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

Surfactants accumulate at an interface and orient their two chemically distinct ends toward their preferred fluid components. The relaxation found in part c lowers the coefficient of the gradient energy, and hence generally lowers the surface tension. If the reduction makes $\tilde{\kappa }<0$, the positive fourth-gradient term proportional to $\gamma$ can stabilize structure at a nonzero wavevector, producing the modulated correlations characteristic of a microemulsion.