Meander and Pearling of Single-Curvature Bilayer Interfaces in the Functionalized Cahn--Hilliard Equation

Abstract

The functionalized Cahn--Hilliard (FCH) free energy models interfacial energy in amphiphilic phase separated mixtures. Its minimizers encompass a rich class of morphologies with detailed inner structure, including bilayers, pore networks, pearled pores, and micelles. We address the existence and linear stability of $\alpha$-single curvature bilayer structures in $d\geq2$ space dimensions for a family of gradient flows associated to the strong functionalization scaling. The existence problem requires the construction of homoclinic solutions in a perturbation of a fourth-order integrable Hamiltonian system, while a negative index argument reduces the linear stability analysis to the characterization of the meander and pearling modes of the second variation of the FCH energy on a family of invariant subspaces, independent of the choice of mass-preserving gradient flow.