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C: Surfaces, Interfaces, Porous Materials, and Catalysis
Oscillatory Motion of a Droplet Cluster Alexander A. Fedorets, Nurken E. Aktaev, Dmitrii N Gabyshev, Edward Bormashenko, Leonid A. Dombrovsky, and Michael Nosonovsky J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b08194 • Publication Date (Web): 04 Sep 2019 Downloaded from pubs.acs.org on September 7, 2019
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The Journal of Physical Chemistry
Oscillatory Motion of a Droplet Cluster Alexander A. Fedorets a, Nurken E. Aktaev a, Dmitrii N. Gabyshev a, Edward Bormashenko c, Leonid A. Dombrovsky a,b, and Michael Nosonovsky* a,d, a
University of Tyumen, 6 Volodarskogo St, Tyumen, 625003, Russia. Tel. +7-3452-597425 ru.mail@alex_fedorets
b
Joint Institute for High Temperatures, 17A Krasnokazarmennaya St, Moscow, 111116, Russia. Tel. +7 910 408 0186,
[email protected] c
Department of Chemical Engineering, Biotechnology and Materials, Engineering Science Faculty, Ariel University, Ariel, 40700, Israel. Tel.: +972-074-7296863,
[email protected] d
Department of Mechanical Engineering, University of Wisconsin–Milwaukee,
3200 North Сramer St, Milwaukee, WI 53211, USA, Tel. +1-414-229-2816,
[email protected] Abstract Horizontal oscillations of small droplet clusters (from one to four droplets) levitating over a locally heated water layer in upward vapor-air flow are investigated experimentally. These oscillations are caused by a complex dynamic interaction between the droplets and the non-steady gas flow. The path of the center of the droplet cluster is similar to a random walk in a potential well. The vibrational spectra of clusters’ centers were obtained by Fourier analysis showed several frequency peaks between f=1.41 Hz and 5.96 Hz found in all clusters, which shows that the cluster tends to oscillate as a whole. The possibility of decoupling of the aerodynamic interaction between the gas flow and the droplets and the interaction between individual droplets is discussed.
*
Corresponding author. Tel. +1-414-229-2816,
[email protected] ACS Paragon Plus Environment
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Introduction The phenomenon of microdroplets levitating over a locally heated layer of water or other liquids in an upward vapor-air flow has been reported for the first time in 20041. Such droplets often form a hexagonally ordered monolayer, the so-called droplet cluster, which behavior became a subject of extensive research1-11. It was shown that the forces responsible for the cluster formation are of aerodynamic nature rather than electrostatic one2. Alternative methods of cluster stabilization and generation of clusters with a desired number of monodisperse droplets3, as well as methods to control droplets size, the distance between them, to trace the droplets, and to use them as chemical and biological reactors, e.g., for the study of microbial life in bioaerosols, have been suggested4. The droplet cluster is similar to other self-organizing systems including colloid and dust crystals12, 13, water breath figures, foams, Rayleigh–Bénard cells, and even Wigner crystals. Despite these similarities, the droplet cluster is a dissipative structure and, unlike the colloidal crystals, it does not emerge at a phase equilibrium surface. The self-assembled droplet cluster is built of water micro-droplets (typically, 10–100 m diameter), which are condensed over a locally heated layer of water. Typical water temperatures are 60–95 ºC, although the phenomenon has been observed at temperatures as low as 27 ºC5. The droplets levitate at an equilibrium height, usually of the same order as their radii, where the drag force from the upward vapor flow is equilibrated by the droplet weight. The droplets do not coalesce due to complex aerodynamic repulsion forces between them6. Changing the temperature of the heated spot at the surface of a water layer under the cluster enables controlling the number of droplets, their density, and size. Furthermore, using infrared irradiation or the modulation of water heating power, it is possible to suppress droplet growth and stabilize the cluster for extended periods of time8, 9. In the previous study, we demonstrated that clusters with an arbitrary small number of almost monodisperse droplets produce stable configurations depending on the number of droplets3, 7.
For a small number of droplets in the cluster, N, the structure deviates from a hexagonal one and
depends on N3, 7. In the past, we proposed a Langevin computational model based on the use of phenomenological potential of aerodynamic forces between the droplets7. The model showed a good agreement with experimental observations for clusters with a small number of droplets. One can characterize the horizontal component of the resulting aerodynamic force acting upon a single droplet by an effective potential W(x, y):
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The Journal of Physical Chemistry 𝑘
𝑊(𝑥, 𝑦) = 2(𝑥2 + 𝑦2)
(1)
where the phenomenological stiffness parameter of the potential was estimated as k=10-7 N/m. Note that this estimate was obtained from the fit of the experimental data and computational modeling of small droplet structure, rather than from oscillation frequencies of the droplets. Small horizontal oscillations (chattering) of the cluster with the amplitude smaller than droplet’s radius are inherent to the interactions, which are responsible for the generation of the cluster. These oscillations are observed even when strict measures to suppress vibration of the experimental equipment are undertaken. By studying the frequency spectrum of these oscillations one can determine the stiffness constant in Eq. 1 directly from the frequency data using the relationship 𝜔=
𝑘
(2)
𝑚
where m is the mass of the droplet. In the present paper, we determine the stiffness constant from observations of the oscillations of a small cluster. Experimental Small clusters with the number of droplets from one to four were generated using the twostage method and the experimental setup described in our earlier work3. A laser beam was used as a heat source (Fig. 1). At the first stage, the power of the laser source was maintained at a low level, and the clusters were generated. At the main (observation) stage, the relatively high constant power of the laser heat source was applied, corresponding to the temperature T=62±1 ºC at the water layer surface. This temperature was measured with a pyrometric sensor. Outside of the heated spot, the water temperature was maintained at T=10 ºC. Using an external infrared irradiation from above, the growth of the droplets was suppressed and their diameters were stabilized at d=54±5 m.
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50 mm
8
6
7 1
2
4
5 dw
ds
Figure 1. The experimental setup. Droplet cluster (1) above the water layer (2) in a metal cuvette with a cylindrical cavity (3) with a glass-ceramic substrate (4) of the thickness ds = 400±5 m coated by a light-absorbing graphite-reinforced thermo-resistant paint and attached by the epoxide glue (5). The laser beam (6) heats the substrate, which heats the water layer. A coolant is supplied through channel (7), the cuvette is covered with a plastic membrane (8) with a 10 mm hole at its center. A preliminary upper estimate of the frequency of mechanical oscillations of the clusters was given by substituting into Eq. 2 values of the estimated stiffness and minimum mass of the droplet, 𝜔𝑚𝑎𝑥 = 𝑘 𝑚𝑚𝑖𝑛 = 10 ―7 10 ―10 ≈ 32 s ―1. This provides the maximum time step in measuring cluster vibrations as ∆𝑡 = (2𝜔𝑚𝑎𝑥) ―1 = 0.156 s. The frequency of recording was equal to 100 frames per second to satisfy the maximum time step condition. The total of 12 records was obtained (three with each droplet number from one to four) with 4096 frames in each (Fig 2).
Fig. 2. Small droplet clusters are shown.
Small clusters are subject to oscillations, which have more or less constant amplitude and never cease even in the absence of external fluctuations. Apparently, these oscillations are caused by fluctuations of the vapor-air upward flow, in which clusters levitate. In small clusters, droplets are stiffly connected with each other, and the cluster oscillates as a whole (Fig. 3a, Supporting information). Since the cluster tends to vibrate as a whole, we study the oscillations of the ACS Paragon Plus Environment
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The Journal of Physical Chemistry
geometric center of the cluster, whose position is calculated by software for every frame. For the droplets with approximately equal diameters, the geometric center is close to the center of mass of the cluster. The trajectory of motion of clusters center is presented in Fig. 3(b). The highest density of the trajectory is at the center of the area, and the low density is at the periphery, which is typical of a random walk in a potential well14. Given the central symmetry of the heated spot and, only oscillations in the x-direction were studied (Fig. 3c).
a
b
c
Fig. 3. (a) A superposition of two frames showing the change in position of the cluster center. (b) An example of a trajectory of the center of a three-droplet cluster. (c) Time dependence of the x-coordinate of the cluster.
Results and discussion The aerodynamic interaction of the gas flow with the droplet cluster is quite complex. For two identical spherical particles or droplets fixed side by side in a uniform stream perpendicular to the line connecting their centers, computational analysis exists only for 10 Re 150 15-17, and it shows that the two spheres repel each other when the spacing is of the order of the diameter. Although these values of Re are one-two orders of magnitude larger than in our system, one can expect qualitatively similar behavior in the case of the droplet cluster. The repulsion is stronger at smaller spacing between the particles, while the particles can attract each other at the intermediate separation distances. The numerical results for Re 10 and 5017 were slightly corrected in recent paper18 and additional results for smaller distances between the particles showed a significant increase in the repelling force. The calculations in the Stokes regime showed that the particles weakly repeal each other at all separations17. Several potential causes of droplet oscillations are possible, including capillary vibrations, the interaction of a non-steady flow with the droplets, and oscillations of the flow. First, let us estimate Rayleigh’s natural frequencies of capillary vibrations of individual droplets, which are given by 𝜔𝑛 =
𝑛(𝑛 ― 1)(𝑛 + 2)𝛾 𝜌𝑅
3
𝐽
𝑘𝑔
, n=2, 3, 4…, where 𝛾 = 72 × 10 ―3𝑚2 and 𝜌 = 103 𝑚3 are the
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surface tension and density of water. Assuming 𝑅 = 10 ― 100 𝜇𝑚 yields 𝜔2≅1.7 × 104 ÷ 5 × 105𝑠 ―1, which is far from the observed oscillations occurring with the frequencies in the range of 1 Hz – 50 Hz. Another potential source of vibrations is the interaction of the non-steady vortex flow with the spherical droplets. Note that the Reynolds number can be estimated as 𝑅𝑒 =
2𝑅𝜌𝑔𝑎𝑠𝑉 𝜇
~0.1 ― 1,
where ≈2 × 10-6 Pa s-1 is the typical dynamic viscosity of the air-vapor mixture, gas ≈1 kg m-3 is its density, and V=0.1 m s-1 is the flow velocity. Such small Reynolds numbers are typical of the Stokes flow regime characterized by a linear momentum equation. The experimental values of the Strouhal number are estimated as 𝑆𝑡 =
2𝑅𝑓 𝑉
=
2 × 10 ―5m × 5s ―1 0.1 ms ―1
~0.1.
A recent study has shown that the growth of condensing droplets in the cluster could be stabilized by oscillating amplitude of the laser beam power output9. Qualitatively, one can argue that, based on the Le Chatelier principle, the oscillations of the droplets may be viewed as their reaction suppressing their condensational growth. The oscillations of the gas flow can be random, however the cluster responses at the resonance frequencies. The synchronous vibrations of droplets in the cluster requires an explanation. These may be caused by an almost flat pressure wave generated in the region downstream, which pushes all the droplets simultaneously. In our earlier work6, 7, we used the assumption that the repelling interaction between individual droplets in the gas flow can be decoupled from the action of the flow upon the individual droplet, which is characterized by a phenomenological quasi-elastic potential, e.g., given by Eq. 1. Comparing the oscillations of a single droplet with those of multi-droplet clusters one can estimate these two contributions of a force acting upon a droplet. Using the discrete fast Fourier transform algorithm in the form 𝑁―1
(
𝑋𝑗 = ∑𝑛 = 0 𝑥𝑛exp ―
2𝜋𝑖 𝑁
)
𝑗𝑛 ,
j=0, … N-1
(3)
we obtained amplitude-frequency diagrams of cluster centers oscillations. Fig. 4(a) shows a typical frequency spectrum (magnitude of the discrete fast Fourier transforms) of clusters’ vibrations, while Fig 4(b) shows a frequency spectrum of the water layer surface vibrations, measured with a laser confocal sensor. The water layer vibration characterizes the natural noise in the system where droplet cluster is generated. Comparing these two spectra indicates that the peaks in the cluster’s spectrum are not caused by the water layer vibration. Fig. 4(a) shows the frequencies of the oscillation whose amplitude is significantly higher than the noise are: f1=1.61 Hz, f2=1.99 Hz, f3=2.52 Hz, f4=3.28 Hz, f5=4.15 Hz, f6=5.96 Hz, ACS Paragon Plus Environment
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The Journal of Physical Chemistry
f7=33.00 Hz, and f8=49.00 Hz. All frequencies were found in every of four clusters with the exception of f7=33 Hz, which was not found in the cluster with one droplet (Fig. 4b). It may be speculated that the frequencies f7=33 Hz, and f8=49 Hz are artefacts of recording the cluster at 100 frames per second or that f7=33 Hz is related to the interaction between the droplets and thus absent from the cluster with only one droplet.
a
1
b
2
8
6
45 3
8
7
7
7
7
2
4 3
8
c
6
1
8
8
5 7
8
Fig. 4. Frequency spectrum of oscillations of (a) three-droplet cluster (one measurement) and of the water surface, using the signal of the confocal sensor. (b) Frequencies f7=33 Hz and f8=49 Hz in clusters with different number of droplets (c). The sum of frequency spectra for clusters
Another important observation is that in all series of experimental observations, the peaks of frequency spectra correspond to almost the same frequencies, irrespective of the number of ACS Paragon Plus Environment
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droplets in the cluster. Fig. 4(c) shows the sum of all 12 spectra having characteristic harmonics at f1=1.61 Hz, f2=1.99 Hz, f3=2.52 Hz, f4=3.28 Hz, f5=4.15 Hz, and f6=5.96 Hz. For known natural frequencies and mass of the vibrating particles (droplets), the shape of the potential can be reconstructed as 6
𝑊 = 2𝜋2𝑚𝑥2∑𝑖 = 1𝑓2𝑖
(4)
For a spherical water droplet with the diameter d=54 m, the mass is m=7.8·10-11 kg. Corresponding effective potential is shown in Fig. 5, along with the potential from Ref. 7 obtained from modeling the structure of small clusters.
Fig. 5. Comparison of potentials calculated: (1) – from Eq. 4 and (2) calculated from Ref [7].
It is observed, that the potentials obtained using two different methods are close to each other, in particular, in the area of the droplet cluster localization (|x|
