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

Colloidal Brazil nut effect in microswimmer mixtures induced by motility contrast

We numerically and experimentally study the segregation dynamics in a binary mixture of microswimmers which move on a two-dimensional substrate in a static periodic triangular-like light intensity field. The motility of the active particles is proportional to the imposed light intensity, and they possess a motility contrast, i.e., the prefactor depends on the species. In addition, the active particles also experience a torque aligning their motion towards the direction of the negative intensity gradient. We find a segregation of active particles near the intensity minima where typically one species is localized close to the minimum and the other one is centered around in an outer shell. For a very strong aligning torque, there is an exact mapping onto an equilibrium system in an effective external potential that is minimal at the intensity minima. This external potential is similar to (height-dependent) gravity such that one can define effective "heaviness" of the self-propelled particles. In analogy to shaken granular matter in gravity, we define a "colloidal Brazil nut effect" if the heavier particles are floating on top of the lighter ones. Using extensive Brownian dynamics simulations, we identify system parameters for the active colloidal Brazil nut effect to occur and explain it based on a generalized Archimedes' principle within the effective equilibrium model: heavy particles are levitated in a dense fluid of lighter particles if their effective mass density is lower than that of the surrounding fluid. We also perform real-space experiments on light-activated self-propelled colloidal mixtures which confirm the theoretical predictions.

Active microswimmers in a finite two dimensional trap: The role of hydrodynamic interaction

We investigate the dynamics of two identical artificial active particles suspended in a free-standing fluid film with a trap of finite radius in an acoustic tweezer. In the two dimensional Oseen approximation, their hydrodynamic coupling is long ranged, which naturally raises the question as under what conditions they can simultaneously reside in the trap. We determine a critical value of the hydrodynamic coupling below which that happens and study the ensuing active pair dynamics inside the trap. For larger couplings, only one particle sits in the trap, while the other diffuses freely until it eventually replaces the particle in the trap. Such a mechanism repeats itself with a characteristic noise-dependent mean residence-retrapping time.

Active matter invasion of a viscous fluid: Unstable sheets and a no-flow theorem

We investigate the dynamics of a dilute suspension of hydrodynamically interacting motile or immotile stress-generating swimmers or particles as they invade a surrounding viscous fluid. Colonies of aligned pusher particles are shown to elongate in the direction of particle orientation and undergo a cascade of transverse concentration instabilities, governed at small times by an equation that also describes the Saffman-Taylor instability in a Hele-Shaw cell, or the Rayleigh-Taylor instability in a two-dimensional flow through a porous medium. Thin sheets of aligned pusher particles are always unstable, while sheets of aligned puller particles can either be stable (immotile particles), or unstable (motile particles) with a growth rate that is nonmonotonic in the force dipole strength. We also prove a surprising "no-flow theorem": a distribution initially isotropic in orientation loses isotropy immediately but in such a way that results in no fluid flow everywhere and for all time.

Trapping and sorting active particles: Motility-induced condensation and smectic defects

We present an experimental realization of the collective trapping phase transition [Kaiser et al., Phys. Rev. Lett. 108, 268307 (2012)PRLTAO0031-900710.1103/PhysRevLett.108.268307], using motile polar granular rods in the presence of a V-shaped obstacle. We offer a theory of this transition based on the interplay of motility-induced condensation and liquid-crystalline ordering and show that trapping occurs when persistent influx overcomes the collective expulsion of smectic defect structures. In agreement with the theory, our experiments find that a trap fills to the brim when the trap angle θ is below a threshold θ_{c}, while all particles escape for θ>θ_{c}. Our simulations support a further prediction, that θ_{c} goes down with increasing rotational noise. We exploit the sensitivity of trapping to the persistence of directed motion to sort particles based on the statistical properties of their activity.

Motility and cell shape roles in the rheology of growing bacteria cultures

Cell shape, size and self-motility appear as determinant intrinsic cell factors in the rheological behavior of living bacterial cultures during the growth process. In this work three different species were considered due to their differences on these intrinsic characteristics: two different strains of Staphylococcus aureus - strain COL and its isogenic cell wall autolysis mutant, RUSAL9 - both non-motile and Escherichia coli and Bacillus subtilis - both presenting intrinsic motility. In situ real-time rheology, was used to characterize the activity of growing bacteria, under steady-shear conditions, in particular the viscosity growth curve was measured, for a constant shear flow rate, presenting for all studied cultures, different and rich flow curves. These complex rheological behaviors are a consequence of two coupled effects: the cell density continuous increase and its changing interacting properties, where cell size and shape and intrinsic motility are major players.

Lamellar ordering, droplet formation and phase inversion in exotic active emulsions

We study numerically the behaviour of a two-dimensional mixture of a passive isotropic fluid and an active polar gel, in the presence of a surfactant favouring emulsification. Focussing on parameters for which the underlying free energy favours the lamellar phase in the passive limit, we show that the interplay between nonequilibrium and thermodynamic forces creates a range of multifarious exotic emulsions. When the active component is contractile (e.g., an actomyosin solution), moderate activity enhances the efficiency of lamellar ordering, whereas strong activity favours the creation of passive droplets within an active matrix. For extensile activity (occurring, e.g., in microtubule-motor suspensions), instead, we observe an emulsion of spontaneously rotating droplets of different size. By tuning the overall composition, we can create high internal phase emulsions, which undergo sudden phase inversion when activity is switched off. Therefore, we find that activity provides a single control parameter to design composite materials with a strikingly rich range of morphologies.

Diffusion of active particles in a complex environment: Role of surface scattering

Experiments have shown that self-propelled particles can slide along the surface of a circular obstacle without becoming trapped over long times. Using simulations and theory, we study the impact of boundary conditions on the diffusive transport of active particles in an obstacle lattice. We find that particle dynamics with sliding boundary conditions result in large diffusivities even at high obstacle density, unlike classical specular reflection. These dynamics are very well described by a model based on run-and-tumble particles with microscopically derived reorientation functions arising from obstacle-induced tumbles. This model, however, fails to describe fine structure in the diffusivity at high obstacle density predicted by simulations for pusherlike collisions. Using a simple deterministic model, we show that this structure results from particles being guided by the lattice. Our results thus show how nonclassical surface scattering introduces a dependence on the lattice geometry at high densities. We discuss implications for the study of bacteria in complex environments.

Which interactions dominate in active colloids?

Despite mounting evidence that the same gradients, which active colloids use for swimming, induce important cross-interactions (phoretic interactions), they are still ignored in most many-body descriptions, perhaps to avoid complexity and a zoo of unknown parameters. Here we derive a simple model, which reduces phoretic far-field interactions to a pair-interaction whose strength is mainly controlled by one genuine parameter (swimming speed). The model suggests that phoretic interactions are generically important for autophoretic colloids (unless effective screening of the phoretic fields is strong) and should dominate over hydrodynamic interactions for the typical case of half-coating and moderately nonuniform surface mobilities. Unlike standard minimal models, but in accordance with canonical experiments, our model generically predicts dynamic clustering in active colloids at a low density. This suggests that dynamic clustering can emerge from the interplay of screened phoretic attractions and active diffusion.

Spontaneous assembly of colloidal vesicles driven by active swimmers

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.

Strictures of a microchannel impose fierce competition to select for highly motile sperm

Investigating sperm locomotion in the presence of external fluid flow and geometries simulating the female reproductive tract can lead to a better understanding of sperm motion during fertilization. Using a microfluidic device featuring a stricture that simulates the fluid mechanical properties of narrow junctions inside the female reproductive tract, we documented the gate-like role played by the stricture in preventing sperm with motilities below a certain threshold from advancing through the stricture to the other side (i.e., fertilization site). All the slower sperm accumulate below (i.e., in front of) the stricture and swim in a butterfly-shaped path between the channel walls, thus maintaining the potential for penetrating the stricture and ultimately advancing toward the fertilization site. Accumulation below the stricture occurs in a hierarchical manner so that dense concentrations of sperm with higher velocities remain closer to the stricture, with more sparsely distributed arrays of lower-velocity sperm lagging behind.

Dynamics of sedimenting active Brownian particles

We investigate the stochastic dynamics of one sedimenting active Brownian particle in three dimensions under the influence of gravity and passive fluctuations in the translational and rotational motion. We present an analytical solution of the Fokker-Planck equation for the stochastic process which allows us to describe the dynamics of one active Brownian particle in three dimensions. We address the time evolution of the density, the polarization, and the steady-state solution. We also perform Brownian dynamics simulations and study the effect of the activity of the particles on their collective motion. These results qualitatively agree with our model. Finally, we compare our results with experiments (J. Palacci et al., Phys. Rev. Lett. 105, 088304 (2010)) and find very good agreement.

Swimming of Spermatozoa in a Maxwell Fluid

It has been suggested that the swimming mechanism used by spermatozoa could be adopted for self-propelled micro-robots in small environments and potentially applied to biomedical engineering. Mammalian sperm cells must swim through a viscoelastic mucus layer to find the egg cell. Thus, understanding how sperm cells swim through viscoelastic liquids is significant not only for physiology, but also for the design of micro-robots. In this paper, we developed a numerical model of a sperm cell in a linear Maxwell fluid based on the boundary element slender-body theory coupling method. The viscoelastic properties were characterized by the Deborah number (De), and we found that, under the prescribed waveform, the swimming speed decayed with the Deborah number in the small-De regime (De < 1.0). The swimming efficiency was independent of the Deborah number, and the decrease in the swimming speed was not significantly affected by the wave pattern.

Ideal circle microswimmers in crowded media

Microswimmers are exposed in nature to crowded environments and their transport properties depend in a subtle way on the interaction with obstacles. Here, we investigate a model for a single ideal circle swimmer exploring a two-dimensional disordered array of impenetrable obstacles. The microswimmer moves on circular orbits in the freely accessible space and follows the surface of an obstacle for a certain time upon collision. Depending on the obstacle density and the radius of the circular orbits, the microswimmer displays either long-range transport or is localized in a finite region. We show that there are transitions from two localized states to a diffusive state each driven by an underlying static percolation transition. We determine the non-equilibrium state diagram and calculate the mean-square displacements and diffusivities by computer simulations. Close to the transition lines transport becomes subdiffusive which is rationalized as a dynamic critical phenomenon.

Learning the space-time phase diagram of bacterial swarm expansion

Most living systems, from individual cells to tissues and swarms, display collective self-organization on length scales that are much larger than those of the individual units that drive this organization. A fundamental challenge is to understand how properties of microscopic components determine macroscopic, multicellular biological function. Our study connects intracellular physiology to macroscale collective behaviors during multicellular development, spanning five orders of magnitude in length and six orders of magnitude in time, using bacterial swarming as a model system. This work is enabled by a high-throughput adaptive microscopy technique, which we combined with genetics, machine learning, and mathematical modeling to reveal the phase diagram of bacterial swarming and that cell–cell interactions within each swarming phase are dominated by mechanical interactions.

Coordinated dynamics of individual components in active matter are an essential aspect of life on all scales. Establishing a comprehensive, causal connection between intracellular, intercellular, and macroscopic behaviors has remained a major challenge due to limitations in data acquisition and analysis techniques suitable for multiscale dynamics. Here, we combine a high-throughput adaptive microscopy approach with machine learning, to identify key biological and physical mechanisms that determine distinct microscopic and macroscopic collective behavior phases which develop as Bacillus subtilis swarms expand over five orders of magnitude in space. Our experiments, continuum modeling, and particle-based simulations reveal that macroscopic swarm expansion is primarily driven by cellular growth kinetics, whereas the microscopic swarming motility phases are dominated by physical cell–cell interactions. These results provide a unified understanding of bacterial multiscale behavioral complexity in swarms.

Stability of a Dumbbell Micro-Swimmer

A squirmer model achieves propulsion by generating surface squirming velocities. This model has been used to analyze the movement of micro-swimmers, such as microorganisms and Janus particles. Although squirmer motion has been widely investigated, motions of two connected squirmers, i.e., a dumbbell squirmer, remain to be clarified. The stable assembly of multiple micro-swimmers could be a key technology for future micromachine applications. Therefore, in this study, we investigated the swimming behavior and stability of a dumbbell squirmer. We first examined far-field stability through linear stability analysis, and found that stable forward swimming could not be achieved by a dumbbell squirmer in the far field without the addition of external torque. We then investigated the swimming speed of a dumbbell squirmer connected by a short rigid rod using a boundary element method. Finally, we investigated the swimming stability of a dumbbell squirmer connected by a spring. Our results demonstrated that stable side-by-side swimming can be achieved by pullers. When the aft squirmer was a strong pusher, fore and aft swimming were stable and swimming speed increased significantly. The findings of this study will be useful for the future design of assembled micro-swimmers.

Inhomogeneous assembly of driven nematic colloids

We present a quantitative analysis of the nonequilibrium assembly of colloidal particles dispersed in a nematic liquid crystal. The driven particles assemble into reconfigurable circular clusters by liquid-crystal-enabled electrokinetic phenomena generated by an AC electric field that provides propulsion along the local director. We identify the coexistence of different aggregation states, including a central, jammed core, where short-range elastic attraction dominates, surrounded by a liquid-like corona where particles retain their mobility but reach a mechanical equilibrium that we rationalize in terms of a balance between centripetal phoretic drive and pairwise repulsion. An analysis of the compressible liquid-like region reveals a linear density profile that can be tuned with the field frequency, and a bond-orientational order that reaches a maximum at intermediate packing densities, where elastic effects are minimized. Since the phoretic propulsion force acts also on assembled particles, we compute the mechanical pressure and show that a hard-disk equation of state can be used to describe the assembly of this driven system.

Chronology of motor-mediated microtubule streaming

We introduce a filament-based simulation model for coarse-grained, effective motor-mediated interaction between microtubule pairs to study the time-scales that compose cytoplasmic streaming. We characterise microtubule dynamics in two-dimensional systems by chronologically arranging five distinct processes of varying duration that make up streaming, from microtubule pairs to collective dynamics. The structures found were polarity sorted due to the propulsion of antialigned microtubules. This also gave rise to the formation of large polar-aligned domains, and streaming at the domain boundaries. Correlation functions, mean squared displacements, and velocity distributions reveal a cascade of processes ultimately leading to microtubule streaming and advection, spanning multiple microtubule lengths. The characteristic times for the processes extend over three orders of magnitude from fast single-microtubule processes to slow collective processes. Our approach can be used to directly test the importance of molecular components, such as motors and crosslinking proteins between microtubules, on the collective dynamics at cellular scale.

Memory-based mediated interactions between rigid particulate inclusions in viscoelastic environments

Many practically relevant materials combine properties of viscous fluids and elastic solids to viscoelastic behavior. Our focus is on the induced dynamic behavior of damped finite-sized particulate inclusions in such substances. We explicitly describe history-dependent interactions that emerge between the embedded particles. These interactions are mediated by the viscoelastic surroundings. They result from the flows and distortions of the viscoelastic medium when induced by the rigid inclusions. Both viscoelastic environments of terminal fluidlike flow and of completely reversible damped elastic behavior are covered. For illustration and to highlight the role of the formalism in potential applications, we briefly address the relevant examples of dragging a rigid sphere through a viscoelastic environment together with subsequent relaxation dynamics, the switching dynamics of magnetic fillers in elastic gel matrices, and the swimming behavior of active microswimmers in viscoelastic solutions. The approach provides a basis for more quantitative and extended investigations of these and related systems in the future.

A statistical physics view of swarming bacteria

Bacterial swarming is a collective mode of motion in which cells migrate rapidly over surfaces, forming dynamic patterns of whirls and jets. This review presents a physical point of view of swarming bacteria, with an emphasis on the statistical properties of the swarm dynamics as observed in experiments. The basic physical principles underlying the swarm and their relation to contemporary theories of collective motion and active matter are reviewed and discussed in the context of the biological properties of swarming cells. We suggest a paradigm according to which bacteria have optimized some of their physical properties as a strategy for rapid surface translocation. In other words, cells take advantage of favorable physics, enabling efficient expansion that enhances survival under harsh conditions.

Approximating microswimmer dynamics by active Brownian motion: Energetics and efficiency

We consider the dynamics of a microswimmer and show that they can be approximated by active Brownian motion. The swimmer is modeled by coupled overdamped Langevin equations with periodic driving. We compare the energy dissipation of the real swimmer to that of the active Brownian motion model, finding that the latter can massively underestimate the complete dissipation. This discrepancy is related to the inability to infer the full dissipation from partial observation of the complete system. We introduce an efficiency that measures how much of the dissipated energy is spent on forward propulsion.

On the limitations of some popular numerical models of flagellated microswimmers: importance of long-range forces and flagellum waveform

For a sperm-cell-like flagellated swimmer in an unbounded domain, several numerical models of different fidelity are considered based on the Stokes flow approximation. The models include a regularized Stokeslet method and a three-dimensional finite-element method, which serve as the benchmark solutions for several approximate models considered. The latter include the resistive force theory versions of Lighthill, and Gray and Hancock, as well as a simplified approximation based on computing the hydrodynamic forces exerted on the head and the flagellum separately. It is shown how none of the simplified models is robust enough with regards to predicting the effect of the swimmer head shape change on the swimmer dynamics. For a range of swimmer motions considered, the resulting solutions for the swimmer force and velocities are analysed and the applicability of the Stokes model for the swimmers in question is probed.

Data-driven quantitative modeling of bacterial active nematics

Active matter comprises individual units that convert energy into mechanical motion. In many examples, such as bacterial systems and biofilament assays, constituent units are elongated and can give rise to local nematic orientational order. Such "active nematics" systems have attracted much attention from both theorists and experimentalists. However, despite intense research efforts, data-driven quantitative modeling has not been achieved, a situation mainly due to the lack of systematic experimental data and to the large number of parameters of current models. Here, we introduce an active nematics system made of swarming filamentous bacteria. We simultaneously measure orientation and velocity fields and show that the complex spatiotemporal dynamics of our system can be quantitatively reproduced by a type of microscopic model for active suspensions whose important parameters are all estimated from comprehensive experimental data. This provides unprecedented access to key effective parameters and mechanisms governing active nematics. Our approach is applicable to different types of dense suspensions and shows a path toward more quantitative active matter research.

Toward Understanding of Self-Electrophoretic Propulsion under Realistic Conditions: From Bulk Reactions to Confinement Effects

Active matter concerns itself with the study of particles that convert energy into work, typically motion of the particle itself. This field saw a surge of interest over the past decade, after the first micrometer-sized, man-made chemical motors were created. These particles served as a simple model system for studying in a well-controlled manner complex motion and cooperative behavior as known from biology. In addition, they have stimulated new efforts in understanding out-of-equilibrium statistical physics and started a revolution in microtechnology and robotics. Concentrated effort has gone into realizing these ambitions, and yet much remains unknown about the chemical motors themselves. The original designs for self-propelled particles relied on the conversion of the chemical energy of hydrogen peroxide into motion via catalytic decomposition taking place heterogeneously over the surface of the motor. This sets up gradients of chemical fields around the particle, which allow it to autophorese. That is, the interaction between the motor and the heterogeneously distributed solute species can drive fluid flow and the motor itself. There are two basic designs: the first relies on redox reactions taking place between the two sides of a bimetal, for example, a gold-platinum Janus sphere or nanorod. The second uses a catalytic layer of platinum inhomogeneously vapor-deposited onto a nonreactive particle. For convenience's sake, these can be referred to as redox motors and monometallic half-coated motors, respectively. To date, most researchers continue to rely on variations of these simple, yet elegant designs for their experiments. However, there is ongoing debate on the exact way chemical energy is transduced into motion in these motors. Many of the experimental observations on redox motors were successfully modeled via self-electrophoresis, while for half-coated motors there has been a strong focus on self-diffusiophoresis. Currently, there is mounting evidence that self-electrophoresis provides the dominant contribution to the observed speeds of half-coated motors, even if the vast majority of the reaction products are electroneutral. In this Account, we will summarize the most common electrophoretic propulsion model and discuss its strengths and weaknesses in relation to recent experiments. We will comment on the possible need to go beyond surface reactions and consider the entire medium as an "active fluid" that can create and annihilate charged species. This, together with confinement and collective effects, makes it difficult to gain a detailed understanding of these swimmers. The potentially dominant effect of confinement is highlighted on the basis of a recent study of an electro-osmotic pump that drives fluid along a substrate. Detailed analysis of this system allows for identification of the electro-osmotic driving mechanism, which is powered by micromolar salt concentrations. We will discuss how our latest numerical solver developments, based on the lattice Boltzmann method, should enable us to study collective behavior in systems comprised of these and other electrochemical motors in realistic environments. We conclude with an outlook on the future of modeling chemical motors that may facilitate the community's microtechnological ambitions.

Effective Interactions between Chemically Active Colloids and Interfaces

Chemically active colloids can achieve force- and torque-free motility ("self-propulsion") via the promotion, on their surface, of catalytic chemical reactions involving the surrounding solution. Such systems are valuable both from a theoretical perspective, serving as paradigms for nonequilibrium processes, as well as from an application viewpoint, according to which active colloids are envisioned to play the role of carriers ("engines") in novel lab-on-a-chip devices. The motion of such colloids is intrinsically connected with a "chemical field", i.e., the distribution near the colloid of the number densities of the various chemical species present in the solution, and with the hydrodynamic flow of the solution around the particle. In most of the envisioned applications, and in virtually all reported experimental studies, the active colloids operate under spatial confinement (e.g., within a microfluidic channel, a drop, a free-standing liquid film, etc.). In such cases, the chemical field and the hydrodynamic flow associated with an active colloid are influenced by any nearby confining surfaces, and these disturbances couple back to the particle. Thus, an effective interaction with the spatial confinement arises. Consequently, the particle is endowed with means to perceive and to respond to its environment. Understanding these effective interactions, finding the key parameters which control them, and designing particles with desired, preconfigured responses to given environments, require interdisciplinary approaches which synergistically integrate methods and knowledge from physics, chemistry, engineering, and materials science. Here we review how, via simple models of chemical activity and self-phoretic motion, progress has recently been made in understanding the basic physical principles behind the complex behaviors exhibited by active particles near interfaces. First, we consider the occurrence of "interface-bounded" steady states of chemically active colloids near simple, nonresponsive interfaces. Examples include particles "sliding" along, or "hovering" above, a hard planar wall while inducing hydrodynamic flow of the solution. These states lay the foundations for concepts like the guidance of particles by the topography of the wall. We continue to discuss responsive interfaces: a suitable chemical patterning of a planar wall allows one to bring the particles into states of motion which are spatially localized (e.g., within chemical stripes or along chemical steps). These occur due to the wall responding to the activity-induced chemical gradients by generating osmotic flows, which encode the surface-chemistry of the wall. Finally, we discuss how, via activity-induced Marangoni stresses, long-ranged effective interactions emerge from the strong hydrodynamic response of fluid interfaces. These examples highlight how in this context a desired behavior can be potentially selected by tuning suitable parameters (e.g., the phoretic mobility of the particle, or the strength of the Marangoni stress at an interface). This can be accomplished via a judicious design of the surface chemistry of the particle and of the boundary, or by the choice of the chemical reaction in solution.

Nutrient Transport Driven by Microbial Active Carpets

We demonstrate that active carpets of bacteria or self-propelled colloids generate coherent flows towards the substrate, and propose that these currents provide efficient pathways to replenish nutrients that feed back into activity. A full theory is developed in terms of gradients in the active matter density and velocity, and applied to bacterial turbulence, topological defects and clustering. Currents with complex spatiotemporal patterns are obtained, which are tunable through confinement. Our findings show that diversity in carpet architecture is essential to maintain biofunctionality.

Weak Shape Anisotropy Leads to a Nonmonotonic Contribution to Crowding, Impacting Protein Dynamics under Physiologically Relevant Conditions

The effect of a nonspherical particle shape on the dynamics in crowded solutions presents a significant challenge for a comprehensive understanding of interaction and structural relaxation in biological and soft matter. We report that small deviations from a spherical shape induce a nonmonotonic contribution to the crowding effect on the short-time cage diffusion compared with spherical systems, using molecular dynamics simulations with mesoscale hydrodynamics of a multiparticle collision dynamics fluid in semidilute systems with volume fractions smaller than 0.35. We show that the nonmonotonic effect due to anisotropy is caused by the combination of a reduced relative mobility over the entire concentration range and a looser and less homogeneous cage packing of nonspherical particles. Our finding stresses that nonsphericity induces new complexity, which cannot be accounted for in effective sphere models, and is of great interest in applications such as formulations as well as for the fundamental understanding of soft matter in general and crowding effects in living cells in particular.

Effective squirmer models for self-phoretic chemically active spherical colloids

Various aspects of self-motility of chemically active colloids in Newtonian fluids can be captured by simple models for their chemical activity plus a phoretic-slip hydrodynamic boundary condition on their surface. For particles of simple shapes (e.g., spheres) --as employed in many experimental studies-- which move at very low Reynolds numbers in an unbounded fluid, such models of chemically active particles effectively map onto the well studied so-called hydrodynamic squirmers (S. Michelin and E. Lauga, J. Fluid Mech. 747, 572 (2014)). Accordingly, intuitively appealing analogies of "pusher/puller/neutral" squirmers arise naturally. Within the framework of self-diffusiophoresis we illustrate the above-mentioned mapping and the corresponding flows in an unbounded fluid for a number of choices of the activity function (i.e., the spatial distribution and the type of chemical reactions across the surface of the particle). We use the central collision of two active particles as a simple, paradigmatic case for demonstrating that in the presence of other particles or boundaries the behavior of chemically active colloids may be qualitatively different, even in the far field, from the one exhibited by the corresponding "effective squirmer", obtained from the mapping in an unbounded fluid. This emphasizes that understanding the collective behavior and the dynamics under geometrical confinement of chemically active particles necessarily requires to explicitly account for the dependence of the hydrodynamic interactions on the distribution of chemical species resulting from the activity of the particles.

Inertial delay of self-propelled particles

The motion of self-propelled massive particles through a gaseous medium is dominated by inertial effects. Examples include vibrated granulates, activated complex plasmas and flying insects. However, inertia is usually neglected in standard models. Here, we experimentally demonstrate the significance of inertia on macroscopic self-propelled particles. We observe a distinct inertial delay between orientation and velocity of particles, originating from the finite relaxation times in the system. This effect is fully explained by an underdamped generalisation of the Langevin model of active Brownian motion. In stark contrast to passive systems, the inertial delay profoundly influences the long-time dynamics and enables new fundamental strategies for controlling self-propulsion in active matter.

Coaxial differential dynamic microscopy for measurement of Brownian motion in weak optical field

Weak optical fields cause less damage to active cells and are easier to realize than traditional and tightly focused optical fields. While these fields are promising for biomedical science and particle manipulation applications, they lack a method for precise particle diffusion measurement because the weak fields cause the small changes in particle motion caused by weak fields. In this paper, we present a coaxial differential dynamic microscopy (CDDM) technique that uses a differential dynamic microscopy system, combined with an adjustable optical field. We use this technique to study Brownian motion of colloidal particles in weak optical fields. CDDM can quantitatively measure both the intensity and the pattern of the weak optical field and the diffusion coefficient of the particles. While the light paths of both the weak optical field and the illumination are coaxially incident on the sample cell, they remain independent. The optical field can be designed to have any pattern and adjusted to any intensity, while the measurements' sample illumination requirements are also satisfied. To verify the accuracy of the technique, we measured particle diffusion in weak Gaussian optical fields of different strengths. The diffusion coefficient was found to decrease with increasing field strength. These experimental results agree well with those results predicted using the Fokker-Planck equation and Euler algorithm simulations. This technique is expected to provide an efficient tool for research into particle manipulation by using weak optical fields, particularly for delicate systems, such as colloidal particles and biological cells.

Globulelike Conformation and Enhanced Diffusion of Active Polymers

We study the dynamics and conformation of polymers composed by active monomers. By means of Brownian dynamics simulations we show that, when the direction of the self-propulsion of each monomer is aligned with the backbone, the polymer undergoes a coil-to-globulelike transition, highlighted by a marked change of the scaling exponent of the gyration radius. Concurrently, the diffusion coefficient of the center of mass of the polymer becomes essentially independent of the polymer size for sufficiently long polymers or large magnitudes of the self-propulsion. These effects are reduced when the self-propulsion of the monomers is not bound to be tangent to the backbone of the polymer. Our results, rationalized by a minimal stochastic model, open new routes for activity-controlled polymers and, possibly, for a new generation of polymer-based drug carriers.

Magnetic Active Matter Based on Helical Propulsion

Micro- and nanomotors are nonliving micro- and nanoparticles that are rendered motile by supplying energy from external sources, for example, through asymmetric chemical reactions or the application of electric, magnetic, optical, or acoustic fields. Their study is interesting for two reasons. First, nanomotors can impact future biomedical practices, where one envisions intelligent multifunctional nanomachines swarming toward a diseased site and delivering therapeutics with high accuracy. The second motivation stems from the prevalence of self-powered systems in nature, ranging from intracellular transport to human migration, which are nonequilibrium phenomena yet to be completely understood. Nanomotors provide a promising route toward the study of complex active matter phenomena with a well-defined and possibly reduced set of variables. Among different ways of powering nanomotors, magnetic field deserves a special mention because of its inherent biocompatibility, minimal dependence on properties of the surrounding medium, and remote powering mechanism. In particular, magnetically actuated propellers (MAPs), which are helical structures driven by rotating fields in fluids and gels, have been demonstrated to be highly suitable for various microfluidic and biotechnology applications. Unfortunately, this method of actuation requires direct application of mechanical torque by the applied field, implying that the system is driven and therefore cannot be considered self-propelled. To overcome this fundamental limitation, we discuss an alternate magnetic drive where the MAPs are powered by oscillating (not rotating) magnetic fields. This technique induces motility in the form of back-and-forth motion but allows the directionality to be unspecified, and therefore, it represents a zero-force, zero-torque active matter where the nanomotors behave effectively as self-propelled entities. The MAPs show enhanced diffusivity compared with their passive counterparts, and their motility can be tuned by altering the external magnetic drive, which establishes the suitability of the MAPs as model active particles. Enhancement of the diffusivity depends on the thermal noise as well as the inherent asymmetries of the individual motors, which could be well-understood through numerical simulations. In the presence of small direct-current fields and interactions with the surface, the swimmers can be maneuvered and subsequently positioned in an independent manner. Next, we discuss experimental results pertaining to the collective dynamics of these helical magnetic nanoswimmers. We have studied nonmagnetic tracer beads suspended in a medium containing many swimmers and found the diffusivity of the beads to increase under magnetic actuation, akin to measurements performed in dense bacterial suspensions. In summary, we envision that rendering the system of MAPs active will not only provide a new model system to investigate fundamental nonequilibrium phenomena but also play a vital role in the development of intelligent theranostic probes for futuristic biomedical applications.

Cellular geometry controls the efficiency of motile sperm aggregates

Sperm that swim collectively to the fertilization site have been observed across several vertebrate and invertebrate species, with groups ranging in size from sperm pairs to massive aggregates containing hundreds of cells. Although the molecular mechanisms that regulate sperm-sperm adhesion are still unclear, aggregation can enhance sperm motility and thus offer a fertilization advantage. Here, we report a thorough computational investigation on the role of cellular geometry in the performance of sperm aggregates. The sperm head is modelled as a persistent random walker characterized by a non-trivial three-dimensional shape and equipped with an adhesive region where cell-cell binding occurs. By considering both, a simple parametric head shape and a computer reconstruction of a real head shape based on morphometric data, we demonstrate that the geometry of the head and the structure of the adhesive region crucially affects both the stability and motility of the aggregates. Our analysis further suggests that the apical hook commonly found in the sperm of muroid rodents might serve to shield portions of the adhesive region and promote efficient alignment of the velocities of the interacting cells.

Transport of particles driven by the traveling obstacle arrays

Transport of three types of particles (passive particles, active particles without polar interaction, and active particles with polar interaction) is numerically investigated in the presence of traveling obstacle arrays. The transport behaviors are different for different types of particles. For passive particles, there exists an optimal traveling speed (or the translational diffusion) at which the average velocity of particles takes its maximum value. For active particles without polar interaction, the average velocity of particles is a peaked function of the obstacle traveling speed. The average velocity decreases monotonically with increase of the rotational diffusion for large driving speed, while it is a peaked function of the rotational diffusion for small driving speed. For active particles with polar interaction, interestingly, within particular parameter regimes, active particles can move in the opposite direction to the obstacles. The average velocity of particles can change its direction by changing the system parameters (the obstacles driving speed, the polar interaction strength, and the rotational diffusion).

Clustering of microswimmers: interplay of shape and hydrodynamics

The spatiotemporal dynamics in systems of active self-propelled particles is controlled by the propulsion mechanism in combination with various direct interactions, such as steric repulsion and hydrodynamics. These direct interactions are typically anisotropic, and come in different "flavors", such as spherical and elongated particle shapes, pusher and puller flow fields, etc. The combination of the various aspects is expected to lead to new emergent behavior. However, it is a priori not evident whether shape and hydrodynamics act synergistically or antagonistically to generate motility-induced clustering (MIC) and phase separation (MIPS). We employ a model of prolate spheroidal microswimmers-called squirmers-in quasi-two-dimensional confinement to address this issue by mesoscale hydrodynamic simulations. For comparison, non-hydrodynamic active Brownian particles (ABPs) are considered to elucidate the contribution of hydrodynamic interactions. For spherical particles, the comparison between ABPs and hydrodynamic-squirmer ensembles reveals a suppression of MIPS due to hydrodynamic interactions. Yet, our analysis shows that dynamic clusters exist, with a broad size distribution. The fundamental difference between ABPs and squirmers is attributed to an increased reorientation of squirmers by hydrodynamic torques during their collisions. In contrast, for elongated squirmers, hydrodynamics interactions enhance MIPS. The transition to a phase-separated state strongly depends on the nature of the swimmer's flow field-with an increased tendency toward MIPS for pullers, and a reduced tendency for pushers. Thus, hydrodynamic interactions show opposing effects on MIPS for spherical and elongated microswimmers, and details of the propulsion mechanism of biological microswimmers may be very important to determine their collective behavior.

Active dumbbells: Dynamics and morphology in the coexisting region

With the help of molecular dynamics simulations we study an ensemble of active dumbbells in purely repulsive interaction. We derive the phase diagram in the density-activity plane and we characterise the various phases with liquid, hexatic and solid character. The analysis of the structural and dynamical properties, such as enstrophy, mean-square displacement, polarisation, and correlation functions, shows the continuous character of liquid and hexatic phases in the coexisting region when the activity is increased starting from the passive limit.

Hydrodynamic Clustering of Human Sperm in Viscoelastic Fluids

We have numerically investigated sperm clustering behaviours, modelling cells as superpositions of regularised flow singularities, coarse-grained from experimentally obtained digital microscopy of human sperm, both in watery medium and a highly viscous-weakly elastic, methylcellulose medium. We find that the cell yaw and cell pulling dynamics inhibit clustering in low viscosity media. In contrast clustering is readily visible in simulations modelling sperm within a methylcellulose medium, in line with previous observations that bovine sperm clustering is much more prominent in a rheological polyacrylamide medium. Furthermore, the fine-scale details of sperm flagellar movement substantially impact large-scale collective behaviours, further motivating the need for the digital microscopy and characterization of sperm to understand their dynamics.

Swarming and Aggregation in the Parasitic Diplomonad Flagellate Spironucleus vortens

Pathogenicity, evolutionary history, and unusual cell organization of diplomonads are well known, particularly for Giardia and Spironucleus; however, behavior of these aerotolerant anaerobes is largely unknown. Addressing this deficit, we studied behavior of the piscine diplomonad Spironucleus vortens (ATCC 50386) in in vitro culture. Spironucleus vortens trophozoites from Angelfish, Pterophyllum scalare, were maintained axenically in modified liver digest, yeast extract, and iron (LYI) medium, at 22 °C in the dark, and subcultured weekly. Cultures were monitored every 1-2 d, by removing an aliquot, and loading cells into a hemocytometer chamber, or onto a regular microscope slide. We observed three distinct swimming behaviors: (i) spontaneous formation of swarms, reaching 200 μm in diameter, persisting for up to several min in situ, (ii) directional movement of the swarm, via collective motility, and (iii) independent swimming of trophozoites to form a band (aggregation), presumably at the location of optimal environmental conditions. These behaviors have not previously been reported in Spironucleus. The observation that flagellate motility can change, from individual self-propulsion to complex collective swarming motility, prompts us to advocate S. vortens as a new model for study of group behavioral dynamics, complementing emerging studies of collective swimming in flagellated bacteria.

Synthetic Nanomotors: Working Together through Chemistry

Active matter, some of whose constituent elements are active agents that can move autonomously, behaves very differently from matter without such agents. The active agents can self-assemble into structures with a variety of forms and dynamical properties. Swarming, where groups of living agents move cooperatively, is commonly observed in the biological realm, but it is also seen in the physical realm in systems containing small synthetic motors. The existence of diverse forms of self-assembled structures has stimulated the search for new applications that involve active matter. We consider active systems where the agents are synthetic chemically powered motors with various shapes and sizes that operate by phoretic mechanisms, especially self-diffusiophoresis. These motors are able to move autonomously in solution by consuming fuel from their environment. Chemical reactions take place on catalytic portions of the motor surface and give rise to concentration gradients that lead to directed motion. They can operate in this way only if the chemical composition of the system is maintained in a nonequilibrium state since no net fluxes are possible in a system at equilibrium. In contrast to many other active systems, chemistry plays an essential part in determining the properties of the collective dynamics and self-assembly of these chemically powered motor systems. The inhomogeneous concentration fields that result from asymmetric motor reactions are felt by other motors in the system and strongly influence how they move. This chemical coupling effect often dominates other interactions due to fluid flow fields and direct interactions among motors and determines the form that the collective dynamics takes. Since we consider small motors with micrometer and nanometer sizes, thermal fluctuations are strong and cannot be neglected. The media in which the motors operate may not be simple and may contain crowding agents or molecular filaments that influence how the motors assemble and move. The collective motion is also influenced by the chemical gradients that arise from reactions in the surrounding medium. By adopting a microscopic perspective, where the motors, fluid environment, and crowding elements are treated at the coarse-grained molecular level, all of the many-body interactions that give rise to the collective behavior naturally emerge from the molecular dynamics. Through simulations and theory, this Account describes how active matter made from chemically powered nanomotors moving in simple and more complicated media can form different dynamical structures that are strongly influenced by interactions arising from cooperative chemical reactions on the motor surfaces.

Stokes velocity generated by a point force in various geometries

In this short review, we visualize the fluid velocity generated by a point force close to a plane free surface or a plane rigid wall. We present separately contributions from all the multipoles which form the corresponding classical systems of images. Such graphical images might be useful in the theoretical and numerical modeling of the dynamics of micro-objects moving close to an interface.

The swimming of a deforming helix

Many microorganisms and artificial microswimmers use helical appendages in order to generate locomotion. Though often rotated so as to produce thrust, some species of bacteria such Spiroplasma, Rhodobacter sphaeroides and Spirochetes induce movement by deforming a helical-shaped body. Recently, artificial devices have been created which also generate motion by deforming their helical body in a non-reciprocal way (A. Mourran et al. Adv. Mater. 29, 1604825, 2017). Inspired by these systems, we investigate the transport of a deforming helix within a viscous fluid. Specifically, we consider a swimmer that maintains a helical centreline and a single handedness while changing its helix radius, pitch and wavelength uniformly across the body. We first discuss how a deforming helix can create a non-reciprocal translational and rotational swimming stroke and identify its principle direction of motion. We then determine the leading-order physics for helices with small helix radius before considering the general behaviour for different configuration parameters and how these swimmers can be optimised. Finally, we explore how the presence of walls, gravity, and defects in the centreline allow the helical device to break symmetries, increase its speed, and generate transport in directions not available to helices in bulk fluids.

Phase coexistence in a monolayer of active particles induced by Marangoni flows

Thermally or chemically active colloids generate thermodynamic gradients in the solution in which they are immersed and thereby induce hydrodynamic flows that affect their dynamical evolution. Here we study a mean-field model for the many-body dynamics of a monolayer of spherically symmetric active particles located at a fluid-fluid interface. Due to the spherical symmetry, the particles do not self-propel. Instead, the dynamics is driven by the long-ranged Marangoni flows, due to the response of the interface to the activity of the particles, which compete with the direct interaction between particles. We demonstrate analytically that, in spite of the intrinsic out-of-equilibrium character of the system, the monolayer evolves to a "pseudoequilibrium" state, in which the Marangoni flows force the coexistence of the thermodynamic phases associated to the direct interaction. In particular, we study the most interesting case of a r-3 soft repulsion that models electrostatic or magnetic interparticle forces. For a sufficiently large average density, two-dimensional phase transitions (freezing from liquid to hexatic, and melting from solid to hexatic) should be observable in a radially stratified, "onion-like" structure within the monolayer. Furthermore, the analysis allows us to conclude that, while the activity may be too weak to allow direct detection of such induced Marangoni flows, it is relevant as a collective effect in the emergence of the experimentally observable spatial structure of phase coexistences noted above. Finally, the relevance of these results for potential experimental realizations is critically discussed.

Load response of shape-changing microswimmers scales with their swimming efficiency

External forces acting on a microswimmer can feed back on its self-propulsion mechanism. We discuss this load response for a generic microswimmer that swims by cyclic shape changes. We show that the change in cycle frequency is proportional to the Lighthill efficiency of self-propulsion. As a specific example, we consider Najafi's three-sphere swimmer. The force-velocity relation of a microswimmer implies a correction for a formal superposition principle for active and passive motion.

Current reversals of active particles in time-oscillating potentials

Rectification of interacting active particles is numerically investigated in a two-dimensional time-oscillating potential. It is found that the oscillation of the potential and the self-propulsion of active particles are two different types of nonequilibrium driving, which can induce net currents with opposite directions. For a given asymmetry of the potential, the direction of the transport is determined by the competition of the self-propulsion and the oscillation of the potential. There exists an optimal oscillating angular frequency (or self-propulsion speed) at which the average velocity takes its maximal positive or negative value. Remarkably, when the oscillation of the potential competes with the self-propulsion, the average velocity can change direction several times due to the change in the oscillating frequency. Especially, particles with different self-propulsion velocities will move in opposite directions and can be separated. Our results provide a novel and convenient method for controlling and manipulating the transport (or separation) of active particles.