Splitting, linking, knotting, and solitonic escape of topological defects in nematic drops with handles

A Monte Carlo simulation study of a Janus discotic liquid crystal droplet

The study of discotic liquid crystals (DLCs) under spherical confinement has gained considerable significance due to its relevance in the design and optimization of advanced materials with tailored properties. The unique characteristics of DLC fluids, coupled with confinement within a spherical Janus surface, offer a compelling avenue for exploring novel behaviors and emergent phenomena. In this study, Monte Carlo simulations within the NpT ensemble are employed to investigate the behavior of a DLC fluid confined by a spherical Janus surface. The Janus surface is characterized by distinct hemispheres, with one promoting homeotropic (face-on) anchoring and the other planar (edge-on) anchoring. Our analysis reveals the emergence of two topological defects: one exclusively on the edge-anchoring hemisphere and the other at the boundary of both anchorings. Each topological defect possessing a topological charge ofk= +1/2. We observe that as the temperature transitions the central region of the droplet into a nematic phase, a disclination line forms, linking the two surface defects. By investigating droplets of three different sizes, we confirm that the isotropic-nematic transition is first-order for the larger droplet studied. However, this transition becomes continuous under strong confinement conditions. In contrast, the nematic-columnar transition remains first order even for smaller systems.

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.

Continuous generation of topological defects in a passively driven nematic liquid crystal

Synthetic active matter is emerging as the prime route for the realisation of biological mechanisms such as locomotion, active mixing, and self-organisation in soft materials. In particular, passive nematic complex fluids are known to form out-of-equilibrium states with topological defects, but their locomotion, activation and experimental realization has been developed and understood to only a limited extent. Here, we report that the concentration-driven flow of small molecules triggers turbulent flow in the thin film of a nematic liquid crystal that continuously generates pairs of topological defects with an integer topological charge. The diffusion results in the formation of counter-rotating vortex rolls in the liquid crystal, which above a velocity threshold transform into a turbulent flow with continuous generation and annihilation of the defect pairs. The pairs of defects are created by the self-amplifying splay instability between the vortices, until a pair of oppositely charged defects is formed.

It has been known that spontaneous defect formation and annihilation can be triggered by turbulent flows in active nematic liquid crystals. Here, Mur et al. show a complementary mechanism induced by the flow of foreign organic molecules into the liquid crystal following the concentration gradient.

Structural transformations of nematic disclinations

Topological defects (TDs) are a consequence of symmetry breaking phase transitions and are ubiquitous in nature. An ideal testbed for their study are liquid crystals (LCs) owing to their large response to external stimuli and their large electrical and optical anisotropies. In this paper, we perform numerical simulations of topological defects of [Formula: see text] or [Formula: see text] enforced by the confining boundary. We use the Landau-de Gennes phenomenological model in terms of the tensor nematic order parameter and the Jones beam propagation model to simulate polarized optical microscopy images. We demonstrate the structure of closed disclination loops near the boundary known as boojums that can be topologically charged or chargeless. We show that pairs of chargeless disclination loops can interact repulsively or attractively depending on if they are arranged parallel or antiparallel, respectively. Sufficiently closely spaced antiparallel pairs can rewire while parallel pairs simply exhibit stronger bending due to the repulsion.

Numerical modeling of optical modes in topological soft matter

Vector and vortex laser beams are desired in many applications and are usually created by manipulating the laser output or by inserting optical components in the laser cavity. Distinctly, inserting liquid crystals into the laser cavity allows for extensive control over the emitted light due to their high susceptibility to external fields and birefringent nature. In this work we demonstrate diverse optical modes for lasing as enabled and stablised by topological birefringent soft matter structures using numerical modelling. We show diverse structuring of light-with different 3D intensity and polarization profiles-as realised by topological soft matter structures in radial nematic droplet, in 2D nematic cavities of different geometry and including topological defects with different charges and winding numbers, in arbitrary varying birefringence fields with topological defects and in pixelated birefringent profiles. We use custom written FDFD code to calculate emergent electromagnetic eigenmodes. Control over lasing is of a particular interest aiming towards the creation of general intensity, polarization and topologically shaped laser beams.

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.

Reconfigurable Multicompartment Emulsion Drops Formed by Nematic Liquid Crystals and Immiscible Perfluorocarbon Oils

Liquid crystalline (LC) oils offer the basis of stimuli-responsive LC-in-water emulsions. Although past studies have explored the properties of single-phase LC emulsions, few studies have focused on complex multicompartment emulsions containing co-existing isotropic and LC domains. In this paper, we report a study of multiphase emulsions using LCs and immiscible perfluoroalkanes dispersed in water or glycerol (the latter continuous phase is used to enable characterization). We found that the nematogen 4'-pentyl-4-biphenylcarbonitrile (5CB) anchors homeotropically (perpendicularly) and weakly at liquid perfluorononane (F9) interfaces, consistent with the smectic layering of 5CB molecules. The proposed role of smectic layering is supported by experiments performed with 4-(trans-4-pentylcyclohexyl)benzonitrile, a nematogen that possesses a cyclohexyl group that frustrates the smectic packing and leads to tilted orientations at the F9 interface. By employing perfluorocarbon and hydrocarbon surfactants in combination with multiphase 5CB and F9 emulsion droplets dispersed in a continuous water or glycerol phase, we observe a range of emulsion droplet morphologies to form, including core-shell and Janus structures, with internal organizations that reflect an interplay of interfacial (anchoring energies; F9 and glycerol) and elastic energies within the confines of the geometry of the emulsion droplet. By comparing experimental observations to simulations of the LC-perfluorocarbon droplets based on a Landau-de Gennes model of the free energy, we place bounds on the orientation-dependent interfacial energies that underlie the internal ordering of these complex emulsions. Additionally, by forming core-shells emulsion droplets from 5CB (shell) and perfluoroheptane (cores), we demonstrate how a liquid-to-vapor phase transition in the perfluorocarbon core can be used to actuate the droplet and rapidly thin the nematic shell. Overall, the results reported in this paper demonstrate that multiphase LC emulsions formed from mixtures of perfluoroalkanes and LCs provide new opportunities to engineer hierarchical and stimuli-responsive emulsion systems.

From Point to Filament Defects in Hybrid Nematic Films

We have studied nematic hybrid films with homeotropic alignment at the top surface and various controlled degrees of in plane ordering, going from a random degenerate organization to a completely uniform alignment along one direction, at the bottom one. We show, by Monte Carlo (MC) computer simulations and experiments on photopatterned films with the bottom support surface fabricated with in-plane order similar to the simulated ones, that the point defects observed in the case of random planar orientations at the bottom tend to arrange along a filament as the surface ordering is sufficiently increased. MC simulations complement the polarized microscopy texture observations allowing to inspect the 3D structure of the defects and examine the role of elastic constants.

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.

Nematic colloidal knots in topological environments

The role of environment in shaping material properties is of great significance, but less is known about how non-trivial topology of the environment couples to material states, which can be of non-trivial topology themselves. In this paper, we demonstrate the role of the topology of the environment on the formation of complex nematic fields and defect structures, specifically in the system of nematic colloidal knots. The topological environments around knotted colloidal particles are suggested to exist as spherical surface-patterned nematic cavities imposing radial, uniform or hyperbolic nematic profiles. We show that topologically different nematic environments significantly affect and create differences in the colloidal field structure created within the environment, such as the location, profile and number of topological defects. Specifically, we demonstrate that topological environments in combination with knotted colloidal particles of non-trivial topology lead to the formation of diverse nematic knotted and linked fields. These fields are different adaptations of the knotted shape of the colloidal particles, creating knots and links of topological defects as well as escaped-core defect-like solitonic structures. These are observed in chiral nematic media but here are stabilised in achiral nematic media as a result of the distinct shape of the knotted colloidal particle, with a double helix segment and nematic environmental patterns. More generally, this paper is a contribution towards understanding the role of environment, especially its topology, on the response and defect formation in elastic fields, such as in nematic liquid crystal colloids.

Nematic Liquid-Crystal Colloids

This article provides a concise review of a new state of colloidal matter called nematic liquid-crystal colloids. These colloids are obtained by dispersing microparticles of different shapes in a nematic liquid crystal that acts as a solvent for the dispersed particles. The microparticles induce a local deformation of the liquid crystal, which then generates topological defects and long-range forces between the neighboring particles. The colloidal forces in nematic colloids are much stronger than the forces in ordinary colloids in isotropic solvents, exceeding thousands of kBT per micrometer-sized particle. Of special interest are the topological defects in nematic colloids, which appear in many fascinating forms, such as singular points, closed loops, multitudes of interlinked and knotted loops or soliton-like structures. The richness of the topological phenomena and the possibility to design and control topological defects with laser tweezers make colloids in nematic liquid crystals an excellent playground for testing the basic theorems of topology.

Colloidal liquid crystals in square confinement: isotropic, nematic and smectic phases

We report on the confinement of colloidal liquid crystals in three dimensional chambers with a square footprint. To this end we use colloidal silica rods and exploit their relatively large density difference with respect to the dispersing solvent to study isotropic, nematic and smectic phases confined into a single chamber. Combining laser scanning confocal microscopy and soft-lithography techniques enables us to characterize the configurations down to the single particle level. We will focus on the smectic phase and compare to recent theories and simulations.

Global defect topology in nematic liquid crystals

We give the global homotopy classification of nematic textures for a general domain with weak anchoring boundary conditions and arbitrary defect set in terms of twisted cohomology, and give an explicit computation for the case of knotted and linked defects in R3, showing that the distinct homotopy classes have a 1–1 correspondence with the first homology group of the branched double cover, branched over the disclination loops. We show further that the subset of those classes corresponding to elements of order 2 in this group has representatives that are planar and characterize the obstruction for other classes in terms of merons. The planar textures are a feature of the global defect topology that is not reflected in any local characterization. Finally, we describe how the global classification relates to recent experiments on nematic droplets and how elements of order 4 relate to the presence of τ lines in cholesterics.

Finite particle size drives defect-mediated domain structures in strongly confined colloidal liquid crystals

When liquid crystals are confined to finite volumes, the competition between the surface anchoring imposed by the boundaries and the intrinsic orientational symmetry-breaking of these materials gives rise to a host of intriguing phenomena involving topological defect structures. For synthetic molecular mesogens, like the ones used in liquid-crystal displays, these defect structures are independent of the size of the molecules and well described by continuum theories. In contrast, colloidal systems such as carbon nanotubes and biopolymers have micron-sized lengths, so continuum descriptions are expected to break down under strong confinement conditions. Here, we show, by a combination of computer simulations and experiments with virus particles in tailor-made disk- and annulus-shaped microchambers, that strong confinement of colloidal liquid crystals leads to novel defect-stabilized symmetrical domain structures. These finite-size effects point to a potential for designing optically active microstructures, exploiting the as yet unexplored regime of highly confined liquid crystals.

Liquid crystals confined to micrometre-sized geometries can be well described by a continuum theory, where the size effect of constituent mesogens is negligible. Here, the authors show how the continuum theory breaks down in colloidal liquid crystal, leading to the formation of defect-mediated domains.

Colloidal spirals in nematic liquid crystals

One of the central experimental efforts in nematic colloids research aims to explore how the interplay between the geometry of particles along with the accompanying nematic director deformations and defects around them can provide a means of guiding particle self-assembly and controlling the structure of particle-induced defects. In this work, we design, fabricate, and disperse low-symmetry colloidal particles with shapes of spirals, double spirals, and triple spirals in a nematic fluid. These spiral-shaped particles, which are controlled by varying their surface functionalization to provide tangential or perpendicular boundary conditions of the nematic molecular alignment, are found inducing director distortions and defect configurations with non-chiral or chiral symmetry. Colloidal particles also exhibit both stable and metastable multiple orientational states in the nematic host, with a large number of director configurations featuring both singular and solitonic nonsingular topological defects accompanying them, which can result in unusual forms of colloidal self-assembly. Our findings directly demonstrate how the symmetry of particle-generated director configurations can be further lowered, or not, as compared to the low point group symmetry of solid micro-inclusions, depending on the nature of induced defects while satisfying topological constraints. We show that achiral colloidal particles can cause chiral symmetry breaking of elastic distortions, which is driven by complex three-dimensional winding of induced topological line defects and solitons.

Nematic droplets on fibers

The emergence of new techniques for the fabrication of nematic droplets with nontrivial topology provides new routes for the assembly of responsive devices. Here we explore some of the properties of nematic droplets on fibers, which constitute the basic units of a type of device that is able to respond to external stimuli, including the detection of gases. We perform a numerical study of spherical nematic droplets on fibers. We analyze the equilibrium textures for homogeneous and hybrid boundary conditions and find that in some cases the nematic avoids the nucleation of topological defects, which would provide a different optical response. We consider in detail a homeotropic nematic droplet wrapped around a fiber with planar anchoring. We investigate the effect of an electric field on the texture of this droplet. In the presence of a dc field, the system undergoes an orientational transition above a given threshold E(c), for which a ring defect is transformed into a figure-eight defect. We also consider ac fields, at high and low frequencies, and find that the textures are similar to those observed for static fields, in contrast with recently reported experiments.

Particles with changeable topology in nematic colloids

We show that nematic colloids can serve as a highly variable and controllable platform for studying inclusions with changeable topology and their effects on the surrounding ordering fields. We explore morphing of toroidal and knotted colloidal particles into effective spheres, distinctively changing their Euler characteristic and affecting the surrounding nematic field, including topological defect structures. With toroidal particles, the inner nematic defect eventually transitions from a wide loop to a point defect (a small loop). Trefoil particles become linked with two knotted defect loops, mutually forming a three component link, that upon tightening transform into a two-component particle-defect loop link. For more detailed topological analysis, Pontryagin-Thom surfaces are calculated and visualised, indicating an interesting cascade of defect rewirings caused by the shape morphing of the knotted particles.

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.