Degenerate Diffusion in Complex Amphiphilic Network Structures

主要研究者在包含简并扩散和功能化Cahn-Hilliard （FCH）能量的框架下研究了两亲性混合物中的复杂网络结构。复杂系统中的网络结构与相变、材料科学和数学生物学有着密切的关系。他开发了模拟和分析两亲性混合物及其性质的结构发展的方法。这样的系统包括生物膜和聚合物材料，如聚合物电解质膜。生物膜在蛋白质运输、药物包封和递送等生物过程中起着至关重要的作用。聚合物电解质膜是燃料电池的关键成分，能够有效地产生清洁能源，是传统电池的有前途的替代品。它们的有效性与它们的精细结构直接相关。在这个项目的过程中，培养了一名研究生和一名本科生。主要研究了简并扩散对复杂两亲结构动力学的影响、功能化Cahn-Hilliard能量的变分性质及其与两亲结构的关系。他将双层、丝状孔隙、胶束和缺陷结构（如开边、端帽和三重结）描述为FCH能量的最小值，并将它们与各种几何约束下FCH能量的变化特性联系起来。通过将简并扩散建立到FCH方程中，他能够模拟生物学上重要的现象，如各种大小和形状的双层囊泡共存，脂质分子沿着脂质双层膜的扩散，以及双层膜的不渗透性。他研究了缺陷的演化及其能量对参数变化的敏感性，并确定了缺陷形成的能量代价。本项目开发的工具在其他物理和生物系统的建模以及与非线性偏微分方程相关的其他应用数学问题的研究中具有广泛的应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The principal investigator studies complex network structures in amphiphilic mixtures under a framework that includes degenerate diffusion and functionalized Cahn-Hilliard (FCH) energy. Network structures in complex systems have close relations to phase transitions, materials science, and mathematical biology. He develops methods to model and analyze the development of structures in amphiphilic mixtures and their properties. Such systems include biological membranes and polymer materials like polymer electrolyte membranes. Biological membranes play critical roles in biological processes such as protein transportation, drug encapsulation and delivery. Polymer electrolyte membranes are key ingredients of fuel cells that are able to efficiently produce clean energy and are a promising replacement for traditional batteries. Their effectiveness is directly related to their fine structures. In the course of this project, a graduate student and an undergraduate student are trained.The principal investigator concentrates on the influence of degenerate diffusion on the dynamics of complex amphiphilic structures, the variational properties of the functionalized Cahn-Hilliard energy, and their relation to amphiphilic structures. He formulates bilayers, filamentous pores, micelles, and defect structures such as open edges, end caps, and triple junctions, as minimizers of the FCH energy, and relates them to variational properties of the FCH energy under various geometric constraints. By building degenerate diffusion into the FCH equation, he is able to model biologically significant phenomena such as the co-existence of bilayer vesicles of various sizes and shapes, the diffusion of lipid molecules along lipid bilayer membranes, and the impermeability of bilayers. He studies the evolution of the defects and the sensitivity of their energies to parametric variation, and identifies the energetic cost for the defects to form. The tools developed in this project have wide applications in the modeling of other physical and biological systems, and in the study of other problems in applied mathematics related to nonlinear PDEs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.