Growth Rates and Spontaneous Navigation of Condensate Droplets

Growth Rates and Spontaneous Navigation of Condensate Droplets Through Randomly Structured Textures Chander Shekhar Sharma, Juliette Combe, Markus Giger, Theo Emmerich, and Dimos Poulikakos* Laboratory of Thermodynamics in Emerging Technologies, Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland S Supporting Information *

ABSTRACT: Dropwise condensation is a phenomenon of common occurrence in everyday life, the understanding and controlling of which is of great interest to applications ranging from technology to nature. Scalable superhydrophobic textures on metals are of direct relevance in improving phase change heat transport in realistic industrial applications. Here we reveal important facets of individual droplet growth rate and droplet departure during dropwise condensation on randomly structured hierarchical superhydrophobic aluminum textures, that is, surfaces with a microstructure consisting of irregular re-entrant microcavities and an overlaying nanostructure. We demonstrate that precoalescence droplet growth on such a surface can span a broad range of rates even when the condensation conditions are held constant. The fastest growth rates are observed to be more than 4 times faster as compared to the slowest growing droplets. We show that this variation in droplet growth on the hierarchical texture is primarily controlled by droplet growth dynamics on the nanostructure overlaying the microstructure and is caused by condensation-induced localized wetting nonuniformity on the nanostructure. We also show that the droplets nucleating and growing within the microcavities are able to spontaneously navigate the irregular microcavity geometry, climb the microtexture, and finally depart from the surface by coalescence-induced jumping. This self-navigation is realized by a synergistic combination of self-orienting Laplace pressure gradients induced within the droplet as it dislodges itself and moves through the texture, as well as multidroplet coalescence. KEYWORDS: spontaneous, superhydrophobic, self-navigation, condensation, dislodging, metal surface, random

H

large number of such surfaces, both nanostructured as well as hierarchical, have been studied.5 Attention has been focused on the wetting state of microdroplets on nanostructured surfaces where droplets in a partially wetting intermediate Cassie state have been shown to result in higher individual droplet growth rate and hence higher overall heat transfer.6−9 Additionally, it has been reported that small droplets, below O(10 μm), maintaining a constant base area during precoalescence growth phase, grow faster than the droplets that maintain a constant contact angle during growth.10,11 In contrast to nanostructured surfaces, hierarchically textured surfaces can achieve stable and robust superhydrophobicity.12,13 However, in such textures, the droplets can nucleate on the top as well as within the microstructure. A number of studies have focused on dynamics of droplet coalescence and droplet departure on hierarchical surfaces14−18 and have shown that droplet ejection from the

eterogeneous condensation of water vapor is ubiquitous in a wide range of natural phenomena.1−3 It also constitutes an important class of industrial processes involving phase-change energy transport, such as many types of heat exchangers, fogwater harvesting, and water desalination.4 Dropwise condensation on partially wetted substrates, wherein vapor condenses in the form of droplets that are regularly shed from the surface, has been long established as an efficient means of phase change heat transfer. Nearly an order of magnitude higher heat transfer coefficients can be achieved during dropwise condensation as compared to filmwise condensation that involves formation of continuous condensate film on fully wetted substrates.4,5 The dropwise condensation cycle starts with nucleation, followed by growth, coalescence, and periodic droplet departure from the surface. The associated droplet dynamics are linked to the wetting state of the condensed droplets. Superhydrophobic surfaces are regarded as a promising pathway to achieve efficient dropwise condensation due to inherently low contact angle hysteresis, the resulting small droplet departure diameters, and high surface renewal rates. A © 2017 American Chemical Society

Received: November 6, 2016 Accepted: February 7, 2017 Published: February 7, 2017 1673

DOI: 10.1021/acsnano.6b07471 ACS Nano 2017, 11, 1673−1682

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Figure 1. (a) SEM image of hierarchical superhydrophobic surface showing the overall irregular texture with re-entrant cavities. (b, c) Zoomed in views showing the details of the typical microfeatures, microcavities, and overlaying nanostructure. (d) FIB section of planar nanostructured surface with details of the boehmite nanostructure (also see Supporting Information, Figure S9).

microstructure is critical to avoid surface flooding and loss of superhydrophobicity during condensation.15,19 Hierarchical surfaces consisting of well-defined regular, “laboratory-type” microfeatures, such as micropillars, micropyramids, or microcones, have been explored to trigger droplet departure by using Laplace pressure gradients within the droplet and coalescenceinduced droplet jumping. 14,16−18,20 Additionally, a few examples of biphilic hierarchical textures have also been presented that can spatially control droplet nucleation as well as well as droplet departure by including hydrophilic areas over an otherwise superhydrophobic surface.21−23 Although such hierarchical textures with precisely controlled microfeature dimensions can be optimized to achieve the desired droplet departure characteristics and may also be realizable in metals through fabrication processes such as laser ablation and templating, among others,22,24,25 such approaches do not lend themselves to be particularly suited for scalability to applications outside the laboratory environment. In contrast, we consider a scalable hierarchical texture in aluminum with randomly structured microfeatures fabricated by a facile approach. We demonstrate important facets of the mechanisms of condensate droplet growth, spontaneous motion, and departure dynamics on hierarchical superhydrophobic surfaces. Through careful measurements of precoalescence droplet growth, we demonstrate that condensation on such surfaces proceeds through droplets that span a broad range of growth rates, even when constant condensation conditions are maintained. We show that this variation in growth rate originates from condensation driven localized variation in effective surface wettability of the nanostructure overlaying the microstructure. In addition, we demonstrate that the droplets nucleating inside the microstructure are able to travel and eventually depart from the surface despite the absence of any regular topography, specifically engineered for this purpose,14,16−18 in the microstructure. The droplets spontaneously navigate the intricate reentrant geometry of the microcavities constituting the microstructure, dislodge from the microcavity, climb the microstructure through multiple coalescence events, and are finally removed by coalescence-induced droplet jumping.16,17,20 We explain the droplet self-navigation as resulting from the deformation of the growing condensate droplet due to the random, re-entrant microcavity geometry. This deformation induces favorable Laplace pressure gradients within the droplet that change orientation in three-dimensions according to the location of the droplet along its path within the geometry of the microcavity. We estimate that the droplet, during its motion through the cavity, contacts the microfeatures only over a fraction of its area, which facilitates its upward motion and

eventual dislodging. Taken together, the results show that it is possible to achieve droplet climbing and departure from hierarchical surfaces without the need of often tediously fabricated topographies, based on the repetition of optimized single cavity designs, which while serving well at the laboratory level, are difficult to upscale in applications and with industrially relevant surface materials.

RESULTS AND DISCUSSION Two scalable textures were fabricated on aluminum substrates: hierarchical textured substrate shown in Figure 1a−c and planar nanostructured surface shown in Figure 1d. The hierarchical texture consists of a microstructure formed by dislocationselective etching12,26 of aluminum. The overlying nanostructure is composed of boehmite (Al2O3·H2O(s)) obtained through the boehmitage process.27−29 All substrates were functionalized with trichloro-1H,1H,2H,2H-perfluorodecylsilane (FDTS) resulting in a superhydrophobic hierarchical aluminum surface.30 Figure 1a shows an SEM image of the hierarchically textured substrate revealing an irregular microstructure with re-entrant microcavities.26 Figure 1b shows details of the typical microfeatures, and Figure 1c shows a magnified view of a microfeature to reveal the details of the nanostructure overlaying the microstructure. Figure 1d shows a focused ion beam (FIB) milled cross-section of the planar nanostructured surface showing a morphology consisting of a nanostructure with a height of ∼300 nm (see Methods and Supporting Information for details of surface fabrication and cross-sectional image of the hierarchical texture). Spatially Varying Droplet Growth Dynamics on Hierarchically Structured Surfaces. The condensation dynamics on the hierarchically textured substrate are explored through in situ ESEM observations. During the ESEM experiments, the surface is maintained at a constant temperature by using a cooling stage, while the pressure is increased until a stable growth of condensed droplets is established. Individual droplets were followed to obtain quantitative data on the temporal evolution of droplet size prior to coalescence with neighboring droplets. We observed significant variation in the precoalescence growth rate of droplets. Figure 2 shows an example of this variation by comparing two droplets d1 and d2 that, starting from similar size, grow at different rates under the same saturation conditions. Figure 2a shows two image sequences illustrating the evolution of drop sizes for d1 and d2. Figure 2b shows the growth curves for the two droplets. Growth of condensed droplets follows a power law r = r0tα, where r is droplet radius,31 and r0 and α are constants. It is evident from Figure 2b that the growth of the two droplets 1674


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Figure 2. Droplet growth dynamics on hierarchical aluminum surface. (a) Image sequences showing growth of d1 and d2 over a period of 40 s and starting from same size of 6 μm. The initial times for image sequences, t0,1 and t0,2, are marked on the corresponding curves for d1 and d2 in (b). The scale bar represents 10 μm. Number in each image indicates droplet diameter in μm. (b) Individual droplet growth curves prior to coalescence for two droplets: d1 and d2. (c) Average droplet diameter growth rate curves. The dashed lines indicate corresponding power law fits. Numbers in parentheses indicate power law exponents.

cannot be described by the same power law fit. Starting from a droplet diameter of 6 μm, while d1 grows to 8.6 μm in 40 s, d2 grows much faster and achieves a size of 13.4 μm in the same time period, indicating a nearly 2.7 times higher growth rate for d2 as compared to d1. When measurements were repeated for multiple droplets, the resulting data led to five distinct average droplet growth rate curves as shown in Figure 2c (refer Supporting Information for details of data analysis). These growth rates are labeled as H1 to H5, and the associated power law exponent α ranges from a low of 0.38 to a high value of 0.54. This translates to the highest growth rate (H5) being nearly 4.5 times higher than the lowest observed growth rate (H1) (see Supporting Information). The droplet growth rates have also important implications to the associated heat transfer since precoalescence growth of small drops, up to a size of O(10 μm),32 contributes most of the heat transfer during dropwise condensation of water. The average heat transfer rate for individual droplets can be estimated as Q =

1 t

information for such droplets due to possibility of coalescence within the cavity, the obtained average growth curves illustrate a significant variation in droplet growth rates on hierarchical superhydrophobic substrates. Mechanism of Droplet Growth Nonuniformity. On the hierarchical texture, the droplets can nucleate and grow both within as well as on top of the texture. It can be reasonably speculated that the droplet growth rate is a function of the location of the droplet in the microstructure as the droplets growing on top of the microstructure may grow slowly due to the heat transfer resistance contributed by the vapor trapped inside any microcavities within the texture. However, we found no relationship between the rate of droplet growth and its location on the microstructure. Additionally, no evidence was found of a higher growth rate of droplets growing on convex microfeatures. This also eliminated any contribution from variation in diffusion flux toward variability in droplet growth rate.33 This is also expected as ESEM creates a water vapor environment with minimal concentration of noncondensable gases.34,35 With no observable dependence of the droplet growth rate on its location on the microstructure, we hypothesized that the difference in growth rate must be related to the wettability of the microdroplets on the nanostructure overlaying the microstructure. However, due to the varying orientation of the droplets growing on random microfeatures, it was not possible to measure the contact angles of the droplets and explore the local wettability for such droplets. In order to resolve this unknown important factor, we looked at condensation growth dynamics on a planar aluminum surface nanostructured by the same boehmitage process as used to create the nanostructure for the hierarchical surface.

d

∫2r max ρw hfg dV min

where t is the time for droplet to grow from critical droplet nucleation radius (rmin) to a maximum diameter dmax before coalescence, ρw is the condensate density, and hfg is the latent heat, respectively.7 Using the fitted power law exponents, this estimation shows that H5 droplets individually transfer more than 4 times more heat as compared to H1 droplets (see Supporting Information). It is evident that the net heat transfer during condensation will depend upon the spatial distribution of these growth rates on the surface. We emphasize that these average droplet growth rate curves are based on the data for droplets growing outside the microcavities. Additional growth rates may be revealed if precoalescence growth of droplets inside the microcavity could be measured. Despite the lack of 1675


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evaporating droplets in between condensation cycles (consisting of alternating condensation and evaporation for multiple ESEM experiments on a substrate)35 and are formed by the coalescence of minute droplets within the nanostructure, which do not dry completely when the water vapor pressure falls below saturation pressure.10,11 Along with unavoidable microscopic defects in the silane layer, such wetted spots act as preferential sites for nucleation in the next condensation cycle10 as shown in Figure 3a (see Supporting Information, Video S1). We observed that there is no unique size of such wetted spots and that there are significant local size variations (see Figure S3 and Video S2). During a condensation cycle, the entire wetted area acts as the site for the growth of a new droplet. This results in a significant difference in contact angles of the growing droplet during the early stages of its growth. Droplets growing from larger wetted spots have smaller contact angles which increase with droplet size.11 Unlike hierarchical surface, the contact angles of the growing droplets on the nanostructured surface can be measured (refer Supporting Information). Figure 3b shows evolution of contact angle for two droplets: dN1 and dN2 with power law exponents αN1 < αN2 (see Figure S2). The contact angles of the droplets change as the droplets grow and stabilize in the range of 150°−160°.11 However, the droplet dN2 shows significantly lower contact angles during its growth as it evolves from a comparatively larger wetted spot. The lower contact angles translate into the lower interfacial area for any given droplet volume. Additionally, the larger base area available for heat transfer reduces the overall heat transfer resistance resulting in faster droplet growth rate for dN2. Hence, the formation of the wetted spots results in a condensationinduced wettability variation on the nanostructure, which in turn results in droplet growth variation on the hierarchically textured surface. It should be noted that this droplet growth variation on hierarchical texture has been observed in the absence of noncondensable gases. However, we expect that the presence of noncondensable gases will further enhance droplet growth nonuniformity on the hierarchical texture due to the additional contributing factor of local diffusion flux nonuniformity.33 Droplet Self-Dislodging from Irregular Hierarchical Surface. Superhydrophobic hierarchical surfaces usually lose their superhydrophobic property during condensation, because of the unavoidable nucleation and growth of the droplets inside the fine features of the topography. Small generated droplets become trapped in the texture leading to the flooding of the texture and loss of superhydrophobicity. This behavior has been reported for both natural (lotus leaf)19,36,37 as well artificial hierarchical textures.15,38 However, a few laboratory-type hierarchical substrates consisting of micropillars and microcones have been carefully designed wherein the geometrical parameters of the regular, periodic micro features are optimized to cause condensate droplet ejection through droplet coalescence and Laplace pressure gradients.16−18 In contrast to the existing studies, we demonstrate here that it is possible for the droplets to eject even irregular, random, microtextures. The hierarchical surface formed by dislocationselective etching of aluminum results in an irregular microstructure (Figure 1), wherein the growing condensate droplets are still able to spontaneously self-navigate the irregular geometry of microcavities. In addition to droplet dislodging from microcavities, we also observe that the droplets show a general tendency to climb the microtexture through a

The nanostructured aluminum substrate was tested under the same conditions as the hierarchical surface. Droplet growth data were obtained for multiple droplets, and using the same procedure as for the hierarchical sample, we found four average droplet diameter growth curves (see Figure S2). The power law exponent for the droplet growth curve ranges from 0.48 for the slowest group to 0.62 for the fastest growing group, thus showing a significant degree of overlap with the growth rates on the hierarchical surface (see Figures 2 and S2). This overlap emphasizes the fact that the droplet growth dynamics on the hierarchical surface are primarily governed by the corresponding dynamics on the nanostructure overlaying the microstructure. The wetting state of the individual droplet on the microfeatures cannot be directly observed, as the height of the nanostructures lies below the resolution achievable in ESEM. However, the condensate droplets are expected to wet the nanostructure through the formation of wetted spots9,10 because the critical nucleation radius r min ≈ 2γT sat / (ρlhfgΔTsubcooling) is expected to be below 1 μm and of the same order in size as the width of and spacing between the nanostructures overlaying the microstructure (shown in Figure 1d). We observed distinct wetted spots on the nanostructured sample as shown in Figure 3a, thus supporting the above argument that the condensing droplets wet the nanostructure (also see Figure S3). These wetted spots are left by the

Figure 3. (a) Preferential nucleation and growth of condensate droplets on wetted spots as marked for four droplets. The scale bar in the images indicates 10 μm. (b) Contact angle evolution with droplet radius for two droplets dN1 and dN2 with αN1 < αN2. Inset figures show the droplets and the respective contact angles at the start of the contact angle growth curves. The scale bars in the two inset figures represent 5 μm. 1676


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Figure 4. Two modes of droplet dislodging from a microcavity. (a) Mode 1A: Droplet 1 grows from within the cavity, while droplet 2 grows outside the cavity (panels i−ii). Droplet 1 is distorted and bulges outward (panel iii) until droplets 1 and 2 coalesce, and the resulting droplet is dislodged from cavity (panel iv). (b) Mode 1B: Droplet 1 grows from within the cavity (panel i) until it is constrained by cavity walls, leading to droplet distortion (panel ii). The resulting Laplace pressure gradient within the droplet dislodges the droplet out of the cavity (panels iii−iv). The scale bars in the panels indicate 20 μm. The two dislodging modes are elucidated by corresponding schematics in (c) and (d).

that the sudden increase in droplet size induces instability in the already deformed droplet, which helps in overcoming any pinning inside the cavity, thus aiding the dislodging of the droplet from the cavity. Figure 4a shows an example of a droplet dislodging from a cavity as a result of coalescence with a droplet in the vicinity of cavity opening (see Video S3). Figure 4c shows a schematic for Mode 1A. Mode 1B: In rare cases, there are no droplets nucleating and growing in the vicinity of the cavity opening. In such situations, the droplet dislodging from cavity is purely driven by the gradually increasing Laplace pressure differential in the droplet as the droplet growth continues (bottom part restrained by the cavity and the top part not). Figure 4b shows an example of such a droplet. The droplet labeled as ‘1’ nucleates and grows from within a microcavity. As the droplet grows, it eventually dislodges itself out of the cavity due to Laplace pressure gradient caused by the deformation of the droplet (see Video S4). Mode 1B is illustrated by a schematic in Figure 4(d). Step 2: Droplets departure f rom surface. Following the droplet dislodging from microcavity, the droplet departure from the surface can also be classified into the following two distinct modes: Mode 2A: Droplet coalesces with neighboring droplets growing in the vicinity of the cavity leading to multiple droplet coalescence-induced jumping from the surface.16,17,20 Droplet 1 from Figure 4b combines with surrounding droplets and departs the surface by coalescence-induced jumping as shown in Figure 5a (see Video S4). Figure 5b shows another example of Mode 2A droplet departure from a relatively deeper microcavity, wherein a droplet bulging out of cavity (droplet 1) coalesces with droplets 2, 3, and 4 present on the side walls

synergistic combination of coalescence driven droplet jumping. At the top of the texture, the droplets either depart from the surface through recoalescence with neighboring droplets and subsequent jumping16,17,20 or can be removed by gravity or vapor shear.39 A careful examination of a number of droplet dislodging events has revealed two major steps comprising the removal of the condensate droplets nucleating within the microcavities: (1) droplet growth inside microcavity followed by outward motion and dislodging from the microcavity (Figure 4) and (2) subsequent departure from the surface (Figure 5). Step 1: Growth of the droplet inside a microcavity and subsequent dislodging f rom the cavity: A droplet, that nucleates inside a microcavity, grows freely until the size of the droplet becomes similar to that of the formation cavity. Beyond this point, the growing droplet is constrained by the walls of the cavity. As droplet growth continues, the constrained droplet becomes distorted, and a Laplace pressure difference is setup within the droplet. The droplet distortion continues to increase with droplet growth until the droplet is dislodged from the cavity when sufficient favorable Laplace pressure gradient is setup. Two distinct modes of droplet dislodging from microcavity were observed: Mode 1A: While a droplet grows from within a microcavity, there are other droplets nucleating and growing outside the cavity and in the vicinity of the cavity opening. In most cases, the bulging interface of the droplet growing from within the cavity coalesces with one or more of such droplets. This coalescence event causes a sudden increase in the droplet size and sets up a favorable pressure gradient within the droplet that dislodges the droplet from the cavity. Additionally, we speculate 1677


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Figure 5. Modes of droplet departure from surface. (a) Mode 2A in shallow cavity: Droplet 1 coalesces with large droplets 2, 3, 4, and 5 (along with a few unmarked small droplets) in panel i and departs by coalescence-induced jumping in panel ii. (b) Mode 2A in deep cavity: Droplet 1 growing from within a microcavity coalesces with droplets 2, 3 and 4 and departs by coalescence-induced jumping. (c) Mode 2B droplet departure from surface. Droplets 1 and 2 coalesce with droplet 3 at top of the texture (panels i−ii). The resulting coalesced droplet 4 sits at the top of the texture (panel iii). Later droplets 4, 5, and 6 coalesce and jump along with a few smaller droplets (panel iv). The scale bars indicate 20 μm (see Videos S4, S5, and S6). The two droplet departure modes are elucidated by corresponding schematics in (d) and (e).

along a nearly perpendicular direction, p2, as shown in panels (iv−vi) as the droplet keeps growing contacting different segments of the random re-entrant cavity (see Supporting Information, Video_S7). Figure 6b relates the droplet movement to evolution in droplet size and curvature differences induced in the droplet along p1 and p2. These differences are 1 1 1 1 given by ΔK12 = R − R along p1 and ΔK34 = R − R along

of a large and deep microcavity, resulting again in a coalescence-induced jumping departure (see Video S5). A schematic for Mode 2A droplet departure is shown in Figure 5c. Mode 2B: Droplets present on the side walls of the texture coalesce with droplets on top of the texture and climb the texture in the process. The resulting coalesced droplet sits at the top of the texture indicating that the droplets on the side walls of microcavities are less pinned as compared to droplets at the top of the texture. As a result, a net mass transport toward the top of the microstructure can be observed. At the top of the texture, droplets depart either by subsequent coalescenceinduced jumping or remain pinned and grow. In the latter case, we expect that such droplets can be removed through gravity or vapor shear. Figure 5d shows an example of droplet departure following Mode 2B. Droplets 1 and 2 grow on the deeper microfeatures and coalesce with droplet 3 growing on top of microstructure (panels i−ii). The coalesced droplet 4 sits at the top of the texture. Eventually, droplets 4, 5, and 6 depart by coalescence-induced jumping (see Video S6). Figure 5e illustrates Mode 2B by a schematic. How Droplets Navigate through Intricate Microcavity Geometry. We next investigate how droplets are able to navigate and dislodge from the highly irregular geometry of the microcavities. The droplet self-navigation process through the cavity geometries is shown as an image sequence in Figure 6a. The droplet movement in this image sequence progresses through two stages. The first stage, shown in Figure 6a, panels (i−iii), consists of a droplet moving primarily along a direction p1 (marked in all panels), which is followed by movement

1

2

3

4

p2, where R1, R2, R3, and R4 are the radii of four interfaces of the droplet as illustrated in the insets in Figure 6b. The droplet size is shown in terms of an equivalent diameter of the circular droplet deq. Additionally, the droplet displacement, S, from initial position in panel (i) is also plotted (refer to Supporting Information for details on measurements). We start our observation at panel (i) that shows the growing droplet constrained on two sides by the microcavity. Prior to this time instant, the droplet grows from within the microcavity similar to as described for droplet 1 in Figure 4b. This motion is normal to the plane of the image and the driven by curvature gradient in the third dimension (not measurable in the twodimensional image). Starting from the deformed shape in panel (i), as the droplet grows, this normal curvature gradient leads to the droplet depinning from the deforming edge of the cavity wall and relaxing to a comparatively less deformed shape, as shown in panel (ii). No significant droplet movement is observed along p1 in panels (i−ii) due to the unfavorable curvature difference along p1 (i.e., ΔK12 > 0, refer to blue curve in Figure 6b). However, at 10 s, ΔK12 turns negative, and the resulting favorable Laplace pressure gradient in the droplet pushes the droplet along p1 and free from the constraining 1678


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Figure 6. A case of droplet self-navigation. (a) Major steps in ejection of the droplet: The numbers mark the position of droplet centroid. The profile of the droplet is traced to highlight the droplet shape. Each frame includes droplet profile in the previous step for reference. Panels i− iii: Droplet grows and ejects itself from the inner part of cavity along direction p1 (path 1−2−3). Panels iv−vi: Droplet grows and ejects itself from the outer part of the cavity along p2 (path 3−4−5−6). The scale bars represent 10 μm. Direction of net force is marked as FL by arrows of corresponding color. (b) Evolution of droplet curvature differences (ΔK12 and ΔK34), droplet size (deq), and droplet displacement from initial position (S). Numbers along the S curve correspond to area-centroid positions marked in (a). Horizontal dashed line corresponds to ΔK12 = ΔK34 = 0 (spherical droplet). Vertical dotted line separates regions of droplet motion along p1 (panels i−iii in (a)) and p2 (panels iv− vi in (a)). Inset figures show definitions of four radii of curvatures R1, R2, R3, and R4 (see Video S7).

steps that correspond to step jumps in droplet size resulting from coalescence with neighboring smaller droplets. Such coalescence events create instability that depins the droplet from the surrounding constraining walls, and the already existing favorable Laplace pressure gradient pushes the droplet outward from the cavity. As the droplet comes out of the cavity, the deformation in the droplet diminishes, and both ΔK12 and ΔK34 return to zero beyond 60 s. This movement of the droplet in Figure 6 provides an indirect visualization of what we speculated for the droplet growing from within a microcavity and coalescing with droplets in the vicinity of cavity opening (see droplet 1 in Figure 4a). The Laplace pressure gradient arising due to the curvature 2σΔK difference in the droplet can be estimated as ∇p ∼ d where

walls into a nearly undistorted spherical shape. This movement is shown in Figure 6a, panel (iii), and reflected by the large step jump in droplet displacement S in Figure 6b (black curve). Notice that at this time instant (17 s after the start of observation), ΔK12 attains a nearly zero value. It should be noted that up to this point, ΔK34 does not affect droplet movement, as the droplet is constrained to move only along direction p1. Subsequent to panel (iii), the droplet gets constrained along p1 but is free to move along p2. Hence, for this part of the droplet navigation, the droplet movement is controlled by ΔK34. ΔK34 turns negative at 29 s and stays negative for subsequent time instants, thus creating a sustained favorable Laplace pressure gradient. Consequently, the droplet is gradually pushed out of the cavity along p2 (panels iv−vi) in

eq

deq is the equivalent droplet diameter. Integrating the pressure 1679


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entrainment in vapor flowing toward the surface.8,15,43 However, this limitation can be significantly overcome in the presence of vapor shear where vapor shear can force a departure of droplets coming out of the microtexture and thus increase the surface renewal rate to delay flooding even at a high degree of subcooling.39 Although the surface under consideration here has shown effective droplet jumping departure under both in situ ESEM as well as large-scale sample investigations, it would be interesting to investigate in the future the self-navigation behavior of droplets in similar hierarchical textures at high subcooling and in the presence of vapor shear. Additionally, such hierarchical superhydrophobic textures can also be compared against biphilic surfaces fabricated for the harvesting of water from fog. It has been recently shown that biphilic surfaces may collect more water as compared to uniformly superhydrophobic surfaces, as droplet jumping departure may not be ideal for such applications due to entrainment and loss of small jumping droplets to the air flow in the vicinity of the surface.23 However, for an irregular hierarchical texture such as the one considered here, it is also important to consider the direction of droplet jumping and the size distribution of the jumping droplets, as all drops will not jump normal to the surface and thus not lost to air flow. In essence, the two kinds of surfaces, hierarchically structured uniformly hydrophobic surfaces and biphilic surfaces, will need to be tested under the same operating conditions relevant to the specific application concerned, especially the degree of subcooling, presence or absence of noncondensable gases, and amount of vapor shear available. Lastly, scalability of fabrication and applicability to metals will also be a key consideration. In contrast to structured hierarchical textures, etching-based random hierarchical textures on metals, that trigger self-removal of condensate droplets, promise a cost-effective and scalable means of achieving enhanced dropwise condensation heat transfer and implementation in realistic technological applications.

gradient over the droplet volume yields an estimate of the π capillary force acting on the droplet40,41 FL ∼ − 3 σ ΔKdeq 2 , where the negative sign ensures a positive force in the direction of droplet movement when ΔK < 0 (as indicated in Figure 6a). Since the droplet size is comparable to the size of microfeatures, the force resisting the motion of the droplet arises from contact angle hysteresis on the nanostructure and can be estimated as FR ∼ fπσdeq(cos(θr) − cos(θa)), where θr and θa are the advancing and receding angles on the nanostructure and f is the fraction of the droplet circumference in contact with the hierarchical texture. Figure 6 shows that ΔK needs to turn negative to trigger movement of the droplet. However, the droplet moves when the capillary force FL exceeds FR, which is a necessary and sufficient condition for the droplet movement. This translates to a critical value for f as fc ∼

−ΔKdeq 3(cos(θr) − cos(θa))

with the droplet moving when f < fc. This condition means that for any droplet size deq, lower hysteresis on the microfeatures will ensure droplet ejection even at small curvature gradients, thus emphasizing the need for nanostructure. Conversely, for any given nanostructure, a microstructure that generates larger values of favorable (negative) curvature gradients will ensure easier droplet ejection. For hierarchical textures consisting of structured microfeatures such as micropillars or microcones, fc is a constant, and a critical size of droplets ejecting from the hierarchical texture can be estimated.18,42 However, for an irregular microstructure as the one considered here, fc depends on the local geometry of the microcavity leading to site-specific droplet departure radius. Even for a single microcavity, the irregular geometry leads to change in fc depending on the location of the droplet inside the cavity geometry. For example, for the droplet considered in Figure 6, fc ranges from a low value of ∼0.15 to a high of ∼0.94, indicating that the droplet contact area changes during its movement through the cavity geometry (see Figure S6 and associated text in Supporting Information). This example illustrates how the Laplace pressure gradient in the droplet changes orientation synergistically along the location of the droplet in a three-dimensional geometry of the irregular microcavity. In essence, the geometry of the cavity itself steers the droplet by changing the direction of the Laplace pressure gradient and satisfying the necessary and sufficient condition for droplet movement as discussed above. The net result is spontaneous droplet self-navigation through the cavity until it is completely ejected out of the cavity without any need for design of regular micro features, with precisely controlled geometrical parameters, attainable for only limited classes of materials in a facile, upscalable manner. In order to look at the large-scale behavior of the hierarchical texture, we have performed a preliminary investigation of the droplet dynamics on a 2 cm × 2 cm surface under pure water vapor as well as humid air environment. We have observed clear evidence of coalescence-induced droplet jumping departure, thus reconfirming that the droplets dislodging out of the irregular microcavities can depart the surface through droplet jumping (refer to Video S8). Superhydrophobic surfaces are susceptible to flooding at (a) large degrees of subcooling due to increased nucleation density and faster droplet growth rate, even when the means of droplet departure additional to gravity, such as droplet jumping, are active, as well as (b) increased pinning due to return of departed drops back to the surface under gravity and

CONCLUSIONS We demonstrate, through in situ experimental investigations, important aspects of droplet growth, dislodging, navigation, and departure dynamics on randomly structured hierarchical superhydrophobic textures consisting of a network of microcavities formed by dislocation-selective etching of aluminum. We report that precoalescence individual droplet growth rates vary significantly on these textures, with the fastest average droplet growth rate recorded being more than 4 times faster than the slowest growth rate. The associated individual droplet heat transfer is estimated to be 3.5−4 times higher for the fastest growing droplets as compared to the slowest ones. We show that this variation in growth rate on the hierarchical texture is primarily controlled by droplet growth dynamics on the nanostructure overlaying the microstructure. Droplets nucleate and grow on top as well as inside the hierarchical texture. However, we show that, in spite of the irregular and reentrant geometry of the microcavities, the droplets nucleating inside the cavities are able to spontaneously dislodge themselves, navigate the cavity geometry, climb the microtexture, and depart the surface by coalescence-induced droplet jumping. During droplet ejection from cavity, the capillary forces created by deformation of the droplets change orientation according to the cavity geometry and force the droplets out of the cavity as the droplets grow by direct condensation or coalescence with neighboring droplets. The 1680


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ACS Nano spontaneous droplet departure from the irregular hierarchical surface is also observed in macroscale condensation experiments. These facets of droplet dynamics can be generally useful for engineering scalable hierarchical textures for enhanced dropwise condensation.

bulging interface coalesces with droplet 2 growing in the vicinity of cavity opening, leading to dislodging of the coalesced droplet from the cavity (AVI) Video S4: Droplet dislodging from a microcavity driven purely by Laplace pressure gradients (Mode 1B). Droplet 1 gets distorted by the microcavity walls, and the resulting Laplace pressure gradient dislodges the droplet from the microcavity. The dislodged droplet coalesces with droplets growing in the vicinity of the cavity opening and departs the surface by coalescence-induced jumping (Mode 2A) (AVI) Video S5: Mode 2A droplet departure from deep microcavity. Droplet growing from within a deep microcavity coalesces with already de-pinned droplets along the side-walls of the microcavity and dislodges from the surface by coalescence-induced jumping (AVI) Video S6: Mode 2B droplet dislodging from hierarchical texture. Droplets 1 and 2, that are growing on deeper microfeatures, climb the microtexture by coalescing with a droplet that is growing on top of the microtexture (droplet 3). Eventually the coalesced droplet coalesces with other neighboring droplets on top of the microtexture and dislodges from the surface by coalescenceinduced jumping (AVI) Video S7: Droplet self-navigation through a microcavity. Droplet first moves normal to image plane, followed by movement along direction p1 and subsequently along a nearly perpendicular direction p2. Eventually the droplet dislodges from the microcavity by coalescence with a larger droplet growing outside the microcavity (AVI) Video S8: Macroscale condensation observations on hierarchically textured surface. Results for condensation in the absence as well as presence of air are shown for a 2cm × 2cm sample. For both kind of experiments, close up views of two droplet jumping events are also presented (AVI)

METHODS Fabrication of Nanostructured and Hierarchical Superhydrophobic Surfaces. 99.5% Aluminum sheets (AW1085) from Metall Service Menziken AG were used as the substrate. The asreceived sheets were ultrasonically cleaned for 10 min each in acetone, isopropyl alcohol, and deionized water. To prepare nanostructured surface, the substrate was cleaned in 1 wt % solution of NaOH for 2 min followed by etching with boiling water for 10 min (boehmitage process).29 For hierarchically textured surface, the sheets were cleaned in NaOH solution for 10 min, followed by dissociation-selective etching with 1 M FeCl3 solution for 25 min and at 25 °C, accompanied by ultrasonic cleaning with isopropyl alcohol every 2.5 min.12 This step created the microstructure. This was followed by the boehmitage process to create the overlaying nanostructure. Finally, superhydrophobicity was achieved by immersing both kinds of samples in 1.43 m(M) solution of trichloro-1H,1H,2H,2H-perfluorodecylsilane (FDTS) (Sigma-Aldrich) in n-hexane for 2 h and then baking for 45 min at 120 °C12 (refer Supporting Information for further details). Microcondensation Investigation in ESEM. FEI Quanta 600 ESEM equipped with a gaseous back scattered electron detector was used to study droplet nucleation and growth on the superhydrophobic surfaces. The sample temperature was controlled through a Peltier cooling stage (Emott AG). The samples were mounted using custommade copper stubs that allowed imaging at beam incidence angles of 45° and ∼87.5°. In order to minimize the beam heating effects,7,44 a beam voltage of 20 kV and a spot current of 0.16 nA were used, and the viewing area was kept above 100 μm × 100 μm for all the experiments. The cooling stage was set to 2 °C, and the chamber pressure was slowly increased until the onset of condensation, following which the chamber pressure was kept constant in the range of 0.7−0.8 kPa. The obtained images were analyzed using ImageJ and MATLAB (refer Supporting Information).

ASSOCIATED CONTENT S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.6b07471. Details on optimization of surface fabrication parameters to obtain nanostructured and hierarchically textured superhydrophobic surfaces; procedure for determination of average droplet growth rates; details on calculation of growth rate and heat transfer rate ratios between fastest and slowest growing droplets; average droplet diameter growth rate curves for nanostructured surface; preferential droplet nucleation observed over large area; details on measurement of droplet contact angles; details on estimation of radii of curvature, size, and position of deformed droplet as it navigates a microcavity; details on relationship between area fraction and droplet movement; EDS analysis; cross-sectional views of hierarchical surface and experimental details for macroscale condensation experiments (PDF) Video S1: Wetted spots on a nanostructured sample and preferential droplet growth from the wetted spots (AVI) Video S2: Preferential growth of droplets from wetted spots on the nanostructured sample (larger observation area) (AVI) Video S3: Mode 1A droplet dislodging from a microcavity. Droplet 1 grows from within a microcavity, and its

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. ORCID

Dimos Poulikakos: 0000-0001-5733-6478 Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS We gratefully acknowledge the funding from Commission for Technology and Innovation (CTI) under the Swiss Competence Centers for Energy Research (SCCER) program (grant no. KTI.2014.0148) and European Research Council (ERC) Advanced Grant (grant no. 669908 INTICE). We thank Asel Maria Aguilar Sanchez and Gabriele Peschke from the Institute for Building Materials, ETH Zurich for their support of ESEM measurements and Dr. Karsten Kunze and Dr. Joakim Reuteler from Scientific Center for Optical and Electron Microscopy, ETH Zurich for their help in surface characterization with FIB, BIB, and EDS analysis. We also thank Dr. Tanmoy Maitra of Stanford University for his input on surface fabrication and Jovo Vidic and Peter Feusi, ETH Zurich, for their assistance in construction of macroscale condensation experimental setup. 1681


Article

ACS Nano

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Growth Rates and Spontaneous Navigation of Condensate Droplets

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