## 4 Results and Discussion
### Prediction Results Compared with Experiments  
#### 4.3.1 Droplets on an inclined Rough SiO\({}_{2}\) Surface  
As is shown in Table 2 and Figure 11, when the tilt angles of the rough SiO\({}_{2}\) hydrophilic surface change from 12\({}^{\circ}\) to 52\({}^{\circ}\), the final motion states of droplets on the inclined surface change from \"pinned\" to \"sliding\". The results from Model 1 show that the \(\theta_{\rm a}\) and \(\theta_{\rm r}\) of the SiO\({}_{2}\) rough surface, decorated by microstructures (\(d=6\) um, \(h=12\) um, and \(a=60\) um), are 75.61\({}^{\circ}\) and 42.91\({}^{\circ}\), respectively.  
For the surface with tilt angle 12\({}^{\circ}\), results from Model 2 and the prediction were \(\theta_{1-0}=64.85^{\circ}<\theta_{\rm a}\) and \(\theta_{2-0}=58.74^{\circ}>\theta_{\rm r}\), respectively. We predicted that the motion state of the droplet would be \"pinned\" on the 12\({}^{\circ}\) inclined SiO\({}_{2}\) surface.  
For the surface with tilt angle 39\({}^{\circ}\), the results from Models 2 and 3 and the prediction, in the droplet's initial state, were that \(\theta_{1-0}=77.71^{\circ}>\theta_{\rm a}\) and \(\theta_{2-0}=46.43^{\circ}>\theta_{\rm r}\) lead to drop stretching. At one point (*), \(\theta_{1-*}=75.48^{\circ}\approx\theta_{\rm a}\) and \(\theta_{2-*}=44.05^{\circ}>\theta_{\rm r}\) led to drop pinning. We predicted the motion state of the droplet would be \"stretching-to-pinned\" on a 39\({}^{\circ}\) inclined SiO\({}_{2}\) surface.  
For the surface with tilt angle 44\({}^{\circ}\), results from Models 2 and 3 and the prediction, in the droplet's initial state, was that \(\theta_{1-0}=79.01^{\circ}>\theta_{\rm a}\) and \(\theta_{2-0}=44.39^{\circ}>\theta_{\rm r}\) lead to drop stretching, respectively. At one point *, \(\theta_{1-*}=76.42^{\circ}>\theta_{\rm a}\) and \(\theta_{2-*}=41.51^{\circ}<\theta_{\rm r}\) lead to drop sliding. We predicted the motion state of droplet is \"stretching-to-sliding\" on the 44\({}^{\circ}\) inclined SiO\({}_{2}\) surface.  
For the surface with tilt angle 52\({}^{\circ}\), results from Model 2 and the prediction in the droplet's initial state were \(\theta_{1-0}=81.21^{\circ}>\theta_{\rm a}\) and \(\theta_{2-0}=40.19^{\circ}<\theta_{\rm r}\), respectively. We predicted the motion state of the droplet is \"sliding\" on the 52\({}^{\circ}\) inclined SiO\({}_{2}\) surface.  
We did experiments to verify predictions. Experimental results showed that when the surfaces tilt angles were in the range 40\({}^{\circ}<\phi<\) 42\({}^{\circ}\), the motion state of the droplet is \"contracting-to-pinned\", which has 1-2\({}^{\circ}\) errors with the prediction results 39\({}^{\circ}\leq\phi<\) 44\({}^{\circ}\). It was showed that the experimental data agreed well with the prediction results (for video, see Video S1-S4 in the Supplementary Materials).  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Tilt Angle\({}^{\circ}\)** & \(\theta_{\rm a}\)\({}^{\circ}\) & \(\theta_{\rm r}\)\({}^{\circ}\) & \(\theta_{1-0}\)\({}^{\circ}\) & \(\theta_{2-0}\)\({}^{\circ}\) & **Initial Motion State** & \(\theta_{1-*}\)\({}^{\circ}\) & \(\theta_{2-*}\)\({}^{\circ}\) & **Final Motion State** \\ \hline
12 & 75.61 & 42.91 & 64.85 & 58.74 & Pinned & - & - & Pinned \\
39 & 75.61 & 42.91 & 77.71 & 46.43 & Stretching & 75.48 & 44.05 & Stretching-to-pinned \\
44 & 75.61 & 42.91 & 79.01 & 44.39 & Stretching & 76.42 & 41.51 & Stretching-to-sliding \\
52 & 75.61 & 42.91 & 81.21 & 40.19 & Sliding & - & - & Sliding \\ \hline \end{tabular}  
* Microstructure parameters are \(d=6\) μm, \(h=12\) μm, and \(a=60\) μm; \(U=2.3\times 10^{-5}\) (N); the volume is 40 μm\(\lambda_{\rm r}\)\(\gamma_{\rm rv}=72.75\) mN/m; and \(\theta_{\rm e}=65.57^{\circ}\).  
\end{table}
Table 2: Numerical results and motion state predictions of droplets on the inclined rough SiO\({}_{2}\) surface \({}^{\rm a}\).