Network formation and ion transport in polymer electrolyte membranes

PromislowDMS-1109127 该项目集中在功能化的Cahn-Hilliard（FCH）能量的分析和扩展。 这种高阶能量结合了溶剂化能和弹性能，以模拟由官能化聚合物在溶剂中产生的丰富的网络结构。 FCH能量支持更广泛的新型界面，包括薄双层，小半径孔结构和小胶束网络的多参数家庭。 研究者解决了FCH能量收敛到经典的Canham-Helfrich锐界面极限的问题。 他减少了一类质量保持梯度流的FCH能源曲率驱动的演变表面的余维1和2。 在初始能量临界点结构的展开过程中构造的结构化界面驱动着几何流动，界面结构耦合着界面的演化。 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，选择性地驱动有利结构的生长。 研究人员通过分析线性化结构和改编他的重整化群方法严格推导出这些几何流减少。 他扩展了FCH能量模型的多组分混合物的官能化和中性聚合物在溶剂中，展开的分歧双层临界点从异宿连接的潜力与多个全球最小值。 将小半径孔隙的构造作为非对称威尔斯井间的连接问题，通过多威尔斯井中行波的约束最小化方法来实现。 美国对21世纪清洁能源的需求日益增长。 这涉及从环境中捕获能量，无论是从太阳能还是风能，并将这些能量储存在电池中或以浓缩的化学形式储存。 这些过程需要允许一种电荷的离子穿过而不允许相反电荷的离子穿过的膜分离器。 研究人员已经开发出一种模型，该模型有可能优化这些膜分离器的纳米级结构，当它们从溶剂形式铸造时。 该项目中进行的分析和数值研究阐明了纳米级网络形成的机制，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.