Curvature-Induced Twist in Homeotropic Nematic Tori

We confine a nematic liquid crystal with homeotropic anchoring to stable toroidal droplets and study how geometry affects the equilibrium director configuration. In contrast to the case of cylindrical confinement, we find that the equilibrium state is chiral—a twisted and escaped radial director configuration. Furthermore, we find that the magnitude of the twist distortion increases as the ratio of the ring radius to the tube radius decreases; we confirm this with computer simulations of optically polarized microscopy textures. In addition, numerical calculations also indicate that the local geometry indeed affects the magnitude of the twist distortion. We further confirm this curvature-induced twisting using bent cylindrical capillaries.

Figure 1

Director schematics of nematic tori with planar anchoring for an (a) axial and a (b) doubly twisted director field.

Figure 2

(a) Schematic detailing the $\{r,\theta ,\phi \}$ toroidal coordinate system, where $r$ is measured from the central ring, while $\theta$ and $\phi$ are the polar and azimuthal angles in the torus, respectively. We characterize the slenderness, or aspect ratio of the torus with $\xi =R_0/a$, the ratio of the central ring radius $R_0$, and the tube radius $a$. (b,c) Bright-field and associated crossed-polar microscopy images of an example nematic toroidal droplet with homeotropic anchoring, where $\xi =5$. Scale bar is $250\mu \mathrm{m}$. (d) Gray-scale intensity profile for the region highlighted in (c), where 1 corresponds to white and 0 to black.

Figure 3

(a,e) Director schematics, (b,f) bright-field, and (c,g) associated crossed-polar microscopy images of an (a–c) escaped radial (ER) configuration and a (e–g) twisted escaped radial (TER) configuration in a cylindrical capillary. The capillaries in (b,c) and (f,g) are filled with 5CB and SSY, respectively, and the scale bar is $100\mu \mathrm{m}$. The nails in (e) represent the director coming out of the page. (d,h) Gray-scale intensity profiles for the regions highlighted in (c,g), respectively, where an intensity of 1 corresponds to white and 0 to black.

Figure 4

(a) Intensity ratio $I_{\max }/I_{\min }$ as a function of the aspect ratio for the homeotropic nematic structures in this work: toroidal droplets filled with 5CB (open circles), a straight cylindrical capillary filled with SSY in the transient ER configuration (filled square) and stable TER configuration (open square), and bent and straight cylindrical capillaries filled with 5CB (triangles). An infinite aspect ratio corresponds to a straight cylindrical capillary. (b) Bright-field microscopy image of a bent cylindrical capillary filled with 5CB under homeotropic anchoring. The scale bar is $200\mu \mathrm{m}$. (c,d) Crossed-polar microscopy images of the highlighted regions in (b), with the polarizer and analyzer orientations indicated schematically in the bottom left of each image. (e,f) Gray-scale intensity profiles across the capillary for the highlighted regions in (c,d), respectively. (g,h) Cross-section director schematic of an ER and a TER configuration, respectively, in a toroidal droplet.

Figure 5

(a,b) The upper half of simulated crossed-polar images of an ER and TER configuration, respectively, in a torus. The twist angle in (b) is 47°. (c) Intensity profiles for the (circles, open squares) highlighted regions of the tori in (a,b), respectively. (d) Intensity ratio $I_{\max }/I_{\min }$ as a function of the twist angle for the simulated crossed-polar images of TER configurations in a torus.