LICI

Objective:

"The world of materials is undergoing a revolution and its main drivers are the complex materials that have a special composition at small scales, exhibiting microstructure. Liquid crystals are a paradigmatic example of complex material withmicrostructure, with extraordinarily successful applications, which is nevertheless still poorly understood at a basic, fundamental level. Nematic liquid crystals are the most relevant technologically and the most studied mathematically, being the center of interest of an active and diverse community. Their complexity raisese a number of challenging issues at the intersection of partial differential equations, topology, algebra, numerical analysis and modelling. Thus, because of these intricacies, almost all the current mathematical studies have focused on the simplest models. The current project aims to systematically address the next level of complexity studying the interactions between liquid crystals and other materials (surfaces: liquid or solid) or fields (light, electric or magnetic). These interactions are the most relevant technologically yet a systematic study of them is missing because of the complicated models that describe them. We will address key questions that provide crucial entry points in the understanding of these interactions. Specifically in the directions of interactions with materials we will study the surface energies, the effect they produce (in a droplet, free boundary problem) and how they can be designed to achieve apriorily desired form through manipulating rugosity of the boundary. Moreover, we will study the interactions between colloidal particles and the surrounding liquid crystal environment focusing both on the pairwise interactions and also on the large-scale homogenization effects. In the directions of interactions with fields we will study the effects of light in the creation of defects (shadow and anisotropic vortices) as well as geometrical optics in a nematic environment. Furthermore we will provide a general study of interactions between nematics, electric and magnetic field, and of the physically relevant reduced theories. From an applications point of view we will provide numerical simulations that will allow to implement, in collaboration with physicists, experiments checking the relevance of our mathematical studies to the design of surface energies, homogenized colloidal mixtures and optical lens."