Modeling, Computation, and Analysis of Complex Liquid Crystal Systems and Transitions

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GartlandDMS-0608670 The investigator carries out numerical modeling of problemsin two important new areas of the physics of liquid crystals: (1)mean-field theories for phases of biaxial and bent-coreliquid-crystal materials, and (2) numerical solution ofmacroscopic models for the dynamics and statics of point defectsin liquid-crystal systems (in the absence or presence of fluidflow). The investigator uses techniques from modern numericalbifurcation analysis together with the theories of singularitiesand bifurcations in the presence of symmetry to study the fullbifurcation and phase behavior of recent mean-field models forbiaxial materials, along with models that he and collaboratorsare developing for the important new class of bent-corematerials. The investigator adapts and extends methods fromcomputational fluids with moving and deforming bodies ("movingoverset grid methods") to provide effective techniques toaccurately model challenging problems involving the dynamics andstatics of point defects in macroscopic liquid-crystal directormodels, with application to surprising recent physicalexperiments on defect-pair annihilation in capillaries filledwith a liquid-crystal material. Liquid crystals are complex fluids that possess partialmolecular orientational order. They can be used to controllight, and this feature underlies many of their significanttechnological applications. In recent years, new liquid-crystalmolecules have been synthesized that exhibit certain specialphases. These new phases are termed "biaxial," because they havesecondary ordering properties in addition to those of the morecommon "uniaxial" materials. The liquid-crystal community hashigh hopes for potential new applications of these new materials. The theoretical and experimental study of biaxial liquid crystalsis at a relatively early stage. In this project the investigatoranalyzes a mathematical model for such materials thatcharacterizes the bulk equilibrium state of the system andassociated transitions, as a function of control parameters suchas temperature. This study requires both mathematical andnumerical techniques. In addition, the investigator andcollaborators develop new models for the special sub-class of"bent core" liquid crystals, which are of high current interest. The merits of this work are two-fold: (1) validation of themodels and (2) the development of predictive tools to aid in thedesign of experiments and in the identification of parameterranges for technological applications. The second main topic ofthe project concerns the numerical modeling of the dynamics ofdefects in liquid crystals (isolated singularities in the fieldsthat characterize the orientational properties of the medium). Defects are ubiquitous in liquid-crystal systems, and they causeconsiderable difficulty both from mathematical and numericalpoints of view. The investigator and co-workers apply numericalprocedures from other areas of computational fluid dynamics, bothto model numerically a target problem involving the annihilationof a pair of point defects in a capillary tube filled with liquidcrystal and to compare these numerical results to recentlypublished experimental data. These techniques (new to the areaof liquid crystals) should open the door to a number of othernumerical-modeling problems involving defects that havepreviously been hampered by unphysical numerical artifacts.