Rotational synchronization of camphor ribbons

Experiments on interacting pinned self-propelled rotators are presented. The rotators are made from paper with camphor infused in its matrix. The ribbons rotate due to Marangoni effect driven forces arising by virtue of surface tension gradients. Two such self-rotating camphor ribbons are observed to experience a repulsive coupling via the camphor layer in the common water medium. Lag synchronization in both corotating (same sense of rotation) and counterrotating (opposite sense of rotation) ribbons is reported for the experiments. This synchronization is found to be dependent on the pivot to pivot distance $l$. For distances less than the span of both the ribbons, $l_c$, the rotators successfully synchronize. Furthermore, it is experimentally perceived that synchronization in the counterrotating ribbons is more robust than in the corotating ribbons. We rationalize the mechanism of this synchronization via a theoretical model involving a Yukawa type interaction which is analyzed numerically.

Figure 1

The schematic side view of the experimental setup (left) and the upper view of the camphor ribbons (right). The white (upper) dots in the ribbons (right) are used to track the ribbon trajectories, while the red (lower) dots indicate the position of the pivot points.

Figure 2

Time variation of $x$ position (left) and $y$ position (right) of the ribbons for (a), (b) unsynchronized, (c), (d) corotating, and (e), (f) counterrotating scenarios. Red (dotted line) and black (solid line) curves represent the positions of the first and second ribbons, respectively. At a distance larger than $l_c$ both $x$ and $y$ positions of the ribbons are unsynchronized with respect to each other but, while at a distance less than $l_c$, both $x$ and $y$ positions are lag synchronized for the corotating as well as for the counterrotating scenario. The slowdown of rotations as one goes from unsynchronized (top) to counterrotating (bottom) is due to the depletion of camphor from the ribbons and the gradual decrease in surface tension gradients as time progresses.

Figure 3

Phase portraits of (a) unsynchronized, (b) corotating, and (c) counterrotating camphor ribbons. Data for 60 s are plotted for each panel. Red (circles) and black (solid) curves represent the $x$ and $y$ positions of the ribbons, respectively. The line of slope 1 (red) and of slope $-1$ (black) are drawn for reference. (a) All the available phase space is filled, indicating unsynchronized dynamics. (b) Both $x$ and $y$ positions are synchronized in phase with a constant phase lag for corotating ribbons. (c) For counterrotating ribbons the $x$ and $y$ positions are respectively in and out of phase synchronized. This type of synchronization is termed mirror synchronization [17].

Figure 4

Time variation of $x$ position (left) and $y$ position (right) of (a), (b) unsynchronized, (c), (d) corotating, and (e), (f) counterrotating particles. Red (dotted line) and black (solid line) curves represent the positions of the first and second particles, respectively. $x_i=cos{\theta }_i$ and $y_i=sin{\theta }_i$ for $i=1,2$. Cosines of the phase of the particles are synchronized in phase for both (c) corotating and (e) counterrotating particles. In contrast, the $y$ positions synchronize (d) in phase for the corotating particles and (f) out of phase for the counterrotating particles.

Figure 5

Phase portraits of (a) unsynchronized, (b) corotating, and (c) counterrotating particles. Red and black curves represent the $x$ and $y$ positions of the particles, respectively. The line of slope 1 (blue) and of slope $-1$ (magenta) are drawn for reference. (c) The red curve and the black curve are along the line of slope 1 and $-1$, respectively. (a) All the available phase space is filled, indicating unsynchronized dynamics. (b) Both $x$ and $y$ positions are synchronized in phase with a constant phase lag for corotating particles. (c) For counterrotating ribbons, $x$ and $y$ positions are respectively in and out of phase synchronized. This is termed mirror synchronization [17].