Anisotropic diffusion dynamics in vector models for liquid crystal
Vector-based continuous models for nematic liquid crystals such as the Oseen-Frank model and the Ericksen model are relatively simpler compared with tensor-based models such as the Landau-de Gennes model.However,these vector models do not respect head-to-tail symmetry.As a result,they cannot predict configurations corresponding to non-orientable line fields,particularly the half-integer defects.This paper confirms a significant discrepancy between the transition dynamics predicted by the Oseen-Frank vector model and Landau-de Gennes tensor model for liquid crystals confined in a two-dimensional square well.The so-called inner product weighted Laplacian operator is introduced as an anisotropic diffusion operator to evolve the Euler-Lagrange equations corresponding to the modified Oseen-Frank model.Numerical results show that both the predicted equilibrium configurations and the transition dynamics from one equilibrium states to another satisfies head-to-tail symmetry and can accommodate half-integer defects.The connections of anisotropic diffusion operator to the graph Laplacian and the discrete Lebwohl-Lasher model are also discussed.The numerical trick proposed in this paper can be considered a simple remedy to restore head-to-tail symmetry in vector models of liquid crystals,making them more applicable in situations such as systems containing half-integer defects where the traditional numerical approach would fail.
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