|
Since their discovery at the end of the nineteenth century, liquid
crystals have been the source of important technological advances.
Liquid crystals consist of elongated and rigid molecules. In the
nematic phase, such molecules exhibit orientational order, whereas the
lower temperature smectic phases present orientational as well as
(one-dimensional) positional order of the centers of mass. Some liquid
crystals may also present chirality (e.g., the smectic~A* phase).
From a different point of view, the nature of the surface interaction
between the liquid crystal and the surrounding media is at the center
of most technological applications of such materials.
A great deal of mathematical effort in liquid crystal research has been devoted to the study of the nematic phase. New mathematical research addresses questions such as the modeling of chiral smectic phases, and the role of the boundary conditions and surface interactions of the liquid crystal with neighboring materials. Such properties are at the core of important new applications such as large video display devices and organic semiconductors. This talk will deal with both such questions, the modeling of chiral smectic phases as well as free boundary problems for nematic droplets. In particular, in the former case, analogies between the conductor-superconductor transition and the nematic-smectic~A* will be mathematically justified. |