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MIT Open Access Articles How Coalescing Droplets Jump

Surface engineering at the nanoscale is a rapidly developing field that promises to impact a range of applications including energy production, water desalination, self-cleaning and anti-icing surfaces, thermal management of electronics, microfluidic platforms, and environmental pollution control. As the area advances, more detailed insights of dynamic wetting interactions on these surfaces are needed. In particular, the coalescence of two or more droplets on ultra-low adhesion surfaces leads to droplet jumping. Here we show, through detailed measurements of jumping droplets during water condensation coupled with numerical simulations of binary droplet coalescence, that this process is fundamentally inefficient with only a small fraction of the available excess surface energy ( ≲ 6%) convertible into translational kinetic energy. These findings clarify the role of internal fluid dynamics during the jumping droplet coalescence process and underpin the development of systems that can harness jumping droplets for a wide range of applications.

When two or more droplets coalesce on a ultra low adhesion nanostructured surface, the resulting droplet can jump away from the surface in a process that has been termed "coalescence-induced droplet jumping". [1][2][3][4][5][6] Droplet jumping has recently received significant attention due its fundamental relevance in understanding dynamic droplet processes, [7][8][9][10][11][12][13][14][15][16][17][18][19] evolved surface structures found in nature, 20 and surfacecharge separation. 3 Jumping droplets have also been used to enhance performance in a variety of applications including condensation heat transfer, 2,21-23 self-cleaning surfaces, 24 thermal diodes, 25,26 antiicing surfaces, 6,27,28 and energy harvesting. 29 Beyond these demonstrated applications, droplet jumping also has potential for use in thermally-driven water desalination systems, 30 in combination with acoustophoretic droplet manipulation to realize novel microfluidic platforms 31 and in mitigating environmentally polluting gas and particle emissions by providing a convenient waste-heat-driven source of microscopic water droplets for wet scrubbing systems. 32 At the basic level, droplet jumping is a process governed by the conversion of excess surface energy into kinetic energy when two or more droplets coalesce 7,33 and involves a symmetry-breaking surface. 8 For the simplest case of two, equally-sized inviscid spherical droplets coalescing on a surface with no adhesion, an energy-balance gives a characteristic jumping speed that follows an inertial-capillary where R, γ and ρ are the initial radii, surface tension and density of the droplets, respectively. Previous water condensation experiments on a hierarchical superhydrophobic surface have shown an approximately constant scaled jumping speed of for jumping droplets with diameters larger than ≈100 μm. For smaller diameters, jumping speeds rapidly decreased with no droplet jumping observed below diameters of ≈ 20 μm. 7 Several subsequent studies have attempted to explain these observations by adding terms to the energy-balance approach to account for viscous dissipation, surface adhesion effects and participating velocity components, 9,13,16 but have demonstrated limited success. While a recent numerical simulation replicated experimental observations, 1 this study was limited to a single droplet diameter. Despite the significant research efforts, there continues to be a lack of quantitative experimental data characterizing droplet jumping and the precise mechanism governing droplet jumping remains not well-understood.
In this work, we studied the process of coalescence-induced droplet jumping over a wide parameter space using a combination of water condensation experiments on nanostructured superhydrophobic surfaces and detailed numerical simulations. In order to experimentally isolate the fundamental energy conversion process associated with droplet jumping, we fabricated surfaces demonstrating apparent contact angles approaching 180° and almost zero contact angle hysteresis, thus mitigating the influence of surface adhesion. 4,11 Using high-speed imaging, we measured droplet-jumping speeds normal to the condensing surface as high as m/s for jumping droplet diameters as small as ≈ 10 μm. However, while the measured velocities were up to approximately six times larger than the highest velocities previously reported, 7,9 they were still approximately one-fifth of the speed indicated by Eq. (1).
Meanwhile, using 2D axisymmetric numerical simulations, we obtained detailed information of the internal flow momentum generated during droplet coalescence. By recognizing that only unbalanced momentum components can contribute to droplet translational motion, a fact neglected in the derivation of Eq. (1), we found that coalescence-induced droplet jumping is an inherently inefficient process with only a small fraction of the available excess surface energy ΔE (≲ 6%) being converted to useful internal flow momentum to provide the jumping droplet with translational kinetic energy E j , i.e., η j = E j /ΔE ≲ 0.06.Contrary to previous conclusions drawn from theoretical estimates, 9,13 we found that internal viscous dissipation plays a limited role in the jumping process for the experimentally accessible low Oh number regime studied (

√ ⁄
, where µ is the droplet dynamic viscosity). 34 With our developed understanding, we determined a dimensionless droplet-size-dependent correction to Eq. (1) that demonstrated good agreement with our experimental data. Our results elucidate the coupled nature of surface-mediated symmetry breaking and the internal flow momentum on jumping droplet dynamics. Furthermore, our mechanistic framework provides a starting point for more detailed analysis involving complicating factors such as finite surface adhesion, unequal coalescing droplet geometries and complex surface architectures.

RESULTS AND DISCUSSION
Experiments. To minimize the influence of the underlying substrate on the droplet jumping process, we prepared fluoropolymer-coated carbon nanotube (CNT)-based superhydrophobic surfaces with a structure spacing length scale of l ~ 100 nm that was much smaller than the typical jumping droplet radius the droplet radius-dependent advancing angle, ( ), indicated that all measured droplet jumping events were in a constant contact angle growth mode. 4 We also fabricated and tested fluoropolymercoated CuO nanostructures, which we have previously shown to demonstrate good jumping droplet behavior, 2-4 with macroscopic advancing contact angle and contact angle hysteresis behavior similar to that of the functionalized CNT surfaces (see Methods). Since the characteristic roughness spacing length scales of the surfaces (l ≈ 0.1 -1 µm) were significantly smaller than the smallest droplet jumping diameter measured (≈ 10 μm), we were able access a droplet growth regime well above the flooding limit with droplets growing in a constant apparent contact angle mode. 2,4,5 Since the work of adhesion scales as ( ), the contact angle measurements suggested that the effect of surface adhesion on the droplet jumping process could be neglected.
Condensation experiments were carried out under humid atmospheric conditions using a twocamera arrangement and in a pressure-regulated environmental chamber using a single-camera arrangement (see Methods, supporting information S2 -S4). In the former experimental set-up, an overhead CCD camera was used to capture images of the droplets prior to coalescence to provide the initial conditions, i.e., the droplet radii and number of droplets involved in the jumping event. Knowledge of these initial conditions allowed us to isolate binary droplet jumping events for droplets of equal radii with near surface-normal trajectories that maximized the jumping velocity. 7,11 The latter experimental setup allowed us to study droplet jumping at condensing surface temperatures higher than the laboratory ambient conditions and under varying gas densities to rule out the possibility that external drag plays a significant role during the droplet coalescence process before jumping, as assumed in our numerical simulations. In both experimental set-ups, a high-speed camera was used to capture the out-of-plane jumping droplet trajectory. Figure 2a shows representative time-lapse images capturing a jumping droplet following the coalescence of two equally sized droplets (R ≈ 19.5 µm) obtained using the two-camera arrangement. The center of the streak was taken as the average y position of the jumping droplet in each frame relative to the initial location of the droplets on the condensing surface (y = 0), while the average velocity of the jumping droplet was found by measuring the length of the streak and dividing this by the shutter speed of the camera. These two measures together defined the experimental trajectory of the jumping droplets. To obtain an accurate estimate of the jumping velocity associated with the converted excess surface energy, the theoretical droplet trajectory was calculated (see supporting information S5).
In Figure 2b the theoretical trajectory is shown fitted to the experimental data to determine the droplet jumping velocity as it departed the surface. Based on the fitted trajectory, we determined a jumping velocity of v j = 0.41±0.01 m/s (inset of Figure 2b), which was 0.21× the characteristic jumping speed given by the capillary-inertial scaling, √ ( ) m/s (Figure 2c). This result was generally repeatable over a number of experiments for a wide range of droplet sizes and is consistent with the previously identified size-independent region. 7 In order to better understand the reason for jumping velocity magnitudes significantly smaller than those given by Eq. (1) in view of the ultra-low surface adhesion characteristic of the condensing surfaces in our study, we considered the nature of the droplet coalescence process and performed numerical simulations to determine the jumping velocity in the absence of surface adhesion effects. Analysis of the simulations improved our understanding of how the excess surface energy was transferred to the translational kinetic energy of the jumping droplet by considering the details of the internal flow developed during the coalescence process.

Theory & Simulations.
A schematic of the coalescence process for two, equally-sized, condensed droplets defined by a radius, R, resting on a surface characterized by and is shown in Figure 3a. Soon after the interfaces of the two droplets make contact, a liquid bridge develops that is accelerated radially from the point of contact due to the curvature difference (in the longitudinal direction) between the bridge radius (1/r b ) and the droplets (1/R). Initially, viscous forces control the bridge flow dynamics. However, inertia begins to dominate the flow dynamics when the bridge radius, r b , exceeds the critical bridge radius ( ) ⁄ , 34,35 where is a constant previously determined from simulation 36 and experimental measurement 35,37,38 to have a value ranging between 1.39 and 1.62.
For water at ambient conditions and based on the range of C, r c ≈ 165 -311 nm. Considering the inertial regime, scaling of the bridge flow yields a time-varying bridge radius ( ) formed during coalescence given by 36,39 √ where √ is the inertial-based time scale of the coalescence process. Differentiating Eq. (2) with respect to time gives Noting the definition of , Eq. (3) shows that √ ( ); corresponding to the scaling given by Eq.
(1). This equivalence between the energy state scaling and the scaling behavior of the radial bridging flow highlights the link between the jumping droplet speed and the excess surface energy being transferred to the internal flow during the coalescence process. Specifically, by considering the symmetry To determine the y-component of momentum, p y , generated during the coalescence process, 3D axisymmetric numerical simulations were performed using a finite-element code specifically designed to accurately model unsteady micro-capillary flows in which both viscous and inertial forces are present in the liquid. This code has previously been shown to accurately capture the coalescence process 40 , and was originally developed for dynamic wetting flows 41 (see Methods).
In Figure 3b The change in droplet shape evolution and internal flow field was reflected in the cumulative viscous dissipation of the flow during the coalescence process, ( ), normalized by the available excess surface energy budget, , shown in Figure 5a. As expected, an increasing proportion of the excess surface energy budget was consumed by viscous losses with increasing Oh. Figure 5b shows the normalized viscous dissipation evaluated from our simulations at r b /R = 2 1/3 as a function of Oh and previous estimations. 9,13 The dissipation estimates are seen to either under predict or over predict the viscous losses compared to our simulations. More importantly though, both estimates predict a linear increase in viscous dissipation with increasing Oh, which is due to the fact that only a single inertial timescale was used to characterize droplet coalescence for all Oh. 9,13 The simulation results, however, show a decreasing rate of viscous dissipation growth with increasing Oh. This can be understood by considering the limiting case of very large Oh where the normalized viscous dissipation must asymptote to unity when evaluated up to r b /R = 2 1/3 resulting in a smooth transition to a sphere during coalescence. 40 As we show next, simply adding a viscous dissipation term into the energy-balance analysis cannot capture the subtle effect of viscosity on the development of the internal flow momentum component normal to the condensing surface with varying Oh.
To extract the y-component (surface-normal) of momentum from the simulations, we considered axisymmetric motion about the z-axis of a cylindrical coordinate system (r, z, β) for z > 0 due to the symmetry between the two coalescing drops. The velocity component perpendicular to the surface is u y = u r sin β, where u r is the velocity in the radial direction and β is the azimuthal coordinate.
Accordingly, the momentum perpendicular to the surface in the upper half of the two coalescing droplets is ∬ (4) The first integral in Eq. (4) gives the y-component of the radial velocity, while the second integral operates over the r-z plane. The total momentum given by Eq. (4) can be further broken down into positive, , and negative, , contributions to p y depending on whether the associated radial flow is moving away from or towards the z-axis. we have defined the excess momentum as the maximum momentum at . This provides consistency for the situation where the peak in momentum may occur at r b /R < 2 1/3 for Oh numbers larger than those simulated here and in the viscous limit ( ⁄ ) where we expect all momentum in the system to tend to zero as r b /R→ 2 1/3 .
As Oh was increased in the simulations, the momentum growth rate in the bridge phase slowed down and the characteristic kink in momentum growth beyond r b /R = 1 observed for smaller Oh (= 0.012) was smoothed out. This corresponded with a reduction in the excess momentum obtained at r b /R ≈ 2 1/3 and a shift to smaller values of r b /R where the maximum momentum occurred. In the region, the minimum found at r b /R ≈ 1 for low Oh shifted to smaller values of r b /R and decreased in magnitude. The change in momentum generated with increasing Oh is consistent with the coalescence dynamics moving into the transition region between the inertially and viscously-dominated coalescence regimes as To complete the picture of flow momentum available for droplet jumping, we considered the lower region of the coalescing droplets (y < 0) where the sign of the momentum components are reversed.
Distinct from the upper region (y > 0), the bridging flow interacts with the condensing surface. At the moment when the bridge impacts the surface ( ), the flow in the negative y-direction has a momentum | . The bridge impact develops a region of high pressure as the dynamic pressure of the flow is converted to static pressure 1 , imposing a force on the liquid and the interface. For simplicity, we assumed that the flow momentum induced by this pressure is equal, but opposite, to the momentum in the negative y-direction just prior to impact, | . We also considered the contribution to the ycomponent of momentum of the internal flow in the flow neighboring the bridge, | . While detailed analysis of 3D simulations are required to confirm these assumptions, we can bound the momentum available for jumping by | ( ) and | ( ) . Verification that the experimental data lay between these bounds will support our developed approach, with the specific expression derived below as a starting point for a more complex description of the full three-dimensional dynamics.
Next, we balanced the y-momentum of the coalescing droplet system by summing the three identified y-components of momentum generated by the internal flow and compared them to the total momentum of the jumping droplet, In Figure  with the remainder of the contributing momentum originating from the bridge impact region. 1 A fit to the numerical data resulted in the following expression for the jumping velocity (see supporting information S9 for fit statistics). Towards the inviscid limit (Oh→0), the jumping velocity is . Calculating the kinetic energy of the jumping droplet based on this velocity and comparing to the available excess surface energy in the system (ΔE), only ≈ 6% of the excess surface energy budget was converted to the translational kinetic energy of the jumping droplet (E j ). As Oh increased, the energy conversion efficiency (η j = E j /ΔE) decreased further reaching η j ≈ 1.8% for Oh = 0.12 (see supporting information S10). Interestingly, the conversion efficiencies of coalescence-induced droplet jumping are characteristically smaller than that of a solid disc transitioning to a sphere as it melts, where η j ≈ 20% have been found. 17 This characteristically larger η j further highlights the importance of the internal flow momentum developed during the capillary-driven jumping process, which is intimately linked to the initial geometry of the system. Next, we compared the results of the numerical simulations to the experimentally measured jumping velocities.
Comparison to experiment. In Figure 6a & b, our experimentally measured droplet jumping velocities and the predictions of our numerical simulations (Eq. (6) & Figure 5d) are plotted as a function of the jumping droplet diameter, 2R j . Also shown in Figure 6a is a previously obtained experimental data set. 7 For experiments performed under low pressure in the environmental chamber (Figure 6b), where vapor density was reduced, but viscosity remained approximately unchanged ( ), a simple linear fit was applied to the measured droplet trajectory to determine the jumping speed due to the potential for molecular slip effects on the drag force. Unlike the two-camera arrangement (Figure 6a), the initial conditions before coalescence were difficult to ascertain using a single-camera arrangement (Figure 6b), especially with decreasing droplet radii. However, we classified jumping events based on the angular deviation of the droplet trajectories from the surface normal, rejecting jumping events deviating more than 5° from the normal. Also rejected from the analysis were droplet jumping events triggered by incoming droplets (see supporting information S1 for movie details). 7,9 In general, data obtained from the one-camera arrangement showed more scatter due to our inability to completely filter coalescing droplets with non-equal radii and jumping events triggered by more than two droplets.
We demonstrated excellent agreement between our numerical predictions and experimental data with the latter well-bounded by the limits placed on the jumping speed by | ( ) and | ( ) , giving us confidence in our interpretation of the physics. In addition, we found that the data measured using the two-camera arrangement, which removes the ambiguity surrounding the initial coalescence conditions associated with the one-camera arrangement, followed the numerical predictions exceptionally well. Indeed, for the experiment presented in Figure 2 using the two-camera arrangement, the numerical simulations predicted , which was within 10% of the experimental jumping velocity of . For the one-camera arrangement, we measured jumping speeds as high as 1.4 m/s, approximately 6× larger than previously reported 7 , due to a combination of the elevated temperature of the environmental chamber with respect to ambient and the smaller jumping droplet diameters allowed by our nanostructured surfaces. Experiments performed in the environmental chamber also confirmed that the surrounding gas pressure played no role on the jumping speed (see supporting information S11).
Interestingly, previous experimental data are well-captured by our numerical results without the need to introduce a surface adhesion term for 2R j > 100 µm, 7 indicating that droplets in this size range had reached a constant contact angle growth wetting state characterized by macroscopic contact angle measurements on this surface, 42 However, for droplets smaller than 100 µm, there was an increasing discrepancy in comparison to our numerical results. While it has previously been suggested that the observed peak and then decreasing jumping speed with decreasing droplet size arises as a result of internal viscous dissipation, 7,9,13,43 the jumping speeds fall off faster than our experimental and numerical data in this range of Oh (Figure 6c). This observation implicates a surface interaction mechanism rather than a fundamental hydrodynamic limitation and highlights that the details of the evolving droplet morphology need to be carefully considered when interpreting experimental data of droplet jumping. Indeed, complex wetting interactions associated with condensed droplet growth on hierarchical surfaces have been shown to influence the ability of coalescing droplets to jump, 10 a fact not captured in our present analysis. However, we believe that the mechanistic framework we have developed to understand droplet jumping can be readily extended to deal with these non-idealities.

CONCLUSIONS
In summary, we demonstrated experimentally and numerically that coalescence-induced droplet jumping on ultra-low adhesion surfaces is fundamentally inefficient. Through the use of detailed

Methods
Surface synthesis. Carbon nanotubes were grown by chemical vapor deposition (CVD) 44 . Silicon growth substrates were prepared by sequentially depositing a 20 nm thick Al 2 O 3 diffusion barrier and a 5 nm thick film of Fe catalyst layer using electron-beam deposition. Growth was performed in a 2.54 cm quartz furnace tube. Following a 15 min purge in a H 2 /He atmosphere, the growth substrate was annealed by ramping the furnace temperature to 750 °C followed by a 3 minute anneal at temperature, while maintaining a flow of H 2 and He at 400 sccm and 100 sccm, respectively. CNT growth was then initiated by flowing C 2 H 4 at 200 sccm. The flow of C 2 H 4 was stopped after a period of 1 minute. The thermallygrown CNT had a typical outer diameter of d ≈ 7 nm. Due to the short growth time (~5 min.) the CNT did not form a well-aligned forest, but rather a tangled turf.
To create the CuO nanostructures, commercially available oxygen-free Cu tabs (20 mm x 2 mm x 1 mm) were used (99.9 % purity), as the test samples for the experiments. Each Cu tab was cleaned in an ultrasonic bath with acetone for 10 minutes and rinsed with ethanol, isopropyl alcohol and de-ionized (DI) water. The tabs were then dipped into a 2.0 M hydrochloric acid solution for 10 minutes to remove the native oxide film on the surface, then triple-rinsed with DI water and dried with clean nitrogen gas. Nanostructured CuO films were formed by immersing the cleaned tabs into a hot (96 ± 3 °C) alkaline solution composed of NaClO 2 , NaOH, Na 3 PO 4 •12H 2 O, and DI water (3.75 : 5 : 10 : 100 wt.%) 45 . During the oxidation process, a thin (≈300 nm) Cu 2 O layer was formed that then re-oxidized to form sharp, knifelike CuO oxide structures with heights of h ≈ 1 μm, solid fraction φ ≈ 0.023 and roughness factor r ≈ 10.
To functionalize the surfaces, a proprietary fluorinated polymer was deposited using plasma enhanced vapor deposition (P2i). The process occurs under low pressure within a vacuum chamber at room temperature. The coating is introduced as a vapor and ionized. This process allows for the development of a highly conformal (≈ 30 nm thick) polymer layer, which forms a covalent bond with the surface, making it extremely durable. Goniometric measurements (Kyowa Interface Science Ltd., MCA-3) of ≈100 nL droplets on a smooth P2i coated silicon wafer surface showed advancing and receding contact angles of θ a = 124.3° ± 3.1° and θ r = 112.6° ± 2.8°, respectively. Using the values of the advancing angles on the rough and smooth P2i surfaces, we estimated the effective solid fraction of both the CNT and CuO surfaces to be ( ) ( ) ⁄ .

Jumping droplet experiments.
Condensation experiments carried out under humid atmospheric conditions used two cameras that were orthogonally aligned and focused at the same point in space (MCA-3, Kyowa Interface Ltd.) (see Supporting information S2). An overhead CCD camera was used to capture images of the droplets prior to coalescence to provide the initial conditions. The out-of-plane trajectory of the jumping droplets was captured using a high-speed camera (Photron, Fastcam SA2) at frame capture rates of 5, 7.5 and 18 kHz corresponding to shutter speeds of 185, 133 and 56 μs, respectively. Back illumination was provided by a fiber light positioned behind the sample with respect to the high-speed camera. The experiment was initiated by first cooling the surface to a temperature of T w = 5 °C, corresponding to a local saturation pressure of p w = 0.9 kPa on a cold-stage under a dry N 2 stream. The dry N 2 stream was then shut-off to initiate condensation from the ambient atmosphere with a relative humidity of ≈ 55% and a dry-bulb temperature of ≈ 20 °C corresponding to a vapor saturation pressure of p v ≈ 1.28 kPa and a supersaturation of S = p v /p w ≈ 1.42.
Condensation experiments were also carried out under saturated conditions in a pressure regulated environmental chamber using a single-camera arrangement (see supporting information S3 & S4). The out-of-plane trajectory of the jumping droplets was captured using a high-speed camera (Vision Research, Phantom v7.1) at frame capture rates of 7.2, 10, and 20 kHz corresponding to shutter speeds of 139, 100, and 50 μs, respectively. The camera was mounted outside of the environmental chamber and fitted with an extended macro lens assembly. Illumination was supplied by light emitting diodes installed inside the chamber and providing back lighting to the sample. The sample tabs were mounted to a flattened copper tube connected to an external cooling loop and was maintained at a temperature of T w = 26 °C (p w = 3.33 kPa). The water vapor supply was vigorously boiled before the experiments to remove non-condensable gases. The experiment was initiated by first evacuating the environmental chamber to medium-vacuum levels (=0.5 ± 0.025 Pa). Water vapor was then introduced into the environmental chamber via a bellows valve set to maintain the chamber pressure at p v = 3.6 ± 0.175 kPa corresponding to a supersaturation of S ≈ 1.05.

Numerical modeling.
A purpose-built finite-element-based computational code was used to capture the dynamics of the coalescence process. The bulk flow of the liquid is governed by the incompressible Navier-Stokes equations whilst the boundary conditions are given by either the conventional model or by the interface formation model, which generalizes the usual boundary conditions for flows in which interfaces are formed or destroyed 46 . In the range of parameter space studied in this paper, the differences in the predictions of the two models for the global dynamics of the drops was small; however, larger differences may be expected when considering smaller drops.
A full description of the models used, benchmark simulations confirming the codes accuracy and a comparison to recent experimental data are provided in ref. 40 Furthermore, a step-by-step user-friendly guide to the development of the code can be found in ref. 47 and the extension to interface formation dynamics is in ref. 41 . Therefore, here we only briefly recapitulate the main details.
The code uses an arbitrary Lagrangian Eulerian approach, so that the free surface dynamics are captured with high accuracy. The mesh is based on the bipolar coordinate system, and is graded so that exceptionally small elements can be placed near the bridge front; consequently, in contrast to many previous works, both local and global physical scales of the coalescence process are properly resolved. Triangular finite elements of V6P3 type are used and the result of our spatial discretization is a system of non-linear differential algebraic equations of index two which are solved using the secondorder backward differentiation formula using a time step which automatically adapts during a simulation to capture the appropriate temporal scale at that instant.

Supporting information available
One video showing incoming droplet triggered jumping, as well as further information on the jumpingdroplet experiments, data collection methodology, and supplementary analysis. This material is available free of charge via the Internet at http://pubs.acs.org.

Competing financial interests
The authors declare no competing financial interests.   /R on a superhydrophobic surface characterized by . A cylindrical coordinate system (r, z, β) is defined with the origin at the point where the two droplets meet. The z-coordinate is parallel to the surface, while the radial r-coordinate is resolved into yand x-coordinates at β = π/2 (pointed to the top of the page) and β = 0 (pointed into the page), respectively. (b-f) Numerically simulated flow field of the coalescence process for Oh = 0.012 (R = 100 μm) and 1/4 < r b /R ≤ 2 1/3 in the region of the droplet indicated by the dot-dashed box in (a). Simulated fluid properties: γ = 72x10 -3 N/m, ρ = 1x10 3 kg/m 3 , μ = 1x10 -3 Pa.s. The contours depict the radial velocity component with red and blue coloring indicating flow away and towards the z-axis, respectively. The grey lines depict the streamlines of the flow. At the earlier stage of coalescence (b), the generated radial flow is restricted primarily to the local bridge region (z/R = 0). This flow is fed by a region of negative radial flow that can be clearly seen in (c) due to the differences in the local curvature along the droplet interface. In (d), the positive and negative radial flow regions expand as the capillary wave moves away from the bridge region. In (e), the negative radial flow feeding the positive radial flow centered around z/R = 0 has arrived at the edge of the droplet. (f) The negative radial flow has vanished and the entire upper portion of the droplet has a positive radial velocity driven by the curvature difference between the interface at z/R ≈ 1.75 (r/R ≈ 0) and z/R ≈ 0 (r/R ≈ 2 1/3 ).

S3. CONDENSATION CHAMBER SETUP
The custom environmental chamber used for this work (Kurt J. Lesker) consists of a stainless steel frame with a door (sealed with a rubber gasket), two viewing windows, and apertures for various components.
Resistive heater lines were wrapped around the exterior of the chamber walls to prevent condensation at the inside walls and then insulated on the exterior walls. The output power of the resistive heater lines was controlled by a voltage regulator (Variac). Two insulated stainless steel water flow lines (Swagelok) were fed into the chamber via a KF flange port (Kurt J. Lesker) to supply cooling water to the chamber from a large capacity chiller (System III, Neslab).
A secondary stainless steel tube line was fed into the chamber via a KF adapter port that served as the flow line for the incoming water vapor supplied from a heated steel water reservoir. The vapor line was wrapped with a rope heater (60 W, Omega) and controlled by a power supply (Agilent). The vapor reservoir was wrapped with another independently-controlled rope heater (120 W, Omega) and insulated to limit heat losses to the environment. The access tubes were welded to the vapor reservoir, each with independently-controlled valves. The first valve (Diaphragm Type, Swagelok), connecting the bottom of the reservoir to the ambient, was used to fill the reservoir with water. The second valve (BK-60, Swagelok), connecting the top of the reservoir to the inside of the chamber, provided a path for vapor inflow. K-type thermocouples were located along the length of the water vapor reservoir to monitor temperature.
A bellows valve (Kurt J. Lesker) was attached to the chamber to serve as a leak port between the ambient and inside of the chamber. In order to monitor temperatures within the chamber, K-type thermocouple bundles were connected through the chamber apertures via a thermocouple feed through (Kurt J. Lesker).
To provide electrical connections inside the chamber for LED lighting and electric field generation,  Figure S2a.
Images of the front and rear of the experimental setup are shown in Figure S2b and c, respectively.

S4. SINGLE CAMERA CONDENSATION PROCEDURE
For each experimental trial, a set of strict procedures were followed to ensure consistency throughout the experiments. The first step of the process was to turn on the voltage regulator to heat up the environmental chamber walls, which prevented condensation on the chamber walls. Simultaneously, the water vapor reservoir was filled with approximately 3.5 liters of DI water (99% full) using a syringe through the vapor release valve. After opening the vapor inflow valve and closing the vapor release valve, the rope heater around the water vapor reservoir was turned on with the heater controller set to maximum output (120 W). Then the rope heater connected to the vapor inflow valve was turned on. The temperature of the water reservoir was monitored with the installed thermocouples; the temperature at the top of the reservoir was higher than that of the middle/bottom of the reservoir due to the water thermal-mass present at the middle/bottom section. Hence, we ensured that the regions of the water reservoir of higher thermal capacity were brought to a sufficiently high temperature for boiling. During the boiling process, aluminum foil was placed on the bottom surface of the inner chamber to collect any of the water leaving the vapor inflow line. Once boiling was achieved and all thermocouples on the reservoir were > 95˚C for at least 10 minutes, the vapor inflow valve was closed. The excess water that spilled inside the chamber during de-gassing of the reservoir was removed.
In order to install the samples onto the rig (Figure S3), the Swagelok female adapters at the ends of the tube samples were connected to the 90 degree male elbow connecters on the rig. Before installing the entire sample setup in the chamber, all adapters/connecters were tightened to ensure that there were no leaks that could affect vacuum performance. The setup was then placed on top of the steel supports and the bellows tubes (for the water inflow/outflow) were connected to the water lines. Then the insulating wet bulb wick was placed near the sample and in contact with the bottom surface of the chamber.
The next step was to begin the vacuum pump-down procedure. Initially, the liquid nitrogen cold trap was filled to about half capacity. The ambient exposed valves connecting the chamber and the vacuum pump were both closed and the valve connected to the liquid nitrogen cold trap was opened. The vacuum pump was then turned on, initiating the pump-down process. The pressure inside the chamber was monitored during the pump-down process. This process took approximately one hour in order to achieve the target vacuum conditions (0.5 Pa < P < 1 Pa). The experimental operating pressure of non-condensable was set to be a maximum of 0.25% of the operating pressure. Non-condensable gas content of above 0.5% (pressure) was shown to significantly degrade performance during dropwise condensation. In our experiments, extreme care was taken to properly de-gas the vacuum chamber and water vapor reservoir prior to experimental testing. In addition, the chamber leak rate was characterized prior to each run in order to estimate the maximum time available for acquiring high fidelity data with non-condensable content of less than 0.25%.
The setup of the water flow-loop is described as follows. The Neslab water pump reservoir was filled and turned on to a flow rate of 5 L/min (0 < ∆T LMTD < 15 K). The flow rate was monitored with the flow meter integrated in the inflow water line. In order to bring the chilled water into the flow loop and to the tube sample, the external chilled water lines were opened.
Prior to beginning experiments, the high-speed camera was turned on for visual recording of the sample during condensation. Afterwards, the rope heater around the water reservoir was turned off and the vapor inflow valve was slowly turned open until the operating pressure was reached. Steady state conditions were typically reached after 2 minutes of full operation.
The droplet ejection process was captured under saturated conditions inside the environmental chamber using a single-camera setup 3 . The out-of-plane trajectory of the ejected droplets was captured using a high-speed camera (Phantom v7.1, Vision Research) at frame capture rates of 7200, 10000 and 20000 Hz corresponding to shutter speeds of 139 µs, 100 µs and 50 µs, respectively. The camera was mounted outside the environmental chamber and fitted with an extended macro lens assembly ( Figure S3a). The lens assembly consisted of a fully extended 5X optical zoom macro lens (MP-E 65 mm, Canon), connected in series with 3 separate 68 mm extension tubes (Auto Extension Tube Set DG, Kenko). The DG extension tubes enable the lens to focus closer than its normal set minimum focal distance, which has the effect of magnifying the image (making it appear larger in the viewfinder). The DG extension tubes have no optics. They were mounted in between the camera body and lens to create more distance between the lens and film plane. By moving the lens further away from the film or CCD sensor in the camera, the lens was allowed to focus much closer than normal. Illumination was supplied by light emitting diodes installed inside the chamber and providing back lighting to the sample.
The experiment was initiated by first evacuating the environmental chamber to medium-vacuum levels (= 0.5 ± 0.025 Pa). The sample was mounted to a flattened copper tube connected to an external cooling loop and was maintained at a temperature of T w = 26 °C (p w = 3.33 kPa) ( Figure S3b,c). The water vapor supply was vigorously boiled before the experiments to remove non-condensable gases. Water vapor was introduced into the environmental chamber via a metering valve set to maintain the chamber pressure at p v = 3.6 ± 0.175 kPa. Figure S4 shows a typical high speed time-lapse obtained during jumping-droplet condensation.

S5. MEASURING JUMPING VELOCITY FROM FITTED DROPLET TRAJECTORY & ESTIMATING ENTRAINING VAPOR FLOW FOR AMBIENT EXPERIMENT
The general form of the vertical jumping droplet trajectory can be found by considering the following force balance where R j , ݉ = ߩሺ4/3ሻߨܴ ଷ and g are the jumping droplet radius, jumping droplet mass and the gravitational constant (g = 9.81 m/s 2 ), respectively. The inertial force on the left-hand side of Eq. S1 is balanced by the gravitational force and a drag force characterized by a drag coefficient C d that dissipates the kinetic energy of the jumping droplet traveling at a velocity v(y). The drag term on the right-hand side of Eq. S1 also accounts for an entraining vapor flow with a velocity v f associated with vapor moving towards the condensing surface to satisfy mass conservation 1,2 . The magnitude of this entrainment flow velocity for the two-camera arrangement at ambient conditions was estimated from experimentally observed droplet growth rates and found to be negligible for the range of condensation conditions explored here, i.e., ห‫ݒ‬ ห = ‫01ݔ2.1‬ ିଷ m/s ≪ ‫,|ݒ|‬ (see Figure S5). For the one-camera arrangement in the environmental chamber, the heat flux could be directly measured and was found to be q" < 0.1W/cm 2 for all measurements, corresponding to ห‫ݒ‬ ห = ‫"ݍ‬ ൫ߩ ௩ ℎ ൯ ⁄ < 0.016 m/s; at least an order of magnitude smaller than typical jumping droplet speeds. Thus, ‫ݒ‬ was approximated as zero in Eq. S1. A possible additional drag force due to viscous interactions in the gap opening up between the departing droplet and the surface discussed in ref. 3 was neglected due to the large Knudsen number expected in the gap between the departing droplet and the condensing surface. Electrostatic effects due to the charged nature of the jumping droplets on the force balance were also neglected on the basis that no external electric fields were imposed on the system 4 .
Using the characteristic speed given by Eq. 1, the Reynolds (Re) number under ambient conditions (twocamera arrangement) was calculated to be as large as 10 over the range of experimental droplet diameters (10 ≲ 2R j < 200 µm), indicating the possibility of a non-linear drag regime, ܴ݁ = ߩ ܷ2ܴ ߤ ⁄ ≲ 10, where ߩ and ߤ are the density and dynamic viscosity of the surrounding gas, respectively. Thus, the drag coefficient in Eq. S1 was calculated from an expression accounting for a Re-dependent drag coefficient given by 5 , where ߣ = ߤ/ߤ is the dynamic viscosity ratio between the liquid and gas phases, under the assumption of a spherical droplet shape during flight. For R j ~10 µm, a liquid density ρ ≈1000 kg/m 3 , a gas density range ρ g ≈ 0.02 -1.2 kg/m 3 , a gas dynamic viscosity µ g ≈ 9.8 µPa·s and γ ≈ 72 mN/m, this assumption is justified given that the Bond number ‫ܤ‬ = ൫ߩܴ݃ ଶ ൯ ߛ ⁄ ≪ 1, Weber number ܹ݁ = ൫ߩ ܷ ଶ ܴ ൯ ߛ ⁄ ≪ 1, and Capillary number ‫ܽܥ‬ = ൫ߤ ܷ൯ ߛ ⁄ ≪ 1, suggesting minimal deformation of the water droplet by gravitational, inertial and viscous forces with respect to the surrounding gas 6 . Eq. S1 was solved numerically using an explicit Runge-Kutta method with a variable time step. The initial jumping droplet velocity, ‫ݒ‬ = ‫ݕ‪ሺ‬ݒ‬ = 0ሻ, was then found by fitting the calculated result to the measured droplet trajectory.

S7. ESTIMATING ା OF THE BRIDGING FLOW
To develop a model description of the momentum developed by the bridge flow (r b /R < 1), we consider the following. As the bridge develops, the flow moving perpendicular to the bridge, i.e., fed by the negative radial flow near the bridge front, expands the radial flow region behind the bridge front such that the width of the radial bridging flow varies from a maximum at the bridge front (r = r b ) to approximately zero at r = 0. A simple estimate of the volume of the induced radial flow can be given by defining a linearly varying width where, from geometric considerations valid for small r b /R, the width of the bridge front at r = r b is ‫ݓ‬ = ‫ݎ‬ ଶ ܴ ⁄ 7,8 . Furthermore, we approximate the radial velocity as decreasing linearly from a maximum of v b at r = r b to zero at r b = 0 such that ‫ݒ‬ሺ‫ݎ‬ሻ = ‫ݎ/ݎ‪ሺ‬‬ ሻ‫ݒ‬ where ‫ݒ‬ is given by Eq. 5 in the manuscript.
Using this simple flow field approximation, we obtain an expression for ‫‬ ௬ ା developed during the bridge formation phase of droplet coalescence (r b /R < 1), Eq. S4 By non-dimensionalizing Eq. S4 by the characteristic momentum of the system, ݉ ܷ, we can observe that the bridging flow momentum follows a volumetric scaling, ‫‬ ௬ ା ݉ ܷ ൗ~ሺ‫ݎ‬ ܴ ⁄ ሻ ଷ .

S10. CONVERTED ENERGY CALCULATION
The jumping speed is given by where ‫ܦ‬ = 3.4026ܱℎ ଶ − 1.5285ܱℎ 0.2831 (Figure 5d), ρ is the liquid density and R is the radius of the two equally-sized droplets prior to coalescence. The mass of the jumping droplet is Eq. S6 From Eq. S5 and Eq. S6, we can calculate the kinetic energy of the jumping droplet as Eq. S7 Next, we calculate the excess surface energy released during the coalescence process for two, equallysized droplets in the limit that ߠ → 180°, Eq. S8 We define the energy conversion efficiency of the process by taking the ratio of Eq. S7 and Eq. S8, Eq. S9 Figure S10 plots Eq. S9 as a function of D. We see that as D decreases (increasing Oh), the amount of excess surface energy transferred to the kinetic energy of the jumping droplet decreases quadratically, reducing from ~6% at D ≈ 0.28 (Oh → 0) down to ~1.8% at D ≈ 0.15 (Oh = 0.12).
Figure S10 -Fraction of excess surface energy converted to the kinetic energy of the jumping droplet as a function of the Oh dependent pre-factor, D, in Eq. S5 for the coalescence of two, equally sized droplets the limit that ߠ → 180°. Figure S11 -Experimental jumping velocities normalized by the theoretical predictions (Eq. 10) plotted as a function of non-condensable gas pressure (p NCG ) in the environmental chamber. A linear fit to the P2i-CuO sample data gives a slope of m = -1.33x10-3 with a norm of the residuals of 1.45. A linear fit to the P2i-CNT sample data gives a slope of m = 6.87x10-4 with a norm of the residuals of 1.12. Despite the scatter, the data does not suggest that the jumping velocity is dependent on the surrounding pressure of the gas.