4.1.1. Centre of mass trajectory and interaction time

For each single bounce the centre of mass (CM) motion can be subdivided into two phases, a free flight and ‘quasi-contact’: see figure 6. During the flight the interaction force is zero and the CM motion is fully decoupled from the droplet oscillation. The CM trajectory is parabolic in time with constant gravitational acceleration, so the speed increases linearly in time upon approaching the substrate. In contrast, during the ‘quasi-contact’ phase the droplet interacts via the squeezed air layer with the substrate (the grey line in figure 5 is practically zero), and the motion of the centre of mass is directly linked to the droplet deformation. The droplet can be viewed as a liquid resonator powered by energy of the centre of mass that is converted into internal kinetic and elastic surface energy. The droplet stretches laterally while the CM height decreases down to about half the droplet radius for the first bounce: see the middle snapshot in figure 1. The maximum speed $v=\text{max}|({\dot{y}}_{CM})|$ is reached at the end of the flight phase and defines the $\mathit{We}$ number of the impact: see figure 6. During the deformation, the CM velocity decreases to zero. Subsequently the velocity is reversed and the droplet bounces off with velocity $v^{\prime }=\max ({\dot{y}}_{CM})$ . During the ‘quasi-contact’ phase the droplet thus experiences an acceleration that must be caused by the interaction with the air film. The blue line in figure 6 shows the interaction force $F$ with a time resolution of about 1 ms, as calculated with (3.5). The force is double (sometimes single) peaked and is always positive, as would be the case for a normal force in the case of elastic contact. We will discuss this observation when analysing the nature of the force in a later section.

Figure 6.

Velocity and estimated interaction force for the experiment shown in figure 5. (a) Time evolution of the CM height (red) and interaction force (blue, peaked during ‘quasi-contact’) derived from the CM acceleration. (b) CM velocity (dark red). The first two bounces are shown ( $t<0.10$ ), as well as a few hovers ( $0.70<t<0.75$ ). The blue shaded area indicates interaction with the substrate, i.e. ‘quasi-contact’.

What happens during the subsequent bounces? Due to energy dissipation and transfer to internal modes, the droplet bounces back to a smaller height in the next flight phase, and its subsequent impact velocity will be lower. Ultimately the free-flight phase is fully eliminated and the drop hovers on an air film of micrometric thickness. Now, the motion of the centre of mass is permanently linked to the shape oscillations. The interaction force does not decrease to zero any more as the droplet is continuously in ‘quasi-contact’ with the substrate via the air film. The (late) time-averaged interaction force converges to the gravitational force ( $46~{\rm\mu}\text{N}$ in figure 6) when the droplet approaches its equilibrium shape. This equilibrium shape is not spherical but flattened, as it constitutes a small gravitational sag ${\it\delta}$ which is the result of a balance between gravity and surface tension, ${\it\delta}/R_{0}\sim \mathit{Bo}$ (Mahadevan & Pomeau Reference Mahadevan and Pomeau1999). The observed sag indeed increases with $\mathit{Bo}$ and ranges from ${\it\delta}/R_{0}\approx 0.15$ for drops of water ( $\mathit{Bo}=0.16$ ; figure 5) to 0.23 for drops of FC-40 ( $\mathit{Bo}=0.30$ ), in good agreement with the prediction.

The overall bouncing behaviour can be described by the restitution coefficient. The restitution coefficient is defined as the ratio between maximum velocities after and before the bounce, ${\it\varepsilon}=v^{\prime }/v=\sqrt{\mathit{We }^{\prime }/\mathit{We}}$ : see figure 6. It is plotted in figure 7 as function of $\mathit{We}$ for water and glycerol drops, for three bouncing series each with different initial impact speeds. For a water droplet with $R=1.03~\text{mm}$ the restitution coefficient is approximately constant over most of the $\mathit{We}$ range and has a value of $0.96\pm 0.04$ , similar to the restitution coefficient found by Richard & Quéré (Reference Richard and Quéré2000) for bouncing on a superhydrophobic substrate – which may also involve an air film. The restitution coefficient in figure 7 shows a characteristic fluctuation pattern versus $\mathit{We}$ , which is remarkably independent of the initial fall height. The fluctuation in restitution is related to a coupling between the oscillation time of the dominant mode $n=2$ (see § 4.1.2) and the flight time. The restitution is maximal when at the moment of impact the droplet is stretching towards oblate, since the bottom interface velocity is then minimized (Biance et al. Reference Biance, Chevy, Clanet, Lagubeau and Quéré2006). On the other hand, when at impact the drop is contracting from a prolate shape, the restitution is expected to show a local minimum. The latter is indeed the case for the impacts indicated with an arrow in figure 7.

Figure 7.

The dependence of the restitution coefficient ${\it\varepsilon}$ on impact $\mathit{We}$ for experiments with 1.03 mm water droplets (initially released from various needle heights: red, 5 mm; blue, 11 mm; black, 15 mm) and 0.95 mm glycerol droplets (in the lower part of the graph: dark red, 5 mm; dark blue, 8 mm; grey, 10 mm). Solid symbols denote the bouncing phase, while open symbols denote the hovering phase. The red arrows denote the sixth and ninth bounce of the experiment in figure 9, for which coupling of the oscillation and flight time (i.e. impact within the prolate shape) leads to a relatively low restitution coefficient.

While the restitution coefficient suddenly drops to zero when solid–liquid contact is made at low $\mathit{We}$ , at the other limit of $\mathit{We}\sim 1$ it decreases more gradually. This can be explained by the large droplet deformation: at higher impact velocity the restitution coefficient is largely determined by the transfer of energy to large-amplitude droplet oscillations rather than dissipation in the liquid or the squeeze layer. With a simple liquid spring free of dissipation, Biance et al. (Reference Biance, Chevy, Clanet, Lagubeau and Quéré2006) showed that for $\mathit{We}>1$ the speed after take-off does not depend on the impact speed or the compression of the spring, and is given by $v^{\prime }\sim \sqrt{{\it\sigma}/({\it\rho}R)}$ . It can then be easily shown that the second bounce should have $\mathit{We}\sim 1$ ; this jump is indeed observed in figure 7. This implies that for each initial impact condition $\mathit{We}>1$ the drop falls into a very similar trajectory after the first bounce. As a result the number of bounces remains remarkably constant, i.e. 16–18 for water droplets. Moreover, from figure 7 we observe that the restitution coefficient and thus the number of bounces depends on the droplet viscosity: the restitution coefficient of highly viscous glycerol droplets is about 0.7 and leads to only two bounces.

In the absence of real liquid–solid contact during the ‘quasi-contact’ or bounce phase, its interaction time is defined as the period during which the interaction force is non-zero. Within the experimental time resolution, this is equivalent to the period during which the interference pattern of the air film is visible. The interaction times for the impact of a 1.03 mm water droplet are shown in figure 8(a): for the successive bounces the interaction time increases while $\mathit{We}$ decreases. Again, the results from the various experiments at different initial impact velocity collapse perfectly. Also, for hydrophilic and hydrophobic substrates we observe no difference. The interaction time is approximately 9.1 ms for bounces at $\mathit{We}\sim 1$ , and increases towards 18.5 ms for the last bounce. Subsequently, the interaction time abruptly becomes infinite at $\mathit{We}\sim 0.02$ , which is the onset of the hover stage. During the hovering we can instead measure the oscillation time of the droplet (and the dimple), which is 20.7 ms.

Figure 8.

The dependence of interaction time on $\mathit{We}$ . (a) A 1.03 mm water droplet is released from various initial heights (red, blue and black data set) onto a glass substrate, or a hydrophobic cured silicon oil layer (grey). In the bouncing phase the interaction time of individual bounces can be obtained (closed symbols), while at the onset of the hovering phase (shaded blue) the interaction time suddenly becomes infinite (see arrow) and we instead measure the oscillation time (open symbols). (b) Normalization of the interaction time by ${\it\tau}=({\it\rho}R_{0}^{3}/{\it\sigma})^{1/2}$ for experiments with droplets of water (black), 85 wt% glycerol (green), sunflower oil (blue), 90 wt% propanol with $R_{0}=0.79~\text{mm}$ and 1.04 mm (red and pink respectively), decane (grey), silicone oil (olive), and FC-40 (dark blue). Solid line, Rayleigh free oscillation time; dashed line, numerical results from Moláček & Bush (Reference Moláček and Bush2012) in the absence of gravity ( $\mathit{We}\gg \mathit{Bo}^{2}$ ); short dashed lines, analytical solutions from Moláček & Bush (Reference Moláček and Bush2012) for a droplet oscillating around its equilibrium position ( $\mathit{We}\ll \mathit{Bo}^{2}$ ).

Simple scaling theory of a weakly deformed liquid drop ( $\mathit{We}\ll 1$ ) shows that the contact time of a non-wetting bounce is described by a characteristic time ${\it\tau}=({\it\rho}R_{0}^{3}/{\it\sigma})^{1/2}$ (Richard, Clanet & Quéré Reference Richard, Clanet and Quéré2002; Okumura et al. Reference Okumura, Chevy, Richard, Quéré and Clanet2003). The contact time then scales as $t_{c}/{\it\tau}\approx A(\mathit{Bo},\mathit{We},\mathit{Oh})$ , where $A$ is a function of the three dimensionless numbers relating the importance of gravity, inertia and viscosity to surface tension. In our experiments we consider only two of them since viscosity effects are assumed negligible at all times, $\mathit{Oh}\ll 1$ . In figure 8(b) the rescaled interaction time $t_{c}/{\it\tau}$ is plotted for eight bounce series with different liquid droplets. In the bounce phase all data collapse, and the abrupt transition to hovering is observed for all liquids – yet the obtained oscillation time and the $\mathit{We}$ number at the transition both vary with $\mathit{Bo}$ . We think this is due to gravity. Gravity effects are negligible at the beginning of the bounce series, but are expected to become dominant when $\mathit{We}$ decreases and the droplet oscillates around its equilibrium shape. Okumura et al. (Reference Okumura, Chevy, Richard, Quéré and Clanet2003) show that the transition between these two regimes is given by $\mathit{We}\sim \mathit{Bo}^{2}$ . Within our small range of $\mathit{Bo}$ numbers studied (0.16–0.31) the abrupt transition to hovering is in exact correspondence with this scaling; see also figure 4(b) (with small deviation at higher $\mathit{Oh}$ ). We can now explain the limiting behaviour as follows: for large $\mathit{We}$ the droplet is expected to approach the Rayleigh free oscillation time of the dominant $n=2$ mode, $t_{free}/{\it\tau}=2{\rm\pi}/\sqrt{8}\sim 2.22$ (see (3.12)). This is indeed the case, as indicated by the black horizontal line. On the other hand, in the weak deformation limit of $\mathit{We}\rightarrow 0$ the calculations of Chevy et al. (Reference Chevy, Chepelianskii, Quéré and Raphael2012) show that the oscillation time depends on $\mathit{Bo}$ : for increasing influence of gravity, the deformation of the droplet is stronger and it behaves in a stiffer way, i.e. the oscillation time decreases in the limit of $\mathit{We}\rightarrow 0$ . This decrease is confirmed in figure 8(b): see arrow. In fact, the global behaviour shows fair agreement with the limiting behaviour in the absence (black dashed line) and presence (coloured small dashed lines) of gravity for the logarithmic spring behaviour derived by Moláček & Bush (Reference Moláček and Bush2012). The remarkable difference, however, is the abrupt transition towards hovering, yielding a constant oscillation time very early on (at $\mathit{We}=\mathit{Bo}^{2}$ ). This may be a consequence of the air film dynamics that were excluded from any previous models. Finally we stress here again that real contact is never made. Air-film-mediated bouncing thus adds another perspective to reducing the real contact time below the Rayleigh free oscillation time, as was achieved very recently using cleverly designed macrostructures (Bird et al. Reference Bird, Dhiman, Kwon and Varanasi2013; Liu et al. Reference Liu, Moevius, Xu, Qian, Yeomans and Wang2014).

4.1.2. Droplet oscillations

So far we have mainly considered the CM motion of the droplet. In addition, during ‘quasi-contact’ the droplet becomes deformed to buffer the energy needed for recoil. Since not all energy is transferred back to the centre of mass, the drop subsequently oscillates during the flight. The magnitude of the oscillation depends on impact velocity and droplet properties such as size, surface tension and viscosity. Let us consider the oscillations associated with the trajectory of an $R_{0}=1.03~\text{mm}$ water droplet impacting at $0.22~\text{m}~\text{s}^{-1}$ as in figure 5. The most significant coefficients $c_{n}$ of the mode decomposition, i.e. up to $n=4$ , are shown in the main panel of figure 9. During the bounce phase the decrease in the mean radius $R_{0}+c_{0}$ and a small $c_{1}$ contribution show that the small-amplitude approximation is not valid here. Thus we need some caution in using the $c_{n}$ values to describe the potential flow inside the droplet. For now, we look at the surface decomposition itself. Amongst the orders ${\geqslant}2$ , the dominant mode is $n=2$ with a oscillation time of 9.1 ms ( $\pm 0.1$ ). Due to its clear dominance, the second mode sets the free oscillation time of the overall drop, that is, the variation of the drop height as shown in the right inset of figure 9. The other oscillation times are 4.7 and 3.0 ms for the $n=3$ and 4 modes, and all show perfect agreement with their respective Rayleigh frequencies, (3.12). The independence of modes is only observed during the free-flight phase; during the ‘quasi-contact’ phase all modes up to $n=4$ are abruptly forced into the same phase, as the droplet becomes deformed by interaction with the squeezed air layer. The result is a single peak for the $n=2$ mode and a double peak for the $n=3$ mode. With each subsequent bounce the amplitudes of all modes during the ‘quasi-contact’ decrease. Gradually the frequencies of the modes change such that in the hover phase all modes are synchronized with the motion of the centre of mass. The transition is most clear for the $n=3$ mode, which initially shows two peaks during a bounce, gradually merged into one. For the $n=2$ mode the characteristic time for ‘quasi-contact’ merely increases. In the final hover stage shown in figure 9 the synchronized oscillation time is 20.7 ms.

Figure 9.

Shape mode decomposition using Legendre polynomials for the bouncing series shown in figure 5, for $n=0$ (black dotted line, shifted by constant $R_{0}$ ),  $n=1$ (black solid line), and in order of decreasing amplitude: $n=2$ (red), $n=3$ (blue), and $n=4$ (grey) versus time. The first two bounces are shown ( $t<0.10$ ), as well as a few hovers with equilibrated gravitational sag ( $0.70<t<0.75$ ). The free oscillation time depends on the mode, $t_{free}=9.1$ , 4.7, 3.0 ms for $n=2\dots 4$ , whereas during hovering the modes are synchronized, $t_{bound}=20.7~\text{ms}$ . Left inset, moment of lift-off after the first bounce. Right inset, overall drop deformation in terms of the half droplet height (magenta).

It is important to note that the oscillation modes with $n>2$ are critical for the bouncing dynamics. The left inset of figure 9 shows the lift-off after the first bounce. Near the contact point the droplet interface has a high curvature due to a favourable superposition of positive even modes $n=2$ and 4 and negative odd mode $n=3$ . Thus, the superposition of higher modes can lead to a significant decrease in the ‘quasi-contact’ area that is critical for a high-restitution lift-off.

From the mode decomposition the half droplet height $B=\sum _{n=0}^{\infty }c_{n}(t)[P_{n}(1)+P_{n}(-1)]/2$ has been calculated and is shown in the right inset (magenta) of figure 9 for the first two bounces and the last few hover oscillations. During the first few bounces, the droplet deforms in the vertical direction to about half its original size, while the oscillations during the intermediary free-flight phase are significantly smaller. They also show a slight decay due to viscous dissipation in the droplet. During the successive bounces the $\mathit{We}$ number decreases and so does the droplet deformation. Consequently the intermediate flight time gets shorter and the bouncing pattern gradually transforms into the single oscillation pattern that can be observed in the hover phase.

Figure 10.

Energy conversion and dissipation for the experiment in figure 5. (a,b) First bounce showing the conversion of the kinetic (dark solid line) and potential (dark dashed line; the thin dashed line shows the linear approximation) parts of (a) the total CM energy and (b) the total internal energy. (c) Total energy (black solid line), total energy plus the calculated internal dissipation (black dotted line; ${\rm\Delta}E_{drop}^{visc}$ is indicated by the grey region), and excess viscous dissipation ${\rm\Delta}E_{excess}^{visc}$ (green region) to explain the difference from the total initial energy $E_{0}$ . Partly reproduced from de Ruiter et al. (Reference de Ruiter, Lagraauw, van den Ende and Mugele2015a , figure 5).

4.1.3. Energy transfer and dissipation

The transfer during the bounce of CM energy to internal energy (i.e. oscillation energy) and subsequent dissipation in the internal flow have previously been described as the primary source of dissipation in non-contact bouncing of droplets (Richard & Quéré Reference Richard and Quéré2000). The main reason for this is the relatively long flight time compared to the contact time. However, in our experiments the free oscillations have small amplitude, and their weak decay in figure 9 shows that there is little dissipation during the flight. We thus need to take into account the dissipation during the ‘quasi-contact’ phase. Using the side view experiments we cannot only derive the CM energy but also the internal energy of the drop from the shape mode decomposition. Figure 10(a,b) shows the energy conversion during the first bounce. The CM energy in figure 10(a) is converted from gravitational to kinetic energy without any losses during the approach flight. A very small scatter is observed in the constant CM energy (162 nJ before the first bounce) because of the inaccuracy in the position determination of the centre of mass. Only during ‘quasi-contact’ do the CM dynamics couple to the oscillation dynamics, and CM energy is temporarily buffered into internal energy while the drop is strongly decelerated. However, a large part of the energy is restored in the CM motion while the drop lifts off, again yielding a constant but slightly lower CM energy of 128 nJ. The internal energy is shown in figure 10(b). During the ‘quasi-contact phase’ the internal flow field has two peaks in the kinetic energy – one before and one after the maximum deformation of the droplet – while the potential energy is maximum when the flow reverses. In the transition to the subsequent flight phase the drop keeps a small residual oscillation energy of 6 nJ, continuously converting kinetic and potential energy with a weak decay due to dissipation.

Figure 10(c) shows the sum of the droplet energy $E_{tot}=E_{CM}+E_{int}$ plotted in black. We calculate the internal dissipation ${\rm\Delta}E_{drop}^{visc}$ (grey area) by time-integration of (3.11), and add this term to the droplet energy (resulting in the black dotted line). During the flight phases this sum is constant (plateaus in the black dotted line) which implies that the small decay in internal drop energy (blue) is indeed due to ${\rm\Delta}E_{drop}^{visc}$ , while the CM energy (red) is conserved. However, accumulated over an entire bounce series, the above-mentioned viscous dissipation during the flight phases amounts to only 20 % of the total energy loss of the droplet. During the short ‘quasi-contact’ phases another 10 % is dissipated via this mechanism but most of the energy (70 %) is dissipated via another mechanism, not yet identified. After each ‘quasi-contact’ or bounce the CM energy has decreased significantly while the internal energies just before and after a bounce are nearly equal. This loss in CM energy cannot be fully accounted for by the internal dissipation ${\rm\Delta}E_{drop}^{visc}$ , as evidenced by the clear jump in the black dotted line. (The scatter during each ‘quasi-contact’ phase is due to limitations of the linear decomposition of the drop shape. As dissipation is always negative, the black curves should be monotonically decreasing.) The green area in figure 10(c) indicates the ‘excess’ dissipation ${\rm\Delta}E_{excess}^{visc}$ , cumulative over the entire bouncing series. It represents the missing 70 % of the total energy loss, dissipated during the ‘quasi-contact’ phases. Inaccuracies of the linear approximation during ‘quasi-contact’, as suggested by the 10 % reductions in $R_{0}+c_{0}$ in figure 9, are expected to cause deviations of the order of 20 %, and yet the amount of ${\rm\Delta}E_{excess}^{visc}$ shown in figure 10(c) suggests a second dissipation channel, namely dissipation in the lubricating air layer.

4.2.1. Interface shape

The interaction of the droplet with the squeezed air layer during ‘quasi-contact’ can be analysed from the film thickness profiles as obtained from the bottom view images using interferometry. As an example, figure 11 shows the evolution of the air film for the first impact of a water drop bouncing series at $\mathit{We}=0.76$ . A height plot of the air film evolution in time is shown in figure 11(a), while the corresponding thickness profiles are plotted in two separate figures: figure 11(b1) shows the spreading of the confined film up to the maximum lateral extension, while figure 11(b2) shows the contraction phase. The transient film has a typical height of only $1~{\rm\mu}\text{m}$ while the lateral extension is about 1 mm, resulting in a small $O(10^{-3})$ height-to-radius aspect ratio. The air film is thus remarkably flat. However, if we stretch the $y$ -axis as in figure 11(b), we observe the typical thickness profile of the air film. The largest film thickness is attained in the centre of the film (i.e. the droplet–air interface is dimpled), and the film is bounded by a ‘kink’ of high interfacial curvature where the film thickness is minimum (near the rim of the film). Outside this kink the interfacial slope diverges.

Figure 11.

Time evolution of the squeezed air film during the first bounce phase for an $R_{0}=1.01~\text{mm}$ water droplet ( $v=0.22~\text{m}~\text{s}^{-1}$ ) shown in figure 13. (a) Contour plot $h(r,t)$ of the film height as a function of radial position and time. The corresponding interface profiles are shown with ${\rm\Delta}t=0.1~\text{ms}$ for (b1) the spreading stage up to 3.6 ms, and (b2) the contraction stage up to 8.6 ms. As the minimum is hard to follow in these diagrams, we show the time evolution of this minimum explicitly in (c), which clearly reveals the curling motion at the turning point. Partly reproduced from de Ruiter et al. (Reference de Ruiter, Lagraauw, van den Ende and Mugele2015a , figure 4).

First of all, an obvious condition for non-contact bouncing is the presence of an air film during the full ‘quasi-contact’ phase. This requires a minimum film thickness larger than ${\sim}200~\text{nm}$ on glass substrates (de Ruiter et al. Reference de Ruiter, Oh, van den Ende and Mugele2012, Reference de Ruiter, Lagraauw, van den Ende and Mugele2015a ): when the film thickness becomes smaller, liquid–solid contact is formed and bouncing is inhibited. It was shown that for larger $\mathit{We}\gtrsim 1$ the minimum film thickness develops around the shoulder near $r=300~{\rm\mu}\text{m}$ : see figure 11(b1). For increasing $\mathit{We}$ it is at this location where the film thins to the critical value needed to obtain liquid–solid contact. Mandre et al. (Reference Mandre, Mani and Brenner2009) demonstrated the influence of surface tension on the minimum film thickness: the strong curvature near the shoulder generates a capillary pressure that prevents further thinning of the air film. This leads to an equilibrium film thickness $h_{c}=5R~\mathit{Oh}_{g}^{8/9}\mathit{We}^{-10/9}$ , where the Ohnesorge number $\mathit{Oh}_{g}={\it\mu}_{g}({\it\sigma}{\it\rho}R_{0})^{-1/2}$ is the ratio between gas viscosity and surface tension and inertia. Hence, $h_{c}$ decreases with $\mathit{We}$ . We previously verified this scaling in our experiments (de Ruiter et al. Reference de Ruiter, Oh, van den Ende and Mugele2012), and use it here to estimate the critical Weber number for which $h_{c}<200~\text{nm}$ , and bouncing is inhibited. Since $\mathit{Oh}_{g}$ only varies by a factor of two for millimetre-sized droplets, the film is expected to collapse when $\mathit{We}\gtrsim 4{-}6$ , which is a narrow range independent of liquid properties. These predictions are plotted in figure 4(b) and are in good agreement with experiments in which $\mathit{We}$ is pushed to its maximum limits while still obtaining a bounce series.

From figure 11 one observes that the evolution of the air film during spreading and contraction of the drop is not time-symmetric: the height plot in figure 11(a) shows that the spreading phase is significantly shorter, and the size of the confined air film grows monotonically up to its maximum extension. During the contraction phase however, the air film shows repeated small contractions and expansions. Figures 11(b1, b2) provide more detail on this asymmetry. During the spreading phase in figure 11(b1) the interface unfolds ever more outwards, while the inner region remains almost unchanged. The initial stagnation area under the centre of the droplet expands into an ever larger region, and the interface moves almost vertically outside this stagnation region. In contrast, during the contraction phase shown in figure 11(b2), the outer kink moves inward towards the centre of the film. Simultaneously the minimum thickness gradually increases after some initial wiggles that coincide with the contraction/expansion cycles. The typical ‘curling’ behaviour of the kink is shown by plotting the air film thickness versus the position of the kink in figure 11(c). It is important to note that the air film thickness inside the kink still decreases but it increases outside the kink as the drop ‘peels off’ the substrate. The pronounced asymmetry between the spreading and the receding phase turns out to be crucial for the bouncing process, as we will see in the next section.

The air film shapes obtained for subsequent bounces can be described by a few simple characteristic parameters, despite the film complexity. As discussed in § 3.2, we use the volume of the air film $V_{sq}$ , its average height $\overline{h}$ , and its minimum film thickness $h_{min}$ , which determines whether the squeezed air layer is stable so that no liquid–solid contact is formed. Figure 12 shows the time evolution of these parameters for a few selected bounces. As expected, all parameters vanish or diverge outside the ‘quasi-contact’ phase, i.e. when the drop is not interacting with the substrate, and we verify again that the ‘quasi-contact’ time increases during a bouncing series until it becomes infinite at the onset of the hover stage (thin black lines). In the hover stage each oscillation continuously passes into the next one as the droplet interface is permanently deformed.

Figure 12.

Time evolution for successive bounces of (a) squeezed film volume, (b) average and (c) minimum film thickness. Three bounces are shown from the experiment in figure 13, $\mathit{We}=0.76$ , 0.26, 0.05 (grey, bounce 1; red, 5; blue, 12), and one hover oscillation, $\mathit{We}=0.007$ (black, 10).

For successive bounces the maximum volume of trapped air decreases from typically 3 to 0.2 nl: see figure 12(a). We plot the maximum entrapped volume as a function of the Weber number of the bounce in figure 13(b). The increase in dimple volume with $\mathit{We}$ is characteristic for the low-velocity regime in which capillarity smooths out the dimple. When the inertia of the drop becomes important, for $\mathit{We}>1$ , the dimple volume again decreases with $\mathit{We}$ , leading to a maximum in dimple volume (Bouwhuis et al. Reference Bouwhuis, van der Veen, Tran, Keij, Winkels, Peters, van der Meer, Sun, Snoeijer and Lohse2012; Klaseboer, Manica & Chan Reference Klaseboer, Manica and Chan2014; de Ruiter et al. Reference de Ruiter, van den Ende and Mugele2015c ). The latter regime is not reached in the experiment of figure 13. We obtain the scaling $\tilde{V}_{sq}\sim \mathit{We}^{0.83}$ in the hovering phase (i.e. the limit of small $\mathit{We}$ ), which is slightly stronger than the exponent of $1/2$ predicted in the low-velocity regime (Bouwhuis et al. Reference Bouwhuis, van der Veen, Tran, Keij, Winkels, Peters, van der Meer, Sun, Snoeijer and Lohse2012). The volume decrease is strongly linked to the decrease in spreading radius $r_{sq}$ from ${\sim}0.8~\text{mm}$ in the first bounce to only ${\sim}0.4~\text{mm}$ in the hover stage, as shown in figure 13(a).

Figure 13.

Interference signals and shape of the air film at minimum thickness for successive bounces. (a) The same bounces as in figure 12, as well as hover oscillations 1, 3, 5, 7, and 9 ( $\mathit{We}=0.02\dots 0.008$ ). Insets: snapshots of the film at maximum lateral extension for the first bounce (left) and the ninth hover (right). The 1 mm scale bar relates to both snapshots. (b) Absolute minimum of the film thickness (black circles) with accompanying dimple volume (red triangles) for all bounces and hover oscillations (closed and open symbols respectively).

The average film thickness decreases as shown in figure 12(b), thus the droplet interface on average approaches the substrate closer and closer with each bounce. However, the minimum film thickness exhibits a rather random behaviour. For each bounce in figure 12(c) it shows several oscillations in time, indicating that the kink comes down repeatedly or new kinks are formed. For the stability of the air film we are interested in the spatiotemporally obtained absolute minimum thickness during each bounce. The air film profiles obtained at the instant of minimum film thickness are shown in figure 13(a). No trend is observed in the minimum film thickness during the bouncing stage, which is confirmed for a full bouncing series in figure 13(b): it varies between 300 and 900 nm depending on the details of the squeeze-out. This is sufficiently large ( ${>}200~\text{nm}$ ) to prevent solid–liquid contact during all subsequent ‘quasi-contacts’.

The dynamics in the hover stage are different. With decreasing $\mathit{We}$ the influence of surface tension becomes ever more prominent and the droplet interface shows smaller fluctuations (both in time and with radial position): see figures 12(c) and 13(a). In fact the vertical impulse on the droplet is too low to replenish the air film, and the centre of the dimple suddenly collapses to a height of, here, approximately 500 nm. From then on the centre of the dimple is more or less stationary and the droplet hovers on a micrometric air layer. The outer kink gradually approaches the substrate more closely with each oscillation until the critical film height of ${\sim}200~\text{nm}$ is reached and liquid–solid contact is established. Figure 13(b) shows the transition between the bouncing and the hover phase: the minimum film thickness in the hover phase shows a clear trend with $\mathit{We}$ . Empirically we find $\tilde{h}_{min}\sim \mathit{We}^{0.36}$ .

4.2.2. Momentum transfer and dissipation

Although the air viscosity is small, the effects of the viscous flow in the squeezed air film may become substantial if the shear rate and pressure gradient diverge, due to the small film thickness. Indeed we find a significant lower number of bounces than the maximum of 1000 that can be obtained in a Leidenfrost situation where the air film has a much larger thickness of about $100~{\rm\mu}\text{m}$ (Biance et al. Reference Biance, Chevy, Clanet, Lagubeau and Quéré2006). This strongly suggests that air film dissipation is substantial. The build-up of a viscous pressure simultaneously leads to an upward force that decelerates the droplet and should provide the impulse to let the droplet bounce, i.e. reverse the drop velocity before continued air film thinning leads to formation of solid–liquid contact. For the experimental air film profiles we calculate the viscous squeeze force and the local dissipation rate, (3.20) and (3.18) respectively, using the lubrication equations.

We notice that the squeezed air film has an extreme radius-to-height aspect ratio, leading to the first naive approach of assuming a flat disk with time-dependent radius and radially invariant height – either $h_{min}(t)$ for an upper limit, or $\overline{h}(t)$ for a lower limit of the force and dissipation. However, the flat disk approximation fails because the viscous force is always oriented opposite to the direction of motion, implying a reversal of the force towards attractive as soon as the drop starts to move away from the substrate (lift-off) and air is flowing radially inward. For the bouncing drops this is evidently not the case as the force is repulsive at all times, as shown from the CM analysis in figure 6. This implies that the interface – at least locally – should exhibit a downward motion, to obtain a net outward air flow. An effective momentum transfer thus arises from the fact that the air film is not flat! The details of the kink motion as described in figure 11(b) are in fact essential to understanding the momentum reversal during the air cushioning quantitatively. Unfortunately, we miss crucial shape information outside the kink due to the steepness of the interface compared to the lateral resolution of the recording: all profiles abruptly end just outside the kink (see figure 11 b). As a result the squeeze force $F_{sq}$ cannot be determined from the experimental data as it relies on the unknown reference pressure just outside the measurable region. To overcome this problem we use a simple parametrization of the interface: figure 14(a) shows a sketch of the interface evolution that is generic for all our experiments. At each time, the radial description of the interface shape is given by a straight line from dimple centre to kink inside the kink, matched to a relatively steep parabola outside the kink, described with respect to the kink by ${\rm\Delta}h=a({\rm\Delta}r)^{2}$ (where $a=1.45\times 10^{-3}~{\rm\mu}\text{m}^{-1}$ for the experiment in figure 11). The kink moves along straight lines: for the experiment in figure 11 the kink approaches the substrate from $[r,z]=[0,4.0]$ to $[800,0.4]$ during the spreading phase (see figure 14 a1) and retracts along another straight line from $[r,z]=[800,0.4]$ to $[0,3.67]~{\rm\mu}\text{m}$ during the contraction phase (see figure 14(a2), while the dimple centre remains fixed at a height of $4~{\rm\mu}\text{m}$ . The approach and retraction of the kink is assumed to be parabolic in time (see figure 14 c, black line).

Figure 14.

Role of air film asymmetry on interaction force. (a) Sketch of the idealized profiles showing the asymmetry in the spreading (a1) and contraction (a2) of the squeezed air layer: red, parabolic outer kink profile; black and blue, linear inner profile during spreading and contraction respectively. To stress the asymmetry, the kink height retracts in the sketch (blue dashed) to a lower value than in the actual parametrization. (b1,2) Examples of pressure profiles calculated from the idealized air film shape. (c) Integrated squeeze force (blue, right axis), together with the minimum air film height $h_{min}$ in the idealized case (black) and the actual experimental evolution (grey dotted). Partly reproduced from de Ruiter et al. (Reference de Ruiter, Lagraauw, van den Ende and Mugele2015a , figures 2 and 4).

In this description the interface unfolds ever more outwards during the spreading phase, while the inner region is stationary. Hence, inside the kink no flow occurs, while outside the kink the flow radiates outwards, and thus the pressure inside the kink is positive. During the contraction phase the outer kink moves inwards but does not follow the slope of the interface, as indicated in figure 14(a2). Now there are two contributions to the flow field: outside the kink the interface moves back up, thus the flow is radially inwards, and the pressure gradient is positive. In addition the region inside the kink is shrinking, still leading to an effectively downward motion of the interface and thus leading to a radially outward flow, resulting in a negative pressure gradient. As a result the pressure is positive for the largest part of the dimple. The calculated pressure profiles are shown in figure 14(b1) during the spreading phase and in figure 14(b2) during the contraction phase. The integrated squeeze force is shown in figure 14(c). The calculated force $F_{sq}$ thus shows two positive peaks with a minimum around maximum drop extension. This is in good agreement with the interaction force $F$ obtained from the drop acceleration described in figure 6. Thus the squeeze force indeed provides the impulse necessary to reverse the drop momentum during the bounce. Further refinement of the result (e.g. elimination of the very small attractive force at the final lift-off) may be obtained by considering a more complex interface evolution: the idealized profiles in figure 14 now assume a simple parabolic approach of the kink in time: see the black line in figure 14(c). Instead one could also implement the repetitive up–down motion of the kink, as observed in the experiment (see grey dotted line), while still forcing the kink to move in a straight line in the $(r,h)$ space as in figure 14(a).

The corresponding viscous dissipation has been calculated from the original experimental air film data, but is restricted to the contribution within the measurable region. First we determine the local height-averaged velocity $\overline{v}$ of the squeezed air using (3.14). From this velocity and the film profile we calculate the local dissipation rate using (3.18). Very high local outward velocities up to ${\sim}0.1~\text{m}~\text{s}^{-1}$ are obtained, in particular in the narrow gap near the outer kink. The velocity is not monotonically increasing in the direction of the kink, in contrast to previous results by van der Veen et al. (Reference van der Veen, Tran, Lohse and Sun2012) where radially averaged rather than local velocities were considered. Instead, after formation of the first kink we also observe regions of moderately negative velocity in the inner region of the squeezed air film. However, the highest velocities are observed at the kink and are positive. This allows for the above-mentioned net outward flow. The local dissipation rate ${\dot{w}}_{sq}\,\text{d}r~(\text{J}~\text{s}^{-1})$ , integrated over the film height, is shown in a contour plot versus radial position and time in figure 15. Obviously, the dissipation rate is diverging (indicated in dark red) at the location where the air gap is both narrow and fast thinning, i.e. close to the outer kink, while the dissipation is negligible in the centre of the dimple which is almost stationary (cf. figure 11). This divergence is consistently observed for all experiments, although to a different extent, depending purely on the shape evolution of the dimple. For the bounce series of a water droplet the divergence at the kink is strongest for intermediate $\mathit{We}$ as in figure 15(b), while for lower and higher $\mathit{We}$ also significant dissipation is obtained in the inner region, as shown most clearly in figure 15(a).

Figure 15.

Local dissipation rate in the air film. The local dissipation rate ${\dot{w}}_{sq}\,\text{d}r~(\text{J}~\text{s}^{-1})$ , with $\text{d}r=3.3~{\rm\mu}\text{m}$ , is plotted versus radial position and time for (a–c) the impact of an $R_{0}=1.03~\text{mm}$ droplet with $\mathit{We}=3.79$ , 0.76, 0.007, (d) the impact of an $R_{0}=0.79~\text{mm}$ 90 wt% propanol droplet with $\mathit{We}=1.18$ , and (e) the impact of an $R_{0}=0.52~\text{mm}$ FC-40 droplet with $\mathit{We}=1.93$ (shaded: no data). The colour scale is logarithmic. The time axis in (c) is compressed by a factor of 3.

The local dissipation rate can be integrated in space and time to obtain the total energy $W_{sq}$ dissipated in the air film. Results are given in table 2. As expected $W_{sq}$ decreases during the bounce series of a water droplet because air is squeezed out more gently at lower drop momentum. For high-viscosity droplets (85 wt% glycerol), $W_{sq}$ is relatively high at similar $\mathit{We}$ due to the small air film thickness during the bounce. This is caused by the ‘stiffer’ droplet shape on short time scales due to the higher droplet viscosity. It is important to note that all measured dissipation values are a lower bound estimate, due to the divergence of the dissipation rate at the kink and the restricted field of view of the interference pattern at that location. Using rough parabolic extrapolations of the interface profiles in the neighbourhood of the kink, we expect that the region of high dissipation rate extends a bit further outside the kink, before it decreases steeply when the local film height diverges. So we miss the contribution from a small region just outside the kink. Nevertheless, we compare the estimated contribution of the air film dissipation with the excess dissipation determined from the side view analysis in § 4.1.3. Table 2 shows that the percentage of air film dissipation roughly increases during a bouncing series. Aside from the underestimation of air film dissipation just discussed, it can be expected that for high $\mathit{We}$ impacts part of the excess dissipation is accounted for by nonlinearities in the drop shape decomposition. At low $\mathit{We}$ the drop oscillations are extremely small and this contribution is negligible. Overall, a conservative estimate yields that air film dissipation accounts only partially for the excess energy loss at the first high- $\mathit{We}$ impacts and for the complete loss during later low- $\mathit{We}$ bouncing events.

Table 2.

Squeeze dissipation in various bounces: total dissipation measured in the confined region, and percentage of excess dissipation found in figure 10.

a For high-viscosity glycerol the viscous dissipation in the drop cannot be estimated by (3.11), thus the excess dissipation cannot be determined.

4.3.1. Model description

To explain the bouncing due to a purely dissipative squeeze force at least qualitatively, we model an impacting droplet as a deformable cylindrical pill box. The droplet has radius $a$ and height $2b$ . The minimal energy surface then demands $a=b=R_{c}$ , where $2{\rm\pi}R_{c}^{3}=V$ is the volume of the droplet. The mass is given by $=2{\rm\pi}R_{c}^{3}{\it\rho}$ , and the relative height by ${\it\beta}=b/R_{c}$ : see figure 16(a). The flow field within the drop is given by ${\dot{z}}=\dot{{\it\varepsilon}}z$ and ${\dot{r}}=-(\dot{{\it\varepsilon}}r)/2$ , where $\dot{{\it\varepsilon}}={\dot{b}}/b$ . Here $r$ and $z$ are constrained to $0\leqslant r\leqslant a=\sqrt{R_{c}^{3}/b}$ and $-b\leqslant z\leqslant b$ . For a bouncing drop we find the following expressions for the kinetic energy $K$ , the potential energy $U$ , and the dissipation rate ${\dot{W}}$ , in terms of the vertical droplet length $2b$ and the height $z$ of the centre of mass:

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle K={\textstyle \frac{1}{2}}M{\dot{b}}^{2}\left({\textstyle \frac{1}{3}}+{\textstyle \frac{1}{8}}{\it\beta}^{-3}\right)+{\textstyle \frac{1}{2}}M{\dot{z}}^{2}, & \displaystyle\end{eqnarray}$$

(4.2) $$\begin{eqnarray}\displaystyle & U=2{\rm\pi}{\it\sigma}R_{c}^{2}({\it\beta}^{-1}+2{\it\beta}^{1/2}-3)+Mgz, & \displaystyle\end{eqnarray}$$

(4.3) $$\begin{eqnarray}\displaystyle & {\dot{W}}=-6{\rm\pi}{\it\mu}R_{c}{\dot{b}}^{2}{\it\beta}^{-2}+({\dot{z}}-{\dot{b}})F_{sq}(b,{\dot{b}},z,{\dot{z}}), & \displaystyle\end{eqnarray}$$

${\dot{h}}F_{sq}$

$h=z-b$

Figure 16.

Properties of the ‘pill box’ droplet with height $2b$ and radius  $a$ . (a) Sketch of the equilibrium droplet with the radius of the squeezed layer $a_{sq}$ shaded in grey. (b)  $a$ (black dashed) and $a_{sq}$ according to (4.5) (red solid) versus  $b$ . Insets: compressed and elongated droplet.

In our pill box model the bottom of the droplet is a flat disk, thus the squeeze force $F_{sq}$ is given by

(4.4) $$\begin{eqnarray}F_{sq}(b,{\dot{b}},z,{\dot{z}})=-\frac{3}{2}{\rm\pi}{\it\mu}a_{sq}^{4}(b)\frac{{\dot{z}}-{\dot{b}}}{(z-b)^{3}},\end{eqnarray}$$

where the radius of the squeezed layer $a_{sq}(b)$ in principle is given by the radius $a(b)=\sqrt{R_{c}^{3}/b}$ of the pill box. In the experiments, the droplet interface is flattening during the impact, which cannot be captured by our simple model. Instead, we implement a flat interface from the start as it ensures that the pressure increases rapidly enough to sustain a micrometre-thick air layer, in contrast to a spheroid model droplet where the pressure increases too gradually ( ${\rm\Delta}p\sim h^{-1}$ only). This simple flat disk shape obviously has its limitations in capturing the exact mechanism of momentum transfer: we demonstrated experimentally that the asymmetric downward and inward motion of the kink is critical to sustaining a repulsive force. However, in the model the asymmetry will be induced via an asymmetry of the ‘quasi-contact’ radius during approach and lift-off. A completely symmetric bounce would lead to a zero momentum transfer; bouncing can only be achieved if the ‘quasi-contact’ area is on average larger during the approach phase than during the contraction. In the model this is achieved by the large elongation in the vertical direction (see figure 16 b) of the drop at lift-off, at the cost of large oscillations during the free-flight phase. To suppress these large oscillations we implemented an additional asymmetry, $a_{sq}(b)=f(b/R_{c})\sqrt{R_{c}^{3}/b}$ , where

(4.5) $$\begin{eqnarray}f({\it\beta})=\left\{{\it\epsilon}+\frac{1-{\it\epsilon}}{1+c{\it\beta}^{25}}\right\}^{1/4}\end{eqnarray}$$

with ${\it\epsilon}=0.001$ and $c=32.38$ : see the red curve and the grey shaded ‘quasi-contact’ areas in figure 16(b). Here the function $f({\it\beta})$ and the values for ${\it\epsilon}$ and $c$ are pragmatically chosen to optimize the model for the experimentally observed restitution.

As the drop makes no direct contact with the substrate, the squeeze force is, besides gravity, the only external force acting on the droplet. Hence, the momentum equation reads

(4.6) $$\begin{eqnarray}F_{sq}=M(\ddot{z}+g).\end{eqnarray}$$

Because $\partial _{t}(K+U)={\dot{W}}$ , we obtain a second differential equation which together with (4.6) describes the behaviour of both the CM height $z$ and the deformation of the droplet in terms of $b$ (see appendix B for details):

(4.7) $$\begin{eqnarray}{\it\alpha}M\ddot{b}=-F_{sq}-6{\rm\pi}{\it\mu}R_{c}{\dot{b}}-3{\rm\pi}{\it\sigma}R_{c}{\textstyle \frac{2}{3}}({\it\beta}^{-1/2}-{\it\beta}^{-2}),\end{eqnarray}$$

where $a\approx 11/24=0.46$ . In free flight when $F_{sq}=0$ , and for small deviations of ${\it\beta}$ from unity, (4.7) reduces to a damped harmonic oscillator equation (see appendix B). The resonance frequency and quality factor of this oscillator are given by ${\it\omega}_{ff}=(3{\rm\pi}{\it\sigma}/{\it\alpha}M)^{1/2}$ and $Q=(3{\rm\pi}{\it\sigma}{\it\alpha}M)^{1/2}/6{\rm\pi}{\it\mu}R_{c}$ , respectively. In the hovering state when $z=b$ , it again reduces to a damped harmonic oscillator equation, now with ${\it\omega}_{ho}=(3{\rm\pi}{\it\sigma}/(1+{\it\alpha})M)^{1/2}$ and $Q=(3{\rm\pi}{\it\sigma}(1+{\it\alpha})M)^{1/2}/6{\rm\pi}{\it\mu}R_{c}$ . So the ratio between the oscillation frequencies in the flight and hovering phase is equal to ${\it\omega}_{ff}/{\it\omega}_{ho}=\sqrt{1/{\it\alpha}+1}=1.8$ , which is in reasonable agreement with the observed ratio $T_{ho}/T_{ff}=2.3$ (cf. figure 9). Based on last value of this ratio we should estimate $a=0.23$ instead of $a=0.46$ .

Equations (4.6) and (4.7) can be written in dimensionless form, where the dimensionless heights $Z,H,B$ are obtained by scaling $z,h$ , and $b$ with $R_{c}$ and the dimensionless time $T$ by scaling $t$ with $t_{0}=(R_{c}/g)^{1/2}$ ( ${\approx}10~\text{ms}$ for a millimetric drop):

(4.8a,b ) $$\begin{eqnarray}\ddot{Z}=-{\it\lambda}_{A}\frac{{\dot{H}}}{B^{2}H^{3}}-1\quad \text{and}\quad {\it\alpha}\ddot{B}={\it\lambda}_{A}\frac{{\dot{H}}}{B^{2}H^{3}}-{\it\lambda}{\dot{B}}-{\it\kappa}\frac{2}{3}(B^{-1/2}-B^{-2}),\end{eqnarray}$$

which depend on the four parameters ${\it\alpha},{\it\kappa},{\it\lambda}$ and ${\it\lambda}_{A}$ . Here, ${\it\kappa}=3{\it\sigma}/(2{\it\rho}gR_{c}^{2})=3/(2\mathit{Bo}_{c})$ represents the stiffness of the liquid spring, ${\it\lambda}=(3{\it\mu}/{\it\rho})(gR_{c}^{3})^{-1/2}$ the damping coefficient in the droplet and ${\it\lambda}_{A}=(3{\it\mu}_{air}/4{\it\rho})(gR_{c}^{3})^{-1/2}=1.58\times 10^{-4}$ the damping coefficient in the air.

To get better agreement with the experiments, the parameters ${\it\alpha},{\it\kappa}$ and ${\it\lambda}$ are determined as follows. The restitution of the bounce is largely determined by the effective mass in vibration; ${\it\alpha}$ is adapted using the effective mass in the potential flow assumption, ${\it\alpha}=3/(2n^{2}+n)$ . We choose $n=3$ , i.e. the lowest mode resulting in a flattened interface, leading to $a=1/7$ . Both ${\it\sigma}$ and ${\it\mu}$ are optimized to obtain the observed resonance frequency and quality factor. This yields estimates for ${\it\lambda}$ and ${\it\kappa}$ , i.e. ${\it\lambda}={\it\alpha}({\it\omega}_{0}t_{0})/Q$ and ${\it\kappa}={\it\alpha}({\it\omega}_{0}t_{0})^{2}$ . Experimentally we measure the oscillation and damping times of the second mode and find that ${\it\omega}_{0}t_{0}=6.61$ and $Q=97$ . This leads to ${\it\lambda}=1.01\times 10^{-2}$ and ${\it\kappa}=6.24$ .

4.3.2. Results: energy transfer and air film dynamics

We have solved (4.6) and (4.7) for $z$ and $b$ numerically, using the scaling just described. The result for an initial height $z_{0}/R_{c}=4$ is given in figure 17(a). The calculations show six bounces before the droplet starts and continues to hover; this phase extends infinitely as wetting is not implemented. Our main observation from figure 17(b) is that the oscillation amplitude during the flight phases is considerably larger than observed in the experiments (see also the inset of figure 9). This implies a significant transfer of energy to internal modes during lift-off of the droplet.

Figure 17.

Calculated bouncing series for a pill box droplet. (a) Time evolution of CM height $Z$ (red, upper curve) and air layer thickness $H$ (grey, lower curve). The bouncing series can be subdivided into a bouncing phase and a hovering phase (shaded blue) that shows an equilibrium sag ${\it\delta}$ . This particular simulation extends to only $T=30$ , but more hover oscillations can be obtained afterwards. (b) Half droplet height  $B$ . The first two bounces are shown, as well as a few hover oscillations with equilibrated gravitational sag. The time axis has been scaled on $t_{0}=10.2~\text{ms}$ . Compare to figures 5 and 9 respectively.

The energy transfer from the centre of mass to internal modes and backwards are shown in figure 18. Figure 18(c) shows the entire bouncing series. Similar to the experiments, the total droplet energy (black solid) shows a step decrease during each contact phase, and a gradual decrease during each flight. The latter is fully attributed to a decrease in internal droplet energy (blue) associated with a single dissipation channel, being dissipation in the internal fluid flow (grey shaded area). Indeed, the sum of the total droplet energy and the internal dissipation (black dotted) is conserved during the flight phases. However, the in-flight internal dissipation amounts to only 12 % of the total energy loss. The other 88 % is lost during the bounce phases, and fully attributed to squeeze dissipation (green shaded area). This squeeze dissipation is fully localized in time at the initial impact of each bounce. This is in line with the energy evolution near the first bounce in figure 18(a,b), where we observe a sudden conversion (to kinetic internal energy, dark blue) and dissipation of kinetic CM energy (dark red) at the approach near $T\sim 2.4$ . The abruptness is due to the diverging build-up of pressure in the air film resulting in a strongly peaked repulsive force. This divergence (see the black line in figure 18 d at $T\sim 2.4$ ) is a consequence of the ‘flat plate’ description of the air film. The force decelerates the droplet interface almost instantaneously to a height $H\approx 0.003$ which remains almost constant afterwards while the force suddenly drops, preluding a gradual squeeze-out phase. In reality the droplet interface flattens from an initial spherical shape, which smooths the transition: both the dissipation (figure 15) and the interaction force (figure 6) show a much broader distribution. After the maximum lateral stretch of the model droplet at $T\sim 2.8$ the CM energy initially increases towards its former value, but decreases again shortly before detachment at $T\sim 3.2$ : see the red line in figure 18(a). Because the lower side of the drop moves upwards now, a small attractive force converts CM energy back into internal oscillations and the drop stretches vertically: see the blue line in figure 18(b) at $T\sim 3.2$ . This attractive force is indicated by the black dashed line in figure 18(d). In reality a very small attractive force occurs only at the final lift-off, if at all (see figures 6 and 14 c), resulting in a smaller oscillation amplitude and a higher restitution coefficient.

Figure 18.

Energy conversion and air film dynamics. (a,b) Details of the first bounce showing the conversion of the kinetic (dark solid) and potential (dark dashed) contributions to (a) the total CM energy (red) and (b) the total internal energy (blue); (a) also shows the squeeze dissipation $\int {\dot{W}}\,\text{d}t|_{squeeze}$ (green, lower curve). (c) Total energy (black solid) and the sum (black dotted) of the total energy plus the internal dissipation $\int {\dot{W}}\,\text{d}t|_{drop}$ (grey shaded region), and squeeze dissipation (green shaded region). $E_{0}$ is the initial total energy. (d) Time evolution of the air film thickness $H$ (green), CM height $Z$ (red), and squeeze force $F$ (black) during the first bounce. The squeeze force becomes attractive in the final stage so $-F$ is plotted instead (dashed).

The single oscillation mode model thus successfully captures the energy conversion and dissipation during a non-wetting bounce; and at the same time it highlights the importance of asymmetry in the air film evolution for obtaining a high-restitution bounce. In the model the asymmetry is introduced by (4.5), which suppresses the attractive force at lift-off. This attraction would otherwise lead to large free oscillations at the cost of droplet momentum.