Self-propulsion of symmetric chemically active particles: Point-source model and experiments on camphor disks

Abstract

Solid undeformable particles surrounded by a liquid medium or interface may propel themselves by altering their local environment. Such nonmechanical swimming is at work in autophoretic swimmers, whose selfgenerated field gradient induces a slip velocity on their surface, and in interfacial swimmers, which exploit unbalance in surface tension. In both classes of systems, swimmers with intrinsic asymmetry have received the most attention but self-propulsion is also possible for particles that are perfectly isotropic. The underlying symmetry-breaking instability has been established theoretically for autophoretic systems but has yet to be observed experimentally for solid particles. For interfacial swimmers, several experimental works point to such a mechanism, but its understanding has remained incomplete. The goal of this work is to fill this gap. Building on an earlier proposal, we first develop a point-source model that may be applied generically to interfacial or phoretic swimmers. Using this approximate but unifying picture, we show that they operate in very different regimes and obtain analytical predictions for the propulsion velocity and its dependence on swimmer size and asymmetry. Next, we present experiments on interfacial camphor disks showing that they indeed self-propel in an advection-dominated regime where intrinsic asymmetry is irrelevant and that the swimming velocity increases sublinearly with size. Finally, we discuss the merits and limitations of the point-source model in light of the experiments and point out its broader relevance.

Kolmogorovian Active Turbulence of a Sparse Assembly of Interacting Marangoni Surfers

“…More precisely, we use agar gel disks loaded with precipitated camphor [21], with radius a ¼ 2.5 mm and height h ¼ 0.6 mm. When individually deposited at an air-water interface, the disks self-propel (with typical swim velocity U s in the order of 10 mm s −1 [22]) by a Marangoni effect arising from the camphor spreading at the interface. The individual hydrodynamical Reynolds number of such swimmers, Re p ¼ U s a=ν (with ν ¼ 1 × 10 −6 m 2 s −1 the kinematic viscosity of water), is of the order of 25, and no fluid turbulence is induced in the particles' wakes.…”

Alignment and scattering of colliding active droplets

“…Instead, we present and validate here a simplified method to obtain the droplet dynamics while still retaining the essential physical ingredients, inspired by recent modelling of camphor boat swimmers. 36 Each droplet has radius R, density r (i) and viscosity Z (i) , and is immersed in a second (outer) fluid of density r (o) and viscosity Z (o) . The droplets emit a chemical solute of diffusivity D with a uniform and steady flux A 4 0 per unit area.…”

“…To do so, the moving singularity model proposed here describes the effect of each droplet on the concentration distribution, c(r,t), solely as a moving point source singularity, 35,36 so that the chemical transport dynamics is governed by an unsteady diffusion equation:…”

“…35 Note that a similar approach was also used recently to analyse the self-propulsion of isotropic camphor boats. 36 A complete modeling of the chemo-hydrodynamic interactions of active droplets was recently proposed for axisymmetric head-on collisions, 37 providing for any Pe a detailed and quantitative characterization of the role of chemical transport and hydrodynamic flow in the rebound and subsequent dynamics. Complex dynamical regimes were identified, including a delay of the droplets' rebound at larger Pe or the emergence of bound states of chasing droplets of different radii.…”

“…To this end, the present study builds upon the observation that hydrodynamic interactions play a limited role for moderate Pe, 35,37 and constructs a moving singularity model for active droplets that retains the fundamental role of convective transport in self-propulsion, neglects any hydrodynamic influence of the droplets on each other and is able to reproduce the single-droplet dynamics exactly, 15,18 in essence improving upon a simpler version of this approach, which significantly over-estimated the self-propulsion velocity. 36 In experiments, droplets evolve and interact within a plane, as a result of buoyancy differences 30 or confined geometry. 25,39 We therefore purposely limit our analysis here to co-planar interactions of the droplets, although the model itself remains entirely three-dimensional and its application to generic 3D trajectories is straightforward.…”