Excitable patterns in active nematics

Vorticity phase separation and defect lattices in the isotropic phase of active liquid crystals

We use numerical simulations and linear stability analysis to study the dynamics of an active liquid crystal film on a substrate in the regime where the passive system would be isotropic. Extensile activity builds up local orientational order and destabilizes the quiescent isotropic state above a critical activity, eventually resulting in spatiotemporal chaotic dynamics akin to the one observed ubiquitously in the nematic state. Here we show that tuning substrate friction yields a variety of emergent structures at intermediate activity, including lattices of flow vortices with associated regular arrangements of topological defects and a new state where flow vortices trap pairs of +1/2 defect that chase each other's tail. These chiral units spontaneously pick the sense of rotation and organize in a hexagonal lattice, surrounded by a diffuse flow of opposite rotation to maintain zero net vorticity. The length scale of these emergent structures is set by the screening length of the flow, controlled by the shear viscosity η and the substrate friction Γ, and can be captured by simple mode selection of the vortical flows. We demonstrate that the emergence of coherent structures can be interpreted as a phase separation of vorticity, where friction plays a role akin to that of birth/death processes in breaking conservation of the phase separating species and selecting a characteristic scale for the patterns. Our work shows that friction provides an experimentally accessible tuning parameter for designing controlled active flows.

Data-Driven Discovery of Active Nematic Hydrodynamics

Active nematics can be modeled using phenomenological continuum theories that account for the dynamics of the nematic director and fluid velocity through partial differential equations (PDEs). While these models provide a statistical description of the experiments, the relevant terms in the PDEs and their parameters are usually identified indirectly. We adapt a recently developed method to automatically identify optimal continuum models for active nematics directly from spatiotemporal data, via sparse regression of the coarse-grained fields onto generic low order PDEs. After extensive benchmarking, we apply the method to experiments with microtubule-based active nematics, finding a surprisingly minimal description of the system. Our approach can be generalized to gain insights into active gels, microswimmers, and diverse other experimental active matter systems.

Coexistence of Active and Hydrodynamic Turbulence in Two-Dimensional Active Nematics

In active nematic liquid crystals, activity is able to drive chaotic spatiotemporal flows referred to as active turbulence. Active turbulence has been characterized through theoretical and experimental work as a low Reynolds number phenomenon. We show that, in two dimensions, the active forcing alone is able to trigger hydrodynamic turbulence leading to the coexistence of active and inertial turbulence. This type of flow develops for sufficiently active and extensile flow-aligning nematics. We observe that the combined effect of an extensile nematic and large values of the flow-aligning parameter leads to a broadening of the elastic energy spectrum that promotes a growth of kinetic energy able to trigger an inverse energy cascade.

Active nematic gel with quenched disorder

With quenched disorder, we introduce two-dimensional active nematics suspended in an incompressible fluid. We write the coarse-grained hydrodynamic equations of motion for slow variables, viz. density, orientation, and flow fields. The quenched disorder is introduced such that it interacts with the local orientation at every point with some strength. Disorder strength is tuned from zero to large values. We numerically study the defect dynamics and system kinetics and find that the finite disorder slows the ordering. The presence of fluid induces large fluctuation in the orientation field, further disturbing the ordering. The large fluctuation in the orientation field due to the fluid is so dominant that it reduces the effect of the quenched disorder. We have also found that the disorder effect is almost the same for both the contractile and extensile nature of active stresses in the system. This study can help to understand the impact of quenched disorder on the ordering kinetics of active gels with nematic interaction among the constituent objects.

Spontaneous population oscillation of confined active granular particles

Spontaneous collective oscillation may emerge from seemingly irregular active matter systems. Here, we experimentally demonstrate a spontaneous population oscillation of active granular particles confined in two chambers connected by a narrow channel, and verify the intriguing behavior predicted in simulation [M. Paoluzzi, R. Di Leonardo and L. Angelani, Self-sustained density oscillations of swimming bacteria confined in microchambers, Phys. Rev. Lett., 2015, 115(18), 188303]. During the oscillation, the two chambers are alternately (nearly) filled up and emptied by the self-propelled particles in a periodic manner. We show that the stable unidirectional flow induced due to the confined channel and its periodic reversal triggered by the particle concentration difference between two chambers jointly give rise to the oscillatory collective behavior. Furthermore, we propose a minimal theoretical model that properly reproduces the experimental results without free parameters. This self-sustained collective oscillation could serve as a robust active granular clock, capable of providing rhythmic signals.

Bridging microscopic cell dynamics to nematohydrodynamics of cell monolayers

It is increasingly being realized that liquid-crystalline features can play an important role in the properties and dynamics of cell monolayers. Here, we present a cell-based model of cell layers, based on the phase-field formulation, that connects cell-cell interactions specified at the single cell level to large-scale nematic and hydrodynamic properties of the tissue. In particular, we present a minimal formulation that reproduces the well-known bend and splay hydrodynamic instabilities of the continuum nemato-hydrodynamic formulation of active matter, together with an analytical description of the instability threshold in terms of activity and elasticity of the cells. Furthermore, we provide a quantitative characterisation and comparison of flows and topological defects for extensile and contractile stress generation mechanisms, and demonstrate activity-induced heterogeneity and spontaneous formation of gaps within a confluent monolayer. Together, these results contribute to bridging the gap between cell-scale dynamics and tissue-scale collective cellular organisation.

Quadrupolar active stress induces exotic patterns of defect motion in compressible active nematics

A wide range of living and artificial active matter exists in close contact with substrates and under strong confinement, where in addition to dipolar active stresses, quadrupolar active stresses can become important. Here, we numerically investigate the impact of quadrupolar non-equilibrium stresses on the emergent patterns of self-organisation in non-momentum conserving active nematics. Our results reveal that beyond having stabilising effects, the quadrupolar active forces can induce various modes of topological defect motion in active nematics. In particular, we find the emergence of both polar and nematic ordering of the defects, as well as new patterns of self-organisation that comprise topological defect chains and transient topological defect asters. The results contribute to further understanding of emergent patterns of collective motion and non-equilibrium self-organisation in active matter.

Role of Advective Inertia in Active Nematic Turbulence

Suspensions of active agents with nematic interactions exhibit complex spatiotemporal dynamics such as mesoscale turbulence. Since the Reynolds number of microscopic flows is very small on the scale of individual agents, inertial effects are typically excluded in continuum theories of active nematic turbulence. Whether active stresses can collectively excite inertial flows is currently unclear. To address this question, we investigate a two-dimensional continuum theory for active nematic turbulence. In particular, we compare mesoscale turbulence with and without the effects of advective inertia. We find that inertial effects can influence the flow already close to the onset of the turbulent state and, moreover, give rise to large-scale fluid motion for strong active driving. A detailed analysis of the kinetic energy budget reveals an energy transfer to large scales mediated by inertial advection. While this transfer is small in comparison to energy injection and dissipation, its effects accumulate over time. The inclusion of friction, which is typically present in experiments, can compensate for this effect. The findings suggest that the inclusion of inertia and friction may be necessary for dynamically consistent theories of active nematic turbulence.

Submersed micropatterned structures control active nematic flow, topology, and concentration

Coupling between flows and material properties imbues rheological matter with its wide-ranging applicability, hence the excitement for harnessing the rheology of active fluids for which internal structure and continuous energy injection lead to spontaneous flows and complex, out-of-equilibrium dynamics. We propose and demonstrate a convenient, highly tunable method for controlling flow, topology, and composition within active films. Our approach establishes rheological coupling via the indirect presence of fully submersed micropatterned structures within a thin, underlying oil layer. Simulations reveal that micropatterned structures produce effective virtual boundaries within the superjacent active nematic film due to differences in viscous dissipation as a function of depth. This accessible method of applying position-dependent, effective dissipation to the active films presents a nonintrusive pathway for engineering active microfluidic systems.

Defect Spirograph: Dynamical Behavior of Defects in Spatially Patterned Active Nematics

Topological defects in active liquid crystals can be confined by introducing gradients of activity. Here, we examine the dynamical behavior of two defects confined by a sharp gradient of activity that separates an active circular region and a surrounding passive nematic material. Continuum simulations are used to explain how the interplay among energy injection into the system, hydrodynamic interactions, and frictional forces governs the dynamics of topologically required self-propelling +1/2 defects. Our findings are rationalized in terms of a phase diagram for the dynamical response of defects in terms of activity and frictional damping strength. Different regions of the underlying phase diagram correspond to distinct dynamical modes, namely immobile defects, steady rotation of defects, bouncing defects, bouncing-cruising defects, dancing defects, and multiple defects with irregular dynamics. These dynamic states raise the prospect of generating synchronized defect arrays for microfluidic applications.

Viscoelastic control of spatiotemporal order in bacterial active matter

Active matter consists of units that generate mechanical work by consuming energy¹. Examples include living systems (such as assemblies of bacteria^2-5 and biological tissues^6,7), biopolymers driven by molecular motors^8-11 and suspensions of synthetic self-propelled particles^12-14. A central goal is to understand and control the self-organization of active assemblies in space and time. Most active systems exhibit either spatial order mediated by interactions that coordinate the spatial structure and the motion of active agents^12,14,15 or the temporal synchronization of individual oscillatory dynamics². The simultaneous control of spatial and temporal organization is more challenging and generally requires complex interactions, such as reaction-diffusion hierarchies¹⁶ or genetically engineered cellular circuits². Here we report a simple technique to simultaneously control the spatial and temporal self-organization of bacterial active matter. We confine dense active suspensions of Escherichia coli cells and manipulate a single macroscopic parameter-namely, the viscoelasticity of the suspending fluid- through the addition of purified genomic DNA. This reveals self-driven spatial and temporal organization in the form of a millimetre-scale rotating vortex with periodically oscillating global chirality of tunable frequency, reminiscent of a torsional pendulum. By combining experiments with an active-matter model, we explain this behaviour in terms of the interplay between active forcing and viscoelastic stress relaxation. Our findings provide insight into the influence of bacterial motile behaviour in complex fluids, which may be of interest in health- and ecology-related research, and demonstrate experimentally that rheological properties can be harnessed to control active-matter flows^17,18. We envisage that our millimetre-scale, tunable, self-oscillating bacterial vortex may be coupled to actuation systems to act a 'clock generator' capable of providing timing signals for rhythmic locomotion of soft robots and for programmed microfluidic pumping¹⁹, for example, by triggering the action of a shift register in soft-robotic logic devices²⁰.

Machine learning forecasting of active nematics

Active nematics are a class of far-from-equilibrium materials characterized by local orientational order of force-generating, anisotropic constitutes. Traditional methods for predicting the dynamics of active nematics rely on hydrodynamic models, which accurately describe idealized flows and many of the steady-state properties, but do not capture certain detailed dynamics of experimental active nematics. We have developed a deep learning approach that uses a Convolutional Long-Short-Term-Memory (ConvLSTM) algorithm to automatically learn and forecast the dynamics of active nematics. We demonstrate our purely data-driven approach on experiments of 2D unconfined active nematics of extensile microtubule bundles, as well as on data from numerical simulations of active nematics.

Confinement-induced self-organization in growing bacterial colonies

We investigate the emergence of global alignment in colonies of dividing rod-shaped cells under confinement. Using molecular dynamics simulations and continuous modeling, we demonstrate that geometrical anisotropies in the confining environment give rise to an imbalance in the normal stresses, which, in turn, drives a collective rearrangement of the cells. This behavior crucially relies on the colony's solid-like mechanical response at short time scales and can be recovered within the framework of active hydrodynamics upon modeling bacterial colonies as growing viscoelastic gels characterized by Maxwell-like stress relaxation.

Active nematics with quenched disorder

We introduce a two-dimensional active nematic with quenched disorder. We write the coarse-grained hydrodynamic equations of motion for slow variables, viz. density and orientation. Disorder strength is tuned from zero to large values. Results from the numerical solution of equations of motion as well as the calculation of two-point orientation correlation function using linear approximation shows that the ordered steady state follows a disorder dependent crossover from quasi-long-range order to short-range order. Such crossover is due to the pinning of ±1/2 topological defects in the presence of finite disorder, which breaks the system in uncorrelated domains. Finite disorder slows the dynamics of +1/2 defect, and it leads to slower growth dynamics. The two-point correlation functions for the density and orientation fields show good dynamic scaling but no static scaling for the different disorder strengths. Our findings can motivate experimentalists to verify the results and find applications in living and artificial apolar systems in the presence of a quenched disorder.

Spectral energy analysis of bulk three-dimensional active nematic turbulence

We perform energy spectrum analysis of the active turbulence in a 3D bulk active nematic using continuum numerical modelling. Specifically, we calculate the spectra of the two main energy contributions - kinetic energy and nematic elastic energy - and combine this with the geometrical analysis of the nematic order and flow fields, based on direct defect tracking and calculation of autocorrelations. We show that the active nematic elastic energy is concentrated at scales corresponding to the effective defect-to-defect separation, scaling with activity as ∼ζ0.5, whereas the kinetic energy is largest at somewhat larger scales of typically several 100 nematic correlation lengths. Nematic biaxiality is shown to have no role in active turbulence at most length scales, but can affect the nematic elastic energy by an order of magnitude at scales of the active defect core size. The effect of an external aligning field on the 3D active turbulence is explored, showing a transition from an effective active turbulent to an aligned regime. The work is aimed at providing a contribution towards understanding active turbulence in general three-dimensions, from the perspective of main energy-relevant mechanisms at different length scales of the system.

Pattern Formation and Defect Ordering in Active Chiral Nematics

Many biological systems display intriguing chiral patterns and dynamics. Here, we present an active nematic theory accounting for individual spin to explore the collective handedness in chiral rod-shaped aggregations. We show that coordinated individual spin and motility can engender a vortex-array pattern with chirality and drive ordering of topological defects. During this chiral process, the stationary trefoil-like defects self-organize into a periodic, hexagon-dominated polygonal network, which segregates persistently rotating cometlike defects in pairs within each polygon, leading to a translation symmetry at the global scale while a broken reflection symmetry at the local scale. Such defect ordering agrees exactly with the Voronoi tiling of two-dimensional space and the emergence of the hexagonal symmetry is deciphered in analogy with topological charge neutralization. We calculate energy barriers to the topological transition of the defect ordering and explain the existing metastable states with nonhexagonal polygons. Our findings shed light on the chiral morphodynamics in life processes and also suggest a potential route towards tuning self-organization in active materials.

Steady-state distributions and nonsteady dynamics in nonequilibrium systems

We search for steady states in a class of fluctuating and driven physical systems that exhibit sustained currents. We find that the physical concept of a steady state, well known for systems at equilibrium, must be generalized to describe such systems. In these, the generalization of a steady state is associated with a stationary probability density of microstates and a deterministic dynamical system whose trajectories the system follows on average. These trajectories are a manifestation of nonstationary macroscopic currents observed in these systems. We determine precise conditions for the steady state to exist as well as the requirements for it to be stable. We illustrate this with some examples.

Spontaneous helical flows in active nematics lying on a cylindrical surface

Within the framework of the two-dimensional Ericksen-Leslie model, we explore the effect of geometric confinement on the spontaneous flow of active nematic gels. The nematic particles are assumed to flow on a cylindrical surface, while a degenerate tangential anchoring is enforced. Using the linear approximation of the motion equations, we show that there is a close interplay among extrinsic curvature, flow, director alignment, and activity. We find that the extrinsic curvature promotes the director alignment parallel to the cylindrical axis and is responsible for raising the critical threshold with respect to the flat case. Our analysis reveals a very rich scenario where the key quantities are the activity coefficient, the tumbling parameter, and the anisotropic viscosity ratio. Thus, solutions can exhibit a double periodicity in both the azimuthal and axial variables. As a consequence, the velocity field can make a finite angle with the cylinder axis and the active flow winds on the surface with a helical pattern, while the director oscillates around the cylinder generators. Our results can be validated on thin layers of nematic gels placed between two concentric cylinders and suggest which material properties are most suited for the design of active microfluidic devices.

Understanding Dense Active Nematics from Microscopic Models

We study dry, dense active nematics at both particle and continuous levels. Specifically, extending the Boltzmann-Ginzburg-Landau approach, we derive well-behaved hydrodynamic equations from a Vicsek-style model with nematic alignment and pairwise repulsion. An extensive study of the phase diagram shows qualitative agreement between the two levels of description. We find in particular that the dynamics of topological defects strongly depends on parameters and can lead to "arch" solutions forming a globally polar, smecticlike arrangement of Néel walls. We show how these configurations are at the origin of the defect ordered states reported previously. This work offers a detailed understanding of the theoretical description of dense active nematics directly rooted in their microscopic dynamics.

Self-propulsion of an active polar drop

We investigate the self-propulsive motion of a drop containing an active polar field. The drop demonstrates spontaneous symmetry breaking from a uniform orientational order into a splay or bend instability depending on the types of active stress, namely, contractile or extensile, respectively. We develop an analytical theory of the mechanism of this instability, which has been observed only in numerical simulations. We show that both contractile and extensile active stresses result in the instability and self-propulsive motion. We also discuss asymmetry between contractile and extensile stresses and show that extensile active stress generates chaotic motion even under a simple model of the polarity field coupled with motion and deformation of the drop.

Low-noise phase of a two-dimensional active nematic system

We consider a collection of self-driven apolar particles on a substrate that organize into an active nematic phase at sufficiently high density or low noise. Using the dynamical renormalization group, we systematically study the two-dimensional fluctuating ordered phase in a coarse-grained hydrodynamic description involving both the nematic director and the conserved density field. In the presence of noise, we show that the system always displays only quasi-long-ranged orientational order beyond a crossover scale. A careful analysis of the nonlinearities permitted by symmetry reveals that activity is dangerously irrelevant over the linearized description, allowing giant number fluctuations to persist although now with strong finite-size effects and a nonuniversal scaling exponent. Nonlinear effects from the active currents lead to power-law correlations in the density field, thereby preventing macroscopic phase separation in the thermodynamic limit.

Topological defects in epithelia govern cell death and extrusion

Epithelial tissues (epithelia) remove excess cells through extrusion, preventing the accumulation of unnecessary or pathological cells. The extrusion process can be triggered by apoptotic signalling, oncogenic transformation and overcrowding of cells. Despite the important linkage of cell extrusion to developmental, homeostatic and pathological processes such as cancer metastasis, its underlying mechanism and connections to the intrinsic mechanics of the epithelium are largely unexplored. We approach this problem by modelling the epithelium as an active nematic liquid crystal (that has a long range directional order), and comparing numerical simulations to strain rate and stress measurements within monolayers of MDCK (Madin Darby canine kidney) cells. Here we show that apoptotic cell extrusion is provoked by singularities in cell alignments in the form of comet-shaped topological defects. We find a universal correlation between extrusion sites and positions of nematic defects in the cell orientation field in different epithelium types. The results confirm the active nematic nature of epithelia, and demonstrate that defect-induced isotropic stresses are the primary precursors of mechanotransductive responses in cells, including YAP (Yes-associated protein) transcription factor activity, caspase-3-mediated cell death, and extrusions. Importantly, the defect-driven extrusion mechanism depends on intercellular junctions, because the weakening of cell-cell interactions in an α-catenin knockdown monolayer reduces the defect size and increases both the number of defects and extrusion rates, as is also predicted by our model. We further demonstrate the ability to control extrusion hotspots by geometrically inducing defects through microcontact printing of patterned monolayers. On the basis of these results, we propose a mechanism for apoptotic cell extrusion: spontaneously formed topological defects in epithelia govern cell fate. This will be important in predicting extrusion hotspots and dynamics in vivo, with potential applications to tissue regeneration and the suppression of metastasis. Moreover, we anticipate that the analogy between the epithelium and active nematic liquid crystals will trigger further investigations of the link between cellular processes and the material properties of epithelia.

Emergent Structures in an Active Polar Fluid: Dynamics of Shape, Scattering, and Merger

Spatially localized defect structures emerge spontaneously in a hydrodynamic description of an active polar fluid comprising polar "actin" filaments and "myosin" motor proteins that (un)bind to filaments and exert active contractile stresses. These emergent defect structures are characterized by distinct textures and can be either static or mobile-we derive effective equations of motion for these "extended particles" and analyze their shape, kinetics, interactions, and scattering. Depending on the impact parameter and propulsion speed, these active defects undergo elastic scattering or merger. Our results are relevant for the dynamics of actomyosin-dense structures at the cell cortex, reconstituted actomyosin complexes, and 2D active colloidal gels.

Negative stiffness and modulated states in active nematics

We examine the dynamics of an active nematic liquid crystal on a frictional substrate. When frictional damping dominates over viscous dissipation, we eliminate flow in favor of active stresses to obtain a minimal dynamical model for the nematic order parameter, with elastic constants renormalized by activity. The renormalized elastic constants can become negative at large activity, leading to the selection of spatially inhomogeneous patterns via a mechanism analogous to that responsible for modulated phases arising at an equilibrium Lifshitz point. Tuning activity and the degree of nematic order in the passive system, we obtain a linear stability phase diagram that exhibits a nonequilibrium tricritical point where ordered, modulated and disordered phases meet. Numerical solution of the nonlinear equations yields a succession of spatial structures of increasing complexity with increasing activity, including kink walls and active turbulence, as observed in experiments on microtubule bundles confined at an oil-water interface. Our work provides a minimal model for an overdamped active nematic that reproduces all the nonequilibrium structures seen in simulations of the full active nematic hydrodynamics and provides a framework for understanding some of the mechanisms for selection of the nonequilibrium patterns in the language of equilibrium critical phenomena.

Classical Nucleation Theory Description of Active Colloid Assembly

Nonaligning self-propelled particles with purely repulsive excluded volume interactions undergo athermal motility-induced phase separation into a dilute gas and a dense cluster phase. Here, we use enhanced sampling computational methods and analytic theory to examine the kinetics of formation of the dense phase. Despite the intrinsically nonequilibrium nature of the phase transition, we show that the kinetics can be described using an approach analogous to equilibrium classical nucleation theory, governed by an effective free energy of cluster formation with identifiable bulk and surface terms. The theory captures the location of the binodal, nucleation rates as a function of supersaturation, and the cluster size distributions below the binodal, while discrepancies in the metastable region reveal additional physics about the early stages of active crystal formation. The success of the theory shows that a framework similar to equilibrium thermodynamics can be obtained directly from the microdynamics of an active system, and can be used to describe the kinetics of evolution toward nonequilibrium steady states.

Correlation lengths in hydrodynamic models of active nematics

We examine the scaling with activity of the emergent length scales that control the nonequilibrium dynamics of an active nematic liquid crystal, using two popular hydrodynamic models that have been employed in previous studies. In both models we find that the chaotic spatio-temporal dynamics in the regime of fully developed active turbulence is controlled by a single active scale determined by the balance of active and elastic stresses, regardless of whether the active stress is extensile or contractile in nature. The observed scaling of the kinetic energy and enstrophy with activity is consistent with our single-length scale argument and simple dimensional analysis. Our results provide a unified understanding of apparent discrepancies in the previous literature and demonstrate that the essential physics is robust to the choice of model.

Defect-Mediated Morphologies in Growing Cell Colonies

Morphological trends in growing colonies of living cells are at the core of physiological and evolutionary processes. Using active gel equations, which include cell division, we show that shape changes during the growth can be regulated by the dynamics of topological defects in the orientation of cells. The friction between the dividing cells and underlying substrate drives anisotropic colony shapes toward more isotropic morphologies, by mediating the number density and velocity of topological defects. We show that the defects interact with the interface at a specific interaction range, set by the vorticity length scale of flows within the colony, and that the cells predominantly reorient parallel to the interface due to division-induced active stresses.

Instabilities, defects, and defect ordering in an overdamped active nematic

We consider a phenomenological continuum theory for an extensile, overdamped active nematic liquid crystal, applicable in the dense regime. Constructed from general principles, the theory is universal, with parameters independent of any particular microscopic realization. We show that it exhibits two distinct instabilities, one of which arises due to shear forces, and the other due to active torques. Both lead to the proliferation of defects. We focus on the active torque bend instability and find three distinct nonequilibrium steady states including a defect-ordered nematic in which +½ disclinations develop polar ordering. We characterize the phenomenology of these phases and identify the relationship of this theoretical description to experimental realizations and other theoretical models of active nematics.

Control of active liquid crystals with a magnetic field

Living cells sense the mechanical features of their environment and adapt to it by actively remodeling their peripheral network of filamentary proteins, known as cortical cytoskeleton. By mimicking this principle, we demonstrate an effective control strategy for a microtubule-based active nematic in contact with a hydrophobic thermotropic liquid crystal. By using well-established protocols for the orientation of liquid crystals with a uniform magnetic field, and through the mediation of anisotropic shear stresses, the active nematic reversibly self-assembles with aligned flows and textures that feature orientational order at the millimeter scale. The turbulent flow, characteristic of active nematics, is in this way regularized into a laminar flow with periodic velocity oscillations. Once patterned, the microtubule assembly reveals its intrinsic length and time scales, which we correlate with the activity of motor proteins, as predicted by existing theories of active nematics. The demonstrated commanding strategy should be compatible with other viable active biomaterials at interfaces, and we envision its use to probe the mechanics of the intracellular matrix.

Emergent self-organization in active materials

Biological systems exhibit large-scale self-organized dynamics and structures which enable organisms to perform the functions of life. The field of active matter strives to develop and understand microscopically driven nonequilibrium materials, with emergent properties comparable to those of living systems. This review will describe two recently developed classes of active matter systems, in which simple building blocks-self-propelled colloidal particles or extensile rod-like particles-self-organize to form macroscopic structures with features not possible in equilibrium systems. We summarize the recent experimental and theoretical progress on each of these systems, and we present simple descriptions of the physics underlying their emergent behaviors.

Multiscale modeling and simulation of microtubule-motor-protein assemblies

Microtubules and motor proteins self-organize into biologically important assemblies including the mitotic spindle and the centrosomal microtubule array. Outside of cells, microtubule-motor mixtures can form novel active liquid-crystalline materials driven out of equilibrium by adenosine triphosphate-consuming motor proteins. Microscopic motor activity causes polarity-dependent interactions between motor proteins and microtubules, but how these interactions yield larger-scale dynamical behavior such as complex flows and defect dynamics is not well understood. We develop a multiscale theory for microtubule-motor systems in which Brownian dynamics simulations of polar microtubules driven by motors are used to study microscopic organization and stresses created by motor-mediated microtubule interactions. We identify polarity-sorting and crosslink tether relaxation as two polar-specific sources of active destabilizing stress. We then develop a continuum Doi-Onsager model that captures polarity sorting and the hydrodynamic flows generated by these polar-specific active stresses. In simulations of active nematic flows on immersed surfaces, the active stresses drive turbulent flow dynamics and continuous generation and annihilation of disclination defects. The dynamics follow from two instabilities, and accounting for the immersed nature of the experiment yields unambiguous characteristic length and time scales. When turning off the hydrodynamics in the Doi-Onsager model, we capture formation of polar lanes as observed in the Brownian dynamics simulation.

Kinetic attractor phase diagrams of active nematic suspensions: the dilute regime

Large-scale simulations by the authors of the kinetic-hydrodynamic equations for active polar nematics revealed a variety of spatio-temporal attractors, including steady and unsteady, banded (1d) and cellular (2d) spatial patterns. These particle scale activation-induced attractors arise at dilute nanorod volume fractions where the passive equilibrium phase is isotropic, whereas all previous model simulations have focused on the semi-dilute, nematic equilibrium regime and mostly on low-moment orientation tensor and polarity vector models. Here we extend our previous results to complete attractor phase diagrams for active nematics, with and without an explicit polar potential, to map out novel spatial and dynamic transitions, and to identify some new attractors, over the parameter space of dilute nanorod volume fraction and nanorod activation strength. The particle-scale activation parameter corresponds experimentally to a tunable force dipole strength (so-called pushers with propulsion from the rod tail) generated by active rod macromolecules, e.g., catalysis with the solvent phase, ATP-induced propulsion, or light-activated propulsion. The simulations allow 2d spatial variations in all flow and orientational variables and full spherical orientational degrees of freedom; the attractors correspond to numerical integration of a coupled system of 125 nonlinear PDEs in 2d plus time. The phase diagrams with and without the polar interaction potential are remarkably similar, implying that polar interactions among the rodlike particles are not essential to long-range spatial and temporal correlations in flow, polarity, and nematic order. As a general rule, above a threshold, low volume fractions induce 1d banded patterns, whereas higher yet still dilute volume fractions yield 2d patterns. Again as a general rule, varying activation strength at fixed volume fraction induces novel dynamic transitions. First, stationary patterns saturate the instability of the isotropic state, consisting of discrete 1d banded or 2d cellular patterns depending on nanorod volume fraction. Increasing activation strength further induces a sequence of attractor bifurcations, including oscillations superimposed on the 1d and 2d stationary patterns, a uniform translational motion of 1d and 2d oscillating patterns, and periodic switching between 1d and 2d patterns. These results imply that active macromolecular suspensions are capable of long-range spatial and dynamic organization at isotropic equilibrium concentrations, provided particle-scale activation is sufficiently strong.

Physics of microswimmers--single particle motion and collective behavior: a review

Locomotion and transport of microorganisms in fluids is an essential aspect of life. Search for food, orientation toward light, spreading of off-spring, and the formation of colonies are only possible due to locomotion. Swimming at the microscale occurs at low Reynolds numbers, where fluid friction and viscosity dominates over inertia. Here, evolution achieved propulsion mechanisms, which overcome and even exploit drag. Prominent propulsion mechanisms are rotating helical flagella, exploited by many bacteria, and snake-like or whip-like motion of eukaryotic flagella, utilized by sperm and algae. For artificial microswimmers, alternative concepts to convert chemical energy or heat into directed motion can be employed, which are potentially more efficient. The dynamics of microswimmers comprises many facets, which are all required to achieve locomotion. In this article, we review the physics of locomotion of biological and synthetic microswimmers, and the collective behavior of their assemblies. Starting from individual microswimmers, we describe the various propulsion mechanism of biological and synthetic systems and address the hydrodynamic aspects of swimming. This comprises synchronization and the concerted beating of flagella and cilia. In addition, the swimming behavior next to surfaces is examined. Finally, collective and cooperate phenomena of various types of isotropic and anisotropic swimmers with and without hydrodynamic interactions are discussed.

Multiscale polar theory of microtubule and motor-protein assemblies

Microtubules and motor proteins are building blocks of self-organized subcellular biological structures such as the mitotic spindle and the centrosomal microtubule array. These same ingredients can form new "bioactive" liquid-crystalline fluids that are intrinsically out of equilibrium and which display complex flows and defect dynamics. It is not yet well understood how microscopic activity, which involves polarity-dependent interactions between motor proteins and microtubules, yields such larger-scale dynamical structures. In our multiscale theory, Brownian dynamics simulations of polar microtubule ensembles driven by cross-linking motors allow us to study microscopic organization and stresses. Polarity sorting and cross-link relaxation emerge as two polar-specific sources of active destabilizing stress. On larger length scales, our continuum Doi-Onsager theory captures the hydrodynamic flows generated by polarity-dependent active stresses. The results connect local polar structure to flow structures and defect dynamics.

Rheological signatures in limit cycle behaviour of dilute, active, polar liquid crystalline polymers in steady shear

We consider the dilute regime of active suspensions of liquid crystalline polymers (LCPs), addressing issues motivated by our kinetic model and simulations in Forest et al. (Forest et al. 2013 Soft Matter 9, 5207-5222 (doi:10.1039/c3sm27736d)). In particular, we report unsteady two-dimensional heterogeneous flow-orientation attractors for pusher nanorod swimmers at dilute concentrations where passive LCP equilibria are isotropic. These numerical limit cycles are analogous to longwave (homogeneous) tumbling and kayaking limit cycles and two-dimensional heterogeneous unsteady attractors of passive LCPs in weak imposed shear, yet these states arise exclusively at semi-dilute concentrations where stable equilibria are nematic. The results in Forest et al. mentioned above compel two studies in the dilute regime that complement recent work of Saintillan & Shelley (Saintillan & Shelley 2013 C. R. Physique 14, 497-517 (doi:10.1016/j.crhy.2013.04.001)): linearized stability analysis of the isotropic state for nanorod pushers and pullers; and an analytical-numerical study of weakly and strongly sheared active polar nanorod suspensions to capture how particle-scale activation affects shear rheology. We find that weakly sheared dilute puller versus pusher suspensions exhibit steady versus unsteady responses, shear thickening versus thinning and positive versus negative first normal stress differences. These results further establish how sheared dilute nanorod pusher suspensions exhibit many of the characteristic features of sheared semi-dilute passive nanorod suspensions.

Defect dynamics in active nematics

Topological defects are distinctive signatures of liquid crystals. They profoundly affect the viscoelastic behaviour of the fluid by constraining the orientational structure in a way that inevitably requires global changes not achievable with any set of local deformations. In active nematic liquid crystals, topological defects not only dictate the global structure of the director, but also act as local sources of motion, behaving as self-propelled particles. In this article, we present a detailed analytical and numerical study of the mechanics of topological defects in active nematic liquid crystals.

Vorticity, defects and correlations in active turbulence

We describe a numerical investigation of a continuum model of an active nematic, concentrating on the regime of active turbulence. Results are presented for the effect of three parameters, activity, elastic constant and rotational diffusion constant, on the order parameter and flow fields. Defects and distortions in the director field act as sources of vorticity, and thus vorticity is strongly correlated to the director field. In particular, the characteristic length of decay of vorticity and order parameter correlations is controlled by the defect density. By contrast, the decay of velocity correlations is determined by a balance between activity and dissipation. We highlight the role of microscopic flow generation mechanisms in determining the flow patterns and characteristic scales of active turbulence and contrast the behaviour of extensile and contractile active nematics.

Spontaneous motility of passive emulsion droplets in polar active gels

We study by computer simulations the dynamics of a droplet of passive, isotropic fluid, embedded in a polar active gel. The latter represents a fluid of active force dipoles, which exert either contractile or extensile stresses on their surroundings, modelling for instance a suspension of cytoskeletal filaments and molecular motors. When the polarisation of the active gel is anchored normal to the droplet at its surface, the nematic elasticity of the active gel drives the formation of a hedgehog defect; this defect then drives an active flow which propels the droplet forward. In an extensile gel, motility can occur even with tangential anchoring, which is compatible with a defect-free polarisation pattern. In this case, upon increasing activity the droplet first rotates uniformly, and then undergoes a discontinuous nonequilibrium transition into a translationally motile state, powered by bending deformations in the surrounding active medium.

Topological structure dynamics revealing collective evolution in active nematics

Topological defects frequently emerge in active matter like bacterial colonies, cytoskeleton extracts on substrates, self-propelled granular or colloidal layers and so on, but their dynamical properties and the relations to large-scale organization and fluctuations in these active systems are seldom touched. Here we reveal, through a simple model for active nematics using self-driven hard elliptic rods, that the excitation, annihilation and transportation of topological defects differ markedly from those in non-active media. These dynamical processes exhibit strong irreversibility in active nematics in the absence of detailed balance. Moreover, topological defects are the key factors in organizing large-scale dynamic structures and collective flows, resulting in multi-spatial temporal effects. These findings allow us to control the self-organization of active matter through topological structures.

Topological defects are observed in a range of active systems, but their dynamical properties are largely unknown. Here, the authors use a simulation of self-propelled hard-rods to generate topological defects in active nematics, finding that their anomalous dynamics may lead to large-scale collective motions.

Spontaneous division and motility in active nematic droplets

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 hydrodynamic flows. These behaviors can be selectively activated by controlling a single physical parameter, namely, an active variant of the capillary number.

Globally aligned states and hydrodynamic traffic jams in confined suspensions of active asymmetric particles

Strongly confined active liquids are subject to unique hydrodynamic interactions due to momentum screening and lubricated friction by the confining walls. Using numerical simulations, we demonstrate that two-dimensional dilute suspensions of fore-aft asymmetric polar swimmers in a Hele-Shaw geometry can exhibit a rich variety of novel phase behaviors depending on particle shape, including coherent polarized density waves with global alignment, persistent counterrotating vortices, density shocks and rarefaction waves. We also explain these phenomena using a linear stability analysis and a nonlinear traffic flow model, both derived from a mean-field kinetic theory.

Velocity correlations in an active nematic

The flow properties of a continuum model for an active nematic are studied and compared with recent experiments on suspensions of microtubule bundles and molecular motors. The velocity correlation length is found to be independent of the strength of the activity while the characteristic velocity scale increases monotonically as the activity is increased, both in agreement with the experimental observations. We interpret our results in terms of the creation and annihilation dynamics of a gas of topological defects.

Transitions through critical temperatures in nematic liquid crystals

We obtain estimates for critical nematic liquid crystal (LC) temperatures under the action of a slowly varying temperature-dependent control variable. We show that biaxiality has a negligible effect within our model and that these delay estimates are well described by a purely uniaxial model. The static theory predicts two critical temperatures: the supercooling temperature below which the isotropic phase loses stability and the superheating temperature above which the ordered nematic states do not exist. In contrast to the static problem, the isotropic phase exhibits a memory effect below the supercooling temperature in the dynamic framework. This delayed loss of stability is independent of the rate of change of temperature and depends purely on the initial value of the temperature. We also show how our results can be used to improve estimates for LC material constants.

Defect annihilation and proliferation in active nematics

Liquid crystals inevitably possess topological defect excitations generated through boundary conditions, through applied fields, or in quenches to the ordered phase. In equilibrium, pairs of defects coarsen and annihilate as the uniform ground state is approached. Here we show that defects in active liquid crystals exhibit profoundly different behavior, depending on the degree of activity and its contractile or extensile character. While contractile systems enhance the annihilation dynamics of passive systems, extensile systems act to drive defects apart so that they swarm around in the manner of topologically well-characterized self-propelled particles. We develop a simple analytical model for the defect dynamics which reproduces the key features of both the numerical solutions and recent experiments on microtubule-kinesin assemblies.

Long-range ordering of vibrated polar disks

Vibrated polar disks have been used experimentally to investigate collective motion of driven particles, where fully ordered asymptotic regimes could not be reached. Here we present a model reproducing quantitatively the single, binary, and collective properties of this granular system. Using system sizes not accessible in the laboratory, we show in silico that true long-range order is possible in the experimental system. Exploring the model's parameter space, we find a phase diagram qualitatively different from that of dilute or pointlike particle systems.

Swarming, swirling and stasis in sequestered bristle-bots

The collective ability of organisms to move coherently in space and time is ubiquitous in any group of autonomous agents that can move and sense each other and the environment. Here, we investigate the origin of collective motion and its loss using macroscopic self-propelled bristle-bots, simple automata made from a toothbrush and powered by an onboard cell phone vibrator-motor, that can sense each other through shape-dependent local interactions, and can also sense the environment non-locally via the effects of confinement and substrate topography. We show that when bristle-bots are confined to a limited arena with a soft boundary, increasing the density drives a transition from a disordered and uncoordinated motion to organized collective motion either as a swirling cluster or a collective dynamical stasis. This transition is regulated by a single parameter, the relative magnitude of spinning and walking in a single automaton. We explain this using quantitative experiments and simulations that emphasize the role of the agent shape, environment and confinement via boundaries. Our study shows how the behavioural repertoire of these physically interacting automatons controlled by one parameter translates into the mechanical intelligence of swarms.

Active chiral processes in thin films

We develop a generic description of thin active films that captures key features of flow and rotation patterns emerging from the activity of chiral motors which introduce torque dipoles. We highlight the role of the spin rotation field and show that fluid flows can occur in two ways: by coupling of the spin rotation rate to the velocity field via a surface or by spatial gradients of the spin rotation rate. We discuss our results in the context of patches of bacteria on solid surfaces and groups of rotating cilia. Our theory could apply to active chiral processes in the cell cytoskeleton and in epithelia.

Liquid crystal microfluidics for tunable flow shaping

We explore the flow of a nematic liquid crystal in microfluidic channels with a rectangular cross section through experiments and numerical modeling. The flow profile and the liquid crystal orientational profile show three distinct regimes of weak, medium, and strong flow as the driving pressure is varied. These are identified by comparing polarizing optical microscopy experiments and numerical solutions of the nematofluidic equations of motion. The relative stability of the regimes is related to the de Gennes characteristic shear-flow lengths e(1) and e(2), together with the channel's aspect ratio w/d. Finally, we show that the liquid crystalline microfluidic flow can be fully steered from left to right of a simple microchannel by applying transverse temperature gradients.