ABSTRACT. Discoveries of new materials and their novel applications have brought fascinating new perspectives into the modeling and mathematical studies of liquid crystals. This lectures will begin with a survey of the calculus of variations methods applied to studying the non-convex energy of nematic configurations and their defects. Riding on symmetry changes in the material as phase transitions to increasingly ordered phases occur, we will explore mathematical problems arising in studies of chiral liquid crystals and the layered smectic phases. The underlying property of molecular polarization and its macroscopic effects will follow our developments. Some materials with bent-shape molecules present smectic phases with polarization strengths comparable to solid ferroelectric compounds. A new feature of such materials is the ability to change shape upon decreasing the temperature. We will also explore liquid crystal elastomers. These are materials consisting of polymeric networks able to deform upon alignment, and by application of electric field. Finally, we will address models of liquid crystal gels and their role in biological systems.
1. Nematic liquid crystals: Modeling and Mathematical
Some ordinary fluids have the capability of becoming orientationally ordered upon decreasing the temperature. Such materials consist of elongated and rigid molecules with some degree of polarization. The Van der Waals interaction between dipole moments together with steric effects, yields the orientationally aligned nematic phase. In this lecture, we will survey the modeling approaches to nematic, from the work of Oseen (1933) through the free energy formulation of Frank (1963), and the later contributions of de Gennes and Ericksen. We will present the mathematical studies dealing with existence of energy minimizing configurations corresponding to nematic phases and explore the role of defects. Stability analyses of nematic configurations with respect to applied electric fields make the modeling background of the Fredericks transition relevant to display applications. We will explore such topics in the context of composite polymer-liquid crystal systems.
2. Phase transitions in smectic liquid crystals
Nematic liquid crystals experience a transition to the positionally ordered smectic~A phase upon lowering the temperature. In this lecture, we will explore the phase transition from nematic to smectic~A in the case of chiral molecules. We will discuss the problem of minimizing the free energy emphasizing the competition between the tendency of molecules to form chiral configurations and that of becoming positionally ordered in layers. For certain values of the temperature, the outcome of such a competition gives rise to the twist grain boundary defects. These are analogs to the vortex phases of superconductors. We will present mathematical studies of twist grain boundaries within the modeling contexts of Landau, de Gennes and Lubensky.
3. Ferroelectric liquid crystals
4. Liquid crystal elastomers
This lecture addresses the effects of change of shape of molecules as orientation takes place. Many of the optic phenomena involving liquid crystals can be modeled under the assumption that liquid crystals are fluids with stiff, rod-like molecules that change very little upon aligning. In particular, this is the case of polymeric nematic liquid crystals, whose molecules posses anisotropic components that are able to form nematic and smectic phases as the concentration of solvent changes. We now consider the case that the polymer chains link together into gel networks, and the melt becomes an elastic solid, very much like a rubber. We will explore the properties that emerge from the ability of molecules to change their shape while in the solid states. We will explore the nature of the free energy and study the coupling between elastic deformation and nematic alignment.
5. Polymers and gels