Instability of point defects in a two-dimensional nematic liquid crystal model

Ignat, Radu; Nguyen, Luc; Slastikov, Valeriy; Zarnescu, Arghir

doi:10.1016/j.anihpc.2015.03.007

Cited by 40 publications

(90 citation statements)

References 30 publications

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“…Thus defects in the Landau -de Gennes theory are not described by singularities in Q. This has been explored for the hedgehog defect by many authors (see, for example, [75,86,91,98,107,109,128]), and for defects of other index in [99,100].…”

Section: Description Of Defects In the Landau -De Gennes Modelmentioning

confidence: 99%

Mathematics and liquid crystals

Ball

2017

Molecular Crystals and Liquid Crystals

127

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Section: Description Of Defects In the Landau -De Gennes Modelmentioning

confidence: 99%

Mathematics and liquid crystals

Ball

2017

Molecular Crystals and Liquid Crystals

127

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“…In this case, if a = 0, i.e., if the escape points coincide at the origin, then the conformal boundary condition n b is m-radial [22,26], and…”

Section: 1mentioning

confidence: 99%

Landau-de Gennes Corrections to the Oseen-Frank Theory of Nematic Liquid Crystals

Fratta¹,

Robbins²,

Slastikov³

et al. 2020

Arch Rational Mech Anal

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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 Γ-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.Landau-de Gennes corrections to the Oseen-Frank theory of nematic liquid crystals 2 subject to Dirichlet boundary conditions n| ∂Ω = n b , where the K j 's are material-dependent constants. For mathematical analysis, the one-constant approximation, K 1 = K 2 = K 3 , is often adopted, according to which the Oseen-Frank energy reduces to the Dirichlet energy, with harmonic maps as critical points.One shortcoming of this description is that in certain domains, the director field n is more appropriately represented by an RP 2 -valued map, stemming from the fact that orientations n and −n are physically indistinguishable. In simply-connected domains, a continuous RP 2 -valued map n can be lifted to a continuous S 2 -valued map, in which case we say that n is orientable. However, in non-simply-connected domains, this may not hold, in which case we say that n is non-orientable; see [3] for further discussion, where the notion of orientability is extended to n ∈ W 1,p (Ω, RP 2 ).Another difficulty is the description of defect patterns. These are singularities in the director field, which correspond physically to sharp changes in orientational ordering on a microscopic length scale. It is well known that boundary conditions can force the director field to have singularities. This occurs, for example, when Ω is a three-dimensional domain with boundary homeomorphic to S 2 and the boundary map n b : ∂Ω → S 2 has nonzero degree. In this case, in spite of the singularity, the infimum Oseen-Frank energy is finite. The difficulty is more acute when the boundary data n b : ∂Ω → RP 2 is non-orientable. In this case, the Oseen-Frank energy is

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“…with r(s), θ(t) as in (18), and ε := ξ/η → 0. The boundary conditions induced from Q ξ | ∂Ω 3 , given in terms of the director field, are:…”

Section: The Case λ = ∞mentioning

confidence: 99%

Spherical Particle in Nematic Liquid Crystal Under an External Field: The Saturn Ring Regime

2018

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We consider a nematic liquid crystal occupying the exterior region in R 3 outside of a spherical particle, with radial strong anchoring. Within the context of the Landau-de Gennes theory, we study minimizers subject to a strong external field, modelled by an additional term which favors nematic alignment parallel to the field. When the external field is high enough we obtain a scaling law for the energy. The energy scale corresponds to minimizers concentrating their energy in a boundary layer around the particle, with quadrupolar symmetry. This suggests the presence of a Saturn ring defect around the particle, rather than a dipolar director field typical of a point defect.

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Instability of point defects in a two-dimensional nematic liquid crystal model

Cited by 40 publications

References 30 publications

Mathematics and liquid crystals

Mathematics and liquid crystals

Landau-de Gennes Corrections to the Oseen-Frank Theory of Nematic Liquid Crystals

Spherical Particle in Nematic Liquid Crystal Under an External Field: The Saturn Ring Regime

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