Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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A non-traditional view on the modeling of nematic disclination dynamics


Authors: Chiqun Zhang, Xiaohan Zhang, Amit Acharya, Dmitry Golovaty and Noel Walkington
Journal: Quart. Appl. Math. 75 (2017), 309-357
MSC (2010): Primary 76A15
DOI: https://doi.org/10.1090/qam/1441
Published electronically: August 18, 2016
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Abstract: Non-singular disclination dynamics in a uniaxial nematic liquid crystal is modeled within a mathematical framework where the kinematics is a direct extension of the classical way of identifying these line defects with singularities of a unit vector field representing the nematic director. It is well known that the universally accepted Oseen-Frank energy is infinite for configurations that contain disclination line defects. We devise a natural augmentation of the Oseen-Frank energy to account for physical situations where, under certain conditions, infinite director gradients have zero associated energy cost, as would be necessary for modeling half-integer strength disclinations within the framework of the director theory. Equilibria and dynamics (in the absence of flow) of line defects are studied within the proposed model. Using appropriate initial/boundary data, the gradient-flow dynamics of this energy leads to non-singular, line defect equilibrium solutions, including those of half-integer strength. However, we demonstrate that the gradient flow dynamics for this energy is not able to adequately describe defect evolution. Motivated by similarity with dislocation dynamics in solids, a novel 2D-model of disclination dynamics in nematics is proposed. The model is based on the extended Oseen-Frank energy and takes into account thermodynamics and the kinematics of conservation of defect topological charge. We validate this model through computations of disclination equilibria, annihilation, repulsion, and splitting. We show that the energy function we devise, suitably interpreted, can serve as well for the modeling of equilibria and dynamics of dislocation line defects in solids, making the conclusions of this paper relevant to mechanics of both solids and liquid crystals.


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Additional Information

Chiqun Zhang
Affiliation: Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email: chiqunz@andrew.cmu.edu

Xiaohan Zhang
Affiliation: Department of Mechanical Engineering, Stanford University, Stanford, California, 94305
Email: xzhang11@stanford.edu

Amit Acharya
Affiliation: Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email: acharyaamit@cmu.edu

Dmitry Golovaty
Affiliation: Department of Mathematics, University of Akron, Akron, Ohio 44325
Email: dmitry@uakron.edu

Noel Walkington
Affiliation: Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email: noelw@math.cmu.edu

DOI: https://doi.org/10.1090/qam/1441
Received by editor(s): March 11, 2016
Published electronically: August 18, 2016
Additional Notes: The print version of this article is in black and white. Color is available online.
Article copyright: © Copyright 2016 Brown University


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