Relaxation phase during droplet impact on superhydrophobic surfaces with high contact angle hysteresis

Droplet impact has motivated vast research for more than a century^1–4 due to its relevance in several scientific and engineering applications such as spray-cooling,^5–7 pesticide deposition,⁸ and inkjet printing.^9,10 The development of novel surfaces with designed wetting properties has attracted attention toward superhydrophobic surfaces due to their low adhesion, high mobility of impacting droplets, and low contact angle hysteresis, all of which minimize the droplet contact time and maximize droplet mobility and total rebound.^11–15 Substantial experimental, analytical, and numerical research has been focused on the different phases of droplet impact, namely the spreading and the recoiling phase, which together determine a total contact time of the droplet with the solid surface. With the goal of increasing or minimizing the contact time depending on the application, research works have focused on the different stages of droplet impact dynamics and on how the process depends on the liquid properties,^16–18 impact velocity, surface vibration,¹⁹ temperature,^20,21 surface wettability,^22,23 and surrounding pressure.^24,25

The first stage of the droplet impact corresponds to the spreading regime where the droplet spreads until reaching a maximum diameter that can be described in terms of the spreading factor $β max ≡ D max / D 0$⁠, where D_max is the maximum spreading diameter after impact, and D₀ is the initial droplet diameter.^1,23,26,27 The spreading stage is mainly controlled by inertia while wettability effects are negligible. After the droplet has reached the maximum spreading, the retraction stage starts. Two retracting regimes have been identified,²⁸ namely a first inertial–capillary regime where capillary forces are countered by inertial forces and a second viscous-capillary regime where viscous forces are mainly responsible for opposing the retracting movement. The transition between these two stages has been found to be determined by an Ohnesorge number value of $O h = 0.05$ (⁠ $O h = μ / ρ σ D 0$⁠, where μ is the fluid viscosity, ρ the density, σ the gas–liquid surface tension, and D₀ the initial droplet diameter).

Although it has been assumed that the spreading stage is immediately followed by the retraction stage, Antonini et al.²⁹ identified the existence of a relaxation time between the end of the spreading and start of the receding phase. The relaxation phase between the spreading and the receding regimes is assumed to be related to the time, or relaxation time τ, that it takes for the droplet to transition from the advancing to the receding contact angle and, thus, to the contact angle hysteresis (CAH). The relaxation time τ has been found to be between 1.5 and 8 ms and, thus, even longer than the time spent in the spreading phase, reason for why the authors suggested introducing this intermediate phase into the analysis of droplet impact.²⁹ However, this relaxation phase has received very little attention in the literature as compared to the rest of the recoiling phase. The time τ spent in the relaxation phase has been observed to decrease with increasing Weber number $W e = ρ V 2 D 0 / σ$ and with decreasing surface wettability^29,30 for surfaces ranging from hydrophilic to superhydrophobic with contact angle hysteresis larger than 8°. It has been observed that surface wettability plays an important role in the relaxation phase even at high $W e$ numbers [up to $W e = 685$ (Refs. 29 and 31)] where usually inertia forces are more relevant than capillary effects. This can be attributed, as discussed in the results section, to the fact that the relaxation stage is controlled by capillary forces, whereas the dependence of the relaxation time with impact $W e$ number is due to the contribution of inertia to the maximum deformation of the droplet and consequent strength of the restitution capillary force. Lin et al.²³ identified a relaxation time τ in lyophobic surfaces with contact angle in the range of 99°–106° and CAH larger than 10°. It was observed that τ decreases with the droplet impact velocity and increases with the liquid viscosity, which can imply that viscous forces and inertia affect this relaxation phase. Unfortunately, no further attention was given to this relaxation phase in their work. Motivated on the effect of the receding contact angle on the retraction phase and subsequent droplet rebound, Wang and Fang³⁰ performed a systematic study of droplet retraction dynamics in a wide range of $W e$ numbers and surface wettabilities from hydrophilic to superhydrophobic. Similar to previous studies, they reported that the relaxation period decreases with increasing impact $W e$ number and surface wettability. However, while receding contact angles ranged from 21° to 153°, the contact angle hysteresis was similar for all surfaces where the relaxation phase was observed.

Although the relaxation period can be longer than the droplet spreading phase, indicating that the relaxation phase plays an important role for the total contact time of the droplet with the solid surface, there is a lack of further studies focusing on this intermediate regime. One reason is that most studies focusing on droplet retraction and rebound have been performed on superhydrophobic surfaces due to their low adhesion and good repellent properties, ideal conditions for minimizing the droplet contact time. Superhydrophobic surfaces present low contact angle hysteresis and the contact line begins to retract almost immediately after the spreading phase is over, preventing the existence of a relaxation phase. The few available studies observing this relaxation regime have shown that the relaxation period depends on the impact $W e$ number, surface wettability, and contact angle hysteresis. However, the relative role of inertia, viscous, and capillary forces in this regime is still not well understood.

In this work, we show that superhydrophobic surfaces with large contact angle hysteresis can also show a relaxation phase before the droplet starts to retract. For the purpose of this study, surfaces with increasing contact angle hysteresis while maintaining a high hydrophobicity were designed and fabricated. The results indicate that the contact angle hysteresis, in addition to the capillary forces, plays a major role in defining the relaxation time τ. The relaxation time depends directly on the droplet maximum deformation, which is related to the impact Weber number and explains the dependency of τ with the $W e$ number observed previously in the literature. It is further shown that the relaxation time scales with the inertial–capillary time when using the droplet relative deformation as the characteristic length scale for this relaxation regime.

The first step was to design and fabricate surfaces with high contact angle ranging from 134° to 165° and varying contact angle hysteresis ranging from 7° to 85°. This was achieved by four different types of rough surfaces, namely square pillars (PI), square pillars with nanowires (PIN), micro-pyramids (PY), and micro-pyramids with nanowires (PYN). The samples were prepared on 2-in. single-side polished silicon wafers (100, P-type) by a combination of photolithography, wet, and dry etching. All the substrates were initially rinsed with ethanol, acetone, and isopropanol (IPA), and dried with nitrogen (N₂) gas. All the surfaces were rendered hydrophobic by vapor deposition of 1H,1H,2H,2H-perfluorooctyl-trichlorosilane (97%) in a vacuum reservoir for more than 4 hours. Square pillars were fabricated by photolithography using a maskless aligner (MLA 150, Heidelberg) followed by a dry etching step in an ICP-RIE Chiller (Plasmalab System 100 ICP-RIE 180, Oxford Instruments, UK) using sulfur hexafluoride (SF₆) and trichloromethane (CHF₃) for 15 min. The resulting square pillars had a 5.5 μm side length and a center-to-center distance (or pitch) of 10 μm. Micro-pyramids were fabricated by a one-step anisotropic wet etching process by immersing the wafer in a mixed solution of 2.5 wt. % potassium hydroxide (KOH) and 4 vt% isopropanol (IPA) at 80 °C for 20 min. Nanowires were fabricated on top of the micro-pillars and micro-pyramids using metal-assisted etching. A thin film of 2 nm thickness of gold particles was deposited on the cleaned wafers at a rate of 5 Å/s using an electron-beam evaporator (AJA International Inc. Custom ATC-2200V). The wafers were soaked in a mixed solution of hydrofluoric acid (HF), hydrogen peroxide (⁠ $H 2 O 2$⁠), and de-ionized (DI) water for 2 min. The gold particles were removed by soaking the wafers in a gold etchant for 20 s. The SEM images of the four test surfaces can be seen in Fig. 1. The surfaces wettability properties for the different fluids used (as described below) can be found in Table I.

Deionized water (DI water) was used as the reference fluid. A 25 vt% ethanol–DI water mixture (E25p) was used in order to vary the fluid surface tension, while mixtures of 30 and 50 wt. % glycerol–DI water solutions (G30p and G50p, respectively,) were used to vary the fluid viscosity. The properties of the fluid mixtures used in this study are found in Table II.

The experimental study consisted in impacting liquid droplets on the surfaces from different heights to control the impacting velocity. The droplets' diameter, which was varied by using different needle diameters, was always below the capillary length scale, so that gravitational effects can be neglected. The droplet impact process was visualized with a high-speed camera from the side, as shown in Fig. 2, at a rate of 4000 fps. Contact angles, droplet diameter, and droplet impact velocities were obtained from these images analysis. Figure 2 also shows the typical temporal evolution of the non-dimensional droplet diameter $D ( t ) / D 0$ during the impact process on surface PI, with D₀, the droplet diameter, before impact. After the maximum diameter is reached, the retraction process starts. However, due to the large CAH of the surface PI (CAH = 38°), it is possible to observe a plateau region in the maximum diameter vs time plot that corresponds to the relaxation time. The relaxation regime and relaxation time τ were defined from the time when the maximum diameter has been reached until the time when the slope of the retraction regime abandons its relatively linear behavior. This was done by considering a linear fit starting at the point of occurrence of the maximum diameter and ending at the point where the spreading coefficient deviated from this linear behavior and the linear fit could not be further adjusted. The relaxation regime shows a much slower retraction rate of the three-phase contact line as compared to the rest of the receding stage. During this period, the advancing contact angle reached during the spreading stage changes to the receding contact angle before starting the receding stage. At the same time, a significant change in the droplet shape takes place, while the three-phase contact line shows only a slight motion. The dynamic change in the receding contact angle while the contact line is in motion is referred to as dynamic contact angle hysteresis.³² The contact line starts moving significantly faster after the contact angle reaches a minimum value that is able to depin the contact line. It is also worth noting here that, as shown in Fig. 2, the relaxation time is comparable or larger than the time spent in the spreading and receding regimes. This means that the relaxation phase should be given more attention in studies of contact time during droplet impact processes, in particular for surfaces with high contact angle hysteresis even when dealing with superhydrophobic surfaces.

To investigate the effect of the contact angle hysteresis on the relaxation time, droplet impact experiments at a constant impact $W e$ number on surfaces with four different CAH were performed, as shown in Fig. 3. The relaxation time τ clearly increases for increasing contact angle hysteresis, even when the surface wettability remains similar between the surfaces. It is seen from Fig. 3(b) how the relaxation time corresponds to the time it takes for the dynamic contact angle to go from its maximum to its minimum value.

The effect of the $W e$ number on the relaxation time τ for constant surface CAH is shown in Fig. 3(c) with data for DI water droplets on the PI surface. Here, the $W e$ number was varied by changing the droplet impact velocity through the droplet release height. The Weber number is defined as $W e = ρ V 2 D 0 / σ$⁠, with ρ and σ being the fluid density and surface tension, respectively, D₀ the initial droplet diameter before impact, and V the droplet impact velocity. As it can be seen, the relaxation time decreases for increasing $W e$ number, as also reported previously in the literature.^23,29–31

To further investigate the effect of the $W e$ number on the relaxation process, the relaxation time for all the experiments are plotted in terms of the impacting $W e$ number in Fig. 4(a). As expected,^23,29–31 the relaxation time decreases with increasing $W e$ number. Note, however, that while the impact velocity will determine the maximum spreading diameter, it is not clear how the impact kinetic energy will directly affect the rate of change of the dynamic contact angle from its advancing to its receding value when the droplet is lying still on the surface. Drawing an analogy to a stretched spring, it is logical to think that the rate of change of the dynamic contact angle during the relaxation phase will depend on the restitutive capillary force due to the droplet deformation and not directly on the impact velocity. Figure 4(b) shows indeed that the same behavior is observed for the relaxation time when plotted either against the Weber number or the relative droplet deformation $( D max − D 0 ) / D 0$⁠, where D_max is the maximum spreading diameter, and D₀ is the initial droplet diameter before impact. For large droplet deformations, there is a larger restitution force that pulls the contact angle from the advancing to the receding value as compared to a smaller droplet deformation, resulting in shorter relaxation times. The experimental data also show a vertical shift correlated with the different contact angle hysteresis. This dependence of the relaxation time with the CAH is expected, since before starting the retraction regime, the contact angle needs to go from its advancing to its receding value, taking longer to do this for larger differences. In particular, it is observed that for the lowest CAH of 7° (surface PYN), the relaxation time is negligible and independent of the deformation. This effect suggests that there is a threshold minimum value of the CAH for the relaxation phase to be present, explaining why this relaxation phase is rarely seen on superhydrophobic surfaces, which most of the time has a small or no CAH.

Going back to the analogy of a stretched spring, capillary forces will control the rate of change of the dynamic contact angle when the droplet is at its maximum spreading and therefore the relaxation time τ. Once the three phase contact line starts moving, surface energy will be converted into inertia, with viscous dissipation and energy loss due to CAH¹¹ also expected to affect the relaxation phase. Figure 4(c) shows the relaxation time non-dimensionalized with the inertial–capillary time $τ cap = ρ ( D max − D 0 ) 3 / σ$ as a function of the non-dimensional droplet deformation at maximum spreading. The latter represents the driving force for the change in dynamic contact angle during the relaxation phase. Data corresponding to droplet impact experiments performed with mixtures of water/ethanol and water/glycerol at 30 and 50 wt. % have also been included, to show the effect of varying surface tension and viscosity, respectively, while keeping the other fluid properties relatively constant. Figure 4(d) shows the same data with the relaxation time non-dimensionalized with the viscous time $τ visc = μ ( D max − D 0 ) / σ$⁠, where μ is the fluid viscosity. It can be seen that the relaxation time for all data with different viscosities, surface tension, and surface wettabilities collapses with the inertia–capillary time, while the scaling with the viscous timescale does not manage to collapse all the data, in particular those with higher viscosity. This result suggests that the inertial–capillary time is the proper scaling time for the relaxation stage. As pointed out previously,^1,11,27 the fact that viscous forces are not a limiting factor can be expected for superhydrophobic surfaces, since these surfaces minimize viscous losses due to their high contact angle.

The fact that the relaxation time collapses with the inertial–capillary time indicates that the relaxation time is independent of the impact velocity. Note that the data are collapsed when the relative droplet deformation $D max − D 0$ is used as the characteristic length scale in the definition of the capillary time, instead of the usually used initial droplet diameter D₀ when studying the retracting regime during droplet impact.^28,30,33 This length scale seems more appropriate than the usually used initial droplet diameter D₀ in this case, since when looking at the dynamics in the relaxation phase, the deformation of the droplet is expected to be more relevant than the droplet diameter before impact. The relaxation time τ was not seen to collapse when using the initial droplet diameter D₀ for the inertial–capillary time, indicating once again that the initial droplet inertia is not affecting the relaxation phase. This is because while the initial droplet inertia and impact We number will determine the spreading phase and maximum spreading diameter, the history of how the droplet reached that maximum spreading point is lost in the retraction regime. During the first part of the retraction regime, namely the relaxation phase, the velocity of the contact line is still very small so that inertia and viscous dissipation are negligible, and capillary forces proportional to the droplet relative deformation $D max − D 0$ are the governing mechanism in this stage. The characteristic time at which the contact angle will go from its advancing to its receding value will depend on the restoring force given by the surface tension and droplet deformation at its maximum diameter, which justifies the choice of $D max − D 0$ as the characteristic length scale for the relaxation phase.

The collapse of the relaxation time with the inertial–capillary time can be attributed to the dominance of the capillary forces over viscous forces. From an energy budget point of view, at the maximum spreading diameter, the initial energy of the droplet has been converted into surface energy. Once the retraction regime starts, part of the stored energy is converted back to inertia and the rest dissipated via viscous losses, contact angle hysteresis, and vibration modes due to capillary waves at the droplet surface.¹¹ Viscous dissipation can be considered negligible for superhydrophobic surfaces due to the large contact angle and the very small contact time,^1,11 which is also the case in this study. In addition, during the relaxation phase, the velocity of the contact line is small, again minimizing any viscous dissipation. Energy losses also occur due to the damping of fast capillary waves oscillating at the droplet surface,^11,34 where viscous dissipation cannot be neglected due to the large number of oscillations in the short impact time of the droplet. The presence of contact angle hysteresis is another way of energy dissipation, since energy loss occurs during the pinning and depinning of a moving contact line. Given that the relaxation time for our data does not collapse with the viscous time τ_visc, we can assume that viscous dissipation is not the determining factor. Effects of viscosity can still be seen from Fig. 4(d), where droplets with glycerol and therefore higher viscosity show a slower relaxation time. Still, as seen in Fig. 4(c), the same data collapse when the relaxation time is scaled with the inertial–capillary time, indicating that capillary effects dominate the dynamics. Effects of CAH can be observed in the slight upward shift of the data with larger CAH in Fig. 4(c), namely, water on PY with CAH of 85°, and G50 on PI with CAH of 49°. Based on these observations, it can easily be assumed that most of the dissipation on the tested superhydrophobic surfaces happens due to the presence of CAH. At the same time as energy is dissipated in the contact line motion, the speed at which the contact angle goes from a maximum to a minimum value in the relaxation phase depends on the droplet relative deformation, as observed in Fig. 4(c) from the much shorter relaxation times for larger droplet deformations, independent of the impact We number. The fact that more stretched droplets retract faster can once again be explained by making an analogy to a stretched spring or Hooke's law,²⁷ where a larger deformation results in a larger restitution force, in this case, the capillary force, and in turn a faster retraction rate or shorter relaxation time. The observed effect of the impact We is only due to the larger deformation attained by the droplets and not because of the dependency of the relaxation time on the droplet initial inertia.

In summary, we have shown that the relaxation time for superhydrophobic surfaces with large contact angle hysteresis is comparable to the spreading and retracting time. This study was possible by designing and fabricating superhydrophobic surfaces with varying contact angle hysteresis. Our results indicate that both contact angle hysteresis and capillary forces play a major role in the dynamics of the relaxation regime, while viscous forces do not appear to be a limiting factor. It was shown that the relaxation time scales with the inertial–capillary time when using the droplet relative deformation as the characteristic length scale for this relaxation regime. These results contribute to a more detailed description of the dynamics of droplet impact on superhydrophobic surfaces with potential application in scientific and engineering applications.

The Research Council of Norway is acknowledged for the support to the Norwegian Micro- and Nano-Fabrication Facility, NorFab, Project No. 295864.