Complex Amphiphilic Structures and the Functionality of Biomaterials

复杂的两亲性结构与相变、材料科学和数学生物学密切相关。本项目主要研究纳米粒子、蛋白质和疏水表面对两亲性结构形态的影响。这项研究极大地促进了对纳米颗粒、生物启发材料和生物膜功能的理解。它帮助我们了解蛋白质运输、药物包裹和药物输送等基本过程。它还为聚合物材料中其他复杂结构的建模和性能分析提供了新的方法和技术。教育活动包括对研究生研究的监督和培训。培养目标是对研究生进行数学建模、偏微分方程的分析和数值模拟方面的培训。这项研究项目集中在应用数学与科学、技术和工程相结合的跨学科研究上。派是一名应用数学家，而联合派是一名化学实验学家。研究人员在功能化的Cahn-Hilliard(FCH)能量框架下，结合适当的强弱锚定条件，模拟了纳米粒子在双层脂质体和聚合物胶束中的包裹、两亲性共聚物对生物材料的包覆以及蛋白质引起的形态变化。这种新的数学框架将材料系统转化为具有适当边界条件的非线性偏微分方程组。它在其他物理、材料和生物系统的建模中有着广泛的应用。PI应用现有的和新的数学工具来研究非线性偏微分方程组。这些工具包括三个方面：渐近分析、变分方法和数值模拟。这一奖项反映了NSF的法定使命，通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为是值得支持的。

Complex amphiphilic structures are closely related to phase transitions, materials science, and mathematical biology. This project concentrates on the influence of nanoparticles, proteins and hydrophobic surfaces on the morphology of amphiphilic structures. This research greatly advances the understanding of the functionality of nanoparticles, bio-inspired materials, and biological membranes. It helps us understand essential processes like protein transportation, drug encapsulation, and drug delivery. It also provides new methods and techniques for the modeling and analysis of properties of other complex structures in polymeric materials. The educational activities include the supervision and training of graduate research. The educational goal is for graduate students to be trained in mathematical modeling, and in the analysis and numerical simulation of partial differential equations.This research project concentrates on interdisciplinary research that combines applied mathematics and science, technology, and engineering. The PI is an applied mathematician, and the co-PI is a chemical experimentalist. The investigators model the encapsulation of nanoparticles in bilayer liposomes and polymeric micelles, the coating of biomaterials by amphiphilic copolymers, and the change of morphology caused by proteins, in the framework of the functionalized Cahn-Hilliard (FCH) energy, coupled with proper strong and weak anchoring conditions. This new mathematical framework translates the materials system into nonlinear PDEs coupled with proper boundary conditions. It has wide applications in the modeling of other physical, materials and biological systems. The PIs apply existing and new mathematical tools for the study of nonlinear PDEs. These tools lie in three areas: asymptotic analysis, variational methods, and numerical simulations. They are very useful 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.