Constant-reservoir diffusion-controlled particle growth ($a^2-a_0^2=2D(c_{\infty }-c_s)t/(c_p-c_s)$)

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.

When attachment is much slower than exterior diffusion, the surface concentration stays close to the reservoir value. An attachment coefficient $k_s$ with units of velocity gives nearly constant radial growth in the dilute-feed approximation. For a sphere the ratio $k_{s}a/D$ distinguishes attachment control from transport control. A finite attachment rate leads to a Robin boundary condition rather than a fixed equilibrium surface concentration.