The Cahn-Hilliard free energy has a rich history as a model of coarsening in binary mixtures of incompatible materials. However many of the polymeric materials of interest for energy conversion applications do not coarsen, that is phase separate into larger and larger domains, rather they self-assemble nanoscale networks that serve the charge-selective networks that drive fuel cells, bulk heterojuction solar cells, and membrane separaters in Lithium ion batteries. Self assembly in polymeric materials is mediated by functionalization of polymers, that is the attachment of ionic sidechains which modify their solubility, rendering them amphiphilic. The free energy of functionalized polymers rewards the formation of interface and has complicated relations between local density of the constituent phases and the material pressure.
We present the functionalized Cahn-Hilliard free energy, a higher-order reformulation of the Cahn-Hilliard free energy, which remaps the critical points, in particular it rewards critical points with high interfacial surface area and low mixture pressure. The analysis of the FCH and its gradient flows requires new techniques, in particular dynamical systems plays a much larger role in identifying interfacial structures, while differential geometry and extensions of Ricci flows determine morphological evolution. On the other hand, the usual framework of Γ-convergence is up-ended.
We will present an overview of the Cahn-Hilliard free energy, provide sufficient chemical background to motivate the FCH free energy. Explain the role of dynamical systems in interface selection, present the underpinnings of the differential geometric reductions, and derive some simple examples of morphological competition.