Finite element analysis of a nematic liquid crystal Landau-de Gennes model with quartic elastic terms

TL;DR

This work analyzes a quartic Landau–Gennes Q-tensor model for nematic liquid crystals and studies its gradient-flow dynamics. It introduces an energy-stable, fully discrete finite element scheme and proves a discrete energy dissipation law; it also establishes Γ-convergence of the discrete energies $\mathcal{F}_h$ to the continuous energy $\mathcal{F}$ when $L_5>0$, ensuring convergence of discrete minimizers as $h\to 0$. The authors prove well-posedness of the nonlinear scheme via a fixed-point contraction for small time steps and demonstrate convergence of the numerical method through 2D simulations of isotropic-to-nematic transitions and tactoid dynamics. The numerical results validate the approach, reveal the impact of quartic elastic terms, and provide a robust framework for simulating phase transitions in nematic LC with strongly anisotropic elastic constants.

Abstract

In arXiv:1906.09232v2, Golovaty et al. present a $Q$-tensor model for liquid crystal dynamics which reduces to the well-known Oseen-Frank director field model in uniaxial states. We study a closely related model and present an energy stable scheme for the corresponding gradient flow. We prove the convergence of this scheme via fixed-point iteration and rigorously show the $Γ$-convergence of discrete minimizers as the mesh size approaches zero. In the numerical experiments, we successfully simulate isotropic-to-nematic phase transitions as expected.

Figures (4)

• Figure 1: Simulated evolution of a degree 1 tactoid, in which the initial and boundary conditions are defined by $\mathbf{n}_0 = (-\sin \theta, \cos \theta)$ in (\ref{['tactoid_boundary']}).
• Figure 2: Simulated evolution of a degree -1 tactoid, in which the initial and boundary conditions are defined by $\mathbf{n}_0 = (-\cos \theta, \sin \theta)$ in (\ref{['tactoid_boundary']}).
• Figure 3: Simulated evolution of a degree 0 tactoid, in which the initial and boundary conditions are defined by $\mathbf{n}_0 = (1, 0)$ in (\ref{['tactoid_boundary']}).
• Figure 4: Energies of decaying tactoids over time for experiments outlined in Sections \ref{['sec:tactOne']} (blue), \ref{['sec:tactMinusOne']} (orange), and \ref{['sec:tactZero']} (green).

Theorems & Definitions (24)

• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• proof