General observation of mixing in droplets
Two droplets were levitated in air by applying acoustic radiation pressure (one at each of the two ultrasound focal points). Then, the distance between the two focal points was reduced, resulting in the coalescence of the droplets to form a single droplet (Fig. 1). From t = −1.5 to 12.4 ms, this process could be seen in the observation window. However, owing to the displacement caused by coalescence, the droplet began to move out of the frame at t = 12.4–49.5 ms. It was mostly unobservable from 55.7 to 74.3 ms but returned to view at 80.4–117.6 ms.
There were bright and dark areas within the droplet owing to the fluorescence of the particles. At t = −1.5 ms, immediately before they merged, the two droplets were clearly distinguishable by their brightness difference. From 6.19 to 30.9 ms, it was still possible to distinguish the brightness distributions in the merged droplet. However, from 105.2 to 136.1 ms, the brightness distribution exhibited a complex pattern. These changes indicate that the component originating from each of the two droplets spread over the entire droplet with time. Therefore, the local concentration of each component could be quantitatively identified from the particle distribution.
The droplet achieved a distorted shape at t = 6.19 ms. From 12.4 to 30.9 ms, it periodically changed its elongation direction. From 99.0 to 136.0 ms, the degree of elongation gradually decreased.
Droplet coalescence
The behaviour of droplets immediately before and after coalescence is shown in Fig. 2. In Fig. 2-i (and in similar figures shown later in the paper), the image that captured light with a short wavelength is coloured green while the image that captured light with a long wavelength is shown in red. The interface shapes of the ethanol and glycerol-water solution droplets were distorted after coalescence. On the other hand, the interface shape of the pure glycerol droplet remained circular. From Fig. 2a-i, at t = -1.5 ms, it can be seen that the two component droplets were clearly distinguishable by the red and green particles. During coalescence, at t = -2.5 ms, the green and red particles were distributed on the left and right sides, respectively, of the droplet. Thus, the components of the droplet could be visualized using the two monochrome cameras and optical filters.
The distribution of the particle-number fraction at t = 5.0 ms is shown in Fig. 2-ii for each case in a-c. As shown in Fig. 2-i, until t = 13.5 ms, the green and red particles were distributed on the left and right sides, respectively, of each droplet. This was reflected in the number-density distributions (except in the middle of each droplet, i.e., the contact area of the two original droplets, wherein cells with number fractions close to 50% were predominant). Therefore, both the images of the particles detected and number-fraction distributions reflected the expected distribution of particles.
Fig. 2-iii shows the velocity field inside each droplet. In all three cases, coalescence produced opposing horizontal flow. Moreover, the flow sometimes changed direction from left–right to up–down. Considering the similarity of this to the case of a droplet with interfacial oscillation22,35, the flow was probably caused by the deformation of the droplet interface.
The maximum velocity in Fig. 2a-iii and b-iii was 0.15 m/s while that in Fig. 2c-iii was 0.10 m/s. Therefore, the maximum velocity was different for droplets with different compositions.
Droplet interface oscillation
The behaviour of the droplets from t = 14.7 ms to 30.0 ms is shown in Fig. 3. The direction of elongation changed periodically (Fig. 3a-i and b-i). By contrast, there was no deformation of the pure glycerol droplet (Fig. 3c-i). The time dependence of the amplitude, A, of interfacial oscillation of the droplets is51
where t represents the time, n, the kinematic viscosity, r, the volume-equivalent radius, and n, the mode of oscillation. Thus, increasing the viscosity of the droplets also increases the damping effect of interfacial oscillation. It can be seen from Fig. 3-i that for the 2nd mode oscillation, the width of each droplet was 2–3 mm. Owing to the acoustic radiation pressure, they were oblate horizontally; therefore, the height had a weaker effect on their volume than width. Thus, the droplets in this study were not expected to have significant differences in volume or volume-equivalent diameter R. In contrast, under the experimental conditions, the kinematic viscosities were 1.387 × 10-6, 2.715 × 10-6, and 720.0 × 10-6 m2/s for ethanol, 33 wt% glycerol–water solution, and pure glycerol, respectively. Therefore, the damping effect of interfacial oscillation was twice as strong for the 33 wt% glycerol–water solution and 720 times stronger for pure glycerol than that for ethanol. These results confirmed that the difference in interfacial oscillation during coalescence was due to the difference in the droplets’ kinematic viscosities.
The distribution of the particle-number fraction at t = 21.3 ms (Fig. 3-ii) was different from that at t = 5.0 ms (Fig. 2-ii). For all three cases in Fig. 3-i, although the green and red particles were concentrated on the left and right sides, respectively, of the droplet, the area with number fraction close to 50% was much larger than that observed in Fig. 2. This indicates that the mixing progressed with time.
Fig. 3-iii shows the velocity field inside the droplets. A radial flow still existed in ethanol and the 33 wt% glycerol–water solution (Fig. 3a-iii and Fig. 3b-iii, respectively), and the maximum velocity of the flow was 0.15 m/s for both droplets. However, in the pure glycerol droplet shown in Fig. 3c-iii, there was only circumferential flow caused by the rotation of the droplet. Therefore, the velocity of the flow decreased near the centre. In a previous study25,35, a relationship was found between the interfacial oscillation and internal flow of a droplet. Thus, in this study, the internal flow of the droplet was attributable to the interfacial oscillation.
Stable levitation after oscillation
The behaviour of the droplet after droplet oscillation reached steady state is shown in Fig. 4. In ethanol (Fig. 4a-i), the distribution of the green and red particles was more complex at t = 125 ms than at t = 21.3 ms (Fig. 3a-ii), and the complexity increased as time passed. Finally, at t = 300 ms, the two types of particles were randomly distributed. In contrast, in the 33 wt% glycerol–water solution and pure glycerol (Fig. 4b-ii and Fig. 4c-ii, respectively), the two types of particles were still separated even at t = 300 ms. Specifically, in the pure glycerol droplet (Fig. 4c-ii), they remained segregated into opposite hemispheres.
The shape deformation of the droplet interface still existed in the ethanol droplet at t = 175 ms shown in Fig. 4a-i. In contrast, the droplets of the 33 wt% glycerol–water solution and pure glycerol (Fig. 4b-i and Fig. 4c-i, respectively) were nearly spherical.
Different rates of mixing were also reflected in the number-fraction distribution of the particles shown in Fig. 4-ii. For ethanol (Fig. 4a-ii), cells with number fractions close to 50% were observed throughout the entire droplet. However, for the 33 wt% glycerol–water solution (Fig. 4b-ii), cells with number fractions close to 50% had a bow-shaped distribution in the lower right corner of the droplet; the change in the distribution was due to the internal flow. Finally, for pure glycerol (Fig. 4c-ii), the distribution was the same as that observed at t = 21.3 ms shown in Fig. 3c-ii.
Finally, Fig. 4-iii shows the velocity field inside each droplet. For all three cases, there was only circumferential flow.
Quantification of mixing performance
To quantify the degree of mixing, the Lacey mixing index, Mc52,53, was calculated for each droplet from the number-fraction distribution (Fig. 5); Mc = 0.0 corresponds to a completely separated state while Mc = 1.0 corresponds to a completely mixed one. For all three droplets, Mc < 0.1 immediately after coalescence. The Mc of ethanol was very close to 1 at 300 ms, which indicates that the two components originating from separate ethanol droplets had become randomly distributed. In contrast, the Mc’s of the glycerol–water solution and pure glycerol droplets were 0.5 and 0.2, respectively, at 300 ms. This trend was qualitatively similar to that observed for the degree of mixing after coalescence, at 300 ms (Fig. 4). In addition, all three Mc’s increased with time. This indicates that the Mc’s reflected the actual mixing states of the component droplets, i.e., the degree of mixing was successfully quantified.
For ethanol, Mc increased from 0.1 to 0.4 until t = 30 ms, then gradually increased from 0.4 to 1.0 until t = 300 ms. For t < 30 ms, the droplet interface exhibited significant oscillation (Fig. 2a-i and Fig. 3a-i). Although the interfacial oscillation subsided with time, it still existed at t = 175 ms, as shown in Fig. 4a-i.
For the 33 wt% glycerol–water solution droplet, Mc increased from 0.1 to 0.5 until t = 50 ms, after which it stayed relatively constant. As in the case of ethanol droplets, the interface in this case also exhibited significant oscillation for t < 30 ms (Fig. 2a-i and Fig. 3a-i). However, the oscillation of the interface was negligible at t = 175 ms (Fig. 4b-i).
Finally, for the pure glycerol droplet, Mc increased from 0.1 to 0.2 until t = 10 ms, maintaining that value thereafter. From t = 0.0 to 7.5 ms, the contact area of the original droplet component increased (Fig. 2.c-i). Thereafter, no interfacial oscillation was observed (Fig. 2c-i, Fig. 3c-i, Fig. 4c-i).
The aforementioned contrasting observations made between the Mc and droplet behaviour suggests three possible factors that led to the increase of the former. The first factor was the increase in the contact area of the original droplet components due to coalescence of the droplets. In this study, no interfacial oscillation was observed in the pure glycerol droplet. However, a comparison of Fig. 5 with Fig. 2c-i shows that for the pure glycerol droplet, Mc increased from 0.1 to 0.2 as the two droplets coalesced into a single droplet from t = 0.0 to 7.5 ms. The second factor was the intensity of interfacial oscillations in the droplets: for t < 30.0 ms (Fig. 2 and Fig. 3), interfacial oscillations were present in the ethanol and 33 wt% glycerol–water solution droplets, with Mc = 0.4 at around t = 30.0 ms. In contrast, interfacial oscillation was absent in the pure glycerol droplet with Mc = 0.2. The final factor was the duration of the interfacial oscillation. Specifically, at t = 300.0 ms, Mc = 1.0 for the ethanol droplet, for which interfacial oscillations were still observed at t = 175 ms. By contrast, Mc = 0.5 for the 33 wt% glycerol–water solution droplet, for which interfacial oscillations were no longer observed at t = 125 ms. Therefore, it is clear that the aforementioned three factors are important for droplet mixing.
Comparison of observed and expected mixing
As discussed in the previous section, the degree of mixing was successfully quantified from the distribution of particles. However, it was yet to be determined whether the degree of mixing measured from the particle distribution reflected the actual state of mixing and whether the mixing of the droplet resulted from its internal flow. Thus, we characterized the mixing dynamics using physical timescales.
Fig. 6 shows the comparison of the various physical timescales considered for droplet mixing. To better understand the mixing dynamics in Fig. 5, we first introduced the diffusion timescale of the tracer particle, which is calculated as follows:
where kB is the Boltzmann constant, T, the absolute temperature, η, the viscosity of liquid, and rp, the radius of the tracer particle. Fig. 6a shows the comparison between tmc and ttr; for all cases, ttr ≫ tmc. Therefore, the mixing of tracer particles in the droplet was enhanced by the internal flow due to droplet coalescence.
As the internal flow was driven by the momentum transfer within the droplet, we chose the viscous dissipation timescale as the second timescale. Fig. 6b shows the relationship between tmc and r2/ν. It is evident that tmc decreased with the decrease in kinematic viscosity (increase of r2/ν). This indicates that the momentum transfer was enhanced for droplets with low viscosity such that tmc was shortened owing to the effect of internal flow. Based on present results, active mixing by mode oscillation can play a critical role in the mixing of viscous droplets as reported in a previous paper23,34. Thus, the combination of natural mixing by droplet coalescence and active mixing by mode oscillation would enable us to merge and mix diverse droplets with a wide range of properties, including viscosity, surface tension, and density. These results demonstrate that the novel development of contactless mixing in droplets using acoustic fields can be realized.
Summery
In this study, the fluid distribution and internal flow were measured simultaneously to understand the mixing dynamics of acoustically levitated airborne droplets. The proposed method can be useful for investigating the relationship between the flow field and mixing for the contactless quantitative study of physicochemical phenomena. Furthermore, it can be beneficial to the understanding of unsteady-state fluid dynamics with time-induced volume and concentration changes.
Although we identified the interfacial oscillation and degree of mixing of acoustically levitated droplets, the direct and precise measurement of the concentration distribution of liquid components in a droplet with different liquid properties is challenging. The present results provide insights into the dynamics of acoustically levitated airborne droplets and will be useful in the study of microfluidic interfacial dynamics, flow fields, heat/mass transfer, and chemical reactions. Such investigations will allow the contactless selection and manipulation and optimal sample handling for future lab-on-a-drop applications.