Active particles in complex and crowded environments

Clemens Bechinger, Roberto Di Leonardo, Hartmut Löwen, Charles Reichhardt, Giorgio Volpe, and Giovanni Volpe

Rev. Mod. Phys. 88, 045006 – Published 23 November 2016

Abstract

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 be explained and understood only 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 particles that feature similar swimming behaviors based on different mechanisms. These man-made micromachines and nanomachines hold a great potential as autonomous agents for health care, sustainability, and security applications. With a focus on the basic physical features of the interactions of self-propelled Brownian particles with a crowded and complex environment, this comprehensive review will provide a guided tour through its basic principles, the development of artificial self-propelling microparticles and nanoparticles, and their application to the study of nonequilibrium phenomena, as well as the open challenges that the field is currently facing.

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Received 30 January 2016

DOI:https://doi.org/10.1103/RevModPhys.88.045006

Physics Subject Headings (PhySH)

Active Brownian particles Living matter & active matter Non-Newtonian fluids

Physics of Living SystemsStatistical Physics & Thermodynamics

Authors & Affiliations

Clemens Bechinger

Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany and Max-Planck-Institut für Intelligente Systeme, Heisenbergstraße 3, 70569 Stuttgart, Germany

Roberto Di Leonardo

Dipartimento di Fisica, Università “Sapienza,” I-00185, Roma, Italy and NANOTEC-CNR Institute of Nanotechnology, Soft and Living Matter Laboratory, I-00185 Roma, Italy

Hartmut Löwen

Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany

Charles Reichhardt

Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

Giorgio Volpe

Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, United Kingdom

Giovanni Volpe^*

Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden and Soft Matter Lab, Department of Physics, and UNAM—National Nanotechnology Research Center, Bilkent University, Ankara 06800, Turkey

^*giovanni.volpe@physics.gu.se

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Vol. 88, Iss. 4 — October - December 2016

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Figure 1
Self-propelled Brownian particles are biological or man-made objects capable of taking up energy from their environment and converting it into directed motion. They are microscopic and nanoscopic in size and have propulsion speeds (typically) up to a fraction of a millimeter per second. The letters correspond to the artificial microswimmers in Table 1. The insets show examples of biological and artificial swimmers. For the artificial swimmers four main recurrent geometries can be identified so far: Janus rods, Janus spheres, chiral particles, and vesicles.
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Figure 2
Active Brownian particles in two dimensions. (a) An active Brownian particle in water ( $R = 1 μ m$ , $η = 0.001 Pa s$ ) placed at position $[x, y]$ is characterized by an orientation $φ$ along which it propels itself with speed $v$ while undergoing Brownian motion in both position and orientation. The resulting trajectories are shown for different velocities (b) $v = 0 μ m s^{- 1}$ (Brownian particle), (c) $v = 1 μ m s^{- 1}$ , (d) $v = 2 μ m s^{- 1}$ , and (e) $v = 3 μ m s^{- 1}$ . With increasing values of $v$ , the active particles move over longer distances before their direction of motion is randomized; four different 10-s trajectories are shown for each value of velocity.
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Figure 3
Biological and artificial chiral active Brownian motion. (a) Phase-contrast video microscopy images showing E. coli cells swimming in circular trajectories near a glass surface. Superposition of 8-s video images. From [221]. (b) Circular trajectories are also observed for E. coli bacteria swimming over liquid-air interfaces but the direction is reversed. From [95]. Trajectories of (c) dextrogyre and (d) levogyre artificial microswimmers driven by self-diffusiophoresis: in each plot, the red bullet corresponds to the initial particle position and the two blue squares to its position after 1 and 2 minutes. The insets show microscope images of two different swimmers with the Au coating (not visible in the bright-field image) indicated by an arrow. From [216].
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Figure 4
Chiral active Brownian motion in two dimensions. (a) A two-dimensional chiral active Brownian particle has a deterministic angular velocity $ω$ that, if the particle’s speed $v > 0$ , entails a rotation around an effective external axis. (b)–(d) Sample trajectories of dextrogyre (red, dark gray) and levogyre (yellow, light gray) active chiral particles with $v = 30 μ m s^{- 1}$ , $ω = 10 rad s^{- 1}$ , and different radii [ $R = 1000$ , 500, and 250 nm for (b), (c), and (d), respectively]. As the particle size decreases, the trajectories become less deterministic because the rotational diffusion, responsible for the reorientation of the particle direction, scales according to $R^{- 3}$ [Eq. (2)].
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Figure 5
Sample trajectories of active Brownian particles corresponding to different mechanisms generating active motion: (a) rotational diffusion dynamics, (b) run-and-tumble dynamics, and (c) Gaussian noise dynamics. The dots correspond to the particle position sampled every 5 s.
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Figure 6
Mean square displacement (MSD) of active Brownian particles and effective diffusion coefficients. Numerically calculated (symbols) and theoretical (lines) MSD for an active Brownian particle with velocity $v = 0 μ m s^{- 1}$ (circles), $v = 1 μ m s^{- 1}$ (triangles), $v = 2 μ m s^{- 1}$ (squares), and $v = 3 μ m s^{- 1}$ (diamonds). For a passive Brownian particle ( $v = 0 μ m s^{- 1}$ , circles) the motion is always diffusive [ $MSD (τ) \propto τ$ ], while for an active Brownian particle the motion is diffusive with diffusion coefficient $D_{T}$ at very short time scales [ $MSD (τ) \propto τ$ for $τ ≪ τ_{R}$ ], ballistic at intermediate time scales [ $MSD (τ) \propto τ^{2}$ for $τ \approx τ_{R}$ ], and again diffusive but with an enhanced diffusion coefficient at long time scales [ $MSD (τ) \propto τ$ for $τ ≫ τ_{R}$ ].
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Figure 7
The scallop theorem. Schematic drawing of a scallop performing a reciprocal motion: by quickly closing its two shells, the scallop ejects two fluid jets behind its hinge so that its body recoils in the opposite direction; then, by slowly opening the shells, the scallop comes back to its initial shape with little net displacement. The scallop theorem states that such a reciprocal motion cannot generate propulsion in a (viscous) fluid at a low-Reynolds-number regime ([318]). From [321].
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Figure 8
Flow singularities. (a) Stokeslet corresponding to the far field of an active particle driven by an external force. (b), (c) Stokes dipoles corresponding to active particles driven by an internal force corresponding to (b) pushers and (c) pullers, both moving horizontally.
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Figure 9
Flow fields created by swimmers at low Reynolds numbers. (a) Pushers have a positive force dipole ( $P > 0$ ) and induce an outgoing flow field directed along their swimming direction (repulsion) and an incoming flow field from their sides (attraction). In all panels, the solid red arrows represent the local forcing from the swimmer on the surrounding fluid and the dashed blue arrows represent the fluid flows around the swimmer. (b) Pullers have a negative force dipole ( $P < 0$ ), inducing an incoming flow field along their swimming direction and an outgoing flow field along their sides. (c) Two pushers on a converging course reorient each other, tending toward a configuration where they are parallel and swimming side by side ( $h$ is the separation distance between the swimmers, and $θ$ is the angle between the swimmers direction and the direction normal to their separation). (d) Two pullers on a diverging course reorient each other, tending toward a configuration in which they are antiparallel and swimming away from each other. From [223].
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Figure 10
Scattering of microswimmers by a wall: (a) the surface scattering of Chlamydomonas reinhardtii is governed by ciliary contact interactions (cilia manually marked in red). From [199]. (b) Time series of snapshots demonstrating the approach to, contact with, and detachment from a wall of a self-propelled Janus particle. From [426].
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Figure 11
Wall-induced rotation of microswimmers. A microswimmer is located at a distance $h$ from a solid surface and oriented at an angle $θ$ with respect to the direction parallel to the surface: (a) pushers are reoriented hydrodynamically in the direction parallel to the surface [equilibrium $θ = 0$ , see also Fig. 9]; (b) pullers are reoriented in the direction perpendicular to the surface [equilibrium $θ = \pm π / 2$ , see also Fig. 9]. The solid red arrows represent local forcing from the microswimmer on the surrounding fluid, and the dashed blue arrows represent the torque acting on the microswimmer. From [223].
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Figure 12
Reciprocal swimming in a low-Reynolds-number non-Newtonian medium. (a) A 3D model of a submillimeter-sized “microscallop” capable of swimming in a low-Reynolds-number non-Newtonian fluid. (b), (c) Displacement of the microscallop in a shear-thickening and a shear-thinning fluid. (b) Forward net displacement of the microscallop in a shear-thickening fluid and asymmetric actuation (blue curve). The image is a time-lapse composite picture of five frames at intervals of 50 s, with the net displacement occurring along the $x$ direction. (c) Corresponding image of the microscallop in shear-thickening fluid with symmetric actuation (blue curve) and no discernible net displacement. From [321].
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Figure 13
Clustering and living crystals. Qualitative explanation of the clustering process: (a) when two active particles collide head on, (b) they block each other and form a two-particle cluster; (c) the cluster breaks apart on the time scale of the rotational diffusion; and (d) depending on the particle speed and density, another particle might collide before the two-particle cluster has broken apart and, thus, the cluster grows into a three-particle cluster. (e)–(h) Clusters assembled from a homogeneous distribution of active particles (solid area fraction $ϕ = 0.14$ ). The false colors show the time evolution of particles belonging to different clusters. The clusters are not static but rearrange, exchange particles, and merge. From [298]. (i)–(k) Consecutive snapshots of a cluster of active Janus particles. The small dark gray (red) arrows indicate the projected orientations of the caps. Particles along the rim mostly point inward. The particle indicated by the large black arrow in (i) leaves the cluster (j) and is replaced by another particle (k). From [57].
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Figure 14
Self-jamming. (a) Particle locations for an active matter system [dark gray (blue) particles] in a uniform liquid state at density $ϕ \approx 0.2$ with an externally driven probe particle [light gray (red) particle]. The arrow indicates the direction of the probe driving force $F_{d}$ . (b) Phase-separated cluster state at $ϕ \approx 0.5$ , consisting of a high-density phase with local crystal ordering coexisting with a low-density liquid phase. From [336].
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Figure 15
Doping of a passive particle solution with active particles. Experimental snapshots of the temporal evolution of a mixture of passive ( $ϕ_{p} = 0.40$ ) and active ( $ϕ_{a} \approx 0.01$ ) particles at (a) 0, (b) 600, (c) 900, and (d) 1200 s for Péclet number $Pe \approx 20$ . The passive particles belonging to clusters are represented as light gray (red) circles, while those not belonging to clusters are represented as open circles. Active particles are shown as dark gray (blue) circles and their trajectories over 300 s prior to each snapshot are represented as solid lines. From [215].
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Figure 16
Superdiffusion of passive particles in an active bath. (a) Experimental setup and fluorescence image of a solution with E. coli mixed with $10 − μ m$ polystyrene spheres. (b) MSD measurements of polystyrene spheres with diameters 4.5 (squares) and $10 μ m$ (circles) at a particle concentration of $\approx 10^{- 3} μ^{- 2}$ in an active bath. The solid lines with slopes $α = 2.0$ , 1.5, and 1.0 are guides to the eyes. The two dashed lines correspond to the thermal diffusion of $4.5 − μ m$ and $10 − μ m$ particles. From [447].
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Figure 17
Interactions mediated by an active bath. (a) Snapshot of a simulation of passive (spherical) particles immersed in an active bath. The active particles are motile bacteria (represented by spherocylinders with white heads pointing in the direction of self-propulsion). (b) Microscopy image from experiment. The bacteria are visible in the enlarged image in the inset. (c) Simulated and (d) experimental radial distribution functions $g (r)$ . Typical two-bead configurations corresponding to the first two peaks are also shown in the insets in (c). Distances are in units of particle diameter $2 R$ (bacteria width is about 0.25 in such a unit). The dashed line is the $g (r)$ obtained in simulations with a mixture of bacteria interacting with only one of the particles, i.e., without depletion. From [13].
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Figure 18
Motion of a passive wedge in an active bath. (a) If the wedge is surrounded by passive particles, there is no directed motion, because the pressure due to the passive particles is equal on all sides of the object. (b) However, in an active bath, the active particles accumulate in the corner of the wedge, producing its net propulsion.
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Figure 19
Bacteria-driven micromotors. (a), (b) Sketch of the collision of a single bacterium with the rotor boundary: (a) bacteria coming from the left area with respect to the normal leave the gear, while (b) bacteria from the right get stuck at the corner exerting a torque on the rotor. The arrows represent the forces exerted by the bacteria on the rotor. (c) Angular velocity $ω$ of the micromotor as a function of time: the black line refers to a single run; the red (lighter) line is the average over 100 independent runs. After a short transient regime (due to the initial configuration of bacteria), a fluctuating velocity around a mean value $ω_{0} \approx 0.21 rad s^{- 1}$ is observed. Inset: The same as the main plot for a symmetrically shaped micromotor, which does not rotate (on average). From [12]. (d) A nanofabricated asymmetric gear ( $48 μ m$ external diameter, $10 μ m$ thickness) rotates clockwise at 1 rpm when immersed in an active bath of motile E. coli cells, visible in the background. The gear is sedimented at a liquid-air interface to reduce friction. The circle points to a black spot on the gear that is used for visual angle tracking. From [94].
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Figure 20
Active-depletion interactions between two plates in an active bath. (a) A schematic of the system containing run-and-tumble particles (spheres) with some particle trajectories indicated by lines and arrows. The run length is $R_{l}$ . The two parallel walls (bars) of length $l$ are separated by a distance $d$ . When a particle moves along a wall, it imparts a force against the wall. From [326]. (b)–(e) Typical snapshots of systems for density distributions with a wall-to-wall distance $d / R$ equal to (b) 2.1, (c) 5, (d) 10, and (e) 20, where $R$ is the radius of the active particles. If the particles are modeled as hard spheres, repulsive as well as attractive active-depletion forces can emerge. From [288].
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Figure 21
Emergence of repulsive active-depletion forces in active baths. (a) Schematic representation of two colloidal disks in a bath of active particles. The persistent force $F_{a}$ acts along a defined axis, which is shown by the arrow as well as the colors, where dark gray (red) corresponds to the back of the particle and light gray (yellow) to its front. (b) Effective rescaled forces $⟨ F / F_{a} ⟩$ experienced by two colloidal disks as a function of their separation for different values of depletant’s activity for $R_{c} = 10 R_{a}$ , where $R_{c}$ is the radius of the colloids and $R_{a}$ is the radius of the active particles ( $ϕ = 0.1$ ). Rescaling is applied only as long as $β F_{a} 2 R_{a} \neq 0$ . Positive values correspond to a repulsion, which dominates any depletion-driven interaction when the bath is active. The larger the active force and the larger the colloid-to-depletant size ratio, the stronger the repulsion. From [163].
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Figure 22
Flexible chain in an active bath. (a)–(f) Sequence of snapshots from simulations depicting a filament at various stages of folding and unfolding. The propelling force acts along the axis connecting the poles of the two hemispheres used to depict the active particles in the light-to-dark gray (blue-to-red) direction. For the sake of clarity, active particles away from the filament have been rendered with a semitransparent filter. In (a), the convention for the definition of the inner and outer regions of a bent filament are explicitly shown. From [164].
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Figure 23
Numerical implementation of reflective boundary conditions. At each time step, (a) the algorithm checks whether a particle has moved inside an obstacle. If this is the case, (b) the boundary of the obstacle is approximated by its tangent $l$ at the point $p$ where the particle entered the obstacle and (c) the particle position is reflected on this line. In (b), $\hat{t}$ and $\hat{n}$ represent the tangential and normal unitary vectors to the surface at point $p$ . Note that the orientation of the particle is not flipped.
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Figure 24
Non-Boltzmann position distributions for active particles in a pore. (a)–(c) Simulated trajectories (10 s, solid lines) of active Brownian particles (radius $R = 1 μ m$ ) moving within a circular pore (radius $20 μ m$ ) with reflective boundaries at velocity (a) $v = 0$ , (b) $v = 5$ , and (c) $v = 10 μ m s^{- 1}$ . The histograms on the bottom show the probability distribution along a diameter of the circular pore: while the probability is uniform across the whole pore in the case of passive Brownian particles, the probability increases toward the walls in the case of active Brownian particles together with the particle velocity and the associated persistence length $L$ [Eq. (6)].
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Figure 25
Capturing active particles with a wedge. (a) Phase diagram marking three different collective trapping states of self-propelled rods at a wedge upon variation of the reduced rod packing fraction ( $ϕ_{rod}$ ): no trapping at large apex angle $α$ , complete trapping at medium $α$ , and partial trapping at small $α$ . Phase boundaries are shown for two different values of the area fraction occupied by the wedge ( $ϕ_{wedge}$ ). The region of complete trapping is bounded by a triple point at larger rod concentration beyond which a smooth transition from no trapping to partial trapping occurs. (b)–(d) Snapshots depicting the examples of the three stationary states. From [197].
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Figure 26
Concentration of active particles. (a) Schematic drawing of the interaction of bacteria with a funnel opening: on the left side, bacteria may (trace 1) or may not (trace 2) get through the gap, depending on the angle of attack. On the right, almost all bacteria colliding with the wall are diverted away from the gap (traces 3 and 4). (b) Scanning electron micrograph of the device. (c) Uniform distribution of bacteria in a structure with a funnel wall after injection and (d) steady-state distribution after 80 min. (e) Ratio of densities in the right and left compartments vs time: the circles are experimental data, and the dashed line is a fit to the relative theory. From [137].
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Figure 27
Observation of particle concentration and depletion by bacteria. Snapshots of particles and bacteria (a) at $t = 0$ (the initial state), where particles are randomly distributed, and (b) at $t = 20 \min$ , where particle distributions have been strongly affected by bacterial transport over asymmetric barriers. The colloidal particles that are not stuck on the surface are highlighted. Particle distributions averaged over a steady state (c) for particles in the bacterial bath between $t_{1} = 15$ and $t_{2} = 20 \min$ ( $Δ t = 5 \min$ ) and (d) for particles in an experiment without bacteria, undergoing simple Brownian motion for $Δ t = 10 \min$ . In the absence of bacteria, the colloidal particles remain trapped within the structures’ compartments. From [207].
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Figure 28
Rectification of active Brownian motion in an asymmetric ratchetlike microchannel. The distribution of passive [dark gray (red) histograms] and active [light gray (yellow) histograms] Brownian particles (radius $R = 1 μ m$ ) released at time $t = 0 s$ from position $x = 0 mm$ are plotted at times (a) $t = 100$ , (b) $t = 500$ , and (c) $t = 1000 s$ . The higher the active particle velocity, the farther the active particles travel along the channel. Every histogram is calculated using 1000 particle trajectories. A segment of the channel, whose dent is $10 μ m$ long, is represented by the gray structure in the inset of (c).
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Figure 29
Effect of a disordered landscape on the driving velocity of active particles. (a) Positions of pinned disks [light gray (blue) circles], active disks [dark gray (red) circles], and disk trajectories over a period of time (lines) in a small section of a sample with obstacle area fraction $ϕ_{p} = 0.0235$ , total area fraction $ϕ = 0.188$ , and run length $R_{l} = 0.004$ . (b) The average drift velocity $⟨ v_{d} ⟩$ as a function of $R_{l}$ passes through a maximum for all the curves; $ϕ_{p} = 0.055$ , 0.094, 0.1413, and 0.188, from top to bottom. From [334].
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Figure 30
Cluster formation induced by obstacles. Snapshots of the positions of active disk and obstacles in a system with area fraction $ϕ = 0.363$ and number of obstacles $N_{p} = 2$ . The arrows indicate the direction of the motor force for each disk. Dark particles with light arrows have a net motion in the positive $y$ (vertical) direction; light particles with dark arrows have a net motion in the negative $y$ direction. Red disks are immobile obstacles. (a) The initial fluctuating state contains small transient clusters. (b) After some time, transient clusters are nucleated by the obstacles. (c) At a still later time, the transient cluster shown in (b) has broken apart. (d) Finally, the dynamically frozen steady state appears where a faceted crystal forms around the obstacles. From [333].
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Figure 31
Trapping of active particles. (a) In a system where the particle orientation is affected by the presence of obstacles, spontaneous particle trapping occurs for large values of the turning speed and of the obstacle density. Obstacles are indicated by gray (red) dots while black arrows correspond to active particles. From [76]. (b) Sample trajectory of a self-propelled rod orbiting around a passive sphere. The sphere has diameter $6 μ m$ , the rod length is $2 μ m$ , and its speed is on the order of $20 μ m s^{- 1}$ . From [392].
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Figure 32
Sorting of microswimmers in a periodic environment. (a)–(e) Typical trajectories of self-propelled particles moving through a triangular lattice (lattice constant $L_{c} = 35 μ m$ ) of elliptical obstacles when a drift force $F_{d} = 0.12 pN$ is applied along the $y$ direction. The characteristic swimming length is (a) $L = 0$ (Brownian particle, no propulsion), (b) 16, (c) 24, (d) 33, and (e) $83 μ m$ . (f)–(j) Corresponding histograms of the experimentally measured [light gray (red)] and simulated [dark gray (blue)] directions of the particle trajectories as defined by two points in the trajectory separated by $100 μ m$ . The experimental histograms were obtained considering more than 100 trajectories in each case. (k) Measured probability that particles are deflected by more than 30° after a traveling length of $100 μ m$ as a function of $L$ for various imposed drift forces $F_{d} = 0.06 \pm 0.02$ (diamonds), $0.12 \pm 0.05$ (squares), and $0.28 \pm 0.12 pN$ (triangles). The solid lines are the results of numerical calculations. From [426].
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Figure 33
Sorting of chiral microswimmers ( $R = 1 μ m$ , $v = 31 μ m s^{- 1}$ , and $ω = \pm 3.14 rad s^{- 1}$ ) with chiral flowers (rectangles). (a) At $t = 0 s$ a racemic mixture of active particles is released inside two chiral flowers with opposite chiralities. (b), (c) As time progresses, the levogyre [dark gray (red) squares] active particles are trapped in the left chiral flower, while the dextrogyre ones [light gray (yellow) circles] are trapped in the right chiral flower.
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