Some properties of the nematic radial hedgehog in the Landau–de Gennes theory

Lamy, Xavier

doi:10.1016/j.jmaa.2012.08.011

Cited by 30 publications

(36 citation statements)

References 13 publications

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“…The boundary-value problem (24) has been studied in detail, see for example in [19,23]. The RH solution is locally unstable with respect to biaxial perturbations, as has been demonstrated in [22,26].…”

Section: Statement Of Resultsmentioning

confidence: 97%

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Uniaxial versus biaxial character of nematic equilibria in three dimensions

2017

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We study global minimizers of the Landau-de Gennes (LdG) energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. Our results are specific to an asymptotic limit coined in terms of a dimensionless temperature and material-dependent parameter, t and some constraints on the material parameters, and we work in the t → ∞ limit that captures features of the widely used Lyuksyutov constraint (Kralj and Virga in J Phys A 34:829-838, 2001). We prove (i) that (re-scaled) global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting map; (ii) we have points of maximal biaxiality and uniaxiality near each singular point of the limiting map; (iii) estimates for the size of "strongly biaxial" regions in terms of the parameter t. We further show that global LdG minimizers in the restricted class of uniaxial Q-tensors cannot be stable critical points of the LdG energy in this limit.

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Section: Statement Of Resultsmentioning

confidence: 97%

“…with I [n] and A n b as in (19), for each j ∈ N. For j sufficiently large, Q j cannot be a stable critical point of the LdG energy in (17) in the admissible space,Ā =…”

Section: Statement Of Resultsmentioning

confidence: 99%

Uniaxial versus biaxial character of nematic equilibria in three dimensions

2017

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“…The nematics confined in a ball B R (0) := {|x| ≤ R} with homeotropic anchoring boundary condition is an important example to understand the local structures and locations of point defects with degree +1 in dimension three. There are many studies on the profiles of the solutions as well as their stabilities both theoretically and numerically [3,8,13,14,18,20] in this setting. It is known that one can find a radially symmetric uniaxial solution, which is explicitly given by…”

Section: Boundary Conditionsmentioning

confidence: 99%

On equilibrium configurations of nematic liquid crystals droplet with anisotropic elastic energy

Wang

Zhang

2017

Res Math Sci

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We investigate the effect of anisotropic elastic energy on defect patterns of liquid crystals confined in a three-dimensional spherical domain within the framework of Landau-de Gennes model. Two typical strong anchoring boundary conditions, namely homeotropic and mirror-homeotropic anchoring conditions, are considered. For the homeotropic anchoring, we find three different configurations: uniaxial hedgehog, ring and split-core, in both cases with or without the anisotropic energy. For the mirror-homeotropic anchoring, there are also three analogue solutions: the uniaxial hyperbolic hedgehog, ring and split-core for the isotropic energy case. However, when the anisotropic energy is taken into account, the numerical results and rigorous analysis reveal that the uniaxial hyperbolic hedgehog is no longer a solution. Indeed, we find ring solution only for negative L 2 (the elastic coefficient of the anisotropic energy), while both split-core and ring solutions can be stable minimizers for positive L 2 . More precisely, the uniaxial hyperbolic hedgehog for L 2 = 0 bifurcates to a split-core solution when L 2 increases and to a ring solution when L 2 decreases. This example shows that the anisotropic energy may significantly affect the symmetry of point defects with degree −1 whenever it is introduced.

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“…However for lower temperatures the melting hedgehog is not a minimizer (Gartland & Mkaddem [45]) and numerical evidence suggests a biaxial torus structure for the defect without melting. For other work on the description of the hedgehog defect according to the Landau -de Gennes theory see, for example, [51,52,57,60,66].…”

Section: Point Defectsmentioning

confidence: 99%