Effect of Latent Heat Released by Freezing Droplets during Frost

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Effect of Latent Heat Released by Freezing Droplets During Frost Wave Propagation Shreyas Chavan, Deokgeun Park, Nitish Singla, Peter Sokalski, Kalyan Boyina, and Nenad Miljkovic Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b00916 • Publication Date (Web): 07 May 2018 Downloaded from http://pubs.acs.org on May 7, 2018

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Effect of Latent Heat Released by Freezing Droplets During Frost Wave Propagation Shreyas Chavan, 1 Deokgeun Park, 1 Nitish Singla, 1 Peter Sokalski, 1 Kalyan Boyina, 1 and Nenad Miljkovic1,2,3,4,* 1

Department of Mechanical Science and Engineering, University of Illinois, Urbana, 61801, USA 2 Frederick Seitz Materials Research Laboratory, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801, USA 3 Department of Electrical and Computer Engineering, University of Illinois at UrbanaChampaign, Urbana, Illinois 61801, USA 4 International Institute for Carbon Neutral Energy Research (WPI-I2CNER), Kyushu University, 744 Moto-oka, Nishi-ku, Fukuoka 819-0395, Japan

*Author to whom correspondence should be addressed. Electronic mail: Nenad Miljkovic: [email protected]

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ABSTRACT Frost spreads on non-wetting surfaces during condensation frosting via an inter-droplet frost wave.

When a supercooled condensate water droplet freezes on a hydrophobic or

superhydrophobic surface, neighboring droplets still in the liquid phase begin to evaporate. Two possible mechanisms govern the evaporation of neighboring water droplets: (1) the difference in saturation pressure of the water vapor surrounding the liquid and frozen droplets induces a vapor pressure gradient, and (2) the latent heat released by freezing droplets locally heats the substrate, evaporating nearby droplets. The relative significance of these two mechanisms is still not understood. Here, we study the significance of the latent heat released into the substrate by freezing droplets, and its effect on adjacent droplet evaporation, by studying the dynamics of individual water droplet freezing on aluminum, copper, and glass-based hydrophobic and superhydrophobic surfaces. The latent heat flux released into the substrate was calculated from the measured droplet sizes and the respective freezing times ( ), defined as the time from initial ice nucleation within the droplet to complete droplet freezing. To probe the effect of latent heat release, we performed 3D transient finite element simulations showing that the transfer of latent heat to neighboring droplets is insignificant, and accounts for a negligible fraction of evaporation during microscale frost wave propagation. Furthermore, we studied the effect of substrate thermal conductivity on the transfer of latent heat transfer to neighboring droplets by investigating the velocity of ice bridge formation.

The velocity of the ice bridge was

independent of the substrate thermal conductivity, indicating that adjacent droplet evaporation during condensation frosting is governed solely by vapor pressure gradients. This study not only provides key insights into the individual droplet freezing process but also elucidates the negligible role of latent heat released into the substrate during frost wave propagation.


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INTRODUCTION Understanding the mechanisms of frost formation is essential to a variety of industrial applications including refrigeration,1-3 aviation,4 wind energy,5 and power transmission.6 Ice accretion and frost formation is a multi-billion dollar problem in the United States alone.7 It has been observed that hydrophobic surfaces offer lower frost density and decreased condensation frost growth rate as compared to hydrophilic surfaces.8

To further delay frost formation,

researchers have proposed the use of suitably designed superhydrophobic surfaces9-12 to delay the heterogeneous nucleation of ice13-16, and thus condensation frosting.17 By exploiting the ultra-low adhesion offered by superhydrophobicity,18 droplets forming during condensation frosting can undergo spontaneous coalescence induced droplet jumping at micrometric length scales (~1 µm) prior to supercooling and freezing on the surface.19-20 In addition to condensate removal prior to freezing, previous researchers have shown superhydrophobic surfaces to facilitate delayed freezing of individual droplets,21-22 as well as bulk water layers due to the delay in ice nucleation and higher thermal resistance at the liquid-solid interface.23-24 The dominant mechanism of condensation frost formation on hydrophobic surfaces is interdroplet frost wave propagation,19, 25-29 or ice bridging. When a supercooled condensate water droplet freezes on a hydrophobic or superhydrophobic surface, neighboring droplets still in the liquid phase immediately begin to evaporate.19, 25 The evaporated water molecules deposit on the frozen droplet and initiate the growth of ice bridges directed toward the droplets being depleted. The neighboring liquid droplets freeze as soon as the ice bridge connects and provides a heterogeneous nucleation site.

Coalescence induced droplet jumping on superhydrophobic

surfaces decreases the droplet distribution density,30-33 thereby decreasing the frost wave propagation speeds, delaying frosting.19 However, given enough time, frost formation still takes


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place due to the propagation of an inter-droplet frost wave from neighboring edge defects, which are present at the edges of the substrate where ice nucleation generally imitates first.19, 34 During inter-droplet ice bridging, two mechanisms govern the evaporation of neighboring water droplets: (1) the difference in saturation pressure of the water vapor surrounding the liquid droplet and the frozen droplet induces a vapor pressure gradient,8, 25, 35 and (2) the latent heat released by freezing droplets locally heats the substrate, evaporating nearby droplets.19 The relative significance of these two mechanisms is currently not well understood. A simple scaling analysis sheds some light on the mechanism. The latent heat of fusion of water at 0°C (ℎ ≈ 334 kJ/kg) is approximately an order of magnitude smaller than the latent heat of vaporization at 0°C (ℎ ≈ 2500 kJ/kg). Upon freezing, a microscale liquid water droplet releases the majority of its latent heat into the substrate,36 which will hemispherically diffuse. Thus, we can estimate that similarly sized neighboring water droplets will receive much less thermal energy than what is required for complete evaporation (ℎ ≪ ℎ ). However, if freezing water droplets have larger length scales ( ~ mm), the latent heat of freezing might be significant enough to cause evaporation of smaller ( ~ µm) neighboring droplets. Thus, depending on the substrate thermal conductivity, the evaporation rate of neighboring water droplets due to conduction of latent heat into the substrate may be significant. Indeed, the substrate thermal conductivity dependent droplet evaporation phenomenon has been observed during the formation of frost halos,37 indicating that the latent heat released may play an important role at a particular, and currently ill-defined, length scale. In this paper, we establish the length scale at which the latent heat released into the substrate is important via the study of individual water droplet freezing and bridge growth dynamics for varying droplet sizes (10 < < 100 µm), on aluminum and copper based hydrophilic ( < 90°),


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hydrophobic (110° <

< 140°), and superhydrophobic (

> 160°) surfaces. Using optical

microscopy, we characterized the latent heat flux released into the substrate by measuring the freezing droplet radius and the respective droplet freezing time ( ), defined as the time from initial ice nucleation within the droplet to complete droplet freezing. To calculate the latent heat flux transferred from the freezing droplet to neighboring droplets, we combined the experimentally characterized heat fluxes with 3D transient finite element method heat transfer simulations. The simulation results showed that the transfer of latent heat to neighboring droplets is indeed insignificant, and accounts for a negligible fraction of evaporation during microscale frost wave propagation. In order to further validate our simulations, we conducted additional experiments and simulations on substrates having varying thermal conductivity (1 to 400 W/m·K) to show that both individual droplet freezing times and adjacent droplet evaporation dynamics remain invariant. Furthermore, we measured a negligible difference between interdroplet bridging velocities for similar droplet arrangements on varying thermal conductivity substrates having identical wettability, pointing to vapor diffusion dynamics as the governing driving force for bridging. The experiments show that the wettability of the substrate has the largest effect on individual droplet freezing time, as the interfacial contact area between the droplet and the surface is inversely proportional to the advancing contact angle of the droplet, governing conduction heat transfer through the bulk of the droplet. Indeed, the disparity between measured individual droplet freezing time scales (~1 ms) to frost wave propagation time scales (~1 s) indicates that the observed bridging phenomena is not affected by individual droplet freezing. This study not only provides vital information on the role of latent heat released into the substrate during frost wave propagation but also explains the effect of substrate thermal conductivity and wettability during ice bridge formation.


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EXPERIMENTAL SECTION Substrate Fabrication and Characterization. Six different types of superhydrophobic surfaces

( > 150°) and four hydrophobic surfaces (110° <

< 140°) were fabricated in order to

study the effect of substrate thermal conductivity ( ), structure length scale ( ), and wettability on the latent heat released by individual droplets into the substrate. Please refer to Section S1 of the Supplementary Information for a detailed description of the surface fabrication and functionalization techniques. Table 1 shows a summary of the different substrate characteristics. Superhydrophobic copper (Cu – SHP) and superhydrophobic copper oxide (CuO – SHP) represent substrates with differing and identical . Superhydrophobic bronze (Bronze – SHP) and CuO-SHP represent substrates with differing and similar (Figure 1a and 1b). Superhydrophobic boehmite (AlO(OH) – SHP) and CuO – SHP represent substrates with both differing and .

The wettability of the substrates was varied by using either

micro/nanostructured (SHP), tool finished (Cu – HP(I)) and polished (Cu – HP(II) and Si – HP) substrates functionalized with hydrophobic coatings.

Experimental Methods. Individual droplet freezing was studied using a custom built top-view optical light microscopy set-up shown diagrammatically in a side view schematic in Figure 1(a). Briefly, samples were placed horizontally on the cold stage (Instec, TP104SC-mK2000A) with a thin film of water underneath in order to provide good thermal communication between the sample and stage. The cold stage was cooled to the test temperature of = -20 ± 0.5°C in a laboratory environment having air temperature, = 22 ± 0.5°C, and relative humidity (RH), = 50 ± 1% (Roscid Technologies, RO120). Individual droplet freezing movies were recorded


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from the top view at 7,000 to 23,000 frames per second with a high speed camera (Phantom, V711, Vision Research) attached to an upright microscope (Eclipse LV100, Nikon). For a droplet of diameter , the freezing time, , was characterized as the time from initial ice nucleation within the droplet to complete droplet freezing. Top view imaging was performed with a 20X and 50X (TU Plan Fluor EPI, Nikon) objective. For the 20X, lens, the working distance was measured to be 5.8 ± 0.5 mm, and for 50X, 11.8 ± 0.5 mm. Illumination was supplied by an LED light source (SOLA SM II Light Engine, Lumencor). The LED light source was specifically chosen for its high-intensity and low power consumption (2.5 W) in order to minimize heat generation at the surface due to light absorption and minimize its influence on the droplet evaporation rates during condensation frosting. Furthermore, by manually reducing the condenser aperture diaphragm opening size and increasing the camera exposure time, we were able to minimize the amount of light energy needed for illumination and hence minimize local heating effects during freezing experiments.38 Figure 2(a) shows the top view time-lapse optical microscopy images of individual droplet freezing on the nanostructured CuO surface (CuO – SHP). Visual inspection of the high-speed videos showing the freezing process clearly delineate the differing stages of freezing due to pixel intensity changes (see Videos S1 to S3, Section S2 of the Supplementary Information). For all the droplet freezing videos taken in this study, it was observed that the image intensity of ice was higher than that of water. In order to mimic the human eye and quantify when the freezing starts and stops, the intensity (average RGB value) of every frame in the droplet freezing video was plotted. Figure 2(b) shows the intensity of the optical microscopy images as a function of time as the water droplet transitions from liquid to solid (see Video S1). While calculating the intensity, a moving average was taken of five neighboring data points to minimize noise.


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Initially, when the droplet was in liquid phase, the intensity was constant. When an ice bridge approaches the liquid droplet and appears in the field of view (at = 1.8 ms), the intensity of the image increases slightly, as observed at the local maxima near = 2.0 ms. As soon as the ice bridge touches the liquid droplet, freezing initiates ( = 2.0 to 4.0 ms). The image intensity oscillates, and then becomes stable at a higher intensity. The oscillation represents the different stages of freezing: i) recalescence, defined as the formation of a porous ice scaffold inside the supercooled liquid droplet21, 39-40 (which starts at the onset of freezing, = 2.0 ms and ends at = 2.4 ms when the droplet is at 0°C), and ii) isothermal (at 0°C) freeze front propagation, which starts at = 2.4 ms and ends at = 4.0 ms. The initiation and end of freezing was identified by the change in intensity from the average value of 27.6 ± 0.1 (peak to peak) before freezing and 29.6 ± 0.1 after freezing due to change of phase from liquid to solid. Typical videos taken of the freezing process (21,000 FPS) resulted in ≈150 images from start to finish, ensuring enough granularity to capture the whole process. Although the relation between the image intensity and phases of water (solid, liquid or mixed) is qualitative, small changes in intensity provide a clear and quantifiable window into freezing. The developed intensity-tracking method works particularly well for microscale droplets due to the significantly smaller radius when compared to the depth of field of the imaging optics. To verify our measurements, we performed additional experiments via focal plane shift imaging (FPSI) for focal planes located at the droplet base instead of the mid-plane. The FPSI results were invariant with a focal plane located within the droplet due to the significantly larger depth of field compared to the droplet length scale. To further verify the developed method, the total freezing time was also characterized by manual image processing of 30 randomly selected high-


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speed videos from the entire data set, showing the accuracy of our results to be within an error bound of ±5 % when compared to image processing with the human eye. RESULTS AND DISCUSSION Microscale Droplet Freezing. Figure 3(a) shows the individual droplet freezing time, as a function of droplet diameter, = 2, for surface temperature = -20ºC. The individual droplet freezing time, on superhydrophobic (SHP) surfaces was an order of magnitude greater than hydrophobic (HP) surfaces due to the higher characteristic length scales of droplets (volume/heat transfer area), and presence of air gaps beneath water droplets which may act as thermal insulation to heat transfer. To study the effect of air gaps of droplet freezing times, freezing experiments were performed on surfaces having different structure length scales, (60 nm to 1 µm). Interestingly, for all droplet sizes (10 µm < < 100 µm) analyzed on the varying length scale SHP surfaces, showed the identical trend and magnitude, indicating that for the length scales examined here, the presence of air gaps beneath supercooled water droplets contributed negligibly to delayed individual droplet freezing, consistent with previous studies on macroscale water films.24 Note, for smaller droplet sizes not examined here ( < 10 µm), the air gap thermal resistance is important when compared to the droplet conduction resistance during freezing, resulting in a structure length scale dependent (see Supplementary Information Section S3 for more information). To help explain the underlying physics governing individual droplet freezing time, we examine the governing analytical heat transfer equations. When ice nucleation initiates in a water droplet, the temperature of the water droplet rises to 0°C due to recalescence (stage i in our results, Fig. 2).37 During recalescence, only a fraction of the liquid volume freezes into an ice scaffold while the latent heat of freezing is almost entirely absorbed by the resulting ice-water mixture,


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resulting in an increase of its temperature from the supercooled state (-20°C) to 0°C. When the ice-water droplet reaches 0°C, freeze front propagation ensues isothermally (at 0°C) within the droplet, converting the ice-water droplet to a homogenous ice droplet and rejecting the latent heat released into the substrate. The recalescence time was observed to be less than 20% of the total freezing time for the water droplets analyzed here. For scaling analysis, we assume that the recalescence time is ≈0.2 and that the process is quasi-steady. Note, the ≈0.2 recalescence time is a conservative estimate appropriate at smaller length scales ( < 100 µm).

Prior

experiments have shown the recalescence duration to be 1 mm).21 Recalescence is a rapid, kinetically controlled process, wherein the fast temperature rise is due to the local nuclei growth, rather than heat transfer across the droplet.39-41 The time taken for recalescence (~ 1 ms) is similar for all droplet sizes (10 µm < < 1 mm). For large droplets ( ~ 1 mm) with ~ 10 s, the recalescence time is negligible as compared to . Whereas, for the microscale droplets studied here ( ~ 10 ms), the recalescence approaches 0.2 . The latent heat released during the recalescence freezing stage will be completely absorbed by the freezing droplet, to sensibly heat from -20°C to 0°C. The fraction of ice formed in the recalescence stage of freezing, can be obtained by a comparison of heat capacities of the supercooled liquid, , and the solid phase, , with the enthalpy of fusion, ℎ :42

∙ ℎ = ( ∙ , + (1 − ) ∙ , )Δ ! , where Δ

!

(1)

is the temperature difference between the supercooled water droplet and 0°C.

Therefore, the total energy released into the substrate during freeze front propagation inside the droplet is equal to the heat transfer release rate, "# $%$ , multiplied by the time taken for freeze front propagation inside the ice-water droplet during freezing, 0.8 :

"# $%$ (0.8 ) = )* +,% ℎ (1 − ) ,

(2)


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where )* is the density of supercooled liquid water, +,% is the droplet volume. To better understand the fundamental physics governing the rate of freezing, we developed an analytical thermal resistance model to analyze the rate-limiting steps for varying droplet morphologies. The thermal resistance analysis yields Δ $%$ = -$%$ "# $%$ , where Δ $%$ is the temperature difference between the just-recalesced droplet (0°C) and far-field substrate temperature, and -$%$ is the total thermal resistance, defined as the summation of the substrate spreading, -

! ,

air gap, - , and droplet conduction, -,% thermal resistances. The droplet

conduction thermal resistance, -,% was calculated via the method employed by Stefan (see Supplementary information, section S3).43-44 For all superhydrophobic substrates tested here, - dominated at smaller droplet length scales ( < 15 µm), while the spreading resistance (-

! )

was negligible at all length scales, indicating that the freezing kinetics were governed

mainly by heat conduction through the water/ice droplet (for a thermal resistance breakdown, see Supplementary Information, section S3). Normalizing the experimentally measured freezing time by the length scale dependent parameters in our thermal resistance analysis (+,% and -$%$ ), results in a surface wettability and droplet length scale independent freezing parameter ( ):

=

0.8 )* (ℎ − ./ Δ ! ) = . +,% -$%$ Δ $%$

(3)

Figure 3(b) shows as a function of for all experimentally measured data points, showing that all data from Figure 3(a) collapse to the same order of magnitude. The collapse of the data indicates that for the substrates and droplet sizes (10 µm < < 100 µm) studied here, the contact angle and droplet length scale dependent heat conduction within the droplet governs the individual droplet freezing time. It is important to note, the frost wave propagation timescale (~1s) observed on the superhydrophobic surfaces was much slower than the individual droplet


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freezing timescale (~ 10ms). Thus, frost wave propagation via ice bridge growth dominates surface condensation frosting dynamics, and delay in individual microscale droplet freezing time has a negligible direct effect on frosting time on non-wetting surfaces. However, individual droplet freezing may have an indirect effect on droplet ice bridging via the conduction of latent heat through the substrate to neighboring evaporating droplets that help grow the liquid bridge. Heat Transfer Modeling. To elucidate the effect of latent heat released into the substrate by freezing of supercooled droplets, we calculated the latent heat transferred to neighboring droplets by performing transient numerical simulations. A 3D numerical model based on finite element method (ANSYS) was used to solve the heat transfer from the freezing droplet to a neighboring water droplet via the chilled substrate. The substrate was modeled (Figure 4) with a constant temperature boundary condition at the left, bottom and right surface. Utilization of insulated boundary conditions negligibly affected the results (< 4%) due to the large domain size 0 (discussed later). An insulated boundary condition was used at the top surface excluding the areas where heat flux is input and output (base areas of freezing and neighboring water droplets respectively). Using a convective boundary condition at the top surface instead of insulated resulted in < 1.5% change in the heat transfer results, at the price of significant convergence time increases. Thus, the insulated boundary condition was used at the top surface where droplets were not present. The heat flux input, "" $ into the substrate at the base of freezing droplet was calculated by:

"" $ =

)* +,% ℎ , 2 3 sin3

(4)

where )* is the water density, +,% is the droplet volume, and is droplet apparent contact angle. For substrates with very low thermal conductivity (< 1 W/m·K), the latent heat released by the freezing droplet can lead to significant evaporation of the remaining liquid in the freezing


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droplet, resulting in a smaller heat flux being released into the substrate (see Section S4 of the Supplementary Information). A convective heat transfer boundary condition is used at the base of the neighboring water droplet with an overall heat transfer coefficient, ℎ! , representing the conduction thermal resistance through the droplet (~ 20,000 W/m2K, Fig. 4a). The neighboring water droplet was assumed to be supercooled at the substrate temperature, 789,,% = = 20°C. To verify the sensitivity of this assumption, additional simulations were conducted to show that a change in neighboring droplet temperature to 789,,% = -10°C resulted in D7 ), which is governed by the time taken after recalescence, F to complete freeze front propagation.37 With an increase in the substrate thermal conductivity , F decreases.37 Numerical Model. To probe the effect of latent heat released into the substrate by freezing droplets at larger length scales, we calculated the latent heat transferred to neighboring droplets, "”* by performing additional 3D transient numerical simulations.

As previously

aforementioned, for substrates having low thermal conductivity (< 1 W/m·K), the latent heat released by the freezing droplet can lead to significant evaporation of the remaining liquid in the freezing (mother) droplet, resulting in a smaller heat flux released into the substrate.37 We used analytical calculations to estimate the fraction of latent heat lost via evaporation of the mother droplet and the remaining latent heat released into the substrate (for calculation details, please see Supplementary Information, Section S4).36

For substrates having very low thermal

conductivity (< 1 W/m·K), more than 50% of the latent heat released during freezing goes to


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evaporating the mother droplet while the remaining heat is released into the substrate. For substrates with higher thermal conductivity (> 10 W/m·K), a negligible amount of heat is lost via evaporation of the mother droplet, hence all latent heat is released into the substrate.36 For the simulations conducted herein, the percentage of latent heat released into the substrate was a conservative (theoretically maximum) estimate. The experimental data of individual droplet freezing times for larger droplet length scales ( = 1.1 mm) on substrates with varying thermal conductivity was obtained from a previous study,37 and utilized in the transient numerical simulations via the application of an appropriate conductivity-dependent heat flux during droplet freezing. Figure 7 shows the heat flux, "”* (left axis), and computed evaporation radius, 8B (right axis), as a function of substrate thermal conductivity, , for freezing droplets having = 12.2 µm and = 1.1 mm. Note, for clarity and to attain a conservative estimate, the computations neglect frost halo microdroplet condensation and subsequent freezing effects. The distance between the edge of the base circle of the freezing droplet and the center of neighboring evaporating droplet, 8 = 8 µm was fixed. For = 1.1 mm, 8B ~ 1 µm, indicating that millimeter sized freezing water droplets have the capability of evaporating micron sized neighboring water droplets. The neighboring water droplets can be pre-existing neighboring supercooled condensate, or the small droplets that consist of the halo itself. As decreases (401 to 0.19 W/m·K) the transferred heat flux from the freezing droplet to a neighboring water droplet increases due to the larger semi-infinite body resistance, -

! =

1/(4 ! ), where ! is the

droplet base radius, resulting in greater spatially confined heat flow to adjacent droplets. However, as previously discussed, for sufficiently thermally insulating surfaces more than 50% of the droplet latent heat content can be lost via evaporation as opposed to substrate conduction.


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For the purpose of numerical simulations, on thermally insulating surfaces assuming no frost halo microdroplet condensation effects, two competing effects are established: (1) the reduction of latent heat transferred to the substrate that acts to lower adjacent microdroplet evaporation, and (2) the localization of latent heat transferred from the freezing droplet to the neighboring droplets which act to increase adjacent microdroplet evaporation.

For these simulations

conducted here (Figure 7), the effect of heat localization dominated (see Section S4 of the Supplementary Information) resulting in larger 8B with lower . Results. The results of the numerical model revealed that a decrease in the substrate thermal conductivity led to an increase in the evaporation of neighboring water droplets, thereby suppressing condensation or frost halo formation. To understand this counter intuitive result which opposes previous experiments,37 we need to examine the main invalid assumption made during our simulations: neglecting frost halo mediated microdroplet condensation and subsequent freezing. The third and dominant factor affecting neighboring microscale droplet formation or evaporation is the pressure gradient (D,E > D7 ), and how long it persists, resulting in diffusion of water vapor from the freezing ice-water droplet to the surrounding substrate. The discrepancy between our computational results and experiments clearly indicate that latent heat transfer to neighboring droplets via conduction through the substrate is not a dominating factor driving frost halo formation with large scale (~ 1mm) droplets. Instead, vapor pressure gradients constitute the main driving force for bridge formation and local droplet evaporation. Indeed, previous results have shown frost halo condensed microdroplets to be ≈ 10 µm ≫ 8B ,37 clearly pointing to vapor diffusion effects dominating droplet freezing dynamics across length scales.

Although, Fig. 7 indicates that millimeter scale droplets on sufficiently insulating

substrates would enable the latent heat released during freezing to evaporate neighboring


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microscale droplets, energy transfer due to vapor-side diffusional gradients dominate substrate conduction and result in no evaporation of neighboring droplets during a frost halo. The experimental and numerical results presented here clearly show that latent heat transfer through the substrate during frost formation has a negligible effect on condensation frosting dynamics on non-wetting substrates. While individual droplet freezing dynamics are governed by liquid/solid heat conduction within the droplet, ice bridge growth and frost halo formation across length scales are governed by vapor diffusion dynamics. In the future, coupling 3D numerical simulations of vapor diffusion dynamics with heat conduction within the supercooled freezing droplet will be key to fully capturing the underlying physics of frost formation on arbitrary surfaces. Furthermore, the development of multi-droplet simulations in the frozen and supercooled state will be important to better represent the physical phenomena where dropletdroplet coupling has the potential to significantly alter the temperature fields inside the substrate and resulting frost wave propagation dynamics. In addition, future studies should examine the important effect of relative humidity on ice bridge formation. Although key for a variety of phase change processes, we do not expect the key message to change based on relative humidity due to the minute contribution of latent heat on adjacent droplet evaporation.

CONCLUSIONS In conclusion, we have demonstrated that latent heat released into the substrate during freezing of individual droplets does not affect frost wave propagation or frost halo formation. Furthermore, we showed that the ice bridge velocities are independent of the substrate thermal conductivity. Ice bridge formation during frost wave propagation is governed solely by the vapor pressure gradient caused by the difference in saturation pressure of the water vapor surrounding the liquid droplet and the frozen droplet. We showed that at length scales tested


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here, the individual droplet freezing time depends on the substrate wettability, since it governs the droplet thermal resistance (which is the most dominant thermal resistance), increasing with higher intrinsic and apparent advancing contact angles. By characterizing the individual droplet freezing (~1ms) and frost wave propagation (~1s) timescales, we show that delay in individual droplet freezing (due to wettability effects) does not affect the overall frosting speed due to the disparate timescales of the two processes. This study not only elucidates the role of latent heat released into the substrate during frost wave propagation but also explains the effect of substrate thermal conductivity and wettability during ice bridge formation and individual droplet freezing.

ASSOCIATED CONTENT Supporting Information The following files are available free of charge. Substrate Fabrication Techniques, Droplet Freezing Videos, Thermal resistance network analysis, and calculation of latent heat released into the substrate. (PDF)

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] Address: 105 South Mathews Avenue, MEL 2136, Urbana, IL 61801, USA. Notes The authors declare no competing financial interest.


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ACKNOWLEDGMENT We acknowledge fruitful discussions with Dr. Jonathan Boreyko of Virginia Tech University and his graduate student Saurabh Nath and thank Dr. Boreyko for reviewing the manuscript. We also acknowledge insightful discussions with Prof. Amy Betz of Kansas State University. We gratefully acknowledge funding support from the Air Conditioning and Refrigeration Center (ACRC), an NSF-founded I/UCRC at UIUC. The authors gratefully acknowledge the support of the International Institute for Carbon Neutral Energy Research (WPI-I2CNER), sponsored by the Japanese Ministry of Education, Culture, Sports, Science and Technology. Electron microscopy was carried out at the Frederick Seitz Materials Research Laboratory Central Facilities, University of Illinois.


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20. Zhang, Y.; Yu, X.; Wu, H.; Wu, J., Facile fabrication of superhydrophobic nanostructures on aluminum foils with controlled-condensation and delayed-icing effects. Appl Surf Sci 2012, 258 (20), 8253-8257. 21. Chaudhary, G.; Li, R., Freezing of water droplets on solid surfaces: An experimental and numerical study. Exp Therm Fluid Sci 2014, 57, 86-93. 22. Zhang, H.; Zhao, Y.; Lv, R.; Yang, C., Freezing of sessile water droplet for various contact angles. International Journal of Thermal Sciences 2016, 101, 59-67. 23. Tourkine, P.; Le Merrer, M.; Quere, D., Delayed Freezing on Water Repellent Materials. Langmuir 2009, 25 (13), 7214-7216. 24. Chavan, S.; Carpenter, J.; Nallapaneni, M.; Chen, J.; Miljkovic, N., Bulk water freezing dynamics on superhydrophobic surfaces. Appl Phys Lett 2017, 110 (4), 041604. 25. Dooley, J. B., Determination and characterization of ice propagation mechanisms on surfaces undergoing dropwise condensation. TEXAS A&M UNIVERSITY: 2010. 26. Guadarrama-Cetina, J.; Mongruel, A.; Gonzalez-Vinas, W.; Beysens, D., Percolationinduced frost formation. Epl-Europhys Lett 2013, 101 (1). 27. Zhang, Y.; Klittich, M. R.; Gao, M.; Dhinojwala, A., Delaying Frost Formation by Controlling Surface Chemistry of Carbon Nanotube-Coated Steel Surfaces. ACS applied materials & interfaces 2017, 9 (7), 6512-6519. 28. Esmeryan, K. D.; Castano, C. E.; Mohammadi, R.; Lazarov, Y.; Radeva, E. I., Delayed condensation and frost formation on superhydrophobic carbon soot coatings by controlling the presence of hydrophilic active sites. Journal of Physics D: Applied Physics 2018, 51 (5), 055302. 29. Kim, M.-H.; Kim, H.; Lee, K.-S.; Kim, D. R., Frosting characteristics on hydrophobic and superhydrophobic surfaces: A review. Energy Conversion and Management 2017, 138, 1-11. 30. Weisensee, P. B.; Wang, Y.; Hongliang, Q.; Schultz, D.; King, W. P.; Miljkovic, N., Condensate droplet size distribution on lubricant-infused surfaces. International Journal of Heat and Mass Transfer 2017, 109, 187-199. 31. Miljkovic, N.; Enright, R.; Nam, Y.; Lopez, K.; Dou, N.; Sack, J.; Wang, E. N., JumpingDroplet-Enhanced Condensation on Scalable Superhydrophobic Nanostructured Surfaces. Nano Letters 2013, 13 (1), 179-187. 32. Rose, J.; Glicksman, L., Dropwise condensation—the distribution of drop sizes. International journal of heat and mass transfer 1973, 16 (2), 411-425. 33. Chavan, S.; Cha, H.; Orejon, D.; Nawaz, K.; Singla, N.; Yeung, Y. F.; Park, D.; Kang, D. H.; Chang, Y.; Takata, Y., Heat transfer through a condensate droplet on hydrophobic and nanostructured superhydrophobic surfaces. Langmuir 2016, 32 (31), 7774-7787. 34. Kim, A.; Lee, C.; Kim, H.; Kim, J., Simple approach to superhydrophobic nanostructured al for practical antifrosting application based on enhanced self-propelled jumping droplets. ACS applied materials & interfaces 2015, 7 (13), 7206-7213. 35. Nath, S.; Boreyko, J. B., On Localized Vapor Pressure Gradients Governing Condensation and Frost Phenomena. Langmuir 2016, 32 (33), 8350-8365. 36. Graeber, G.; Schutzius, T. M.; Eghlidi, H.; Poulikakos, D., Spontaneous self-dislodging of freezing water droplets and the role of wettability. Proceedings of the National Academy of Sciences 2017, 114 (42), 11040-11045. 37. Jung, S.; Tiwari, M. K.; Poulikakos, D., Frost halos from supercooled water droplets. Proceedings of the National Academy of Sciences 2012, 109 (40), 16073-16078.


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38. Rykaczewski, K.; Scott, J. H. J.; Fedorov, A. G., Electron beam heating effects during environmental scanning electron microscopy imaging of water condensation on superhydrophobic surfaces. Appl Phys Lett 2011, 98 (9). 39. Aliotta, F.; Giaquinta, P. V.; Ponterio, R. C.; Prestipino, S.; Saija, F.; Salvato, G.; Vasi, C., Supercooled water escaping from metastability. Sci Rep-Uk 2014, 4, 7230. 40. Feuillebois, F.; Lasek, A.; Creismeas, P.; Pigeonneau, F.; Szaniawski, A., Freezing of a subcooled liquid droplet. J Colloid Interf Sci 1995, 169 (1), 90-102. 41. Angell, C., Supercooled water. Annual Review of Physical Chemistry 1983, 34 (1), 593630. 42. Buttersack, T.; Weiss, V. C.; Bauerecker, S., Hypercooling Temperature of Water Is About 100 K Higher Than Calculated Before. The journal of physical chemistry letters 2018. 43. Stefan, J., Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere. Annalen der Physik 1891, 278 (2), 269-286. 44. Nauenberg, M., Theory and experiments on the ice–water front propagation in droplets freezing on a subzero surface. European Journal of Physics 2016, 37 (4), 045102. 45. Carey, V. P., Liquid-Vapor Phase-Change Phenomena. 2nd ed.; CRC Press: 2008.


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TABLES Table 1. Surface wetting characteristics. For structure fabrication and chemical functionalization details, please see Section S1 of the Supplementary Information. ks [W/m·K ]

Surface ID

Substrat e

Structure s

Hydrophobi c Coating

Structur e Height [um]

AlO(OH) – SHP

Al

AlO(OH)

HTMS

~ 0.1

205

Cu – SHP

Cu

Ag

HDFT

~ 0.06 to 0.2

401

CuO – SHP

Cu

CuO

HTMS

~ 1.0

401

Bronze – SHP

Bronze

CuO

HTMS

~ 1.0

110

Cu – HP (I)

Cu

none

HTMS

NA

401

Cu – HP (II)

Cu

none

HTMS

NA

401

Si - HP

Si

none

HTMS

NA

148

Cu

none

Glaco

< 0.1

401

Si

none

Glaco

< 0.1

1

Cu_Glaco SHP Glass_Glac o - SHP

Advancin g angle [degrees] 170.4 ± 1.4° 173.4 ± 1.3° 170.5 ± 7.2° 165.2 ± 4.3° 146.0 ± 2.1° 118.9 ± 1.7° 110.4 ± 0.3° 165.7 ± 0.7° 163.5 ± 1.9°

Receding angle [degrees] 165.4 ± 4.1° 164.6 ± 4.7° 162.7 ± 3.4° 160.8 ± 3.9° 124.2 ± 3.2° 74.2 ± 0.9° 69.7 ± 1.1° 164.3 ± 0.6° 161.3 ± 0.9°


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FIGURES

Figure 1. (a) Side view schematic of the freezing experiment (not to scale). Scanning electron micrograph of a (b) 10 min oxidized nanostructured CuO surface coated with a ≈ 5 nm thick layer of HTMS and (c) 10 min oxidized nanostructured Bronze surface coated with a ≈ 5 nm thick layer of HTMS. The length scale of structures on Cu and Bronze substrates are same with different substrate thermal conductivity (K = 401 W/m·K, L%7M8 = 110 W/m·K). The sharp, knife-like CuO structures have characteristic heights, ℎ ≈ 1 µm, and solid fraction, N ≈ 0.04. The cold stage was kept at a temperature = -20 ± 0.5°C. Movies were recorded at 7000 to 23000 frames per second. Experimental conditions: ambient air temperature = 22 ± 0.5°C, vapor temperature B = $ OP $ ( )Q = 11.1 ± 0.5°C, relative humidity = 50 ± 1%.


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Figure 2. (a) Time-lapse optical microscopy images of individual droplet freezing on a nanostructured CuO – SHP surface (

≈ 170.5 ± 7.2°). (b) Intensity (average RGB value) of the optical microscopy images as a function of time as the water droplet transitions from liquid to solid. At the beginning of the video at = 0, the droplet is still in liquid state. Freezing initiates at = 2.0 ms and it ends at = 4.0 ms. The initiation and end of freezing was identified by the increase in intensity from the average value of 27.63 ± 0.04 (peak to peak) before freezing and 29.59 ± 0.05 after freezing. Please refer to Video S1 in the Supplementary Information.


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Figure 3. (a) Individual droplet freezing time, , defined as the time from initial ice nucleation within the droplet to complete droplet freezing and (b) reduced freezing parameter, as a function of droplet diameter, for surface temperature = -20ºC. For the hydrophobic samples (Cu – HP (I), Cu – HP (II), and Si – HP), frost wave propagated on the whole surface before the droplets could grow greater than 60 μm in diameter. Thus, there is no data for > 60 μm for hydrophobic samples. The individual droplet freezing time, for superhydrophobic (SHP) surfaces is an order of magnitude greater than hydrophobic (HP) surfaces due to the higher characteristic length scales (volume/heat transfer area) for > 20 μm, and presence of air gaps beneath water droplets which act as thermal insulation to heat transfer for < 20 μm (see Supplementary information, section S3). To normalize the results, the freezing time is divided by the droplet volume and the total thermal resistance, = /(+,% -$%$ ) (see Supplementary information, Section S3), resulting in a constant reduced freezing parameter ss a function of droplet diameter, . Error bars are not included for clarity (maximum error in freezing time and diameter ≈ ±5% of the measured value).


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Figure 4. (a) Schematic of the simulation domain. (b) The hexahedral meshing of the domain (substrate) used in ANSYS Fluent. Inset: Zoomed-in view of the input and output boundary conditions. The domain was chosen based on the calculation of the thermal penetration depth of the released latent heat, estimated as 0 ~

Effect of Latent Heat Released by Freezing Droplets during Frost

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