Curvature driven flow of bi-layer interfaces

Abstract

We introduce the Functionalized Cahn–Hilliard (FCH) energy, a negative multiple of the Cahn–Hilliard energy balanced against the square of its own variational derivative, as a finite width regularization of the sharp-interface Canham–Helfrich energy. Mass-preserving gradient flows associated to the FCH energy are higher-order phase field models which develop not only single-layer, or front-type interfaces, but also bi-layer, or homoclinic interfaces with associated endcap and multi-junction structures. The single-layer interfaces manifest a fingering instability which grows into endcapped bi-layers. The meandering growth of the bi-layer interfaces and the subsequent merging lead to a multi-junction dominated network that bears a striking similarity to the phase separated domains of both perfluorosulfonic membranes and amphiphilic di-block co-polymer solutions. The bi-layers generated by the gradient flows of the FCH energy have an interfacial width which scales with ε≪1, however for fixed ε, there is a class of bi-layers parameterized by width and background state. Our primary result is the asymptotic derivation of the normal velocity of a closed bi-layer hypersurface in Rd (d≥2) coupled to the evolution for the surface width, curvature, and background state. We also show the convergence of the FCH energy to a scaled Canham–Helfrich type energy for both single and bi-layer interfaces, with the surface area coefficient of the limiting Canham–Helfrich energy coupling to the bi-layer width. Thus the bi-layer networks grow to maximize surface area while minimizing the square of curvature, up to the point that the increase in surface area stretches the bi-layers too thin.