2.1 Experimental set-up and procedure

Our experimental set-up consists of a linear three-axis translation stage placed on an optical table, exchangeable flat substrates in Plexiglas boxes, a high-speed camera (Photron Fastcam SA1.1 Model 675K-M1) and a syringe pump (KD Scientific KDS 210 CE). To better visualize the liquid surface, boxes with square cross-section were employed. The box contains a layer of Milli-Q water, having a contact angle of close to $90^{\circ }$ with the Plexiglas walls (see table 1). A sketch of the set-up is displayed in figure 1(a).

Figure 1.

(a) Schematic illustration of the experimental set-up with a needle connected to a syringe pump controlling the flow rate. Time lapse images of the collapse of a hole at the centre of a liquid layer on a bounded superhydrophobic Al plate are shown in top (b) and side (c) view, which are results from two different experiments. The top view indicates that the hole collapses with a circular shape. The side view (the dark regions are air and the light regions are water) suggests that the thickness of the liquid layer does not change appreciably relative to the situation before the collapse of the hole. The scale bars represent 10 mm.

We first carried out experiments using etched and silanized aluminium (Al) plates (Song et al. Reference Song, Xu, Liu, Lu and Sun2012) to obtain a stable superhydrophobicity. On this surface, the advancing contact angle for water is $\unicode[STIX]{x1D703}_{a}=167.5^{\circ }\pm 0.7^{\circ }$ with a contact angle hysteresis of $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}=3.5^{\circ }\pm 0.9^{\circ }$ , which guarantees a small contact line friction. In an experiment, we fix the Al plate at the bottom of the box and add a thin layer of water with an initial height ${\sim}h_{\infty }/2$ , after which we create a comparatively large hole at the centre of the substrate employing clean compressed air flow through a nozzle having an inner diameter of 3 mm. Here, $h_{\infty }=2l_{c}\sin (\unicode[STIX]{x1D703}/2)$ is the thickness of an infinite unbounded liquid layer (de Gennes et al. Reference de Gennes, Brochard-Wyart and Quéré2003), $l_{c}\approx 2.73$  mm is the capillary length of water and $\unicode[STIX]{x1D703}$ is the contact angle on the substrate. The outlet of the nozzle is typically ${\sim}2$  cm away from the substrate and faces its central spot. When the hole has been created and reaches a circular shape, we stop the air flow. Shortly after that, the liquid film with the hole becomes static. Subsequently, using the syringe pump, we smoothly (flow rate $Q=0.1~\text{ml}~\text{min}^{-1}$ ) add more water without any appreciable perturbation to the liquid layer. Six different boxes with a side length $L$ ranging from 7 cm to 19.5 cm were used. In the smallest box ( $L=7$  cm), the height of the liquid layer only increases by $Q/L^{2}\approx 20.4~\unicode[STIX]{x03BC}\text{m}~\text{min}^{-1}$ (or ${\approx}0.3~\unicode[STIX]{x03BC}\text{m}~\text{s}^{-1}$ ), which suggests that the liquid layer is very close to its static configuration while liquid is supplied. Therefore, the influence of the flow rate on the liquid layer is negligible.

2.2 Experimental observations

The time evolution of the hole is shown in figure 1(b,c) in top and side view, respectively. $t=0$ s is defined as the moment when the recording starts. During the experiments, imaging is typically performed with 500 frames per second (f.p.s.) to guarantee a long-time recording. In the regime of high-speed processes, i.e. after the collapse sets in, 73 000 f.p.s. are usually applied, but with a smaller field of view (fov). First, we observe that the size of the hole decreases smoothly maintaining a circular shape. In this state, if we stop adding water, the hole remains open and static. However, when the hole diameter $d$ reaches a specific value  $d_{c}$ , the hole suddenly collapses (see supplementary movie 1 available at https://doi.org/10.1017/jfm.2018.680). This scenario is very different from the one studied by Taylor & Michael (Reference Taylor and Michael1973) or Bankoff et al. (Reference Bankoff, Johnson, Miksis, Schluter and Lopez2003), in which the authors first produced a liquid layer with a specific thickness, and then created a hole by removing an object which was placed in advance at the centre of the liquid layer or by blowing air using a capillary tube.

Before we go over to study the closure of the liquid film in more detail, it will be useful to provide an intuitive understanding of the key factors the hole collapse depends on. There are two principal curvatures of the liquid surface. The meridional curvature (see the cross-section in figure 1 c) creates a pressure jump favouring the opening of the hole. However, the azimuthal curvature or ‘hoop curvature’ (see the top view in figure 1 b) and the hydrostatic pressure push the liquid to the centre. The side views in figure 1(c) show that the liquid surface profiles are similar to each other at different instances, which indicates that the variations of the meridional curvature and the hydrostatic pressure are not pronounced. Therefore, the collapse of the liquid layer is caused by the pronounced increase of the hoop curvature as additional liquid is supplied.

As can be seen in side view (figure 1 c, see supplementary movie 2), when the hole closes, the remaining air is divided into two parts, i.e. an air bubble remains at the surface ( $t=1.855$  s). Subsequently, the air bubble is compressed to a pancake shape, which is very similar to the spreading of air bubbles on superhydrophobic substrates reported very recently (de Maleprade, Clanet & Quéré Reference de Maleprade, Clanet and Quéré2016). In the following, the air bubble disappears. However, in experiments carried out on other substrates (e.g. Teflon, see supplementary movie 3) the air bubble remains, which indicates that on the Al surface the air goes to the micro- and nanoscopic cavities and might be dissolved. Finally, the liquid film at the coalescence point grows, and the whole liquid layer reaches a uniform thickness. The thickness after hole collapse is not significantly different from the thickness before the onset of the collapse.

Figure 2.

Time evolution of holes in liquid layers. The plot shows the instantaneous diameter $d$ of the hole as a function of time $(t_{0}-t)$ on superhydrophobic Al surfaces. $t_{0}$ is defined as the moment when $d=0$ . All the data points are average values of five experiments with error bars representing the standard deviation. Six boxes with a side length $L$ ranging from 7 cm to 19.5 cm have been employed.

In order to quantify the dynamics of the hole, we plot the relationship between the instantaneous value of $d$ and time $(t_{0}-t)$ for a large range of box sizes, denoting $t_{0}$ the moment when $d=0$ , as shown in figure 2. The instantaneous diameter $d$ of the hole is determined as the minimum value of the neck of the air phase (the dark regions visible in figure 1 c). The data points in the relationship between $d$ and $(t_{0}-t)$ were obtained by averaging five experiments, with corresponding error bars. Figure 2 clearly demonstrates the existence of two regimes, namely a region with a small slope and a region with a higher slope. In the former regime, since the hole remains open with a specific static diameter $d$ when the feeding is stopped, the shrinking is caused by the flow due to the pump. We refer to this situation as the hole being stable. On the contrary, in the latter regime the hole starts to close even as the feeding is stopped, so the hole is unstable. The methods employed in most of the previous research only capture the latter regime, for example by removing a barrier in the central region of the liquid film (e.g. Taylor & Michael Reference Taylor and Michael1973; Diez et al. Reference Diez, Gratton and Gratton1992; Bankoff et al. Reference Bankoff, Johnson, Miksis, Schluter and Lopez2003; Zheng et al. Reference Zheng, Fontelos, Shin, Dallaston, Tseluiko, Kalliadasis and Stone2018a ,Reference Zheng, Fontelos, Shin and Stone b ).

The universality of the above results was checked through experiments with substrates of different wettabilities. Apart from the superhydrophobic Al plates, three other substrates, namely liquid infused glass ( $\unicode[STIX]{x1D703}_{a}=97.6\pm 2.3^{\circ }$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}=4.6\pm 2.0^{\circ }$ ) (Eifert et al. Reference Eifert, Paulssen, Varanakkottu, Baier and Hardt2014) Teflon ( $\unicode[STIX]{x1D703}_{a}=125.6\pm 0.7^{\circ }$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}=31.9\pm 1.5^{\circ }$ ) and silicon ( $\unicode[STIX]{x1D703}_{a}=55.9\pm 1.8^{\circ }$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}=16.8\pm 1.4^{\circ }$ ), were employed (see table 1). For this purpose, all experiments were performed in the same square box ( $L=7$  cm). For the liquid-infused glass and Teflon surfaces, the diameter $d$ as the function of time $t$ shows a similar behaviour as in figure 2. As already indicated, after the closure of the hole, an air bubble remains on the Teflon (see supplementary movie 3) and glass substrates.

Figure 3.

Evolution of $d$ with time $(t_{0}-t)$ on a silicon substrate in a square box of $L=7$  cm. (a) The red dots are experimental results, and the dashed line is the prediction for $d_{c}$ based on the asymptotic analytical solution (see § 3.2). (b) Selected frames from another experiment in top view. The scale bar represents 10 mm. The brightness (in sharp contrast to the other parts) around the three-phase contact line results from the reflection of the light source.

Particularly, we would like to take a more detailed look at the hole on the silicon wafer (see supplementary movie 4). The experimental procedure is the same as described in § 2.1. In figure 3(a), $d$ is plotted as a function of ( $t_{0}-t$ ) for the $L=7$  cm box. Owing to the higher contact angle hysteresis (see table 1), the curve for the silicon wafer is not as smooth as for the Al surface (see figure 2). Before $d$ reaches $d_{c}$ , several pinning/depinning events occur (i.e.  $(t_{0}-t)\in [2,4]$ s, $[8,10]$ s, $[12,14]$ s). However, after reaching $d_{c}$ (the dashed line in figure 3 a), a smoother dynamics is observed. Selected frames are displayed in figure 3(b), showing the hole shapes on the silicon substrate in top view, indicating that the solid–liquid–gas three-phase contact lines deviate from a circular shape. This is the reason for the larger error bars in the experiments on silicon wafers in comparison to the other substrates, as shown in figure 3(b). Surprisingly, even if the contact angle hysteresis on Teflon is higher than that on silicon, a smoother evolution of the contact line was observed on Teflon (see supplementary movie 3). So far, the reasons behind that are unclear.

3.1 Numerical solution of the Young–Laplace equation

Considering that the contact angle between water and the box walls is close to $90^{\circ }$ , we assume on the one hand that the perturbation caused by the boundaries/edges of the box does not influence the shape of the liquid surface very much. On the other hand, we assume that the shape (square or circular) of the box does not affect the stability of the liquid layer. This assumption is verified in appendix B.

Including gravity, the Young–Laplace equation takes the following form in an axisymmetric coordinate system,

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle -\left(\frac{\text{d}\unicode[STIX]{x1D711}}{\text{d}s}+\frac{\sin \unicode[STIX]{x1D711}}{r}\right)\unicode[STIX]{x1D6FE}=\unicode[STIX]{x0394}p_{0}-\unicode[STIX]{x1D70C}gz, & \displaystyle\end{eqnarray}$$

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}r}{\text{d}s}=\cos \unicode[STIX]{x1D711}, & \displaystyle\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}z}{\text{d}s}=\sin \unicode[STIX]{x1D711}, & \displaystyle\end{eqnarray}$$

where $s$ is the arc-length along the liquid surface, and $\unicode[STIX]{x1D711}$ is the tangential angle (i.e. $\unicode[STIX]{x1D711}=\arctan (\text{d}z/\text{d}r)$ ), as shown in figure 4(a). Here, we choose $s$ and $\unicode[STIX]{x1D711}$ as the only two variables parametrizing the surface. For convenience, by introducing the capillary length $l_{c}$ , we define dimensionless forms of the relevant geometrical and physical quantities,

(3.5a-d ) $$\begin{eqnarray}\tilde{r}=\frac{r}{l_{c}},\quad \tilde{z}=\frac{z}{l_{c}},\quad \tilde{s}=\frac{s}{l_{c}},\quad \unicode[STIX]{x0394}\tilde{p}=\frac{\unicode[STIX]{x0394}pl_{c}}{\unicode[STIX]{x1D6FE}},\end{eqnarray}$$

and we rewrite (3.2)–(3.4) into

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle -\left(\frac{\text{d}\unicode[STIX]{x1D711}}{\text{d}\tilde{s}}+\frac{\sin \unicode[STIX]{x1D711}}{\tilde{r}}\right)=\unicode[STIX]{x0394}\tilde{p}_{0}-\tilde{z}, & \displaystyle\end{eqnarray}$$

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}\tilde{r}}{\text{d}\tilde{s}}=\cos \unicode[STIX]{x1D711}, & \displaystyle\end{eqnarray}$$

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}\tilde{z}}{\text{d}\tilde{s}}=\sin \unicode[STIX]{x1D711}, & \displaystyle\end{eqnarray}$$

with three initial conditions $\unicode[STIX]{x1D711}|_{\tilde{s}=0}=\unicode[STIX]{x1D703}$ , $\tilde{r}|_{\tilde{s}=0}=\tilde{d}/2$ , $\tilde{z}|_{\tilde{s}=0}=0$ . To solve these three first-order ordinary nonlinear and coupled differential equations, a shooting method was employed. Our shooting method is based on changing the values of the two unknown parameters $\tilde{s}_{tot}$ and $\unicode[STIX]{x0394}\tilde{p}_{0}$ until the following two boundary conditions are satisfied,

(3.9a,b ) $$\begin{eqnarray}\tilde{V}|_{\tilde{s}=\tilde{s}_{tot}}=\tilde{V}_{0},\quad \unicode[STIX]{x1D711}|_{\tilde{s}=\tilde{s}_{tot}}=0,\end{eqnarray}$$

in which $V_{0}$ is the volume of the liquid for a specific configuration (which corresponds to a specific value of $d$ for a static state), and $s_{tot}$ is the total arc-length. The calculations were performed using the Matlab function ode45. Figure 4(b) demonstrates good agreement between the theory and the experimental results (more comparisons for various values of $\unicode[STIX]{x1D703}$ are shown in the supplementary material). After the solutions for the surface profile have been determined, other relevant geometrical and physical quantities such as the curvature at an arbitrary point and each contribution to the total energy of the system (i.e. surface energy and gravitational energy) can also be obtained.

Figure 4.

(a) Axisymmetric liquid layer profile shown in side view, $d$ is the diameter of the hole and $s$ is the arc-length along the liquid surface. As shown in the inset, the tangential angle $\unicode[STIX]{x1D711}$ is defined as $\unicode[STIX]{x1D711}=\arctan (\text{d}z/\text{d}r)$ . (b) Comparison between the numerical results of § 3.1 (red hollow circles) and the experimental results. The substrate is the superhydrophobic Al plate with $\unicode[STIX]{x1D703}_{a}=167.5^{\circ }$ , the box size is $L=14.5$  cm and $d=26.1$  mm in the current frame. The scale bar represents 5 mm.

In figure 5, two cases are studied: (i) liquid layers on a completely non-wetting surface (i.e.  $\unicode[STIX]{x1D703}=180^{\circ }$ ) in six circular boxes with various diameters $L$ ranging from 7 cm to 19.5 cm, which is much larger than the capillary length  $l_{c}$ ; (ii) liquid layers in a (circular) box with diameter $L=14.5$  cm, but with varying contact angles ranging from $\unicode[STIX]{x1D703}=5^{\circ }$ to $\unicode[STIX]{x1D703}=180^{\circ }$ . The data in figure 5 are normalized by  $l_{c}$ . For each curve, when the hole diameter is large, the liquid volume $V$ increases with decreasing  $d$ . However, a maximum volume $V_{c}$ exists, corresponding to a critical value of  $d_{c}$ . Based on the minimum-volume theorem formulated in Langbein (Reference Langbein2002), the point ( $V_{c},d_{c}$ ) corresponds to a point of instability. Up to now, the critical diameter has only been obtained based on a small contact angle approximation (Moriarty & Schwartz Reference Moriarty and Schwartz1993). If $d<d_{c}$ , solutions of (3.1) exist mathematically, but they are unstable (Moriarty & Schwartz Reference Moriarty and Schwartz1993) and do not correspond to any physically observable configuration. In our experiments, the volume of the liquid film is increased monotonously until it reaches the maximum  $V_{c}$ . Since the syringe pump is always running, the liquid volume finally exceeds  $V_{c}$ , reaching a regime where no stable solution of the Young–Laplace equation exists, so the hole collapses. Furthermore, our results indicate the existence of a finite value of  $d_{c}$ , no matter what the size of the box or the contact angle on the substrate is. If we increase the box size beyond all limits, our results converge to the results obtained by Taylor & Michael (Reference Taylor and Michael1973) with $h=2l_{c}\sin (\unicode[STIX]{x1D703}/2)$ , suggesting a stable infinite hole diameter (see figure 8 in Taylor & Michael Reference Taylor and Michael1973).

Figure 5.

Numerical solutions of (3.1) showing the dependence of $V$ on $d$ when varying the size of the box and the wettability of the substrate. The data are normalized by the capillary length  $l_{c}$ . (a) The contact angle is fixed, i.e.  $\unicode[STIX]{x1D703}=180^{\circ }$ . Circular boxes with six different diameters $L$ are employed, i.e. $L=7.0$  cm, 9.5 cm, 12.0 cm, 14.5 cm, 17.0 cm, 19.5 cm. $V_{c}$ and $d_{c}$ denote the critical values of the volume and the hole diameter when the instability occurs. (b) The box diameter is fixed to $L=14.5$  cm, and contact angles of $\unicode[STIX]{x1D703}=5^{\circ }$ , $30^{\circ }$ , $60^{\circ }$ , $90^{\circ }$ , $120^{\circ }$ , $150^{\circ }$ , $180^{\circ }$ are considered. The black hollow circles are numerical solutions obtained by surface evolver using a square box with a side length of $L=14.5$  cm (see appendix B).

3.2 Asymptotic model for small contact angles

If the contact angle $\unicode[STIX]{x1D703}$ is very small $(\unicode[STIX]{x1D703}\ll 1)$ , on the basis of the following approximations,

(3.10a,b ) $$\begin{eqnarray}\sin \unicode[STIX]{x1D711}\approx \unicode[STIX]{x1D711}\approx \frac{\text{d}\tilde{z}}{\text{d}\tilde{r}},\quad \frac{\text{d}\unicode[STIX]{x1D711}}{\text{d}\tilde{s}}\approx \frac{\text{d}^{2}\tilde{z}}{\text{d}\tilde{r}^{2}}\end{eqnarray}$$

we can rewrite (3.6) into a linear second-order ordinary differential equation,

(3.11) $$\begin{eqnarray}\frac{\text{d}^{2}\tilde{z}}{\text{d}\tilde{r}^{2}}+\frac{1}{\tilde{r}}\frac{\text{d}\tilde{z}}{\text{d}\tilde{r}}-\tilde{z}=-\unicode[STIX]{x0394}\tilde{p}_{0}.\end{eqnarray}$$

Considering the boundary conditions $\tilde{z}|_{\tilde{r}=\tilde{d}/2}=0$ , $\tilde{z}^{\prime }|_{\tilde{r}=\tilde{d}/2}=\tan \unicode[STIX]{x1D703}$ , $\tilde{z}^{\prime }|_{\tilde{r}=\tilde{L}/2}=0$ ( $\unicode[STIX]{x0394}\tilde{p}_{0}$ is a Lagrange multiplier, which can be obtained using the boundary conditions), an analytical solution of the above equation can be obtained,

(3.12) $$\begin{eqnarray}\tilde{z}(\tilde{r})=\frac{\text{I}_{1}\left({\displaystyle \frac{\tilde{L}}{2}}\right)\left[\text{K}_{0}(\tilde{r})-\text{K}_{0}\left({\displaystyle \frac{\tilde{d}}{2}}\right)\right]+\text{K}_{1}\left({\displaystyle \frac{\tilde{L}}{2}}\right)\left[\text{I}_{0}(\tilde{r})-\text{I}_{0}\left({\displaystyle \frac{\tilde{d}}{2}}\right)\right]}{\text{I}_{1}\left({\displaystyle \frac{\tilde{d}}{2}}\right)\text{K}_{1}\left({\displaystyle \frac{\tilde{L}}{2}}\right)-\text{I}_{1}\left({\displaystyle \frac{\tilde{L}}{2}}\right)\text{K}_{1}\left({\displaystyle \frac{\tilde{d}}{2}}\right)}\tan \unicode[STIX]{x1D703},\end{eqnarray}$$

where $\text{I}_{n}$ is the modified Bessel function of the first kind and $\text{K}_{n}$ is the Bessel function of the second kind (Magnus, Oberhettinger & Soni Reference Magnus, Oberhettinger and Soni1966). Note that this solution has already been obtained by Moriarty & Schwartz (Reference Moriarty and Schwartz1993) and used by López et al. (Reference López, Miksis and Bankoff2001). It further leads to an analytical solution for the volume  $V$

(3.13) $$\begin{eqnarray}\tilde{V}\left(\frac{\tilde{d}}{2}\right)=(\unicode[STIX]{x1D6E4}_{1}+\unicode[STIX]{x1D6E4}_{2})\unicode[STIX]{x03C0}\tan \unicode[STIX]{x1D703},\end{eqnarray}$$

with

(3.14) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{1}=\frac{\left[{\displaystyle \frac{1}{4}}(\tilde{d}^{2}-\tilde{L}^{2})\text{I}_{0}\left({\displaystyle \frac{\tilde{d}}{2}}\right)-\tilde{d}\text{I}_{1}\left({\displaystyle \frac{\tilde{d}}{2}}\right)\right]\text{K}_{1}\left({\displaystyle \frac{\tilde{L}}{2}}\right)}{\text{I}_{1}\left({\displaystyle \frac{\tilde{d}}{2}}\right)\text{K}_{1}\left({\displaystyle \frac{\tilde{L}}{2}}\right)-\text{I}_{1}\left({\displaystyle \frac{\tilde{L}}{2}}\right)\text{K}_{1}\left({\displaystyle \frac{\tilde{d}}{2}}\right)}, & \displaystyle\end{eqnarray}$$

(3.15) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{2}=\frac{\left[{\displaystyle \frac{1}{4}}(\tilde{d}^{2}-\tilde{L}^{2})\text{K}_{0}\left({\displaystyle \frac{\tilde{d}}{2}}\right)+\tilde{d}\text{K}_{1}\left({\displaystyle \frac{\tilde{d}}{2}}\right)\right]\text{I}_{1}\left({\displaystyle \frac{\tilde{L}}{2}}\right)}{\text{I}_{1}\left({\displaystyle \frac{\tilde{d}}{2}}\right)\text{K}_{1}\left({\displaystyle \frac{\tilde{L}}{2}}\right)-\text{I}_{1}\left({\displaystyle \frac{\tilde{L}}{2}}\right)\text{K}_{1}\left({\displaystyle \frac{\tilde{d}}{2}}\right)}. & \displaystyle\end{eqnarray}$$

To test the quality of the small-angle approximation the analytical solution is based on, we choose $L=7$  cm. Figure 6(b) shows a series of analytical results for $d$ versus $V$ (solid lines) together with the numerical solutions (red dots) for different contact angles. It becomes clear that the asymptotic solution is still accurate for contact angles up to $\unicode[STIX]{x1D703}\approx 35^{\circ }$ .

Figure 6.

(a) Axisymmetric liquid profile shown in side view ( $\unicode[STIX]{x1D703}\ll 1$ ). The geometrical parameters are the same as those defined in figure 4(a). (b) Dependence of the diameter $d$ on the volume $V$ in dimensionless form for different contact angles in a circular box with $L=7$  cm. The black solid lines represent the asymptotic theory, and the red dots are numerical results obtained as described in § 3.1.

To find the critical diameter $d_{c}$ , we have to determine the maximum value of the function given in (3.13) by solving $\text{d}V/\text{d}d=0$ , which has been done numerically. In figure 7(a), we show the dependency of $d_{c}$ on $\unicode[STIX]{x1D703}$ in dimensionless form for box sizes ranging from 7 to 19.5 cm (which correspond to the values chosen in our experiments). The dashed lines are the values of $d_{c}$ obtained from the asymptotic solution, and the dots are the numerical results obtained in § 3.1.

Figure 7(a) suggests a weak contact angle dependence of $d_{c}$ but a stronger dependence on the size of the box. First we focus on the case when $\unicode[STIX]{x1D703}$ is small. In this regime the asymptotic model given in the above follows the numerical results (figures 6 b, 7 a) very well. Based on figures 5 and 6(b), we know that $d_{c}$ is determined from $\text{d}V/\text{d}d=0$ . From (3.13) we can see that $\text{d}V/\text{d}d=0$ corresponds to $\text{d}(\unicode[STIX]{x1D6E4}_{1}+\unicode[STIX]{x1D6E4}_{2})/\text{d}d=0$ , because $\unicode[STIX]{x1D703}$ is a material parameter but not a function of $d$ or  $L$ . Equations (3.14) and (3.15) tell us that $\unicode[STIX]{x1D6E4}_{1}$ and $\unicode[STIX]{x1D6E4}_{2}$ are just functions of $d$ and  $L$ . In other words, if $L$ is given, $d_{c}$ can be uniquely determined and does not depend on the contact angle  $\unicode[STIX]{x1D703}$ . This is the reason why in figure 7(a), when $L$ is given, $d_{c}$ does not depend on  $\unicode[STIX]{x1D703}$ . Furthermore, $d_{c}$ decreases for larger contact angles, which means that the asymptotic solution represents an upper limit for  $d_{c}$ . Although the asymptotic result is obtained based on small contact angles, the comparison shows that it agrees well with the numerical results even for $\unicode[STIX]{x1D703}\approx 90^{\circ }$ . Secondly, by employing a simplified model in which it is assumed that the fluid film has a rectangular cross-section (see the supplementary material), the slight decrease of $d_{c}$ with $\unicode[STIX]{x1D703}$ and the dependence of $d_{c}$ on the size of the container are qualitatively explained.

Figure 7(b) shows the importance of considering finite box sizes. For a specific contact angle, the approach of $h_{c}/h_{\infty }$ towards 1 becomes slower and slower with increasing  $L$ , where $h_{c}$ is the film thickness at the boundary of the box in the configuration where the critical hole diameter is reached. As one example, when $\unicode[STIX]{x1D703}=180^{\circ }$ and $L=19.5$  cm, $h_{c}/h_{\infty }\approx 0.9476$ . In other words, for obtaining values of $h_{c}$ very close to  $h_{\infty }$ , the size of the box needs to be extremely large. Simultaneously, $d_{c}$ increases with increasing box size. In the limiting case, i.e.  $L\rightarrow \infty$ , our results converge to the results obtained by Taylor & Michael (Reference Taylor and Michael1973) leading to an infinite critical diameter  $d_{c}$ .

Figure 7.

(a) Dependence of $d_{c}$ on $\unicode[STIX]{x1D703}$ for six different box sizes. The dashed lines represent the asymptotic result ( $\unicode[STIX]{x1D703}\ll 1$ ), and the symbols are numerical solutions obtained in § 3.1. (b) Numerical results for the critical height $h_{c}$ (corresponding to $d_{c}$ ) scaled by $h_{\infty }$ as a function of the contact angle  $\unicode[STIX]{x1D703}$ . $h_{c}$ is the film thickness at the boundary of the box.

3.3 Comparison between experimental results and model results

In figure 8(a), results obtained on superhydrophobic Al substrates in square boxes of different sizes are displayed. The red dots represent the experimental values of $d_{c}$ (see appendix C) that have been obtained as the average of five measurements. The green triangles are numerical results extracted from the data shown in figure 5(a) (i.e. the critical values $d_{c}$ and $V_{c}$ corresponding to the maximum volume). Even though the asymptotic model is constructed based on the assumption $\unicode[STIX]{x1D703}\ll 1$ and cannot be justified for large contact angles a priori, the numerical simulations suggest a weak dependence of $d_{c}$ on the contact angle. Therefore, we also show the results of the asymptotic theory, represented by the black solid line. Figure 8(a) shows that $d_{c}$ increases with  $L$ . The experimental and the numerical results are consistent with each other. In spite of its inherent assumption of small contact angles, even the asymptotic method provides a reasonable approximation to the experimental data.

Figure 8(b) clearly shows that stable holes exist for a wide range of contact angles. Moreover, $d_{c}$ is virtually independent of the contact angle, and the experimental results are consistent with the numerical and theoretical results according to §§ 3.1 and 3.2.

Figure 8.

(a) Dependence of $d_{c}$ on $L$ on superhydrophobic Al surfaces. The experiments were performed in square boxes with $L=7.0$  cm, 9.5 cm, 12.0 cm, 14.5 cm, 17.0 cm and 19.5 cm. The green triangles are numerical results (in circular boxes with different box diameters  $L$ , see § 3.1) with the green dashed line as a guide for the eye. The black solid line is the theoretical result based on an asymptotic method for small contact angles (in circular boxes with different box diameters  $L$ , see § 3.2). (b)  $d_{c}$ as a function of the contact angle. Four different contact angles (red dots) have been studied experimentally in the square box with $L=7$  cm. The green triangles are numerical results. The black solid line is the asymptotic solution for  $d_{c}$ . All the red dots are average values of five experiments with error bars representing the standard deviation.

4.1 Dynamics of hole collapse

To better resolve the final moments of hole collapse, additional experiments were performed with a high time resolution when $t\rightarrow t_{0}$ . The complete time evolution of  $d$ , starting from the initial moment when $d\gg d_{c}$ is shown in figure 9 as a log–log plot. We employed a high-speed camera with mainly two different frame rates to capture the initial dynamics (500 f.p.s., right-hand side) and the final stages (73 000 f.p.s., left-hand side), respectively. The first set of experiments was carried out on Al superhydrophobic substrates with different box sizes. To study the influence of the wettability of the substrate, we performed experiments with two different substrates. In figure 9(b), data obtained with a further increased frame rate of 100 000 f.p.s. are shown: the red squares and blue dots represent experiments carried out using superhydrophobic Al plates ( $L=7$  cm) and liquid-infused glass ( $L=12$  cm), respectively. It is interesting to note that the hole collapse predominantly obeys a scaling law $d=(t_{0}-t)^{0.55}$ , similar to experimental results recently reported for converging fluid films driven by both surface tension and gravitational forcing (Dijksman et al. Reference Dijksman, Mukhopadhyay, Gaebler, Witelski and Behringer2015). The values for the coefficient $c_{1}$ are $0.045~\text{m}~\text{s}^{-0.55}$ and $c_{1}\approx 0.047~\text{m}~\text{s}^{-0.55}$ for the Al and liquid-infused substrates, respectively. Such a scaling has been recognized as a result of the viscous limit of liquid spreading (Bostwick et al. Reference Bostwick, Dijksman and Shearer2017). However, in contrast to our work, Bostwick et al. (Reference Bostwick, Dijksman and Shearer2017) performed experiments with silicon oil, having a much higher viscosity than water. A rough estimation of the typical Reynolds number ( $Re\approx 550$ ) in our experiments reveals that inertial forces dominate.

Figure 9.

Time evolution of the liquid surface around the point when the hole closes. (a) Superhydrophobic Al substrates have been used in square boxes of different size. $d$ as a function of $(t_{0}-t)$ is shown in a log–log plot. $t_{c}$ and $d_{c}$ denote the quantities corresponding to the onset of instability in a box with $L=7$  cm. (b) Results on Al surfaces ( $L=7$  cm) obtained with water (red squares) and water–glycerol mixture (green triangles), together with results on liquid-infused glass ( $L=12$  cm) obtained with water (blue circles), recorded at 100 000 f.p.s.

In fact, the hole collapse in our experiments is reminiscent of the final stages of the pinch-off of air bubbles in an inviscid liquid, which has been studied quite extensively. Longuet-Higgins, Kerman & Lunde (Reference Longuet-Higgins, Kerman and Lunde1991) demonstrated that the pinch-off dynamics can be captured by a two-dimensional model. It shows that the inertia of the water dominates the asymptotic dynamics, so that the surface tension, viscosity and the influence of the gas may be neglected. In this limit the evolution should approximately follow the power law $d\sim t^{0.5}$ , which is supported by simulations and experiments (Longuet-Higgins et al. Reference Longuet-Higgins, Kerman and Lunde1991; Oguz & Prosperetti Reference Oguz and Prosperetti1993; Burton, Waldrep & Taborek Reference Burton, Waldrep and Taborek2005). Thoroddsen, Etoh & Takehara (Reference Thoroddsen, Etoh and Takehara2007) carried out a systematic experimental study of the pinch-off of a bubble in water and found that the radius $d/2$ of the neck of the bubble shrinks with a power law behaviour $(d/2)\sim t^{0.57\pm 0.03}$ over more than two decades. Their results are in good agreement with the numerical studies of Leppinen & Lister (Reference Leppinen and Lister2005), who carried out numerical simulations of the break-up of a bubble in an inviscid liquid and observed a pinch-off power law with an exponent $0.55\pm 0.01$ . In the analytical works of Gordillo et al. (Reference Gordillo, Sevilla, Rodríguez-Rodríguez and Martinez-Bazan2005) and Eggers et al. (Reference Eggers, Fontelos, Leppinen and Snoeijer2007), the collapse of an axisymmetric cavity in a low-viscosity fluid such as water was studied, in which the surface tension, gas density and viscosity were neglected. They obtained the time-dependent scaling exponents as $\unicode[STIX]{x1D6FC}\approx 1/2+1/[-4\ln (t_{0}-t)]$ and $\unicode[STIX]{x1D6FC}\approx 1/2+1/[4\sqrt{-\ln (t_{0}-t)}]$ , respectively. Based on the latter expression, Eggers et al. (Reference Eggers, Fontelos, Leppinen and Snoeijer2007) obtained a small increase of the scaling coefficient from 0.55 to 0.57 during collapse. These results also indicate that the dynamics of inviscid bubble pinch-off is not universal, the exact value of the exponent weakly depends on the details of each experiment. In the theoretical model built by Eggers et al. (Reference Eggers, Fontelos, Leppinen and Snoeijer2007), the only two remaining terms in the dynamic equation are inertial. In agreement with Eggers et al. (Reference Eggers, Fontelos, Leppinen and Snoeijer2007), our results indicate that viscosity, the capillary forces and the presence of air inside the hole can be neglected. $d\sim (t_{0}-t)^{0.55}$ is consistent with the corresponding theoretical prediction. The capillary forces therefore just trigger the instability, and the final collapse is governed by inertia only. We derived a simplified model (see appendix D) which contains the essentials of the hole closure dynamics.

To further check the influence of viscosity on the dynamics, we measured $d$ as a function of $(t_{0}-t)$ using a mixture of water and glycerol that is 2.6 times more viscous than pure water. The corresponding data display weak viscous corrections. The details of the results are given in the supplementary material. On the other hand, the value of the Ohnesorge number $Oh=\unicode[STIX]{x1D702}/(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FE}d)^{1/2}$ , which relates the viscous forces to inertial and surface tension forces, can be estimated. Typically, $Oh\approx 0.7\times 10^{-3}$ in our experiments, in which the material properties of pure water at ${\sim}25\,^{\circ }\text{C}$ ( $\unicode[STIX]{x1D702}=0.891\times 10^{-3}$  Pa s, $\unicode[STIX]{x1D70C}=997.1~\text{kg}~\text{m}^{-3}$ , $\unicode[STIX]{x1D6FE}=0.072~\text{N}~\text{m}^{-1}$ ) were used and $d=20$  mm was chosen. Even when the diameter of the collapsing hole decreases to $d=0.1$  mm, i.e. the resolution limit of our camera, we obtain $Oh\approx 0.01$ . These analyses suggest that the influence of viscosity on the hole dynamics can be ignored.

4.2 Evolution of the liquid column after hole collapse

When the hole closes, a growing liquid column separates the gas above from the gas inside the surface bubble. Finally, the liquid gets redistributed into a layer of uniform thickness to minimize the potential energy. Experimental results plotted in figure 10 show that the height of the liquid column $h$ shortly after the hole has closed obeys

(4.1) $$\begin{eqnarray}h(t)\approx c_{2}\left(\frac{\unicode[STIX]{x1D6FE}l_{c}}{\unicode[STIX]{x1D70C}}\right)^{1/4}(t-t_{0})^{1/2}\end{eqnarray}$$

in which the unknown coefficient has been determined to be $c_{2}=3.1$ by fitting.

Compared with the hole collapse discussed in § 4.1, the evolution of the height of the liquid column $h$ after hole closure/bubble pinch-off has been studied to a lesser extent. Thoroddsen et al. (Reference Thoroddsen, Etoh and Takehara2007) reported that after pinch-off, the vertical separation of the air cones as a function of time obeys a power law with a fitted exponent of $0.49\pm 0.03$ , which is in agreement with the numerics of Leppinen & Lister (Reference Leppinen and Lister2005) who found an exponent of $0.48\pm 0.05$ for the evolution of the separation length scale in the breakup of a bubble in an inviscid liquid. The closing of the hole bears a geometrical similarity to the coalescence of droplets or air bubbles (Thoroddsen et al. Reference Thoroddsen, Etoh, Takehara and Ootsuka2005; Ristenpart et al. Reference Ristenpart, McCalla, Roy and Stone2006; Bird et al. Reference Bird, Ristenpart, Belmonte and Stone2009; Courbin et al. Reference Courbin, Bird, Reyssat and Stone2009; Eddi, Winkels & Snoeijer Reference Eddi, Winkels and Snoeijer2013; Paulsen et al. Reference Paulsen, Carmigniani, Kannan, Burton and Nagel2014), which has been intensely studied in recent years. For instance, Duchemin, Eggers & Josserand (Reference Duchemin, Eggers and Josserand2003) studied the inviscid coalescence of drops using a boundary integral method. They proposed a power-law scaling for the neck radius as a function of time with an exponent of 0.5. In the work of Thoroddsen et al. (Reference Thoroddsen, Etoh, Takehara and Ootsuka2005), the growth rate of the neck formed after the coalescence of two air bubbles was measured and modelled based on a capillary/inertial balance. A scaling for the neck radius as a function of time with an exponent of 0.5, valid for a comparatively short time interval, was found. Previous results (Eggers, Lister & Stone Reference Eggers, Lister and Stone1999) indicate that already a one-dimensional model captures the underlying physics. For convenience, we model the dynamics in close analogy to the work reported on the coalescence of drops or air bubbles. As shown in the inset of figure 10, during hole closure $h^{2}\approx 4\unicode[STIX]{x1D6FF}R$ , denoting by $\unicode[STIX]{x1D6FF}$ and $R$ the radii of curvature at the ends of the liquid column and of the liquid surface before hole closure, respectively. In our case we have $R\approx l_{c}$ . The balance of capillary and inertial forces for a corresponding liquid column leads to $2\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FF}\approx (\unicode[STIX]{x1D70C}(h/2t)^{2})/2$ . This results in the functional form of (4.1) with $c_{2}=64^{1/4}\approx 2.8$ , showing good agreement to the value obtained from fitting the experimental data.

Figure 10.

Time evolution of the liquid column after hole closure. $h$ is represented as a function of $(t-t_{0})$ in a log–log plot. Each dot represents the average value of five experiments with error bars indicating the standard deviation. The red squares and green triangles represent results obtained on Al surfaces with water and a water–glycerol mixture, respectively. The solid line is the best fit to the experimental data based on the scaling law of (4.1). The snapshots added to specific sections of the curves give an impression of the corresponding shape of the liquid surface. The sketch shows the geometry used for the scaling analysis.

Furthermore, we experimentally studied the influence of viscosity on the evolution of the liquid column (see supplementary material). With a mixture 2.6 time more viscous than water, very similar results were obtained, from which we conclude that the viscosity does not play an important role.