Two-dimensional skyrmions and other solitonic structures in confinement-frustrated chiral nematics

Hidden topological constellations and polyvalent charges in chiral nematic droplets

Topology has an increasingly important role in the physics of condensed matter, quantum systems, material science, photonics and biology, with spectacular realizations of topological concepts in liquid crystals. Here we report on long-lived hidden topological states in thermally quenched, chiral nematic droplets, formed from string-like, triangular and polyhedral constellations of monovalent and polyvalent singular point defects. These topological defects are regularly packed into a spherical liquid volume and stabilized by the elastic energy barrier due to the helical structure and confinement of the liquid crystal in the micro-sphere. We observe, for the first time, topological three-dimensional point defects of the quantized hedgehog charge q=−2, −3. These higher-charge defects act as ideal polyvalent artificial atoms, binding the defects into polyhedral constellations representing topological molecules.

Once a purely mathematical discipline, topology has become an essential tool to investigate physical phenomena such as topological states in liquid crystals. Posnjak et al. observe the existence of 3D point defects of higher than unit topological charge in thermally quenched chiral nematic droplets.

Cholesteric liquid crystals in rectangular microchannels: skyrmions and stripes

In this paper, we present experimental and numerical studies on the microstructures of a cholesteric liquid crystal (CLC) confined in rectangular micron-channels. By using a sequence of microfabrication techniques we fabricated the micro-sized channels with accurately controlled size, aspect ratio and homeotropic surface anchoring. Through optical microscopic studies we established a phase diagram for the liquid crystal defect textures as a function of the channel depth and width. For the channel width larger than ∼2 times the cholesteric pitch p, the LC molecules are oriented primarily vertical to the channel when the channel depth is below 0.75p, form bubble domain defects when the channel depth is around 0.75p, and form stripe textures when the cell depth is above the cholesteric pitch p. In addition, the bubble domain size and the stripe texture periodicity are found to grow with the increase of the channel width. For the channel width smaller than ∼2p and the channel depth between 0.6p to 1.1p, no textures can be observed in the channels. Numerical simulations based on a director tensor relaxation approach yield detailed molecular director fields, and show that the bubble domain defects are baby-skyrmions and that the stripes are the first type of cholesteric fingerprints. A comparison with previous experiments and numerical simulations indicates that the size of the microchannels also influences what type of soliton-like topological textures form in the CLCs confined in the channels.

Points, skyrmions and torons in chiral nematic droplets

Chiral nematic droplets with perpendicular surface alignment of liquid crystalline molecules frustrate the helical structure into convoluted 3D textures with complex topology. We observe the droplets with fluorescent confocal polarising microscopy (FCPM), and reconstruct and analyse for the first time the topology of the 3D director field using a novel method of director reconstruction from raw data. We always find an odd number of topological defects, which preserve the total topological charge of the droplet of +1 regardless of chirality. At higher chirality, we observe up to 5 point hedgehog defects, which are elastically stabilized with convoluted twisted structures, reminiscent of 2D skyrmions and toron-like structure, nested into a sphere.

Energy surface and minimum energy paths for Fréedericksz transitions in bistable cholesteric liquid crystals

The multidimensional energy surface of a cholesteric liquid crystal in a planar cell is investigated as a function of spherical coordinates determining the director orientation. Minima on the energy surface correspond to the stable states with particular director distribution. External electric and magnetic fields deform the energy surface and positions of minima. It can lead to the transitions between states, known as the Fréedericksz effect. Transitions can be continuous or discontinuous depending on parameters of the liquid crystal which determine an energy surface. In a case of discontinuous transition when a barrier between stable states is comparable with the thermal energy, the activation transitions may occur, and it leads to the modification of characteristics of the Fréedericksz effect with temperature without explicit temperature dependencies of liquid crystal parameters. A minimum energy path between stable states on the energy surface for the Fréedericksz transition is found using the geodesic nudged elastic band method. Knowledge of this path, which has maximal statistical weight among all other paths, gives the information about a barrier between stable states and configuration of director orientation during the transition. It also allows one to estimate the stability of states with respect to the thermal fluctuations and their lifetime when the system is close to the Fréedericksz transition.

Reversal of helicoidal twist handedness near point defects of confined chiral liquid crystals

Handedness of the director twist in cholesteric liquid crystals is commonly assumed to be the same throughout the medium, determined solely by the chirality of constituent molecules or chiral additives, albeit distortions of the ground-state helicoidal configuration often arise due to the effects of confinement and external fields. We directly probe the twist directionality of liquid crystal director structures through experimental three-dimensional imaging and numerical minimization of the elastic free energy and show that spatially localized regions of handedness opposite to that of the chiral liquid crystal ground state can arise in the proximity of twisted-soliton-bound topological point defects. In chiral nematic liquid crystal confined to a film that has a thickness less than the cholesteric pitch and perpendicular surface boundary conditions, twisted solitonic structures embedded in a uniform unwound far-field background with chirality-matched handedness locally relieve confinement-imposed frustration and tend to be accompanied by point defects and smaller geometry-required, energetically costly regions of opposite twist handedness. We also describe a spatially localized structure, dubbed a "twistion," in which a twisted solitonic three-dimensional director configuration is accompanied by four point defects. We discuss how our findings may impinge on the stability of localized particlelike director field configurations in chiral and nonchiral liquid crystals.

Electric field generation of Skyrmion-like structures in a nematic liquid crystal

Skyrmions are particle-like topological objects that are increasingly drawing attention in condensed matter physics, where they are connected to inversion symmetry breaking and chirality. Here we report the generation of stable Skyrmion-like structures in a thin nematic liquid crystal film on chemically patterned patchy surfaces. Using the interplay of material elasticity and surface boundary conditions, we use a strong electric field to quench the nematic liquid crystal from a fully aligned phase to vortex-like nematic liquid crystal structures, centered on patterned patches, which carry two different sorts of topological defects. Numerical calculations reveal that these are Skyrmion-like structures, seeded from the surface boojum topological defects and swirling towards the second confining surface. These observations, supported by numerical methods, demonstrate the possibility to generate, manipulate and study Skyrmion-like objects in nematic liquid crystals on patterned surfaces.

Uniaxial Pressure Dependence of Magnetic Order in MnSi

We report comprehensive small angle neutron scattering measurements complemented by ac susceptibility data of the helical order, conical phase, and Skyrmion lattice phase (SLP) in MnSi under uniaxial pressures. For all crystallographic orientations uniaxial pressure favors the phase for which a spatial modulation of the magnetization is closest to the pressure axis. Uniaxial pressures as low as 1 kbar applied perpendicular to the magnetic field axis enhance the Skyrmion lattice phase substantially, whereas the Skyrmion lattice phase is suppressed for pressure parallel to the field. Taken together we present quantitative microscopic information on how strain couples to magnetic order in the chiral magnet MnSi.

Photoswitchable and dye-doped bubble domain texture of cholesteric liquid crystals

We demonstrate control of the transmittance of the naturally formed bubble domain (BD) texture of cholesteric liquid crystals (CLC) with negative dielectric anisotropy confined into a cell with homeotropic surface anchoring. By using a photosensitive chiral dopant with variable helical twisting power under light irradiation, control of packing density of bubbles, spatial patterning, and all optical switching between bistable states with different optical densities is achieved. By introducing dichroic dye into the CLC mixture, a bistable and switchable by applied electric field guest-host system is obtained. The light dimming properties of dye-doped BD CLC systems may lead to development of a wide range of applications.

Ferromagnetic Switching of Knotted Vector Fields in Liquid Crystal Colloids

We experimentally realize polydomain and monodomain chiral ferromagnetic liquid crystal colloids that exhibit solitonic and knotted vector field configurations. Formed by dispersions of ferromagnetic nanoplatelets in chiral nematic liquid crystals, these colloidal ferromagnets exhibit spontaneous long-range alignment of magnetic dipole moments of individual platelets, giving rise to a continuum of the magnetization field M(r). Competing effects of surface confinement and chirality prompt spontaneous formation and enable the optical generation of localized twisted solitonic structures with double-twist tubes and torus knots of M(r), which exhibit a strong sensitivity to the direction of weak magnetic fields ∼1  mT. Numerical modeling, implemented through free energy minimization to arrive at a field-dependent three-dimensional M(r), shows a good agreement with experiments and provides insights into the torus knot topology of observed field configurations and the corresponding physical underpinnings.

Creation and manipulation of topological states in chiral nematic microspheres

Topology is a universal concept that is encountered in daily life and is known to determine many static and dynamical properties of matter. Taming and controlling the topology of materials therefore constitutes a contemporary interdisciplinary challenge. Building on the controllable spatial properties of soft matter appears as a relevant strategy to address the challenge, in particular, because it may lead to paradigmatic model systems that allow checking theories experimentally. Here we report experimentally on a wealth of complex free-standing metastable topological architectures at the micron scale, in frustrated chiral nematic droplets. These results support recent works predicting the formation of free-standing knotted and linked disclination structures in confined chiral nematic fluids. We also demonstrate that various kinds of external fields (thermal, electrical and optical) can be used to achieve topological remote control. All this may foster the development of new devices based on topologically structured soft media.

Self-assembly and electrostriction of arrays and chains of hopfion particles in chiral liquid crystals

Some of the most exotic condensed matter phases, such as twist grain boundary and blue phases in liquid crystals and Abrikosov phases in superconductors, contain arrays of topological defects in their ground state. Comprised of a triangular lattice of double-twist tubes of magnetization, the so-called ‘A-phase’ in chiral magnets is an example of a thermodynamically stable phase with topologically nontrivial solitonic field configurations referred to as two-dimensional skyrmions, or baby-skyrmions. Here we report that three-dimensional skyrmions in the form of double-twist tori called ‘hopfions’, or ‘torons’ when accompanied by additional self-compensating defects, self-assemble into periodic arrays and linear chains that exhibit electrostriction. In confined chiral nematic liquid crystals, this self-assembly is similar to that of liquid crystal colloids and originates from long-range elastic interactions between particle-like skyrmionic torus knots of molecular alignment field, which can be tuned from isotropic repulsive to weakly or highly anisotropic attractive by low-voltage electric fields.

Topological defects can be spontaneously generated to thermodynamically stabilize a variety of peculiar condensed matter phases for technological applications. Here, Ackerman et al. show electrically controllable self-assembly of knotted defects into periodic arrays in chiral liquid crystals.

Topology and self-assembly of defect-colloidal superstructure in confined chiral nematic liquid crystals

We describe formation of defect-colloidal superstructures induced by microspheres with normal surface anchoring dispersed in chiral nematic liquid crystals in confinement-unwound homeotropic cells. Using three-dimensional nonlinear optical imaging of the director field, we demonstrate that some of the induced defects have nonsingular solitonic nature while others are singular point and line topological defects. The common director structures induced by individual microspheres have dipolar symmetry. These topological dipoles are formed by the particle and a hyperbolic point defect (or small disclination loop) of elementary hedgehog charge opposite to that of a sphere with perpendicular boundary conditions, which in cells with thickness over equilibrium cholesteric pitch ratio approaching unity are additionally interspaced by a looped double-twist cylinder of continuous director deformations. The long-range elastic interactions are probed by holographic optical tweezers and videomicroscopy, providing insights to the physical underpinnings behind self-assembled colloidal structures entangled by twisted solitons. Computer-simulated field and defect configurations induced by the colloidal particles and their assemblies, which are obtained by numerically minimizing the Landau-de Gennes free energy, are in agreement with the experimental findings.

Theory of skyrmion states in liquid crystals

Within the Oseen-Frank theory we derive numerically exact solutions for axisymmetric localized states in chiral liquid crystal layers with homeotropic anchoring. These solutions describe recently observed two-dimensional skyrmions in confinement-frustrated chiral nematics [P. J. Ackerman et al., Phys. Rev. E 90, 012505 (2014)]. We stress that these solitonic states arise due to a fundamental stabilization mechanism responsible for the formation of skyrmions in cubic helimagnets and other noncentrosymmetric condensed-matter systems.