Transport of the moving barrier driven by chiral active particles
aa r X i v : . [ c ond - m a t . s o f t ] M a y Transport of the moving barrier driven by chiral active particles
Jing-jing Liao , , Xiao-qun Huang , and Bao-quan Ai ∗ Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,School of Physics and Telecommunication Engineering,South China Normal University, Guangzhou 510006, China. and College of Applied Science, Jiangxi University ofScience and Technology, Ganzhou 341000, China. (Dated: May 29, 2018)
Abstract
Transport of a moving V-shaped barrier exposed to a bath of chiral active particles is investigatedin a two-dimensional channel. Due to the chirality of active particles and the transversal asymmetryof the barrier position, active particles can power and steer the directed transport of the barrierin the longitudinal direction. The transport of the barrier is determined by the chirality of activeparticles. The moving barrier and active particles move in the opposite directions. The averagevelocity of the barrier is much larger than that of active particles. There exist optimal parameters(the chirality, the self-propulsion speed, the packing fraction and the channel width) at which theaverage velocity of the barrier takes its maximal value. In particular, tailoring the geometry of thebarrier and the active concentration provides novel strategies to control the transport propertiesof micro-objects or cargoes in an active medium. ∗ Email: [email protected] . INTRODUCTION In recent years, active particle transport in complex environments has attracted widelyattention and much interest in biology, chemistry and nanotechnology [1–3]. Different fromthe features of passive colloids, the intrinsic nonequilibrium property of active particles, alsoknown as self-propelled Brownian particles or microswimmers and nanoswimmers, can takeenergy from their environment and produce a force which pushes them forward [4]. When anactive particle features a common symmetry axis of body and self-propelling force, it swimslinearly only [5]. Otherwise it experiences a constant torque, which is called ’chiral activeparticle’, and performs circular motion in two dimensions and helicoidal motion in threedimensions due to the self-propulsion force being not aligning with the propulsion direction[6]. Chiral active matter can exhibit intriguing phenomena [7–35], such as self-organizationand collective behaviors. Instances of such new active matter system can be found in activechiral fluids [36–39] and many biological micro-swimmers ranging from spermatozoa [40] andEscherchia Coli [41, 42] to Listeria monocytogenes [43].In particular, the behaviors and dynamics of obstacles exposed to an active fluid andthe transport properties of active particles take on great importance in several practicalapplications [12–19, 48], such as driving microscopic gears and motors [16, 19], the cap-ture and rectification of active particles [12–14, 17, 18], and using active suspensions topropel wedge-like carriers [15, 48]. The interactions of active particles with obstacles havebeen investigated by using theoretical studies, simulations and experiments [11–15, 44–53].Potiguar and coworkers [11] found a vortex-type motion of self-propelled particles aroundconvex symmetric obstacles and an steady particle current in an array of non-symmetricconvex obstacles. Galajda et al. [12] observed a ratchet motion of the swimming bacteriaplaced in a confined area containing an array of funnel shapes. Kaiser et al. [13] showedthe interaction between active self-propelled rods and stationary wedges was as a functionof the wedge angle. In a subsequent work, Kaiser et al. [15] demonstrated that the directedtransport of mesoscopic carriers through the suspension could be powered and steered bycollective turbulentlike motion in a bacterial bath. C. Reichhardt and C. J. O. Reichhardt[44] found that a ratchet effect produced by chirality was observed for circularly movingparticles interacting with a periodic array of asymmetric L-shaped obstacles. They alsostudied active particles which are placed in an asymmetric array of funnels could produce2 ratchet effect even in the absence of an external drive [45]. Ratchet reversals producedby collective effects and the use of active ratchets to transport passive particles were in-vestigated and reviewed in Ref. [46] and Ref. [47]. Angelani et al. [48] showed activeparticles powered the asymmetric arrow-shaped barriers to move in one dimension. Malloryand coworkers [49] numerically studied the transport of a asymmetric tracer immersed in ahigh dilution suspension of self-propelled nanoparticles. Marconi et al. [51] studied the roleof self-propulsion in active particles interacting with a moving semipermeable membranewith a constant velocity. Chiral active particles can be rectified in the longitudinal directionwhen the potentials or the fixed obstacles are asymmetric along the transversal direction inthe periodic channel [52, 53].In the previous studies, active particles are considered to interact with a fixed obstacle,or an obstacle with a constant velocity or a non-symmetric obstacle driven by active par-ticles. However, the directed transport of the moving symmetric barrier driven by chiralactive particles has not been considered yet, which results in a fascinating wealth of newnonequilibrium phenomena actually. In this paper, we expose a V-shaped barrier to a bathof chiral active particles. We emphasize on studying the interplay between active particlesand the barrier, finding the directed transport of the barrier powered and steered by activeparticles and investigating how the system parameters and the moving barrier affect the rec-tification of chiral active particles. We also focus on comparing the transport of chiral activeparticles between the cases of the barrier fixed and moving. It is found that the transportspeed of the barrier is much larger than that of active particles. The scaled average velocityof active particles in the moving case is reduced much than in the fixed case. The movingbarrier and active particles move in the opposite directions. We can obtain maximal scaledaverage velocity of the barrier when the system parameters are optimized. Our results canbe applied practically in powering and steering carriers and motors by a bath of bacteria orartificial microswimmers.
II. MODEL AND METHODS
We consider n a chiral active particles with radius r moving in a two-dimensional straightchannel with periodic boundary conditions (the period L x ) in the x -direction and hard wallboundary conditions (the width L y ) in the y -direction as shown in Fig. 1. A V-shaped3arrier with angle α is exposed to the bottom of the channel. In order to restrict theV-shaped barrier moving only along the x -direction, two parallel tracks (active particlescannot feel the tracks) are settled in the channel, one is fixed at the bottom of channel andthe other is fixed at the top of the barrier. Each side of the V-shaped barrier consists of n p particles with radius r . The total particle number of chiral active particles and the V-shapedbarrier is N = n a + 2 n p + 1. The position of particle i is described by r i ≡ ( x i , y i ), and itsspeed direction is denoted by the orientation θ i of the polar axis n i ≡ (cos θ i , sin θ i ). Wedefine F i = F xi e x + F yi e y = P j F ij and G i = G xi e x + G yi e y = P j G ij as the forces actingon particle i from other active particles and from the V-shaped barrier, respectively. Theparticle i obeys the following overdamped Langevin equations: a v c v x L y L upper channel walllower channel wall (cid:68) FIG. 1. Schematic of chirality-powered motor. A V-shaped barrier with angle α is exposed to thebottom of the channel. The barrier consists of (2 n p + 1) particles with radius r . The V-shapedbarrier has two cases: fixed and moving. Periodic boundary conditions are imposed in the x -direction, and hard wall boundaries in the y -direction. v a and v c denote the average velocity ofchiral active particles and the center of the V-shaped barrier along the x -direction, respectively. dx i dt = µ [ F xi + G xi ] + v cos θ i + p D ξ xi ( t ) , (1)4 y i dt = µ [ F yi + G yi ] + v sin θ i + p D ξ yi ( t ) , (2) dθ i dt = Ω + p D θ ξ θi ( t ) , (3)where v denotes the magnitude of self-propelled velocity and µ is the mobility. Ω is theangular velocity and its sign determines the chirality of active particles. Particles are definedas the clockwise particles for Ω < >
0. Thetranslational and rotational diffusion coefficients are denoted by D and D θ , respectively. ξ xi ( t ), ξ yi ( t ), and ξ θi ( t ) are the unit-variance Gaussian white noises with zero mean.The force F ij between active particle i and j , and the force G ij between active particle i and the barrier particle j are taken as the linear spring form with the stiffness constant k and k , respectively. F ij = k (2 r − r ij ) e r , if r ij < r ( F ij = 0 otherwise), and r ij isthe distance between active particle i and j . G ij = k (2 r − r ij ) e r , if r ij < r ( G ij = 0otherwise), and r ij is the distance between active particle i and the barrier particle j . Weuse large values of k and k to imitate hard particles. It ensures that particle overlaps decayquickly.We can rewrite Eqs.(1)-(3) in the dimensionless forms by introducing the characteristiclength scale and the time scale: ˆ x = xr , ˆ y = yr , and ˆ t = µkt , d ˆ x i d ˆ t = ˆ F xi + ˆ G xi + ˆ v cos θ i + q D ˆ ξ xi (ˆ t ) , (4) d ˆ y i d ˆ t = ˆ F yi + ˆ G yi + ˆ v sin θ i + q D ˆ ξ yi (ˆ t ) , (5) dθ i d ˆ t = ˆΩ + q D θ ˆ ξ θi (ˆ t ) , (6)and the other parameters can be rewritten as ˆ L x = L x r , ˆ L y = L y r ,ˆ v = v µkr , ˆ D = D µkr , andˆ D θ = D θ µk . In the following discussions, only the dimensionless variables will be used, andthe hats for all quantities appearing in the above equations shall be omitted .By integration of the Langevin Eqs.(4)-(6) using the second-order stochastic Runge-Kutta algorithm, we can get the transport behaviors of the quantities. To quantify theratchet effect, we only calculate average velocity in the x -direction because particles alongthe y -direction are confined and directed transport only occurs in the x -direction. In theasymptotic long-time regime, we can obtain the average velocity of chiral active particles5long the x -direction using the following formula v a = 1 n a n a X i =1 lim t →∞ x i ( t ) − x i (0) t . (7)The forces acting on the V-shaped barrier particle j from the chiral active particle i aredefined as G j = G xj e x + G yj e y = P i G ij . It leads to the barrier moving when the V-shapedbarrier is not fixed. The motion equation for the center of the V-shaped barrier is as follows: dx c dt = γG x , (8)where γ is the coefficient we set, the barrier is fixed when γ = 0 and can be driven to movealong the x -direction when γ = 1 . x c is the center position of the barrier in the x -direction, G x = P j G xj / (2 n p + 1) is the average force acting on the center of the V-shaped barrieralong the x -direction. In the asymptotic long-time regime, the average velocity of the centerof the V-shaped barrier along the x -direction can be obtained from the following formula v c = lim t →∞ x c ( t ) − x c (0) t . (9)We define the ratio between the area occupied by particles and the total available areaas the packing fraction φ = π (2 n p + 1 + n a ) r / ( L x L y ). In addition, we define the scaledaverage velocity as η a = v a /v and η o = v c /v which respectively stand for rectification ofchiral particles and the barrier. III. NUMERICAL RESULTS AND DISCUSSION
In our simulations, we have considered more than 100 realizations to improve accuracyand minimize statistical errors. The total integration time was chosen to be more than 10 and the integration step time was smaller than 10 − . Unless otherwise noted, our simulationsare under the parameter sets: L x = 24 . L y = 16 .
0. We vary Ω, D , D θ , n p , v , α , φ ,and L y to calculate average velocity of chiral active particles and the V-shaped barrier whenthe barrier is fixed and moving.Actually, there are two critical elements of ratchet setup in nonlinear systems [54]. Oneis (a) Asymmetry (temporal and/or spatial), which can violate the left-right symmetry ofthe response. The other is (b) fluctuating input zero-mean force: it should break thermody-namical equilibrium, which forbids a directed transport appearing due to the second law of6hermodynamics. For our system, the asymmetry comes from the upper-lower asymmetry ofthe channel due to the position of the V-shaped barrier and the fluctuating input zero-meanforce comes from the self-propulsion of active particles. Because the circular trajectory ra-dius of chiral particles v / | Ω | is much larger than the channel cell, chiral particles slide alongthe walls rather than move circularly. The channel is upper-lower asymmetric, therefore,the motion time along the upper wall is significantly smaller than along the lower wall. Thecounterclockwise particles Ω > < -0.2 -0.1 0.0 0.1 0.2-0.15-0.10-0.050.000.050.100.15 (a) -0.2 -0.1 0.0 0.1 0.2-0.004-0.0020.0000.0020.004 (b) -0.2 -0.1 0.0 0.1 0.2-0.4-0.20.00.20.4 (c) FIG. 2. The scaled average velocity η a and η o as a function of the angular velocity Ω. (a) Activeparticles for the fixed case at α = π/ α = π/ π/ π/
2, and 2 π/
3. (c) The moving barrier at α = π/ π/ π/
2, and 2 π/
3. The other parameters are D = 0 . D θ = 0 . v = 1 . N = 150, L y = 16 .
0, and n p = 9. Figure 2 shows the scaled average velocity as a function of the angular velocity Ω. When7he V-shaped barrier is fixed, the average velocity of the obstacle is zero. For active particles(see Fig. 2(a)), η a is negative for Ω >
0, zero at Ω = 0, and positive for Ω <
0. The sign ofΩ completely determine the transport direction of active particles. That is to say, we canseparate active particles with different chiralities due to their different directions of motion.Additionally, when Ω →
0, the chirality can be neglected, and the ratchet effect disappearsbecause the symmetry of the system cannot be broken, thus η a →
0. When Ω → ∞ , theself-propelled angle changes very fast, particles will experience a zero averaged force, so η a tends to zero. Therefore, there exists an optimal value of | Ω | at which η a takes its maximalvalue.When the V-shaped barrier can move, the scaled average velocity of active particles (seeFig. 2(b)) is reduced much than that in the fixed case, while the transport speed of the barrier(see Fig. 2(c)) is about much larger than that of active particles. The movement direction ofthe obstacle is also completely determined by the sign of Ω and is opposite to the direction ofactive particles. The transport behaviors which are the same as the above are demonstratedin the following results (see Fig. 2-Fig. 8). Now we explain the underlying reason forthe barrier and chiral particles transport. The nonequilibrium driving which comes fromthe chiral particles breaks thermodynamical equilibrium and power the V-shaped barrier tomove in the x -direction. Because the driving forces on the barrier come from chiral particles,their transport behavior are similar and in opposite directions. In our simulation, we choose n p = 9 and N = 150. In other words, the barrier consists of 19 particles and there are 131chiral active particles. Similar to the collision between a large mass of moving object anda small mass of stationary object, all self-propelled particles ( n a = 131) act on the barrier( n p = 9) resulting in much larger transport speed of the barrier and much smaller velocity ofactive particles. Additionally, the velocity of the barrier is about 131 times larger than thatof active particles. That is to say, the velocity ratio between the barrier and active particlesis decided by the number of active particles. The force acting on the center of the barrierincreases as the increasing number of active particles, then the velocity ratio between thebarrier and active particles increases. When Ω → → ∞ , the scaled average velocityof active particles η a →
0, thus the driving effect can be neglected and the scaled averagevelocity of the barrier η o goes to zero. Therefore, there exists an optimal value of | Ω | atwhich η o takes its maximal value. Additionally, we can control the movement direction ofthe barrier by tuning the angular velocity of chiral particles which is a new technique and8dvantage in contrast to using achiral particles. D (a) D (b) (c) D FIG. 3. The scaled average velocity η a and η o as a function of the translational diffusion coefficient D . (a) Active particles for the fixed case at α = π/
2. (b) Active particles for the moving case at α = π/ π/ π/
2, and 2 π/
3. (c) The moving barrier at α = π/ π/ π/
2, and 2 π/
3. The otherparameters are Ω = − . D θ = 0 . v = 1 . N = 150, L y = 16 .
0, and n p = 9. Figure 3 displays the scaled average velocity versus the translational diffusion coefficient D . As we know, the translational diffusion coefficient D can cause two results: (A) reducingthe self-propelled driving which blocks the ratchet transport when the particles can easilypass across the barrier. (B) Facilitating particles to cross the barrier which promotes therectification when the particles cannot easily stride over the barrier. When the barrier isfixed (see Fig. 3(a)), active particles can easily cross the barrier, the factor A dominates thetransport, thus the rectification η a decreases monotonically with increasing D . Comparedwith the fixed case, active particles cannot easily stride over the barrier when the barrier can9ove. When D increases from zero, the factor A firstly dominates the transport at α = π/ π/
3, and π/ α = 2 π/ α is, the more particles are trapped in the corner of the barrier. When D → ∞ , thetranslational diffusion is very large, the effect of the asymmetric barrier disappears and thescaled average velocity η a goes to zero. Therefore, in the moving case, the scaled averagevelocity η a decreases monotonously with increasing D at α = π/ π/
3, and π/
2, whilethere exists an optimal D value where the rectification is maximal at α = 2 π/ D from zero, the magnitude | η o | of the average velocity ofthe moving barrier decreases monotonously at α = π/ π/
3, and π/
2, while the magnitude | η o | is a peaked function of D at α = 2 π/ D θ isillustrated in Fig. 4. It is found that the curves are similar when the barrier is fixed or canmove (shown in Figs. 4(a) and 4(b)). When D θ →
0, the self-propelled angle θ almost doesnot change, and the scaled average velocity approaches its maximal value. As D θ increasesto be large, the particles cannot feel the self-propelled driving and the ratchet effect reduces,so η a and | η o | decreases and tends to zero. Similarly to the previous figures, the transportspeed of the barrier is much larger than that of active particles in the both two cases (seeFig. 4(c)). Due to the small scaled average velocity of chiral particles in the moving case,the curves in Fig.4 (b) are not smooth in the presence of statistical errors. We can getsmoother curves by increasing the number of realizations or the total integration time.In Figure 5, we present the scaled average velocity as a function of the particle number n p of the V-shaped barrier. In the case of the barrier fixed (see Fig. 5(a)), the curve ofactive particles is observed to be bell shaped, and there exists an optimal value of n p atwhich η a takes its maximal value. It can be explained as follows. When n p is very small,the channel is near to symmetric and the effect of the asymmetric barrier disappears. Thescaled average velocity tends to zero. As n p increases, the channel becomes asymmetric,and η a increases. When n p increases enough to block the channel, the barrier separates thechannel into two parts and particles cannot pass across the barrier, thus η a goes to zero.Therefore, the optimal n p can facilitate the ratchet transport. When increasing the channelwidths L y , the magnitude of the scaled average velocity η a decreases because the effect of theasymmetric barrier is reduced. And the position of the optimal number n p shifts to small n p .01 0.1 1 10 1000.000.030.060.090.12 D (a) D (b) o D (c) FIG. 4. The scaled average velocity η a and η o as a function of the rotational diffusion coefficient D θ . (a) Active particles for the fixed case at α = π/
2. (b) Active particles for the moving case at α = π/ π/ π/
2, and 2 π/
3. (c) The moving barrier at α = π/ π/ π/
2, and 2 π/
3. The otherparameters are Ω = − . D = 0 . v = 1 . N = 150, L y = 16 .
0, and n p = 9. slightly. Namely, the position of the optimal number n p is insensitive to the channel widths L y . When the barrier can move (shown in Figs. 5(b) and 5(c)), for very small n p , a largenumber of particles act on the barrier which consists of few particles, resulting in maximaltransport speed of the barrier (see Fig. 5(c)). On increasing n p , because the driving effectdecreases, the scaled average velocity decreases monotonically and finally tends to zero forlarge n p .Figure 6 shows the scaled average velocity versus the self-propulsion speed v . For the caseof the barrier fixed (see Fig. 6(a)), the fluctuating input and the ratchet effect disappear as v tends to zero, thus η a is nearly equal to zero. As v increases, the rectification approaches11 n p L y =16 L y =80 L y =150 L y =400 (a) n p (b) n p (c) FIG. 5. The scaled average velocity η a and η o as a function of the particle number of the V-shapedbarrier n p . (a) Active particles for the fixed case at α = π/ L y . (b) Active particles for the moving case at L y = 16 for different values of the barrierangle α . (c) The moving barrier at L y = 16 for different values of the barrier angle α . The otherparameters are Ω = − . D = 0 . v = 1 . n a = 140, and D θ = 0 . its maximal value. With a further increase in v , the scaled average velocity decreasesgradually and then tends to a constant. However when v is large enough, the asymmetriceffect can be negligible, thus the scaled average velocity decreases and tends to zero (notshown in the figure). When the barrier can move (see Fig. 6(b) and Fig. 6(c)), η a goes tozero as v →
0. For large values of v , the asymmetric effect disappears more easily thanthat in the fixed case due to the motion of the barrier, thus the directed transport decreasessharply. Therefore, the optimal self-propulsion speed can facilitate the rectification of activeparticles. 12 v (a) v (b) v (c) FIG. 6. The scaled average velocity η a and η o as a function of the self-propulsion speed v . (a)Active particles for the fixed case at α = π/
2. (b) Active particles for the moving case at α = π/ π/ π/
2, and 2 π/
3. (c) The moving barrier at α = π/ π/ π/
2, and 2 π/
3. The other parametersare Ω = − . D = 0 . n p = 9, N = 150, L y = 16 .
0, and D θ = 0 . The dependence of the scaled average velocity on the barrier angle α is shown in Figure7. When the barrier is fixed (shown in Fig. 7(a)), there exists an optimal value at which η a takes its maximal value. When α →
0, particles can pass through the V-shaped barrierbecause the height of the V-shaped barrier is smaller than the height of channel L y and thechannel is not blocked. Thus, η a is small but does not tend to zero. For very large value of α , the asymmetry effect disappear and no directed transport occurs, thus η a →
0. Whenthe barrier can move (shown in Figs. 7(b) and 7(c)), η a and | η o | increase slightly with anincrease in α . In other words, the scaled average velocity is insensitive to the angle α . Thisis consistent with the other figures in the moving case. In particular, the scaled average13
30 60 90 120 150 1800.020.040.060.080.100.120.14 v =1.0 (a) (b) v =1.0 v =3.0 v =5.0 (c) v =1.0 v =3.0 v =5.0 FIG. 7. The scaled average velocity η a and η o as a function of the barrier angle α . (a) Activeparticles for the fixed case at v = 1 .
0. (b) Active particles for the moving case at v = 1 .
0, 3 . .
0. (c) The moving barrier at v = 1 .
0, 3 .
0, and 5 .
0. The other parameters are Ω = − . D = 0 . n p = 9, N = 150, L y = 16 .
0, and D θ = 0 . velocity reaches to the maximum in the limit case α = π . We can explain as follows: when α = π , the V-shaped barrier has two sides and each side has n p particles. The driving forcesact on every particle of each side from both the positive and negative direction of x . When α = π , the pushing effect on the barrier which becomes a straight stick is bigger than thatin the case of α = π . Because the average force exerted by active particles is mainly alongthe negative direction of x . Thus, maximal rectification is achieved when α = π .Figure 8 depicts the scaled average velocity as a function of the packing fraction φ . Whenthe barrier is fixed (see Fig. 8(a)), the rectification of active particles decreases slowly withthe increasing φ . For a large φ , the particles are jammed, thus η a tends to zero. When14 .0 0.2 0.4 0.6 0.8 1.00.000.040.080.120.16 (a) (b) o (c) FIG. 8. The scaled average velocity η a and η o as a function of the packing fraction φ . (a) Activeparticles for the fixed case at α = π/
2. (b) Active particles for the moving case at α = π/ π/
2. (c) The moving barrier at α = π/ π/
2. The other parameters are Ω = − . D = 0 . v = 1 . n p = 9, L y = 16 .
0, and D θ = 0 . the barrier can move (shown in Figs. 8(b) and 8(c)), the rectification of active particlesdecreases sharply with the increasing φ because the driving effect increases sharply (see Fig.8(b)). η a also tends to zero as φ → φ tends to zero, thus | η o | →
0. For high concentration, the active bath is jammed,which leaves no mobility for the barrier, so | η o | →
0. Therefore, there exists an optimalpacking fraction that maximizes the scaled average velocity of the barrier. As the aboveresults, the velocity ratio between the barrier and active particles is decided by the numberof active particles. 15
20 40 60 80 100 1200.030.060.090.120.15 L y = /2 (a) L y = /6 = /3 = /2 =2 /3 (b) L y = /6 = /3 = /2 =2 /3 (c) FIG. 9. The scaled average velocity η a and η o as a function of the channel width L y . (a) Activeparticles for the fixed case at α = π/
2. (b) Active particles for the moving case at α = π/ π/ π/
2, and 2 π/
3. (c) The moving barrier at α = π/ π/ π/
2, and 2 π/
3. The other parameters areΩ = − . D = 0 . v = 1 . n p = 9, n = 150, and D θ = 0 . The dependence of the scaled average velocity on the channel width L y is shown in Fig.9. In the present system, the arm length of the barrier is 9 and L y must be larger than9. In the case of the barrier fixed (see Fig. 9(a)), there exists an optimal value of L y atwhich η a takes its maximal value. It can be explained as follows. When L y is very small,the barrier blocks the channel, particles cannot pass across the barrier, thus η a goes to zero.When L y → ∞ , the channel is near to symmetric and the effect of the asymmetric barrierdisappears, thus, η a →
0. When the barrier can move (shown in Figs. 9(b) and 9(c)), for verysmall L y , particles are difficult to stride over the obstacle, thus η a and | η o | tends to zero. Onincreasing L y , η a and | η o | increase monotonically and reach the maximum because particles16ross the barrier more and more easily. However, when L y → ∞ , most of particles do notinteract with the barrier and particle-barrier interaction becomes insignificant. Therefore,the ratchet effect disappears, η a and | η o | tends to zero (not shown in the figure).Finally, we discuss the possibility of realizing our model in experimental setups. Considera system of Bacillus subtilis (diameter about 1 µm ) moving in a two-dimensional channel atroom temperature. The suspension of bacteria is grown for 8-12h in Terrific Broth growthmedium (Sigma Aldrich). We can continuously measure the optical scattering of the mediumusing an infrared proximity sensor to monitor the concentration of bacteria during the growthphase [15]. A V-shaped barrier is fabricated by photolithography [55, 56]. To control theorientation of the barrier with an external magnetic field, we can mix a liquid photoresistSU-8 with micron-size magnetic particles before spin coating [15]. In order to restrict theV-shaped barrier moving only along the x -direction, two parallel tracks (active particlescannot feel the tracks) are settled in the channel, one is fixed at the bottom of channeland the other is fixed at the top of the barrier. The influence of gravity is negligible. Dueto the chirality of active particles and the transversal asymmetry of the barrier position,active particles can power and steer the directed transport of the barrier in the longitudinaldirection. The motion of the barrier and active particles are captured by a digital high-resolution microscope camera, from which the average velocity can be calculated. IV. CONCLUDING REMARKS
In conclusion, we numerically studied the transport of a moving V-shaped barrier exposedto a bath of chiral active particles in a two-dimensional channel. It is found that the barriercan be driven to move directly along the bottom of the channel by chiral active particles.When the barrier is fixed at the bottom of the channel, the upper-lower asymmetric dueto the position of the V-shaped barrier and the intrinsic property of chiral particles canbreak thermodynamical equilibrium and induce the rectified transport of active particles.Chiralities determine the transport direction of active particles. By choosing suitable systemparameters, the transport efficiency of active particles can reach the maximum. When theV-shaped barrier can move along the bottom of the channel, the nonequilibrium drivingwhich comes from the chiral particles breaks thermodynamical equilibrium and power thebarrier to move in the x -direction. The transport of the barrier is determined by the chirality17f active particles. The moving barrier and active particles move in the opposite directions.Comparing the transport of chiral active particles between the cases of the barrier fixedand moving, the rectified efficiency of active particles in the moving case is reduced muchthan that in the fixed case. The velocity ratio between the barrier and active particles isdecided by the number of active particles. Maximal transport velocities of active particlesand the barrier are obtained when the system parameters are optimized. In particular,changing n p , α , and φ in the moving case lead to the transport behaviors more differentthan that in the fixed case. In other words, tailoring the geometry of the barrier and theactive concentration provides novel strategies to control the transport properties of micro-objects or cargoes in an active medium. Maybe our results can be applied practically inpropelling carriers and motors by a bath of bacteria or artificial microswimmers, such ashybrid micro-device engineering, drug delivery, micro-fluidics and lab-on-chip technology. ACKNOWLEDGMENTS
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