Chiral nematic liquid crystals in torus-shaped and cylindrical cavities

We present a Monte Carlo simulation study of chiral nematic liquid crystals confined in torus-shaped and cylindrical cavities. For an achiral nematic with planar degenerate anchoring confined to a toroidal or cylindrical cavity, the ground state is defect free, with an untwisted director field. As chirality is introduced, the ground state remains defect free but the director field becomes twisted within the cavity. For homeotropic anchoring, the ground state for an achiral nematic within a toroidal cavity consists of two disclination rings, one large and one small, that follow the major circumference of the torus. As chirality is introduced and increased, this ground state becomes unstable with respect to twisted configurations. The closed nature of the toroidal cavity requires that only a half integer number of twists can be formed and this leads to the ground state being either a single disclination line that encircles the torus twice or a pair of intertwined disclination rings forming stable, knotted defect structures.

Figure 1

(a) The director field for an achiral nematic (${\varepsilon }_c^{*}=0.00$) with planar anchoring inside a cylinder of dimensions $l^{*}=48, r_c^{*}=16$. (b) The director field in the defect-free ground state for a torus of dimensions $r_t^{*}=28, r_c^{*}=10$ and (c) in a higher energy metastable state with two pairs of boojum defects. Each boojum consists of a $s=+1$ defect on the outer waist of the surface and a $s=-1$ defect on the inner waist, giving a total surface topological charge of 0.

Figure 2

Orientation of the aligning field in a torus of dimensions $r_t^{*}=30, r_c^{*}=20$ when (a) ${\xi }_t=0.0$ and (b) ${\xi }_t=1.0$. Streamlines at the center of the tube are colored red, then become green, then blue as the surface is approached. (c) Average energy per particle as a function of field strength ${\varepsilon }_f$ for ${\xi }_t=0.0$; the line indicates the best fit, with the extrapolation to zero field, ${\varepsilon }_f=0.0$. (d) Average energy per particle, extrapolated to ${\varepsilon }_f=0.0$, as a function of the imposed twist ${\xi }_t$.

Figure 3

Chiral nematics with planar anchoring inside a cylinder of dimensions $l^{*}=36, r_c^{*}=16$ with ${\varepsilon }_c^{*}$ equal to (a) 0.00, (b) 0.06, (c) 0.12, (d) 0.18, and (e) 0.24. A torus of dimensions $r_t^{*}=30, r_c^{*}=20$ with ${\varepsilon }_c^{*}$ equal to (f) 0.06 and (g) 0.18.

Figure 4

(a) An achiral nematic with homeotropic anchoring inside a cylinder of dimensions $l^{*}=48, r_c^{*}=16$ and (b) inside a torus of dimensions $r_t^{*}=30, r_c^{*}=15$. (c) A cut-through view of the torus, showing the planarity of the disclination rings.

Figure 5

Representative configuration for chiral nematics with homeotropic anchoring inside cylindrical cavities of dimensions $l^{*}=96, r_c^{*}=16$ with (a) $\xi =0.0$, (b)$\xi =0.5$, (c) $\xi =1.0$, and (d) $\xi =1.5$.

Figure 6

(a) The average energy per particle as a function of $\xi$ for ${\varepsilon }_c^{*}=0.18$ for fixed $r_c^{*}=16, l^{*}=48$ (triangles, dashed line) $l^{*}=72$ (circles, dotted line), and $l^{*}=96$ (squares, solid line). (b) The average energy per particle as a function of $\xi$ for ${\varepsilon }_c^{*}=0.18$ for fixed $l^{*}=48, r_c^{*}=12$(triangles, dashed line), $r_c^{*}=16$ (circles, dotted line) and $r_c^{*}=20$ (squares, solid line). (c) The inverse of the pitch $(1/p^{*})$ for the bulk chiral nematic (solid circles) and the inverse of the repeat distance for cylindrically confined systems as a function of chirality; data from all systems are investigated. The open symbols correspond to the cylindrically confined nematics with $l^{*}=48$ (triangles), 72 (squares), 96 (circles), and $r_c^{*}=16$. The solid line is a fit to the bulk pitch chiral nematic, while the dashed line is a fit to the combined chiral nematic cases.

Figure 7

Snapshots of the director field and disclination line(s) for chiral nematics with homeotropic anchoring inside a toroidal cavity of dimensions $r_t^{*}=30, r_c^{*}=15$ with $\xi =$(a) 0.0, (b) 0.5, (c) 1.0, (d) 1.5, (e) 2.0, (f) 2.5, and (g) 3.0. (h) The average energy per particle as a function of twist for ${\varepsilon }_c^{*}=0.00$ to 0.24 (from top to bottom). (i) The expected twist (solid circles) based on the bulk pitch length ($2\pi r_t^{*}/p^{*}$) and the actual twist in the toroidal cavity (open symbols); $r_t^{*}=12$ (circles), 15 (squares), and 20 (triangles). The solid line is a fit to the bulk pitch length case, while the dashed line is a fit to the confined nematic cases.