Dynamics of Coalesced Droplet Jumping on Superhydrophobic Surface with Asymmetrically Wettable Ridge

: Spontaneous detachment from superhydrophobic surfaces can be induced by the coalescence of two or more adjacent droplets. The phenomena have provided implications for the self-removal of droplets in the fields of self-cleaning, anti-icing, and heat transfer. However, many studies focus mainly on the theoretical jumping direction perpendicular to the substrate, although the velocity in the horizontal direction must be involved in practical applications due to various scenarios. This study analyzes numerically the effect of the distribution in ridge structure’s wettability on the performance of coalesced droplet jumping. The jumping dynamics are discussed for varying contact angle ratios and the aspect ratios of the ridge, which are the initial values for the current model. We obtain the height of the jumping and the offset distance in the horizontal direction under the several initial values. In addition, the characteristics of the asymmetric behavior are discussed based on the temporal evolution of the average velocities of the jumping droplets for each direction. Numerical results show that the horizontal offset distance is significantly pronounced at both the high asymmetry in wettability and the high aspect ratio of the ridge geometry. The phenomenon occurs when the droplet detaches from the ridge surface in the retraction process. We determine the role of the distribution within the ridge structure on its wettability, as well as the role of the aspect ratios of the ridge in facilitating the efficient transport of droplets.


Introduction
When two or more adjacent droplets merge on a superhydrophobic surface, spontaneous droplet removal can be induced.This droplet jumping phenomenon was reported by Kollera and Grigull in 1969 during experiments with mercury droplets and was examined by Boreyko and Chen in 2009, observing the jumping phenomenon in droplet coalescence [1].Coalesced droplet jumping has garnered significant attention in various industrial fields, such as self-cleaning [2,3], anti-icing/frosting [4,5], and enhancement of condensation heat transfer [6,7].Many researchers have investigated the mechanism and applicability of droplet jumping, developing applied technologies by introducing various factors such as droplet-solid interactions and various surface structures [8].
Superhydrophobic surfaces can rapidly detach when colliding droplets, similar to lotus leaves [9,10].This surface was able to promote the spontaneous removal of droplets by maintaining the contact angle between droplet and solid at over 150 • and keeping the contact angle hysteresis very small.Numerous experiments and theoretical studies have been conducted on the microstructure of superhydrophobic surfaces, and research on improving superhydrophobic surface structures continues.The performance of superhydrophobic surfaces is expressed by the very high contact angle or residence time of droplets, where the residence time can be approximated by τ ~(ρD 3 /8σ) 1/2 under inviscid fluid conditions [11], where ρ, D, and σ correspond to density, equivalent diameter, and surface tension, respectively.For small droplets, behavior is dominated by inertia force and surface tension.Important dimensionless numbers to express this are the Weber number (We) and the Reynolds number (Re): the Weber number, We = ρDV 2 /σ, represents the inertia force relative to the surface tension, whereas the Reynolds number, Re = ρDV/µ, represents the inertia force relative to the viscous force, where V is velocity and µ is the viscosity of the liquid.
A few research studies on droplet coalescence jumping have been investigated from various aspects, such as jumping speed and direction, as well as energy conversion.On a flat surface without introduced structures, the jumping speed when two droplets merge is roughly proportional to the inertia-capillary speed of [σ/(ρD)] 1/2 [12].Regarding energy conversion, when two droplets combine, surface energy is converted into kinetic energy as part of the motion, inducing a vertical component of velocity on the substrate.On flat surfaces, the jumping speed of droplets can only be limited to a certain maximum, so introducing structures on the surface to increase the energy conversion rate has been proposed [13][14][15][16][17][18].Wang et al. [13] introduced triangular prisms to increase jumping speed and reported a high energy conversion rate.Yuan et al. [14] verified an increase in rebound speed and energy conversion rate by introducing an egg-tray structure.Vahabi et al. [15] investigated the energy conversion mechanism by observing internal flow during droplet coalescence.Peng et al. [16] and Liu et al. [17] investigated jumping speed and energy conversion by introducing micro-grooves and U-shaped grooves, respectively, and discussed the horizontal component of jumping direction.Gao et al. [18] proposed asymmetric ridge structures to investigate horizontal movement of droplets on superhydrophobic surfaces and validated the hypothesis through experiments and simulations.
On a horizontal superhydrophobic surface, the dynamics of droplet coalescencejumping exhibit variations in velocity and droplet morphology, depending on factors such as surface properties, structures, and temperature.Despite these differences, the fundamentals of coalesced droplet jumping remain similar, as follows.Droplets in close proximity form a small liquid bridge.The merging of the droplets is induced by the capillary pressure, along with pressure differences across the interface, which causes droplets to expand.The liquid bridge can induce a vertical upward force near the top of the ridge.The merged droplet exhibits an upward motion, which initiates oscillations.The retraction of the merged droplet begins, and the contact area between the droplet and the substrate surface progressively diminishes.The merged droplet ascends into the air while oscillating horizontally in both directions.
The rebound direction of combined multiple droplets still predominantly remains perpendicular to the plane [12].When droplets re-approach the surface, they can be removed again due to collisions with other droplets, but horizontal movement can still be restricted [19,20].The process of droplets re-approaching the surface is very similar to the process of colliding with the surface at low collision speeds.Structures such as point-like defects [21,22] and tapered post arrays [23] have had a significant impact on droplet collision dynamics.It has been reported that adjusting surface wettability during droplet collision can increase the horizontal displacement distance by several tens of times the droplet size [24,25].However, most studies have implemented changes in the micro/nanostructures of flat surfaces to achieve asymmetric wettability, without considering the influence of wettability on droplet behavior on macro-structures.Furthermore, the durability of micro/nanostructures and surface coatings on macro-structures to maintain superhydrophobicity under practical conditions (such as extreme temperatures) imposes limitations on anti-icing performance.This would lead to asymmetric wettability phenomena on macro-structures and potentially cause defects in vertical jumping performance.Therefore, it is essential to investigate the influence of macro-structure wettability on the horizontal movement of droplets.
This study analyzed numerically the influence of ridge's wettability distribution on droplet coalescence jumping performance using the volume of fluid (VOF) method [26].As shown in Figure 1, spontaneous merging of two droplets is induced, and due to the presence of superhydrophobic ridges, the droplets completely detach from the surface and recoil.The dynamics of droplet recoil were investigated based on the initial contact angle Appl.Sci.2024, 14, 3584 3 of 10 ratio (β = θ R /θ L ) and aspect ratio (AR) of the ridges.Changes in these two factors were examined with respect to the height of the jumping, the horizontal offset distance, and behavioral characteristics, based on the average speed of jumping droplets along each axis.The role of ridge's wettability distribution and geometric aspect ratio in efficient droplet transportation was analyzed.This study can provide strategies for controlling the direction of droplet jumping and increasing the coalescence frequently between adjacent droplets in related applications.
Appl.Sci.2024, 14, x FOR PEER REVIEW 3 of 10 presence of superhydrophobic ridges, the droplets completely detach from the surface and recoil.The dynamics of droplet recoil were investigated based on the initial contact angle ratio (β = θR/θL) and aspect ratio (AR) of the ridges.Changes in these two factors were examined with respect to the height of the jumping, the horizontal offset distance, and behavioral characteristics, based on the average speed of jumping droplets along each axis.The role of ridge's wettability distribution and geometric aspect ratio in efficient droplet transportation was analyzed.This study can provide strategies for controlling the direction of droplet jumping and increasing the coalescence frequently between adjacent droplets in related applications.

Numerical Methods
This study analyzed the effect of asymmetry in ridge wettability on droplet coalescence jumping using the VOF numerical analysis method.The behavior of droplet coalescence jumping was investigated by varying the contact angle ratio and the aspect ratio of the ridges.Liquids and gases, namely water and air, were used at room temperature and atmospheric pressure, respectively.The two phases were denoted by subscripts 1 and 2, and the volume fraction was represented by ψ.Numerical analysis was conducted based on previous studies that modeled collision phenomena on solid surfaces [27,28].The three-dimensional unsteady mass and momentum equations were employed in the computational domain, where  =   + (1 −  ) ,  =   + (1 −  ) .The curvature of the interface was defined as  = −(∇ •  ⃗), where  ⃗ is the unit vector normal to the interface, and  ⃗ is the gravitational acceleration.Volume forces were introduced based on a continuous surface tension model [29], and the volume fraction was calculated by employing the equation / +  ⃗ • ∇ = 0, with a surface tracking method based on Rider and Kothe [30].The discretization model for spatial and temporal derivatives was based on basic convection models and research by Leonard [31], respectively.Residuals were constrained to be less than 10 −5 to improve convergence of velocity and pressure fields.A time step of 1 µs and a maximum internal iteration of 25 were applied.A minimum of 50 grid cells were assigned per droplet size to ensure analysis accuracy, and the computational domain introduced a rectangular channel (10 × 8 × 8 mm 3 ) with ridges on the bottom surface.In this study, as shown in Figure 1a, the aspect ratio (AR) of the ridge was defined as the ratio of height (H) to width (W), denoted as H/W, with W fixed at 0.2 mm.Three ridge

Numerical Methods
This study analyzed the effect of asymmetry in ridge wettability on droplet coalescence jumping using the VOF numerical analysis method.The behavior of droplet coalescence jumping was investigated by varying the contact angle ratio and the aspect ratio of the ridges.Liquids and gases, namely water and air, were used at room temperature and atmospheric pressure, respectively.The two phases were denoted by subscripts 1 and 2, and the volume fraction was represented by ψ.Numerical analysis was conducted based on previous studies that modeled collision phenomena on solid surfaces [27,28].The three-dimensional unsteady mass and momentum equations were employed in the computational domain, ∂ ∂t ∂ ∂t where The curvature of the interface was defined as κ = −(∇• → n ), where → n is the unit vector normal to the interface, and → g is the gravitational acceleration.Volume forces were introduced based on a continuous surface tension model [29], and the volume fraction was calculated by employing the equation ∂ψ/∂t + → v •∇ψ = 0, with a surface tracking method based on Rider and Kothe [30].The discretization model for spatial and temporal derivatives was based on basic convection models and research by Leonard [31], respectively.Residuals were constrained to be less than 10 −5 to improve convergence of velocity and pressure fields.A time step of 1 µs and a maximum internal iteration of 25 were applied.A minimum of 50 grid cells were assigned per droplet size to ensure analysis accuracy, and the computational domain introduced a rectangular channel (10 × 8 × 8 mm 3 ) with ridges on the bottom surface.
In this study, as shown in Figure 1a, the aspect ratio (AR) of the ridge was defined as the ratio of height (H) to width (W), denoted as H/W, with W fixed at 0.2 mm.Three ridge aspect ratios of 1.0, 1.5, and 2.0 were introduced for analysis.Additionally, droplet coalescence jumping on a flat surface without ridges was analyzed to obtain comparative data.Previous research reported that the advancing and receding contact angles of the droplets were nearly identical at 160 • , with negligible contact angle hysteresis [32].Based on this, static contact angles on the surface (θ S ) and left side (θ L ) were set to 160 • to predict jumping behavior.To impart asymmetric contact angles, a static contact angle condition of θ R = 90-160 • was applied to the right side.For symmetrical wetting, a relationship of β = θ R /θ L = 1.0 (i.e., θ R = 160 • ) was satisfied.
The droplet size and properties were defined as D = 2.0 mm, ρ = 998.2kg/m 3 , σ = 0.072 N/m, and µ = 1 mPa•s, respectively.The diameter of the merged droplet can be calculated using the relationship D e = 2 1/3 D. The average velocities along each axis were calculated accordingly: where v represents velocity components, sgn(z) is the sign function, and Ω corresponds to volume.Dimensionless mean velocities (U x , U y , U z ) were defined by dividing the velocity values obtained from Equations ( 3)-( 5) by the inertia-capillary velocity [σ/(ρD)] 1/2 [12,18].
To validate the current model, we compared the jumping velocity with the previous theoretical and experimental results on the flat surface without any ridges.The merging of two droplets of the same size on a surface can be characterized by a dimensionless jumping velocity, which follows the inertial-capillary scale of U c * = [σ/(ρR)] 1/2 , where R is a radius of the droplet.The relationship between the jumping velocity and inertial-capillary scale can be as follows: U y */U c * = 0.2-0.3[1,12].Experimental results on the flat surface without any ridges revealed that the relation was U y */U c * = 0.23, which was made by superhydrophobic nano-coating [18].Our numerical data with different initial droplet sizes on the surface agreed with the experimental results, as shown in Figure 2.
Appl.Sci.2024, 14, x FOR PEER REVIEW 4 of 10 aspect ratios of 1.0, 1.5, and 2.0 were introduced for analysis.Additionally, droplet coalescence jumping on a flat surface without ridges was analyzed to obtain comparative data.Previous research reported that the advancing and receding contact angles of the droplets were nearly identical at 160°, with negligible contact angle hysteresis [32].Based on this, static contact angles on the surface (θS) and left side (θL) were set to 160° to predict jumping behavior.To impart asymmetric contact angles, a static contact angle condition of θR = 90-160° was applied to the right side.For symmetrical wetting, a relationship of β = θR/θL = 1.0 (i.e., θR = 160°) was satisfied.
To validate the current model, we compared the jumping velocity with the previous theoretical and experimental results on the flat surface without any ridges.The merging of two droplets of the same size on a surface can be characterized by a dimensionless jumping velocity, which follows the inertial-capillary scale of Uc* = [σ/(ρR)] 1/2 , where R is a radius of the droplet.The relationship between the jumping velocity and inertial-capillary scale can be as follows: Uy*/Uc* = 0.2-0.3[1,12].Experimental results on the flat surface without any ridges revealed that the relation was Uy*/Uc* = 0.23, which was made by superhydrophobic nano-coating [18].Our numerical data with different initial droplet sizes on the surface agreed with the experimental results, as shown in Figure 2.

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

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 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 (β) 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 ∆x, and its effect becomes more pronounced as β decreases.For instance, at the lowest β, no jumping occurs (Figure 3a), while jumping occurs at slightly higher values of β = 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.
Appl.Sci.2024, 14, x FOR PEER REVIEW 5 of 10 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 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 (β) 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 Δx, and its effect becomes more pronounced as β decreases.For instance, at the lowest β, no jumping occurs (Figure 3a), while jumping occurs at slightly higher values of β = 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 (hmax) and the horizontal offset distance (Δx) increase.As seen in Figure 3d-f, during droplet contraction, significant horizontal velocity components are observed at low β, while high recoil results are observed at high β.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 β.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 For ridges with high aspect ratios (AR), both the maximum height of the jumping (h max ) and the horizontal offset distance (∆x) increase.As seen in Figure 3d-f, during droplet contraction, significant horizontal velocity components are observed at low β, while high recoil results are observed at high β.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 β.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 β = 1.0 (i.e., θ R = 160 • ), the lowest β 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 distance generally increases.For instance, compared to weak asymmetry, with β = 0.84 (θ R = 135 • ), the lowest β results in an approximately 2.7-fold increase in offset distance (for AR = 2.0).Additionally, for the lowest ridge (AR = 1.0), an approximately 8-fold increase in height is predicted.Therefore, ensuring uniform wettability distribution on the ridges seems crucial for droplet vertical mobility.When the jumping droplets can be moved again upon re-approaching the surface, due to collisions with solid structures or other droplets, their horizontal movement can occur.The process of droplets re-approaching the surface closely resembles that of collisions with the surface at low speeds.The current prediction of the asymmetric wettability phenomena on macrostructures potentially provides vertical jumping performance in practical applications.
aspect ratios.Generally, as asymmetry increases, the maximum height decreases.For example, compared to symmetric wettability with β = 1.0 (i.e., θR = 160°), the lowest β 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 distance generally increases.For instance, compared to weak asymmetry, with β = 0.84 (θR = 135°), the lowest β results in an approximately 2.7-fold increase in offset distance (for AR = 2.0).Additionally, for the lowest ridge (AR = 1.0), an approximately 8-fold increase in height is predicted.Therefore, ensuring uniform wettability distribution on the ridges seems crucial for droplet vertical mobility.When the jumping droplets can be moved again upon re-approaching the surface, due to collisions with solid structures or other droplets, their horizontal movement can occur.The process of droplets re-approaching the surface closely resembles that of collisions with the surface at low speeds.The current prediction of the asymmetric wettability phenomena on macrostructures potentially provides vertical jumping performance in practical applications.The observed changes in droplet bounce and trajectory variation due to wettability highlight the characteristics of droplet coalescence jumping influenced by wettability.These features can be explained by the horizontal and vertical velocity components of the droplet.Figure 5 depicts the predicted horizontal average velocity (Ux) of the merged droplets over time.For analysis, specific droplet shapes at certain time intervals are included below the graph.Consequently, increased wettability asymmetry induces significant changes in Ux, particularly during contraction, when the droplet detaches from the ridge surface around 10-15 ms.At this moment, a significant difference in Ux is observed due to the disparity in droplet detachment from the left and right sides of the ridge.This difference in detachment maintains a significant disparity in Ux, even after 15 ms.For instance, compared to weak asymmetry with β = 0.84, a 7.2-fold and a 4.5-fold increase in Ux are predicted at β = 0.56 and 0.66, respectively.Moreover, the maximum Ux is predicted around 20-25 ms, coinciding with the maximum height of the jumping.These results suggest that adjusting the distribution of ridge wettability is necessary to control the horizontal velocity of detaching droplets.The observed changes in droplet bounce and trajectory variation due to wettability highlight the characteristics of droplet coalescence jumping influenced by wettability.These features can be explained by the horizontal and vertical velocity components of the droplet.Figure 5 depicts the predicted horizontal average velocity (U x ) of the merged droplets over time.For analysis, specific droplet shapes at certain time intervals are included below the graph.Consequently, increased wettability asymmetry induces significant changes in U x , particularly during contraction, when the droplet detaches from the ridge surface around 10-15 ms.At this moment, a significant difference in U x is observed due to the disparity in droplet detachment from the left and right sides of the ridge.This difference in detachment maintains a significant disparity in U x , even after 15 ms.For instance, compared to weak asymmetry with β = 0.84, a 7.2-fold and a 4.5-fold increase in U x are predicted at β = 0.56 and 0.66, respectively.Moreover, the maximum U x is predicted around 20-25 ms, coinciding with the maximum height of the jumping.These results suggest that adjusting the distribution of ridge wettability is necessary to control the horizontal velocity of detaching droplets.
The vertical velocity component is crucial in droplet coalescence jumping, as it directly influences self-cleaning and heat transfer performance [12].Figure 6 shows the temporal variation and maximum values of U y under wettability asymmetry and the ridge's aspect ratio.Figure 6a,b represent the flat surface without ridges using dashed lines and a star symbol, respectively.From the start of droplet coalescence at 0 s, U y gradually increases.During the contraction phase, where the droplet shape transitions from elliptical to contracted, U y sharply increases (5-10 ms).The maximum value of U y is reached fastest on surfaces without ridges, and next fastest at the lowest β.As the wettability ratio increases, the time taken to reach the maximum vertical velocity increases.U y starts decreasing around 10-15 ms, when the droplet detaches from the ridge surface.The rate of decrease in U y (i.e., the slope of the graph) is highest on surfaces without ridges and next highest at the lowest β.As the difference between θ R and θ L decreases, the rate of decrease in U y relatively diminishes.After approximately 15 ms, the droplet recontacts the ridge, and the velocity approaches zero.The vertical velocity component is crucial in droplet coalescence jumping, as it directly influences self-cleaning and heat transfer performance [12].Figure 6 shows the temporal variation and maximum values of Uy under wettability asymmetry and the ridge's aspect ratio.Figure 6a,b represent the flat surface without ridges using dashed lines and a star symbol, respectively.From the start of droplet coalescence at 0 s, Uy gradually increases.During the contraction phase, where the droplet shape transitions from elliptical to contracted, Uy sharply increases (5-10 ms).The maximum value of Uy is reached fastest on surfaces without ridges, and next fastest at the lowest β.As the wettability ratio increases, the time taken to reach the maximum vertical velocity increases.Uy starts decreasing around 10-15 ms, when the droplet detaches from the ridge surface.The rate of decrease in Uy (i.e., the slope of the graph) is highest on surfaces without ridges and next highest at the lowest β.As the difference between θR and θL decreases, the rate of decrease in Uy relatively diminishes.After approximately 15 ms, the droplet recontacts the ridge, and the velocity approaches zero.As seen in Figure 6a, the maximum value of Uy decreases as wettability asymmetry increases.This is evident in the graph depicted in Figure 6b, where the maximum value of Uy sharply increases from β = 0.75 (i.e., θR = 120°).In summary, when β is less than 0.75,  The vertical velocity component is crucial in droplet coalescence jumping, as it directly influences self-cleaning and heat transfer performance [12].Figure 6 shows the temporal variation and maximum values of Uy under wettability asymmetry and the ridge's aspect ratio.Figure 6a,b represent the flat surface without ridges using dashed lines and a star symbol, respectively.From the start of droplet coalescence at 0 s, Uy gradually increases.During the contraction phase, where the droplet shape transitions from elliptical to contracted, Uy sharply increases (5-10 ms).The maximum value of Uy is reached fastest on surfaces without ridges, and next fastest at the lowest β.As the wettability ratio increases, the time taken to reach the maximum vertical velocity increases.Uy starts decreasing around 10-15 ms, when the droplet detaches from the ridge surface.The rate of decrease in Uy (i.e., the slope of the graph) is highest on surfaces without ridges and next highest at the lowest β.As the difference between θR and θL decreases, the rate of decrease in Uy relatively diminishes.After approximately 15 ms, the droplet recontacts the ridge, and the velocity approaches zero.As seen in Figure 6a, the maximum value of Uy decreases as wettability asymmetry increases.This is evident in the graph depicted in Figure 6b, where the maximum value of Uy sharply increases from β = 0.75 (i.e., θR = 120°).In summary, when β is less than 0.75, 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 β = 0.75 (i.e., θ R = 120 • ).In summary, when β 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 β 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 β, there was little difference in the maximum values of U y based on AR, but there was a difference for high β.For example, at β = 0.66 (i.e., θ R = 105 • ), 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 β = 1.0 (i.e., θ R = 160 • ).From Figure 6b, 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 β values are added above and below the graph.Regardless of β, 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 β.For example, at the lowest β (i.e., θ R = 90 • ), a long spreading length occurred due to the low receding angle on the ridge (26-36 ms).In contrast, with intermediate values of β, a shorter spreading length occurs due to the high receding angle (26-36 ms).Thus, for β values greater than or equal to 0.75 (i.e., θ R = 120 • ), the similar variations of U z are observed.In summary, when β 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 β greater than 0.75, where the influence of ridge wettability difference is less significant, as discussed earlier.
under the changes in the ridge's aspect ratio.For AR = 1.0, 1.5, and 2.0, the maximum values of Uy were approximately 1.4, 1.5, and 1.6 times greater, respectively, compared to surfaces without ridges.Furthermore, for low β, there was little difference in the maximum values of Uy based on AR, but there was a difference for high β.For example, at β = 0.66 (i.e., θR = 105°), the maximum value of Uy showed a maximum difference of 1.03 times, while the maximum difference in Uy was 1.1 times at β = 1.0 (i.e., θR = 160°).From Figure 6b, 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 β values are added above and below the graph.Regardless of β, 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, Uz becomes dependent on the initial value of β.For example, at the lowest β (i.e., θR = 90°), a long spreading length occurred due to the low receding angle on the ridge (26-36 ms).In contrast, with intermediate values of β, a shorter spreading length occurs due to the high receding angle (26-36 ms).Thus, for β values greater than or equal to 0.75 (i.e., θR = 120°), the similar variations of Uz are observed.In summary, when β 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 β greater than 0.75, where the influence of ridge wettability difference is less significant, as discussed earlier.

Conclusions
Previous research into controlling droplet jumping through macrostructures has primarily emphasized the coalescence stage, with less exploration into how macrostructures affect the motion of a descending jumping droplet.Moreover, many studies focused mainly on the theoretical jumping direction perpendicular to the substrate, although the

Conclusions
Previous research into controlling droplet jumping through macrostructures has primarily emphasized the coalescence stage, with less exploration into how macrostructures affect the motion of a descending jumping droplet.Moreover, many studies focused mainly on the theoretical jumping direction perpendicular to the substrate, although the velocity in the horizontal direction must be involved in practical applications due to various scenarios.This study analyzed numerically the effect of the distribution in ridge structure's wettability on the performance of coalesced droplet jumping.The jumping dynamics were discussed for varying contact angle ratios and the aspect ratios of the ridge, which were the initial values for the current model.The influence of the ridge's wettability asymmetry on droplet coalescence jumping was numerically analyzed.In the current study, while varying the contact angle ratio and the geometric aspect ratio of the ridges, the maximum height of the jumping and horizontal offset distance were predicted.The results showed that at the lowest β, there was approximately a 30% decrease in the height compared to the case with homogenous symmetric wettability.Furthermore, at the lowest β, there was an approximately 2.7-fold increase in the offset distance compared to the case with weak

Figure 1 .
Figure 1.(a) Schematics of droplet coalescence and droplet jumping on a ridged surface with different apparent contact angles of θL and θR.The wettability of the flat surface is represented as the contact angle of θS (=θL = 160°).The ratio of wettability is defined as β = θR/θL in the current study.(b) Shape evolution in the coalescence process.The single droplet has a diameter of the droplet D, and the aspect ratio of the ridge is denoted by a ratio between the height and width (i.e., AR = H/W).

Figure 1 .
Figure 1.(a) Schematics of droplet coalescence and droplet jumping on a ridged surface with different apparent contact angles of θ L and θ R .The wettability of the flat surface is represented as the contact angle of θ S (=θ L = 160 • ).The ratio of wettability is defined as β = θ R /θ L in the current study.(b) Shape evolution in the coalescence process.The single droplet has a diameter of the droplet D, and the aspect ratio of the ridge is denoted by a ratio between the height and width (i.e., AR = H/W).

Figure 2 .
Figure 2. Model validation based on the jumping velocity on the flat surface.

Figure 2 .
Figure 2. Model validation based on the jumping velocity on the flat surface.

Figure 3 .
Figure 3. Jumping behavior of the coalesced droplet under two ARs of the ridges.(a-c) Droplet shapes on ridges with (a) β = 0.56, (b) 0.66, and (c) 0.75, under a constant AR of 1.0.(d-f) Droplet shapes on ridges with (a) β = 0.56, (b) 0.66, and (c) 0.75, under a constant AR of 2.0.Ux and Uy represent the averaged jumping velocities of the merged droplets in the x and y directions, respectively.Δx denotes the offset distance of the drop's center from the center of the ridge.

Figure 3 .
Figure 3. Jumping behavior of the coalesced droplet under two ARs of the ridges.(a-c) Droplet shapes on ridges with (a) β = 0.56, (b) 0.66, and (c) 0.75, under a constant AR of 1.0.(d-f) Droplet shapes on ridges with (a) β = 0.56, (b) 0.66, and (c) 0.75, under a constant AR of 2.0.U x and U y represent the averaged jumping velocities of the merged droplets in the x and y directions, respectively.∆x denotes the offset distance of the drop's center from the center of the ridge.

Figure 4 .
Figure 4. Jumping feature of the coalesced droplet: the maximal height (filled symbol; left axis) and the offset distance (open symbol; right axis), investigated as a function of θR, under three ARs of the ridges.

Figure 4 .
Figure 4. Jumping feature of the coalesced droplet: the maximal height (filled symbol; left axis) and the offset distance (open symbol; right axis), investigated as a function of θ R , under three ARs of the ridges.

10 Figure 5 .
Figure 5. Temporal evolution of averaged jumping velocities of the merged droplets in the x direction for varying β.The snapshots of the inset represent the morphological behavior of the coalesced droplet at several times.

Figure 6 .
Figure 6.(a) Temporal evolution of averaged jumping velocities of the merged droplets in the y direction for varying β.(b) Maximal Uy as a function of θR.

Figure 5 .
Figure 5. Temporal evolution of averaged jumping velocities of the merged droplets in the x direction for varying β.The snapshots of the inset represent the morphological behavior of the coalesced droplet at several times.

Figure 5 .
Figure 5. Temporal evolution of averaged jumping velocities of the merged droplets in the x direction for varying β.The snapshots of the inset represent the morphological behavior of the coalesced droplet at several times.

Figure 6 .
Figure 6.(a) Temporal evolution of averaged jumping velocities of the merged droplets in the y direction for varying β.(b) Maximal Uy as a function of θR.

Figure 6 .
Figure 6.(a) Temporal evolution of averaged jumping velocities of the merged droplets in the y direction for varying β.(b) Maximal U y as a function of θ R .

Figure 7 .
Figure 7. Temporal evolution of averaged jumping velocities of the merged droplets in the z direction for varying β.The snapshots of the insets represent the morphological behavior of the coalesced droplet in the yz plane at three distinct times.

Figure 7 .
Figure 7. Temporal evolution of averaged jumping velocities of the merged droplets in the z direction for varying β.The snapshots of the insets represent the morphological behavior of the coalesced droplet in the yz plane at three distinct times.