Experiments have recently shown the feasibility of utilising bacteria as micro-scale robotic devices, with special attention paid to the development of bacteria-driven micro-swimmers taking advantage of built-in actuation and sensing mechanisms of cells. Here we propose a stochastic fluid dynamic model to describe analytically and computationally the dynamics of microscopic particles driven by the motion of surface-attached bacteria undergoing run-and-tumble motion. We compute analytical expressions for the rotational diffusion coefficient, the swimming speed and the effective diffusion coefficient. At short times, the mean squared displacement (MSD) is proportional to the square of the swimming speed, which is independent of the particle size (for fixed density of attached bacteria) and scales linearly with the number of attached bacteria; in contrast, at long times the MSD scales quadratically with the size of the swimmer and is independent of the number of bacteria. We then extend our result to the situation where the surface-attached bacteria undergo chemotaxis within the linear response regime. We demonstrate that bacteria-driven particles are capable of performing artificial chemotaxis, with a chemotactic drift velocity linear in the chemical concentration gradient and independent of the size of the particle. Our results are validated against numerical simulations in the Brownian dynamics limit and will be relevant to the optimal design of micro-swimmers for biomedical applications.
Bacteria commonly live in structured communities that affect human health and influence ecological systems. Heterogeneous populations, such as motile and non-motile populations, often coexist in bacteria communities. Motile subpopulations in microbial communities are believed to be important to dispersal, quest for food, and material transport. However, except in circumstances where motile cells drive colony expansion (e.g. bacterial swarming), the physiological functions of motile subpopulations in bacterial communities are largely unclear. Here we discovered that motile cells in routinely cu…
We study a synthetic system of motile Escherichia coli bacteria encapsulated inside giant lipid vesicles. Forces exerted by the bacteria on the inner side of the membrane are sufficient to extrude membrane tubes filled with one or several bacteria. We show that a physical coupling between the membrane tube and the flagella of the enclosed cells transforms the tube into an effective helical flagellum propelling the vesicle. We develop a simple theoretical model to estimate the propulsive force from the speed of the vesicles, and demonstrate the good efficiency of this coupling mechanism. Togeth…
Flagellated bacteria exploiting helical propulsion are known to swim along circular trajectories near surfaces. Fluid dynamics predicts this circular motion to be clockwise (CW) above a rigid surface (when viewed from inside the fluid) and counter-clockwise (CCW) below a free surface. Recent experimental investigations showed that complex physicochemical processes at the nearby surface could lead to a change in the direction of rotation, both at solid surfaces absorbing slip-inducing polymers and interfaces covered with surfactants. Motivated by these results, we use a far-field hydrodynamic m…
We show, using differential dynamic microscopy, that the diffusivity of non-motile cells in a three-dimensional (3D) population of motile E. coli is enhanced by an amount proportional to the active cell flux. While non-motile mutants without flagella and mutants with paralysed flagella have quite different thermal diffusivities and therefore hydrodynamic radii, their diffusivities are enhanced to the same extent by swimmers in the regime of cell densities explored here. Integrating the advective motion of non-swimmers caused by swimmers with finite persistence-length trajectories predicts our …
We study catalytic Janus swimmers and Escherichia coli bacteria swimming in a two-dimensional colloidal crystal. The Janus swimmers orbit individual colloids and hop between colloids stochastically, with a hopping rate that varies inversely with fuel (hydrogen peroxide) concentration. At high fuel concentration, these orbits are stable for 100s of revolutions, and the orbital speed oscillates periodically as a result of hydrodynamic, and possibly also phoretic, interactions between the swimmer and the six neighbouring colloids. Motile E.~coli bacteria behave very differently in the same colloi…
We consider the tracer diffusion $D_{rr}$ that arises from the run-and-tumble motion of low Reynolds number swimmers, such as bacteria. Assuming a dilute suspension, where the bacteria move in uncorrelated runs of length $λ$, we obtain an exact expression for $D_{rr}$ for dipolar swimmers in three dimensions, hence explaining the surprising result that this is independent of $λ$. We compare $D_{rr}$ to the contribution to tracer diffusion from entrainment.
A growing body of work aims at designing and testing micron-scale synthetic swimmers. One method, inspired by the locomotion of flagellated bacteria, consists of applying a rotating magnetic field to a rigid, helically-shaped, propeller attached to a magnetic head. When the resulting device, termed an artificial bacteria flagellum, is aligned perpendicularly to the applied field, the helix rotates and the swimmer moves forward. Experimental investigation of artificial bacteria flagella shows that at low frequency of the applied field, the axis of the helix does not align perpendicularly to the…
Self-propelled microorganisms, such as unicellular algae or bacteria, swim along their director relative to the fluid velocity. Under a steady shear flow the director rotates in close orbit, a periodic structure that is preserved under an oscillatory shear flow. If the shear flow is subjected to small fluctuations produced by small irregularities in the microchannel or by other swimmers nearby, the director dynamics becomes stochastic. Numerical integration of the swimmer motion shows that there is stochastic resonance: The displacement in the vorticity direction is maximized for a finite nois…
We propose minimal models of one-, two- and three-dimensional micro-swimmers at low Reynolds number with a periodic non-reciprocal motion. These swimmers are either "pushers" or "pullers" of fluid along the swimming axis, or combination of the two, depending on the history of the swimming motion. We show this with a linear three-bead swimmer by analytically evaluating the migration speed and the strength of the dipolar flow induced by its swimming motion. It is found that the distance traveled per cycle and the dipolar flow can be obtained from an integral over the area enclosed by the traject…
Micro-swimmers such as bacteria perform random walks known as run-and-tumbles to move up chemo-attractant gradients and as a result aggregate with others. It is also known that such micro-swimmers can self-organize into macroscopic patterns due to interactions with neighboring cells through the fluidic environment they live in. While the pattern formation resulting from chemotactic and hydrodynamic interactions separately and together have been previously investigated, the effect of the anisotropy in the tumbles of micro-swimmers has been unexplored. Here we show through linear analysis and fu…
Inspired by bacterial chemotaxis we propose a hydrodynamic molecular scale hunter that can swim and find its target. The system is essentially a stochastic low Reynolds swimmer with ability to move in two dimensional space and sense the local value of the chemical concentration emitted by a target. We show that by adjusting the geometrical and dynamical variables of the swimmer we can always achieve a swimmer that can navigate and search for the region with higher concentration of a chemical emitted from a source. The system discussed here can also be considered as a theoretical framework for …
Microorganisms can preferentially orient and move along gradients of a chemo-attractant (i.e., chemotax) while colonies of many microorganisms can collectively undergo complex dynamics in response to chemo-attractants that they themselves produce. For colonies or groups of micro-swimmers we investigate how an "auto-chemotactic" response that should lead to swimmer aggregation is affected by the non-trivial fluid flows that are generated by collective swimming. For this, we consider chemotaxis models based upon a hydrodynamic theory of motile suspensions that are fully coupled to chemo-attracta…
One of the principal mechanisms by which surfaces and interfaces affect microbial life is by perturbing the hydrodynamic flows generated by swimming. By summing a recursive series of image systems we derive a numerically tractable approximation to the three-dimensional flow fields of a Stokeslet (point force) within a viscous film between a parallel no-slip surface and no-shear interface and, from this Green's function, we compute the flows produced by a force- and torque-free micro-swimmer. We also extend the exact solution of Liron & Mochon (1976) to the film geometry, which demonstrates tha…
Micro-swimmers can serve as cargo carriers that move deep inside complex flow networks. When a school collectively entrains the surrounding fluid, their transport capacity can be enhanced. This effect is quantified with good agreement between experiments with self-propelled droplets and a confined Brinkman squirmer model. The volume of liquid entrained can be much larger than the droplet itself, amplifying the effective cargo capacity over an order of magnitude, even for dilute schools. Hence, biological and engineered swimmers can efficiently transport materials into confined environments.
Small organisms (e.g., bacteria) and artificial microswimmers move due to a combination of active swimming and passive Brownian motion. Considering a simplified linear three-sphere swimmer, we study how the swimmer size regulates the interplay between self-driven and diffusive behavior at low Reynolds number. Starting from the Kirkwood-Smoluchowski equation and its corresponding Langevin equation, we derive formulas for the orientation correlation time, the mean velocity and the mean square displacement in three space dimensions. The validity of the analytical results is illustrated through nu…
Suspensions of unicellular microswimmers such as flagellated bacteria or motile algae exhibit spontaneous density heterogeneities at large enough concentrations. Based on the relative location of the biological actuation appendages i.e. flagella or cilia) microswimmers' propulsion mechanism can be classified into two categories: (i) pushers, like \textit{E. coli} bacteria or spermatozoa, that generate thrust in their rear, push fluid away from them and propel themselves forward; (ii) pullers, like the microalgae \textit{Chlamydomonas reinhardtii}, that have two flagella attached to their front…
Understanding the stochastic dynamics of tracer particles in active fluids is important for identifying the physical properties of flow generating objects such as colloids, bacteria or algae. Here, we study both analytically and numerically the scattering of a tracer particle in different types of time-dependent, hydrodynamic flow fields. Specifically, we compare the tracer motion induced by an externally driven colloid with the one generated by various self-motile, multi-sphere swimmers. Our results suggest that force-free swimmers generically induce loop-shaped tracer trajectories. The speci…
We use a three-bead-spring model to investigate the dynamics of bi-flagellate micro-swimmers near a surface. While the primary dynamics and scattering are governed by geometric-dependent direct contact, the fluid flows generated by the swimmer locomotion are important in orienting it toward or away from the surface. Flagellar noise and in particular cell spinning about the main axis help a surface-trapped swimmer escape, whereas the time a swimmer spends at the surface depends on the incident angle. The dynamics results from a nuanced interplay of direct collisions, hydrodynamics, noise and th…
Swimming bacteria in passive nematics in the form of lyotropic liquid crystals are defined as a new class of active matter known as living liquid crystals in recent studies. It has also been shown that liquid crystal solutions are promising candidates for trapping and detecting bacteria. We ask the question, can a similar class of matter be designed for background nematics which are also active? Hence, we developed a minimal model for the mixture of polar particles in active nematics. It is found that the active nematics in such a mixture are highly sensitive to the presence of polar particles…
We study the effect of magnetic field on the rheology of magnetic micro-swimmers suspensions. We use a model of a dilute suspension under simple shear and subjected to a constant magnetic field. Particle shear stress is obtained for both pusher and puller types of micro-swimmers. In the limit of low shear rate, the rheology exhibits a constant shear stress, called actuated stress, which only depends on the swimming activity of the particles. This stress is induced by the magnetic field and can be positive (brake state) or negative (motor state). In the limit of low magnetic fields, a scaling r…
We study the collective dynamics of elongated swimmers in a very thin fluid layer by devising long, filamentous, non-tumbling bacteria. The strong confinement induces weak nematic alignment upon collision, which, for large enough density of cells, gives rise to global nematic order. This homogeneous but fluctuating phase, observed on the largest experimentally-accessible scale of millimeters, exhibits the properties predicted by standard models for flocking such as the Vicsek-style model of polar particles with nematic alignment: true long-range nematic order and non-trivial giant number fluct…
The effect of crowding on the run-and-tumble dynamics of swimmers such as bacteria is studied using a discrete lattice model of mutually excluding particles that move with constant velocity along a direction that is randomized at a rate $α$. In stationary state, the system is found to break into dense clusters in which particles are trapped or stopped from moving. The characteristic size of these clusters predominantly scales as $α^{-0.5}$ both in 1D and 2D. For a range of densities, due to cooperative effects, the stopping time scales as ${\cal T}_{1d}^{0.85}$ and as ${\cal T}_{2d}^{0.8}$, wh…
We present an automated procedure for the design of optimal actuation for flagellar magnetic microswimmers based on numerical optimization. Using this method, a new magnetic actuation method is provided which allows these devices to swim significantly faster compared to the usual sinusoidal actuation. This leads to a novel swimming strategy which makes the swimmer perform a 3D figure-8 trajectory. This shows that a faster propulsion is obtained when the swimmer is allowed to go out-of-plane. This approach is experimentally validated on a scaled-up flexible swimmer.
Self-propelled particles move along circles rather than along a straight line when their driving force does not coincide with their propagation direction. Examples include confined bacteria and spermatozoa, catalytically driven nanorods, active, anisotropic colloidal particles and vibrated granulates. Using a non-Hamiltonian rate theory and computer simulations, we study the motion of a Brownian "circle swimmer" in a confining channel. A sliding mode close to the wall leads to a huge acceleration as compared to the bulk motion, which can further be enhanced by an optimal effective torque-to-fo…
Surface interactions provide a class of mechanisms which can be employed for propulsion of micro- and nanometer sized particles. We investigate the related efficiency of externally and self-propelled swimmers. A general scaling relation is derived showing that only swimmers whose size is comparable to, or smaller than, the interaction range can have appreciable efficiency. An upper bound for efficiency at maximum power is 1/2. Numerical calculations for the case of diffusiophoresis are found to be in good agreement with analytical expressions for the efficiency.
We use boundary element simulations to study the interaction of model microswimmers with a neutrally buoyant spherical particle. The ratio of the size of the particle to that of the swimmer is varied from $R^\mathrm{P} / R^\mathrm{S} \ll 1$, corresponding to swimmer--tracer scattering, to $R^\mathrm{P} / R^\mathrm{S} \gg 1$, approximately equivalent to the swimmer interacting with a fixed, flat surface. We find that details of the swimmer and particle trajectories vary for different swimmers. However, the overall characteristics of the scattering event fall into two regimes, depending on the r…
Biological systems often involve the self-assembly of basic components into complex and function- ing structures. Artificial systems that mimic such processes can provide a well-controlled setting to explore the principles involved and also synthesize useful micromachines. Our experiments show that immotile, but active, components self-assemble into two types of structure that exhibit the fundamental forms of motility: translation and rotation. Specifically, micron-scale metallic rods are designed to induce extensile surface flows in the presence of a chemical fuel; these rods interact with ea…
Using a fluid-particle dynamics approach, we numerically study the effects of hydrodynamic interactions on the collective dynamics of active suspensions within a simple model for bacterial motility: each microorganism is modeled as a stroke-averaged dumbbell swimmer with prescribed dipolar force pairs. Using both simulations and qualitative arguments, we show that, when the separation between swimmers is comparable to their size, the swimmers' motions are strongly affected by activity-induced hydrodynamic forces. To further understand these effects, we investigate semidilute suspensions of swi…
This article presents a computational approach for determining the optimal slip velocities on any given shape of an axisymmetric micro-swimmer suspended in a viscous fluid. The objective is to minimize the power loss to maintain a target swimming speed, or equivalently to maximize the efficiency of the micro-swimmer. Owing to the linearity of the Stokes equations governing the fluid motion, we show that this PDE-constrained optimization problem reduces to a simpler quadratic optimization problem, whose solution is found using a high-order accurate boundary integral method. We consider various …
Active (i.e., self-propelled or swimming) particles moving through an isotropic fluid exhibit conventional diffusive behavior. We report anomalous diffusion of an active particle moving in an anisotropic, nematic background. Whilst the translational motion parallel to the nematic director shows ballistic behavior, the long-time transverse motion is super-diffusive, with an anomalous scaling $\propto t \ln t$ of the mean squared displacement with time $t$. This behavior is predicted by an analytical theory that we present here, and is corroborated by numerical simulation of active particle diff…
Magnetically driven artificial microswimmers have the potential to revolutionize many biomedical technologies, such as minimally-invasive microsurgery, micro-particle manipulation, and localized drug delivery. However, many of these applications will require the controlled dynamics of teams of these micro-robots with minimal feedback. In this work, we study the motion and fluid dynamics produced by groups of artificial microswimmers driven by a torque induced through a uniform, rotating magnetic field. Through Stokesian dynamics simulations, we show that the swimmer motion produces a rotationa…
The swimming of a pair of spherical bladders that change their volumes and mutual distance is efficient at low Reynolds numbers and is superior to other models of artificial swimmers. The change of shape resembles the wriggling motion known as {\it metaboly} of certain protozoa.
Collective behaviour in suspensions of microswimmers is often dominated by the impact of long-ranged hydrodynamic interactions. These phenomena include active turbulence, where suspensions of pusher bacteria at sufficient densities exhibit large-scale, chaotic flows. To study this collective phenomenon, we use large-scale (up to $N=3\times 10^6$) particle-resolved lattice Boltzmann simulations of model microswimmers described by extended stresslets. Such system sizes enable us to obtain quantitative information about both the transition to active turbulence and characteristic features of the t…
Fission and fusion processes of particles clusters occur in many areas of physics and chemistry from subnuclear to astronomic length scales. Here we study fission and fusion of magnetic microswimmer clusters as governed by their hydrodynamic and dipolar interactions. Rich scenarios are found which depend crucially on whether the swimmer is a pusher or a puller. In particular a linear magnetic chain of pullers is stable while a pusher chain shows a cascade of fission (or disassembly) processes as the self-propulsion velocity is increased. Contrarily, magnetic ring clusters show fission for any …
Small objects can swim by generating around them fields or gradients which in turn induce fluid motion past their surface by phoretic surface effects. We quantify for arbitrary swimmer shapes and surface patterns, how efficient swimming requires both surface ``activity'' to generate the fields, and surface ``phoretic mobility.'' We show in particular that (i) swimming requires symmetry breaking in either or both of the patterns of "activity" and ``mobility,'' and (ii) for a given geometrical shape and surface pattern, the swimming velocity is size-independent. In addition, for given available …
Marine microorganisms must cope with complex flow patterns and even turbulence as they navigate the ocean. To survive they must avoid predation and find efficient energy sources. A major difficulty in analysing possible survival strategies is that the time series of environmental cues in non-linear flow is complex, and that it depends on the decisions taken by the organism. One way of determining and evaluating optimal strategies is reinforcement learning. In a proof-of-principle study, Colabrese et al. [Phys. Rev. Lett. (2017)] used this method to find out how a micro-swimmer in a vortex flow…
We study the shear induced migration of microswimmers (primarily, active Brownian particles or ABP's) in a plane Poiseuille flow. For wide channels characterized by $U_b/HD_r \ll 1$, the separation between time scales characterizing the swimmer orientation dynamics (of O($D^{-1}_r$)) and those that characterize migration across the channel (of O($H^{2}D_r/U^{2}_b$)), allows for use of the method of multiple scales to derive a drift-diffusion equation for the swimmer concentration profile; here, $U_b$ is the swimming speed, $H$ is the channel half-width, and $D_r$ is the swimmer rotary diffusiv…
Cooperation is vital for the survival of a swarm$^1$. Large scale cooperation allows murmuring starlings to outmaneuver preying falcons$^2$, shoaling sardines to outsmart sea lions$^3$, and homo sapiens to outlive their Pleistocene peers$^4$. On the micron-scale, bacterial colonies show excellent resilience thanks to the individuals' ability to cooperate even when densely packed, mitigating their internal flow pattern to mix nutrients, fence the immune system, and resist antibiotics$^{5-14}$. Production of an artificial swarm on the micro-scale faces a serious challenge $\frac{\;\;}{\;\;}$ whi…
We explore the self-assembly process of colloidal structures immersed in active baths. By considering low-valence particles we numerically investigate the irreversible aggregation dynamics originated by the presence of run-and-tumble swimmers. We observe the formation of long closed chains -- vesicles -- densely filled by active swimmers. On the one hand the active bath drives the self-assembly of closed colloidal structures, and on the other hand the vesicles formation fosters the self-trapping of swimmers, suggesting new ways both to build structured nanomaterials and to trap microorganisms.
The presence of obstacles is intuitively expected to hinder the diffusive transport of micro-swimmers. However, for chiral micro-swimmers, a low density of obstacles near a surface can enhance their diffusive behavior, due to the rectification of the chiral motion by the obstacles. Here, we study numerically the role that disorder plays in determining the transport dynamics of chiral micro-swimmers on surfaces with obstacles. We consider different densities of regularly spaced obstacles and distinct types of disorder: noise in the dynamics of the micro-swimmer, quenched noise in the positions …
We study the effect of a nearby planar wall on the propulsion of a spherical phoretic micro-swimmer driven by reactions on its surface. An asymmetric coverage of catalysts on its surface which absorb reactants and generate products gives rise to an anisotropic interfacial flow that propels the swimmer. We analyse the near-wall dynamics of such a self-phoretic swimmer as a function of the asymmetric catalytic coverage of the surface. By an analysis of the fundamental singularities of the flow and concentration or electrostatic potential gradients generated we are able to obtain and rationalise …
Recently, the Brownian dynamics of self-propelled (active) rod-like particles was explored to model the motion of colloidal microswimmers, catalytically-driven nanorods, and bacteria. Here, we generalize this description to biaxial particles with arbitrary shape and derive the corresponding Langevin equation for a self-propelled Brownian spinning top. The biaxial swimmer is exposed to a hydrodynamic Stokes friction force at low Reynolds numbers, to fluctuating random forces and torques as well as to an external and an internal (effective) force and torque. The latter quantities control its sel…
The run-and-tumble (RT) dynamics followed by bacterial swimmers gives rise first to a ballistic motion due to their persistence, and later, through consecutive tumbles, to a diffusive process. Here we investigate how long it takes for a dilute swimmer suspension to reach the diffusive regime as well as what is the amplitude of the deviations from the diffusive dynamics. A linear time dependence of the mean-squared displacement (MSD) is insufficient to characterize diffusion and thus we also focus on the excess kurtosis of the displacement distribution. Four swimming strategies are considered: …
A simple mean-field micro-swimmer model is presented. The model is inspired by the nonequilibrium thermodynamics of multi-component fluids that undergo chemical reactions. These thermodynamics can be rigorously described in the context of the GENERIC (general equation for the nonequilibrium reversible-irreversible coupling) framework. More specifically this approach was recently applied to non-ideal polymer solutions (Indei and Schieber, J. Chem. Phys, 146, 184902, 2017). One of the species of the solution is an unreactive polymer chain represented by the bead-spring model. Using this detailed…
Artificial microswimmers, nano and microrobots, are essential in many applications from engineering to biology and medicine. We present a Stokesian Dynamics study of the dynamical properties and efficiency of one of the simplest artificial swimmer, the three linked spheres swimmer (TLS), extensively shown to be an excellent and model example of a deformable micromachine. Results for two different swimming strokes are compared with an approximate solution based on point force interactions. While this approximation accurately reproduces the solutions for swimmers with long arms and strokes of sm…
We report an experimental study on ion-exchange based modular micro-swimmers in low-salt water. Cationic ion-exchange particles and passive cargo particles assemble into self-propelling complexes, showing self-propulsion at speeds of several microns per second over extended distances and times. We quantify the assembly and speed of the complexes for different combinations of ion exchange particles and cargo particles, substrate types, salt types and concentrations, and cell geometries. Irrespective of experimental boundary conditions, we observe a regular development of the assembly shape with…
We use moving light patterns to control the motion of {\it Escherichia coli} bacteria whose motility is photo-activated. Varying the pattern speed controls the magnitude and direction of the bacterial flux, and therefore the accumulation of cells in up- and down-stream reservoirs. We validate our results with two-dimensional simulations and a 1-dimensional analytic model, and use these to explore parameter space. We find that cell accumulation is controlled by a competition between directed flux and undirected, stochastic transport. We articulate design principles for using moving light patter…
The majority of studies on self-propelled particles and microswimmers concentrates on objects that do not feature a deterministic bending of their trajectory. However, perfect axial symmetry is hardly found in reality, and shape-asymmetric active microswimmers tend to show a persistent curvature of their trajectories. Consequently, we here present a particle-scale statistical approach of circle-swimmer suspensions in terms of a dynamical density functional theory. It is based on a minimal microswimmer model and, particularly, includes hydrodynamic interactions between the swimmers. After deriv…
We report simulations of a spherical Janus particle undergoing exothermic surface reactions around one pole only. Our model excludes self-phoretic transport by design. Nevertheless, net motion occurs from direct momentum transfer between solvent and colloid, with speed scaling as the square root of the energy released during the reaction. We find that such propulsion is dominated by the system's short-time response, when neither the time dependence of the flow around the colloid nor the solvent compressibility can be ignored. Our simulations agree reasonably well with previous experiments.
We present a detailed numerical study of a microscopic artificial swimmer realized recently by Dreyfus et al. in experiments [R. Dreyfus et al., Nature 437, 862 (2005)]. It consists of an elastic filament composed of superparamagnetic particles that are linked together by DNA strands. Attached to a load particle, the resulting swimmer is actuated by an oscillating external magnetic field so that it performs a non-reciprocal motion in order to move forward. We model the superparamagnetic filament by a bead-spring configuration that resists bending like a rigid rod and whose beads experience fri…
We studied a collection of chiral active particles (CAP) on a two dimensional substrate using extensive numerical study. Particles interact through soft repulsive interaction. The activity and chirality of particles is tuned by varying their self-propulsion speed and angular velocity respectively. Kinetics and steady state properties of particles are studied for different chirality and activity. The phase diagram of system on the plane of activity and chirality shows three distinct phases. For small chirality when activity is dominant, particles show enhanced dynamics and macroscopic phase sep…
