Geometric Evolution of Multicomponent Amphiphilic Networks

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The development of efficient energy conversion and storage devices is a fundamental goal of 21st century science. Many of the most prominent devices, such as dye-sensitized and bulk-heterojunction solar cells, lithium ion batteries, and polymer electrolyte membrane fuel cells, rely upon ionomer membranes with charge-selective transport properties that arise from nanoscale networks of charge-lined pores. Progress in the improvement of these devices requires new materials based upon a fundamental understanding of the formation of these pore networks, which typically self-assemble during casting processes. The investigator presents a mathematical framework that characterizes the evolution of network morphologies arising in blends of charged polymer (amphiphile) with solvent. A better understanding of how these networks shape themselves can enable the development of better devices. The framework is based upon an incorporation of the powerful descriptive properties of dynamical systems within a higher-order free energy landscape. Students are trained in the course of the project. The Functionalized Cahn-Hilliard (FCH) free energy unites the languages of dynamical systems and calculus of variations to produce a model of network self-assembly in amphiphilic polymers that play a central role in the cost-efficient self-assembly of materials with nano-structured interfacial morphology. The investigator studies the mechanisms of geometric evolution and competition-induced bifurcation of network morphologies of binary and multicomponent amphiphilic systems within the framework of gradient flows of the FCH free energy. For binary mixtures of amphiphilic polymer and solvent, the investigator analyzes the pearling bifurcation induced by competition between codimension one bilayer morphologies and codimension two pore morphologies. Rigorous existence of pearled morphologies for curved codimension one interfaces are derived via spatial dynamics center manifold reduction, and rigorous sharp-interface reductions of FCH gradient flows to mass-preserving Willmore flows via renormalization group techniques are pursued. End cap and line defects play a significant role in the evolution of network morphologies, and the investigators characterizes reduced representations for these defect structures and quantifies their impact on roll-up of bicelles (a bicelle is a flat bilayer with an end-cap boundary). For multicomponent mixtures, the investigator follows three approaches to the bilayer existence problem: a novel billiard limit involving a piece-wise smooth potential with sharp transitions, a homomorphic reduction, and a singular perturbation limit. Each approach permits explicit construction of coexisting symmetric and asymmetric bilayer morphologies, and an analysis of the resulting bifurcation structure. The investigator derives a sharp-interface competitive geometric evolution for families of distinct, coexisting bilayer networks and a non-zero intrinsic curvature limit for asymmetric bilayers subject to non-gradient form perturbations.