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

In the exterior write $\varphi =-{\varphi }_B+\tilde{\varphi }$. Linearization at the bulk minimum givesFor 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 $\mu \simeq \alpha \tilde{\varphi }$. The conserved order-parameter dynamics then becomesDiffusion relaxes the exterior profile much faster than the droplet radius changes. In this quasistatic limit ${\partial }_t\tilde{\varphi }\simeq 0$, and therefore