Chiral structures and defects of lyotropic chromonic liquid crystals induced by saddle-splay elasticity

An experimental and theoretical study of lyotropic chromonic liquid crystals (LCLCs) confined in cylinders with degenerate planar boundary conditions elucidates LCLC director configurations. When the Frank saddle-splay modulus is more than twice the twist modulus, the ground state adopts an inhomogeneous escaped-twisted configuration. Analysis of the configuration yields a large saddle-splay modulus, which violates Ericksen inequalities but not thermodynamic stability. Lastly, we observe point defects between opposite-handed domains, and we explain a preference for point defects over domain walls.

Figure 1

(a) Coordinate system used for director configuration and defect energy calculations. The director $\mathbf{n}$ is described by the angle $\alpha$ between the director projection ${\mathbf{n}}_{xy}$ and $\widehat{\mathbf{r}}$, and by the angle $\beta$ between $\mathbf{n}$ and the capillary axis, which is parallel to $\widehat{\mathbf{z}}$. (b) 3D cutaway view of the capillary and the ground-state director field using $K_2/K_3=0.1$ and $K_{24}/K_3=4.6$, which approximates the Frank moduli of 30% wt./wt. SSY at $25{}^{\circ }\mathrm{C}$. Notice the large twist angle at the capillary surface close to ${90}^{\circ }$. $\alpha$ is independent of position and $\beta$ depends only on the radial coordinate $r$. This configuration has right-handed chirality.

Figure 2

Phase diagram of ${\beta }_1$, the angle between the nematic director and the capillary axis at the capillary surface, as a function of elastic moduli ratios of saddle splay $\left(K_{24}\right)$ to twist $\left(K_2\right)$ and bend $\left(K_3\right)$. Inset: an example capillary with streamlines indicating a surface director field at angle ${\beta }_1$ with left-handed chirality.

Figure 3

Representative flickering measurements of 30% wt./wt. SSY in a $90.6\mu \text{m}$ capillary at $25{}^{\circ }\mathrm{C}$. (a) Single frame from a movie cropped to a $20\mu \text{m}$ square and after background subtraction (see main text). (b) Averaged FFT of a movie containing many images of fluctuations and a contour plot of a 2D Gaussian fit to the averaged FFT. The long axis of the fit is perpendicular to the dominant fluctuation direction and yields a measurement of $\beta$ for the image slice. (c) Fit of Eq. (4) to $\beta \left(r\right)$ obtained by fluctuation measurements along the capillary radius. Error bars are the standard deviations in degrees of the angle found for the 2D Gaussian fits as in (b). A nonlinear least-squares fit of the parameters $K_2/K_3$ and $K_{24}/K_3$ gives estimates of the elastic constant ratios ($6.5$ and $0.15$, respectively) for the sample. Across all measurements, the average $K_{24}/K_3=6.6$ and has a bounding interval $[3.8,9.4]$.

Figure 4

(a) Left: cross-polarized quasimonochromatic optical images of singular point defects bordering ET regions of opposite handedness in a $90\mu \text{m}$ capillary. Yellow arrows indicate the polarizer pass axis directions. Right: images reconstructed using Jones matrix calculations from numerically computed director fields of defects. (b) A 3D cutaway view of a capillary with opposite-handedness ET regions separated by a wall defect. (c) A 3D cutaway view of a capillary with opposite-handedness ET regions separated by a point defect as imaged and simulated in (a). In both (b) and (c), the director field represents an LC with $K_2/K=0.1$ and $K_{24}/K=4.6$, where $K_1=K_3\equiv K$ and the color scale is the same as Fig. 1. (d) Energies of the point and domain-wall defects relative to the ET energy as a function of either $K_{24}/K$ or equivalently ${\beta }_1$, with $K_2/K=0.1$. Points indicate numerical calculations and lines indicate analytical approximations [47]; the latter have higher energy than the former but demonstrate similar qualitative behaviors.