## 3 Results  
To investigate droplet coalescence jumping behavior on ridges with asymmetric wettability, we first predicted the jumping behavior induced by ridges with symmetric wettability. Figure 1b presents the numerical analysis results illustrating the recoil phenomenon due to the surface tension of the merged droplet. A droplet of elliptical shape forms upon coalescence (4.0 ms), and strong recoil forces occur at the ends, where a large curvature is formed, causing the droplet to retract. Due to the presence of hydrophobic ridges, the droplet detaches completely and jumps (12.5 ms). The recoiling droplet reaches its maximum height and falls freely before contacting the ridge again. The assumptions of  
Figure 2: Model validation based on the jumping velocity on the flat surface. this study are as follows: breaking the symmetry of ridge surface wettability leads to the formation of horizontal jumping velocity, and the final landing position of the droplet may vary. This investigation analyzed the dynamics of droplet jumping, focusing on the initial contact angle ratio (\(\beta\)) and the aspect ratio (\(AR\)) of the ridges. Variations in these factors were scrutinized concerning the jumping height, horizontal offset distance, and behavioral traits determined by the average speed of jumping droplets along each axis. This study also analyzed the impact of ridge's wettability distribution and geometric aspect ratio on effective droplet transportation. Predictions of droplet coalescence jumping due to increasing asymmetry in ridge's surface wettability are presented in Figure 3. Figure 3a-c correspond to a geometric aspect ratio (\(AR\)) of 1.0, while Figure 3d-f correspond to 2.0. The elliptical shape formed upon droplet fusion was similar in all cases (4.0 ms). During contraction, droplet separation from the surface is slower on ridges with significant wettability asymmetry. Here, the horizontal offset distance is defined as \(\Delta x\), and its effect becomes more pronounced as \(\beta\) decreases. For instance, at the lowest \(\beta\), no jumping occurs (Figure 3a), while jumping occurs at slightly higher values of \(\beta\) = 0.66 and 0.75 (Figure 3b). Upon droplet recontact with the surface, significant differences in droplet volume between the right-hand side (RHS) and left-hand side (LHS) are observed at 30 ms.  
For ridges with high aspect ratios (\(AR\)), both the maximum height of the jumping (\(h_{max}\)) and the horizontal offset distance (\(\Delta x\)) increase. As seen in Figure 3d-f, during droplet contraction, significant horizontal velocity components are observed at low \(\beta\), while high recoil results are observed at high \(\beta\). Additionally, as the ridge's height increases, the vertical velocity component of the droplet also increases. Comparing Figure 3a,d, an increase in ridge height results in a higher horizontal displacement at the same \(\beta\). Similarly to cases with low aspect ratios, it is observed that the volume of the droplet on the RHS upon recontact with the surface is greater than that on the LHS (Figure 3d at 30 ms). The effect of asymmetry in wettability on maximum height of the jumping and horizontal offset distance is quantitatively analyzed in Figure 4. The left axis represents the maximum height, while the right axis represents the offset distance, under three ridge's aspect ratios. Generally, as asymmetry increases, the maximum height decreases. For example, compared to symmetric wettability with \(\beta\) = 1.0 (i.e., \(\theta_{R}\) = 160\({}^{\circ}\)), the lowest \(\beta\) results in approximately 30% lower height (for \(AR\) = 2.0). Similarly, for the lowest ridge (\(AR\) = 1.0), a 21% decrease in height is predicted. Conversely, as asymmetry increases, the offset  
Figure 3: Jumping behavior of the coalesced droplet under two \(AR\)s of the ridges. (**a–c**) Droplet shapes on ridges with (**a**) \(\beta\) = 0.56, (**b**) 0.66, and (**c**) 0.75, under a constant \(AR\) of 1.0.