Back-and-forth micromotion of aqueous droplets in a dc electric field

Recently, it was reported that an aqueous droplet in an oil phase exhibited rhythmic back-and-forth motion under stationary dc voltage on the order of 100 V. Here, we demonstrate that the threshold voltage for inducing such oscillation is successfully decreased to the order of 10 V through downsizing of the experimental system. Notably, the threshold electric field tends to decrease with a nonlinear scaling relationship accompanied by the downsizing. We derive a simple theoretical model to interpret the system size dependence of the threshold voltage. This model equation suggests the unique effect of additional noise, which is qualitatively characterized as a coherent resonance by an actual experiment as a kind of coherent resonance. Our result would provide insight into the construction of micrometer-sized self-commutating motors and actuators in microfluidic and micromechanical devices.

Figure 1

Schematic representation of the experimental setup. Stationary dc voltage was applied to the oil phase containing microwater droplets. $V$, applied dc voltage; $L$, distance between the electrodes.

Figure 2

Spatiotemporal diagram of the motion of a droplet with a diameter of 34 $\mu$m at (a) $L=210\mu$m and (b) $L=140\mu$m. An increase in the applied dc voltage results in bifurcation from the static stationary to an oscillatory state.

Figure 3

(a) Phase diagram for mode bifurcation between rhythmic motion and a stationary state as observed for droplets with different diameters, where each point represents the threshold value for bifurcation. The right top point is adapted from the last paper of our group [10]. The arrows correspond to the spatiotemporal behavior given in Fig. 2. (b) Plot of Log $L$ vs Log $V$ for the experimental data given in (a). The blue solid line is $L^{1.3}$, which is a best fit line to the experimental data.

Figure 4

Spatiotemporal diagram (right) of the motion of a droplet together with the time trace of its center of mass (left) under the dc potential of 6 V. The droplet first moved toward the right electrode, then returned back, and finally stayed still at the position near the middle of the couple of electrodes.

Figure 5

Spatiotemporal diagram deduced from the numerical calculations based on Eqs. (2) and (5), where $t_0=0.6,\alpha =0.075,\beta =0.025,k=0.4$. The corresponding value of voltage $v$ changes linearly as in the graph beside the $x-\tau$ diagram. The distance between the electrodes is $l=6.0$ in (a) and $l=8.0$ in (b). The applied voltage is changed from $v=11.0$ to $v=26.0$, whereas in (b), the voltage is from $v=14.0$ to $v=32.0$, as shown in the time traces shown on the right. The time ($\tau$) and space ($x$) scales are arbitrary.

Figure 6

Phase diagram showing the numerical results with changes in $l$ and $v$. The parameters are $t_0=0.6$, $\alpha =0.075$, $\beta =0.025$, $k=0.4$. $v$ and $l$ are dimensionless variables. At the open circles, the droplet exhibits oscillatory motion. At the solid circles, oscillatory and stationary states coexist. At the cross marks, the droplet falls to the fixed point $x=0$. At the square, the droplet is trapped at fixed points near the electrodes or it oscillates. It depends on the initial condition.

Figure 7

Spatiotemporal diagram of the motion of a droplet with a diameter of 24 $\mu$m at $L$ $=$ 56 $\mu$m. For the period from $t$ $=$ 0 to $t$ $=$ 100 s, the stationary dc voltage is decreased in a stepwise manner as shown in the figure. From $t$ $=$ 100 s, we started adding the noise to dc 5 V.