DMREF/Collaborative Research: Materials engineering of chromonic and colloidal liquid crystals via mathematical modeling and simulation

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This research project is at the intersection of the fields of physics of nonlinear phenomena, applied mathematics, nonlinear analysis, and computation. Knowledge of liquid-crystal-based suspensions is currently advancing quite rapidly, motivated by applications in materials science as well as in biological systems. At a fundamental level, and in contrast with the disordered nature of normal suspending fluids, nematic order in a liquid crystalline matrix leads to long range elastic interactions either among colloidal particles or with bounding walls, resulting in a variety of unexpected phenomena. Furthermore, order in the matrix is distorted by the suspended particles, resulting in unavoidable topological defects that must move with the particles. On the one hand, the existence of structure in the liquid matrix affords new opportunities for flow control, processing, and suspension stability. At the same time, and for the same reasons, efficient engineering of these systems requires major advances to our current understanding of simple fluid colloids. From proposals for new display technologies and nanofluidic devices to more fundamental questions about the mechanisms of clustering and de-clustering in systems of particles, new experimental findings call for major modeling and analysis efforts. For example, studies of electrophoresis in structured media can facilitate related efforts in biology to model and control nano-fluidic transport as well as contribute towards understanding of motion of cancer cells and their clustering in tumor metastasis. This project addresses these important challenges through the formulation, analysis, and simulation of variational models of liquid crystalline colloids that allow for the presence of defects. Technology transfer is another component of the proposed research. Improved understanding of liquid crystal anchoring and defect dynamics will allow for higher resolution, faster display devices. The research aims to develop a predictive theory of transport in suspensions within an anisotropic liquid crystalline matrix, including electrostatically charged particles and ions. The particles can have arbitrary shapes, be rigid or soft, charged or electrically neutral, or be domains of the isotropic-nematic phase transition (chromonics). Analysis and computation will be used to explore both static and time dependent problems. Primarily variational methods will be employed, either within energy minimization for static problems or within the minimum dissipation principle for time dependent problems. Novel theoretical aspects include comprehensive models of colloidal systems in structured media that incorporate elasticity of the nematic matrix, surface anchoring, electric field, ions, and flow and their interplay. The relative importance of these effects will be established via the feedback between the modeling and experimental components of this project. A significant feature that determines the behavior of nematic liquid crystalline colloids is that the suspended particles are accompanied by topological defects in the nematic matrix. As singular structures, defects are inherently difficult to handle from a mathematical point of view, however they must be incorporated into any physically correct model. The principal challenge and contribution of the proposed work is formulation, analysis, and simulation of variational models of liquid crystalline colloids that allow for the presence of defects. Among the questions to be addressed are those of modeling of observed nonlinear electrophoresis and particle levitation, investigation of nematic domains in isotropic lyotropic chromonic liquid crystal, and modeling of the experimentally observed motion of disclination curves accompanied by negligible or no flow.