Microdroplet Growth Mechanism during Water Condensation on

May 1, 2012 - The droplet height does not change appreciably during this steplike base diameter increase, leading to a small decrease of the contact a...
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Microdroplet Growth Mechanism during Water Condensation on Superhydrophobic Surfaces Konrad Rykaczewski* Material Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, United States ABSTRACT: By promoting dropwise condensation of water, nanostructured superhydrophobic coatings have the potential to dramatically increase the heat transfer rate during this phase change process. As a consequence, these coatings may be a facile method of enhancing the efficiency of power generation and water desalination systems. However, the microdroplet growth mechanism on surfaces which evince superhydrophobic characteristics during condensation is not well understood. In this work, the sub10 μm dynamics of droplet formation on nanostructured superhydrophobic surfaces are studied experimentally and theoretically. A quantitative model for droplet growth in the constant base (CB) area mode is developed. The model is validated using optimized environmental scanning electron microscopy (ESEM) imaging of microdroplet growth on a superhydrophobic surface consisting of immobilized alumina nanoparticles modified with a hydrophobic promoter. The optimized ESEM imaging procedure increases the image acquisition rate by a factor of 10−50 as compared to previous research. With the improved imaging temporal resolution, it is demonstrated that nucleating nanodroplets coalesce to create a wetted flat spot with a diameter of a few micrometers from which the microdroplet emerges in purely CB mode. After the droplet reaches a contact angle of 130−150°, its base diameter increases in a discrete steplike fashion. The droplet height does not change appreciably during this steplike base diameter increase, leading to a small decrease of the contact angle. Subsequently, the drop grows in CB mode until it again reaches the maximum contact angle and increases its base diameter in a steplike fashion. This microscopic stick-and-slip motion can occur up to four times prior to the droplet coalescence with neighboring drops. Lastly, the constant contact angle (CCA) and the CB growth models are used to show that modeling formation of a droplet with a 150° contact angle in the CCA mode rather than in the CB mode severely underpredicts both the drop formation time and the average heat transfer rate through the drop.

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on a few natural45−47 and artificial42,43,48−57 surfaces. Dietz et al.50 studied the drop size distribution during this process using environmental scanning electron microscopy (ESEM) and estimated that condensation on the nanostructured superhydrophobic surface is twice as effective in transferring heat as condensation on a flat hydrophobic surface. Subsequently, other experimental investigations have focused on the development and the characterization of superhydrophobic promoters for DWC.51−54,57−63 In contrast, only Kim and Kim64 and Miljkovic et al.65 focused on the development of heat transfer theory suitable for modeling DWC on superhydrophobic surfaces. Kim and Kim’s64 model generalizes DWC heat transfer theory2,6−9,66 to account for effects of nonhemispherical droplet shapes. Their model is founded on the assumption that heat transfer occurs only through the condensed drops. The total heat transfer rate is found by integrating the product of heat flux through an individual drop and drop size distribution over all possible drop radii. Miljkovic et al.65 focused on growth of developed high contact angle droplets with diameters above 10 μm and found that the

INTRODUCTION Water condensation is a common phenomenon in natural and industrial settings. When condensation takes place on a surface that is not wet by the condensate, water beads up into droplets and rolls off the surface. This process is referred to as dropwise condensation (DWC) and is a much more effective heat transfer mode than its counterpart, filmwise condensation.1 DWC has been extensively studied2−20 since 1930, when Schmidt et al.21 recognized the efficiency benefits associated with the use of hydrophobic coatings.1 The development of artificial superhydrophobic surfaces22−27 generated a lot of interest in using such coatings to promote DWC. However, initial reports on the subject were quite disappointing. Namely, condensation on surfaces with macroscopic water contact angles above 150° resulted in complete wetting.28−41 Chen et al.42 were the first to show that the presence of nanoscale topographical features is necessary for a surface to sustain superhydrophobic characteristics during water condensation. In a follow-up work, Boreynko et al.43 demonstrated that release of excess surface energy during coalescence of microdroplets growing on the nanostructured superhydrophobic surfaces leads to self-propelled motion of the new droplet.44 In the past few years similar condensation dynamics have been observed This article not subject to U.S. Copyright. Published 2012 by the American Chemical Society

Received: February 6, 2012 Published: May 1, 2012 7720

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thermal resistance of the gap below suspended microdroplets (Cassie state) can be significant and detrimental to the condensation heat transfer efficiency. They extended Kim and Kim’s64 model to account for the gap thermal resistance. On the basis of the extended DWC model, the authors also developed a droplet growth model which was validated using ESEM experiments. However, recently, Rykaczewski et al.53,54,62 used an optimized ESEM imaging procedure which significantly improves the temporal and the spatial resolution of the technique to demonstrate that the physical mechanism of microdroplet formation on nanostructured superhydrophobic surfaces is significantly different from that assumed in the Kim and Kim64 and Miljkovic et al.65 models. To this end, in this work the sub-10 μm dynamics of microdroplet formation on nanostructured superhydrophobic surfaces are studied experimentally and theoretically. A quantitative model for droplet growth in the constant base (CB) area mode is developed. The model is validated using optimized ESEM imaging of microdroplet growth on a superhydrophobic surface consisting of immobilized alumina (Al2O3) nanoparticles modified with a hydrophobic promoter. Lastly, the CB and Kim and Kim’s constant contact angle (CCA) growth models are used to quantify the impact of the growth mode on the microdroplet formation time and average heat transfer rate through the drop.

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Figure 1. (a) AFM and (b) top-down SEM images of the superhydrophobic surface consisting of immobilized alumina nanoparticles. (c) SEM images of the FIB-milled cross section of the same coating. beam (FIB) cross sectioning and SEM imaging are used to determine the thickness of the nanoparticle coating. To avoid shape distortion due to direct ion beam exposure and indirect effects such as material redeposition, the coating is capped with a platinum layer using focused ion beam induced deposition (FIBID) of C9H16Pt gas precursor, an ion beam energy of 30 keV, an ion current of 93 pA, and a dwell time of 200 ns per pixel. The cross section is obtained by FIB milling at an ion current of 0.92 nA and polishing of the surfaces at a lower current of 93 pA, with both steps performed at an ion beam energy of 30 keV. The cross section, shown in Figure 1c, is imaged at 52° tilt using an electron beam current of 0.4 nA and an electron beam energy of 5 keV. The SEM image of the sample cross section reveals that the nanoparticles are immobilized by an about 30 nm thick uniform base layer. Lastly, water contact angles are measured using a camerabased system (first 10 Å) with vendor-supplied image capture and analysis software. The superhydrophobic coating has static, receding, and advancing contact angles of 151.0° ± 0.3°, 142.1° ± 0.8°, and 154.8° ± 0.8°, respectively. The reported contact angles are averages of goniometer measurements in five locations across the surface; the calculated uncertainty is expressed with a coverage factor of 1. Optimized ESEM Imaging Procedure. The dynamics of water condensation are imaged using an FEI Quanta 200 field emission gun (FEG) environmental scanning electron microscope.67 The sample is mounted using carbon tape at 85−90° on a custom-made brass sample holder62 to prevent electron beam damage to the hydrophobic coating55 and to provide a clear view of the growing droplets.54,62 The environmental scanning electron microscope chamber water vapor pressure is directly controlled using the microscope’s differential pumping system, while the sample temperature is indirectly controlled by cooling the brass sample holder using a vendor-supplied watercooled thermoelectric (Peltier) cooling stage. The chamber pressure and cooling stage temperature are recorded automatically with each taken image. Unfortunately, in the current setup direct measurement of the sample surface temperature is not possible. This temperature is likely to be higher than that of the cooling stage because of the thermal resistance of the sample, the carbon tape, and the brass holder. The electron beam working distance is set to 6−8 mm. After two chamber purging cycles, the sample is chilled at −10 to −15 °C at a vapor pressure of 150 Pa for 2 min. Sustained water condensation is achieved by a step increase in the pressure to about 800−1000 Pa. Because the Peltier cooling stage cannot remove all the heat released by the condensation process, the cooling stage temperature increases quickly with the pressure increase (see Figure 2b). Miljkovic et al.65 estimated that in this setup the sample surface temperature is likely to be about 0.01−0.1 K below the saturation temperature corresponding to the set chamber vapor pressure. To avoid transient effects of the pressure and the temperature equilibration,55 imaging is performed at least 1 min

EXPERIMENTAL METHODS

Hydrophobic and Superhydrophobic Surface Fabrication Procedure. Water condensation dynamics on a flat hydrophobic silicon wafer and a superhydrophobic surface consisting of immobilized Al2O3 nanoparticles are studied. To fabricate the flat hydrophobic surface, silicon wafers are first rinsed with 2-propanol and distilled water prior to being placed in a UV ozone cleaner for approximately 15 min for the removal of adventitious hydrocarbon. After being cleaned and subsequently rinsed with 2-propanol, samples are blown dry with nitrogen and placed in a desiccator, which is pumped down with house vacuum and brought back to atmospheric pressure with house N2. Monolayer formation on the wafers by vapor deposition is accomplished by exposure to (1H,1H,2H,2Hperfluorodecyl)trichlorosilane (Alfa Aesar) in the desiccator under house vacuum for 24−48 h. The modified silicon wafers have a contact angle of 106° ± 2°. The superhydrophobic surface consisting of immobilized Al2O3 nanoparticles is fabricated using RPX-540 manufactured by Integrated Surface Technologies.55 The Al2O3 nanoparticles are formed in the gas-phase reaction of trimethylaluminium (TMA) and water vapor: 2Al(CH3)3 + 3H2O → Al2O3 + 6CH4. The reaction takes place in a low-pressure reaction chamber, with injection of TMA at about 27 Pa (200 mTorr) lasting about 3 s followed sequentially by injection of a water−alcohol mixture at about 270 Pa (2 Torr) lasting about 20 s. The injection sequence followed by a pump/purge is repeated six times. To create a durable nanostructured superhydrophobic surface, the nanoparticles are encapsulated in a thin glasslike matrix using atomic layer deposition (ALD) of SiO2. The SiO2 ALD process consists of two self-limiting surface reactions: SiOH* + SiCl4 → SiOSiCl3* + HCl and SiCl* + H2O → SiO* + HCl and is catalyzed using pyridine (C5H5N).55 A total of 400 ALD cycles produce an about 30 nm thick SiO2 layer. In the final fabrication step, the nanostructured surface is modified with a functionalization exposure to (tridecafluoro-1,1,2,2-tetrahydrooctyl)trichlorosilane (FOTS). Prior to any characterization, the nanostructured samples are ultrasonicated in 2-propanol for 1 min to remove any loose nanoparticles. Top-down SEM of the resulting nanostructured surface is shown in Figure 1b. The topography of the superhydrophobic surface is characterized using tapping mode atomic force microscopy (AFM). Drift in this measurement is corrected for by leveling individual traces. The flat regions are used to fit a line and level the trace (see Figure 1a). The average roughness of the nanoparticle coating is 50.1 nm. Focus ion 7721

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The corresponding standard error,σV, is calculated according to the formula 2 ⎞1/2 ⎛⎛ πhσ d ⎞2 ⎛⎛ πdbase 2 πh2 ⎞ ⎞ ⎟ dbase base ⎜ ⎟σh⎟⎟ ⎟ + ⎜⎜⎜ σV = ⎜ + ⎜⎝ 4 2 ⎠ ⎠ ⎟⎠ ⎠ ⎝⎝ 8 ⎝

(6)

All reported uncertainties are calculated with a coverage factor of 1.

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THEORY The schematic in Figure 3 shows that a drop is assumed to be a spherical cap with radius of curvature r, base diameter dbase,

Figure 2. (a) Varied magnification ESEM images of a typical condensation experiment showing sub-10 μm microdroplet growth between larger droplets. (b) ESEM chamber pressure and cooling stage temperature recorded during a typical condensation experiment. The red vertical dashed line indicates that imaging is only performed after both the pressure and the temperature equilibrate. after the beginning of the condensation process. As shown in Figure 2b, this procedure ensures that both the cooling stage temperature and the chamber pressure equilibrate before the beginning of imaging. This work focuses on imaging of microdroplets with diameters below 10 μm forming between larger droplets during sustained condensation (see Figure 2a). To prevent any electron beam heating effects,53 the drops are imaged with an electron beam energy and a current of 10 keV and 0.16 nA, respectively. The dynamics of the condensation process are imaged with 512 pixel × 471 pixel frame sizes and a dwell time of 250 μs per pixel. The corresponding images are saved every 0.2 s. The ESEM images are analyzed using FIJI image analysis software.68 All reported values of the base diameter, dbase, and the height, h, are averages of six measurements with associated standard errors (σdbase and σh corresponding to the standard deviations of dbase and h). The radii and contact angles are calculated with the assumption that the drop is a spherical cap. Specifically, the radius, r, is calculated as

r=

dbase 2 h + 8h 2

Figure 3. Schematic of a drop of liquid with density ρl and thermal conductivity kl represented as a spherical cap with radius of curvature r, base diameter dbase, and contact angle θ condensed on a substrate with a hydrophobic coating of thickness δ.

contact angle θ, and volume V = πr3(2 + cos θ)(1 − cos θ)2/ 3.6,64,65,69,70 Such droplet growth is affected by both the properties of the substrate and the surrounding conditions. Under numerous scenarios, droplet growth is due to the incorporation of vapor molecules at the perimeter of the drop.10,11,16,32−35 However, Anand and Son69 and Miljkovic et al.65 found that droplet growth in the ESEM is primarily due to direct mass deposition from the vapor onto the drop surface. Accordingly, to model microdroplet condensation in the ESEM, it is assumed that this mass transport mode is dominant. The heat transferred to the drop from the vapor is assumed to be negligible as compared to the energy released during the phase change process, Q = HfgρlV(r,θ) (Hfg is the latent heat of vaporization of water, and ρl is the density of the liquid). As a result, the heat transfer rate through the drop, qd, is equal to the rate at which the enthalpy of the newly condensed vapor changes:

(1)

and its standard error, σr, as 2 ⎞1/2 ⎛⎛ σ d ⎞2 ⎛ ⎛ dbase 2 ⎞⎞ ⎟ 1 dbase base ⎜ ⎜ ⎟ ⎟⎟ ⎟ + ⎜σh⎜ − σr = ⎜ ⎜⎝ 4h ⎠ 8h2 ⎠⎠ ⎟⎠ ⎝ ⎝2 ⎝

(2)

The contact angle is calculated as

⎛ ⎞ 8h2 ⎟ θ = cos−1⎜1 − 2 2 dbase + 4h ⎠ ⎝

d V (r , θ ) dQ = Hfgρl (7) dt dt The heat transfer rate through the drop is calculated using the DWC heat transfer model1,6−8,71,72 which was recently generalized by Kim and Kim64 to account for effects of nonhemispherical droplet shapes. Specifically, it is assumed that conduction through the drop is the dominant heat transfer mode.1,7,8,71 Since the conduction time scale,73 tconduction ≈ r2/α ≈ 7 μs, is much shorter than the experimentally observed drop growth time scale, tgrowth ≈ 1 ms, quasi-steady-state heat

(3)

qd =

and its standard error, σθ, as

⎛ 4σ 2h3 + 4σ 2d 2h ⎞1/2 d h base ⎟ σθ = ⎜⎜ base ⎟ 2 d 4 h2 + ⎝ ⎠ base The volume of the spherical cap, V, is calculated according to

⎛d 2 h2 ⎞ V = πh⎜ base + ⎟ 6⎠ ⎝ 8

(4) 62

(5) 7722

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conduction is modeled (α is the thermal diffusivity of water). The drop surface is assumed to be at a uniform temperature, which leads to an assumption of a uniform interfacial heat transfer coefficient, hi. According to Carey71 2 ⎞ ⎛ 2σ ̂ ⎞ pl Hfg ⎛ M̅ ⎜⎜ ⎟ ⎜ ⎟ hi = ⎝ 2 − σ ̂ ⎠ Tvapor ⎝ 2πTvaporR̅ ⎟⎠

qd(r , θ ) =

1/2

(8)

dr = dt 2Hfgρl

2πh ir (1 − cos θ )

ΔTdrop =

ΔTcap =

ΔTcoat =

2Tsatσlg ρl Hfgr

=

rmin ΔT r

δcoat kcoat sin 2 θ

)

(15)

(

ΔT 1 −

(

1 2h i

+

rmin r

rθ(1 − cos θ ) 4kl sin θ

)

+

δcoat(1 − cos θ ) kcoat sin 2 θ

)

(16)

Hfgρl dbase (4δcoath ik l + 2k lkcoat + 0.5θh ikcoatdbase + θhi + 2k lkcoat cos θ)

(17)

Equations 16 and 17 demonstrate that the drop growth rate can be expressed as a function of a single geometrical parameter in both the CCA and CB growth modes. The plot in Figure 4a shows the relation between dr/dt and r for the case of CCA growth with a contact angle of 100°. The presented trend matches well with previously published reports.17,75−77 Specifically, dr/dt decreases rapidly with increasing drop radius until saturating for values of r greater than 50 μm. The plots in Figure 4b,c show the relation between dθ/dt and θ for the CB growth mode. For all levels of surface subcooling, dθ/dt decreases with increasing θ in three distinct phases. First, dθ/dt decreases rapidly until θ ≈ 10°. Subsequently, dθ/dt decreases steadily until θ ≈ 150°, after which it falls sharply. Equations 16 and 17 are solved numerically using Euler’s method:65

(9)

(10)

(11)

(12)

rn + 1 = rn +

dr dt

Δt (18)

rn

qdδcoat πr 2kcoat sin 2 θ

(13)

θn + 1 = θn +

⎛ rθ 1 ⎜ + 4kl sin θ πr (1 − rmin /r ) ⎝ 2h i(1 − cos θ ) qd

⎞ ⎟ sin 2 θ ⎠

δcoat kcoat

dθ dt

Δt θn

(19)

Equation 18 is solved with a starting value of r0 = 1.1rmin and an adaptive time step of Δt = 10−9/(dr/dt)rn. In turn, eq 19 is solved with a starting θ0 = 0.1° and an adaptive time step of Δt = 10−4/(dθ/dt)θn. For growth on the superhydrophobic surface, δcoat = 50 nm and thermal conductivity kcoat = 16 W/mK are used. For all simulations Tsurf = 279 K and Tsat is varied to achieve the desired level of surface subcooling. The approach used by Lauri et al.,77 Anand and Son,69 and Miljkovic et al.65 is taken to validate the derived CB growth model. Specifically, the predictions of eq 19 are compared directly with ESEM observations of the time evolution of the droplet’s contact angle, diameter, and height.

2

+

+

128ΔTh ik l(cos 4 0.5θ)(0.5dbase 2 − rmin sin θ)

where kl is the water thermal conductivity, σlg is the liquid water surface tension, and rmin is the critical radius, which is equal to 2Tsatσlg/Hfgρ1ΔT. Substitution of eqs 10−13 into eq 9 leads to64 ΔT =

rθ 4kl sin θ

3

qdθ 4πrkl sin θ

+

dθ = dt

qd 2

1 2h i(1 − cos θ )

)

For purely CB growth, the base diameter of the drop is a constant that is related to r and θ through r = dbase/(2 sin θ).69 As a consequence, the volume of the drop can be calculated as a function of θ, V = π[di3/(sin3 θ)](2 + cos θ)(1 − cos θ)2/24, leading to dV(θ)/dt = (dV/dθ)(dθ/dt). Substitution of these expressions into eqs 7 and 15 yields an ordinary differential equation relating dθ/dt to θ:

The individual temperature drops are calculated using the following equations: ΔTi =

(

rmin r

By combining eqs 7 and 15, the drop volumetric growth rate can be calculated as dV(r,θ)/dt=qd(r,θ)/Hfgρl. For the constant contact angle (CCA) growth mode, dV(r)/dt = (dV/dr)(dr/ dt) and dr/dt can be related to r through the following ordinary differential equation:64

where σ̂ is the accommodation coefficient which for ESEM conditions approximately equals 0.8,69 M̅ is the molecular weight of the gas, R̅ is the universal gas constant, and Tvapor is the vapor temperature, which is assumed to be equal to the saturation temperature, Tsat. As opposed to the work of Miljkovic et al.,65 the contribution of the thermal resistance of the water and surface interface is neglected because the average roughness of the surface is only about 50 nm. It is important to note that on superhydrophobic coatings consisting of more complex surface architecture the interfacial resistances can be significant if the condensed microdroplets are in the nonwetting Cassie−Baxter state.65,74 If the base surface of the drop is assumed to be at uniform temperature, Tsurf, the total temperature drop between the vapor and the surface, ΔT = Tsat − Tsurf, is equal to the sum of temperature differences due to all the contributing thermal resistances.64 Namely, ΔT is equal to the sum of the temperature drops due to the thermal resistance of the vapor−liquid interface (ΔTi), the conduction across the drop (ΔTdrop), the capillary depression of the equilibrium saturation temperature (ΔTcap), and the conduction through the hydrophobic coating (ΔTcoat) with thickness δcoat and thermal conductivity kcoat: ΔT = ΔTi + ΔTdrop + ΔTcap + ΔTcoat

(

ΔTπr 2 1 −

(14)

The heat transfer rate through the drop is related to θ and r by rearranging eq 14: 7723

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Figure 5. Time evolution of the radii of four drops observed in the environmental scanning electron microscope during condensation on a flat silicon substrate with a macroscale water contact angle of 106° ± 2° and simulated radius vs time for CCA growth with a contact angle of 100° with surface subcooling of 0.015, 0.02, and 0.025 K.

When significant nanoscale roughness is introduced onto the hydrophobic surface, the water droplet growth mode changes dramatically. The nucleating nanodroplets wet and fill a volume between the nanostructures, creating a wetted flat spot with a diameter of a few micrometers.53,54 The sequence of ESEM images in Figure 6a illustrates that subsequently the droplet emerges out of the wetted spot by predominantly increasing its contact angle.53,54,62 The time evolution of the contact angle for 10 droplets with initial base diameters around 2.5, 3.5, and 6 μm growing on the superhydrophobic surface consisting of immobilized alumina nanoparticles55,62,80 matches well the numerical predictions of eq 19 with subcooling of 0.04 K (see Figure 6b). Within the first 1.5 s of growth, droplets with initial base diameters of 2.5−3.5 and 6 μm reach a maximum contact angle of about 150° and 140°, respectively. In this initial time period, the simulated and the experimentally observed droplets' volumes also match well (see Figure 6c). However, after about 1.5 s of growth the contact angle saturates around the maximum value, leading to a constant simulated volumetric growth rate. This is contrary to the experimentally observed volumetric growth rate, which increases at the same time. This discrepancy is explained by the steplike increase in the base diameter of the droplet corresponding to the volumetric growth rate increase (see Figure 6d). The simulations clearly confirm that an increase in the droplet base diameter increases the volumetric growth rate. Therefore, after formation of the wetted flat spot, the droplet grows in purely CB mode. When the droplet reaches its maximum contact angle, its base diameter increases in a discrete steplike fashion. The droplet height does not change appreciably during the steplike base diameter increase, leading to a small decrease of the contact angle. Subsequently, the drop grows in CB mode until it again reaches the maximum contact angle and increases its base diameter in a steplike fashion. This growth mechanism is schematically illustrated in the inset in Figure 6b and clearly observed in Figure 6d, which shows the time evolution of an individual droplet’s base diameter, height, and contact angle. It was observed that 80% of all studied droplets underwent at least two steplike base diameter increases prior to coalescing with other drops. An additional third discrete diameter increase was observed in 40% of cases (see Figure 6e). A fourth diameter slip was observed only for one drop. As in the case of its macroscopic equivalent during droplet evaporation on

Figure 4. (a) Constant contact angle mode growth rate (dr/dt) vs droplet radius (r) for the growth of a water droplet on a surface with a contact angle of 100° and varied levels of surface subcooling (ΔT). (b, c) Constant base area mode growth rate (dθ/dt) vs contact angle (θ) for the growth of water with varied levels of ΔT with (b) dbase = 2 μm and (c) dbase = 4 μm.

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RESULTS AND DISCUSSION After a drop nucleates on a flat surface, it grows unrestrained by increasing its radius at a nearly constant contact angle.65,69,77,78 This CCA growth mode is illustrated by the sequence of ESEM images in Figure 5, which shows a few examples of droplet growth on the flat silicon wafer modified with a perfluorinated coating (see ref 79 for a short discussion of CCA droplet growth dynamics). Figure 5 also shows that, using the optimized ESEM imaging procedure, images are acquired at a 5 fps (frames per second) rate, which is 10−50 times faster than the 0.1−0.5 fps rate used previously by Lauri et al.,77 Anand and Son.,69 and Miljkovic et al.65 It will be demonstrated below that the enhanced imaging temporal resolution is necessary to accurately capture the microdroplet growth mechanism on the superhydrophobic surface. 7724

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Figure 6. (a) Sequence of 90° ESEM images illustrating nearly constant base area growth of an isolated droplet. (b) Experimentally observed contact angle and (c) volume time evolution of 10 drops condensing on the nanostructured superhydrophobic surface and (b) simulated contact angle and (c) volume vs time with assumed confined base diameters of 2.5, 3.5, and 6 μm and surface subcooling of 0.04 K. (d) Time evolution of the base diameter, the height, and the contact angle of one of the drops. The blue and red arrows are used to indicate a discrete diameter slip and related contact angle decrease. (e) Contact angles and corresponding base diameters observed prior to diameter slips of all studied drops. The sequence of each the slip is also identified (the points indicated as “2nd” correspond to the second observed slip of a drop).

superhydrophobic surfaces, this microscopic stick-and-slip motion is likely due to pinning and depinning dynamics of the triple line.81 Next the growth models are used to compare the dynamics of high contact angle microdroplet formation in the CB and the CCA growth modes. Figure 7 illustrates formation of identical microdroplets with a contact angle of 150° through the CB and the CCA modes. Taking Kim and Kim’s64 and Miljkovic et al.'s65 approach, it is assumed that in the CCA mode the droplet formation begins with a nucleus with a radius equal to the critical radius and a contact angle of 150°. The analysis is limited to growth of droplets with base diameters below 10 μm,

because larger drops grow predominantly by coalescence with other drops.53,54,62 To achieve a quantitative comparison of the drop formation dynamics in the two growth modes, the time evolution of the drop volume is simulated. The plots in Figure 8a−c illustrate that it takes significantly longer to reach a volume of 30 μm3 (dbase ≈ 2 μm) in the CCA mode than in the CB mode, regardless of the surface subcooling. To further facilitate comparison between the two growth modes, the total droplet formation times, ΔtCB and ΔtCCA, are calculated as a function of the base diameter (see illustration marked in red in Figure 8a). As shown in Figure 8d, the total formation times can be conveniently compared for different levels of surface subcooling 7725

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with other droplets until reaching a base diameter of about 10 μm. Figure 9 illustrates that ΔQ/Δt in the CCA mode is just

Figure 7. Schematic contrasting high contact angle microdroplet formation in CB and CCA growth modes.

by plotting the product of ΔT and Δt. It is evident that ΔTΔtCB is nearly independent of dbase and equal to 0.007 K s and 0.018 K s for surface subcooling of 0.02 and above 1 K, respectively. In contrast, ΔTΔtCCA increases rapidly with increasing base diameter, reaching a value of about 1 K s for dbase ≈ 8 μm. The ratio ΔtCCA/ΔtCB plotted in Figure 8e reveals that it takes 10−150 times longer for a 150° contact angle drop to form in the CCA than in the CB mode. The microdroplets grow slower in the CCA mode than in the CB mode because of the lower heat transfer rate during the process. To quantify this effect, the average heat transfer rate during droplet formation, ΔQ/Δt, is evaluated. ΔQ/Δt is approximated as the ratio of total energy released during condensation of the drop (ΔQ = HfgρlΔV) and the total droplet formation time. In this formulation, is it implicitly assumed that the unwetted surface area in the CCA growth mode is inactive. This assumption is a reasonable approximation because most microdroplets do not begin to coalesce

Figure 9. Ratio of the average heat transfer rate during individual microdroplet condensation in CCA and CB area growth modes.

7−10%, 2−4%, and 0.7−1.4% of that in the CB mode for dbase values of 2.5, 5, and 10 μm, respectively. The dramatic difference between (ΔQ/Δt)CCA and (ΔQ/Δt)CB stems from the difference in the base contact areas during droplet formation. By definition, the contact area in the CB mode is constant and equal to the final value of the contact area in the CCA mode. The discrepancy between ΔQ/Δt in the CCA and CB modes is more pronounced at higher levels of surface subcooling, because the critical radius is only a few nanometers for such conditions.

Figure 8. Volume vs time for individual droplets growing in the CCA and the CB modes with surface subcooling of (a) ΔT = 0.02 K, (b) ΔT = 1 K, and (c) ΔT = 10 K. (d) Total time required for a droplet to reach the size corresponding to the given base diameter (Δt). To simplify comparison between different subcooling levels and growth modes, the product of Δt and ΔT is presented. (e) Ratio of Δt for CCA and CB growth modes as a function of the final base diameter. 7726

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CONCLUSIONS In this work, the sub-10 μm mechanism of microdroplet formation on superhydrophobic surfaces was revealed through optimized ESEM imaging and growth simulations. The microdroplets initially grow in purely CB mode until reaching a contact angle of 130−150°. Subsequently, the droplet base diameter increases in a discrete steplike fashion, leading to a slight decrease of the contact angle. Next the drop gradually grows in CB mode until the next steplike base diameter increase. This microscopic stick-and-slip motion can be repeated up to four times and leads to periodic contact angle oscillations near its maximum value. Lastly, the CCA and the CB growth models were used to quantify the impact of the growth mode on the microdroplet formation time and average heat transfer rate through the drop. It was demonstrated that modeling formation of a droplet with a 150° contact angle in the CCA mode severely underpredicts both the formation time and average heat transfer rate through the drop. These results highlight that properly describing the microdroplet formation mechanism is crucial to modeling the overall heat transfer during dropwise condensation on superhydrophobic substrates.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS This research was performed while I held a National Research Council American Recovery and Reinvestment Act (NRC ARRA) Research Associateship Award at the National Institute of Standards and Technology in Gaithersburg, MD. I kindly acknowledge Dr. Jeff Chinn from Integrated Surface Technologies for providing the superhydrophobic sample, Dr. William A. Osborn from NIST for the AFM surface characterization, Dr. Marlon L. Walker from NIST for silicon wafer surface modification, and Mr. David M. Anderson, Dr. Andrei G. Fedorov, and Dr. Peter A. Kottke from Georgia Tech for insightful conversations and comments.

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REFERENCES

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power exponent for this power law, the growth dynamics of an ensemble of condensing drops should be evaluated. Nevertheless, a short discussion of the power exponent estimated from the growth of the four imaged drops is included here for completion. In agreement with the work of Miljkovic et al., in the current experiments the power exponent for the growth of the individual drops is initially 0.8, but after 10 s of growth, it decreases to 0.5. At higher levels of surface subcooling, the calculated power exponent is constant during the growth and is equal to about 0.5. The values of the power exponent obtained in this work match well with those reported by McCormick et al.2 (α ≈ 0.5 for water condensation on copper) and Ichikawa et al.75 (α ≈ 0.66−0.5 for water condensation on silanized glass) but are higher than α ≈ 0.33 reported by Beysens and co-workers for the growth of water droplets on a variety of different substrates.5,10,11,16,78 According to Anand and Son69 and Miljkovic et al.,65 the small discrepancy among the power law exponents implies that in the environmental scanning electron microscope vapor diffusion is not a drop-growth-limiting factor. (80) Chinn, J.; Helmrich, F.; Guenther, R.; Wiltse, M.; Hurst, K.; Ashurst, R. W. Durable Super-Hydrophobic Nano-Composite Films. NSTI-Nanotech 2010, Anaheim, CA, June 21−24, 2010; Nano Science and Technology Institute: Austin, TX, 2010; Vol. 1. (81) Bormashenko, E; Musin, A; Whyman, G; Zinigrad, M. Wetting Transitions and Depinning of the Triple Line. Langmuir 2012, 28, 3460−3464.

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