Figure 1

(a) Illustration of the difference between writhe and geometric twist. The writhe measures the amount of three-dimensional variation of the ribbon's tangent, while the geometric twist measures rotation of the director field around the tangent. (b) For $-1/2$ nematic disclinations, the geometric twist that represents the rotation of its cross section (marked with gray sticks) is linearly proportional to the local elastic twist that appears in the free energy expansion (green ellipsoids).Reuse & Permissions

Figure 2

(a) Threefold profile of a $-1/2$ disclination line, visualized using the splay-bend parameter $S_{\text{SP}}$. The positive values of this parameter highlight high-splay regions (blue), while the negative values highlight high-bend regions (yellow). The core of the disclination exhibits low values of the order parameter $S$ (red). (b–e) The well-known Saturn rings, theta, figure-eight, and omega dimer structures, respectively. The threefold ribbons formed by the high-bend and high-splay isosurfaces can be traced around the loops to see if they end up with the same orientation. This reflects that the self-linking number is $\text{Sl}=0$ for (b,c) and $\text{Sl}=-2/3$ for (d,e). Numerically integrated writhe $\text{Wr}$ deviates from its ideal value of $\text{Sl}$ for (d,e). Free energy differences $\Delta F$ from the ground state (b) are shown in arbitrary units. The interparticle separation is 0.116 nm.Reuse & Permissions

Figure 3

Influence of the interparticle separation on the writhe and total disclination length. (a) With increasing separation, the writhe of the omega structure increases and the writhe of the figure-eight structure decreases. Linear extrapolation suggests that the writhe converges to the ideal value of $-2/3$ when the particles are touching. (b) The disclination length increases monotonously with the interparticle distance, except for the theta structure, where the central ring shrinks with the increasing gap (bottom data set). The data for the omega structure stops at 200 nm separation when it collapses into two Saturn rings.Reuse & Permissions

Figure 4

Dependence of the free energy, total disclination length, and writhe on the twist angle for all four dimer structures. The horizontal axis measures the angle between the top and and bottom rubbing directions. The interparticle separation is fixed at 116 nm. (a) The free energy contribution of the structures is quadratic with the twist angle. The Saturn rings structure is the ground state in a parallel cell and twisted cells with very small twist, but at largest twist angles, the figure-eight structure is the most stable. The free energy scale is in arbitrary units. (b) A similar dependence is observed for disclination lengths. The chiral structures (figure-eight and omega) have the shortest disclinations at a nonzero angle, while the other two structures exhibit a symmetric dependence. For twist angles greater than ${30}^{\circ }$, the figure-eight structure has shorter disclinations than the Saturn rings. The quadratic terms of the fitted curves are all in the interval $(0.12\pm 0.005)\mu \mathrm{m}{\mathrm{rad}}^{-2}$. (c) Changing the nematic cell from parallel to twisted causes changes in writhe. Even the symmetric structures like theta and Saturn rings exhibit nonzero writhe in a chiral environment. For the theta structure, writhe of the central ring is negligible.Reuse & Permissions

Figure 5

The geometric twist of the entangled dimer states in a parallel nematic cell, calculated from the theoretical value of the self-linking number and the numerically obtained writhe. The geometric twist has a universal tendency to decrease with the inverse cholesteric pitch, a consequence of the relationship between the geometric and elastic twist. For the theta structure, the twist contribution of the central ring is shown separately. The $x$ axis is the inverse cholesteric pitch in the units of $\pi$ turns per cell thickness. The interparticle separation is $116\mathrm{n}\mathrm{m}$.Reuse & Permissions