Curvature-Induced Twist in Homeotropic Nematic Tori

Confinement twists achiral liquid crystals and causes chiral liquid crystals to twist in the opposite handedness: cases in and around sessile droplets

We study the chiral symmetry breaking and metastability of confined nematic lyotropic chromonic liquid crystals (LCLCs) with and without chiral dopants. The isotropic-nematic coexistence phase of the LCLC renders two confining geometries: sessile isotropic (I) droplets surrounded by the nematic (N) phase and sessile nematic droplets immersed in the isotropic background. In the achiral system with no dopants, LCLC's elastic anisotropy and topological defects induce a spontaneous twist deformation to lower the energetic penalty of splay deformation, resulting in spiral optical textures under crossed polarizers both in the I-in-N and N-in-I systems. While the achiral system exhibits both handednesses with an equal probability, a small amount of the chiral dopant breaks the balance. Notably, in contrast to the homochiral configuration of a chirally doped LCLC in the bulk, the spiral texture of the disfavored handedness appears with a finite probability both in the I-in-N and N-in-I systems. We propose director field models explaining how chiral symmetry breaking arises by the energetics and the opposite-twist configurations exist as meta-stable structures in the energy landscape. These findings help us create and control chiral structures using confined LCs with large elastic anisotropy.

Flow-induced periodic chiral structures in an achiral nematic liquid crystal

Supramolecular chirality typically originates from either chiral molecular building blocks or external chiral stimuli. Generating chirality in achiral systems in the absence of a chiral input, however, is non-trivial and necessitates spontaneous mirror symmetry breaking. Achiral nematic lyotropic chromonic liquid crystals have been reported to break mirror symmetry under strong surface or geometric constraints. Here we describe a previously unrecognised mechanism for creating chiral structures by subjecting the material to a pressure-driven flow in a microfluidic cell. The chirality arises from a periodic double-twist configuration of the liquid crystal and manifests as a striking stripe pattern. We show that the mirror symmetry breaking is triggered at regions of flow-induced biaxial-splay configurations of the director field, which are unstable to small perturbations and evolve into lower energy structures. The simplicity of this unique pathway to mirror symmetry breaking can shed light on the requirements for forming macroscopic chiral structures.

Curvature and confinement effects on chiral liquid crystal morphologies

Chiral liquid crystals (ChLCs) exhibit an inherent twist that originates at the molecular scale and can extend over multiple length scales when unconstrained. Under confinement, the twist is thwarted, leading to formation of defects in the molecular order that offer distinct optical responses and opportunities for colloidal driven assembly. Past studies have explored spheroidal confinement down to the nanoscopic regime, where curved boundaries produce surface defects to accommodate topological constraints and restrict the propagation of cuboidal defect networks. Similarly, strict confinement in channels and shells has been shown to give rise to escaped configurations and skyrmions. However, little is known about the role of extrinsic curvature in the development of cholesteric textures and Blue Phases (BP). In this paper, we examine the palette of morphologies that arises when ChLCs are confined in toroidal and cylindrical cavities. The equilibrium morphologies are obtained following an annealing strategy of a Landau-de Gennes free energy functional. Three dimensionless groups are identified to build phase diagrams: the natural twist, the ratio of elastic energies, and the circumscription of a BP cell. Curvature is shown to introduce helical features that are first observed as a Double Twist, and progress to Chiral Ribbons and, ultimately, Helical BP and BP. Chiral ribbons are examined as useful candidates for driven assembly given their tunability and robustness.

Giant director fluctuations in liquid crystal drops

We report the discovery and elucidation of giant spatiotemporal orientational fluctuations in nematic liquid crystal drops with radial orientation of the nematic anisotropy axis producing a central "hedgehog" defect. We study the spatial and temporal properties of the fluctuations experimentally using polarized optical microscopy, and theoretically, by calculating the eigenspectrum of the Frank elastic free energy of a nematic drop of radius R_{2}, containing a spherical central core of radius R_{1} and constrained by perpendicular boundary conditions on all surfaces. We find that the hedgehog defect with radial orientation has a complex excitation spectrum with a single critical mode whose energy vanishes at a critical value μ_{c} of the ratio μ=R_{2}/R_{1}. When μ<μ_{c}, the mode has positive energy, indicating that the radial hedgehog state is stable; when μ>μ_{c}, it has negative energy indicating that the radial state is unstable to the formation of a lower-energy state. This mode gives rise to the large-amplitude director fluctuations we observe near the core, for μ near μ_{c}. A collapse of the experimental data corroborates model predictions for μ<μ_{c} and provides an estimate of the defect core size.

Rods in a lyotropic chromonic liquid crystal: emergence of chirality, symmetry-breaking alignment, and caged angular diffusion

In lyotropic chromonic liquid crystals (LCLCs), twist distortion of the nematic director costs much less energy than splay or bend distortion. This feature leads to novel mirror-symmetry breaking director configurations when the LCLCs are confined by interfaces or contain suspended particles. Spherical colloids in an aligned LCLC nematic phase, for example, induce chiral director perturbations ("twisted tails"). The asymmetry of rod-like particles in an aligned LCLC offer a richer set of possibilities due to their aspect ratio (α) and mean orientation angle (〈θ〉) between their long axis and the uniform far-field director. Here we report on the director configuration, equilibrium orientation, and angular diffusion of rod-like particles with planar anchoring suspended in an aligned LCLC. Video microscopy reveals, counterintuitively, that two-thirds of the rods have an angled equilibrium orientation (〈θ〉 ≠ 0) that decreases with increasing α, while only one-third of the rods are aligned (〈θ〉 = 0). Polarized optical video-microscopy and Landau-de Gennes numerical modeling demonstrate that the angled and aligned rods are accompanied by distinct chiral director configurations. Angled rods have a longitudinal mirror plane (LMP) parallel to their long axis and approximately parallel to the substrate walls. Aligned rods have a transverse and longitudinal mirror plane (TLMP), where the transverse mirror plane is perpendicular to the rod's long axis. Effectively, the small twist elastic constant of LCLCs promotes chiral director configurations that modify the natural tendency of rods to orient along the far-field director. Additional diffusion experiments confirm that rods are angularly confined with strength that depends on α.

Cholesteric Shells: Two-Dimensional Blue Fog and Finite Quasicrystals

We study the phase behavior of a quasi-two-dimensional cholesteric liquid crystal shell. We characterize the topological phases arising close to the isotropic-cholesteric transition and show that they differ in a fundamental way from those observed on a flat geometry. For spherical shells, we discover two types of quasi-two-dimensional topological phases: finite quasicrystals and amorphous structures, both made up of mixtures of polygonal tessellations of half-skyrmions. These structures generically emerge instead of regular double twist lattices because of geometric frustration, which disallows a regular hexagonal tiling of curved space. For toroidal shells, the variations in the local curvature of the surface stabilizes heterogeneous phases where cholesteric patterns coexist with hexagonal lattices of half-skyrmions. Quasicrystals and amorphous and heterogeneous structures could be sought experimentally by self-assembling cholesteric shells on the surface of emulsion droplets.

Layering transitions and metastable structures of cholesteric liquid crystals in cylindrical confinement

Our study of cholesteric lyotropic chromonic liquid crystals in cylindrical confinement reveals the topological aspects of cholesteric liquid crystals. The double-twist configurations we observe exhibit discontinuous layering transitions, domain formation, metastability, and chiral point defects as the concentration of chiral dopant is varied. We demonstrate that these distinct layer states can be distinguished by chiral topological invariants. We show that changes in the layer structure give rise to a chiral soliton similar to a toron, comprising a metastable pair of chiral point defects. Through the applicability of the invariants we describe to general systems, our work has broad relevance to the study of chiral materials.

Extremely small twist elastic constants in lyotropic nematic liquid crystals

Recent measurements of the elastic constants in lyotropic chromonic liquid crystals (LCLCs) have revealed an anomalously small twist elastic constant compared to the splay and bend constants. Interestingly, measurements of the elastic constants in the micellar lyotropic liquid crystals (LLCs) that are formed by surfactants, by far the most ubiquitous and studied class of LLCs, are extremely rare and report only the ratios of elastic constants and do not include the twist elastic constant. By means of light scattering, this study presents absolute values of the elastic constants and their corresponding viscosities for the nematic phase of a standard LLC composed of disk-shaped micelles. Very different elastic moduli are found. While the splay elastic constant is in the typical range of 1.5 pN as is true in general for thermotropic nematics, the twist elastic constant is found to be one order of magnitude smaller (0.30 pN) and almost two orders of magnitude smaller than the bend elastic constant (21 pN). These results demonstrate that a small twist elastic constant is not restricted to the special case of LCLCs, but is true for LLCs in general. The reason for this extremely small twist elastic constant very likely originates with the flexibility of the assemblies that are the building blocks of both micellar and chromonic lyotropic liquid crystals.

Geometry of Bend: Singular Lines and Defects in Twist-Bend Nematics

We describe the geometry of bend distortions in liquid crystals and their fundamental degeneracies, which we call β lines; these represent a new class of linelike topological defect in twist-bend nematics. We present constructions for smecticlike textures containing screw and edge dislocations and also for vortexlike structures of double twist and Skyrmions. We analyze their local geometry and global structure, showing that their intersection with any surface is twice the Skyrmion number. Finally, we demonstrate how arbitrary knots and links can be created and describe them in terms of merons, giving a geometric perspective on the fractionalization of Skyrmions.

Binary mixtures of active and passive particles on a sphere

We study the cooperation and segregation dynamics of binary mixtures of active and passive particles on a sphere. According to the competition between rotational diffusion and polar alignment, we find three distinct phases: a mixed phase and two different demixed phases. When rotational diffusion dominates the dynamics, the demixing is due to the aggregation of passive particles, where active and passive particles respectively occupy two hemispheres. When polar alignment is dominated, the demixing is caused by the aggregation of active particles, where active particles occupy the equator of the sphere and passive particles occupy the two poles of the sphere. In this case, there exist a circulating band cluster and two cambered surface clusters, which is a purely curvature-driven effect with no equivalent in the planar model. When rotational diffusion and polar alignment are comparable, particles are completely mixed. Our findings are relevant to the experimental pursuit of segregation dynamics of binary mixtures on curved surfaces.

Microfluidic control over topological states in channel-confined nematic flows

Compared to isotropic liquids, orientational order of nematic liquid crystals makes their rheological properties more involved, and thus requires fine control of the flow parameters to govern the orientational patterns. In microfluidic channels with perpendicular surface alignment, nematics discontinuously transition from perpendicular structure at low flow rates to flow-aligned structure at high flow rates. Here we show how precise tuning of the driving pressure can be used to stabilize and manipulate a previously unresearched topologically protected chiral intermediate state which arises before the homeotropic to flow-aligned transition. We characterize the mechanisms underlying the transition and construct a phenomenological model to describe the critical behaviour and the phase diagram of the observed chiral flow state, and evaluate the effect of a forced symmetry breaking by introduction of a chiral dopant. Finally, we induce transitions on demand through channel geometry, application of laser tweezers, and careful control of the flow rate.

It is interesting phenomenon that chiral order can emerge in intrinsically achiral liquid crystals. Here Čopar et al. demonstrate achiral-to-chiral transition of the nematic liquid crystals flow in microfluidic channels and their behaviour, stability, and dependence on geometric and material parameters.

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.

Curved boundaries and chiral instabilities - two sources of twist in homeotropic nematic tori

Many liquid crystalline systems display spontaneous breaking of achiral symmetry, as achiral molecules aggregate into large chiral domains. In confined cylinders with homeotropic boundary conditions, chromonic liquid crystals - which have a twist elastic modulus which is at least an order of magnitude less than their splay and bend counter parts - adopt a twisted escaped radial texture (TER) to minimize their free energy, whilst 5CB - which has all three elastic constants roughly comparable - does not. In a recent series of experiments, we have shown that 5CB confined to tori and bent cylindrical capillaries with homeotropic boundary conditions also adopts a TER structure resulting from the curved nature of the confining boundaries [P. W. Ellis et al., Phys. Rev. Lett., 2018, 247803]. We shall call this microscopic twist, as the twisted director organization not only depends on the confinement geometry but also on the values of elastic moduli. Additionally, we demonstrate theoretically that the curved geometry of the boundary induces a twist in the escaped radial (ER) texture. Moving the escaped core of the structure towards the center of the torus not only lowers the splay and bend energies, but lowers the energetic cost of this distinct source for twist that we shall call geometric twist. As the torus becomes more curved, the ideal location for the escaped core approaches the inner radius of the torus.