= Solution
Integrate the <diffusion-controlled droplet growth> equation: $d(R^3)/dt=-3M\gamma/(2\phi_B^2)$. Thus
$$
R(t)^3=R_0^3-\frac{3M\gamma}{2\phi_B^2}t
=R_0^3\left(1-\frac t{\tau_\infty}\right).
$$
Setting the radius to zero gives
$$
\boxed{\tau_\infty(R_0)=\frac{2\phi_B^2R_0^3}{3M\gamma}.}
$$
This is the lifetime predicted by the sharp-interface quasi-static model. The final stage at a radius comparable to the interface width falls outside that approximation, while the leading lifetime for a large initial droplet is dominated by the larger radii.
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