Geometric Evolution of Multicomponent Amphiphilic Networks

多组分两亲网络的几何演化

• 批准号: 1409940
• 负责人: Keith Promislow
• 金额: $ 32.84万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1409940&HistoricalAwards=false
• 关键词: Geometric; Evolution; Multicomponent; Amphiphilic; Networks

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.

开发高效的能量转换和存储设备是21世纪科学的根本目标。许多最著名的器件，如染料敏化和体相异质结太阳能电池、锂离子电池和聚合物电解质膜燃料电池，都依赖于离聚体膜，这些离聚体膜具有电荷选择性传输特性，这些离聚体膜是由电荷衬里的纳米网络形成的。在改进这些装置方面的进展需要新的材料，其基础是对这些气孔网络的形成有基本的了解，这些气孔网络通常在铸造过程中自组装。这位研究人员提出了一个数学框架，描述了带电聚合物(两亲性)与溶剂共混时网络形态的演变。更好地了解这些网络是如何塑造自身的，可以帮助开发更好的设备。该框架基于高阶自由能图景中动力系统强大的描述性属性的结合。学生们在这个项目的过程中接受了培训。功能化的Cahn-Hilliard(FCH)自由能将动力学系统和变分的语言结合在一起，产生了一个两亲性聚合物的网络自组装模型，该模型在具有纳米结构界面形态的材料的低成本自组装中发挥了核心作用。在FCH自由能梯度流的框架下，研究了二元和多组分两亲体系的几何演化和竞争诱导的网络形态分叉机制。对于两亲性聚合物和溶剂的二元混合物，研究人员分析了共维一双层结构和共维两孔结构竞争引起的珠光分叉现象。通过空间动力学中心流形约化得到了弯曲余维1界面珍珠形态的严格存在性，并利用重整化群技术研究了FCH梯度流到保质量Willmore流的严格锐界面约化.端帽和线缺陷在网络形态的演变中扮演着重要的角色，研究人员表征了这些缺陷结构的简化表示，并量化了它们对双极的卷曲的影响(双极是具有端帽边界的平坦双层)。对于多组分混合物，研究者采用了三种方法来解决双层结构的存在性问题：一种新的台球极限，它包含一个具有尖锐跃迁的分段光滑势，一个同态约化，以及一个奇异摄动极限。每种方法都允许显式构建共存的对称和非对称双层结构，并分析由此产生的分叉结构。研究人员得到了不同共存双层网络族的尖锐界面竞争几何演化和非对称双层膜在非梯度形式扰动下的内禀曲率极限。