= Diffusion-controlled droplet growth
For a three-dimensional droplet with concentration jump $\Delta\phi$, constant <mobility> $M$, and quasi-static exterior <chemical potential>,
$$
\mu(r)=\mu_\infty+(\mu_s-\mu_\infty)\frac Rr,
\qquad
\dot R=-\frac{M(\mu_s-\mu_\infty)}{\Delta\phi\,R}.
$$
The first formula solves the exterior <Laplace equation>. The second follows from conservation: the total outward flux $4\pi MR(\mu_s-\mu_\infty)$ removes excess composition at the rate $-4\pi R^2\Delta\phi\,\dot R$. For a symmetric mixture, $\Delta\phi=2\phi_B$ and the <Gibbs--Thomson relation> gives $\mu_s=\gamma/(\phi_B R)$.
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