On Ericksen’s model for liquid crystals

Lin, Fucai; Poon, Chi Cheung

doi:10.1007/bf02921587

Cited by 16 publications

(11 citation statements)

References 7 publications

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“…Certainly it is not unreasonable to think that flows or other influences could convert a rather stable nematic configuration to one of the biaxial type, etc. I [19] am one of those who have argued that, near isotropic-nematic phase transitions, it should be quite easy to induce such changes. Accounting for such possibilities does add significant complications to the equations and the problems of analyzing them.…”

Section: Introductionmentioning

confidence: 99%

Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond

Majumdar¹,

Zarnescu²

2009

Arch Rational Mech Anal

227

419

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We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in W 1,2 , to a global minimizer predicted by the Oseen-Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen-Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau-De Gennes global minimizer.We also study the interplay between biaxiality and uniaxiality in Landau-De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions.

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Section: Introductionmentioning

confidence: 99%

Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond

Majumdar¹,

Zarnescu²

2009

Arch Rational Mech Anal

227

419

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“…Hence, as in Ericksen [12] and Lin-Poon [33,34], we can reorganize the expression of W 2 into the form…”

Section: Descriptions Of Main Theoremsmentioning

confidence: 99%

“…either under well prepared Dirichlet boundary values (t ǫ , g ǫ ) when Ω ⊂ R 3 is a bounded smooth domain, or under the volume constraint for nematic region when Ω = R 3 , as ǫ → 0. Notice that for any fixed ǫ > 0, the existence and regularity of minimizer (s ǫ , n ǫ ) to (1.15) have been studied by Lin [29,30], Lin-Poon [33] and Ambrosio [2,3]. For a bounded smooth Ω ⊂ R 3 , we prescribe (t ǫ , g ǫ ) : ∂Ω → R × S 2 as follows.…”

Section: Descriptions Of Main Theoremsmentioning

confidence: 99%

Isotropic-Nematic Phase Transition and Liquid Crystal Droplets

Lin¹,

Wang²

2020

Preprint

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Liquid crystal droplets are of great interest from physics and applications. Rigorous mathematical analysis is challenging as the problem involves harmonic maps (and in general the Oseen-Frank model), free interfaces and topological defects which could be either inside the droplet or on its surface along with some intriguing boundary anchoring conditions for the orientation configurations. In this paper, through a study of the phase transition between the isotropic and nematic states of liquid crystal based on the Ericksen model, we can show, when the size of droplet is much larger in comparison with the ratio of the Frank constants to the surface tension, a Γ-convergence theorem for minimizers. This Γ-limit is in fact the sharp interface limit for the phase transition between the isotropic and nematic regions when the small parameter ε, corresponding to the transition layer width, goes to zero. This limiting process not only provides a geometric description of the shape of the droplet as one would expect, and surprisingly it also gives the anchoring conditions for the orientations of liquid crystals on the surface of the droplet depending on material constants. In particular, homeotropic, tangential, and even free boundary conditions as assumed in earlier phenomenological modelings arise naturally provided that the surface tension, Frank and Ericksen constants are in suitable ranges.

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“…One can translate the problem of minimizers (s, d) of Ericksen's energy functional (1.6) into the problem of minimizing harmonic maps (s, u), with potentials w 0 , min Ω ((k − 1)|∇s| 2 + |∇u| 2 + w 0 (s)) dx (1.7) subject to the constraint |s| = |u| and (s, u) = (s 0 , u 0 ) on ∂Ω. Then the problem (1.7) is essentially a minimizing harmonic map into a circular cone [18] (see also [19][20][21][22][23] for further related works). Theorem 1.6.…”

Section: Example 13 ([1012]mentioning

confidence: 99%

Recent developments of analysis for hydrodynamic flow of nematic liquid crystals

Lin

Wang

2014

Phil. Trans. R. Soc. A.

115

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The study of hydrodynamics of liquid crystals leads to many fascinating mathematical problems, which has prompted various interesting works recently. This article reviews the static Oseen-Frank theory and surveys some recent progress on the existence, regularity, uniqueness and large time asymptotic of the hydrodynamic flow of nematic liquid crystals. We will also propose a few interesting questions for future investigations.

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On Ericksen’s model for liquid crystals

Cited by 16 publications

References 7 publications

Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond

Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond

Isotropic-Nematic Phase Transition and Liquid Crystal Droplets

Recent developments of analysis for hydrodynamic flow of nematic liquid crystals

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