Many 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).
