Dewetting Transitions of Dropwise Condensation on Nanotexture

Cunjing Lv, Pengfei Hao,* Xiwen Zhang, and Feng He

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Dewetting Transitions of Dropwise Condensation on NanotextureEnhanced Superhydrophobic Surfaces Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

ABSTRACT Although realizing dewetting transitions of droplets

spontaneously on solid textured surfaces is quite challenging, it has become a key research topic in many practical applications that require highly efficient removal of liquid. Despite intensive efforts over the past few decades, due to impalement of vapor pockets inducing strong pinning of the contact lines, how to realize the selfremoval of small droplets trapped in the textures remains an urgent problem. We report an in situ spontaneous dewetting transition of condensed droplets occurring on pillared surfaces with two-tier roughness, from the valleys to the tops of the pillars, owing to the nanotexture-enhanced superhydrophobicity, as well as the topology of the micropillars. Three wetting transition modes are observed. It is found that a further decreased Laplace pressure on the top side of the individual droplets accounts for such a surprising transition and self-removal of condensed water. An explicit model is constructed, which quite effectively predicts the Laplace pressure of droplets trapped by the textures. Our model also reveals that the critical size of the droplet for transition scales as the spacing of the micropillars. These findings are expected to be crucial to a fundamental understanding, as well as a remarkable strategy to guide the fabrication, of optimum super-water-repellant surfaces. KEYWORDS: spontaneous transition . nano/microstructures . superhydrophobic . dropwise condensation . individual droplets . Laplace pressure

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nderstanding and realizing spontaneous wetting transitions (e.g., from the sticky Wenzel state to the nonsticking water-repellent Cassie
Baxter state) and self-removal of droplets on waterrepellent materials is highly desired and of critical importance for a wide range of practical applications, such as self-cleaning, anti-icing, antifouling, water harvest, and drag reduction,1
14 and particularly in heat exchange technologies.15
18 It is well known that dropwise condensation achieves heat and mass transfer coefficients over an order of magnitude higher than its filmwise counterpart15,16 because small individual droplets can regularly form and shed off the surface before a thick liquid is formed, thereby minimizing the thermal resistance to heat transfer across the condensate layer. Recently, there has been significant interest in developing superhydrophobic surfaces for promoting dropwise condensaLV ET AL.

tion.2,3,15
31 Such surfaces benefit greatly from the combination of nano/microstructures and the inherent hydrophobicity of their chemistry, which allows attaining extreme nonwetting properties with vapor trapped underneath (Cassie state) and coalescenceinduced self-propelled dropwise condensation.2,3,15,19
23,28
31 Unfortunately, even the most optimal natural superhydrophobic material, lotus leaves, ultimately become sticky (Wenzel state) to condensed water,4 which strongly suppresses its water repellence and use in practical applications. It is well known from experiments, due to impalement of the vapor pockets, inducing strong pinning of the contact lines, that the transition from the Cassie to Wenzel state is an irreversible event.32,33 Although the transition from Wenzel to Cassie state can be induced by some external assistance (e.g., applying pressure/force to the droplet,34,35 electric voltage36
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* Address correspondence to [email protected]. Received for review September 6, 2015 and accepted November 13, 2015. Published online 10.1021/acsnano.5b05607 C XXXX American Chemical Society

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RESULTS Silicon wafer substrates with square-shaped micropillars (Figure 1A, B), with side length L and spacing S LV ET AL.

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mechanical vibrations,4,40 heating the substrate41), these methods suffer from some inherent disadvantages, e.g., the heating and current pulse36,41 result in additional heat input and may be undesirable in biological systems; the mechanical vibration4,40 is difficult to implement in lab-on-a-chip applications; and the magnetic method39 has special requirements for the materials. For these reasons, how to realize wetting transitions to promote self-removal naturally and spontaneously on rough surfaces remains a key challenge, being particularly difficult for smaller droplets due to the dominant barrier that derives from the pinning of the contact lines, scaling as γr
2 per mass6 (r characterizes the size of the droplet, and γ is the surface tension of the liquid). Although dewetting transitions of condensed droplets on superhydrophobic materials have obtained some significant achievements, these phenomena were employed by droplet coalescence occurring on the surface of the textured substrates3,20,22
24,29,30 or the tops of the pillars17,19,21,28 due to a small contact area. Expelling droplets trapped in materials via spontaneous dewetting transition has been not fully attained, and our fundamental understanding of the underlying mechanism remains elusive. Significantly, a novel self-removal of condensed microdroplets from the legs of water striders was very recently reported,2 owing to the elastic deformation of the network of setae by growing drops, as well as extremely desirable wetting properties of the hierarchical micro- and nanostructures.42,43 Here, by employing excellent superhydrophobicity at the nanoscale and topologies of the microstructures (i.e., two-tier hierarchical rough surfaces), we not only suppress the pinning of the contact lines to a minimum but also stimulate a Laplace pressure difference on the top and bottom sides of droplets, allowing unexpected, spontaneous squeezing out of small individual droplets (e10 μm) from the valleys of the micropillars. Different from the recent three-step self-removal of condensed water on the legs of water striders,2 three unexpected dewetting transition modes are observed in our experiments. An explicit model is developed that not only is exceptionally effective at predicting the Laplace pressure of the droplet deformed by the pillars, but also establishes a direct link between the critical size of the droplet for transition and the geometry of the substrates. We also systematically investigate the effect of the geometry on the probability of different transition modes, which reveals that decreasing the space of the pillars constitutes a good strategy to promote the transition event. We believe that these novel phenomena, as well as our fundamental understanding of them, shed light on the design and practical applications of superhydrophobic nanomaterials.

Figure 1. Topology of the two-tier nano/microstructured surface (L = 3 μm, S = 4.5 μm, H = 5 μm) and its roughness characterized by SEM in micro- (A) and nanoscale (B, C), respectively. (C) and (D) are measured on flat surfaces. (D) is characterized by AFM, and the roughness at the nanoscale is Ra = 24.5 nm, which corresponds to a 1 μm  1 μm area.

of the neighbors, are employed. The height H of the micropillars in this work is fixed at 5 μm. The key feature is that the surface of the micropillars and the bottom of the substrate are treated with a commercial coating agent,23,44,45 in which hydrophobic nanoparticles (Figure 1C,D) are contained to guarantee excellent superhydrophobicity; that is, even on flat one-tier nanotextured surfaces (Figure 1C), the apparent contact angle and hysteresis are found to be θ = 159.2° ( 1.5° and Δθ = 10.2° ( 2.1°, respectively. Figure 1D is characterized using AFM on a 1 μm  1 μm flat area (Ra = 24.5 nm) and shows that the coating is composed of self-assembled nanoparticles with a fractal-type structure, whose space and depth range from about 100 to 300 nm and 50 to 200 nm, respectively. There are four types of samples investigated, with L = 2, 3, 4, and 8.4 μm, respectively, and S = 3, 4.5, 6.5, and 13 μm, respectively. Under an ambient environment, all of them demonstrate high apparent contact angles (>160°) and low hysteresis ( 60 s, when the size of the droplet reaches approximately 10 μm, some of them unexpectedly climb to the top of the micropillars from the valley (e.g., one droplet marked as A from t = 60 s to t = 100 s and two droplets marked as B from t = 100 s to t = 120 s); that is, dewetting transitions occur spontaneously without any external assistance. What we observed is completely different from the partial Wenzel to Cassie transition for quite big droplets, whose size is much larger than the pillars on the substrate,25 or external force induced dewetting (e.g., intense electric pulse36
38 or vibration4,40). Subsequently (e.g., t g 120 s), coalescences are observed, mainly occurring at the tops of the pillars, allowing most of the droplets to merge with each other and shed off the surface (e.g., droplet marked as C from t = 100 s to t = 120 s and droplets marked as D, E, and F from t = 120 s to t = 121 s). During the cooling period, the above wetting phenomena occur continuously and alternately. Moreover, the dewetting transition and self-removal are remarkable, and the former plays a key role in the self-removal of water. On one hand, spontaneously squeezing out LV ET AL.

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Figure 2. Cascade of dropwise condensation and wetting transitions on two-tier substrates, which corresponds to L = 3 μm, S = 4.5 μm, and H = 5 μm. Droplets with a lighter contrast indicate a direct contact on the substrate among the neighboring pillars. On the contrary, droplets with a darker contrast indicate a noncontact with the substrate; that is, such droplets attach to the side wall of the micropillars (small ones) or sit on the top of multiple micropillars (large ones).

of droplets inducing the dewetting transition avoids a further accumulation of water in the valleys, which would lead to further wetting (e.g., sticky Wenzel state). On the other hand, the chance of coalescence will be significantly increased after many droplets climb to the top of the pillars. On the basis of the above observations, we generally summarize these droplet condensation phenomena as the following three processes: (i) continuous growth of individual droplets; (ii) spontaneous dewetting transitions between different wetting states, i.e., from the valley to the top of the micropillars; and (iii) coalescence and departure. This paper will focus on the second process. Spontaneous Dewetting Transition Modes. In the spontaneous dewetting transition process, three unexpected dewetting transition modes are observed and can be generalized below. Mode I: Dewetting Transition of an Individual Droplet. As shown in Figure 3A, at the very beginning, the diameter of the droplet (e.g., t = 10 s) is smaller than S, and it is supposed to stay at the valley or the side of the pillars. When the size of the droplet exceeds 21/2S, however, the gap between the neighboring pillars is filled, and the droplet is squeezed by the surrounding four pillars and deformed naturally away from a spherical shape (see also Figure 2, droplets marked as A and B at t = 60 s). While the droplet in the valley continues to grow, a distinct phenomenon occurs: a dewetting transition happens suddenly. As a result of this, the droplet will stay on the top of the four pillars (from t = 109 s to t = 110 s in Figure 3A), which is supposed to be attributed to the upward motion of the contact lines along the micropillars. Furthermore, the apparent diameter of the droplet is 13 μm at t = 110 s, which is visibly larger than 10.3 μm at t = 109 s (Figure 3A,C) because of the release of the constraint. Furthermore, even though all of the observations are captured from the top view, such a dewetting transition could also be discerned by the distinct change of the contrast of the droplet.25 The lighter appearance (t e 109 s) of the droplet indicates a direct contact between the liquid and the substrate because water is transparent, and it obtains a better reflection from the substrate. On the contrary, the darker appearance (t g 110 s) of the droplet demonstrates that a gap exists between the substrate and the lower side of the droplet, which indicates that the droplet attaches to the top of the micropillars. In Figure 3B, we plot the corresponding pixel grayscale value as a function of each line along the horizontal direction in Figure 3A (120 pixels  120 pixels for each frame), in which the straight lines go through the center of the droplets at each moment. Thus, the lengths of the additional lines with double-headed arrows in Figure 3B represent the diameter (in pixels) of the droplets in Figure 3A. Comparing between t = 109 s and t = 110 s (the red and blue lines in Figure 3B, respectively), the large gap of the

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ARTICLE Figure 3. Spontaneous dewetting transition mode I on the two-tier substrates (L = 3 μm, S = 4.5 μm, and H = 5 μm). (A) Cascade (120 pixels  120 pixels for each frame) of dropwise condensation and the dewetting transition. The straight solid lines with different colors go through the center of the individual droplet, and the dotted line (when t = 133 s) is kept in the same position as the other lines. (B) Relationship between the pixel grayscale value and the position of different lines in (A). The colored double-headed arrows represent the diameters (in pixels) of the droplet at each moment. (C) Time evolution of the diameter of an individual droplet during condensation (blue circles), in which the growth rate is scaled as D ∼ t0.47 (the red solid line). The inset covers t e 109 s in the log
log graph. The inserted individual frames along the growing processes correspond to t = 20, 109, 110, 132, and 133 s, respectively. There is a dewetting transition at t = 109 s and a jump at t = 132 s. (D) Additional details are displayed with the assistance of schematic diagrams.

pixel grayscale value strongly supports the occurrence of the dewetting transition. Because of the limitation of the observation technique, we were unable to access the nanoscale directly to elucidate the details of the contact. However, we assume that a sudden detachment of the pinning contact lines (Figure 3D) leads to the jump from a slight sticking wetting state to air pockets, forming a composite wetting state. Different from self-removal of condensed droplets on the legs of water striders, the droplets can even be ejected from the textures (see Figure 4B in ref 2) under the assistance of the accumulated elastic energy of the flexible hairs; in our experiment, the droplets still attach to the pillars after dewetting (Figure 3A). Moreover, Figure 3C shows the variation of the diameter D of individual droplets with time, which obeys a scaling law D ∼ tR,26,27 and R is determined as R ≈ 0.47 using the method of least-squares. It is surprising to notice that the diameter of individual droplets increases with the square root of time instead of the cubic root of time,26,27 which is attributed to the constraints exerted by the surrounding micropillars (see Supporting Information). Mode II: Dewetting Transition Triggered by a Coalescence of Two Neighboring Droplets. As illustrated in Figure 4A, initially, two neighboring droplets marked as LV ET AL.

Figure 4. Dewetting transition modes on the two-tier substrates (L = 3 μm, S = 4.5 μm, and H = 5 μm) induced by multiple droplets. (A) Mode II: A dewetting transition results from the merging of two isolated droplets marked as 1 and 2. (B) Mode III: A dewetting transition results from a flying droplet from another location, i.e., droplet 2 þ 3 þ 4.

1 and 2 are constrained in the surrounding micropillars and grow independently (indicated by the lighter contrast of their appearance). However, they touch and merge with each other immediately when they become large enough, and the coalescence occurs in the VOL. XXX

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LV ET AL.

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micropillars (not on the tops) and results in a dewetting transition simultaneously (indicated by the darker contrast of the subsequent droplet). After the transition, the merged droplet will sit on the tops of the multiple pillars. The remarkable feature of our observation is that before and after the coalescence the droplets are static, which is quite different from the previous coalescence phenomena,3 i.e., out-of-plane jumping motion. Mode III: Dewetting Transition Triggered by Coalescence of a Flying Droplet. As displayed in Figure 4B, initially, many individual droplets form, and they are constrained by the neighboring pillars (e.g., droplets marked as 1, 2, 3, and 4). During condensation, if the droplets are close to each other (2, 3, and 4) occasionally, they will merge,23 trigger a dewetting transition (as in mode II), and subsequently lead to a jumping. When this flying droplet touches another constrained droplet (marked as 1), a transition will again be stimulated. Figure 4B demonstrates a new formed droplet and a significant increase of volume. Furthermore, the sudden change of the contrast supports the occurrence of the dewetting transition. To the best of our knowledge, this is the first report of the spontaneous dewetting transition of individual droplets occurring on two-tier nano/microstructured rigid surfaces, as well as the three distinct transition modes. Although various nano/micro-multiscale hierarchical structured surfaces were widely employed in the past, most of them focused on coalescence behaviors occurring on the tops of the textures or struggled with the impalement of the vapor pockets, inducing strong pinning of the contact line. Specifically, our observations are distinguished from the previous experimental results in the following respects: (1) the nanostructures with extreme water-repellent properties on the wall and the top of the micropillars are crucial for the transition, and they become the key reason that the meniscus of the droplet is never fully pierced (e.g., Wenzel wetting state in the nanoscale), from the condensation stage to the transition stage (including the coalescence phase in mode II and mode III); (2) the low value of contact angle hysteresis at the nanoscale also plays a key role, which guarantees that the contact lines detach from the substrate easily instead of inducing strong pinning, even though the droplets are deformed remarkably by the constraint of the neighboring pillars; (3) the self-removal of water trapped inside the materials is not only possible by employing extreme water-repellent materials and topology of the structures but can also constitute a quite advantageous way of removing small droplets (e.g., ∼6 μm in diameter; see more results in the next section). Using traditional methods, because the droplet is smaller, it is more difficult for it to shed off; for example, even jumping simulated by coalescence could not occur in the case of very small droplets.3,4 However, our nanotexture-enhanced superhydrophobic

Figure 5. (A) Definitions of the relevant geometrical parameters for a droplet constrained by four micropillars from the side and top views. The red and green colors represent the liquid
vapor and solid
liquid areas, respectively. hcon is the distance between the upper solid
liquid
vapor contact boundary and the substrate; h is the height of the droplet; r is half of the maximum width of the droplet; θ is the apparent contact angle between the liquid and the wall of the micropillars; and κ1 and κ2 are the two main curvatures. (B) Relationship between the normalized curvature and height of the droplet. Here, the normalized spacing is S/a = 3. The red circles and black squares represent numerical results of θ = 180° and θ = 150°, respectively. The red and black solid lines are the corresponding theoretical results of eq 1.

surfaces offer an effective strategy in which we can force smaller droplets to be transited to the top of the textures first. In this way, the probability of coalescence will be significantly increased and, at the same time, the coalescence leads to an increase in the size of the droplet, thereby resulting in easier jumping and self-removal of water. (4) The spontaneous wetting transition occurs continuously and steadily during our experiment period (i.e., 600 s) without any degradation. DISCUSSION Theory and Simulations. How do such unexpected dewetting transition phenomena occur spontaneously? To elucidate the underlying mechanism and obtain a clear answer, we are interested in how the Laplace pressure, ΔP = 2κγ, of a droplet varies with its configurations in the transition processes. Here, γ and 2κ = κ1 þ κ2 are the surface tension and the curvature of the liquid
vapor interface,1,47 respectively. The curvature is constant for a certain volume because gravity can be ignored due to the small size. κ1 and κ2 represent the two main curvatures illustrated in Figure 5A from the side and top views. The importance of the Laplace pressure in wetting transitions on structured substrates has been previously discussed in depth and in connection with other relevant phenomena and aspects, such as pinning, aspect ratio of pillars, and energetics of the surface properties.28,48,49 Unfortunately, since all of these previous works dealt with droplets that are much larger than the scale of the micropillars, these constructed models are not suitable for the novel phenomena occurring in our experiments. Therefore, a completely new insight is required, and urgent exploration is necessary. It is worth noting that there is no known analytical solution for such a complicated configuration of a droplet deformed by four micropillars (marked as A VOL. XXX

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at t = 60 s in Figure 2). Here, we find a first-order approximate solution (see Supporting Information): pffiffiffi ΔP 2 cos θ 1 ¼ 2K ¼
þ (4 þ 2 2cos θ) γ S h

(1)

where θ is the contact angle between water and the side walls (nanotextured surfaces) of the pillars and h is the height of the droplet. For convenience, all parameters are normalized by γ and a length scale factor a (a = 1 μm, which is comparable with both the size of the micropillar and the size of the droplet during the transition), i.e., ΔP = ΔPa/γ, hκ = κa, and h = h/a. We first aim to check the validity of eq 1. In the numerical calculations displayed in Figure 5A,B, we choose S = S/a = 3, but we do not limit the height of the pillars (see Supporting Information) in order to obtain the value of the Laplace pressure (i.e., the curvature) over a wide range. In Figure 5B, the red circles and black squares represent numerical results (using Surface Evolver50) of θ = 180° and θ = 150°, respectively, and the red and black solid lines are corresponding theoretical results deduced from eq 1. These comparisons confirm that eq 1 is extremely useful and can be utilized to predict ΔP precisely enough with no fitting parameters. To the best of our knowledge, this is the first time that such a first-order approximate solution has been obtained. Moreover, eq 1 reveals that, for a constrained droplet, the curvature of the meniscus ranges from a maximum (2κmax = 2/rmin = 23/2/S) to a minimum value (2κmin f
2cos θ/S), which also could be perceived directly from the moment the droplet touches the edges of the four pillars (hmin = 2rmin = 21/2S) until it grows sufficiently large (1/h f 0). When condensation lasts until hcon f H, the lower side of the droplet is still constrained by the four micropillars. However, the solid
liquid
vapor contact lines begin to spread to the top of these pillars, which results in an expansion of the droplet in the upper side, and subsequently a pressure difference inside of the droplet will arise, i.e., ΔPlower > ΔPupper. As a result, the liquid is driven to move upward, and the droplet starts to escape from the constraint of the pillars. Thus, hcon = H can be treated as the critical condition for discerning when the dewetting transition occurs. Simultaneously, the other geometrical parameters also approach the critical values, i.e., h = hc, r = rc, and 2κ = 2κc (or ΔP = ΔPc). Here, rc is defined as the critical radius of the droplet immediately before the moment of transition from the top view (e.g., t = 109 s in Figure 3). Thus, we have 2κc =
2cos θ/S þ (4 þ 23/2 cos θ)/hc. We define 2κc = 2/R*, where R* is employed to characterize the critical size of the droplet when transition occurs. Thus, we have pffiffiffi 2 2 cos θ 1 þ (4 þ 2 2cos θ)  ¼
S hc R

(2)

Using two geometrical coefficients β = R*/rc and η = hc/H, we obtain a quantitative relationship LV ET AL.

Figure 6. (A) Relationship between the critical droplet radius rc and S. The black squares and red circles represent experimental measurements of mode I and mode II, respectively, and the black and red solid lines are predictions using eq 3. (B) Number of dewetting transitions vs transition modes. All of the statistics are carried out on a 350 μm  114 μm area, for a condensation duration of 600 s.

that accounts for critical radius prediction and is of the form pffiffiffi 1 1 β 1 ¼
(β cos θ) þ (2 þ 2cos θ) rc S η H

(3)

Comparison to Experiment. Figure 6A presents comparisons of the analytical results predicted by eq 3 and more experimental results performed on samples with various spacings (see Supporting Information). The black squares with error bars are the average values of five experimental measurements of the dewetting transition of mode I. The black solid line is the result of eq 3 using the method of least-squares, i.e., 1/rc = 0.73/S þ 0.24/H. Without loss of generality, let θ = 160°. Then, we get β = 0.78 and η = 2.2. We attribute the large value of η to the pinning of contact lines, which results in larger deformation of the droplet. Although the contact angle hysteresis is quite small for our substrates, line pinning is still supposed to exist in a moist ambient environment. Here, we further verify the validity of β in the following. Based on eq 2, we know that R* ∈ [S/21/2,
S/cos θ] ≈ [0.7S, S]. On the other hand, we get 1/rc = 0.73/S þ 0.24/H directly from the experimental data, as previously mentioned. Thus, we roughly estimate that R*/rc ∈ [0.5, 0.75] þ 0.2 S/H, and so the fitted coefficient β = 0.78 is covered by the estimation, which further indicates that our theoretical analysis and experiments are self-consistent. Furthermore, for mode II, when two droplets with R*1 and R*2 (R*1 ≈ R*2) merge with each other, we will have a new droplet with radius R* ≈ 21/3R*1. Thus, we naturally have rc (mode II) = 21/3rc (mode I) (plotted as the red solid line in Figure 6A), which agrees well with the experimental results. Concerning that the probability of mode III (see Figure 6B) is quite low, as well as the kinetic energy that becomes involved, we do not compare the experimental data in Figure 6A with eq 3, which is constructed on the basis of static force analysis. We also display the statistical results between the number of dewetting transitions and transition modes in Figure 6B, as well as the influence of the spacing of the micropillars. From the perspective of the specific VOL. XXX

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ARTICLE Figure 7. Numerical results regarding the transition processes of a droplet. The geometrical parameters correspond to L = 2 μm, S = 3 μm, H = 5 μm, and θ = 150°. (a) Calculated geometrical ratio h/2r as a function of the normalized distance (hb/a) from the bottom of the droplet to the substrate. (b) Variation of the reduced r/r0 and h/2r0 with hb/a. (c) Variation of the reduced surface energy (E/E0) and Laplace pressure (ΔP/ΔP0) with hb/a.

transition modes, it is clear that the wetting transition is dominant for the microstructures with the smallest value of space (S = 3 μm). On the contrary, the transition phenomenon is weakest for the microstructures with the largest value of space (S = 13 μm in mode I and mode II). For mode I, the number of transitions decreases monotonically with S, and our findings suggest that decreasing the space of the micropillars constitutes a favorable strategy to decrease the critical size of the droplet for transitions. However, in mode II and mode III, the difference of the number of transitions for S = 4.5 and 6.5 μm is not so obvious. For mode III, the number of transitions on the surface S = 13 μm is slightly larger than the surface with S = 4.5 and 6.5 μm. Furthermore, from the perspective of the specified values of S, for S = 3, 4.5, and 6.5 μm, the number of transitions generally decreases from mode I to mode II, to mode III, while this feature reverses for S = 13 μm. As indicated by Figure 2, we can see that after the running of the cooling system, small droplets appear and distribute randomly. Moreover, if the space of the pillars is small, mode I will occur promptly when eq 2 is satisfied, and it is not necessary to wait for the stimulations of mode II and mode III because not all condensed droplets are close to each other. However, if the space of the pillars is too large (S = 13 μm), more time is needed to wait until the individual droplet reaches the critical size. The probability of the wetting transition is shown to be quite low for these three modes. Scaling Analysis for the Transition. We now extend our arguments to the variation of the relevant geometrical and physical quantities during a well-controlled quasistatic transition process. As shown in Figure 7, the distinct change of the vertical deformation h and Laplace pressure ΔP are remarkably different from the slight change of the lateral deformation r and surface energy E (which corresponds to surface area). hb = hb /a denotes the normalized distance between the bottom of the droplet to the substrate, and r0 is used to characterize the volume of the droplet, i.e., LV ET AL.

V0 = 4πr03/3. The value of ΔPc and Ec correspond to hb = 0. For a droplet deformed by the pillars, the volume and surface energy scale as V ∼ r03 ∼ Sh2 and E ∼ γrh, respectively (see Supporting Information). Thus, the ratio of the surface energy between the spherical droplet after transition (scaling as Es ∼ γr02) and Ec can be obtained as Es/Ec ∼ S1/2V
1/3(rc/r0)
1 ∼ (1 þ δ/r0) ∼ δ/r0, denoting δ = r0
rc, which is further verified from Figure 7B, C (the green triangles and red circles, respectively); that is, variations of both surface energy and lateral deformation of the droplet are at the same level (∼5%). Furthermore, the ratio of the Laplace pressure between the spherical droplet after transition (scaling as ΔPs ∼ γ/r0) and ΔPc can be obtained as ΔPs/ΔPc ∼ (r0/hc)
1 ∼ (1þε/2r0) ∼ ε/2r0, denoting ε = hc
2r0. This scaling analysis is also confirmed in Figure 7B, C (the yellow triangles and blue diamonds, respectively; both variations are about ∼16%). These important analyses also inform us that increasing δ or ε will lead to more accumulations of surface energy or Laplace pressure, and thereby is beneficial for wetting transitions (e.g., transfer to more kinetic energy to escape the resisting force). Our theoretical and scaling analyses not only are consistent with ref 2 but also further confirm that the underlying mechanism for the spontaneous dewetting is attributed to a gradient in the droplet curvature (i.e., Laplace pressure) and surface energy, for both flexible hairs and rigid pillars. However, the occurring processes of self-removal on these two kinds of substrates are different: for the flexible hairs, it is attributed to the growing and the self-propulsion of the droplets along the conical hairs, elastic deformation of the setae array, and the competition between the elastic force and the resisting force. For our rigid substrate, the condensation will first increase the volume of the droplet; subsequently the deformation of the droplet induced by the constraint of the rigid pillars will lead to a pressure gradient when the droplet is large enough, and then the dewetting is simulated by this pressure gradient (i.e., surface energy gradient) VOL. XXX

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CONCLUSION To summarize, we report spontaneous dewetting transitions of condensed microdroplets (e10 μm) on rigid superhydrophobic substrates owing to excellent superhydrophobicity at the nanoscale and the geometrical topologies at the microscale, which has not been

METHODS Fabrication of Superhydrophobic Two-Tier Nano/Microstructured Surfaces. Silicon wafer substrates with square-shaped micropillars are fabricated by photolithography and etching of inductively coupled plasma (ICP).23 Then, they are produced by treatment with a commercial coating agent (Glaco Mirror Coat “Zero”, Soft 99, Co.) containing nanoparticles and an organic reagent.44,45 The superhydrophobic coating was applied on the smooth silicon wafer by pouring the Glaco liquid over the substrate. A thin liquid film wets the silicon surface and dries in less than 1 min. The silicon surfaces are then put into an oven and kept at 200 °C for 30 min. The pouring and heating processes will be performed three to four times. Surface Characterization. Surface topographies are analyzed using a scanning electron microscope (SEM, JSM 6330 from JEOL) and an atomic force microscope (AFM, NanoScope 5 from DI). Apparent contact angles, sliding angles, and advancing and receding contact angles for all samples were measured and analyzed using a microgoniometer (JC2000CD1). Experimental Setup. All of the experiments are performed using an optical microscopy technique under a moist ambient environment allowing for focusing on the in situ dynamic characteristics of the condensed droplets. All of the samples are placed horizontally on a Peltier cooling stage, which is fixed on the slider of an optical microscope (BX 51, Olympus, Japan). The optical microscope that we employ enables in situ detection of the details with a high resolution of 0.2 μm. The laboratory temperature is measured at 29 °C with a relative humidity of 40% (the corresponding dew point is 14 °C). During the running of the cooling system, the temperature of the sample is well maintained at 10 ( 1 °C. Top-down images of the processes are captured using a CCD camera (MegaPlus, Redlake, USA) installed on the microscope. Conflict of Interest: The authors declare no competing financial interest. Acknowledgment. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11072126, 11172156) and the Foundation of State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources (Grant No. LAPS14018).

LV ET AL.

previously attainable. Three dewetting transition modes are observed. We not only develop a theoretical expression that can be utilized to predict the Laplace pressure of the droplet with no fitting parameters, but also construct an explicit model that links the critical size of the droplet and the spacing and height of the micropillars. Our experimental and theoretical results indicate that a further decreased value of Laplace pressure on the top side of the individual droplet leads to instability and subsequently an unexpected spontaneous dewetting transition without any external force. We further reveal that the spacing of the micropillars is essential for determining the critical size of the droplet for transition. It is important to emphasize that the contact angle hysteresis of the material/structure systems plays an essential role in accounting for the motion of the droplets. Only if it is further suppressed and overcome can the spontaneous transitions be realized effectively. Thus, the design of devices with not only highly efficient removal of water but also highly enhanced heat transfer performance in real applications can be inspired by our findings.

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generated on the droplet itself (without any other energy assistance) in the case where the resisting force is overcome. Finally, the driving force Fd = δPS2 ∼ γS is on the same order of magnitude as the resisting force Fr ∼ γSΔθ sin θ,2 denoting δP = ΔPlower
ΔPupper and Δθ as the pressure difference and contact angle hysteresis, respectively. As a result, only a high enough contact angle with a low enough contact angle hysteresis can guarantee such wetting transition behaviors, such as excellent superhydrophobicity at the nanoscale, which are critical in our experiments (i.e., the wall of the micropillars). This is impossible to manage via its conventional hydrophobic counterparts4,17
22,25,28,33,51 even though they possess excellent water-repellent properties on a microscopic level.

Supporting Information Available: The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.5b05607. Geometrical parameters and wetting properties of the samples employed; detailed deduction of the Laplace pressure of the individual droplet constrained by the neighboring pillars; modeling of the numerical simulations and comparisons with theoretical analysis; scaling laws obeyed by the droplets under condensation; and possible wetting states and transitions (PDF)

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