Knot theory realizations in nematic colloids

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.

Controlling liquid crystal boojum defects on fixed microparticle arrays via capillarity-assisted particles assembly

HYPOTHESIS

Colloidal particles in nematic liquid crystals (LCs) are of high interest for self-assembly of soft matter systems. When two free particles approach within a uniaxially-oriented nematic LC, an elastic force is generated due to the distorted nematic director configuration around them, allowing particles to self-assemble by an attractive force. We hypothesize that if particles are immobilized, repulsive forces emerge instead, causing the deflection of the interacting defects to compensate for the energy increase.

EXPERIMENTS

We fabricated tailored arrays of spherical silica microparticles via capillarity-assisted particle assembly (CAPA) to investigate the interactions of defects as a function of particle separation. By transferring the particle arrays from the CAPA templates to a glass substrate, we studied interacting boojum defect textures within thin LC films sandwiched between two substrates using polarized optical microscopy (POM).

FINDINGS

We observed deflected boojum defects on arrays of fixed silica particles, confirming our hypothesis that the elastic repulsive force between the particles affects the defect orientation. The nematic director configuration is reconstructed by Landau-de Gennes q-tensor modeling, and simulated POM images are obtained by the Jones-Matrix method. Our results provide a new platform for controlling defect interactions and pave the way for future work to study topology and implement new defect based applications in LC films.

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.

Design and Preparation of Nematic Colloidal Particles

Colloidal systems are abundant in technology, in biomedical settings, and in our daily life. The so-called "colloidal atoms" paradigm exploits interparticle interactions to self-assemble colloidal analogs of atomic and molecular crystals, liquid crystal glasses, and other types of condensed matter from nanometer- or micrometer-sized colloidal building blocks. Nematic colloids, which comprise colloidal particles dispersed within an anisotropic nematic fluid host medium, provide a particularly rich variety of physical behaviors at the mesoscale, not only matching but even exceeding the diversity of structural and phase behavior in conventional atomic and molecular systems. This feature article, using primarily examples of works from our own group, highlights recent developments in the design, fabrication, and self-assembly of nematic colloidal particles, including the capabilities of preprogramming their behavior by controlling the particle's surface boundary conditions for liquid crystal molecules at the colloidal surfaces as well as by defining the shape and topology of the colloidal particles. Recent progress in defining particle-induced defects, elastic multipoles, self-assembly, and dynamics is discussed along with open issues and challenges within this research field.

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.

Topological braiding and virtual particles on the cell membrane

Braiding of topological structures in complex matter fields provides a robust framework for encoding and processing information, and it has been extensively studied in the context of topological quantum computation. In living systems, topological defects are crucial for the localization and organization of biochemical signaling waves, but their braiding dynamics remain unexplored. Here, we show that the spiral wave cores, which organize the Rho-GTP protein signaling dynamics and force generation on the membrane of starfish egg cells, undergo spontaneous braiding dynamics. Experimentally measured world line braiding exponents and topological entropy correlate with cellular activity and agree with predictions from a generic field theory. Our analysis further reveals the creation and annihilation of virtual quasi-particle excitations during defect scattering events, suggesting phenomenological parallels between quantum and living matter.

Electric response of topological dipoles in nematic colloids with twist-bend nematic droplets as the dispersed phase

Colloidal systems comprising solid or fluid particles dispersed in nematic monodomains are known to be a convenient means to study topological defects. Recent experiments have shown that twist-bend nematic (N_{TB}) droplets in a nematic matrix act as colloidal particles that lead to the formation of elastic dipoles, quadrupoles, and their ordered clusters. In this study, we examine the effect of low-frequency (f∼mHz) electric fields on such defect configurations. We find that (i) the hyperbolic hedgehogs of elastic dipoles shift toward the negative electrode in static fields and perform oscillatory motion in AC fields, indicating the presence of nonvanishing flexoelectric polarization in the field-free state; (ii) the elastic dipoles, propelled by forces of backflow due to coupled flexoelectric and dielectric distortions, drift uniformly along their axes with the N_{TB} drops in lead; (iii) the translational velocity v_{d} increases linearly with both f and the diameter of N_{TB} drops; and (iv) with increasing applied voltage U, v_{d}(U) exhibits a monotonic, slightly nonlinear variation at f≤200mHz, tending toward linearity at higher frequencies.

Isogonal piecewise linear embeddings of 1-periodic weaves and some related structures

Crystallographic descriptions of isogonal piecewise linear embeddings of 1-periodic weaves and links (chains) are presented. These are composed of straight segments (sticks) that meet at corners (2-valent vertices). Descriptions are also given of some plaits - woven periodic bands, three simple periodic knots and isogonal interwoven rods.

Connecting particle clustering and rheology in attractive particle networks

The structural properties of suspensions and other multiphase systems are vital to overall processability, functionality and acceptance among consumers. Therefore, it is crucial to understand the intrinsic connection between the microstructure of a material and the resulting rheological properties. Here, we demonstrate how the transitions in the microstructural conformations can be quantified and correlated to rheological measurements. We find semi-local parameters from graph theory, the mathematical study of networks, to be useful in linking structure and rheology. Our results, using capillary suspensions as a model system, show that the use of the clustering coefficient, in combination with the coordination number, is able to capture not only the agglomeration of particles, but also measures the formation of groups. These phenomena are tightly connected to the rheological properties. The present sparse networks cannot be described by established techniques such as betweenness centrality.

Topological defects due to twist-bend nematic drops mimicking colloidal particles in a nematic medium

Colloids formed of solid/fluid particle dispersions in oriented nematic liquid crystals are known to be an ideal means of realizing fundamentally significant topological defect geometries. We find, experimentally, that twist-bend nematic (NTB) droplets formed in the N-NTB biphasic regime, either of pure compounds or mesogenic mixtures, completely mimic colloidal particles in their ability to generate a rich variety of defects. In the biphasic regime, the topological features of both liquid crystal colloids and chiral nematic droplets are revealed by (i) topological dipoles, quadrupoles and their patterned clusters formed in planar nematic liquid crystals orientationally perturbed by coexisting NTB drops, (ii) the transformation of hyperbolic hedgehogs into knotted Saturn rings encircling the NTB drops dispersed in a 90°-twisted nematic matrix and (iii) the Frank-Pryce defect texture evident in smaller (relative to sample thickness) NTB drops. In larger drops with fingerlike outgrowths, additional line defects appear; most of these are deemed to be pairs of disclinations to which are attached pairs of screw dislocations intervening in the growth process of the NTB droplets.

Degenerate conic anchoring and colloidal elastic dipole-hexadecapole transformations

The defect structure associated with a colloid in a nematic liquid crystal is dictated by molecular orientation at the colloid surface. Perpendicular or parallel orientations to the surface lead to dipole-like or quadrupole-like defect structures. However, the so-called elastic hexadecapole discovered recently, has been assumed to result from a conic anchoring condition. In order to understand it at a fundamental level, a model for this anchoring is introduced here in the context of a Landau-de Gennes free energy functional. We investigate the evolution of defect configurations, as well as colloidal interactions, by tuning the preferred tilt angle (θe). The model predicts an elastic dipole whose stability decreases as θe increases, along with a dipole-hexadecapole transformation, which are confirmed by our experimental observations. Taken together, our results suggest that previously unanticipated avenues may exist for design of self-assembled structures via control of tilt angle.

Topological entanglement of interlocked knotted-unknotted polymer rings

Topological entanglements in biopolymers could drive them to certain internal statics and dynamics with important implications for biological functions. In this study, by means of molecular dynamics simulations, we demonstrate that the minimal crossing pattern of a braid plays a major role in its structural and dynamical properties; the braid consists of a knotted ring and an interlocked entwined unknotted polymer ring. In particular, we show that depending on the bending rigidity of the chains, the conformational energy of the braid can be either lower or higher than the unlocked polymer rings. Additionally, we find that a non-identical crossing pattern in the braid could distinctly enforce concerted internal conformational fluctuations between the interlocked rings.

Curvature-Induced Twist in Homeotropic Nematic Tori

We confine a nematic liquid crystal with homeotropic anchoring to stable toroidal droplets and study how geometry affects the equilibrium director configuration. In contrast to the case of cylindrical confinement, we find that the equilibrium state is chiral-a twisted and escaped radial director configuration. Furthermore, we find that the magnitude of the twist distortion increases as the ratio of the ring radius to the tube radius decreases; we confirm this with computer simulations of optically polarized microscopy textures. In addition, numerical calculations also indicate that the local geometry indeed affects the magnitude of the twist distortion. We further confirm this curvature-induced twisting using bent cylindrical capillaries.

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.

Shape control of surface-stabilized disclination loops in nematic liquid crystals

Recent studies on topological defects in conventional and active nematic liquid crystals have revealed their potential as sources of advanced functionality whereby the collective behavior of the constituent molecules or cells is controlled. On the other hand, the fact that they have high energies and are metastable makes their shape control a nontrivial issue. Here, we demonstrate stabilization of arbitrary-shaped closed disclination loops with 1/2 strength floating in the bulk by designing the twist angle distribution in a liquid crystal cell. Continuous variation of the twist angle from below to above |π/2| allows us to unambiguously position reverse twist disclinations at will. We also analyze the elastic free energy and uncover the relationship between the twist angle pattern and shrink rate of the surface-stabilized disclination loop.

Harmonic field in knotted space

Knotted fields enrich a variety of physical phenomena, ranging from fluid flows, electromagnetic fields, to textures of ordered media. Maxwell's electrostatic equations, whose vacuum solution is mathematically known as a harmonic field, provide an ideal setting to explore the role of domain topology in determining physical fields in confined space. In this work, we show the uniqueness of a harmonic field in knotted tubes, and reduce the construction of a harmonic field to a Neumann boundary value problem. By analyzing the harmonic field in typical knotted tubes, we identify the torsion driven transition from bipolar to vortex patterns. We also analogously extend our discussion to the organization of liquid crystal textures in knotted tubes. These results further our understanding about the general role of topology in shaping a physical field in confined space, and may find applications in the control of physical fields by manipulation of surface topology.

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.

Nematic line defects in microfluidic channels: wedge, twist and zigzag disclinations

We report an experimental study of structural transformations of disclination lines in nematic liquid crystals in microfluidic channels. The anchoring conditions of the channel walls enforce the generation of a disclination line of the wedge type in the absence of flow. The wedge disclination is transformed to a twist disclination by the flow of the nematic liquid crystal in the channel. The application of an electric field perpendicular to the channel axis induces a second transformation to a zigzag shape. The threshold field strength for the second transformation increases with increasing flow velocity. The experimental results are compared to predictions based on model director fields of the different disclination structures.

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.

Bipolar configuration with twisted loop defect in chiral nematic droplets under homeotropic surface anchoring

Optical textures and appropriate orientational structures have been studied within droplets of chiral nematic dispersed in polymer assigning the homeotropic anchoring. The helix axis of the chiral structure inside droplets forms the bipolar configuration. The optical droplet textures were analysed in the unpolarised light, analyser switching-off scheme and in crossed polarisers. The twisted loop defect reveals itself convincingly in all schemes. Its appearance at the optical patterns of the chiral nematic droplets has been examined depending on their size and the aspect direction. The existence of the defect has been verified by the structural and optical calculations. The effect of an electric field on both the defect line shape and the orientational structure of chiral nematic has been studied.

Topological defects in an unconfined nematic fluid induced by single and double spherical colloidal particles

We present numerical solutions to the Landau-de Gennes free-energy model under the one-constant approximation for systems of single and double spherical colloidal particles immersed in an otherwise uniformly aligned nematic liquid crystal. A perfect homeotropic surface anchoring of liquid-crystal molecules on the spherical surface is considered. A large parameter space is carefully examined, including those in the free-energy model and those describing the dimer configurations and the background liquid-crystal orientation. The stability of the resulting liquid-crystal defects appearing in the neighborhood of the colloidal dimer pair is analyzed in light of the numerical results for their free energies. A number of scenarios are considered: a free dimer pair in a nematic fluid where the free-energy ground states are described in terms of a phase diagram, and a constrained dimer pair where the interparticle distance and the relative orientation of the distance vector to the nematic director can be manipulated. We pay particular attention to the nonsymmetric solutions, which yield several metastable defect states that can be observed in real systems. The high-precision numerical calculations are based on a spectral method, which is an enabling factor that allows us to compare the subtle difference in the free energies of different defect structures.

Lattice topology dictates photon statistics

Propagation of coherent light through a disordered network is accompanied by randomization and possible conversion into thermal light. Here, we show that network topology plays a decisive role in determining the statistics of the emerging field if the underlying lattice is endowed with chiral symmetry. In such lattices, eigenmode pairs come in skew-symmetric pairs with oppositely signed eigenvalues. By examining one-dimensional arrays of randomly coupled waveguides arranged on linear and ring topologies, we are led to a remarkable prediction: the field circularity and the photon statistics in ring lattices are dictated by its parity while the same quantities are insensitive to the parity of a linear lattice. For a ring lattice, adding or subtracting a single lattice site can switch the photon statistics from super-thermal to sub-thermal, or vice versa. This behavior is understood by examining the real and imaginary fields on a lattice exhibiting chiral symmetry, which form two strands that interleave along the lattice sites. These strands can be fully braided around an even-sited ring lattice thereby producing super-thermal photon statistics, while an odd-sited lattice is incommensurate with such an arrangement and the statistics become sub-thermal.

Nanoscale interfacial defect shedding in a growing nematic droplet

Interfacial defect shedding is the most recent known mechanism for defect formation in a thermally driven isotropic-to-nematic phase transition. It manifests in nematic-isotropic interfaces going through an anchoring switch. Numerical computations in planar geometry established that a growing nematic droplet can undergo interfacial defect shedding, nucleating interfacial defect structures that shed into the bulk as +1/2 point defects. By extending the study of interfacial defect shedding in a growing nematic droplet to larger length and time scales, and to three dimensions, we unveil an oscillatory growth mode involving shape and anchoring transitions that results in a controllable regular distributions of point defects in planar geometry, and complex structures of disclination lines in three dimensions.

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.

Linked and knotted chimera filaments in oscillatory systems

While the existence of stable knotted and linked vortex lines has been established in many experimental and theoretical systems, their existence in oscillatory systems and systems with nonlocal coupling has remained elusive. Here, we present strong numerical evidence that stable knots and links such as trefoils and Hopf links do exist in simple, complex, and chaotic oscillatory systems if the coupling between the oscillators is neither too short ranged nor too long ranged. In this case, effective repulsive forces between vortex lines in knotted and linked structures stabilize curvature-driven shrinkage observed for single vortex rings. In contrast to real fluids and excitable media, the vortex lines correspond to scroll wave chimeras [synchronized scroll waves with spatially extended (tubelike) unsynchronized filaments], a prime example of spontaneous synchrony breaking in systems of identical oscillators. In the case of complex oscillatory systems, this leads to a topological superstructure combining knotted filaments and synchronization defect sheets.

Tailoring Anisotropic Interactions between Soft Nanospheres Using Dense Arrays of Smectic Liquid Crystal Edge Dislocations

We investigated composite films of gold nanoparticles (NPs)/liquid crystal (LC) defects as a model system to understand the key parameters, which allow for an accurate control of NP anisotropic self-assemblies using soft templates. We combined spectrophotometry, Raman spectroscopy, and grazing incidence small-angle X-ray scattering with calculations of dipole coupling models and soft sphere interactions. We demonstrate that dense arrays of elementary edge dislocations can strongly localize small NPs along the defect cores, resulting in formation of parallel chains of NPs. Furthermore, we show that within the dislocation cores the inter-NP distances can be tuned. This phenomenon appears to be driven by the competition between "soft (nano)sphere" attraction and LC-induced repulsion. We evidence two extreme regimes controlled by the solvent evaporation: (i) when the solvent evaporates abruptly, the spacing between neighboring NPs in the chains is dominated by van der Waals interactions between interdigitated capping ligands, leading to chains of close-packed NPs; (ii) when the solvent evaporates slowly, strong interdigitation between the is avoided, leading to a dominating LC-induced repulsion between NPs associated with the replacement of disordered cores by NPs. The templating of NPs by topological defects, beyond the technological inquiries, may enable creation, investigation, and manipulation of unique collective features for a wide range of nanomaterials.

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.