Run-and-Tumble Particles with Hydrodynamics: Sedimentation, Trapping, and Upstream Swimming

Abstract

We simulate by lattice Boltzmann the nonequilibrium steady states of run-and-tumble particles (inspired by a minimal model of bacteria), interacting by far-field hydrodynamics, subject to confinement. Under gravity, hydrodynamic interactions barely perturb the steady state found without them, but for particles in a harmonic trap such a state is quite changed if the run length is larger than the confinement length: a self-assembled pump is formed. Particles likewise confined in a narrow channel show a generic upstream flux in Poiseuille flow: chiral swimming is not required.PACS numbers: 47.63. Gd, 87.10.Mn, 87.17.Jj The motility of microorganisms raises basic physics questions that range from local swimming mechanisms [1-3] to many-body emergent phenomena [4,5,7]. In the latter context, even grossly simplified models represent a challenging and active area of nonequilibrium statistical mechanics [4]. In some cases experimental nearcounterparts to these models can be devised in which various complicating factors (cell division, chemotaxis, etc.) are environmentally or genetically suppressed [8].Indeed certain bacteria, including E. coli, exhibit motion which can be idealized as a 'run-and-tumble' model. Here straight 'runs' at constant speed v are punctuated by sudden, rapid and complete randomizations in direction, or 'tumbles', occurring stochastically with rate α [8]. The mean run length is = v/α and duration 1/α; at larger length and time scales Fick's law is obeyed, with diffusivity D = v 2 /dα in d dimensions [9]. This model offers an important paradigm for a diffusion process that is fundamentally non-Brownian. Subtle consequences of this are manifest for particles in external force fields, such as gravity or a harmonic trap [10]. In the first case, the gravitational decay length λ falls strictly to zero when the gravitational force f exceeds the propulsive force f p , in contrast to Brownian particles for which λ = D/f [10]. In a harmonic trap (f = −kr), particles are strictly confined within a radius r * = f p /k; and for > ∼ r * the maximum density occurs at r ∼ r * not r = 0. In this limit, a particle in the trap interior rapidly swims out to r * and stays there a long time until its next tumble [10].The qualitative physics of the aforementioned results is robust to both a distribution in v, or a residual true Brownian diffusivity. On the other hand, because there is no underlying free energy (which would give a Boltzmann distribution as the unique steady state), long-range hydrodynamic interactions (HI) between the particles could have major consequences, even for steady-state behavior. Several computational approaches to address hydrodynamics have been developed [5], but none have addressed the basic physics problems considered below: (a) sedimentation in a container with a solid bottom; (b) confinement by a harmonic trap; and (c) Poiseuille flow between parallel plates. These we consider at small but finite particle density, so that in (a,b) only the far-field hydrodynamics are important. In (c),...

Dynamics of self-propelled particles under strong confinement

“…When the persistence length of active motion becomes comparable to the mean free path, uniquely active effects arise that transcend the thermodynamically allowed behaviors of equilibrium systems, including giant number fluctuations and spontaneous flow [3,14,[16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Importantly, a sufficient active persistence length is the only requirement for macroscopic manifestations of activity, as revealed by athermal phase separation of nonaligning, repulsive self-propelled particles [31][32][33][34][35][36][37][38][39][40][41].When boundaries and obstacles are patterned on the scale of the active correlation length, they dramatically alter the dynamics of the system, and striking macroscopic properties emerge [42][43][44][45][46][47][48][49]; for example, ratchets and funnels drive spontaneous flow in active fluids [42][43][44][45][46]. This effect has been used to direct bacterial motion [50] and harness bacterial power to propel microscopic gears …”

“…More generally, any real-world device necessarily includes boundaries, and thus the effects of boundary size and shape are essential design parameters. Although recent studies have explored confinement in simple geometries [43,47,[54][55][56], there is no general theory for the effect of boundary shape.In this Letter, we study the dynamics of non-aligning and non-interacting self-propelled particles confined to two-dimensional convex containers, such as ellipses and polygons. We find that the boundary shape dramatically affects the active fluid's dynamics and thermomechanical properties in the limit of "strong confinement", in which the container size is small compared to the active persistence length (the distance a particle travels before its orientation decorrelates).…”

Self-Induced Polar Order of Active Brownian Particles 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].…”

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.…”