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

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 self-generated 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.

Figure 1

Schematic of the swimmer investigated. The motion is steady with velocity $v$. The point source is shifted by a distance $b$ behind the center and has position $\mathbf{R}\left(t\right)=(vt-b){\mathbf{e}}_z$. At time $t=0$, the swimmer center coincides with the origin.

Figure 2

$\mathrm{Pe}\left(M\right)$ relation in the point-source model in the small and large $\mathrm{Pe}$ regimes (left and right, respectively) for 3D autophoretic (a), 3D interfacial (b), and 2D swimmers (c). Full lines give the solution computed numerically for asymmetry parameter $\chi =0$, 0.1, 0.2, 0.5, and 0.9, from bottom to top. The purple line is the 2D symmetric case with evaporation and $A^2=0.1$. Approximations are shown in dashed lines. Going from left to right, they correspond to (a) Eqs. (14), (13), and (15); (b) Eqs. (20), (19), and (21) with $\chi =0.5$; and (c) Eqs. (24), (23), (27), and (25).

Figure 3

Measurement of camphor release by 25 fully immersed 4-mm camphor disks (see Sec. C of S.M. [62] for details). $C\left(t\right)$ is the concentration in the solution at time $t$. The continuous line is a fit to Eq. (31) including a time shift $t_o=450\mathrm{s}$ and giving a characteristic time $\tau =1.2\times {10}^5\mathrm{s}$. The dashed line is the concentration predicted if camphor does not accumulate in the solution ($C\left(t\right)\ll C_{\mathrm{sat}}$), as given by Eq. (32). (Inset) Expected flux for a single swimmer as a function of time.

Figure 4

Growing diffusive layer model. The transport of camphor is governed by diffusion in the gel layer and an advancing front that moves in the solid domain. Notations are introduced in the text.

Figure 5

Time dependence of the velocity. The swimmer has a radius $a=4\mathrm{mm}$. Left: The line is the data averaged over a 1-min sliding window and the blue area around corresponds to one standard deviation. Right: Close-up on the 700- to 900-s interval over which the velocity is computed. Note that the average value decreases by only a few percent.

Figure 6

Depth dependence of the velocity. The swimmers are $4\mathrm{mm}$ in radius.

Figure 7

Swimming velocity of camphor disks with controlled asymmetry. For each hole size $a_{\mathrm{h}}$, the velocity is normalized by the average value $\overline{v}$. The dashed line corresponds to a unit ratio $v/\overline{v}=1$. A few swimmers are shown as insets. In all cases, $a=4\mathrm{mm}$ and $h=10\mathrm{mm}$.

Figure 8

Size dependence of the swimming velocity. The pool depth is $h=10\mathrm{mm}$. The continuous line is a fit to $v\sim a^{1/3}$.