Self-Arranged Levitating Droplet Clusters: A Reversible Transitionfrom Hexagonal to Chain StructureAlexander A. Fedorets,† Mark Frenkel,‡ Irina Legchenkova,‡ Dmitry V. Shcherbakov,†

Leonid A. Dombrovsky,†,§ Michael Nosonovsky,*,†,∥ and Edward Bormashenko‡

†University of Tyumen, 6 Volodarskogo St., Tyumen 625003, Russia‡Department of Chemical Engineering, Biotechnology and Materials, Engineering Sciences Faculty, Ariel University, Ariel 407000,Israel§Joint Institute for High Temperatures, 17A Krasnokazarmennaya St., Moscow 111116, Russia∥Mechanical Engineering, University of WisconsinMilwaukee, 3200 North Cramer St., Milwaukee, Wisconsin 53211, UnitedStates

*S Supporting Information

ABSTRACT: Water microdroplets condense over locally heated water-vapor interfacesand levitate in an ascending vapor-air flow forming self-assembled ordered monolayerclusters. The droplets do not coalesce due to complex aerodynamic interactions betweenthem. The droplet cluster formation is governed by the condensation/evaporation balanceand by coupling of heat flux and vapor flow with aerodynamic forces. Here, we report theobservations of a reversible structural transition from the ordered hexagonal-structurecluster to the chain-like structure and provide an explanation of its mechanism andconditions under which the transition occurs. The phenomenon provides new insights onthe fundamental physical and chemical processes with microdroplets including their rolein reaction catalysis in nature and their potential for aerosol and microfluidic applications.

■ INTRODUCTION

Self-assembled microdroplet clusters levitating over locallyheated water-vapor interfaces have been already reported inthe literature.1−4 Theoretical analysis and accurate predictionof physical behavior of cluster-like systems consisting of a smallnumber of particles that interact with one another are quitechallenging because they constitute many-body physicalsystems in which accurate implementation of mechanical andstatistical methods is difficult.5 Such clusters appear in colloidalsystems6,7 including nematic colloid crystals,8 dusty plas-mas,9,10 sedimentation,11,12 and combustion.13 Clusters arebuilt from rigid particles6−8,12,13 and droplets1−4,14 anddemonstrate ordered1−10 and random15 structures. Verydifferent physical interactions may be responsible forclustering. Coulomb (and screened Coulomb) interactionsare responsible for clustering in colloidal systems7,16 and dustyplasmas.9,10 Dipole−dipole interactions are responsible forclustering of floating particles.17 Capillary18 and hydro-dynamic19,20 interactions give rise to clustering of colloidalparticles and microdroplets. The interaction between levitatingdroplets was also studied for the inverse Leidenfrost state.21,22

Clusters are more than just crude models for atomicsystems;23 they are themselves a form of matter withinteresting collective behavior not seen at the atomic andmacroscopic scales.16 A particularly interesting kind of clusteris the “chain cluster”.7,24,25

When a spot on the surface of a thin (about 0.5 mm) waterlayer is heated by a laser beam or another source of heat totemperatures of 60−95 °C, microdroplets are condensed abovesuch a spot, and they can form a hexagonally orderedmonolayer referred to as the “droplet cluster”. The heatingresults in an air-vapor flow rising above the heated spot. Due tothe temperature gradient, spherical droplets with typicaldiameters between 0.01 and 0.05 mm condense in theascending air-vapor flow. The vertical component of the dragforce in the upward flow is equilibrated by the droplet’s weightat a certain equilibrium levitation height usually comparablewith the droplet’s radius. While the droplets are draggedtoward the center of the heated spot, they do not merge witheach other due to an aerodynamic repulsive pressure forcefrom the gas flow between the droplets. The repulsion forcebetween the droplets is balanced by the horizontal componentof the drag force, which drives the droplets in the directiontoward the center of the heated spot where the temperatureand the intensity of the ascending flow are the highest. Forlarge clusters, this may result to a hexagonal arrangementsimilar to packing of identical rigid balls at the bottom of a

Received: October 8, 2019Revised: October 28, 2019Published: October 30, 2019

Article

pubs.acs.org/LangmuirCite This: Langmuir 2019, 35, 15330−15334

© 2019 American Chemical Society 15330 DOI: 10.1021/acs.langmuir.9b03135Langmuir 2019, 35, 15330−15334

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bowl,1,2,4,26 while the structure of small clusters may be quitediverse.3

The growth of the droplets can be controlled by infraredirradiation, which shifts the condensation−evaporation bal-ance, while the number of droplets in the cluster and their sizecan be controlled by other methods.3,4

■ EXPERIMENTAL SECTIONThe experimental procedure described in our previous publicationswas used in the present study.3 The droplet cluster was self-assembledover the locally heated surface of a thin layer of distilled water. Thethickness of the water layer (400 ± 2 μm) was controlled using theconfocal chromatic sensor IFC2451 made by the company Micro-Epsilon (USA). The cuvette temperature was equal to 10 ± 0.5 °C.The water layer was placed on a flat sitall substrate of 400 μmthickness. The lower surface of this substrate was blackened to absorbthe visible radiation at the 808 ± 5 nm wavelength emitted by thelaser BrixX 808-800HP (Omicron Laserage, Germany). The diameterof the laser beam at the substrate surface was equal to 1 mm. The laserpower was controlled by a PM200 device equipped with an S401csensor (Thorlabs, USA). Video images of the cluster were taken usinga Zeiss AXIO Zoom.V16 stereomicroscope (Germany) and high-speed (1000 frames per second) PCO.EDGE 5.5C video camera(Germany) with a spatial resolution of 0.6 μm. The region of thedroplet cluster is schematically presented in Figure 1 where the size ofdroplets and distances between them are specified.

Since the number of droplets in the cluster determines thepossibility of the transition, the clusters containing different numbersof almost identical monodisperse droplets were generated with the useof a specific two-stage procedure developed elsewhere.3 At the firststage of this procedure, the water layer was heated up to temperatureT1 so that the growing small droplets migrated toward the centerregion of the cluster. After the accumulation of the required numberof droplets, the laser heating power was increased sharply, and thesurface temperature of the water layer under the cluster increased toT2 (10−20° above T1). At this stage, the small droplets located on theperiphery of the cluster were carried away by the stronger vapor-airjet, in contrast to the relatively heavy droplets at the center of thecluster generated at the first stage of the process, and the cluster’sgrowth decelerated due to the lack of arrival of new droplets.

■ RESULTS AND DISCUSSIONThe hexagonally ordered droplet clusters have been observedfor the first time in 2004.1 Recently, it has been found that achain-like structure of a cluster can form under certaincircumstances following a transition from the hexagonallyordered state.4 In the chain state, droplets almost touch eachother and form linear chains in the central part of the cluster(Figure 2). Dozens of reproducible experiments have beenperformed with the chain cluster, and a methodology wasdeveloped to generate the chain cluster and observe thetransition. The lifetime of the chain cluster is about 20−30 son average. After that, the cluster collapses due to the

coalescence of growing condensing droplets with the waterlayer, which is typical also for the hexagonally ordered clusters.The transition to the chain cluster is abrupt, and it is

observed at a certain critical ratio of droplet size and distancebetween their centers or, in other words, packing density. Themaximum size of the cluster depends on the area of the locallyheated spot, which was constant in our experiments; thenumber of droplets in the cluster depends on their size. If alarge cluster was assembled at the initial stage, the criticalpacking density is achieved for relatively small-sized droplets,as a result of their condensational growth. However, if a smallcluster was assembled, they can stretch during the condensa-tional growth, and the critical density is achieved for relativelylarger droplets.The transition is also reversible since the cluster can turn

back to the hexagonal structure by turning on infrared heating,which suppresses the droplet growth due to condensation ofwater vapor.Two types of cluster evolution leading to the hexagonally

ordered (A) and chain cluster (B) are shown in Figure 2.Cluster A had only about 70 droplets, while cluster B hadabout 200 droplets. This size difference and packing densitydifference resulted in different evolutions during the condensa-tional growth of water droplets. While cluster A consisted ofindividual droplets tending to form a hexagonal structureduring the entire time of its existence, the structure of cluster Bchanged dramatically over time with the formation of chains ofdroplets almost touching each other (Movie S1). The latteroccurs as soon as water droplets in the central part (core) ofthe cluster exceed a certain critical diameter.While the hexagonally cluster is more ordered than the chain

cluster in terms of its Voronoi entropy, the chain cluster is alsonot completely disordered or chaotic. The transition betweenthe two states is analogous to the second-order phasetransition between ordered and disordered states in dustyplasma,9−11 colloidal crystals, microfluidic dipoles,5 andpolymer-like droplet structures;25 however, there is asignificant difference. The droplet cluster is a dissipativestructure, and unlike the colloidal crystals, it does not exist atphase equilibrium.Note that the Reynolds number can be estimated by

= ∼ −ρ

μRe 0.1 1

R V2 gas , where μ ≈ 2 × 10−6 Pa s−1 is the

typical dynamic viscosity of the air-vapor mixture, ρgas ≈ 1 kgm−3 is its density, and V = 0.1 m s−1 is the flow velocity. Suchsmall Reynolds numbers are typical for the Stokes flowconditions.27 The noncoalescence of water droplets constitut-ing the chains in Figure 2c,d is similar to the noncoalescence ofmiscible liquids reported in the literature,14,28 which is usuallyexplained by the air layer, separating droplets,28 or by theMarangoni instabilities at the boundary separating thedroplets.29 The characteristic time required for a droplet to

attain thermal equilibrium is estimated by τ ≅ ≅α

0.1 sR2

(where R = 0.1 mm is the typical radius of a droplet and

α ≅ × −1.4 10 7 ms

2

is the thermal diffusivity of water). This

time is much smaller than the characteristic time of the stablelife of “chains” (on the order of dozens of seconds). Therefore,the Marangoni flow is hardly responsible for the non-coalescence of droplets, and it is reasonable to relate theeffect to the vapor layer separating the droplets.29

Figure 1. Schematic side view of a droplet cluster experimental setup.

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Experimental results are summarized as a phase diagramfunction of droplet diameters and distances in Figure 3a. Thephase diagram line was obtained using 8 different chainclusters and about 20 clusters, for which the transition did notoccur. Experimental points in the figure correspond to differentclusters. There are two examples of cluster evolution: cluster B,which became a chain cluster, and cluster A, which neverbecame a chain cluster and coalesced. Only one typical curve isshown in the diagram. The values of the average dropletdiameter Dav and average distances between droplet centers Lavand between neighboring droplets lav were measured for 19droplets in the cluster core to characterize the cluster structureup to the chain-like structure formation.The evolution of two sample clusters is shown in Figure

3a,b, with cluster A retaining its hexagonal structure and clusterB undergoing the transition to the chain-like structure. Bothclusters A and B were generated at almost the sameexperimental conditions. However, cluster A was relativelysmall, and the distance between droplets of this clusterincreased with growing droplets. On the contrary, the core of alarger cluster B was sandwiched by the outer layers of droplets.As a result, the distance between the neighboring droplets incluster B decreased with increasing droplet size (Figure 3b).Consequently, the structure of cluster B changes upon reachingthe critical value of ratio lav/Dav.The transition is reversible, and if the droplets are forced to

decrease, the chains again disassemble into separate droplets.By turning on the infrared heating at t = 15 s, the condensationgrowth of the droplets was suppressed30 so that the size of thedroplets started decreasing and the cluster returned to the

hexagonal arrangement. The time dependencies of both theaverage diameter of droplets and the average length of dropletchains in the cluster are presented in Figure 3c,d.The hexagonal-to-chain structural transition depends mainly

on the number of droplets in the cluster or on its size. Thecluster levitates at a small height above the water layer, and theascending gas stream passes between the central waterdroplets. When there are few droplets in a cluster, the coreof the cluster is not constrained by the outer layers of dropletsand the growing droplets have the opportunity to move apart.In a large cluster, droplets in the central are constrained by theouter droplets and the distance between them decreases withthe growing diameter of the droplets. The resistance of thecluster to the upward gas stream increases, which causesstructure reorganization. Instead of uniformly distributeddroplets of the hexagonal cluster, chains of droplets withrelatively wide gaps between them emerge.The laboratory observations of small particles of dust on the

surface of the water layer showed that the dust particles swimfreely under relatively small droplets. However, when thedroplets grow above a certain critical diameter, a vortex flowdevelops in the space between the water layer and the cluster.The vortex is observed by the motion of dust particles, whichbegin to move along closed trajectories clearly distinguishablein the video given in Movie S2.The Fourier analysis of the spectra of kink angles of the

droplet chains did not reveal any features of the distribution ofthese angles, which turned out to be almost chaotic. Thus, ourchain clusters may be represented by the free-joint model ofthe polymer (oligomer) chain, which shows the random walk

Figure 2. Evolution of clusters A (a,b) and B (c,d) with time: (a,c) initial structures; (b,d) resulting structures.

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dependency of the average end-to-end distance: ⟨r2⟩ = ND2,where N is the number of monomers (droplets) and D is theirdiameter.29

The orderliness of droplet clusters is quantified by theVoronoi entropy of the cluster images.2,3 The Voronoidiagrams of droplet cluster before and after its transformationinto the cluster comprising chains are shown in Figure 4. Theexpectable jump in the Voronoi entropy, corresponding toformation of pseudo-oligomer chains, is clearly recognized.

■ CONCLUSIONS

To summarize, we reported the observations of a reversiblestructural transition in a self-assembled droplet clusterlevitating over the locally heated water layer from the ordinaryhexagonal to a chain structure. The conditions of such atransition are explained by a qualitative physical modelsuggested in the paper. The reported droplet cluster enabledin situ control of its parameters resulting in the phasetransition.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.lang-muir.9b03135.

Video S1. Transition from the hexagonal to the chainstructure of the droplet cluster (AVI)Video S2. Observation of vortices below the dropletcluster (MP4)

■ AUTHOR INFORMATIONCorresponding Author*E-mail: nosonovs@uwm.edu.ORCIDAlexander A. Fedorets: 0000-0001-6595-3927Michael Nosonovsky: 0000-0003-0980-3670Edward Bormashenko: 0000-0003-1356-2486Author ContributionsA.A.F. conceived the research and conducted experiments.M.F., I.L., and E.B. calculated Voronoi entropy. D.V.S.designed the software for image analysis. L.A.D., M.N., andE.B. provided theoretical analysis and wrote the manuscript.All authors read the manuscript.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThe authors gratefully acknowledge the Russian ScienceFoundation (project 19-19-00076) for the financial supportof this work.

Figure 3. (a) Phase diagram of transition from hexagonal to chain-likestructure of a cluster: 1 − the transition boundary (data pointscorrespond to different clusters); 2 − example of a phase trajectory(cluster B); 3 − example of a phase trajectory (cluster A). (b) Relativedistance between the centers of neighboring droplets as a function ofincreasing droplet diameter (superscript 0 corresponds to the initialtime moment). Evolution of a cluster under the external infraredheating: variation of (c) average diameter of droplets and (d) averagelength of droplet chains.

Figure 4. Voronoi tessellations and temporal evolution of the Voronoientropy during the cluster transformation into the chain structure.The Voronoi diagrams of the central area of the cluster (a) before and(b) after the transition. (c) The temporal evolution of the Voronoientropy; blue dashed line corresponds to panel (a); red dashed linecorresponds to panel (b). Colors in panel (b) correspond to thehexagons (gray), pentagons (yellow), heptagons (blue), and octagons(brown).

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