Can wicking control droplet cooling? - Langmuir (ACS Publications)

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Can wicking control droplet cooling? Manuel Auliano, Damiano Auliano, Maria Fernandino, Pietro Asinari, and Carlos A. Dorao Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.9b00548 • Publication Date (Web): 30 Apr 2019 Downloaded from http://pubs.acs.org on May 5, 2019

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Can wicking control droplet cooling? ∗,†

Manuel Auliano,

†

Damiano Auliano,

†

Maria Fernandino,

Carlos A. Dorao

†Department

Pietro Asinari,

‡

and

†

of Energy and Process Engineering, Norwegian University of Science and Technology, Trondheim, Norway

‡Department

of Energy Engineering, Politecnico di Torino, Torino, Italy

E-mail: [email protected]

Abstract Wicking, dened as absorption and passive spreading of liquid into a porous media, has been identied as key mechanism to enhance the heat transfer and prevent the thermal crisis. Reducing the evaporation time and increasing the Leidenfrost point (LFP) are important for an ecient and safe design of thermal management applications, such as electronics, nuclear and aeronautics industry. Here we report the eect of the wicking of super-hydrophilic nanowires on the droplet vaporization from low temperatures to temperatures above the Leidenfrost transition. By tuning the wicking capability of the surface, we show that the most wickable nanowire results in the fastest evaporation time (reduction of 82 %, 76 % and 68 % compared to a bare surface at respectively 51 C, 69 °C and 92 °C) and in one of the highest shift of the Leidenfrost point of a water

°

droplet (5 µl) in the literature (about 260 °C ).

Keywords wicking, droplet cooling, Si nanowires, Leidenfrost phenomenon, evaporation 1

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INTRODUCTION Heat production accounts for most of the world's primary energy consumption. 1 A better thermal management in several applications would result in a more ecient and safer design. In this context, evaporation and boiling phenomena are ubiquitous for several applications, such as thermal management of power electronics 2 , power generation 3,4 , electronic cooling 5 , seawater desalination 6 , distillation 7 and biosciences 8 . Among the dierent technologies, spray cooling is a promising approach because it seems to oer the best balance in term of uid inventory, surface iso-thermality and high heat uxes removal capability. 9 It can remove heat in excess of 100 W/cm2 using uorinerts 10 and more than 500 W/cm2 using water 11 . Due to the complex thermal-uid interaction between the population of droplet of dierent sizes, the study is usually limited to the case of a single droplet. When a droplet impinges on a heated surface, the droplet can either cool the substrate down eciently or not depending on the surface temperature. Figure 1a) sketches the dierent regimes which a droplet undergoes after impacting on a heated surface. For enhancing the droplet cooling heat transfer, it is desirable for the droplet to spread as much as possible, as fast as possible. 12 Increasing the safety margins by delaying the occurrence of the Leidenfrost point is also crucial for high-heat-ux applications, as in the case of nuclear reactors where the release of radioactive substances due to insucient cooling can result in global disaster (see Fukushima accident 13 ). An ideal surface (represented by the green continuous curve in Figure 1a)) would decrease the evaporation time in the evaporation and nucleate boiling regime, broaden the transition boiling region without signicant deterioration and shift the Leidenfrost point towards higher temperatures compared to a plain surface (represented by the red dashed line).

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Figure 1: (a) Sessile droplet evaporation curve. At surface temperature below the saturation point, evaporation at the vapour-liquid interface occurs. When the surface temperature is suciently high, nucleation sites are activated and bubbles are released. The phase change improves the heat transfer in the nucleate boiling regime. Once bubble merge and form a continuous cushion, the droplet hovers over the surface in a frictionless regime, and the cooling eciency is reduced. This phenomenon found in daily life occasions, was rst described by J.G. Leidenfrost in 1756 and it is thus commonly referred as the Leidenfrost phenomenon.(b) Droplet spreading and wicking on a super-hydrophilic surface. When a droplet is deposited on a super-hydrophilic and porous surface, it is absorbed and spreads as a thin lm. This phenomenon is known as wicking and its dynamics was rst theoretically described by Washburn in 1921. 14 From the literature it is known that the experimental data n in these stages show a dynamic scaling behaviour as r α t 2 . 15,16 Since the wicking distance (dened as the dierence between the precursor rim and contact line) is consistent with the Washburn's law, the wicking coecient w is usually reported and dened as the average slope of the wicking distance versus the square root of time. 17 In this regard, texturing the surface with micro/nanostructures is promising since it controls the surface wettability and the heat transfer performance. 18 This approach is the most appealing since the others are usually constrained. In particular, super-hydrophilic surfaces have shown to enhance the nucleate boiling heat transfer and critical heat ux for pool boiling, 19 delay the occurrence of the dryout in ow boiling, 20 enhance droplet spread and evaporation heat transfer 21,22 and increase the Leidenfrost point 23,24 . These enhancements are mainly attributed to the capillary wicking identied as the main mechanism to keep the surface wetted at high surface superheat and to re-wet the dry spots avoiding the burnout of the surface. A brief description of the wicking phenomenon in the case of a droplet as 3


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source is provided in Figure 1b). Recent research studies have shown that wicking on structured surfaces in pool and ow boiling experiments enhances the CHF by increasing the liquid supply to dry patches and heat dissipation from the wicking coolant. For instance, Rahman et al.(2014) showed that the wickability of super-hydrophilic nanostructures fabricated by biological templates is the key mechanism dictating CHF on structured superhydrophilic surfaces in pool boiling experiments. 25 Kim et al. (2016) reported that randomly distributed vertical Si nanowires with higher wicking enhance more the CHF in ow boiling experiment. 26 Moreover, among the possible inuential parameters aecting the Leidenfrost point, the capillary wicking has been cited as one of the possible explanations for delaying the occurrence of the lm boiling regime. Indeed, Kim et al. (2013) reported a delay in the occurrence of the cutback phenomenon and of the LFP of a water droplet on superheated superhydrophilic zircaloy nanotubes. 27 This was explained considering the enhancement of the wettability and spreading induced by the nanostructures. Kim et al. (2014) reported a higher Leidefrost point of water droplet on micro and nanostructured zirconium alloy surfaces attributed to the enhanced wettability and increase of nucleation site density responsible of an explosive behaviour that triggered the liquid-vapor interface collapse of the droplet. Recently, the LFP shift of water droplet observed on titanium oxide nanotubes was attributed to the induced capillary pressure. 23 Moreover, recently we concluded that a super nano-wicking surface will allow maximizing the cooling in the transition boiling region due to the lift-o process and suppressing the Leidenfrost phenomenon even at higher temperatures. 28

Despite all these studies, an interesting question still remains: can we control the droplet cooling by tuning the wicking capability of a surface? The role of the wicking on the droplet cooling (in both the evaporation and boiling regimes) remains not well understood, which motivates research on how the wicking property of the surface can aect the heat transfer process. This corresponds in Figure 1a) to control the evaporation time of the droplet and 4


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the shift of the Leidenfrost point. The answer to this question would have fundamental signicance for applications desiring to maximize the heat transfer performance and to wide the transition boiling region such as quenching of hot metals, cooling of nuclear reactors during emergency shutdown, heat exchangers.

EXPERIMENTAL SECTION In this work, we employed a simple approach to control the wicking performance, which is to tailor a surface with Si stochastic nanowires. These nanostructures have already shown excellent performance for droplet cooling, 24 pool boiling 19 and ow boiling 20,29 , and moreover are scalable, low-cost to fabricate 30 and robust to droplet impact 24 . In particular, Kim et al.(2016) evaluated experimentally the hemi-wicking behavior on Si random nanowires with dierent heights and observed that increasing the NWs height enhances the hemi-wicking (up to 15.8 µm). 26 Therefore, we fabricated three Si nanowire surfaces with dierent height by employing the Metal Assisted Chemical Etching. SEM top and cross section views are shown in Figure 2 (a),(b),(c). Details regarding the fabrication and characterization of Si nanowires are reported in Supporting Information in the section S1.

After a careful and critical analysis of the wicking metric employed in the literarure, we developed and validated our wicking and wettability method for characterizing the droplet spreading on super-hydrophilic nanostructures. Indeed, although several studies 12,16,17,25,26,3136 investigated the capillary wicking in porous media, there is no a unique common method for characterizing the wicking properties of the dierent surfaces. This yields a inconsistent comparison between the wicking parameters of the surface and a inconsistent use of them in predictive models in relation to the observed phenomenon. Some works 17,33 immerse the sample vertically in a becker lled with the wetting uid. Others 25,36 bring a small tube

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lled with the wetting uid in contact with the sample and measure the wicking volume ux by tracking the transient liquid meniscus in the tube. Despite the simplicity of the approach that captures the global wetting behaviour of the surface, this method does not allow to capture the evolution of the precursor rim (not visible from a front camera), contact line and the wicking length. Most of them 16,31,32,37 place the surface horizontally and measure the radial spreading of the droplet impinged at a certain point on the surface. In this case the wicking properties were investigated by taking into account only one radial spreading direction. Only Fan et al. 31 and Kim et al. 17 have taken into account more wicking directions (two perpendicular and octangular radial lines, respectively). Recently, Wemp et al. 12 derived the mean radius of the spreading droplet from the measured wetted area (processed by ImageJ), thus taking into account all the wicking directions. The eective spreading droplet area (referred to the precursor rim) is an important parameter to be evaluated since it is related to the eciency of the droplet cooling. 38 In our case, in order to be consistent with the phenomenon under investigation, we choose a droplet as wicking source. In addition, since both wicking and wettability are needed for fully describing the wetting behavior of extremely wetting surfaces 36 , we develoved a synchronized and reliable metric for characterizing both the wettability and wicking of super-hydrophilic surfaces. We implemented a wicking and wettability in-house Matlab codes to track simultaneously the wicking of the entire droplet (both the contact line and the precursor rim) and the transient evolution of the contact angle. The main advantage of this method is the capability of tracking automatically the spreading of the droplet, regardless of the number of frames to be digitally processed. The possibility of acquiring a large number of data points can improve the accuracy for determining the power law coecient (as it will discussed in the section S2 and Supporting Information ).

The method allows to assess the predictive capability of the model

proposed by Wemp et al.(2017) 12 . Indeed, the hemispreading relation 12 can be employed to predict the transient spreading of the precursor rim once the height of the nanostructures is measured and the transition point from the synchronous to the hemispreading stage is 6


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determined by the experimental wicking curves. Moreover, it allows an in-situ measurement of the wicking and wettability during the droplet cooling experiments compared with other methods. Figures 2 (d) and (e) show respectively the experimental setup and the in-house calibration tool used in the wicking characterization.

Figure 2: SEM top views (top) and cross-section (bottom) of the Si nanowire surfaces (d is the NW diameter, h is the NW height): (a) NW-A (d = 153 ± 9 nm, h = 2.506 ± 0.008 µm); (b) NW-B (d = 173 ± 8 nm, h = 6.324 ± 0.019 µm)(c) NW-C (d = 226 ± 25 nm, h = 25.267 ± 0.082 µm);(d) Experimental setup ;(e) Snapshot of the calibration tool during the calibration (only 4 of the 5 patterned squares are visible in the eld view of the high speed camera). 7


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The front camera provides information about the contact angle and contact line evolution in the time, while the top camera provides information on the wetted area (that is not homogeneous for real surfaces) and on the equivalent radius of the precursor rim that cannot be detected from the front camera since it spreads within the micro/nanostructures. Calibration, validation, uncertainty analysis and wicking characterization are available in Supporting Information

in the section S2.

RESULTS AND DISCUSSION Wicking and wettability on Si nanowires Figure 3(a) shows the synchronous evolution of the contact line and the precursor rim for a 5 µl DI water droplet on a nanowire surface at room temperature. As suggested by Wemp et al. 12 , the equivalent radius is computed as the square root of the wetted area divided by π . The onset of the precursor rim propagation divides the overall process in two stages. Figure 3(b) shows the eect of the nanowire height on the spreading of the precursor rim. It can be observed that the precursor rim of the tallest NWs (NW-C) surface permeates faster into the nano-forest compared to the ones of the other NWs surfaces. This can be explained considering that the wicking could depend on two contrasting forces such as the capillary force and the viscous force of the ow. 32 By looking at the topography of the NW surface from the SEM top view in Fig. 2 (a),(b) and (c), as the height increases the bundling eect becomes more prominent (as reported also by Yamaguchi et al.(2014) 39 ), thus increasing the spacing between the NWs bundles and reducing the ow resistance. The shortest nanowire NW-A has the lowest power law coecient np (as shown in Table 1). Although the short spacing between the dierent nano-channels increase the capillary pressure, at the same time the increase in ow resistance plays a more dominant role. As more space is available, the reduction of the viscous force becomes more dominant to the reduction of the capillary force caused by the increase of the spacing between the nano-channels. Figure 3(b) shows that the 8


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NW-C has the steepest line, that results in the highest power law coecient (as shown in Table 1). This method allows to process several frames of the wicking video, thus improving the accuracy of the power law coecient (see Figure S3 in Supporting Information ). Figure 3(c) depicts the evolution of the mean apparent contact angle. The droplet spreading on the surface NW-C reaches within 1 s nearly zero mean apparent contact angle that cannot be detected anymore by the digital image processing code.

Figure 3: Wicking and wettability characterization of the NWs surfaces:(a) Transient evolution of the droplet precursor rim and contact line (the subset shows the boundaries of the objects tracked by the in-house Matlab code); (b) Comparison of the evolution of the precursor rim on the NW surface; (c) Transient evolution of the mean apparent contact angle on the NW surface. Consistently with previous studies, 40 we report an increase of the wicking coecient (shown in Table 1) with the nanowire height.

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Table 1: Wicking performance of the nanowire arrays: np and w are respectively the precursor rim coecient and the wicking coecient introduced in the caption of Figure 1b)and dened in the Supporting Information in equations S7 and S13. NW-A NW-B NW-C

h [µm] 2.506 ± 0.008 6.324 ± 0.019 25.267 ± 0.082

np [-] 0.206 ± 0.014 0.246 ± 0.009 0.348 ± 0.016

w [mm/s0.5 ] 0.419 ± 0.041 0.5154 ± 0.049 0.888 ± 0.076

Droplet cooling on Si nanowires In order to characterize the droplet cooling performance of the surface, we employed the droplet evaporation method based on the sessile droplet evaporation curve. 41 The experimental procedure and analysis of the uncertainties are reported in the section S3 in the Supporting Information

and in our previous work. 24 To be consistent with the wicking char-

acterization, we employed the same droplet size. Figure 4(a) shows a signicant reduction in the evaporation regime of the droplet lifetime compared to the plain Si. This can be attributed to the wicking occurring in the NWs surface that yields the complete spreading of the droplet and the formation of a thin precursor rim. In this way, the total heat transfer area and the conductive thermal resistance increase and decrease respectively, thus resulting in a reduction of the cooling time. In addition to the wicking, the improvement of surface wettability is responsible for a decrease of the evaporation time, in agreement with previous studies 42 . Figure 4(b) shows the inuence of the surface temperature on the precursor rim wicking coecient np . Due to the long duration of the heat transfer experiments, only two videos are considered for measuring the coecient. It can be observed that the evaporation decreases the wicking eect, in accordance with the previous work of Fries et al.(2018) 43 that compared the capillary rise of a liquid column (HFE 7500) into a weave material at low and high evaporation rate. However, the eect is less signicant for the tallest NW surface. Both the surface tension and the liquid viscosity decrease with temperature (as shown in Figure S14 in the Supporting Information ), which can result in a reduction of the capillary pressure and

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viscous loss respectively. Due to the loss of liquid by evaporation, the wicking front travels more slowly than in the case without evaporation. 43

Figure 4: (a) Ratio of the droplet lifetime on the Si NWs compared to the bare surface from the evaporation to nucleate boling regime; 44 (b) Eect of the surface temperature on the precursor rim coecient; 45 (c) Sessile droplet evaporation curve from nucleate boiling to lm boiling regime (the full dots represent the Leidenfrost points and the orange arrow highlights the broadening of the transition boiling region); 45 (d) Eect of the precursor rim coecient on the Leidenfrost temperature. 45 As the temperature increases above saturated conditions, the precursor rim still forms, but the digital image processing becomes more challenging due to the nucleation of bubbles. The droplet dynamics in the evaporation and nucleate boiling regimes is reported for the tallest nanowire surface (NW-C) (see section S4 in Supporting Information ). For all the three cases, the formation of the precursor rim occurs even at high temperatures (T = 164°C), while the total bulk drainage occurs only at low wall superheat (T = 51°C and T = 69°C). As the temperature increases, the spreading area of the precursor rim is lower. It is important to observe that the precursor rim and an upper droplet with low contact angle are noticeable 11


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only from the top view camera (as shown in Figure S15 in section S4 in Supporting Information ).

This implies that using a top camera is important to have an accurate measurement

of the evaporation time. Figure S12 (see section S4 in Supporting Information ) shows the eect of the surface temperature in the evaporation regime on the apparent contact angle of the upper droplet. A large dierence in wettability is observed between the heated bare surface and the nanowire. In particular, the tallest nanowire surface (NW-C) shows the lowest contact angles for all the three temperatures. As the wall superheat increases, the dierence in wettability between the nanowires decreases. However, the dierence is not remarkable within the experimental uncertainties. Above saturated conditions, nucleate boiling occurs, thus aecting the determination of contact angle evolution due to bubble nucleation. At high temperature (T ∼ 165 °C), the droplet evaporation time on the nanowire surfaces becomes larger compared to the plain Si surface (see Figure 4 (a)). The high-speed video (see Figure S16 in Supporting Information ) shows that rst nucleate boiling occurs, followed by an evaporation stage where no continuous release of bubbles is observed. At higher temperature (T ∼ 183 °C), the evaporation stage is reduced compared to the nucleate boiling stage for the NW-A and NW-B surfaces (see Figure S17 in Supporting Information), while the droplet undergoes the lift-o phenomenon on the NW-C surface. In the case of the plain Si, by increasing the surface temperature yields a more vigorous nucleate boiling accompanied by the ejection of small satellite droplets. At T ∼ 183 °C, the CHF occurs for the plain Si, which means the lowest evaporation time (or alternatively the maximum heat ux) is reached. Although at T ∼ 183 °C the evaporation time decreases on NW-A and NW-B, their ratio compared to the plain Si increases further beyond 1 because of the minimum evaporation time reached in the case of the latter (CHF condition). Figure 4 (c) shows the sessile droplet evaporation curve for all the surfaces at high surface temperatures. The broadening of the transition boiling region results in a shift of the Leidenfrost point, that is higher with taller nanostructures (that present higher wicking). Figure 4 (d) shows the eect of the wicking of the NW surface (quantied by the precursor 12


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rim coecient) on the Leidenfrost temperature. It can be observed that higher wicking results in a higher Leidenfrost temperature. This could be explained considering that in the case of higher wicking, the liquid droplet is able to penetrate into the nanowire array and be lifted up from the substrate in the lift-o phenomenon. The occurrence of the lift-o (shown in Figure S18 the Supporting Information ) prevents the establishment of a vapour cushion layer and therefore delays the occurrence of the Leidenfrost point. The sessile droplet evaporation curve by the tallest nanowire surface (NW-C) is compared with other recent micro/nanostructured surfaces (see Figure S19 in Supporting Information ). As the droplet touches the nanowire, the surface temperature below the droplet will decrease. However, the droplet cannot overcome the spinodal decomposition temperature of water at ambient pressure (321 ± 17 °C). 46 More details about this aspect are discussed in the section S4 in Supporting Information.

In addition to the wicking, size and distribution of active nucleation sites play a signicant role in establishing the continuous vapour layer between the liquid and the substrate. The LFP model described by Bernardin and Mudawar (2002) predicts the surface superheat temperature required to initiate the growth of hemispherical vapor bubbles from the existing cavities that act as sites for heterogeneous nucleation of bubbles. 47 According to the YoungLaplace equation, the pressure drop across a spherical bubble interface of radius r can be estimated as:

pv − pl =

2σ r

(1)

where σ is the liquid-vapor surface tension. Combined with the Clausius-Clayperon equation, it predicts the temperature required to initiate the nucleation of a hemispherical vapour bubble as: 2συlv

Tnucl = Tsat e rhlv

(2)

where Tsat is the saturation temperature,υlv is the dierence between the vapour and

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liquid specic volumes, hlv is the latent heat. All the uid properties are the values corresponding to the saturated conditions. For millimetric droplets, the use of Eq.(2) is acceptable instead of the Kelvin equation 48 . Indeed, the latter is based on the assumption that gv = gl + 2σ/(R ρl ) , where g is the Gibbs free energy, R is the droplet radius, is the liquid density. 49 Considering our case (R ∼ 1.1 mm) and assuming the uid properties at saturation, then it can be observed that the term 2σ/(R ρl ) is neglibible compared to gv , which means that the starting assumption of the Kelvin equation reduces to the derivation of Clausius-Clayperon equation(gv = gl ) 49 used for predicting the activation temperature of the nucleation sites with Eq. (2). Moreover, Kim et al.(2011) 50 reported that starting from the Rayleigh equation for the inertia-controlled phase of bubble growth, the pressure dierence across the vapour interface at the point of nucleation can be estimated as

∆p ∼ ρl v 2

(3)

where ρl is the liquid density and v is the velocity of the expanding vapour-liquid interface. Assuming that the pressure dierence across the vapour interface at the point of nucleation can be approximated to the pressure drop across the spherical bubble interface (the evaporating liquid can be considered at the nucletion temperature in the beginning of the nucleation process 51 ) , the velocity of the expanding vapour-liquid interface can be estimated. When the vapour phase velocity is greater than the critical velocity of the Kelvin-Helmholtz instability, the liquid-vapor interface can be disrupted. Kim et al.(2011) 50 reported that nanoporosity is essential to delay the LFP by preventing the formation of a continuous vapour cushion due to the heterogeneous nucleation of bubbles that disrupt the interface. Figure 5 depicts the temperature required for the activation of nucleation sites (see Equation (2))) and the inuence of the cavity size on the velocity of bubbles released by the nucleation sites (computed by equating Eq. 1 and 3). It can be observed that bubbles released from sub-micron cavities (that mostly characterize the nanowire surfaces 24 ) overcome 14


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the critical velocity of the Kelvin-Helmholtz instability, therefore are fast enough to disrupt the interface and result in the explosive lift-o of the droplet. A combination of the fast velocity of the bubble released by the nucleation sites and the liquid capillary pumping can prevent the formation of a stable continuous vapour layer. The occurrence of evaporation regime on the nanowire surface at higher superheat (as shown in Figure S16 and S17) could be explained by the delay of the onset of nucleate boiling on nanostructured hydrophilic surfaces (as reported by Van Carey et al.(2018) 52 ). Indeed, nanocavities require larger superheat to be activated as shown in Figure 5(a). The activation of more cavities at higher temperature (T ∼ 183 °C) results in a vigorous vapour bubble growing and ejection with a distortion and almost breaking of the droplet interface (see Figure S21 in the Supporting Information ).

As the superheat increases (T ∼ 231 °C), the lift-o regime occurs due a

combination of droplet wicking and vigorous bubble shooting. After the impact, the droplet keeps a dome shape at the top while instability occurs at the bottom with ejection of bubbles. The lift-o from the substrate is followed by a maximum spreading and a simultaneous disintegration at the bottom of the droplet (see Figure S22 in the Supporting Information ).

In addition to the wettability and cavity size, several other parameters aect the LFP 41 . Among these, the eect of the surface thermal properties has been poorly investigated, especially in the case of rough surfaces tailored with micro/nano-structures. Recently, Nair et al. (2014) 53 proposed a timescale approach by including the eect of thermal diusivity in the heat transport timescale, which is the time for the heat to ow through the nanostructures. A lower thermal conductivity increases the heat transport timescale. If this is much longer than the timescale required for the cooling of the nanowires and the exposure of the nanowires to the vapor ow, then the droplet comes in contact with the nanostructures at a temperature much lower than the initial one. This implies that the onset of the lm boiling regime requires a higher temperature, therefore the LFP is delayed. This eect was commented in our previous work 24 and extended also to more micro/nano-structures available in the literature. 15


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In addition to the diusivity, the change of thermal conductivity of the surface with nanowires aects also the thermal eusivity that in turn determines the contact temperature of two bodies that touch each other, which might be decisive for this phenomenon. Indeed, the theoretical contact temperature Tc proposed by Seki et al. 54 might be estimated as weighted average of liquid and solid eusivities by analogy with two semi-innite solids in contact: 55

Figure 5: (a) Velocity of the bubbles and temperature required for the activation of the nucleation sites as function of the cavity size. The blue dashed-dot line represents the critical velocity of Kelvin-Helmholtz instability 56 The SEM image of a NW surface highlights the presence of sub-micron cavities within the NWs bundles and of micro-cavities between the NWs bundles. (b) Estimation of the theoretical contact temperature between water droplet and surface at dierent temperatures. The green curly bracket identies the interval of thermal conductivity of Si nanowires fabricated with a Silver Assisted Chemical Etching 57 .

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Tc =

Ts s + Tl l s + s

(4)

where the subscripts s and l denote the solid surface and the liquid, Ts and Tl are the temperatures of solid and liquid, and is the eusivity computed as:

=

p

ρck

(5)

where ρ, c and k are the mass density, the specic heat and the thermal conductivity, respectively. Considering the droplet initially at room temperature (20 °C) and dierent temperatures of the substrates (80 and 300 °C), we can observe in Figure 5(b) that the thermal conductivity of the substrate can aect the contact temperature especially at high surface temperatures. In particular, coating a surface with Si nanowires etched with Metal Assisted Chemical Etching (k ∼ 1-50 W/m/K 57 ) can lower signicantly the contact temperature, which could be important for delaying the onset of lm boiling. In the case of micro/nanostructured surface, the eective thermal conductivity of the surface can be approximated to the thermal conductivity of the nanowires (see section S4 in Supporting Information). The thermal conductivity of Si nanowires depends on dierent factors such as internal porosity, doping concentration and diameter 57,58 . In this case, the nanowire internal porosity (depending on the etchant solution) and the doping concentration (P+ Boron 3.2 x 1018 atoms/cm3 - 8 x 1018 atoms/cm3 ) can be considered the same for all the samples. Although the diameter distribution is variable, the thermal conductivity of our nanowires could be restricted to a limited range (k ∼ 1-4 W/m/K) based on previous studies. Indeed, Hochbaum et al. (2008) reported that nanowires etched from Si wafer with a resistivity of 0.01-0.02 ohm cm (heavily doped) have a much lower thermal conductivity than the other nanowires, for instance k = 1.6 and 4 W/m/K at room temperature for an average diameter of d = 52 and 115 nm. 58 Moreover, Weisse et al.(2012) reported that the thermal conductivity of porous SiNWs fabricated with MaCE (k ∼ 1-50 W/m/K) drops to about 1 W/m/K when the dop-

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ing concentration is increased to 1018 atoms/cm3 . 57 Based on these considerations, the eect of the thermal eusivity might be neglected in this study since the thermal conductivity is expected to not change signicantly among the dierent samples. Figure 6 reports an overview of the LFP temperature shifts reported with water droplets compared to a plain surface (including the results from this work). 13,24,46,50,5962 Compared to our previous work 24 , the LFP shift normalized to the droplet volume is not reported since it might not be an accurate metric.

Figure 6: LFP shifts of water droplet on micro-nanostructured 13,24,46,50,5962 compared to the corresponding plain surface. (The red bars with dashed edges correspond to the maximum value of the experimental setup. The transparent and solid bars indicate that the value reported is the temperature inside the heating block or on the surface, respectively). Indeed, although Hassebrook et al. 63 reported a linear trend of the LFP versus the droplet size for the same processed surface (see Figure S12 in the Supporting Information ), this might not hold for all the micro/nanostructures and the droplet weight could not be the dominant contribution to the LFP shift (as discussed in subsection Validation (S3) in the Supporting Information ).

Moreover, the eect of the gravity was not considered in the physical model

proposed by Kwon et al. 13 . The deviation of the predictions from the experimental results 18


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was attributed mainly to a dependence of the contact patch length (assumed constant for all the structures) on the pitch between the pillars. In addition, Adera et al. 2 neglected the droplet weight in their model based on the balance between wetting (gravity and surface tension) and anti-wetting (induced vapour pressure) forces. However, even considering the absolute LFP shift without taking account the droplet volume, the results show that random Si nanowires represent a promising candidate for shifting the LFP. Indeed, they provide one of the highest Leidenfrost temperature shifts in the case of a water droplet and represent an inexpensive and scalable solution for industrial applications.

CONCLUSIONS In summary, we have investigated the eect of the wicking on the droplet cooling. We tuned the wicking capability of a bare surface by tailoring it with stochastic Si nanowires with three dierent heights. By increasing the wicking, one of the highest LFP shifts was achieved in the literature in the case of a water droplet of 5 µl (about 260 °C), a signicant reduction of the evaporation time was observed (up to 82 %) and a large broadening of the transition boiling region (of about 265 °C) was observed without a signicant heat transfer deterioration. The linear relationship found between the Leidenfrost temperature and the wicking conrmed the key role of the capillary wicking in delaying the onset of the lm boiling regime. As ongoing work, well-organized Si nanowires will be investigated to assess the eect of the surface topography on the wicking and the droplet cooling.

Author contributions M. A. and D. A. contributed equally to this work. M.A. conceived the idea. C.A.D. provided support and feedback in the development of the work. M.A. fabricated and characterized the NWs surfaces. M.A. and D.A. developed the wicking code, D.A. developed the contact angle code and validated both of them. D.A. performed the wicking characterization of the 19


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samples, conceived and developed the error analysis. M.A. and D.A. carried out the droplet cooling experiments. M.A. wrote the manuscript. All the co-authors revised and gave their feedback on the manuscript.

Acknowledgement The Research Council of Norway is acknowledged for the support under the FRINATEK project 231529 "Fundamental study of hydrodynamics and mass transfer in annular-mist ow". The Research Council of Norway is acknowledged for the support to the Norwegian Micro- and Nano-Fabrication Facility, NorFab (197411/V30). GE Healthcare is is acknowledged for providing the reference paper "Grade 3 Chr Cellulose Chromatography "for the validation of the wicking codes.

Supporting Information Available The following le is available free of charge. Supporting Information: Fabrication and characterization process of the nanowire surfaces; Validation of the wicking and wettability codes developed to characterize the super-hydrophilic surfaces, including a systematic analysis on the experimental uncertainties; Calibration and uncertainties on the droplet cooling experiments; Additional results and discussion.

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Can wicking control droplet cooling? - Langmuir (ACS Publications)

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