Applied Mathematics, Modeling, and Experimental and Computational Analysis of Liquid Crystals

Calderer 9704714 This project being jointly funded by the Office of Multidisciplinary Activities and by the Division of Mathematical Sciences of the Directorate for Mathematical & Physical Sciences and also by the Division of Civil and Mechanical Systems of the Directorate for Engineering. The proposed research addresses and combines mathematical and computational issues in the study of 'liquid crystals' and 'ferroic materials', both solids and fluids, within the context of industrial applications. One underlying theme of this proposal is to develop an understanding of static and flow patterns observed in liquid crystals and ferroic materials in the presence of external fields. (For instance, in dealing with ferroic solids, a goal is to achieve lower critical values of coercive fields in order to curb dissipation; in ferroelectric liquid crystals one also seeks to increase device switching speed.) While the occurrence of 'texture' and 'defects' tends to hinder the outcome of structural manufacturing processes, it may also bring out desirable features in optic applications, such as improvement of the accuracy and display memory of the devices. Within this framework, one can formulate a relevant common problem in terms of the control of texture and defects in solid and liquid crystal systems, with either the intent to enhance such structures or to eliminate them. It is the goal of this research team to contribute towards the understanding of such questions by exploring a broad spectrum of methods and techniques: modeling, mathematical, computational and experimental in liquid crystal and ferroic systems as a whole. It is planned to revise and, perhaps, unify some of the available descriptions so that the ferroelectric nature can be understood in a more universal manner. Such a study could ultimately suggest a way of designing new and better ferroelectric materials (solids and/or liquid crystals) with 'large polarization' and 'small coerci ve fields'. This proposal also addresses aspects of (non-Newtonian) liquid crystal flow such as defects and instabilities of various regimes. Special attention will be devoted to the analysis of processing flows of thin films as well as to the study of free-boundary problems. Results from singularity and bifurcation theory, and an appreciation of the role played by symmetry, have been integrated within the finite-element technique to establish the fundamental instability mechanisms of a number of classical Newtonian flows. These numerical techniques are now mature enough to apply to the study of more technologically important flows with more complex physics and multiple parameters. The proposed research deals with mathematical and computational modeling of 'liquid crystals' and 'ferroic materials' (both, solids and liquids), within the context of their industrial applications. In dealing with liquid crystals we intend to construct and analyze mathematical models of 'smart display devices', involving ferroelectric systems as well as composites. The main goal is to preserve good optical resolution with increasing display sizes, and achieve low switching values of electricor magnetic fields. As part of our research plan, we intend to draw already existing mathematical and physical information from the field of ferroic solids, in order to help us in the studies of ferroelectric liquid crystals. In turn, the ordering properties and larger degrees of freedom of liquid molecules may provide good feedback for improving the modeling and design of ferroelectric 'sensors' and 'transducers', with the special aim towards miniaturization of such devices. One of the main issues of our research is to study how to use polymeric liquid crystal materials to design typically solid devices. (The former materials are often cheaper to manufacture and to activate). From a different point of view, mathematical problems in flow modeling of ferroelectric liquids arise in the manufacturing processes of such materials, and bring a whole new class of mathematical questions that relate to those that traditionally arise in studies of Newtonian fluid flow. Overall, we intend to bring together independent fields of research and carry out a transfer of mathematical and computational methods among them based on exploiting physical and phenomenological analogies.