Bouncing Oil Droplet in a Stratified Liquid and its Sudden Death

Droplets can self-propel when immersed in another liquid in which a concentration gradient is present. Here we report the experimental and numerical study of a self-propelling oil droplet in a vertically stratified ethanol-water mixture: At first, the droplet sinks slowly due to gravity, but then, before having reached its density matched position, jumps up suddenly. More remarkably, the droplet bounces repeatedly with an ever increasing jumping distance, until all of a sudden it stops after about 30 min. We identify the Marangoni stress at the droplet-liquid interface as responsible for the jumping: its strength grows exponentially because it pulls down ethanol-rich liquid, which in turn increases its strength even more. The jumping process can repeat because gravity restores the system. Finally, the sudden death of the jumping droplet is also explained. Our findings have demonstrated a type of prominent droplet bouncing inside a continuous medium with no wall or sharp interface.

Figure 1

Continuous jumps and the eventual sudden death of the oil drop. (a) Successive snapshots of the $0.5\mu \mathrm{L}$ oil drop for the first two cycles. The scale bar is 1 mm. (b) Ethanol weight fraction $w_e$ as a function of height $h$ is measured by laser deflection. Error of $w_e$ measurement is $\pm 2\mathrm{wt}\%$. (c) The oil drop’s center position $h$ versus time $t$ for the first 2 cycles, with the final height being taken as 0. The inset shows the vertical oscillation, with frequency 2.15 Hz, during the sinking of the drop. (d) The oil drop’s center position $h$ for all the cycles of the jumping process. After each jump, the drop sinks to a lower position, but at each jump, it still reaches almost the same height $h\approx 6\mathrm{mm}$, thus even increasing the jumping amplitude. This particular drop jumps 26 times within 30 min. (e) The interfacial tension between oil and ethanol-water mixture. Error bars are the standard deviation over 5 measurements.

Figure 2

Shadowgraph (left) and PIV (right) measurements of a $0.5\mu \mathrm{L}$ oil drop [top row, (a)–(e)] and the numerical simulations [bottom row, (f)–(j)] during the drop’s first jump. Light intensity gradient in the background of the shadowgraph indicates the ethanol concentration in the surrounding liquid. Velocities are shown in the laboratory frame. Scale bar is 1 mm, and color bars denote velocity magnitude as well as ethanol fraction. (a),(f) Shortly after release, the buoyant liquid in the droplet’s wake generates a relatively strong jet—the buoyancy jet. A drifted uniform layer of ethanol-rich liquid leads to a very weak Marangoni flow. (b),(g) The Marangoni flow becomes relatively strong; thus the flow directly above the drop is pointing downwards; we refer to it as “replenishing flow.” Frames (c),(h) are close to the time when the buoyancy jet vanishes. Ethanol-rich liquid above the drop is being brought downwards to its apex, and $V_M$ starts to increase. (d),(i) This downward flow brings more ethanol to the apex, further increasing $V_M$ in the upper half of the droplet, which then will move as a “puller” [1]. (e),(j) The Marangoni flow has increased by 2 orders of magnitude in less than 1 s, pulling the drop upwards.

Figure 3

Measured velocities as a function of time. (a) $V_M$ (red), $V_R$ (blue) and sinking velocity $V_{\text{drop}}$ (black) during the first jumping cycle. $V_M$ and $V_R$ are measured at the positions shown in Fig. 2. Positive velocity is upward. The absolute value of $V_M$, $V_{\text{drop}}$ and $V_R$ all decrease. $V_R$ vanishes and then changes direction. This leads to the sharp increase of $V_M$ around $t\approx 55\mathrm{s}$, until it is large enough to pull the droplet up. (b) $V_M$ in logarithmic scale as a function of time in the later stage. $V_M$ increases exponentially between 58.6 and 59.1 s.

Figure 4

(a) $V_{\text{drop}}$ and $V_{\text{buo}}$ (measured 1.5 radius above the drop) of a sinking droplet after the addition of surfactant to the bulk liquid. The inset shows the definition of $V_{\text{buo}}$. (b) $V_{\text{drop}}$ at $h=2.5\mathrm{mm}$ as a function of the number of the subsequent sinking event. (c) Sketch of the balance between buoyancy jet and Marangoni flow. For later sinking events $n>1$, $V_{\text{buo}}$ changes from the blue dash-dotted line, to the dashed line and finally to the solid line. Because the last line shown does not intersect with $V_{M,R}$, so $V_R=V_{\text{buo}}-V_{M,R}$ does not change direction and the drop does not jump. (d) Measured $V_R(t)$ for different sinking cycles. The starting point (highest point) of each cycle is at $t=0$, but the initial stages of the cycles could not be measured because the camera could not record the entire cycle at the necessary frame rate.