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

Abstract

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...

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

“…In contrast, our method has the following advantages: (a) no assumption is made on the mesh structure (other than being shape regular); (b) the Ericksen energy can be very general (not just the one-constant approximation); (c) the method non-linearly couples full electro-statics, which was not done previously. Therefore, our result is more general than in [33,70,71].…”

“…We show that the finite element approximation of the discrete energy (3.18) Γ-converges to the continuous energy (2.42). The result presented here is not the same as the result shown in [70,71] or in [33], all of which used a special discretization of the Ericksen energy that is limited to the one constant approximation (5.2). Furthermore, their discretization requires the underlying mesh to be weakly acute in order to prove Γ-convergence of their method; the weakly acute assumption is quite severe for three-dimensional meshes [51,52,86].…”

“…We use an alternating direction minimization algorithm, similar to what is in [33,63,70,71], for finding discrete (local) minimizers of ℎ . In addition, we use a line search to ensure that the energy decreases at each step.…”

“…which is needed to ensure that does not trivially vanish on the interface, and so cause (2.30) to vanish as well [33,63,71]. The parameters ⊥ , ‖ , ori : Γ → [0, ∞) allow for different weighting and the ability to model more general physical settings; throughout the paper, we assume ⊥ , ‖ , ori in ∞ (Γ).…”

“…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]. The two main limitations of the approach in [33,63,70,71] are: (1) it is for the one-constant Ericksen model, and (2) the method requires the computational mesh to be weakly acute to guarantee convexity of the discrete energy.…”