Nematic liquid crystals on spherical surfaces: Control of defect configurations by temperature, density, and rod shape

Recent experiments have shown that defect conformations in spherical nematic liquid crystals can be controlled through variations of temperature, shell thickness, and other environmental parameters. These modifications can be understood as a result of the induced changes in the effective elastic constants of the system. To characterize the relation between defect conformations and elastic anisotropy, we carry out Monte Carlo simulations of a nematic on a spherical surface. As the anisotropy is increased, the defects flow from a tetrahedral arrangement to two coalescing pairs and then to a great circle configuration. We also analyze this flow using a variational method based on harmonic configurations.

Figure 1

(a) Nematic molecules are represented as rods made up of connected spheres. The interaction between the rods is modeled as the sum of the interactions between beads. (b) Scheme to describe the defect structure in a spherical nematic. The defect positions and orientation are defined by three angles: $\theta$, $\phi$, and $\gamma$. (c) Simulation configuration at a very high density. The defects in each pair on either hemisphere are close to each other.

Figure 2

Evolution with temperature of defect configurations in a spherical nematic. Results correspond to simulations at $u/k_{B}T=1.37$, 2.22, 4.0, and 40.0 [(a)–(d), respectively]. The small spheres represent a defect location, and defect orientations are shown as red vectors. As temperature is lowered in the nematic phase, there is a continuous transition from a high-temperature tetrahedral configuration characterized by $\epsilon \approx 0$ to a low-temperature great circle configuration with much stronger elastic anisotropy $\epsilon \approx 1$. We pair nearest-neighbor defects and associate a great circle to each of the pairs. At high temperatures, we observe a tetrahedral arrangement with the angle between great circles $\approx \pi /2$. The angle continuously decreases with temperature, and at very low temperatures the great circles coincide and the defect directors are perpendicular to them.

Figure 3

Comparisons of nematic textures around a defect core: on a sphere around a +1/2 defect (snapshot from simulations) and on the plane with an enforced boundary condition to have a +1/2 defect at the center for $\epsilon =0.74$. We match the texture on the sphere with that of the plane to estimate the elastic anisotropy in our simulations.

Figure 4

Asymptotic nematic textures on a plane around a defect core for different values of elastic anisotropy: (a) $\epsilon =0.0$, (b) $\epsilon =0.30$, (c) $\epsilon =0.60$, and (d) $\epsilon =0.90$. These textures have angular but not radial dependence.

Figure 5

Monte Carlo simulation results: (a)–(c) Different angles as functions of temperature for a system of $N=1600,l=15\sigma$ at a fixed surface density ($\rho \approx 0.42)$. The final droplet size is $59.99\sigma$ in LJ units. (d) Elastic anisotropy as functions of temperature. The solid lines are merely guides to the eye.

Figure 6

Variation of $\theta$ and $\phi$ as a function of $l/R$ at $u/k_{B}T=1.33$. The great circle angle ($\phi$) decreases smoothly with density ($l/R$). However, the distance between the pair of defects ($\theta$) on each hemisphere initially decreases and finally increases with $l/R$. In these simulations, we consider three different rod lengths: $l=13\sigma ,15\sigma$, and $17\sigma$. The corresponding droplet sizes range from $52.96\sigma$ to $63.85\sigma$ in LJ units. The solid lines are simply guides to the eye.

Figure 7

Comparison of simulation and variational results. (a) Polar angle ($\theta$) as functions of $\epsilon$. (b) Great circle angle ($\phi$) as functions of $\epsilon$. (c) Energy contribution from $\epsilon$ and non-$\epsilon$ term. The total energy is plotted as a solid line.

Figure 8

Harmonic nematic textures that minimize the free energy for different $\epsilon$: (a) $\epsilon =0.06$, (b) $\epsilon =0.30$, (c) $\epsilon =0.60$, and (d) $\epsilon =0.90$. The top row shows their projection on the complex plane while the bottom row shows their realization in the sphere. The small spheres indicate the position of the defects, and their directors are marked as red rods.