Uniaxial versus biaxial character of nematic equilibria in three dimensions

Henao, Duvan; Majumdar, Apala; Pisante, Adriano

doi:10.1007/s00526-017-1142-8

Cited by 41 publications

(37 citation statements)

References 35 publications

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“…Many research works, e.g. [1], [6], [8]- [10], [13]- [15], [22]- [25], [32], [34]- [38], [46]- [47], [52], [55], [59], focus on equilibrium solutions of the theories. These equilibrium solutions (particularly their core structures near disclinations) play key roles in understanding phases of liquid crystal materials at different temperatures.…”

Section: Introductionmentioning

confidence: 99%

Disclinations in Limiting Landau–de Gennes Theory

2020

Arch Rational Mech Anal

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In this article we study the low-temperature limit of a Landau-de Gennes theory. Within all S 2 -valued R-axially symmetric maps (see Definition 1.1), the limiting energy functional has at least two distinct energy minimizers. One minimizer has biaxial torus structure, while another minimizer has split-core segment structure on the z-axis.(1). u is isotropic at x if x is a point singularity of u (see Definition 1.2);(2). u is uniaxial at x if two of the three eigenvalues determined by u are identical and different from zero at x;(3). u is biaxial at x if all the three eigenvalues determined by u are different at x.Moreover a vector field is called director field determined by u if its values are normalized eigenvectors corresponding to the largest eigenvalue in λ 1 , λ 2 , λ 3 .With the above definitions, we can define the so-called biaxial torus and split-core line segment.Definition 1.4. Suppose that T is an open torus in B 1 and u is a given 3-vector field on T . T is called biaxial torus with uniaxial core determined by u if u is uniaxial on a closed simple loop in T . Meanwhile u is biaxial in T except the points on the closed simple loop. In this case the torus T is also called a biaxial torus of the augmented map L rus. Given a segment, we have a straight line, denoted by l, passing across the segment. The segment is called split-core segment associated with a 3-vector field u if there is a neighborhood, denoted by O, containing the segment so that u is uniaxial on O X l except the two end-points of the segment. Meanwhile u is isotropic at the two end-points of the segment and biaxial on O " l. Here and in what follows, we use " to denote the set minus. In this case L rus has split-core segment structure on l.

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Section: Introductionmentioning

confidence: 99%

Disclinations in Limiting Landau–de Gennes Theory

2020

Arch Rational Mech Anal

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“…for i, j, k = 1, 2, 3. The C 1,α -regularity of the weak solutions of the system (21) was first proven in the p-Laplacian case for p > 2 [30], for 1 < p < 2 see [1], for convex functions of general growth, see [25], and [11] where there is an excess decay. See Section 3.2 below for problems with a right-hand side.…”

Section: Preliminariesmentioning

confidence: 99%

“…We multiply both sides of the Euler-Lagrange equations (21) by L −q |Q − Π(Q)| q ν ij (Q) to get at the right-hand side (dropping the subscript L for brevity)…”

Section: Anmentioning

confidence: 99%

“…the limiting map is an excellent approximation of the LdG minimizers in this asymptotic limit, away from defects. In Contreras & Lamy [7] and Henao, Majumdar & Pisante [21], the authors study a different asymptotic limit, namely, the low temperature limit of minimizers of the LdG energy (with a one-constant elastic energy density) and prove that minimizers cannot have purely isotropic points with Q = 0 in this limit. Henao, Majumdar & Pisante demonstrate the uniform convergence of LdG minimizers to a minimizing harmonic map, away from the singularities of the limiting map, in this asymptotic limit.…”

Section: Introductionmentioning

confidence: 99%

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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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“…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%