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1. Geometry and mechanics of disclination lines in 3D nematic liquid crystals

   https://doi.org/10.1039/D0SM01899F  

   Long, Cheng; Tang, Xingzhou; Selinger, Robin L.; Selinger, Jonathan V. (March 2021, Soft Matter)

   null (Ed.)

   In 3D nematic liquid crystals, disclination lines have a range of geometric structures. Locally, they may resemble +1/2 or −1/2 defects in 2D nematic phases, or they may have 3D twist. Here, we analyze the structure in terms of the director deformation modes around the disclination, as well as the nematic order tensor inside the disclination core. Based on this analysis, we construct a vector to represent the orientation of the disclination, as well as tensors to represent higher-order structure. We apply this method to simulations of a 3D disclination arch, and determine how the structure changes along the contour length. We then use this geometric analysis to investigate three types of forces acting on a disclination: Peach–Koehler forces due to external stress, interaction forces between disclination lines, and active forces. These results apply to the motion of disclination lines in both conventional and active liquid crystals. 

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2. A tensor density measure of topological charge in three-dimensional nematic phases

   https://doi.org/10.1098/rspa.2023.0564  

   Schimming, Cody D; Vinals, Jorge (May 2024, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   A path-independent measure in order parameter space is introduced such that, when integrated along any closed contour in a three-dimensional nematic phase, it yields the topological charge of any line defects encircled by the contour. A related measure, when integrated over either closed or open surfaces, reduces to known results for the charge associated with point defects (hedgehogs) or Skyrmions. We further define a tensor density, the disclination density tensor $\mathbf{D}$, from which the location of a disclination line can be determined. This tensor density has a dyadic decomposition near the line into its tangent and its rotation vector, allowing a convenient determination of both. The tensor $\mathbf{D}$may be non-zero in special configurations in which there are no defects (double-splay or double-twist configurations), and its behaviour there is provided. The special cases of Skyrmions and hedgehog defects are also examined, including the computation of their topological charge from $\mathbf{D}$. 

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3. Kinematics and dynamics of disclination lines in three-dimensional nematics

   https://doi.org/10.1098/rspa.2023.0042  

   Schimming, Cody D.; Viñals, Jorge (May 2023, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   An exact kinematic law for the motion of disclination lines in nematic liquid crystals as a function of the tensor order parameter $\text{Q}$is derived. Unlike other order parameter fields that become singular at their respective defect cores, the tensor order parameter remains regular. Following earlier experimental and theoretical work, the disclination core is defined to be the line where the uniaxial and biaxial order parameters are equal, or equivalently, where the two largest eigenvalues of $\text{Q}$cross. This allows an exact expression relating the velocity of the line to spatial and temporal derivatives of $\text{Q}$on the line, to be specified by a dynamical model for the evolution of the nematic. By introducing a linear core approximation for $\text{Q}$, analytical results are given for several prototypical configurations, including line interactions and motion, loop annihilation, and the response to external fields and shear flows. Behaviour that follows from topological constraints or defect geometry is highlighted. The analytic results are shown to be in agreement with three-dimensional numerical calculations based on a singular Maier–Saupe free energy that allows for anisotropic elasticity. 

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4. Tunable three-dimensional architecture of nematic disclination lines

   https://doi.org/10.1073/pnas.2300833120  

   Modin, Alvin; Ash, Biswarup; Ishimoto, Kelsey; Leheny, Robert L; Serra, Francesca; Aharoni, Hillel (July 2023, Proceedings of the National Academy of Sciences)

   Disclination lines play a key role in many physical processes, from the fracture of materials to the formation of the early universe. Achieving versatile control over disclinations is key to developing novel electro-optical devices, programmable origami, directed colloidal assembly, and controlling active matter. Here, we introduce a theoretical framework to tailor three-dimensional disclination architecture in nematic liquid crystals experimentally. We produce quantitative predictions for the connectivity and shape of disclination lines found in nematics confined between two thinly spaced glass substrates with strong patterned planar anchoring. By drawing an analogy between nematic liquid crystals and magnetostatics, we find that i) disclination lines connect defects with the same topological charge on opposite surfaces and ii) disclination lines are attracted to regions of the highest twist. Using polarized light to pattern the in-plane alignment of liquid crystal molecules, we test these predictions experimentally and identify critical parameters that tune the disclination lines’ curvature. We verify our predictions with computer simulations and find nondimensional parameters enabling us to match experiments and simulations at different length scales. Our work provides a powerful method to understand and practically control defect lines in nematic liquid crystals. 

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5. Reconfiguration of Nematic Disclinations in Plane-Parallel Confinements

   https://doi.org/10.3390/cryst13060904  

   Harkai, Saša; Rosenblatt, Charles; Kralj, Samo (June 2023, Crystals)

   We study numerically the reconfiguration process of colliding m=1/2 strength disclinations in an achiral nematic liquid crystal (NLC). A Landau–de Gennes approach in terms of tensor nematic-order parameters is used. Initially, different pairs m1,m2 of parallel wedge disclination lines connecting opposite substrates confining the NLC in a plane-parallel cell of a thickness h are imposed: {1/2,1/2}, {−1/2,−1/2} and {−1/2,1/2}. The collisions are imposed by the relative rotation of the azimuthal angle θ of the substrates that strongly pin the defect end points. Pairs {1/2,1/2} and {−1/2,−1/2} “rewire” at the critical angle θc1=3π4 in all cases studied. On the other hand, two qualitatively different scenarios are observed for {−1/2,1/2}. In the thinner film regime hhc, the colliding disclinations at θc2 reconfigure into boojum-like twist disclinations. 

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