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Project supported by JST, ACT-T Grant No. JPMJAX190S Japan and Multidisciplinary Research Laboratory System for Future Developments (MIRAI LAB)
We investigate the collective motion of rotlets that are placed in a single plane. Due to the hydrodynamic interactions, the particles move through the two-dimensional (2D) plane and we analyze these diffusive motions. By analyzing the scaling of the values, we predict that the diffusion coefficient scales with ϕ0.5, the average velocity with ϕ, and relaxation time of the velocity autocorrelation function with ϕ–1.5, where ϕ is the area fraction of the particles. In this paper, we find that the predicted scaling could be seen only when the initial particle position is homogeneous. The particle collective motions are different by starting the simulation from random initial positions, and the diffusion coefficient is the largest at a minimum volume fraction of our parameter range, ϕ = 0.05. The deviations based on two initial positions can be explained by the frequency of the collision events. The particles collide during their movements and the inter-particle distances gradually increase. When the area fraction is large, the particles will result in relatively homogeneous configurations regardless of the initial positions because of many collision events. When the area fraction is small (ϕ < 0.25), on the other hand, two initial positions would fall into different local solutions because the rare collision events would not modify the inter-particle distances drastically. By starting from the homogeneous initial positions, the particles show the maximum diffusion coefficient at ϕ ≈ 0.20. The diffusion coefficient starts to decrease from this area fraction because the particles start to collide and hinder each other from a critical fraction ∼ 23 %. We believe our current work contributes to a basic understanding of the collective motion of rotating units.
The collective motion of the rotating unit can be seen in the artificial or natural systems, and a number of interesting phenomena have been reported. By applying a rotating external magnetic to magnetic colloids, the particles rotate and show complex collective motions due to the hydrodynamic and magnetic interactions between the colloids. Previous works applied in-plane[1–4] or out-of-plane[5–8] rotational magnetic field to magnetic colloids that locate in a single plane and analyse their collective motions. Recently, we reported that the magnetic rotors that are placed in a square grid show interesting rotational patterns.[9,10] We found a quarter pattern, in which the rotors undergo full rotations with different quadrants of the array turning in different directions when the external magnetic field is dominant. When the dipolar interaction between the rotors is dominant, a stripe pattern in which the rotors swing upwards or downwards in alternating stripes emerges. In recent years, we also proposed that this magnetic torque is useful to control a single magnetic object.[11–13]
In nature, a group of bacteria that locates close to a rigid wall shows a crystal pattern[14] or clustering phenomenon.[15,16] When the bacteria are swimming toward the bottom wall, they have attractive interactions between each other due to the hydrodynamic interactions of rotating doublets. Previous studies[16,17] reported that the clustering of magnetotactic bacteria can be controlled by changing the external magnetic field. There is also a report that rotating units, such as bacteria flagella, that are fixed to a wall would create a metachronal beating pattern.[18]
When there are a number of rotating units under small but finite Reynolds number, they form a cluster and rotate together as a group.[19,20] There are also interesting reports that a binary mixture, which is two groups of rotating units that have opposite rotational directions, shows a phase separation.[21–23] All these physics can be categorized to the active matter physics, which is a research area that is growing in recent years.
In these previous studies, they reported the collective motions of rotating units that have not only hydrodynamic interactions but also the magnetic interactions, inertial force, binary mixtures, or other factors. In this paper, we set a goal to understand the basic character of collective motions of rotating units, under a simple problem setup. We assume Stokes flow in the present work and investigate the collective motion of rotlets (torques), which is a flow field that is driven by a point torque,[24] that are placed in a single 2D plane with changing the area fraction of the particles ϕ. When a particle is under the two extremes conditions, a dilute limit ϕ ≪ 1 or maximum packing fraction, the particle has no movement because there is no flow that translates the particle. Therefore, the particle would obtain the maximum velocity and diffusion coefficient at an intermediate area fraction, and we discuss this point in this paper.
Consider N particles with the radius a that are placed in a single monolayer xy-plane, with an area fraction ϕ as shown in Fig.
In an ideal hexagonal lattice, the centre-to-centre distance ℓ can be described as a function of ϕ as
In this paper, we consider this length as the average inter-particle distance under homogeneous distributions. Note that the maximum area fraction of spheres in 2D is
In this paper, we use far-field hydrodynamic interactions of the Stokesian dynamics.[25,26] In a dilute regime ϕ ≪ 1, the i-th particle would rotate about z-axis with an angular velocity
![](cpb_29_6_064705/cpb_29_6_064705_ieqn2.gif)
Note that this model stands only for the dilute to semi-dilute regime, and it should be revised when the particles are in the touching distance. In this paper, we keep this far-field assumption in order to understand the basic character of the collective motions of rotlets.
By defining
There is only one dimensionless parameter ka2/T in this setup, which is the strength of the repulsive force compared to the rotlet interactions. In this paper, we kept this parameter as a constant and set it small ka2/T = 0.1 since our main focus is on the rotlet contribution. The number of particles N = 2000 is kept constant and the domain size L × L is varied to control the volume fraction ϕ, where
In this paper, we utilize two different initial positions of the particles: random or homogeneous configurations. The particle positions are initially random for the random configurations, while the particles are positioned in a square grid for the homogeneous configurations.
Figure
Figure
![]() | Fig. 3. Particle trajectories of eight particles for dimensionless time ![]() |
Many previous papers reported that the rotating units, such as rotating disks[19] or bacteria swimming close to a surface,[14] form hexagonal lattice structures. In these previous studies, they form a crystal-like structure due to the existences of the attractive and repulsive forces. In the case of rotating disks under finite Reynolds number,[19] the Magnus effect contributes to repulsive interactions while hydrodynamic interactions contribute to attractive interactions. In the present study, the steric repulsive force F drives the crystallization for ϕ ≥ 0.40. Figure
Before further reporting the numerical results, we show here estimations of the scalings of three parameters in this subsection: average velocity 〈 v 〉 = 〈 |v| 〉, diffusion constant D, and relaxation time of the velocity auto-correlation function τ0. Note that we ignore the effect of the steric force F in this estimation.
Coming back to the equation of translational velocity that is generated by the rotlets vT, Eq. (
We now consider a new dimensionless system: fixing the distance ℓ and varying the area fraction ϕ to change the particle size a(ϕ). If we rewrite the velocity (
![](cpb_29_6_064705/cpb_29_6_064705_ieqn12.gif)
![](cpb_29_6_064705/cpb_29_6_064705_ieqn13.gif)
![](cpb_29_6_064705/cpb_29_6_064705_ieqn14.gif)
In the same matter, scaling of other values can be evaluated as
Figure
![](cpb_29_6_064705/cpb_29_6_064705_ieqn15.gif)
Figure
By starting from the homogeneous positions, on the other hand, the diffusion coefficient scales with ϕ0.5 for small ϕ as we predicted, and the coefficient reaches a maximum around ϕ ∼ 0.20. The deviation from the scaling ϕ0.5 at ϕ ∼ 0.20 can be explained by the steric effect. When a particle would go through space between two particles i and j, the center-to-center distance between two particles needs to be larger than rij/a = 4. Solving ℓ/a = 4 gives the critical area fraction ϕc ≈ 0.23, and the particles hinder each other if the area fraction is larger than this value. Note we evaluate the diffusion coefficient with other homogeneous initial position (hexagonal lattice with a small spacial inhomogeneity) just for a single case (ϕ = 0.15), and find that the diffusion is nearly the same (
In order to investigate the difference between two initial configurations, we show the radial distribution functions in Fig.
These distributions are drastically different for small area fractions ϕ < 0.20 by starting the simulation from random initial positions. Compared to the homogeneous initial positions, firstly, the particles locate closer to each other. At ϕ = 0.05 (Fig.
Contrary to the case of small area fractions, the distributions are the same for large area fractions. This contrast can be explained by the frequency of collision events. As we discussed in the previous sections, the inter-particle distance gradually increases due to the collision events. The radial distribution function is the same regardless of the initial positions for large area fractions since many collision events would quickly modify the particle configurations. For small area fractions, on the other hand, the radial distribution function at small distances would remain from the initial random positions, because the rare collision events would not modify the inter-particle distances drastically. As a result, the radial distribution function of the random initial positions fall into a different local solution from the homogeneous one.
The average velocity of the particle 〈v〉 also depends on the initial positions, as shown in Fig.
If we assume that the translational velocity of a particle is mainly determined by a single neighboring particle at a distance ℓ/a, the translational velocity is evaluated by substituting Eq. (
![](cpb_29_6_064705/cpb_29_6_064705_ieqn18.gif)
![](cpb_29_6_064705/cpb_29_6_064705_ieqn19.gif)
Figure
In this paper, we investigate the collective motion of far-field rotlets that are placed in a single plane. Due to the hydrodynamic interactions, the particles move through the 2D plane and we analyze these diffusive motions. By analyzing the scaling of the values, we predict that the diffusion coefficient scales with ϕ0.5, the average velocity with ϕ, and relaxation time of the velocity autocorrelation function with ϕ–1.5, where ϕ is the area fraction of the particles. In this paper, we find that the predicted scaling could be seen only when the initial particle position is homogeneous. The particle collective motions are different by starting the simulation from random initial positions, and the diffusion coefficient is the largest at a minimum volume fraction of our parameter range, ϕ = 0.05.
The deviations based on two initial positions can be explained by the frequency of the collision events. The particles collide during their movements and the particle–particle distance gradually increases. When the area fraction is large, the particles will result in relatively homogeneous configurations regardless of the initial positions because of many collision events. When the area fraction is small (ϕ < 0.25), on the other hand, two initial positions would fall into different local solutions because the rare collision events would not modify the inter-particle distances drastically.
By starting from the homogeneous initial positions, the particles show the maximum diffusion coefficient at ϕ ≈ 0.20. The diffusion coefficient starts to decrease from this area fraction because the particles start to collide and hinder each other from a critical fraction ∼ 23 %.
We believe our current work contributes to a basic understanding of the collective motion of rotating units.
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