Numerics for Liquid Crystals with Variable Degree of Orientation

Nochetto, Ricardo H.; Walker, Shawn W.; Zhang, Wujun

doi:10.1557/opl.2015.159

Cited by 7 publications

(11 citation statements)

References 27 publications

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“…A method was developed in [69,70] by the author and collaborators to solve the (one-constant) Ericksen model of nematic liquid crystals (summarized in Sect. 2.1.1).…”

Section: Introductionmentioning

confidence: 99%

“…2.1.1). A discrete form of the energy (5.2) was developed in [69,70] and shown to Γ-converge to (5.2); in addition, a method for computing discrete minimizers was given. This method was later extended to account for colloidal particle effects and external electric fields [71], as well as simulating liquid crystal droplets with anisotropic surface tension effects [33,63].…”

Section: Introductionmentioning

confidence: 99%

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A finite element method for the generalized Ericksen model of nematic liquid crystals

Walker

2020

ESAIM: M2AN

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We consider the generalized Ericksen model of liquid crystals, which is an energy with 8 independent “elastic”constants that depends on two order parameters n (director) and s (variable degree of orientation). In addition, we present a new finite element discretization for this energy, that can handle the degenerate elliptic part without regularization, with the following properties: it is stable and it Γ-converges to the continuous energy. Moreover, it does not require the mesh to be weakly acute (which was an important assumption in our previous work). Furthermore, we include other effects such as weak anchoring (normal and tangential), as well as fully coupled electro-statics with flexo-electric and order-electric effects. We also present several simulations (in 2-D and 3-D) illustrating the effects of the different elastic constants and electric field parameters.

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“…A method was developed in [69,70] by the author and collaborators to solve the (one-constant) Ericksen model of nematic liquid crystals (summarized in Sect. 2.1.1).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A finite element method for the generalized Ericksen model of nematic liquid crystals

Walker

2020

ESAIM: M2AN

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“…This section describes the mathematical formulation of the minimization problem for the one-constant Landau-deGennes energy E LdG under the uniaxiality constraint (2.4) (Section 3.1). The model we obtain has similarities with the Ericksen model [9,[43][44][45], but it has the advantage of allowing for non-orientable minimizers that exhibit half-integer order defects. This model is mainly of interest when d=3, since when d=2, Q necessarily has the form of a uniaxial tensor.…”

Section: The Uniaxially Constrained Q-modelmentioning

confidence: 99%

Accelerated Gradient Descent Methods for the Uniaxially Constrained Landau-de Gennes Model

Chukwuemeka¹,

Walker²

2022

AAMM

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“…with finite energy: E erk (s, n) < ∞. The parameter κ in (2.2) can influence the appearance of defects; see [37,38] for examples of this effect.…”

Section: Introductionmentioning

confidence: 99%

A Finite Element Method for a Phase Field Model of Nematic Liquid Crystal Droplets

Diegel¹,

Walker²

2019

CiCP

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We develop a novel finite element method for a phase field model of nematic liquid crystal droplets.The continuous model considers a free energy comprised of three components: the Ericksen's energy for liquid crystals, the Cahn-Hilliard energy representing the interfacial energy of the droplet, and a weak anchoring energy representing the interaction of the liquid crystal molecules with the surface tension on the interface (i.e. anisotropic surface tension). Applications of the model are for finding minimizers of the free energy and exploring gradient flow dynamics. We present a finite element method that utilizes a special discretization of the liquid crystal elastic energy, as well as mass-lumping to discretize the coupling terms for the anisotropic surface tension part. Next, we present a discrete gradient flow method and show that it is monotone energy decreasing. Furthermore, we show that global discrete energy minimizers Γ-converge to global minimizers of the continuous energy. We conclude with numerical experiments illustrating different gradient flow dynamics, including droplet coalescence and break-up. arXiv:1708.02513v2 [math.NA] 12 Sep 2017 into three parts: the mixing energy, the bulk free energy for liquid crystals, and an anchoring energy. For instance, in [58], Zhao et. al. develop an energy-stable scheme for a binary hydrodynamic phase field model of mixtures of nematic liquid crystals and viscous fluids where they use the Cahn-Hilliard energy to describe the mixing energy and the Oseen-Frank energy to describe the bulk free energy for liquid crystals. Defects are effectively regularized by penalizing the unit length constraint.The work presented herein is unique in the following sense: the Cahn-Hilliard energy is combined directly with Ericksen's energy in order to develop a phase field model for nematic liquid crystal droplets in a pure liquid crystal substance. The model considers a free energy which is comprised of three components: the Ericksen's energy for liquid crystals, the Cahn-Hilliard energy representing the interfacial energy of the droplet, and a weak anchoring energy representing the interaction of the liquid crystal molecules with the surface tension on the interface (which gives rise to anisotropic surface tension). The goal is to find minimizers of this free energy. To this end, we present a finite element discretization of the energy and apply a modified time-discrete gradient flow method to compute minimizers. In this way, the numerical scheme considered herein combines the finite element approximation of the Ericksen model of nematic liquid crystals in [37], which captures point and line defects and requires no regularization, and the technique considered in [24] which follows a convex splitting gradient flow strategy for modeling the Cahn-Hilliard equation.An outline of the paper is as follows. Section 2 describes the continuous energy model for the liquid crystal/surface tension system. In Section 3, we present a discretization of the total energy (2.12) followed by the deve...

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Numerics for Liquid Crystals with Variable Degree of Orientation

Cited by 7 publications

References 27 publications

A finite element method for the generalized Ericksen model of nematic liquid crystals

A finite element method for the generalized Ericksen model of nematic liquid crystals

Accelerated Gradient Descent Methods for the Uniaxially Constrained Landau-de Gennes Model

A Finite Element Method for a Phase Field Model of Nematic Liquid Crystal Droplets

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