## 1 Introduction
Our research is also for Newtonian fluid droplets. We used the minimum energy theory and the minimum energy dissipation theory to analyze the motion states of a droplet on an inclined, hydrophilic rough surface in gravity, and gave the corresponding prediction method. We did the following steps. First, we set up Model 1 to calculate \(\theta_{\rm a}\) and \(\theta_{\rm r}\). Second, we set up Model 2 to calculate out the initial droplet profile \(\Omega_{0}\), the initial droplet front contact angle \(\theta_{1-0}\), and the initial droplet rear contact angle \(\theta_{2-0}\), when a droplet begins to stay on an inclined hydrophilic rough surface. Third, we set up Model 3 to calculate out the dynamic droplet profile \(\Omega_{*}\) (* represents every position during droplet motion), the dynamic front contact angle \(\theta_{1-s}\), and the dynamic rear contact angle \(\theta_{2-s}\) when the droplet stretches or contracts its three-phase contact line on the inclined, rough surface. Fourth, we gave out the prediction for the motion states of a droplet on an inclined, hydrophilic rough surface. Finally, we did many experiments to verify the predictions, and found that the predictions are in good agreement with the experimental results. We gave a simple description for the methods of Models 1-3 and the prediction, which will be described in detail in the following sections. Model 1 was based on the minimum energy theory. As is shown in Figure 1a, a droplet was imaged to stay on a flat hydrophilic rough surface. When we imaged to continuously add the volume of the droplet, but fixed the three-phase contact line, we could gain the potential energy \(\Delta E\). When \(\Delta E\) is equal to the energy barrier \(E_{\rm barr}\), preventing the contact line from moving, the contact angle is \(\theta_{\rm a}\). In contrast, we could calculate \(\theta_{\rm r}\) by imaging to continuously decrease the volume. Model 2 was based on the minimum energy theory. As is shown in Figure 1b, when a droplet was initially plated on an inclined surface, the sum of gravitational energy and the interface free energy is minimal. We gave out the integral expression of the sum energy, numerically dispersed the droplet, minimized, and got the initial shape \(\Omega_{0}\), \(\theta_{1-0}/\theta_{2-0}\). Model 3 was based on the minimum energy dissipation theory. As is shown in Figure 1c, when a droplet moves on an inclined surface, the droplet has the local minimum sum energy on each point of droplet inchworm motion. We dispersed the droplet, minimized the sum energy with the constraint of contact line length, and calculated out the dynamic shape \(\Omega_{*}\) and \(\theta_{1-*}/\theta_{2-*}\). As is shown in Figure 1d,e, the prediction was based on comparing \(\theta_{1-0}/\theta_{2-0}\) and \(\theta_{1-*}/\theta_{2-*}\) with \(\theta_{\text{a}}/\theta_{\text{r}}\). We thought that the front contact line of droplet moves if \(\theta_{1-0}(\theta_{1-*})\geq\theta_{\text{a}}\) and the rear contact line of droplet moves if \(\theta_{2-0}(\theta_{2-*})\leq\theta_{\text{r}}\); otherwise, they are pinned. Furthermore, the droplet can keep sliding if both the front and the rear contact lines move. Otherwise, the droplet will be pinned finally when both the front and the rear contact lines stop. We gave a simple example to predict the droplet motion state of \"stretching-to-pinned\" on an inclined rough SiO\({}_{2}\) surface, which will be described in detail in Section 4.3.1. A SiO\({}_{2}\) surface was patterned by circular microstructures (\(d=6\) um, \(h=12\) um, and \(a=60\) um). The droplet had the volume of 40 umL, and the surface was \(39^{\circ}\) inclined to the horizontal plane. As shown in Table 1, using Models 1-3 we got \(\theta_{\text{a}}=75.61^{\circ}\), \(\theta_{\text{r}}=42.91^{\circ}\), \(\theta_{1-0}=77.71^{\circ}\), \(\theta_{2-0}=46.43^{\circ}\), \(\theta_{1-*}=75.48^{\circ}\), and \(\theta_{2-*}=44.05^{\circ}\).