Chemosensitive running droplet

Chemical control of the spontaneous motion of a reactive oil droplet moving on a glass substrate under an aqueous phase is reported. Experimental results show that the self-motion of an oil droplet is confined on an acid-treated glass surface. The transient behavior of oil-droplet motion is also observed with a high-speed video camera. A mathematical model that incorporates the effect of the glass surface charge is built based on the experimental observation of oil-droplet motion. A numerical simulation of this mathematical model reproduced the essential features concerning confinement of oil droplet motion within a certain chemical territory and also its transient behavior. Our results may shed light on physical aspects of reactive spreading and a chemotaxis in living things.

Figure 1

Schematic representation of the experimental setup (a) on a Y-shaped path on a glass substrate and (b) on an acid-treated narrow glass substrate. Prior to the experiment, both ends of the narrow glass substrate $(\approx 15\mathrm{mm})$ were treated with acid.

Figure 2

Experiments on spontaneous droplet motions. (a) Confined motion of an oil droplet. The path of an oil droplet $\left(100\mu \mathrm{l}\right)$ was enclosed within the acid-treated surface. The left image shows the actual appearance of a substrate. The outside of the Y-shaped region was treated with acid. In the middle image, the path of the CM of an oil droplet is shown. The superposition of both images is shown on the right. (b),(c) Motion of a $15\text{−}\mu \mathrm{l}$ oil droplet on a narrow acid-treated glass substrate. An image from the experiment and a spatiotemporal image of oil-droplet motion are shown. An oil droplet showed (b) shuttling motion and (c) intermittent shuttling motion. When an oil droplet did not intrude into the acid-treated region it showed turning-back motion, which resulted in shuttling motion.

Figure 3

Experiments on spontaneous droplet motion. Motion of a $15\text{−}\mu \mathrm{l}$ oil droplet on a narrow acid-treated glass substrate. An image from the experiment and a spatiotemporal image of oil-droplet motion are shown. An oil droplet showed (a) stopping and (b) slowing. When an oil droplet intruded into the acid-treated region, it showed stopping (a) or slowing (b).

Figure 4

Time traces of the apparent length $r$ and velocity $v$ of an oil droplet for the motion shown in Fig. 2c.

Figure 5

Schematic diagram of oil-droplet motion and glass surface conditions. ${\mathrm{STA}}^{+}$ is represented as a hydrophobic bar with a hydrophilic head.

Figure 6

Schematic diagram of the model of oil-droplet motion on a glass substrate. The lower arrow represents the spatial coordinate, where 0 to ${\ell }_0$ and $\ell -{\ell }_0$ to ℓ are treated with acid. $u_0$ and $u_1$ are the saturated concentration of the ${\mathrm{STA}}^{+}$ ions on the glass substrate on bare and acid-treated glass substrate, respectively. The oil droplet is modeled as two lines (perpendicular to the $x$ axis), with a mass density of $\sigma$, connected to a spring with an elastic constant of $\beta \left(r_0\right)$ (see also Appendix ).

Figure 7

Spatiotemporal plot of the oscillatory motion of a droplet given by numerical calculations using Eqs. (11, 12, 13, 14, 15, 16, 17, 18). Only the parameter $U_1$ was changed: (a) $U_1=0.65$ and (b) $U_1=0.658$. The other parameters are $\mu =0.2$, $R_0=0.05$, $E_0=5\times {10}^{-5}$, $A=1$, $n=4$, $W=0.1$, $D=2.5\times {10}^{-5}$, $L_0=0.25$, $U_0=1$, $K_2=1$, $K_3=1$, $X_{10}=0.23$, and $X_{20}=0.27$. The thick lines along the $x$ axis correspond to the regions treated with acid.

Figure 8

Spatiotemporal plot of the slowing of a droplet in the acid-treated region given by numerical calculations using Eqs. (11, 12, 13, 14, 15, 16, 17, 18). Only the parameter $U_1$ was changed: (a) $U_1=0.68$ and (b) $U_1=0.75$. The other parameters are the same as those in Fig. 7. The thick lines along the $x$ axis correspond to the regions treated with acid.

Figure 9

Temporal changes in (a) the length $X_1-X_2$ and (b) the velocity of the oil droplet undergoing the oscillatory motion shown in Fig. 7b. The high-frequency oscillation is attributed to the eigenfrequency of the mathematical model, which did not appear in the experimental results.

Figure 10

Schematic representation of the framework of the “beads-spring” model. The hemisphere-shaped oil droplet as shown in (a) is regarded as a rectangular solid. Since we consider only the $x$ dimension, the conservation law for the volume corresponds to the elastic constant of the spring.