Pair creation, motion, and annihilation of topological defects in two-dimensional nematic liquid crystals

Topological defects in multi-layered swarming bacteria

Topological defects, which are singular points in a director field, play a major role in shaping active systems. Here, we experimentally study topological defects and the flow patterns around them, that are formed during the highly rapid dynamics of swarming bacteria. The results are compared to the predictions of two-dimensional active nematics. We show that, even though some of the assumptions underlying the theory do not hold, the swarm dynamics is in agreement with two-dimensional nematic theory. In particular, we look into the multi-layered structure of the swarm, which is an important feature of real, natural colonies, and find a strong coupling between layers. Our results suggest that the defect-charge density is hyperuniform, i.e., that long range density-fluctuations are suppressed.

Hierarchical defect-induced condensation in active nematics

Topological defects play a central role in the formation and organization of various biological systems. Historically, such nonequilibrium defects have been mainly studied in the context of homogeneous active nematics. Phase-separated systems, in turn, are known to form dense and dynamic nematic bands, but typically lack topological defects. In this paper, we use agent-based simulations of weakly aligning, self-propelled polymers and demonstrate that contrary to the existing paradigm phase-separated active nematics form -1/2 defects. Moreover, these defects, emerging due to interactions among dense nematic bands, constitute a novel second-order collective state. We investigate the morphology of defects in detail and find that their cores correspond to a strong increase in density, associated with a condensation of nematic fluxes. Unlike their analogs in homogeneous systems, such condensed defects form and decay in a different way and do not involve positively charged partners. We additionally observe and characterize lateral arc-like structures that separate from a band's bulk and move in transverse direction. We show that the key control parameters defining the route from stable bands to the coexistence of dynamic lanes and defects are the total density of particles and their path persistence length. We introduce a hydrodynamic theory that qualitatively recapitulates all the main features of the agent-based model, and use it to show that the emergence of both defects and arcs can be attributed to the same anisotropic active fluxes. Finally, we present a way to artificially engineer and position defects, and speculate about experimental verification of the provided model.

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.

Defect loops in three-dimensional active nematics as active multipoles

We develop a description of defect loops in three-dimensional active nematics based on a multipole expansion of the far-field director and show how this leads to a self-dynamics dependent on the loop's geometric type. The dipole term leads to active stresses that generate a global self-propulsion for splay and bend loops. The quadrupole moment is nonzero only for nonplanar loops and generates a net "active torque," such that defect loops are both self-motile and self-orienting. Our analysis identifies right- and left-handed twist loops as the only force- and torque-free geometries, suggesting a mechanism for generating an excess of twist loops. Finally, we determine the Stokesian flows created by defect loops and describe qualitatively their hydrodynamics.

Active Extensile Stress Promotes 3D Director Orientations and Flows

We use numerical simulations and linear stability analysis to study an active nematic layer where the director is allowed to point out of the plane. Our results highlight the difference between extensile and contractile systems. Contractile stress suppresses the flows perpendicular to the layer and favors in-plane orientations of the director. By contrast extensile stress promotes instabilities that can turn the director out of the plane, leaving behind a population of distinct, in-plane regions that continually elongate and divide. This supports extensile forces as a mechanism for the initial stages of layer formation in living systems, and we show that a planar drop with extensile (contractile) activity grows into three dimensions (remains in two dimensions). The results also explain the propensity of disclination lines in three dimensional active nematics to be of twist type in extensile or wedge type in contractile materials.

Orientational Correlations in Active and Passive Nematic Defects

We investigate the emergence of orientational order among +1/2 disclinations in active nematic liquid crystals. Using a combination of theoretical and experimental methods, we show that +1/2 disclinations have short-range antiferromagnetic alignment, as a consequence of the elastic torques originating from their polar structure. The presence of intermediate -1/2 disclinations, however, turns this interaction from antialigning to aligning at scales that are smaller than the typical distance between like-sign defects. No long-range orientational order is observed. Strikingly, these effects are insensitive to material properties and qualitatively similar to what is found for defects in passive nematic liquid crystals.

Emergent spatiotemporal population dynamics with cell-length control of synthetic microbial consortia

The increased complexity of synthetic microbial biocircuits highlights the need for distributed cell functionality due to concomitant increases in metabolic and regulatory burdens imposed on single-strain topologies. Distributed systems, however, introduce additional challenges since consortium composition and spatiotemporal dynamics of constituent strains must be robustly controlled to achieve desired circuit behaviors. Here, we address these challenges with a modeling-based investigation of emergent spatiotemporal population dynamics using cell-length control in monolayer, two-strain bacterial consortia. We demonstrate that with dynamic control of a strain's division length, nematic cell alignment in close-packed monolayers can be destabilized. We find that this destabilization confers an emergent, competitive advantage to smaller-length strains-but by mechanisms that differ depending on the spatial patterns of the population. We used complementary modeling approaches to elucidate underlying mechanisms: an agent-based model to simulate detailed mechanical and signaling interactions between the competing strains, and a reductive, stochastic lattice model to represent cell-cell interactions with a single rotational parameter. Our modeling suggests that spatial strain-fraction oscillations can be generated when cell-length control is coupled to quorum-sensing signaling in negative feedback topologies. Our research employs novel methods of population control and points the way to programming strain fraction dynamics in consortial synthetic biology.

Design of nematic liquid crystals to control microscale dynamics

The dynamics of small particles, both living such as swimming bacteria and inanimate, such as colloidal spheres, has fascinated scientists for centuries. If one could learn how to control and streamline their chaotic motion, that would open technological opportunities in the transformation of stored or environmental energy into systematic motion, with applications in micro-robotics, transport of matter, guided morphogenesis. This review presents an approach to command microscale dynamics by replacing an isotropic medium with a liquid crystal. Orientational order and associated properties, such as elasticity, surface anchoring, and bulk anisotropy, enable new dynamic effects, ranging from the appearance and propagation of particle-like solitary waves to self-locomotion of an active droplet. By using photoalignment, the liquid crystal can be patterned into predesigned structures. In the presence of the electric field, these patterns enable the transport of solid and fluid particles through nonlinear electrokinetics rooted in anisotropy of conductivity and permittivity. Director patterns command the dynamics of swimming bacteria, guiding their trajectories, polarity of swimming, and distribution in space. This guidance is of a higher level of complexity than a simple following of the director by rod-like microorganisms. Namely, the director gradients mediate hydrodynamic interactions of bacteria to produce an active force and collective polar modes of swimming. The patterned director could also be engraved in a liquid crystal elastomer. When an elastomer coating is activated by heat or light, these patterns produce a deterministic surface topography. The director gradients define an activation force that shapes the elastomer in a manner similar to the active stresses triggering flows in active nematics. The patterned elastomer substrates could be used to define the orientation of cells in living tissues. The liquid-crystal guidance holds a major promise in achieving the goal of commanding microscale active flows.

Memory effects, arches and polar defect ordering at the cross-over from wet to dry active nematics

We use analytic arguments and numerical solutions of the continuum, active nematohydrodynamic equations to study how friction alters the behaviour of active nematics. Concentrating on the case where there is nematic ordering in the passive limit, we show that, as the friction is increased, memory effects become more prominent and +1/2 topological defects leave increasingly persistent trails in the director field as they pass. The trails are preferential sites for defect formation and they tend to impose polar order on any new +1/2 defects. In the absence of noise and for high friction, it becomes very difficult to create defects, but trails formed by any defects present at the beginning of the simulations persist and organise into parallel arch-like patterns in the director field. We show aligned arches of equal width are approximate steady state solutions of the equations of motion which co-exist with the nematic state. We compare our results to other models in the literature, in particular dry systems with no hydrodynamics, where trails, arches and polar defect ordering have also been observed.

Morphological transformation in the supramolecular assembly of discotic liquid crystal molecules using silver nanoparticles and its sensing application

Triphenylene based discotic liquid crystal (TP) molecules are rich in π-electrons which facilitate π-stacking interaction of the molcules leading to formation of one dimensional nanowires. These nanowires can assemble to form nanoribbons due to a lateral cohesive force among the nanowires. The flat nanoribons undergo a morphological transformation due to incorporation of silver nanoparticles (SNP) into the matrix of TP molecules. The presence of SNP induces a chiral twisting to the nanoribbons and therefore the flat nanoribbons transform into a helical nanoribbon structure. The global chiral structure exhibited by the composition of achiral constituents is due to the creation of topological defects like disclination and dislocation. These defects can lead to a geometrical frustration in the nanoribbons which relaxes with the formation of twisted helical nanoribbons. A minor change in morphology of the supramolecular assembly can have a remarkable effect on the physicochemical properties of the nanoribbons. In this article, we demonstrate that even a minor change in the geometry of aliphatic chains on the surface of nanoribbons can be employed for sensing organic solvents such as acetone and ethanol. The sensing was performed at room temperature. Relative humidity has no effect on the sensing response.

Dynamics of active nematic defects on the surface of a sphere

A nematic liquid crystal confined to the surface of a sphere exhibits topological defects of total charge +2 due to the topological constraint. In equilibrium, the nematic field forms four +1/2 defects, located at the corners of a regular tetrahedron inscribed within the sphere, since this minimizes the Frank elastic energy. If additionally the individual nematogens exhibit self-driven directional motion, the resulting active system creates large-scale flow that drives it out of equilibrium. In particular, the defects now follow complex dynamic trajectories which, depending on the strength of the active forcing, can be periodic (for weak forcing) or chaotic (for strong forcing). In this paper we derive an effective particle theory for this system, in which the topological defects are the degrees of freedom, whose exact equations of motion we subsequently determine. Numerical solutions of these equations confirm previously observed characteristics of their dynamics and clarify the role played by the time dependence of their global rotation. We also show that Onsager's variational principle offers an exceptionally transparent way to derive these dynamical equations, and we explain the defect mobility at the hydrodynamics level.

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.

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.

Active matter invasion

Biologically active materials such as bacterial biofilms and eukaryotic cells thrive in confined micro-spaces. Here, we show through numerical simulations that confinement can serve as a mechanical guidance to achieve distinct modes of collective invasion when combined with growth dynamics and the intrinsic activity of biological materials. We assess the dynamics of the growing interface and classify these collective modes of invasion based on the activity of the constituent particles of the growing matter. While at small and moderate activities the active material grows as a coherent unit, we find that blobs of active material collectively detach from the cohort above a well-defined activity threshold. We further characterise the mechanical mechanisms underlying the crossovers between different modes of invasion and quantify their impact on the overall invasion speed.

Activity Driven Orientational Order in Active Nematic Liquid Crystals on an Anisotropic Substrate

We investigate the effect of an anisotropic substrate on the turbulent dynamics of a simulated two-dimensional active nematic. This is introduced as an anisotropic friction and an effective anisotropic viscosity, with the orientation of the anisotropy being defined by the substrate. In this system, we observe the emergence of global nematic order of topological defects that is controlled by the degree of anisotropy in the viscosity and the magnitude of the active stress. No global defect alignment is seen in passive liquid crystals with anisotropic viscosity or friction confirming that ordering is driven by the active stress. We then closely examine the active flow generated by a single defect to show that the net kinetic energy of the flow is dependent on the orientation of the defect relative to the substrate, resulting in a torque on the defect to align it with the anisotropy in the substrate.

Self-organized dynamics and the transition to turbulence of confined active nematics

We study how confinement transforms the chaotic dynamics of bulk microtubule-based active nematics into regular spatiotemporal patterns. For weak confinements in disks, multiple continuously nucleating and annihilating topological defects self-organize into persistent circular flows of either handedness. Increasing confinement strength leads to the emergence of distinct dynamics, in which the slow periodic nucleation of topological defects at the boundary is superimposed onto a fast procession of a pair of defects. A defect pair migrates toward the confinement core over multiple rotation cycles, while the associated nematic director field evolves from a distinct double spiral toward a nearly circularly symmetric configuration. The collapse of the defect orbits is punctuated by another boundary-localized nucleation event, that sets up long-term doubly periodic dynamics. Comparing experimental data to a theoretical model of an active nematic reveals that theory captures the fast procession of a pair of [Formula: see text] defects, but not the slow spiral transformation nor the periodic nucleation of defect pairs. Theory also fails to predict the emergence of circular flows in the weak confinement regime. The developed confinement methods are generalized to more complex geometries, providing a robust microfluidic platform for rationally engineering 2D autonomous flows.

Theory of defect motion in 2D passive and active nematic liquid crystals

The motion of topological defects is an important feature of the dynamics of all liquid crystals, and is especially conspicuous in active liquid crystals. Understanding defect motion is a challenging theoretical problem, because the dynamics of orientational order is coupled with backflow of the fluid, and because a liquid crystal has several distinct viscosity coefficients. Here, we suggest a coarse-grained, variational approach, which describes the motion of defects as effective "particles". For passive liquid crystals, the theory shows how the drag depends on defect orientation, and shows the coupling between translational and rotational motion. For active liquid crystals, the theory provides an alternative way to describe motion induced by the activity coefficient.

The interplay between activity and filament flexibility determines the emergent properties of active nematics

Active nematics are microscopically driven liquid crystals that exhibit dynamical steady states characterized by the creation and annihilation of topological defects. Motivated by differences between previous simulations of active nematics based on rigid rods and experimental realizations based on semiflexible biopolymer filaments, we describe a large-scale simulation study of a particle-based computational model that explicitly incorporates filament semiflexibility. We find that energy injected into the system at the particle scale preferentially excites bend deformations, reducing the apparent filament bend modulus. The emergent characteristics of the active nematic depend on activity and flexibility only through this activity-renormalized bend 'modulus', demonstrating that apparent values of material parameters, such as the Frank 'constants', depend on activity. Thus, phenomenological parameters within continuum hydrodynamic descriptions of active nematics must account for this dependence. Further, we present a systematic way to estimate these parameters from observations of deformation fields and defect shapes in experimental or simulation data.

Defect Unbinding in Active Nematics

We formulate the statistical dynamics of topological defects in the active nematic phase, formed in two dimensions by a collection of self-driven particles on a substrate. An important consequence of the nonequilibrium drive is the spontaneous motility of strength +1/2 disclinations. Starting from the hydrodynamic equations of active nematics, we derive an interacting particle description of defects that includes active torques. We show that activity, within perturbation theory, lowers the defect-unbinding transition temperature, determining a critical line in the temperature-activity plane that separates the quasi-long-range ordered (nematic) and disordered (isotropic) phases. Below a critical activity, defects remain bound as rotational noise decorrelates the directed dynamics of +1/2 defects, stabilizing the quasi-long-range ordered nematic state. This activity threshold vanishes at low temperature, leading to a reentrant transition. At large enough activity, active forces always exceed thermal ones and the perturbative result fails, suggesting that in this regime activity will always disorder the system. Crucially, rotational diffusion being a two-dimensional phenomenon, defect unbinding cannot be described by a simplified one-dimensional model.

Twist-induced crossover from two-dimensional to three-dimensional turbulence in active nematics

While studies of active nematics in two dimensions have shed light on various aspects of the flow regimes and topology of active matter, three-dimensional properties of topological defects and chaotic flows remain unexplored. By confining a film of active nematics between two parallel plates, we use continuum simulations and analytical arguments to demonstrate that the crossover from quasi-two-dimensional (quasi-2D) to three-dimensional (3D) chaotic flows is controlled by the morphology of the disclination lines. For small plate separations, the active nematic behaves as a quasi-2D material, with straight topological disclination lines spanning the height of the channel and exhibiting effectively 2D active turbulence. Upon increasing channel height, we find a crossover to 3D chaotic flows due to the contortion of disclinations above a critical activity. Above this critical activity highly contorted disclination lines and disclination loops are formed. We further show that these contortions are engendered by twist perturbations producing a sharp change in the curvature of disclinations.