Toward vanishing droplet friction on repellent surfaces

Significance A small raindrop will easily slide off the surface of a lotus leaf or the feathers of a bird. These natural systems are beautiful examples of superhydrophobic surfaces. Today, there is a strong focus on mimicking this natural surface design to manufacture artificial liquid-repellent substrates for a wide range of applications, such as self-cleaning and nonwetting materials. The physics describing how water drops move on these surfaces has been studied extensively to find even better surface designs. Here, we developed a technique to measure the tiny friction of drops moving on extremely slippery superhydrophobic surfaces. We found an important plastron friction force that needs to be considered when designing the next generation of ultraslippery water-repellent coatings.

We carefully analyzed the effect of the aerodynamic dissipative force in our oscillating MFS experiments.The drag force from the air on the oscillating micropipette and drop was calculated as  air ∼  a Re 1/2  2 (4), where  a is the density of air, Re is the Reynolds number of the pipette (Re p ∼  a 2 p  p / a , where  p is the pipette radius and  p the pipette speed) or drop (Re d ∼  a 2/ a ), S is the reference area of the pipette ( p ∼ 2 p  p , where  p is the cantilever length) or drop ( d ∼  2 ).We calculated the sum of the air resistance on the pipette and drop as a function of time from many oscillating MFS experiments with water or carbonated water.However, we found that the effect from this is several orders of magnitude lower than the effect of the viscous dissipation of  (Fig. S12).This finding is in agreement with the work by Mouterde et al. (4), showing that air resistance becomes relevant at drop speeds of order 1 m/s.The typical max speed of drops in our oscillating MFS experiments is ~10 to 100 mm/s, and air resistance can thus safely be left out of the ODT equation., where subscript sim refers to used simulation parameter and fit refers to value from the fitted ODT model.The inaccurate area the lower left corner of Fig. S11d-f is due to the poor fitting, which could be improved with adjusting the starting guesses in the fitting.In addition, a longer simulation time could improve the accuracy since the oscillations continue longer than the simulation time (8529 ms for the smaller mass and 9404 ms for larger mass, with 8000 ms simulation time for decaying).The total simulation time is different, since the total time consists of a short waiting at the start (100 ms), movement of the potential based on resonance frequency (varied) and the decaying (8000 ms).Overall, the accurate range of ODT model coincides with the measured parameters in the main experiments, which gives confidence in the measured results and the used model.As the used simulation friction force decreases, the solved friction force decreases faster resulting in larger negative error (Fig. S11b).These increasing errors hinder the accurate measurement of vanishing line friction (  ≪ 1 nN), which is the case for the levitating carbonated droplets.For the levitating carbonated drop, the friction force is zero as the drop does not touch the surface.However, the analytical fit will always give a non-zero value for the friction force (on the order of   = 10 −20 N), even if the value is clearly incorrect.In these cases, the friction force is estimated to be zero instead of the extremely low number, since the accuracy of the low force is uncertain, and estimating the force as zero is more accurate.
The two types of dissipation can be distinguished from each other when the other one is dominating, since the viscous force dominated dissipation leads to clear exponential decay of the oscillations, while the friction force dominated dissipation leads to linear dissipation (Fig. S13).While the friction and viscous forces are of similar magnitude, then the dissipation is a linear combination of these two types of decay.This holds true even when the resonance frequency of the system changes, which can be seen by comparing simulated droplet data with the two different masses.The effect of time step interpolation was investigated by fitting the ODT model for the same location data set with different time steps (Fig. S14).The different time steps were done by interpolating the original simulated data (1000 fps) to different time steps from 1000 fps to 30 fps using smoothing spline.Some of the data points near 30 fps are not visible in the figures, since the values are vastly off from the used simulation input value such as the smallest friction force value 2.1 ⋅ 10 −19 N with 35 fps capture rate.Then this larger time step data was returned to 1000 fps using smoothing spline and the ODT model was fit to the data.The results of these fits show that the 60 fps is needed for good fit, which is lower than the experimentally used 120 fps.5A in main text) or glycerol (bottom,  = 0.88 ± 0.01 mm/s,  = 290 ± 10 m).Drops are similar, but the behaviours are quite different, since the deviations from the plateau value at low speed appear earlier with glycerol, above 0.2 mm/s instead of 2 mm/s for water.We also note that the plateau value is slightly higher for glycerol with smaller surface tension, thus of larger contact angle hysteresis.When we add to the graph the calculated plastron friction (orange line) and drop viscous friction (dark blue line), the origin of the difference becomes clear.Both the frictions are linear in velocity, but, logically, the viscous friction in the drop (dark line) becomes dominant at much smaller velocity when the liquid viscosity is higher (glycerol  1500 water), so as to dominate the plastron friction (orange line), whatever the velocity.No oscillating MFS experiments could be performed with glycerol drops and we simply assume that the plastron friction (independent of ) is the same as for water.The dashed blue line is a fit to the plateau at low speed, and the solid dark line is not a fit but just the calculated Mahadevan-Pomeau friction   ∼  2 /2 with the parameters of the experiment.S3 | Data used for Fig. 1C.The scanning MFS black silicon data for Fig. 1C.The errors for the dimensionless force are error propagations using the time-averages and standard deviations of the force F and contact area diameter 2l taken over the time frame used for analysing the sliding friction (see example in Fig. S2).The surface tension is that of water ( = 72 mN/m).The relative error for the speed is 8% as stated by the manufacturer of the motor.The measured k and calculated kt spring constants of the oscillating MFS system plotted in Fig. 2C with spring constant ( p = 7.7 ± 0.2 mN/m).The errors for k are the 95% confidence intervals of the ODT fits and the error for kt is the error propagations of the components (and their standard deviations) in Eq. ( The measured k and calculated kt spring constants of the oscillating MFS system plotted in Fig. 2C with spring constant ( p = 10.8 ± 0.4 mN/m).The errors for k are the 95% confidence intervals of the ODT fits and the error for kt is the error propagations of the components (and their standard deviations) in Eq. (1).k, mN/m kt, mN/m 5.8 ± 1. 4 5.2 ± 0.4 6.6 ± 1.9 5.8 ± 0.4 8.0 ± 2.2 7.0 ± 0.6 8.0 ± 0.8 5.9 ± 1.9 6.7 ± 0.6 6.2 ± 0.2 6.0 ± 0.9 5.7 ± 1.0 4.6 ± 0.3 5.0 ± 0.3 5.1 ± 0.4 5.6 ± 0.2 5.6 ± 0.5 5.0 ± 0.5 4.4 ± 0.7 4.7 ± 1.0 4.1 ± 1.1 5.0 ± 0.7 4.8 ± 0.9 4.8 ± 0.3 4.9 ± 0.7 4.8 ± 1.1 4.9 ± 0.5 5.5 ± 0.3 7.6 ± 0.6 6.3 ± 0.4 7.0 ± 0.5 6.5 ± 0.5 5.9 ± 0.4 5.8 ± 0.   5D.The scanning MFS bSi data from sample µA+bSi A in Fig. 5D.The errors for the dimensionless force are error propagations using the time-averages and standard deviations of the force F and contact area diameter 2l taken over the time frame used for analysing the sliding friction (see example in Fig. S2).The surface tension is that of water ( = 72 mN/m).The relative error for the speed is 8% as stated by the manufacturer of the motor.5D.The scanning MFS bSi data from sample µB+bSi A in Fig. 5D.The errors for the dimensionless force are error propagations using the time-averages and standard deviations of the force (F) and contact area diameter 2l taken over the time frame used for analysing the sliding friction (see example in Fig. S2).The surface tension is that of water ( = 72 mN/m).The relative error for the speed is 8% as stated by the manufacturer of the motor.

Fig. S1 .
Fig. S1.Contact angles on bSi A. a) Schematic drawing of the advancing ( adv ) and receding ( rec ) contact angles for a drop moving with a speed V on a black silicon superhydrophobic surface.b) The contact angles from the experiments in main Fig. 1B-C remain constant for a water drop moving at different speeds on bSi A. The error for the contact angles (±4 deg) is evaluated from previous simulations for superhydrophobic substrates(6).c) The contact angle hysteresis (from the data in b) remains constant, indicating that there is no viscous effect on the contact angles and that the contact-line friction remains constant within this drop speed range.

Fig. S4 .
Fig. S4.Drop damping coefficient for carbonated drops levitating on Glaco.The data in main Fig. 3C collapse using a) two different micropipette spring constants, as well as b) two different wavelengths of the LED.

Fig
Fig. S5.Contact-line friction measured with scanning and oscillating MFS on bSi A. The   from the ODT fit performed on the oscillating MFS data for differently sized water drops (red crosses) are in excellent agreement with the linear fit (solid black line) done to the low-velocity scanning MFS data (data points not plotted for clarity, see data in Fig. 5A).The dashed black lines are the error lines for the scanning MFS experiments.

Fig. S6 .
Fig. S6.Friction of water and glycerol drops on bSi A. The dimensionless total friction measured using scanning MFS with drops of either water (top,  = 0.92 ± 0.05 mm and  = 270 ± 30 m, same as Fig.5Ain main text) or glycerol (bottom,  = 0.88 ± 0.01 mm/s,  = 290 ± 10 m).Drops are similar, but the behaviours are quite different, since the deviations from the plateau value at low speed appear earlier with glycerol, above 0.2 mm/s instead of 2 mm/s for water.We also note that the plateau value is slightly higher for glycerol with smaller surface tension, thus of larger contact angle hysteresis.When we add to the graph the calculated plastron friction (orange line) and drop viscous friction (dark blue line), the origin of the difference becomes clear.Both the frictions are linear in velocity, but, logically, the viscous friction in the drop (dark line) becomes dominant at much smaller velocity when the liquid viscosity is higher (glycerol  1500 water), so as to dominate the plastron friction (orange line), whatever the velocity.No oscillating MFS experiments could be performed with glycerol drops and we simply assume that the plastron friction (independent of ) is the same as for water.The dashed blue line is a fit to the plateau at low speed, and the solid dark line is not a fit but just the calculated Mahadevan-Pomeau friction   ∼  2 /2 with the parameters of the experiment.

Fig. S7 .
Fig. S7.Etched micropillared samples.Side-view scanning electron microscopy (SEM) images of µA+bSi (left) and µB+bSi (right) samples, where the microstructure is bSi A. The top width of the µ-pillars is 10 µm, which provides the scale.

Fig. S8 .
Fig. S8.Friction of water on black silicon (bSi) and µ+bSi samples.The damping coefficient from oscillating droplet tribology (ODT) as a function of the contact area radius of differently sized drops.The solid lines are fits of  ∼  2 to the data.The error bars for  are the 95% confidence intervals for the ODT fit and the error for the contact area radius is the standard deviations from the time-averages of l.Frictions on the etched pillared samples µ + bSi are significantly lower than on normal black silicon bSi.

Fig. S9 .
Fig. S9.Expected regimes of friction for different contact-line frictions.The dominating friction as a function of plastron thickness and drop speed for a  = 1 mm water drop (with  = 0.2 mm) on surfaces with   = 1, 10, 100 and 1000 nN.The grey area (marked with ) denotes the regime where the viscous dissipation in the drop dominates.The plastron friction is especially relevant on highly-slippery materials (low Fµ), and it appears, at fixed H, above a threshold velocity, as discussed in the main text.

Fig. S10 .
Fig. S10.Expected regimes of friction for drops with various viscosities.The dominating friction as a function of plastron thickness and drop speed for a  = 1 mm drop (with  = 0.2 mm) on a   = 10 nN surface at different liquid viscosities ( = 1, 10, 100, 1000 mPa.s).The grey area marked with  denotes the region where viscous dissipation in the drop dominates.The plastron friction appears to be relevant when the drop viscosity is below 10 mPa.s; conversely, bulk viscosity logically imposes the friction for liquids such as glycerol (  1000 mPa.s), as seen in Fig S6.

Fig. S11 .
Fig. S11.Accuracy maps of ODT analysis based on simulated data.The difference between the simulation input value and the solved ODT value is used as estimation on the error in a, b, d, and e. Estimation the total accuracy of ODT analysis is calculated by the combined errors of viscous and friction coefficients in c and f.The error magnitudes larger than 15% are capped to 15% in magnitude for increased readability.
Fig. S12 Friction in oscillating MFS experiments.The total air resistance on the pipette and drop ( air , red), viscous force () from the ODT (blue), and contact-line friction (  ) from the ODT (black) for a water drop with  = 1.3 mm and  = 0.4 mm oscillating on bSi A. The air resistance is several orders of magnitude smaller than the other frictions and can thus be neglected in the ODT.

Fig. S13 .
Fig. S13.Example simulated oscillations with different dissipation forces.The ac show example simulated oscillations with droplet mass 1 mg and the df is for 10 mg.The friction force is dominating in a and d, viscous force is dominating in b and e, while c and f have equal contribution from viscous and frictional force.At the start of the oscillations, the potential is moving to cause the droplet to move.

Fig. S14 .
Fig. S14.Analysis of one simulated data set with different frame rates.The solved spring coefficient (a), viscous coefficient (b) and friction force (c) for one simulated data set with different frame rates.The solved values are accurate within error bars for captured frame rates higher than 60 frames per second (fps) when the frame rate is interpolated to 1000 fps for fitting.

Table S1 | Thickness of black silicon samples.
The average thickness of the etched black silicon (bSi) microstructures as measured with sideview SEM.

Table S2 |
Data used for Fig.1C.The scanning MFS micropillar data for Fig.1C.The errors are standard deviations from the timeaverages of the force F and contact area diameter 2l taken over the time frame used for analysing the sliding friction (see example in Fig.S2).The error for the speed comes from the manufacturer of the motor (8% relative error).

Table S4 |
Data used for Fig.2C.The measured k and calculated kt spring constants of the oscillating MFS system plotted in Fig.2Cwith spring constant ( p = 2.48 ± 0.06 mN/m).The errors for k are the 95% confidence intervals of the ODT fits and the error for kt is the error propagations of the components (and their standard deviations) in Eq. (1).

Table S7 | Data used for Fig. 3B.
The gas film height as a function of radial distance as measured with RICM for Fig.3B.

Table S8 | Data used for Fig. 3C.
The calculated  cushion and measured  damping coefficient of the oscillating MFS measurements on the carbonated drops plotted in Fig.3C.The errors for  are the 95% confidence intervals of the ODT fits and the errors for  cushion are the error propagations of the components (and their standard deviations) in Eq. (2).

Table S9 | Data used for Fig. 4F.
The contact area radius  and damping coefficient  data from the oscillating MFS measurements with water drops on bSi A plotted in Fig.4F.The errors for  are the 95% confidence intervals of the ODT fits and the errors for  are the standard deviation of the time-averaged distance measurement.

Table S10 | Data used for Fig. 4F.
The contact area radius  and damping coefficient  data from the oscillating MFS measurements with water drops on bSi B plotted in Fig.4F.The errors for  are the 95% confidence intervals of the ODT fits and the errors for  are the standard deviation of the time-averaged distance measurement.

Table S11 | Data used for Fig. 4F.
The contact area radius  and damping coefficient  data from the oscillating MFS measurements with water drops on bSi C plotted in Fig.4F.The errors for  are the 95% confidence intervals of the ODT fits and the errors for  are the standard deviation of the time-averaged distance measurement.

Table S12 | Data used for Fig. 4F.
The contact area radius  and damping coefficient  data from the oscillating MFS measurements with water drops on bSi D plotted in Fig.4F.The errors for  are the 95% confidence intervals of the ODT fits and the errors for  are the standard deviation of the time-averaged distance measurement.

Table S13 | Data used for Fig. S8.
The contact area radius  and damping coefficient  data from the oscillating MFS measurements with water drops on µA+bSi A plotted in Fig.S8.The errors for  are the 95% confidence intervals of the ODT fits and the errors for  are the standard deviation of the time-averaged distance measurement.

Table S14 | Data used for Fig. S8.
The contact area radius  and damping coefficient  data from the oscillating MFS measurements with water drops on µB+bSi A plotted in Fig.S8.The errors for  are the 95% confidence intervals of the ODT fits and the errors for  are the standard deviation of the time-averaged distance measurement.