## 3 Results and Discussion
### Droplet Impact on Horizontally Moving Micro-Pillar Surfaces
#### 3.2.2 Velocity Restitution Coefficient and Energy Dissipation  
When the droplet bounces off from the surface, the speed of takeoff \(v^{\prime}\) is a key parameter and the restitution coefficient \(\varepsilon_{n}\) is introduced as \(v^{\prime}/v_{n}\) to illustrate the droplet dynamic characteristics. [PERSON] et al. [58] studied the velocity restitution coefficient \(\varepsilon_{n}\) of droplet impact on heated silicon surfaces and derived it by establishing the spring model \(\varepsilon_{n}=aWe_{n}^{-0.5}\). [PERSON] and [PERSON] [59] established a similar correlation for droplet impact on stationary smooth superhydrophobic surfaces as \(\varepsilon_{n}=aWe_{n}^{-0.25}\). Subsequently, [PERSON] and [PERSON] [60] studied the droplet impact on inclined surfaces and proposed that \(\varepsilon_{n}=1.2 We_{n}^{-0.12}\). Additionally, [PERSON] et al. [54] summarized a correlation for the longitudinal restitution coefficient of droplet impact on horizontally moving grooved surfaces as \(\varepsilon_{n}=1.45 We_{n}^{-0.5}\). However, due to the difference in the microstructure on the surface, the variation in the velocity restitution coefficient during droplet impact in this study exhibits a different trend, as shown in Figure 11. Specifically, when micrometer-sized droplets impact micro-pillar surfaces, the velocity restitution coefficient does not follow the form of \(\varepsilon_{n}=aWe_{n}^{b}\). The velocity restitution coefficient is mainly divided into two stages. First, as \(We_{n}\) increases, the velocity restitution coefficient \(\varepsilon_{n}\) rapidly decreases (as indicated by the pink region), dropping from approximately 0.5 to around 0.25. As \(We_{n}\) further increases further, the restitution coefficient \(\varepsilon_{n}\) fluctuates between 0.2 and 0.3 (as indicated by the light blue region). This trend differs from the impact behavior of millimeter-sized droplets in which the velocity restitution coefficient of millimeter-sized droplets typically continues to decrease with increasing \(We_{n}\). The occurrence of this phenomenon is attributed to the fact that, within this range of impact Weber numbers (\(We_{n}=\) 3-5.4), the penetration depth of the droplet remains almost constant, as shown in Figure 6a. Therefore, the differences in viscous dissipation caused by variations in micro-pillar height can be neglected, and the velocity restitution coefficient remains within a relatively stable range. The fluctuations in  
Figure 10: Pressure distribution on the \(x\)-\(y\) cross-section inside the droplet (\(t^{*}=1.62\), \(z=0\), \(We_{n}=1.8\)). (a) \(We_{t}=0\); (b) \(We_{t}=6.4\). the velocity restitution coefficient are primarily due to changes in the spreading diameter and the varying degrees of droplet rolling. The previously mentioned contact time and velocity restitution coefficient are related to the competition between kinetic energy and dissipated energy, prompting us to conduct an energy analysis of the model in this study. The total energy during droplet impact mainly includes kinetic energy \(E_{k}\), surface energy \(E_{s}\), gravitational potential energy \(E_{g}\), and dissipated energy \(E_{\mu}\). According to energy conservation, the following relationship should be satisfied:  
\[E_{0}=E_{k}+E_{s}+E_{g}+E_{\mu}, \tag{12}\]  
where \(E_{0}\) is the total energy of the droplet in its initial state. \(E_{k}\) can be calculated by an integral of each mass element's kinetic energy. The surface energy \(E_{s}\) of the droplet is given by  
\[E_{s}=A_{\rm{lv}}\sigma_{\rm{lv}}+A_{\rm{ls}}(\sigma_{\rm{ls}}-\sigma_{\rm{sv }}). \tag{13}\]  
According to [PERSON]'s equation that \(\sigma_{\rm{sv}}-\sigma_{\rm{ls}}=\sigma_{\rm{lv}}\cos\theta\), we have \(E_{s}=\sigma_{\rm{lv}}(A_{\rm{lv}}-A_{\rm{ls}}\cos\theta)\), where \(A\) represents the contact area and the subscripts \(l\), \(v\) and \(s\) are liquid, gas and solid, respectively.