Straight-to-Curvilinear Motion Transition of a Swimming Droplet Caused by the Susceptibility to Fluctuations

In this Letter, a water-in-oil swimming droplet’s transition from straight to curvilinear motion is investigated experimentally and theoretically. An analysis of the experimental results and the model reveal that the motion transition depends on the susceptibility of the droplet’s direction of movement to external stimuli as a function of environmental parameters such as droplet size. The simplicity of the present experimental system and the model suggests implications for a general class of transitions in self-propelled swimmers.

Figure 1

(a) Solubilization of water around a swimming droplet. The water is dyed using Acid Red 52 (Tokyo Chemical Industry Co., Ltd.). The red color gradient around the droplet shows that the stained water inside the droplet was being emitted. The black arrow represents the direction of the droplet’s movement. The scale bar represents $200\mu \mathrm{m}$. (b) Schematic illustration of the swimming mechanism of a droplet. The droplet is driven by the Marangoni flow (green arrows) generated by the solubilization of water (blue arrows).

Figure 2

Trajectories of the centroids of the swimming droplets over 60 seconds (see videos 1–3 in the Supplemental Material). (a) The typical trajectories of the droplet centroids. The radii of the water droplets are $56\mu \mathrm{m}$ (left), $157\mu \mathrm{m}$ (middle), and $281\mu \mathrm{m}$ (right). The scale bars represent $200\mu \mathrm{m}$. (b) Trajectories of the centroids of the droplets for three categories of droplet radii. The starting points are set to the origin. The radii categories are $30\le R<100\mu \mathrm{m}$ (left), $100\le R<200\mu \mathrm{m}$ (middle), and $200\le R<400\mu \mathrm{m}$ (right). Color variation depicts different droplets.

Figure 3

Analysis of the experimental results. (a) Angular autocorrelation function ${⟨C(t,\tau )⟩}_t$ of the direction of movement of the droplets. The thin lines correspond to the individual droplets, and the thick lines show the ensemble averages of the droplets in the respective size categories. (b) Droplet radius dependence of the decay time $1/a_1$. Each black dashed line represents an exponential fitting line for the data points with $R<100\mu \mathrm{m}$ and $\ge 100\mu \mathrm{m}$. (c) Droplet radius dependence of the time-averaged speed of the droplets (red dots). Error bars represent the standard deviations of the speed. The blue line is a fitting curve using Eq. (26) in H of the Supplemental Material with $R<250\mu \mathrm{m}$.

Figure 4

Results of PIV experiments. The flow modes were extracted from PIV data of the internal flow and were calculated into the concentration mode of the chemical. (a) 2D density map between the difference angle $\mathrm{\Psi }=\lambda -2\nu$ and the degree of curved motion as $\arccos C(t,{\tau }_{\min })$. The color indicates the density. (b) The average ratio of flow modes observed in the PIV experiments, i.e., $(l,m)=(1,\pm 1)$: dipolar, (2, 0): radial, $(2,\pm 2)$: quadrupolar, and the residual flow fields, given by the norm of each vector field, and the droplet radius. Error bars represent the standard deviations of the ratios. (c) Scatter diagram between the speed of the droplet $|\mathbf{v}|$ and model suggested parameter $\gamma \rho$ estimated from the PIV results. The green dots represent $30<R<200\mu \mathrm{m}$ and red dots represent $200\le R<400\mu \mathrm{m}$. The model denotes the blue line as $|\mathbf{v}|=7.77\times {10}^6\times \gamma \rho \mu \mathrm{m}{\mathrm{s}}^{-1}$. (d) Scatter diagram between angular velocity $\arccos C(t,{\tau }_{\min })/{\tau }_{\min }$ and $\gamma \mu \sin \mathrm{\Psi }/R$. The green and red dots represent the same range as those of (c). The blue line is predicted by the present model without free fitting parameters as $d\nu /dt=2.20\times {10}^7\times (\gamma \mu \sin \mathrm{\Psi }/R)\mathrm{rad}{\mathrm{s}}^{-1}$.

Figure 5

Analysis of solutions of straight motion for the diffusion coefficient $D={10}^{-10}{\mathrm{m}}^2/\mathrm{s}$. (a) Values and stability of solutions of $\mathrm{\Psi }=0$: puller (a-1) and $\mathrm{\Psi }=\pi$: pusher (a-2). (b) Multiplier coefficient $M$ of motion direction against stimulus versus droplet radius $R$. (c) Time development of the direction of movement of the droplet from the initial direction when the external stimulus 1° is applied to the direction of movement. For the stable solution of straight motion at radius $R=18\mu \mathrm{m}$ (blue) and $R=121\mu \mathrm{m}$ (magenta). (d) Schematic illustrations of motions of droplets under stimuli in real space with $M<1$ (d-1) and $>1$ (d-2). (e) Schematic diagram of the internal flow modes of the droplet.