ARTICLE pubs.acs.org/Langmuir
Are Superhydrophobic Surfaces Best for Icephobicity? Stefan Jung,†,‡ Marko Dorrestijn,† Dominik Raps,‡ Arindam Das,§ Constantine M. Megaridis,§ and Dimos Poulikakos*,† †
Laboratory of Thermodynamics in Emerging Technologies, Mechanical and Process Engineering Department, ETH Zurich, 8092 Zurich, Switzerland ‡ Dept. IW-MS, EADS Innovation Works, 81663 Munich, Germany § Mechanical and Industrial Engineering Department, University of Illinois at Chicago, Chicago, Illinois 60607, United States
bS Supporting Information ABSTRACT: Ice formation can have catastrophic consequences for human activity on the ground and in the air. Here we investigate water freezing delays on untreated and coated surfaces ranging from hydrophilic to superhydrophobic and use these delays to evaluate icephobicity. Supercooled water microdroplets are inkjet-deposited and coalesce until spontaneous freezing of the accumulated mass occurs. Surfaces with nanometer-scale roughness and higher wettability display unexpectedly long freezing delays, at least 1 order of magnitude longer than typical superhydrophobic surfaces with larger hierarchical roughness and low wettability. Directly related to the main focus on heterogeneous nucleation and freezing delay of supercooled water droplets, the observed ensuing crystallization process consisted of two distinct phases: one very rapid recalescent partial solidification phase and a subsequent slower phase. Observations of the droplet collision process employed for the continuous liquid mass accumulation up to the point of ice formation reveal a previously unseen atmospheric-pressure, subfreezing-temperature regime for liquid-on-liquid bounce. On the basis of the entropy reduction of water near a solid surface, we formulate a modification to the classical heterogeneous nucleation theory, which predicts the observed freezing delay trends. Our results bring to question recent emphasis on super water-repellent surface formulations for ice formation retardation and suggest that anti-icing design must optimize the competing influences of both wettability and roughness.
’ INTRODUCTION The safety and performance of modern aircraft are significantly reduced even by light, scarcely visible ice on airfoils, compression inlets of air-breathing engines, and air flow measurement instruments. In-flight icing occurs mainly under certain weather conditions during the holding time before landing, usually at altitudes 9000-20 000 ft, when the aircraft exterior might be subjected to impact of supercooled water droplets present in the upper troposphere and cirrus clouds.1,2 These droplets, which range in size3,4 from 0 to 500 μm, collide with the cool skin of the aircraft and may cause ice accretion, thus compromising safety. In another example, ice buildup on wind turbine blades adversely affects blade aerodynamics, thus increasing drag and, in turn, robbing the energy output of the turbine. Ice formation on power cables causes thousands of power grid disruptions each year in the US alone. It thus comes to no surprise that significant effort has been expended to develop surfaces that facilitate the removal of ice or retard its formation. The term icephobic has been established to describe surfaces that prevent ice from forming and adhering to it. In pursuit of materials with icephobic properties, extensive research has been performed on superhydrophobic surfaces, with recent results5,6 reporting anti-icing behavior for such surfaces. Freezing has been studied for room temperature r 2011 American Chemical Society
droplets placed upon subfreezing surfaces6,7 and for supercooled water poured onto surfaces.5 However, the impact and accumulation of supercooled droplets on surfaces, a common mechanism of ice formation in the environment, has not been investigated yet and thus lacks a thorough fundamental understanding.8,9 Droplet-on-liquid impact ultimately affects ice buildup and thus must be investigated in the context of icephobicity studies. Under certain conditions, a film of gas can be trapped between the two approaching liquid volumes, thus preventing coalescence.10 So far, bouncing of colliding water droplets has been observed only at pressures 2.7-6 bar.11,12 This suggests that droplets hitting a surface covered by a water film in a terrestrial atmosphere (1 bar or less) would stick to this surface, thus enhancing ice buildup. New ways to encourage droplet bounce are of great interest because of their potential to reduce ice buildup. We report a basic study on delayed freezing of inkjet-generated supercooled water microdroplets accumulating sequentially on various surfaces, including superhydrophobic coatings, ultrasmooth hydrophilic surfaces, and films with icephobic potential. Received: November 30, 2010 Revised: January 13, 2011 Published: February 14, 2011 3059
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Figure 1. Representative scanning electron micrographs (a, b) of a nearly superhydrophobic coating (sample 2, similar images obtained for samples 1 and 3) and (c) a low-roughness sol-gel coating (sample 7, similar images obtained for samples 6 and 9) at different magnifications. (d) Roughness values and water advancing contact angles from Supporting Information. The error bars indicate standard deviations based on five measurement points.
In order to achieve the same temperature (-20 °C) for both the target surface and the falling droplets, a low-temperature dropon-demand system has been constructed. The results suggest that target surface roughness and its effect on ice nucleation and nuclei growth have a stronger influence on freezing delay statistics than the presence of the superhydrophobic texture itself. We also demonstrate that bouncing of water microdroplets on accumulated liquid volumes is possible even at atmospheric pressure when the liquid temperature is sufficiently low. Although classical nucleation theory has been successful in explaining freezing statistics qualitatively, doubt has been cast over its quantitative predictability for heterogeneous nucleation.13 We propose a modification to the classical theory that yields a satisfactory fit to the present data. The modification accounts for the reduction in excess entropy of water in the proximity of a surface, as compared to “bulk liquid”.14 We speculate that the spatial gradient in molecular disorder near a solid surface is of the same nature as the order-disorder transition found in bulk supercooled water toward lower temperatures.15 This is implemented in the classical nucleation theory by means of a lowered ice-water surface free energy.
’ MATERIALS AND METHODS Superhydrophobic Coatings. Five (samples 1-5) superhydrophobic (water contact angles above 150°) or nearly superhydrophobic surfaces were produced in-house by coating smooth aluminum plates with polymer-based, large-area, water-repellent coatings.16,17 The polymer matrix of the coatings was a mixture of Poly(methyl methacrylate) (PMMA) and poly(vinylidene fluoride) (PVDF), both obtained from Sigma-Aldrich. Poly(tetrafluoroethylene) (PTFE) particles (260 ( 54.2 nm; Sigma-Aldrich) were added to the mixture to create a texture that induces high water repellency. These five samples were prepared using three different types of coating formulation (varying PTFE particle wt %) and two different types of substrate surfaces. The coatings of samples 1-3 were applied on sand-blasted (“sb”) aluminum plates. The remaining two (samples 4 and 5) were applied on smooth aluminum plates. Detailed information on the preparation of these coatings and other surfaces is given in the Supporting Information.
Icephobic Surfaces. Samples 6 and 8 (coated smooth aluminum substrate surfaces) were obtained from commercial suppliers and were designed to have icephobic properties. The coating of sample 6 consisted of two layers: a two-component epoxy primer and a polyurethane clear top coat. The coating of sample 8 was an epoxy paint containing different salts (e.g., ethoxy silicates) acting as freezing point depressants. Sample 7 (produced in-house) consisted of an aluminum plate treated with an epoxy-based inorganic-organic hybrid sol-gel coating. The coating was based on 3-glycidyloxypropyltrimethoxysilane (GPTMS), including phenyltrimethoxysilane (PhTMS) and 3-aminopropyltrimethoxysilane (APTES), and aluminum sec-butoxide, Al(OBu)3. Further details of the sol-gel preparation are given in the Supporting Information. With respect to aircraft operation the coatings of samples 6 and 7 show high abrasion and erosion resistance and remain stable in a wide temperature range. Figure 1 shows a morphological comparison of sample 2, a nearly superhydrophobic PMMA/PVDF/PTFE coating applied on a sandblasted surface (a, b), and sample 7, a sol-gel coated, low-roughness, icephobic surface (c). The SEM image in Figure 1b reveals typical hierarchical micro-to-nanoscale texture, a necessary requirement for high liquid repellency. The copper sample 9 was polished using colloidal silica (OP-S, Struers), with an average grain size of 0.04 μm and a pH of 9.8. The reference surfaces 10 and 11 were the uncoated aluminum backsides of samples 4 and 3 (smooth and sandblasted, respectively). Finally, the three silicon wafers (Advanced Diamond Technologies) comprised one untreated (sample 12) and two different post-treated surfaces (samples 13 and 14). The last two wafer surfaces were first pretreated in a hot filament chemical vapor deposition (HF-CVD) process. The diamond deposition occurred from a CH4-H2 gas mixture at elevated temperature. In a postdeposition step, the roughness of both ultrananocrystalline diamond (UNCD) surfaces was reduced by replacing hydrogen with OH groups, which made these surfaces hydrophilic. To recover a degree of hydrophobicity, sample plate 14 was heated up to ∼800 °C in a hydrogen atmosphere containing hot tungsten filaments. The hot filaments cracked the hydrogen molecules, thus increasing the atomic hydrogen species concentration, which then reacted with the surface carbon atoms, replacing the OH terminations and, in turn, reducing hydrophilicity. Sample Surface Characterization. Advancing water contact angles (Figure 1d) were obtained by means of a drop shape analysis 3060
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Langmuir setup with a computer-controlled liquid dispensing system (Kr€uss G10/ DSA10). The advancing contact angles were recorded during expansion of droplets from 3 to 5 μL volume by continuously dispensing water through a needle (0.5 mm diameter) placed inside them. Each contact angle measurement was repeated five times on five different locations on each sample plate at an ambient temperature of 21 °C and a relative humidity of 60%. Roughness measurements (Figure 1d) were carried out using a stylus profilometer (Dektak 3ST, diamond tip radius 2.5 μm) and, for the very smooth surfaces, an atomic force microscope AFM (TopometrixExplorer, contact mode). The roughness values Ra and Rt (Supporting Information) are the arithmetic average of the roughness profile and the peak-to-valley distance, respectively, in accordance with DIN 4768. Experimental Procedure and Apparatus. A cryogenic cooler (KGW-Isotherm TG-LKF) was used to cool down a styrofoam chamber of inner dimensions 40 cm 50 cm 25 cm (wall thickness 5 cm). As shown in Figure 2a, a separate inner chamber was created by adding a U-shaped copper plate (thickness 1 mm), which separated it from the outer chamber with a 5 mm gap between the copper plate and the inner sidewall of the styrofoam chamber (schematic of Figure 2a not shown to scale). The cool nitrogen gas was led through the outer chamber and bled into the inner chamber through the gap. No condensation of water vapor was observed because the air—initially present in the chamber— was eventually displaced by nitrogen gas through the gap. At steady state, the absolute humidity near the sample surface was measured to be 0.20 g/m3. The plate samples were placed on an automated rotary stage (Micro Controle-ITL09); the center of the rotary stage was positioned eccentrically to the copper tube in order to place each accumulating droplet (axis of the copper tube) on a circle with droplet-to-droplet distances of 14 mm. Visual access was possible through a sidewall observation window, using a CCD Camera (Sony DSC-F717, 25 fps) mounted on a binocular microscope (Olympus SZ1145TR). For studying collision behavior of smaller droplets impacting onto the larger sessile droplets, a high-speed camera (Olympus-Encore, 1000 fps) was mounted onto the microscope. A piezo-electric jetting device (MicroFab MJ-SF-01-60, 60 μm orifice diameter) was attached to a stainless steel reservoir (100 mL) filled with DI water. Droplets (56-60 μm diameter or 0.09-0.11 nL volume) were generated at a rate of 80 Hz. The jetting device was mounted onto the 0.4 m long copper tube (diameter 20 mm, wall thickness 1 mm) that extended into the chamber (Figure 2a). For the investigation of droplet coalescence vs bouncing, the inkjet parameters could be tuned as to produce larger droplets in the diameter range of 68-80 μm. During the experiments, the temperature was continuously monitored with K-type thermocouples (“T1-T3” in Figure 2a). “T1” was placed immediately next to the lower end of the copper tube and served as a feedback to the cryogenic cooler. “T2” and “T3” were attached directly onto the sample surface and on the inside of the copper tube in the vicinity of the inkjet, respectively. The plate sample was cooled by the flowing nitrogen gas and equilibrated at a temperature of -20 °C 35 min after the initiation of each experiment. During all measurements, “T1” and “T2” read -20 °C ( 0.5 °C and -20 °C ( 0.2 °C, respectively. In order to calculate the impact temperature of a falling droplet, the temperature distributions along the tube axis and the inner tube wall were measured at stable operating temperature using a K-type thermocouple inserted from above through the insertion point of the jetting device (Figure 2a). The time delay between measurements was 2 min to ensure equilibration. On the basis of these measurements (Figure 2b), the temperature of a falling droplet at the exit of the tube (located at a distance of 3 cm above the testing sample) was calculated to be -20 °C (Figure 2b). The initial temperature of the generated droplets was conservatively taken to be 20 °C (the temperature at “T3” was -3.5 °C). The reader is referred to the Supporting Information for details concerning the simulation of heat transfer from a flying droplet through the cooler ambient gas.
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Figure 2. (a) Schematic drawing of the apparatus showing flow of nitrogen gas (blue arrows) and thermocouples T1-T3 (red dots). (b) O, measured gas temperature profile along the axis (z) of the copper tube; 4, measured wall temperature profile of the inner copper tube wall along z; —, calculated temperature distribution for a 60 μm diameter water droplet (ejected just above the chamber with initial temperature 20 °C) falling through the 0.4 m long copper tube in the z-direction. Each data point shown by a symbol is the average of five measurements. The error bars indicate one standard derivation. Droplets continuously impinged, one after the other, onto the same sample location, so as to form, by successive coalescence, a growing sessile droplet until freezing occurred. While rotating the stage to the next position, the inkjet was turned off.
’ RESULTS AND DISCUSSION Freezing Delay Times. Inkjet deposited supercooled droplets (∼60 μm diameter) impinged with frequency 80 Hz onto the same sample location within an area of 300-500 μm across and coalesced into a single growing droplet before eventually freezing. During all measurements, the ambient temperature, the temperature of the water droplet, and the sample surface were kept constant within -20 °C ( 0.5 °C. Freezing delay times were defined as the time between the first droplet impact and the onset of freezing, as determined by direct imaging through a microscope. The onset of freezing coincided with clouding of the water volume, as caused by spontaneous crystallization. Figure 3a shows a box plot of the time until freezing on each surface, based on a minimum of 18 data points per sample plate. An illustration of the box plot concept is given in the Supporting 3061
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for water at -20 °C is in the same range as the roughness of samples 12-14, but 3 orders of magnitude smaller than the roughness of samples 1-5 as well as sample 11. The polished copper sample (no. 9) had a roughness of 21 nm, which resulted in a significant delay (5) compared to the rougher reference sample 10. Yet the freezing delay for sample 9 is O(10) times below the delay for samples 6 and 7, where “O(...)” denotes order of magnitude. Since the roughnesses are very similar for these three samples (6, 7, and 9), surface energies seem to play a significant role in delaying the onset of freezing. Indeed, according to the classical heterogeneous nucleation theory,18 the freezing probability depends not only on surface roughness but also on contact angle (i.e., wettability). Employing the rate at which critical nuclei are generated within the growing droplet18 JðRa , θ, tÞ ¼ KAint ðθ, tÞ expð-ΔGc ðRa , θIW Þ=kTÞ
ð1Þ
and showing that its time dependence can be approximated as J(Ra,θ,t) J0(Ra,θ)t2/3, we have estimated (see Supporting Information for mathematical derivation) the median freezing delay time for this distribution to be 3=5 5 ln 2 ð2Þ tmed ðRa , θÞ ¼ 3J0 ðRa , θÞ
Figure 3. (a) Logarithmic box plot (see Supporting Information for definition) of the time to freezing (i.e., freezing delay) of a growing (through successive impacts of inkjetted droplets) sessile droplet on each of the 14 surfaces considered herein. (b) Geometric area of the liquid-solid interface at the onset of freezing.
Information. All surfaces exhibited delayed freezing compared to the two untreated reference surfaces (10 and 11), as based on the median values of freezing delay. Samples 12-14 performed best; for example, on surface 12, droplets remained liquid 150 times longer than on sample 10. In addition, samples 12-14 performed much better than the superhydrophobic or nearly superhydrophobic coatings (samples 1-5). These three surfaces (12-14) were clearly the smoothest (average roughness 1.4, 3.8, and 5.6 nm, respectively; see Figure 1d). For roughnesses comparable to or smaller than the critical ice nucleus radius, freezing has been found to be significantly delayed.5 The critical ice nucleus radius is the minimum size an incipient ice crystal needs to reach in order to maintain a stable freezing process. For a water temperature of -20 °C, the radius of the critical nucleus rc = 2.2 nm was calculated from18 rc = 2γIW/ΔGf,ν, where γIW is the water-ice interfacial tension (23 mJ/m2 at 253 K)1 and ΔGf,ν = ΔHf,ν(Tm - T)/Tm = 21 MJ/m3 is the volumetric free energy of bulk ice vs bulk liquid, which follows from the GibbsHelmholtz equation; ΔHf,v = 287 MJ/m3 is the water volumetric enthalpy of fusion, T = 253 K (-20 °C), and Tm = 273 K (0 °C) the ice melting temperature at 1 atm. According to the above, rc
where the symbols K, k, t, θ, and Aint denote a kinetic constant, Boltzmann’s constant, the time to ice formation, the advancing water contact angle, and the geometric substrate-liquid “apparent” contact area (see Supporting Information for mathematical derivation), respectively. ΔGc(Ra,θIW) (θIW is the ice/ water contact angle) denotes the free energy barrier for formation of a critically sized nucleus. Figure 4 shows median freezing delay times vs contact angle (Figure 4a) and roughness (Figure 4b). The data are separated according to roughness of the corresponding surface represented in the theory by the ratio x = Ra/rc (values are listed in the Supporting Information). For rough surfaces, x . O(10), shown by closed symbols in Figure 4a, the freezing delay time is independent of roughness (see Supporting Information). The model is fitted to the data with K (eq 2, Figure 4a) as the single fitting parameter (K1 = 1.3 10-26); K is embedded in the expression of J0(Ra,θ) (see Supporting Information). The classical heterogeneous nucleation theory (using quantities for the actual temperature of 253 K) is shown by fitting curve 1 in Figure 4a and deviates significantly from the experimental data. A similar discrepancy was reported by Gorbunov et al.13 To remedy this, we modify the classical theory taking into account the decrease in excess entropy associated with the reduced disorder of the water molecules near a surface, as compared to “bulk liquid”. Even at room temperature, the “bulk” water structure can be altered in the vicinity of solid surfaces, thus forming a hydration layer with significantly different properties.14 For example, Kim et al.19 found that the viscosity of water within 5 nm of an oligo(ethylene glycol) surface at room temperature is 6 orders of magnitude larger than for bulk water. A similar transition has been observed for bulk supercooled water in the range of 190-250 K.20 As for hydration layers, the increase in viscosity is caused by a decrease in entropy according to A η ¼ η0 exp ð3Þ TSconf 3062
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Figure 4. (a) Freezing delay time tmed (median values) vs advancing contact angle for “highly” corrugated surfaces, x . O(10). Contrary to the classical nucleation theory (fitting curve 1) the modified model (fitting curve 2) offers a better fit for the experimental data (symbols). (b) Freezing delay time (median values) vs roughness for smooth surfaces, x e O(10). The value beside each symbol indicates the measured contact angle of the respective surface. The modified model is indicated by fitting curve 3. The error bars indicate standard deviation based on five measurement points.
where η0 = 1.64 10-5 Pa s and A = 31.9 kJ/mol.20 The configurational entropy Sconf can be approximated as Sconf ≈ Sex Sliquid - Scrystal, the difference between the total entropy and the crystal entropy at the same temperature.20 Equation 3 shows that for constant Sconf water behaves as an Arrhenius liquid. This is the case for T < 190 K, where Sex is known to level off.20 Above this temperature, a transition occurs from Arrhenius-type (“strong”) liquid to non-Arrhenius (“fragile”) liquid.21 Because the excess entropy in hydration layers is significantly smaller than that of bulk liquid (the results of Kim et al.19 suggest a factor of 4.4 at room temperature), the water in hydration layers at 253 K is conceivably closer to being Arrhenius type. This implies that the local fluid properties could be close to those of bulk water for T far below 253 K. At the temperature of 229 K, the enthalpy of fusion is ΔHf,v = 180 MJ/m3 and the ice-water surface energy is γIW = 15 mJ/m2,1 which is much lower than the corresponding value (23 mJ/m2) at 253 K. In order to satisfy the validity of Young’s equation (Supporting Information) within the hydration layer, the surface tension of water γW must be adjusted as well. According to Antonoff’s rule,22
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γW|229K = γI|229K - γIW|229K = 91 mJ/m2, where the surface energy of ice (γI = 106 ( 3 mJ/m2)23 does not vary significantly with temperature. Antonoff’s rule is consistent with the cited values at 253 K, taking into account that the ice surface exhibit a disordered, quasi-liquid layer at temperatures considerably below 0 °C.24 This layer can further reduce the surface energy of ice, as suggested by Furukawa et al.25 The adjusted free energy values, based on the reduction of entropy within the hydration layer, affect the free energy barrier ΔGc (Supporting Information), which leads to a rate of critical nucleation that better represents the experimental data (eq 2, fitting curve 2, K2 = 1.78 10-5, Figure 4a). The reduced γIW and ΔGf,v also affect the critical nucleus radius rc, which thus becomes 1 nm. Figure 4b shows the median freezing time vs roughness and contact angle as measured for the smooth surfaces (x e O(10)). The adjusted model (eq 2 with modified values of ΔGc and rc) is fitted to the data and produces “fitting curve 3” in Figure 4b (K3 = 3.55 10-3), which confirms the significant roughness dependency of freezing delay for very small values of Ra (samples 12-14). These results indicate that for rough surfaces (Figure 4a) the freezing delay rises with increased hydrophobicity (higher θ), while for smooth surfaces (Figure 4b) the delay rises with decreased roughness (lower Ra). Most importantly, surfaces with surface roughness Ra close to the critical nucleus radius rc resist icing considerably longer than typical hierarchical rough (“rough” means in this context: roughness values are fare away from the critical nucleus radius) superhydrophobic surfaces. The agreement in trends between experiment and theory shown in Figure 4 is noteworthy, despite several uncertainties involved in the modification of the classical nucleation theory, namely concerning hydration layer thickness,26-28 hydrodynamic properties,28-31 and quantitative atomic-scale information.26 In order to evaluate the influence of the interfacial contact area Aint(θ,t) and the free energy barrier ΔGc(Ra,θIW) on the freezing delay times (see Figure 3a), Figure 3b shows the interfacial contact areas (see Supporting Information for details) at the onset of freezing. A comparison (median values) with Figure 3a shows that for several surfaces freezing typically occurred for similar contact area but at different times (samples 4 and 5, samples 8, 9, 10, 11), likely due to differences in the free energy barrier. When both contact areas and freezing times are similar, one may be tempted to infer that the free energy barriers are also similar (samples 1-3, 6, 7). However, Aint(θ,t) generally differs from the actual contact area because of surface roughness and entrapped gas micro- or nanobubbles between liquid and the solid. The actual contact area is bounded between two extreme cases:32,33 either the liquid follows the contours of the solid surface (Wenzel state, actual area larger than apparent) or it rests on the crests of the roughness (superhydrophobic or Cassie state, actual area typically smaller than apparent, samples 1-5). Hence, the values plotted in Figure 3b may only be viewed as a first estimate of this important quantity. Crystallization Dynamics. Video recordings of the crystallization process (excerpts shown in Figure 5) revealed that freezing occurred in two distinct stages: a rapid (fraction of a second) quasi-adiabatic, recalescent, partial solidification of the droplet (heat production . heat dissipation), resulting in a mixed liquid/ice phase occupying the entire droplet domain,34 and a subsequent isothermal freezing stage of the remaining liquid, in agreement with previous work on rapid ice formation.35,36 Smaller droplets impacted and accumulated into a 3063
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Figure 5. Drop-by-drop accumulation and freezing of a growing supercooled water volume on sample surface 8: The time from the onset of deposition is indicated on each frame. (a) First impacted supercooled droplets have merged together to form one major unit and some secondary satellite units. (b) The main drop has grown by the sequential impaction of inkjetted droplets from above and remains all liquid. (c) Frozen drop shortly after the first freezing stage (shown in more detail in Figure 6), which rapidly converts the supercooled liquid into a mushy liquid plus ice phase. (d, e) The crystallization front (horizontal dashed line) starts at the liquid-solid contact area and gradually advances upward. (f) Completely frozen drop.
larger drop (Figure 5a,b) until the first rapid freezing stage revealed by the instantaneous loss of water clarity (Figure 5c); the heterogeneous nucleation invariably initiated at the substrate-liquid interface, thus indicating that possible impurities in the bulk liquid were not a factor. The second crystallization phase proceeded more slowly with the ice front moving parallel to and away from the sample plate (Figure 5d-f). After the first crystallization phase, additional small droplets landing on the large deposit freeze upon impact, forming a vertical protrusion (Figure 5d-f), a feature distinct from the cusp reported in Anderson et al.37 where quiescent freezing was studied. The additional deformation seen lower in the periphery of the freezing volume in Figure 5e,f is attributed to the different specific volumes of ice and water38 or the possible freezing of surface waves generated by rearrangement of the mushy phase above. High-speed imaging of the first freezing phase revealed further details of the rapid crystallization process. For a sessile 70 nL droplet with a height of 423 μm, the durations of the quasiadiabatic crystallization and isothermal freezing phases were 13 ms and 3 s, respectively. This corresponds to average crystallization front velocities of 32 and 0.14 mm/s. In the first phase, crystallization proceeds much faster, since the enthalpy of fusion is laterally absorbed by adjacent liquid zones. As shown in Figure 6, the front propagation velocity decreases with time due to recalescence and the gradually diminishing mass available to absorb the heat released. The measurements thus confirmed a rapid transition to the solid-liquid equilibrium temperature (∼0 °C) followed by a slower heat transfer limited phase. The second phase also commenced at the substrate-liquid interface, which cooled down more rapidly than the rest of the droplet after the release of recalescent heat. The frozen mass fraction j after the initial freezing phase can be expressed as j = cW/cIS = 0.27,35 where cW
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Figure 6. High-speed imaging sequence (2 ms apart) of phase one of crystallization, which, in this case, initiated at the lower right corner. This sequence was observed on sample surface 8. The white angled lines mark the instantaneous position of the freezing front. The bright triangular spot at the top left of the droplet is due to the light reflection of the stroboscope.
and cI are the specific heat capacity of the liquid water (4458 J/ (kg K)) and the ice (2031 J/(kg K)), respectively.38,39 The Stefan number S, a measure of the droplet supercooling, can be defined as S = cIΔΘ/ΔGf,m, where ΔΘ = ΘW - ΘI, and ΘW ≈ 0 °C,40 and ΘI = -20 °C are the equilibrium freezing temperature and the initial droplet temperature, respectively; ΔGf,m is the specific enthalpy of fusion (334 kJ/kg). On the basis of the calculated frozen mass fraction j after the initial freezing phase and the measured freezing durations, the average crystallization velocities for initial and subsequent freezing were 79 and 0.9 μmol/s, respectively. The crystallization front velocity can be estimated by using the measured time of freezing (13 ms and 3 s) and modeling the sessile droplet as cylinders of the same height but different volumes according to the frozen mass fractions j and 1 - j for the first and second phase of crystallization, respectively. The calculated crystallization front velocities of 32.2 and 0.141 mm/s match very well the measured values (32 and 0.14 mm/s). Droplet-on-Liquid Coalescence vs Bouncing. For certain ranges of droplet size, impact velocity, and ambient gas pressure, droplets are known to bounce off a liquid surface rather than coalescing. We found a previously unseen transition to a bouncing regime for water at ambient pressure. Droplets in the range of 56-60 μm were invariably seen to coalesce with sessile droplets, whereas others in the range of 68-80 μm always bounced, independent of the “off-center distance” of the droplet-droplet impact (61-67 μm droplets could not be reliably produced). All the 305 coalescence events observed correspond to all-liquid droplet collisions that occurred before freezing commenced. 3064
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Figure 7. Deformation gap (scattered data) and corresponding calculated critical deformation gap (curves) vs ln(We-1/4) for four measured droplet sizes with the corresponding simulated terminal velocities. The Stokes numbers of the critical gaps correspond to four different measured droplet diameters (merged droplets: open symbols; bounced droplets: closed symbols). All experiments were conducted with supercooled water (-20 °C).
According to our observation, the transition between coalescence and bouncing lies in the range of We = 0.005-0.009 at 20 °C. The Weber number We for the falling drops is defined as We = FlU2r/γw, where Fl = 994 kg/m3 is the water density,41 γW = 78 mJ/m2 is the surface tension at -20 °C (supercooled water),42 and r is the droplet radius. The velocities of the falling droplets U were terminal velocities, as obtained from the simulation in the Supporting Information. We calculated a theoretical boundary between the coalescence and bouncing regimes according to Bach et al.11 Their theory considered the distance between an impacting droplet and a flat liquid surface at which the droplet starts to deform due to compression of the gaseous medium in the gap. This deformation gap is defined as rCa1/2, where Ca = μgU/γw is the capillary number and μg the viscosity of the gas. A dimensionless deformation gap is defined by δ ¼ rCa1=2 =λ
ð4Þ
where λ is the mean free path of the gas molecules. Bach et al.11 found the following critical deformation gap above which droplets were predicted to bounce: δc ¼ 8:09 - 0:254St 1=2 þ 0:0088St þ 0:582 lnðWe1=4 Þ þ 0:102St 1=2 lnðWe-1=4 Þ
ð5Þ
which was obtained using a fit to numerical results in the range of 17 e St1/2 e 37 and 5.5 10-4 e Kn e 6.9 10-4, where St We/Ca = FlUr/μg is the Stokes number characterizing the droplet inertia and kn = λ/r is the Knudsen number. Figure 7 shows the dimensionless deformation gaps δ (eq 4, symbols) calculated for four different measured droplet diameters using simulated terminal velocities (see Supporting Information) and λ = 46 nm for nitrogen gas; the mean free path was calculated from the variable hard sphere (VHS) model with nitrogen gas viscosity μg µ T0.7167.43,44 Also plotted (lines) are the critical deformation gaps δc from eq 5 for values of (St)1/2 corresponding
to the simulated terminal velocities for these same measured droplet diameters mentioned above. All fluid properties were obtained at -20 °C. The water vapor content of the nitrogen gas (0.015 wt %) was negligible. All four deformation gaps δ (symbols) lie below their corresponding critical line (δc), which would suggest that all droplet sizes should have coalesced. According to the present experiments, however, 68 and 80 μm droplets (closed symbols in Figure 7) bounce under these conditions. The discrepancy can be explained by the fact that for the present λ and droplet diameter range Kn is in the regime 9 10-4 e Kn e 15 10-4, i.e., about twice the Knudsen values for which the model was developed. Indeed, Bach et al. observed that for higher Kn, the predicted critical gaps were shifted downward versus their experimental data. The present results suggest that for water droplets the presence of a bouncing regime depends not only on pressure (as previously established) but also on temperature. This implies that more fundamentally, the number density n of the gas molecules trapped in the gap between the colliding liquid volumes is the determining variable, as follows from the ideal gas law, n = p/kT, where p is the pressure. This is important because in ice-forming applications, droplet bouncing can reduce the total amount of liquid collected on surfaces, in which case bouncing droplets can be carried away by flowing air, for example, before freezing occurs.
’ CONCLUSIONS Delayed freezing of supercooled (-20 °C) water droplets impacting and accumulating on surfaces also at -20 °C and ranging from hydrophilic to superhydrophobic has been studied to evaluate icephobicity. All samples exhibited significant freezing delays compared to basic untreated aluminum surfaces (relatively smooth or roughened). Unexpectedly, the highest icing delay was obtained for a hydrophilic untreated silicon wafer and two hydrophilic ultrananocrystalline diamond coatings, all having very low roughness (1.4-6 nm). A clear correlation was observed between median freezing delay time, roughness, and contact angle: while hydrophobic surfaces show higher resistance against icing than rough hydrophilic surfaces, hydrophilic surfaces with roughness values close to the critical nucleus radius display considerably higher icephobicity (i.e., by an order of magnitude longer freezing delay times) than typical hierarchically rough superhydrophobic surfaces. A modification to the classical heterogeneous nucleation theory yields a satisfactory fit to the freezing time delay data for both rough and smooth surfaces. The modification is based on the hypothesis of reduced entropy in the hydration layer near a solid surface. Observations of droplet collision dynamics revealed a previously not reported temperature dependence of the impact regime in which water droplets bounced off larger sessile droplets due to the increased density of gas molecules in the film between the two droplets at subfreezing temperatures. Droplets in the diameter range 68-80 μm impacting at terminal velocity invariably recoiled. Previously, bouncing of water-on-water droplets had only been demonstrated at elevated pressures. In antiicing applications, this bouncing behavior is beneficial because it can reduce the total amount of water collected on surfaces exposed to water-droplet-laden airflows. The present results indicate that superhydrophobic surfaces may not necessarily offer the best choice (in terms of freezing 3065
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Langmuir delay) for icephobic applications. Nanometer-scale smooth surfaces (irrespectively of wettability) showed much better icephobic properties under the present controlled conditions. Separately, liquid droplet roll-off on superhydrophobic surfaces may offer an additional mechanism against ice buildup in terms of reducing the amount of liquid that stays in contact with the surface. To this end, the selection of the appropriate icephobic surface for a specific technological application requires a multifaceted evaluation of freezing delay and liquid-shedding ability and their competing effects.
’ ASSOCIATED CONTENT
bS
Supporting Information. Detailed description of the sample materials, simulation of heat transfer from a falling droplet, box plot concept, the derivations of a Young’s-type equation and the geometric substrate-liquid “apparent” contact area, and the theoretical estimation of the median freezing delay time. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
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