We develop an exact Green-Kubo formula relating nonequilibrium averages in systems of interacting active Brownian particles to equilibrium time-correlation functions. The method is applied to calculate the density-dependent average swim speed, which is a key quantity entering coarse grained theories of active matter. The average swim speed is determined by integrating the equilibrium autocorrelation function of the interaction force acting on a tagged particle. Analytical results are validated using Brownian dynamics simulations.
Microswimmers exhibit noisy circular motion due to asymmetric propulsion mechanisms, their chiral body shape, or by hydrodynamic couplings in the vicinity of surfaces. Here, we employ the Brownian circle swimmer model and characterize theoretically the dynamics in terms of the directly measurable intermediate scattering function. We derive the associated Fokker-Planck equation for the conditional probabilities and provide an exact solution in terms of generalizations of the Mathieu functions. Different spatiotemporal regimes are identified reflecting the bare translational diffusion at large w…
Typically, in the description of active Brownian particles, a constant effective propulsion force is assumed, which is then subjected to fluctuations in orientation and translation leading to a persistent random walk with an enlarged long-time diffusion coefficient. Here, we generalize previous results for the swimming path statistics to a time-dependent and thus in many situations more realistic propulsion which is a prescribed input. We analytically calculate both the noise-free and the noise-averaged trajectories for time-periodic propulsion under the action of an additional torque. In the …
Active Brownian Particles (ABPs) transmit a swim pressure $Π^{swim}=nζD^{swim}$ to the container boundaries, where $ζ$ is the drag coefficient, $D^{swim}$ is the swim diffusivity and $n$ is the uniform bulk number density far from the container walls. In this work we extend the notion of the isotropic swim pressure to the anisotropic tensorial swim stress $\mathbfσ^{swim} = - n ζ\mathbf{D}^{swim}$, which is related to the anisotropic swim diffusivity $\mathbf{D}^{swim}$. We demonstrate this relationship with ABPs that achieve nematic orientational order via a bulk external field. The anisotrop…
Self-propelled particles move along circles rather than along a straight line when their driving force does not coincide with their propagation direction. Examples include confined bacteria and spermatozoa, catalytically driven nanorods, active, anisotropic colloidal particles and vibrated granulates. Using a non-Hamiltonian rate theory and computer simulations, we study the motion of a Brownian "circle swimmer" in a confining channel. A sliding mode close to the wall leads to a huge acceleration as compared to the bulk motion, which can further be enhanced by an optimal effective torque-to-fo…
Free chiral self-propelled robots compared to active Brownian circle swimmers
Institut für Theoretische Physik, Universität Innsbruck（因斯布鲁克大学理论物理研究所）；The Raymond and Beverley School of Chemistry, Tel Aviv University（特拉维夫大学雷蒙德和贝弗利化学学院）
Hot microswimmers are self-propelled Brownian particles that exploit local heating for their directed self-thermophoretic motion. We provide a pedagogical overview of the key physical mechanisms underlying this promising new technology. It covers the hydrodynamics of swimming, thermophoresis and -osmosis, hot Brownian motion, force-free steering, and dedicated experimental and simulation tools to analyze hot Brownian swimmers.
We consider chirality in active systems by exemplarily studying the phase behavior of planar systems of interacting Brownian circle swimmers with a spherical shape. Continuing previous work presented in [G.-J. Liao, S. H. L. Klapp, Soft Matter, 2018, 14, 7873-7882], we derive a predictive field theory that is able to describe the collective dynamics of circle swimmers. The theory yields a mapping between circle swimmers and noncircling active Brownian particles and predicts that the angular propulsion of the particles leads to a suppression of their motility-induced phase separation, being in …
Active Brownian particles are capable of taking up energy from their environment and converting it into directed motion; examples range from chemotactic cells and bacteria to artificial micro-swimmers. We have recently demonstrated that Janus particles, i.e. gold-capped colloidal spheres, suspended in a critical binary liquid mixture perform active Brownian motion when illuminated by light. In this article, we investigate in some more details their swimming mechanism leading to active Brownian motion. We show that the illumination-borne heating induces a local asymmetric demixing of the binary…
Suspensions of motile active particles with space dependent activity form characteristic polarization and density patterns. Recent single-particle studies for planar activity landscapes identified several quantities associated with emergent density-polarization patterns, including swim pressure, that are solely determined by bulk variables and thus constitute state functions. We show that for radially symmetric activity steps these variables depend on the curvature of the active-passive interface. In the specific case of total polarization and swim pressure, we generalize this result for arbit…
A self-propelled artificial microswimmer is often modeled as a ballistic Brownian particle moving with constant speed aligned along one of its axis, but changing direction due to random collisions with the environment. Similarly to thermal noise, its angular randomization is described as a memoryless stochastic process. Here, we speculate that finite-time correlations in the orientational dynamics can affect the swimmer's diffusivity. To this purpose we propose and solve two alternative models. In the first one we simply assume that the environmental fluctuations governing the swimmer's propul…
A micron-sized droplet of bromine water immersed in a surfactant-laden oil phase can swim (S. Thutupalli, R. Seemann, S. Herminghaus, New J. Phys. 13 073021 (2011)). The bromine reacts with the surfactant at the droplet interface and generates a surfactant mixture. It can spontaneously phase-separate due to solutocapillary Marangoni flow, which propels the droplet. We model the system by a diffusion-advection-reaction equation for the mixture order parameter at the interface including thermal noise and couple it to fluid flow. Going beyond previous work, we illustrate the coarsening dynamics o…
The concept of swim pressure quantifies the average force exerted by microswimmers on confining walls in non-equilibrium. Here we explore how the swim pressure depends on the wall curvature and on the presence of sharp corners in the wall. For active Brownian particles at high dilution, we present a coherent framework which describes the force and torque on passive particles of arbitrary shape, in the limit of large particles compared to the persistence length of the swimmer trajectories. The resulting forces can be used to derive, for example, the activity-induced depletion interaction betwee…
While the behavior of vesicles in thermodynamic equilibrium has been studied extensively, how active forces control vesicle shape transformations is not understood. Here, we combine theory and simulations to study the shape behavior of vesicles containing active Brownian particles. We show that the combination of active forces, dimensionality and membrane bending free energy creates a plethora of novel phase transitions. At low swim pressure, the vesicle exhibits a discontinuous transition from a spherical to a prolate shape, which has no counterpart in two dimensions. At high swim pressure it…
Within Kirkwood theory, we study the translational diffusion coefficient of a single polymer chain in dilute solution, and focus on the small difference between the short--time Kirkwood value $D^{(K)}$ and the asymptotic long--time value $D$. We calculate this correction term by highly accurate large--scale Brownian Dynamics simulations, and show that it is in perfect agreement with the rigorous variational result $D < D^{(K)}$, and with Fixman's Green--Kubo formula, which is re--derived. This resolves the puzzle posed by earlier numerical results (Rey {\em et al.}, Macromolecules 24, 4666 (19…
Swimming microorganisms can influence the diffusion of passive particles. The effect of this swimmer-particle interaction depends on different properties, such as the hydrodynamic field of the swimmer and the relative sizes of microorganisms and particles. We investigated an enhancement of the diffusion of silica doublets in a suspension of microalgae Chlamydomonas reinhardtii in a flat capillary. Depending on the concentration of microswimmers, the translational and rotational diffusion constants increase by several orders of magnitude in the presence of the swimming algae. For low concentrat…
Recently, the Brownian dynamics of self-propelled (active) rod-like particles was explored to model the motion of colloidal microswimmers, catalytically-driven nanorods, and bacteria. Here, we generalize this description to biaxial particles with arbitrary shape and derive the corresponding Langevin equation for a self-propelled Brownian spinning top. The biaxial swimmer is exposed to a hydrodynamic Stokes friction force at low Reynolds numbers, to fluctuating random forces and torques as well as to an external and an internal (effective) force and torque. The latter quantities control its sel…
The dynamics of self-locomotion of active particles in aligned or liquid crystalline fluids strongly deviates from that in simple isotropic media. We explore the long-time dynamics of a swimmer moving in a three-dimensional smectic liquid crystal and find that the mean-square displacement (MSD) transverse to the director exhibits a distinct logarithmic tail at long times. The scaling is distinctly different from that in an isotropic or nematic fluid and hints at the subtle but important role of the director fluctuation spectrum in governing the long-time motility of active particles. Our findi…
The active Brownian particle (ABP) model describes a swimmer, synthetic or living, whose direction of swimming is a Brownian motion. The swimming is due to a propulsion force, and the fluctuations are typically thermal in origin. We present a 2D model where the fluctuations arise from nonthermal noise in a propelling force acting at a single point, such as that due to a flagellum. We take the overdamped limit and find several modifications to the traditional ABP model. Since the fluctuating force causes a fluctuating torque, the diffusion tensor describing the process has a coupling between tr…
We investigate motility-induced phase separation of active Brownian particles, which are modeled as purely repulsive spheres that move due to a constant swim force with freely diffusing orientation. We develop on the basis of power functional concepts an analytical theory for nonequilibrium phase coexistence and interfacial structure. Theoretical predictions are validated against Brownian dynamics computer simulations. We show that the internal one-body force field has four nonequilibrium contributions: (i) isotropic drag and (ii) interfacial drag forces against the forward motion, (iii) a sup…
Chiral active particles (or self-propelled circle swimmers) feature a rich collective behavior, comprising rotating macro-clusters and micro-flock patterns which consist of phase-synchronized rotating clusters with a characteristic self-limited size. These patterns emerge from the competition of alignment interactions and rotations suggesting that they might occur generically in many chiral active matter systems. However, although excluded volume interactions occur naturally among typical circle swimmers, it is not yet clear if macro-clusters and micro-flock patterns survive their presence. Th…
We generalize the Green-Kubo approach, previously applied to bulk systems of spherically symmetric active particles [J. Chem. Phys. 145, 161101 (2016)], to include spatially inhomogeneous activity. The method is applied to predict the spatial dependence of the average orientation per particle and the density. The average orientation is given by an integral over the self-part of the van Hove function and a simple Gaussian approximation to this quantity yields an accurate analytical expression. Taking this analytical result as input to a dynamic density functional theory approximates the spatial…
Magnetotactic swimmers tend to align along magnetic field lines against stochastic reorientations. We show that the swimming strategy, e.g. active Brownian motion versus run-and-tumble dynamics, strongly affects the orientation statistics. The latter can exhibit a velocity condensation whereby the alignment probability density diverges. As a consequence, we find that the swimming strategy affects the nature of the phase transition to collective motion, indicating that Lévy run-and-tumble walks can outperform active Brownian processes as strategies to trigger collective behavior.
Active Brownian particles (ABPs) represent a minimal model of active matter consisting of self-propelled spheres with purely repulsive interactions and rotational noise. Here, we examine the pressure of ABPs in two dimensions in both closed boxes and systems with periodic boundary conditions and show that its nonmonotonic behavior with density is a robust feature, independent of system size. We correlate the time evolution of the mean pressure towards its steady state value with the kinetics of motility-induced phase separation. For parameter values corresponding to phase separated steady stat…
In a theoretical and simulation study, active Brownian particles (ABPs) in three-dimensional bulk systems are exposed to time-varying sinusoidal activity waves that are running through the system. A linear response (Green-Kubo) formalism is applied to derive fully analytical expressions for the torque-free polarization profiles of the particles. The activity waves induce fluxes that strongly depend on the particle size and may be employed to de-mix mixtures of ABPs or to drive the particles into selected areas of the system. Three-dimensional Langevin dynamics simulations are carried out to ve…
Microswimmers often exhibit surprising patterns due to the nonequilibrium nature of their dynamics. Collectively, suspensions of microswimmers appear as a liquid whose properties set it apart from its passive counterpart. To understand the impact of hydrodynamic interactions on the basic statistical features of a microswimmer's liquid, we investigate its structure by means of the pair distribution function. We perform particle-based simulations of microswimmers that include steric effects, shape anisotropy, and hydrodynamic interactions. We find that hydrodynamic interactions considerably alte…
Hydrodynamically interacting active particles in an external harmonic potential form a self-assembled fluid pump at large enough Péclet numbers. Here, we give a quantitative criterion for the formation of the pump and show that particle orientations align in the self-induced flow field in surprising analogy to ferromagnetic order where the active Péclet number plays the role of inverse temperature. The particle orientations follow a Boltzmann distribution $Φ(\mathbf{p}) \sim \exp(A p_z)$ where the ordering mean field $A$ scales with active Péclet number and polar order parameter. The mean flow…
We review recent advances in rectification control of artificial microswimmers, also known as Janus particles, diffusing along narrow, periodically corrugated channels. The swimmer self-propulsion mechanism is modeled so as to incorporate a nonzero torque (propulsion chirality). We first summarize the effects of chirality on the autonomous current of microswimmers freely diffusing in channels of different geometries. In particular, left-right and upside-down asymmetric channels are shown to exhibit different transport properties. We then report new results on the dependence of the diffusivity …
Combining experiments on active colloids, whose propulsion velocity can be controlled via a feedback loop, and theory of active Brownian motion, we explore the dynamics of an overdamped active particle with a motility that depends explicitly on the particle orientation. In this case, the active particle moves faster when oriented along one direction and slower when oriented along another, leading to an anisotropic translational dynamics which is coupled to the particle's rotational diffusion. We propose a basic model of active Brownian motion for orientation-dependent motility. Based on this m…
By combining confocal microscopy and Stress Assessment from Local Structural Anisotropy (SALSA), we directly measure stresses in 3D quiescent colloidal liquids. Our non-invasive and non-perturbative method allows us to measure forces $\lesssim$ 50 fN with a small and tunable probing volume, enabling us to resolve the stress fluctuations arising from particle thermal motions. We use the Green-Kubo relation to relate these measured stress fluctuations to the bulk Brownian viscosity at different volume fractions and comparing against simulations and conventional rheometry measurements. We demonst…
We study quasi two-dimensional, monodisperse systems of active Brownian particles (ABPs) for a range of activities, stiffnesses, and densities. We develop a microscopic, analytical method for predicting the dense phase structure formed after motility-induced phase separation (MIPS) has occurred, including the dense cluster's area fraction, interparticle pressure, and radius. Our predictions are in good agreement with our Brownian dynamics simulations. We, then, derive a continuum model to investigate the relationship between the predicted interparticle pressure, the swim pressure, and the macr…
Starting from the microscopic Smoluchowski equation for interacting Brownian particles under stationary shearing, exact expressions for shear-dependent steady-state averages, correlation and structure functions, and susceptibilities are obtained, which take the form of generalized Green-Kubo relations. They require integration of transient dynamics. Equations of motion with memory effects for transient density fluctuation functions are derived from the same microscopic starting point. We argue that the derived formal expressions provide useful starting points for approximations in order to des…
We find that crystalline states of repulsive active Brownian particles at high activity melt into a hexatic state but this transition is not driven by an unbinding of bound dislocation pairs as suggested by the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory. Upon reducing the density, the crystalline state melts into a high-density hexatic state devoid of any defects. Decreasing the density further, the dislocations proliferate and introduce plasticity in the system, nevertheless maintaining the hexatic state, but eventually melting into a fluid state. Remarkably, the elastic constan…
Biological microswimmers often inhabit a porous or crowded environment such as soil. In order to understand how such a complex environment influences their spreading, we numerically study non-interacting active Brownian particles (ABPs) in a two-dimensional random Lorentz gas. Close to the percolation transition in the Lorentz gas, they perform the same subdiffusive motion as ballistic and diffusive particles. However, due to their persistent motion they reach their long-time dynamics faster than passive particles and also show superdiffusive motion at intermediate times. While above the criti…
Micro- and nano-swimmers moving in a fluid solvent confined by structures that produce entropic barriers are often described by overdamped active Brownian particle dynamics, where viscous effects are large and inertia plays no role. However, inertial effects should be considered for confined swimmers moving in media where viscous effects are no longer dominant. Here, we study how inertia affects the rectification and diffusion of self-propelled particles in a two-dimensional asymmetric channel. We show that most of the particles accumulate at the channel walls as the masses of the particles in…
We discuss recent advances in developing a mode-coupling theory of the glass transition (MCT) of two-dimensional systems of active Brownian particles (ABP). We specifically discuss the case of a single ABP tracer in a glass-forming passive host suspension; a case that has recently been studied in experiments on colloidal Janus particles. We employ event-driven Brownian dynamics (ED-BD) computer simulations to test the ABP-MCT, and find good agreement between the two for the MSD. The ED-BD simulation results also compare well to experimental data, although a peculiar non-monotonic mapping of se…
The spatial and orientational distribution in a dilute active suspension of non-Brownian run-and-tumble spherical swimmers confined between two planar hard walls is calculated theoretically. Using a kinetic model based on coupled bulk/surface probability density functions, we demonstrate the existence of a concentration wall boundary layer with thickness scaling with the run length, the absence of polarization throughout the channel, and the presence of sharp discontinuities in the bulk orientation distribution in the neighborhood of orientations parallel to the wall in the near-wall region. O…
Oscillatory Active Brownian Motion: A Minimal Model for Sperm Dynamics
School of Biomedical and Chemical Engineering, Colorado State University（科罗拉多州立大学生物医学与化学工程学院）；Instituto de Biología y Medicina Experimental (IBYME-CONICET)（实验生物学与医学研究所 (IBYME-CONICET)）
Suspensions of self-propelled bodies generate a unique mechanical stress owing to their motility that impacts their large-scale collective behavior. For microswimmers suspended in a fluid with negligible particle inertia, we have shown that the virial `swim stress' is a useful quantity to understand the rheology and nonequilibrium behaviors of active soft matter systems. For larger self-propelled organisms like fish, it is unclear how particle inertia impacts their stress generation and collective movement. Here, we analyze the effects of finite particle inertia on the mechanical pressure (or …
Self-propelled Brownian particles show rich out-of-equilibrium physics, for instance, the motility-induced phase separation (MIPS). While decades of studying the structure of liquids have established a deep understanding of passive systems, not much is known about correlations in active suspensions. In this work we derive an approximate analytic theory for three-body correlations and forces in systems of active Brownian disks starting from the many-body Smoluchowski equation. We use our theory to predict the conditional forces that act on a tagged particle and their dependence on the propulsio…
Most classical work on the hydrodynamics of low-Reynolds-number swimming addresses deterministic locomotion in quiescent environments. Thermal fluctuations in fluids are known to lead to a Brownian loss of the swimming direction. As most cells or synthetic swimmers are immersed in external flows, we consider theoretically in this paper the stochastic dynamics of a model active particle (a self-propelled sphere) in a steady general linear flow. The stochasticity arises both from translational diffusion in physical space, and from a combination of rotary diffusion and run-and-tumble dynamics in …
We study the steady-state behavior of active, dipolar, Brownian spheroids in a planar channel subjected to an imposed Couette flow and an external transverse field, applied in the 'downward' normal-to-flow direction. The field-induced torque on active spheroids (swimmers) is taken to be of magnetic form by assuming that they have a permanent magnetic dipole moment, pointing along their self-propulsion (swim) direction. Using a continuum approach, we show that a host of behaviors emerge over the parameter space spanned by the particle aspect ratio, self-propulsion and shear/field strengths, and…
Self-propelled particles in an external gravitational field have been shown to display both an increased sedimentation length and polar order even without particle interactions. Here, we investigate self-propelled particles which additionally are bottom-heavy, that is they feel a torque aligning them to swim against the gravitational field. For bottom-heavy particles the gravitational field has the two opposite effects of i) sedimentation and ii) upward alignment of the particles' swimming direction. We perform a multipole expansion of the one-particle distribution with respect to orientation …
We present a mode-coupling theory (MCT) for the high-density dynamics of two-dimensional spherical active Brownian particles (ABP). The theory is based on the integration-through-transients (ITT) formalism and hence provides a starting point for the calculation of non-equilibrium averages in active-Brownian particle systems. The ABP are characterized by a self-propulsion velocity $v_0$, and by their translational and rotational diffusion coefficients, $D_t$ and $D_r$. The theory treats both the translational and the orientational degrees of freedom of ABP explicitly. This allows to study the e…
We perform a coarse-graining analysis of the paradigmatic active matter model, Active Brownian Particles, yielding a continuum description in terms of balance laws for mass, linear and angular momentum, and energy. The derivation of the balance of linear momentum reveals that the active force manifests itself directly as a continuum-level body force proportional to an order parameter-like director field, which therefore requires its own evolution equation to complete the continuum description of the system. We derive this equation, demonstrating in the process that bulk currents may be sustain…
Particles kicked by external forces to produce mobility distinct from thermal diffusion are an iconic feature of the active matter problem. Here, we map this onto a minimal model for experiment and theory covering the wide time and length scales of usual active matter systems. A particle diffusing in a harmonic potential generated by an optical trap is kicked by programmed forces with time correlation at random intervals following the Poisson process. The model's generic simplicity allows us to find conditions for which displacements are Gaussian (or not), how diffusion is perturbed (or not) b…
We present a predictive local field theory for the nonequilibrium dynamics of interacting active Brownian particles with a spherical shape in two spatial dimensions. The theory is derived by a rigorous coarse-graining starting from the Langevin equations that describe the trajectories of the individual particles. For maximal accuracy and generality of the theory, it includes configurational order parameters and derivatives up to infinite order. In addition, we discuss possible approximations of the theory and present reduced models that are easier to apply. We show that our theory contains pop…
Differently from passive Brownian particles, active particles, also known as self-propelled Brownian particles or microswimmers and nanoswimmers, are capable of taking up energy from their environment and converting it into directed motion. Because of this constant flow of energy, their behavior can only be explained and understood within the framework of nonequilibrium physics. In the biological realm, many cells perform directed motion, for example, as a way to browse for nutrients or to avoid toxins. Inspired by these motile microorganisms, researchers have been developing artificial partic…
Ratchet effects can arise for single or collectively interacting Brownian particles on an asymmetric substrate when a net dc transport is produced by an externally applied ac driving force or by periodically flashing the substrate. Recently, a new class of active ratchet systems has been realized through the use of active matter, which are self-propelled units that can be biological or non-biological in nature. When active materials such as swimming bacteria interact with an asymmetric substrate, a net dc directed motion can arise even without external driving, opening a wealth of possibilitie…
