Contact Topology and the Classification of Disclination Lines in Cholesteric Liquid Crystals

We give a complete topological classification of defect lines in cholesteric liquid crystals using methods from contact topology. By focusing on the role played by the chirality of the material, we demonstrate a fundamental distinction between “tight” and “overtwisted” disclination lines not detected by standard homotopy theory arguments. The classification of overtwisted lines is the same as nematics, however, we show that tight disclinations possess a topological layer number that is conserved as long as the twist is nonvanishing. Finally, we observe that chirality frustrates the escape of removable defect lines, and explain how this frustration underlies the formation of several structures observed in experiments.

Figure 1

(a) Schematic of the tubular neighborhood of a defect line (blue) with the director (green cylinders) shown on some cross-sections and part of the dividing curve (red). The frame $\{{\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3\}$ is shown on one cross section. The schematic corresponds to the representative texture ${\mathbf{n}}_{-1/2,1}$. (b) Examples from the family of tight cholesteric disclinations (2) for a selection of values of $(k,q)$, shown in abstracted, standardized form. The examples include the “exceptional” cases $k=1$ and $q=0$ (3), where the dividing curve has either zero or infinite slope. Where the dividing curve has more than one component, one of them has been displayed in darker shade to aid visualization. The dividing curve facilitates the computation of invariants associated with the tight disclination in a manner described in the text.

Figure 2

Illustrations of convex surfaces, their dividing curves, and the computation of the Thurston-Bennequin (TB) number. (a) A convex torus (gray) cutting across a cholesteric texture exhibiting a change in the number of layers. The director, shown as white sticks, points out from the surface in orange regions and into the surface in blue regions. The topological content of this data is entirely captured by the dividing curve (red), the set of points where the director is tangent to the convex surface. The green line indicates an example calculation of a Thurston-Bennequin number. (b) On a convex surface cutting across a Skyrmion, the dividing curve is a closed contractible loops. By Giroux’s criterion the contact structure is overtwisted. (c) Convex surface tomography of a Hopfion. As the convex surface is slid across the Hopfion, changes in the dividing curve track the changes in the layer structure. The presence of a closed component bounding a disk (top right) reveals the presence of a Skyrmion, with a local structure equivalent to that shown in panel (b) and shows that the Hopfion is overtwisted. The pale blue curves indicate the linked ${\lambda }^{+1}$ lines of the Hopfion.

Figure 3

Three motifs occur in the dividing curve (red) on a section of a convex torus. (a) The dividing curve turns everywhere counterclockwise and the winding $k_{\varphi }$ is constant. (b) The dividing curve turns back on itself, indicating a region (green) in which $k_{\varphi }=-1/2$, while $k_{\varphi }=+1/2$ in the remainder of the panel. (c) There is a separate component of the dividing curve which bounds a disk on the torus, also indicating the presence of a region (green) with $k_{\varphi }=-1/2$.