Modeling and Computation of Liquid Crystals

Wang, Wei; Zhang, Lei; Zhang, Pingwen

doi:10.48550/arxiv.2104.02250

“…The simplest kernel of the Onsager functional is the dipolar potential k(t) = −σt. K has an only negative eigenvalue − 4 3 πσ, with three linearly independent eigenvectors v(p) = p i , (i = 1, 2, 3); all the other eigenvalues are zero (with infinite multiplicity). Therefore, the isotropic state ρ 0 is a minimizer for σ < 3, and a 3-saddle for σ > 3.…”

Section: Solution Landscapementioning

confidence: 99%

“…In 1949, Onsager proposed a free-energy model to describe the isotropic-nematic phase transition in three-dimensional rod-like liquid crystals, and related the equilibrium states to critical points of the functional [1]. Since then, this molecular model has been applied to describe static and dynamic phenomena of liquid crystals [2,3,4]. By including the positional dependence of the distribution function, the Onsager model can describe complicated phenomena of liquid crystals [5,6,7,8].…”

Section: Introductionmentioning

confidence: 99%

Solution landscape of the Onsager model identifies non-axisymmetric critical points

Yin,

Zhang,

Zhang

2021

Preprint

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We investigate the solution landscapes of the Onsager free-energy functional with different potential kernels, including the dipolar potential, the Maier-Saupe potential, the coupled dipolar/Maier-Saupe potential, and the Onsager potential. A uniform sampling method is implemented for the discretization of the Onsager functional, and the solution landscape of the Onsager functional is constructed by using the saddle dynamics coupled with downward/upward search algorithms. We first compute the solution landscapes with the dipolar and Maier-Saupe potentials, for which all critical points are axisymmetric. For the coupled dipolar/Maier-Saupe potential, the solution landscape shows a novel non-axisymmetric critical point, named as tennis, which exists for a wide range of parameters. We further demonstrate various non-axisymmetric critical points in the Onsager functional with the Onsager potential, including square, hexagon, octahedral, cubic, quarter, icosahedral, and dodecahedral states. The solution landscape provides an efficient approach to show the global structures as well as the bifurcations of critical points, which can not only verify the previous analytic results but also propose several conjectures based on the numerical findings.

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“…In recent decades, several mathematical theories for NLCs, from microscopic models to macroscopic models, have been proposed [1,6,7]. Microscopic Onsager models with different potential kernels, for instance, have been applied to describe the static and dynamic phenomena of liquid crystals [8,9,10,11,12].…”

Section: Introductionmentioning

confidence: 99%

Nematic Liquid Crystals in a Rectangular Confinement: Solution Landscape, and Bifurcation

Shi

¹

,

Han

²

,

Zhang

³

2022

SIAM J. Appl. Math.

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We study the solution landscape and bifurcation diagrams of nematic liquid crystals confined on a rectangle, using a reduced two-dimensional Landau-de Gennes framework in terms of two geometry-dependent variables: half short edge length λ and aspect ratio b. First, we analytically prove that, for any b with a small enough λ or for a large enough b with a fixed domain size, there is a unique stable solution that has two line defects on the opposite short edges. Second, we numerically construct solution landscapes by varying λ and b, and report a novel X state, which emerges from saddle-node bifurcation and serves as the parent state in such a solution landscape. Various new classes are then found among these solution landscapes, including the X class, the S class, and the L class. By tracking the Morse indices of individual solutions, we present bifurcation diagrams for nematic equilibria, thus illustrating the emergence mechanism of critical points and several effects of geometrical anisotropy on confined defect patterns.

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“…In recent decades, several mathematical theories for NLCs, from microscopic models to macroscopic models, have been proposed [1,6,7]. Microscopic Onsager models with different potential kernels, for instance, have been applied to describe the static and dynamic phenomena of liquid crystals [8,9,10,11,12].…”

Section: Introductionmentioning

confidence: 99%

Nematic liquid crystals in a rectangular confinement: solution landscape and bifurcation

Shi¹,

Han²,

Zhang³

2021

Preprint

Self Cite

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We study the solution landscape and bifurcation diagrams of nematic liquid crystals confined on a rectangle, using a reduced two-dimensional Landau-de Gennes framework in terms of two geometry-dependent variables: half short edge length λ and aspect ratio b. First, we analytically prove that, for any b with a small enough λ or for a large enough b with a fixed domain size, there is a unique stable solution that has two line defects on the opposite short edges. Second, we numerically construct solution landscapes by varying λ and b, and report a novel X state, which emerges from saddle-node bifurcation and serves as the parent state in such a solution landscape. Various new classes are then found among these solution landscapes, including the X class, the S class, and the L class. By tracking the Morse indices of individual solutions, we present bifurcation diagrams for nematic equilibria, thus illustrating the emergence mechanism of critical points and several effects of geometrical anisotropy on confined defect patterns.

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Modeling and Computation of Liquid Crystals

Cited by 3 publications

References 169 publications

Solution landscape of the Onsager model identifies non-axisymmetric critical points

Solution landscape of the Onsager model identifies non-axisymmetric critical points

Nematic Liquid Crystals in a Rectangular Confinement: Solution Landscape, and Bifurcation

Nematic liquid crystals in a rectangular confinement: solution landscape and bifurcation

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