## 3 Results
After approximately 15 ms, the droplet recontacts the ridge, and the velocity approaches zero. As seen in Figure 6a, the maximum value of \(U_{y}\) decreases as wettability asymmetry increases. This is evident in the graph depicted in Figure 6b, where the maximum value of \(U_{y}\) sharply increases from \(\beta\) = 0.75 (i.e., \(\theta_{R}\) = 120\({}^{\circ}\)). In summary, when \(\beta\) is less than 0.75, it can be classified as a region where the influence of ridge wettability difference on jumping is significant (wettability-dependent regime), whereas the opposite is a region for \(\beta\) greater than 0.75, where the influence of ridge wettability difference is less significant (wettability-independent regime). The magnitude of the vertical velocity was predicted under the changes in the ridge's aspect ratio. For \(AR\) = 1.0, 1.5, and 2.0, the maximum values of \(U_{y}\) were approximately 1.4, 1.5, and 1.6 times greater, respectively, compared to surfaces without ridges. Furthermore, for low \(\beta\), there was little difference in the maximum values of \(U_{y}\) based on \(AR\), but there was a difference for high \(\beta\). For example, at \(\beta\) = 0.66 (i.e., \(\theta_{R}\) = 105\({}^{\circ}\)), the maximum value of \(U_{y}\) showed a maximum difference of 1.03 times, while the maximum difference in \(U_{y}\) was 1.1 times at \(\beta\) = 1.0 (i.e., \(\theta_{R}\) = 160\({}^{\circ}\)). From Figure 6b,  
Figure 5: Temporal evolution of averaged jumping velocities of the merged droplets in the \(x\) direction for varying \(\beta\). The snapshots of the inset represent the morphological behavior of the coalesced droplet at several times. Figure 6: (**a**) Temporal evolution of averaged jumping velocities of the merged droplets in the \(y\) direction for varying \(\beta\). (**b**) Maximal \(U_{y}\) as a function of \(\theta_{R}\). it can be inferred that even if uniform hydrophobicity cannot be maintained due to surface damage, designing for efficient jumping could be achieved by ensuring a large \(AR\). Investigating the spreading phenomenon when droplets recontact the ridges after jumping may aid in analyzing the final trajectory of the droplets. Figure 7 shows the spreading velocity in the \(z\)-direction of the droplet over time. Two cases of droplet shape change for two \(\beta\) values are added above and below the graph. Regardless of \(\beta\), the variation (period) in the \(z\)-directional average velocity up to about 20 ms remained almost the same. However, after the droplet starts spreading on the ridges, around 20 ms, \(U_{z}\) becomes dependent on the initial value of \(\beta\). For example, at the lowest \(\beta\) (i.e., \(\theta_{R}=90^{\circ}\)), a long spreading length occurred due to the low receding angle on the ridge (26-36 ms). In contrast, with intermediate values of \(\beta\), a shorter spreading length occurs due to the high receding angle (26-36 ms). Thus, for \(\beta\) values greater than or equal to 0.75 (i.e., \(\theta_{R}=120^{\circ}\)), the similar variations of \(U_{z}\) are observed. In summary, when \(\beta\) is less than 0.75, it can be classified as a region where the influence of ridge wettability difference on jumping is significant, whereas the opposite is true for regions with \(\beta\) greater than 0.75, where the influence of ridge wettability difference is less significant, as discussed earlier.