Elastic interactions between topological defects in chiral nematic shells

We present a self-consistent and robust theoretical model to investigate elastic interactions between topological defects in liquid crystal shells. Accounting for the nonconcentric nature of the shell in a simple manner, we are able to successfully and accurately explain and predict the positions of the defects, most relevant in the context of colloidal self-assembly. We calibrate and test our model on existing experimental data and extend it to all observed defects configurations in chiral nematic shells. We perform experiments to check further and confirm the validity of the present model. Moreover, we are able to obtain quantitative estimates of the energies of $+1$ or $+3/2$ disclination lines in cholesterics, whose intricate nature was only reported recently [A. Darmon, et al. Proc. Natl. Acad. Sci. USA 113, 9469 (2016)].

Figure 1

(a) and (b) Side view optical microscopy pictures of an eccentric cholesteric liquid crystal shell. The scale bar is $10\mu \mathrm{m}$. (c) Side view schematics of an eccentric shell. The concentric cholesteric layers are represented by thin solid gray lines. The thick gray line on the inner sphere signifies the region where the director field becomes slightly distorted to satisfy the tangential anchoring. (d) Schematics of defects on the outer sphere.

Figure 2

Configuration with four +1/2 defects. Shown on the top left is the top view image between crossed polarizers of a $4[+1/2]$ nematic shell. The bottom left shows the schematics of the defect arrangement. On the right is the angular distance $\beta$ between nearby defects as a function of $u$. The blue squares correspond to experimental data from [13] and the solid red line is the result of the minimization of the free energy in Eq. (7) with $h_0=0.02$.

Figure 3

Configuration with two +1 defects. Shown on the top left is the top view image between crossed polarizers of a $2[+1]$ cholesteric shell. The bottom left shows the schematics of defect arrangement. On the right is the angular position $\theta$ as a function of $u$. Green squares show the experimental data from [16] for which a rolling average was performed. The inset displays a picture of the intricate structure of the $+1$ disclination in a cholesteric shell.

Figure 4

Configuration with one +1 and two +1/2 defects. Shown on the left are angular positions ${\theta }_i$ as a function of $u$. The inset displays the vertex angle $\alpha$ of the isosceles triangle as a function of $u$ (experimental results are shown as gray squares). The top right is top view image between crossed polarizers of a $[+1]+2[+1/2]$ cholesteric shell with a zoom-in on the defects (inset). The bottom right shows the schematics of the defect arrangement.

Figure 5

Configuration with one +3/2 defect and one +1/2 defect. Shown on the left are angular positions ${\theta }_i$ as a function of $u$. The top right is the top view image between crossed polarizers of a $[+3/2]+[+1/2]$ cholesteric shell. The bottom right shows the schematics of the defect arrangement.