Mathematical Modeling of Advanced Liquid Crystal Systems: Ferroelectricity and Chirality

DMS奖摘要奖#：0128832PI：Calder，Maria-Carme研究所：明尼苏达大学双子城项目：应用数学项目经理：Catherine Mavriplis标题：高级液晶系统的数学建模：铁电性和手性液态晶体是介于固体和液体之间的中间相，具有丰富的流变性和电光性质。新的技术应用涉及高度有序的相，如铁电近晶相~C*。液晶中的铁电性是相的手性的结果(即，分子以螺旋图案排列)。手性也被证明是所有生命有机体的核心属性。这一事实为液晶研究带来了一个生物学的范围。这项建议在光学和流变学的应用背景下，涉及先进液晶材料的数学和建模问题。铁电性和手性是提出的模型的核心。铁电显示器的建模是本课题的研究内容之一。这种显示器的切换时间比其他液晶技术要短得多(例如，快门能够提供70微秒的过渡时间)；它还允许顺序着色。该项目还涉及液晶薄膜研究的相关当前趋势，以期实现生物应用。一种这样的趋势涉及对设备小型化的需求，例如制造非常薄的铁电显示器(厚度从10到100纳米)。在交互式视频应用中尤其需要这些组件，其中一些显示组件旨在可穿戴。这项研究的一个相关方面是研究液晶和各向同性流体之间的界面和接触面，以及近晶液晶相作为表面活性剂的作用。这些问题在生物膜和过程的研究中也是相关的，例如药物通过细胞壁的运输，以及蛋白质与DNA菌株的附着。流动问题的分析将包括单轴和双轴向列相、手性和近晶液晶和聚合物。该研究所还将开展与液晶在光开关和电信方面的应用相关的研究，并将在这些课题上寻找产业合作伙伴关系。这项工作将结合现代的偏微分方程组和变分理论。还将对其中的一些问题进行数值模拟。国际数学联合会将通过以下方式继续积极地将研究活动与本科教育联系起来：夏季REU计划，组织工业研讨会，参与和组织面向本科生和高中生的开放参观活动，第一年研讨会，以及将最先进的计算机软件和技术引入本科数学。自19世纪末发现液晶以来，液晶一直是最令人兴奋的科学事件的核心，并带来了20世纪真正非凡的技术进步之一。它们存在于最基本的家居用品中，如手表和计算器，到计算机屏幕和仪器显示面板(LCD)。随着20世纪中叶第一个显示设备的商业化，液晶的物理和数学模型在20世纪最后25年的技术发展中发挥了重要作用。此外，连续介质力学的液晶模型提出的数学挑战带来了大量的分析和偏微分方程式的活动。这些成就反过来又推动了进一步的技术发展。液晶研究的未来预示着科学技术进步的新水平，但也充满了挑战。提出的研究目标是积极参与这一努力，走在现代液晶研究的前沿。这项建议在光学和流变学的应用背景下，涉及先进液晶材料的数学和建模问题。这项工作将结合现代的偏微分方程组和变分理论。一些问题也将进行数值模拟。日期：2001年6月25日

DMS Award AbstractAward #: 0128832PI: Calderer, Maria-CarmeInstitution: University of Minnesota, Twin Cities Program: Applied MathematicsProgram Manager: Catherine MavriplisTitle: Mathematical Modeling of Advanced Liquid Crystal Systems: Ferroelectricity and ChiralityLiquid crystal phases are intermediate between solid and liquid and present a wealth of rheological and electro-optical properties. Novel technological applications involve highly ordered phases such as the ferroelectric smectic~C*. Ferroelectricity in liquid crystals is a consequence of the chirality of the phase (i.e., molecules are arranged in helical patterns). Chirality also turns out to be a central attribute of all living organisms. This fact brings a biological scope to liquid crystals research. This proposal deals with mathematical and modeling issues of advanced liquid crystal materials, in the context of applications to optics and rheology. Ferroelectricity and chirality are at the core of the proposed models. One of the issues of the project is the modeling of ferroelectric display. The switching time of such a display is much less than that of other liquid crystal technologies (e.g., shutters are capable of a 70 microseconds transition time); it also allows for sequential coloring. The project also addresses related current trends of thin film research for liquid crystals, with a view towards biological applications. One such trend relates to the need of miniaturization of devices, such as the manufacturing of very thin ferroelectric displays (from 10 to 100 nano-meters thickness). These are particularly needed in interactive video applications, where some of the display components are intended to be wearable. One related aspect of the proposed research is the study of interfaces and contact surfaces between liquid crystals and isotropic fluids, and the role of smectic liquid crystal phases as surfactants. Such questions are also relevant in studies of biological membranes and processes, such as the transport of drugs through cell walls, and the attachment of proteins to strains of DNA. The analysis of flow problems will encompass uniaxial and biaxial nematic, chiral and smectic liquid crystals and polymers. The PI will also carry out research related to applications of liquid crystals to optical switching and telecommunications, and will search for industrial partnership in such topics. The work will incorporate modern theories of partial differential equations and calculus of variations. Numerical simulations will also be carried out for some of the problems. The PI will continue the very active program of connecting research activities with undergraduate education through the following: the summer REU program, organization of industrial seminars, participation and organization of open houses for undergraduate students as well as for high school students, first year seminars, and bringing state of the art computer software and technology to undergraduate mathematics. Since their discovery at the end of the nineteen century, liquid crystals have been at the core of the most exciting scientific events, and have brought one of the truly remarkable technological advances of the twentieth century. They are found in the most basic household items, such as watches and calculators, to computer screens and instrument display panels (LCD). Following the commercialization of the first display device towards the mid 20th century, the physical and mathematical modeling of liquid crystals has played a major role in their technological development, during the last quarter of the 20th century. Moreover, the mathematical challenges posed by the liquid crystal models of continuum mechanics brought a wealth of activity in analysis and partial differential equations. These achievements, in turn, prompted further technological developments. The future of the liquid crystal research promises new levels of scientific and technological advances, but it is also filled with challenges. The goal of the proposed research is to actively participate in this endeavor and work on the forefront of the modern liquid crystal research. This proposal deals with mathematical and modeling issues of advanced liquid crystal materials, in the context of applications to optics and rheology. The work will incorporate modern theories of partial differential equations and calculus of variations. Numerical simulations will also be carried out for some of the problems.Date: June 25, 2001