Liquid Crystals and Their Defects

Ball, J. M.

doi:10.1007/978-3-319-67600-5_1

Cited by 22 publications

(32 citation statements)

References 91 publications

(87 reference statements)

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“…This condition is compatible with (4) and large values of t if one works with materials for which B = o (C) and A < 0, C > 0 are fixed and consistent with (4). In particular, the physicality constraints on the eigenvalues can be respected with such an interpretation of the t → ∞ limit, for domains with ξ u ξ b D. Next, we briefly summarize the main results of this paper.…”

Section: Introductionmentioning

confidence: 77%

“…if and only if |Q| 6 = 6 trQ 3 2 [4] and we have maximal biaxiality β 2 = 1 if and only if trQ 3 = 0 which necessarily requires a vanishing eigenvalue. The LdG theory is a variational theory and has an associated LdG free energy.…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…The bulk potential, f B , is minimized by uniaxial tensors of a constant norm for low temperatures i.e. for A < 0, f B is minimized on the set of uniaxial tensors Q min , with |Q min | 2 = 2 3 s 2 + , and s + can be explicitly worked out in terms of A, B, C [4,22,27]. We work with large domains, ⊂ R 3 , whose characteristic length D is much larger than the temperature-dependent biaxial correlation length, ξ b = L Bs + [17,18] i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Let h + = 3+ √ 9+8t 4 and a direct calculation shows that s + = B 3C h + . Then the condition ξ u ξ b is equivalent to t h + or t 1, which in turn is equivalent to |T − T I N | |T I N − T * | where T I N is the isotropic-nematic transition temperature and T * is another critical temperature [17,18].…”

Section: Introductionmentioning

confidence: 99%

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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: Introductionmentioning

confidence: 77%

Section: Statement Of Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Uniaxial versus biaxial character of nematic equilibria in three dimensions

2017

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“…Notice that the assumptions (H 1 )-(H 3 ), (H 5 ) guarantee the excess decay estimate for local minimizers of functional of Uhlenbeck type with φ-growth, see [11]. Further, f B is the usual quartic thermotropic potential that dictates the isotropic-nematic phase transition as a function of the temperature [23,2]:…”

Section: Setting Of the Problem And Statement Of The Main Resultsmentioning

confidence: 99%

Minimizers of a Landau–de Gennes energy with a subquadratic elastic energy

Canevari

Majumdar

Stroffolini

2019

Arch Rational Mech Anal

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Ball in the occasion of his 70th birthday. Abstract We study a modified Landau-de Gennes model for nematic liquid crystals, where the elastic term is assumed to be of subquadratic growth in the gradient. We analyze the behaviour of global minimizers in two-and three-dimensional domains, subject to uniaxial boundary conditions, in the asymptotic regime where the length scale of the defect cores is small compared to the length scale of the domain. We obtain uniform convergence of the minimizers and of their gradients, away from the singularities of the limiting uniaxial map. We also demonstrate the presence of maximally biaxial cores in minimizers on two-dimensional domains, when the temperature is sufficiently low.

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Variational Analysis of Nematic Shells

Canevari

Segatti

2018

Springer INdAM Series

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In this paper we discuss the behavior of the Oseen-Frank model for nematic liquid crystals in the limit of vanishing thickness. More precisely, in a thin slab Ω × (0, h) with Ω ⊂ R 2 and h > 0 we consider the one-constant approximation of the Oseen-Frank model for nematic liquid crystals. We impose Dirichlet boundary conditions on the lateral boundary and weak anchoring conditions on the top and bottom faces of the cylinder Ω × (0, h). The Dirichlet datum has the form (g, 0), where g : ∂Ω → S 1 has non-zero winding number. Under appropriate conditions on the scaling, in the limit as h → 0 we obtain a behavior that is similar to the one observed in the asymptotic analysis (see [7]) of the two-dimensional Ginzburg-Landau functional. More precisely, we rigorously prove the emergence of a finite number of defect points in Ω having topological charges that sum to the degree of the boundary datum. Moreover, the position of these points is governed by a Renormalized Energy, as in the seminal results of Bethuel, Brezis and Hélein [7].

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Liquid Crystals and Their Defects

Abstract: Summary. These lectures describe some classical models of liquid crystals, the relations between them, and the different ways in which these models describe defects.

Cited by 22 publications

References 91 publications

Uniaxial versus biaxial character of nematic equilibria in three dimensions

Uniaxial versus biaxial character of nematic equilibria in three dimensions

Minimizers of a Landau–de Gennes energy with a subquadratic elastic energy

Variational Analysis of Nematic Shells

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