Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 344 / 3 / b / Solution

Apply force balance to one hemisphere. The pressure force is $\pi R^2({\Pi }_{\mathrm{in}}-{\Pi }_{\mathrm{out}})$, while the interfacial tension around its equator gives $2\pi R\gamma$. The Young–Laplace equation therefore yieldsTo leading order in curvature, let the two bulk compositions be ${\varphi }_{+}={\varphi }_B+{\delta }_{+}$ and ${\varphi }_{-}=-{\varphi }_B+{\delta }_{-}$. Their curvatures agree, $\alpha =f^{\prime \prime }(\pm {\varphi }_B)=-2a$, so equal chemical potentials require $\alpha {\delta }_{+}=\alpha {\delta }_{-}$ and hence a common shift $\delta$.

Since $d\Pi /d\varphi =\varphi f^{\prime \prime }(\varphi )$, expansion about the flat coexistence compositions gives ${\Pi }_{+}-{\Pi }_{-}=2{\varphi }_B\alpha \delta +O({\delta }^2)$. Matching the capillary pressure proves the Gibbs--Thomson relation in these variables:The pressure jump balances interfacial forces; it does not permit unequal chemical potentials. A common shift in both compositions supplies a common interfacial chemical potential while their pressures differ. The expansion assumes $R$ large compared with the phi-four diffuse interface width.