Influence of coalescence-induced droplet jumping by W-shaped groove structures on superhydrophobic surfaces

To explore the influence exerted by the structural features of the W-shaped groove on coalescence-induced droplet jumping, we fabricated a large number of samples characterized by three structural parameters in Fig. 1 (w-central convexity angle, h-groove depth, and b-groove width = 0.9 mm).

The samples were printed with a 3D printer using photopolymer resin. Because of the hydrophilic surface of the printed samples, a commercial hydrophobic aqueous coating solution, NeverWet Multisurface, was applied to change the surface wettability. The sample was ultrasonically washed in deionized water for 20 min, then immersed in a hydrophobic aqueous solution for about 10 min, and air-dried for at least 40 min under standard laboratory conditions. For optimal results, it is necessary to repeat the process, which will lead to the creation of a superhydrophobic surface on the sample to achieve optimal superhydrophobicity. Finally, the surface-state contact angles of the samples were measured to validate their hydrophobicity. The Contact Angle plug-in based on ImageJ was selected as the measurement software, and the five-point method was adopted as the measurement approach. The results were then averaged. The static contact angles of a 1.5 μL volume droplet on the superhydrophobic surface were measured to be (161.3 ± 2)°. The advancing contact angle was (162.2 ± 2)°, and the receding contact angle was (156.4 ± 2)°.

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We utilized a modified hydrophobic-coated pusher needle to actively manipulate droplets and observe the coalescence phenomenon. To ensure the accuracy of the experimental results, we selected an underground laboratory maintained at a constant temperature of (24 ± 3) °C and a relative humidity of 60 ± 5%. Figure 2 displays the droplet coalescence-induced jumping test device, which consists of a rapid-fire camera system and a motion system. The rapid-fire camera system is composed of a high-speed camera (ACS-1 M60), a backlight (Luxpad 23), and an electric control system. The moving system comprises two Z-axis displacement device, one YZ-axis displacement device, two three-axis displacement devices, and one electric displacement device. The first Z-axis displacement device is used to mount the YZ-axis displacement device with a high-speed camera, ensuring that the sample surface and the focus plane of the camera remain at the same level during the shooting process and eliminating shooting position mistakes. The second Z-axis displacement device is installed with two three-axis displacement devices, which are respectively used to install the electric displacement device (fixed to the hydrophobic needle) and the sample. The combination of the two three-axis displacement devices allows the hydrophobic needle to flexibly adjust its position to propel the droplet. The hydrophobic needle pushes droplets along one of the grooves at a constant speed to induce coalesce with the droplets in the other groove. (If the droplet is propelled perpendicular to the groove, it will fall from the surface into the groove before coalescing with the droplet in the other groove, resulting in an unstable initial velocity and compromising the reliability of the results.)

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This experiment investigates coalescence behavior between dual droplets within microfluidic grooves. As shown in Fig. 3 of the Comparison of Test Choice Perspectives. In Method 1, the hydrophobic needle is moved along the groove to propel one droplet while keeping the other droplet stationary in an adjacent groove. The high-speed camera is positioned perpendicular to the groove orientation to avoid visual obstruction by the needle, though this approach does not reveal the groove structure. However, it ensures stable low-speed coalescence while maintaining both droplets within the grooves. In Method 2, droplets are initially positioned on the groove's external surface. The hydrophobic needle is oriented perpendicular to the groove direction to initiate coalescence. During this process, droplets descend into the groove before merging. While this setup enables groove structure visualization, it introduces droplet motion instability during initialization, compromising experimental reliability. The characteristics of water at a temperature of 24°C are shown in Table 1.

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The operation steps are described as follows: Initially, we employed a micropipette to distribute two identical droplets (R is between 0.5 and 0.9 mm, volume is between 0.52 and 3.05 μL), respectively, located on the double groove of the W groove. Turn on the lamp source and operate the three-axis displacement device to control the position of the hydrophobic needle to ensure that the center connection of the two droplets is at the same level as the hydrophobic needle. Operate the Z-axis displacement device and the YZ-axis displacement device to adjust the camera position and ensure that the focus plane of the camera is at the same level as the surface of the sample. Activate the rapid-fire camera system with a rate of 15,000 frames per second. Subsequently, turn on the electric displacement device and propel the droplet in one groove to the stationary droplet in the other groove at a constant low speed (v_t= 3 × 10^–5 m·s⁻¹). Moving the droplet at a slow speed resulted in negligible kinetic energy compared to the energy generated during coalescence. Eventually, the velocities of droplet jumping were determined by extracting data from the trajectories of the droplet centroid using Image-Pro Plus processing results obtained from several photos.

Our numerical simulations are carried out using the laminar flow model and the conservation form phase field model in COMSOL. The gas–liquid phase is distinguished by the Phase Field function φ, where φ = -1 denotes the air phase and φ = 1 denotes the liquid phase. The Cahn–Hilliard equation of the Phase Field model in conservation form is [19]:where Ψ is phase-field auxiliary variable, φ is phase-field function, λ is mixed energy density, γ is surface tension, and ε_pf is initial interface thickness. Fluid physical property parameters can be expressed as [19]:where ρ represents density, μ represents dynamic viscosity, and Vf is volume fraction of phase. For the liquid phase, use the subscript l, while for the air phase, use the subscript k. Since it is an incompressible fluid, the continuity and momentum equations of the fluid can be stated as [19]:where P represents pressure, U is velocity vector, and g is gravitational acceleration.

The boundary is an open pressure outlet. In the realm of computation, the superhydrophobic surface is considered a non-slip wall, which disregards the hysteresis of the contact angle. The computational realm is set to 10 mm × 8 mm. The computational domain for numerical simulation is shown in Fig. 4.

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Coalescence-induced droplet jumping on the superhydrophobic flat surface: A series of images showing the droplet coalescence jumps (R = 0.6 mm) on the superhydrophobic flat surface are presented in Figs. 5, 6. Here, we can break down the four-step process of the droplet coalescence jump process: (I) Upon collision of drops, the droplets form a liquid bridge that expands rapidly, (II) The liquid bridge expands rapidly until its base impacts the surface, (III) Liquid bridges impact the surface and then rapidly form large droplets until maximizing the interfacial area between the solid and liquid, (IV) The interfacial area between the solid and liquid is diminished until the droplet separates and flows away from the surface.

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Stage 1, the two droplets begin to contact, and forming a liquid bridge (Fig. 5 and Fig. 6–0.20 ms). The rapidly expanding liquid bridge has the shape of a dumbbell. Stage 2, the liquid on both sides flows rapidly to the center, the top of the liquid bridge changes from dumbbell-shape to convex-shape, and the bottom impacts the surface (Fig. 5–1.6 ms and Fig. 6–1.4 ms). Stage 3, droplets coalesce into large droplets, and the interfacial area between the solid and liquid reaches the maximum (Fig. 5–4.2 ms and Fig. 6–4.5 ms). Stage 4, the droplet shrinks its interfacial area between the solid and liquid, converting excess surface energy into upward kinetic energy. When the interfacial area between the solid and liquid shrinks to point of disappearance, the droplet detaches from the surface (Fig. 5–6.2 ms, and Fig. 6–6.5 ms).

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Coalescence-induced droplet jumping on the superhydrophobic surface with the W-shaped groove: A sequence of images demonstrating the droplet coalescence jumps (R = 0.6 mm) on the W-shaped groove is presented in Fig. 7 and Fig. 8. The four stages of droplet jumping are consistent with those on the flat surface. However, the W-shaped groove structure modifies stages 2 and 3. In stage 2, the liquid bridge on the W-shaped groove structure takes less time to impact the surface (Fig. 7 -1.1 ms and Fig. 8 -0.9 ms) than it does on the flat surface (Fig. 5 -1.6 ms and Fig. 6 -1.4 ms). The liquid bridge's base impacts the surface when its top curves outward on the flat surface. In contrast, when the top surface of the liquid bridge is still dumbbell-shaped on the W-shaped groove structure, the bottom of the bridge impacts the central convexity. Due to the short coalesce time of the liquid bridge in the W-shaped groove structure, it is faster when it impacts the central convexity, which creates a stronger impact on the central convexity, resulting in a larger impact force than on the flat surface, and the droplet jumping speed will be larger. In stage 3, as the two droplets on the flat surface start to coalesce into large droplets, the liquid bridge bottom area only shrinks in the horizontal direction (Fig. 5 -1.6 ms ~ 4.2 ms and Fig. 6 -1.4 ms ~ 4.5 ms). In contrast, when two droplets coalesce into a larger droplet on the W-shaped groove structure (Fig. 7 -1.1 ms ~ 3.2 ms and Fig. 8 -0.9 ms ~ 3.4 ms), the bottom of the liquid bridge was consistently influenced by the extrusion of the central convexity, which caused the droplet to simultaneously constrict in the horizontal and vertical surface axes. The increased deformation at the liquid bridge's base leads to the buildup of extra excess surface energy, hence increasing the upward kinetic energy.

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To analyze the energy conversion during droplet coalescence-induced jumping, we numerically investigated the energy transition dynamics on both W-shaped grooves and flat surfaces. The coordinate system is defined as follows: droplet coalescence occurs along the X-axis, and the resultant motion of the merged droplet aligns with the Z-axis.

Two droplets of identical radius R coalesce to form a larger droplet of radius R_c. The primary driving force for droplet jumping is the excess surface energy E_s,ex, expressed as:where E_s(t) = γ_sA_surf (t) represents the droplet surface energy, γ_s represents the water surface tension, A_surf (t) symbolizes the total area on the outer surface of the droplet, and E_s,c = γ_s4πR2 c represents the surface energy of the droplet that has coalesced together and reached its final size.

Initially, the coalesce process involves the excess surface energy E_s,ex(0) = E_s(0)-E_s,c, where E_s(0) = γ_s8πR2 0. Throughout the process of coalescence jump, the excess surface energy is progressively dissipated, with a portion of it being transformed into upward kinetic energy E_k,up, that are calculable as:where m_c = ρ_s4πR3 c/3 denotes the cumulative mass of the merged droplet, ρ_s represents the water density, and V_d(t) denotes the droplet upward velocity. The remaining portion is dispersed through adhesion work E_w and viscous dissipation of the droplets E_vis. The adhesion work E_w is dependent on the solid–liquid contact area, which can be computed by [15]:where θ represents the droplet state contact angles, A_s represents the interface area between the solid and liquid. It is very difficult to accurately calculate the viscous dissipation of droplets. Since the viscous dissipation is primarily caused by the direction of the two droplets coalescing with each other, it may be represented as follows [9]:where Φ denotes the dissipation function and ω designates the volume of an individual droplet. The coalescence time, τ_t, is defined as the duration between the start of coalescence and the moment when droplets jump. u represents the mean combined speed of every droplet and can be determined using the following formula [9]:where F_t = 2γ_s /R represents the interfacial tension. By substituting Eqs. (10)-(12) into Eq. (9), Eq. (9) can be reformulated as follows [9]:

The required parameters were transformed into nondimensional form using the following procedure: nondimensional time t^* = t/t_ic, where \(t_{ic} = \sqrt {\rho_{s} R_{0}^{3} /\gamma_{s} }\) indicates the inertial-capillary time scale[23], the nondimensional excess surface energy E* s,ex(t^*) = E_s,ex(t)/E_s,ex(0), and the nondimensional upward kinetic energy E* k,up(t^*) = E_k,up(t)/E_s,ex(0). When first starting to coalesce, E* s,ex(0) = 1, E* k,up(0) = 0. In order to provide a clearer explanation of the physical phenomena, we have condensed the process of droplet coalescence jumps into three distinct stages: During stage 1, two droplets collide and create a liquid bridge in the proximity region. In stage 2, after the liquid bridge impacts the surface, large droplets are rapidly formed through coalescence until the interface area between the solid and liquid reaches its maximum. In stage 3, the interface area between the solid and liquid at the bottom shrinks until it disappears, and the droplet detaches from the surface.

First, consider two droplets (R = 0.6 mm) that coalesced on the flat surface. At the coalescing beginning, the liquid bridge underwent expansion along the X direction, resulting in a decrease in the excess surface energy E* s,ex(t^*), the upward velocity V_d = 0, and the upward kinetic energy E* k,up = 0 (stage I in Fig. 9a). At t^* = 1.3, the lower end of the droplet impacted the surface (Fig. 9c), causing a disruption of the Z-direction symmetry, which compelled the droplet to move upward, producing an upward velocity V_d and upward kinetic energy E* k,up (stage II in Fig. 9a). Simultaneously, the droplet reaches its highest level expansion along the X-axis, and two symmetrical high-pressure regions appear near the extreme positive and negative X positions (Fig. 9d). As a result, the fluid inside the droplet moves from the areas with high pressure at both ends to the areas with low pressure toward the center, causing the droplet to shrink on the X-axis. The interfacial area between the solid and liquid reached its maximum at t^* = 2.5 (Fig. 9e). The largest solid–liquid interfacial area provided the droplet with the greatest upward force, causing the droplet to move in the positive Z-direction. However, a vortex region existed at the bottom of the droplet that always interfered with the velocity field and prevented the droplet from moving upward (Fig. 9e). Finally, as the upward force overcame the interference from the vortex region and other factors, the droplet detached from the superhydrophobic surface (stage III in Fig. 9a, Fig. 9f).

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Next, we will examine the two droplets (R = 0.6 mm) that coalesce on the W-shaped groove. Similar to droplet coalescence on the plane, droplet coalescence begins at t^* = 0.1, and the liquid bridge underwent symmetric deformation in the X-direction by capillary forces. The excess surface energy E* s,ex(t^*) decreases, the upward velocity V_d = 0, and the upward kinetic energy E* k,up = 0 (stage I in Fig. 10a). However, at t^* = 0.5, the existence of the central convex structure causes the lower end of the drop to impact the surface early, resulting in early breaking of Z-direction symmetry deformation (Fig. 10c), forcing the droplet to move upwards and generating an upward velocity V_d and an upward kinetic energy E* k,up (stage II in Fig. 10a). Due to the early impact, the droplet coalescing time τ_t is reduced, and from Eq. 13 it can be inferred that while the fluid characteristics, contact angle, excess surface energy, and droplet radius are held constant, the coalescing time of the droplet is directly proportionate to the viscous dissipation. Therefore, when the droplet coalescence time in the W-shaped groove was reduced, the viscous dissipation decreased, and the excess surface energy is transformed into upward kinetic energy E* k,u_p and upward velocity V_d to a higher extent than on the flat surface. Subsequently, due to the extrusion of the central convexity, the droplet expands from the center to the sides and reaches the maximum extension length in the X direction at t^* = 1.3. Since both droplets are located in the groove, the expansion of the droplet will be compressed by the groove boundary, resulting in the emergence of two symmetrical high-pressure zones at the extreme positive and negative X positions, which exceed the pressure on the flat surface (Fig. 10d). The liquid moves from the areas of high pressure at both ends towards the area of low pressure in the center, causing the droplet to gradually contract into a large droplet. Contrary to the flat surface, when the droplet contracts, the presence of a convex obstruction at the center causes the velocity vector to be mostly aligned with the + Z direction (Fig. 10d). The alteration in the velocity vector leads to a substantial augmentation in E* k,up (stage II in Fig. 10a). The high-pressure region moves down to the vicinity of the central convexity, and the droplets "wrap" around the central convexity, which has the largest solid–liquid interfacial area at t^* = 2.1 (Fig. 10e). The combined synergistic effect of the upward reaction force from the greatest interface between the solid and liquid and the high-pressure region at the droplet base forces it to jump, which increases the upward kinetic energy E* k,up (stage III in Fig. 10a). Eventually, the droplet left the surface at t^* = 3.0. When it leaves the superhydrophobic surface (Fig. 10f), it has a high upward kinetic energy E* k,up (stage III in Fig. 10a).

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We conducted experiments on two equal-diameter droplets (R = 0.5 mm–0.9 mm, volume 0.52 – 3.05μL) whose sizes matched the single groove width b = 0.9 mm, thereby preventing undersized droplets from failing to contact and coalesce. Tests were performed with four groove depths h (0.5 mm–0.8 mm) and a flat surface, while fixing the central convexity angle w at 60° to isolate the influence of h on jumping velocity v_j and dimensionless jumping velocity v* j. For each configuration, 20 trials were conducted to calculate averaged values. The result is shown in Fig. 11. On a flat surface, the inertia-capillary scaling law governs the leaping velocity of droplets, which can be calculated by [7]:where u_ic is the inertia-capillary velocity. The dimensionless jumping velocity v* j is able to be calculated by:where v_j is jumping velocity, which is the constant velocity of the aggregated droplets' center of mass after they leave the surface in the experiment, u_ic is the inertia-capillary velocity of the coalesced droplets with a radius of R. If the coalescing droplet radius is 2^1/3R, while the mass and gravity remain unchanged, but the droplet outside area decreases, then the energy conversion efficiency η can be determined using the following calculation:

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Figure 11 demonstrates that the droplet jumping velocity on flat surfaces adheres to the inertia-capillary scaling law, with an upper dimensionless jumping velocity limit of v* j = 0.22, corresponding to an energy conversion efficiency η ≈ 4%. In contrast, droplet jumping velocities on W-shaped grooves universally exceeded this scaling law. The velocity exhibited a negative correlation with droplet radius: as R increased from 0.5 to 0.9 mm, v_j decreased rapidly due to increased droplet mass and groove width b mismatch. At R = 0.5 mm, the peak jumping velocity v_j = 0.25 m/s was threefold higher than that on flat surfaces. Figure 11b illustrates the effect of groove depth h on dimensionless jumping velocity. The maximum v* j = 0.65 at h = 0.8 mm corresponded to η ≈ 35.04%, whereas the peak v* j = 0.57 at h = 0.5 mm yielded η ≈ 26.89%. This indicates that η at h = 0.8 mm was 8.15% higher than at h = 0.5 mm and 8.76 times greater than on flat surfaces. Notably, the velocity-enhancement effect diminished with decreasing h.

To interpret the observed physical trends, we employ scaling arguments based on an energy balance framework. During droplet coalescence-induced jumping, the upward kinetic energy primarily originates from excess surface energy generated by droplet deformation, which is typically uniformly distributed across the droplet surface. However, in W-shaped grooves, the central convexity obstructs coalescence, resulting in localized surface energy accumulation near the convexity region. This concentrated energy, quantified as γ_sRh, is converted into upward kinetic energy during droplet detachment. One way to express the related scaling argument in terms of an energy balance is as follows [30]:where C₁ and C₂ are constants that represent the geometric factors and the partial energy conversion, respectively. Nondimensionalizing the groove depth as h^* = h/R, Eq. 17 can be rewritten as follows:where We_j represents the Weber number at the moment of droplet-surface separation, which is defined as a metric quantifying the conversion of surface energy into upward kinetic energy. As explicitly demonstrated in Eq. 18, We_j increases with the dimensionless groove depth h^*. Physically, this relationship indicates that an increase in groove depth h enhances the surface-to-kinetic energy conversion efficiency, thereby accelerating droplet jumping velocity.

To investigate the impact of the central convexity angle w on droplet jumping velocity, we selected a fixed groove depth h = 0.8 mm with central convexity angle w varying between 60 and 90°. As shown in Fig. 12, the maximum droplet jumping velocity occurred at w = 60°, corresponding to a dimensionless jumping velocity v* j = 0.65 and an energy conversion efficiency η ≈ 35.04%. In contrast, at w = 90°, the peak dimensionless jumping velocity decreased to v* j = 0.603, yielding η ≈ 29.43%. This indicates that the energy conversion efficiency at w = 60° was 5.61% higher than at w = 90°.The results demonstrate that reducing w increases both jumping velocity and energy conversion efficiency, though the enhancement effect remains limited. We hypothesize that smaller w values lower the central mass positions of pre-coalescence droplets closer to the groove and to each other. However, the energy conversion efficiency did not improve significantly, likely due to insufficient positional adjustment caused by minor angular variations in w.

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The effects of the groove depth h and the central convexity angle w of the middle section on droplet coalescence jumping behaviors were systematically investigated in the preceding sections. To comprehensively assess the effects of these two groove parameters on droplet coalescence jumping phenomena, a coupled parameter analysis was performed by simultaneously varying the groove depth h and the central convexity angle w of the intermediate groove section to observe the effects on droplet coalescence jumping behavior.

As shown in Fig. 13, the lowest droplet jumping velocity was observed at groove depth h = 0.5 mm and central convexity angle w = 90° in the middle section, with a maximum dimensionless jumping velocity v* j = 0.41, corresponding to an energy conversion efficiency η ≈ 22.95%. In contrast, the jumping velocity increased with larger h and smaller w, reaching its maximum value at h = 0.8 mm and w = 60°, where v* j = 0.48 and η ≈ 28.32%. These experimental results demonstrate that synergistic variations in structural parameters (groove depth h and central convexity angle w) collectively govern jumping velocity and energy conversion efficiency by modulating energy redistribution and momentum transfer direction during the coalescence of droplets.

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When the groove depth h increased, lateral droplet spreading was significantly constrained, while vertical fluid contraction became more focused, enhancing the release of surface energy into upward kinetic energy. Concurrently, reducing the central convexity angle w amplified this process by altering the hydrodynamic characteristics of the droplet contraction pathway. Specifically, the raised structure induced vortex formation with normal acceleration during the contraction phase of the droplet secondary lobes. Furthermore, geometric groove constraints minimized energy losses by shortening the triple-phase line contraction time and suppressing viscous dissipation and surface pinning effects. This synergy resulted in a homogeneous internal pressure distribution within the droplet, suppressed lateral momentum dispersion, and concentrated kinetic energy transfer in the vertical direction.

At h = 0.5 mm and w = 90°, extensive lateral spreading and dispersed contraction paths caused partial conversion of surface energy into lateral oscillatory kinetic energy, reducing η to 22.95%. However, when h increased to 0.8 mm and w decreased to 60°, the tilted groove sidewalls accelerated side-valve reflux through reactive forces, precisely guiding droplet contraction vertically. This geometric configuration concentrated the post-impact liquid bridge along the vertical axis, shortened the liquid bridge spreading time, optimized the surface-to-kinetic energy conversion pathway, and thereby improved η to 28.32%.

While prior droplet jumping studies predominantly addressed symmetric coalescence, asymmetric coalescence is more common in real-world scenarios. In this work, groove depth h = 0.8 mm and central convexity angle w = 60° were held constant. With R₁ fixed at 0.7 mm, R₂ was systematically varied to quantify the effects of asymmetric droplet coalescence on jumping dynamics and energy conversion efficiency.

As shown in Fig. 14, the jumping velocity and energy conversion efficiency initially increase and then decrease with increasing R₂, reaching the highest jumping velocity v_j and the highest energy conversion efficiency η when R₁ is equal to R₂. In this study, we argue that the mechanism of droplet size asymmetry affecting jump velocity and energy conversion efficiency fundamentally stems from the synergistic interplay between surface energy release and energy redistribution modes during coalescence. When R₁ = R₂ = 0.7 mm (symmetric droplet coalescence), the coalescence produces highly symmetric capillary contraction waves. The spatial homogeneity of the Laplace pressure gradient, induced by the synchronized retraction of the liquid–solid contact line, results in an efficient directional conversion of surface energy into upward kinetic energy. This optimal momentum superposition at the center point of the merged droplet establishes the dynamical basis for the observed peak jump velocity (v_j = 0.175 m/s) and maximum energy conversion efficiency (η = 23%).

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When R₂ deviates from the symmetric condition (R₂ ≠ R₁), asymmetric coalescence induces asynchronous contraction of the droplets. The inertia of the larger droplet suppresses effective capillary force transmission from the smaller droplet, generating energy-dissipating vortical flows. According to the law of surface energy, although the total surface energy increases monotonically with R₂, the asymmetric energy distribution causes partial energy to be converted into internal vortices within the droplet rather than upward kinetic energy. For R₂ < 0.7 mm, the rapid spreading of the smaller droplet prematurely triggers contact line pinning, producing a reverse pressure gradient that hinders liquid bridge expansion. Consequently, smaller R₂ values result in lower jumping velocities and energy conversion efficiencies. Conversely, for R₂ > 0.7 mm, the inertial delay of the larger droplet weakens the transient capillary-driven impulsive force, leading to shear layers that cancel out axial momentum components during neck contraction. Thus, larger R₂ values similarly reduce both jumping velocity and energy conversion efficiency. These dual mechanisms collectively explain the unimodal trend observed in the parameter curves.