Ricocheting Droplets Moving on Super‐Repellent Surfaces

Abstract Droplet bouncing on repellent solid surfaces (e.g., the lotus leaf effect) is a common phenomenon that has aroused interest in various fields. However, the scenario of a droplet bouncing off another droplet (either identical or distinct chemical composition) while moving on a solid material (i.e., ricocheting droplets, droplet billiards) is scarcely investigated, despite it having fundamental implications in applications including self‐cleaning, fluid transport, and heat and mass transfer. Here, the dynamics of bouncing collisions between liquid droplets are investigated using a friction‐free platform that ensures ultrahigh locomotion for a wide range of probing liquids. A general prediction on bouncing droplet–droplet contact time is elucidated and bouncing droplet–droplet collision is demonstrated to be an extreme case of droplet bouncing on surfaces. Moreover, the maximum deformation and contact time are highly dependent on the position where the collision occurs (i.e., head‐on or off‐center collisions), which can now be predicted using parameters (i.e., effective velocity, effective diameter) through the concept of an effective interaction region. The results have potential applications in fields ranging from microfluidics to repellent coatings.


Video Captions
Video S1. 1 mM SDS droplets head-on bouncing collision (We = 20.0, B = 0.01). The video was recorded at 5,155 frames per second and plays at 30 frames per second.
Video S2. 1 mM SDS droplets off-center bouncing collision (We = 8.9, B = 0.43). The video was recorded at 5,155 frames per second and plays at 30 frames per second.  in Asahiklin 225 (Asahi Glass Co.) for ca. 5 min using a Paasche airbrush at a distance of ~20 cm with a N2 pressure of 58 psi and subsequently cross-linked in an oven at 70 °C for 2 h. The substrate was stored in a dust-free desiccator, and any unbound particles on the super-repellent coating were removed using compressed air before the coated mesh super-repellent surface was subjected to the droplet collision experiments.

Characterization.
A Philips XL30 scanning electron microscope was used to image the surface morphology of the super-repellent surface at 5 kV. A Ramé-Hart 200-F1 goniometer was used to measure contact angles, roll-off angles and liquid surface tension. Averages from N ≥ 5 independent measurements were reported and the measured errors in contact angle, roll-off angle and surface tension were ±1°, ±0.5° and ±1 mN m −1 , respectively. The surface durability was demonstrated by continuous droplet impinging and rolling tests using 1 mM SDS aqueous droplets. Liquid droplets were released from a height of 10 cm above the substrate and rolled off after impinging on the surface. The roll-off angle was reassessed after a set numbers of tests ( Figure S1). High-speed movies were obtained using a Fastec Hispec1 camera.
Droplet rolling. We determined the friction between the super-repellent copper mesh and the contacting liquid by droplet roll-off on a slope. The minimum tilt angle ω required for droplet roll-off was experimentally demonstrated to be <3° for all liquids tested (Table S1), thus the friction was determined as 10 -7 -10 -6 N by balancing with the gravity force which gives = (see details in Abbreviations and Definitions). Such minute forces enable the droplets to roll on the surface in a way close to the ideal case, which could be concluded by comparing the accelerations as a mass body (i.e., liquid droplet) rolling off a given slope β > ω.
For the ideal case, the acceleration is given as a = gsin β. By recording the distance s that a droplet travels in a period t on a super-repellent slope β, the practical acceleration a′ along the slope can be computed as ′ = 2 2 . When a liquid droplet (e.g., n-pentane) rolls off a surface tilted at 37°, the measured acceleration was 5.85 m s -2 , in good agreement with the ideal case Droplet collision. A super-repellent copper mesh (10 cm × 5 cm) was bent at the midway point to have a slope of ~37° for droplet acceleration followed by a flat section for collisions ( Figure   S2). Shallow depressions and tracks (~0.1 mm) were forged by pressing with a metal ball to enable the deposition of droplets and to guide the direction of the rolling droplets. A droplet was first placed in a depression and another droplet was subsequently released on the slope.
Droplet collisions were recorded vertically from above the surface using a manually triggered camera at 5,155 frames per second (unless specified otherwise). Photron FASTCAM viewer software was used to analyze the droplet size r0, impact parameter B and impact velocity u0 upon collisions, and the droplet deformation and contact time for bouncing collisions.
Collision boundary. Three collision regimes can be defined as (1) bouncing collision, (2) permanent coalescence and (3) stretch separation ( Figure S3 and S4). At high impact parameters, only a small region of the droplets will come into contact: the width of the interaction region can be computed as h = 2r0(1 -B) if the colliding droplets are identical. Thus, the boundary between permanent coalescence and stretch separation can be computed by balancing the total effective stretching kinetic energy and the surface energy of the interaction region: [S2] (S1) As is shown in Figure S5, though Equation S1 is applicable for droplet collisions in air, the prediction works reasonably well with other experimental results even for our collision study on a solid surface. This indicates that the porous super-repellent platform makes the droplet collision acts as if it took place in air. In contrast, when droplet collision occurs on a non-porous surface, [S3] film drainage slows as the air flow is hindered, which results in a high probability of bouncing collisions.
By comparison, the occurrence of bouncing collisions is largely dependent on the maintenance of the layer of air, which originates from the local dynamics of the fluid (e.g., air and droplet).
A possible criterion for bouncing is that the effective kinetic energy only produces limited deformation of the colliding droplets: Though our observations are mostly within the range predicted by this criterion, understanding the mechanism of maintaining a vapour layer remains challenging. [S5] In some cases, the probabilities of bouncing and coalescence are comparable. For perspective, a higher chance of bouncing collisions requires a more repulsive interaction interface, and a locally high air pressure may prevent permanent coalescence. [S6] As for collisions of binary composition droplets (i.e., identical size), same regimes can be observed ( Figure 1g, Figure S4). The boundaries of bouncing, coalescing, and stretch separation can also be described by the above Equation S1 and Equation S2, respectively. The We of these events can also be determined by the moving droplets on the super-repellent platform.
In cases including pesticide spraying, which is usually through an aqueous media, and spray coating, which requires either an organic or an aqueous solvent, coalescence in the sprayed droplets is desirable. We thus investigated the coalescence efficiency of droplet collisions on the super-repellent surface ( Figure S6). The coalescence efficiency e of polar aqueous droplets was experimentally determined to be directly related to the impact Weber number (i.e. ~3 ).
This result is in good agreement with droplet collisions in air. [S7] However, non-polar droplets, such as n-pentane, behave completely differently within the same Weber number range. This might be due to the comparably weaker intermolecular forces of non-polar liquids, which result in a higher chance to bounce off before coalescence.
Maximum deformation. At the time corresponding to roughly half of the oscillation period (i.e., ~0.5τ0), the bouncing droplets reach maximum deformation lmax, which is due to the excess of inertial forces over the surface tension of the droplet, similar to the case observed for droplets impacting on a super-repellent surface, [S8,S9] where maximum deformation occurs at ~0.3τ0 due to the non-compensatory blocking exerted by the non-movable hard-solid super-repellent surface. In comparison, the droplet-droplet interfaces during bouncing collisions are much more flexible considering the soft nature of liquid surfaces, as well as the high mobility of droplets on the near-zero friction super-repellent platform. However, gravity plays a negligible role in horizontal collisions (e.g., droplet-droplet bouncing collisions). In contrast, for a bouncing droplet impacting vertically on a solid surface (i.e., single droplet bouncing collision), the vertical impact force is in the same direction as the gravitational force, which allows the droplet to deform to a greater extent (~10%) (Figure 2b).

Effective parameters.
For off-center collisions, droplet-droplet interaction mainly occurs within a reduced region, while the other regions of each droplet tend to maintain the initial trajectory of the droplet. Therefore, there is a practical need to define effective parameters to quantify the bouncing dynamics of off-center collisions. As is illustrated in Figure 3b Contact time. First, we consider droplet-droplet head-on bouncing collisions analytically.
Bouncing droplets' transient contact is composed of two parts-deformation (τd) and retraction (τr)-that are expected to be functions of the droplet oscillation period (i.e., 0 = d + r ).
According to a study on binary liquid collisions, [S10] the timescale for maximum droplet deformation can be empirically determined to be d ≈ 0.5 0 for low Weber numbers (i.e. We  Figure S11). Taking into account the effective droplet diameter, we can deduce the contact time for off-center bouncing collisions as: Oscillator. Two droplets involving bouncing collisions can be seen as oscillators, in which their deformation can be written as L1(t) and L2(t), respectively. For identical droplet collisions, both droplets undergo similar deformation (i.e. L1(t) = L2(t)) and the contact time can be estimated by the droplet oscillation period as c ≈ 1 = 2 . As for the contact time of binary liquid droplet-droplet bouncing, theoretically we have to consider the expansion rate and the retraction rate in the precinct of droplet departure. For example, considering the head-on bouncing collision of a water-hexadecane system (i.e., a water droplet impacts onto a hexadecane drop), at the end of contact, the water droplet experiences retraction while the hexadecane droplet is still expanding. After the time point where these two events reach the same rate, the two droplets depart from each other permanently. Thus, the contact time can be computed as     Table S1 for more details of liquids used.  of liquid permittivity in (e) is n-pentane < n-hexadecane < n-octanol < n-butanol < cyclopentanol < ethanol < dimethylformamide < 0.5 mM aqueous SDS < water. Insets in (e) are representative outcomes for low-(top: head-on and off-center bouncing) and high-permittivity (bottom: head-on and off-center coalescence) colliding liquids. Liquids that have a low static permittivity have a higher probability to bounce, whereas liquids that have a high static permittivity have a higher probability to coalesce. This is likely related to the dispersion forces within the liquids. [S12]    The dimensionless contact times are also plotted against the impact Weber number (d) and the Ohnesorge number (e). As is shown, slightly decreased contact times are observed but they are still largely scaled with the corresponding inertia-capillary timescales. This might be due to the slightly higher viscosity of cyclopentanol droplets (11.7 mPa s) and the viscous dissipation might start to play a part in determining the contact time of bouncing droplet-droplet collisions.   Table S2 and S3 for more information.