Oscillatory droplet dissolution from competing Marangoni and gravitational flows

The dissolution or growth of a droplet in a host liquid is an important part of processes like chemical extraction, chromatography, or emulsification. In this work we look at the dissolution of a pair of vertically aligned droplets immersed in water, both experimentally and numerically. The liquids used for the droplets are long chain alcohols with a low but finite solubility in water and a significantly lower density than that of the host liquid. Therefore, a solutal plume is formed above of the bottom droplet and natural convection dominates the dissolution process. We monitor the volume of the droplets and the velocity field around them over time. When the liquids of the two droplets are the same, our previously found scaling laws for the Sherwood and Reynolds numbers as functions of the Rayleigh number [Dietrich et al., J. Fluid Mech. 794, 45 (2016)] can be applied to the lower droplet. However, remarkably, when the liquid of the top droplet is different than that of the bottom droplet the volume as function of time becomes nonmonotonic, and an oscillatory Marangoni flow at the top droplet is observed. We identify the competition between solutal Marangoni flow and density-driven convection as the origin of the oscillation and numerically model the process.

Figure 1

(a) Sketch of the cuvette and the capillaries connected to it. (b) Typical example of two droplets just after infusion with tracer particles. The definitions of three length scales are shown. A laser sheet (parallel to the plane of the image) is coming from the right to illuminate the tracer particles. (c) Sketch of the experimental setup from a top view. The glass cuvette was illuminated from the back (from the right in the image) with a white light source. A diffuser was used to have an approximately homogenous illumination. The laser sheet illuminated the central plane normal to the direction of white light. A long distance microscope was used to observe the droplets. Each capillary was connected to a syringe mounted to a syringe pump. A cage of acrylic was built around the cuvette to reduce air currents and heat exchange with the surroundings. The cage was open from the top.

Figure 2

Schematic of the numerical setup with the boundary conditions. The subscripts $j$ and $k$ take the values $j=2,3$ and $k=1,3$.

Figure 3

Dissolution of pentanol droplets. (a) Diagrams showing the three configuration used in this figure. A single droplet down (SD), a single droplet up (SU), and double droplets down (DD) and up (DU). (b) Volume vs time of four pentanol droplets in the three configurations described in panel (a). Inset: Rate of volume change over time showing a local maximum. (c) Sherwood number compensated with ${\mathrm{Ra}}^{-1/4}$ vs the $\mathrm{Ra}$ number. The horizontal dashed black line is a guide to the eye to compare with the data. Data with $\mathrm{Ra}<70$ have been omitted because the influence of the tube increases the noise significantly (the complete data set can be found in the SM [48]). The vertical colored lines in panels (b) and (c) indicate the time $t_{\mathrm{tube}}$ when the radius of curvature $R_{\mathrm{c}}$ is equal to the radius of the tubes $R_{\mathrm{t}}$.

Figure 4

Velocity caused by natural convection. (a) Velocity fields around the three different configurations. Two times are shown: 7 s after the droplets were infused (top) and when $t=t_{\mathrm{tube}}$ (bottom). The velocity fields are averaged over a few frames, equivalent to $\mathrm{\Delta }t=1$ $\mathrm{s}$. (b) Time evolution of the velocity measured at the axis of symmetry and approximately one initial radius away from the edges of the droplets. The locations where the velocity was obtained are marked with a small green square in the velocity fields of panel (a). The vertical black arrows indicate $t=7$ $\mathrm{s}$. ($c$) Reynolds number compensated with ${\mathrm{Ra}}^{-5/8}$ vs the $\mathrm{Ra}$ number. The horizontal solid black line is a guide to the eye to compare with the data. The dashed black line corresponds to a power law of 1/2. The vertical colored lines in panels (b) and (c) indicate $t=t_{\mathrm{tube}}$. A plot with the complete range of $\mathrm{Ra}$ numbers can be seen in the SM [48].

Figure 5

Volume vs time for a 1-pentanol droplet dissolving below a second droplet of 1-pentanol or a different alcohol (1-hexanol, 1-heptanol, or 1-octanol). The initial separation is $L_0\sim 2R_{e,0}$. (a) Volume of the lower droplet. The dissolution of the bottom pentanol droplet is hardly affected by changing the alcohol used for the upper droplet. Inset: $\mathrm{Sh}$ number compensated with ${\mathrm{Ra}}^{-1/4}$ vs the $\mathrm{Ra}$ number, calculated with the volume of the lower droplet. For large values of the $\mathrm{Ra}$ the curve is approximately flat. The vertical colored lines in both plots indicate the time $t_{\mathrm{tube}}$ at which $R_c=R_t$. (b) Volume of the top droplet as function of time. When the top droplet is made of a different alcohol, it first increases in volume before it starts to dissolve. The diagrams in both panels (a) and (b) indicate the droplet from which the volume is plotted by the darker color and the arrows.

Figure 6

Velocity-related data for pairs of droplets dissolving simultaneously with an initial separation of $L_0\sim 2R_{e,0}$. (a) Velocity magnitude taken at the middle point between the two droplets. When the liquid of the upper droplet is not 1-pentanol the velocity temporarily oscillates once the plume created by the lower droplet reaches the upper droplet. Inset: Zoom of the first 100 s. During the oscillations, the velocity can reach more than 1 mm/s and then goes down to close to zero in each cycle. For the first cycles the vertical velocity can even become momentarily negative. (b) Reynolds number compensated with ${\mathrm{Ra}}^{-5/8}$ vs the $\mathrm{Ra}$ number. In all cases, there is a region over which the data are approximately constant. The horizontal flat dashed line is used only as a reference. Inset: Sketch of the two droplets. The black star indicates the position where the velocity was measured. A plot with the complete range of $\mathrm{Ra}$ numbers is shown in the SM [48].

Figure 7

Times series of the vertical velocity at the middle point between a 1-octanol droplet (top) and a 1-pentanol droplet (below). The numerical results are shown in panel (a) for the whole life time of the 1-pentanol droplet and in panel (b) for the first 20 $\mathrm{s}$. The six times marked in panel (b) indicate the corresponding times of the snapshots shown in Fig. 8. The first 20 $\mathrm{s}$ of the experimental velocity are shown in panel (c). The experimental data were shifted horizontally to match the moment at which the velocity suddenly increases for the first time. The velocity fields corresponding to the red points in panel (c) are shown in Fig. 10. The initial separation between the edges of the droplets is $L_0=R_{e,0}$ in all plots.

Figure 8

Snapshots of the numerical results at six different moments during one oscillation cycle. At (1) the plume is building up and moves towards the top droplet. At (2) the plume has reached the top droplet, creating a gradient in interfacial tension and triggering an upwards Marangoni flow. At (3) the Marangoni flow has brought 1-pentanol rich liquid all around the droplet in a relatively thin concentration boundary layer. At (4) pentanol rich liquid has accumulated close to the contact line between the top droplet and the top capillary, causing a secondary Marangoni flow. At (5) the secondary Marangoni flow has created a thicker concentration boundary layer all around the droplet. The velocity at the axis of symmetry and in between the droplets points down, causing the plume to “detach” from the upper droplet. At (6) buoyancy competes with the secondary Marangoni flow. Eventually the plume is able to get in contact with the upper droplet again and the process restarts. The insets show the interfacial tension as a function of $s$, the coordinate tangential to the interface of the upper droplet.

Figure 9

Central velocity for (a) two experiments and (b) two simulations with different initial distances. (c) Time of oscillation $t_{\mathrm{osc}}$ vs the initial distance $L_0/R_{e,0}$ between the droplets, for 1-heptanol and 1-octanol in the top droplet in experiments and for 1-octanol in the top droplet in simulations. For the three smallest values of $L_0/R_{e,0}$ used in the simulations, the oscillations last for the whole life span of the pentanol droplet.

Figure 10

Velocity fields obtained from experiments (left) and simulations (right) at three different moments during the first oscillation cycle: (a) before the plume reaches the upper droplet, (b) when there is a Marangoni flow all around the upper droplet, and (c) when the velocity between the droplets has changed direction. The scale bar represents 300 $\mathrm{\mu m}$. Notice that the color bars have different ranges in each panel. The times shown here correspond to red points in Figs. 7 and 7. Movies 2 and 3 in the SM [48] correspond to the simulated and experimental fields respectively.

Figure 11

Results from a numerical simulation of a 1-pentanol droplet below and a 1-octanol droplet on top, but the interfacial tension at the top droplet increases with increasing 1-pentanol concentration. The velocity measured at the middle point between the droplets for (a) the whole lifespan of the lower droplet and (b) for the first 90 $\mathrm{s}$ of the dissolution process. (c) The interfacial tension as function of the tangential coordinate $s$ along the interface of the top droplet at five different times. (d) Five snapshots of the concentration and velocity fields at increasing times. The five times are marked in panel (b) with red points and correspond with the five curves in panel (c). The four snapshots inside the black rectangle correspond to different moments of a cycle.