Collective motion of symmetric camphor papers in an annular water channel

We investigate the collective motion of symmetric self-propelled objects that are driven by a difference in the surface tension. The objects move around an annular water channel spontaneously and interact through the camphor layer that develops on the water surface. We found that two collective motion modes, discrete and continuous density waves, are generated depending on the number of self-propelled objects. The two modes are characterized by examining the local and global dynamics, and the collective motion mechanism is discussed in relation to the distribution of camphor concentration in the annular water channel. We conclude that the difference between these two modes originates from that of the driving mechanism that pushes a camphor paper away from a cluster, through which mechanism density waves are generated and maintained.

Figure 1

Experimental setup. Filter paper cut into 10-mm-diam circles was soaked in a saturated solution of camphor in methanol for several seconds. The dried filter papers are referred to as “camphor papers.” To form the annular channel, 11 ml of distilled water was poured onto a ring-shaped plastic film placed on a Teflon plate.

Figure 2

Snapshots of the collective motion for $N_{\mathrm{c}}$ $=$ (a) 1, (b) 7, and (c) 14 (top view). The time intervals of the snapshots were (a) 0.30, (b) 0.36, and (c) 0.30 s. The white arrow denotes the direction of motion of the camphor paper. The papers without a white arrow do not move, or hardly move. The number ($i$ $=$ 1, 2, or 3) on the camphor paper indicates the different camphor papers (P${}_i$).

Figure 3

Different collective motion from the local point of view. (a) θ${}_j$ ($j=1-N_{\mathrm{c}}$) is the central angle of the camphor paper (P${}_j$) relative to the annular channel. (b) The upper figures show time-space ($t$-θ${}_j$) diagrams, and the lower figures show the relationship between θ${}^{+}$ and θ${}^{-}$, corresponding to the bold lines in the upper figures, for $N_{\mathrm{c}}$ $=$ 1, 7, and 14.

Figure 4

Dependence of the number of camphor papers on collective motion for global dynamics. (a) θ${}_{\min }$ is the angle between two papers in contact, and $r{\delta }_{N_{\mathrm{c}}}$ is the center mass of P${}_j$. (b) The relationship between the number of camphor papers $N_{\mathrm{c}}$ and the normalized order parameter Δ. The diamonds show the average of Δ over 100 s for each experiment. The open diamonds indicate that mode II is the dominant mode of the collective motion.

Figure 5

Schematic illustration of the mechanism for (a) mode II and (b) mode I. The camphor papers are represented by the black ellipses and the gray line shows the camphor concentration on the water surface.