Active matter comprised of self-propelled interacting units holds a major promise for extraction of useful work from its seemingly chaotic out-of-equilibrium dynamics. Streamlining active matter to produce work is especially important at microscale, where the viscous forces prevail over inertia and the useful modes of transport require very specific non-reciprocal type of motion. Here we report that microscopic active droplets representing aqueous dispersions of swimming bacteria Bacillus subtilis show unidirectional propulsion when placed in an inactive nematic medium. Random motion of bacteria inside the droplet is rectified into a directional self-locomotion of the droplet by the polar director structure that the droplet itself creates in the surrounding nematic through anisotropic molecular interactions at its surface. Droplets without swimming bacteria show no net displacement. The trajectory of the active droplet can be predesigned as rectilinear or curvilinear by patterning the molecular orientation of the nematic medium. The effect demonstrates that swimming at microscale can be achieved at the expense of broken spatial symmetry of the medium; it can be used in development of micromachines.
We study the response of optically trapped achiral and chiralised nematic liquid crystal droplets to linear as well as circular polarised light. We find that there is internal dissipation in rotating achiral nematic droplets trapped in glycerine. We also demonstrate that some chiralised droplets rotate under linearly polarised light. The best fit to our data on chiralised droplets indicates that rotational frequency of these droplets with radius R is approximately proportional to1/R^2, rather than to 1/R^3.
Inverse nematic emulsions in which surfactant-coated water droplets are dispersed in a nematic host fluid have distinctive properties that set them apart from dispersions of two isotropic fluids or of nematic droplets in an isotropic fluid. We present a comprehensive theoretical study of the distortions produced in the nematic host by the dispersed droplets and of solvent mediated dipolar interactions between droplets that lead to their experimentally observed chaining. A single droplet in a nematic host acts like a macroscopic hedgehog defect. Global boundary conditions force the nucleation o…
The emergence of new techniques for the fabrication of nematic droplets with nontrivial topology provides new routes for the assembly of responsive devices. Here we perform a numerical study of spherical nematic droplets on fibres. We analyse the equilibrium textures and find that, under certain conditions,the nematic can avoid the nucleation of topological defects. We consider in detail a homeotropic nematic droplet wrapped around a fibre with planar anchoring. We investigate the effect of an electric field on the texture of this droplet such type of system. In the presence of a DC field, the…
We stabilize nematic droplets with handles against surface-tension-driven instabilities using a yield-stress material as outer fluid and study the complex nematic textures and defect structures that result from the competition between topological constraints and the elasticity of the nematic liquid crystal. We uncover a surprisingly persistent twisted configuration of the nematic director inside the droplets when tangential anchoring is established at their boundaries, which we explain after considering the influence of saddle-splay on the elastic free energy. For toroidal droplets, we find th…
Aqueous dispersions of exfoliated, bile-salt stabilized single-wall carbon nanotubes exhibit a first order transition to a nematic liquid-crystalline phase. The nematic phase presents itself in the form of micron-sized nematic droplets also known as tactoids, freely floating in the isotropic host dispersion. The nematic droplets are spindle shaped and have an aspect ratio of about four, irrespective of their size. We attribute this to a director field that is uniform rather than bipolar, which is confirmed by polarization microscopy. It follows that the ratio of the anchoring strength and the …
Recently, it was observed that water droplets suspended in a nematic liquid crystal form linear chains (Poulin et al., Science 275, 1770 (1997)). The chaining occurs, e.g., in a large nematic drop with homeotropic boundary conditions at all the surfaces. Between each pair of water droplets a point defect in the liquid crystalline order was found in accordance with topological constraints. This point defect causes a repulsion between the water droplets. In our numerical investigation we limit ourselves to a chain of two droplets. For such a complex geometry we use the method of finite elements …
One objective of active matter science is to unveil principles by which chaotic microscale dynamics could be transformed into useful work. A nematic liquid crystal environment offers a number of possibilities, one of which is a directional motion of an active droplet filled with an aqueous dispersion of swimming bacteria. In this work, using the responsiveness of the nematic to the electric field and light, we demonstrate how to control the direction and speed of active droplets. The dielectric response of nematic to the electric field causes two effects: (i) reorientation of the overall direc…
To model a nematic emulsion consisting of a surfactant-coated water droplet dispersed in a nematic host, we performed a molecular dynamics simulation of a droplet immersed in a system of 2048 Gay-Berne ellipsoids in a nematic phase. Strong radial anchoring at the surface of the droplet induced a Saturn ring defect configuration, consistent with theoretical predictions for very small droplets. A surface ring configuration was observed for lower radial anchoring strengths, and a pair of point defects was found near the poles of the droplet for tangential anchoring. We also simulated the falling …
Suspended droplets of cholesteric (chiral nematic) liquid crystals spontaneously rotate in the presence of a heat flux due to a temperature gradient, a phenomenon known as Lehmann effect. So far, it is not clear whether this effect is due to the chirality of the phase and the molecules or only to the chirality of the director field. Here, we report the continuous rotation in a temperature gradient of nematic droplets of a lyotropic chromonic liquid crystal featuring a twisted bipolar configuration. The achiral nature of the molecular components leads to a random handedness of the spontaneous t…
Using computer simulations we investigate the microscopic structure of the singular director field within a nematic droplet. As a theoretical model for nematic liquid crystals we take hard spherocylinders. To induce an overall topological charge, the particles are either confined to a two-dimensional circular cavity with homeotropic boundary or to the surface of a three-dimensional sphere. Both systems exhibit half-integer topological point defects. The isotropic defect core has a radius of the order of one particle length and is surrounded by free-standing density oscillations. The effective …
We investigated the effect of a photovoltaic field generated on the surface of iron-doped lithium niobate crystals on droplets of a ferroelectric nematic liquid crystalline and a standard nematic liquid crystalline material deposited on this surface. When such assembly is illuminated with a laser beam, a wide range of dynamic phenomena are initiated. Droplets located outside the laser spot are dragged in the direction of the illuminated area, while droplets located inside the illuminated region tend to bridge each other and rearrange into tendril-like structures. In the ferroelectric nematic p…
For the first time we observed formation of small satellite droplets from the bridge at droplet coalescence. Investigations were made using a Hele-Shaw cell in the two-phase region at nematic-isotropic phase transition. In previous works on coalescence it was considered that before start of coalescence there exists a bridge between the outer fluid connecting regions on the two sides of the droplets (outer bridge). After start of coalescence a bridge connecting the two droplets appears (droplet bridge) and the outer bridge is broken. For the first time we have shown that there are coalescence p…
In this paper, a Freedericksz-like positional transition is found for a spherical micro-droplet suspended in a nematic liquid crystal cell in the presence of an external electric field. Based on the numerical calculation of elastic energy using Green function method, the equilibrium position of micro-droplet is decided through a competition between the buoyant force and the effective force built by the elastic energy gradient existing inside the nematic liquid crystal(NLC) cell. It is shown that the elastic energy dominates the kinetics of micro-droplet until the external field applied reaches…
The quasistationary spreading of a circular liquid drop on a solid substrate typically obeys the so-called Tanner law, with the instantaneous base radius R(t) growing with time as R ~ t^{1/10} -- an effect of the dominant role of capillary forces for a small-sized droplet. However, for droplets of nematic liquid crystals, a faster spreading law sets in at long times, so that R ~ t^alpha with alpha significantly larger than the Tanner exponent 1/10. In the framework of the thin film model (or lubrication approximation), we describe this "acceleration" as a transition to a qualitatively differen…
Here we study theoretically the dynamics of a 2D and a 3D isotropic droplet in a nematic liquid crystal under a shear flow. We find a large repertoire of possible nonequilibrium steady states as a function of the shear rate and of the anchoring of the nematic director field at the droplet surface. We first discuss homeotropic anchoring. For weak anchoring, we recover the typical behaviour of a sheared isotropic droplet in a binary fluid, which rotates, stretches and can be broken by the applied flow. For intermediate anchoring, new possibilities arise due to elastic effects in the nematic flui…
Active forces occurring within cells can drive crucial biological processes that involve spontaneous organization and shape change, such as cell division. Motivated by recent in vitro experiments of nematic droplets of cytoskeletal filaments and motors that self-organize and divide, we present a minimal hydrodynamic model that combines the nonequilibrium kinetics of motor-filament interactions with equilibrium nematic phase separation. The motors organize within droplets and structure filaments into polarized aster defects. At large motor activity, they can even deform or divide the droplet, o…
We study droplets of chromonic liquid crystals squeezed between parallel plates inducing degenerate tangential anchoring on the nematic director. In the coexistence regime, where droplets in the nematic phase are at equilibrium with the surrounding melt, our two-dimensional theoretical model predicts a regime of shape bistability for sufficiently large bipolar droplets, where tactoids (pointed shapes) and discoids (smooth shapes) coexist in equilibrium. This phenomenon has not yet been observed. Two-dimensional droplets of disodium cromoglycate (DSCG) have been the object of a thorough experim…
From the mitotic spindle up to tissues and biofilms, many biological systems behave as active droplets, which often break symmetry and change shape spontaneously. Here, I show that active nematic droplets can experience a fingering instability. I consider an active fluid that acquires nematic order through anchoring at the droplet interface, and I predict its morphological stability in terms of three dimensionless parameters: the anchoring angle, the penetration length of nematic order compared to droplet size, and an active capillary number. Droplets with extensile (contractile) stresses and …
We investigate the mechanics of an active droplet endowed with internal nematic order and surrounded by an isotropic Newtonian fluid. Using numerical simulations we demonstrate that, due to the interplay between the active stresses and the defective geometry of the nematic director, this system exhibits two of the fundamental functions of living cells: spontaneous division and motility, by means of self-generated hydrodynamics flows. These behaviors can be selectively activated by controlling a single physical parameter, namely, an active variant of the capillary number.
Dynamic simulations of the isotropic-nematic phase transformation of liquid crystal droplets under homeotropic anchoring are found to predict chiral symmetry-breaking dynamics. These observations occur when using material parameters for pentyl-cyanobiphenyl (5CB) but not under the single elastic constant approximation of this material, frequently used in simulation. The twisting dynamic process occurs during the relaxation of the domain from an unstable radial texture to a stable uniform texture and involves simultaneous defect loop motion and twisting of the bulk nematic texture.
We use molecular dynamics to study the ordering of a nematic liquid crystal around a spherical particle or droplet. Homeotropic boundary conditions and strong anchoring create a hedgehog director configuration on the particle surface and in its vicinity; this topological defect is cancelled by nearby defect structures in the surrounding liquid crystal, so as to give a uniform director field at large distances. We observe three defect structures for different particle sizes: a quadrupolar one with a ring defect surrounding the particle in the equatorial plane; a dipolar one with a satellite def…
Microorganisms often encounter anisotropy, for example in mucus and biofilms. We study how anisotropy and elasticity of the ambient fluid affects the speed of a swimming microorganism with a prescribed stroke. Motivated by recent experiments on swimming bacteria in anisotropic environments, we extend a classical model for swimming microorganisms, the Taylor swimming sheet, actuated either by transverse or longitudinal traveling waves in a three-dimensional nematic liquid crystal without twist. We calculate the swimming speed and entrained volumetric flux as a function of the swimmer's stroke p…
Tactoids are pointed, spindle-like droplets of nematic liquid crystal in an isotropic fluid. They have long been observed in inorganic and organic nematics, in thermotropic phases as well as lyotropic colloidal aggregates. The variational problem of determining the optimal shape of a nematic droplet is formidable and has only been attacked in selected classes of shapes and director fields. Here, by considering a novel class of admissible solutions for a bipolar droplet, we study the prevalence in the population of all equilibrium shapes of each of the three that may be optimal (tactoids primar…
We theoretically investigate under what conditions the director field in a spindle-shaped nematic droplet or tactoid obtains a twisted, parity-broken structure. By minimizing the sum of the bulk elastic and surface energies, we show that a twisted director field is stable if the twist and bend elastic constants are small enough compared to the splay elastic constant, but only if the droplet volume is larger than some minimum value. We furthermore show that the transition from an untwisted to a twisted director-field structure is a sharp function of the various control parameters. We predict th…
An isolated charge-neutral droplet in a uniform electric field experiences no net force. However, a droplet pair can move in response to field-induced dipolar and hydrodynamic interactions. If the droplets are identical, the center of mass of the pair remains fixed. Here, we show that if the droplets have different properties, the pair experiences a net motion due to nonreciprocal interactions. We analyze the three-dimensional droplet trajectories using asymptotic theory, assuming spherical droplets and large separations, and numerical simulations based on a boundary integral method. The dynam…
We propose an efficient numerical scheme, based on the method of lines, for solving the Landau-de Gennes equations describing the relaxational dynamics of nematic liquid crystals. Our method is computationally easy to implement, balancing requirements of efficiency and accuracy. We benchmark our method through the study of the following problems: the isotropic-nematic interface, growth of nematic droplets in the isotropic phase and the kinetics of coarsening following a quench into the nematic phase. Our results, obtained through solutions of the full coarse-grained equations of motion with no…
Active nematics contain topological defects which under sufficient activity move, create and annihilate in a chaotic quasi-steady state, called active turbulence. However, understanding active defects under confinement is an open challenge, especially in three-dimensions. Here, we demonstrate the topology of three-dimensional active nematic turbulence under the spherical confinement, using numerical modelling. In such spherical droplets, we show the three-dimensional structure of the topological defects, which due to closed confinement emerge in the form of closed loops or surface-to-surface s…
Nematic drops suspended in the isotropic phase of the same substance were subjected to alternating electrical fields of varying frequency. The system was carefully kept in the isotropic-nematic coexistance region, which was broadened due to small amounts of non-mesogenic additives. Whereas the droplets remained spherical at low (order of 10 Hz) and high frequencies (in the kHz range), at intermediate frequencies, we observed a marked flattening of the droplet in the plane perpendicular to the applied field. The deformation of the liquid crystal (LC) droplets occurred both in substances with po…
In experiments, an individual chemically active liquid crystal (LC) droplet submerged in the bulk of a surfactant solution may self-propel along a straight, helical, or random trajectory. In this paper, we develop a minimal model capturing all three types of self-propulsion trajectories of a drop in the case of a nematic LC with homeotropic anchoring at LC-fluid interface. We emulate the director field within the drop by a single preferred polarization vector that is subject of two reorientation mechanisms, namely, the internal flow-induced displacement of the hedgehog defect and the droplet's…
Inspired by recent experiments using synthetic microswimmers to manipulate droplets, we investigate the low-Reynolds-number locomotion of a model swimmer (a spherical squirmer) encapsulated inside a droplet of comparable size in another viscous fluid. Meditated solely by hydrodynamic interactions, the encaged swimmer is seen to be able to propel the droplet, and in some situations both remain in a stable co-swimming state. The problem is tackled using both an exact analytical theory and a numerical implementation based on boundary element method, with a particular focus on the kinematics of th…
Sessile droplets of a ferroelectric nematic liquid crystalline material were exposed to surface electric fields produced by pyroelectric and photogalvanic (photovoltaic) effects in X-cut iron-doped lithium niobate crystals. The resulting dynamic processes were monitored by polarization optical (video)microscopy (POM). During heating/cooling cycles, at first, the droplets change their shape from spherical to extended ellipsoidal. Then they start to move rapidly along the surface electric field, i.e., along the crystal's polar axis (c-axis). During this motion, several droplets merge into runnin…
We numerically investigate the morphology and disclination line dynamics of active nematic droplets in three dimensions. Although our model only incorporates the simplest possible form of achiral active stress, active nematic droplets display an unprecedented range of complex morphologies. For extensile activity finger-like protrusions grow at points where disclination lines intersect the droplet surface. For contractile activity, however, the activity field drives cup-shaped droplet invagination, run-and-tumble motion or the formation of surface wrinkles. This diversity of behaviour is explai…
Building on the striking similarity between the structure of the spindle during mitosis in living cells and nematic textures in confined liquid crystals, we use a continuum model of two-dimensional nematic liquid crystal droplets, to examine the physical aspects of cell division. The model investigates the interplay between bulk elasticity of the microtubule assembly, described as a nematic liquid crystal, and surface elasticity of the cell cortex, modelled as a bounding flexible membrane, in controlling cell shape and division. The centrosomes at the spindle poles correspond to the cores of t…
By means of computer simulations we study how droplets of hard, rod-like particles optimize their shape and internal structure under the influence of the osmotic compression caused by the presence of spherical particles that act as depletion agents. At sufficiently high osmotic pressures the rods that make up the drops spontaneously align to turn them into uniaxial nematic liquid crystalline droplets. The nematic droplets or "tactoids" that are formed this way are not spherical but elongated, resulting from the competition between the anisotropic surface tension and the elastic deformation of …
Emerging technologies based on liquid crystal (LC) materials increasingly leverage the presence of nanoscale defects, unlike the canonical application of LCs -- LC displays. The inherent nanoscale characteristics of LC defects present both significant opportunities and barriers for the application of this fascinating class of materials. Simulation-based approaches to the study of the effects of confinement and interface anchoring conditions on LC domains has resulted in significant progress over the past decade, where simulations are now able to access experimentally-relevant micron-scales whi…
Coalescence-induced alignment of anisotropic particles in drying sessile droplets
Forschungszentrum Jülich（于利希研究中心）；Helmholtz Institute Erlangen-Nürnberg for Renewable Energy (IET-2)（亥姆霍兹埃尔朗根-纽伦堡可再生能源研究所（IET-2））；Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡弗里德里希·亚历山大大学）
Recent experiments have shown how nematically-ordered tactoid shaped actin droplets can be reorganized and divided by the action of myosin molecular motors. In this paper, we consider how similar morphological changes can potentially be achieved under equilibrium conditions. Using simulations, both atomistic and continuum, and a phenomenological model, we explore how the nucleation dynamics, shape changes, and the final steady state of a nematic tactoid droplet can be modified by interactions with model adhesive colloids that mimic a myosin motor cluster. Our results provide a prescription for…
Cells are fundamental building blocks of living organisms displaying an array of shapes, morphologies, and textures that encode specific functions and physical behaviors. Elucidating the rules of this code remains a challenge. In this work, we create biomimetic structural building blocks by coating ellipsoidal droplets of a smectic liquid crystal with a protein-based active cytoskeletal gel, thus obtaining core-shell structures. By exploiting the patterned texture and anisotropic shape of the smectic core, we were able to mold the complex nematodynamics of the interfacial active material and i…
Emulsions comprising isotropic fluid drops within a nematic host are of interest for applications ranging from biodetection to smart windows, which rely on changes of molecular alignment structures around the drops in response to chemical, thermal, electric and other stimuli. We show that absorption or desorption of trace amounts of common surfactants can drive continuous transformations of elastic multipoles induced by the droplets within the uniformly aligned nematic host. Out-of-equilibrium dynamics of director structures emerge from a controlled self-assembly or desorption of different sur…
Droplet breakup is a well studied phenomena in Newtonian fluids. One property of this behavior is that, independent of initial conditions, the minimum radius exhibits power law scaling with the time left to breakup tau. Because they have additional structure and shear dependent viscosity, liquid crystals pose an interesting complication to such studies. Here, we investigate the breakup of a synthetic nematic liquid crystal known as MBBA. We determine the phase of the solution by using a cross polarizer setup in situ with the liquid bridge breakup apparatus. Consistent with previous studies of …
Rheotaxis is a well-known phenomenon among microbial organisms and artificial active colloids, wherein the swimmers respond to an imposed flow. We report the first experimental evidence of upstream rheotaxis by spherical active droplets. It is shown that the presence of a nearby wall and the resulting strong flow-gradient at the droplet level is at the root of this phenomenon. Experiments with optical cells of different heights reveal that rheotaxis is observed only for a finite range of shear rates, independent of the bulk flow-rate. We conjecture that the flow induced distortion of an otherw…
We analyze the phase behavior of lyotropic nematic liquid crystals in the self-organizing flow, viz. so called active nematics (AN). Their elastic properties are mutually caused by evolution of topological defects (for instance, disclinations and boojums) and the flow regime. Such changes in elasticity of AN comparing with conventional inactive ones set the new working characteristics of these materials, have an influence on their switchable and tunable properties. In this work, we study the uniaxial droplet AN phases with topological defects in their collective flow, we apply the gauge string…
In the last years self-motile droplets attracted the attention of scientists from different fields ranging from applied biology to theoretical physics, because of their promising technological applications and important biological implications. In this Chapter we review the state of the art of the research on active droplets with a particular focus on theoretical and numerical studies. In particular, we reviewed the active gel theory, namely a generalization of the standard Landau-de Gennes theory for liquid crystals adapted to take into account internal active injection due to the presence of…
We study spreading dynamics of nematic liquid crystal droplets within the framework of the long-wave approximation. A fourth order nonlinear parabolic partial differential equation governing the free surface evolution is derived. The influence of elastic distortion energy and of imposed anchoring variations at the substrate are explored through linear stability analysis and scaling arguments, which yield useful insight and predictions for the behaviour of spreading droplets. This behaviour is captured by fully nonlinear time-dependent simulations of three dimensional droplets spreading in the …
