Landau-de Gennes numerical simulation of nematic liquid crystals utilizing radial basis functions

Abstract

Numerical simulations based on radial basis functions have been developed for systems with complex geometries and have been successfully applied across various fields, including seismology, coastal hydrodynamics, and biology. However, examples in liquid crystal modeling are limited. In this study, we present a Landau-de Gennes numerical simulation of nematic liquid crystals utilizing radial basis functions, emphasizing its advantages over traditional cubic grid calculations, such as enhanced geometric flexibility and improved computational efficiency. Through simulations of liquid crystal-colloid systems with diverse geometries, we demonstrate that our approach effectively captures the essential topological and energetic features of liquid crystal equilibrium structures. Additionally, we introduce an adaptive node refinement scheme that is crucial for resolving the fine structure of singular defects in nematic liquid crystals.

Figures (5)

• Figure 1: Example superpositions of radial basis functions in one- (a) and two- (b) dimensional systems.
• Figure 2: Radial-basis-function simulation of nematic LCs with handlebody geometries. (a) Simulation of a nematic LC initially aligned along $\hat{\bf{n}}_0$ and then perturbed by a torus-shaped particle with planar degenerate surface anchoring $\theta_\rm{e}=90 °$. Nematic director field is rendered using cylinder rods, colored by their $x$ projection. (b,c) THe corresponding polarizing optical microscopy (POM) simulations with (b) our without (c) a half-wave plate. (d-f) Simulations with a handlebody of genus-5 with planar degenerate boundary conditions, visualized using cylinders (d) and POMs (e,f). (g,h) The corresponding experimental POM micrographs of nematic LCs near the genus-5 surfaces. Polarizers (P) and anayzers (A) are marked in each optical images and the green lines at 45° ° indicate the optical axis of half-wave plates when used.
• Figure 3: Radial-basis-function simulation of nematic LCs droplets confined in genus 2 and 4 surfaces. (a) RBFFD calculation of a nematic LC confined to the volume of a genus-2 handlebody with the LC-channel interfaces have planar surface anchoring $\theta_\rm{e}=90 °$. Inset shows the colorscheme for the $x$-$y$ azimuthal angle $\phi$ of the nematic director. (b,c) The corresponding POM simulations with (b) or without (c) a half-wave plate. (d) The experimental realization of the nematic LC droplet within the figure-8-shaped geometry imaged using POM. (e-h) Similarly presented simulation (e-g) and experiment (h) of LC in tubes with genus-4 surface. Polarizers (P) and anayzers (A) are marked in each optical images and the green lines at 45° ° indicate the optical axis of half-wave plates if used.
• Figure 4: Radial-basis-function-based numerical simulation of LC with colloidal spheres of different anchoring conditions. (a-c) Computed nematic director $\hat{\bf{n}}$ field around a sphere immersed in a uniformly aligned nematic director field $\hat{\bf{n}}_0$, showing elastic quadrupoles for homeotropic $\theta_\rm{e}=0 °$ (a) and degenerate planar $\theta_\rm{e}=90 °$ (c) anchoring, and a hexadecapole for degenerate conic anchoring $\theta_\rm{e}=45 °$ (b). Equilibrium surface anchoring angle $\theta_\rm{e}$ between molecular director $\hat{\bf{n}}$ and surface normal vector are marked for each case. Local orientations of the director field are visualized on the spherical surfaces as well as on $n_0$-$x$ cross-sections, as shown by the inset in (a). Director field local orientations are represented by cylinders colored according to their $x$ component $n_x$ with the colormap in (a). Surface anchoring coefficient $W=2\times10^{-4}~\rm{Jm}^{-2}$ and sphere radius $r_\rm{c}=5~\upmu\rm{m}$. (d-f) The corresponding local average of node separation distance on the $n_0$-$x$ cross-sections for each simulation with different $\theta_\rm{e}$ colored by rescaled node separation distances $r_0$.
• Figure 5: Nematic LC simulations based on regular grid points and comparison to RBFFD simulations. (a-c) Numerical simulation in regular grids of LC surrounding a spherical particle. Particle surfaces and director fields are colored by the $x$ projection of $\hat{\bf{n}}$, and blackened regions of the surface correspond to singular defects. Far-field LC director orientation $\hat{\bf{n}}_0$ and $x$ axis are labeled in (a). Surface anchoring coefficient $W=2\times10^{-4}~\rm{Jm}^{-2}$ and sphere radius $r_\rm{c}=5~\upmu\rm{m}$. (d-f) Corresponding particle-induced LC elastic distortions at a radius $r = 1.5r_\rm{c}$, whose distance is marked in (a). (g) Elastic multipole moment $b_l$ calculated for LC systems shown above. Dashed lines are results obtained from regular grid sets whereas solid lines with circles are from radial-basis-function finite difference (RBFFD) calculations (Fig. \ref{['sphere']}). (h) Rescaled LC free energies $f$ over number of iterations during numerical energy relaxation. All numerical simulations are performed under one-constant approximation (see Sec. \ref{['method']}).