The project centers on an analysis and extension of the functionalized Cahn-Hilliard (FCH) energy. This higher order energy incorporates solvation and elastic energies to model the rich network structures produced by functionalized polymers in solvent. The FCH energy supports a wider class of novel interfaces, including multi-parameter families of thin bi-layers, small radius pore structures, and small micelle networks. The investigator addresses the convergence of the FCH energy to the classic Canham-Helfrich in sharp interface limit. He reduces a class of mass-preserving gradient flows of the FCH energy to curvature-driven evolution for surfaces of co-dimension one and two. The structured interfaces constructed in the unfolding of the original energies' critical point structures drive the geometric flows, with the interfacial structure coupling the interface's evolution. The flexibility of the FCH energy lies in the freedom to assign positive or negative energies to different critical points, selectively driving growth of favored structures. The investigator rigorously derives these geometric flow reductions via analysis of the linearized structures and an adaptation of his renormalization group approach. He extends the FCH energy to model multicomponent mixtures of functionalized and neutral polymers in solvents, unfolding the bifurcation of bi-layer critical points from heteroclinic connections for potentials with multiple global minima. The construction of small-radius pores as a connection problem between asymmetric wells is achieved via the constrained minimization approach developed for traveling waves in multi-wells.
The US has a growing need for clean energy sources for the 21st century. This involves capturing energy from the environment, either from solar or wind sources, and storing this energy either in batteries or in a concentrated chemical form. These processes require membrane separators that permit ions of one charge to cross, but not ions of the opposite charge. The investigator has developed a model that has the possibility to optimize the nanoscale structure of these membrane separators when they are cast from solvent form. The analysis and numerical studies performed in this project elucidate the mechanisms underlying the formation of the nanoscale networks, and lead to the development of more energy-efficient conversion processes. The project is funded by the Division of Mathematical Sciences and the Division of Materials Research.