We present a numerical method, based on a tensor order parameter description of a nematic phase, that allows fully anisotropic elasticity. Our method thus extends the Landau-de Gennes Q-tensor theory to anisotropic phases. A microscopic model of the nematogen is introduced (the Maier-Saupe potential in the case discussed in this paper), combined with a constraint on eigenvalue bounds of Q. This ensures a physically valid order parameter Q (i.e., the eigenvalue bounds are maintained), while allowing for more general gradient energy densities that can include cubic nonlinearities, and therefore elastic anisotropy. We demonstrate the method in two specific two dimensional examples in which the Landau-de Gennes model including elastic anisotropy is known to fail, as well as in three dimensions for the cases of a hedgehog point defect, a disclination line, and a disclination ring. The details of the numerical implementation are also discussed.
Bibliographical noteFunding Information:
C.D. Schimming acknowledges financial support by NSF DMR-1838977.J. Vi?als acknowledges financial support by NSF DMR-1838977.S.W. Walker acknowledges financial support by the NSF DMS-1555222 (CAREER).
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- Finite element method
- Landau-de Gennes
- Liquid crystals
- Singular bulk potential