Chirality-induced directional rotation of a symmetric gear in a bath of chiral active particles

We conduct a numerical study exploring the rotation of a symmetric gear driven by chiral particles in a two-dimensional box with periodic boundary conditions. The symmetric gear is submerged in a sea of chiral active particles. Surprisingly, even though the gear is perfectly symmetric, the microscopic random motion of chiral active particles can be converted into macroscopic directional rotation of the gear. (i) In the case of zero alignment interaction, the direction of rotation of the gear is determined by the chirality of active particles. Optimal parameters (the chirality, self-propelled speed, and packing traction) exist, at which the rotational speed reaches its maximum value. (ii) When considering a finite alignment interaction, alignment interactions between particles play an important role in driving the gear to rotate. The direction of rotation is dictated by the competition between the chirality of active particles and the alignment interactions between them. By tuning the system parameters, we can observe multiple rotation reversals. Our findings are relevant to understanding how the macroscopic rotation of a gear connects to the microscopic random motion of active particles.

In the last few years, the directed transport of active particles has attracted the interest of the research community. Due to their self-propelled force, active particles in periodic structures can display markedly different behavior than that of passive particles [25][26][27][28]. In a non-equilibrium bath of active particles, ratchet devices can extract energy from the bath and perform work. Experiments show the capability of these devices to induce the rotation of gears when immersed in a bath of swarming bacteria or Janus particles [29][30][31][32]. Asymmetric structures are typically needed to direct the random motion of active particles into a consistent directional rotation for the gear. Angelani and coworkers observed this directional motion of a rotary microdevice when placed in a bath of motile microorganisms [33]. Lugo and coworkers reported an inverse λ-like transition in the dynamics of ratchet gears in an active bath of self-propelling rods [34]. Jerez and coworkers studied the dynamics of these ratchets and established that their angular velocity displayed a nonmonotonic dependence with particle concentration [35]. Rotation reversal of a ratchet gear was also observed in these baths of active particles [36]. In the previous works [29][30][31][32][33][34][35][36], the asymmetry of the gear is necessary to obtain the directional rotation. Recently, Liao and coworkers [37] have discovered that a gear can rotate in a bath of chiral active particles. However, they offered only a brief introduction and did not conduct a systematic study. Consequently, further research is necessary to explore the rotation of the symmetrical gear in a bath of chiral active particles. Understanding the underlying mechanism of this phenomenon could lead to potential future applications in engineering and other industries. Therefore, it is of great interest to develop a comprehensive and detailed understanding of this phenomenon.
In this work, we address the question of whether a symmetric gear can be powered by chiral active particles. We demonstrate that the random motion of chiral active particles can be converted into directional rotation of the gear. Specifically, we find that: (i) in the absence of alignment interaction, the rotation direction of the gear is completely determined by the chirality of active particles. By altering the relevant system parameters (i.e. chirality, self-propelled speed, and packing traction), the gear's rotation can be striking. (ii) For finite alignment interaction, the collective motion of chiral active particles leads to multiple rotation reversals. In this case, the rotation direction of the gear is under the control of a balance between particle chirality and alignment interactions.

Model and methods
We consider N p chiral particles (red disks) moving in a two-dimensional box of size L × L with periodic boundary conditions. In the center of the chamber in figure 1, a symmetric gear with six teeth, whose center of mass is fixed, consists of N g passive particles (blue disks). Both active particles and passive particles are modeled as disks with radius a. The coordinates of chiral particle i can be described by its center⃗ r i ≡ (x i , y i ) and it is oriented at an angle θ i of the polar axis ⃗ n i ≡ (cos θ i , sin θ i ). The dynamics of chiral particle i can be governed by the overdamped Langevin equations [38][39][40][41][42], where v 0 is the self-propulsion speed and µ the mobility. ξ i (t) is a Gaussian white noise with zero mean and unit variance. D r denotes the rotation diffusion coefficient. The chiral particles experience a constant torque leading to circular motion. The corresponding dynamics is affected by the angular velocity Ω i (the sign of Ω i determines the chirality of the particle). Here we define particles as the clockwise particles for positive Ω and the counterclockwise particles for negative Ω.
The interactions between particles are considered as short-ranged repulsive force: ⃗ F ij = k(2a − r ij )⃗ r ij if particle-particle overlap (r ij < 2a) and ⃗ F ij = 0 otherwise, where k is the spring constant. r ij = |⃗ r i −⃗ r j | denotes the distance between the chiral particle i and the object (active or passive particle) j and⃗ r ij = (⃗ r i −⃗ r j )/r ij . Note that a soft repulsive potential is used in soft matter to prevent molecules from coming too close together and clumping. This potential can accurately model interactions between atoms and molecules and is computationally efficient. Replacing the soft potential with a hard repulsive potential will not result in a qualitative difference in outcomes.
We also consider the polar velocity alignment interactions between active particles in equation (2), and their strength is controlled by the coupling constant g ⩾ 0. Such an interaction mechanism constitutes a smooth version of the interaction defining in the Vicsek model. The sum in equation (2) runs over all the neighbors j at a distance less than the interaction radius R from the ith, and ∂ i is the set of neighboring particles.
In this work, the gear is regarded as a symmetric rigid body with two teeth sides of equal length, so that the distance between any two gear particles remains fixed. The force exerted on the gear by the chiral particle i produces the net torque ⃗ M i = − ∑ Ng j ⃗ r j × ⃗ F ij , which can cause the rotation of the gear. The dynamics of the gear can be described by the following equation, where θ g is the angular position of the gear. I is the moment of inertia of the gear and γ the rotational friction coefficient. Note that the size of the gear is larger, so we adopt underdamped dynamics; whereas the size of the particles is smaller, so we adopt overdamped dynamics. Moreover, the result is not qualitatively dependent on the choice of dynamics. By introducing characteristic length scale⃗ r i = ⃗ r i a and time scalet = a 2 t µI , equations (1)-(3) can be rewritten in the dimensionless form, and the other parameters areL From now on, we will only use the dimensionless variables and omit the hat for all quantities.
In the long-time regime, the average angular velocity of the gear can be calculated using the formula, and for convenience, we use the scaled average angular velocity ω = aω v0 . The gear rotates counterclockwise for ω > 0 and clockwise for ω < 0. In addition, we define a packing fraction which means the ratio between the area occupied by the chiral particles and the total available area as ϕ = Npπ a 2 L 2 −Ngπ a 2 −Sg with S g being the area enclosed by the passive particles.
In our simulations, we integrate equations (4)-(6) by using the second-order stochastic Runge-Kutta algorithm with an integration step time of 10 −4 and a total integration of 10 5 (which is sufficient to ensure that the system reaches a steady state). Unless otherwise stated, the parameters used are: L = 50.0, k = 100.0, γ = 1.0, D r = 0.01, and N g = 48. Particle positions are initialized with a uniform random distribution inside the box, and orientations are distributed randomly over the interval [0, 2π].

Results and discussion
In the following, we explore the directional rotation of the gear for two cases: (A) without alignment interaction, and (B) with alignment interaction. We focus on calculating the scaled average angular velocity of the gear by varying the self-propulsion speed v 0 , the angular velocity Ω, the packing fraction ϕ, and the alignment interaction strength g.

Without alignment interaction (g = 0)
First of all, we study the rotation dynamics of the gear in the absence of alignment interactions between particles. Figure 2(a) depicts the scaled average angular velocity ω as a function of the chirality Ω. It can be seen that ω is negative for Ω < 0, zero at Ω = 0, and positive for Ω > 0, thus the chirality determines the rotation direction of the gear. For example, when particles rotating clockwise collide with edges l 2 , they can traverse the edges of the gear until getting stuck in the corner, thus propelling the gear in the counterclockwise direction. In contrast, when particles rotating counterclockwise collide with edges l 1 , they are able to push the edges, thus inducing a negative net torque and causing the gear to rotate in the clockwise direction. Figure 2(b) shows the angular position θ g of the gear as a function of the time t in order to make the gear rotation more comprehensible. It can be seen that the gear rotates counterclockwise at Ω = 0.4 and rotates clockwise at Ω = −0.4. For the remainder of the study, we will only consider the case of Ω > 0.
The dependence of the scaled average angular velocity ω on the chirality Ω, for different v 0 at ϕ = 0.1, is displayed in figure 3. With increasing Ω, ω increases to its maximum before decreasing to zero. There exists an optimal value of Ω at which the scaled average angular velocity is maximized. This can be understood by noting that when Ω → 0, the chirality of particles can be disregarded and the system is symmetrical, resulting in no net gear rotation (i.e. ω → 0). On the other hand, when Ω → ∞, the orientation θ i changes quickly, causing the particles to be in almost-fixed rotation, which again results in ω → 0. Thus, the optimal chirality facilitates the rotation of the gear. In addition, the position of the peak in the curves shifts to a larger value of Ω with an increase in v 0 . This phenomenon can be explained as follows. The direction of rotation of the gear is dependent on the radius R c = v 0 /Ω of the circular motion of the driving particles. When R c is very small, the particle is rotating almost in place and cannot adequately drive the gear, thus producing a small ω. Conversely, when R c is very large, relative to the gear, the particle is essentially moving in a straight line, and the gear cannot transform the chirality of the particle into directional rotation, thus producing a very small Ω. The peak in the curves corresponds to the optimal R c , where ω takes on its maximal value. Due to the relationship R c = v 0 /Ω, the larger the value of v 0 , the larger the value of Ω must be to obtain the optimal R c . Therefore, increasing v 0 results in a shift of the peak position to larger values of Ω. Figure 4 describes the scaled average angular velocity ω as a function of the self-propelled speed v 0 for different values of Ω at ϕ = 0.1. When Ω is small, ω is a peaked function of v 0 , with the peak shifting to larger values of v 0 as Ω increases. For very large values of Ω, such as Ω = 5.0, the peak in the curve disappears, and ω increases monotonously with v 0 . Additionally, the trajectory radii of chiral particles become small for v 0 < 5, resulting in a low scaled average angular velocity. As v 0 increases, however, the trajectory radii become larger and the scaled average angular velocity gradually increases for v 0 < 10. Note that the position of the peak shifts towards larger values of v 0 as Ω increases. This illustrates that the optimal circular motion  radius leads to the optimal directional rotation. This phenomenon can be explained by analogy to the corresponding explanation in figure 3.
To study the dependence of ω on v 0 and Ω in more detail, we plot a phase diagram of ω in v 0 − Ω representation at ϕ = 0.1 in figure 5. The scaled average angular velocity is always positive, indicating that the chiral particles with clockwise rotation can break the symmetry of the system and make the gear to rotate counterclockwise. Interestingly, very small and large values of Ω are not conducive to the rotation of the gear. For an intermediate value of Ω (e.g. Ω = 1.0), the scaled average angular velocity increases monotonically with the self-propelled speed for v 0 < 10. However, for larger speeds, the angular velocity reduces asymptotically and tends towards zero. The figure clearly indicates that there exists an optimal R c = v 0 /Ω that results in the best directional rotation. Figure 6 shows the scaled average angular velocity ω as a function of the packing fraction ϕ at v 0 = 4.0 and Ω = 0.4. The scaled average angular velocity ω displays a peaked shape as a function of the packing fraction ϕ. As the packing fraction ϕ approaches zero, the chiral particles hardly impact the gear and the gear is unable to rotate, thereby leading to ω tending towards zero. As ϕ approaches 1, the chiral particles fill the  whole space, thus obstructing the rotation of the gear, and consequently resulting in ω tending to zero. Hence, there exists an optimal packing fraction that yields a maximum scaled average angular velocity.

With polar alignment interaction (g ̸ = 0)
Besides the steric interactions between particles, we also consider the polar velocity alignment interactions. These alignment interactions can lead to complex and striking behaviors [38][39][40][41][42]. Chiral active particles with polar alignment interactions can feature long-range polar order, which violates rotational invariance even for monofrequent rotations, and induces new patterns. Slow rotations lead to the coarsening of macro-clusters, with an enhanced polarization, while faster rotations induce micro-flock patterns with a characteristic size [38]. The polar alignment interactions can also induce simultaneous phase separation in chiral active mixtures [40]. These systems display more collective behaviors due to the polar alignment interactions. Thus, it would be very interesting to study the directional rotation of a symmetrical gear in the presence of polar alignment interactions. We shall focus, next, on how the polar alignment affects the directional rotation of the gear.   Figure 7 shows the typical snapshots of the gear in a bath of chiral active particles for various g values at ϕ = 0.1, v 0 = 4.0, and Ω = 0.05. It is clear that polar alignment interactions affect the way in which chiral active particles interact with the gear. When g = 0, the particles interact with the gear independently due to the low particle density (shown in figure 7(a)). When the alignment interaction strength is small (e.g. g = 0.01), some particles accumulate in small clusters, leading to both clusters and individual particles colliding with the gear (shown in figure 7(b)). When the alignment interaction strength is strong (e.g. g = 1), all particles are gathered into a large cluster, moving together and subsequently colliding with the gear as a single entity (shown in figure 7(c)). Thus, the colliding behavior of the particles with the gear will affect the rotational direction of the gear. It should be noted that the density of particles also affects the collective behavior caused by the polar alignment. When the particle density is very small, due to the large average distance between particles, the effect of the alignment interaction is minimal and can practically be ignored. When the particle density is very large, however, the average distance between particles is significantly reduced, leading to a large effect of the alignment interaction. In other words, when the alignment interaction is fixed, the greater the density of particles, the easier it is for them to move synchronously.
In figure 8, we plot the scaled average angular velocity ω versus the alignment interaction strength g for different v 0 at Ω = 0.05 and ϕ = 0.1. When v 0 is small (e.g. v 0 = 0.5), the scaled average angular velocity initially increases and then decreases, indicating that an optimal alignment interaction intensity can promote the rotation of the gear. When v 0 is large (e.g. v 0 = 6.0), the scaled average angular velocity decreases from positive values to negative values as g increases. When both v 0 and g are large enough, the trajectory radius surpasses the system length L and the particles perform collective motion, thus driving the clockwise rotation of the gear and resulting in negative net torque. Figure 9 depicts the scaled average angular velocity ω as a function of the chirality Ω for different v 0 at g = 1.0 and ϕ = 0.1. When v 0 is small (e.g. v 0 = 0.5), the corresponding trajectory radius is small. The scaled average angular velocity is always positive and the gear rotates counterclockwise. However, when v 0 is large (e.g. v 0 > 2), the direction of rotation of the gear becomes very sensitive to the angular velocity Ω. To explain the phenomenon of multiple rotation reversals clearly, the collisions between the particles and the gears can be analyzed by looking at a four-gear configuration in figure 10. In it, the red and blue dashed lines represent the trajectories of the collective motion of particles corresponding to the points A and B in figure 9(a), respectively. At point A, the particles' clockwise rotation induces the gear to rotate clockwise (i.e. ω < 0) for small Ω (e.g. Ω = 0.07). At point B, where Ω is larger (e.g. Ω = 0.2), the corresponding trajectory surrounds four gears. Here, the particle swarm rotates clockwise and thus the gear will rotate counterclockwise (i.e. ω > 0). In summary, when the chiral particles move in the space between the gears (see the blue trajectory in figure 10), they will drive the gears to rotate counterclockwise; when the particles move around gears (see the red trajectory in figure 10), they will drive the gears to rotate clockwise. Thus, the multiple rotation reversals are caused by the competition between those two types of driving mechanisms. Figure 11 shows the scaled average angular velocity ω as a function of the self-propelled speed v 0 at Ω = 0.1, g = 1.0, and ϕ = 0.1. As v 0 increases, we can observe the rotation reversals of the gear. When v 0 is small, the scaled average angular velocity is positive. As v 0 becomes larger, the rotation direction of the gear  depends on the competition between two different driving mechanisms. As illustrated in figure 10, if the particles move within the space between the gears, the gears are driven to rotate counterclockwise. On the other hand, when the particles move around the gears, the gears are driven to rotate clockwise.

Concluding remarks
In conclusion, we have numerically studied the rotation of a gear in a bath of chiral particles. Through this study, it has been shown that the chirality of active particles can power and steer the macroscopic directional rotation of a symmetric gear. Specifically, when zero polar interactions are present, the direction of the gear's rotation is observed to depend on the chirality of the active particles, with a clockwise rotation occurring for Ω < 0 and a counterclockwise rotation occurring for Ω > 0. Furthermore, it was found that there exists a set of optimal parameters consisting of the chirality, self-propelling speed, and packing traction at which the rotational speed would reach its maximum. In the case of polar alignment interaction, chiral active particles perform collective motion, which results in the rotation behavior of the gear becoming much more complex. In this situation, the direction of the gear's rotation is determined by the competition between the chirality of active particles and the polar alignment interactions; multiple rotation reversals can occur upon changing the system parameters. Ultimately, our findings are relevant to understanding how microscopic random motion of active particles is converted into macroscopic directional rotation of a gear, which could potentially (in the future) be implemented in experiments such as a Bacillus subtilis powered submillimeter gear [29] or a ratchet gear rotation involving self-propelling catalytic Janus particles [32]. Finally, we need to point out that there is another type of system related to our research: odd-viscosity fluids [43][44][45][46]. Examples of this include active spinners or active chiral particles. Upon analysis, we have speculated that a symmetric gear can rotate in a flow of an odd-viscosity system. More specifically, skyrmions, fluid vortices, and active spinners passing by a symmetric obstacle can induce rotation, whereas non-odd viscosity fluids cannot. This could be utilized as a novel way to extract work from odd-viscosity systems.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.