Network formation and ion transport in polymer electrolyte membranes

该项目集中于功能化Cahn-Hilliard （FCH）能的分析和扩展。这种高阶能量结合了溶剂化和弹性能来模拟溶剂中功能化聚合物产生的丰富网络结构。FCH能量支持更广泛的新型界面，包括多参数薄双层、小半径孔隙结构和小胶束网络。研究人员将FCH能量收敛到经典Canham-Helfrich的锐界面极限。他将一类FCH能量的保质量梯度流简化为协维1和协维2曲面的曲率驱动演化。原始能量临界点结构展开形成的结构界面驱动几何流动，界面结构耦合界面演化。FCH能量的灵活性在于可以自由地将正能量或负能量分配给不同的临界点，选择性地驱动所青睐结构的生长。研究者通过对线性化结构的分析和他的重整化群方法的适应，严格地推导出这些几何流量减少。他将FCH能量扩展到溶剂中功能化聚合物和中性聚合物的多组分混合物模型，揭示了具有多个全局最小值的电位的异斜连接的双层临界点的分岔。采用多井行波约束最小化方法求解非对称井间的连接问题——小半径孔隙的形成。21世纪，美国对清洁能源的需求日益增长。这包括从环境中获取能量，无论是太阳能还是风能，并将这些能量储存在电池中或以浓缩的化学形式储存。这些过程需要膜分离器，它允许带一种电荷的离子穿过，但不允许带相反电荷的离子穿过。研究人员开发了一个模型，当这些膜分离器从溶剂形式铸造时，该模型有可能优化这些膜分离器的纳米级结构。在这个项目中进行的分析和数值研究阐明了纳米尺度网络形成的机制，并导致更节能的转换过程的发展。该项目由数学科学部和材料研究部资助。

PromislowDMS-1109127 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.