Numerical Investigation of Single-Core to Split-Core Transitions in Nematic Liquid Crystals

TL;DR

This work analyzes the stability and transitions between single-core and split-core defects in nematic liquid crystals by minimizing the two-dimensional Landau–de Gennes energy with a one-constant approximation. The authors identify a domain-size–dependent temperature threshold $\theta_{tr}(D)$ marking the disappearance of split-core configurations, and they derive a scaling law $\theta_{tr} \approx -(10^7 D)^{-2}$ in the small-domain regime. They quantify defect-core sizes and show that split-core cores merge into a larger central core as temperature increases, with the single-core core size increasing with domain size under confinement. The results provide quantitative guidance on defect stability under confinement and temperature, with implications for defect engineering in LC-based devices.

Abstract

We analyze single-core and split-core defect structures in nematic liquid crystals within the Landau-de Gennes framework by studying minimizers of the associated energy functional. A bifurcation occurs at a critical temperature threshold, below which both split-core and single-core configurations are solutions to the Euler-Lagrange equation, with the split-core defect possessing lower energy. Above the threshold, the split-core configuration vanishes, leaving the single-core defect as the only stable solution. We analyze the dependence of such temperature threshold on the domain size and characterize the nature of the transition between the two defect types. We carry out a quantitative study of defect core sizes as functions of temperature and domain size for both single and split core defects.

Figures (7)

• Figure 1: Total energy for domain diameter $D = 1 \times 10^{-7}\ $m. A: Single core for $\theta = -0.8 ,$ (12.18 °C); B: Split core for $\theta = -1.2 ,$ (7.016 °C); C: Single core for $\theta = -1.5,$ (3.145 °C); D: Split core for $\theta = -1.5,$ (3.145 °C). The total energy $E_{LDG}$ is calculated using Eq. \ref{['Engeq']}. Planar radial (single core) initial guess is used for A and C, and split core initial guess is used for B and D.
• Figure 2: Energy calculated over 2-D disc, $D=1 \times 10^{-7}$ m. Single core and split core results. For this domain size, the split core defect was not observed for $\theta > -1.10\approx 8.306$ °C. The total energy $E_{LDG}$ is calculated using Eq. \ref{['Engeq']}.
• Figure 3: Plots of scalar order parameter $s_h = 2 \lambda_h$ for different temperatures with domain diameter $1 \times 10^{-7}$ m. A: $\theta = -1.4$ (4.435 °C), B: $\theta = -1.17$ (7.403 °C), C: $\theta = -1.1$ (8.306 °C), D: $\theta = -1.05$ (8.952 °C). Split core initial guess is used.
• Figure 4: Plots of scalar order parameter $s_h = 2 \lambda_h$ for different temperatures with $D = 1.52 \times 10^{-6}$ m. A: $\theta = -0.30$ (18.63 °C), B: $\theta = -0.04$ (21.98 °C), C: $\theta = -0.02$ (22.24 °C), D: $\theta = -0.01$ (22.37 °C). Split core initial guess is used.
• Figure 5: Interpolation of split core threshold for different domain diameters $D$ and linear asymptote calculated via Eq. \ref{['st2']}.