Out-of-equilibrium interactions and collective locomotion of colloidal spheres with squirming of nematoelastic multipoles

Many living and artificial systems show similar emergent behavior and collective motions on different scales, starting from swarms of bacteria to synthetic active particles, herds of mammals, and crowds of people. What all these systems often have in common is that new collective properties like flocking emerge from interactions between individual self-propelled or driven units. Such systems are naturally out-of-equilibrium and propel at the expense of consumed energy. Mimicking nature by making self-propelled or externally driven particles and studying their individual and collective motility may allow for deeper understanding of physical underpinnings behind collective motion of large groups of interacting objects or beings. Here, using a soft matter system of colloids immersed into a liquid crystal, we show that resulting so-called nematoelastic multipoles can be set into a bidirectional locomotion by external oscillating electric fields. Out-of-equilibrium elastic interactions between such colloidal objects lead to collective flock-like behaviors emerging from time-varying elasticity-mediated interactions between externally driven propelling particles. Repulsive elastic interactions in the equilibrium state can be turned into attractive interactions in the out-of-equilibrium state under applied external electric fields. We probe this behavior at different number densities of colloidal particles and show that particles in dense dispersions collectively select the same direction of a coherent motion due to elastic interactions between near neighbors. In our experimentally implemented design, their motion is highly ordered and without clustering or jamming often present in other colloidal transport systems, which is promising for technological and fundamental-science applications, like nano-cargo transport, out-of-equilibrium assembly, and microrobotics.

Programmable Self-Assembly of Nanoplates into Bicontinuous Nanostructures

Self-assembly is the process by which individual components arrange themselves into an ordered structure by changing the shapes, components, and interactions. It has enabled us to construct an extensive range of geometric forms on many length scales. Nevertheless, the potential of two-dimensional polygonal nanoplates to self-assemble into extended three-dimensional structures with compartments and corridors has remained unexplored. In this paper, we show coarse-grained Monte Carlo simulations demonstrating self-assembly of hexagonal/triangular nanoplates via complementary interactions into faceted, sponge-like "bicontinuous polyhedra" (or infinite polyhedra) whose flat walls partition space into a pair of mutually interpenetrating labyrinths. Two bicontinuous polyhedra can be self-assembled: the regular (or Platonic) Petrie-Coxeter infinite polyhedron (denoted {6,4|4}) and the semi-regular Hart "gyrangle". The latter structure is chiral, with both left- and right-handed versions. We show that the Petrie-Coxeter assembly is constructed from two complementary populations of hexagonal nanoplates. Furthermore, we find that the 3D chiral Hart gyrangle can be assembled from identical achiral triangular nanoplates decorated with regioselective complementary interaction sites. The assembled Petrie-Coxeter and Hart polyhedra are faceted versions of two of the simplest triply periodic minimal surfaces, namely, Schwarz's primitive and Schoen's gyroid surfaces, respectively, offering alternative routes to those bicontinuous nanostructures, which are widespread in synthetic and biological materials.

Rectified Rotational Dynamics of Mobile Inclusions in Two-Dimensional Active Nematics

We investigate the dynamics of mobile inclusions embedded in 2D active nematics. The interplay between the inclusion shape, boundary-induced nematic order, and autonomous flows powers the inclusion motion. Disks and achiral gears exhibit unbiased rotational motion, but with distinct dynamics. In comparison, chiral gear-shaped inclusions exhibit long-term rectified rotation, which is correlated with dynamics and polarization of nearby +1/2 topological defects. The chirality of defect polarities and the active nematic texture around the inclusion correlate with the inclusion's instantaneous rotation rate. Inclusions provide a promising tool for probing the rheological properties of active nematics and extracting ordered motion from their inherently chaotic motion.

Thin pyramidal cones in nematic liquid crystals

The present study investigates the arrangement of hollow pyramidal cone shells and their interactions with degenerate planar anchoring on the inner and outer surfaces of particles within the nematic host. The shell thickness is in order of the nematic coherence length. The numerical behavior of colloids is determined by minimizing the Landau-de Gennes free energy in the presence of the Fournier surface energy and using the finite element method. Colloidal pyramidal cones can orient parallel and perpendicular with the far director orientation. In the parallel alignment, we found the splay director distortion into the pyramid with two boojum defects at the inner and outer tips. The director shows bending distortion without defect patterns when the pyramid is aligned perpendicularly. They induce long-range dipolar interaction and can form nested structures in close contact.

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.

A Nanoparticle-Based Model System for the Study of Heterogeneous Nucleation Phenomena

Heterogeneous nucleation processes are involved in many important phenomena in nature, including devastating human diseases caused by amyloid structures or the harmful frost formed on fruits. However, understanding them is challenging due to the difficulties of characterizing the initial stages of the process occurring at the interface between the nucleation medium and the substrate surfaces. This work implements a model system based on gold nanoparticles to investigate the effect of particle surface chemistry and substrate properties on heterogeneous nucleation processes. Using widely available techniques such as UV-vis-NIR spectroscopy and light microscopy, gold nanoparticle-based superstructure formation was studied in the presence of substrates with different hydrophilicity and electrostatic charges. The results were evaluated on grounds of classical nucleation theory (CNT) to reveal kinetic and thermodynamic contributions of the heterogeneous nucleation process. In contrast to nucleation from ions, the kinetic contributions toward nucleation turned out to be larger than the thermodynamic contributions for the nanoparticle building blocks. Electrostatic interactions between substrates and nanoparticles with opposite charges were crucial to enhancing the nucleation rates and decreasing the nucleation barrier of superstructure formation. Thereby, the described strategy is demonstrated advantageous for characterizing physicochemical aspects of heterogeneous nucleation processes in a simple and accessible manner, which could be potentially explored to study more complex nucleation phenomena.

Nanoparticle localization within chiral liquid crystal defect lines and nanoparticle interactions

Self-assembly of colloidal particles into predefined structures is a promising way to design inexpensive manmade materials with advanced macroscopic properties. Doping of nematic liquid crystals (LCs) with nanoparticles has a series of advantages in addressing these grand scientific and engineering challenges. It also provides a very rich soft matter platform for the discovery of unique condensed matter phases. The LC host naturally allows the realization of diverse anisotropic interparticle interactions, enriched by the spontaneous alignment of anisotropic particles due to the boundary conditions of the LC director. Here we demonstrate theoretically and experimentally that the ability of LC media to host topological defect lines can be used as a tool to probe the behavior of individual nanoparticles as well as effective interactions between them. LC defect lines irreversibly trap nanoparticles enabling controlled particle movement along the defect line with the use of a laser tweezer. Minimization of Landau-de Gennes free energy reveals a sensitivity of the ensuing effective nanoparticle interaction to the shape of the particle, surface anchoring strength, and temperature, which determine not only the strength of the interaction but also its repulsive or attractive character. Theoretical results are supported qualitatively by experimental observations. This work may pave the way toward designing controlled linear assemblies as well as one-dimensional crystals of nanoparticles such as gold nanorods or quantum dots with tunable interparticle spacing.

Uniformly aligned flexible magnetic films from bacterial nanocelluloses for fast actuating optical materials

Naturally derived biopolymers have attracted great interest to construct photonic materials with multi-scale ordering, adaptive birefringence, chiral organization, actuation and robustness. Nevertheless, traditional processing commonly results in non-uniform organization across large-scale areas. Here, we report magnetically steerable uniform biophotonic organization of cellulose nanocrystals decorated with superparamagnetic nanoparticles with strong magnetic susceptibility, enabling transformation from helicoidal cholesteric (chiral nematic) to uniaxial nematic phase with near-perfect orientation order parameter of 0.98 across large areas. We demonstrate that magnetically triggered high shearing rate of circular flow exceeds those for conventional evaporation-based assembly by two orders of magnitude. This high rate shearing facilitates unconventional unidirectional orientation of nanocrystals along gradient magnetic field and untwisting helical organization. These translucent magnetic films are flexible, robust, and possess anisotropic birefringence and light scattering combined with relatively high optical transparency reaching 75%. Enhanced mechanical robustness and uniform organization facilitate fast, multimodal, and repeatable actuation in response to magnetic field, humidity variation, and light illumination.

Naturally derived biopolymers attracted great interest to construct photonic materials but traditional processing commonly results in non-uniform organization across largescale areas. Here, the authors report a uniform biophotonic organization of cellulose nanocrystals decorated with superparamagnetic nanoparticles enabling transformation from helicoidal cholesteric to uniaxial nematic phase with near-perfect orientation.

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.

Defect patterns of two-dimensional nematic liquid crystals in confinement

A two-dimensional or quasi-two-dimensional nematic liquid crystal refers to a surface-confined system. When such a system is further confined by external line boundaries or excluded from internal line boundaries, the nematic directors form a deformed texture that may display defect points or defect lines, for which winding numbers can be clearly defined. Here, a particular attention is paid to the case when the liquid crystal molecules prefer to form a boundary nematic texture in parallel to the wall surface (i.e., following the homogeneous boundary condition). A general theory, based on geometric argument, is presented for the relationship between the sum of all winding numbers in the system (the total winding number) and the type of confinement angles and curved segments. The conclusion is validated by comparing the theoretical defect rule with existing nematic textures observed experimentally and theoretically in recent years.

Nontrivial topological defects of micro-rods immersed in nematics and their phototuning

Combinations of different geometries and surface anchoring conditions give rise to the diversity of topological structures in nematic colloid systems. Tuning these parameters in a single system offers possibilities for observing the evolution of the topological transformation and for manipulating colloids through topological forces. Here we investigate the nontrivial topological properties of micro-rods dispersed in nematic liquid crystals through experimental observation and computer simulation. The topological variation is driven by photodynamically changing the surface anchoring using azobenzene-based surface-commander molecules, the majority of which are localized on both the substrates and the surface of micro-rods. By comparing experimental and simulation results, we show previously unidentified topological properties of the two-body LC-rod-colloid system. Moreover, unlike the traditional photoresponsive liquid crystal systems, the localization of azobenzene molecules on the surfaces makes it possible to change only the direction of the surface orientation, not disordering of the bulk structures. The results assist in the development of photo-driven micro-robotics in fluids.

Colloidal Self-Assembly Approaches to Smart Nanostructured Materials

Colloidal self-assembly refers to a solution-processed assembly of nanometer-/micrometer-sized, well-dispersed particles into secondary structures, whose collective properties are controlled by not only nanoparticle property but also the superstructure symmetry, orientation, phase, and dimension. This combination of characteristics makes colloidal superstructures highly susceptible to remote stimuli or local environmental changes, representing a prominent platform for developing stimuli-responsive materials and smart devices. Chemists are achieving even more delicate control over their active responses to various practical stimuli, setting the stage ready for fully exploiting the potential of this unique set of materials. This review addresses the assembly of colloids into stimuli-responsive or smart nanostructured materials. We first delineate the colloidal self-assembly driven by forces of different length scales. A set of concepts and equations are outlined for controlling the colloidal crystal growth, appreciating the importance of particle connectivity in creating responsive superstructures. We then present working mechanisms and practical strategies for engineering smart colloidal assemblies. The concepts underpinning separation and connectivity control are systematically introduced, allowing active tuning and precise prediction of the colloidal crystal properties in response to external stimuli. Various exciting applications of these unique materials are summarized with a specific focus on the structure-property correlation in smart materials and functional devices. We conclude this review with a summary of existing challenges in colloidal self-assembly of smart materials and provide a perspective on their further advances to the next generation.

Nematic Order, Plasmonic Switching and Self-Patterning of Colloidal Gold Bipyramids

Dispersing inorganic colloidal nanoparticles within nematic liquid crystals provides a versatile platform both for forming new soft matter phases and for predefining physical behavior through mesoscale molecular-colloidal self-organization. However, owing to formation of particle-induced singular defects and complex elasticity-mediated interactions, this approach has been implemented mainly just for colloidal nanorods and nanoplatelets, limiting its potential technological utility. Here, orientationally ordered nematic colloidal dispersions are reported of pentagonal gold bipyramids that exhibit narrow but controlled polarization-dependent surface plasmon resonance spectra and facile electric switching. Bipyramids tend to orient with their C₅ rotation symmetry axes along the nematic director, exhibiting spatially homogeneous density within aligned samples. Topological solitons, like heliknotons, allow for spatial reorganization of these nanoparticles according to elastic free energy density within their micrometer-scale structures. With the nanoparticle orientations slaved to the nematic director and being switched by low voltages ≈1 V within a fraction of a second, these plasmonic composite materials are of interest for technological uses like color filters and plasmonic polarizers, as well as may lead to the development of unusual nematic phases, like pentatic liquid crystals.

Nematoelasticity of hybrid molecular-colloidal liquid crystals

Colloidal rods immersed in a thermotropic liquid-crystalline solvent are at the basis of so-called hybrid liquid crystals, which are characterized by tunable nematic fluidity with symmetries ranging from conventional uniaxial nematic or antinematic to orthorhombic [Mundoor et al., Science 360, 768 (2018)SCIEAS0036-807510.1126/science.aap9359]. We provide a theoretical analysis of the elastic moduli of such systems by considering interactions between the individual rods with the embedding solvent through surface-anchoring forces, as well as steric and electrostatic interactions between the rods themselves. For uniaxial systems, the presence of colloidal rods generates a marked increase of the splay elasticity, which we found to be in quantitative agreement with experimental measurements. For orthorhombic hybrid liquid crystals, we provide estimates of all 12 elastic moduli and show that only a small subset of those elastic constants play a relevant role in describing the nematoelastic properties. The complexity and possibilities related to identifying the elastic moduli in experiments are briefly discussed.

Nonequilibrium Dynamics of a Magnetic Nanocapsule in a Nematic Liquid Crystal

Colloidal particles in nematic liquid crystals show a beautiful variety of complex phenomena with promising applications. Their dynamical behaviour is determined by topology and interactions with the liquid crystal and external fields. Here, a nematic magnetic nanocapsule reoriented periodically by time-varying magnetic fields is studied using numerical simulations. The approach combines Molecular Dynamics to resolve solute–solvent interactions and Nematic Multiparticle Collision Dynamics to incorporate nematohydrodynamic fields and fluctuations. A Saturn ring defect resulting from homeotropic anchoring conditions surrounds the capsule and rotates together with it. Magnetically induced rotations of the capsule can produce transformations of this topological defect, which changes from a disclination curve to a defect structure extending over the surface of the capsule. Transformations occur for large magnetic fields. At moderate fields, elastic torques prevent changes of the topological defect by tilting the capsule out from the rotation plane of the magnetic field.

Elastomeric nematic colloids, colloidal crystals and microstructures with complex topology

Control of physical behaviors of nematic colloids and colloidal crystals has been demonstrated by tuning particle shape, topology, chirality and surface charging. However, the capability of altering physical behaviors of such soft matter systems by changing particle shape and the ensuing responses to external stimuli has remained elusive. We fabricated genus-one nematic elastomeric colloidal ring-shaped particles and various microstructures using two-photon photopolymerization. Nematic ordering within both the nano-printed particle and the surrounding medium leads to anisotropic responses and actuation when heated. With the thermal control, elastomeric microstructures are capable of changing from genus-one to genus-zero surface topology. Using these particles as building blocks, we investigated elastomeric colloidal crystals immersed within a liquid crystal fluid, which exhibit crystallographic symmetry transformations. Our findings may lead to colloidal crystals responsive to a large variety of external stimuli, including electric fields and light. Pre-designed response of elastomeric nematic colloids, including changes of colloidal surface topology and lattice symmetry, are of interest for both fundamental research and applications.

Dynamics of defects around anisotropic particles in nematic liquid crystals under shear

Nematic multiparticle collision dynamics is used to simulate disclination ring defects around spherocylinders suspended in a liquid crystal. A solvent-solute interaction potential is integrated over a short-time scale by an auxiliary molecular dynamics procedure that updates the translational and angular coordinates of the spherocylinders. For suspended particles with length in the range ∼(60,160)nm and a fixed aspect ratio, this method is able to simulate static defects reported previously in the literature. It also simulates orientation fluctuations of the elongated colloids that exhibit a broad distribution and a slow relaxation rate. Finally, a nematic host driven from equilibrium by shear flow is simulated, and the consequent dynamic behavior of the colloid-defect pair is studied. Defects under shear present significant structural transformations from chairlike disclination rings to extended defects that cover most of the cylindrical surface of the colloid. This effect results from the hydrodynamic torque on the nematic field caused by the distorted flow around the spherocylinder, and it is present for small Reynolds and Ericksen numbers of order unity.

In Situ Quantification of Interactions between Charged Nanorods in a Predefined Potential Energy Landscape

Quantitative understanding of nanoscale interactions is a prerequisite for harnessing the remarkable collective properties of nanoparticle systems. Here, we report the combined use of liquid-phase transmission electron microscopy and electron beam lithography to elucidate the interactions between charged nanorods in a predefined potential energy landscape. In situ site-selective lift-off of surface-functionalized lithographed gold nanorods is achieved by patterning them with adhesion layer materials that undergo etching at different rates. Analysis of the subsequent nanorod motion, which is two-dimensionally confined as a result of the particle-substrate attraction, allows quantification of interparticle interactions in a lithographically engineered environment. For lithographed nanorods patterned with the same adhesion layer material, their self-assembly behavior following lift-off is tuned by changing their starting spatial arrangement. Our approach facilitates investigation of interparticle interactions in designed nanoparticle systems and affords fundamental insights into the role of the potential energy landscape in determining the kinetic pathway for nanoparticle self-assembly.

Anisotropic electrostatic screening of charged colloids in nematic solvents

The physical behavior of anisotropic charged colloids is determined by their material dielectric anisotropy, affecting colloidal self-assembly, biological function, and even out-of-equilibrium behavior. However, little is known about anisotropic electrostatic screening, which underlies all electrostatic effective interactions in such soft or biological materials. In this work, we demonstrate anisotropic electrostatic screening for charged colloidal particles in a nematic electrolyte. We show that material anisotropy behaves markedly different from particle anisotropy. The electrostatic potential and pair interactions decay with an anisotropic Debye screening length, contrasting the constant screening length for isotropic electrolytes. Charged dumpling-shaped near-spherical colloidal particles in a nematic medium are used as an experimental model system to explore the effects of anisotropic screening, demonstrating competing anisotropic elastic and electrostatic effective pair interactions for colloidal surface charges tunable from neutral to high, yielding particle-separated metastable states. Generally, our work contributes to the understanding of electrostatic screening in nematic anisotropic media.

Defects and defect engineering in Soft Matter

Soft matter covers a wide range of materials based on linear or branched polymers, gels and rubbers, amphiphilic (macro)molecules, colloids, and self-assembled structures. These materials have applications in various industries, all highly important for our daily life, and they control all biological functions; therefore, controlling and tailoring their properties is crucial. One way to approach this target is defect engineering, which aims to control defects in the material's structure, and/or to purposely add defects into it to trigger specific functions. While this approach has been a striking success story in crystalline inorganic hard matter, both for mechanical and electronic properties, and has also been applied to organic hard materials, defect engineering is rarely used in soft matter design. In this review, we present a survey on investigations on defects and/or defect engineering in nine classes of soft matter composed of liquid crystals, colloids, linear polymers with moderate degree of branching, hyperbranched polymers and dendrimers, conjugated polymers, polymeric networks, self-assembled amphiphiles and proteins, block copolymers and supramolecular polymers. This overview proposes a promising role of this approach for tuning the properties of soft matter.

Structure and rheology of soft hybrid systems of magnetic nanoparticles in liquid-crystalline matrices: results from particle-resolved computer simulations

Hybrid mixtures composed of magnetic nanoparticles (MNP) in liquid crystalline (LC) matrices are a fascinating class of soft materials with intriguing physical properties and a wide range of potential applications, e.g., as stimuli-responsive and adaptive materials. Already in the absence of an external stimulus, these systems can display various types of orientationally disordered and ordered phases, which are enriched by self-assembled structures formed by the MNPs. In the presence of external fields, one typically observes highly nonlinear macroscopic behavior. However, an understanding of the structure and dynamics of such systems on the particle level has, so far, remained elusive. In the present paper we review recent computer simulation studies targeting the structure, equilibrium dynamics and rheology of LC-MNP systems, in which the particle sizes of the two components are comparable. As a numerically tractable model system we consider mixtures of soft spherical or elongated particles with a permanent magnetic dipole moment and ellipsoidal non-magnetic particles interacting via a Gay-Berne potential. We address, first, equilibrium aspects such as structural organization and self-assembly (cluster formation) of the MNPs in dependence of the orientational state of the matrix, the role of the size ratio, the impact of an external magnetic field, and the translational and orientational diffusion of the two components. Second, we discuss the non-equilibrium dynamics of LC-MNP mixtures under planar shear flow, considering both, spherical and non-spherical MNPs. Our results contribute to a detailed understanding of these intriguing hybrid materials, and they may serve as a guide for future experiments.

Interplay of Electrostatic Dipoles and Monopoles with Elastic Interactions in Nematic Liquid Crystal Nanocolloids

Doping of nematic liquid crystals with colloidal nanoparticles presents a rich soft matter platform for controlling material properties and discovering diverse condensed matter phases. We describe nematic nanocolloids that simultaneously exhibit strong electrostatic monopole and dipole moments and yield competing long-range anisotropic interactions. Combined with interactions due to orientational elasticity and order parameter gradients of the nematic host medium, they lead to diverse forms of self-assembly both in the bulk of an aligned liquid crystal and when one-dimensionally confined by singular topological defect lines. Such nanocolloids exhibit facile responses to electric fields. We demonstrate electric reconfigurations of nanocolloidal pair-interactions and discuss how our findings may lead to realizing ferroelectric and dielectric molecular-colloidal fluids with different point group symmetries.

A review of shaped colloidal particles in fluids: anisotropy and chirality

This review treats asymmetric colloidal particles moving through their host fluid under the action of some form of propulsion. The propulsion can come from an external body force or from external shear flow. It may also come from externally-induced stresses at the surface, arising from imposed chemical, thermal or electrical gradients. The resulting motion arises jointly from the driven particle and the displaced fluid. If the objects are asymmetric, every aspect of their motion and interaction depends on the orientation of the objects. This orientation in turn changes in response to the driving. The objects' shape can thus lead to a range of emergent anisotropic and chiral motion not possible with isotropic spherical particles. We first consider what aspects of a body's asymmetry can affect its drift through a fluid, especially chiral motion. We next discuss driving by injecting external force or torque into the particles. Then we consider driving without injecting force or torque. This includes driving by shear flow and driving by surface stresses, such as electrophoresis. We consider how time-dependent driving can induce collective orientational order and coherent motion. We show how a given particle shape can be represented using an assembly of point forces called a Stokeslet object. We next consider the interactions between anisotropic propelled particles, the symmetries governing the interactions, and the possibility of bound pairs of particles. Finally we show how the collective hydrodynamics of a suspension can be qualitatively altered by the particles' shapes. The asymmetric responses discussed here are broadly relevant also for swimming propulsion of active micron-scale objects such as microorganisms.

Nature-Inspired Emerging Chiral Liquid Crystal Nanostructures: From Molecular Self-Assembly to DNA Mesophase and Nanocolloids

Liquid crystals (LCs) are omnipresent in living matter, whose chirality is an elegant and distinct feature in certain plant tissues, the cuticles of crabs, beetles, arthropods, and beyond. Taking inspiration from nature, researchers have recently devoted extensive efforts toward developing chiral liquid crystalline materials with self-organized nanostructures and exploring their potential applications in diverse fields ranging from dynamic photonics to energy and safety issues. In this review, an account on the state of the art of emerging chiral liquid crystalline nanostructured materials and their technological applications is provided. First, an overview on the significance of chiral liquid crystalline architectures in various living systems is given. Then, the recent significant progress in different chiral liquid crystalline systems including thermotropic LCs (cholesteric LCs, cubic blue phases, achiral bent-core LCs, etc.) and lyotropic LCs (DNA LCs, nanocellulose LCs, and graphene oxide LCs) is showcased. The review concludes with a perspective on the future scope, opportunities, and challenges in these truly advanced functional soft materials and their promising applications.

Electrically Driven Rotation and Nonreciprocal Motion of Microparticles in Nematic Liquid Crystals

Dispersion of microparticles in nematic liquid crystals offers a novel means for controlling both their orientation and position through the combination of topology and external stimuli. Here, cuboidal and triangular prism shaped microparticles in parallel plate capacitor cells filled with a nematic liquid crystal are studied. Experimental observations are compared with numerical simulations to show that the optimal orientation of the particles is determined by their aspect rations, the relative separation gap of their containers and the applied voltage. It is observed that in systems that allow unrestricted particle rotation, the long axes of the particles are able to fully align themselves with the external electric field. However, when particle rotation is geometrically restricted, it is found that increasing the voltage past a critical value causes the short axis of the particle to realign with the electric field due to anchoring breaking. It is shown that symmetry of the particles then plays a key role in their dynamics following the removal of the electric field, allowing the triangular prisms to travel perpendicular to the applied electric field, whereas only rotation is possible for the cuboidal particles.

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.

Identification and sorting of particle chirality using liquid crystallinity

Particles dispersed in liquid crystals (LCs) have been shown to assemble due to the elastic interactions arising from the molecular anisotropy. Studies have shown that the alignment of the particles within LCs were strongly dependent on the surface director of LCs on particles. Different from the past studies involving particles with degenerate planar anchoring of LCs, this study shows that the azimuthal surface director can be used to control and finely tune the positioning of the particles in LCs. Specifically, polymeric particles with two flat surfaces that mediate parallel or non-parallel (chiral) anchoring were synthesized and dispersed in nematic 5CB with spatial variations in the director profile. Besides demonstration of their positioning, it was observed that the particles with same chiral handedness with the LC twist were distributed within the LC film, whereas particles with opposite handedness were repelled from the LC medium due to the elastic energy contributions. In addition, a pronounced effect of the surface anchoring of the particles were present on their orientation during non-equilibrium events such as sedimentation. Overall, the studies presented here will find potential use in sensors, separations, optics or soft robotic applications that will take advantages of chirality or chiral interactions.

Chaining of hard disks in nematic needles: particle-based simulation of colloidal interactions in liquid crystals

Colloidal particles suspended in liquid crystals can exhibit various effective anisotropic interactions that can be tuned and utilized in self-assembly processes. We simulate a two-dimensional system of hard disks suspended in a solution of dense hard needles as a model system for colloids suspended in a nematic lyotropic liquid crystal. The novel event-chain Monte Carlo technique enables us to directly measure colloidal interactions in a microscopic simulation with explicit liquid crystal particles in the dense nematic phase. We find a directional short-range attraction for disks along the director, which triggers chaining parallel to the director and seemingly contradicts the standard liquid crystal field theory result of a quadrupolar attraction with a preferred [Formula: see text] angle. Our results can be explained by a short-range density-dependent depletion interaction, which has been neglected so far. Directionality and strength of the depletion interaction are caused by the weak planar anchoring of hard rods. The depletion attraction robustly dominates over the quadrupolar elastic attraction if disks come close. Self-assembly of many disks proceeds via intermediate chaining, which demonstrates that in lyotropic liquid crystal colloids depletion interactions play an important role in structure formation processes.

Field-triggered vertical positional transition of a microparticle suspended in a nematic liquid crystal cell

In this paper, based on the numerical calculation of total energy utilizing the Green's function method, we investigate how a field-triggered vertical positional transition of a microparticle suspended in a nematic liquid crystal cell is influenced by the direction of the applied field, surface anchoring feature, and nematic's dielectric properties. The new equilibrium position of the translational movement is decided via a competition between the buoyant force and the effective force built on the microparticle by the elastic energy gradient along the vertical direction. The threshold value of external field depends on thickness L and Frank elastic constant K and slightly on the microparticle size and density, in a Fréedericksz-like manner, but by a factor. For a nematic liquid crystal cell with planar surface alignment, a bistable equilibrium structure for the transition is found when the direction of the applied electric field is (a) perpendicular to the two plates of the cell with positive molecular dielectric anisotropy or (b) parallel to the two plates and the anchoring direction of the cell with negative molecular dielectric anisotropy. When the electric field applied is parallel to both plates and perpendicular to the anchoring direction, the microparticle suspended in the nematic liquid crystal tends to be trapped in the midplane, regardless of the sign of the molecular dielectric anisotropy. Such a phenomenon also occurs for negative molecular dielectric anisotropy if the external field is applied perpendicular to the two plates. Explicit formulas proposed for the critical electric field agree extremely well with the numerical calculation.

Steady-state rheology and structure of soft hybrid mixtures of liquid crystals and magnetic nanoparticles

Using non-equilibrium molecular dynamics simulations, we study the rheology of a model hybrid mixture of liquid crystals (LCs) and dipolar soft spheres (DSS) representing magnetic nanoparticles. The bulk isotropic LC-DSS mixture is sheared with different shear rates using Lees-Edwards periodic boundary conditions. The steady-state rheological properties and the effect of the shear on the microstructure of the mixture are studied for different strengths of the dipolar coupling, λ, among the DSS. We find that at large shear rates, the mixture shows a shear-thinning behavior for all considered values of λ. At low and intermediate values of λ, a crossover from Newtonian to non-Newtonian behavior is observed as the rate of applied shear is increased. In contrast, for large values of λ, such a crossover is not observed within the range of shear rates considered. Also, the extent of the non-Newtonian regime increases as λ is increased. These features can be understood via the shear-induced changes of the microstructure. In particular, the LCs display a shear-induced isotropic-to-nematic transition at large shear rates with a shear-rate dependent degree of nematic ordering. The DSS show a shear-induced nematic ordering only for large values of λ, where the particles self-assemble into chains. Moreover, at large λ and low shear rates, our simulations indicate that the DSS form ferromagnetic domains.

Electrostatic interactions between spheroidal dielectric particles

Theory is developed to address the significant problem of electrostatic interactions between charged polarizable dielectric spheroids. The electrostatic force is defined by particle dimensions and charge, dielectric constants of the interacting particles and medium, and the interparticle separation distance; and it is expressed in the form of an integral over the particle surface. The switching behavior between like charge repulsion and attraction is demonstrated as depending on the ratio of the major and minor axes of spheroids. When the major and minor axes are equal, the theory yields a solution equivalent to that obtained for spherical particles. Limiting cases are presented for nonpolarizable spheroids, which describe the electrostatic behavior of charged rods, discs, and point charges. The developed theory represents an important step toward comprehensive understanding of direct interactions and mechanisms of electrostatically driven self-assembly processes.

Colloidal interactions and unusual crystallization versus de-mixing of elastic multipoles formed by gold mesoflowers

Colloidal interactions in nematic liquid crystals can be described as interactions between elastic multipoles that depend on particle shape, topology, chirality, boundary conditions and induced topological defects. Here, we describe a nematic colloidal system consisting of mesostructures of gold capable of inducing elastic multipoles of different order. Elastic monopoles are formed by relatively large asymmetric mesoflower particles, for which gravity and elastic torque balancing yields monopole-type interactions. High-order multipoles are instead formed by smaller mesoflowers with a myriad of shapes corresponding to multipoles of different orders, consistent with our computer simulations based on free energy minimization. We reveal unexpected many-body interactions in this colloidal system, ranging from de-mixing of elastic monopoles to a zoo of unusual colloidal crystals formed by high-order multipoles like hexadecapoles. Our findings show that gold mesoflowers may serve as a designer toolkit for engineering colloidal interaction and self-assembly, potentially exceeding that in atomic and molecular systems.

Elasticity-mediated particle interaction in a hosting medium holds promise for material engineering of unusual structures. Yuan et al. show that the gold microparticles can induce elastic multipoles of different symmetries when dispersed in a nematic liquid crystal as building blocks for various crystals.

Soft matter from liquid crystals

Liquid crystals (LCs) are fluids within which molecules exhibit long-range orientational order, leading to anisotropic properties such as optical birefringence and curvature elasticity. Because the ordering of molecules within LCs can be altered by weak external stimuli, LCs have been widely used to create soft matter systems that respond optically to electric fields (LC display), temperature (LC thermometer) or molecular adsorbates (LC chemical sensor). More recent studies, however, have moved beyond investigations of optical responses of LCs to explore the design of complex LC-based soft matter systems that offer the potential to realize more sophisticated functions (e.g., autonomous, self-regulating chemical responses to mechanical stimuli) by directing the interactions of small molecules, synthetic colloids and living cells dispersed within the bulk of LCs or at their interfaces. These studies are also increasingly focusing on LC systems driven beyond equilibrium states. This review presents one perspective on these advances, with an emphasis on the discovery of fundamental phenomena that may enable new technologies. Three areas of progress are highlighted; (i) directed assembly of amphiphilic molecules either within topological defects of LCs or at aqueous interfaces of LCs, (ii) templated polymerization in LCs via chemical vapor deposition, an approach that overcomes fundamental challenges related to control of LC phase behavior during polymerization, and (iii) studies of colloids in LCs, including chiral colloids, soft colloids that are strained by LCs, and active colloids that are driven into organized states by dissipation of energy (e.g. bacteria). These examples, and key unresolved issues discussed at the end of this perspective, serve to convey the message that soft matter systems that integrate ideas from LC, surfactant, polymer and colloid sciences define fertile territory for fundamental studies and creation of future transformative technologies.

Electrostatically controlled surface boundary conditions in nematic liquid crystals and colloids

Differing from isotropic fluids, liquid crystals exhibit highly anisotropic interactions with surfaces, which define boundary conditions for the alignment of constituent rod-like molecules at interfaces with colloidal inclusions and confining substrates. We show that surface alignment of the nematic molecules can be controlled by harnessing the competing aligning effects of surface functionalization and electric field arising from surface charging and bulk counterions. The control of ionic content in the bulk and at surfaces allows for tuning orientations of shape-anisotropic particles like platelets within an aligned nematic host and for changing the orientation of director relative to confining substrates. The ensuing anisotropic elastic and electrostatic interactions enable colloidal crystals with reconfigurable symmetries and orientations of inclusions.

Controlled Multistep Self-Assembling of Colloidal Droplets at a Nematic Liquid Crystal-Air Interface

We present a controlled cascade of self-assemblings of colloidal droplets at a nematic liquid crystal-air interface into large-scale ordered structures. Changing the tilt of the droplet-induced elastic dipoles via its dependence on the nematic film thickness, we are able to control the dipole-dipole interaction and thus the self-assembling regime. For a progressively large tilt, droplets form anisotropic lattices, which then transform into arrays of repulsive chains, then to bands of half-period-shifted densely bound chains. These structures with chain order at the inner scale aggregate into different large-scale clusters that have a pronounced circular pattern and are stabilized by the many-body elastocapillary attraction.

Designed self-assembly of metamaterial split-ring colloidal particles in nematic liquid crystals

The fabrication of orientationally and positionally ordered colloidal clusters is of interest to several fields from materials science to photonics. An interesting possibility to obtain such colloidal crystalline structures is by the self-assembly of colloidal particles in a liquid crystal matrix. This work demonstrates the self-assembly in a nematic liquid crystal of a specific type of colloidal particle, split ring resonators (SRRs), which are well known in the field of photonic metamaterials and chosen for their ability to obtain resonances in response to a magnetic field. Using free energy minimisation calculations, we specifically optimise geometrical parameters of the SRR particles to reduce and prevent formation of irregular metastable colloidal states, which in more general view corresponds to concepts of pre-designed self-assembly. Using the pre-designed particles, we then show self-assembly into two- and three-dimensional nematic colloidal crystals of split-ring particles. Our work is a contribution to the development of designed large-scale colloidal crystals, the properties of which could be finely tuned with external parameters, and are of high interest for photonic applications, specifically as tunable metamaterials.

Optical properties of metamaterial split ring nematic colloids

The fabrication of 3D bulk metamaterials, optical materials with sub-wavelength building blocks, is an open challenge, along with the tuning of their optical properties, such as transmissivity or exit polarization where a possible approach is to embed liquid crystalline materials into metamaterials and use their tunable birefringence. In this work, we explore using numerical modelling the photonic properties of a composite of split ring resonator colloidal particles, dispersed in nematic liquid crystal, which was optimised to enable self-assembly fully. Specifically, using generalised FDTD simulations for light propagation in birefringent profiles, we demonstrate the photonic response of single particles, 2D and 3D colloidal crystals. The material transmittance is shown to exhibit clear resonant behaviour with the resonances tunable with the birefringence in the order of ~5%. Electric and magnetic field modes emergent on the particles are shown, as affected by the surrounding nematic birefringence, both the in the slit region of the split ring resonator (SRR) particles as well as around the particles. Observed photonic response is further explained by introducing basic equivalent LC circuits. Finally, this work is aimed at developing soft and fluid metamaterials, which exhibit optical anisotropy in the photonic response as a potent mechanism for controlling the flow of light at wavelength and even sub-wavelength scales.

High-order elastic multipoles as colloidal atoms

Achieving and exceeding diversity of colloidal analogs of chemical elements and molecules as building blocks of matter has been the central goal and challenge of colloidal science ever since Einstein introduced the colloidal atom paradigm. Recent advances in colloids assembly have been achieved by exploiting the machinery of DNA hybridization but robust physical means of defining colloidal elements remain limited. Here we introduce physical design principles allowing us to define high-order elastic multipoles emerging when colloids with controlled shapes and surface alignment are introduced into a nematic host fluid. Combination of experiments and numerical modeling of equilibrium field configurations using a spherical harmonic expansion allow us to probe elastic multipole moments, bringing analogies with electromagnetism and a structure of atomic orbitals. We show that, at least in view of the symmetry of the "director wiggle wave functions," diversity of elastic colloidal atoms can far exceed that of known chemical elements.

Colloidal analogues of polymer chains, ribbons and 2D crystals employing orientations and interactions of nano-rods dispersed in a nematic liquid crystal

Robust control over the position, orientation and self-assembly of nonspherical colloids facilitate the creation of new materials with complex architecture that are important from technological and fundamental perspectives. We study orientation, elastic interaction and co-assembly of surface functionalized silica nano-rods in thin films of nematic liquid crystal. With homeotropic boundary condition, the nano-rods are predominantly oriented perpendicular to the nematic director which is different than the mostly parallel orientation of the micro-rods. The percentage of perpendicular nano-rods are significantly larger than the parallel nano-rods. The perpendicular nano-rods create very weak elastic deformation and exhibit unusual, out-of-plane, attractive interaction. On the other hand, the nano-rods oriented parallel to the director create strong elastic deformation and shows anisotropic, in-plane, dipolar interaction. In both orientations, the induced defects reside in the nano-rods. With the help of a dynamic laser tweezers and using nano-rods as building blocks we demonstrate colloidal analogues of linear polymer chains, ribbons and two-dimensional binary crystals.

Modifying Nobili-Durand surface energy for conically degenerate anchorings at the interface of liquid crystal colloids

We propose a surface energy for conically degenerate anchorings of uniaxial liquid crystal mesogens by modifying tensorial Nobili-Durand surface energy that is usually employed for fixed anchoring orientations with preferred polar angles. By minimizing Landau-de Gennes free energy and the proposed surface energy, we obtain the equilibrium director configuration around a spherical colloid in the uniform nematic liquid crystal. Our calculations show that the proposed surface energy can cause boojum or/and Saturn-ring defect textures depending on the equilibrium conic angle. We also study the interactions between two spherical colloids with the equilibrium conic angle 45^{∘}, where the surface energy provides both boojum and Saturn-ring defects on the surface of particles. We compare the calculated anisotropic colloidal interactions with experimental observations [B. Senyuk et al., Nat. Commun. 7, 10659 (2016)2041-172310.1038/ncomms10659]. In agreement with experiment, our results show two stable angular assemblies in the close particle-particle separations. Also, the long-range elastic interactions are almost consistent with the hexadecapolar elastic distortion.

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.

Anomalous transport of magnetic colloids in a liquid crystal-magnetic colloid mixture

We report an extensive molecular dynamics study on the translational dynamics of a hybrid system composed of dipolar soft spheres (DSS), representing ferromagnetic particles, suspended in a liquid crystal (LC) matrix. We observe that the LC matrix strongly modifies the dynamics of the DSS. In the isotropic regime, the DSS show a crossover from subdiffusive to normal diffusive behavior at long times, with an increase of the subdiffusive regime as the dipolar coupling strength is increased. In the nematic regime, the LC matrix, due to the collective reorientation of LC particles, imposes a cylindrical confinement on the DSS chains. This leads to a diffusive dynamics of DSS along the nematic director and a subdiffusive dynamics (with an exponent of ∼0.5) in the perpendicular direction. The confinement provided by the LC matrix is also reflected by the oscillatory behavior of the components of the velocity autocorrelation function of the DSS in the nematic phase.

Self-assembly of fractal liquid crystal colloids

Nematic liquid crystals are anisotropic fluids that self-assemble into vector fields, which are governed by geometrical and topological laws. Consequently, particulate or droplet inclusions self-assemble in nematic domains through a balance of topological defects. Here, we use double emulsions of water droplets inside radial nematic liquid crystal droplets to form various structures, ranging from linear chains to three-dimensional fractal structures. The system is modeled as a formation of satellite droplets, distributed around a larger, central core droplet and we extend the problem to explain the formation of fractal structures. We show that a distribution of droplet sizes plays a key role in determining the symmetry properties of the resulting geometric structures. The results are relevant to a variety of inclusions, ranging from colloids suspensions to multi-emulsion systems. Such systems have potential applications for novel switchable photonic structures as well as providing wider insights into the packing of self-assembled structures.

Self-assembled nematic colloidal motors powered by light

Biological motors are marvels of nature that inspire creation of their synthetic counterparts with comparable nanoscale dimensions, high efficiency and diverse functions. Molecular motors have been synthesized, but obtaining nanomotors through self-assembly remains challenging. Here we describe a self-assembled colloidal motor with a repetitive light-driven rotation of transparent micro-particles immersed in a liquid crystal and powered by a continuous exposure to unstructured ~1 nW light. A monolayer of azobenzene molecules defines how the liquid crystal's optical axis mechanically couples to the particle's surface, as well as how they jointly rotate as the light's polarization changes. The rotating particle twists the liquid crystal, which changes polarization of traversing light. The resulting feedback mechanism yields a continuous opto-mechanical cycle and drives the unidirectional particle spinning, with handedness and frequency robustly controlled by polarization and intensity of light. Our findings may lead to opto-mechanical devices and colloidal machines compatible with liquid crystal display technology.

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.

Complex electric double layers in charged topological colloids

Charged surfaces in contact with liquids containing ions are accompanied in equilibrium by an electric double layer consisting of a layer of electric charge on the surface that is screened by a diffuse ion cloud in the bulk fluid. This screening cloud determines not only the interactions between charged colloidal particles or polyelectrolytes and their self-assembly into ordered structures, but it is also pivotal in understanding energy storage devices, such as electrochemical cells and supercapacitors. However, little is known to what spatial complexity the electric double layers can be designed. Here, we show that electric double layers of non-trivial topology and geometry -including tori, multi-tori and knots- can be realised in charged topological colloidal particles, using numerical modelling within a mean-field Poisson-Boltzmann theory. We show that the complexity of double layers -including geometry and topology- can be tuned by changing the Debye screening length of the medium, or by changing the shape and topology of the (colloidal) particle. More generally, this work is an attempt to introduce concepts of topology in the field of charged colloids, which could lead to novel exciting material design paradigms.