Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 1 / b / Solution

The composition changes most rapidly across an interface, so $\nabla \varphi$ points along its normal. The termtherefore couples the surfactant's head-to-tail orientation to the interface normal. Minimization aligns $\mathbf{p}$ antiparallel to $\nabla \varphi$ when $\lambda >0$ and parallel when $\lambda <0$. The sign is fixed by which fluid component is called positive $\varphi$ and by which molecular end defines positive $\mathbf{p}$, together with the preferential affinity of the head and tail for the two components.