Many-defect solutions in planar nematics: interactions, spiral textures and boundary conditions.

From incompressible flows to electrostatics, harmonic functions can provide solutions to many two-dimensional problems and, similarly, the director field of a planar nematic can be determined using complex analysis. We derive a closed-form solution for a quasi-steady state director field induced by an arbitrarily large set of point defects and circular inclusions with or without fixed rotational degrees of freedom, and compute the forces and torques acting on each defect or inclusion. We show that a complete solution must include two types of singularities, generating a defect winding number and its spiral texture, which have a direct effect on defect equilibrium textures and their dynamics. The solution accounts for discrete degeneracy of topologically distinct free energy minima which can be obtained by defect braiding. The derived formalism can be readily applied to equilibrium and slowly evolving nematic textures for active or passive fluids with multiple defects present within the orientational order.

Effect of rotational shear on the dielectric dispersion of a nematic liquid crystal above the Freedericksz threshold field.

Rheo-dielectric studies of soft materials provide important information on the dynamic structure and electric polarization. We study the dielectric dispersion of a nematic liquid crystal by applying a high AC probe field without a DC bias and a low AC probe field with a high DC bias under steady rotational shear. The dielectric anisotropy of the nematic is positive and the applied electric field is parallel to the velocity gradient with a magnitude larger than the Freedericksz threshold field. We find that the dielectric dispersion and the relaxation frequencies are strongly shear rate dependent. The analysis of the results based on a simple physical model shows that the effective dielectric constant of the nematic with non-uniform director tilt in the shear plane can be modelled as a series combination of parallel and perpendicular components. Our experiments demonstrate changes in dielectric dispersion are due to molecular reorientation under the influence of the competing effects of hydrodynamic and dielectric torques.

Self-shaping liquid crystal droplets by balancing bulk elasticity and interfacial tension.

The shape diversity and controlled reconfigurability of closed surfaces and filamentous structures, universally found in cellular colonies and living tissues, are challenging to reproduce. Here, we demonstrate a method for the self-shaping of liquid crystal (LC) droplets into anisotropic and three-dimensional superstructures, such as LC fibers, LC helices, and differently shaped LC vesicles. The method is based on two surfactants: one dissolved in the LC dispersed phase and the other in the aqueous continuous phase. We use thermal stimuli to tune the bulk LC elasticity and interfacial energy, thereby transforming an emulsion of polydispersed, spherical nematic droplets into numerous, uniform-diameter fibers with multiple branches and vice versa. Furthermore, when the nematic LC is cooled to the smectic-A LC phase, we produce monodispersed microdroplets with a tunable diameter dictated by the cooling rate. Utilizing this temperature-controlled self-shaping of LCs, we demonstrate life-like smectic LC vesicle structures analogous to the biomembranes in living systems. Our experimental findings are supported by a theoretical model of equilibrium interface shapes. The shape transformation is induced by negative interfacial energy, which promotes a spontaneous increase of the interfacial area at a fixed LC volume. The method was successfully applied to many different LC materials and phases, demonstrating a universal mechanism for shape transformation in complex fluids.

Dense packings of geodesic hard ellipses on a sphere.

Packing problems are abundant in nature and have been researched thoroughly both experimentally and in numerical models. In particular, packings of anisotropic, elliptical particles often emerge in models of liquid crystals, colloids, and granular and jammed matter. While most theoretical studies on anisotropic particles have thus far dealt with packings in Euclidean geometry, there are many experimental systems where anisotropically-shaped particles are confined to a curved surface, such as Pickering emulsions stabilized by ellipsoidal particles or protein adsorbates on lipid vesicles. Here, we study random close packing configurations in a two-dimensional model of spherical geodesic ellipses. We focus on the interplay between finite-size effects and curvature that is most prominent at smaller system sizes. We demonstrate that on a spherical surface, monodisperse ellipse packings are inherently disordered, with a non-monotonic dependence of both their packing fraction and the mean contact number on the ellipse aspect ratio, as has also been observed in packings of ellipsoids in both 2D and 3D flat space. We also point out some fundamental differences with previous Euclidean studies and discuss the effects of curvature on our results. Importantly, we show that the underlying spherical surface introduces frustration and results in disordered packing configurations even in systems of monodispersed particles, in contrast to the 2D Euclidean case of ellipse packing. This demonstrates that closed curved surfaces can be effective at introducing disorder in a system and could facilitate the study of monodispersed random packings.

Spotting plants' microfilament morphologies and nanostructures.

The tracheary system of plant leaves is composed of a cellulose skeleton with diverse hierarchical structures. It is built of polygonally bent helical microfilaments of cellulose-based nanostructures coated by different layers, which provide them high compression resistance, elasticity, and roughness. Their function includes the transport of water and nutrients from the roots to the leaves. Unveiling details about local interactions of tracheary elements with surrounding material, which varies between plants due to adaptation to different environments, is crucial for understanding ascending fluid transport and for tracheary mechanical strength relevant to potential applications. Here we show that plant tracheary microfilaments, collected from Agapanthus africanus and Ornithogalum thyrsoides leaves, have different surface morphologies, revealed by nematic liquid crystal droplets. This results in diverse interactions among microfilaments and with the environment; the differences translate to diverse mechanical properties of entangled microfilaments and their potential applications. The presented study also introduces routes for accurate characterization of plants' microfilaments.

Long-range order in quadrupolar systems on spherical surfaces.

The interplay between curvature, confinement and ordering on curved manifolds, with anisotropic interactions between building blocks, takes a central role in many fields of physics. In this paper, we investigate the effects of lattice symmetry and local positional order on orientational ordering in systems of long-range interacting point quadrupoles on a sphere in the zero temperature limit. Locally triangular spherical lattices show long-range ordered quadrupolar configurations only for specific symmetric lattices as strong geometric frustration prevents general global ordering. Conversely, the ground states on Caspar-Klug lattices are more diverse, with many different symmetries depending on the position of quadrupoles within the fundamental domain. We also show that by constraining the quadrupole tilts with respect to the surface normal, which models interactions with the substrate, and by considering general quadrupole tensors, we can manipulate the ground state configuration symmetry.

Three-Dimensional Active Defect Loops.

We describe the flows and morphological dynamics of topological defect lines and loops in three-dimensional active nematics and show, using theory and numerical modeling, that they are governed by the local profile of the orientational order surrounding the defects. Analyzing a continuous span of defect loop profiles, ranging from radial and tangential twist to wedge ±1/2 profiles, we show that the distinct geometries can drive material flow perpendicular or along the local defect loop segment, whose variation around a closed loop can lead to net loop motion, elongation, or compression of shape, or buckling of the loops. We demonstrate a correlation between local curvature and the local orientational profile of the defect loop, indicating dynamic coupling between geometry and topology. To address the general formation of defect loops in three dimensions, we show their creation via bend instability from different initial elastic distortions.

Waltzing route toward double-helix formation in cholesteric shells.

Liquid crystals, when confined to a spherical shell, offer fascinating possibilities for producing artificial mesoscopic atoms, which could then self-assemble into materials structured at a nanoscale, such as photonic crystals or metamaterials. The spherical curvature of the shell imposes topological constraints in the molecular ordering of the liquid crystal, resulting in the formation of defects. Controlling the number of defects, that is, the shell valency, and their positions, is a key success factor for the realization of those materials. Liquid crystals with helical cholesteric order offer a promising, yet unexplored way of controlling the shell defect configuration. In this paper, we study cholesteric shells with monovalent and bivalent defect configurations. By bringing together experiments and numerical simulations, we show that the defects appearing in these two configurations have a complex inner structure, as recently reported for simulated droplets. Bivalent shells possess two highly structured defects, which are composed of a number of smaller defect rings that pile up through the shell. Monovalent shells have a single radial defect, which is composed of two nonsingular defect lines that wind around each other in a double-helix structure. The stability of the bivalent configuration against the monovalent one is controlled by c = h/p, where h is the shell thickness and p the cholesteric helical pitch. By playing with the shell geometry, we can trigger the transition between the two configurations. This transition involves a fascinating waltz dynamics, where the two defects come closer while turning around each other.

Sensing surface morphology of biofibers by decorating spider silk and cellulosic filaments with nematic microdroplets.

Probing the surface morphology of microthin fibers such as naturally occurring biofibers is essential for understanding their structural properties, biological function, and mechanical performance. The state-of-the-art methods for studying the surfaces of biofibers are atomic force microscopy imaging and scanning electron microscopy, which well characterize surface geometry of the fibers but provide little information on the local interaction potential of the fibers with the surrounding material. In contrast, complex nematic fluids respond very well to external fields and change their optical properties upon such stimuli. Here we demonstrate that liquid crystal droplets deposited on microthin biofibers--including spider silk and cellulosic fibers--reveal characteristics of the fibers' surface, performing as simple but sensitive surface sensors. By combining experiments and numerical modeling, different types of fibers are identified through the fiber-to-nematic droplet interactions, including perpendicular and axial or helicoidal planar molecular alignment. Spider silks align nematic molecules parallel to fibers or perpendicular to them, whereas cellulose aligns the molecules unidirectionally or helicoidally along the fibers, indicating notably different surface interactions. The nematic droplets as sensors thus directly reveal chirality of cellulosic fibers. Different fiber entanglements can be identified by depositing droplets exactly at the fiber crossings. More generally, the presented method can be used as a simple but powerful approach for probing the surface properties of small-size bioobjects, opening a route to their precise characterization.

Knot theory realizations in nematic colloids.

Nematic braids are reconfigurable knots and links formed by the disclination loops that entangle colloidal particles dispersed in a nematic liquid crystal. We focus on entangled nematic disclinations in thin twisted nematic layers stabilized by 2D arrays of colloidal particles that can be controlled with laser tweezers. We take the experimentally assembled structures and demonstrate the correspondence of the knot invariants, constructed graphs, and surfaces associated with the disclination loop to the physically observable features specific to the geometry at hand. The nematic nature of the medium adds additional topological parameters to the conventional results of knot theory, which couple with the knot topology and introduce order into the phase diagram of possible structures. The crystalline order allows the simplified construction of the Jones polynomial and medial graphs, and the steps in the construction algorithm are mirrored in the physics of liquid crystals.

Action of fields on captive disclination loops.

We report on two effects observed in experiments with captive disclination loops on polymeric fibers immersed in nematics and submitted to electric and/or magnetic fields. We show that the magnetic field oblique to a fiber with axial or helicoidal anchoring on its surface induces translation of disclination loops. Fields orthogonal to fibers with helicoidal anchoring make disclination loops rotate around the field direction. In the linear regime of this last chirogyral effect, the angle of rotation is proportional to the helix wave vector and its sense unveils the chirality of the helix. We propose a model explaining the origin and all features of these two effects.

Topological defects in cholesteric liquid crystal shells.

We investigate experimentally and numerically the defect configurations emerging when a cholesteric liquid crystal is confined to a spherical shell. We uncover a rich scenario of defect configurations, some of them non-existent in nematic shells, where new types of defects are stabilized by the helical ordering of the liquid crystal. In contrast to nematic shells, here defects are not simple singular points or lines, but have a large structured core. Specifically, we observe five different types of cholesteric shells. We study the statistical distribution of the different types of shells as a function of the two relevant geometrical dimensionless parameters of the system. By playing with these parameters, we are able to induce transitions between different types of shells. These transitions involve interesting topological transformations in which the defects recombine to form new structures. Surprisingly, the defects do not approach each other by taking the shorter distance route (geodesic), but by following intricate paths.

Hedgehogs in the dowser state.

We show how to easily generate point defects called hedgehogs, in the so-called quasi-planar texture --the dowser state-- of a nematic layer confined between surfaces with homeotropic anchoring conditions. We point out that the dowser texture can be preserved infinitely in spite of its higher energy with respect to the homogeneous homeotropic texture. For topological reasons the dowser state in a squeezed droplet must contain at least one hedgehog. We submitted this hedgehog to a rotating magnetic field and controlled the continuous evolution, transitioning continuously between radial, hyperbolic and circular hedgehogs, which, just as in previous experiments by Lavrentovich et al., are topologically equivalent states. The dynamics of this transformation is shown to be directly sensitive to energy costs of different geometric configurations of the hedgehog defect and therefore can be used as a rough probe for elastic constants; knowing the principal elastic constants K1,2,3, one can retrieve information about the K24 constant. We propose also a method of generation of hedgehog pairs by application of a Poiseuille flow to a dowser state wound by a rotating magnetic field.

Exploiting imperfections in the bulk to direct assembly of surface colloids.

We exploit the long-ranged elastic fields inherent to confined nematic liquid crystals (LCs) to assemble colloidal particles trapped at the LC interface into reconfigurable structures with complex symmetries and packings. Spherical colloids with homeotropic anchoring trapped at the interface between air and the nematic LC 4-cyano-4'-pentylbiphenyl create quadrupolar distortions in the director field causing particles to repel and consequently form close-packed assemblies with a triangular habit. Here, we report on complex open structures organized via interactions with defects in the bulk. Specifically, by confining the nematic LC in an array of microposts with homeotropic anchoring conditions, we cause defect rings to form at well-defined locations in the bulk of the sample. These defects source elastic deformations that direct the assembly of the interfacially trapped colloids into ring-like assemblies, which recapitulate the defect geometry even when the microposts are completely immersed in the nematic. When the surface density of the colloids is high, they form a ring near the defect and a hexagonal lattice far from it. Because topographically complex substrates are easily fabricated and LC defects are readily reconfigured, this work lays the foundation for a versatile, robust mechanism to direct assembly dynamically over large areas by controlling surface anchoring and associated bulk defect structure.

Magnetocontrollable droplet mobility on liquid crystal-infused porous surfaces.

Magnetocontrollable droplet mobility on surfaces of both solids and simple fluids have been widely used in a wide range of applications. However, little is understood about the effect of the magnetic field on the wettability and mobility of droplets on structured fluids. Here, we report the manipulation of the dynamic behaviors of water droplets on a film of thermotropic liquid crystals (LCs). We find that the static wetting behavior and static friction of water droplets on a 4'-octyl-4-biphenylcarbonitrile (8CB) film strongly depend on the LC mesophases, and that a magnetic field caused no measurable change to these properties. However, we find that the droplet dynamics can be affected by a magnetic field as it slides on a nematic 8CB film, but not on isotropic 8CB, and is dependent on both the direction and strength of the magnetic field. By measuring the dynamic friction of a droplet sliding on a nematic 8CB film, we find that a magnetic field alters the internal orientational ordering of the 8CB which in turn affects its viscosity. We support this interpretation with a scaling argument using the LC magnetic coherence length that includes (i) the elastic energy from the long-range orientational ordering of 8CB and (ii) the free energy from the interaction between 8CB and a magnetic field. Overall, these results advance our understanding of droplet mobility on LC films and enable new designs for responsive surfaces that can manipulate the mobility of water droplets. Electronic Supplementary Material: Supplementary material (further details of the stability of LCIPS against water-induced dewetting, the interfacial tension and contact angle measurement using a goniometer, the estimation of the thickness of LC wrapping layer at air-water interface on droplets, SEM measurements, the average sliding velocity of a water droplet on 5CB, E7, silicone oil, and mineral oil films with and without a magnetic field, representative force diagram (Fd versus time) of a 3-µL water droplet moving at a speed of 0.1 mm/s on a nematic 8CB film, Fdynamic acting on 3 µL water droplets moving at speeds of 0.1-1 mm/s on an isotropic 8CB film, the calculated magnetic coherence length as a function of the magnitude of the magnetic field applied to the nematic LCIPS, and the apparent advancing and receding contact angles of a moving water droplet on nematic LCIPS as a function of time, and polarized light micrographs (top view) of a nematic 8CB film between two DMOAP-functionalized glass slides before and after applying a horizontal magnetic field) is available in the online version of this article at 10.1007/s12274-022-5318-y.

Measure of overlap between two arbitrary ellipses on a sphere.

Various packing problems and simulations of hard and soft interacting particles, such as microscopic models of nematic liquid crystals, reduce to calculations of intersections and pair interactions between ellipsoids. When constrained to a spherical surface, curvature and compactness lead to non-trivial behaviour that finds uses in physics, computer science and geometry. A well-known idealized isotropic example is the Tammes problem of finding optimal non-intersecting packings of equal hard disks. The anisotropic case of elliptic particles remains, on the other hand, comparatively unexplored. We develop an algorithm to detect collisions between ellipses constrained to the two-dimensional surface of a sphere based on a solution of an eigenvalue problem. We investigate and discuss topologically distinct ways two ellipses may touch or intersect on a sphere, and define a contact function that can be used for construction of short- and long-range pair potentials.

From coffee stains to uniform deposits: Significance of the contact-line mobility.

HYPOTHESIS: Contact-line motion upon drying of a sessile droplet strongly affects the solute transport and solvent evaporation profile. Hence, it should have a strong impact on the deposit formation and might be responsible for volcano-like, dome-like and flat deposit morphologies. EXPERIMENTS: A method based on a thin-film interference was used to track the drop height profile and contact line motion during the drying. A diverse set of drying scenarios was obtained by using inks with different solvent compositions and by adjusting the substrate wetting properties. The experimental data was compared to the predictions of a phenomenological model. FINDINGS: We highlight the essential role of contact-line mobility on the deposit morphology of solution-based inks. A pinned contact line produces exclusively ring-like deposits under normal conditions. On the contrary, drops with a mobile contact line can produce ring-, flat- or dome-like morphology. The developed phenomenological model shows that the deposit morphology depends on solvent evaporation profile, evolution of the drop radius relative to its contact angle, and the ratio between initial and maximal (gelling) solute concentration. These parameters can be adjusted by the ink solvent composition and substrate wetting behaviour, which provides a way for deposition of uniform and flat deposits via inkjet printing.

Nematic braids: topological invariants and rewiring of disclinations.

The conventional topological description given by the fundamental group of nematic order parameter does not adequately explain the entangled defect line structures that have been observed in nematic colloids. We introduce a new topological invariant, the self-linking number, that enables a complete classification of entangled defect line structures in general nematics, even without particles, and demonstrate our formalism using colloidal dimers, for which entangled structures have been previously observed. We also unveil a simple rewiring scheme for the orthogonal crossing of two -1/2 disclinations, based on a tetrahedral rotation of two relevant disclination segments, that allows us to predict possible nematic braids and calculate their self-linking numbers.

Microfluidic control over topological states in channel-confined nematic flows.

Compared to isotropic liquids, orientational order of nematic liquid crystals makes their rheological properties more involved, and thus requires fine control of the flow parameters to govern the orientational patterns. In microfluidic channels with perpendicular surface alignment, nematics discontinuously transition from perpendicular structure at low flow rates to flow-aligned structure at high flow rates. Here we show how precise tuning of the driving pressure can be used to stabilize and manipulate a previously unresearched topologically protected chiral intermediate state which arises before the homeotropic to flow-aligned transition. We characterize the mechanisms underlying the transition and construct a phenomenological model to describe the critical behaviour and the phase diagram of the observed chiral flow state, and evaluate the effect of a forced symmetry breaking by introduction of a chiral dopant. Finally, we induce transitions on demand through channel geometry, application of laser tweezers, and careful control of the flow rate.

Periodic Arrays of Chiral Domains Generated from the Self-Assembly of Micropatterned Achiral Lyotropic Chromonic Liquid Crystal.

Achiral building blocks forming achiral structures is a common occurrence in nature, while chirality emerging spontaneously from an achiral system is usually associated with important scientific phenomena. We report on the spontaneous chiral symmetry-breaking phenomena upon the topographic confinement of achiral lyotropic chromonic liquid crystals in periodically arranged micrometer scale air pillars. The anisotropic fluid arranges into chiral domains that depend on the arrangement and spacing of the pillars. We characterize the resulting domains by polarized optical microscopy, support their reconstruction by numerical calculations, and extend the findings with experiments, which include chiral dopants. Well-controlled and addressed chiral structures will be useful in potential applications like programmable scaffolds for living liquid crystals and as sensors for detecting chirality at the molecular level.