= Solution
The far-field composition $-\phi_B$ has <chemical potential> zero. Local curved-interface coexistence imposes $\mu(R)=\mu_s=\gamma/(\phi_BR)$. The decaying spherically symmetric solution of the exterior <Laplace equation> is
$$
\boxed{\mu(r)=\mu_s\frac Rr=\frac\gamma{\phi_Br},\qquad r\geq R.}
$$
With constant <mobility>, its outward current is
$$
\boxed{J_r(r)=-M\frac{d\mu}{dr}=\frac{M\gamma}{\phi_Br^2},\qquad J_r(R)=\frac{M\gamma}{\phi_BR^2}.}
$$
The linearized exterior composition is $\phi(r)=-\phi_B+\mu(r)/\alpha$, consistent with the surface shift in the previous part.
For <diffusion-controlled droplet growth>, the interior is uniform to leading order and has no diffusive current. Replacing exterior material by interior material requires the composition jump $2\phi_B$, so conservation gives $2\phi_B\dot R=-J_r(R)$. Hence
$$
\boxed{\dot R=-\frac{M\gamma}{2\phi_B^2R^2}.}
$$
The negative sign represents evaporation. The total outward current is $4\pi R^2J_r(R)=4\pi M\gamma/\phi_B$, independent of $R$ in this leading quasi-static capillary approximation.
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