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DROPLET CLUSTERS LEVITATING OVER THE HEATED WATER SURFACE

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

Edward Bormashenko
Department of Chemical Engineering, Biotechnology, and Materials, Ariel University, Ariel, Israel

Michael Nosonovsky
Department of Mechanical Engineering, University of Wisconsin-Milwaukee, Milwaukee, WI


Self-assembled clusters of regularly spaced small droplets condensing from humid air may appear above the locally heated water surface. This phenomenon was discovered by Alexander Fedorets in 2003–2004 (Fedorets, 2004). Over the next several years, the behavior of levitating droplet clusters was studied in detail. The milestones of this research were described in a keynote lecture by Fedorets and Dombrovsky (2018). The work in this direction continues and practical applications of droplet clusters to the study of biochemical processes in stably levitating droplet clusters are being considered. This article reviews the current understanding of the conditions and process of cluster formation and discusses the different types of droplet clusters studied by the authors and their colleagues.

A layer of microdroplets over a uniformly heated surface of water (Schaefer, 1971; Ienna et al., 2012) and a droplet cluster levitating in an air–vapor flow over a locally heated area of the surface differ significantly in their dynamics and structure. In the first case, the droplets were positioned randomly and the effects typical of a droplet cluster (Arinstein and Fedorets, 2010) were not observed: the self-assembly of an ordered structure, condensational growth of droplets, as well as displacement of the cluster as a whole while maintaining its structure. The relatively strong local heating of the water surface is the main factor responsible for the formation of droplet clusters. The transition from chaotically moving droplets to a cluster has been studied in Fedorets et al. (2022a), and the main results obtained are presented here. We also discuss the observed clusters, which are unusual in shape and structure, to form a complete picture of the diversity of droplet clusters.

All the experiments were performed in the laboratory of microhydrodynamic technologies at the University of Tyumen (Russia). A detailed description of the modified laboratory setup can be found in Fedorets et al. (2022a). The schematic of the experiment is presented in Fig. 1. A droplet cluster (1) is formed over a heated area of a layer of distilled water (2) containing the surfactant impurity. A substrate (3) is glued to the metal bottom of the cuvette. The cuvette body has channels (4) connected to a cryothermostat, which allows stabilizing the temperature of water in the range of T0 from 9 to 60°C. The water layer is locally heated by a laser beam (5) directed to the lower blackened surface of the substrate. In the experiments, the thickness of the water layer was maintained at equal to 400 ± 2 μm. The axisymmetric temperature field of the water surface was recorded with a thermal imager. Video recording of the cluster was carried out using a stereomicroscope equipped with a high-speed camera.

Schematic of local laser heating of water

Figure 1. Schematic of local laser heating of water

The experiments were performed with a steady-state surface temperature profile of the water layer but without external infrared irradiation (Dombrovsky et al., 2016) to stabilize the droplet cluster. The temperature profiles were axisymmetric, with a monotonic decrease in temperature with distance from the axis. The parameter K = T1/2/R1/2 [T1/2 = (TmaxT0)/2 and R1/2, shown in Fig. 1] was used to characterize the temperature profile. In a series of experiments, the parameter K decreased due to an increase in T0 at a constant temperature Tmax = 60°C. In each experiment, the time required for cluster formation was maintained.

Figure 2 shows two typical images of clusters and the corresponding Voronoi tessellation illustrating the ordering of droplets in the cluster (Fedorets et al., 2017a,b, 2022a; Bormashenko et al., 2018). The transition from a layer of droplets to a cluster was observed in the range of 1 K/mm < K < 2.5 K/mm. This regime shown in Fig. 3 is characterized by a significant decrease in the Voronoi entropy, S, to values typical of regular clusters of water droplets. At larger values of K, the Voronoi entropy decreased linearly with the increase in K.

Change of droplet cluster with the parameter K: (a, b)  K = 17.2 K/mm, S = 0.24; (a) image of clusters; and (b) Voronoi tessellations (pentagons, hexagons, and heptagons, highlighted in yellow, gray, and blue, respectively)
(a)

(b)

Change of droplet cluster with the parameter K: (c, d) K = 2.9 K/mm, S = 0.91; (c) image of clusters; and (d) Voronoi tessellations (pentagons, hexagons, and heptagons, highlighted in yellow, gray, and blue, respectively)
(c)(d)

Figure 2. Change of droplet cluster with the parameter K: (a, b) K = 17.2 K/mm, S = 0.24; (c, d) K = 2.9 K/mm, S = 0.91; (a, c) images of clusters; and (b, d) Voronoi tessellations (pentagons, hexagons, and heptagons, highlighted in yellow, gray, and blue, respectively)


Effect of local heating of water surface on ordering the levitating droplets

Figure 3. Effect of local heating of water surface on ordering the levitating droplets

The experimental setup enabled us to change parameter K at constant temperatures T0 and Tmax. In the series of experiments at T0 = 29 ± 0.2°C and Tmax = 70 ± 0.2°C, five different regimes of heating were used, corresponding to the range of 15.5 ≤ K ≤ 16.6 K/mm, when water droplets form a hexagonal cluster. The images of the cluster were processed by a computer code that allows measuring the position and diameters of the droplets, as well as calculating the average values of the droplet diameter and the distance between the centers of neighboring droplets in the selected group. The ratio of  / was used to evaluate the density of “packing” the droplets in the cluster.

The results for clusters with the number of droplets from 60 to 80 and the average diameter of central droplets from 31 to 33 μm from two video recordings for each of the five clusters showed that the distance between the neighboring droplets in a cluster is sensitive to changes in K, even in the case of a constant temperature of the water surface under the cluster. As one might expect, a decrease in the area of the hot spot leads to an increase in the gas flow rate and a decrease in the “packing” density of droplets in the cluster.

Analysis of conditions for the formation of a levitating droplet cluster would be incomplete without a physical model of cluster evolution based on numerous laboratory observations. At significant local heating of the water surface, when conditions for droplet cluster formation arise, water evaporates intensively and an upward gas flow from a mixture of water vapor and air is formed. In other words, we are dealing with a flow of humid air, not just water vapor. This statement becomes clear when one recalls that the pressure of saturated vapor, even at 80°C, is < 50% of the normal atmospheric pressure. Since the room temperature is much lower than the humid air temperature at the water surface and the air contains dust particles, the water vapor, when cooled, condenses on these particles and small water droplets are formed. Some of these droplets are carried away by the gas flow (then, when mixed with the main flow of relatively dry ambient air, these droplets evaporate); other droplets manage to become larger due to condensation of supersaturated vapor and descend by gravity, approaching the water surface. The droplets cannot escape the flow of air, because the pressure of the surrounding still air is noticeably higher. Moreover, the falling water droplets collect in the central zone of the flow, where gas velocity is higher and pressure is lower, forming the basis of a future cluster.

Water droplets cannot remain at a significant height, because then they will be carried away by the gas flow and evaporate. As a result, a relatively thin cloud of a large number of droplets is formed. The larger droplets in the central part of the droplet cloud are responsible for the local increase in the velocity of the gas flow around them, so droplets located at a greater distance from the axis tend not only to remain in the same plane as the larger droplets but also to be as close to them as possible. Thus, the cloud of droplets gathers into a flat cluster. The distance between the droplets is determined by the gas flow rate.

It is known that a droplet cluster continues to grow both due to the joining of peripheral small droplets and due to vapor condensation, which leads to an increase in the size of the central droplets. There comes a moment when the most massive droplets coalesce with the water layer and disappear. Typically, the diameter of the droplet increases at a rate of ∼ 1 μm/s, which limits the cluster lifetime to a few tens of seconds. However, for example, external infrared irradiation can completely suppress condensational growth and then the cluster lifetime is limited by the period of maintaining the right conditions (Dombrovsky et al., 2016, 2020).

The above description of cluster evolution is based on the concepts of gas dynamics. At the same time, it is known that evaporating water droplets have their own electric charge and therefore, the behavior of a droplet cluster can manifest the electrostatic interaction of droplets. It was shown by Fedorets et al. (2020a) that the parameters of the cluster are determined by aerodynamic forces and the presence of a small electric charge of water droplets is a minor factor. Note that electrical effects should be taken into account when analyzing the behavior of the hierarchical clusters (see the final paragraph).

It is interesting that the change in the properties of a thin surface layer of water, the variation of intensity of local heating of water, as well as the use of special devices for independently generating droplets placed above the heated area of the water surface can be used to obtain a wide variety of droplet clusters of different size, shape, structure, and behavior over time. For a general idea of the possible options, some amazing examples of such clusters are subsequently demonstrated.

It turns out that the droplet cluster is very sensitive to thermo-capillary flows in the thin surface layer of water below the cluster. These flows may form due to significant temperature gradients in the locally heated spot on the layer surface. A typical, relatively stable droplet cluster is only obtained with natural or specially added surfactants that suppress thermo-capillary flows. In Fedorets et al. (2020b), it was shown for the first time that, in the absence of surfactant, the ordinary cluster collapses and, at a certain stage of this process, an unusual ring-shaped cluster is formed. One of the images of this amazing cluster during the beginning of the formation of its second ring is shown in Fig. 4. A ring of droplets surrounds the toroidal vortex area. As a rule, only one ring is formed. However, it depends on the number of droplets in the cluster, and it is possible to form both an open ring (an arc if there are few droplets) and more than one ring if the number of droplets is large.

Photograph of a ring-shaped cluster being formed

Figure 4. Photograph of a ring-shaped cluster being formed

Droplet clusters usually contain a large number of droplets of various sizes. At the same time, considering the possible use of levitating droplets as microreactors for studying chemical and biochemical processes, it would be better to have clusters of a small number of the largest droplets of almost the same size. It turns out that such clusters, called “small clusters,” can be generated using a simple method developed and implemented by the first author of this paper. For this, at first, a relatively low power of local heating of water is used, which makes it possible to obtain a droplet cluster from a large number of small droplets. These droplets migrate continuously to the center of the cluster. Upon reaching the desired number of droplets in the central zone, the heating power increases sharply. As a result, a much more intense ascending gas flow carries away the smallest droplets from the cluster periphery, while the central droplets increase in size and become nearly identical (Fedorets et al., 2017b).

Images of some small clusters are shown in Fig. 5. It should be noted that small clusters have a structure that differs from the classical hexagonal structure of ordinary large clusters. It is interesting that, by using the above-described method, it is possible to generate small clusters with a specified number of droplets from one to several dozen. Moreover, starting from 33 droplets, the variety of cluster structures degenerates into a universal hexagonal structure, which does not depend on the number of droplets (Fedorets et al., 2020c).

Typical small clusters of nearly identical droplets—clusters of 11, 14, and 16 droplets

Figure 5. Typical small clusters of nearly identical droplets—clusters of 11, 14, and 16 droplets

Droplet clusters like the one shown in Fig. 2(a) are best known. Because of the mutual arrangement of the nearest droplets, such clusters are called hexagonal. It turns out that the structure of the central part of an ordinary hexagonal cluster consisting of a large number of droplets changes with a significant increase in the local heating of the water layer (Fedorets et al., 2019). This change is explained by an increase in the flow rate of humid air under the center of the cluster. Recall that the height of cluster levitation above the water layer is approximately equal to the diameter of large droplets, that is, almost two orders of magnitude less than the diameter of a large cluster. Therefore, the gas flow under the central part of the cluster cannot go around the cluster, and the only possible path for it is between the water droplets in the center of the cluster. In a small cluster, the droplets could simply move apart, letting the humid air flow through the cluster. However, this is not possible in a large cluster. As a result, some of the large droplets get closer (but do not merge) with each other, forming branching chains. Figure 6 shows a fragment of a growing chain cluster with chains of droplets highlighted in different colors for clarity.

Part of an asymmetric cluster containing growing branched chains of water droplets

Figure 6. Part of an asymmetric cluster containing growing branched chains of water droplets

The average width of the gaps between the droplet chains in a chain cluster is greater than the distance between droplets in a hexagonal cluster. Therefore, the hydraulic drag of the central part of the chain cluster is less, which makes it possible for the increased amount of humid air to flow. Interestingly, the transition from a hexagonal to a chain cluster, like other second-order phase transitions, is reversible. For example, under external infrared irradiation, the chain cluster transforms it into the former hexagonal cluster.

After several years of research, the recently discovered fundamentally new hierarchical cluster structure came as a big surprise (Fedorets et al., 2022b). In the central region of the hierarchical cluster, groups of several almost merging droplets are formed (Fig. 7), separated by a submicron layer of gas. Interestingly, both aerodynamic and electrostatic forces are responsible for droplet interaction within each group. Groups of droplets with different electric charges are constantly rearranged, droplets are exchanged, and individual droplets merge. At the same time, the outer layers of the hierarchical cluster retain a stable hexagonal structure. The droplet merging effect characteristic of the hierarchical cluster is of interest for microbiological and chemical experiments, since it is expected to allow in situ studies of merging the droplets of different compositions.

Fragment of the axisymmetric hierarchical cluster of water droplets

Figure 7. Fragment of the axisymmetric hierarchical cluster of water droplets


REFERENCES

Arinstein, E.A. and Fedorets, A.A. (2010) Mechanism of Energy Dissipation in a Droplet Cluster, JETP Lett., 92(10): 658–661.

Bormashenko, E., Frenkel, M., Vilk, A., Legchenkova, I., Fedorets, A.A., Aktaev, N.E., Dombrovsky, L.A., and Nosonovsky, M. (2018) Characterization of Self-Assembled 2D Patterns with Voronoi Entropy, Entropy, 20: 956.

Ienna, F., Yoo, H., and Pollack, G.H. (2012) Spatially Resolved Evaporative Patterns from Water, Soft Matter, 8(47): 11850–11856.

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. (2020) 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. (2004) Droplet Cluster, JETP Lett., 79(8): 372–374.

Fedorets, A.A. and Dombrovsky, L.A. (2018) Self-Assembled Stable Clusters of Droplets over the Locally Heated Water Surface: Milestones of the Laboratory Study, Proc. of 16th Int. Heat Transfer Conf., Beijing, KN-02, August 10–15.

Fedorets, A.A., Bormashenko, E., Dombrovsky, L.A., and Nosonovsky, M. (2020c) Symmetry of Small Clusters of Levitating Water Droplets, Phys. Chem. Chem. Phys., 22(21): 12233–12244.

Fedorets, A.A., Dombrovsky, L.A., Bormashenko, E., and Nosonovsky, M. (2022b) A Hierarchical Levitating Cluster Containing Transforming Small Aggregates of Water Droplets, Microfluid. Nanofluidics, 26: 52.

Fedorets, A.A., Dombrovsky, L.A., Gabyshev, D.N., Bormashenko, E., and Nosonovsky, M. (2020a) Effect of External Electric Field on Dynamics of Levitating Water Droplets, Int. J. Therm. Sci., 153: 106375.

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., Bormashenko, E., and Nosonovsky, M. (2017b) Small Levitating Ordered Droplet Clusters: Stability, Symmetry, and Voronoi Entropy, J. Phys. Chem. Lett., 8(22): 5599–5602.

Fedorets, A.A., Frenkel, M., Legchenkova, I., Shcherbakov, D., Dombrovsky, L., Nosonovsky, M., and Bormashenko, E. (2019) Self-Arranged Levitating Droplet Clusters: A Reversible Transition From Hexagonal to Chain Structure, Langmuir, 35: 15330–15334.

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

Fedorets, A.A., Shcherbakov, D.V., Dombrovsky, L.A., Bormashenko, E., and Nosonovsky, M. (2020b) Impact of Surfactants on the Formation and Properties of Droplet Clusters, Langmuir, 36(37): 11154–11160.

Schaefer, V.J. (1971) Observations of an Early Morning Cup of Coffee, Am. Sci., 3(7): 534–535.

References

  1. Arinstein, E.A. and Fedorets, A.A. (2010) Mechanism of Energy Dissipation in a Droplet Cluster, JETP Lett., 92(10): 658–661.
  2. Bormashenko, E., Frenkel, M., Vilk, A., Legchenkova, I., Fedorets, A.A., Aktaev, N.E., Dombrovsky, L.A., and Nosonovsky, M. (2018) Characterization of Self-Assembled 2D Patterns with Voronoi Entropy, Entropy, 20: 956.
  3. Ienna, F., Yoo, H., and Pollack, G.H. (2012) Spatially Resolved Evaporative Patterns from Water, Soft Matter, 8(47): 11850–11856.
  4. 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.
  5. Dombrovsky, L.A., Fedorets, A.A., Levashov, V.Yu., Kryukov, A.P., Bormashenko, E., and Nosonovsky, M. (2020) Stable Cluster of Identical Water Droplets Formed Under the Infrared Irradiation: Experimental Study and Theoretical Modeling, Int. J. Heat Mass Transf., 161: 120255.
  6. Fedorets, A.A. (2004) Droplet Cluster, JETP Lett., 79(8): 372–374.
  7. Fedorets, A.A. and Dombrovsky, L.A. (2018) Self-Assembled Stable Clusters of Droplets over the Locally Heated Water Surface: Milestones of the Laboratory Study, Proc. of 16th Int. Heat Transfer Conf., Beijing, KN-02, August 10–15.
  8. Fedorets, A.A., Bormashenko, E., Dombrovsky, L.A., and Nosonovsky, M. (2020c) Symmetry of Small Clusters of Levitating Water Droplets, Phys. Chem. Chem. Phys., 22(21): 12233–12244.
  9. Fedorets, A.A., Dombrovsky, L.A., Bormashenko, E., and Nosonovsky, M. (2022b) A Hierarchical Levitating Cluster Containing Transforming Small Aggregates of Water Droplets, Microfluid. Nanofluidics, 26: 52.
  10. Fedorets, A.A., Dombrovsky, L.A., Gabyshev, D.N., Bormashenko, E., and Nosonovsky, M. (2020a) Effect of External Electric Field on Dynamics of Levitating Water Droplets, Int. J. Therm. Sci., 153: 106375.
  11. 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
  12. Fedorets, A.A., Frenkel, M., Bormashenko, E., and Nosonovsky, M. (2017b) Small Levitating Ordered Droplet Clusters: Stability, Symmetry, and Voronoi Entropy, J. Phys. Chem. Lett., 8(22): 5599–5602.
  13. Fedorets, A.A., Frenkel, M., Legchenkova, I., Shcherbakov, D., Dombrovsky, L., Nosonovsky, M., and Bormashenko, E. (2019) Self-Arranged Levitating Droplet Clusters: A Reversible Transition From Hexagonal to Chain Structure, Langmuir, 35: 15330–15334.
  14. Fedorets, A.A., Frenkel, M., Shulzinger, E., Dombrovsky, L.A., Bormashenko, E., and Nosonovsky, M. (2017a) Self-Assembled Levitating Clusters of Water Droplets: Pattern-Formation and Stability, Sci. Rep., 7: 1888.
  15. Fedorets, A.A., Shcherbakov, D.V., Dombrovsky, L.A., Bormashenko, E., and Nosonovsky, M. (2020b) Impact of Surfactants on the Formation and Properties of Droplet Clusters, Langmuir, 36(37): 11154–11160.
  16. Schaefer, V.J. (1971) Observations of an Early Morning Cup of Coffee, Am. Sci., 3(7): 534–535.
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