The collective dynamics of self-propelled particles

Abstract

We have proposed a method for the dynamic simulation of a collection of self-propelled particles in a viscous Newtonian fluid. We restrict attention to particles whose size and velocity are small enough that the fluid motion is in the creeping flow regime. We have proposed a simple model for a self-propelled particle, and extended the Stokesian Dynamics method to conduct dynamic simulations of a collection of such particles. In our description, each particle is treated as a sphere with an orientation vector p, whose locomotion is driven by the action of a force dipole S p of constant magnitude S 0 at a point slightly displaced from its centre. To simplify the calculation, we place the dipole at the centre of the particle, and introduce a virtual propulsion force F p to effect propulsion. The magnitude F 0 of this force is proportional to S 0 . The directions of S p and F p are determined by p. In isolation, a self-propelled particle moves at a constant velocity u 0 p, with the speed u 0 determined by S 0 . When it coexists with many such particles, its hydrodynamic interaction with the other particles alters its velocity and, more importantly, its orientation. As a result, the motion of the particle is chaotic. Our simulations are not restricted to low particle concentration, as we implement the full hydrodynamic interactions between the particles, but we restrict the motion of particles to two dimensions to reduce computation. We have studied the statistical properties of a suspension of self-propelled particles for a range of the particle concentration, quantified by the area fraction φ a . We find several interesting features in the microstructure and statistics. We find that particles tend to swim in clusters wherein they are in close proximity. Consequently, incorporating the finite size of the particles and the near-field hydrodynamic interactions is of the essence. There is a continuous process of breakage and formation of the clusters. We find that the distribution of particle velocity at low and high φ a are qualitatively different; it is close to the normal distribution at high φ a , in agreement with the experimental measurements of Wu & Libchaber (2000). The motion of the particles is diffusive at long time, and the self-diffusivity decreases with increasing φ a . The pair correlation function shows a large anisotropic buildup near contact, which decays rapidly with separation. There is also an anisotropic orientation correlation near contact, which decays more slowly with separation.

Suspension biomechanics of swimming microbes

“…These tendencies were in satisfactory qualitative agreement with previous experimental observations. In the model of Mehandia & Nott (2008), however, coherent structures similar to former experimental observations were not mentioned. The difference may have come from the near-field hydrodynamics between interacting particles.…”

Collective motion and density fluctuations in bacterial colonies

“…2 confirm the core assumption of local alignment in self-propelled particle models (8)(9)(10)(11)(12)(13) and demonstrate that short-range correlations extending to the third nearest neighbor are sufficient to produce collective motion. The spatial correlations observed in our system originate from hydrodynamic (25)(26)(27)(28)(29) and excluded-volume (29-31) interactions between bacteria, and from physical intertwining of flagella of neighboring bacteria (20,(32)(33)(34). It has been shown that both excluded-volume (35) and hydrodynamic (28) interactions can lead to local orientational order.…”

“…The spatial correlations observed in our system originate from hydrodynamic (25)(26)(27)(28)(29) and excluded-volume (29-31) interactions between bacteria, and from physical intertwining of flagella of neighboring bacteria (20,(32)(33)(34). It has been shown that both excluded-volume (35) and hydrodynamic (28) interactions can lead to local orientational order. Physical intertwining of flagella has been directly demonstrated with fluorescent imaging (33,34), but the interaction due to intertwining has not been quantified in experiments.…”

Emergent behavior in active colloids

“…2D simulations showed that long-range hydrodynamic interactions result in strong reorientation rates Ω i HI that are sufficient to entirely suppress motility-induced phase separation of squirmers [346]. Simpler non-squirming swimmers simulated with the lattice Boltzmann method in 2D [348,349] and with Stokesian dynamics in a monolayer in 3D [350] showed some clustering. We performed three-dimensional MPCD simulations of squirmers and strongly confined them between two parallel plates, such that they could only move in a monolayer (quasi-2D geometry) [161].…”