The <Young–Laplace equation> gives a pressure excess $2\sigma/R$ inside a spherical droplet. Near coexistence, the common chemical-potential shift is $\mu\simeq\alpha\delta$, where $\alpha=f''(\phi_B)$, while changing phase changes the composition by approximately $2\phi_B$. Balancing the capillary pressure against this thermodynamic shift gives the <Gibbs--Thomson relation>
$$
2\phi_B\alpha\delta\simeq\frac{2\sigma}{R}.
$$
Hence the compositions immediately outside and inside are
$$
\phi(R^+)=-\phi_B+\delta(R),
\qquad
\phi(R^-)=+\phi_B+\delta(R),
$$
with
$$
\boxed{\delta(R)=\frac{\sigma}{\alpha\phi_BR}\propto\frac\sigma R.}
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-344/3/a/solution