Lassoing saddle splay and the geometrical control of topological defects

Director Response of Liquid Crystals in Spatially Varying Magnetic Fields with Antagonistic Anchoring Conditions

In the presence of a magnetic field, a liquid crystal (LC) director can be distorted from a ground state set by a combination of LC elasticity and surface anchoring at any relevant interfaces. Uniform magnetic fields are often used to produce simple LC distortions on demand, but producing more spatially complex distortions is practically challenging. We develop a strategy for the spatially resolved control of the LC director by leveraging field patterns induced by ferromagnetic materials. Patterned magnetic fields are generated from high-permeability ferromagnetic microstructures embedded into nematic liquid crystals (NLCs) to manipulate the LC director's orientation. Each ferromagnetic microstructure produces a unique spatially varying magnetic field. In turn, tuning magnetic field strength in competition with NLC elasticity can pattern a range of spatially complex director configurations. Simulations relate the spatial variation induced in a magnetic field by a ferromagnetic geometry and the resultant director. Our predictive models can inform the inverse design of ferromagnetic microstructures to generate bespoke director patterns. We also link changes in the magnetic field to the migration of elastically driven periodic extinctions in birefringence near the edges of ferromagnetic structures.

Confinement and magnetic-field effect on chiral ferroelectric nematic liquid crystals in Grandjean-Cano wedge cells

We explore the structure and magnetic-field response of edge dislocations in Grandjean-Cano wedge cells filled with chiral mixtures of the ferroelectric nematic mesogen DIO. Upon cooling, the ordering changes from paraelectric in the cholesteric phase N^{*} to antiferroelectric in the smectic SmZ_{A}^{*} and to ferroelectric in the cholesteric N_{F}^{*}. Dislocations of the Burgers vector b equal to the helicoidal pitch P are stable in all three phases, while dislocations with b=P/2 exist only in the N^{*} and SmZ_{A}^{*}. The b=P/2 dislocations split into pairs of τ^{-1/2}λ^{+1/2} disclinations, while the thick dislocations b=P are pairs of nonsingular λ^{-1/2}λ^{+1/2} disclinations. The polar order makes the τ^{-1/2} disclinations unstable in the N_{F}^{*} phase, as they should be connected to singular walls in the polarization field. We propose a model of transformation of the composite τ^{-1/2} line-wall defect into a nonsingular λ^{-1/2} disclination, which is paired up with a λ^{+1/2} line to form a b=P dislocation. The SmZ_{A}^{*} behavior in the in-plane magnetic field is different from that of the N_{F}^{*} and N^{*}: the dislocations show no zigzag instability, and the pitch remains unchanged in the magnetic fields up to 1 T. The behavior is associated with the finite compressibility of smectic layers.

Moiré effect enables versatile design of topological defects in nematic liquid crystals

Recent advances in surface-patterning techniques of liquid crystals have enabled the precise creation of topological defects, which promise a variety of emergent applications. However, the manipulation and application of these defects remain limited. Here, we harness the moiré effect to engineer topological defects in patterned nematic liquid crystal cells. Specifically, we combine simulation and experiment to examine a nematic cell confined between two substrates of periodic surface anchoring patterns; by rotating one surface against the other, we observe a rich variety of highly tunable, novel topological defects. These defects are shown to guide the three-dimensional self-assembly of colloids, which can conversely impact defects by preventing the self-annihilation of loop-defects through jamming. Finally, we demonstrate that certain nematic moiré cells can engender arbitrary shapes represented by defect regions. As such, the proposed simple twist method enables the design and tuning of mesoscopic structures in liquid crystals, facilitating applications including defect-directed self-assembly, material transport, micro-reactors, photonic devices, and anti-counterfeiting materials.

Nematic-isotropic phase transitions in thin slabs of liquid crystals with topological defect arrays

We study the nematic-to-isotropic phase transitions in thin slabs of nematic liquid crystals with photopatterned director fields of topological defect arrays at constant heating rates and show that the transition kinetics is significantly impacted by both the heating rate and the topological strengths of these defects. Specifically, with ±1/2 defect arrays, the isotropic domains emerge from the defect cores when the heating rate is high, while from random places when the heating rate is low. With ±1 defect arrays, the isotropic domains always emerge from the defect cores regardless of the heating rate. Furthermore, the isotropic domains show significant movements at slow heating rates, and the total area of the isotropic domains grows with the temperature T following a simple power law (T - T')γ, where the exponent γ is approximately 1 in most cases and is 2/3 for the ±1 defect arrays at low heating rates when the isotropic domains are pinned on the defect cores. We attribute this phenomenon to an interplay between the surface tension and bulk free energy.

Role of Stimuli on Liquid Crystalline Defects: From Defect Engineering to Switchable Functional Materials

Achieving tunable physical properties is currently one of the most exciting research topics. In order to realize this goal, a medium that is responsive to external stimuli and can undergo a change in its physical property is required. Liquid crystal (LC) is a prominent candidate, as its physical and optical properties can be easily manipulated with various stimuli, such as surface anchoring, rubbing, geometric confinement, and external fields. Having broken away from the past devotion to obtaining a uniform domain of LCs, people are now putting significant efforts toward forming and manipulating ordered and oriented defect structures with a unique arrangement within. The complicated molecular order with tunability would benefit the interdisciplinary research fields of optics, physics, photonics, and materials science. In this review, the recent progress toward defect engineering in the nematic and smectic phases by controlling the surface environment and electric field and their combinational methods is introduced. We close the review with a discussion of the possible applications enabled using LC defect structures as switchable materials.

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.

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.

Optical Textures and Orientational Structures in Cholesteric Droplets with Conical Boundary Conditions

Cholesteric droplets dispersed in polymer with conical boundary conditions have been studied. The director configurations are identified by the polarising microscopy technique. The axisymmetric twisted axial-bipolar configuration with the surface circular defect at the droplet’s equator is formed at the relative chirality parameter N0≤2.9. The intermediate director configuration with the deformed circular defect is realised at 2.9<N0<3.95, and the layer-like structure with the twisted surface defect loop is observed at N0≥3.95. The cholesteric layers in the layer-like structure are slightly distorted although the cholesteric helix is untwisted.

Topological structure and dynamics of three-dimensional active nematics

Topological structures are effective descriptors of the nonequilibrium dynamics of diverse many-body systems. For example, motile, point-like topological defects capture the salient features of two-dimensional active liquid crystals composed of energy-consuming anisotropic units. We dispersed force-generating microtubule bundles in a passive colloidal liquid crystal to form a three-dimensional active nematic. Light-sheet microscopy revealed the temporal evolution of the millimeter-scale structure of these active nematics with single-bundle resolution. The primary topological excitations are extended, charge-neutral disclination loops that undergo complex dynamics and recombination events. Our work suggests a framework for analyzing the nonequilibrium dynamics of bulk anisotropic systems as diverse as driven complex fluids, active metamaterials, biological tissues, and collections of robots or organisms.

Programming emergent symmetries with saddle-splay elasticity

The director field adopted by a confined liquid crystal is controlled by a balance between the externally imposed interactions and the liquid's internal orientational elasticity. While the latter is usually considered to resist all deformations, liquid crystals actually have an intrinsic propensity to adopt saddle-splay arrangements, characterised by the elastic constant [Formula: see text]. In most realisations, dominant surface anchoring treatments suppress such deformations, rendering [Formula: see text] immeasurable. Here we identify regimes where more subtle, patterned surfaces enable saddle-splay effects to be both observed and exploited. Utilising theory and continuum calculations, we determine experimental regimes where generic, achiral liquid crystals exhibit spontaneously broken surface symmetries. These provide a new route to measuring [Formula: see text]. We further demonstrate a multistable device in which weak, but directional, fields switch between saddle-splay-motivated, spontaneously-polar surface states. Generalising beyond simple confinement, our highly scalable approach offers exciting opportunities for low-field, fast-switching optoelectronic devices which go beyond current technologies.

Woven Nematic Defects, Skyrmions, and the Abelian Sandpile Model

We show that a fixed set of woven defect lines in a nematic liquid crystal supports a set of nonsingular topological states which can be mapped on to recurrent stable configurations in the Abelian sandpile model or chip-firing game. The physical correspondence between local skyrmion flux and sandpile height is made between the two models. Using a toy model of the elastic energy, we examine the structure of energy minima as a function of topological class and show that the system admits domain wall skyrmion solitons.

Mosaics of topological defects in micropatterned liquid crystal textures

Topological defects in the orientational order that appear in thin slabs of a nematic liquid crystal, as seen in the standard schlieren texture, behave as a random quasi-two-dimensional system with strong optical birefringence. We present an approach to creating and controlling the defects using air pillars, trapped by micropatterned holes in the silicon substrate. The defects are stabilized and positioned by the arrayed air pillars into regular two-dimensional lattices. We explore the effects of hole shape, lattice symmetry, and surface treatment on the resulting lattices of defects and explain their arrangements by application of topological rules. Last, we show the formation of detailed kaleidoscopic textures after the system is cooled down across the nematic-smectic A phase transition, frustrating the defects and surrounding structures with the equal-layer spacing condition of the smectic phase.

Defect transitions in nematic liquid-crystal capillary bridges

We use experiment and computational modeling to understand the defect structure and director configuration in a nematic liquid crystal capillary bridge confined between two parallel plates. We find that tuning of the aspect ratio of the bridge drives a transition between a ring defect and a point defect. This transition exhibits hysteresis, due to the metastability of the point-defect structure. In addition, we see that the shape of the capillary-bridge surface determines whether the defect is hyperbolic or radial, with waistlike bridges containing hyperbolic defects and barrel-like bridges containing radial defects.

In Silico Measurement of Elastic Moduli of Nematic Liquid Crystals

Experiments on confined droplets of the nematic liquid crystal 5CB have questioned long-established bounds imposed on the elastic free energy of nematic systems. This elasticity, which derives from molecular alignment within nematic systems, is quantified through a set of moduli which can be difficult to measure experimentally and, in some cases, can only be probed indirectly. This is particularly true of the surfacelike saddle-splay elastic term, for which the available experimental data indicate values on the cusp of stability, often with large uncertainties. Here, we demonstrate that all nematic elastic moduli, including the saddle-splay elastic constant k_{24}, may be calculated directly from atomistic molecular simulations. Importantly, results obtained through in silico measurements of the 5CB elastic properties demonstrate unambiguously that saddle-splay elasticity alone is unable to describe the observed confined morphologies.

Artificial web of disclination lines in nematic liquid crystals

Disclinations are topological singularities of molecular arrangement in liquid crystals, which typically occur when the average orientation of molecules makes a π rotation along a fictitious closed loop taken inside the liquid crystal. Depending on the sense of molecular rotation, the disclination lines are either of 1/2 or −1/2 strength. When two disclination lines with the opposite strength meet, they are annihilated without trace. It is hence generally considered difficult in the nematic phase to stabilize a condensed array of free-standing disclination lines without the aid of topological objects like colloidal inclusions. Here we show that a free-standing web of 1/2-strength twist disclination lines can be stably formed in thin liquid crystal cells by means of a judicious combination of orientationally patterned confining surfaces fabricated by the micropatterned photoalignment technique. Theoretical model indicates that disclination lines are held apart at the intersection by a repulsive force generated by the Frank elasticity.

Disclination lines are topological defects in molecular orientation widely found in liquid crystals. Here Wang et al. use a surface patterning technique to produce a very stable freestanding 3D array of ½ twist disclinations, which could be exploited in a variety of nanometre scale applications.