After successful completion of the course, students are able to master several models for liquid crystalline materials, which involve nonlinear partial differential equations, nonlocal effects, and nonconvex constraints. The analytical and numerical techniques presented in the lecture can be applied to treat other mathematical models (e.g., phase-field models or models for magnetic materials).
Liquid crystals (LCs) are materials which exhibit properties intermediate between isotropic liquids and crystalline solids. For example, they flow like liquids, but their constituent molecules retain orientational order, which generates optical properties typical for solids. The study of LCs began in 1888, when the Austrian botanist Friedrich Reinitzer observed that cholesteryl benzoate exhibited a LC phase. Nowadays, LCs play an important role in modern technology, e.g., they find wide use in LC displays (LCDs), which rely on the optical properties of LCs in the presence or absence of an electric field. LCDs are used in a variety of devices such as digital clocks, mobile phones, calculating machines, appliances, and televisions.
Physical processes involving LCs are multiphysics phenomena and the mathematical tools developed for their comprehensive understanding combines methods from various fields, e.g., solid mechanics, elasticity, topology, partial differential equations, calculus of variations, and geometric measure theory. Moreover, the need of reliable numerical software to perform large-scale simulations of LC systems gave rise to the design and the analysis of several numerical approaches. In this lecture, we give an overview on the mathematics behind liquid crystals, touching on several topics, mostly in the fields of mathematical modeling, analysis, and numerics.
Topics and keywords:
Physical background and applications of LCs, classification of LCs, molecular vs. continuum theories, order parameters, defects, anchoring conditions, interaction with electric and magnetic fields. Director theory of liquid crystals: Oseen-Frank energy, Frank’s formula, Ericksen’s inequalities, harmonic mappings. Liquid crystals with variable degree of orientation: Ericksen energy, double-well potentials. Q-tensor theory of liquid crystals: Landau-deGennes energy, uniaxiality vs. biaxiality. Dynamic theories of liquid crystals: Ericksen-Leslie system, Beris-Edwards system. Numerical methods for nematic liquid crystals: finite element method, energy minimization, discrete gradient flows, structure-preserving methods, time-stepping schemes for LC dynamics.
Further topics could be also addressed depending on students' interests.
