Singular limits, saturation, and defects in block copolymer morphology

RENDMS-0907777 A两嵌段共聚物分子是由ANA-单体嵌段共价接枝到B-单体嵌段上的线性链。由于不同单体之间的排斥力，不同类型的亚链往往会分离，但由于它们是链分子中的化学键，亚链的分离不会导致宏观相分离。只发生局部微相分离：产生富含A单体的微区和富含B单体的微区。由微区形成的Anano尺寸的图案被称为非晶相。研究人员开发了一种奇异极限方法来研究形态相，这些相集中在空间中的点、曲线和曲面上，而现有的自由能方法可能无法找到这些相。这些方法将复杂的非线性、非局部、变分和偏微分方程问题简化为更简单的几何问题。他们分析了饱和现象：少量大物体的伸长、变形和断裂形成大量小物体的过程。它们还解释了由局部-非局部竞争或拓扑约束引起的形态相缺陷。在嵌段共聚泡囊的情况下，研究了自由能下的弯曲刚性。奇异极限技术被推广到没有变分结构的问题，如发育中的生物形态发生的Gierer-Meinhardt系统。嵌段共聚物是一种柔软的凝聚材料，与晶态固体不同，它在分子尺度上具有类流体无序性，在较长尺度上具有高度有序性。现代纳米技术可以设计出几乎无限数量的分子结构，以制造具有特殊机械、电学、势垒、离子和其他物理性质的材料。普通盒式胶带使用三嵌段共聚物来实现压敏粘接。嵌段共聚物在道路施工中与沥青混合，以减少路面在低温和高温极端情况下的开裂和车辙。在这个项目中，研究人员研究嵌段共聚物中与材料形态相变化相关的图案形成，从而与更大规模的材料性能有关。该项目包括研究生，他们发展数学和材料科学方面的技能和知识。

RenDMS-0907777 A diblock copolymer molecule is a linear chain of anA-monomer block grafted covalently to a B-monomer block. Becauseof the repulsion between the unlike monomers, the different typesub-chains tend to segregate, but as they are chemically bondedin chain molecules, segregation of sub-chains cannot lead to amacroscopic phase separation. Only a local micro-phaseseparation occurs: micro-domains rich in A monomers andmicro-domains rich in B monomers emerge as a result. Anano-sized pattern formed from micro-domains is known as amorphology phase. The investigator develops singular limitmethods to study morphology phases that concentrate on points,curves, and surfaces in space, and that might not be found byexisting free energy methods. These methods reduce complicatednonlinear, nonlocal, variational, and partial differentialequation problems to simpler geometric problems. They analyzethe saturation phenomenon: a process of elongation, deformation,and breaking off of a small number of large objects to form alarge number of small objects. They also explain defects inmorphological phases caused by local-nonlocal competition ortopological constraints. In the case of block copolymer vesiclesthe investigator studies the bending rigidity in the free energy. The singular limit techniques are extended to problems withoutvariational structures, such as the Gierer-Meinhardt system forbiological morphogenesis in development. Block copolymers are soft condensed materials that incontrast to crystalline solids, are characterized by fluid-likedisorder on the molecular scale and a high degree of order on alonger length scale. An almost unlimited number of moleculararchitectures can be designed by modern nano-technologies toproduce materials with particular mechanical, electric, barrier,ionic and other physical properties. Common box tapes usetriblock copolymers to achieve pressure-sensitive adhesion. Block copolymers are blended with asphalt in road construction toreduce pavement cracking and rutting at low and high temperatureextremes. In this project the investigator studies patternformations within block copolymers that are related to changes inthe morphology phase of the material, and hence to larger-scalematerial properties. The project includes graduate students, whodevelop skills and knowledge in both mathematics and materialsscience.