The cluster droplets levitating above the water surface grow rapidly due to the condensation of water vapor, and the natural lifetime of the cluster is only ∼1 min. These are not the conditions under which chemical or biological studies in the droplets can be carried out. Therefore, it was important to switch from studying clusters to suppressing the growth of droplets (stabilizing the clusters). Positive results were obtained for the first time by Fedorets et al. (2015) and Dombrovsky et al. (2016) using the near-infrared irradiation of the cluster. Theoretical predictions based on the calculations of the radiation power absorbed by semi-transparent droplets were confirmed experimentally. The calculations were based on the Mie theory (Bohren and Huffman, 1998; Dombrovsky and Baillis, 2010) and took into account the spectrum of the miniature radiative sources used in the experiments. Fortunately, the required infrared radiation power turned out to be only ∼6% of the laser power used for water heating. Attempts to use laser power modulation to prevent droplet growth have not been as successful (Fedorets et al., 2018).

Some of the experimental results reported by Dombrovsky et al. (2016) are subsequently presented. The experimental setup is schematically presented in Fig. 1. The laser heating of a thin polished glass plate at the bottom of the water volume was used. The heated lower surface of the glass plate was coated with a paint containing microparticles of graphite. A continuous wave laser with the wavelength of \(\lambda =\) 0.808 μm and maximum power \(W^{\rm max}_{\rm L} =\) 600 mW was used. The laser beam was focused on the central part of the lower surface of the glass plate.

Scheme of the experiment (Reprinted from Dombrovsky et al. with permission from Elsevier, Copyright 2016)

Figure 1. Scheme of the experiment (Reprinted from Dombrovsky et al. with permission from Elsevier, Copyright 2016)

The estimated time of heating of the water layer is ∼2 min. Therefore, the laser was turned on not later than 2 min before the observations. The integral radiative flux from four infrared radiation sources, \(q_{\rm IR}\), was changed in the range of 0–12.2 mW/mm2. It is convenient to use the following power parameter of the cluster infrared heating:

\(W_{\rm IR}=\pi R^2_{\rm h}\ q_{\rm IR}\) (1)

The maximum value of this power in the experiments was equal to 18.8 mW and it is relatively small: \(W^{\rm max}_{\rm IR}\ll W_{\rm L}\). The duration of every video record was equal to 60 s, including the following three periods: (i) the initial period of \(t\lt\) 20 s without infrared irradiation, (ii) the “active” period of 20 \(\lt t \lt\) 40 s with infrared heating the droplet cluster, and (iii) the last period of 40 \(\lt t \lt\) 60 s without infrared heating.

The observed effect of infrared irradiation is illustrated in Fig. 2, where the time variation of the droplet surface area \(S = 4\pi a^2\) and \(W_{\rm L} =\) 163 mW is presented. The parameter \(S\) was calculated as an average value for the five largest droplets in the central region of the droplet cluster. The ordinary time dependence of the ratio \(\overline{S} =\) \(S(W_{\rm IR})/S(0)\) without infrared irradiation of the cluster is also shown in Fig. 2. One can see that every part of the time dependence \(\overline{S}(t)\) is almost linear and the rates of the droplet growth at the first and third periods are the same. This means that the droplet growth can be well described by the known d-squared law (Sirignano, 1999) or its modification, called the elliptic law (Dombrovsky and Sazhin, 2003). This statement appears to be true also in the case of infrared irradiation, at least in the range shown in Fig. 2. The infrared irradiation leads to a significant decrease in the growth rate of water droplets. Note that it takes ∼2 s for the increase of the infrared radiation power from zero to the nominal value and ∼1 s for the complete turn off of the infrared radiation source. Therefore, the kinks on green and blue curves in Fig. 2 are not exactly positioned at \(t=\) 20 and \(t=\) 40 s.

Time variation of relative area of droplet surface at the initial droplet radius, a = 17.1 µm: 1) without infrared irradiation, 2) W_IR = 9.3 mW, and 3) W_IR = 19.1 mW (Reprinted from Dombrovsky et al. with permission from Elsevier, Copyright 2016)

Figure 2. Time variation of relative area of droplet surface at the initial droplet radius, \(a=\) 17.1 μm: 1) without infrared irradiation, 2) \(W_{\rm IR} =\) 9.3 mW, and 3) \(W_{\rm IR} =\) 19.1 mW (Reprinted from Dombrovsky et al. with permission from Elsevier, Copyright 2016)

It turns out that the ratio \({W^0_{\rm IR}}/{W_{\rm L}}\), where \(W^0_{\rm IR}\) corresponds to the case of a negligible time variation of the droplet radius while infrared heating the cluster, is weakly sensitive to the laser power in the range of 131 \(\lt W_{\rm L}\lt\) 195 mW and can be estimated as follows:

\({\overline{W}}^0_{\rm IR}=\dfrac{W^0_{\rm IR}}{W_{\rm L}}\approx 6\%\) (2)

Approximate relation (2) is convenient to obtain the required infrared radiation power to stabilize the levitating clusters at different conditions of the laser heating of the water layer. The correct choice and possible correction of the infrared irradiation is a relatively simple way to radically increase the lifetime of levitating droplet clusters.

Following a recent paper by Dombrovsky et al. (2020), let us consider a long-time stabilization of small clusters generated using the procedure developed by Fedorets et al. (2017). For clarity, from a series of laboratory experiments we selected the results obtained for a cluster consisting of six droplets, the sizes of which initially differed significantly from each other. The photographs in Fig. 3 taken at various stages of the process illustrate the transition to an equilibrium cluster. The measured current radii of all six droplets are presented in Fig. 4. The stepwise nature of the curves is a result of the limited spatial resolution of digital images with a pixel size of ∼0.6 μm. One can see that infrared irradiation leads to a decrease in the radius of droplets 1 and 2 and an increase in the radius of droplets 4–6. The radius of droplet 3 remains constant and equal to the equilibrium value.

Evolution of droplet cluster during the long-term infrared irradiation (Reprinted from Dombrovsky et al. with permission from Elsevier, Copyright 2020)

Figure 3. Evolution of droplet cluster during the long-term infrared irradiation (Reprinted from Dombrovsky et al. with permission from Elsevier, Copyright 2020)

Time variation of radius of water droplets with numbers from 1 to 6 (see Fig. 3); the stepwise nature of the curves is a result of an insufficiently high resolution of the digital image (Reprinted from Dombrovsky et al. with permission from Elsevier, Copyright 2020).

Figure 4. Time variation of radius of water droplets with numbers from 1 to 6 (see Fig. 3); the stepwise nature of the curves is a result of an insufficiently high resolution of the digital image (Reprinted from Dombrovsky et al. with permission from Elsevier, Copyright 2020).

It is reasonable to assume that the theoretical model for the infrared radiative heating of water droplets can be based on the assumption of the isothermal droplet. This assumption supported by the account for the conductive and convective heat transfer inside the droplet has been confirmed by successful computational modeling of the abovementioned consideration of the evolution of small clusters under infrared irradiation (Dombrovsky et al., 2020).

It is convenient to characterize the heating of the water surface under the cluster with the temperature without infrared irradiation, \(T_{\rm s0}\), and also with the increase in this temperature, \(\Delta T_{\rm s} =T_{\rm s}- T_{\rm s0}\), due to absorption of the infrared radiation. Experiments at different powers of infrared heating showed that there is a boundary separating the region, in which the condensational growth only slows down (but there is a coalescence with the water layer), from the region where the asymptotic stabilization of clusters is observed. One can see in Fig. 5(a) that this boundary is in the narrow range of 6 \(\lt\Delta T_{\rm s}/T_{\rm s0}\lt\) 9%. As one can expect, the minimum equilibrium radius of water droplets, \(a_{\rm eq}\), at this boundary decreases monotonically with the increasing intensity of infrared irradiation. This is illustrated in Fig. 5(b), where the dots show the results of a series of laboratory experiments. Note that the ratio \(\Delta T_{\rm s}/T_{\rm s0}\) shown on the abscissa in Fig. 5(b) is proportional to the infrared radiative flux. For better understanding the results presented in Fig. 5(a), it should be noted that one can obtain different values of the equilibrium droplet radius in the region above the boundary curve at the same temperature \(T_{\rm s0}\). Obviously, the value of \(a_{\rm eq}\) increases at relatively high values of both \(T_{\rm s0}\) and the infrared radiative flux. Of course, one should distinguish the small droplets at the boundary curve and relatively large droplets above the boundary curve with the greater equilibrium radius.

Equilibrium of irradiated droplet clusters: (a) region of possible equilibrium clusters (Reprinted from Dombrovsky et al. with permission from Elsevier, Copyright 2020)
Equilibrium of irradiated droplet clusters: (b) equilibrium radius of levitating water droplets (Reprinted from Dombrovsky et al. with permission from Elsevier, Copyright 2020)
(a)(b)

Figure 5. Equilibrium of irradiated droplet clusters: (a) region of possible equilibrium clusters and (b) equilibrium radius of levitating water droplets (Reprinted from Dombrovsky et al. with permission from Elsevier, Copyright 2020)

A series of experiments showed that the droplets reach the same equilibrium radius, not only in the case of small clusters, but also during infrared heating of clusters consisting of several dozen droplets. Moreover, the effect of the number of droplets in the cluster on this process is insignificant. It is only important that the distance between the relatively small equilibrium droplets of water turns out to be significantly larger than the size of the droplets themselves. An example of such a relatively large cluster at the final stage of its infrared heating is shown in Fig. 6. It is seen that the droplets in the central zone of the cluster are almost identical. The relatively small size of some droplets at the periphery of the cluster is partly due to the limited diameter of the almost isothermal spot of heating the surface of water layer, as well as to the fact that the peripheral droplets only recently joined the cluster and did not manage to reach the equilibrium.

Typical upper view of the droplet cluster consisting of a large number of droplets; the photograph was taken after prolonged infrared irradiation of the cluster (Reprinted from Dombrovsky et al. with permission from Elsevier, Copyright 2020).

Figure 6. Typical upper view of the droplet cluster consisting of a large number of droplets; the photograph was taken after prolonged infrared irradiation of the cluster (Reprinted from Dombrovsky et al. with permission from Elsevier, Copyright 2020).

REFERENCES

Bohren, C.F. and Huffman, D.R. (1998) Absorption and Scattering of Light by Small Particles, Hoboken: Wiley.

Dombrovsky, L.A. and Sazhin, S.S. (2003) A Simplified Non-Isothermal Model for Droplet Heating and Evaporation, Int. Commun. Heat Mass Transf., 30(6): 787–796.

Dombrovsky, L.A. and Baillis, D. (2010) Thermal Radiation in Disperse Systems: An Engineering Approach, Danbury, CT: Begell House.

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., Dombrovsky, L.A., and Medvedev, D.N. (2015) Effect of Infrared Irradiation on the Suppression of the Condensation Growth of Water Droplets in a Levitating Droplet Cluster, JETP Lett., 102(7): 452–454.

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., 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.

Sirignano, W.A. (1999) Fluid Dynamics and Transport of Droplets and Sprays, 2nd Ed., New York: Cambridge University Press.

Referencias

  1. Bohren, C.F. and Huffman, D.R. (1998) Absorption and Scattering of Light by Small Particles, Hoboken: Wiley.
  2. Dombrovsky, L.A. and Sazhin, S.S. (2003) A Simplified Non-Isothermal Model for Droplet Heating and Evaporation, Int. Commun. Heat Mass Transf., 30(6): 787–796.
  3. Dombrovsky, L.A. and Baillis, D. (2010) Thermal Radiation in Disperse Systems: An Engineering Approach, Danbury, CT: Begell House.
  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., Dombrovsky, L.A., and Medvedev, D.N. (2015) Effect of Infrared Irradiation on the Suppression of the Condensation Growth of Water Droplets in a Levitating Droplet Cluster, JETP Lett., 102(7): 452–454.
  7. 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.
  8. 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.
  9. Sirignano, W.A. (1999) Fluid Dynamics and Transport of Droplets and Sprays, 2nd Ed., New York: Cambridge University Press.
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