Landau-de Gennes corrections to the Oseen-Frank theory of nematic liquid crystals

Abstract

We study the asymptotic behavior of the minimisers of the Landau-de Gennes model for nematic liquid crystals in a two-dimensional domain in the regime of small elastic constant. At leading order in the elasticity constant, the minimum-energy configurations can be described by the simpler Oseen-Frank theory. Using a refined notion of $\Gamma$-development we recover Landau-de Gennes corrections to the Oseen-Frank energy. We provide an explicit characterisation of minimizing $Q$-tensors at this order in terms of optimal Oseen-Frank directors and observe the emerging biaxiality. We apply our results to distinguish between optimal configurations in the class of conformal director fields of fixed topological degree saturating the lower bound for the Oseen-Frank energy.

References in corpus (10)

• Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond
• Orientability and energy minimization in liquid crystal models
• Stability of the melting hedgehog in the Landau-de Gennes theory of nematic liquid crystals
• Analysis of Nematic Liquid Crystals with Disclination Lines
• On Minimizers of the Landau-de Gennes Energy Functional on Planar Domains
• Refined approximation for a class of Landau-de Gennes energy minimizers
• Half-integer point defects in the $Q$-tensor theory of nematic liquid crystals
• Scalings and Limits of Landau-deGennes Models for Liquid Crystals: A Comment on Some Recent Analytical Papers
• Line defects in the small elastic constant limit of a three-dimensional Landau-de Gennes model
• Uniaxial symmetry in nematic liquid crystals

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• Spherical Particle in Nematic Liquid Crystal with a Magnetic Field and Planar Anchoring
• Tetrahedral frame fields via constrained third order symmetric tensors