Topological solitons, cholesteric fingers and singular defect lines in Janus liquid crystal shells

Abrikosov clusters in chiral liquid crystal droplets

Self-organizing triangular lattices of topological vortices have been observed in type-II superconductors, Bose-Einstein condensates, and chiral magnets under external forcing. Liquid crystals exhibit vortex self-organization in dissipative media. In this study, we experimentally investigate the formation of vortex clusters, analogous to Abrikosov lattices, in temperature-driven chiral liquid crystal droplets. Based on a Ginzburg-Landau-like equation, we derive the interaction laws underlying the formation of these Abrikosov clusters of chiral domains. The origin of these is elucidated due to the competition between the repulsive interaction and the spatial effect of the confinement within the droplet. Our results advance the theoretical understanding of localized vortex self-organization in liquid crystals and open up possibilities for controlling the clustering of these topological defects.

Topologically frustrated structures in inkjet printed chiral nematic liquid crystal droplets - experiments and simulations

Director field alignment in inkjet printed droplets of chiral nematic liquid crystalline materials is investigated using both experiments and numerical simulations. Experimental investigations are performed by depositing droplets of varying sizes and pitches on homeotropic alignment layers. The competition between the bulk behaviour of the chiral nematic liquid crystal and the boundary conditions imposed by the droplet surface leads to the formation of a range of possible internal director configurations. Numerical investigations are performed using a free energy minimisation approach, and the resultant simulated polarising optical microscope images are found to agree well with experimental observations.

Tunable templating of photonic microparticles via liquid crystal order-guided adsorption of amphiphilic polymers in emulsions

Multiple emulsions are usually stabilized by amphiphilic molecules that combine the chemical characteristics of the different phases in contact. When one phase is a liquid crystal (LC), the choice of stabilizer also determines its configuration, but conventional wisdom assumes that the orientational order of the LC has no impact on the stabilizer. Here we show that, for the case of amphiphilic polymer stabilizers, this impact can be considerable. The mode of interaction between stabilizer and LC changes if the latter is heated close to its isotropic state, initiating a feedback loop that reverberates on the LC in form of a complete structural rearrangement. We utilize this phenomenon to dynamically tune the configuration of cholesteric LC shells from one with radial helix and spherically symmetric Bragg diffraction to a focal conic domain configuration with highly complex optics. Moreover, we template photonic microparticles from the LC shells by photopolymerizing them into solids, retaining any selected LC-derived structure. Our study places LC emulsions in a new light, calling for a reevaluation of the behavior of stabilizer molecules in contact with long-range ordered phases, while also enabling highly interesting photonic elements with application opportunities across vast fields.

Temperature reconfigurable skyrmionic solitons in cholesteric liquid crystals

In this work, a reversible transformation between torons and cholesteric fingers is realized by continuously changing the pitch through temperature variation of the chiral nematic liquid crystal twist inversion system. By decreasing the pitch, the torons act as seeds from which cholesteric fingers gradually grow. By increasing the pitch, the cholesteric fingers gradually shorten and transform back to the initial state. We find that although the morphology of the torons is severely deformed and cannot be distinguished during the heating-cooling loops, the torons are very well topologically protected and can hardly be destroyed.

Topological phases and curvature-driven pattern formation in cholesteric shells

We study the phase behaviour of cholesteric liquid crystal shells with different geometries. We compare the cases of tangential anchoring and no anchoring at the surface, focussing on the former case, which leads to a competition between the intrinsic tendency of the cholesteric to twist and the anchoring free energy which suppresses it. We then characterise the topological phases arising close to the isotropic-cholesteric transition. These typically consist of quasi-crystalline or amorphous tessellations of the surface by half-skyrmions, which are stable at lower and larger shell sizes, respectively. For ellipsoidal shells, defects in the tessellation couple to a local curvature, and according to the shell size, they either migrate to the poles or distribute uniformly on the surface. For toroidal shells, the variations in the local curvature of the surface stabilise heterogeneous phases where cholesteric or isotropic patterns coexist with hexagonal lattices of half-skyrmions.

Optical Imaging and Analytical Design of Localized Topological Structures in Chiral Liquid Crystals

We combine numerical modeling and analytical design techniques to study several of the most common localized topological structures in frustrated chiral nematic liquid crystal cells. An energy minimization procedure is applied to the lattice model to simulate the director field distributions. These distributions are also approximated using the suitably designed analytical ansatz. We present both simulated and approximated results for optical polarizing microscopy textures and different visualizations of director field structure such as distributions of the azimuthal director angle and isolines for the normal component of the director in coordinate planes. The ansatz correctly mimicked the geometry and optical properties of the solitonic structures under consideration.

Elastic Instability of Cubic Blue Phase Nano Crystals in Curved Shells

Many crystallization processes, including biomineralization and ice-freezing, occur in small and curved volumes, where surface curvature can strain the crystal, leading to unusual configurations and defect formation. The role of curvature on crystallization, however, remains poorly understood. Here, we study the crystallization of blue phase (BP) liquid crystals under curved confinement, which provides insights into the mechanism by which BPs reconfigure their three-dimensional lattice structure to adapt to curvature. BPs are a three-dimensional assembly of high-chirality liquid crystal molecules arranged into body-centered (BPI) or simple cubic (BPII) symmetries. BPs with submicrometer cubic-crystalline lattices exhibit tunable Bragg reflection and submillisecond response time to external stimuli such as an electric field, making them attractive for advanced photonic materials. In this work, we have systematically studied BPs confined in spherical shells with well-defined curvature and boundary conditions. The optical behavior of shells has also been examined at room temperature, where the cholesteric structure forms. In the cholesteric phase, perpendicular anchoring generates focal conic domains on the shell's surface, which transition into stripe patterns as the degree of curvature increases. Our results demonstrate that both higher degrees of curvature and strong spatial confinement destabilize BPI and reconfigure that phase to adopt the structure and optical features of BPII. We also show that the coupling of curvature and confinement nucleates skyrmions at greater thicknesses than those observed for a flat geometry. These findings are particularly important for integrating BPs into miniaturized and curved/flexible devices, including flexible displays, wearable sensors, and smart fabrics.

Fabrication of Arrays of Topological Solitons in Patterned Chiral Liquid Crystals for Real-Time Observation of Morphogenesis

Topological solitons have knotted continuous field configurations embedded in a uniform background, and occur in cosmology, biology, and electromagnetism. However, real-time observation of their morphogenesis and dynamics is still challenging because their size and timescale are enormously large or tiny. Liquid crystal (LC) structures are promising candidates for a model-system to study the morphogenesis of topological solitons, enabling direct visualization due to the proper size and timescale. Here, a new way is found to rationalize the real-time observation of the generation and transformation of topological solitons using cholesteric LCs confined in patterned substrates. The experimental demonstration shows the topologically protected structures arise via the transformation of topological defects. Numerical modeling based on minimization of free energy closely reconstructs the experimental findings. The fundamental insights obtained from the direct observations pose new theoretical challenges in understanding the morphogenesis of different types of topological solitons within a broad range of scales.

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.

Control of Monodomain Polymer-Stabilized Cuboidal Nanocrystals of Chiral Nematics by Confinement

Liquid crystals are important components of optical technologies. Cuboidal crystals consisting of chiral liquid crystals-the so-called blue phases (BPs), are of particular interest due to their crystalline structures and fast response times, but it is critical that control be gained over their phase behavior as well as the underlying dislocations and grain boundaries that arise in such systems. Blue phases exhibit cubic crystalline symmetries with lattice parameters in the 100 nm range and a network of disclination lines that can be polymerized to widen the range of temperatures over which they occur. Here, we introduce the concept of strain-controlled polymerization of BPs under confinement, which enables formation of strain-correlated stabilized morphologies that, under some circumstances, can adopt perfect single-crystal monodomain structures and undergo reversible crystal-to-crystal transformations, even if their disclination lines are polymerized. We have used super-resolution laser confocal microscopy to reveal the periodic structure and the lattice planes of the strain and polymerization stabilized BPs in 3D real space. Our experimental observations are supported and interpreted by relying on theory and computational simulations in terms of a free energy functional for a tensorial order parameter. Simulations are used to determine the orientation of the lattice planes unambiguously. The findings presented here offer opportunities for engineering optical devices based on single-crystal, polymer-stabilized BPs whose inherent liquid nature, fast dynamics, and long-range crystalline order can be fully exploited.

Design of nematic liquid crystals to control microscale dynamics

The dynamics of small particles, both living such as swimming bacteria and inanimate, such as colloidal spheres, has fascinated scientists for centuries. If one could learn how to control and streamline their chaotic motion, that would open technological opportunities in the transformation of stored or environmental energy into systematic motion, with applications in micro-robotics, transport of matter, guided morphogenesis. This review presents an approach to command microscale dynamics by replacing an isotropic medium with a liquid crystal. Orientational order and associated properties, such as elasticity, surface anchoring, and bulk anisotropy, enable new dynamic effects, ranging from the appearance and propagation of particle-like solitary waves to self-locomotion of an active droplet. By using photoalignment, the liquid crystal can be patterned into predesigned structures. In the presence of the electric field, these patterns enable the transport of solid and fluid particles through nonlinear electrokinetics rooted in anisotropy of conductivity and permittivity. Director patterns command the dynamics of swimming bacteria, guiding their trajectories, polarity of swimming, and distribution in space. This guidance is of a higher level of complexity than a simple following of the director by rod-like microorganisms. Namely, the director gradients mediate hydrodynamic interactions of bacteria to produce an active force and collective polar modes of swimming. The patterned director could also be engraved in a liquid crystal elastomer. When an elastomer coating is activated by heat or light, these patterns produce a deterministic surface topography. The director gradients define an activation force that shapes the elastomer in a manner similar to the active stresses triggering flows in active nematics. The patterned elastomer substrates could be used to define the orientation of cells in living tissues. The liquid-crystal guidance holds a major promise in achieving the goal of commanding microscale active flows.

Review: knots and other new topological effects in liquid crystals and colloids

Humankind has been obsessed with knots in religion, culture and daily life for millennia, while physicists like Gauss, Kelvin and Maxwell already involved them in models centuries ago. Nowadays, colloidal particles can be fabricated to have shapes of knots and links with arbitrary complexity. In liquid crystals, closed loops of singular vortex lines can be knotted by using colloidal particles and laser tweezers, as well as by confining nematic fluids into micrometer-sized droplets with complex topology. Knotted and linked colloidal particles induce knots and links of singular defects, which can be interlinked (or not) with colloidal particle knots, revealing the diversity of interactions between topologies of knotted fields and topologically nontrivial surfaces of colloidal objects. Even more diverse knotted structures emerge in nonsingular molecular alignment and magnetization fields in liquid crystals and colloidal ferromagnets. The topological solitons include hopfions, skyrmions, heliknotons, torons and other spatially localized continuous structures, which are classified based on homotopy theory, characterized by integer-valued topological invariants and often contain knotted or linked preimages, nonsingular regions of space corresponding to single points of the order parameter space. A zoo of topological solitons in liquid crystals, colloids and ferromagnets promises new breeds of information displays and a plethora of data storage, electro-optic and photonic applications. Their particle-like collective dynamics echoes coherent motions in active matter, ranging from crowds of people to schools of fish. This review discusses the state of the art in the field, as well as highlights recent developments and open questions in physics of knotted soft matter. We systematically overview knotted field configurations, the allowed transformations between them, their physical stability and how one can use one form of knotted fields to model, create and imprint other forms. The large variety of symmetries accessible to liquid crystals and colloids offer insights into stability, transformation and emergent dynamics of fully nonsingular and singular knotted fields of fundamental and applied importance. The common thread of this review is the ability to experimentally visualize these knots in real space. The review concludes with a discussion of how the studies of knots in liquid crystals and colloids can offer insights into topologically related structures in other branches of physics, with answers to many open questions, as well as how these experimentally observable knots hold a strong potential for providing new inspirations to the mathematical knot theory.

Swelling Cholesteric Liquid Crystal Shells to Direct the Assembly of Particles at the Interface

Cholesteric liquid crystals can exhibit spatial patterns in molecular alignment at interfaces that can be exploited for particle assembly. These patterns emerge from the competition between bulk and surface energies, tunable with the system geometry. In this work, we use the osmotic swelling of cholesteric double emulsions to assemble colloidal particles through a pathway-dependent process. Particles can be repositioned from a surface-mediated to an elasticity-mediated state through dynamically thinning the cholesteric shell at a rate comparable to that of colloidal adsorption. By tuning the balance between surface and bulk energies with the system geometry, colloidal assemblies on the cholesteric interface can be molded by the underlying elastic field to form linear aggregates. The transition of adsorbed particles from surface regions with homeotropic anchoring to defect regions is accompanied by a reduction in particle mobility. The arrested assemblies subsequently map out and stabilize topological defects. These results demonstrate the kinetic arrest of interfacial particles within definable patterns by regulating the energetic frustration within cholesterics. This work highlights the importance of kinetic pathways for particle assembly in liquid crystals, of relevance to optical and energy applications.

Commensurate states and pattern switching via liquid crystal skyrmions trapped in a square lattice

Using continuum based simulations we show that a rich variety of skyrmion liquid crystal states can be realized in the presence of a periodic obstacle array. As a function of the number of skyrmions per obstacle we find hexagonal, square, dimer, trimer and quadrimer ordering, where the n-mer structures are a realization of a molecular crystal state of skyrmions. As a function of external field and obstacle radius we show that there are transitions between the different crystalline states as well as mixed and disordered structures. We discuss how these states are related to commensurate effects seen in other systems, such as vortices in type-II superconductors and colloids interacting with two dimensional substrates.

Electrically powered motions of toron crystallites in chiral liquid crystals

Malleability of metals is an example of how the dynamics of defects like dislocations induced by external stresses alters material properties and enables technological applications. However, these defects move merely to comply with the mechanical forces applied on macroscopic scales, whereas the molecular and atomic building blocks behave like rigid particles. Here, we demonstrate how motions of crystallites and the defects between them can arise within the soft matter medium in an oscillating electric field applied to a chiral liquid crystal with polycrystalline quasi-hexagonal arrangements of self-assembled topological solitons called "torons." Periodic oscillations of electric field applied perpendicular to the plane of hexagonal lattices prompt repetitive shear-like deformations of the solitons, which synchronize the electrically powered self-shearing directions. The temporal evolution of deformations upon turning voltage on and off is not invariant upon reversal of time, prompting lateral translations of the crystallites of torons within quasi-hexagonal periodically deformed lattices. We probe how these motions depend on voltage and frequency of oscillating field applied in an experimental geometry resembling that of liquid crystal displays. We study the interrelations between synchronized deformations of the soft solitonic particles and their arrays, and the ensuing dynamics and giant number fluctuations mediated by motions of crystallites, five-seven defects pairs, and grain boundaries in the orderly organizations of solitons. We discuss how our findings may lead to technological and fundamental science applications of dynamic self-assemblies of topologically protected but highly deformable particle-like solitons.