# Spreading Dynamics of Droplet Impact on a Wedge-Patterned Biphilic Surface
## 3 Results and Discussion  
### 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 \(t=20\) 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 \begin{table}
\begin{tabular}{c c c c c c c} \hline  & **Case 1–4** & **Case 5–11** & **Case 12** & **Case 13** & **Case 14** & **Case 15** \\ \hline \(H\) (mm) & 3.54 & 3.82 & 3.17 & 5.31 & 6.48 & 9.51 \\ \(v\) (m/s) & 0.14 & 0.16 & 0.11 & 0.23 & 0.28 & 0.37 \\ \(We\) (-) & 0.70 & 0.89 & 0.45 & 1.90 & 2.70 & 4.75 \\ \(Re\) (-) & 398 & 448 & 319 & 654 & 778 & 1032 \\ \(Ca\) (-) & \(1.8\times 10^{-3}\) & \(2.0\times 10^{-3}\) & \(1.4\times 10^{-3}\) & \(2.9\times 10^{-3}\) & \(3.5\times 10^{-3}\) & \(4.6\times 10^{-3}\) \\ \(\tilde{t}\) (ms) & 17.5 & 21.4 & 21.8 & 10.7 & 9.0 & 6.8 \\ \hline \end{tabular}
\end{table} 
Table 1: Different initial impact states. Figure 4: Images of a droplet impact for different apex angles. 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 \(\tau=t/\tilde{t}\). 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 \(\tau=tv_{0}/D_{0}\). 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\({}^{\circ}\), is the largest, which is beyond 1.6 \(D_{0}\) at 4 \(\tau\). The maximum spreading diameter for 36.9\({}^{\circ}\) and 18.9\({}^{\circ}\), are very close to that of 67.4\({}^{\circ}\) which are also 1.6 \(D_{0}\). The droplet spreads only a little at 11.4\({}^{\circ}\), only reaching at about 1.0 \(D_{0}\). 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 [PERSON] (2013) and [PERSON] (2014), the driving surface tension force could be measured as:  
\[F\ =\ \gamma_{\text{SG}}-\gamma_{\text{SL}}-\gamma\cos\theta_{d} \tag{1}\]  
where \(\gamma_{\text{SG}}\), \(\gamma_{\text{SL}}\), and \(\gamma\) represent the surface tension between the solid-gas interface, solid-liquid interface and liquid-gas interface, separately.