3.1. Effect of Apex Angle
Figure 4 shows the typical images of a droplet impinging on a wedge-patterned biphilic surface, with different apex angles. In
Figure 4, the red arrow shows the transportation direction of the droplet. There are six instances selected for demonstration. At 0 ms, the droplet impacts on the solid surface. After the droplet contacts the solid surface, the bottom of the droplet formed a layer which quickly spreads on the solid surface as shown at
= 20 ms. Then, the droplet spread radially onto the solid surface with the tendency toward the more hydrophilic area. The hydrophilic area is also designed to be wider, so the droplet has a motivation subjected to the surface tension difference. Due to the relatively low release height, there is little oscillation in the normal direction. There are two main phenomena, in that the droplet spreads on the surface, as well as transporting it to the wider hydrophilic edge. The transportation properties along the wedge-patterned surface are studied somewhere else, using the force balance and energy conservation [
22]. In this paper, attention is paid to the spreading characteristics. Comparing images at the same instance for different apex angles, the droplet is found to spread more quickly, and it reaches a larger attached diameter for a larger apex angle. The amplitude of droplet oscillation along the spreading direction is larger as well.
The initial kinetic energy is dissipated, partly by the oscillation along the solid surface, and the viscosity dissipation inside the droplet. The increased surface energy and the reduced gravitational potential energy are balanced by the kinetic energy reduction and the viscous dissipation along the solid surface and the droplet inside.
Here, a dimensionless time is defined as
. The attached diameter versus time of the whole process is plotted in
Figure 5. The y axis is the dimensionless spreading diameter, and x axis is the dimensionless time, defined as
.
Figure 5 shows that the advancing edge shows a close velocity, with the back edge at the initial state. The advancing edge that spreads on the hydrophilic area shows a faster velocity. The spreading diameter was deduced as the distance between the two edges. The equilibrium spreading diameter for 67.4°, is the largest, which is beyond 1.6
at
. The maximum spreading diameter for 36.9° and 18.9°, are very close to that of 67.4° which are also 1.6
. The droplet spreads only a little at 11.4°, only reaching at about 1.0
. For the advancing edge, the diameter oscillation is larger than that of the back edge. The surface tension helps the droplet to spread faster toward the more hydrophilic area, if the apex angle is larger.
From the analysis of de Pierre [
23] and Brochard–Wyard [
24], the driving surface tension force could be measured as:
where
,
, and
represent the surface tension between the solid–gas interface, solid–liquid interface and liquid-gas interface, separately.
is the dynamic contact angle when spreads and approximately equals to the static contact angle
. Due to the superhydrophobic surface coating, the surface tensions related to the solid surface are different. The driving forces toward the coating part and the silicon hydrophilic surface varied.
In order to analyze the effect of the apex angle of wedged patterns on droplet spreading, the spreading model should be rebuilt. The initial diameter of the droplet was supposed to be , and the initial velocity before impact on the solid surface was .
After the impact on a solid surface, droplet spreads radially. When it reaches the maximum diameter, the droplet topology could be regarded as a cylinder whose bottom has a maximum spreading diameter , and its height is .
According to the mass conservation, one can obtain:
If the droplet flows into the cylinder with a velocity of
through a circle area with a diameter
, then the spreading velocity
could be calculated, based on the mass conservation:
Assuming the diameter
, and combining Equations (3) and (4), gives:
By integrating Equation (5), the spreading diameter evolution can be given:
As the spreading diameter consists of the superhydrophobic part and the hydrophilic part within the wedged patterns, the spreading diameter has an expression that is roughly described according to the different apex angles:
where
is the spreading area when reaching the maximum diameter
. As the spreading area contains the hydrophilic part, as well as the superhydrophobic part, the exact expression should be:
where
is the apex angle in rad on wedge patterned surface, and
and
are the spreading diameters on the superhydrophobic surface and on hydrophilic surface, separately.
For simplicity, a linear parameter is proposed here:
From Equations (6)–(8), this gives:
is constant for certain surfaces, and then Equation (10) can be changed into:
where:
Combining Equations (6) and (11), gives:
Thus, a relation between the spreading diameter and to time can be derived:
3.2. Effect of the Tilting Angle
The inclined effect of the droplet impact on the solid surface has been investigated, by changing the tilting angle to estimate the ability for harvest droplet and transportation properties.
At a low Weber number, the droplet spreading is dominated by surface tension, and it cannot spread completely at the initial state. When the Weber number increases, the inertia effect becomes more important, and the droplet could spread during the first oscillation. The surface-tension effect on droplet spreading and migration becomes dominant, the more attached the area of the liquid–solid interface becomes. For a low Weber number, each oscillation has a similar period, until the droplet stops.
For droplet impact on a solid surface with a −15° tilting angle, where minus means that the driving force from the hydrophilic area is designed along the gravitational direction, the spreading diameter changes more obviously at the same instant. The period of each oscillation for each different tilting angle is quite similar. It is quite reasonable that the spreading is a combination of a normal spread, and a spring-like oscillation along the spreading direction. The oscillation is too complicated to research if the normal direction is considered. The main focus of this paper is on the final equilibrium spreading diameter.
The first oscillation along the spreading direction is quite similar for different tilting angles. The period of the first oscillation lasts for about , and the amplitude is about 1.2 . When the droplet vibrates along the solid surface, the droplet has an obviously inclined posture as the droplet bottom spreads rather fast along the hydrophilic area, but the bulk and top parts of the droplet still stick to the initial site. When the tilting angle becomes larger, the inclined posture is more obvious. Considering the positions of −10° and −5°, the tilting angle has little influence on the final equilibrium diameter and the vibrating frequency of droplet spreading. It comes out differently for the positive tilting angles 5°, 10°, and 15°. When the droplet impacts the solid surface at a 15° angle, the final equilibrium spreading diameter is reduced by less than 1.5 . The difference of the tilting angles has a minor effect on the spreading diameter, as well as the frequency of oscillation.
The spreading factors for the different tilting angle are presented in
Figure 6 and
Figure 7. From
Figure 6,
,
and
are roughly equal for each tilting angle. Thus, the conclusion can be made that the effects of gravitational force show no impact when the droplet spreads onto a solid surface at a low Weber number.
3.3. Effects of Impact Velocity
The main harvesting ability of the wedge patterned biphilic surface is investigated through droplet impact experiments with different initial velocities. The release height is an easy way to adjust the initial impact velocity. The air viscosity and drag efficiency are neglected when calculating the critical time. The initial impact velocity of a droplet from a settled height can be roughly estimated by
, which has little deviation when the Weber number is relatively low (<100). The release height is set as 3.2 mm, 5.3 mm, 6.5 mm, and 9.5 mm and the corresponding Weber number and Reynolds numbers are listed in
Table 1. As the initial velocity increases, the final spreading diameter increases accordingly. It is noticed that the final spreading diameter for
= 2.7 is the largest. For a proper Weber number of around 3, and an apex angle of 36.9°, the spreading diameter reaches the maximum.
For a 3.2 mm release height, the front edge spreads faster than the back one before 2 . The time for when advancing edge spreads faster than the back edge occurred at 2 , 3 , and 3 , for each height. Before that time, the spreading has the same velocity in all directions. For = 2.7 the spreading diameter is much larger than the left.
One can see in
Figure 8 that a low-Weber-number droplet spreads linearly with time. As ther Weber number increases, the spreading diameter increases during the first oscillation. The maximum spreading diameter is reached within the first two oscillations. For
= 4.8, the maximum spreading diameter is reached within the first spreading.
Comparing the dimensionless spreading diameters for each Weber number versus time, the periodic time of each oscillation changes little. During the same time scale, the driving surface tension shows dominance. For a low Weber number, the surface tension becomes dominant, with a spreading diameter that increased constantly. For a relatively high Weber number, the surface tension came into effect on migration at an earlier time than that of a low Weber number as the larger contact area.
The initial kinetic energy:
The interfacial energy of the solid-gas interface is:
When the droplet reaches the maximum spreading diameter, the interfacial energy of the cylinder liquid is:
The interfacial energy of solid surface is
The final kinetic energy is
The work of deforming the droplet by viscosity is [
25]:
where
is the viscous dissipation function,
is the droplet volume, and
is the time taken to spread.
From the knowledge of [
25], the viscous dissipation function has a rough expression:
where
is the liquid viscosity and
is the characteristic length in height.
As Pasandideh-Fard et al. [
20] mentioned, the characteristic length should be approximately the boundary layer thickness:
The boundary layer then can be measured from White [
26]:
Combining Equations (23)–(27), gives:
As the energy conservation, there should be:
Solve Equation (30), the maximum spreading factor can be derived:
For distilled water on the silicon surface
= 68°, the maximum spreading factor can be calculated from the model of Pasandideh-Fard et al. [
20] and Zhao et al. [
21]. The resolutions are shown in
Table 2.
The static contact angle has an effect on the maximum spreading diameter. For the static contact angle
= 68° and
= 142°, the maximum spreading factor varies widely. As the apex angle is relatively small, the homogeneous surface model should be analyzed first. Taking the static contact angle
= 142°, the comparison between the different models are listed in
Figure 9.
In
Figure 10a, there is not much difference between the different models. The model of Zhao et al. [
21] agrees well with the experimental data for a homogeneous superhydrophobic surface.
For a wedge-patterned biphilic surface with apex angle
, the equivalent diameter is promoted to be estimated according to the spreading area
:
where
and
represent the maximum spreading diameters on the hydrophilic surfaces and superhydrophobic surfaces, separately.
Combining Equations (33)–(36), one can get:
Taking the apex angle
= 36.9°, the equivalent maximum spreading diameters, calculated on different models are listed in
Figure 10b.
Based on Equation (37), the equivalent maximum spreading diameters are calculated for apex angle
= 36.9°, as shown in
Figure 10b. From
Figure 10b, the maximum spreading diameters based on Equation (31) agree well with the experimental data, better than those two models of Pasandideh-Fard et al. [
20] and Zhao et al. [
21].
In order to derive a relation for the maximum spreading factor of the impact on wedge patterned surfaces, the first oscillation has been paid more attention. The maximum spreading factor during the first oscillation is shown in
Figure 11 for different Weber numbers.
Taking account of the expression of Eggers et al. [
27], an exponential function is promoted, then
. Using the least squares procedure on the experimental results for maximum spreading diameters, one can find that
.
The experimental data of the maximum spreading diameters under different Weber numbers and the fitting exponential curve are plotted in
Figure 11.
Based on the limited experimental data during the first oscillation under different Weber numbers, the maximum diameter then has a relation with the Weber number: