 Diffusion-controlled droplet growth

For a three-dimensional droplet with concentration jump $\Delta \varphi$, constant mobility $M$, and quasi-static exterior chemical potential,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 $-4\pi R^2\Delta \varphi \dot{R}$. For a symmetric mixture, $\Delta \varphi =2{\varphi }_B$ and the Gibbs--Thomson relation gives ${\mu }_s=\gamma /({\varphi }_{B}R)$.