Influence of added dye on Marangoni-driven droplet instability

Multiphase flows are challenging systems to study, not only from a fundamental point of view, but also from a practical one due to the difficulties in visualizing phases with similar refraction indices. An additional challenge arises when the multiphase flow to be observed occurs in the submillimetric range. A common solution in experimental fluid dynamics is the addition of a color dye for contrast enhancement. However, added dyes can act as surface-active agents and significantly influence the observed phenomena. In the recently reported Marangoni bursting phenomenon, where a binary droplet on top of an oil bath is atomized into thousands of smaller droplets by Marangoni stresses, visualization is particularly challenging and the use of color dyes is naturally tempting. In this work, we quantify experimentally the significant influence of added methyl blue dye, as well as the obtained contrast enhancement. We find that the reaction of the system to such an additive is far from trivial and brings an opportunity to learn about such a complex phenomenon. Additionally, we propose a simple method to analyze the contrast enhancement as a function of dye concentration, with the aim of finding the minimum concentration of dye for successful imaging in similar systems.

Figure 1

(a) Schematic of the experimental setup. (b) Schematic of the experiment in an instance in time (not to scale). Illustrated are the mother (injected) droplet and the daughter droplets in light blue, on the sunflower oil bath in the background in yellow. The outer dashed circle denotes the outer bound of the experiment, i.e., the furthest the daughter droplets travel radially outwards. The inner dashed circle denotes the mother droplet radius, $R_{CL}$, which changes in time, as more and more alcohol evaporates. The unstable rim of the droplet and the ligament formation and droplet breakup are omitted in this schematic for the sake of simplicity. The approximate position of the field of view of the microscope/camera is shown in red. We place a marker at the center on the bottom side of the Petri dish, to align our experiments such that the field of view position is comparable between different repetitions and experiments. (c) Partial image of the mother droplet and the emitted daughter droplets at high concentration of methyl blue dye (5.6 mg/ml).

Figure 2

Selected images of a typical experiment with medium dye concentration (0.56 mg/ml). Note the time stamps in every panel. The scale bar in panel (a) also applies to the other two panels. In each panel, the mother droplet contact line is indicated with a dashed red line. In panel (a) $t=t_0$ as it is the first frame of the recording which is in focus. The daughter droplets as well as the contact line of the mother droplet are both spreading outwards (indicated with the red arrow). In panels (b) and (c) we can see the outer bounds of the experiment in the top left corners. The daughter droplets and the pinch-off sites along the unstable rim are visible. The contact line of the mother droplet has come to a halt in panel (b) and is receding in panel (c).

Figure 3

Example images of experiments with varying methyl blue concentrations, at comparable time instances of experiments. From left to right, as ${\text{c}}_{\text{MB}}$ increases, we see (i) ligament formation at the contact line and a change in droplet pinch-off, as well as (ii) a significant increase in imaging contrast. Note that, for better comparison of the original imaging contrast, we choose to show all images in this figure without any postprocessing. As a visual aid, we added dashed lines at the contact line positions in the first three images for which the imaging contrast is low. The images capture 4.9 mm in width by 4.9 mm in height. The background in each panel is consistent with the chosen concentration color map of later plots.

Figure 4

Probability density functions of daughter droplet diameters for different dye concentrations. From panels (a) to (d), ${\text{c}}_{\text{MB}}$ corresponds to 0.28 mg/ml, 0.56 mg/ml, 2.8 mg/ml, and 5.6 mg/ml. The different colors of individual distribution plots indicate the time [see the color map in panel (a), valid for all panels]. For panel (d), $t_{\text{final}}=12.4\mathrm{s}$, and the length of the experiments is comparable for all concentrations (also see Fig. 5).

Figure 5

(a) Instability wavelength $\lambda$ at the unstable droplet rim over time, for different MB concentrations. Plots show the average of at least two experiments, with the standard deviation as error bar. The increase in wavelength in the beginning results from the outward spreading contact line. Added dye increases the instability wavelength. Between $t=3$ s and $t=8$ s (orange shaded plot area) we calculate a plateau wavelength ${\lambda }_p$ as an average of data points. (b) Plateau wavelength ${\lambda }_p$ for varying MB concentration. (c) Contact line radius $R_{CL}$ over time. The same color code applies as in the other panels. $R_{CL}$ for all dye concentration is smaller than the reference case (${\text{c}}_{\text{MB}}=0)$. (d) Instability wavelength scaled with the circumference, $2\pi R_{CL}$, corresponds to the estimated total number of pinch-off sites $N$ around the mother droplet. Added dye leads to a significant drop in $N$.

Figure 6

Example for imaging contrast quantification with a distance transformation based on the droplet edge. An example image is shown in panel (a) of an experiment with ${\text{c}}_{\text{MB}}=5.60\text{mg/ml}$. In panel (b) we show the two masks created from the cropped original, in order to get the distance transforms for inside the mother droplet, as well as outside; see panel (c). In panel (d) we plot the average image intensity $I_{\text{av,norm}}$ over the distance from the edge, with the edge position denoted with a vertical red line. The intensity is averaged over all pixels with the same distance from the edge (sign sensitive). For better comparison with other dye concentrations, we normalize the averaged intensity with the mean intensity inside the mother droplet.

Figure 7

(a) Average overall imaging contrast as a function of dye concentration. We normalize the overall imaging contrast found as intensity difference with the bit depth of 16 bit. The leftmost data point represents a reference experiment with no dye. The contrast enhancement with dye leads to a rise in overall intensity to $\sim 60\%$. After reaching this plateau value, adding more dye does not yield higher overall intensity any more. (b) Contrast quantification based on distance transformation for different dye concentrations. The colors of the plots correspond to the legend in panel (a). Note that it was not possible to compute this parameter in the absence of dye due to the lack of contrast for the masking image. We observe a significant contrast enhancement across the droplet edge with increasing dye concentration.

Figure 8

(a) Interfacial tension between water and sunflower oil, for an increasing concentration of methyl blue. (b) Interfacial tension for water and IPA mixtures (with the initial mother drop concentration in this work, 60 vol % IPA, as well as the critical concentration of IPA, present at the unstable rim, of 45 vol % IPA), for an increasing concentration of methyl blue. (c) Interfacial tension for water and IPA mixtures (60 vol % and 45 vol % IPA) for an increasing methyl blue concentration.