Geometric Evolution of Multicomponent Amphiphilic Networks

开发高效的能量转换和储存装置是21世纪世纪科学的基本目标。 许多最突出的装置，如染料敏化和本体异质结太阳能电池、锂离子电池和聚合物电解质膜燃料电池，依赖于具有电荷选择性传输性质的离聚物膜，所述电荷选择性传输性质产生于电荷内衬孔的纳米级网络。 这些设备的改进的进展需要新的材料的基础上，这些孔网络的形成，通常在铸造过程中自组装的基本理解。 研究人员提出了一个数学框架，其特点的网络形态的演变所产生的带电聚合物（两亲物）与溶剂的共混物。 更好地了解这些网络如何塑造自己，可以开发出更好的设备。 该框架是基于一个强大的描述性动力系统的高阶自由能景观内的属性。 学生在项目过程中接受培训。官能化Cahn-Hilliard（FCH）自由能结合动力学系统和变分法的语言，产生两亲聚合物中的网络自组装模型，该模型在具有纳米结构界面形态的材料的成本效益自组装中发挥核心作用。 研究人员在FCH自由能梯度流的框架内研究了二元和多元两亲体系网络形态的几何演化和竞争诱导分岔机制。 对于两亲性聚合物和溶剂的二元混合物，研究者分析了由余维1双层形态和余维2孔形态之间的竞争引起的珠光分叉。 通过空间动力学中心流形的减少，严格存在的珍珠状形态弯曲余维一接口推导出，严格的尖锐接口减少FCH梯度流质量保持Willmore流通过重整化群技术追求。 端盖和线缺陷在网络形态的演变中起着重要的作用，研究人员表征了这些缺陷结构的简化表示，并量化了它们对bicelle卷起的影响（bicelle是具有端盖边界的平坦双层）。 对于多组分的混合物，调查如下三种方法的双层存在问题：一个新的台球限制，涉及一个分段光滑的潜在的急剧转变，同态减少，和奇异摄动限制。 每种方法允许明确的建设共存的对称和不对称的双层形态，并分析所产生的分叉结构。 研究人员推导出一个尖锐的接口竞争的几何演化的家庭不同的，共存的双层网络和非零的非对称双层固有曲率限制非梯度形式的扰动。

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.