Waltzing route toward double-helix formation in cholesteric shells

Abstract

Liquid crystals, when confined to a spherical shell, offer fascinating possibilities for producing artificial mesoscopic atoms, which could then self-assemble into materials structured at a nanoscale, such as photonic crystals or metamaterials. The spherical curvature of the shell imposes topological constraints in the molecular ordering of the liquid crystal, resulting in the formation of defects. Controlling the number of defects, that is, the shell valency, and their positions, is a key success factor for the realization of those materials. Liquid crystals with helical cholesteric order offer a promising, yet unexplored way of controlling the shell defect configuration. In this paper, we study cholesteric shells with monovalent and bivalent defect configurations. By bringing together experiments and numerical simulations, we show that the defects appearing in these two configurations have a complex inner structure, as recently reported for simulated droplets. Bivalent shells possess two highly structured defects, which are composed of a number of smaller defect rings that pile up through the shell. Monovalent shells have a single radial defect, which is composed of two nonsingular defect lines that wind around each other in a double-helix structure. The stability of the bivalent configuration against the monovalent one is controlled by c = h/p, where h is the shell thickness and p the cholesteric helical pitch. By playing with the shell geometry, we can trigger the transition between the two configurations. This transition involves a fascinating waltz dynamics, where the two defects come closer while turning around each other.iquid crystals offer fascinating possibilities for producing materials that are organized at a mesoscopic scale, such as photonic crystals or metamaterials (1). In a seminal paper, Nelson (2) proposed to induce valency into simple spherical particles by coating their surfaces with a nematic liquid crystal shell. In this elegant approach, the spherical symmetry of the particle is broken by the presence of topological defects, which stem from frustrations in the liquid crystal orientational order due to the curvature of the particle surface. These defects, once functionalized, could act as sticky surface patches inducing directional particle bonding through patchpatch interactions, which would make possible the fabrication of complex materials by spontaneous self-assembly. The number of surface defects would set the valence of the particle, whereas their position would determine the directionality of the eventual bonds. The type of defect configuration in the shell, and thus its valence, results from a very subtle interplay between topological constraints and elastic free-energy minimization. For a two-dimensional nematic shell, theory predicts a tetravalent configuration where four defects are organized in a tetrahedral fashion (3). These defects have a winding number s i = 1=2, indicating a π-rotation of n, the average molecular orientation, around the defect (4). This is consistent with...

Elastic interactions between topological defects in chiral nematic shells

“…Besides, unexpected symmetries/valencies have been recently reported in cholesteric shells [16,17], in which there is a spontaneous helical arrangement of the director field. Remarkably, it is possible to achieve a good control over the equilibrium defect positions by tuning the shell thickness heterogeneity [13,16,18]. This feature could then be further exploited to produce shells with a variable bonding directionality.…”

“…Since Nelson's seminal paper, many experimental studies focused on nematic shells have demonstrated the applicability of such concepts [12][13][14][15]. Besides, unexpected symmetries/valencies have been recently reported in cholesteric shells [16,17], in which there is a spontaneous helical arrangement of the director field. Remarkably, it is possible to achieve a good control over the equilibrium defect positions by tuning the shell thickness heterogeneity [13,16,18].…”

“…After performing self-consistency checks, we use our model to estimate the energies of recently reported non trivial defect structures. [16,17,22]. On a sphere, a two-dimensional in-plane director field must fulfill the topological requirements of the Poincaré-Hopf theorem [23][24][25].…”

“…In our experiments, we observe that h 0 is constant and independent of the shell nature and geometry. On physical grounds, this can be explained by the fact that it is the disjoining pressure between the inner and outer interfaces which sets the value of h 0 [16] (see [18] for a recent numerical study on the effects of varying h 0 ). Hence, we are only left with the parameter u, which actually measures the thickness gradient within the shell, and shall thus rigorously write h(θ; u) for the local thickness.…”

“…Minimizing the free energy with respect to θ 1 and θ 2 , and noting that β = θ 2 1 + θ 2 2 , we obtain the equilibrium [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.…”