Modeling And Computation Of Flowing Polymer-particulate Nanocomposites

Computational mathematics is the critical link between mathematics and the rest of the academic disciplines. Given the ever-increasing complexity in our research ob-jects, Computational mathematics requires significant amount of interdisciplinary inter-action with other disciplines in order to make impact to the general area of mathematics, science and engineering.Materials research is a typical interdisciplinary field involving the study of ma-terial's properties and their applications to various areas of science and engineering. Material research has become the next technological frontier nowadays. Rather than haphazardly looking for and discovering particular materials and exploiting their prop-erties, the modern material research aims at understanding so as to modifying materials and, furthermore, optimizing material combinations and their synergistic functions so that blur the distinction between a material and a functional device comprised of dis-tinct materials. Material research requires large amount of computations and numerical simulations.One class of the promising composite materials are made from blending polymers of distinctive properties in the liquid phase. Blends consisting of flexible polymers with rodlike or platelet nanoparticles (NPs) or nematic liquid crystal polymers (NL-CPs) have many industrial and military applications due to their potentially ultrahigh modulus and stiffness as well as chemical, biological and thermal properties [1,2]. Polymer-dispersed liquid crystals and polymer-stabilized small molecule liquid crys-tals have been used in electro-optical devices such as liquid crystal display devices, light shutters/switches, and protective goggles [3]. Polymer particulate nanocomposites (PNCs) such as clay/polymer nanocomposites, metal/polymer nanocomposites, carbon-nanotube/polymer nanocomposites etc. have emerged as a new class of materials with a great deal of promise for potential applications as high-performance and the most versa-tile industrial advanced materials [4-10]. For this reason polymer-particulate nanocom-posites have been a cult research area for the past 20 years. The main feature of polymer-particulate nanocomposites, in contrast to conven-tional composites, is that the particulate/reinforcement on the order of nanometer can deeply affect final macroscopic properties. More precisely, it is the formation of meso-scopic structures by the nano-size particles that largely dictates the extraordinary prop-erties of the nanocomposites. Given the promising applications of PNCs, a thorough understanding of their mesoscale dynamics, mesoscopic morphology development and evolution, and the full spectrum of rheological behavior in processing conditions be-comes important. Yet theoretical studies of these aspects of polymer-nanoparticulate blends are sparse. Most of the related studies have been on blends of polymers with a focus on thermodynamical phases [11]. A decade ago, Liu and Fredrickson devel-oped a mean field thermodynamic theory to study phase separation kinetics in polymer blends focusing on low frequency and long wave behavior [12]. Muratov and E derived a kinetic theory for the incompressible mixture of flexible polymers and rodlike liquid crystalline polymers, neglecting the conformational detail of the flexible polymer chain [13]. In both theories, the detailed conformational dynamics of the flexible polymers are ignored. Ginzburg et al investigated the phases of polymer clay mixtures using density function theories and arrived at various phase diagrams for various possible thermodynamical equilibrium phases in the nanocomposites [15,16]. A few years ago, Forest and Wang derived a kinetic theory for semi-dilute dispersions of rigid spheroidal macromolecules in a polymeric solvent to study dynamics of blends of polymers and liquid crystal polymers, in which conformational dynamics of the polymer chains are accounted for [14].In this Ph.D. dissertation, I will develop the kinetic theory for monodomains of PNCs accounting for nano-particle Brownian motion, excluded volume interactions, conformational dynamics of the polymer phase, semiflexibility of the nano-particle phase, and surface interaction between the flexible polymer and the nano-particle phase, extending the work of Forest and Wang. I will employ a closure approximation to solve the equation in the regime of weak semiflexibility of the nanoparticle and study the model prediction of the orientational phases. Furthermore, I will use direct numerical methods to solve the Smoluchowski equation for PNCs, and then explore the phase diagram and phase transitions predicted by the kinetic theory.