The investigator and his collaborators study the Smoluchowski equation arising in the kinetic molecular theory of nematic liquid crystalline polymers. The intermolecular interaction of excluded volume effects, which is responsible for the isotropic-nematic phase transition, is modeled by a mean-field potential. Both the Mayer-Saupe and the Onsager potentials are used. The main purpose of the project is a rigorous study of the dynamics and rheological properties of such systems. The problems to be addressed include bifurcation analysis of both the equilibrium and the shear flow case, and questions arising in the treatment of the equations by means of the classical theory of infinite dimensional systems. One of the central problems is the question of the existence of inertial manifolds on which the partial differential equation reduces to a finite system of ordinary differential equations, the so-called inertial form. The starting point of the project is recent results in which the isotropic-nematic phase transition was resolved for the Smoluchowski equation using the Mayer-Saupe potential in the equilibrium case (no external flow), as well as the existence of absorbing balls (cones) in Gevrey spaces of analytic functions. These questions remain open for the Onsager potential, and are addressed in the study. In the absence of the flow, the Smoluchowski equation is a gradient system that preserves uniaxial symmetry, and as such it exhibits fairly simple dynamical behavior. However, even a passive interaction with a symmetry-breaking shear flow introduces complicated dynamical features, such as different kinds of periodic solutions (tumbling, kayaking, log-rolling), and even chaos. These are characteristic for intermediate shear rates. At low and high shear rates flow aligning takes place. A rigorous investigation of these dynamical features is conducted in the project.
The main goal of the project is a mathematical understanding of the transition that occurs in a class of polymeric materials containing anisotropic molecules known as mesogens to a so-called liquid crystal phase. Due to their anisotropy, the molecules interact with each other, resulting in alignment. This long-range orientational order resembles the one found in solid crystals, while, unlike solid crystals, the material retains fluidity due to the absence of the long-range positional order of molecules. The transition to such a phase is induced by changes in the temperature and the concentration of the mesogens in the solution. The distribution of the molecules can be described by a partial differential equation known as the Smoluchowski equation, which is the main subject of this project. It was already employed successfully to explain the transition to the liquid crystal phase under some very special conditions, with many unanswered questions still remaining. The significance of these liquid crystalline polymers lies in the fact that crystal-like properties, combined with many of the useful and versatile properties of polymers, make these materials suitable for a wide range of important applications. For example, liquid crystalline polymers are abundant in living systems such as DNA, polypeptides, and cell membranes. Accordingly, they attract particular attention in the field of biomimetic chemistry. An application of liquid crystalline polymers that has been successfully developed for industry is the area of high strength fibers. Kevlar, which is used to make such things as helmets and bullet-proof vests, is just one example of the use of polymer liquid crystals in applications calling for strong, light-weight materials. The optical properties of liquid crystalline polymers lead to applications in optical imaging and the display industry. Liquid crystalline polymers can be used to coat drugs in the pharmaceutical industry. However, as very complex materials, liquid crystalline polymers exhibit very complex behavior, and many of the actual and potential applications of these materials depend on multidisciplinary research. A better mathematical understanding of the process of aligning is essential for better understanding of the behavior of liquid crystalline polymers. This project addresses some of the central questions regarding this process.
