Challenges in Systems with Semctic and Nematic Order

提议：DMS-0405853PI：Georg Dolzmann研究所：马里兰大学学院公园标题：近晶和向列相系统中的挑战摘要许多在自然界和现代技术应用中都可以找到的物理系统在微观尺度上具有内部自由度，对系统的宏观性质有巨大影响。在过去的几十年里，许多基础性研究都致力于发展数学理论，使人们能够“跨越尺度”，即以分析的方式理解与微观和宏观结构有关的机制，预测这些材料对外力的响应，并设计允许对系统进行有效模拟的算法策略。这项计划的成功完成将最终使人们能够为关键技术应用“量身定制--制造智能材料”。首席调查员计划在这一总框架内解决关键问题，特别强调两个具有极大技术应用潜力的特殊系统。第一个体系是一类特殊的弹性体，它结合了聚合物网络的熵响应和液晶的取向不稳定性。另一个系统是脂质双层膜(生物膜)，它构成了生物系统中所有膜的绝大多数。每一层中的分子倾向于具有向列有序，并且均匀地排列在宏观尺寸的区域中。关键问题涉及两层膜的内部结构对膜的宏观形状和弹性的影响、膜内区的形成及其稳定性。潜在的应用包括人造血细胞和药物输送机制。从一般数学框架的观点出发，这一建议涉及到对这些系统建模过程中出现的变分问题的分析，并解决了与变分积分的结构性质密切相关的解的存在性和正则性问题。更好地理解分析方面也是上述系统数值模拟的有效算法的核心，预计分析和数值相结合的方法将导致对复杂物理系统的理解取得重大进展。许多现代材料具有显著的弹性性质，允许新的技术应用。这些令人惊讶的特性通常与内部自由度和在材料中形成的图案(所谓的微结构)有关，这些图案的尺度比样品本身的尺寸(纳米尺度)小得多。这项建议旨在开发用于预测和模拟这些效应的分析和数值工具，特别强调两个特殊系统：一类橡胶类材料(可能用作人造肌肉或导光装置)和生物膜(可能应用包括药物输送机制)。

Proposal: DMS-0405853PI: Georg DolzmannInstitution: University of Maryland College ParkTitle: Challenges in Systems with Smectic and Nematic OrderABSTRACTMany physical systems that can be found both in nature and in modern technological applications possess internal degrees of freedom at a microscopic scale with tremendous impact on the macroscopic properties of the system. A lot of fundamental research has been directed in the past decades towards developing mathematical theories that allow one to `bridge the scales', i.e., to understand analytically the mechanisms that relate microscopic and macroscopic structures, to predict the response of these materials to applied forces, and to design algorithmic strategies that allow an efficient simulation of the system. A successful completion of this program would ultimately allow one to `tailor--make smart materials' for key--technological applications. The principal investigator plans to addresses key questions within this general framework with special emphasis on two particular systems with excellent potential for technological applications. The first system is a special class of elastomers that combine the entropic response of polymer networks with the orientational instabilities of liquid crystals. The other systems are lipid bilayer membranes (bio-membranes), which form the vast majority of all membranes in biological systems. The molecules in each of the layers tend to have a nematic order and are uniformly arranged in domains of macroscopic sizes. Key issues concern the influence of the internal structure of the two layers on the macroscopic shape and elasticity of the membranes, the formation of domains within the layer, and their stability. Potential applications include artificial blood cells and mechanisms for drug delivery. From the point of view of the general mathematical framework, this proposal is concerned with the analysis of variational problems that arise in the modeling of these systems and addresses questions of existence and regularity of solutions which are closely related to structural properties of the variational integrals. An improved understanding of the analytical aspects is also at the heart of efficient algorithms for the numerical simulation of the systems mentioned above, and it is expected that a combined analytical and numerical approach will lead to significant progress in the understanding of the complex physical systems.Many modern materials have striking elastic properties that allow novel technological applications. Frequently these surprising properties are related to internal degrees of freedom and patterns (so-called microstructures) that form in the materials at a scale much smaller than the size of the sample itself (nanoscale). This proposal aims at developing analytical and numerical tools for the prediction and simulation of these effects, with special emphasis on two particular systems, a class of rubber-type materials (with potential applications as artificial muscles or light guiding devices) and bio-membranes (possible applications include drug delivery mechanisms).