= Constant-reservoir diffusion-controlled particle growth
{title2=$a^2-a_0^2=2D(c_\infty-c_s)t/(c_p-c_s)$}
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.
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