Solution landscapes of the simplified Ericksen--Leslie model and its comparison with the reduced Landau--de Gennes model

Yucen Han, Jianyuan Yin, Yucheng Hu, Apala Majumdar, Lei Zhang

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Abstract

We investigate the solution landscapes of a simplified Ericksen--Leslie (sEL) vector model for nematic liquid crystals, confined in a two-dimensional square domain with tangent boundary conditions. An efficient numerical algorithm is developed to construct the solution landscapes by utilizing the symmetry properties of the model and the domain. Since the sEL model and the reduced Landau--de Gennes (rLdG) models can be viewed as Ginzburg--Landau functionals, we systematically compute the solution landscapes of the sEL model, for different domain sizes, and compare with the solution landscapes of the corresponding rLdG models. There are many similarities, including the stable diagonal and rotated states, bifurcation behaviors, and sub-solution landscapes with low-index saddle solutions. Significant disparities also exist between the two models. The sEL vector model exhibits the stable solution $C\pm$ with interior defects, high-index "fake defects" solutions, novel tessellating solutions, and certain types of distinctive dynamical pathways. The solution landscape approach provides a comprehensive and efficient way for model comparison and is applicable to a wide range of mathematical models in physics.
Original languageEnglish
Article number20210458
Number of pages21
JournalProceedings of the Royal Society Part A - Mathematical, Physical and Engineering Sciences
Volume477
Issue number2253
Early online date8 Sep 2021
DOIs
Publication statusPublished - 29 Sep 2021

Keywords

  • saddle point
  • Ericksen-Leslie model
  • model comparison
  • solution landscape
  • nematic liquid crystals
  • Landau-de Gennes model

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