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SELF-STABILIZATION OF DROPLET CLUSTERS

Alexander A. Fedorets
Microdynamics Technologies Laboratory, X-BIO Institute, University of Tyumen, Tyumen, Russia

Leonid A. Dombrovsky
Heat Transfer Laboratory, Joint Institute for High Temperatures, Moscow, Russia

Vladimir Yu. Levashov
Institute of Mechanics of Moscow State University, Moscow, Russia


Following from: Droplet clusters levitating over the heated water surface


It is known that various organic reactions are accelerated when reagents are present in microdroplets or thin films. The reaction rate increases by orders of magnitude with decreasing droplet size. The rate constant is usually inversely proportional to the square root of the droplet volume. This phenomenon has long attracted the attention of researchers but remains understudied, despite its great practical importance (Wei et al., 2020). Laboratory research requires the generation of steadily levitating small water droplets of a given constant size (with a diameter of about 20–200 μm). The droplet clusters we study usually consist of water droplets of this size (Fedorets et al., 2022a). It should be also noted that fine water droplets suspended in the air may contain microorganisms, and the transfer of such droplets in the atmosphere is one of the mechanisms for the spread of plant pathogens as well as animal and human diseases (Joung et al., 2017; Nath et al., 2019; Dombrovsky et al., 2020a). Therefore, the study of biochemical processes in small droplets is of important practical value.

Obviously, for biochemical experiments, it is necessary to have steadily levitating water droplets of constant size. However, the droplet clusters are usually not stable and exist for a short time (no more than a few minutes); as a result of the condensational growth of droplets, the cluster coalesces with the water layer. Therefore, a method for cluster stabilization using the external infrared radiation, which heats the droplets and increases their evaporation, has been developed (Dombrovsky et al., 2016, 2020b). Unfortunately, the technical characteristics of compact sources of the near-infrared radiation do not ensure their reliable long-term operation and the droplets are additionally heated, which may be undesirable in experiments with living microorganisms. The use of modulation of the heating power to prevent the condensational growth of droplets made it possible to reduce the droplet growth rate by about half (Fedorets et al., 2018), but this turned out to be insufficient for the long-time stabilization of clusters.

The foregoing motivated the development of a fundamentally different method for suppressing the growth of cluster droplets using self-regulation of the evaporation rate of the water layer. The use of an alternative technology for generating an initial cluster, where the droplets are supplied by an external source, made it possible, in particular, to work with a layer of water in which a nonevaporating substance is dissolved. An interesting solution to such a problem was recently proposed by Fedorets et al. (2022b) with the use of a small amount of sodium chloride (NaCl, hereafter simply called “salt”) dissolved in the water layer.

Understanding the process of the gradual formation of steady-state parameters of a layer of water salt dissolved in it requires a comparison of the characteristic time of salt diffusion with the thermal relaxation time. This can be done using the following relations:

(1) (1)

where d is the thickness of the water layer, κw is the thermal diffusivity of water (a small concentration of salt is neglected), and D is the diffusivity of salt in water. In the temperature range of 40°C < T < 80°C, κw is almost constant and equal to 1.6 × 10–7 m2/s, whereas D increases with temperature from about 2.1 × 10–9 m2/s to 4.4 × 10–9 m2/s. For d = 0.4 mm, we obtain the values theat ≈ 1 s and 36 s < tdiff < 76 s. This means that the temperature profile in the water layer under the cluster can be established in 1 s, and it takes about 1 min to establish the salt concentration profile.

Obviously, at the beginning of water evaporation, the salt concentration increases in a thin surface layer. As a result, the surface salt concentration, csurf, becomes greater than its average value, cav, in the water layer. At the same time, rather slow diffusion does not have time to counteract this local increase in salt concentration. Therefore, in the case of low-salinity water, the cluster droplets first grow until the evaporation rate decreases considerably. This growth of water droplets slows down because, according to Raoult's law, the rate of evaporation decreases as the surface concentration of salt increases. After a time exceeding tdiff, one should expect a decrease in the size of water droplets in the cluster to a certain equilibrium value. At the same time, a steady-state profile of salt concentration over the thickness of the water layer is achieved.

The schematic of the experiment reported by Fedorets et al. (2022b) is shown in Fig. 1(a). A sitall substrate (3) is glued to the metal bottom of the cuvette. The cryothermostat with a coolant temperature of 8 ± 1°C allows for stabilizing the temperature of the water layer, which is heated by a laser beam (5) aimed at the lower blackened surface of the substrate. The radial temperature profile of the water layer surface [Fig. 1(b)] was recorded with a thermal imager. Throughout the experiment, the bottom of the cuvette (4) was cooled by pumping a coolant. As a result, the temperature of water (2) away from the heated central zone was low, which led to the condensation of water vapor from the humid air. Considering that the processes proceeded in a hermetically sealed working volume, this condensation was practically sufficient to compensate for water evaporation in the central zone. Video recording of the cluster image was carried out using a stereomicroscope equipped with a camera (6). The thickness of the water layer (2) is equal to 0.4 mm. It was controlled by a laser confocal sensor (7) and maintained with an accuracy of ± 2 μm.

(a) Schematic of the experimental setup: 1, the droplet cluster; 2, the layer of water with a surfactant and salt; 3, the substrate; 4, cuvette; 5, the laser beam; 6, microscope and thermal imager; and 7, water feeder; and (b) typical temperature profile at the water surface. The pink stripe marks the cluster localization.
(a)(b)

Figure 1. (a) Schematic of the experimental setup: 1, the droplet cluster; 2, the layer of water with a surfactant and salt; 3, the substrate; 4, cuvette; 5, the laser beam; 6, microscope and thermal imager; and 7, water feeder; and (b) typical temperature profile at the water surface. The pink stripe marks the cluster localization.

The experiments were carried out with salt solutions (impurity content of NaCl was < 0.1%) in distilled water. Analytical scales with an accuracy of 0.1 mg were used to weigh the salt, making it possible to obtain solutions with a mass concentration of salt differing from the set value by only a hundredth of a percent. The mass concentration of salt in water varied from one experiment to another. The solution always contained a surfactant (sodium lauryl sulfate) with a concentration of 0.02 g/l, which is necessary to suppress the thermocapillary flow in the water layer. The water surface temperature under the cluster was monitored with a pyrometric sensor (6). The spectral range of the sensor sensitivity was from 8 to 14 μm, and the resulting error is equal to ∓ 1 °C.

A small cluster with the number of droplets from 10 to 20 (1) was “printed” from distilled water by injection of microdroplets with a piezoelectric dispenser. The nozzle of the dispenser generated a chain of very close-sized separately flying droplets. With the right choice of parameters (nozzle position, velocity, and size of the droplets, etc.), these droplets were captured by the gas flow over a locally heated area of the water layer and formed a cluster. Intensive heating of the water layer eliminates the penetration of spontaneously condensed microdroplets into the cluster, so the cluster contains only droplets with a given initial concentration of the dissolved substance. The use of small clusters is explained by the fact that such clusters are located near the maximum temperature of the surface of the water layer [Fig. 1(b)], which provides almost the same external conditions for all droplets in the cluster. The designation Tsurf is hereafter used for the constant temperature under the cluster.

Note that a pause of ∼ 300 s was maintained, which was necessary to stabilize the droplet size, and the cluster image was recorded. Then, the laser heating power was increased by 20 mW. The equilibrium droplet radius, aeq, was measured for several values of the laser heating power. A special computer code analyzed all the droplets in each frame and measured the diameter of each droplet. After that, the average diameter of the droplets in the cluster was calculated. It was more convenient for us to work with the average radius of the droplets. Each aeq value was then averaged over 50 consecutive frames of all droplets in the cluster. The experiments were carried out for pure water and also at the following values of the average mass concentration of salt in the water layer: cav = 0.1, 0.2, 0.3, and 0.4%, but the first two of these values appeared to be insufficient to obtain the self-stabilized cluster of levitating water droplets in a wide temperature range.

Let us consider typical results of the laboratory experiments in which the formation of stable/equilibrium droplet clusters was observed. At the Tsurf = 60 ∓ 0.5°C and two values of the salt concentration of cav = 0.3 and 0.4%, small clusters of 11 and 13 droplets were formed, respectively. Photographs of these clusters at different points in time are shown in Fig. 2. The regular arrangement of the cluster droplets—the distance between which is approximately the same and they are all in the same plane—was studied and received a physical explanation in Dombrovsky et al. (2020) and Fedorets et al. (2017, 2022a).

Images of small clusters of water droplets: (a—c) at c_av = 0.3%, where (a) is t = 0 s, (b) is t = 90 s, and (c) is t = 290 s
(a)

(b)

(c)

Images of small clusters of water droplets: (d—f) at c_av = 0.4%, where (d) is t = 0 s, (e) is t = 90 s, and (f) is t = 290 s
(d)(e)(f)

Figure 2. Images of small clusters of water droplets: (a–c) at cav = 0.3% and (d–f) at cav = 0.4%, where (a, d) is t = 0 s, (b, e) is t = 90 s, and (c, f) is t = 290 s

The change in the radius of water droplets over time is clearer in Fig. 3(a). The radii of all droplets of each of the two clusters were measured at different points in time, and all results of the measurements are shown in Fig. 3. That is why the number of points is so large and there is a small scatter of radii of different droplets of the same cluster. Obviously, the experimental results presented in Fig. 3(a) confirm the abovementioned theoretical predictions concerning nonmonotonic changes in time of the radius of levitating water droplets and the transition of this radius to an equilibrium value.

Time variation of radius of water droplets in two clusters: (a) in the physical coordinates and (b) in the transformed coordinates
(a)(b)

Figure 3. Time variation of radius of water droplets in two clusters: (a) in the physical coordinates and (b) in the transformed coordinates

As one might expect, the dependences a(t) at different values of cav are similar to each other. However, at a higher salt concentration in the water layer, the cluster droplets are smaller and their initial growth stops earlier, after which the droplet radius decreases to the equilibrium value aeq. It is interesting that the results for two values of salt concentration practically coincide with each other (except for the regime close to equilibrium) when the physical coordinates (t, a) are replaced by the coordinates (, ) = [cavt, (√cav) a] [see Fig. 3(b)], according to the usual criteria for diffusion-like processes when we are dealing with the ratio of t/d2.

It turns out that an increase in water temperature is favorable for the self-stabilization of the cluster. As shown in Fig. 4(a), at Tsurf = 74°C this effect is observed even at cav = 0.2%. For any values of cav, the equilibrium radius of droplets aeq increases almost linearly with an increasing water temperature. The experimentally determined domain of self-stabilization of droplet clusters levitating over a layer of salt water is shown in Fig. 4(b).

Equilibrium water droplets in stabilized droplet clusters: (a) dependences of droplet radius on water surface temperature and (b) domain of self-stabilized droplet clusters. The dashed line is an approximate calculation of the domain boundary.
(a)(b)

Figure 4. Equilibrium water droplets in stabilized droplet clusters: (a) dependences of droplet radius on water surface temperature and (b) domain of self-stabilized droplet clusters. The dashed line is an approximate calculation of the domain boundary.

Strictly speaking, the boundary of this domain is not a straight line, which is the result of interpolation over a limited number of experimental points. Perhaps, the calculated boundary [dashed line in Fig. 4(b)] is a more realistic one. The details of the approximate calculations are discussed in Fedorets et al. (2022b). In all cases, this diagram seems to be physically correct and convenient for choosing parameters at which droplet clusters are stabilized.

Noted that the self-stabilized equilibrium droplet clusters, formed over a layer of salt water for ∼ 5 min, did not undergo any changes for at least another 30 min, which is quite sufficient for conducting biochemical experiments in the individual droplets. This experimental result looks especially spectacular compared to clusters over a layer of pure water or slightly salted water, when the droplets grow rapidly and then almost instantly coalesce with the layer of water (see Fig. 5).

The radii of droplets levitating over pure water and slightly salted water at T_surf = 60°C

Figure 5. The radii of droplets levitating over pure water and slightly salted water at Tsurf = 60°C

Following Fedorets et al. (2022b), we do not consider a general theoretical model that could claim to obtain a necessary and sufficient condition for spontaneous stabilization of a droplet cluster levitating over a thin layer of salt water. The experimental results obtained allow us to limit ourselves to determine the conditions under which such spontaneous stabilization of the cluster is physically impossible.

Obviously, cluster stabilization cannot occur at very low salt concentration when the droplet cluster evolution is the same as over a layer of pure (unsalted) water. Therefore, it is a matter of theoretical determination of some threshold salt concentration, at which spontaneous stabilization of the cluster becomes possible. A stable droplet cluster can exist only over a layer of water with a steady salt distribution when the salt concentration on the water surface is maximum. It is this constant surface salt concentration that provides the necessary reduction of water evaporation, which prevents excessive intensive condensation of water vapor on the surface of water droplets and an increase in the droplet size of the cluster.

It was shown that the steady-state temperature field in the water layer is reached much faster than the steady-state distribution of the dissolved salt. Since the small concentration of salt does not affect the thermophysical properties of water, the calculation of the temperature field can be done independently. For this purpose, it is sufficient to solve the axisymmetric heat conduction problem (Fedorets and Dombrovsky, 2017; Fedorets et al., 2022b). Of course, the resulting temperature profile in the water layer on the symmetry axis of the problem (under the cluster) is linear. However, the difference between the temperatures at the bottom and at the surface of the layer is large enough that the temperature change of the salt diffusion coefficient should be taken into account.

According to Fedorets et al. (2022b), we consider only the steady-state solution—even for the diffusion problem—in order to obtain the condition of the desired self-stabilization of the cluster. The obtained analytical solution contains a parameter that is directly proportional to the evaporation rate calculated using a modified evaporation model (Levashov and Kryukov, 2017), which takes into account the effect of a relatively high salt concentration on the surface of the water layer. This solution allowed the authors to derive an approximate condition for the self-stabilization of the cluster. The obtained boundary of the self-stabilization region is represented by the dashed line in Fig. 4(b). This theoretical estimate of the minimum required salt concentration in the aqueous layer agrees well with the experimental results.

The new method of self-stabilization of droplet clusters opens up promising possibilities for studying chemical and biochemical processes in small droplets, which can be used as natural micro-reactors. The results obtained can also be interesting for studies of processes in the surface layer of the ocean and the analysis of the remote sensing results.


REFERENCES

Dombrovsky, L.A., Fedorets, A.A., and Medvedev, D.N. (2016) The Use of Infrared Irradiation to Stabilize Levitating Clusters of Water Droplets, Infrared Phys. Technol., 75: 124–132.

Dombrovsky, L.A., Fedorets, A.A., Levashov, V.Yu., Kryukov, A.P., Bormashenko, E., and Nosonovsky, M. (2020a) Modeling Evaporation of Water Droplets as Applied to Survival of Airborne Viruses, Atmosphere, 11(9): 965.

Dombrovsky, L.A., Fedorets, A.A., Levashov, V.Yu., Kryukov, A.P., Bormashenko, E., and Nosonovsky M. (2020b) Stable Cluster of Identical Water Droplets Formed under the Infrared Irradiation: Experimental Study and Theoretical Modeling, Int. J. Heat Mass Transf., 161: 120255.

Fedorets, A.A., Aktaev, N.E., and Dombrovsky, L.A. (2018) Suppression of the Condensational Growth of Droplets of a Levitating Cluster Using the Modulation of the Laser Heating Power, Int. J. Heat Mass Transf., 127A: 660–664.

Fedorets, A.A. and Dombrovsky L.A. (2017) Generation of Levitating Droplet Clusters above the Locally Heated Water Surface: A Thermal Analysis of Modified Installation, Int. J. Heat Mass Transf., 104: 1268–1274.

Fedorets, A.A., Dombrovsky, L.A., Shcherbakov, D.V., Bormashenko, E., and Nosonovsky, M. (2022a) Thermal Conditions for the Formation of Self-Assembled Cluster of Droplets over the Water Surface and Diversity of Levitating Droplet Clusters, Heat Mass Transf. DOI: 10.1007/s00231-022-03261-8

Fedorets, A.A., Frenkel, M., Shulzinger, E., Dombrovsky, L.A., Bormashenko, E., and Nosonovsky, M. (2017) Self-Assembled Levitating Clusters of Water Droplets: Pattern-Formation and Stability, Sci. Rep., 7: 1888.

Fedorets, A.A., Shcherbakov, D.V., Levashov, V.Yu., and Dombrovsky, L.A. (2022b) Self-Stabilization of Droplet Clusters Levitating over Heated Salt Water, Int. J. Thermal Sci., 182: 107822.

Joung, Y.S., Ge, Z., and Buie, S.R. (2017) Bioaerosol Generation by Raindrops on Soil, Nature Commun., 8: 14668.

Levashov, V.Yu. and Kryukov, A.P. (2017) Numerical Simulation of Water Droplet Evaporation into Vapor–Gas Medium, Colloid J., 79: 647–653.

Nath, S., Ahmadi, S.F., Gruszewski, H.A., Budhiraja, S., Jung, S., Schmale III, D.G., and Boreyko, J.B. (2019) “Sneezing” Plants: Pathogen Transport via Jumping-Droplet Condensation, J. R. Soc. Interface, 16: 20190243.

Wei, Z., Li, Y., Cooks, R.G., and Yan, X. (2020) Accelerated Reaction Kinetics in Microdroplets: Overview and Recent Developments, Ann. Rev. Phys. Chem., 71: 31–51.

References

  1. Dombrovsky, L.A., Fedorets, A.A., and Medvedev, D.N. (2016) The Use of Infrared Irradiation to Stabilize Levitating Clusters of Water Droplets, Infrared Phys. Technol., 75: 124–132.
  2. Dombrovsky, L.A., Fedorets, A.A., Levashov, V.Yu., Kryukov, A.P., Bormashenko, E., and Nosonovsky, M. (2020a) Modeling Evaporation of Water Droplets as Applied to Survival of Airborne Viruses, Atmosphere, 11(9): 965.
  3. Dombrovsky, L.A., Fedorets, A.A., Levashov, V.Yu., Kryukov, A.P., Bormashenko, E., and Nosonovsky M. (2020b) Stable Cluster of Identical Water Droplets Formed under the Infrared Irradiation: Experimental Study and Theoretical Modeling, Int. J. Heat Mass Transf., 161: 120255.
  4. Fedorets, A.A., Aktaev, N.E., and Dombrovsky, L.A. (2018) Suppression of the Condensational Growth of Droplets of a Levitating Cluster Using the Modulation of the Laser Heating Power, Int. J. Heat Mass Transf., 127A: 660–664.
  5. Fedorets, A.A. and Dombrovsky L.A. (2017) Generation of Levitating Droplet Clusters above the Locally Heated Water Surface: A Thermal Analysis of Modified Installation, Int. J. Heat Mass Transf., 104: 1268–1274.
  6. Fedorets, A.A., Dombrovsky, L.A., Shcherbakov, D.V., Bormashenko, E., and Nosonovsky, M. (2022a) Thermal Conditions for the Formation of Self-Assembled Cluster of Droplets over the Water Surface and Diversity of Levitating Droplet Clusters, Heat Mass Transf. DOI: 10.1007/s00231-022-03261-8
  7. Fedorets, A.A., Frenkel, M., Shulzinger, E., Dombrovsky, L.A., Bormashenko, E., and Nosonovsky, M. (2017) Self-Assembled Levitating Clusters of Water Droplets: Pattern-Formation and Stability, Sci. Rep., 7: 1888.
  8. Fedorets, A.A., Shcherbakov, D.V., Levashov, V.Yu., and Dombrovsky, L.A. (2022b) Self-Stabilization of Droplet Clusters Levitating over Heated Salt Water, Int. J. Thermal Sci., 182: 107822.
  9. Joung, Y.S., Ge, Z., and Buie, S.R. (2017) Bioaerosol Generation by Raindrops on Soil, Nature Commun., 8: 14668.
  10. Levashov, V.Yu. and Kryukov, A.P. (2017) Numerical Simulation of Water Droplet Evaporation into Vapor–Gas Medium, Colloid J., 79: 647–653.
  11. Nath, S., Ahmadi, S.F., Gruszewski, H.A., Budhiraja, S., Jung, S., Schmale III, D.G., and Boreyko, J.B. (2019) “Sneezing” Plants: Pathogen Transport via Jumping-Droplet Condensation, J. R. Soc. Interface, 16: 20190243.
  12. Wei, Z., Li, Y., Cooks, R.G., and Yan, X. (2020) Accelerated Reaction Kinetics in Microdroplets: Overview and Recent Developments, Ann. Rev. Phys. Chem., 71: 31–51.
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