=s(D0t)½, where ρ is the radius, D0 is a constant having dimensions of area/time and s is a number, the quantity of particular interest. A comparison theorem is established which, in part, states that s increases with increasing diffusion coefficient. A consequence of the theorem is that a unique value of s is associated with a prescribed diffusion coefficient. The comparison theorem also leads to bounds on s. Some other bounds, involving functionals of the diffusion coefficient, are also obtained for s.