## 4 Results and Discussion
### Prediction Results Compared with Experiments
#### 4.3.2 Droplets on an Inclined, Smooth PMMA Surface  
As is shown in Table 3 and Figure 12, when the tilt angles of the smooth PMMA hydrophilic surface change from 10\({}^{\circ}\) to 28\({}^{\circ}\), the final motion states of droplets on the inclined surface change from \"pinned\" to \"sliding\". Results from Model 1 indicate that the \(\theta_{\rm a}\) and \(\theta_{\rm r}\) of the smooth PMMA surface are 81.62\({}^{\circ}\) and 65.10\({}^{\circ}\), respectively.  
Figure 12: The simulations, predictions, and experiments of droplets on the inclined, smooth PMMA surface, where \(V=60\) uL: (**a**) 10\({}^{\circ}\), (**b**) 14\({}^{\circ}\), (**c**) 19\({}^{\circ}\), and (**d**) 28\({}^{\circ}\).  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Tilt** & \multirow{2}{*}{\(\theta_{a}\)/\({}^{c}\)} & \multirow{2}{*}{\(\theta_{a}\)/\({}^{c}\)} & \multirow{2}{*}{\(\theta_{a}\)/\({}^{c}\)} & \multirow{2}{*}{\(\theta_{1-0}\)/\({}^{c}\)} & \multirow{2}{*}{\(\theta_{2-0}\)/\({}^{c}\)} & **Initial Motion** & \multirow{2}{*}{\(\theta_{1-}\)/\({}^{c}\)} & **Final Motion** \\
**Angle/\({}^{c}\)** & & & & & & & **State** \\ \hline
10 & 81.62 & 65.10 & 76.20 & 67.56 & Pinned & - & - & Pinned \\
14 & 81.62 & 65.10 & 77.48 & 63.89 & Contracting & 80.71 & 65.13 & Contracting-to- \\  & & & & & & & pinned \\
19 & 81.62 & 65.10 & 80.42 & 62.37 & Contracting & 82.64 & 64.86 & Contracting-to- \\
28 & 81.62 & 65.10 & 86.85 & 55.71 & Sliding & - & - & Sliding \\ \hline \hline \end{tabular}  
* \({}^{\mathrm{a}}\)_U = 2.1 \(\times\) 10\({}^{-5}\)_ (N), the volume is 60 uL, \(\gamma_{\mathrm{lv}}\) = 72.75 mN/m, and \(\theta_{e}\) = 74.73\({}^{\circ}\).  
\end{table}
Table 3: Numerical results and motion state predictions of droplets on the inclined, smooth PMMA surface *.  
For the surface with tilt angle \(10^{\circ}\), the results from Model 2 and the prediction were that \(\theta_{1-0}=76.20^{\circ}<\theta_{\rm a}\) and \(\theta_{2-0}=67.56^{\circ}>\theta_{\rm r}\), respectively. We predicted that the motion state of droplet is \"pinned\" on the \(10^{\circ}\) inclined PMMA surface.  
For the surface with tilt angle \(14^{\circ}\), the results from Models 2 and 3 and the prediction, in the droplet's initial state, were \(\theta_{1-0}=77.48^{\circ}<\theta_{\rm a}\) and \(\theta_{2-0}=63.89^{\circ}<\theta_{\rm r}\), leading to drop contracting, respectively. At one point (*), \(\theta_{1-*}=80.71^{\circ}<\theta_{\rm a}\) and \(\theta_{2-*}=65.13^{\circ}\approx\theta_{\rm r}\) lead to drop pinning. We predicted the motion state of droplet would be \"contracting-to-pinned\" on the \(14^{\circ}\) inclined PMMA surface.  
For the surface with tilt angle \(19^{\circ}\), the results from Models 2 and 3 and the prediction, in the droplet's initial state, was that \(\theta_{1-0}=80.42^{\circ}<\theta_{\rm a}\) and \(\theta_{2-0}=62.37^{\circ}<\theta_{\rm r}\), leading to drop contracting, respectively. At one point (*), \(\theta_{1-*}=82.64^{\circ}>\theta_{\rm a}\) and \(\theta_{2-*}=64.86^{\circ}<\theta_{\rm r}\), leading to drop sliding. We predicted the motion state of droplet is \"contracting-to-sliding\" on the \(19^{\circ}\), inclined PMMA surface.  
For the surface with a tilt angle of \(28^{\circ}\), the results from Model 2 and the prediction, in the droplet's initial state, were that \(\theta_{1-0}=86.85^{\circ}>\theta_{\rm a}\) and \(\theta_{2-0}=55.71^{\circ}<\theta_{\rm r}\), respectively. We predicted the motion state of droplet would be \"sliding\" on the \(28^{\circ}\) inclined PMMA surface.  
We did experiments to verify our predictions. Experimental results showed that when the surface tilt angles are in the range \(19^{\circ}<\phi<23^{\circ}\), the motion state of droplet is \"contracting-to-pinned\"; this has 4-\(5^{\circ}\) errors, with prediction results at \(14^{\circ}<\phi<19^{\circ}\). It was shown that the experimental data agreed well with the prediction results (for video, see Video S5-S8 in the Supplementary Materials).