Collective motion of polar active particles on a sphere∗

Yi Chen(陈奕) Jun Huang(黄竣) Fan-Hua Meng(孟繁华)Teng-Chao Li(李腾超) and Bao-Quan Ai(艾保全)

1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,School of Physics and Telecommunication Engineering,South China Normal University,Guangzhou 510006,China

2Guangdong–Hong Kong Joint Laboratory of Quantum Matter,South China Normal University,Guangzhou 510006,China

Keywords: clustering motion of Brownian particles,polar active particles,sphere

1. Introduction

In nature, the collective motion of various living systems (for example, bird flocks,[1]fish schools,[2]bacterial colonies,[3]and mixtures of motor and cytoskeletal proteins,[4])and the synthesis of slender living particle swarms have interesting common features, such as swirling patterns and swarming-related to the alignment of motion with their neighbors.[5]Vicseket al.proposed an agent-based minimal model for flocking,accounting for the interplay between fluctuations and simultaneous interactions of multiple agents.[6]On the basis of these, there have been several works modeling the motion of micrometric on simple curved surface like sphere. Sanchezet al.[7]constructed a model of active particles confined to move on a sphere. Sknepnek carried out relevant research on the movement of active entities on the surface, and Henkes carried out computational research on the interaction of active particles moving on the sphere.[8]Along the same line, Brusset al.[9]found the steady state profile of a flock on a sphere and on a catenoid, as well as topological modes due to curvature and activity in the system. Apaza and Sandoval[10]generalized the dynamics of active Brownian particles (ABPs) moving on the surface of a sphere by adding both translational and rotational Brownian motion. In addition, Janssenet al.[11]studied the motion of self-propelled particles(spheres and rods)on a sphere,and for various particle densities. A theoretical study on the dynamics of active particles performing rotational Brownian motion on curved surfaces has also been presented by Filyet al.[12]Cuiet al.proposed a strategy for separating micro-suspents by using micro-suspents to surf on the substrate chemical wave front.[13]Chenet al. found that new collective dynamics and structures appeared when the dimer’s translational and rotational self-propulsion occurred simultaneously,[14]and made it possible for the chemical self-propulsion motor to operate in an active chemical medium far from equilibrium.[15]Shiet al.found that the self-organization pattern of cortical microtubules in plant cells can be regulated by controlling the assembly dynamics of individual microtubules.[16]These studies provide more possibilities for our theoretical research.

Beyond swarming, self-propelled particles (SPPs) also form dense clusters via motility-induced phase separation,whereby SPPs phase separate into coexisting dense and dilute phases.[17,18]When we observe the motion of particles on a curved surface,the existence of inherent surface curvature disrupts the local order,resulting in a phenomenon different from that in a two-dimensional plane.[11,19–34]It has been proved that the degree of phase separation of SPPs on the surface is largely caused by the microgroup effect and depended on the gaussian curvature of the closed surface nonlinearly.[9]Focusing on the case of self-propelled particles confined to the surface of a sphere, they found that for high curvature, particles converge to a common orbit to form symmetry-breaking microswarms. In the Vicsek model, particles are point like and align instantaneously. The model can be extended to include particle volume which has not been considered before,but its effects remain poorly understood,especially at high densities.In this paper, we discussed the motion of Brownian particles on a sphere with a combination of short-range repulsion and angular diffusion. In the model, we added a polar alignment strength that affects particles motion,along with gaussian curvature,affecting particle swarm motion.

2. Model and methods

We consider thatNspherical particles of diameterσare confined to a sphere surface of radiusR. The dynamics of particleiis described by the positionri ≡(xi,yi,zi)of its center and the orientationni. The particle velocityv0and the directionniare constrained at each point in the tangent plane. The surface of the sphere is algebraically described byg(ri)=0,where

We assume translational and rotational diffusion to be independent and translational diffusion to be negligibly small, so that the limit of small persistence lengths exists in the model effectively. In the overdamped limit, the dynamics of particleiare described by the following Langevin equations with holonomic constraint:[8,20,22]

whereFijis the short-range nonactive two-body force andµis the mobility.G(r)=▽rg(r) is the gradient ofg(r).λis the Lagrange multiplier which is determined by the RATTLE algorithm.[35]Note thatg(r) can be interpreted as a potential, and the constraint trajectories will then lie on the isopotential surface with potential value 0. Therefore, there exists an effective forceG(ri)Tλithat penalizes any deviations from the isopotential surface,which prevents particles falling down from the surface.

Fig.1. Particles interact via a short-range soft potential Eint,which is finite for any value of rij.

In order to fully describe motion of the particles,we need to also specify dynamics for the internal degree of freedom,i.e.,the direction vectorni. Usually,the dynamics ofnicontains two contributions to the torque:[22](i)polar alignment of neighboring directors and (ii) polar alignment of the directorniwith the direction of its own velocity.The second alignment in our model takes effect only when the activity of particles is very low. Therefore, we only consider the first polar alignment. So theXY-like angular dynamics on the sphere is given by[8]

whereξi(t)is a scalar delta-correlated angular noise with distribution〈ξi(t)ξj(t′)〉=Drσi jσ(t −t′),andDris the rotational diffusion constant. The summation in Eq. (2) is carried over all neighbors within a 4.0σcut off radius. WhereJis polar alignment strength and we only consider the ferromagnetic case(J>0)and ˆriis the local surface normal. ThePNis the normal projection of a vector on the unit normal to the tangent plane asPN( ˆNi,a)=(a·ˆNi). The orientational order of active particles on a sphere can be characterized by the polar order parameter

Active particles move in any direction with equal probability whenP →0,while active particles move together along a circulating ring whenP →1. We define the ratio between the area occupied by particles and the total available area as the packing fractionϕ=Nπ(σ/2)2/4πR2.

3. Results and discussion

In our simulations, equation (1) has been numerically solved by using RATTLE discretization.[35]Particles are constrained to the curved surface by projecting the positions and force vectors back onto the local tangent plane after every time step.Equation(2)has been integrated numerically,the torques were projected onto the surface normal atri, and finally,niwas rotated by a random angle around the same normal.[36]The integration step time ∆t=0.01 and the total integration time was more than 106. The stochastic averages were obtained as ensemble averages over 300 trajectories with random initial conditions. Unless otherwise noted, our simulations were under the parameter sets:µk=50,σ=0.5,ϕ=0.12.We tested that the presented results are robust against reasonable changes in these parameters. In the following,we explore the collective behavior of particles by the rotational diffusion coefficientDr, the polar alignment strengthJ, and the selfpropelled velocityv0.

Figure 2 simulates the motion of particles on the sphere.When the radius of sphere and packing fraction remain unchanged, with the increase of the polar alignment strengthJ,the collective motion of particles gradually changes from disorder to order. When the polar alignment strengthJ=0.001,as shown in Fig. 2(a), the active particles move randomly on the sphere. When the polar alignment strengthJ=0.02, as shown in Fig.2(b),the active particles exhibit a small range of clustered and ordered collective motion.When the polar alignment strengthJ=5,as shown in Fig.2(c),the active particles move in an ordered group around the equator.

Fig. 2. The schematic motion of particles on a sphere when the radius of sphere and packing fraction remain unchanged(the radius of sphere R=6 and the packing fraction ϕ =0.12). (a)When the polar alignment strength J=0.001,the active particles move randomly on the sphere. (b)When the polar alignment strength J =0.06, the active particles have a small range of clustered and ordered collective motion. (c) When the polar alignment strength J=5,the active particles move in ordered groups around the equator.

Figure 3 shows the order parameterPas the ratio of the radius of the sphere to the diameter of the particleR/σfor different values of the polar alignment strengthJ. When the polar alignment strengthJ=0(represented by the black line),the order of particles decreases as the radius of the sphere increases. In this case, the collective behavior of the particles is only related to the gaussian curvature.[9]The larger the polar alignment strengthJis, the larger the order parameterPis, and the smallerR/σis, the larger the order parameterPis. Therefore, there is a competitive relationship between the polar alignment strengthJandR/σ. When 01,the polar alignment strengthJbecomes the main factor affecting the collective motion of particles.

Fig.3. The order parameter P versus the ratio of the radius of the sphere to the diameter of the particle R/σ for different J at ϕ =0.12,Dr=0.01,and v0=1.

Fig.4. The collective motion of particles when the polar alignment strength J=0.02. (a)The order parameter P=0.30 with the radius of sphere R=6.(b)The order parameter P=0.36 with the radius of sphere R=8. (c)The order parameter P=0.629 with the radius of sphere R=16.

When the polar alignment strengthJ=0.02, we change the radius of the sphereR.As can be seen from Fig.4,with the decrease of the gaussian curvature (in a sphere, the Gaussian curvature is the reciprocal of the radius of the sphere) of the sphere, the collective motion of particles gradually changes from disorder to order. When the radius of sphereR=6, the order parameterP=0.30, and the active particles move randomly in small clusters on the sphere. When the radius of sphereR=8,the order parameterP=0.36,and the active particles move in an ordered manner with small clusters;when the radius of sphereR=16, the order parameterP=0.629, and the active particles move in an ordered manner with clusters.

Fig. 5. The order parameter P versus the polar alignment strength J for different R at ϕ =0.12,Dr=0.01,and v0=1.

The collective motion of particles is affected by the radiusRand the polar alignment strengthJ. Figure 5 shows the order parameterPcorresponding to the continuously changing the polar alignment strengthJwhen the radiusRis different.When the radiusRremains unchanged, the order parameterPincreases with the increase of the polar alignment strengthJand eventually approaches 1, that is, the collective motion of particles is in order. When the polar alignment strengthJis relatively small (J<0.01), the main factor affecting the collective motion of particles is the size of radius, that is, the order parameterPof particles decreases with the increase of the radiusR. When 0.01

Fig.6. Different states of motion of particles at ϕ=0.12,Dr=0.01,v0=1,and R=6.(a)The particles move in long chains around the equator.(b)Particles clumped together and moved around the equator.

Fig. 7. The phase diagram of P versus J for different R at ϕ = 0.12,Dr=0.01,and v0=1.

In addition,figure 8 shows the effects of the polar alignment strengthJand velocity of particlesv0on the order parameterP. As shown in Fig.8(a),we observed the change ofPwith the continuous change ofv0in the case of differentJ.The function ofJis to make particles move collectively, and the function ofv0is to disperse particles, so there is a competitive relationship between the two on the influence of the collective motion of particles. When 0

Fig.8. (a)P versus v0 of the particles for different J at ϕ=0.12,Dr=0.01,and R=8. (b) P versus v0 of the particles for different R at ϕ =0.12,Dr=0.01,and J=0.1.

Fig.9. (a)The order parameter P versus Dr of the particles for different J at ϕ =0.12,v0=1,and R=8. (b)The order parameter P versus Dr of the particles for different R at ϕ =0.12,v0=1,and J=0.1.

Figure 9(a) shows the continuous change ofPwithDrat differentJ. As can be seen from the figure, the higher the rotational diffusion coefficient is,the more disordered are the particles. This is because the higher the rotational diffusion coefficient is,the more active particles will be. The largerJis,the less influence the rotational diffusion coefficient has on the

At last,we observed the change ofPunder differentRandϕ. It can be seen from Fig.10 that whenϕis small(ϕ<0.3),the influence of different radiusRonPis constant under the sameϕ. Whenϕis relatively large(ϕ>0.3),under the sameϕ, the larger the spherical radius, the larger theP. Thus it can be seen that under the sameϕ, both the radiusRand the packing fractionϕinfluence the effect of the polar alignment strengthJ.Obviously,the biggerRis,the more space particles move,the stronger the effect ofJis.

Fig.10. The order parameter P versus the packing fraction ϕ of the particles for different R at Dr=0.01,v0=1,and J=0.1.

4. Concluding remarks

In conclusion, we discussed the motion of active particles on a sphere under the influence of the polar alignment strengthJand the radius of sphereR. Particles moving on the sphere are affected by the radius of sphereRand the polar alignment strengthJ, and they compete with each other.WhenJincreases,the influence of curvature on the collective motion of particles will decrease. WhenJ>1,Jbecomes the main factor affecting the collective motion of particles. In addition,we also observed the particle clustering under differentDrand different particle velocitiesv0. From the results,the velocity of active particles has little effect on the motion of particle clusters, and the rotational diffusion coefficient of particles will affect the motion of particles clusters. A relatively large rotational diffusion coefficient is not conducive to the formation of ordered motion of particles. At last,we also discussed the change ofPunder differentRandϕ. Under the influence of the sameJ,particles with smaller gaussian curvature and smaller packing fraction are more likely to form an ordered motion of clustering.