 Constant-reservoir diffusion-controlled particle growth

For an isolated growing sphere, a quasi-steady exterior diffusion equation, constant surface concentration $c_s$, and a maintained far-field concentration $c_{\infty }$ give $c(r)=c_{\infty }-(c_{\infty }-c_s)a/r$. A sharp moving boundary with interior concentration $c_p$ then obeys $(c_p-c_s)\dot{a}=D(c_{\infty }-c_s)/a$. The displayed integrated law assumes all concentrations and the diffusion coefficient are constant. It becomes $a\propto \sqrt{t}$ when the seed radius is negligible. This reservoir-fed law differs from curvature-controlled Ostwald ripening in a depleting many-particle system.