Nematic Liquid Crystal Toroids with Hometropic Anchoring

Perry Ellis, Karthik Nayani, and Alberto Fernandez-Nieves

### Introduction

Nematic liquid crystals (NLC) are birefringent rod-like molecules that prefer to align with their long axis roughly parallel. We use the director n to describe the local average orientation of a group of molecules, as seen schematically in Figure 1A. When confined to a cylindrical capillary with homeotropic boundary conditions, a common equilibrium director field is known as an escaped radial (ER) configuration, shown schematically in Figure 1B. The associated bright field and crossed-polar image of a capilalry filled with a NLC called 5CB can also be seen in Figure 1B, with the polarizer and analyzer directions drawn on the image. Note that the crossed polar image looks almost "striped", with a dark-bright-dark-bright-dark pattern. For some NLC with propensity to twist, the equilibrium director field is instead a twisted escaped radial (TER) configuration, shown schematically in Figure 1C. In this schematic, the nails on the individual director lines represent an out-of-plane component of the director. We demonstate this with capillaries filled with a NLC called SSY; the bright-field and crossed polar images can be seen in Figure 1C. Note how in the crossed polar image for the TER configuration in Figure 1C, the "striped" pattern has signifigantly less contrast than the "striped" pattern in an ER configuration.

Fig. 1: (A), schematic demonstrating the director as a local average of the orientation of the NLC molecules. (B), The escaped raidial director configuration. The configuration is shown schematically with the associated bright field and crossed-polar image below. The polarizer and analyzer directions are marked on the crossed polar image. (C) The twisted escaped raidial director configuration. The configuration is shown schematically with the associated bright field and crossed-polar image below. The polarizer and analyzer directions are marked on the crossed polar image. (D) Capillary intensity profiles. The position coordinates across the capillary as well as a schematic of an intensity cut are shown in director schematic in (B) and (C).

This difference in the crossed-polar intensity between the TER configuration and the ER configuration can be seen by taking an intensty cut across the capillary, as seen in Figure 1D. For the intensty profiles, the position coordinates across the capillary are specified on the director schematics, and the blue and red boxes on the schematics show the regions used to measure the instensity profiles. Note how the intesnity profile for the TER case does not exhibit the same magnitude of intensity fluctuations as the intensity profiles in the ER case.

Here, we use curvature to generate a TER director configuration for 5CB, a NLC that doesn't twist in a straight capillary. We confine 5CB in both toroids and in bent capillaries with homeotropic boundary conditions, as seen for a bright field and crossed polar image in Figure 2A,B, respectively. We characterize the toroids by the aspect ratio, $$R_0/a$$, as defined schematically on the bright field image in Figure 2A. We also find an effective aspect ratio for the bent capillaries by measuring the radius of curvature for the bend and using it as an effective "$$R_0$$". When we look at the intensity profiles for both the toroids and the capillaries, shown in Figure 1A,B respectively, we see that they resemble the intensity profile for a TER capillary more than the intensity profile for an ER configuration. To quantify this, we take the ratio between the maximum of the intensity profile and the central minimum in the intesnity profile, and plot this ratio as a function of aspect ratio, as seen in Figure 2C for the toroids (open blue circles), bent capillaries (black closed triangles), a single straight capillary (infinite aspect ratio) with a TER configureation (open red square).

Fig. 2: (A), homeotropic NLC toroid and intensity profile. The bright field image is above the crossed-polar image. The polarizer and analyzer directions are marked on the crossed-polar image. The intensity profile in the white region highlighted in the crossed-polar image is shown in the bottom plot. (B), homeotropic bent NLC capillary and intensity profile. The bright field image is above the crossed-polar image. The polarizer and analyzer directions are marked on the crossed-polar image. The intensity profile in the white region highlighted in the crossed-polar image is shown in the bottom plot. (C) Intensity ratios for the toroid and the bent capillary. The maximum intensity / minimum intensity for the central portion of the intensity profile is plotted as a function of aspect ratio for the toroids (open blue circles), the bent capillaries (closed black triangles), and a straight capillary with a TER structure (open red square). The effective aspect ratio for the bent capillaries was determined using the radii of curvature of the bend.

The intensity ratio depends only on the aspect ratio, and it varies from the minimum set by the TER capillary to the maximum set by the ER capillary. Now we turn to simulations to relate the intensity ratio to twist in the director field. We use Jones Calculus to simulate crossed-polar images for ER and TER director configuration in toroids and capillaries, as seen in Figure 3A-C. When we plot the intensity ratio versus twist angle in the director, we see that the realtionship is monotonic with the intensity ratio inversely propotional to the twist angle. This means that the the relationship between intensity ratio and aspect ratio plotted in Figure 2D can be recast as a relationship between 1 / twist angle and aspect ratio. Thus, we see that curvature can drive twist and force an ER configuration to twist and become a TER configuration.

Fig. 3: (A), simulated crossed-polar image for a toroid with an ER configuration. (B), simulated crossed-polar image for a toroid with a TER configuration with a twist angle of 45$$^o$$. (C), simulated crossed-polar images of capilalrites with an ER configuration, a TER configuration with twist angle of 22.5$$^o$$ , and a TER configuration with twist angle 45$$^o$$. (D), plot of twist angle versus intensity ratio for the simulated toroids and capillaries.

Soft Condensed Matter Laboratory, School of Physics, Georgia Institute of Technology
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alberto.fernandez [at] physics.gatech.edu