Water Wicking and Droplet Spreading on Randomly Structured Thin

Our layers are both very thin and have nanoscale porosity, making them low-permeability layers that exhibit strong wicking. An added benefit is that t...
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Water Wicking and Droplet Spreading on Randomly Structured Thin Nanoporous Layers Claire K. Wemp and Van P. Carey* Department of Mechanical Engineering, University of California, Berkeley, California 94720, United States ABSTRACT: Growing thin, nanostructured layers on metallic surfaces is an attractive, new approach to create superhydrophilic coatings on heat exchangers that enhance spray cooling heat transfer. This paper presents results of an experimental study of enhanced droplet spreading on zinc oxide, nanostructured surfaces of this type that were thermally grown on copper substrates. The spreading rate data obtained from experimental high speed videos was used to develop a model specifically for this type of ultrathin, nanoporous layer. This investigation differs from previous related studies of droplet spreading on porous surfaces, which have generally considered either ordered, thin, moderately permeable layers, or thicker, microporous layers. Our layers are both very thin and have nanoscale porosity, making them low-permeability layers that exhibit strong wicking. An added benefit is that the thermally grown, stochastic nature of our surfaces make manufacturing easily scalable and particularly attractive for spray-cooled heat exchanger applications. The model presented here can predict the spreading rate for the wetted footprint of a deposited water droplet over two spreading stages: an early synchronous spreading stage, followed by hemispreading. The comparison of experimental data and model predictions confirms the presence of these two specific spreading stages. The model defines the transition conditions between synchronous and hemispreading regimes based on the change in spreading mechanisms, and we demonstrate that the model predictions of spreading rate are in good agreement with the experimental determinations of droplet footprint variation with time. The results indicate that the early synchronous spreading regime is characterized by flow in the porous layer that is primarily localized near the upper droplet contact line. The potential use of these experimental findings and model for optimizing superhydrophilic, nanostructured surface coatings is also discussed, as it pertains to the surface‘s ability to enhance water vaporization processes.

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INTRODUCTION Droplet impingement and spreading of liquid droplets on a porous surface has been studied and modeled in a variety of prior studies. Some have explored the fundamental physics of droplet spreading on porous surfaces,1−11 while others have specifically explored aspects related to boiling, droplet vaporization, or other heat dispersion applications.6,12−24 Most of the earlier work focusing on droplet spreading or bulk liquid spreading on porous surfaces has explored spreading for relatively deep porous media,1−4,6 or spreading on thinner, highly ordered, periodic, microstructured layers created by MEMS-type micromanufacturng technologies.16−20 By contrast, the work summarized here focuses on water droplet spreading on thin, porous, stochastic nanostuctured layers, which vary from 3 to 5 μm thick, based on SEM image analysis. Additionally, the surface production technique used in this study relies on thermal growth from seed particles on a metal substrate, which differs from more structured development done in highly controlled lab or clean room environments. Previous experimental studies23,24 have shown that water droplets deposited on superhydrophilic surfaces of this type quickly spread to form an ultrathin liquid film which evaporates very rapidly, producing extremely efficient heat © XXXX American Chemical Society

transfer. This suggests that surfaces of this type have the potential to enhance heat transfer during water spray cooling of air-cooled heat exchangers, which is commonly done in Rankine cycle power plants to augment heat rejection. Compared to micromanufacturing processes, which produce nearly exact periodic structures, thermal growth of nanoporous coatings results in a more spatially random structure. While this thermal growth allows less precise control of the resulting morphology, it allows the possibility to scale-up the metal surface coating process to the large, complicated surface areas typical of heat exchanger applications. Thin coatings of this type are attractive because they require less processing time to grow, and they add virtually no additional thermal resistance. Figure 1 shows electron micrograph images of the type of coating used in experiments for this study. This surface was fabricated by thermally growing a layer of ZnO nanopillars on a smooth copper substrate. The resulting nanopillars form a layer that is 3−5 μm thick and pillars are nominally hexagonal in cross section, with the maximum dimension across the hexagon Received: October 23, 2017 Revised: November 19, 2017 Published: November 20, 2017 A

DOI: 10.1021/acs.langmuir.7b03687 Langmuir XXXX, XXX, XXX−XXX

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system in which capillary pressure energy is the dominant driving mechanism, all three of these dimensionless parameters will be much less than one: Bo* =

ρl grd2

≪1

bΔPcap σ ≪1 Z* = bΔPcap

We* =

rdρl wd2 bΔPcap

≪1

(1)

Here, ΔPcap is the capillarity pressure difference across the wicking front, b is the layer thickness, ρl is the liquid density, σ is the surface tension, rd is the initial spherical droplet radius, and wd is the impact velocity of the deposited droplet. For our droplet spreading experiments on the thin nanoporous layers described here, we estimate that Bo* is about 3.4 × 10−4, We* is about 6.8 × 10−4, and Z* is about 2.0 × 10−2. Other recent studies have proposed models for droplet spreading on porous layers,6,9,16,17 but our experimental studies indicated that the types of thin, superhydrophilic, nanoporous layers of interest here exhibit some distinct droplet spreading features which are not fully represented in earlier models. For this type of a system, the model analysis of droplet spreading on a dry porous layer must account for the presence of the advancing wicking front in the porous layer near the upper droplet contact line. Similar to the framerwork developed by Young and Buie,6 we develop a model in which the initial stage of spreading in the system is characterized by the contact line and wicking front moving in tandem. In contast, Starov et al.,8,9 developed a model of droplet spreading on a dry porous layer in which they treated the early stage of spreading as if the wicking front is far beyond the upper droplet contact line, and liquid flow to the contact line region is driven by capillary effects and variations of interface curvature in the upper droplet. Because the model of Starov et al. does not account for the close proximity of the wicking front to the contact line, and thus does not include capillary pressure differences across the wicking front as the main driving mechanism, this model does not appear to be well-suited for predicting the capillary pressure dominated spreading of interest in this study. For the later observed spreading of the wicking front within the porous layer beyond the upper droplet contact line, Kim et al.17 proposed a Washburn-type model25 for spreading on a surface layer with periodic, microporous structures. This model was appropriate for the portion of the hemispreading process observed on their surfaces, but it does not encompass all the features of the early synchronized spreading and later hemispreading process that we observed on surfaces, like that shown in Figure 1. These droplet spreading models, then, are limited in their ability to fully model the mechanisms that dictate the nanomorphology effects and the physics of spreading on these advanced surfaces through all stages of the spreading process. Using Darcy analysis of the flow in the porous layer under the droplet, we developed a droplet spreading model that indicates an early stage linear variation of footprint radius with time, consistent with our data, and also predicts the subsequent hemispreading process observedin our experiments, consistent with the treatment proposed by Kim et al.17 In developing their model analysis, Joung and Buie6 did not specifically consider a droplet spreading system dominated by viscous and capillary effects. However, we have shown that if their model is simplified to that case and we use Darcy flow

Figure 1. Surface 1.1 shows zinc oxide nanostructures, created from a 6 nm seeding solution and grown in solution for 8 h. The tightly packed, randomized pillar structure on the copper substrate was imaged with an SEM and shown at two magnifications.

being about 200 nm. These geometry parameters can be varied by adjusting certain thermal growth process parameters, as discussed in the next section. To optimally design nanostructured coatings to enhance spray cooling heat transfer, it is desirable for the droplet to spread as much as possible, as fast as possible. Our objectives in this study were therefore to develop a better understanding of how the upper droplet liquid interacts with the wicking flow in the nanoporous layer, how layer morphology affects the spreading process, and, if possible, to develop the ability to predict the spreading dynamics to aid in nanostructured layer design studies. We emphasize that this study specifically focuses on fluid and surface combinations in which capillary pressure forces and porous layer flow resistance (viscous dissipation effects) dominate the spreading process. In these systems, the very strong capillary pressure difference across the interfaces in the porous layer (at the advancing wicking front of liquid) drives the flow. One way to quantify the dominance of this capillary force is to compare the ratio of pressure forces on the upper droplet with capillary pressure forces within the nanostructured layer. The smaller the radii between pillars within the nanoporous layer, the larger the resulting capillary pressure force will be in that region. Meanwhile, the capillary surface tension effect for the upper droplet is much smaller because the droplet radius in our system is many orders of magnitude larger than the interface radius in the nanoporous layer. Interface capillary pressure difference is inversely proportional to interface curvature, so the ratio of upper droplet capillary pressure difference to capillary pressured difference across the wicking front is proportional to Rd/rnl where Rd is the upper droplet radius of curvature and rnl is half the distance between pillars in the porous structure. In our system, this ratio is typically on the order of 10−2. In addition to force balance, one can also consider the energy associated with the upper droplet gravitational potential energy, kinetic energy, and interfacial surface tension energy compared to the work energy done by capillary forces at the wicking front in the nanoporous layer. The ratio of each of the upper droplet energy quantities to the capillary pressure energy can be expressed in terms of a Bond number Bo*, Weber number We*, and droplet to layer capillary pressure ratio Z*. For a B

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Langmuir Table 1. Surfaces Used for Experimental Study

Figure 2. Droplet spreading process on the ZnO nanostructured surface.

C

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Figure 2a depicts the initial contact of a droplet deposited on a thin nanostructured surface at time t = 0. Because the nanoporous layer is thin, the time required for the liquid to wick across the layer, tp, is very small (see Figure 2b). Liquid then wicks radially outward (Figure 2c) in the nanostructured layer while the upper droplet spreads radially. During this portion of the spreading process, the droplet contact line visually appears to stay within the leading edge portion of the nanoporous layer filled with liquid. We will refer to this radial spreading of the upper droplet in tandem with liquid wicking in the porous layer as synchronized spreading. It would seem logical that the upper droplet may resist spreading its contact line beyond the limit of the liquid-filled porous layer because it would have to transfer to a portion of the surface with a higher apparent contact angle. The surface under the droplet is effectively a composite surface comprised of solid fraction ϕs, wetted at intrinsic contact angle θE, and liquid regions (fraction 1 − ϕs), effectively wetted at zero contact angle. The Cassie−Baxter27 composite surface model suggests that the resulting equilibrium apparent contact angle θapp is given by

analysis to evaluate viscous dissipation, their model predicts a linear variation of droplet footprint radius that agrees completely with the model developed in our investigation. A linear variation of wetted footprint radius with time was also observed in the similar circumstances of wicking front flow from a capillary tube into a microporous layer, observed by Rahman et al.15Descriptions of our droplet spreading experiments, model development, and comparison of model predictions with experimental data are provide in the following sections.

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EXPERIMENTAL SECTION

In order to better understand the nature of the advanced surfaces in this study, we conducted droplet deposition tests in which a water droplet was deposited on a thin ZnO nanostructured surface on a copper substrate. The surface morphology for these experiments can be seen in Figure 1 and is the result of a thermal growth fabrication method known as hydrothermal synthesis. This process, which is described in greater detail by Padilla,23,26 begins with the cleaning and polishing of a copper substrate surface made of 99.9% pure 110 copper. The surface is polished to achieve micron-level uniformity before it is cleaned in successive sonication baths to rid the surface of any oils or residue from polishing. A liquid nanoparticle solution containing 6 nm zinc oxide particles in suspension is then deposited evenly on the clean, dry surface using a microliter deposition technique. The piece is then annealed, cooled, and submerged in a liquid growth solution (details in Padilla23,26) and is placed a 90 °C oven for 8 h. The resulting surface is removed and air-dried. Finally, the surface undergoes a desorption process, which is repeated before any experimenting because superhydrophilic surfaces adsorb some molecular species from their environment over time. The desorption process, consisting of heating the surface to 275 °C for 1 h, can renew the surfaces wetting characteristics if exposure decreases it. Eight different combinations of surface morphology and wetting conditions (apparent contact angle) were produced for this study and are shown in Table 1, where apparent contact angles were calculated using the relation between droplet volume and spread area (where a spherical cap assumption was used). Three different surface samples were created by the thermal growth fabrication process and successive desorption processes for each of these samples produced small variations in wettability, effectively creating surfaces with the same morphology, but slightly different wicking characteristics. All eight surfaces were made using the same manufacturing process. The spreading characteristics for each surface differ slightly, but measured apparent contact angles for these surfaces were consistently in the 4− 10° range, which indicates significant hydrophilicity. The surfaces in Table 1 have nanostructure layer thicknesses that were experimentally measured and verified in the following way. Layer thickness, b, for the different surfaces is a funciton of pillar height, which was measured from SEM images. As seen in Figure 1, the morphology of the surface is made up of pillars at varying angles. We know that the pillars grow outward and upward from seeding particles and we can directly measure the taper of any given pillar from SEM imaging. Measuring the taper angle for two neighboring pillars, it is then possible to calculate at what pillar length these two neighboring pillars will meet up at their mutual base. With a known pillar length, geometry was used to determine layer thickness based on the measured azimuthal angle of the pillar. By averaging measurements taken across the different of SEM surfaces, a resulting thickness of 4.9 μm with a standard deviation of 0.64 μm was determined. For each experiment on the surfaces, a measured volume of liquid was deposited onto a nominally horizontal nanostructured surface and the resulting spreading process was recorded using a high-speed video camera operating at 1000 fps. The characteristics of the droplet spreading process observed in our experiments are depicted in Figure 2. The values for ts and Rs will be discussed in further detail in the section discussing model development.

cos θapp = ϕs cos θE + (1 − ϕs)

(2)

Transfer of the contact line to a higher contact angle dry region would force the interface to at least temporarily become more convex near the contact line, which is expected to increase the pressure in the liquid there (due to capillary effects across the interface), moving liquid away from that location and thus slowing the advance of the contact line. As indicated in Figures 2d,e, we observed spreading of the liquid droplet continuing until the contact angle and interface radius of curvature of the droplet adjusts to be consistent with the total volume of liquid in the droplet and minimize the free energy in the system. Since departure from this equilibrium increases system free energy, the droplet resists leaving that state once it has achieved it. This implies that during the initial stage of droplet spreading, the upper droplet responds as if it is spreading on a composite surface that is a mosaic of dry solid surface and fully wetted liquid surface. The upper droplet stops expanding when it reaches the footprint radius, Rs. In the time period following the arrival at Rs, there is no experimentally observed shrinkage of the droplet. Shrinkage is expected to be negligible because the nanoporous layer is so thin that the liquid transferred from the droplet to the layer during the duration of the experiment is much less than the volume of the upper droplet. This is true even for the smallest droplet (2 μL) and the thickest layer (5 μL) that were tested in our experiments. The virtually stationary upper droplet established at R − Rs is idealized as

⎡ ⎤1/3 3Vd ⎥ sin θapp Rs = ⎢ ⎢⎣ π(2 − 3 cos θapp + cos2 θapp) ⎥⎦

(3)

If the nanoporous layer is permeable enough and the capillary pressure for the layer is high enough, the region of the porous layer that is filled with liquid will continue to expand beyond the contact line of the upper droplet. This circumstance, depicted in Figure 2e, is commonly referred to as hemispreading.28 Frames from high-speed video footage of a droplet spreading experiment are shown in Figure 4. Figure 3 shows two frames of the spreading process for a 2 μL droplet (1.6 mm diameter before deposition) spreading at room temperature. These frames illustrate the appearance of the droplet before (a) and after (b) reaching the transition or separation point, (ts, Rs), where the hemispreading begins. This illustrates the two spreading behaviors leading to two different stages in the spreading model analysis. Values for ts, obtained from our experiments, range from 0.010 to 0.022 s, which results in radial spreading rates of the liquid at these early times which is 20 times higher than reported values by Rahman and McCarthy et al.15,16 for spreading from a capillary tube into micro and nanostructured surfaces. The rapidity of the spreading makes the optimization of this type of surface particularly motivating. The image processing software, ImageJ, was used to extract position data for the contact line and liquid front using frames from digital D

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area was measured for chronological video frames with a known scale ratio of approximately 35 pixels/mm. The upper droplet contact line and wicking limit radius measurements were determined to ±2 to 3 pixels, which resulted in a ± 2% to 3% uncertainty in the radius measurements. An example of the resulting R(t) data is shown in Figure 5 for spreading of a 2 μL droplet deposited on Surface 1.1 (see Table 1) containing unheated ZnO nanostructured surface like that shown in Figure 1 at room temperature.

Figure 3. Spreading of a 2 μL water droplet on a copper surface with a ZnO nanostructured layer. (a) initial spreading at t = 0.014 s; (b) hemi spreading at t = 4 s.

Figure 5. Surface 1.1: experimental variation of liquid front radius with time as a 2 μL droplet spreads on the nanostructured ZnO surface at room temperature. Figure 5 shows the R(t) data on a log−log plot to clearly depict the very early time variation during which the droplet rapidly spreads to the radius of a few millimeters in less than 0.02 s. It should be noted that the R(t) variation of the data in this figure clearly reflects the two spreading stages represented in the process described above. At early times, the synchronous spreading variation of R with t is close to linear. Then there is a clear transition to hemispreading with the upper droplet contact line essentially stopped, and the slower wicking front expansion beyond the contact line of the upper droplet in the nanoporous layer. This transition between stages, at about t = 0.011 s in Figure 5, corresponds approximately to the time when the upper droplet visually is observed to stop expanding. Further data analysis requires an introduction to the model developed in this study. In the next section, the spreading behavior observations, summarized above, will be used to guide the formulation of a model analysis for droplet spreading on thin, superhydrophilic, nanoporous layers.

Figure 4. High speed video frames for a 2 μL water droplet spreading on copper surface with a ZnO nanostructured layer. Necking is observed as the capillary forces pull the droplet across the surface at t = 0.0033 s. video recordings of the droplet spreading process in our experiments. To determine the variation of the spreading droplet‘s mean footprint radius with time, R(t), the wetted area was divided by π, and the square root of the result was taken to be the mean radius of the spread droplet at that point in the spreading process. The corresponding time was computed from the frame number relative to the start and the known frame rate of the video camera. The frame rate for videos in this study was 1000 frames per second, leading to an uncertainty of roughly ±0.5 ms, as movement between frames wasn‘t captured. To obtain the best possible estimate of the time of initial droplet contact with the porous surface, we initially took the frame showing first liquid contact as corresponding to time = 0. We then determined a best linear fit to the raw times and radius values for a few early frames of the video and extrapolated the linear relation to predict the time when the footprint radius R was zero. We then subtracted this predicted first contact time from raw time values for all our data so that time t for each frame is measured from this predicted initial contact. This typically involved an offset of only 0.2−0.8 ms and helped increase accuracy in the early stages of spreading, where there is higher percentage uncertainty due to the movement occurring on a much smaller time scale ( 1 (50)

These changes to nondimensionalize the spreading relations result in a universal curve for the synchronous early spreading stage and a collection of curves for different b/Rs ratios in the hemispreading stage.

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RESULTS AND DISCUSSION The model developed was used to generate a curve for R(t) to compare with experimental data like that in Figure 5. Surface characteristics such as mean permeability, capillary pressure characteristics, porosity, thickness of the nanostructured layer, and water properties would need to be specified in order to J

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Figure 11. Comparison of 2 μL, 3 μL, and 5 μL droplet spread data with the model predicted variation of R̂ = R/Rs with t ̂ = t/ts for Surface 3.2.

regime transition occurs at a spreading radius Rs, which minimizes the upper droplet free energy for its apparent contact angle, was found to be consistent with our experimental data and observations. Because the model analysis presented here treats the liquid transport in nanostructured surfaces as porous medium transport, the framework can be applied empirically for random nanostructured surfaces like the thermally grown ZnO nanopillar surfaces shown in Figure 1. In that case, effective porosity and other porous medium parameters could be used to predict capillary pressure and permeability from empirical correlations or relations developed for more ordered structures. For lithographically produced, highly ordered nanostructures, this same model analysis could be used in tandem with calculations of porosity from the periodic geometry of the layer, and fluid dynamic models that relate permeability to flow passage geometry. The adaptability of the model developed here enhances its potential usefulness as a tool for assessing nanostructured surface coatings for enhanced boiling or droplet evaporation. It should be noted that the model analysis described here was specifically developed for circumstances in which the ratios of the upper droplet gravitational potential energy, kinetic energy and interfacial surface tension energy to the capillary pressure energy are small. As noted above, by constructing relations for these ratios, these requirements can be expressed in terms of a Bond number Bo*, Weber number We*, and droplet-to-layer capillary pressure ratio Z*, in eq 1. We have shown that these conditions are satisfied in the system explored in our experiments, and the model predictions agree well with our spreading data. This suggests that this model analysis of droplet spreading on an ultrathin, nanostructured, hydrophilic layer on a solid substrate should be applicable to other fluid and surface systems that meet these requirements. However, this model may be less accurate for droplet spreading on thick porous layers (where b/R is not small compared to one), low permeability porous structures where flow deviates from the Darcy model, or systems that do not satisfy all the eq 1 requirements for Bo*, We*, and Z*. The fundamental droplet spreading model developed here provides the capability to relate porous medium morphology parameters to spreading performance for a variety of surfaces. This can provide the opportunity to use the modeling

Figure 12. Comparison of droplet radius spread to the droplet volume, emphasizing that the change in volume results in consistent apparent contact angle for a given surface prep.

variations of Rs with droplet volume Vd for various constant θapp values. It can be seen in Figure 12 that the variation of Rs with Vd is consistent with what is expected for each surface at a constant apparent contact angle θapp, with the contact angle varying only slightly between the two surfaces. Overall, our comparisons indicate that with appropriately specified (ts, Rs) values, the model predictions agree well with the nanostructured surface spreading data in both the synchronous spreading and hemispreading stages considered in the model analysis, and the data exhibit trends that are consistent with the postulated behavior in our model.

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SUMMARY AND CONCLUSION The good agreement between our spreading rate data and predictions of the model framework proposed here support the contention that the proposed spreading model appropriately represents the mechanisms of the spreading process for the thin, nanoporous-layer surfaces of interest here. Particular noteworthy is that the model of the early synchronous spreading predicts that spreading is facilitated by very localized flow from the upper droplet to the advancing wicking front in the porous layer near the upper droplet contact line. This shortpath transport is precisely what allows the observed very rapid spreading of the upper droplet and wicking front in this initial regime, even when the nanoporous layer is very thin and/or has a very low permeability. Also, our model’s idealization that the K

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(7) Starov, V.; Kosvintsev, S.; Sobolev, V.; Velarde, M.; Zhdanov, S. Spreading of liquid drops over saturated porous layers. J. Colloid Interface Sci. 2002, 246, 372−379. (8) Starov, V.; Zhdanov, S.; Kosvintsev, S.; Sobolev, V.; Velarde, M. Spreading of liquid drops over dry porous layers: complete wetting case. J. Colloid Interface Sci. 2002, 252, 397−408. (9) Starov, V.; Zhdanov, S.; Kosvintsev, S.; Sobolev, V.; Velarde, M. Spreading of liquid drops over saturated porous substrates. Adv. Colloid Interface Sci. 2003, 104, 123−158. (10) Wang, Z.; Espin, L.; Bates, F.; Kumar, S.; Macosko, C. Water droplet spreading and imbibition on superhydrophilic poly (butylene terephthalate) melt-blown fiber mats. Chem. Eng. Sci. 2016, 146, 104− 114. (11) Espin, L.; Kumar, S. Droplet spreading and absorption on rough, permeable substrates. J. Fluid Mech. 2015, 784, 465−486. (12) Chen, R.; Lu, M.; Srinivasan, V.; Wang, Z.; Cho, H.; Majumdar, A. Nanowires for Enhanced Boiling Heat Transfer. Nano Lett. 2009, 9, 548−553. (13) Lu, M.; Chen, R.; Srinivasan, V.; Carey, V.; Majumdar, A. Critical heat flux of pool boiling on Si Nanowire Array-Coated Surfaces. Int. J. Heat Mass Transfer 2011, 54, 5359−5367. (14) Yao, Z.; Lu, Y.; Kandlikar, S. Effects of nanowire height on pool boiling performance of water on silicon chips. Int. J. Therm. Sci. 2011, 50, 2084−2090. (15) Rahman, M.; Ö lcȩroğlu, E.; McCarthy, M. Scalable Nanomanufacturing of Virus-Templated Coatings for Enhanced Boiling. Adv. Mater. Interfaces 2014, 1, 1300107. (16) Rahman, M.; Ö lcȩroğlu, E.; McCarthy, M. Role of wickability on the critical heat flux of structured superhydrophilic surfaces. Langmuir 2014, 30, 11225−11234. (17) Kim, B.; Lee, H.; Shin, S.; Choi, G.; Cho, H. Interfacial wicking dynamics and its impact on critical heat flux of boiling heat transfer. Appl. Phys. Lett. 2014, 105, 191601. (18) Chu, K.; Soo Joung, Y.; Enright, R.; Buie, C.; Wang, E. Hierarchically structured surfaces for boiling critical heat flux enhancement. Appl. Phys. Lett. 2013, 102, 151602. (19) Yao, Z.; Lu, Y.; Kandlikar, S. Pool boiling heat transfer enhancement through nanostrucutres on silicon microchannels. J. Nanotechnol. Eng. Med. 2012, 3, 031002. (20) Chu, K.; Enright, R.; Wang, E. Structured surfaces for enhanced pool boiling heat transfer. Appl. Phys. Lett. 2012, 100, 241603. (21) O’Hanley, H.; Coyle, C.; Buongiorno, J.; McKrell, T.; Hu, L.; Rubner, M.; Cohen, R. Separate effects of surface roughness, wettability, and porosity on the boiling critical heat flux. Appl. Phys. Lett. 2013, 103, 024102. (22) Zou, A.; Maroo, S. Critical height of micro/nano structures for pool boiling heat transfer enhancement. Appl. Phys. Lett. 2013, 103, 221602. (23) Padilla, J.; Carey, V. Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition (IMECE2014, Motreal, Quebec, Canada, November 2014). (24) LaBrie, R. I.; Padilla, J.; Carey, V. Experimental study of aqueous binary mixture droplet vaporization on nanostructured surfaces. Heat Transfer Eng. 2017, 38, 1260−1273. (25) Washburn, E. The dynamics of capillary flow. Phys. Rev. 1921, 17, 273−283. (26) Padilla, J. Experimental Study of Water Droplet Vaporization on Nanostructured Surfaces. Ph.D. Thesis, University of California, Berkeley, 2014. (27) Cassie, A.; Baxter, S. Wettability of porous surfaces. Trans. Faraday Soc. 1944, 40, 546. (28) Quéré, D. Wetting and roughness. Annu. Rev. Mater. Res. 2008, 38, 71−99. (29) Mitra, S.; Mitra, S. K.. (30) Ruiz, M.; Kunkle, C.; Padilla, J.; Carey, V. Boiling heat transfer performance in a spiraling radial inflow michrochannel cold plate. Heat Transfer Eng. 2017, 38, 1247−1259.

framework to develop optimized nanoporous surfaces that enhance droplet spreading and, as a result, enhance water evaporative spray cooling. It is also noteworthy that, for surfaces of the type considered here, the model framework can be used to determine the value of a wickability parameter ω* = √2κΔPcap/bϵρlνl from simple droplet deposition experiments. Optimizing the choices of ts and Rs to fit the spreading data to the model prediction, determines Rs and ts. The wickability parameter is then simply computed as ω* = √2κΔPcap/bϵρlνl = Rs/ts, allowing use of the model to experimentally evaluate the performance of different wicking surfaces.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; Phone: +1 (510) 642 7177. ORCID

Claire K. Wemp: 0000-0002-5727-2643 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Support for this research was provided by the Department of Energy through the US-China Clean-Energy Research Center (CERC-WET Grant) under Award Number DE-IA0000018. Research support from the National Science Foundation Graduate Research Fellowship Program is also gratefully acknowledged.

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NOMENCLATURE nanostructured layer height radial distance from center of droplet upper droplet radius mean interface radius of liquid−vapor interface in nanostructured layer Pa ambient atmospheric pressure Pd pressure inside upper droplet ΔPcap nanolayer capillary pressure difference ϵ porosity κ permeability ϕs solid fraction at top surface of nanostructured layer θapp apparent contact angle θE intrinsic contact angle b r rd rnl

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REFERENCES

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