Rebound of self-lubricating compound drops

An oil shell encapsulating a water drop promotes rebound after impact on a solid surface, irrespective of substrate wettability.


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Section S1. Lower rebound limit Section S2. Setup for bottom view reflection contrast imaging Section S3. Original images combined bottom and side view Section S4. Acceleration and terminal velocity of the compound drop Section S5. Motion of the core with respect to the shell Section S6. Further notes on the core position model Section S7. Rebounded volume (decomposition) Section S8. Pinch-off height Section S9. Parameter space injection method Section S10. Discussion on the oil film stability and rupture Section S11. Fluid properties Section S12. Supplementary videos captions Fig. S1. Model for the lower rebound limit.    Fig. S7. Volumes of rebounded water-in-oil drops and the jet as a function of the impact height h, the substrate wetting properties, and the compound drop-generation method. Fig. S8. Pinch-off height h po of the rebounding core. Fig. S9. Map of the rebound behavior after impact on a hydrophilic surface as a function of the water volume fraction α and the impact height (left axis) and Weber number We (right axis), for compound drops produced with the injection method. Legends for movies S1 to S8 References (61-76) The translational kinetic energy of the core before impact can be expressed as 1 12 πρ w D 3 w V 2 , where D w = α 1/3 D o . Just above the lower rebound limit, full rebound is observed. Therefore the surface energy of a fully rebounded core is considered, which is πσ o D 2 w . The relevant surface tension is that of the oil, since the rebounded core is still encapsulated by a layer of oil. The interfacial energy of the water core in oil is driving the rebound from the surface. However, it is restored after rebound to the same value as before impact and therefore does not contribute to the energy balance. The thickness of this layer is neglected. Balancing the initial kinetic energy of the core with the surface energy of the rebounded core, gives 1 12 πρ w αD 3 Therefore the minimum Weber number for core rebound is We min ∝ α −1/3 . This equation is illustrated in ig. S1. The equation correctly predicts a decrease of We min with increasing α, though the value of the exponent does not match. Quantitatively, the observed We min is typically an order of magnitude larger than predicted by the equation, since the equation above does not consider other relevant effects, such as viscous dissipation, which is not negligible, as well as the energy transfer between oil and water. An energy balance approach is traditionally used in the literature to understand the drop impact behaviour. We are aware of the limit of such an approach, as already highlighted by the work of Roisman  . Model for the lower rebound limit. The map contains the same experimental data as Fig. 2 in the main text, and includes the predicted lower rebound limit: We min ∝ α −1/3 -see text for details. The reflectance on the interface between glass and another substance is given by Fresnel equation (62) for light at normal incidence R = (n − n s ) 2 /(n + n s ) 2 , where n s is the refractive index of the substrate and n the refractive index of the medium on top of it. The recorded intensity on the camera sensor can be expressed as I = R 0 I 0 + RI 0 , where I 0 is the incoming intensity, R 0 is the constant reflectance of the bottom surface of the glass slide, and R is the reflectance of the top surface of the glass slide, which depends on the medium on top of it.
The different reflectance of water and silicone oil can be used to visualise the water contacting the substrate. Silicone oil has a refractive index of typically n = 1.4 − 1.5 (63, 64), which is higher than the refractive index of water. Glass has a refractive index that is typically higher than both water and silicone oil. The refractive index of the glass is therefore closer to the index of the oil than it is to the index of water. Silicone oil will therefore appear darker than water in the images. The constant reflectance of the bottom surface of the glass results in an offset in the recorded intensity by the camera, limiting the contrast of the images. The contrast was therefore enhanced later on by digital image post-processing in MATLAB, at the expense of having a higher noise level.
In addition to the bottom interfaces of the glass, the liquid-air interface on top of the liquid also causes reflections. Because of the curved nature of this interface, those reflections are generally not directed towards the camera. However, whenever the liquid-air interface is parallel to the glass interfaces, these reflections are recorded by the camera, resulting in overexposed areas in the images. Information about the liquid-substrate interface is lost in these areas. This temporary loss of information is maximal when the drop adopts the shape of a disk at maximal spreading (see Fig. 3A in the main text at 3.9 ms).   The vertical coordinate of the compound drop centre of mass z cm can be expressed as Therefore, we simply refer to the motion of the outer shell of the compound drop as the motion of the compound drop, and neglect the difference between z cm and z o . The compound drop experiences two external force components: a gravitational component and air drag. We assume the drag coefficient to be constant. Thus the force on the outer drop can be expressed as where m tot is the total mass of the compound drop and k is some constant. It is easily shown The acceleration of the compound drop can therefore be expressed as V can be solved analytically as a function of the impact height h (65) where D o is the diameter of the drop, and h 0 , the effective pinch-off length of the drop when it separates from the nozzle. The best fit to our experimental data (D o = 2.4 mm) was found for a terminal velocity V T = 5.88 m/s and h 0 = 5.7 mm. The experimental data and the best fit are shown in ig. S4.
The gravitational force component acting on the core is simply given by: where Ω w is the volume of the core. The buoyancy force for an object in an accelerating medium moving with velocity V is given by Because of the equivalence of inertial and gravitational mass, the acceleration of the medium can simply be added to g. We assume the drag force on the core obeys Stoke's Law, i.e.
dt is the velocity of the core with respect to the surrounding oil medium. After solving for V rel we found that Re ∼ 1 for the internal flow, justifying our approximation.
Adding up all force components and dividing through by the mass of the drop ρ w Ω w , we arrive at the following equation for the velocity V w of the core Substituting Ω w = 4 3 π (D w /2) 3 and subtracting dV dt , we obtain the following equation for the relative velocity An improvement to the model for the core position can be achieved by applying more realistic boundary conditions than those described above. For α < 0.3 the relative velocity of the core can be estimated when the core is approximately concentric with the shell. Under these conditions, the motion of the core can be directly observed. The theoretical transition height found with these initial conditions is given in ig. S1. It should also be noted that during pinchoff from the coaxial needle, the water drop is drawn at high speed into the oil shell, and is subsequently strongly decelerated. Despite strong assumptions, good agreement between the model and the experiments is observed.  In the main text we have neglected, for simplicity, the effect of the added mass on the moving core. Here we show that the effect of the added mass is minimal. As before, we consider the water core to be a non-deformable sphere, for which the added mass is 1 2 ρ o Ω w , where Ω w is the displaced volume, and ρ o is the density of the surrounding liquid. Multiplying Eq. S4 by the f mass ρ w Ω w of the core, and adding the added mass term yields Eq. S5 can now be written as Eq. S7 was solved the same way as in the main text. The result is shown in ig. S6, showing that the effect of the added mass is limited.  The We-axis corresponds to drops produced with the coaxial method, which have a slightly larger diameter than drops produced with the injection method.
h In addition to the rebounded volumes, another parameter was measured to characterise the rebound of the core. We define the pinch-off height h po as the vertical distance between the pinch-off point of the first (upper) oil-encapsulated water drop that escapes. The results are given in ig. S8. Just like the rebounded volumes, the pinch-off heights are independent of the wetting properties of the substrate below the core-substrate contact threshold. Above this threshold, rebound is strongly or completely reduced on hydrophilic substrates.  Fig. 5 in the main text. The magnified symbol corresponds to the image on the right. The We-axis corresponds to α = 0.3, but is representative for all α since the dependence on α is small.
It must be noted that a new regime was observed with the injection method: oil jetting as a function of the water volume fraction α and the impact height (left axis) and Weber number We (right axis), for compound drops produced with the injection method. f no rebound; was observed for heights below the lower core-rebound threshold. This behaviour was only incidentally observed with the coaxial method. Oil jetting can be attributed to the presence of a substantial volume of oil above the impacting compound drop. With the coaxial method, this volume is initially absent. Furthermore, oscillations due to the pinch-off from the needle lift the core drop from its original position at the bottom. This is illustrated in ig. S10. Therefore the core may still need to undergo some gravity-driven translation (Stokes regime) to return to the bottom (lubrication regime). Because of the optical distortion due to the oil-air interface (lensing), it is challenging to quantify this oscillation-driven lifting of the core. Therefore, it is difficult to draw strong conclusions about the impact velocity to break the lubricating oil layer. Before compound drop impact, the water core is always wetted by the oil due to the higher surface tension of water (σ w > σ o + σ ow ), as already explained in the main text. Therefore, an oil layer exists between the water droplet and the solid surface at the time of impact. We have demonstrated in the main text that the stability of this oil layer during impact plays a crucial role in the rebound of the water core on a hydrophilic surface. The transition from rebound to deposition corresponds to the impact height from which the oil layer ruptures.
We can get an order-of-magnitude estimate of the minimum oil thickness below the water core before impact using lubrication theory. Carlson et al. (50) give the evolution of the film thickness below the drop as ∂∆ ∂t = ∂ ∂x ∆ 3 3µo ∂p ∂x , where ∆ is the film thickness and x is the radial cylindrical coordinate. Before impact, the main pressure contribution comes from the air drag p ∼ 1/2ρV 2 . If we assume a uniform thickness ∆(x) = ∆, and take the drop radius R as the relevant horizontal length scale, we get ∂∆ ∂t ∼ ∆ 3 p 3µoR 2 . If we estimate ∂∆ ∂t as ∆ T , with T the relevant time scale, we find ∆ ∼ 3µoR 2 pT . Taking the fall time ∼ 2h/g as T , we find a thickness of order 10 -100 µm.
We have discussed in the manuscript the rebound observed here to the rebound of a water drop on a lubricated surface (36,45,46,51,(66)(67)(68)(69)(70)(71) or due to air cushioning (33,34,(72)(73)(74)(75)(76) Replacing the gas viscosity µ g in the equation by the oil viscosity µ o and using the oil-water interfacial tension σ ow and the water droplet size R w , gives the resulting threshold shown in The solid lines correspond to the break-up limit predicted by the theory in Eq. S8 for different values of the critical film thickness h c (33, 34, 52).
ig. S11 for several values of h c . Although the theory predicts a decreasing trend, as observed in our experiments, the predicted threshold is much higher and decreases more slowly, suggesting a different rupture mechanism than for the air film. In the limit of R w = R o , the theoretically predicted impact height (using h c = 200 nm) would be 125 cm (V = 4.2 m/s, We = 1950), which is much higher than the observed limit.
For comparison, ig. S12 visualises the lines for constant Weber number, based on the water core diameter, We w = ρ w D w V 2 /σ ow . Although a decreasing trend can be observed, the transition does not occur at a constant Weber number. It must be noted that several assumptions were made in deriving this theory, such as a high viscosity ratio (µ o ∼ µ w , whereas for air µ a µ w ), that are not valid in our configuration. The have demonstrated that the air film can cushion the impacting drop over a longer distance as the viscosity of the air increases, suggesting that the rupture of the air film could be due to a shear instability. The exact mechanism responsible for the oil film rupture during the impact of a compound drop therefore requires further investigation.
Same data as in fig. S9. We w = 100 We w = 200 We w = 300 The solid lines are lines of constant We w = ρ w D w V 2 /σ ow .  .