Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation

Barrett, John W.; Feng, Xiaobing; Prohl, Andreas

doi:10.1051/m2an:2006005

Cited by 19 publications

(20 citation statements)

References 6 publications

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“…Some are based on harmonic mappings [7,1,15,19,23] where a unit vector field (called the director field) is used to represent the orientation of liquid crystal molecules; in these methods, they only model the equilibrium state. Some methods for the dynamics of liquid crystals can be found in [18,24,28].…”

Section: Introductionmentioning

confidence: 99%

Numerics for Liquid Crystals with Variable Degree of Orientation

Nochetto

Walker

Zhang³

2015

MRS Proc.

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We consider the simplest one-constant model, put forward by J. Eriksen, for nematic liquid crystals with variable degree of orientation. The equilibrium state is described by a director field n and its degree of orientation s, where the pair (n, s) minimizes a sum of Frank-like energies and a double well potential. In particular, the Euler-Lagrange equations for the minimizer contain a degenerate elliptic equation for n, which allows for line and plane defects to have finite energy. Using a special discretization of the liquid crystal energy, and a strictly monotone energy decreasing gradient flow scheme, we present a simulation of a plane-defect in three dimensions to illustrate our method.

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Section: Introductionmentioning

confidence: 99%

Numerics for Liquid Crystals with Variable Degree of Orientation

Nochetto

Walker

Zhang³

2015

MRS Proc.

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“…Subsequently there have been many developments and refinements. Various mixed methods have been considered in [2,3,13]; spectral schemes have been developed in [6]; methods with estimates independent of the penalty parameter were developed in [3,27]; and "semi-implicit" schemes which only require the solution of a linear system at each time step are developed in [13]. All of these schemes for this simplified model utilize the first order implicit Euler stepping scheme to evolve the solution in time.…”

Section: Related Resultsmentioning

confidence: 99%

“…The numerical approximation of solutions to the Lin-Liu equations has been studied extensively [2,6,13,[27][28][29]. These papers consider approximations of the weak statement…”

Section: Lin-liu Equationmentioning

confidence: 99%

Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations

Walkington¹

2010

ESAIM: M2AN

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Abstract. Numerical approximation of the flow of liquid crystals governed by the Ericksen-Leslie equations is considered. Care is taken to develop numerical schemes which inherit the Hamiltonian structure of these equations and associated stability properties. For a large class of material parameters compactness of the discrete solutions is established which guarantees convergence.Mathematics Subject Classification. 76A15, 65M12, 65M60, 76M10.

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“…At submicrometre and micrometre scales, phenomenological continuum approaches based on the free energy functionals become reliable. They use experimentally measured phenomenological material constants and usually need to be solved numerically, by approaches such as the finite difference (37)(38)(39)(40), finite element (41) or lattice Boltzmann methods (42). Alternatively to free-energy-based approaches, density functional theory can be applied where liquid crystal is considered to be a suspension of hard-core particles, for example ellipsoids (43).…”

Section: Introductionmentioning

confidence: 99%

Colloidal structures in confined nematic liquid crystals

Ravnik

2011

Liquid Crystals Today

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Colloidal structures in confined nematics offer novel routes for designing complex optical materials with micrometre and submicrometre functionality. In this paper, we review some of our recently assembled colloidal structures that form in confined nematic cells. We present effective elastic binding via nematic distortion as a mechanism for the assembly of two-dimensional colloidal crystals of elastic dipoles and elastic quadrupoles. We introduce entangled colloids as novel types of structures, where particles are topologically bound by delocalised defect loops, producing robust and possibly chiral structures. The concept of hierarchical assembly is demonstrated in colloids with particles of various scales. In cholesteric blue phases, the assembly of three-dimensional colloidal crystals is shown based on naturally occurring three-dimensional arrays of trapping sites produced by blue phases.Keywords: colloids; nematic liquid crystal; self-assembly; entanglement; blue phases IntroductionLiquid crystal colloids are novel materials (1-3) with promising applications in optics, including photonic crystals (4), tunable lasers (5) and metamaterials (6). They can be used to design optical patterns at scales ranging from nanometres to micrometres, using selfassembly (7), application of external fields (8, 9) and trapping by domain boundaries (10).The current main research focus in liquid crystal colloids is on nematic-based materials, which can provide for large-scale assembly of chains (2), clusters (11), two-dimensional (2D) lattices at interfaces (12, 13) and 2D colloidal crystals (7). Additionally, such colloidal assemblies have been functionalised by using facetted particles (14, 15), ellipsoids (16), rods (17) and Janus particles (18). Recently, chiral cholesteric and blue phase colloids are also attracting growing interest. Self-organisation of planar particles in cholesteric (19), size-dependent defects and non-Stokesian microrheology (20), stabilisation of blue phases by nanoparticles (21, 22) and stabilisation of blue phases by guest components (23) have been demonstrated. This growing interest in blue phases was started by applications and recent advances in materials synthesis. Blue phase materials with a wide temperature range have been synthesised (24), where flexoelectricity was interpreted as an important contributor to the phase stabilisation (25). Blue phase materials can yield high switching times, which is attractive for display applications (26).

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Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation

Cited by 19 publications

References 6 publications

Numerics for Liquid Crystals with Variable Degree of Orientation

Numerics for Liquid Crystals with Variable Degree of Orientation

Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations

Colloidal structures in confined nematic liquid crystals

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