## 3 Results and Discussion  
### Droplet Impact on a Stationary Micropillar Surface  
When the Weber number of the droplet impact \(We_{n}\) is relatively low, the droplet cannot penetrate the interior of the micro-pillars. In this scenario, the droplet remains in the Cassie state. As the Weber number increases further, the droplet penetrates the micro-pillar array and infiltrates into the inter-pillar gaps, transitioning into a partial penetration state. The transition between these two states is governed by the balance between the wetting pressure (\(P_{wet}\)) and the anti-wetting pressure (\(P_{attinet}\)) [46, 51]. The wetting pressure comprises two components: the water hammer pressure (\(P_{WH}\)) and the dynamic pressure (\(P_{D}\)). The water hammer pressure can be calculated using the formula \(P_{WH}=k\rho Cv_{n}\), where \(k\) is an empirical constant typically set to 0.003, \(\rho\) represents the droplet density, and \(C\) denotes the speed of sound within the droplet (taken as 1480 m s\({}^{-1}\) under standard temperature and pressure conditions). The dynamic pressure is determined by the formula of \(P_{D}=\frac{1}{2}\rho v_{n}^{2}\). On the other hand, the anti-wetting pressure \(P_{attinet}\) corresponds to the capillary pressure (\(P_{C}\)) of the micro-pillar surface, which can be calculated as follows:  
\[P_{attinet}=P_{C}=\frac{4\pi d_{p}\sigma\sin\theta}{\pi{d_{p}}^{2}-4\big{(}d_ {x}+d_{p}\big{)}^{2}}. \tag{10}\]  
Based on the micro-pillar parameters in this study, the anti-wetting pressure \(P_{attinet}\) is 1667.93 Pa. When \(P_{wet}\) is less than \(P_{attinet}\), the droplet remains in the Cassie state, as illustrated in Figure 5a. Conversely, when \(P_{wet}\) is greater than or equal to \(P_{attinet}\), the droplet penetrates the micro-pillar surface and infiltrates downward. Under the operational conditions in this study, the critical velocity for penetration is 0.352 m s\({}^{-1}\), corresponding to a critical Weber number of 0.079. When the droplet penetrates the micro-pillar array but does not reach the bottom surface, it is in a partial penetration state, as shown in Figure 5b. With a further increase in the Weber number, the droplet transitions from partial penetration to the Wenzel state [11] upon contacting the bottom surface, as depicted in Figure 5c. Consequently, the maximum penetration depth (\(H_{max}\)) during the whole impact process serves as a crucial parameter for characterizing the droplet state. We define the dimensionless maximum penetration depth as \(H^{\star}=H_{max}/H\). As shown in Figure 6a, the dimensionless maximum penetration depth gradually increases with increasing impact velocity. Below the critical penetration velocity, the maximum penetration depth is primarily influenced by droplet surface oscillations and curvature [52]. A plateau region is observed within the velocity range of 2 to 3 m s\({}^{-1}\) (corresponding to \(We_{n}\) between 3 and 5.4), where the maximum penetration depth no longer increases. This phenomenon arises because the droplet volume remains constant while the spreading diameter increases, causing the kinetic energy to be converted into the droplet's lateral spreading capability. [PERSON] et al. [53] found that the contact time of a droplet impacting on smooth planar surface is determined solely by its diameter and the properties of the liquid, independent of the impact velocity, i.e., \(\tau_{c}\propto(\rho D_{0}^{3}/8\sigma)^{1/2}\). Based on this relationship, the dimensionless contact time is defined as \(t^{\star}=t_{c}/\tau_{c}\) as a comparison between the smooth and micro-pillar surfaces, where \(t_{c}\) represents the actual contact time of the droplet impact in this study. As shown in Figure 6b, the contact time exhibits various trends under different droplet states, which differs from the trends when droplets impact millimeter-scale microstructured surfaces [54].