2.1. Experimental set-up and procedure

A schematic of the experimental set-up is depicted in figure 2. Prior to conducting our experiments, we first level the set-up to ensure the platform is positioned horizontally. This is achieved by employing level pins with a resolution of 0.01 mm, as depicted in figure 2. To be precise, we place a level with a resolution of $0.1^{\circ }$ onto the substrate and carefully adjust the level pins until the substrate is levelled horizontally. We first prepare a clean glass placed on a platform as a flat substrate, and the wettability of the upper surface of the glass is adjustable by chemical treatment. We then place a cylinder on the top of the glass to form a circular outer boundary. The glass plate and the cylinder form a container. By employing a syringe tube connected to a syringe pump, deionized water is very smoothly added into the container, and a water layer is generated. To get a better visualization, a LED panel is horizontally placed on the top of the container as a uniform light source. By employing a mirror with a tilt angle of $45^{\circ }$, the photographs of the water layer in the container are reflected to a camera, which is placed horizontally.

Figure 2.

Schematic illustration of the experimental set-up, which mainly consists of a light source, a container composing a glass plate and a cylinder, a syringe tube, level pins, a mirror and a high-speed camera.

In our experiment, we use different sizes of the cylinder. Moreover, the wetting properties of the glass plate and the inner wall of the cylinder are easily modified. Therefore, we are able to present experiments with varying parameters, such as the contact angle and the size of the water layer. A summary of the parameters of different experiments is given in table 1, where $\theta$, $\theta _a$ and $\theta _r$ represent the apparent contact angle, advancing and receding contact angles of water on the glass plate and the plastic Petri dish, respectively. Here, $\Delta \theta = \theta _a - \theta _r$ represents the contact angle hysteresis. Five containers with different diameters $d$ varying from 6.6 to 14.2 cm are employed. Typically, we first add a thin layer of water with an initial height $\sim h_{\infty }/2$, after that, we create a comparatively large hole near the wall of the container by employing clean compressed air flow through a nozzle with an inner diameter of 3 mm. Here, $h_{\infty } = 2 l_c \sin (\theta /2)$ is the thickness of an infinite unbounded liquid layer, and $l_c \approx 2.73 \, \mathrm {mm}$ is the capillary length of water (de Gennes et al. Reference de Gennes, Brochard-Wyart and Quéré2003). The outlet of the nozzle is placed approximately 2 cm away from the substrate. When the hole at the wall has been created, we smoothly decline the flow rate of the nozzle until the flow stops. Shortly after that, the hole reaches a stable equilibrium state and the water film becomes static. We then smoothly add water (flow rate $Q = 1\,\mathrm {ml}\,\mathrm {min}^{-1}$ for most cases) to the water film through a syringe tube. The smallest diameter of the container we use is $d = 6.6\,{\rm cm}$, a simple calculation could estimate the growth rate of the water film height, i.e. $Q/({\rm \pi} d^2/4) \approx 292.3\,\mathrm {\mu }\mathrm {m}\,\mathrm {min}^{-1}$. Therefore, we neglect the influence of the disturbance on the water film resulting from the supplied water.

Table 1.

Wetting properties of the materials, obtained for water.

2.2. Experimental observations

2.2.1. Three wetting modes of holes at the wall

In our experiments, the time evolution of the hole is obtained from the bottom view by employing a high-speed camera. Moreover, we use a superhydrophobic glass plate (treated by Glaco Mirror Coat ‘Zero’, Soft 99, Co., see Appendix A) as the substrate, whereas we separately use three different kinds of wettability for the inner wall of the cylinder, i.e. superhydrophobic, hydrophobic and hydrophilic glasses. As illustrated in figure 3, typical experimental results of the wetting modes of holes at the wall are given, and three unexpected wetting modes are observed and can be generalized as follows.

Figure 3.

Three wetting modes of holes at the wall. (a) Mode I; (b) mode II; (c) mode III. For each mode, panels (i–iv) are time-lapse images of the water layer with a hole, while panels (v–viii) are the schematics of the experimental observations corresponding to panels (i–iv), respectively. The red arrows denote the flow direction of the front of the liquid–vapour meniscus. The scale bar represents 10 mm.

In figure 3(a) wetting mode I is presented. In this case, both the glass plate and the inner wall of the container are superhydrophobic $\theta = 158.2 \pm 3.0^{\circ }$. When adding water into the container, the hole is shrinking. Here, limited by the field of view of the camera, the experimental visualization (figure 3a i–iv) contains incomplete photographs of the container. Nevertheless, the main characteristic of the evolution of the hole can be captured. Figure 3(a v–viii) gives the sketches of mode I to demonstrate the complete view of the container, where the red arrows denote the flow direction of the front of the liquid–vapour meniscus. Confined by the container and the liquid–vapour interface, a crescent-shaped hole is formed. When more water is added to the liquid, the hole gradually gets smaller until it disappears (see supplementary movie 1 available at https://doi.org/10.1017/jfm.2023.753). In this case, the liquid film is always in a quasi-static state, which is different from the collapse of a hole located at the centre of a container reported previously (Diez et al. Reference Diez, Gratton and Gratton1992; Wilson & Duffy Reference Wilson and Duffy1996; Bankoff et al. Reference Bankoff, Johnson, Miksis, Schluter and Lopez2003; Dijksman et al. Reference Dijksman, Mukhopadhyay, Gaebler, Witelski and Behringer2015; Bostwick et al. Reference Bostwick, Dijksman and Shearer2017; Lv et al. Reference Lv, Eigenbrod and Hardt2018; Zheng et al. Reference Zheng, Fontelos, Shin, Dallaston, Tseluiko, Kalliadasis and Stone2018a,Reference Zheng, Fontelos, Shin and Stoneb).

In figure 3(b) wetting mode II is presented. In this case, the glass plate is superhydrophobic ($\theta = 158.2 \pm 3.0 ^{\circ }$), while a cylindrical glass with a hydrophobic inner wall ($\theta = 116.0 \pm 1.9 ^{\circ }$) is employed. In contrast to mode I, an eye-shaped hole is formed (see supplementary movie 2). Here, the flow rate of the nozzle is $Q = 0.1\,\mathrm {ml}\,\mathrm {min}^{-1}$ when adding water into the water film, which guarantees a very smooth shrinking of the hole. In the beginning, the time evolution of the appearance of the hole is hardly observed by the naked eye, i.e. from 0 to 13.0 s, therefore, the hole is in a quasi-static state in this regime. However, as time progresses, the situation will change when the hole size approaches a critical value. Since the geometrical shape of the hole is not regular, we use an area $A(t)$ to describe the geometry of the hole, where $A(t)$ is the projected area of the hole. As the hole area $A(t)$ decreases to a critical value $A_c$, the hole is no longer able to hold its stability, but collapses, i.e. the area $A(t)$ suddenly reduces to zero (from 17.0 to 17.1 s). The stability and collapse of the hole in mode II are very similar to the observations of Lv et al. (Reference Lv, Eigenbrod and Hardt2018).

In figure 3(c) wetting mode III is presented. In this case, the glass plate is superhydrophobic ($\theta = 158.2 \pm 3.0 ^{\circ }$), while a cylindrical glass with a hydrophilic inner wall ($\theta = 29.7 \pm 5.8 ^{\circ }$) is employed. The water film wets the inner wall of the cylinder as much as possible but decreases the contact area with the substrate. Different from wetting modes I and II, the front of the water film spreads more quickly along the inner wall of the cylinder than on the glass plate (see supplementary movie 3), and a funnel-shaped hole appears (from 0 to 94.5 s). Eventually, the left and right fronts of the water film merge at the wall, and a transition from a hole at the wall to an inner hole happens, and the hole is pushed a little further from the inner wall of the cylinder. Meanwhile, we observe that the hole reshapes itself to a circle shape soon after the transition. This behaviour has not been observed previously (Lv et al. Reference Lv, Eigenbrod and Hardt2018; Zheng et al. Reference Zheng, Fontelos, Shin, Dallaston, Tseluiko, Kalliadasis and Stone2018a,Reference Zheng, Fontelos, Shin and Stoneb).

To the best of our knowledge, this is the first report of the different wetting behaviours of the hole in the liquid layer. Although liquid layers were widely studied in the past, most of them focused on the hole occurring in the centre of the container (Wilson & Duffy Reference Wilson and Duffy1996; Bankoff et al. Reference Bankoff, Johnson, Miksis, Schluter and Lopez2003; Dijksman et al. Reference Dijksman, Mukhopadhyay, Gaebler, Witelski and Behringer2015; Bostwick et al. Reference Bostwick, Dijksman and Shearer2017; Lv et al. Reference Lv, Eigenbrod and Hardt2018; Zheng et al. Reference Zheng, Fontelos, Shin, Dallaston, Tseluiko, Kalliadasis and Stone2018a,Reference Zheng, Fontelos, Shin and Stoneb), or the dependence of the collapse behaviours on the viscosity of the liquid (Redon et al. Reference Redon, Brochard-Wyart and Rondelez1991; Diez et al. Reference Diez, Gratton and Gratton1992; Zheng et al. Reference Zheng, Fontelos, Shin, Dallaston, Tseluiko, Kalliadasis and Stone2018a,Reference Zheng, Fontelos, Shin and Stoneb; Lu & Corvalan Reference Lu and Corvalan2019; Lu et al. Reference Lu, Ferri, Ubal, Campanella and Corvalan2019) and the wettabilities of the container (Redon et al. Reference Redon, Brochard-Wyart and Rondelez1991; Lv et al. Reference Lv, Eigenbrod and Hardt2018). Here, our observations are distinguished from the previous experimental results.

2.2.2. Experimental analysis

In order to quantify the dynamics of the hole, we plot the relationship between the instantaneous value of $A$ and time $(t_0 - t)$, denoting $t_0$ the moment when $A = 0$. As shown in figures 4 and 5, we plot the results for modes I and II, respectively.

Figure 4.

Wetting mode I. (a) Selected frames from the top view. Glaco-treated glasses are used as the planar substrate and the cylindrical wall of the container. The scale bar represents 10 mm. (b) Time evolution of the hole area in a liquid layer for wetting mode I. Glaco-treated glasses are used as the planar substrate and the cylindrical wall of the container. The plot shows the instantaneous area $A$ of the hole as a function of time $(t_0 - t)$, where $t_0$ is defined as the moment when $A = 0$. All the data points are average values of five experiments with error bars representing the standard deviation.

Figure 5.

Wetting mode II. (a) Selected frames from the experiment of wetting mode II. A planar Glaco-treated glass is used as the substrate, while a cylindrical hydrophobic glass is used as the wall of the container. The scale bar represents 10 mm. (b) Evolution of the instantaneous area $A$ with time $(t_0 - t)$ for wetting mode II. $t_0$ is defined as the moment when $A = 0$. A plastic Petri dish is used as the container. All the data points are average values of five experiments with error bars representing the standard deviation. Five containers with a diameter $d$ ranging from 6.6 to 14.2 cm have been employed.

As shown in figure 4, we first demonstrate the behaviour of wetting mode I. Here, Glaco-treated glasses are used as the planar substrate and the cylindrical wall of the container, which provide an advancing contact angle $\theta _a = 163.3 \pm 1.0^{\circ }$ and a receding contact angle $\theta _r = 151.8 \pm 2.4^{\circ }$. Selected frames are displayed in figure 4(a), showing the time evolution of the hole shape in bottom view, indicating that the solid–liquid–vapour three-phase contact lines gently move forwards. Based on the experimental data shown in figure 4(b), it is found that the hole shrinks uniformly when water is added. Owing to the contact line pinning, the curve is not as smooth as a straight line and there is jerky glide motion observed in supplementary movie 1.

As shown in figure 5(a), we demonstrate the behaviour of wetting mode II. Here, a cylindrical hydrophobic glass is used as the wall of the container, which provides an advancing contact angle $\theta _a = 119.3 \pm 1.3^{\circ }$ and a receding contact angle $\theta _r = 105.8 \pm 2.2^{\circ }$. Moreover, a planar Glaco-treated glass is used as the substrate, which provides an advancing contact angle $\theta _a = 163.3 \pm 1.0^{\circ }$ and a receding contact angle $\theta _r = 151.8 \pm 2.4^{\circ }$. In figure 5(b), $A$ is plotted as a function of time $(t_0 - t)$ for a range of box sizes, where the data points were obtained by averaging five experiments with corresponding error bars. Here, the plastic Petri dish is used as the container, which provides an advancing contact angle $\theta _a = 102.7 \pm 0.7^{\circ }$ and a receding contact angle $\theta _r = 80.5 \pm 1.6^{\circ }$. Five different containers with diameters $d$ ranging from 6.6 to 14.2 cm have been employed. The curves for wetting mode II shown in figure 5(b) contain two regimes. In the first regime, the area of the hole decreases smoothly with time corresponding to the photographs from 0 to 13.0 s (see supplementary movie 2). In this process, the shrinking of the hole is due to the feeding of water. Since water is added at very small flow rate, the shrinking of the hole is not obvious. Here, it is stressed that, when the feed is stopped, the shrinking of the hole stops correspondingly and the liquid is in a static state. In this regard, the hole is considered to be stable. However, when the hole shrinks to a critical size, a subsequent volume increase will trigger the collapse of the hole. The second regime happens at this time. As shown in figure 5(b), the area of the hole steeply decreases with time corresponding to the photographs from 17.0 to 17.1 s. During the second stage, the hole cannot remain static even we stop adding water. In this process, the hole collapse is highly dynamic in contrast to the first regime. Although containers with different sizes are used, the curves in figure 5(b) have the same trend.

Before we carry out systematic theoretical analyses (see § 4) for these three wetting modes, an intuitive understanding could be obtained. For the meniscus of the hole, there are two principal curvatures, i.e. the azimuthal curvature (see the top view in figure 1a) and the meridional curvature (see the side view in figure 1b). Therefore, for the wetting mode I, as shown in figure 3(a), both the two curvatures are positive, which create a pressure jump together to maintain the opening of the hole. As a result, the hole is able to be stable until it closes. However, for the wetting mode II, as shown in figure 3(b), the azimuthal curvature becomes negative to push the liquid to the hole. As additional liquid is supplied, the increase of the azimuthal curvature is pronounced while the variations of the meridional curvature and the hydrostatic pressure are relatively small. Therefore, the collapse of the liquid layer is caused by the variation of the azimuthal curvature. Lastly, for wetting mode III, as shown in figure 3(c), when adding enough liquid, the front liquid menisci must merge together. To further decrease the energy of the system, the hole will trend to become circular and meanwhile it seems the hole is pushed along the central direction of the container.

3.1. Surface evolver simulation

We first employ a public domain software package SE (Brakke Reference Brakke1992) to simulate holes at the wall in liquid layers. The basic concept of SE is to minimize the energy and find the equilibrium shape of a liquid volume surface with given parameters such as the surface tension and stiffness, subjected to external forces (e.g. gravity, centrifugal force, magnetic force) and constraints (e.g. volume conservation, contact angle, pinning of the contact line). The SE has been widely applied to studying various interfacial phenomena (such as morphologies of droplets and particles (Cho et al. Reference Cho, Yi, Lim, Kim, Manoharan, Pine and Yang2005; Crawford, Lim & Gradecak Reference Crawford, Lim and Gradecak2013; Lv et al. Reference Lv, Chen, Chuang, Tseng, Yin, Grey and Zheng2014; Dević et al. Reference Dević, Encarnación, José and Lohse2019; Lv & Hardt Reference Lv and Hardt2021; Chu et al. Reference Chu, Zhao, Hao and Lv2023), foams (Kabla & Debregeas Reference Kabla and Debregeas2007) as well as the equilibrium configuration of biomembranes (Michalet Reference Michalet2007; Sakashita et al. Reference Sakashita, Urakami, Ziherl and Imai2012), with excellent agreement with experimental results. However, for our problem, SE could only simulate static or quasi-static problems and could not take dynamic behaviour into consideration. As we analysed in § 2.1, the influence of the flow rate on the evolution of the liquid layer is negligible and the liquid layer is in the quasi-static state before collapse. Therefore, we propose the convenient simplification of reducing the problem to a succession of quasi-static liquid profiles.

As shown in figure 6(a), we give the results of the configuration of the liquid film simulated by SE. For a given volume $\varOmega$ of the liquid layer, two configurations from different views on the left and the right panels demonstrate the initial and the final configurations of the liquid layer, respectively (for more details of the SE simulations, see S1 in the supplementary information). Moreover, further information such as the areas of the liquid–vapour and the solid–liquid interfaces, as well as the potential of the system, can be extracted based on the final configuration. By the help of SE, a series of stable liquid layers with different volumes $\varOmega$ can be obtained.

Figure 6.

Schematic sketch of SE simulation. (a) The evolution of a liquid layer simulated by SE. The yellow and blue colours represent the liquid–vapour and solid–liquid interfaces, respectively. The two configurations from different views in panels (a i,a ii) and (a iii,a iv) demonstrate the initial and the final configurations of the liquid layer, respectively. (b) Results obtained from SE simulation showing dependency of $A$ on $\varOmega$, with $A$ and $\varOmega$ being the area of the hole and the volume of water, respectively. The contact angles of the substrate and the wall are $\theta _s = 160^{\circ }$ and $\theta _w = 90^{\circ }$, respectively. The diameter of the container is $d = 9 \, \mathrm {cm}$.

As presented in figure 6(b), the relationship between the hole area $A$ and the liquid volume $\varOmega$ is given. Here, the container diameter is fixed at $d = 9 \, \mathrm {cm}$, and the contact angles of the substrate and the wall of the container are set to $\theta _s = 160^{\circ }$ and $\theta _w = 90^{\circ }$, respectively. Each point in figure 6(b) represents a static configuration of the system. It is shown that the area of the hole $A$ first decreases linearly with the volume $\varOmega$ of the liquid. However, when $\varOmega$ exceeds a critical value $\varOmega _c$ (which corresponds to a critical area $A_c$), we can never obtain a static liquid layer with a hole. Actually, we cannot precisely determine the threshold of the collapse of the hole. Therefore, we artificially define the threshold of the collapse of the hole in which a further 1 % increase in liquid volume will trigger the hole closure. Thus, the last point $(\varOmega _c, A_c)$ in figure 6(b) is considered to be the threshold accounting for the collapse of the hole. Close to the critical point, the linear relation between $A$ and $\varOmega$ is not maintained and the slope of the simulation data suddenly increases.

Moreover, the effect of the size of the container is investigated by considering five diameters from $d = 7$ to 15 cm. For each diameter of the container with certain wettability, a series of simulations with different liquid volumes are conducted. Specifically, we first give a sufficiently small volume $\varOmega$ of liquid (in other words, a liquid layer with a larger hole) in our simulations and then obtain the equilibrium wetting state of the system. After that, we continuously increase the liquid volume to find the corresponding equilibrium wetting state, until we find the critical volume $\varOmega _c$. Figure 7(a) shows the top views of outcomes of $d = 9$ cm with different liquid volumes, from which we can see that an eye-shaped hole appears in each case. Similarly, equilibrium wetting states of liquid layers in containers with other diameters can be obtained. Figure 7(b) shows the outcomes of SE simulations with different container diameters, and when given proper liquid volumes, eye-shaped holes with similar shapes appear, which indicates that the wetting state of the liquid is not affected by the size of the container.

Figure 7.

The snapshots of liquid layer. The contact angles of the substrate and the wall are $\theta _s = 160^{\circ }$ and $\theta _w = 90^{\circ }$, respectively. (a) Configuration of the liquid with the volume. The diameter of the container is fixed at $d = 9\,\mathrm {cm}$. (b) The diameter of the container $d$ ranges from 7 to 15 cm. The configurations correspond to the thresholds of the collapse of the hole.

Furthermore, the simulation data for containers with diameters $d$ are shown in figure 8, in which five diameters $d$ ranging from 7 to 15 cm are presented with different colours and shapes. Here, the wettability of the container is fixed, i.e. the contact angles of the substrate and the wall of the container are $\theta _s = 160^{\circ }$ and $\theta _w = 90^{\circ }$, respectively. We can see that all the data have similar trends no matter the value of the diameter of the container. However, each set of the data has a group of thresholds $(\varOmega _c, A_c)$, and these thresholds depend on the container diameter. The threshold values will be analysed theoretically in the following sections.

Figure 8.

Dependency of the hole area $A$ on liquid volume $\varOmega$ for various diameters of the container obtained by SE simulations. In this case, a circular container with $\theta _s = 160^{\circ }$ and $\theta _w = 90^{\circ }$ has been considered, denoting with $\theta _s$ and $\theta _w$ the contact angles of the substrate and the wall of the container, respectively. Different container diameters $d$ = 7 cm, 9 cm, 11 cm, 13 cm and 15 cm are employed.

Next, the influence of the wettability on the wetting property of the liquid layer is systematically investigated. For the sake of simplicity, let the contact angles of the substrate and the wall of the container have the same value, i.e. $\theta _s = \theta _w$. As shown in figure 9, we consider contact angles ranging from $90^{\circ }$ to $180^{\circ }$. Here, the diameter of the container is fixed at $d = 9$ cm. There is a remarkable difference of the wetting behaviour of the liquid layer between different contact angles. Each set of the data has a group of thresholds $(\varOmega _c, A_c)$, which depends on the contact angle. Moreover, the phase diagram of parameter space can be divided into two parts, which correspond to wetting modes I and II, and the boundary between mode I and mode II is $\theta _s = \theta _w = 145^{\circ }$ (see the dashed line). Here, we artificially define the criterion to distinguish mode I and mode II based on the following: when $A_c$ is smaller than 1 % of the projected area of the container, it is mode I; otherwise, it is mode II. In our SE simulations, when $\theta _s=\theta _w>145^{\circ }$, we found $A_c$ is very close to zero and it is not visually discernible. However, when $\theta _s = \theta _w < 145^{\circ }$, when the hole area is smaller than an obvious value of $A_c$ (with a maximum value $\varOmega _c$), we can never find a stable liquid layer, i.e. a slight 1 % increase in liquid volume will trigger the closure of the hole. The results shown in figure 9 demonstrate that the threshold value accounting for the collapse of the hole is closely related to the wetting property of the container.

Figure 9.

(a) Dependency of $A$ on $\varOmega$ for various wettabilities of the container obtained from SE simulations. Here, a circular container with $d$ = 9 cm has been considered, with contact angles of $\theta _s=\theta _w=90^{\circ }$, $100^{\circ }$, $110^{\circ }$, $120^{\circ }$, $130^{\circ }$, $140^{\circ }$, $150^{\circ }$, $160^{\circ }$, $170^{\circ }$ and $180^{\circ }$. The arrow indicates the direction in which the volume $\varOmega$ is increasing. (b) Snapshots of the liquid profile from the top view when the threshold of collapse is reached. Examples with different contact angles $\theta _s=\theta _w=90^{\circ }$, $120^{\circ }$, $150^{\circ }$ and $180^{\circ }$ are demonstrated, which are obtained from SE simulations.

Despite the influence of the wettability on the instability of the hole as discussed in the above, the wettabilities of the substrate and the wall of the container have not been considered separately. As shown in figure 10, we investigate the influence of the wettability on the instability of the hole solely resulting from the substrate or the wall of the container. The snapshots in figure 10 demonstrate the thresholds of instability of the liquid layer in containers. Figure 10(a) shows a series of results where the contact angle of the container substrate is fixed at $\theta _s = 160^{\circ }$ while the contact angle of the container wall is varied as $\theta _w = 60^{\circ }$, $90^{\circ }$ and $160^{\circ }$. We can see that there exists an obvious eye-shaped hole for a small value of $\theta _w$ when the threshold of instability is reached. However, with an increase of $\theta _w$, the size of the crescent-shaped hole becomes smaller and it is even indiscernible when $\theta _w$ is large enough (i.e. $\theta _w = 160^{\circ }$). Moreover, figure 10(b) shows the cases where the contact angle of the container wall is fixed at $\theta _w = 90^{\circ }$ while the contact angle of the container substrate is varied as $\theta _s = 60^{\circ }$, $90^{\circ }$ and $160^{\circ }$. We can see in this case that, even though the value of $\theta _s$ is varied over a wide range, an obvious eye-shaped hole always exists when the threshold of the instability is reached. In this regard, the instability of the liquid layer is mainly influenced by $\theta _w$ rather than $\theta _s$, and this conclusion is consistent with our experimental observations in § 2.2 where the substrate is fixed but the wall of the container is changed.

Figure 10.

Snapshots showing the thresholds of instability of the liquid layer in containers with different contact angles. Here, the diameter of the container is $d = 9$ cm. (a) The contact angle of the substrate is fixed at $\theta _s = 160^{\circ }$ and the contact angle of the wall is varied as $\theta _w = 60^{\circ }$, $90^{\circ }$ and $160^{\circ }$. (b) The contact angle of the wall is fixed at $\theta _w = 90^{\circ }$ and the contact angle of the substrate is varied as $\theta _s = 60^{\circ }$, $90^{\circ }$ and $160^{\circ }$.

3.2. Lattice Boltzmann method

Although SE gives a distinct picture of wetting mode I and wetting mode II, wetting mode III is still not considered. As shown in our experimental observations in § 2.2, wetting mode III happens in a hydrophilic condition of the container and involves a dynamics which is beyond the capability of SE (see the supplementary information). Therefore, we employ another numerical method, i.e. LBM, to simulate wetting mode III. Details of how we realize simulations via LBM is given in the supplementary information. Here, we only demonstrate the main results.

As shown in figure 11, we perform simulations for two-phase flows under gravity and surface tension. The simulations include the three wetting modes we have observed in experiments, which are realized by assigning certain values of the contact angle to the substrate and the wall of the container. Specifically, we assign ($\theta _s = 120^{\circ }$, $\theta _w = 180^{\circ }$), ($\theta _s = 120^{\circ }$, $\theta _w = 90^{\circ }$) and ($\theta _s = 120^{\circ }$, $\theta _w = 35^{\circ }$) in figures 11(a), 11(b) and 11(c), respectively. The snapshots demonstrate the time evolution of the liquid layers with its volume. As shown in figure 11(a), when $\theta _w = 180^{\circ }$, the hole is shrinking very smoothly when feeding more liquid into the container, until the liquid layer entirely fills the container. However, as shown in figure 11(b), when $\theta _w = 90^{\circ }$, the hole is shrinking smoothly in the first stage, and then reaches a critical state. After that, the hole suddenly collapses with a further increase of the liquid volume. Furthermore, as shown in figure 11(c), when $\theta _w = 35^{\circ }$, with a continuous increase of the liquid volume, the liquid first spreads along the junction of the substrate and the wall. When enough liquid is added, the liquid forms a ring-like structure including a hole inside, but meanwhile the hole is suddenly pushed into the central direction of the container. The simulation results (see supplementary movies 4–6) very well reproduce the experimental observations. Here, it is stressed that, in both the numerical simulations and experiments, despite the collapse that causes the hole to be pushed towards the centre of the container, the hole consistently maintains a certain distance from the wall and is not located right in the centre of the container.

Figure 11.

Snapshots showing time evolution of liquid layer with its volume which are simulated by employing LBM. The final volume of the liquid layer for each mode is defined as $\varOmega _0$. Here, the contact angle of the substrate of the container is fixed at $\theta _s = 120^{\circ }$, while three different contact angles of the wall of the container are simulated. The diameter of the container is $d \approx 60$ mm. (a) Wetting mode I ($\theta _s = 120^{\circ }$, $\theta _w = 180^{\circ }$); (b) wetting mode II ($\theta _s = 120^{\circ }$, $\theta _w = 90^{\circ }$); (c) wetting mode III ($\theta _s = 120^{\circ }$, $\theta _w = 35^{\circ }$).

By employing LBM simulations, we systematically investigate the influence of wettability on the wetting modes. The dots represent the simulation results as shown in figure 12. Specifically, we fix the contact angle of the container substrate $\theta _s$ ranging from $15^{\circ }$ to $175^{\circ }$, and we choose a high value of the contact angle of the container wall $\theta _w = 175^{\circ }$, then we obtain the results of wetting mode I. After that, we gradually decrease the contact angle $\theta _w$ for each $\theta _s$ until wetting mode II appears. By employing this method, we not only find the parameter regime of wetting mode I, we also find the boundary (the upper dashed curve) between wetting modes I and II, which is determined by averaging the neighbouring contact angles and the dashed curve is extended by employing a fitting. After further decreasing the value of $\theta _w$, we obtain the parameter regime of wetting mode II, as well as the boundary between wetting modes II and III. After we obtain enough data, we are able to plot a phase diagram in terms of the contact angles $\theta _w$ and $\theta _s$ which includes the three wetting modes. Based on the results shown in figure 12, we can see the wetting mode is mainly determined by the contact angle of the wall of the container, i.e. $\theta _w$. Generally, when $\theta _s$ takes a relatively large value (i.e. $\theta _s > 90^{\circ }$), the two boundaries occur at $\theta _w \approx 45^{\circ }$ and $\theta _w \approx 145^{\circ }$. However, when $\theta _s$ takes a relatively small value (i.e. $\theta _s < 90^{\circ }$), the situation is more complex, the regime of wetting mode II declines but the regimes of wetting modes I and II increase.

Figure 12.

Phase diagram illustrated in terms of $\theta _s$ and $\theta _w$ showing the wetting modes obtained by LBM simulations. The triangles, squares and circles denote wetting modes I, II and III, respectively. The two dashed curves represent the boundaries between the neighbouring wetting modes. The diameter of the container is $d \approx 60$ mm. The dashed line in the region for small $\theta _s$ (i.e. $\theta _s < 15^{\circ }$) is an educated guess.

4.1. Theoretical model

Here, in our theoretical analysis, we consider the case of wetting modes I and II. To obtain the potential of the system from a theoretical point of view, we assume on the one hand that the liquid–vapour interface is perpendicular to both the substrate and the wall, and on the other hand that the edge of the hole is circular, as sketched in figure 13. The former assumption is based on the large ratio between the container diameter and the thickness of the liquid layer. In the smallest container ($d = 6.6$ cm), the thickness of the liquid layer is approximately $2l_c \approx 5.46$ mm. Therefore, the ratio between the thickness and the width of the liquid layer is approximately 0.08 ($\ll 1$), which suggests that the variation of the meniscus thickness around the edge of the hole may have little influence on the energy of the system (see the supplementary information). The latter assumption about a circular shape of the hole is based on the observation in the experiments and simulations.

Figure 13.

Simplified liquid layer profile shown from oblique view (a) and top view (b). Relevant geometric parameters are defined.

When additional liquid is fed into the liquid layer, at a specific time, the liquid has a specific volume $\varOmega$. The free energy of the system can be written as $E = E_s + E_g = A_{lv} \gamma + A_{sl} ( \gamma _{sl} - \gamma _{sv} ) + \rho \varOmega g h/2$. Here, $\rho$ and $g$ represent the mass density of the liquid and the gravitational acceleration respectively, $E_s = A_{lv} \gamma + A_{sl} ( \gamma _{sl} - \gamma _{sv} )$ and $E_g = \rho \varOmega g h/2$ represent the surface energy and the gravity potential, respectively, with $h$ being the thickness of the liquid layer. Further, $A_{lv}$ and $A_{sl}$ represent the liquid–vapour area and the solid–liquid area respectively, and $\gamma$, $\gamma _{sl}$ and $\gamma _{sv}$ represent the liquid–vapour, solid–liquid and solid–vapour interfacial tensions, respectively. The basic idea is that, for given parameters such as $d$, $\rho$, $g$, $\gamma$, $\gamma _{sl}$ and $\gamma _{sv}$, we will find the solution when the total energy $E$ reaches a minimum $E_{min}$ under the constraint of the certain volume $\varOmega$, as well as the geometric relationships as shown in figure 13(b). When we obtain a wetting state which satisfies $E = E_{min}$, we find the equilibrium wetting state of the liquid layer.

The geometry parameters are presented in figure 13(b), in which $\alpha$ and $\beta$ are the half-angles of the liquid and the hole satisfying the relationship $\theta _w + \alpha + \beta = {\rm \pi}$. In addition, $A_1$ represents the area of the upper liquid–vapour interface, which is equal to the corresponding liquid–substrate interface, and $A_2$ and $A_3$ represent the areas of the side liquid–vapour interface at the boundary of the hole and the area of the side liquid–wall interface, respectively. These areas can be written as $A_1 = r_1^2 ( {\rm \pi}-\alpha + \sin \alpha \cos \alpha ) - r_2^2 ( \beta - \sin \beta \cos \beta )$, $A_2 = 2 r_2 \beta h$ and $A_3 = 2r_1 ( {\rm \pi}- \alpha ) h$, where $r_1$ and $r_2$ are the radii of the liquid and the hole satisfying $r_1 \sin \alpha = r_2 \sin \beta$.

Moreover, the surface energy $E_s$ consists of the liquid–vapour surface energy, the solid–liquid and the solid–vapour surface energy, which can be written as

(4.2) \begin{align} E_s & = A_1 \gamma + A_1 ( \gamma_{sl}^s - \gamma_{sv}^s ) + A_2 \gamma +A_3 ( \gamma_{sl}^w - \gamma_{sv}^w ) \nonumber\\ & = A_1 \gamma \left ( 1 - \cos \theta_s \right ) + A_2 \gamma - A_3 \gamma \cos \theta_w , \end{align}

where the second equation uses the Young equation as $\cos \theta _s = ( \gamma _{sv}^s - \gamma _{sl}^s ) / \gamma$ and $\cos \theta _w = ( \gamma _{sv}^w - \gamma _{sl}^w ) / \gamma$. The superscripts $s$ and $w$ represent the substrate and the wall, respectively.

For convenience, by employing the radius of the liquid layer $r_1$ (in other words, the radius of the container), we define dimensionless forms of the relevant geometric and physical quantities

(4.3)\begin{equation} \left. \begin{aligned} & \displaystyle \tilde{h} \equiv \frac{h}{r_1},\quad \tilde{A}_1 \equiv \frac{A_1}{r_1^2},\quad \tilde{A}_2 \equiv \frac{A_2}{r_1^2},\quad\tilde{A}_3 \equiv \frac{A_3}{r_1^2},\\ & \displaystyle \tilde{\varOmega} \equiv \frac{\varOmega}{r_1^3} ,\quad \tilde{E}_s \equiv \frac{E_s}{r_1^2 \gamma},\quad \tilde{E}_g \equiv \frac{E_g}{r_1^2 \gamma}, \end{aligned} \right\} \end{equation}

and we rewrite $\tilde {A}_1$, $\tilde {A}_2$, $\tilde {A}_3$, $\tilde {E}_s$ and $\tilde {E}_g$ into

(4.4)\begin{gather} \tilde{A}_1 = \left( {\rm \pi}-\alpha + \sin \alpha \cos \alpha \right) - \frac{\sin^2\alpha}{\sin^2\beta} \left( \beta - \sin \beta \cos \beta \right), \end{gather}

(4.5)\begin{gather} \tilde{A}_2 = 2 \frac{\sin\alpha}{\sin\beta} \beta \tilde{h}, \end{gather}

(4.6)\begin{gather} \tilde{A}_3 = 2 \left ( {\rm \pi}- \alpha \right ) \tilde{h}, \end{gather}

(4.7)\begin{gather} \tilde{E}_s = \tilde{A}_1\left ( 1 - \cos \theta_s \right ) + \tilde{A}_2 - \tilde{A}_3 \cos \theta_w, \end{gather}

(4.8)\begin{gather} \tilde{E}_g = \frac{1}{2} \tilde{\varOmega} \tilde{h} \left ( \frac{r_1}{l_c} \right )^2. \end{gather}

In order to obtain the equilibrium profile of the liquid, we calculate the derivative of the energy in term as $\alpha$, and let

(4.9) \begin{equation} \frac{\partial \tilde{E}}{\partial \alpha}=\frac{\partial(\tilde{E}_s+\tilde{E}_g)}{\partial \alpha}=0. \end{equation}

In addition, considering the constraint

(4.10)\begin{equation} \tilde{\varOmega}=\tilde{A}_1 \tilde{h}=\mathrm{constant}, \end{equation}

we have $\partial \varOmega / \partial \alpha = 0$, thus, the following relation can be obtained:

(4.11)\begin{equation} \frac{\partial \tilde{h}}{\partial \alpha}={-}\frac{\tilde{h}}{\tilde{A}_1} \frac{\partial \tilde{A}_1}{\partial \alpha}. \end{equation}

Substituting equations (4.4)–(4.8) and (4.11) into (4.9), we obtain the dimensionless value of the liquid layer thickness

(4.12)\begin{align} \tilde{h} & = \frac{\partial \tilde{A}_1}{\partial \alpha}\left(1-\cos \theta_s\right) \left\{ \left[2 \beta \frac{\sin \alpha}{\sin \beta} - 2\left({\rm \pi}-\alpha\right) \cos \theta_w+ \frac{1}{2} \tilde{\varOmega} \left(\frac{r_1}{l_c}\right)^2\right] \frac{1}{\tilde{A}_1} \frac{\partial \tilde{A}_1}{\partial \alpha} \right. \nonumber\\ &\quad \left. +\,2 \frac{\sin \alpha}{\sin \beta} - \frac{2 \beta}{\sin ^2 \beta}\left(\sin \alpha \cos \beta + \cos \alpha \sin \beta\right) - 2 \cos \theta_w \right\}^{{-}1}, \end{align}

where

(4.13)\begin{align} \frac{\partial \tilde{A}_1}{\partial \alpha} & = 2\frac{\sin ^2 \alpha}{\sin ^2 \beta} - 2 \beta\left(\frac{\sin \alpha \cos \alpha}{\sin ^2 \beta} + \frac{\sin ^2 \alpha \cos \beta}{\sin ^3 \beta}\right) + 2\frac{\sin \alpha \cos \alpha \cos \beta}{\sin \beta} \nonumber\\ &\quad +\cos ^2 \alpha-\sin ^2 \alpha-1. \end{align}

The solution of the problem is solved based on the following line of thought. Putting (4.4) and (4.12) into (4.10), and considering $\theta _w + \alpha + \beta = {\rm \pi}$, we can see there is only one unknown parameter $\alpha$. By employing a shooting method carried out by Matlab, $\alpha$ can be obtained. Based on that, $\beta$ can be obtained from $\theta _w + \alpha + \beta = {\rm \pi}$, moreover, the other unknown interested parameters such as $h$, $E_s$ and $E_g$ can be obtained based on the (4.12), (4.7) and (4.8).

It is stressed that, when $\theta _w = 180^{\circ }$ (even though it is difficult to achieve in practical cases, it is meaningful from a theoretical point of view), we cannot find the solution of the problem based on (4.12) because some singularities exist in the algebraic equations. Therefore, the condition for $\theta _w = 180^{\circ }$ should be considered specifically, and the details are shown in Appendix B. For this case, we obtain the following relationship:

(4.14)\begin{equation} \tilde{h}=\frac{{\rm \pi} \tilde{r}^2\left(1-\cos \theta_s\right)}{\displaystyle {\rm \pi}\tilde{r}+\frac{1}{2} \tilde{\varOmega}\left(\frac{r_1}{l_c}\right)^2}. \end{equation}

Here, we emphasize that the theory developed in the above is exclusively applicable to the quasi-static evolution of the hole in wetting modes I and II.