Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 344 / 1 / e / Solution

Far from a planar interface, the equilibrium profile equals a bulk minimum and contributes the uniform density $f({\varphi }_B)$. The localized transition layer contributes an additional free energy proportional to its area $A$, because translation invariance makes the excess per unit area independent of position. By the definition of surface tension as interfacial excess free energy per area,