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 Landaude 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 twodimensional 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.