Departure of Condensation Droplets on Superhydrophobic Surfaces Cunjing Lv,† Pengfei Hao,*,† Zhaohui Yao,† and Fenglei Niu‡ †

Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China

ABSTRACT: This article focuses on the departure of multidroplet coalescence on a superhydrophobic surface with nanoscale roughness. Out-of-plane jumping events triggered by multidroplet coalescence and a single fallen droplet are observed. Experimental data show that the departure of droplets due to the multidroplets coalescence and the jumping modes is dominant for the removal of condensed droplets from the substrate. The energy barrier is easier to overcome and the critical size of the self-propelled droplets could be further decreased in multidroplet coalescence jumping mode. A general theoretical model is developed which accounts quantitatively for determining the jumping velocity and the critical size of the multidroplet coalescence.

INTRODUCTION Vapor condensation is widely observed in nature and plays an essential role in energy and environmental applications including power generation, water harvesting, thermal management, and air conditioning.1−10 When water vapor condenses on surfaces with high or low surface energy, the condensate forms a liquid film or distinct droplets, respectively. Previous studies have shown that the heat-transfer performance of dropwise condensation is much greater than that of filmwise condensation due to the efficient gravity-driven droplet removal mechanism.9−13 In recent years, superhydrophobic surfaces, where a combination of low surface energy and surface texturing is used to enhance the water repellency, have been proposed as a promising approach to promoting dropwise condensation, further enhancing the droplet removal performance by minimizing the pinning force between condensed droplets and surfaces.9,10,14−19 When small water droplets merge on suitably designed superhydrophobic surfaces, they can undergo coalescence-induced jumping due to the release of excess surface energy. This self-propelled droplet jumping has been demonstrated to increase the condensation rate by decreasing the maximum departure diameter. The average droplet radius at steady state was 30 times smaller than the capillary length (2.7 mm for water), which could enhance the condensation heat-transfer performance.9,10,15,19−22 In Hou’s work,10 by employing a bioinspired hybrid surface, the heattransfer coefficient demonstrates a 63% enhancement as compared to conventional dropwise condensation on the flat hydrophobic silicon surface. To achieve droplet jumping, structures on the substrate need to be designed to minimize the adhesive force between droplets and the substrate using hierarchical structures or nanoscale structures with low solid−liquid area fractions. Many © XXXX American Chemical Society

researchers have modeled and fabricated surfaces that show stable droplet jumping.9,10,23−26 Other researchers have worked on the physical mechanism of spontaneous droplet motion. Wang et al.27 derived a simple relation for the coalescenceinduced velocity of two droplets based on the energy conservation of surface energy, viscous dissipation energy, and dynamic energy. Their results show that a 10 μm droplet diameter threshold is necessary for surface energy release during coalescence to dominate viscous dissipation and gravitational potential energy. Liu et al. reported that nanostructures are crucial to the formation of microdroplets with high contact angles.28 Hao et al. studied the dynamic behavior of microdroplets condensed on a lotus leaf and observed the out-of-plane jumping relay of condensed droplets triggered by falling droplets as well as the sustained speed obtained in continuous jumping relays. Their analyses showed that the dissipation of surface energy during droplet transition and that caused by contact angle hysteresis limit the size of the jumping droplet on the microscale.29 Most recently, Rykaczewski et al. reported several new droplet shedding modes, which are aided by the tangential propulsion of mobile droplets. These droplet shedding modes comprise multiple droplets merging during serial coalescence events, which culminate in the formation of a droplet that either departs or remains anchored to the hierarchical superhydrophobic surface.30 However, the details and mechanisms of the droplet jumping induced by multidroplet merging events are still limited to date. In this article, out-of-plane jumping events of condensed droplets triggered by multiple droplet coalescence and a single Received: November 27, 2014 Revised: January 28, 2015


DOI: 10.1021/la504638y Langmuir XXXX, XXX, XXX−XXX


Langmuir fallen droplet were observed. Our experimental results demonstrate that the multidroplet coalescence and the jumping modes are dominant for the removal of condensed droplets from the superhydrophobic surface with nanoscale roughness. We have also developed general theoretical models to explain these results, which account quantitatively for determining the jumping velocity and the critical size of the condensed droplets.


In this work, condensation on nanostructured surfaces is examined. The superhydrophobic surfaces with nanoscale roughness are produced by treatment with a commercial coating agent (Glaco Mirror Coat “Zero”, Soft 99 Co. Japan) containing nanoparticles and an organic reagent.31 The superhydrophobic coating is applied to the smooth silicon wafer by pouring the coating agent over the substrate. A thin liquid film wets the silicon surface and dries in less than 1 min. Then we heat the substrate to 200 °C and sustain it for half an hour and then repeat the pouring and heating processes three to four times. Surface topographies are analyzed using a scanning electron microscope (SEM, JSM 6330 from JEOL) and an atomic force microscope (AFM, NanoScope 5 from DI). The micrograph of the smooth silicon wafer with nanoparticle coatings is shown in Figure 1.

Figure 2. Dropwise condensation on a superhydrophobic substrate with nanoscale structures. The solid and dashed white circles in (c) and (d) show the moment before and after the departure of multidroplets happens in two different areas.

some of the merged droplets detach from the substrate, and dry areas will be formed. We noticed that detachment happens more easily when multiple droplets merge with each other, and then small droplets will appear again in the dry area; such phenomena are repeatable during condensation. In Figure 2, detailed information before coalescence and after departure of the droplets is given. (See the solid and dashed white circles in Figure 2c,d, respectively. The time interval is 1 s.) The coalescence and departure of droplets generate large dry areas, which will significantly reduce the thermal resistance caused by the accumulated liquid film and will be very helpful for heattransfer enhancement. Dropwise condensation and departure processes shown in Figure 3 are obtained by a high-speed digital CCD camera. Droplets grow up during condensation, and then they merge with each other suddenly (seven droplets enclosed in the white circle in Figure 3a). During coalescence, the total surface energy will decrease to overcome the adhesion force and other dissipations and finally transfer to kinetic energy. When the kinetic energy of the merged droplet is larger enough, the droplet leaves the original position and bounces off to a new position (white arrows in Figure 3b−d). In this observation, the merged droplet jumps 45 μm far away from the original center of the seven droplets. What is surprising is that droplets in different areas which are far away from each other could also leave the surface because there will be a second bounce. One can imagine that in one area, coalescence happens and the droplets merge together and bounce up. After that droplet falls (the shadows in Figure 4b,c), droplets in another area (enclosed in the white dashed circles in Figure 4) are triggered by this falling droplet (the shadows in Figure 4b,c) and bounce up again (the large shadow in Figure 4d). These phenomena are different from Rykaczewski’s observations,27 in which the new bounces are caused by

Figure 1. Topology of the Glaco-coated silicon wafer obtained by employing SEM (a) and AFM (b). The roughness of the nanoscale structures in (b) is Ra = 24.5 nm, which corresponds to a 1 μm × 1 μm area. Figure 1a shows a 5 × 5 μm2 SEM topological image of the coating deposited on a flat silicon wafer surface. Figure 1b shows a smaller 1 × 1 μm2 AFM image. The images indicate that the characteristic particle size of the coating ranges from 30 to 50 nm (Figure 1). On a large scale, the coating has self-assembled nanoparticles which form a fractal-type structure with a roughness ranging from 100 to 300 nm in spacing and 50 to 200 nm in depth. The equilibrium contact angle and sliding angle are measured using the sessile drop method with a commercial contact angle meter (JC2000CD1) with a water droplet volume maintained at 5 μL. The contact angle is measured to be 155.5 ± 2.0° on this superhydrophobic surface, the sliding angle is less than 5°, and the droplet can run off easily enough. In the condensation experiments, the sample is placed horizontally on a Peltier cooling stage, which is fixed on the slider of an optical microscope (BX51, Olympus, Japan). The laboratory temperature is measured to be 29 °C with a relative humidity of 40% (the corresponding dew point is 14 °C). After the cooling system is turned on, the temperature of the sample is controlled carefully and maintained at 10 ± 1 °C. Video imaging of water vapor condensing from the air is captured with a computer-controlled digital camera (MegaPlus, RadLake, USA) attached to a BX51 microscope for topdown imaging.

RESULTS AND DISCUSSION Figure 2 shows the dropwise condensation and departure processes on the superhydrophobic substrate. The droplets are growing up when t < 40 s, and then coalescence occurs and some of the droplets start to merge with each other when their sizes exceed 10 μm. Motion is observed during coalescence, B

DOI: 10.1021/la504638y Langmuir XXXX, XXX, XXX−XXX



Figure 3. Spontaneous droplet motion caused by multidroplets merging: (a) initial state, (b) seven droplets merge with each other and bounce off, (c) motion of the merged droplet, and (d) the droplet comes down into a new position.

Figure 4. Multidroplets merge and bounce off because of a falling droplet: (a) initially, there are five droplets inside the white circle, (b) flying of a droplet from somewhere else (the shadow at the arrowhead), (c) falling of the droplet (the shadow at the arrowhead), and (d) droplets on the substrate merge with each other, triggered by the falling droplet, and then the whole new droplet bounces off (the shadow at the arrowhead).

Figure 5. Droplet bouncing triggered by a falling droplet: the size and the velocity of the falling droplet are 20 μm and 0.02 m/s, and the size and the velocity of the rising droplet are 60 μm and 0.12 m/s.

horizontal scanning of the last bounce, but in our experiment, we observed that the new bounce is caused by the falling of the droplet in the vertical direction. In order to check this point further, detailed information on the bouncing behavior is recorded from the side view (Figure 5). We can see that after one falling droplet touches the substrate, coalescence is triggered again and a new bounce happens. The sizes of the original droplet and the merged droplet are 20 and 60 μm, respectively. Moreover, the falling velocity and the rebounding velocity are 0.02 and 0.12 m/s, which strongly support what we observed from the top view (Figure 4). The larger values of size and velocity indicate that the droplet in Figure 5d obtains more kinetic energy in the second coalescence. Different from Rykaczewski’s results,30 we conclude that there are at least three underlying mechanisms for the spontaneous motion of a droplet caused by coalescence: (1) growing up of droplets and merging with their neighbors when the size of the droplets reaches the critical size, (2) triggered by falling droplets, and (3) triggered by horizontal scanning of merged droplets. As shown in Figure 6, after making statistics over a 60 μm × 60 μm reference area, we give the trends in surface coverage ε2 of droplets with time. The surface coverage ε2 is defined as the ratio of the projected surface area covered by the droplets over the reference substrate. The result shows that the surface coverage changes periodically; in other words, when the surface overage reaches a maximum, it will decrease suddenly. Figure 6 clearly shows the trends for condensed droplets in the reference substrate, which is consistent with what we observe in Figures 2−4.

Figure 6. Relationship between the area fraction ε2 and the time t(s) over a 60 μm × 60 μm reference area.

We also give the trends in the probability of jumping events and the minimum value of the critical droplet diameter with the number of merging droplets. The probability of jumping events for “N” is defined as the ratio of the number of jumping times for N-droplet coalescence over the total number of jumping times (71 times) in a period long enough (t ≈ 600 s) in a 300 μm × 300 μm reference area. As shown in Figure 7a, what is surprising is that multidroplet merging and jumping is in the majority. Among them, three and four droplets merging and jumping have the highest values, and even seven droplets merging and jumping obviously appears. On the contrary, the value of the probability of two droplets bouncing has the smallest value but has been most commonly reported.15,27,29,32 In Figure 7b, it is shown that the minimum value of the critical C

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Figure 7. (a) Number of jumping events vs number of merging droplets. (b) Minimum value of droplet diameter vs number of merging droplets. All of the statistics are determined over a 300 μm × 300 μm area with about 600 s duration.

Figure 8. (a) Relationship between the radius of the droplets and the jumping velocity after merging, with various values of U0 presented in (a). (b) Relationship between U0 and the minimum value of the droplet diameter.

⎛1 1⎞ ρV0U 2 = 2V0γLV ⎜ − ⎟ − 2πrf γLV(r1 sin θ1)2 r2 ⎠ ⎝ r1

droplet diameter for jumping decreases with the number of merged droplets. The phenomenon we observed on nanostructured surface is quite similar to the coalescence reported on the micronano-structured surface.29 Our results indicate that thanks to the multidroplet jumping mode, jumping and departure not only happen more easily but also could be realized on a smaller scale, which will contribute greatly to the benefit of the enhancement of heat transfer by employing superhydrophobic surfaces.

⎛ A sec ⎞2 γLVr13 (1 + cos θY ) − 3μπ ⎜ + ΔE k ⎟ ρ ⎝ A0 ⎠


in which the left term is the whole kinetic energy of the whole droplet after merging (∼2ρV0U2/2), U is the droplet velocity after coalescence, and V0 is the volume of a single droplet before merging. The first term on the right is the change in the total potential energy, the second and third terms are total dissipation caused by viscosity and contact angle hysteresis, respectively.29 ΔEk = (ρV0)U02/2 is the kinetic energy of the moving droplet with velocity U0, and ρ is the mass density of water. γLV = 0.074 N/m and μ = 1.306 mPa·s are the surface tension and viscosity of water at 10 °C, respectively. θY is the intrinsic Young contact angle of the surface. r1 and r2 are the radii of the droplets just before and after merging and could be obtained from V0 and apparent contact angles θ1 = 153.5° and θ2 = 157.5° (which correspond to the receding and advancing contact angles). rf = 1.5 is the roughness of the substrate obtained by AFM. Here, the roughness is characterized as the ratio between the solid−liquid contact area and the projected area. A0 = πr12, and Asec = r12(θ1 + sin θ1 cos θ1) is the cross section of a single droplet before merging. Figure 8 demonstrates that because of the initial kinetic energy of one droplet, there will be extra energy accumulated


In order to understand the above phenomena and reveal the underlying mechanism, we will give theoretical analyses below. Very recently, some quantitative relationships have been constructed27,29 and could be used to describe the dynamic wetting behaviors of two static droplets with equal size. However, in real experiments as we observed (Figure 5), one droplet is static but the other one has an initial velocity before coalescence; this case has never been studied systematically. We will investigate this wetting phenomenon first. For convenience, we assume that the volumes of the static and moving droplets are equal. On the basis of the previous model,29 we write the dynamic equation as D

DOI: 10.1021/la504638y Langmuir XXXX, XXX, XXX−XXX



Figure 9. (a) Relationship between the radius of each droplet and the velocity for multiple-N merging. (b) Relationship between the number of merging droplet and 1/rc. The black squares are experimental results with error bars, the red circles are given by eq 3, and the black dashed line is 1/ rc ∼ N.

and finally transferred to the kinetic energy of the merged droplet, which causes the velocity of the merged droplet to be higher. Moreover, the merged droplet could overcome the energy barrier more easily, which is the reason that the minimum value of the size of the jumping droplet decreases. Multidroplet coalescence has the following characteristics. First, the contribution of the obtained extra surface energy to the net kinetic energy should be more and more dominant with the number of satellite droplets. (One can imagine that, on the contrary, the value of the total area will be larger and larger if one droplet is divided into more and more satellite droplets.) This is the main reason that multidroplet coalescence-induced jumping could happen more often (Figure 7a) and the minimum value of the droplet size could be even smaller (as shown in Figure 7b). Second, it is very interesting that there is a plateau in Figure 7a (when N = 4). Qualitatively, if one droplet touches its neighbor droplets, then merging and jumping happen at almost the same moment, so there is probably not enough time and space for them to merge more and more droplets. If some special prerequisites are satisfied, for example, it will be better that all of the neighboring droplets are closed to each other and one is sitting in the center in order to increase the probability of merging (Figure 3a). However, in real experiments, this could not be attained in every jumping event, which is the reason that the number of events will increase first and then decrease in Figure 7a, also demonstrating that N = 4 is the maximum probability for multidroplet coalescence in an ambient environment. Third, the attachment/detachment and the component of energy from different parts are very complicated even for the coalescence of two droplets,32 and the merging behavior could be more complicated for the coalescence of N droplets. Both the contact area and the shape of the droplets may deform greatly during coalescence. In other words, the solid−liquid contact area will become larger and there will be more dissipation caused by the moving of the droplets toward each other along their ways. Moreover, if the value of N is large, then the initial droplets could compose many different arrangements and subsequently merge into each other with different modes as well. In light of that above analysis, it is probably not easy to give a uniform equation to describe all of the possible wetting conditions even if we have a certain value of N (N ≥ 3). In order to take into consideration these influences without a loss of generality, we introduce a geometrical parameter β that is used to reflect the increase in

the solid−liquid contact area along their moving paths. So a general equation to describe the coalescence of N droplets with equal size could be given as ⎛1 1 1⎞ ρV0U 2N = NV0γLV ⎜ − ⎟ − Nπrf γLV(r1 sin θ1)2 2 r2 ⎠ ⎝ r1 ⎛ A sec ⎞2 γLVr13 3 (1 + cos θY )β − Nμπ ⎜ ⎟ 2 ρ ⎝ A0 ⎠


Equation 2 is similar to eq 1, but it is not the overlay of eq 1 because of the analysis that we demonstrated above. Unfortunately, we could not determine the exact value of β theoretically. However, on the basis of our experimental data (as shown in Figure 7b), we could obtain a rough estimation of β by a polynomial fitting. Finally, we get β = 0.007N3 − 0.147N2 + 1.115N − 0.68. The other geometrical and physical parameters are chosen as we used above. The numerical results are shown in Figure 9a. Recently, Rykaczewski systematically studied the microscale water condensation dynamics on nanostructured superhydrophobic surfaces and gave quantitative relationships between the roughness of the substrate and the minimum value of condensed droplets attainable.9 However, after droplets appear, studies on the critical size of coalescence-induced self-propelled droplet bouncing are still limited. On the basis of eq 2, letting U = 0, we could give an estimation of the minimal value of the size of the droplets for jumping, ⎡ ργLV ⎢ 1 − 1 = 2 ⎢ rc μ ⎢ ⎢⎣

r1 r2

3 4

⎤2 rf sin 2 θ1(1 + cos θY )β ⎥ ⎥ 2 3 A sec r1 ⎥ ⎥⎦ A0 rs


() ( )( ) r1 rs 9 8


The results are shown in Figure 9b (solid red circles). We can see that eq 3 could capture the trends between the number of merging droplets and their minimal radii rc. Moreover, we get 1/rc ∼ N if we do a direct fit. Figure 9 could reflect the influence of the number of droplet on the velocity and the critical size, which are consistent with our experimental observations (Figure 7). In other words, the energy barrier could be further overcome and the droplet jumping will benefit greatly from the multidroplet coalescence. However, the contributions of different parts of the energy in eq 2 and E

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how the energy barrier is overcome especially for multidroplet coalescence are still not very clear, so further investigations are well worth carrying out in the near future.


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CONCLUSIONS Different from previous research, in which attention was mainly focused on the self-propelled jumping phenomena triggered by two droplets and their mechanisms, in our study, that departure of multidroplets induced by condensation is revealed to be dominant on the superhydrophobic surfaces during condensation. Moreover, out-of-plane jumping events triggered by a single fallen droplet are also observed. These wetting phenomena happen widely during condensing but were scarcely reported before. In this article, we not only make detailed measurements of the relevant probability of jumping events but also give mathematical models and analyses of the underlying physics systematically. Our experimental and theoretical results show that jumping is much easier to achieve for multidroplet coalescence than for two droplets, the energy barrier is much easier to overcome, and more surface energy will transfer to kinetic energy, which is a direct result of the decrease in the critical size of the droplet. Moreover, the frequency of departure also increases and benefits from the multidroplet coalescence. The general model we constructed not only characterizes the jump velocity but also accounts quantitatively for the relationship between the number of merging droplets and their critical sizes. We put forward a law (1/rc ∼ N) which gives the trend in critical sizes for N-droplet coalescence which is consistent with experimental observation. The experimental observations and theoretical analyses are critically important to deepen our understanding of the underlying physics of droplet jumping phenomena on surperhydrophobic materials. These results will be significant to realize wide industrial applications where thermal conduction via droplet jumping plays a key role, including superhydrophobic substrate design and optimization, rapid cooling/condensation, heat exchanger technologies, and so on. Even though we observed multidroplet coalescence jumping experimentally and give quantitative statistical laws as well (Figure 7), there are still many unknown details, for example, how coalescence and detachment happen, the inner flow field, the ratio of contributions to kinetic energy from different parts, the influence of the original arrangement of the multidroplets on the detaching behavior, and so on. These unknown questions will inspire us to carry out further quantitative investigations in the future.



Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (grant nos. 11072126, 11272176, 91326108, and 51206042) and the Foundation of Stake Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources (grant no. LAPS14018). F

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DOI: 10.1021/la504638y Langmuir XXXX, XXX, XXX−XXX

Departure of condensation droplets on superhydrophobic surfaces.

This article focuses on the departure of multidroplet coalescence on a superhydrophobic surface with nanoscale roughness. Out-of-plane jumping events ...
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