Spraying water droplets on air fin surfaces is often used to augment the performance of air-cooled Rankine power plant condensers and wet cooling tower heat exchangers for building air-conditioning systems. To get the best performance in such processes, the water droplets delivered to the surface should spread rapidly into an extensive, thin film and evaporate with no liquid leaving the surface due to recoil or splashing. This paper presents predictions of theoretical/computational modeling and results of experimental studies of droplet spreading on thin-layer, nanostructured, superhydrophilic surfaces that exhibit very high wicking rates (wickability) in the porous layer. Analysis of the experimental data in the model framework illuminates the key aspects of the physics of the droplet-spreading process and evaporation heat transfer. This analysis also predicts the dependence of droplet-spreading characteristics on the nanoporous surface morphology and other system parameters. The combined results of this investigation indicate specific key strategies for design and fabrication of surface coatings that will maximize the heat transfer performance for droplet evaporation on heat exchanger surfaces. The implications regarding wickability effects on pool boiling processes are also discussed.
Numerous recent investigations have examined the use of nanostructured surfaces to enhance features of water boiling or liquid water evaporation processes. Since prior research has shown that increasing surface wetting by the liquid generally improves boiling performance, it is not surprising that use of nanostructured hydrophilic surfaces has frequently been proposed as a means of enhancing nucleate boiling heat transfer and critical heat flux (CHF) for pool boiling [1–16], suppressing wall dryout in flow boiling , and enhancing droplet spread and evaporation heat transfer in water spray cooling [18,19]. The use of superhydrophilic nanostructured surfaces would appear to offer the promise that substantial enhancement of water pool boiling and liquid evaporation processes, since superhydrophilic surface structures of this type have been found to exhibit rapid spreading of liquid over an initially dry surface [7,8,18]. Recent studies [18,19] have shown, for example, that surfaces of this type can quickly transition a millimeter-sized droplet to a liquid film on the surface only about 100 μm thick. The subsequent rapid transfer of heat from the surface across the film to the interface results in rapid evaporation and a very high heat transfer coefficient over the footprint of the droplet.
Recent studies by Rahman et al.  and Kim et al.  have used a wickability parameter to quantify the tendency for superhydrophilic micro- or nanostructured surfaces to exhibit rapid spreading of liquid over an initially dry surface. Rahman et al.  proposed a simple way of measuring an appropriately defined wickability for micro- and nanostructured surfaces. The results of these investigations [7,8] indicate that augmentation of CHF in boiling processes correlates with the magnitude of the wicking parameter for a variety of micro- and nanostructured surface morphologies.
The prior work described above has demonstrated that use of hydrophilic, strongly wicking, nanostructured surfaces can enhance water vaporization heat transfer processes. However, this raises the question: how should nanostructured surfaces be designed to maximize the benefits of a high wickability surface layer? In the study summarized here, we explored this question using a droplet-spreading model developed in tandem with our experimental studies of droplet spreading on nanostructured surfaces. The model is used here as a means to predict how changes in the morphology of the nanoporous layer affect the spreading mechanisms and heat transfer process during water droplet evaporation.
As will be discussed below, the model used here was specifically developed for droplet spreading in ultrathin, nanostructured layers on metal substrates that generate high capillary forces and exhibit low to moderate permeability. In this regard, this investigation differs from earlier studies that have considered liquid spreading on deep porous layers or larger microstructured surface morphologies (e.g., see Refs.  and [20–22]). In addition, we specifically focused on nonordered, nanostructured surfaces that can be thermally grown on metal substrates because this type of process is scalable and adaptable to complex substrates, making it an ideal approach to putting nanostructured, superhydrophilic coatings on heat exchanger surfaces.
In the Model Formulation section, the theoretical framework of the droplet spreading model is described and its predictions are compared with experimental observations and data. The discussion in those sections demonstrates the connection between the nanoporous layer morphology and the speed and extent of liquid spreading, which strongly impact evaporation heat transfer. The Comparison of Model Predictions With Experimental Data section explores the parametric effects of nanostructure morphology changes on wickability, spreading, and associated phase change heat transfer processes.
Previous studies have developed models for spreading of droplets on flat solid surfaces  and on the surface of a thick porous medium [22,24–26]. Here, we are specifically interested in low Weber number deposition and spreading of a droplet on a specific category of ultrathin, nanostructured, superhydrophilic layers on a metal substrate which exhibits the following key features that cause the spreading to differ from droplet spreading on either a solid flat surface or a thick porous medium:
The nanostructure has large capillary pressure difference across the interface at the ultrasmall pores of the structure.
The thickness of the nanostructured layer is very small, and therefore transport across the structure is very fast compared to radial transport of liquid.
The superhydrophilic nature of these surfaces makes hemi-spreading  possible (hemi-spreading being a phenomenon in which the leading edge of liquid infusion into the porous nanostructure separates and proceeds beyond the contact line of the upper droplet).
The nanostructure surface features listed above are of central interest here because they were characteristics observed experimentally for superhydrophilic ZnO nanostructured surfaces studied by Ruiz et al.  and Padilla and Carey , and they are expected to be representative of the behavior for other similar superhydrophilic nanostructured surfaces. Previous studies of droplet spread on surfaces mentioned above have not considered this specific type of surface (e.g., see Refs.  and [20–22]).
Early Time Synchronous Spreading Process.
Figure 1(a) depicts the initial contact of a droplet deposited on a thin nanostructured surface at time t = 0. Because the layer is thin, the time required for the liquid to wick across the thin nanoporous layer is very small (see Fig. 1(b)). The liquid then wicks radially outward (Fig. 1(c)) in the nanostructured layer while the droplet above spreads radially. During this first stage of the spreading process, the upper droplet contact line is postulated to stay within the leading edge of the nanoporous layer filled with liquid. This postulated synchronized spreading of the upper droplet and the liquid wicking in the porous layer appears justifiable for two reasons:
Transfer of the upper droplet contact line to a higher contact angle dry region would force the interface to at least temporarily become more convex near the contact line, as a result of the expected to increase the pressure in the liquid there (due to capillary effects across the interface from a wet to a dry region). This pressure difference temporarily moves liquid away from that location and prevents the liquid from spreading beyond the contact line within the wick .
Observation of experiments for these conditions indicates that the contact line does tend to stay within the liquid-filled portion of the nanoporous layer for conditions of interest here.
If the nanoporous layer is permeable enough and the capillary pressure for the layer is high enough, the region of the porous layer that is filled with liquid will continue to expand beyond the contact line of the droplet. As discussed above, this circumstance, depicted in Fig. 1(e), is referred to as hemi-spreading .
The spreading sequence depicted in Fig. 1 is consistent with that observed on thin nanostructured layers on metal substrates we tested. Postulated behavior for the model developed here is based on such experimental observations. A basic premise adopted here is that the flow in the nanoporous layer is driven by a pressure field that is determined from a basic transport equation. The proposed model postulates that
The droplet exhibits axisymmetric spreading, although spreading of real droplets may deviate significantly from such symmetry. Spreading in our experimental studies typically is close to radially symmetric, with stronger deviation in the later stages. Also, the transport is idealized as being quasi-steady in the following sense: It is postulated that at a given spread radius, a pressure field is established which is equivalent to that for steady transport of liquid to the outer perimeter of the liquid-filled region of the porous layer. As the spread radius increases, the pressure field is presumed to adjust rapidly to velocity and mass flow changes to sustain the equilibrium pressure field, and once the pressure field is known, the resulting flow quantities and motion of the liquid can be computed from it.
When the droplet first touches the structured surface, the capillary pressure difference draws liquid first across the thickness of the thin nanoporous layer, which takes a very short time, .
Capillary pressure in the layer then drives liquid flow radially in the porous layer.
The upper droplet spreads over the portion of the layer already filled with liquid. The apparent contact angle on this composite surface is less than that for the liquid on the dry surface. The contact angle would have to increase for droplet contact line to move beyond the radial extent of the liquid-occupied portion of the nanoporous layer. The resulting change in curvature would produce a local rise in liquid pressure that would drive liquid away from the contact line. This tends to resist the contact line extending beyond the leading edge of the liquid-filled nanostructured layer. It is expected, however, that the contact line could move beyond it if the droplet fluid momentum and/or stagnation pressure are large enough.
Based on the argument in (iv), the upper droplet can be expected initially to spread within the confines of the liquid saturated portion of the nanostructured layer below, for low impingement velocities and Weber numbers.
Here, we do not invoke modifications similar to Brinkman's  model to account for possible inertia term effects because we are specifically interested in nanoporous low-permeability structures in which inertia effects are expected to be negligible. The flow under the droplet can be modeled as two-dimensional (2D) flow in the porous layer with appropriate pressure boundary conditions. However, for reasons discussed below, a one-dimensional (1D) model was adopted here, which is consistent with the physics and is more straightforward to handle mathematically.
In a typical situation of interest here, r values and R (the spread extent of the droplet) are on the order of a centimeter or two (∼0.02 m), whereas the thickness of the nanostructured layer is several orders of magnitude thinner, with b ∼1 μm. Because the nanostructured layer is so thin, the mass transport due to pressure differences between the droplet and the layer pores will be very fast, and the liquid in the pores will quickly establish pressure equilibrium with the droplet interior. This indicates that two different solutions of the above equation exist in two different region of the porous layer. In the equilibrium layer far from the contact line of the upper droplet (r ≪ R), the solution is simply . In the region close to r = R, will vary between and the value specified by condition (7b): at r = R. This is depicted in Fig. 2. Note that this implies that the new fluid added to the nanostructured layer comes primarily from the region of the upper droplet near the contact line.
Thus, the model predicts a linear variation of R with t for the initial synchronous stage of the spreading process with the droplet contact line closely following the edge of the liquid-filled nanoporous layer under the droplet.
This 1D relation, plotted in dimensionless form in Fig. 3, predicts that the flux of liquid into the layer is zero except in the region close to the contact line ().
This predicted variation of the mass flux into the layer is also plotted in Fig. 3. It can be seen in Fig. 3 that the 2D Darcy flow model and the 1D model are virtually identical away from the contact line region. However, the 2D Darcy solution predicts that the mass flux from the droplet into the layer increases without bound as . This singularity is a consequence of the discontinuity in the pressure boundary condition at the contact line location (r, z) = (R, b) and can be similarly seen in the analogous heat transfer problem (see the discussion in Gebhart ). Here, the discontinuous boundary pressure and the resulting infinite mass flux are not expected to be accurate predictions for a real system. The 1D model, therefore, appears to be a better representation of the physics for this system, predicting a linear variation of R with time for the initial synchronous stage of the spreading process.
Modeling of Hemi-Spreading.
As discussed above, thermodynamic analysis of the upper droplet suggests that the linear variation of R with t will continue until the upper droplet contact line reaches the spread radius and apparent contact angle that minimizes its free energy. Further spreading of the upper droplet beyond that point is not thermodynamically favored. We, therefore, model the upper droplet as stopping its spread at the radius where it minimizes its free energy and achieves its equilibrium apparent contact angle. However, growth of the liquid-filled region of the porous layer can continue beyond that point, leading to a hemi-spreading process.
Here, the wicking spread of liquid in the porous layer beyond the upper droplet contact line is modeled in the same quasi-static manner as for the porous region flow under the droplet. As depicted in Fig. 4, flow in the porous layer beyond the droplet is driven by the difference in pressure between the edge of the upper droplet and the capillary-reduced pressure at the edge of the growing liquid-filled porous layer.
Note that this condenses the spreading relations into a universal curve for the synchronous early spreading regime and a collection of curves for different ratios in the hemi-spreading regime. An obvious question at this point is: does spreading of water droplets on a thin, nanostructured, superhydrophilic layer on a solid substrate exhibit behavior consistent with this predicted two-regime model? Results of the spreading experiments were used to explore this question in detail.
Comparison of Model Predictions With Experimental Data
To assess the spreading model defined in Eqs. (36a) and (36b), we conducted droplet deposition experiments in which a water droplet was deposited on a thin ZnO nanostructured surface on a copper substrate. The development of this surface uses a process known as hydrothermal synthesis. This process is described below and in more detail in Padilla  and Padilla and Carey :
Surface preparation involves: polishing the copper surface to achieve near-uniformity in surface smoothness and cleaning the copper surface using a sonication bath.
ZnO nanoparticles measuring 6 nm in diameter are evenly deposited on the clean surface and this is annealed in a dry oven at 150–160 °C.
Once annealed, the surface is cooled and submerged, coated side facing down, in a liquid growth solution (details in Refs.  and ) and placed in a 90 °C oven for 8 h.
Once removed and cooled, the surface is desorbed on a heating plate at 275 °C for an hour to get rid of any liquid or substances adsorbed onto the surface.
An electron microscope image of the resulting surface morphology is shown in Fig. 5. The desorption process was repeated prior to all experimental tests in order desorb any adsorbed molecular species on the surface between experiments. Each time we do this pre-experiment prep, the resulting surface's intrinsic wetting can vary slightly, so we treat the surface after being prepped this way as a distinct surface. In our results, we refer to “prep 1” and “prep 2” surfaces, which designate two surfaces with the same nanomorphology, but slightly different intrinsic wetting. When comparing parametric changes in the spreading process, we only compare experiments done on an identical morphology from the same prep process. The designation of surface prep is identified in the captions of Figs. 6 and 7.
In the spreading experiments, a measured volume of liquid was deposited on a nominally horizontal nanostructured surface of this type and the resulting spreading process was recorded using a high-speed video camera. Frames from a high-speed video of a droplet-spreading experiment are shown in Fig. 8. The droplet was released from the pipette and gently lowered till it made contact with the surface which released the full volume of the droplet from the pipette. The droplet detaches from the pipette within less than 0.0075 s from first contact of the droplet with the surface and the pipette does not have any further interaction with the liquid after that point. The deposition process is very short in comparison to the full liquid droplet-spreading process, therefore, we ignore any effects of the initial deposition process on the spreading.
Figure 9 shows two frames of a digital video of the spreading process for a 2 μl droplet (1.6 mm diameter before deposition) spreading at room temperature. These frames illustrate the appearance of the droplet before (a) and after (b) reaching the separation point (, ) where the upper droplet stops spreading and the liquid continues to penetrate radially within the nanostructured layer. This illustrates the behavior leading to the onset of hemi-spreading that leads to the two different regimes in the model analysis.
The image processing software imagj was used to extract contact line and nanolayer liquid front position data from frames of the digital video. To determine the variation of the mean radius of the spread droplet with time, the wetted area was divided by π and the square root of the result was taken to be the mean radius of the spread droplet at that point in the spreading process. The corresponding time was computed from the frame number relative to the start, and the known frame rate of the video camera. The frame rate for these videos was 1000 frames per second, leading to an uncertainty of roughly ±0.0005 s, as movement between frames was not captured. While this time interval is very small, it does lead to more uncertainty in the early stage data when movement is occurring on a much smaller timescale. The fast, synchronous stage spreading can lead to ±10% uncertainty in the determination of the time, while later stage spreading causes roughly ±1% uncertainty. Therefore, as the droplet spreading slows, the uncertainty decreases. Uncertainty in radius is a result of image processing. Wetted area from photos was measured with a known scale ratio of 30 pixels/mm. The contact line and wicking limit radius measurements were determined to ±2–3 pixels, which translates to ±2–3% uncertainty in the radius measurements. An example of the resulting R(t) data is shown in Fig. 10 for spreading of a 2 μl droplet deposited on an unheated ZnO nanostructured surface like that shown in Fig. 5 at room temperature.
Figure 10 is a log–log plot to clearly depict the very early time variation during which the droplet rapidly spreads to the radius of a few millimeters in less than 0.02 s. It should be noted that the R(t) variation of the data in this figure clearly reflect the two regimes represented in the model described above. At early times, the variation of R with t is close to linear, corresponding to the synchronous solution for which the droplet contact line expands radially in tandem with the liquid front in the nanoporous layer. The data in Fig. 10 also show a clear transition to the slower expansion of the liquid front beyond the contact line of the upper droplet after the upper droplet stops expanding. This transition, at about t = 0.011 s, corresponds approximately to the time when the upper droplet visually is observed to stop expanding.
To determine this key transition point (, ), we iteratively determined the pair of and values that provided the best fit of the normalized and video frame data to the linear universal curve and transition point predicted by Eqs. (36a) and (36b).
As shown in Fig. 6, for the data shown in Fig. 10 (using surface prep 1), = 3.06 mm and = 0.0090 s provide a best fit. Note that once this is done, the value of b/ that provides the best fit of Eq. (36b) to the data in the hemi-spreading region indicates the b/ ratio that characterizes this nanostructured surface. Since = 3.06 mm, the best fit for b/= 0.0016 implies that the mean surface layer thickness b is 4.9 microns. We also found that the values determined by the model-fitting method agreed well with the time in our video recording where hemi-spreading was observed to begin.
The fitting process to determine (, ) was also applied to droplet spread data obtained for another experiment, using data from surface prep 2, in which both 2 μl and 3 μl droplets were deposited on the surface. These data are shown in Fig. 7. For the 2 μl droplet on this surface, a best fit implied an value of 2.37 mm, whereas for the 3 μl droplet, the results indicate an value of 2.73 mm. Note that the postulated variation proportional to indicates that for this 3 μl droplet case should be about 14% higher than for the 2 μl droplet experiment. The determined value of 2.7 is about 15% higher, which is consistent with the model prediction. Overall our comparisons indicate that with appropriately specified (, ) values, the model predictions agree well with the nanostructured surface-spreading data in both the regimes considered in the model analysis, and the model and data exhibit trends that are consistent with the postulated behavior in our model.
Implications for Enhancing Droplet Evaporation or Boiling
The generally good agreement between the experimentally determined R(t) spreading data and the model prediction suggests that the idealizations in the model may be appropriate for this type of process. Our motivation in developing the model is to better understand how the morphology and material of the nanostructured layer can be chosen to maximize the heat transfer performance associated with droplet spread and evaporation on the surface.
For vaporizing of spreading droplets, two strategies tend to enhance heat transfer. One is to spread the droplet faster. The model solution indicates that faster spreading of the upper droplet results when the wickability is larger. The definition of wickability here indicates that, if other factors are held constant, spreading speed is enhanced when:
the permeability is increased
the porosity is decreased
the capillary pressure difference across the interface in the pores of the nanostructured layer is increased
the viscosity of the liquid is lower
Note that changing the geometry of the nanoporous layer may change more than one of these parameters. And, although water is usually the fluid of interest, its liquid viscosity varies substantially with temperature, and, therefore, changing the temperature can strongly affect the wickability. The key prediction of the model analysis presented here is that some of the parameters affecting wickability may be interdependent, but the overall strategy to enhance spreading speed must be to maximize the combination of these parameters in .
The second strategy to enhance droplet evaporation heat transfer is to enhance the footprint area of the spread droplet. For the thin nanostructured layers considered in this study, the high wickability spreads a droplet over an extensive area in a time that is very short compared to the evaporation time. In effect, the upper droplet spreads first to its maximum extent , followed by evaporation of the resulting thin upper droplet. Furthermore, the extremely thin nanoporous layers considered here will evaporate very little liquid compared to the content of the upper droplet, and virtually all the liquid evaporation occurs at the upper droplet liquid–vapor interface.
The model analysis developed here also indicates that for any specified droplet volume, smaller contact angle for the spread droplet results in greater spread area. The Cassie–Baxter model in Eq. (1) can be used to predict the apparent contact angle for the upper droplet if the intrinsic contact angle for the nanoporous layer solid material, and the fraction of the top surface of the layer that is solid, , are known. If we accept this as a model for the apparent contact angle, Eq. (1) dictates that minimizing requires making as large as possible and making as close to zero as possible. Apparent contact angles for the surfaces in our analysis ranged from 4 deg to 10 deg and were calculated using a spherical cap model for a known spread area and droplet volume.
Our analysis of enhanced droplet spreading on nanostructured surfaces also connects to the heat transfer issues associated with pool boiling CHF conditions. In a recent experimental study, Rahman et al.  presented results indicating that enhancement of pool boiling CHF on heated micro- and nanoporous surfaces correlates with the magnitude of a wickability parameter. The study done by Rahman et al.  relates wickability to a volume flux, , which is experimentally determined in their study wherein they initiated deposition of water onto a hydrophilic microstructured surface from a tube by raising a surface until it contacted a pendent liquid at the bottom of the tube.
Based on images in the study , we estimate a to be in the range of 1–2. For comparison, if a value of a = 1 is chosen to relate the gap height shown in Fig. 1 to the inner tube radius, the reported values from the study of Rahman et al.  (for nanostructured and microstructured surfaces) would result in values of around 1–3 mm/s. For the ultrathin, nanostructured, superhydrophilic surfaces considered in our study, our data indicate that wicking rates ( values) were on the order of 200–300 mm/s, implying that they have a substantially stronger capability to enhance droplet-spreading and liquid-wicking transport in boiling processes.
Our experimental data and observations, together with the model framework described here, support the conclusion that droplet spreading on our highly wicking nanostructured surfaces is characterized by two regimes. Early in the process, localized liquid flow from the upper droplet into the porous layer near the upper droplet contact line facilitates very rapid, synchronized spreading of liquid in the upper droplet and in the porous layer. At a specific mean footprint radius, dictated by the apparent contact angle of the upper droplet and the volume of the deposited droplet, the upper droplet spreading essentially halts and hemi-spreading continues the flow of liquid from the upper droplet contact line into the porous layer.
The predictions of the model framework discussed here agree well with these trends in the observed spreading behavior. In particular, the model provides insight into the mechanism of very rapid spreading early in the process, and we have demonstrated how a wickability parameter can be determined from measurement of the time and mean spread radius at the regime transition point. The model also indicates that for the surfaces tested in this study, the high wickability associated with this early stage is a consequence of the extremely thin nanoporous layer and the high capillary pressure difference generated in its very small interstitial spaces. The early rapid, synchronous spreading process can quickly spread the droplet to a large footprint area, which can strongly enhance the subsequent droplet evaporation heat transfer rate.
It should be noted that the model analysis described here and our experiments correspond to low Weber number droplet spreading dominated by capillary and viscous forces on an ultrathin, nanostructured, hydrophilic layer on a solid substrate. The model is not designed to apply to vaporization on thicker porous layers, low permeability, low capillary pressure porous layers, or for deposition and spreading at higher impact Weber numbers.
The model developed suggests specific strategies for increasing surface wickability and the extent of liquid spreading on a solid surface as a means of enhancing droplet evaporation heat transfer. Our results, together with those from other studies of wickability boiling enhancement, suggest that these same nanostructure parameter strategies also are likely to enhance boiling heat transfer.
Partial funding for this research was provided by the U.S. Department of Energy (DOE) U.S.-China Clean-Energy Research Center's CERC-WET (Grant No. DE-IA0000018).
- b =
nanostructured layer height
ambient atmospheric pressure
pressure inside upper droplet
- r =
a radial distance from center of droplet
upper droplet radius
mean interface radius of liquid–vapor interface in nanostructured layer
apparent contact angle
intrinsic contact angle
solid fraction at top surface of nanostructured layer