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The ordinary hexagonal droplet clusters, such as the one shown in Fig. 1, are best known. Some examples of levitating droplet clusters of different structures were presented in Fedorets et al. (2022a). 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., 2019a). This change is explained by an increase in the flow rate of the 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 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. However, this is not possible in a large cluster. As a result, some of the large droplets close (but do not merge) with each other, forming branching chains (see Fig. 2). 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.

Droplet cluster of a hexagonal structure

Figure 1. Droplet cluster of a hexagonal structure

Typical chain cluster (Reprinted from Frenkel et al. with permission from the American Physical Society, Copyright 2022)

Figure 2. Typical chain cluster (Reprinted from Frenkel et al. with permission from the American Physical Society, Copyright 2022)

Interestingly, the transition from a hexagonal to a chain cluster, like other second-order phase transitions, is reversible. For example, under external infrared irradiation, when the droplets become smaller, the chain cluster transforms into an ordinary hexagonal cluster. The illustration in Fig. 3 shows a fragment of a growing chain cluster with chains of droplets highlighted in different colors for clarity. It was shown by Frenkel et al. (2022) that the Kramers theorem works reasonably well for long droplet chains containing ∼100 droplets as shown in Fig. 4.

Central part of a chain cluster containing branched chains of droplets (Reprinted from Fedorets et al. with permission from the Springer Nature, Copyright 2022)

Figure 3. Central part of a chain cluster containing branched chains of droplets (Reprinted from Fedorets et al. with permission from the Springer Nature, Copyright 2022)

Long-branched droplet chain, built of 101 droplets (Reprinted from Frenkel et al. with permission from the American Physical Society, Copyright 2022)

Figure 4. Long-branched droplet chain, built of 101 droplets (Reprinted from Frenkel et al. with permission from the American Physical Society, Copyright 2022)

A specific time variation of the heating intensity of the water layer can lead to unusual, sometimes very useful types of droplet clusters. In particular, clusters of a given small number of nearly identical droplets can be produced in this way. The small clusters are convenient for laboratory studies of chemical and microbiological processes in individual droplets (Fedorets et al., 2019b). The following method is used to generate such clusters. First, a droplet cluster from a large number of small droplets is produced with relatively low power of local heating of water. These droplets migrate continuously to the center of the cluster. Upon reaching the desired number of droplets in the central zone, the heating power is increased sharply (for example, two to three times). As a result, a much more intense ascending flow of humid air carries away the smallest droplets from the cluster periphery, while the central droplets increase in size and become nearly identical. A description of the experimental method and the equipment used can be found in Fedorets et al. (2017). Images of some small clusters are shown in Fig. 5. These clusters are not necessarily hexagonal in structure. Classification of small clusters according to their symmetry was reported by Fedorets et al. (2020a). It is interesting that using the above-described method, it is possible to generate small clusters with a specified number of droplets from one to several dozen.

Typical images of small droplet clusters

Figure 5. Typical images of small droplet clusters

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 (Rosen and Kunjappu, 2012). 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, when an insufficient surfactant is present, the ordinary cluster collapses and an unusual ring-shaped cluster is formed at some stage of this process. One of the images of this amazing cluster during the beginning of the formation of its second ring is shown in Fig. 6. 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.

Photo of a ring-shaped cluster being formed (Reprinted from Fedorets et al. with permission from the Springer Nature, Copyright 2022)

Figure 6. Photo of a ring-shaped cluster being formed (Reprinted from Fedorets et al. with permission from the Springer Nature, Copyright 2022)

After several years of research, the recently discovered fundamentally new hierarchical cluster structure (Fedorets et al., 2022b) came as a big surprise. 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 chemical compositions.

A fragment of the axisymmetric hierarchical cluster of water droplets (Reprinted from Fedorets et al. with permission from the Springer Nature, Copyright 2022)

Figure 7. A fragment of the axisymmetric hierarchical cluster of water droplets (Reprinted from Fedorets et al. with permission from the Springer Nature, Copyright 2022)

It turns out that even droplets in stable clusters oscillate in the upward flow of moist air. Both vertical and horizontal oscillations of the droplets are observed. Of course, vertical and horizontal oscillations of droplets in a gas stream are interrelated. More details on this effect can be found in Fedorets et al. (2019c, 2021). Interestingly, one can sometimes observe clusters whose droplets perform a surprisingly periodic motion. Examples of such unusual clusters are shown in Videos 1 and 2. Unlike the previously described less mobile clusters, whose properties are satisfactorily described by physical models, the reason for the complex behavior of such mobile clusters is not always clear.

Video 1: Dancing pairs of droplets

Video 2: Everything was fine, but suddenly a chill

REFERENCES

Fedorets, A.A., Aktaev, N.E., Gabyshev, D.N., Bormashenko, E., Dombrovsky, L.A., and Nosonovsky, M. (2019c) Oscillatory Motion of a Droplet Cluster, J. Phys. Chem. C, 123(38): 23572–23576.

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

Fedorets, A.A., Bormashenko, E., Dombrovsky, L.A., and Nosonovsky, M. (2019b) Droplet Clusters: Nature-Inspired Biological Reactors and Aerosols, Philos. Trans. Royal Soc. A, 377: 20190121.

Fedorets, A.A., Bormashenko, E., Dombrovsky, L.A., and Nosonovsky, M. (2020a) 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., Shcherbakov, D.V., Frenkel, M., 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., Legchenkova, I., Shcherbakov, D., Dombrovsky, L., Nosonovsky, M., and Bormashenko, E. (2019a) Self-Arranged Levitating Droplet Clusters: A Reversible Transition from Hexagonal to Chain Structure, Langmuir, 35: 15330–15334.

Fedorets, A.A., Gabyshev, D.N., Shcherbakov, D., Bormashenko, E., Dombrovsky, L.A., and Nosonovsky, M. (2021) Vertical Oscillations of Droplets in Small Droplet Clusters, Coll. Surf. A, 628: 127271.

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.

Frenkel, M., Fedorets, A.A., Shcherbakov, D.V., Dombrovsky, L.A., Nosonovsky, M., and Bormashenko, E. (2022) Branched Droplet Clusters and the Kramers Theorem, Phys. Rev. E, 105(5): 055104.

Rosen, M.J. and Kunjappu, J.T. (2012) Surfactants and Interfacial Phenomena, Hoboken: Wiley.

Referências

  1. Fedorets, A.A., Aktaev, N.E., Gabyshev, D.N., Bormashenko, E., Dombrovsky, L.A., and Nosonovsky, M. (2019c) Oscillatory Motion of a Droplet Cluster, J. Phys. Chem. C, 123(38): 23572–23576.
  2. Fedorets, A.A., Frenkel, M., Bormashenko, E., and Nosonovsky, M. (2017) Small Levitating Ordered Droplet Clusters: Stability, Symmetry, and Voronoi Entropy, J. Phys. Chem. Lett., 8(22): 5599–5602.
  3. Fedorets, A.A., Bormashenko, E., Dombrovsky, L.A., and Nosonovsky, M. (2019b) Droplet Clusters: Nature-Inspired Biological Reactors and Aerosols, Philos. Trans. Royal Soc. A, 377: 20190121.
  4. Fedorets, A.A., Bormashenko, E., Dombrovsky, L.A., and Nosonovsky, M. (2020a) Symmetry of Small Clusters of Levitating Water Droplets, Phys. Chem. Chem. Phys., 22(21): 12233–12244.
  5. 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.
  6. Fedorets, A.A., Dombrovsky, L.A., Shcherbakov, D.V., Frenkel, M., 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., Legchenkova, I., Shcherbakov, D., Dombrovsky, L., Nosonovsky, M., and Bormashenko, E. (2019a) Self-Arranged Levitating Droplet Clusters: A Reversible Transition from Hexagonal to Chain Structure, Langmuir, 35: 15330–15334.
  8. Fedorets, A.A., Gabyshev, D.N., Shcherbakov, D., Bormashenko, E., Dombrovsky, L.A., and Nosonovsky, M. (2021) Vertical Oscillations of Droplets in Small Droplet Clusters, Coll. Surf. A, 628: 127271.
  9. 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.
  10. Frenkel, M., Fedorets, A.A., Shcherbakov, D.V., Dombrovsky, L.A., Nosonovsky, M., and Bormashenko, E. (2022) Branched Droplet Clusters and the Kramers Theorem, Phys. Rev. E, 105(5): 055104.
  11. Rosen, M.J. and Kunjappu, J.T. (2012) Surfactants and Interfacial Phenomena, Hoboken: Wiley.
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