3.1. Shape evolution of the air disc

Figure 2 shows examples of the interference fringes for different impact velocities and drop sizes. Figure 2(a) uses bottom reflection interferometry, which gives us better image contrast, compared to the top transmission-light interference fringes shown in figure 2(b,c). Furthermore, the liquid–solid contact zone in the right-most panel of figure 2(a) appears dark as most of the light passes directly through this liquid–solid interface and shines away from the camera. On the other hand, this contact zone is bright in figure 2(b,c) as the light passes through the liquid–solid interface towards the camera. The wiggled circular lines in the right-most column represent the location of the inner contact lines at the edge of the enclosed air disk. The smooth circular lines in the upper right corners of figure 2(a,b) are the outer edges of the wet region.

Keep in mind that we use the bottom radius of curvature, $R_{b}$ , as the characteristic drop dimension, to account for the large shape oscillations following the release from the nozzle, as has been highlighted in other impact studies (Thoroddsen et al. Reference Thoroddsen, Etoh, Takehara, Ootsuka and Hatsuki2005; Hicks & Purvis Reference Hicks and Purvis2010; Liu et al. Reference Liu, Tan and Xu2013; Thoraval et al. Reference Thoraval, Takehara, Etoh and Thoroddsen2013; Wang, Kuan & Tsai Reference Wang, Kuan and Tsai2013). The shape at impact was monitored from a side view, with a separate high-speed video camera (Phantom V1610), at 20 kf.p.s. and with a $3~{\rm\mu}\text{s}$ exposure. Figure 1(b) shows some of the varied shapes of drops studied. The fringes remain axisymmetric, both for top and bottom illumination. Note that for the largest $V$ , at first contact there are many fewer fringes, which indicates a thinner air disc.

Figure 3. The air-layer thickness profiles under the drops, measured from the interferometric video clips at 5 Mf.p.s. The black dashed horizontal line marks the surface of the solid glass substrate. (a–c) For $V=2.43~\text{m}~\text{s}^{-1}$ and $R_{b}=8.2~\text{mm}$ ( ${\it\epsilon}^{-1}=6.0$ , $St=9.2\times 10^{-7}$ ). (a) The bottom deformation of the approaching drop. The black curves are spaced by 400 ns, with the bottom curve at $2.2~{\rm\mu}\text{s}$ before touchdown, which is shown by the red curve. The dots mark the centre locations of the dark or bright fringes, with interpolations at the centreline and at the tip of the downwards kink in (b, f). (b) Detail of the formation of the bottom dimple and touchdown of the outer kink, to entrap the air disc. Note the different vertical scales. The dashed red curve denotes the last profile in (a). Profiles spaced by 200 ns. (c) Rapid expansion of the air disc, after first contact. Profiles spaced by 400 ns. The thickness of the edge of the contracting air disc cannot be measured and we simply draw it as a flat segment, but it is much thicker than the disc, see Thoroddsen et al. (Reference Thoroddsen, Etoh, Takehara, Ootsuka and Hatsuki2005), Lee et al. (Reference Lee, Weon, Je and Fezzaa2012) and Liu et al. (Reference Liu, Tan and Xu2013). (d) Shapes for $V=4.05~\text{m}~\text{s}^{-1}$ and $R_{b}=3.2~\text{mm}$ ( ${\it\epsilon}^{-1}=14.4$ , $St=1.4\times 10^{-6}$ ). Profiles shown at $t=-2$ , $-1.8$ , $-1.6$ , $-1.4$ , $-1.2$ , $-1$ , $-0.8$ , 0, $2.2~{\rm\mu}\text{s}$ relative to first contact. In this case the last three profiles before contact (not shown) cannot be determined accurately. The open red squares show the shape of the air disc at first contact and the open blue circles at the end of the rapid expansion. (e, f) Shapes for $V=1.06~\text{m}~\text{s}^{-1}$ and $R_{b}=4.1~\text{mm}$ ( ${\it\epsilon}^{-1}=0.69$ , $St=4.3\times 10^{-6}$ ). (e) Bottom deformation of the drop, at $t=-22$ , $-20$ , $-18$ , $-16$ , $-14$ , $-12.6$ , $-11$ and $-10.4~{\rm\mu}\text{s}$ before contact (red curve). (f) Closeup of central dimple and downwards kink. Times shown at $t=-10.4$ , $-9.8$ , $-9.2$ , $-8.6$ , $-8$ , $-7.4$ , $-6.4$ , $-5.6$ , $-4.6$ $-3.8$ , $-2.8$ , $-1.6$ and $0~{\rm\mu}\text{s}$ relative to first contact (red curve).

Figure 4. The radial velocity of the minimum thickness kink in the bottom surface, which first touches the substrate, compared with the theoretically predicted power law from (3.1). The data encompass the entire range of impact velocities, including the compressible regime.

Figure 3 shows the surface shapes, $h(r,t)$ , extracted from the fringes, when the drop approaches, deforms and contacts the glass substrate. The deceleration of the drop bottom is very abrupt, the drop retaining at least 80 % of $V$ to within $10~{\rm\mu}\text{m}$ from the solid surface. The measured deceleration of the bottom of the drop for $V=4.05~\text{m}~\text{s}^{-1}$ is $\simeq 3\times 10^{5}~\text{g}$ . Then the motion at the axis of symmetry reverses and a dimple forms under its centre, which is bounded at the outer edge by a kink in the surface, highlighted schematically in figure 1(a). The kink moves radially outwards at rapid horizontal velocity $U$ . This kink makes the first circular contact with the solid, thereby entrapping the air disc (Korobkin et al. Reference Korobkin, Ellis and Smith2008; Mandre et al. Reference Mandre, Mani and Brenner2009; Duchemin & Josserand Reference Duchemin and Josserand2011; Hicks & Purvis Reference Hicks and Purvis2013). From the video clips we can measure $U$ . In figure 4 we plot this velocity normalized by the drop impact velocity. For the largest impact velocity, $V=6~\text{m}~\text{s}^{-1}$ , we get $U/V\simeq 50$ , which is even larger than the velocity of the ejecta moving along the substrate, see Thoroddsen, Takehara & Etoh (Reference Thoroddsen, Takehara and Etoh2012).

Mandre et al. (Reference Mandre, Mani and Brenner2009) found a self-similar solution of the simplified equations for the shape and radial velocity of this kink at the outer edge of the dimpled region. This solution makes contact with the solid surface in finite time and the horizontal velocity of the kink location is dominated by radial advection within the liquid. The similarity gives the scaling $U\sim V(h_{min}/R)^{-1/2}$ , where $h_{min}$ is the minimum gap under the kink, while $h_{min}/R\sim St^{2/3}$ , resulting in

(3.1) $$\begin{eqnarray}U\sim V\;St^{-1/3}\end{eqnarray}$$

with a prefactor of $O(1)$ . Here $St$ is defined as the inverse of the conventional Stokes number, i.e.

(3.2) $$\begin{eqnarray}\displaystyle St=\frac{{\it\mu}_{g}}{{\it\rho}_{l}VR_{b}}. & & \displaystyle\end{eqnarray}$$

Our data in figure 5 show good correspondence with this scaling with a prefactor of 0.42. Mandre & Brenner (Reference Mandre and Brenner2012) (their figure 3) show the same power law with a similar prefactor of 0.34 for the 2-D case, at lower impact velocities. See also the axisymmetric simulations of the surface shapes by Duchemin & Josserand (Reference Duchemin and Josserand2011).

The shape evolution shown in figure 3(b, f) is very similar to those obtained from solutions of the lubrication theory, see for example figure 2 in Duchemin & Josserand (Reference Duchemin and Josserand2011) and figure 2(a) in Hicks & Purvis (Reference Hicks and Purvis2013).

The subsequent contraction of the air disc, after liquid–solid contact, into a bubble was successfully modelled by Thoroddsen et al. (Reference Thoroddsen, Etoh, Takehara, Ootsuka and Hatsuki2005) assuming an increasing but uniform disc thickness. However, the edge of the contracting air disc grows in size, forming a rim. Liu et al. (Reference Liu, Tan and Xu2013) improved upon this model by taking into account this thickening of the edge.

3.2. Size of the initial air disc

The deformation of the bottom of the drop, leading to the dimple formation and entrapment of the air disc, is governed by a balance between the lubrication pressure in the air trying to escape the oncoming drop $P\sim {\it\mu}_{g}VR_{b}/H^{2}$ , and the inertia of the drop, which must be decelerated before contact, ${\it\rho}_{l}V^{2}/H^{\ast }$ , where $H^{\ast }$ is the thickness of the air layer at the start of drop deformation. This gives a layer height $H^{\ast }\sim R_{b}St^{2/3}$ . Based on this, one can use simple geometric arguments for the horizontal size of the air disc ( $L=\sqrt{R_{b}H}$ , Mandre et al. Reference Mandre, Mani and Brenner2009) to get $L/R_{b}\sim St^{1/3}$ , in the incompressible case. However, the horizontal extent of the air disc, where the drop contacts the solid, is determined by the touchdown of the kink in the free surface, described by the above-mentioned similarity solution from Mandre et al. (Reference Mandre, Mani and Brenner2009). The drop liquid deforms inviscidly, funnelling liquid away from the central stagnation point, thereby increasing the velocity components of the kink, which in turn is resisted from contact by the increased pressure in the air layer. This is repeated on ever smaller length scales until contact is made in finite time, see discussion in Mandre & Brenner (Reference Mandre and Brenner2012).

Theoreticians prefer to tackle the 2-D configuration (Mandre et al. Reference Mandre, Mani and Brenner2009; Hicks & Purvis Reference Hicks and Purvis2013), representing cylindrical drops. While their scaling arguments carry over from 2-D, only the axisymmetric theory allows quantitative comparison to experiments. Hicks & Purvis (Reference Hicks and Purvis2010) formulated such a theory for the incompressible case, where they predicted the initial contact radius of the disc,

(3.3) $$\begin{eqnarray}L_{o}=3.8\left(\frac{4{\it\mu}_{g}}{{\it\rho}_{l}V}\right)^{1/3}R_{b}^{2/3}.\end{eqnarray}$$

Hicks et al. (Reference Hicks, Ermanyuk, Gavrilov and Purvis2012) showed this to be in reasonable agreement with impacting solid spheres and the available droplet data of Thoroddsen et al. (Reference Thoroddsen, Etoh, Takehara, Ootsuka and Hatsuki2005), without any adjustable constants. However, while their compressible theory (Hicks & Purvis Reference Hicks and Purvis2013) is limited to 2-D, they have also shown that $L_{o}$ is not dependent on the gas compressibility. This is indeed born out by our data, in figure 5, which shows a perfect fit of $L_{o}$ to their expression, over a large range of impact values. This contradicts earlier results by Liu et al. (Reference Liu, Tan and Xu2013), where smaller water drops were used and $L_{o}$ was essentially independent of $V$ .

Hicks & Purvis (Reference Hicks and Purvis2013) mention that the adiabatic case of Mandre et al. (Reference Mandre, Mani and Brenner2009) predicts that $L_{o}$ should decrease when ${\it\epsilon}^{-1}$ increases, where ${\it\epsilon}$ is the compressibility factor, defined below. Teasing out these subtle differences will require future experiments at reduced pressure.

Figure 5. The size of the initial air disc, measured at first contact with the wall, versus (3.3), which is based on the theory of Hicks et al. (Reference Hicks, Ermanyuk, Gavrilov and Purvis2012).  $R_{b}$ is the bottom radius of curvature of the drop.

3.3. Thickness of the disc and compression of the gas

Building on previous incompressible theory (Smith et al. Reference Smith, Li and Wu2003; Korobkin et al. Reference Korobkin, Ellis and Smith2008), Mandre et al. (Reference Mandre, Mani and Brenner2009) derived conditions for when compressibility of the gas will come into play for higher impact velocities. This formulation compares the static atmospheric pressure to the lubrication pressure, the ratio of which forms a compressibility factor

(3.4) $$\begin{eqnarray}{\it\epsilon}=\frac{P_{atm}}{(RV^{7}{\it\rho}_{l}^{4}/{\it\mu}_{g})^{1/3}}.\end{eqnarray}$$

If ${\it\epsilon}^{-1}>3$ one expects significant compression of the air disc and thereby smaller values of $H^{\ast }$ . According to Mandre et al. (Reference Mandre, Mani and Brenner2009), when using adiabatic compression, the normalized thinning should scale with the compressibility factor as $H^{\ast }/(R_{b}St^{2/3})$ $=3.2{\it\epsilon}^{1/3}$ . For isothermal compression the power law is different, i.e. ${\sim}{\it\epsilon}^{1}$ . Our values of ${\it\epsilon}^{-1}$ reach ${\sim}40$ for the large impact velocities $V$ and large bottom radii of curvature $R_{b}$ , which takes us well into the compressible regime.

In figure 6 we plot the air-layer height at the centreline $h=H^{\ast }$ , at the instant of first contact. Note that our definition of $H^{\ast }$ , taken at first contact, differs from that used in Smith et al. (Reference Smith, Li and Wu2003) and Mandre et al. (Reference Mandre, Mani and Brenner2009), where they look at $H^{\ast }$ at the onset of deformation. The two should only differ by a small factor, but our definition is much easier to quantify experimentally. The data fit the theory and numerics of Mandre et al. (Reference Mandre, Mani and Brenner2009) reasonably closely, when we use their adiabatic case, ${\it\gamma}=1.4$ (see their figure 2). Our incompressible asymptote is at 3.4 versus 4.3 in their 2-D case. Our measurements show that the thinning of the air layer, as compressibility sets in, follows an empirical power law of $4.2{\it\epsilon}^{0.40}$ . This mirrors closely their numerical results when using the ratio of specific heats ${\it\gamma}=1.4$ , for an adiabatic process, where they see essentially the same exponent of 0.40. This is significantly steeper than their theoretical prediction of an asymptote of ${\it\epsilon}^{1/3}$ , which may be realized at much larger ${\it\epsilon}^{-1}$ than in our experiments. Here again our prefactor differs from the value of 3.2 observed in their 2-D case. Liu et al. (Reference Liu, Tan and Xu2013) observed a much smaller exponent for their impacts of water drops. Note that our results are far from the isothermal ( ${\it\gamma}=1$ ) power law ${\sim}{\it\epsilon}^{1}$ , which is also drawn in the figure.

Selecting the instant of first contact will not show the minimum value of $H^{\ast }$ , as figure 3(b,d, f) shows that the height at the centre has rebounded somewhat from its minimum. This is also seen in theoretical studies (Mandre et al. Reference Mandre, Mani and Brenner2009; Duchemin & Josserand Reference Duchemin and Josserand2011; Hicks & Purvis Reference Hicks and Purvis2013). We estimate the minimum centreline separation for the $V=4.05~\text{m}~\text{s}^{-1}$ video (figure 3 d), as being $\simeq 480~\text{nm}$ . For the highest impact velocity $V\simeq 6~\text{m}~\text{s}^{-1}$ we estimate the minimum height at only one dark and one bright fringe, using top lighting, i.e. $H^{\ast }\simeq 320~\text{nm}$ .

Figure 6. The air-film thickness at the centreline, measured at the instant of contact along the outer ring, versus the compressibility parameter in (3.4). Compared to the theory of Mandre et al. (Reference Mandre, Mani and Brenner2009) for isothermal conditions ( ${\it\gamma}=1$ ) (blue dashed line) and for adiabatic conditions ( ${\it\gamma}=1.4$ ) (red dashed line).

Immediately following the contact, the trapped air disc undergoes a rapid expansion, see figure 3(c,d). For $V=2.43~\text{m}~\text{s}^{-1}$ the height at the centre has risen by a factor of ${\sim}2$ in $5.6~{\rm\mu}\text{s}$ after first contact, while for $V=4.05~\text{m}~\text{s}^{-1}$ it has grown by a factor of 3 in only $2.2~{\rm\mu}\text{s}$ . This highlights the importance of our rapid imaging, to catch the detailed dynamics. At the end of this expansion, we see some overshoot and slight oscillations. For $V=2.43~\text{m}~\text{s}^{-1}$ , in figure 3(c), the time from first contact to first maximum is ${\rm\Delta}T=5.6~{\rm\mu}\text{s}$ , while the duration until the subsequent expansion maximum after the last curve in figure 3(c) is ${\rm\Delta}T=15.6~{\rm\mu}\text{s}$ . For the highest impact velocity ( $V=6.0~\text{m}~\text{s}^{-1}$ ) the expansion is faster and we can observe more oscillation cycles, with each subsequent expansion cycle to subsequent maxima, taking longer, i.e. ${\rm\Delta}T=0.4$ , 3.4, 4.4 and $5.2~{\rm\mu}\text{s}$ .

Figure 7. Compression of the air disc versus the impact parameter in (3.4). The compression of the air disc at the first contact along the ring (▫) and after the first vertical expansion at the centreline (▵). ${\it\Omega}_{disc}$ comes from integrating the thickness profile from the interferometry, whereas ${\it\Omega}_{b}$ is the volume of the final bubble, left behind after the contraction of the air disc.

We integrate the air volume at first entrapment and after the maximum early expansion, from the video frames. This can be compared to the theory of Mandre et al. (Reference Mandre, Mani and Brenner2009), which gives the air-film thickness at the start of compression as $H_{c}\sim R({\it\mu}_{g}V/(RP_{0}))^{1/2}=R\,St^{2/3}{\it\epsilon}^{-1/2}$ and maximal compression at $H_{min}\sim R\,St^{2/3}{\it\epsilon}^{(2-{\it\gamma})/(2{\it\gamma}-1)}=R\,St^{2/3}{\it\epsilon}^{1/3}$ , for ${\it\gamma}=1.4$ ; $P_{0}$ is the ambient pressure. The ratio of these two gives the volume compression ${\it\Omega}_{b}/{\it\Omega}_{disc}\sim {\it\epsilon}^{-5/6}$ , or incorporating our experimentally obtained empirical result $H^{\ast }\sim {\it\epsilon}^{0.40}$ we get a slightly different power law ${\it\Omega}_{b}/{\it\Omega}_{disc}\sim {\it\epsilon}^{-0.9}$ . Figure 7 shows reasonable correspondence with this theory, with the ratio growing as large as a factor of 5 (▵) for the largest $V$ . However, comparing this expanded air volume to that contained in the final spherical bubble detached from the surface, additional volume increase is observed (▫ in figure 7). This was also observed by Liu et al. (Reference Liu, Tan and Xu2013), but their lower frame rate could not catch the earliest part of the expansion. With $\text{d}t=7~{\rm\mu}\text{s}$ their images are essentially snapshots of the earliest evolution. We propose that the conical air disc is still subject to significant dynamic pressure, due to the ongoing impact so soon after first contact. This pressure is released later as the disc pulls into a bubble, further increasing the volume ratios. We observe a total compression as large as a factor of 14 for the largest $V$ .

The very large pressure produced just prior to the first contact could potentially deform the solid locally. To probe possible effects on the air disc, we conducted a few separate experiments using a much thinner, $150~{\rm\mu}\text{m}$ -thick, glass slide (Menzel-Deckglacer), seeing no change in the thickness of the entrapped air disc.