Linked topological colloids in a nematic host

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.

3D printing colloidal crystal microstructures via sacrificial-scaffold-mediated two-photon lithography

The orderly arrangement of nanomaterials’ tiny units at the nanometer-scale accounts for a substantial part of their remarkable properties. Maintaining this orderness and meanwhile endowing the nanomaterials with highly precise and free-designed 3D micro architectures will open an exciting prospect for various novel applications. In this paper, we developed a sacrificial-scaffold-mediated two-photon lithography (TPL) strategy that enables the fabrication of complex 3D colloidal crystal microstructures with orderly-arranged nanoparticles inside. We show that, with the help of a degradable hydrogel scaffold, the disturbance effect of the femtosecond laser to the nanoparticle self-assembling could be overcome. Therefore, hydrogel-state and solid-state colloidal crystal microstructures with diverse compositions, free-designed geometries and variable structural colors could be easily fabricated. This enables the possibility to create novel colloidal crystal microsensing systems that have not been achieved before.

Colloidal crystals are widely applied in the fabrication of optoelectronic devices, but realizing freedom of design, such as in 3D printing, in colloidal crystal fabrication remains challenging. Here, the authors demonstrate a sacrificial-scaffold-mediated two-photon lithography strategy that enables the fabrication of complex 3D colloidal crystal microstructures with orderly arranged nanoparticles in the bulk.

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.

Topological and physical links in soft matter systems

Linking, or multicomponent topological entanglement, is ubiquitous in soft matter systems, from mixtures of polymers and DNA filaments packedin vivoto interlocked line defects in liquid crystals and intertwined synthetic molecules. Yet, it is only relatively recently that theoretical and experimental advancements have made it possible to probe such entanglements and elucidate their impact on the physical properties of the systems. Here, we review the state-of-the-art of this rapidly expanding subject and organize it as follows. First, we present the main concepts and notions, from topological linking to physical linking and then consider the salient manifestations of molecular linking, from synthetic to biological ones. We next cover the main physical models addressing mutual entanglements in mixtures of polymers, both linear and circular. Finally, we consider liquid crystals, fluids and other non-filamentous systems where topological or physical entanglements are observed in defect or flux lines. We conclude with a perspective on open challenges.

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.

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.

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.

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.

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.

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.

Orientation, elastic interaction and magnetic response of asymmetric colloids in a nematic liquid crystal

Colloidal particles in nematic liquid crystals create elastic distortion and experience long-range forces. The symmetry of elastic distortion and consequently the complexity of interaction strongly depends largely on the liquid crystal anchoring, topology and shape of the particles. Here, we introduce a new nematic colloidal system made of peanut-shaped hematite particles. We report experimental studies on spontaneous orientation, mutual interaction, laser assisted self-assembly and the effect of external magnetic fields on the colloids. Majority of the colloids spontaneously orient either parallel or perpendicular to the nematic director. The colloids that are oriented perpendicularly exhibit two types of textures due to the out of plane tilting, which is corroborated by the Landau-de Gennes Q-tensor modelling. The transverse magnetic moment of the peanut-shaped colloids is estimated by using a simple analysis based on the competing effects of magnetic and elastic torques.

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.

Nematic colloidal knots in topological environments

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

Chiral liquid crystal colloids

Colloidal particles disturb the alignment of rod-like molecules of liquid crystals, giving rise to long-range interactions that minimize the free energy of distorted regions. Particle shape and topology are known to guide this self-assembly process. However, how chirality of colloidal inclusions affects these long-range interactions is unclear. Here we study the effects of distortions caused by chiral springs and helices on the colloidal self-organization in a nematic liquid crystal using laser tweezers, particle tracking and optical imaging. We show that chirality of colloidal particles interacts with the nematic elasticity to predefine chiral or racemic colloidal superstructures in nematic colloids. These findings are consistent with numerical modelling based on the minimization of Landau-de Gennes free energy. Our study uncovers the role of chirality in defining the mesoscopic order of liquid crystal colloids, suggesting that this feature may be a potential tool to modulate the global orientated self-organization of these systems.

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.

Key-lock colloids in a nematic liquid crystal

The Landau-de Gennes free energy is used to study theoretically the effective interaction of spherical "key" and anisotropic "lock" colloidal particles. We assume identical anchoring properties of the surfaces of the key and of the lock particles, and we consider planar degenerate and perpendicular anchoring conditions separately. The lock particle is modeled as a spherical particle with a spherical dimple. When such a particle is introduced into a nematic liquid crystal, it orients its dimple at an oblique angle θ_{eq} with respect to the far field director n_{∞}. This angle depends on the depth of the dimple. Minimization results show that the free energy of a pair of key and lock particles exhibits a global minimum for the configuration when the key particle is facing the dimple of the lock colloidal particle. The preferred orientation ϕ_{eq} of the key-lock composite doublet relative to n_{∞} is robust against thermal fluctuations. The preferred orientation θ_{eq}^{(2)} of the dimple particle in the doublet is different from the isolated situation. This is related to the "direct" interaction of defects accompanying the key particle with the edge of the dimple. We propose that this nematic-amplified key-lock interaction can play an important role in self-organization and clustering of mixtures of colloidal particles with dimple colloids present.

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.

Analytical description of the Saturn-ring defect in nematic colloids

We derive an analytical formula for the Saturn-ring configuration around a small colloidal particle suspended in nematic liquid crystal. In particular we obtain an explicit expression for the ring radius and its dependence on the anchoring energy. We work within Landau-de Gennes theory: Nematic alignment is described by a tensorial order parameter. For nematic colloids this model had previously been used exclusively to perform numerical computations. Our method demonstrates that the tensorial theory can also be used to obtain analytical results, suggesting a different approach to the understanding of nematic colloidal interactions.

Topological nanocolloids with facile electric switching of plasmonic properties

Combining topology and plasmonics paradigms in nanocolloidal systems may enable new means of pre-engineering desired composite material properties. Here we design and realize orientationally ordered assemblies of noble metal nanoparticles with genus-one topology and unusual long-range ordering mediated by their interactions with the surrounding nematic fluid host. Facile electric switching of these composites is reminiscent of that of pristine liquid crystals (LCs), but provides a means of reconfiguring the nanoparticle assembly and thus also the ensuing composite medium's optical properties. Our findings may lead to formation of new molecular-colloidal soft matter phases with unusual optical properties, as well as optical metamaterials.