Topological and Geometrical Problems in Soft Matter

软物质中的拓扑和几何问题

• 批准号: 1262047
• 负责人: Randall Kamien
• 金额: $ 57.5万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1262047&HistoricalAwards=false
• 关键词: Topological; Geometrical; Problems; Soft; Matter

TECHNICAL SUMMARYThe Division of Mathematical Sciences and the Division of Materials Research contribute funds to this award. It supports theoretical research and education to advance and elucidate the theory of knotted fields. Though it has been well known for more than three decades that topological defects can act on each other through commutation, it is only in recent times that we can: a) knot defects by design; b) tie and manipulate knotted disclinations; and c) study biaxial nematics, a physical system with a non-Abelian fundamental group. The stream of experimental results that has already started will be ample fodder for studying the topological dynamics of these knots. What makes this most exciting is that these are now questions that can be probed directly through experiment.A second thrust of the research is the topological characterization and classification of defects in translationally ordered media. The PI will focus specially on smectic liquid crystals because translational order is only broken in one direction and thus there is only one associated Nambu-Goldstone mode. In addition to dislocations and disclinations, smectics enjoy a different class of geometrically precise configurations, focal conic domains. The interaction between focal conic domains and topological defects is largely underexplored. Finally, and most importantly, because smectics are so soft, dislocations, disclinations and focal conic domains are easy to observe and manipulate in experiment. Building on the work of the PI and his group over the past three years, the PI will formulate a theoretical description to combine the three types of building blocks in the case of three-dimensional smectics. This award also supports the PI's efforts to convey the science of liquid crystals effectively to K-12 and college students, high school teachers, and the general public.NONTECHNICAL SUMMARYThe Division of Mathematical Sciences and the Division of Materials Research contribute funds to this award. It supports theoretical research and education to advance the theory of soft matter. The PI has a track record of learning and applying modern mathematics and incorporating it into his research. In turn, these insights have been applied directly to experiments that employ traditional microscopy, polarizing filters, and hot plates. The focus of this research project is on liquid crystals, materials that pervade our modern life, from our phones and watches to our computers and cars. The optical properties that make them the backbone of a $100 billion/year industry arise from a beautiful interplay between chemistry, theoretical physics, geometry, and topology. The PI will study the generalization of the fluid vortices that shed off the tip of a canoe paddle or an airplane wing. To do this, the PI will integrate the fields of materials physics, statistical mechanics, topology, and geometry, challenging traditional disciplinary lines with the hope of substantial technological and intellectual progress. Visually compelling, the field of liquid crystals has and will allow the PI to reach out effectively to K-12 and college students, high school teachers, and the general public.

技术总结数学科学部和材料研究部为该奖项提供资金。它支持理论研究和教育，推进和阐明结合场理论。虽然众所周知，拓扑缺陷可以通过对易相互作用，但直到最近，我们才能：a)通过设计来打结缺陷；b)捆绑和操纵打结的错位；以及c)研究双轴向列学，这是一个具有非阿贝尔基本群的物理系统。已经开始的一系列实验结果将为研究这些节点的拓扑动力学提供充足的素材。最令人兴奋的是，这些问题现在可以通过实验直接探索。研究的第二个重点是平移有序介质中缺陷的拓扑表征和分类。PI将特别聚焦于近晶液晶，因为平移有序只在一个方向上被破坏，因此只有一个相关的Nambu-Goldstone模。除了位错和位错外，近晶者还享有一类不同的几何精确构型，焦锥形域。焦锥形结构域与拓扑缺陷之间的相互作用在很大程度上还没有得到充分的研究。最后，也是最重要的是，由于近晶是如此柔软，位错、位错和焦点圆锥域在实验中很容易观察和操纵。在过去三年PI及其小组工作的基础上，PI将制定一个理论描述，在三维近景的情况下结合三种类型的积木。该奖项还支持国际学生联合会努力将液晶科学有效地传播给K-12和大学生、高中教师和普通公众。非技术总结数学科学部和材料研究部为该奖项提供资金。它支持理论研究和教育，推动软物质理论的发展。这位PI在学习和应用现代数学并将其纳入他的研究方面有着良好的记录。反过来，这些洞察力被直接应用于使用传统显微镜、偏振滤光片和加热板的实验。这项研究项目的重点是液晶，从我们的手机和手表到我们的电脑和汽车，液晶是渗透到我们现代生活的材料。使它们成为每年1000亿美元产业支柱的光学特性来自化学、理论物理、几何和拓扑学之间的美丽相互作用。PI将研究从独木舟桨尖或飞机机翼尖端脱落的流体涡流的概化。为了做到这一点，PI将整合材料物理、统计力学、拓扑学和几何领域，挑战传统的学科界限，希望取得实质性的技术和智力进步。在视觉上，液晶领域已经并将使PI能够有效地接触到K-12和大学生、高中教师和普通公众。