Letter Cite This: J. Phys. Chem. Lett. 2018, 9, 4771−4775
pubs.acs.org/JPCL
Vapor-Induced Attraction of Floating Droplets Dongdong Liu and Tuan Tran* School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798 Singapore
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S Supporting Information *
ABSTRACT: Droplets partially submersed in an immiscible liquid pool attract each other. We experimentally study the interaction of floating droplets containing aqueous solution of a volatile liquid. A droplet affects its neighbors by generating a vapor gradient to the surroundings and responds by evaporating asymmetrically over its exposed cap. We show that the induced asymmetric temperature distribution causes a surface tension gradient driving the attracting motion. We highlight that the attracting dynamics starts with an accelerating stage, followed by a decelerating stage. We finally provide a theoretical model that quantitatively captures the interactive forces between droplets and predicts essential features of the attracting motion.
bjects floating on a liquid surface flock together, a phenomenon commonly referred to as the Cheerios effect.1,2 This attractive capillary interaction, which occurs when neighboring objects mutually break symmetrical deformation of the liquid surface supporting their weights, belongs to a class of long-range interactions3 between elements of complex systems.4 Other effects causing autonomous motion of the elements and thus their collective dynamics include chemical potential gradient,5−8 inhomogeneous depletion during solubilization,9,10 and asymmetric evaporation.11,12 Because these effects often result in striking analogous patterns of participating elements, they have been exploited to control aggregate of the elements in applications such as self-assembly and self-patterning of materials.13 Thus beside the apparent interest in fundamental study of such complex systems and their collective dynamics, research efforts have been focusing on exploring novel mechanisms causing interactions and subsequent autonomous motions of the elements. In this Letter, we report the attractive interaction of twocomponent droplets partially submerged in a silicone oil pool (Figure 1a). The attractive interaction is induced without any surfactant, chemical reaction, or external field. The two components are water and a volatile fluid such as isopropyl alcohol (IPA), ethanol, and acetone. This phenomenon is robust for wide ranges of involving parameters such as weight ratio of volatile fluid (from 40 to 80%), droplet diameter (from 0.5 to 5 mm), and oil viscosity (from 5 to 150 cSt). To reveal the mechanism of the attractive interaction, we investigate two floating droplets of an aqueous solution of a volatile liquid (50% IPA with density 798 kg·m−3) and the same radius (Rs = 1.0 mm). The droplets are gently deposited on a 100 cSt silicone oil pool 2 cm deep at an initial separating distance 6.5 mm. The oil density is ρo = 960 kg·m−3. Each droplet almost submerges completely in the oil pool (Figure
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1b), leaving only a small cap exposed to air. We observe that the droplets start moving toward each other shortly after being deposited on the pool (Figure 1c). We quantify the approaching motion by measuring the edge-to-edge distance L between the two droplets and show the time dependence of L in Figure 1d. We find that the droplets first accelerate toward each other and then decelerate as their separating distance crosses a critical value L*. Although L* varies with experimental conditions, we always observe this behavior regardless of the droplet size, weight ratio of IPA, and oil viscosity. We postulate that the approaching motion originates from evaporation-induced thermocapillary flows in the droplets and the pool. We use an infrared camera to measure the temperature distribution of the droplet caps and the surrounding oil surface. For each droplet, the temperature measurement reveals a significant temperature difference (∼2 °C) between the cap and the pool’s far field (∼26.2 °C) (Figure 2a,b), suggesting that the latent heat of evaporation of IPA at the cap is supplied by the surrounding oil. We also note that across the cap there is a measurably small temperature variation (∼0.5 °C) that changes its characteristic with L. In particular, when L > L*, the temperature at the perimeter of a droplet’s cap is lower at the side far from the other droplet (Figure 2a), whereas when L < L*, the temperature difference between two sides of the perimeter becomes insignificant (Figure 2b). The asymmetric temperature distributions across the caps induce asymmetric flows in the droplets and the oil pool. In Figure 3, we show the flow fields inside the droplets as they approach each other. When L > L*, we observe that each Received: June 6, 2018 Accepted: August 3, 2018 Published: August 3, 2018 4771
DOI: 10.1021/acs.jpclett.8b01742 J. Phys. Chem. Lett. 2018, 9, 4771−4775
Letter
The Journal of Physical Chemistry Letters
Figure 1. (a) Top-view time lapse of 22 droplets deposited on an oil pool showing the approaching motions of all droplets and their coalescence. Droplets of different weight ratios of IPA are marked by dyes of different colors: 40% IPA (blue), 60% (red), and 80% (yellow). (b) Side-view snapshot of a droplet of 50% IPA. (c) Bottom-view time lapse of two droplets (50% IPA) approaching one another. The initial positions of the droplets (at t = 0 s) are marked by two black dots. The time interval between two consecutive positions is 30 s. (d) Time dependence of the distance L between two droplets (50% of IPA). The error bars of the measurement are smaller than the marker size. The critical distance L* separates two approaching dynamics: accelerating (v increases) for L > L* and decelerating (v decreases) for L < L*. The scaling bars represent 1 mm.
Figure 2. (a) Representative temperature distribution of the caps of two approaching droplets and the surrounding oil when (L > L*). (b) Representative temperature distribution when (L < L*), showing that their thermally affected regions overlap. The uncertainty of the temperature measurement by infrared camera is ±0.5 K. The dashed circles indicate the perimeters of droplet caps.
droplet has two flows circulating in opposite directions, as indicated by the arrows shown in Figure 3a; typically the intensity of the circulating flow far from the other droplet is higher. When L ≈ L*, the intensities of the two circulating flows switch sides; that is, the one far from the other droplet becomes weaker (Figure 3b). Finally, when L < L*, both circulating flows appear significantly weakened (Figure 3 c). These observations indicate that flow asymmetry inside the droplets plays a key role in understanding the approaching mechanism and the change in approaching dynamics. The observed approaching dynamics of two droplets therefore is caused by two crucial factors. First, the force driving a droplet’s motion originates from asymmetrical evaporation rate across its cap due to the vapor concentration gradient generated by the other one. Second, the change in the approaching dynamics of each droplet, from accelerating to decelerating, is caused by the decrease in energy supply from the surrounding oil to the droplet. In other words, the energy supply for each droplet is hindered by the presence of the other one, and when the energy reduction becomes significant,
the approaching velocity v = dL/dt shifts from increasing to decreasing. To capture the approaching motion of the droplets using a simple analysis, we consider a model system of two initially identical droplets: One is immobile and the other one mobile (Figure 4a). Both droplets have the same cap radius Rc ≈ (0.58 + 0.09c)Rs and droplet radius Rs, which is considered constant during the approaching time (Supporting Information). We also assume that the immobile droplet is unaffected by evaporation of the mobile one. The evaporation rate per unit area of IPA at the cap of the immobile droplet is A ṁ p ≈ cρl Ṙ s Ac (1) where c is the weight ratio of IPA, ρl is the density of IPA, As = 4πR2s is the total surface area of the droplet, Ac is the area of the 4772
DOI: 10.1021/acs.jpclett.8b01742 J. Phys. Chem. Lett. 2018, 9, 4771−4775
Letter
The Journal of Physical Chemistry Letters
oil flow outside the immobile droplet.14 Here Re = 2upRs/ν is the Reynolds number, Pr = ν/a is the Prandtl number, up is the characteristic velocity of the oil flow (Supporting Information), ν is the kinematic viscosity, and a is the thermal diffusivity of oil. The immobile droplet affects evaporation at the cap of the mobile droplet by three mechanisms. First, the thermally affected zone of the immobile droplet hinders the energy supply carried by oil flowing toward the mobile droplet within an angle α. Here α is the viewing angle of the immobile droplet’s thermally affected zone from the center of the mobile one (Figure 4 a). Thus the parameter representing the effective energy supplied by the oil flow to the mobile one is Kq = 1 − α/2π. Second, when the thermally affected zones of two droplets overlap, the temperature of oil flowing toward the mobile droplet starts decreasing, causing an additional reduction in the energy supply. The parameter representing this effect is approximated as KT = 1 for L > 2δ and KT = (L/ 2δ)γ for L ≤ 2δ. The exponent γ = 1.7 is chosen as the optimum value yielding the best fits of the measured distance L(t) for all of the tested weight ratios c. Finally, evaporation of IPA from the immobile droplet imposes a vapor profile φ = Rc/ r, that is, partial vapor pressure of IPA normalized by its saturated vapor pressure, over the cap of the mobile droplet. Here r is the distance from the cap center of the immobile droplet.11 Thus the evaporation rate per unit area over the cap of the mobile droplet ṁ m is estimated as
Figure 3. There are two uneven circulating flows inside each droplet. (a) When L > L*, the stronger circulating flow occurs further away from the other droplet. (b) When L ≈ L*, these circulating flows switch sides; that is, the stronger one occurs more closely to the other droplet. (c) Eventually, when L < L*, the intensities of circulating flows become roughly similar.
cap, and Ṙ is the rate of change in radius of a pure IPA droplet (Supporting Information). We note that evaporation of IPA at the droplet cap induces thermocapillary stresses and subsequently generates circulating flows inside the droplet and flows in the oil pool; the generated oil flows, in turn, supply latent heat of evaporation at the cap from the pool’s far field. Quantitatively, heat flows toward the droplet cap through two stages: (1) Oil flows at room temperature transfer a heat rate Qo to the droplet through a characteristic area Ah = As/2 − Ac (Figure 4b) and (2) convective flows inside the droplet deliver to the cap a heat rate Qd, which eventually is converted to latent heat of evaporation ṁ pAchfg. Thus Q o ≈ Q d ≈ ṁ pAchfg
ṁ m ≈ cρl Ṙ(1 − φ)KqKT
As Ac
(3)
We now calculate the temperature gradient over the cap of the mobile droplet. This temperature gradient is required to calculate the thermocapillary stress contributing to driving the motion of mobile droplet. Consider an area limited by an angle dθ on the droplet cap (Figure 4a); the latent heat of the evaporation at this area is approximated as ṁ m(θ)hfgAcdθ/2π, where ṁ m(θ) is the evaporation rate at the perimeter of the cap (Supporting Information). The rate of convective heat transfer to that area is estimated as (ρcpu(θ)ΔT(θ))dAcdθ/2π, where the subscript d indicates the properties of the droplet. Here ΔTd(θ) is the temperature difference between the center and the rim of the cap and ρd and cpd, respectively, are the density and heat capacity of the solution inside the droplet. The
(2)
The heat transfer toward the immobile droplet is enabled by a temperature gradient across a thermally affected zone surrounding the droplet (Figure 4a). The characteristic length of the thermally affected zone is da = Rs − Rc + δ, where δ = 2Rs/(Re1/2Pr1/3) is the thermal boundary layer thickness of the
Figure 4. (a) Top-view schematic (not to scale) of a model system consisting of an immobile droplet and a mobile one. Each droplet has radius Rs, and its cap has radius Rc. The thermal boundary layer of the surrounding oil has thickness δ. The surrounding oil within an annulus of length da is thermally affected by the cap’s lower temperature. The temperature difference on the cap from the center to the rim is ΔTd, and the temperature difference on the oil surface across the thermally affected area is ΔTo. (b) Side-view schematic of the mobile droplet. The effective area for heat transfer from the oil pool to the droplet is Ah. Dashed arrows indicate convective flows. Solid arrows indicate asymmetrical evaporation across the cap of the mobile droplet. 4773
DOI: 10.1021/acs.jpclett.8b01742 J. Phys. Chem. Lett. 2018, 9, 4771−4775
Letter
The Journal of Physical Chemistry Letters
Figure 5. (a) Comparison between the measured values (circles) and modeled ones (solid line) of the distance L(t) (left axis) between two droplets of radius Rs = 1.0 mm and weight ratio c = 60% IPA. The modeled approaching velocity v(t) (dashed line, right axis) shows that the two droplets accelerate toward each other for L > L* and decelerate for L < L*. The dotted line marks the critical point L = L*. Comparison between experiment and theory for (b) critical distance L*, (c) approaching velocity vc at L = L*, and (d) total approaching time ta for different values of c. The uncertainty results from taking the average of six individual experiments.
characteristic velocity ud of the convective flow inside the droplet is estimated by balancing the viscosity and thermocapillarity, giving ud ≈ (σTΔT/μ)d, where σT = ∂σ/∂T is derivative of the surface tension σ with respect to T.15,16 Hence, balancing the latent heat of evaporation and the rate of convective heat transfer gives ÄÅ É ÅÅ ṁ (θ )h ÑÑÑ1/2 fg Ñ ÅÅ m ÑÑ ΔTd(θ ) ≈ ÅÅÅ Ñ ÅÅ (ρcpσTμ−1)d ÑÑÑ (4) ÅÇ ÑÖ
over An, the thermally affected area of the oil surface, and Ac, the cap area. The force driving the mobile droplet is then resisted by the viscous drag Fv ≈ μoRsU, where U is the velocity of the mobile droplet. The resulting force balance for the mobile droplet is Fo + Fd ≈ Fv
(6)
We numerically solve eq 6 to obtain the distance L as a function of time t for droplets of IPA weight ratio c from 40 to 80%. In Figure 5a, we show the representative distance L(t) and approaching velocity v(t) calculated from eq 6 for two droplets of c = 60%; we select the initial distance L0 by leastsquares fitting to the experimental data. The modeled result of L(t) captures the approaching motion of the two droplets and agrees reasonably well with the measured data. The model also captures the switch in approaching dynamics, from accelerating to decelerating, as L crosses the critical value L*. In our model, we determine L* as the inflection point of L(t), that is, the point at which v switches from increasing to decreasing. Both the calculated critical distance L* and the approaching velocity vc at L = L* are consistent with the measured values for a wide range of IPA weight ratio 40 ≤ c ≤ 80% (Figure 5b,c). Furthermore, the total approaching time ta, defined as the duration for the two droplets to touch from the initial distance L0, is well-captured by the model (Figure 5d). In summary, we report autonomous attractive interaction of droplets floating on another immiscible liquid, for example, silicone oil. We confirm that the phenomenon is robust for various volatile components, for example, IPA, ethanol, and acetone, for wide ranges of weight ratio and droplet size. The attractive interaction originates from evaporation of the volatile component without any additives. We show that while asymmetric evaporation induces the approaching motion of droplets, the surrounding immiscible liquid plays an important role in supplying heat to sustain evaporation and inducing both
Similarly, the temperature difference across the thermally affected zone of the mobile droplet is determined by balancing the rate of convective heat transfer in the oil pool and that in the droplet. We take the upper half of the droplet’s submerged area Ah = As/2 − Ac as the effective area for heat transfer because fluid convection is mostly concentrated in the upper half of the droplet (as suggested by the flow field shown in Figure 3b). The energy balance for a spherical wedge of the droplet with angle dθ is (ρc pu(θ)ΔT(θ)) dA cdθ/2π ≈ (ρcpu(θ)ΔT(θ))oAhdθ/2π, where the subscripts d and o, respectively, indicate the properties of the droplet and oil. Therefore, we obtain ÄÅ É ÅÅ A (ρc σ μ−1) ÑÑÑ1/2 dÑ ÅÅ c p T ÑÑÑ ΔTd(θ ) ΔTo(θ ) ≈ ÅÅÅ ÅÅ Ah (ρcpσTμ−1)o ÑÑÑ (5) ÅÇ ÑÖ We are now ready to write the expressions of forces acting on the mobile droplet. The temperature difference ΔTd(θ) across the droplet cap generates a thermocapillary stress τd(θ) ≈ (σTΔT(θ))d/Rc, whereas the temperature difference ΔTo(θ) across the surrounding thermally affected area with characteristic distance da generates the stress τo(θ) ≈ (σTΔT(θ))o/da (Figure 4a). The total thermocapillary forces acting on the thermally affected area and the cap of the mobile droplet are then determined by integrating the thermocapillary stresses 4774
DOI: 10.1021/acs.jpclett.8b01742 J. Phys. Chem. Lett. 2018, 9, 4771−4775
Letter
The Journal of Physical Chemistry Letters
(9) Herminghaus, S.; Maass, C. C.; Krüger, C.; Thutupalli, S.; Goehring, L.; Bahr, C. Interfacial mechanisms in active emulsions. Soft Matter 2014, 10, 7008−7022. (10) Izri, Z.; Van Der Linden, M. N.; Michelin, S.; Dauchot, O. Selfpropulsion of pure water droplets by spontaneous Marangoni-stressdriven motion. Phys. Rev. Lett. 2014, 113, 248302. (11) Cira, N.; Benusiglio, A.; Prakash, M. Vapour-mediated sensing and motility in two-component droplets. Nature 2015, 519, 446−450. (12) Man, X.; Doi, M. Vapor-induced motion of liquid droplets on an inert substrate. Phys. Rev. Lett. 2017, 119, 044502. (13) Whitesides, G. M.; Grzybowski, B. Self-assembly at all scales. Science 2002, 295, 2418−2421. (14) Ghoshdastidar, P. Heat Transfer, 2nd ed.; Oxford University Press: New Delhi, India, 2012. (15) Davis, S. H.; Homsy, G. M. Energy stability theory for freesurface problems: buoyancy-thermocapillary layers. J. Fluid Mech. 1980, 98, 527−553. (16) Hegseth, J. J.; Rashidnia, N.; Chai, A. Natural convection in droplet evaporation. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1996, 54, 1640.
far-field and near-field behaviors, that is, accelerating and decelerating, respectively. We propose a simple model capturing the essential features of the attracting motion including the critical distance L* separating the far-field and near-field behaviors, the approaching velocity at critical point, and the approaching time. These results may provide a new insight into complex systems consisting of autonomous or interactive elements. Finally, mechanistic understanding of such interactions will also be beneficial to devise novel strategies to control behaviors of collective or self-organized systems.
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ASSOCIATED CONTENT
* Supporting Information S
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.8b01742. Cap radius Rc for droplets with different weight ratios of IPA, rate of change in radius for an ideal IPA droplet (c = 100%), characteristic velocity up of the oil flow around the immobile droplet, evaporation rate at the perimeter of the mobile droplet’s cap ṁ m(θ), comparison between the measured and modeled values of the distance L(t) in the cases of IPA aqueous solutions for several different weight ratios, and comparison between the measured and modeled values of the distance L(t) in the case of ethanol aqueous solution of weight ratio 80%. (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Tuan Tran: 0000-0002-5132-6495 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the Nanyang Technological University (NTU) and A*STAR, Singapore. D.L. acknowledges the research fellowship supported by A*STAR.
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REFERENCES
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DOI: 10.1021/acs.jpclett.8b01742 J. Phys. Chem. Lett. 2018, 9, 4771−4775
