Liquid 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
