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Unidirectional Fast Growth and Forced Jumping of Stretched Droplets on Nanostructured Microporous Surfaces Abulimiti Aili, Hongxia Li, Mohamed H Alhosani, and TieJun Zhang ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.6b05324 • Publication Date (Web): 03 Aug 2016 Downloaded from http://pubs.acs.org on August 4, 2016

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ACS Applied Materials & Interfaces

Unidirectional Fast Growth and Forced Jumping of Stretched Droplets on Nanostructured Microporous Surfaces

Abulimiti Aili, Hongxia Li, Mohamed H. Alhosani, TieJun Zhang* Department of Mechanical and Materials Engineering, Masdar Institute of Science and Technology, P. O. Box 54224, Abu Dhabi, UAE * Corresponding author: [email protected]

Abstract

Superhydrophobic nanostructured surfaces have demonstrated outstanding capability in energy and water applications by promoting dropwise condensation, where fast droplet growth and efficient condensate removal are two key parameters. However, they remain contradictory. Although efficient droplet removal is easily obtained through coalescencejumping on uniform superhydrophobic surfaces, simultaneously achieving fast droplet growth is still challenging. Also, on such surfaces droplets can grow to larger sizes without restriction if there is no coalescence. In this work, we show that superhydrophobic nanostructured microporous surfaces can manipulate the droplet growth and jumping. Microporous surface morphology effectively enhances the growth of droplets in pores owing to large solid-liquid contact area. At low supersaturations, upward growth rate (1—1.5 µm/s) 1 ACS Paragon Plus Environment


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of these droplets in pores is observed to be around 15—25 times that of the droplets outside the pores. Meanwhile, their top curvature radius increases relatively slowly (~0.25 µm/s) due to pore confinement, which results in highly stretched droplet surface. We also observed forced jumping of stretched droplets in pores either through coalescence with spherical droplets outside pores, or through self-pulling without coalescence. Both experimental observation and theoretical modeling reveal that excess surface free energy stored in stretched droplet surface and micropore confinement are responsible for this pore-scale forced jumping. These findings reveal the insightful physics of stretched droplet dynamics, and offer guidelines for the design and fabrication of novel super-repellent surfaces with microporous morphology.

Keywords: stretched droplets, self-/coalescence jumping, nanostructures, microporous, superhydrophobic, condensation

Introduction Droplet jumping is an interesting phenomenon that occurs by taking advantage of the excess surface free energy released upon coalescence of droplets on superhydrophobic nanostructured surfaces.1–3, And, it has received great attention in enhanced condensation heat transfer,4–8 and many other applications such as self-cleaning9 and anti-icing surfaces,10 water11,12 and energy harvesting.13 Previous studies thus have been mainly focused on designing extremely rough surfaces with micro- and nanostructures coated with ultra-lowsurface-energy materials.14–17 This gives significantly small contact area and solid fraction to droplets growing on the condensing surface, greatly minimizing the surface adhesion upon jumping. However, there exists a contradiction that a very small solid-liquid contact area is an obstacle to fast droplet growth and heat transfer while it is beneficial for droplet jumping.6, 2 ACS Paragon Plus Environment

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,18

This contradiction comes from the following assumptions: (a) solid surface is always

underneath the droplet and acts mainly as a heat conductor; (b) condensate droplets are always spherical and release excess surface free energy only through coalescence. Several studies then have been done on hybrid surfaces to enhance droplet growth while ensuring that droplet jumping still take places.19,20,21 However, this kind of surfaces generally require sophisticated and expensive cleanroom technologies for large-scale applications. Despite these intense efforts, simultaneously enhancing both growth and removal of condensate droplets on homogeneously superhydrophobic surfaces is still contradictory and challenging. On traditional superhydrophobic surfaces, spherical droplets can grow to any size until they coalesce and jump. While wetting transition was observed on micropillar surfaces, where deformed droplets within the micropillars self-pull themselves to the top of the structures, no direct droplet jumping occurred.4,22 This is probably due to larger surface adhesion when droplets and surface nanostructures have similar dimensions. In this work, however, we report that droplet growth can be significantly enhanced owing to increased solid-liquid contact area of nanostructured micromesh surfaces. Meanwhile, droplets in micromesh-confined pores are highly deformed and forced to jump at sizes determined by pore opening. The pore confinement also favors the coalescence and unidirectional growth of droplets inside the pore. We used a facile and scalable method to fabricate a microporous superhydrophobic surface by covering a copper plate surface with a microporous copper mesh, followed by heat treatment, oxidation and hydrophobic silanization. Condensation experiments were carried out at low supersaturations (S ≈ 1.06) in an Environmental Scanning Electron Microscopy (ESEM) at different working currents, which enabled us to better observe the dynamics of droplet growth and jumping. Micropores wrap the droplets, providing larger liquid-solid contact area and side confinement. As a result, droplets in pores grow mainly in the upward direction at a rate of 1—1.5 µm/s, which

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is about 15—25 times that (0.6—0.7 µm/s) of the droplets outside micropores or on the surface without micropore. Meanwhile, the top curvature radius of these droplets growing in pores increases relatively slowly at a rate of around 0.25 µm/s, leading to highly stretched droplet surface. Micropores then release these droplets at a range of micropore sizes, leading to coalescence-jumping or even self-jumping without coalescence. Theoretical analysis and modeling results reveal that forced coalescence jumping of stretched droplets in pores is due to the released excess surface energy and abrupt pressure drop in the upper part of the droplets upon coalescence with spherical droplets outside the pores. On the other hand, selfjumping is due to the excess surface free energy stored in the stretched droplet surface, and due to the local high pressure point at droplet base upon detaching from the base substrate. This work offers new insights into enhancement of droplet growth and removal on nanostructured superhydrophobic surfaces with microporous surface morphology. We believe these findings act as guidelines for designing novel super-repellent surfaces for enhanced condensation heat transfer and surface self-cleaning. Experimental Section Superhydrophobic microporous surface fabrication The microporous surface was prepared sequentially through heat treatment, chemical oxidation and silane coating of a copper plate covered with a copper micromesh. The thickness of the copper plate is 0.20 mm, and average pore opening and wire diameter of the copper mesh is 34 µm. In this case, the condensation surface area of mesh-covered microporous surface is up to 5 times larger than that without mesh pores, as indicated in the Supporting Information S5. Prior to treatment, copper mesh and plate were ultrasonically cleaned in acetone for 10 min to remove the organic contaminants, triple rinsed with ethanol and deionized (DI) water, and cleaned with nitrogen gas. They were then immersed into hydrochloric acid solution to remove the oxide layer, which otherwise would impede the


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attachment of copper plate and copper mesh. After that, the mesh was placed on top of the plate, pressed by a strong binder, and put into a vacuum oven at 200 ˚C for 4 hours, which led to welding of the copper plate and copper mesh. Chemical oxidation was used to obtain nanoscale roughness on surfaces of the base plate and attached mesh.23,24 The microporous surface prepared with the above procedures was again cleaned in hydrochloric acid and rinsed with DI water. The sample was then immersed into a hot oxidation bath containing a solution of NaClO2, NaOH, Na3PO4·12H2O, (SigmaAldrich Chemical Co.) and DI water (3.75: 5: 10: 100 wt. %) for at least 10 min at 96 ± 6 ℃ to form an oxide layer of blade-like nanostructures. After that, it was cleaned with DI water for multiple times, and dried with nitrogen flow to forcibly remove the remaining water. Silane coating was used to convert the nanostructured sample from hydrophilic into superhydrophobic.25,26 Before coating, the sample was plasma-cleaned for 30 min to remove any existing organic contaminants and to increase the surface temperature to ensure that coating would be uniform. Then, it was quickly placed in a desiccator with a small amount ~0.1 μl of trichloro(1H,1H,2H,2H-perfluorooctyl) silane (Sigma-Aldrich Chemical Co.) that has an intrinsic contact angle of around 115˚.25 The desiccator was pumped for 15 min to remove the air, and a valve was closed to isolate the chamber to deposit a thin layer of silane in vacuum for another 30 min. Note that insufficient deposition time led to a thick greenish contamination layer upon exposure to air. After deposition, the sample was cleaned with DI water and dried with nitrogen. It took a few minutes for the surface to become completely superhydrophobic in air possibly due to the slow hydrolysis and dehydration reaction between silane chains. Figure 1 is the schematic diagram of the fabrication process described above. For comparison, a superhydrophobic copper plate surface without microporous was also prepared using the same oxidization and silanization processes. Surface morphologies and



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geometries of copper oxide nanostructures were obtained by using scanning electron microscopies (SEM, FEI Quanta 250 & Nova Nano). Static contact angles of the samples were measured with a goniometer (DM-501, Kyowa Interface Science Co., Ltd).

Figure 1. Schematic diagram of the fabrication process of a copper micromesh-covered microporous superhydrophobic surface. A copper plate surface was covered with a copper micro-mesh, followed by heat treatment in vacuum, chemical oxidation, and hydrophobic silane coating.

ESEM condensation experiment Microscale water condensation experiment was carried out using an environmental scanning electron microscopy (ESEM, FEI Quanta 250) with a gaseous secondary electron detector (GSED).27,28 The sample was mounted on a 90° custom-designed copper stub (Figure S1) using high purity silver paint, and the stub was placed on a Peltier coiling stage. Once the camber was completely evacuated, stage temperature was set at 2 C˚, and the vapor pressure was gradually increased to 700 Pa. When the temperature stabilized, the pressure was set to 750 Pa, and water droplet nucleation started. Images and videos were recorded at a working distance of about 5 mm and a constant beam energy of 10 keV. The cooling stage was tilted at angles ranging from 5˚ to 30˚ according to imaging and other special requirements. Smaller spot size of 2.0 (beam current of 0.13 nA) and 2.5 (0.22 nA) was generally used to achieve better imaging resolution and to avoid significant heating of the condensate surface and growing droplets on it. In a special case, larger spot size of 3.0 (0.51 nA) was used to hinder the nucleation and growing of droplets outside the pores.


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Results Surface morphology and wettability Top view of the microporous surface formed by a base substrate and a micromesh with a pore size of around 34 µm is given in Figure 2a, and the insets are the magnified top views of pore wall (top) and pore base (bottom). Figure 2b is the magnified tilted view of the nanostructures covering the whole sample surface. The surface is coated with a hydrophobic silane. Due to high roughness, we would expect a larger apparent contact angle of macroscopic water droplets from the measurement of a goniometer than on the surface without micropore. Interestingly, apparent contact angles of both surfaces turned out to be almost the same: 163°±2 on the microporous surface and 162˚±2 on the surface without micropore, although we admit Cassie-Baxter equation is still valid. Advancing contact angle and contact angle hysteresis of two surfaces are ∗ 166° 2, Δ ∗ 5°, and ∗ 164°

2, Δ ∗ 4°, respectivley. However, our contact angle of interest is that on the surface

without micropore because it provides more reliable information when we deal with microscopic droplets in micropores during condensation experiments (Figure 2c and d).

Figure 2. Surface morphology and wettability of the microporous surface. (a) SEM image of the microporous surface formed by a micromesh and a base plate. Insets are magnified views 7 ACS Paragon Plus Environment


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of pore walls (top) and the pore base (bottom). (b) Magnified view of the surface nanostructures covering both pore wall and base surfaces. (c) Tilted ESEM images of condensate droplets with high contact angles of around 164˚ on the microporous surface. Droplets are 3-dimensionally located in and outside pores. (d) Magnified view of droplets in and outside a pore. Condensation was conducted at T = 2˚C, P = 750 Pa, and 100% humidity which corresponds to supersaturation S ≈ 1.06. ESEM Images were captured at a beam current of 0.22 nA.

Dropwise condensation Microscopic water condensation experiments were carried in environmental scanning electron microscopy. As shown in Movie S1.1 and Figure 3, condensate droplets are 3dimensionally located in and outside pores, and droplets in pores are deformed. It is clearly demonstrated that stretched droplets in micropores are growing at a significantly higher rate than droplets outside the micropores even when there is no coalescence, which is discussed in detail in the following section. Another interesting phenomenon observed is that stretched droplets in pores always jump after growing as per same size of the hosting pores. By taking a snapshot of Movie S1.1. at the 12th second or the TOC graphic (d), we can see several droplets have big tails (in ‘V’ shape) left inside the pore while they are squeezed by surrounding mesh wires before jumping. This is different from the coalescence-jumping of spherical droplets on plane superhydrophobic surfaces widely reported in literatures, where size of the jumping droplet varies from a few to a hundred micrometer or beyond.1,16,29 This observation has a special implication that droplet jumping can be forcibly induced, and size of the droplets upon jumping can be pre-determined by adding micropores with specific sizes to the condensing surface.


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Figure 3. Pore scale forced jumping of stretched droplets. (a) Time-Lapse tilted ESEM images at low magnification. (b) Time-lapse ESEM images taken from the top at high magnification. Stretched droplets leave the pores at size of the hosting pores, implying that micropores have determining effect on the size of the droplets upon jumping. Images were taken at a beam current of 0.22 nA. Yellow circles represent the stretched droplets about to forcibly jump. And, red circles in the same position of the yellow circles represent the pores without the previous stretched droplets.

Individual droplet growth and jumping As previously mentioned, droplets in micropores grow at higher rate that the droplets outside the pores. In order to capture the detailed growth dynamics, we also filmed at a high magnification (Movie S1.2), and time-lapse ESEM images are given in Figure 4a. Note that a small beam current of 0.13 nA was used to avoid overheating the droplets at high magnification.27,28 Visible height evolution, ∗ , of two droplets (Droplet 1 and 2) outside pores and a droplet (Droplet 3) emerging from a pore were measured, and is plotted in Figure 4b. Abrupt increase in the slope of the curves represents droplet growth induced by 9 ACS Paragon Plus Environment


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coalescence. When there is no coalescence corresponding to the smooth section of the curves, droplets grow at a relative lower rate (0.06—0.07 µm/s) outside the pores. Meanwhile, Droplet 3 grows at a significantly higher rate in the upward direction, especially at the last stage of growth where its growth rate (1—1.5 µm/s) is about 15 times that of the droplets outside the pores. This is indeed a common phenomenon according to Movie S1.1. Note that at the last stage (t = 35 s) Droplet 3 takes away many other droplets outside pores, which is called sweeping behavior. 29,30

Figure 4. Droplet growth in and outside micropores. (a) Time-lapse tilted ESEM images of two droplets (D1 and D2) growing outside pores and a droplet (D3) growing inside a pore. Images were captured at a low beam current of 0.13 nA in order to avoid heating effect. (b) Height evolution of the three droplets shown in (a). Without coalescence, Droplet 1 and 2 grow slowly at a rate of around 0.06—0.07 µm/s. Droplet 3 grows at a higher rate, especially at the last stage where its growth rate is 1—1.5 µm/s. (c) Time-lapse images of three droplets 10 ACS Paragon Plus Environment

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D4, D5 and D6 growing in pores. A larger beam current of 0.51 nA and a smaller tilt angle of 10—20˚ was used to hinder droplet nucleation and growth outside pores. (d) Experimental results of droplet growth in (c). Droplet height grows at a significantly higher rate of 1—1.5 µm/s. Top curvature radius of these droplets, on the contrary, is increasing slowly at a rate of about 0.25 µm/s, resulting in a stretched droplet interface. Red circles represent droplet left the pores.

We tried to minimize the coalescence interference and clearly characterize the growth process of droplets in micropores. Images were taken under the same conditions except that higher beam current of 0.51 nA and a lower tilt angle were used for imaging. A higher beam current imposed significant heating effect on droplets growing outside pores; 27,28 and a lower tilt angle enabled scanning and heating the surface mainly in the direction parallel to the surface, with the droplets growing inside pores less affected by the beam. Top curvature radius r and visible height H* of three droplets (Droplets 4, 5, and 6 in Movie S1.3, and in Figure 4c) in pores were measured by using ImageJ with a curve fitting plugin,31 and the results are plotted in Figure 4d. As shown, droplet height is increasing at a rate of 1.0—1.5 µm/s, which is at least 15 times larger than the growth rate of droplets outside pores in Figure 4b or on the surface without micropore in Figure S2. On contrary, top curvature radius r of these droplets is increasing at a relatively lower rate of around 0.25 µm/s. And, their radius upon jumping is all around 25 µm. This confirms the pre-determined droplet size upon jumping previously mentioned. However, their observable time spans are different, with Droplet 4 and 5 having a time span of around 13 s while Droplet 6 having a time span of as long as 32 s. More interestingly, Droplet 5 and 6 leave the surface without significant coalescence effect, implying the possibility of self-jumping without coalescence. Note that size mismatch between coalescing droplets leads to significantly decreased conversion of the excess surface energy to kinetic energy of the resulting jumping droplet, diminishing the 11 ACS Paragon Plus Environment


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probability of jumping.29 This further supports the possibility of self-jumping since there are large droplets around the pores of interest. Moreover, it was also reported in Ref.32 that the directional movement of a single droplet and its departure size were determined by the groove shape angle. In their work,32 the droplet was dispensed by a syringe needle rather than the condensate. In this study, we were able to capture the unidirectional growth of a poreconfined condensing droplet before self-jumping, as shown in the TOC graphic (d). In fact, experimental characterization of self-jumping in condensation process is challenging due to the difficulty of fully eliminating coalescence interference. Theoretical analysis of selfjumping will be shown in the subsequent droplet jumping section. Discussion There are mainly two reasons for the faster growth of the droplets in micropores even when there is no coalescence, corresponding to smoothly increasing sections of the growth lines. The first reason is pore confinement that forces the droplets to grow mainly in the upward direction. The second but most important reason is the larger liquid-solid contact area that leads to higher conduction heat transfer rate and thus faster growth. Droplets in micropores are in contact with the solid surface not only at their base but also around the droplets, which is obviously different from the droplets growing outside the pores. Of course, temperature at the top of the wires could be slightly higher even though condensation started when the temperature stabilized. Although detailed heat transfer analysis is challenging for stretched droplets growing in pores, results shown here can be used to design novel superhydrophobic surfaces for enhanced condensation heat transfer. Significantly higher growth rate of droplet height in the upward direction than top curvature radius leads to stretched droplet surface, which stores more excess surface energy budget for forced droplet removal either through self- or coalescence jumping. In the case of coalescence jumping, a sudden pressure drop in the upper part of the stretched droplet occurs


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upon coalescence with a spherical droplet outside the pore. As the lower part is still confined by the pore and remains in high pressure, a pressure gradient is generated through the droplet, which is the driving force that leads to departure from the pores. Since the droplets in pores need to grow to a specific size at which they can reach the droplets outside pores, coalescence jumping always occurs at sizes determined by the hosting pores. Forced self-jumping of stretched droplets is also similar to the above case. As the droplet becomes increasingly large, its upper part expands and the pressure inside decreases. Meanwhile, lower part of the droplet is confined by the pore walls and remains in high pressure. This pressure difference pulls lower part upward, and at a specific size droplet base detaches from the base substrate, leading to subsequent self-jumping of the stretched droplet without coalescence. A similar case is the dewetting transition on superhydrophobic surfaces. 4, 22

In terms of energy, excess surface free energy stored in the stretched droplet interface is

the driving force for this surprising self-jumping. Theoretical model of droplet growth in micropores In order to build a theoretical model to probe the growth physics of stretched droplets in micropores, we conducted hydrodynamic analysis under the condition that droplet growth is a quasi-static process (see Figure 5a and Supporting Information S3). Before touching the pore walls when the droplet is relatively smaller, its growth process is supposed to be the same as that of spherical droplets, whose related growth mechanism can be found elsewhere.23,33,34 Therefore, our analysis starts at a droplet that is just big enough to touch the pore walls. Upon touching, it rapidly reaches an equilibrium leading to two triple-phase points on each walldroplet contact line (in cross-sectional view). The top point is marked with position angle ϕ1, and the bottom point is marked with position angle ϕ2. Position angle above the line joining the centers of mesh wires is considered positive, while position angle below this centerline is considered negative. Then subsequent quasi-static droplet growth process begins. And,



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contact angles at top and bottom triple phase points are assumed to be the same and remain constant during the whole growth process due to low contact angle hysteresis. A relationship between radius r of the top curvature of the droplet and its corresponding position angle ϕ1 is then obtained as cos 2 cos$

(1)

where R is the mesh wire radius; L is the center-to-center distance of micropores; α is the supplementary contact angle, thus α = 180˚ − θ* ≈ 17˚ in this work. Figure 5b shows, for R = 34 µm and L = 4R, how radius r of the top curvature changes as the top position angle ϕ1 increases in the upward direction along the pore wall. Interestingly, r does not increase monotonically. Initially pore slot in contact with the droplet gradually shrinks, squeezing the growing droplet and thus decreasing r. As the top triple phase point exceeds a specific position, pore slot in contact with the droplet then gradually widens, relaxing the droplet and thus increasing r. This slowly increasing radius corresponds to the experimental results given in Figure 4d. Note that relaxation does not occur where the pore slot is the smallest when ϕ1 is equal to zero. Instead, it occurs where the top curvature has minimum radius as % 0 %

(2)

which yields a characteristic position angle ϕc of the top curvature 2 & arctan + ∙ tan $

(3)

Inserting ϕc into Eq. (1), minimum radius rmin can be found. Note that Eq. (3) is only valid for large contact angles. Otherwise, exact expression should be obtained from Eq. S3.16. Figure 5b also shows that radius r of the top curvature becomes even smaller with decreasing pore size (i.e. pore size 2 2 ). This is because the smaller the pore is, the more confined the droplet is. Meanwhile, the Laplace pressure, which is inversely proportional to


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the radius of curvature (Δ. 2//), first increases and then decreases as shown in Figure 5c. The maximum pressure, corresponding to minimum radius rmin, is given by Δ.1 2

2/34 &

(4)

where σlv is the liquid-vapor surface tension. It is worth mentioning that Δ.1 2 is the socalled breakthrough pressure.35,36 In addition, pressure evolution is more drastic for smaller pore sizes. Furthermore, if a sudden coalescence with droplets outside pores takes place during growth, pressure in the upper part of the droplet decreases drastically and causes a huge pressure difference from the bottom part. This pressure difference is the driving force that immediately pushes the droplet out of the pore, leading to departure from the surface as shown in Figure 3. This phenomenon is called mixed coalescence jumping due to the involvement of stretched and spherical droplets, which is further discussed in droplet jumping section. Varying top curvature and pressure inside the droplet affects the position angle ϕ2 of the bottom curvature. Since the pressure inside the droplet tends to be uniform during a quasistatic process, radius of the top and bottom curvatures also tends to be the same, while hydrostatic pressure inside a microscopic droplet is negligible.37 Thereby, a relationship between radius r of the top curvature and position angle ϕ2 of the bottom curvature can be obtained as < 67 + 1 + sin 5 : ∙ sin ; + 5 = 4 2 5 5 < 5 cos 5 ∙ sin ; $= 4 2

(5)

Thus, 5 5> 5> 6 :

(6)

where D is the average distance from the base substrate to the bottom surface of mesh wires, which is added due to consideration of the wave-like surface profile of copper mesh. Figure



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5d shows, for D = R/2, that ϕ2 is negative within the plotted range of ϕ1, meaning it lies below the centerline of the pore throughout the whole growth process. ϕ2 first decreases (becomes more negative) due to the initial increase in pressure inside the droplet, implying that the bottom curvature is growing sideward and filling the pore corner with more water. Then, it starts to increase (becomes less negative) due to the decrease in the pressure when top position angle ϕ1 exceeds ϕc, implying that the bottom curvature is receding from the pore corner. As for the overlapping curves of bottom position angle ϕ2 for different values of wire radius R, it is because R, L, and D have a linear relationship here, although the effect of L and

D for a fixed value of R is also discussed in Figure S6 and S7. Diameter of the droplet base in contact with the solid surface is another variable which changes with position angles ϕ1 or ϕ2. Normalized base diameter 2b* is given by 7 1 2? ∗ @ 5 1 2 cos 5 + 2 A + 1 + sin 5 B ∙ +tan 5 + cos 5

(7)

where b* is the normalized base radius by interpore distance L. As shown in Figure 5e, 2b* initially increases due to the sideward expansion of the bottom curvature with the increasing Laplace-pressure in Figure 5c. Then it decreases as the pressure starts to decrease until the droplet base is no longer in touch with the solid surface. This offers an important boundary condition under which the droplet pulls itself from the pore and leave the surface—termed as “self-jumping” as mentioned previously. A characteristic position angle ϕ2c of the bottom curvature upon “self-jumping” is then expressed as 5& @> ? ∗ 0

(8)

And, another characteristic angle ϕ1c of the top curvature upon self-jumping is expressed as & > 65 5& :

(9)

which is the upper limit of ϕ1 in all plots in Figure 5. Self-jumping is further discussed in detail in droplet jumping section.


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Figure 5. Theoretical model of droplet growth in a micropore. (a) Experimental droplet morphology in a pore (left) and its schematic (right). (b) Evolution of top curvature radius r with top potions angle ϕ1. r initially decreases due to the shrinking pore slot in contact with the droplet. Then it starts to increase as the pore slot in contact with the droplet widens. (c) Laplace pressure evolution. Δ. is at its maximum when the top curvature has minimum radius. (d) Evolution of the bottom position angle ϕ2. ϕ2 first decreases because the bottom curvature is forced to expand sideward as the Δ. increases until its maximum. (e) Evolution 17 ACS Paragon Plus Environment


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of the droplet base diameter 2b*, in response to base curvature evolution. When 2b* becomes zero, the droplet is no longer bound to the base substrate and leaves the pore due to the suddenly created at the droplet base. In all plots, minus position angle means it is below the line joining the centers of the pore walls. Black, blue, and red curves are for three cases of R = 10 µm, R = 17 µm, and R = 25 µm, respectively.

It is shown in Figure 4d that droplets in pores grow at a significantly higher rate in the upward direction than their top curvature radius. Geometrical height evolution is obtained as ∗ 61 + sin $ : 1 sin

(10)

where ∗ is the visible droplet height—the difference of the actual droplet height and the

distance from the base substrate to the top surface of pores ( ∗ 2 7, where H is the actual droplet height). H* is used because the beam scan direction was almost in parallel to the condensing surface in experiments, thus we were able to see the droplets only when they grew beyond the top surface of pores. Results of Eq. (1) and Eq. (10), for R = 17 µm, are given in Figure 6 showing that H* increases by around 32 µm until the droplet is forced to leave the pore while radius r of the top curvature changes from around 18 to only around 25 µm. This theoretical growth pattern is in agreement with the experimental results given in Figure 4d, although position angle instead of time is taken as the independent variable due to the complication of heat transfer analysis. Growth mainly in the upward direction leads to stretched droplet surface body that stores more excess surface free energy than spherical droplets of the same size.


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Figure 6. Theoretical evolution of the visible height and radius of top curvature of a droplet growing in a micropore. This growth pattern is in agreement with the experimental results in Figure 4d, although independent variable is top position angle instead of time due to the complication of heat transfer analysis. Relatively faster increase in the droplet height than its top curvature radius leads to stretched droplet surface that stores more excess surface free energy than spherical droplets of the same size.

Pore-scale forced droplet jumping If two spherical droplets coalesce on a superhydrophobic surface, the resulting droplet may jump by taking advantage of the excess surface free energy released upon coalescence.2,38,39 We call this “conventional coalescence jumping” (Figure 7a) in order to differentiate the jumping involving stretched droplets in our work. The definition given above does not explicitly state two important reasons behind droplet jumping. (a) Release of the excess surface free energy is the result of a decrease in the total surface area of the droplets after coalescence. (b) Resulting droplet jumps in the direction normal to the condensing surface which breaks the symmetry of two momentum components in the upper and lower half of the resulting droplet. 1,38,39 Because of the second reason together with the incomplete release of the excess surface free energy upon jumping, conversion efficiency of the excess surface free 19 ACS Paragon Plus Environment


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energy to surface-kinetic energy is generally low. Another counteracting factor during coalescence and jumping is surface adhesion, which plays a more or less significant role depending on solid surface energy, solid fraction of the liquid-solid interface, and droplet size.6,40 In addition to this conventional coalescence jumping, droplet jumping can be further extended to self-jumping of a single stretched droplet (Figure 7b), and mixed coalescence jumping of stretched and spherical droplets (Figure 7c) as mentioned in previous sections.

Figure 7. Schematics of three types of droplet jumping. (a) Conventional coalescence jumping of two spherical droplets. (b) Self-jumping of a single stretched droplet without coalescence. (c) Mixed coalescence jumping of a stretched and a spherical droplet. Flow momentum of the spherical coalescing droplets has initially symmetrical up and down components, until this symmetry is broken by the condensing surface. Contrarily, flow momentum of the self-jumping droplet is largely in the upward direction, implying it has higher conversion efficiency of the excess surface energy budget to the kinetic energy. Energy conversion efficiency of the mixed coalescence jumping is expected to be between them.

Self-jumping of a single stretched droplet Experimental results in Figure 4c and the base diameter evolution in Figure 5e shows selfjumping of stretched droplets in micropores is possible due to the detachment of the droplet base at a specific droplet size determined by pore opening. Specifically, the reason for selfjumping is that droplets in micropores possess the characteristics needed for jumping as; (a) 20 ACS Paragon Plus Environment

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they are stretched, thus storing more surface free energy than spherical droplets of the same size. Note that effective solid fraction of the liquid-solid contact area is significantly small: fs ≈ 0.085 obtained from modified Young’s equation CDE ∗ F CDE + F 1.41 (b) If they grow without coalescence jumping to the characteristic angle ϕ1c (Eq.9), their base is no longer bound to the base substrate, which releases the droplets and leads to local high pressure at the lowest point of the droplet base. Since the droplet can expand only outside the pore, the pressure difference further results in unbalanced flow momentum. Moreover, this momentum is directed by the pore wall confinement mainly in the vertical direction, thus resulting in self-jumping (Figure 7b). Using the boundary conditions obtained from Eqs. 8 and 9, the volume of a stretched droplet upon self-jumping can be obtained through geometrical analysis (Figure 8a) as 1 GH 3 5T 5 @ 6sinN

Unidirectional Fast Growth and Forced Jumping of ... - ACS Publications

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