Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 344 / 3 / d

Using $\delta (R)=\sigma /(\alpha {\varphi }_{B}R)$, the growth law isForone has $F^{\prime }=8\pi \sigma R-4\pi \Delta R^2$. Therefore, with $\mathcal{M}(R)=M/(16\pi {\varphi }_B^2{R}^3)$,The first term in $F$ is the positive surface cost and the second is the negative bulk free-energy gain of converting a metastable volume. The nonzero stationary radius and barrier are the classical nucleation theory valuesSince $F^{\prime \prime }(R^{\ast })=-8\pi \sigma <0$, this stationary point is a maximum.