Competition and Complexity in Amphiphilic Polymer Morphology

We analyze the competitive evolution of codimension one and two morphologies within the \(H^{-1}\) gradient flow of the strong Functionalized Cahn-Hilliard equation. On a slow time scale a sharp hypersurface reduction yields a degenerate Mullins-Sekerka evolution for both codimension one and two hypersurfaces, leading to a geometric flow that depends locally on curvatures couples to the dynamic value of the spatially constant far-field chemical potential. Both codimension one and two morphologies admit two classes of bifurcations, one leads to pearling, a short-wavelength in-plane modulation of interfacial width, the other flips motion by curvature to the locally-ill posed motion against curvature, which leads to fingering instabilities. We present a bifurcation diagram for the morphological competition, and compare our results quantitatively to simulations of the full system and qualitatively to simulations of self-consistent mean field models and laboratory experiments; illuminating the role of the pearling bifurcation in the development of complex network morphologies.