Nematic colloids, topology and photonics

Optically Transportable Optofluidic Microlasers with Liquid Crystal Cavities Tuned by the Electric Field

Liquid crystal microdroplets with readily adjustable optical properties have attracted considerable attention for building reconfigurable optofluidic microsystems for sensing, imaging, and light routing applications. In this quest, development of active optical microcavities serving as versatile integrated sources of coherent light and ultra-sensitive environmental sensors has played a prominent role. Here, we study transportable optofluidic microlasers reversibly tunable by an external electric field, which are based on fluorophore-doped emulsion droplets of radial nematic liquid crystals manipulated by optical tweezers in microfluidic chips with embedded liquid electrodes. Full transparency of the electrodes formed by a concentrated electrolyte solution allows for applying an electric field to the optically trapped droplets without undesired heating caused by light absorption. Taking advantage of independent, precise control over the electric and thermal stimulation of the lasing liquid crystal droplets, we characterize their spectral tuning response at various optical trapping powers and study their relaxation upon a sudden decrease in the trapping power. Finally, we demonstrate that sufficiently strong applied electric fields can induce fully reversible phase transitions in the trapped droplets even below the bulk melting temperature of the used liquid crystal. Our observations indicate viability of creating electrically tunable, optically transported microlasers that can be prepared on-demand and operated within microfluidic chips to implement integrated microphotonic or sensing systems.

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.

Thermotropic liquid crystals with low-dimensional carbon allotropes

As display devices based on liquid crystals have matured over the last decades, liquid crystal research has shifted its priorities in slightly different directions, such as sensors, photonics, nanotechnology and even more biologically related fields like drug delivery. This implied a change of emphasis in the development of novel materials, of which a completely new class of liquid crystal based composites emerged, that of nanoparticle-dispersed liquid crystals. The underlying ideas were to add functionality, while maintaining switchability, and the exploitation of liquid crystal self-organisation to build hierarchical nanostructures. Of particular interest for applications are dispersions of carbon nanomaterials, such as fullerenes, nanotubes and the graphene variants, due to their interactions with conventional liquid crystals. While such systems have been investigated for the past two decades, we concentrate in this review on the effects of dimensionality of the dispersed carbon nanoparticles, which goes hand in hand with the more recent developments in this field. Examples are the doping of 0D fullerenes in liquid crystals and implications for Blue Phase stability, or 1D nanotubes in nematic and ferroelectric liquid crystals, questions of dispersibility and applications as alignment media in ITO-free devices. Graphene (2D) and especially graphene oxide are mainly investigated for their formation of lyotropic liquid crystals. We here discuss the more recent aspects of dispersion in thermotropics.

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.

Coalescence of isotropic droplets in overheated free standing smectic films

A theoretical study of the interaction and coalescence of isotropic droplets in overheated free-standing smectic films (FSSF) is presented. Experimentally it is clear that merging of such droplets is extremely rare. On the basis of the general thermodynamic approach to the stability of FSSF, we determined the energy gains and losses involved in the coalescence process. The main contributions to the critical work of drop coalescence are due to the gain related to the decrease of the surface energy of the merging drops, which is opposed by the entropic repulsions of elementary steps at the smectic interface between them. To quantify the evolution of the merging drops, we use a simple geometrical model in which the volume of the smectic material, rearranged in the process of coalescence, is described by an asymmetrical pyramid at the intersection of two drops. In this way, the critical work for drop coalescence and the corresponding energy barrier have been calculated. The probability of the thermal activation of the coalescence process was found to be negligibly small, indicating that droplet merging can be initiated by only an external stimulus. The dynamics of drop merging was calculated by equating the capillary force driving the coalescence, and the Stokes viscous force slowing it down. For the latter, an approximation of moving oblate spheroids permitting exact calculations was used. The time evolution of the height of the neck between the coalescing drops and that of their lateral size are in good agreement with experiments.

Cuboidal liquid crystal phases under multiaxial geometrical frustration

Cuboidal liquid crystal phases - the so-called blue phases - consist of a network of topological defects arranged into a cubic symmetry. They exhibit striking optical properties, including Bragg reflection in the visible range and fast response times. Confining surfaces can interfere with the packing of such a network, leading to structures that have not been explored before. In this work, a Landau-de Gennes free energy formalism for the tensor alignment field Q is used to investigate the behavior of chiral liquid crystals under non-isotropic confinement. The underlying free energy functional is solved by relying on a Monte Carlo method that facilitates efficient exploration of configuration space. The results of simulations are expressed in terms of phase diagrams as a function of chirality and temperature for three families of spheroids: oblate, spherical, and prolate. Upon deformation, blue phases adapt and transform to accommodate the geometrical constraints, thereby resulting in a wider range of thermal stability. For oblate spheroids, confinement interferes with the development of a full blue phase structure, resulting on a combination of half skyrmions. For prolate spheroids, the blue phases are hybridized and exhibit features of blue phases I and II. More generally, it is shown that mechanical deformation provides an effective means to control, manipulate and stabilize blue phases and cholesterics confined in tactoids.

Nucleation and growth of droplets in the overheated free-standing smectic films

We present a theoretical explanation for the formation of nematic droplets in free-standing smectic films (FSSF) overheated above the temperature of the bulk smectic - nematic transition. The conditions for the formation of the nematic droplets in smectic films are studied on the basis of the general thermodynamic approach to the stability of FSSF. It is shown that the formation of droplets in overheated FSSF is only possible in the presence of a certain amount of thermally generated dislocation loops. We determined the gain in the free energy related with the formation of the nematic droplets, the value of the critical work and the critical size of the drops. The initial increase of the drops size is due to release of material from the growing dislocation loops. At the second stage the drops growth occurs through coalescence of the smaller drops with the larger ones. The droplets attract each other by means of capillary forces arising due to gradients of the surface energy in the area between them. Drops size evolution, the dynamics of their growth and merging are in good agreement with experiments.

Hierarchical Self-Assembly of Cellulose Nanocrystals in a Confined Geometry

Complex hierarchical architectures are ubiquitous in nature. By designing and controlling the interaction between elementary building blocks, nature is able to optimize a large variety of materials with multiple functionalities. Such control is, however, extremely challenging in man-made materials, due to the difficulties in controlling their interaction at different length scales simultaneously. Here, hierarchical cholesteric architectures are obtained by the self-assembly of cellulose nanocrystals within shrinking, micron-sized aqueous droplets. This confined, spherical geometry drastically affects the colloidal self-assembly process, resulting in concentric ordering within the droplet, as confirmed by simulation. This provides a quantitative tool to study the interactions of cellulose nanocrystals beyond what has been achieved in a planar geometry. Our developed methodology allows us to fabricate truly hierarchical solid-state architectures from the nanometer to the macroscopic scale using a renewable and sustainable biopolymer.

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.

COLLOIDS. Colloidal matter: Packing, geometry, and entropy

Colloidal particles with well-controlled shapes and interactions are an ideal experimental system for exploring how matter organizes itself. Like atoms and molecules, these particles form bulk phases such as liquids and crystals. But they are more than just crude analogs of atoms; they are a form of matter in their own right, with complex and interesting collective behavior not seen at the atomic scale. Their behavior is affected by geometrical or topological constraints, such as curved surfaces or the shapes of the particles. Because the interactions between the particles are often short-ranged, we can understand the effects of these constraints using geometrical concepts such as packing. The geometrical viewpoint gives us a window into how entropy affects not only the structure of matter, but also the dynamics of how it forms.

Nanosecond control and optical pulse shaping by stimulated emission depletion in a liquid crystal

We study anisotropic Stimulated Emission Depletion (STED) from dye molecules, which are collectively ordered in a host liquid crystal. Due to the ordering of fluorescent emitters, the STED efficiency depends on the polarization of the depletion beam and time-delay of the STED pulse. The depletion efficiency is highest at lower temperatures in the highly ordered smectic-A phase and deteriorates in the higher temperature nematic and isotropic phases. We demonstrate by temporal tuning of STED that it is possible to generate an arbitrary sequence of nanosecond fluorescent pulses with variable width and variable delay. Our results show that the STED mechanism in principle allows for very fast (GHz) and efficient control of light by light, which could in the future be used for all-optical control of the flow of light in photonic microdevices based on liquid crystals. Using STED anisotropy and time-control, new modalities of STED imaging in liquid crystals could be developed.

Topological binding and elastic interactions of microspheres and fibres in a nematic liquid crystal

We present a detailed analysis of topological binding and elastic interactions between a long, and micrometer-diameter fiber, and a microsphere in a homogeneously aligned nematic liquid crystal. Both objects are surface treated to produce strong perpendicular anchoring of the nematic liquid crystal. We use the opto-thermal micro-quench of the laser tweezers to produce topological defects with prescribed topological charge, such as pairs of a Saturn ring and an anti-ring, hyperbolic and radial hedgehogs on a fiber, as well as zero-charge loops. We study the entanglement and topological charge interaction between the topological defects of the fiber and sphere and we observe a huge variety of different entanglement topologies and defect-mediated elastic bindings. We explain all observed phenomena with simple topological rule: like topological charges repel each other and opposite topological charges attract. These binding mechanisms not only demonstrate the fascinating topology of nematic colloids, but also open a novel route to the assembly of very complex topological networks of fibers, spheres and other objects for applications in liquid crystal photonics.

Elastic multipoles in the field of the nematic director distortions

Theory of the interaction between all types of elastic dipoles and quadrupoles and distortions of the nematic director is presented. If a particle is small relative to the characteristic distortion length, the interaction is determined by the director derivatives at the particle location. We consider a spherical particle since, even under the standard assumptions of the multipole theory (weak deformations, one constant approximation), the problem can be solved analytically only in this case. Different dipoles interact with different distortion modes (e.g., isotropic dipole interacts with the splay, chiral dipole with the twist, and so on). In the main order, the interaction of a dipole is linear in the director derivatives, and the interaction of a quadrupole is linear in the second-order director derivatives. The theory goes beyond the main-order terms and predicts an effective distortion-induced dipolar component on a particle. This effect is described by the free energy term quadratic in the director derivatives and has contributions both of a bulk and surface origin. The bulk effect takes place even if the director at the particle surface is fixed, whereas the surface effect appears if the surface director is perturbed by the distortions due to a weak surface anchoring. The theory is illustrated by simple examples of the interaction of elastic dipoles with a disclination line, with cholesteric spiral, and with the director distortions in a hybrid cell.

Effects of flow on topological defects in a nematic liquid crystal near a colloid

We perform molecular dynamics simulations of a nematic liquid crystal flowing around a colloidal particle. We study the flow-induced modifications of the topological defects in the liquid crystal due to the presence of the colloid. We show that flow distorts Boojum defects into an asymmetrically larger downstream lobe, and that Saturn ring defects are convected downstream along the flow direction, which is in agreement with experimental observations. Additionally, for a Janus colloid with both parallel and perpendicular patches, exhibiting a Boojum defect and a Saturn ring defect, we find that the Boojum defect facing the upstream direction is destroyed and the Saturn ring is convected downstream.

Stabilisation of 2D colloidal assemblies by polymerisation of liquid crystalline matrices for photonic applications

Colloidal crystals in anisotropic matrices are extremely stable and versatile, but disassemble as soon as the anisotropy of the matrix disappears. We present an approach to first custom-assemble colloidal structures and subsequently stabilize them through photo-polymerisation of the liquid crystalline matrix. The resulting 2D colloidal assemblies are stable at high temperatures and can even be obtained as free-standing films without a decrease in the degree of organization. This approach could be used to stabilize and extract recently proposed soft-matter photonic microcircuits based on liquid crystal optical microresonators, microlasers and microfibers, and opens up routes towards real soft matter photonic devices that are stable over extended time and temperatures.

Disclination lines at homogeneous and heterogeneous colloids immersed in a chiral liquid crystal

In the present work we perform Monte Carlo simulations in the isothermal-isobaric ensemble to study defect topologies formed in a cholesteric liquid crystal due to the presence of a spherical colloidal particle. Topological defects arise because of the competition between anchoring at the colloidal surface and the local director. We consider homogeneous colloids with either local homeotropic or planar anchoring to validate our model by comparison with earlier lattice Boltzmann studies. Furthermore, we perform simulations of a colloid in a twisted nematic cell and discuss the difference between induced and intrinsic chirality on the formation of topological defects. We present a simple geometrical argument capable of describing the complex three-dimensional topology of disclination lines evolving near the surface of the colloid. The presence of a Janus colloid in a cholesteric host fluid reveals a rich variety of defect structures. Using the Frank free energy we analyze these defects quantitatively indicating a preferred orientation of the Janus colloid relative to the cholesteric helix.

Knots and nonorientable surfaces in chiral nematics

Knots and knotted fields enrich physical phenomena ranging from DNA and molecular chemistry to the vortices of fluid flows and textures of ordered media. Liquid crystals provide an ideal setting for exploring such topological phenomena through control of their characteristic defects. The use of colloids in generating defects and knotted configurations in liquid crystals has been demonstrated for spherical and toroidal particles and shows promise for the development of novel photonic devices. Extending this existing work, we describe the full topological implications of colloids representing nonorientable surfaces and use it to construct torus knots and links of type (p,2) around multiply twisted Möbius strips.