Modeling a spheroidal microswimmer and cooperative swimming in a narrow slit

Collective behavior of squirmers in thin films

Bacteria in biofilms form complex structures and can collectively migrate within mobile aggregates, which is referred to as swarming. This behavior is influenced by a combination of various factors, including morphological characteristics and propulsive forces of swimmers, their volume fraction within a confined environment, and hydrodynamic and steric interactions between them. In our study, we employ the squirmer model for microswimmers and the dissipative particle dynamics method for fluid modeling to investigate the collective motion of swimmers in thin films. The film thickness permits a free orientation of non-spherical squirmers, but constraints them to form a two-layered structure at maximum. Structural and dynamic properties of squirmer suspensions confined within the slit are analyzed for different volume fractions of swimmers, motility types (e.g., pusher, neutral squirmer, puller), and the presence of a rotlet dipolar flow field, which mimics the counter-rotating flow generated by flagellated bacteria. Different states are characterized, including a gas-like phase, swarming, and motility-induced phase separation, as a function of increasing volume fraction. Our study highlights the importance of an anisotropic swimmer shape, hydrodynamic interactions between squirmers, and their interaction with the walls for the emergence of different collective behaviors. Interestingly, the formation of collective structures may not be symmetric with respect to the two walls. Furthermore, the presence of a rotlet dipole significantly mitigates differences in the collective behavior between various swimmer types. These results contribute to a better understanding of the formation of bacterial biofilms and the emergence of collective states in confined active matter.

Simulating dynamics of ellipsoidal particles using lattice Boltzmann method

Anisotropic particles are often encountered in different fields of soft matter and complex fluids. In this work, we present an implementation of the coupled hydrodynamics of solid ellipsoidal particles and the surrounding fluid using the lattice Boltzmann method. A standard link-based mechanism is used to implement the solid-fluid boundary conditions. We develop an implicit method to update the position and orientation of the ellipsoid. This exploits the relations between the quaternion which describes the ellipsoid's orientation and the ellipsoid's angular velocity to obtain a stable and robust dynamic update. The proposed algorithm is validated by looking at four scenarios: (i) the steady translational velocity of a spheroid subject to an external force in different orientations, (ii) the drift of an inclined spheroid subject to an imposed force, (iii) three-dimensional rotational motions in a simple shear flow (Jeffrey's orbits), and (iv) developed fluid flows and self-propulsion exhibited by a spheroidal microswimmer. In all cases the comparison of numerical results shows good agreement with known analytical solutions, irrespective of the choice of the fluid properties, geometrical parameters, and lattice Boltzmann model, thus demonstrating the robustness of the proposed algorithm.

Motion of microswimmers in cylindrical microchannels

Biological and artificial microswimmers often have to propel through a variety of environments, ranging from heterogeneous suspending media to strong geometrical confinement. Under confinement, local flow fields generated by microswimmers, and steric and hydrodynamic interactions with their environment determine the locomotion. We propose a squirmer-like model to describe the motion of microswimmers in cylindrical microchannels, where propulsion is generated by a fixed surface slip velocity. The model is studied using an approximate analytical solution for cylindrical swimmer shapes, and by numerical hydrodynamics simulations for spherical and spheroidal shapes. For the numerical simulations, we employ the dissipative particle dynamics method for modelling fluid flow. Both the analytical model and simulations show that the propulsion force increases with increasing confinement. However, the swimming velocity under confinement remains lower than the swimmer speed without confinement for all investigated conditions. In simulations, different swimming modes (i.e. pusher, neutral, puller) are investigated, and found to play a significant role in the generation of propulsion force when a swimmer approaches a dead end of a capillary tube. Propulsion generation in confined systems is local, such that the generated flow field generally vanishes beyond the characteristic size of the swimmer. These results contribute to a better understanding of microswimmer force generation and propulsion under strong confinement, including the motion in porous media and in narrow channels.

Minimal numerical ingredients describe chemical microswimmers' 3-D motion

The underlying mechanisms and physics of catalytic Janus microswimmers is highly complex, requiring details of the associated phoretic fields and the physiochemical properties of catalyst, particle, boundaries, and the fuel used. Therefore, developing a minimal (and more general) model capable of capturing the overall dynamics of these autonomous particles is highly desirable. In the presented work, we demonstrate that a coarse-grained dissipative particle-hydrodynamics model is capable of describing the behaviour of various chemical microswimmer systems. Specifically, we show how a competing balance between hydrodynamic interactions experienced by a squirmer in the presence of a substrate, gravity, and mass and shape asymmetries can reproduce a range of dynamics seen in different experimental systems. We hope that our general model will inspire further synthetic work where various modes of swimmer motion can be encoded via shape and mass during fabrication, helping to realise the still outstanding goal of microswimmers capable of complex 3-D behaviour.

Methods and Measures for Investigating Microscale Motility

Motility is an essential factor for an organism's survival and diversification. With the advent of novel single-cell technologies, analytical frameworks, and theoretical methods, we can begin to probe the complex lives of microscopic motile organisms and answer the intertwining biological and physical questions of how these diverse lifeforms navigate their surroundings. Herein, we summarize the main mechanisms of microscale motility and give an overview of different experimental, analytical, and mathematical methods used to study them across different scales encompassing the molecular-, individual-, to population-level. We identify transferable techniques, pressing challenges, and future directions in the field. This review can serve as a starting point for researchers who are interested in exploring and quantifying the movements of organisms in the microscale world.

Near- and far-field hydrodynamic interaction of two chiral squirmers

Hydrodynamic interaction strongly influences the collective behavior of microswimmers. With this work, we study the behavior of two hydrodynamically interacting self-propelled chiral swimmers in the low Reynolds number regime, considering both the near- and far-field interactions. We use the chiral squirmer model [see Burada et al., Phys. Rev. E 105, 024603 (2022)2470-004510.1103/PhysRevE.105.024603], a spherically shaped body with nonaxisymmetric surface slip velocity, which generalizes the well-known squirmer model. The previous work was restricted only to the case, while the far-field hydrodynamic interaction was influential among the swimmers. It did not approach the scenario while both the swimmers are very close and lubrication effects become dominant. We calculate the lubrication force between the swimmers when they are very close. By varying the slip coefficients and the initial configuration of the swimmers, we investigate their hydrodynamic behavior. In the presence of lubrication force, the swimmers either repel each other or exhibit bounded motion where the distance between the swimmers alters periodically. We identify the possible behaviors exhibited by the chiral squirmers, such as monotonic divergence, divergence, and bounded, as was found in the previous study. However, in the current study, we observe that both the monotonic convergence and the convergence states are converted into divergence states due to the arising lubrication effects. The lubrication force favors the bounded motion in some parameter regimes. This study helps to understand the collective behavior of dense suspension of ciliated microorganisms and artificial swimmers.

Emergent collective dynamics of pusher and puller squirmer rods: swarming, clustering, and turbulence

We study the interplay of steric and hydrodynamic interactions in suspensions of elongated microswimmers by simulating the full hydrodynamics of squirmer rods in the quasi two-dimensional geometry of a Hele-Shaw cell. To create pusher or puller-type squirmer rods, we concentrate the surface slip-velocity field more to the back or to the front of the rod and thereby are able to tune the rod's force-dipole strength. We study a wide range of aspect ratios and area fractions and provide corresponding state diagrams. The flow field of pusher-type squirmer rods destabilizes ordered structures and favors the disordered state at small area fractions and aspect ratios. Only when steric interactions become relevant, we observe a turbulent and dynamic cluster state, while for large aspect ratios a single swarm and jammed cluster occurs. The power spectrum of the turbulent state shows two distinct energy cascades at small and large wave numbers with power-law scaling and non-universal exponents. Pullers show a strong tendency to form swarms instead of the disordered state found for neutral and pusher rods. At large area fractions a dynamic cluster is observed and at larger aspect ratio a single swarm or jammed cluster occurs.

Alignment and propulsion of squirmer pusher-puller dumbbells

The properties of microswimmer dumbbells composed of pusher-puller pairs are investigated by mesoscale hydrodynamic simulations employing the multiparticle collision dynamics approach for the fluid. An individual microswimmer is represented by a squirmer, and various active-stress combinations in a dumbbell are considered. The squirmers are connected by a bond, which does not impose any geometrical restriction on the individual rotational motion. Our simulations reveal a strong influence of the squirmers' flow fields on the orientation of their propulsion directions, their fluctuations, and the swimming behavior of a dumbbell. The properties of pusher-puller pairs with equal magnitude of the active stresses dependent only weakly on the stress magnitude. This is similar to dumbbells of microswimmers without hydrodynamic interactions. However, for non-equal stress magnitudes, the active stress implies strong orientational correlations of the swimmers' propulsion directions with respect to each other as well as the bond vector. The orientational coupling is most pronounced for pairs with large differences of the active stress magnitude. The alignment of the squirmer propulsion directions with respect to each other is preferentially orthogonal in dumbbells with a strong pusher and weak puller, and antiparallel in the opposite case when the puller dominates. These strong correlations affect the active motion of dumbbells which is faster for strong pushers and slower for strong pullers.

Hydrodynamics of an Elliptical Squirmer

In this paper the propulsion of elliptical objects (called squirmers) by imposed tangential velocity along the surface is studied. For a symmetric velocity distribution (a neutral squirmer), pushers (increased tangential velocity on the downstream side of the ellipse) and pullers (increased tangential velocity on the upstream side of the ellipse), the hydrodynamic characteristics, are simulated numerically using the immersed boundary-lattice Boltzmann method. The accuracy of the numerical scheme and code are validated. The effects of Reynolds number (Re) and squirmer aspect ratio (AR) on the velocity u*, power expenditure P* and hydrodynamic efficiency η of the squirmer are explored. The results show that the change of u* along radial direction r* shows the relation of u*~r*−2 for the neutral squirmer, and u*~r*−1 for the pusher and puller. With the increase of Re, u* of the pusher increases monotonically, but u* of the puller decreases from Re = 0.01 to 0.3, and then increases from Re = 0.3 to 3. The values of P* of the pusher and puller are the same for 0.01 ≤ Re ≤ 0.3; P* of the pusher is larger than that of the puller when Re > 0.3. η of the pusher and puller increases with increasing Re, but the pusher has a larger η than the puller at the same Re. u* and P* decrease with increasing AR, and the pusher and puller have the largest and least u*, respectively. The values of P* of the pusher and puller are almost the same and are much larger than those of the neutral squirmer. With the increase of AR, η increases for the neutral squirmer, but changes non-monotonically for the pusher and puller.

Gyrotactic cluster formation of bottom-heavy squirmers

Squirmers that are bottom-heavy experience a torque that aligns them along the vertical so that they swim upwards. In a suspension of many squirmers, they also interact hydrodynamically via flow fields that are initiated by their swimming motion and by gravity. Swimming under the combined action of flow field vorticity and gravitational torque is called gyrotaxis. Using the method of multi-particle collision dynamics, we perform hydrodynamic simulations of a many-squirmer system floating above the bottom surface. Due to gyrotaxis they exhibit pronounced cluster formation with increasing gravitational torque. The clusters are more volatile at low values but compactify to smaller clusters at larger torques. The mean distance between clusters is mainly controlled by the gravitational torque and not the global density. Furthermore, we observe that neutral squirmers form clusters more easily, whereas pullers require larger gravitational torques due to their additional force-dipole flow fields. We do not observe clustering for pusher squirmers. Adding a rotlet dipole to the squirmer flow field induces swirling clusters. At high gravitational strengths, the hydrodynamic interactions with the no-slip boundary create an additional vertical alignment for neutral squirmers, which also supports cluster formation.

Simulating wet active polymers by multiparticle collision dynamics

The conformational and dynamical properties of active Brownian polymers embedded in a fluid depend on the nature of the driving mechanism, e.g., self-propulsion or external actuation of the monomers. Implementations of self-propelled and actuated active Brownian polymers in a multiparticle collision (MPC) dynamics fluid are presented, which capture the distinct differences between the two driving mechanisms. The active force-free nature of self-propelled monomers requires adaptations of the MPC simulation scheme, with its streaming and collision steps, where the monomer self-propulsion velocity has to be omitted in the collision step. Comparison of MPC simulation results for active polymers in dilute solution with results of Brownian dynamics simulations accounting for hydrodynamics via the Rotne-Prager-Yamakawa tensor confirm the suitability of the implementation. The polymer conformational and dynamical properties are analyzed by the static and dynamic structure factor, and the scaling behavior of the latter with respect to the wave number and time dependence are discussed. The dynamic structure factor displays various activity-induced temporal regimes, depending on the considered wave number, which reflect the persistent diffusive motion of the whole polymer at small wave numbers, and the activity-enhanced internal dynamics at large wave numbers. The obtained simulation results are compared with theoretical predictions.

Hydrodynamics of immiscible binary fluids with viscosity contrast: a multiparticle collision dynamics approach

We present a multiparticle collision dynamics (MPC) implementation of layered immiscible fluids A and B of different shear viscosities separated by planar interfaces. The simulated flow profile for imposed steady shear motion and the time-dependent shear stress functions are in excellent agreement with our continuum hydrodynamics results for the composite fluid. The wave-vector dependent transverse velocity auto-correlation functions (TVAF) in the bulk-fluid regions of the layers decay exponentially, and agree with those of single-phase isotropic MPC fluids. In addition, we determine the hydrodynamic mobilities of an embedded colloidal sphere moving steadily parallel or transverse to a fluid-fluid interface, as functions of the distance from the interface. The obtained mobilities are in good agreement with hydrodynamic force multipoles calculations, for a no-slip sphere moving under creeping flow conditions near a clean, ideally flat interface. The proposed MPC fluid-layer model can be straightforwardly implemented, and it is computationally very efficient. Yet, owing to the spatial discretization inherent to the MPC method, the model can not reproduce all hydrodynamic features of an ideally flat interface between immiscible fluids.

An effective and efficient model of the near-field hydrodynamic interactions for active suspensions of bacteria

Active suspensions of microswimmers demonstrate novel emergent behaviors (self-organizations, active turbulence, etc.) on macroscopic length scales. For such systems with, minimally, thousands of microswimmers, direct numerical simulations of the hydrodynamic interactions are computationally infeasible, and reduced models are needed. We demonstrated that existing models are not satisfactory in describing the hydrodynamic interactions for microswimmers in close proximity with even qualitatively erroneous predictions, indicating a pressing need for an adequate model. We propose a model that is both physically effective and computationally efficient in describing such hydrodynamics. The main novelty of our model is the description of hydrodynamic interactions through a resistance tensor, as opposed to an effective steric interaction in existing models.

Near-field hydrodynamic interactions in active fluids are essential to determine many important emergent behaviors observed, but have not been successfully modeled so far. In this work, we propose an effective model capturing the essence of the near-field hydrodynamic interactions through a tensorial coefficient of resistance, validated numerically by a pedagogic model system consisting of an Escherichia coli bacterium and a passive sphere. In a critical test case that studies the scattering angle of the bacterium–sphere pair dynamics, we prove that the near-field hydrodynamics can make a qualitative difference even for this simple two-body system: Calculations based on the proposed model reveal a region in parameter space where the bacterium is trapped by the passive sphere, a phenomenon that is regularly observed in experiments but cannot be explained by any existing model. In the end, we demonstrate that our model also leads to efficient simulation of active fluids with tens of thousands of bacteria, sufficiently large for investigations of many emergent behaviors.

Multi-ciliated microswimmers-metachronal coordination and helical swimming

The dynamics and motion of multi-ciliated microswimmers with a spherical body and a small number N (with [Formula: see text]) of cilia with length comparable to the body radius, is investigated by mesoscale hydrodynamics simulations. A metachronal wave is imposed for the cilia beat, for which the wave vector has both a longitudinal and a latitudinal component. The dynamics and motion is characterized by the swimming velocity, its variation over the beat cycle, the spinning velocity around the main body axis, as well as the parameters of the helical trajectory. Our simulation results show that the microswimmer motion strongly depends on the latitudinal wave number and the longitudinal phase lag. The microswimmers are found to swim smoothly and usually spin around their own axis. Chirality of the metachronal beat pattern generically generates helical trajectories. In most cases, the helices are thin and stretched, i.e., the helix radius is about an order of magnitude smaller than the pitch. The rotational diffusion of the microswimmer is significantly smaller than the passive rotational diffusion of the body alone, which indicates that the extended cilia contribute strongly to the hydrodynamic radius. The swimming velocity is found to increase with the cilia number N with a slightly sublinear power law, consistent with the behavior expected from the dependence of the transport velocity of planar cilia arrays on the cilia separation.

Multiparticle collision dynamics simulations of a squirmer in a nematic fluid

We study the dynamics of a squirmer in a nematic liquid crystal using the multiparticle collision dynamics (MPCD) method. A recently developed nematic MPCD method [Phys. Rev. E 99, 063319 (2019)] which employs a tensor order parameter to describe the spatial and temporal variations of the nematic order is used to simulate the suspending anisotropic fluid. Considering both nematodynamic effects (anisotropic viscosity and elasticity) and thermal fluctuations, in the present study, we couple the nematic MPCD algorithm with a molecular dynamics (MD) scheme for the squirmer. A unique feature of the proposed method is that the nematic order, the fluid, and the squirmer are all represented in a particle-based framework. To test the applicability of this nematic MPCD-MD method, we simulate the dynamics of a spherical squirmer with homeotropic surface anchoring conditions in a bulk domain. The importance of anisotropic viscosity and elasticity on the squirmer's speed and orientation is studied for different values of self-propulsion strength and squirmer type (pusher, puller or neutral). In sharp contrast to Newtonian fluids, the speed of the squirmer in a nematic fluid depends on the squirmer type. Interestingly, the speed of a strong pusher is smaller in the nematic fluid than for the Newtonian case. The orientational dynamics of the squirmer in the nematic fluid also shows a non-trivial dependence on the squirmer type. Our results compare well with existing experimental and numerical data. The full particle-based framework could be easily extended to model the dynamics of multiple squirmers in anisotropic fluids.

Chlamydomonas reinhardtii swimming in the Plateau borders of 2D foams

Unicellular Chlamydomonas reinhardtii micro-algae cells were inserted into a quasi-2D Hele-Shaw chamber filled with saponin foam. The movement of the algae along the bubble borders was then manipulated and tracked. These self-propelled particles generate flow and stresses in their surrounding matter. In addition, the algae possess the capability of exerting forces that alter bubble boundaries while maintaining an imminent phototactic movement. We find that by controlling the gas fraction of the foam we can change the interaction of the algae and bubbles. Specifically, our data expose three distinct swimming regimes for the algae with respect to the level of confinement due to the Plateau border cross-section: unlimited bulk, transition, and overdamped regimes. At the transition regime we find the speed of the algae to be modeled by a simple force balance equation emerging from the shear inside the Plateau border. Thus, we have shown that it is possible to create an algae-friendly foam while controlling the algae motion. This opens doors to multiple applications where the flow of nutrients, oxygen and recirculation of living organisms is essential.

Hydrodynamic interactions in squirmer dumbbells: active stress-induced alignment and locomotion

Hydrodynamic interactions are fundamental for the dynamics of swimming self-propelled particles. Specifically, bonds between microswimmers enforce permanent spatial proximity and, thus, enhance emergent correlations by microswimmer-specific flow fields. We employ the squirmer model to study the swimming behavior of microswimmer dumbbells by mesoscale hydrodynamic simulations, where the squirmers' rotational motion is geometrically unrestricted. An important aspect of the applied particle-based simulation approach-the multiparticle collision dynamics method-is the intrinsic account for thermal fluctuations. We find a strong effect of active stress on the motility of dumbbells. In particular, pairs of strong pullers exhibit orders of magnitude smaller swimming efficiency than pairs of pushers. This is a consequence of the inherent thermal fluctuations in combination with the strong coupling of the squirmers' rotational motion, which implies non-exponentially decaying auto- and cross-correlation functions of the propulsion directions, and active stress-dependent characteristic decay times. As a consequence, specific stationary-state relative alignments of the squirmer propulsion directions emerge, where pullers are preferentially aligned in an antiparallel manner along the bond vector, whereas pushers are preferentially aligned normal to the bond vector with a relative angle of approximately 60° at weak active stress, and one of the propulsion directions is aligned with the bond at strong active stress. The distinct differences between dumbbells comprised of pusher or pullers suggest means to control microswimmer assemblies for future microbot applications.

A particle-field approach bridges phase separation and collective motion in active matter

Whereas self-propelled hard discs undergo motility-induced phase separation, self-propelled rods exhibit a variety of nonequilibrium phenomena, including clustering, collective motion, and spatio-temporal chaos. In this work, we present a theoretical framework representing active particles by continuum fields. This concept combines the simplicity of alignment-based models, enabling analytical studies, and realistic models that incorporate the shape of self-propelled objects explicitly. By varying particle shape from circular to ellipsoidal, we show how nonequilibrium stresses acting among self-propelled rods destabilize motility-induced phase separation and facilitate orientational ordering, thereby connecting the realms of scalar and vectorial active matter. Though the interaction potential is strictly apolar, both, polar and nematic order may emerge and even coexist. Accordingly, the symmetry of ordered states is a dynamical property in active matter. The presented framework may represent various systems including bacterial colonies, cytoskeletal extracts, or shaken granular media.

Interacting self-propelled particles exhibit phase separation or collective motion depending on particle shape. A unified theory connecting these paradigms represents a major challenge in active matter, which the authors address here by modeling active particles as continuum fields.

Squirmer rods as elongated microswimmers: flow fields and confinement

Microswimmers or active elements, such as bacteria and active filaments, have an elongated shape, which determines their individual and collective dynamics. There is still a need to identify what role long-range hydrodynamic interactions play in their fascinating dynamic structure formation. We construct rods of different aspect ratios using several spherical squirmer model swimmers. With the help of the mesoscale simulation method of multi-particle collision dynamics we analyze the flow fields of these squirmer rods both in a bulk fluid and in Hele-Shaw geometries of different slab widths. Based on the hydrodynamic multipole expansion either for bulk or confinement between two parallel plates, we categorize the different multipole contributions of neutral as well as pusher-type squirmer rods. We demonstrate how confinement alters the radial decay of the flow fields for a given force or source multipole moment compared to the bulk fluid.

Emergent collective dynamics of bottom-heavy squirmers under gravity

We present the results of hydrodynamic simulations using the method of multi-particle collision dynamics for a system of squirmer microswimmers moving under the influence of gravity at low Reynolds numbers. In addition, the squirmers are bottom-heavy so that they experience a torque which aligns them along the vertical. The squirmers interact hydrodynamically by the flow fields of a stokeslet and rotlet, which are initiated by the acting gravitational force and torque, respectively, and by their own flow fields. By varying the ratio of swimming to bulk sedimentation velocity and the torque, we determine state diagrams for the emergent collective dynamics of neutral squirmers as well as strong pushers and pullers. For low swimming velocity and torque we observe conventional sedimentation, while the sedimentation profile becomes inverted when their values are increased. For neutral squirmers we discover convective rolls of circulating squirmers between both sedimentation states, which sit at the bottom of the system and are fed by plumes made of collectively sinking squirmers. At larger torques porous clusters occur that spawn single squirmers. The two latter states can also occur transiently starting from a uniform squirmer distribution and then disappear in the long-time limit. For strong pushers and pullers only weak plume formation is observed.

Axisymmetric spheroidal squirmers and self-diffusiophoretic particles

We study, by means of an exact analytical solution, the motion of a spheroidal, axisymmetric squirmer in an unbounded fluid, as well as the low Reynolds number hydrodynamic flow associated to it. In contrast to the case of a spherical squirmer-for which, e.g. the velocity of the squirmer and the magnitude of the stresslet associated with the flow induced by the squirmer are respectively determined by the amplitudes of the first two slip ('squirming') modes-for the spheroidal squirmer each squirming mode either contributes to the velocity, or contributes to the stresslet. The results are straightforwardly extended to the self-phoresis of axisymmetric, spheroidal, chemically active particles in the case when the phoretic slip approximation holds.

Reconfigurable structure and tunable transport in synchronized active spinner materials

Ensembles of actuated colloids are excellent model systems to explore emergent out-of-equilibrium structures, complex collective dynamics, and design rules for the next generation materials. Here, we demonstrate that ferromagnetic microparticles suspended at an air-water interface and energized by an external rotating magnetic field spontaneously form dynamic ensembles of synchronized spinners in a certain range of the excitation field parameters. Each spinner generates strong hydrodynamic flows, and collective interactions of the multiple spinners promote a formation of dynamic lattices. On the basis of experiments and simulations, we reveal structural transitions from liquid to nearly crystalline states in this novel active spinner material and demonstrate that dynamic spinner lattices are reconfigurable, capable of self-healing behavior and that the transport of embedded inert cargo particles can be remotely tuned by the parameters of the external excitation field. Our findings provide insights into the behavior of active spinner materials with reconfigurable structural order and tunable functionalities.

Enhanced Rotational Motion of Spherical Squirmer in Polymer Solutions

The rotational diffusive motion of a self-propelled, attractive spherical colloid immersed in a solution of self-avoiding polymers is studied by mesoscale hydrodynamic simulations. A drastic enhancement of the rotational diffusion by more than an order of magnitude in the presence of activity is obtained. The amplification is a consequence of two effects, a decrease of the amount of adsorbed polymers by active motion and an asymmetric encounter with polymers on the squirmer surface, which yields an additional torque and random noise for the rotational motion. Our simulations suggest a way to control the rotational dynamics of squirmer-type microswimmers by the degree of polymer adsorption and system heterogeneity.

Particle-resolved lattice Boltzmann simulations of 3-dimensional active turbulence

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 × 106) 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 turbulent state itself. In the dilute limit, we test analytical predictions for a number of static and dynamic properties against our simulation results. For higher swimmer densities, where swimmer-swimmer interactions become significant, we numerically show that the length- and timescales of the turbulent flows increase steeply near the predicted finite-system transition density.

Frequency-dependent higher-order Stokes singularities near a planar elastic boundary: Implications for the hydrodynamics of an active microswimmer near an elastic interface

The emerging field of self-driven active particles in fluid environments has recently created significant interest in the biophysics and bioengineering communities owing to their promising future for biomedical and technological applications. These microswimmers move autonomously through aqueous media, where under realistic situations they encounter a plethora of external stimuli and confining surfaces with peculiar elastic properties. Based on a far-field hydrodynamic model, we present an analytical theory to describe the physical interaction and hydrodynamic couplings between a self-propelled active microswimmer and an elastic interface that features resistance toward shear and bending. We model the active agent as a superposition of higher-order Stokes singularities and elucidate the associated translational and rotational velocities induced by the nearby elastic boundary. Our results show that the velocities can be decomposed in shear and bending related contributions which approach the velocities of active agents close to a no-slip rigid wall in the steady limit. The transient dynamics predict that contributions to the velocities of the microswimmer due to bending resistance are generally more pronounced than those due to shear resistance. Bending can enhance (suppress) the velocities resulting from higher-order singularities whereas the shear related contribution decreases (increases) the velocities. Most prominently, we find that near an elastic interface of only energetic resistance toward shear deformation, such as that of an elastic capsule designed for drug delivery, a swimming bacterium undergoes rotation of the same sense as observed near a no-slip wall. In contrast to that, near an interface of only energetic resistance toward bending, such as that of a fluid vesicle or liposome, we find a reversed sense of rotation. Our results provide insight into the control and guidance of artificial and synthetic self-propelling active microswimmers near elastic confinements.

Active Brownian filaments with hydrodynamic interactions: conformations and dynamics

The conformational and dynamical properties of active self-propelled filaments/polymers are investigated in the presence of hydrodynamic interactions by both, Brownian dynamics simulations and analytical theory. Numerically, a discrete linear chain composed of active Brownian particles is considered, analytically, a continuous linear semiflexible polymer with active velocities changing diffusively. The force-free nature of active monomers is accounted for-no Stokeslet fluid flow induced by active forces-and higher order hydrodynamic multipole moments are neglected. Hence, fluid-mediated interactions are assumed to arise solely due to intramolecular forces. The hydrodynamic interactions (HI) are taken into account analytically by the preaveraged Oseen tensor, and numerically by the Rotne-Prager-Yamakawa tensor. The nonequilibrium character of the active process implies a dependence of the stationary-state properties on HI via the polymer relaxation times. In particular, at moderate activities, HI lead to a substantial shrinkage of flexible and semiflexible polymers to an extent far beyond shrinkage of comparable free-draining polymers; even flexible HI-polymers shrink, while active free-draining polymers swell monotonically. Large activities imply a reswelling, however, to a less extent than for non-HI polymers, caused by the shorter polymer relaxation times due to hydrodynamic interactions. The polymer mean square displacement is enhanced, and an activity-determined ballistic regime appears. Over a wide range of time scales, flexible active polymers exhibit a hydrodynamically governed subdiffusive regime, with an exponent significantly smaller than that of the Rouse and Zimm models of passive polymers. Compared to simulations, the analytical approach predicts a weaker hydrodynamic effect. Overall, hydrodynamic interactions modify the conformational and dynamical properties of active polymers substantially.

A lattice Boltzmann model for squirmers

The squirmer is a simple yet instructive model for microswimmers, which employs an effective slip velocity on the surface of a spherical swimmer to describe its self-propulsion. We solve the hydrodynamic flow problem with the lattice Boltzmann (LB) method, which is well-suited for time-dependent problems involving complex boundary conditions. Incorporating the squirmer into LB is relatively straightforward, but requires an unexpectedly fine grid resolution to capture the physical flow fields and behaviors accurately. We demonstrate this using four basic hydrodynamic tests: two for the far-field flow-accuracy of the hydrodynamic moments and squirmer-squirmer interactions-and two that require the near field to be accurately resolved-a squirmer confined to a tube and one scattering off a spherical obstacle-which LB is capable of doing down to the grid resolution. We find good agreement with (numerical) results obtained using other hydrodynamic solvers in the same geometries and identify a minimum required resolution to achieve this reproduction. We discuss our algorithm in the context of other hydrodynamic solvers and present an outlook on its application to multi-squirmer problems.

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.

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.

Maximum in density heterogeneities of active swimmers

Suspensions of unicellular microswimmers such as flagellated bacteria or motile algae can exhibit spontaneous density heterogeneities at large enough concentrations. We introduce a novel model for biological microswimmers that creates the flow field of the corresponding microswimmers, and takes into account the shape anisotropy of the swimmer's body and stroke-averaged flagella. By employing multiparticle collision dynamics, we directly couple the swimmer's dynamics to the fluid's. We characterize the nonequilibrium phase diagram, as the filling fraction and Péclet number are varied, and find density heterogeneities in the distribution of both pullers and pushers, due to hydrodynamic instabilities. We find a maximum degree of clustering at intermediate filling fractions and at large Péclet numbers resulting from a competition of hydrodynamic and steric interactions between the swimmers. We develop an analytical theory that supports these results. This maximum might represent an optimum for the microorganisms' colonization of their environment.

Beating of grafted chains induced by active Brownian particles

We study the interplay between active Brownian particles (ABPs) and a "hairy" surface in two-dimensional geometry. We find that the increase of propelling force leads to and enhances inhomogeneous accumulation of ABPs inside the brush region. Oscillation of chain bundles (beating like cilia) is found in company with the formation and disassembly of a dynamic cluster of ABPs at large propelling forces. Meanwhile chains are stretched and pushed down due to the effective shear force by ABPs. The decrease of the average brush thickness with propelling force reflects the growth of the beating amplitude of chain bundles. Furthermore, the beating phenomenon is investigated in a simple single-chain system. We find that the chain swings regularly with a major oscillatory period, which increases with chain length and decreases with the increase of propelling force. We build a theory to describe the phenomenon and the predictions on the relationship between the period and amplitude for various chain lengths, and propelling forces agree very well with simulation data.

Simulating squirmers with multiparticle collision dynamics

Multiparticle collision dynamics is a modern coarse-grained simulation technique to treat the hydrodynamics of Newtonian fluids by solving the Navier-Stokes equations. Naturally, it also includes thermal noise. Initially it has been applied extensively to spherical colloids or bead-spring polymers immersed in a fluid. Here, we review and discuss the use of multiparticle collision dynamics for studying the motion of spherical model microswimmers called squirmers moving in viscous fluids.

Reduced-order model for inertial locomotion of a slender swimmer

The inertial locomotion of an elongated model swimmer in a Newtonian fluid is quantified, wherein self-propulsion is achieved via steady tangential surface treadmilling. The swimmer has a length 2l and a circular cross section of longitudinal profile aR(z), where a is the characteristic width of the cross section, R(z) is a dimensionless shape function, and z is a dimensionless coordinate, normalized by l, along the centerline of the body. It is assumed that the swimmer is slender, ε=a/l≪1. Hence, we utilize slender-body theory to analyze the Navier-Stokes equations that describe the flow around the swimmer. Therefrom, we compute an asymptotic approximation to the swimming speed, U, as U/u_{s}=1-β[V(Re)-1/2∫_{-1}^{1}zlnR(z)dz]/ln(1/ε)+O[1/ln^{2}(1/ε)], where u_{s} is the characteristic speed of the surface treadmilling, Re is the Reynolds number based on the body length, and β is a dimensionless parameter that differentiates between "pusher" (propelled from the rear, β<0) and "puller" (propelled from the front, β>0) -type swimmers. The function V(Re) increases monotonically with increasing Re; hence, fluid inertia causes an increase (decrease) in the swimming speed of a pusher (puller). Next, we demonstrate that the power expenditure of the swimmer increases monotonically with increasing Re. Further, the power expenditures of a puller and pusher with the same value of |β| are equal. Therefore, pushers are superior in inertial locomotion as compared to pullers, in that they achieve a faster swimming speed for the same power expended. Finally, it is demonstrated that the flow structure predicted from our reduced-order model is consistent with that from direct numerical simulation of swimmers at intermediate Re.

Flagellated microswimmers: Hydrodynamics in thin liquid films

The hydrodynamics of a flagellated microswimmer moving in thin films is discussed. The fully resolved hydrodynamics is exploited by solving the Stokes equations for the actual geometry of the swimmer. Two different interfaces are used to confine the swimmer: a bottom solid wall and a top air-liquid interface, as appropriate for a thin film. The swimmer follows curved clockwise trajectories that can converge towards an asymptotically stable circular path or can result in a collision with one of the two interfaces. A bias towards the air-liquid interface emerges. Slight changes in the swimmer geometry and film thickness strongly affect the resulting dynamics suggesting that a very reach phenomenology occurs in the presence of confinement. Under specific conditions, the swimmer follows a "crown-like" path. Implications for the motion of bacteria close to an air bubble moving in a microchannel are discussed.

Steady state sedimentation of ultrasoft colloids

The structural and dynamical properties of ultra-soft colloids-star polymers-exposed to a uniform external force field are analyzed by applying the multiparticle collision dynamics technique, a hybrid coarse-grain mesoscale simulation approach, which captures thermal fluctuations and long-range hydrodynamic interactions. In the weak-field limit, the structure of the star polymer is nearly unchanged; however, in an intermediate regime, the radius of gyration decreases, in particular transverse to the sedimentation direction. In the limit of a strong field, the radius of gyration increases with field strength. Correspondingly, the sedimentation coefficient increases with increasing field strength, passes through a maximum, and decreases again at high field strengths. The maximum value depends on the functionality of the star polymer. High field strengths lead to symmetry breaking with trailing, strongly stretched polymer arms and a compact star-polymer body. In the weak-field-linear response regime, the sedimentation coefficient follows the scaling relation of a star polymer in terms of functionality and arm length.

Swimming and rafting of E.coli microcolonies at air-liquid interfaces

The dynamics of swimming microorganisms is strongly affected by solid‐liquid and air‐liquid interfaces. In this paper, we characterize the motion of both single bacteria and microcolonies at an air‐liquid interface. Both of them follow circular trajectories. Single bacteria preferentially show a counter‐clockwise motion, in agreement with previous experimental and theoretical findings. Instead, no preferential rotation direction is observed for microcolonies suggesting that their motion is due to a different physical mechanism. We propose a simple mechanical model where the microcolonies move like rafts constrained to the air‐liquid interface. Finally, we observed that the microcolony growth is due to the aggregation of colliding single‐swimmers, suggesting that the microcolony formation resembles a condensation process where the first nucleus originates by the collision between two single‐swimmers. Implications of microcolony splitting and aggregation on biofilm growth and dispersion at air‐liquid interface are discussed.

Diffusivity of E. coli-like microswimmers in confined geometries: The role of the tumbling rate

We analyzed the effect of confinement on the effective diffusion of a run-and-tumble E. coli-like flagellated microswimmer. We used a simulation protocol where the run phases are obtained via a fully resolved swimming problem, i.e., Stokes equations for the fluid coupled with rigid-body dynamics for the microorganism, while tumbles and collisions with the walls are modeled as random reorientation of the microswimmer. For weak confinement, the swimmer is trapped in circular orbits close to the solid walls. In this case, optimal diffusivity is observed when the tumbling frequency is comparable with the angular velocity of the stable orbits. For strong confinement, stable circular orbits disappear and the diffusion coefficient monotonically decreases with the tumbling rate. Our findings are generic and can be potentially applied to other natural or artificial chiral microswimmers that follow circular trajectories close to an interface or in confined geometries.

Noncontact Cohesive Swimming of Bacteria in Two-Dimensional Liquid Films

Bacterial swimming in confined two-dimensional environments is ubiquitous in nature and in clinical settings. Characterizing individual interactions between swimming bacteria in 2D confinement will help to understand diverse microbial processes, such as bacterial swarming and biofilm formation. Here we report a novel motion pattern displayed by flagellated bacteria in 2D confinement: When two nearby cells align their moving directions, they tend to engage in cohesive swimming without direct cell body contact, as a result of hydrodynamic interaction but not flagellar intertwining. We further found that cells in cohesive swimming move with higher directional persistence, which can increase the effective diffusivity of cells by ∼3 times as predicted by computational modeling. As a conserved behavior for peritrichously flagellated bacteria, cohesive swimming in 2D confinement may be key to collective motion and self-organization in bacterial swarms; it may also promote bacterial dispersal in unsaturated soils and in interstitial space during infections.