Small Levitating Ordered Droplet Clusters: Stability, Symmetry, and

Letter Cite This: J. Phys. Chem. Lett. 2017, 8, 5599-5602

pubs.acs.org/JPCL

Small Levitating Ordered Droplet Clusters: Stability, Symmetry, and Voronoi Entropy Alexander A. Fedorets,† Mark Frenkel,‡ Edward Bormashenko,‡ and Michael Nosonovsky*,†,§ †

University of Tyumen, 6 Volodarskogo St., Tyumen, 625003, Russia Department of Chemical Engineering, Biotechnology and Materials, Engineering Sciences Faculty, Ariel University, Ariel, Israel 40700 § Mechanical Engineering, University of WisconsinMilwaukee, 3200 North Cramer Street, Milwaukee, Wisconsin 53211, United States ‡

S Supporting Information *

ABSTRACT: A method to generate levitating monodisperse microdroplet clusters with an arbitrary number of identical droplets is presented. Clusters with 1−28 droplets levitate over a locally heated water layer in an ascending vapor-air jet. Due to the attraction to the center of the heated area combined with aerodynamic repulsion between the droplets, the clusters form structures that are quite diverse and different from densest packing of hard spheres. The clusters self-organize into stable and reproducible configurations dependent on the number of droplets while independent of the droplets’ size. The central parts of larger clusters reproduce the shape of smaller clusters. The ability to synthesize stable clusters with a given number of droplets is important for tracing droplets, which is crucial for potential applications such as microreactors and for chemical analysis of small volumes of liquid.

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he emergence of new technology, such as biomicro/ nanoelectromechanical systems (bioMEMS/NEMS), microfluidic, and lab-on-a-chip devices, drives miniaturization and requires new methods to trace and manipulate small objects analyzing small volumes of substances. One particularly important area is microdroplet manipulation. Microdroplets can serve as microreactors in devices and systems of rapid in situ liquid analysis and sensing. Various new techniques to trace microdroplets and manipulate them have been proposed recently, including droplet levitation, droplet/bubble-based logical operations,1 etc. One particularly promising area of research is ordered droplet clusters, a phenomenon discovered in 2004 by Fedorets,2−10 who showed that a self-assembled monolayer of hexagonally ordered microdroplets emerges over a locally heated (typically, at 60−95 °C) surface of water. Growing condensing droplets with a typical diameter of 10−200 μm are supported by a vapor-air jet

Figure 2. Condensation rate of the droplet surface area vs the number of droplets.

rising over the heated spot. The height of levitation and the distance between the droplets are of the same order as their diameters. Due to the complex nature of aerodynamic forces between the microdroplets in an ascending jet, the droplets do not coalesce but form hexagonal structures corresponding to the dense packed location of hard spheres and showing similarity with various classical and newly discovered objects, where self-organization Received: October 8, 2017 Accepted: October 31, 2017 Published: October 31, 2017

Figure 1. Notation of the cluster structure. Two clusters are shown (N11 and N16) with central groups G3 and G5 and nuclei S11 and S16. © XXXX American Chemical Society

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DOI: 10.1021/acs.jpclett.7b02657 J. Phys. Chem. Lett. 2017, 8, 5599−5602

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The Journal of Physical Chemistry Letters

because it is easier to trace individual droplets in them, which is crucial for potential applications. Unlike the clusters with a large number of droplets, small clusters cannot always form a hexagonally symmetric structure. Instead, they produce various more or less symmetric configurations depending on the number of droplets. The symmetry, orderliness, and stability of these configurations can be studied with such a measure of selforganization as the Voronoi entropy.9−12 Small clusters with the number of droplets from 1 to 28 were generated using the novel method described below. The generated clusters had a reproducible structure, which in some cases possessed symmetry. This structure depends only on the number of droplets in the cluster, and it is independent of the droplet diameter or distance between the droplets (see the Supporting Information). We use the following notation to describe the structure of the clusters: N is the number of droplets, G is the central group of droplets, and S is the nucleus including the central group and the adjacent droplet layer. For example, we call G3S11N11 and G5S16N16 the two clusters, of 11 and 16 droplets, shown in Figure 1. All clusters observed in the experiment had the central group belonging to one of five types (Figure S1). Clusters from N1 to N5 consisted only of the central groups. Clusters from N6 to N16 consisted of nuclei (Figure S2). Starting from N17, the nuclei reproduce themselves from S6 to S12 in a consistent manner. For example, following the cluster with the S6 nucleus, one with the S7 nucleus is formed, and so on (Figure S3). However, the nuclei S13, S14, S15, and S16 never form in the clusters from N17 to N28. The data in Figures S1−S3 include Voronoi entropy, the group of symmetry, and the number of polygons in the Voronoi diagrams. The central group clusters from N1 to N5 arrange themselves in the shape of a polygon, with the corresponding number of edges. The distance between the droplets tends to remain constant, thus forming regular (equilateral) polygons. When the sixth droplet is added, the cluster does not form a hexagon. Instead, one droplet places itself at the center of the cluster with five others surrounding it, forming the G1S6N6 nucleus structure. When a seventh and an eighth droplet are added, they are packed in the outer row of the cluster, which therefore forms a hexagon G1S7N7 and heptagon G1S8N8. However, as soon as a ninth droplet appears, it fills the vacancy at the center G2S9N9 similar to that of the six-droplet cluster G1S6N6. The further growth of the cluster is described in Figure S3. The Voronoi entropy was at a minimum for symmetric clusters (G1S6N6, G1S7N7, G1S8N8, G3S12N12, G4S14N14, G1S7N19, G2S10N24), as shown in Figure S4. The highest value of the Voronoi entropy was for G5S15N15. One can hypothesize that symmetric cluster structures minimize the energy of the cluster, associated with the effective interaction potential between the droplets. The effective potential is due to complex aerodynamic forces acting between the droplets in the ascending vapor-air jet.3,9 The independency of the small cluster structure of the droplet diameters can provide insights on the scaling of the interdroplet forces. Alternatively, one can suggest that the symmetric structures minimize viscous energy dissipation rates in a similar manner to friction-induced alignment of particles on rotating spinners13 and similar selfsynchronization effects. As the number of droplets in the cluster increased, their condensation rate decreased by more than two times (Figure 2). This may be explained by shifting the vaporization−condensation

Figure 3. (a) Experimental setup. Droplet cluster (1) above the water layer (2) in a metal cuvette with a cylindrical cavity (3) with a glassceramic substrate (4) of thickness ds = 400 ± 5 μm coated by a lightabsorbing graphite-reinforced thermoresistant paint and attached by epoxide glue (5). The laser beam (6) heats the substrate, which heats the water layer. A coolant is supplied through channel (7), and the cuvette is covered with a plastic membrane (8) with a 10 mm hole at its center. (b) Temperature distribution during the initial assembly stage of the cluster and during the observation stage.

is prominent, including water breath figures, colloid and dust crystals, foams, and, to some extent, ice crystals. The droplets tend to pack near the center of the heated area where the temperature and the intensity of the ascending vapor jets are the highest.3−5 At the same time, there are repulsion forces of aerodynamic nature between the droplets. Consequently, the cluster packs itself in the mostly hexagonal structure with a certain distance between the droplets dependent on the repulsion forces.9 By controlling the temperature and temperature gradient, one can control the number of droplets and their density and size.4 Using infrared irradiation, it is possible to suppress droplet growth and stabilize them for extended periods of time.6 It has been suggested that the phenomenon, when combined with a spectrographic study of droplet content, can be used for rapid biochemical in situ analysis.8 Recent studies have shown that the cluster can exist at lower temperatures of about 20 °C, which makes it suitable for biochemical analysis of living objects.10 Previous studies have concentrated on clusters with a large number of droplets. In the present Letter, we demonstrate that clusters with an arbitrary small number of droplets also form ordered structures of interest. Small clusters are important 5600

DOI: 10.1021/acs.jpclett.7b02657 J. Phys. Chem. Lett. 2017, 8, 5599−5602

Letter

The Journal of Physical Chemistry Letters

Figure 4. Assembly stage of the cluster and the observation stage. The two frames show the same 27-droplet cluster separated by 20 s. The droplet diameter grew by 20%; however, the droplet configuration remained the same.

0.05 K, and the pixel size was 50 μm). The temperature distribution was axially symmetric (Figure 3b). To control the optical power of the laser beam, PL, the equipment (Thorlabs) was used including the control block PM200 and power sensor S401C (the spectral diapason from 0.19 to 10.6 μm, measured power from 10 μWt to 1 Wt with ±5% accuracy). The thickness of the water layer was controlled by a laser triangulation sensor Riftec RF603-15/2 with an accuracy of ±2 μm. The experiments were conducted with distilled water. Small amounts of natural surfactants present in water suppressed the thermocapillary flow. A constant thickness of the water layer of dw = 200 ± 5 μm was maintained throughout all experiments. The water temperature outside of the heating zone far away from the cluster was maintained at 3.8 ± 0.3 °C. A microscope (ZeissAxio Zoom.V16) with a camera (pco.edge 5.5) was used to record videos of the clusters. Small Cluster Generation. Previously used methods for droplet cluster generation resulted in continuous addition of new droplets to the cluster.2−9 Furthermore, the droplets obtained by these methods were polydisperse, i.e., they could vary in size. To generate clusters with a constant number of almost identical monodisperse droplets (i.e., with very close diameters), the process was divided into two stages. At the initial assembly stage, the power output of the laser was maintained at the small level

balance in larger clusters as a result of decreasing vapor saturation near the droplet in larger clusters. In addition, the increased aerodynamic resistance of the cluster can also be partially responsible. Note that the cluster structures are quite diverse, and they are different from those corresponding to the densely packed locations of hard spheres. The ability to synthesize monodisperse stable clusters with a given number of droplets employing a purely macroscopic method is likely to be important for future applications. In particular, the ability to trace droplets is crucial for using them as microreactors and for chemical analysis of small volumes of liquid.

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EXPERIMENTAL METHODS Experimental Equipment and Setup. The experimental setup includes a cuvette with a submillimeter-thick layer of water heated locally by laser irradiation from the bottom to reach a water surface temperature of 50−70 °C, Figure 3a. A semiconductor laser BrixX 808-800HP (by Omicron Laserage, wavelength 0.808 μm) was used as a source of heating. The laser beam was focused on the substrate in a round spot with the diameter of 1 mm. The temperature field on the water surface was registered by a thermographic camera FLIR A655sc with a camera lens Close-up IR 2.9× (the spectral diapason was from 7.5 to 14 μm, the temperature resolution was 5601

DOI: 10.1021/acs.jpclett.7b02657 J. Phys. Chem. Lett. 2017, 8, 5599−5602

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The Journal of Physical Chemistry Letters

(10) Fedorets, A. A.; Dombrovsky, L. A.; Ryumin, P. I. Expanding the Temperature Range for Generation of Droplet Clusters over the Locally Heated Water Surface. Int. J. Heat Mass Transfer 2017, 113, 1054−1058. (11) Barthélemy, M. Spatial Networks. Phys. Rep. 2011, 499, 1−101. (12) Senthil Kumar, V.; Kumaran, V. Voronoi Cell Volume Distribution and Configurational Entropy of Hard Spheres. J. Chem. Phys. 2005, 123, 114501. (13) Kazachkov, A.; Multanen, V.; Danchuk, V.; Frenkel, M.; Bormashenko, E. Friction, Free Axes of Rotation and Entropy. Entropy 2017, 19, 123.

of PL1 = 94 mW, which allowed the droplets to migrate toward the center of the heated spot. After the required number of droplets was obtained, the power output was abruptly increased to PL2 = 235 mW, leading to the increased speed of the ascending vapor-air jet (Figure 4). The droplets generated in the heating zone at the initial stage were large enough to levitate, while newly created small droplets were flown away by the strong vapor-air jet. Consequently, the cluster contained only those droplets, which were generated at the initial stage. Because the cluster assembly stage was short, the diameter of the droplets was uniform.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b02657. Photos of clusters of 1−28 droplets (nuclei, central groups, and larger clusters), their Voronoi entropy, and the group symmetry (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +1-414-229-2816. ORCID

Edward Bormashenko: 0000-0003-1356-2486 Michael Nosonovsky: 0000-0003-0980-3670 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge helpful discussions with Dr. Leonid Dombrovsky from Moscow Joint Institute of High Temperatures. The authors are grateful to the Ministry of Education and Science of the Russian Federation (Project No. 3.8191.2017/ BCh) for partial financial support of the present study. M.F. acknowledges partial support from the Israel Ministry of Immigrant Absorption.

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REFERENCES

(1) Epstein, I. R. Can Droplets and Bubbles Think? Science 2007, 315, 775−776. (2) Fedorets, A. A. Droplet Cluster. JETP Lett. 2004, 79, 372−374. (3) Fedorets, A. A. On the Mechanism of Non-coalescence in a Droplet Cluster. JETP Lett. 2005, 81, 437−441. (4) Fedorets, A. A. Mechanism of Stabilization of Location of a Droplet Cluster Above the Liquid−gas Interface. Tech. Phys. Lett. 2012, 38, 988−990. (5) Fedorets, A. A.; Dombrovsky, L. A. Generation of Levitating Droplet Clusters Above the Locally Heated Water Surface: A Thermal Analysis of Modified Installation. Int. J. Heat Mass Transfer 2017, 104, 1268−1274. (6) Dombrovsky, L. A.; Fedorets, A. A.; Medvedev, D. N. The use of Infrared Irradiation to Stabilize Levitating Clusters of Water Droplets. Infrared Phys. Technol. 2016, 75, 124−132. (7) Fedorets, A. A.; Dombrovsky, L. A.; Smirnov, A. M. The use of Infrared Self-emission Measurements to Retrieve Surface Temperature of Levitating Water Droplets. Infrared Phys. Technol. 2015, 69, 238− 243. (8) Fedorets, A. A. Application of a Droplet Cluster to Visualize Microscale Gas and Liquid Flows. Fluid Dyn. 2008, 43, 923−926. (9) Fedorets, A. A.; Frenkel, M.; Shulzinger, E.; Dombrovsky, L. A.; Bormashenko, E.; Nosonovsky, M. Self-assembled Levitating Clusters of Water Droplets: Pattern-formation and Stability. Sci. Rep. 2017, 7, 1888. 5602

DOI: 10.1021/acs.jpclett.7b02657 J. Phys. Chem. Lett. 2017, 8, 5599−5602