Self-Induced Polar Order of Active Brownian Particles in a Harmonic Trap

Abstract

Hydrodynamically interacting active particles in an external harmonic potential form a selfassembled fluid pump at large enough Péclet numbers. Here, we give a quantitative criterion for the formation of the pump and show that particle orientations align in the self-induced flow field in surprising analogy to ferromagnetic order where the active Péclet number plays the role of inverse temperature. The particle orientations follow a Boltzmann distribution Φ(p) ∼ exp(Apz) where the ordering mean field A scales with active Péclet number and polar order parameter. The mean flow field in which the particles' swimming directions align corresponds to a regularized stokeslet with strength proportional to swimming speed. Analytic mean-field results are compared with results from Brownian dynamics simulations with hydrodynamic interactions included and are found to capture the self-induced alignment very well.Introduction Understanding the non-equilibrium behavior of self-propelled particles is one of the major challenges at the interface of physics, biology, and also chemical engineering [1,2]. Interacting active particles may show exotic phenomena such as swirling motion In order to understand the collective dynamics of active particles and their steady-state distributions, they are often mapped onto passive systems that move in effective potentials [24][25][26]. However, for interacting particles there is no general route for identifying an equilibrium counterpart [1,6,10].The system we investigate here is composed of selfpropelled or active Brownian particles whose swimming directions undergo rotational diffusion in a harmonic trap. Bacteria or both active and passive colloids confined in optical traps have attracted experimental [27][28][29] as well as theoretical [24,[30][31][32][33] interest. Passive colloids are operated in non-equilibrium by switching the trapping force [31] while active particles are intrinsically out of equilibrium [24,32,33]. Run-and-tumble particles in lattice Boltzmann simulations develop a pump state which breaks the rotational symmetry of the harmonic trap and cause a macroscopic fluid flow [32]. Here, we demonstrate similar behavior for active Brownian particles which interact by hydrodynamic flow fields. However, more importantly we explain the emerging orientational order of particles by mapping the self-induced alignment of swimmers in a harmonic potential onto an equilibrium system which exhibits ferromagnetic order.To this end we first establish a quantitative criterion for the formation of the pump and then introduce a mean-field description for the fully formed pump state. The mean-field system shows a striking analogy to the

Focusing by blocking: Repeatedly generating central density peaks in self-propelled particle systems by exploiting diffusive processes

“…As briefly demonstrated, steric interactions among the particles can promote the effect. Details of interactions between particles [10,11,30], steric alignment interactions between non-spherical particles and cavity walls [10,11,38,[46][47][48], or hydrodynamic interactions among the particles and with cavity walls [10,11,40,44,46,49,50] may influence the statistics in different ways. The above theoretical description should then be adjusted to the specific situation under investigation.…”

“…1 (c). Such a blocking effect due to a confining potential has been observed in various forms before [27,[38][39][40]: the self-propelling drive pushes the particles towards the walls, and the only way of escape is a reorientation of the self-propulsion direction. In our case, this reorientation is solely due to rotational diffusion.…”

Getting drowned in a swirl: Deformable bead-spring model microswimmers in external flow fields

“…It will not stay in the potential minimum. Instead, it moves against and climbs up the potential walls [35,51,113,114], at least as long as it does not reorient. Being able to work against the walls requires an effective active drive of the swimmer.…”

“…In this "stuck" situation, the active drive becomes visible: it is now transmitted to the fluid, which is set into motion as if a net force were acting on it. An effective hydrodynamic fluid pump can be realized in this way [51,113,114]. We apply this approach to the one active bead on our swimmer body.…”

“…Furthermore, the collective behavior of many such interacting particles has been explored [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Collections of self-propelled particles in liquid environment exhibit fascinating and complex nonequilibrium phenomena emerging from self-organization, where hydrodynamic interactions can play a significant role [11,[36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52].…”