| Issue |
Natl Sci Open
Volume 3, Number 4, 2024
Special Topic: Active Matter
|
|
|---|---|---|
| Article Number | 20230066 | |
| Number of page(s) | 13 | |
| Section | Physics | |
| DOI | https://doi.org/10.1360/nso/20230066 | |
| Published online | 05 January 2024 | |
RESEARCH ARTICLE
Spontaneous unidirectional rotation of a symmetric gear driven by spherical active particles
1
Center for Soft Condensed Matter Physics & Interdisciplinary Research, School of Physical Science and Technology, Soochow University, Suzhou 215006, China
2
School of Physics and Electronic Engineering, Linyi University, Linyi 276005, China
* Corresponding authors (emails: kangchen@suda.edu.cn (Kang Chen); tianwende@suda.edu.cn (Wende Tian))
Received:
23
October
2023
Revised:
7
December
2023
Accepted:
29
December
2023
Time reversal asymmetry and spatial anisotropy are considered two prerequisites for Brownian ratchet. An intriguing realization can be achieved by placing an asymmetric gear in the suspension of motile rod-like bacteria. Usually, alignment interactions caused by anisotropic collisions or hydrodynamics would boost the ratchet effect. Here, we are concerned with a perfectly isotropic system, i.e., symmetric gear immersed in a bath of spherical active Brownian particles. We find that, under certain conditions, kinetic symmetry-breaking arises spontaneously, i.e., the symmetric gear keeps rotating in one direction. Unexpectedly, such ratchet phenomenon does not rely on the direct many-particle interactions and moreover the introduction of alignment interaction would counterintuitively prevent it from happening! Further investigation reveals that such spontaneous symmetry-breaking phenomenon shares similarities with the equilibrium phase transition of the Ising model. Our results provide new insights and enhance our understanding of the fundamental aspects of active ratchet phenomena.
Key words: ratchet effect / spontaneous symmetry breaking / active brownian particles / ising model / hysteresis
© The Author(s) 2024. Published by Science Press and EDP Sciences.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
INTRODUCTION
The Brownian motor or ratchet, also known as the Smoluchowski-Feynman ratchet, is a fascinating topic in thermodynamics [1,2]. The fundamental physics behind the operation of the Brownian ratchets is the second law of thermodynamics, which prohibits extraction of useful work from a single equilibrium heat bath. Generally, to achieve the rectification phenomena, or ratchet effect, both time and spatial symmetries must be broken [1,3]. This requires driving the system out of thermal equilibrium. Numerous models of Brownian ratchets that could break both the symmetries have been developed and studied, including temperature ratchets [4], friction ratchets [5], on-off ratchets [6], travelling potential ratchets [1], fluctuating potential ratchets [7,8], drift ratchets [9], and more. Many of them can be classified as pulsating ratchets or tilting ratchets [1]. In the past two decades, a new class called active Brownian ratchets has attracted significant research interest [3,10]. Unlike conventional Brownian ratchets, the self-propelling motion of agents in active Brownian ratchets inherently breaks detailed balance or time-reversal symmetry. And the breaking of spatial symmetry is then often achieved by introducing asymmetric potentials/structures [2,3,11,12] or material properties [13]. Other possibilities are related to asymmetric dynamics either intrinsic (e.g., chiral active particles) [14–16] or generated spontaneously by alignment-induced collective motion [17,18].
An intriguing realization of active Brownian ratchets is achieved by immersing an asymmetric gear in a bacterial bath, which was able to rotate unidirectionally [19–22]. This demonstrates a simple design of autonomous micromachine powered by microorganisms. A consensus among these researches is that the asymmetric shape of the gear, which breaks spatial symmetry, is essential for the persistent directed rotation. However, in this paper, we focus on a perfectly isotropic system, i.e., we study by numerical simulations a symmetric gear immersed in a bath of spherical active Brownian particles (ABPs). In this simple dry model, there is no shape/property asymmetry, chiral dynamics, hydrodynamics or collision-induced alignment interactions. Overall, there does not seem to be any factor that could cause the breaking of spatial symmetry. Surprisingly, we find that without any additional nonequilibrium perturbations, the perfectly symmetric gear can persistently rotate in one direction under certain conditions. This is a spontaneous symmetry-breaking phenomenon. Simulations also show that such ratchet phenomenon does not rely on the direct many-particle interactions. Moreover, if isotropic spherical ABPs are replaced by rod-like ones (akin to elongated E. coli cells), the thus introduced alignment effect (by collisions) would counterintuitively prevent the ratchet motion from happening. Further investigations reveal similarities between this spontaneous kinetic symmetry-breaking phenomenon and the equilibrium phase transition of the Ising model of the ferromagnetic systems. Our results provide new insights into the roles of asymmetric shape, alignment effect, and many-particle interactions in active-matter-powered rotors, and enhance our understanding of the fundamentals of active ratchet phenomena.
MODEL AND METHODS
In our two-dimensional model, a symmetric gear with eight teeth (Figure 1A–C) is mixed with round ABPs of mass m and size σ. The rim of the gear is composed of highly overlapped beads of the same size σ. The inner and outer radii of the gear are set as 8σ and 14σ, respectively. All particles and beads interact with the repulsive Weeks-Chandler-Andersen (WCA) potential, 
with a cutoff at 
beyond which 
. Each ABP undergoes Brownian dynamics according to the following translational and rotational equations of motion
 |
Figure 1 Rotation of the symmetric gear in the ABP bath. (A)–(C) Snapshots at sequential times of the simulation. The solid cyan circles are ABPs. The symmetric gear is marked with a blue arrow and the black curved arrow indicates the direction of rotation. (A) The aggregation of ABPs around the gear just before the start of rotation. Typical time sequences of |
where 
is the total potential of particle i. 
and 
are the translational friction coefficient and rotational diffusion rate, respectively. The ABPs are non-chiral and propelled by a centric force of strength Fa along the direction 
, which varies diffusively over time. 
and 
are unit-variance Gaussian white noises. We consider the gear as a rigid body. In order to focus on the rotational motion, we fix its center position while allowing it to rotate freely [23,24]. The rotational motion of the gear is governed by
where 
is the rotational inertia, 
the rotational friction coefficient, and 
the total torque on the gear due to the gear-ABP interactions. The simulations are performed in a box of 
with periodic boundary conditions. 
, 
and 
are used as scaling units and the corresponding time unit 
. We set 
, 
, 
and 
. 
and 
are large enough that both the motions of the ABPs and the gear are essentially overdamped [23,24]. Keeping the inertial terms in Eqs. (1) and (3) allows us to use the velocity-verlet algorithm (by the home-modified LAMMPS software) and a large time step.
RESULTS
The phenomenon of unidirectional rotation
The propelling strength 
, the rotational diffusion rate 
and the density 
(area fraction) of ABPs are three key variables that characterize the activity and amount of active agents in the bath. We find that under some conditions such as 
, 
and 
, the symmetric gear can persistently rotate along one direction (see Figure 1A–C and supplementary movie S1). In the beginning, the ABPs aggregate densely around the gear (Figure 1A) and the rotation then starts in a direction stochastically determined by the imbalanced push from the surrounding ABPs. It is worth noting that no shape asymmetry of any kind is introduced into the system unlike prior works where either intrinsic shape asymmetry (asymmetric gear) [19–22,25] or spontaneously generated shape asymmetry (collective deformation of chains grafted on the colloidal disk) [23,24] was found to play a crucial role in persistent unidirectional rotation. The central questions here are why and under what conditions could a symmetrical gear persistently rotate in one direction, and why has this phenomenon not been observed in previous works?
By tuning the rotational diffusion rate 
and/or the density 
of ABPs, we observe three kinds of typical time sequences of the rotating angular velocity of the gear, 
(Figure 1D–F and supplementary movies S1–S3). The time interval for measuring the “instantaneous” angular velocity is 
. We denote the three types of time sequences as three rotating states: unidirectional rotating state (URS), transitional state (TS) and random rotating state (RRS). In the URS, 
fluctuates around a non-zero mean value and the accumulative angle, 
, increases (counterclockwise) or decreases (clockwise) nearly monotonically (Figure 1D, G). ABPs continuously enter and leave the vicinity of the gear during rotation. Unlike the asymmetric gear powered by bacteria or rod-like active particles [19], there is no reversal of rotation at the beginning for the symmetric gear (Figure 1D). In the TS, the gear could rotate in one direction for an extended period (Figure 1E, G), but the direction would occasionally reverse due to sudden, large fluctuations. And, in the RRS, there is no apparent directional rotation at all (Figure 1F, G).
Phase diagram
Figure 2A depicts the “phase diagram” for the states of rotation in 
-
space. We set the upper bound of 
as 0.2, below which ABPs do not spontaneously form large clusters in the bulk. This avoids strong irregular disturbance to the gear rotation from the ABP aggregates in the bulk. 
characterizes the persistence time of the active motion of ABPs. Smaller 
(stronger activity of the ABPs) and higher 
would cause more ABPs to gather around the gear, presumably leading to stronger push on the gear and less impact of fluctuations. Therefore, both small 
and large 
are conducive to persistent directional rotation as demonstrated in Figure 2A (URS appears at the upper left part of the phase diagram). As 
and 
vary, the state of rotation continuously changes among the three states. In Figure 2B, we show the typical probability distributions of 
in the three rotating states. One big peak around 
splits into two symmetric peaks at nonzero values when the gear changes from the RRS to the TS/URS. In the TS, the probability around zero angular velocity 
, is notably above zero, indicating that the rotational direction reverses from time to time. However, it is almost zero in the URS, implying that the change of rotational direction hardly occurs.
 |
Figure 2 (A) Phase diagram in the |
Mechanism for the unidirectional rotation
The symmetric gear in our model is not expected to rotate consistently in one direction, since neither spatial asymmetry nor alignment rules that can cause collective directional motion are included. However, we do observe the URS at small 
and moderate 
. Figure 3A displays the long-time-averaged particle-flow field, 
around the gear, where ρ represents the local particle number density and 
represents the local mean velocity. As anticipated, the ABPs form a circular flow field near the gear, following its rotation. At first glance, the situation seems to resemble a travelling potential ratchet [1], where the rotating gear serves as a traveling potential that rectifies the motion of ABPs. However, the essential difference lies in the fact that the traveling potential (the rotating gear) is not externally controlled but generated by the motion of ABPs themselves. Interestingly, when we choose the rotating gear as the reference frame, the ABPs form small vortices between the teeth with opposite orientations relative to the rotation of the gear (Figure 3B). The formation of these vortices and the maintenance of the unidirectional rotation are closely related. In the schematics of Figure 3C–F, we ignore the change in the propelling direction of an ABP during its interaction with the gear since the URS happens at low 
. The behaviors of ABPs after hitting the two sides of a rotating tooth are different (see supplementary movies S4 and S5). We refer to these two sides as “head-on side” (HOS) and “rear-end side” (RES), respectively (Figure 3C). When an ABP collides onto the HOS, it tends to slide inward in the next moment due to the rotation of the gear (Figure 3D) and is then trapped in the groove (Figure 3E) until the gear rotates by a large enough angle that the propelling direction on the ABP becomes outward. In contrast, when an ABP collides onto the RES, it tends to slide outward and leave the gear (Figure 3D and E). Such asymmetric kinetic trend or biased trapping causes the inverse local particle vortices relative to the rotating gear. In view of the resultant torque on the gear, ABPs hitting the HOS initially generate a torque that impedes the rotation of the gear (Figure 3D). But after the gear rotates for a certain amount of angle, their push on the gear becomes promoting the rotation (Figure 3E). The analysis of the line density of normal force on the rim of the gear (Figure 3F) demonstrates that ABPs trapped near the grooves contribute the major driving torque to the rotation. The rotation of the gear and its consequent asymmetric impacts or biased trapping on incident ABPs form a positive feedback loop that sustains the unidirectional rotation.
 |
Figure 3 Example long-time averaged flow field of ABPs in the lab (A) and the gear (B) frames ( |
Why has the unidirectional rotation of symmetrical gear not been observed in previous studies [19–22,25]? The probable reason, we believe, is the use of rod-like active particles/bacteria rather than spherical ones with the freedom of orientation. The propelling direction of rod-like active objects is related to their asymmetric shape. When they collide each other or a boundary, their bodies and hence propelling directions tend to align with each other or the boundary. When the rod density is low, the behavior of an active rod entering a rotating gear is illustrated in Figure 4A and B. Shortly after touching the rim of the gear, the active rod moves parallel to the rim with the direction dependent mostly on the incident angle rather than whether it hits the HOS or RES. Therefore, the aforementioned positive feedback loop between the rotation of the gear and the asymmetric kinetic behavior of particles no longer exists. When the rod density is high, rod-rod aligned packing also dominates over the impact of gear rotation (see supplemental movie 6). Our simulations on varied situations show that symmetric gear immersed in a bath of active rods does not exhibit unidirectional rotation (Figure 4C–E). These results indicate that alignment effects counterintuitively hinder the breaking of spatial symmetry and hence the emergence of unidirectional rotation in the case of symmetric gear.
 |
Figure 4 (A) and (B) Schematics of collision of a rod-like active object with the gear. Long-time averaged flow field of rod-like active particles (C), time sequence of |
Non-interacting ABPs
Collective motion is usually the key to spontaneous symmetry breaking in active-matter systems. In bulk, with steric repulsion, ABPs can spontaneously form clusters through self-trapping mechanism and [26–30] move collectively, including transient rotation [31]. A question here is whether steric packing of ABPs is essential to the phenomenon of unidirectional rotation. To address it, we perform simulations with non-interacting ABPs by ignoring the ABP-ABP pair interactions in the model. The potential term 
in Eq. (1) now only contains the ABP-gear interaction. The gear is thus immersed in an “idea gas” of ABPs. Our results show that the URS still occurs at small 
and moderate 
(Figure 5A and B). The mechanism illustrated in Figure 3 works as well. The inverse particle vortices relative to the rotating gear become stronger and closer to the rim.
 |
Figure 5 Unidirectional rotation of the gear in the bath of non-interacting ABPs. (A) Time sequence of |
Without the particle-particle steric interaction, trapped ABPs can overlap and accumulate around the cusp of the groove (see supplementary Figure S1). A very good linear relation between the mean angular speed 
(
represents the absolute value and 
averages over time and independent runs) and the propelling strength Fa has been observed in the URS (Figure 5C). This linear relation can be explained by the following analysis [23,24]. The rotation of the gear together with the trapped ABPs depends on the balance of torque. The driving torque arises from the propulsions on the trapped ABPs, while the damping torque results from frictions on both the rotating gear and the trapped moving ABPs. The driving torque 
, where 
is the mean number of trapped ABPs and 
is the inner radius of the gear. The factor α accounts for the mean ratio of the projection of a propelling force normal to the radial direction. The damping torque comprises two parts: 
, arising from the frictions on the gear and the trapped ABPs, respectively. In the overdamped limit, 
, which gives
Equation (4) describes the 
relation very well (Figure 5C). 
is independent of 
and in all situations it varies in a narrow range ~0.31–0.34. 
depends on 
(see supplementary Figure S2), but it appears in both the numerator and the denominator in Eq. (4). As a consequence, the factor in front of 
on the right hand side of Eq. (4) is nearly a constant, implying a linear relation between 
and 
.
Comparison with the Ising model and the phenomenon of hysteresis
Figure 2B shows the example probability distributions of 
in the three typical states. To quantify the transition between the RRS and the TS/URS, we choose 
(the peak values of the distribution curves) as the order parameter. Its variation with 
at 
and 
is shown in Figure 2C. Interestingly, the transition is continuous, akin to the Ising model of ferromagnetic systems at zero external field. 
acts as the “temperature” and the transition occurs at a critical point, 
. Below and near the critical point, the order parameter can be well described by the scaling relation: 
. In Figure 2C, 
, 
and the critical exponent 
. Even though our model is two-dimensional, the exponent 
is close to that of the three-dimensional Ising model. Note that the transition here differs significantly from the Ising model. First, the transition in our system is kinetic and nonequilibrium, while the transition in the Ising model is thermodynamic and equilibrium. Second, the transition in our system is intrinsically a phenomenon of finite size, rather than of thermodynamic limit like the Ising model (the order parameter in our system is the state of rotation of one gear, rather than the magnetization averaged over individual spins in the Ising model). Third, once the spontaneous magnetization occurs below 
in the Ising model, the broken symmetry is then definite (up or down). In contrast, the gear in our system enters the TS (red hollow circles in Figure 2C) right below the critical point 
, in which the gear occasionally reverses its rotational direction. Hence, spatial symmetry does not break down on average over long times in the TS. Only when entering the URS (black hollow squares in Figure 2C), the direction of rotation and the symmetry breaking are stable and definite (at least within the time window of our simulations). Last, we cannot make a comprehensive analogy between the transitions in our system and in the Ising model. For example, we cannot find quantities in our system that corresponds to the heat capacity and the correlation length associated with the transition. Hence, we cannot obtain all the critical exponents and compare them with the Ising model.
To mimic the effect of external magnetic field in the Ising model, we add an extra term, representing external torque, Me to the equation of motion of the gear
The gear can rotate with a stable angular velocity 
in the absence of ABPs. We choose a URS (
and 
) in the phase diagram as an example and add external torque Me of different amplitudes. As shown in Figure 6A, the distribution of 
becomes more and more asymmetric as Me increases. Positive Me not only increases the probability of the gear turning in the positive direction (higher peak for 
), but also increases (for 
)/decreases (
) the angular velocity (shift of the peaks to the right in Figure 6A). It is worth noting that the probability curves at 
remain near zero, indicating that the gear is still in the URS under external torque. In other words, the gear can maintain unidirectional rotation even if the rotational direction is opposite to that of Me. However, when Me is sufficiently large, the rotational direction of the gear becomes completely determined by it and the stochasticity of the rotational direction ceases to exist (see Me = 1000 in Figure 6A).
 |
Figure 6 Rotation of the gear under external torque. Fa = 8, Dr = 0.001 and |
Since the gear remains in the URS in the presence of external torque, we use, for simplicity, the mean angular velocity 
as the new order parameter instead of the peak value 
. 
represents mean angular velocity averaged over time and independent runs of positive (negative) rotations. The relation between 
and Me is shown in Figure 6B. The dashed lines indicate the threshold values of external torque 
, within which the rotational direction of the gear is indefinite, exhibiting a certain degree of stochasticity.
Figure 6B is reminiscent of the magnetic hysteresis loop. In Figure 7, we calculate the mean angular velocity 
averaged over time and independent runs of both positive and negative rotations, taking into account their signs. Starting from random initial conditions, as in all the previous calculations, we obtain the blue curve in Figure 7. The hollow symbols signify that both positive and negative rotations are observed in the 100 independent runs for each case, while the solid symbols indicate that only positive or negative rotations occur. As expected, the symbols on the blue curve become solid after 
reaches 
. The blue curve is analogous to the initial magnetization curve. Then, starting from an initial state of 
(definite positive rotation), we decrease 
to obtain the black curve. Due to the biased initial condition, definite positive rotation extends to around 
(far below 
). Below 
, the probability of negative rotation gradually increases. When the external torque reaches the negative “coercivity” 
, the probabilities of positive and negative rotations become equal. Further decreasing 
to below 
, the rotation becomes definite negative. A similar red curve is obtained, starting from an initial state of 
(define negative rotation). The black and red curves form a hysteresis loop between 
. In reality, the lines of solid symbols in Figure 7 align with those in Figure 6B. The 
of retentivity is around 0.023, which is just the 
or 
at zero external torque.
 |
Figure 7 Hysteresis loop. Mean angular velocity |
The hysteresis phenomenon is also observable for non-interacting ABPs as shown in Figure 8. Since the non-interacting ABPs can overlap each other, they play a stronger role in pushing the gear than the ordinary ones. This can be partly reflected in the difference of mean angular speed at zero external torque (comparing Figures 1D and 5A) or 
’s of retentivity in Figures 7 and 8). Therefore, it is anticipated that the rotation of gear driven by non-interacting ABPs will be more heavily influenced by the initial state and the threshold (
) and coercivity (
) external torques will be greater, i.e., larger hysteresis loop. In Figure 8, 
(
) is more than four (five) times that in Figure 7. It is worth noting that, for example, the gear maintains definite positive rotation until approximately 
, when decreasing from the initial state of 
. The bare angular speed driven by external torque of 
is 
, much larger than 
or 
at zero external torque, which again manifests the strong dependence of the gear rotation on the initial state. Near 
, the number and distribution of trapped ABPs around the gear are not greatly influenced by the external torque. Then we can analyze the relation between the mean angular velocity and the external torque analogous to the derivation of Eq. (4). From the torque balance, we obtain
 |
Figure 8 Hysteresis loop for non-interacting ABPs. (A) Mean angular velocity |
The blue and green lines in Figure 8 are predictions of Eq. (6), which are in good agreement with the simulation data.
SUMMARY
In summary, complementary to current understandings, we find that perfectly symmetric gear can also rotate unidirectionally when powered by simple ABPs. Conventionally, alignment interactions and steric packing between active objects promote collective motions, but we find that they have no or even negative effects on the ratchet phenomena of a symmetric gear. In this paper, we only consider a “dry” model that could be realized and tested through macroscopic granular/robotic experiments [23]. While the impact of shape-induced alignment effects has been discussed, the impact of hydrodynamic interactions remains open for future research. In some situations, the impact is believed to be quantitative rather than qualitative [19,23,24]. However, hydrodynamic effects are complicated and may have significant impacts in our model. Additionally, the translational motion of the gear is turned off in this study, leaving the cooperation between the translational and rotational motion in the ABP-driven kinetics of a symmetric gear as another open topic for future research. Our findings provide new insights into the fundamental aspects of active ratchet phenomena.
Acknowledgments
We thank Ming-cheng Yang for helpful discussion.
Funding
This work was supported by the National Natural Science Foundation of China (21774091 (K.C.) and 21674078 (W.T.)).
Author contributions
K.C. and C.W. conceived and designed the research. C.W., W.L. and H.L. performed the simulations. All the authors participated in the analysis of the data. K.C. and W.T. wrote the manuscript.
Conflict of interest
The authors declare no conflict of interest.
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All Figures
|
Figure 1 Rotation of the symmetric gear in the ABP bath. (A)–(C) Snapshots at sequential times of the simulation. The solid cyan circles are ABPs. The symmetric gear is marked with a blue arrow and the black curved arrow indicates the direction of rotation. (A) The aggregation of ABPs around the gear just before the start of rotation. Typical time sequences of |
| In the text | |
|
Figure 2 (A) Phase diagram in the |
| In the text | |
|
Figure 3 Example long-time averaged flow field of ABPs in the lab (A) and the gear (B) frames ( |
| In the text | |
|
Figure 4 (A) and (B) Schematics of collision of a rod-like active object with the gear. Long-time averaged flow field of rod-like active particles (C), time sequence of |
| In the text | |
|
Figure 5 Unidirectional rotation of the gear in the bath of non-interacting ABPs. (A) Time sequence of |
| In the text | |
|
Figure 6 Rotation of the gear under external torque. Fa = 8, Dr = 0.001 and |
| In the text | |
|
Figure 7 Hysteresis loop. Mean angular velocity |
| In the text | |
|
Figure 8 Hysteresis loop for non-interacting ABPs. (A) Mean angular velocity |
| In the text | |
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Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.