= Solution
In the exterior write $\phi=-\phi_B+\widetilde\phi$. Linearization at the bulk minimum gives
$$
\mu=\frac{\delta F}{\delta\phi}
\simeq\alpha\widetilde\phi-\kappa\nabla^2\widetilde\phi.
$$
For a large droplet, variations occur on scale $R$ much larger than the interfacial <correlation length>, so the second term is smaller by $O(\kappa/(\alpha R^2))$ and $\mu\simeq\alpha\widetilde\phi$. The <conserved order-parameter dynamics> then becomes
$$
\partial_t\widetilde\phi\simeq\alpha M\nabla^2\widetilde\phi.
$$
Diffusion relaxes the exterior profile much faster than the droplet radius changes. In this quasistatic limit $\partial_t\widetilde\phi\simeq0$, and therefore
$$
\boxed{\nabla^2\widetilde\phi=0\qquad(r>R).}
$$
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