The results demonstrate that a Nasturtium leaf, and presumably any similar superhydrophobic surface, can selectively trap water drops that exceed a critical temperature while permitting colder drops to bounce. Furthermore, the experiments indicate that this phenomenon cannot be accounted for by a reduction surface tension alone. Specifically, the results show that a hot drop would stick on the superhydrophobic surface when, at the same surface tension, an ambient-temperature drop would bounce (
Figure 4). Additionally, a heated drop that would stick on an ambient-temperature surface might bounce had the surface been heated (for example
K in
Figure 5). Thus, the temperature of the drop and the surface both appear to be important parameters for the bouncing-sticking transition, independent of the surface tension. This result is noteworthy because it suggests that the temperature of the drop can significantly affect the properties of the superhydrophobic surface. Because the drop and surface only interact during the bounce, the mechanism required to stop a bounce must sufficiently modify the surface in the short time before the bounce is complete.
Drawing from past studies, we focus on two alternative mechanisms: melting of surface microtexture and evaporation–condensation within the superhydrophobic texture. The results in
Figure 5 might appear inconsistent with both mechanisms. Specifically, if a drop at
were able to melt the microtexture, then it would seem improbable that drops of any temperature would bounce if the surface were above this temperature
, yet the results illustrate that they do (thus we estimate
). Similarly, the existence of the yellow region in
Figure 5 is in contrast with previous works for static drops, which predicts sticking for a drop whenever its temperature is larger than that of the surface [
13]. However, if the dynamics are considered, it might be possible that, depending on the conditions, there could be regimes in which these effects are present but insufficient to inhibit bouncing.
4.1. Melting of the Surface Microtexture
The height of surface microtexture is one of the physical conditions that can noticeably influence transition from the Cassie–Baxter state (necessary for drop bouncing) to the Wenzel state [
9]. When the drop temperature is larger than the melting point of microtexture material, melting may shorten and smooth the microtexture surface and thus create conditions that promote the Wenzel state (
Figure 1C). To model this melting process, we approximate the superhydrophobic surface as a semi-infinite body that undergoes phase change. We assume a uniform solid surface temperature
that is cooler the material melting point
. Subsequently, at
, the drop contacts the superhydrophobic surface and resides on the surface over the period of the residence time
. If the drop temperature is cooler the substrate melting point (
), then the substrate remains intact and the drop bounces. However, if the drop temperature exceeds the substrate melting point (
), the energy transfer can induce a phase change. The amount of material that can melt will grow with the amount of time that that heat can transfer from the hot drop. The extent of this melting can be estimated by the position of the self-similar melting front
, which we refer to as the melted length. Thus, this moving boundary problem can be solved by using Neumann’s solution for the melting of a semi-infinite body [
24].
Based on Neumann’s problem, when a solid–liquid interface forms as a result of melting, two regions can be defined that obey the following governing equations: a liquid region (
) where
and a solid region (
) where
. Applying the boundary conditions
and
for the liquid,
and
and the solid, and the initial conditions
,
together with the interface energy equation, leads to:
Here, the subscripts ℓ and denote the properties of the liquid and solid phase, respectively, is the thermal conductivity, c is the thermal capacity, is the thermal diffusivity, is the density and is the latent heat of fusion. This equation provides a value of that can subsequently be used to calculate the melted length, noting . Therefore, if melting occurs over the entire time that the drop resides on the surface (), the length of surface microtexture would be , which—depending on conditions—could transition the drop from the Cassie–Baxter to the Wenzel state.
According to the solution of Equation (
1), the melting mechanism predicts that a drop would more readily melt the surface if the surface were warmer. Specifically, if melting of the surface microtexture were responsible for the observed bouncing-sticking transition, then the threshold drop temperature would be expected to decrease with increasing surface temperature
. This prediction is in stark contrast to what is observed; the threshold temperature increases with surface temperature, at least up to a point. Thus, it seems unlikely that melting of the microstucure would be responsible for the transition for
K in these experiments (
Figure 5).
4.2. Condensation of the Vapor within the Superhydrophobic Texture
Condensation is another mechanism that has been attributed to the transition from the Cassie–Baxter to the Wenzel state (
Figure 1D). Through this mechanism, liquid evaporates from the drop to saturate the air within the microtexture and then condenses. If air pockets within the microtexture fill with water, then the drop transitions to the Wenzel state. Here, we estimate the timescale associated with this evaporation–condensation process. Specifically, we model the air within the microtexture between the surface and the drop, neglecting any fluxes to regions that are not covered by the drop. Noting that condensation occurs only when the relative humidity is maintained at
, we first calculate the timescale for evaporation to fully saturate the air, and then we calculate the timescale for sufficient liquid to condense to fill up the air gap within the microstructure.
To estimate the time needed to saturate the microtexture air, we note that the mass flux across the liquid interface can be modeled as:
. Here,
J is the evaporation rate,
m
s is the diffusion coefficient of water [
25],
is the concentration of the saturated vapor,
is the concentration far from the evaporating liquid,
is relative humidity and
L is a characteristic lengthscale on order of the surface microtexture. By integrating the equation with respect to time, the relative humidity can be computed as a function of time:
, where
is the humidity of the ambient air and
is the characteristic evaporation timescale. The value of
where
V is the total volume of gap between microstructure and
A is its projected area. Thus, the time for the air within the microtexture to become saturated relative to the drop residence time is
. In the experiments, the evaporation time is estimated to be on the order of a microsecond. Because this timescale is more than a thousand times faster than the residence time, the air within the microstructure can be estimated as being fully saturated.
We next estimate the condensation time
, which we define as the characteristic time needed for condensate to grow and fill the superhydrophobic microtexture. If saturated air contacts a surface that is at slightly lower temperature, the saturated air will locally cool and condensation will occur. Condensation will continue as long as warmer saturated air is cooled on the surface. To estimate the condensation rate, we adopt a condensation model proposed by Kim et al. [
26] in which the rate of condensation growth
can be approximated as
. Here,
is the interfacial heat transfer coefficient,
is the saturated air temperature,
is the surface temperature,
is the latent heat of vaporization and
is the water density. This expression can be further reduced by modeling the interfacial heat transfer coefficient
in terms of the vapor properties [
27]. Additionally, we estimate the time to fill a microtexture with characteristic lengthscale
L as
, so that this time is modeled in terms of the temperatures as:
Here,
is the condensation accommodation coefficient [
28],
is the universal gas constant,
is the the molecular weight,
is the liquid density,
is the vapor density, and
is temperature of the saturated vapor within the microstructure, which is modeled to be halfway between the drop temperature
and surface temperature
.
Figure 6 shows plots of the condensation time, predicted from Equation (
2), as a function of the temperature difference between the water drop and surface. Here, the characteristic microtexture length is
, and the different curves represent three different surface temperatures. The results show that this condensation filling timescale
is significantly larger than the evaporation timescale
, and therefore it is reasonable to assume that the drop evaporation keeps the microtexture air fully saturated as the condensate forms. Additionally, the results illustrate the importance of the temperature difference
, rather than the temperatures themselves, in determining the condensation filling timescale
.
For the evaporation–condensation mechanism to trap a drop, the condensate must sufficiently fill the microtexture air before the drop bounces off of the surface. Therefore, it is natural to expect this process to depend on the relative scales of the condensation time
and the drop residence time
. This ratio motivates a dimensionless grouping of parameters, which we define as
:
Here, we have dropped all dimensionless prefactors, as the focus is on the scaling relationship and the grouping of the dimensional parameters. Note that a condensation time requires ; however, our definition for is valid regardless of the values of and , which we have defined as .
The relevance of the parameter
in the drop trapping is illustrated in
Figure 7. Here, a value of
is calculated for each experimental data point in
Figure 5 and plotted in terms of the ratio
(symbols in
Figure 7). Small values of
are associated with drop bouncing and large values of
are associated with drop sticking. Furthermore, the mechanics motivating the parameters suggest the data fall in one of three regimes. When
, the surface is hotter than the drop, and therefore even when the gap air is completely saturated, it will be below saturation directly on the texture surface so that no condensation will form, the surface will remain dry, and drops will bounce (purple region). When
, condensation would be expected to form and, if
is above a critical value, it would sufficiently fill the microtexture and trap the drop (white region). From the data, it appears that the critical value is around unity, although caution should be taken interpreting this value too closely, as estimates of certain parameters, such as the characteristic microtexture lengthscale of the Nasterium leaf, are less precise than others. Finally, our results indicate that if
, bouncing on the Nasterium leaf occur even when
(yellow region). In our model, this region represents drops that bounce on superhydrophobic surfaces that are filled with insufficient condensate to transition the drop into a Wenzel state.