Reconstruct Morphological Phases from Nonlocal Geometric Systems

模式的形成是自组织原则的有序结果。形态相是复杂的几何结构，在物理和生物系统中形成为结构化的模式。例子包括嵌段共聚物的形态阶段、动物皮毛和细胞发育中的皮肤色素沉着。在这些图案形成系统中，常见的是偏离同质性对其进一步增加具有强烈的正反馈。此外，图案的形成为局部自增强过程提供了更长范围的限制。对花样形成的研究揭示了这些体系的力学、光学、电学、离子、势垒等性质。在这个项目中，研究人员研究具有多种成分的生物和物理系统中的形态相。通常，这些阶段是在实验中观察到的，目前的理论或模型都没有预测到它们。研究人员的目标是进行严格的分析，不仅要预测这些结构的存在，还要预测它们的特殊性质。形态相在现代材料中很重要，例如嵌段共聚物，它们被商业用作热塑性弹性体：酒瓶塞、光缆外罩、粘合剂、沥青改性剂或人造器官技术。研究生和本科生参与了该项目的工作。研究人员研究物理和生物系统中的形态阶段。特别感兴趣的体系是Ohta-Kawasaki、Fitzhugh-Nagumo、Gierer-Meinhardt以及三嵌段和四嵌段共聚物。通常，形态相由由狭窄界面隔开的微域组成。通常，窄界面的形态相可以由其尖锐的界面限制来近似，其中成分被没有厚度的界面完全分开。在尖锐界面的情况下，本项目研究圆环、Hopf链接、双气泡和三气泡的组装。发展了以尖锐界面形态相为骨架重构窄界面形态相的分析方法。通过重整化和无限维约化技术，库仑力被视为长程相互作用，得到了低维结构的自由能。研究生和本科生参与了该项目的工作。

Pattern formation is an orderly outcome of self-organization principles. Morphological phases are complex geometric structures that form as structured patterns in physical and biological systems. Examples include morphological phases in block copolymers, animal coats, and skin pigmentation in cell development. Common in these pattern-forming systems is that a deviation from homogeneity has a strong positive feedback on its further increase. In addition, pattern formation provides a longer ranging confinement of the locally self-enhancing process. Studying pattern formation reveals the mechanical, optical, electrical, ionic, barrier, and other properties of these systems. In this project, the investigator studies morphological phases in biological and physical systems having multiple constituents. Often these phases have been observed in experiments where no current theory or models predict them. The investigator aims for a rigorous analysis that not only predicts the existence of these structures but also their particular properties. Morphological phases are important in modern materials such as block copolymers, which are commercially used as thermoplastic elastomers: wine bottle stoppers, outer coverings for optical fibre cables, adhesives, bitumen modifiers, or in artificial organ technology. Graduate and undergraduate students participate in the work of the project.The investigator studies morphological phases in physical and biological systems. Systems of particular interest are the Ohta-Kawasaki, FitzHugh-Nagumo, Gierer-Meinhardt, and tri- and tetrablock copolymers. Typically a morphological phase consists of micro-domains separated by narrow interfaces. Often, narrow-interface morphological phases can be approximated by their sharp interface limits, where constituents are fully separated by interfaces with no thickness. In the sharp interface case this project investigates assemblies of tori, Hopf links, double bubbles, and triple bubbles. Analytical methods are developed to use sharp interface morphological phases as skeletons to reconstruct morphological phases with narrow interfaces. Through renormalization and an infinite-dimensional reduction technique, where Coulomb forces are treated as long-range interactions, free energies of lower-dimensional structures are derived. Graduate and undergraduate students participate in the work of the project.