Oscillatory motion of a camphor grain in a one-dimensional finite region

The motion of a self-propelled particle is affected by its surroundings, such as boundaries or external fields. In this paper, we investigated the bifurcation of the motion of a camphor grain, as a simple actual self-propelled system, confined in a one-dimensional finite region. A camphor grain exhibits oscillatory motion or remains at rest around the center position in a one-dimensional finite water channel, depending on the length of the water channel and the resistance coefficient. A mathematical model including the boundary effect is analytically reduced to an ordinary differential equation. Linear stability analysis reveals that the Hopf bifurcation occurs, reflecting the symmetry of the system.

Figure 1

Schematic illustration of the experimental setup.

Figure 2

Experimental results. (a) Snapshots of the camphor grain at the surface of (a-1) 3.5 mol/L and (a-2) 2.5 mol/L glycerol aqueous solution. The viscosities of the aqueous phases were (a-1) 3.1 $\mathrm{mPa}\mathrm{s}$ and (a-2) 2.3 $\mathrm{mPa}\mathrm{s}$. The length of the channel was 30 mm in both cases. (b) Time evolution of the position. (c) Time evolution of the velocity. The plots colored with blue (dark gray) and orange (light gray) correspond to (a-1) and (a-2), respectively. The corresponding videos are available in Supplemental Material [34].

Figure 3

Dependence of the motion of the camphor grain on the water channel length. (a) Amplitude of the position. (b) Amplitude of the velocity. The results for pure water [blue (dark gray), $1.0 \mathrm{mPa}\mathrm{s}$] and 3.5 mol/L glycerol aqueous solution [orange (light gray), $3.1 \mathrm{mPa}\mathrm{s}$] are shown. Error bars show the standard deviation.

Figure 4

Phase diagram obtained by experiments. Circles and crosses correspond to the oscillatory and rest state, respectively. The concentrations of the glycerol aqueous solutions were 0 mol/L (pure water) for 1.0 $\mathrm{mPa}\mathrm{s}$, 1.0 mol/L for 1.3 $\mathrm{mPa}\mathrm{s}$, 2.5 mol/L for 2.3 $\mathrm{mPa}\mathrm{s}$, 3.5 mol/L for 3.1 $\mathrm{mPa}\mathrm{s}$, and 5.0 mol/L for 5.3 $\mathrm{mPa}\mathrm{s}$.

Figure 5

Bifurcation diagram obtained by the theoretical analysis. The curve describes $\eta =C\left(R\right)$. In the dark and light gray regions, the fixed point is stable and unstable, respectively. Here, the fixed point corresponds to the rest state of the camphor grain at the system center. The oscillatory motion of the camphor grain around the system center can be observed in experiments when the fixed point is unstable.

Figure 6

Representative numerical results. (a) Time series of the camphor grain position $X$. (b) Trajectories on the position-velocity ($X-\dot{X}$) phase space, and the concentration field, $c$, at each time. (a-1,b-1) Oscillatory motion. (a-2,b-2) Damping oscillation to the rest state. The system size was $R=1$ in both cases and the resistance coefficient, $\eta$, was set to (a-1,b-1) $\eta =0.3$ and (a-2,b-2) $\eta =0.5$.

Figure 7

Dependence of the camphor grain motion on the system size, $R$. The amplitudes of (a) the position and (b) the velocity are plotted against the system size, $R$, for each resistance coefficient, $\eta$. The green squares, orange circles, blue triangles, and pink diamonds correspond to $\eta =$ 0.2, 0.3, 0.4, and 0.5, respectively.

Figure 8

Comparison of the phase diagrams obtained numerically and analytically. Circles show the bifurcation points obtained numerically and the solid curve shows the bifurcation curve, $C\left(R\right)$, obtained analytically.

Figure 9

Time evolution of the driving force obtained with the original mathematical model [blue (dark gray) broken curve], and the reduced driving force [orange (light gray) solid curve]. The parameters were (a) $R=1$ and $\eta =0.3$, (b) $R=4$ and $\eta =0.45$, (c) $R=8$ and $\eta =0.45$, and (d) $R=1$ and $\eta =0.25$.

Figure 10

Schematic illustration on the behaviors of the camphor grain depending on the length of water channels with a constant viscosity. (a) Schematic bifurcation diagram showing the relation between the amplitude of motion and the system boundary. The darker-colored areas indicate the regions suffering from the effect of the boundaries, i.e., within the diffusion length from the boundaries. (b–d) Schematic description of the time series of the typical three motions depending on the system size. (b) Rest state. (c) Oscillation. (d) Translation and reflection. It is noted that there is no definite classification between (c) and (d).