Reversal of helicoidal twist handedness near point defects of confined chiral liquid crystals

Handedness of the director twist in cholesteric liquid crystals is commonly assumed to be the same throughout the medium, determined solely by the chirality of constituent molecules or chiral additives, albeit distortions of the ground-state helicoidal configuration often arise due to the effects of confinement and external fields. We directly probe the twist directionality of liquid crystal director structures through experimental three-dimensional imaging and numerical minimization of the elastic free energy and show that spatially localized regions of handedness opposite to that of the chiral liquid crystal ground state can arise in the proximity of twisted-soliton-bound topological point defects. In chiral nematic liquid crystal confined to a film that has a thickness less than the cholesteric pitch and perpendicular surface boundary conditions, twisted solitonic structures embedded in a uniform unwound far-field background with chirality-matched handedness locally relieve confinement-imposed frustration and tend to be accompanied by point defects and smaller geometry-required, energetically costly regions of opposite twist handedness. We also describe a spatially localized structure, dubbed a “twistion,” in which a twisted solitonic three-dimensional director configuration is accompanied by four point defects. We discuss how our findings may impinge on the stability of localized particlelike director field configurations in chiral and nonchiral liquid crystals.

Figure 1

Experimental polarizing optical micrographs of a stable twisted configuration dubbed twistion in a partially polymerizable confined chiral nematic LC as viewed (a) between crossed polarizers, (b) between horizontally oriented parallel polarizers, (c) with only one horizontally and (d) one vertically oriented polarizer, and (e) without polarizers. Double-headed white arrows depict orientations of linear polarization of light passing through these polarizers. The scale bar shown in part (a) applies to all micrographs (a–e).

Figure 2

3D colored surfaces depicting the twisted director field configuration of twistions that were reconstructed from experimental images and numerically simulated data. (a) Experimentally reconstructed and (b) the corresponding computer-simulated isosurface that depicts the director field configuration at constant $n_z=0.8$, with the azimuthal orientations shown using the color-coded scheme in the inset. The 3D coordinate system in the inset attached to the color scheme helps to describe the orientation of surfaces relative to the viewing perspective direction. The colors visualize azimuthal director orientations that are characterized by the projection of the unit director field to the $x-y$ plane, according to the color wheel that is shown with the coordinate system. (c) An experimental configuration shown for $n_z=0.8$ isosurface similar to (a) but when viewed from a $z$ perspective view position, as indicated by the flipped coordinate system in the inset next to it. (d–f) Color-coded isosurfaces visualizing $\mathit{n}\left(\mathit{r}\right)$ for (d) $n_z=0.8$, (e) $n_z=0$, and (f) $n_z=-0.8$ constructed from the numerically simulated director field and viewed from the $z$ perspective viewpoint. The scale bar between (a) and (c) relates to experimental 3D reconstructions of the director structures on isosurfaces shown in (a,c). All experimental images were obtained for the partially polymerized chiral nematic LC system with the composition detailed in the methods section, and 5CB elastic constants were used in the corresponding computer simulations.

Figure 3

Cross-sectional views of the director field within the twistion configurations with point defects. (a) A color-coded isosurface at $n_z=0.8$ that was constructed from computer-simulated director field data, shown along with the cross-sectional planes that denote locations of the lateral $x-y$ cross sections (b) near the top of the structure, (c) near the bottom, (d–f) in-between, and in the two mutually orthogonal vertical cross sections (g,h) that contain point defects. Director field $\mathit{n}\left(\mathit{r}\right)$ local orientations are shown with rods. Locations of the four self-compensating point defects are depicted with orange and blue dots that represent clockwise RGB and BGR color changes of azimuthal orientation of $\mathit{n}\left(\mathit{r}\right)$ around defects for the isosurface shown in (a), respectively. (g,h) Two mutually orthogonal $x-z$ cross sections (g) through all four point defects and (h) $y-z$ cross section through only two point defects shown with the help of enlarged rods depicting $\mathit{n}\left(\mathit{r}\right)$ for clarity. Arrows in (e) mark the locations of these vertical cross sections, in addition to the planes shown in (a). These results of computer simulations are shown for elastic constants of 5CB.

Figure 4

Visualization of twist handedness and energy density of twistions in confined chiral LCs. (a) Vertical $x-z$ cross section of the computer-simulated director field of the twistion (with the director orientations shown using rods) is overlaid on top of a density plot of isotropic handedness normalized to $q_0$ and shown in accord with the color scheme provided in the inset. (b) A 3D perspective view of the isosurfaces of isotropic handedness that correspond to energetically favorable twist ($\mathcal{H}=q_0$, cyan) and the isosurface that encloses the volume with opposite handed twist ($\mathcal{H}=-0.01q_0$, magenta). (c,d) In-plane $x-y$ cross sections of the director structure that are overlaid on top of the density plots of isotropic handedness (c) above and (d) below the bottom point defect, as marked in (a). The director orientation is represented by cylinders with red- and blue-colored ends to help guide the reader's eye when he or she inspects the directional handedness in the corresponding regions. (e) Isosurface ${\chi }_{33}=-0.01q_0$ for the twist along $z$ (depicted in orange color) that is shown along with the same cyan isosurface as in (b) for reference. The reader should note here that the spatial overlap of the two 3D surfaces gives the appearance of green color. (f) Vertical $x-z$ director cross section of the twistion that is overlaid on top of a density plot of the twist term of the free energy density $f_{22}$ normalized to the energy density of the untwisted state $f_0$ and shown with the color scheme that is provided in the right-side inset. (g) Twist free energy isosurface $f_{22}=2f_0$ (yellow) shown along with the same handedness isosurface ($\mathcal{H}=q_0$, cyan) as in (b) for reference. Overlap of the two surfaces again gives the appearance of green. Computer simulations were done for the material parameters for 5CB (Table 1).

Figure 5

Visualization of twist handedness and free energy density within a triple twist toron of the first kind. (a) Vertical $x-z$ director field cross section of a computer-simulated toron structure that is overlaid on top of a density plot of isotropic handedness normalized to $q_0$ and plotted in accord with the color scheme shown in the right-side inset. The bottom-left inset shows an isosurface of the 3D director structure for $n_z=0.5$ of the corresponding computer-simulated toron configuration, where colors depict azimuthal director orientations in accord with a similar color scheme like that used in Fig. 2. (b) 3D perspective view of isosurfaces in isotropic handedness corresponding to ($\mathcal{H}=q_0$, cyan) and ($\mathcal{H}=-0.01q_0$, magenta). (c,d) In-plane $x-y$ director cross sections that are overlaid on top of the corresponding density plots of isotropic handedness for cross sections (c) above and (d) below the bottom point defect, as marked on (a). The local director orientation is represented by cylinders with red- and blue-colored ends to help guide the reader's eye when he or she inspects the directional handedness. (e) Analysis of the 3D patterns of directional handedness represented using orange isosurface with ${\chi }_{33}=-0.01q_0$, for the director twist along $z$, depicted along with the same cyan isosurface as shown in (b) for reference. The spatial overlap of the two 3D surfaces appears green. (f) $x-z$ director cross section of the twistion overlaid on top of a density plot of twist free energy density $f_{22}$ normalized by the energy density of the untwisted state $f_0$; the color-coded free energy density scale is shown in the right-side inset. (g) A 3D perspective view on the yellow/green isosurface with $f_{22}=2f_0$, shown in accord with the cross-sectional plot depicted in (f). This isosurface is shown along with the same cyan isosurface of handedness as in (b) for reference. The yellow isosurface corresponds to the high energy $f_{22}=2f_0$ in the regions with the twist direction opposite from that of the ground-state cholesteric LC and for the same handedness but with the twist rate higher than that of the equilibrium helicoidal structure. Overlap of the two surfaces appears green. Computer simulations are based on material parameters of 5CB (Table 1).