Minimizers of the Landau–de Gennes Energy Around a Spherical Colloid Particle

Alama, Stan; Bronsard, Lia; Lamy, Xavier

doi:10.1007/s00205-016-1005-z

Cited by 33 publications

(54 citation statements)

References 38 publications

(143 reference statements)

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“…By this limit,(2.17),(1) in(2.22) and lower-semi continuity, it followsρ Dσ kˇD ξ u 8ˇ2 ď lim inf nÑ8 ż Dσ kˇD ξ u nˇ2¨`xn,1`rn ξ 1˘. xn,1`rn ξ 1 , y n`ǫn u n˘Àµ, ρ˚ˆr n ǫ n˙2ÝÑ 0, as n Ñ 8.…”

mentioning

confidence: 85%

“…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%

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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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confidence: 85%

Section: Introductionmentioning

confidence: 99%

Disclinations in Limiting Landau–de Gennes Theory

2020

Arch Rational Mech Anal

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“…An interesting area of current activity in which topology plays a role is that of nematic shells (see, for example, [38,39,154,160,161]). Related to this work is the interest in the topology of disclination lines induced by colloidal systems (see, for example, [8,51,132,166,168]). Other interesting connections with topology appear in [3,4].…”

Section: Omissionsmentioning

confidence: 99%

Mathematics and liquid crystals

Ball

2017

Molecular Crystals and Liquid Crystals

127

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“…a. Adding simple boundaries (1:24 -1: 32) The "Add a simple object" button allows the user to easily add a limited number of predefined types of boundary conditions to the simulation. The user can add a spherical colloid (specifying its position and radius) or a flat wall which has a normal in thex,ŷ, orẑ direction and sits on a particular lattice plane.…”

Section: Data Availability Statementmentioning

confidence: 99%

Fast, Scalable, and Interactive Software for Landau-de Gennes Numerical Modeling of Nematic Topological Defects

Sussman

Beller

2019

Front. Phys.

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Numerical modeling of nematic liquid crystals using the tensorial Landau-de Gennes (LdG) theory provides detailed insights into the structure and energetics of the enormous variety of possible topological defect configurations that may arise when the liquid crystal is in contact with colloidal inclusions or structured boundaries. However, these methods can be computationally expensive, making it challenging to predict (meta)stable configurations involving several colloidal particles, and they are often restricted to system sizes well below the experimental scale. Here we present an open-source software package that exploits the embarrassingly parallel structure of the lattice discretization of the LdG approach. Our implementation, combining CUDA/C++ and OpenMPI, allows users to accelerate simulations using both CPU and GPU resources in either single-or multiplecore configurations. We make use of an efficient minimization algorithm, the Fast Inertial Relaxation Engine (FIRE) method, that is well-suited to large-scale parallelization, requiring little additional memory or computational cost while offering performance competitive with other commonly used methods. In multi-core operation we are able to scale simulations up to supra-micron length scales of experimental relevance, and in single-core operation the simulation package includes a user-friendly GUI environment for rapid prototyping of interfacial features and the multifarious defect states they can promote. To demonstrate this software package, we examine in detail the competition between curvilinear disclinations and point-like hedgehog defects as size scale, material properties, and geometric features are varied. We also study the effects of an interface patterned with an array of topological point-defects.

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Minimizers of the Landau–de Gennes Energy Around a Spherical Colloid Particle

Abstract: International audienc

Cited by 33 publications

References 38 publications

Disclinations in Limiting Landau–de Gennes Theory

Disclinations in Limiting Landau–de Gennes Theory

Mathematics and liquid crystals

Fast, Scalable, and Interactive Software for Landau-de Gennes Numerical Modeling of Nematic Topological Defects

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