3.1. Initial wetting state

The initial wetting state for the different surfaces is compared in figure 2 using the bottom view and acoustic reflection at small amplitude. Glass, flat PDMS and SHS 1 surfaces appear black from the bottom view, indicating that the light coming from the bottom is not reflected, and hence the solid surfaces are fully wet. The bottom view of the SHS 2 surface is much brighter, indicating that the bottom light is reflected by air–water interface, and thus an air layer is trapped inside the micro-structure. The stability of the Cassie–Baxter state has been predicted by Bico, Thiele & Quéré (Reference Bico, Thiele and Quéré2002) based on an energy analysis. They show that an air layer will be stable if the contact angle of the constitutive material on a flat surface $\theta _f$ obeys

(3.1)\begin{equation} \theta_f>\theta_c,\quad \textrm{with}\ \cos{\theta_c}=\frac{\varPhi-1}{R-\varPhi}, \end{equation}

where $R=1+{\rm \pi} D H/P^2$ is the roughness, and $\varPhi =({\rm \pi} D^2)/(4P^2)$ is the solid fraction. Our SHS 1 (SHS 2) has a roughness $R=1.43$ ($R=2.5$) and a solid fraction of $\varPhi =0.12$ ($\varPhi =0.41$), leading to a critical contact angle $\theta _c=134^\circ$ ($\theta _c=106^\circ$). The water contact angle on flat PDMS is $\theta _f=110\unicode{x2013}115^\circ$ (Mata, Fleischman & Roy Reference Mata, Fleischman and Roy2005). In agreement with the theoretical prediction, the $\theta _f$-value for the flat PDMS is smaller than $\theta _c$ for the SHS 1 surface so water invades the microstructure. For the SHS 2 surface $\theta _f> \theta _c$, therefore, a stable air layer is trapped between the micropillars, as observed in the experiments.

Figure 2. Comparison of the initial wetting state of the different solid surfaces submerged in water. The first row of images shows the bottom view by a high-speed camera, while the second row of data are the back-scattered sound detected by monitoring the voltage of the transducer. For comparison, the acoustic reflection from the glass surface solely (bottom left) is added in grey for the other surfaces.

The back-scattered sound of the different surfaces is then analysed at a small amplitude. The HIFU transducer voltage averaged over ten pulses is shown in figure 2. For all surfaces, echoes are detected at $85\ \mathrm {\mu }{\rm s}$, corresponding to the time required for an acoustic wave to travel to the focal point and return to the transducer. The reflected pulse has the same shape as the incoming pulse for the glass surface. For the flat PDMS and the SHS 1, the back-scattered pulse is more complex and consists of a first small amplitude reflection followed by a second higher reflection similar to the glass case. More surprisingly, the back-scattered sound from the SHS 2 is only made of a strong pulse with a ${\rm \pi}$-phase change, compared with the incoming pulse.

These observations can be explained using linear acoustics. The pressure reflection coefficient ($r$) of acoustic plane waves on an interface is given by ${r_{i,j} = (Z_j-Z_i)/(Z_i+Z_j)}$, with $Z_i = \rho _i c_i$ the acoustic impedance, $\rho _i$ and $c_i$ the density and sound velocity in the medium $i$. The subscript $i$ ($j$) represents the medium supporting the incident (transmitted) pulse. The different interfaces present in our experiments have the following reflection coefficient: $r_{w,g} = 0.8$ for water/glass interface, $r_{w,p} = -0.18$ for water/PDMS, $r_{w,a} = -1$ for water/air and $r_{p,g} = 0.85$ for PDMS/glass (Xu et al. Reference Xu, Ni, Chen, Tu, Guo, Bruus and Zhang2020). For the water–glass interface, $r_{w,g}$ is positive and close to 1, and the back-scattered pulse is expected to be of high amplitude without phase change, as observed in figure 2. The flat PDMS and SHS 1 surfaces are held by a glass plate, and the incoming pulse will first encounter the water/PDMS interface leading to a first weak reflection since $r_{w,p}$ is small. A non-negligible part of the pulse is transmitted to the PDMS and then reflected by the PDMS/glass interface, generating the second strong echo since $r_{p,g}$ is high.

Finally, the back-scattered sound coming from the SHS 2 surface is solely made of a strong pulse with ${\rm \pi}$-phase change, in agreement with a reflection on a water/air interface as $r_{w,a}=-1$. This confirms the presence of an air layer inside the SHS 2 microstructure, as optically observed, demonstrating that acoustics can be of great use to probe plastron stability. The absolute value of the reflection coefficient of the water/air interface is higher than the coefficient of water/glass ($|r_{w,a}|>|r_{w,g}|$). The amplitude of the back-scattered signal from SHS 2 should therefore be higher than the one from the glass surface, but the measured reflections are of similar amplitude (see figure 2). This discrepancy might be ascribed to the fact that the SHS 2 interface is not a pure water/air interface but a mixture of water/air and PDMS pillars (41 % of the surface is PDMS), which modifies the reflection coefficient.

3.2. Cavitation behaviour

The solid surfaces were subjected to high-intensity pulses of different amplitudes to characterize their cavitation behaviour. The cavitation is monitored by side-view images (see figure 3b–c) taken during the acoustic excitation when cavitation clouds are visible. The area of such cavitation cloud ($A_{cav}$) is measured and compared for the different surfaces under various excitation amplitudes in figure 3(a). Although the data are scattered, which is expected for cavitation experiments, it can be seen that only a few small cavitation bubbles appear on flat PDMS and on the SHS 1. In contrast, significant cavitation is observed on the glass surface and the SHS 2. The flat glass surface exhibits cavitation at low acoustic excitation, while a cavitation cloud appears later for SHS 2.

Figure 3. (a) Area of the cavitation cloud ($A_{cav}$, captured from the side view) as a function of the maximum incident acoustic intensity ($I_{max}$) for the different solid surfaces. Points in (a) represent the area of one experiment, while the solid lines represent the average values. (b–c) Side-view images of the cavitation cloud on glass surface (in b) and on SHS 2 (in c) appearing for an incident maximum intensity of $3.9\ {\rm kW}\ {\rm cm}^{-2}$.

The ability of surfaces or liquids to cavitate under relatively small pressure excitation, well above the spinodal pressure of water (${\sim }-140$ MPa) (Zheng et al. Reference Zheng, Durben, Wolf and Angell1991), is determined by the presence of micro/nanobubbles trapped in surface defects or impurities, called cavitation nuclei (Harvey et al. Reference Harvey, Barnes, McElroy, Whiteley, Pease and Cooper1944). Stabilization of such nuclei on a surface requires a hydrophobic geometrical defect called a crevice (Atchley & Prosperetti Reference Atchley and Prosperetti1989; Bussonniere, Liu & Tsai Reference Bussonniere, Liu and Tsai2020). When subjected to low enough negative pressure (i.e. cavitation threshold), such nuclei become mechanically unstable, and cavitation bubbles appear.

The different cavitation responses are therefore either due to a difference in nuclei population and/or in local negative pressure. The first nuclei source in the experiments comes from tap water, which contains many impurities and thus a large number of nuclei. However, the same water was used in all experiments, and hence these nuclei could not explain the cavitation differences observed. The second source of nuclei comes from the different solid surfaces. The PDMS is naturally hydrophobic, which should favour nuclei trapping compared with hydrophilic glass. However, this soft material is expected to have little, if no, sharp defects (crack like) due to its polymerization from a liquid form, which can explain why almost no cavitation on flat PDMS was detected (figure 3). Similarly, SHS 1 in a Wenzel wetting state, although being micro-textured with cylindrical pillars, is not expected to have sharp crevices, which can explain the small cavitation activity observed. Glass, on the other hand, can have small cracks, potentially leading to significant cavitation activity as observed on glass.

The other important factor for cavitation is the local negative pressure, which is greatly influenced by the acoustic boundary condition. The acoustic wave reflected by a surface with a factor of $r$ interferes with the incident wave, resulting in a local minimum pressure, which can greatly differ from the incident minimum pressure, ${P}_{{min}}^{{inc}}$. For example, at the interface between water and a surface $j$, the minimum pressure is ${P}_{{j}}^{{wall}}=(1+r_{w,j})\ {P}_{{min}}^{{inc}}$. The minimum pressure is therefore amplified on the glass surface (${P}_{glass}^{wall}=1.8\ {P}_{min}^{inc}$), reduced on flat PDMS and SHS 1 (${P}_{pdms,shs1}^{wall}=0.8\ {P}_{min}^{inc}$), or almost cancelled on SHS 2 (${P}_{shs2}^{wall}\approx 0$). This simplified analysis is in agreement with the experiments, where cavitation clouds appear on the surface for glass (figure 3b), whereas no cavitation is observed upon the surface of SHS 2 (figure 3c). Note that the small bubbles on the top of figure 3(b) are actually on the surface, although they appear away from it due to the slight tilt between the camera and the surface.

Interference between the incident and reflected waves also occurs away from the surface. If a continuous excitation were used, a standing wave in the liquid would have been generated. However, here, we use short pulses with only 2 periods with a significant amplitude. Wave interference therefore only occurs in the surface vicinity, in a region of approximately one wavelength ($\lambda$). Local minima generated by the transient standing wave depend on the reflection coefficient sign. If $r>0$, the pressure anti-nodes are located on the surface, at $\lambda /2$ and $\lambda$ away for the surface. Differently, if $r<0$ the minimum pressures are located at $\lambda /4$ and $3\lambda /4$ away from the surface. The above analysis may be consistent with the experimental data, which reveal that the cavitation cloud appears at $\lambda /4$ and $3\lambda /4$ for the SHS 2 experiments. However, with the analysis one would have expected the appearance of cavitation at $\lambda$ and $\lambda /2$ in the experiments with the glass surface and not only on the surface.

To understand this discrepancy for the glass surface, one needs to analyse the high acoustic intensity used in the experiments. Under such strong excitation, the acoustic propagation is nonlinear, and the acoustic waveform departs from a simple cosine function, as shown in figure 4(a). The solid lines represent the results from the nonlinear acoustic simulations. The simulation results are in good agreement with the experimental data (crosses) at low values of $I_{max}$. Note that the measurements at the focal point can only be performed at low intensity to prevent deterioration of the hydrophone by cavitation. As detailed in the Appendix, the acoustic field at high intensity was calibrated by measuring the pressure at the second pressure lobe (after the focal region). As $I_{max}$ increases, the waveform become asymmetric, and the maximum incident pressure ${P}_{max}^{inc}$ increases faster than the absolute minimum pressure $|{P}_{min}^{inc}|$, as shown in figure 4(b).

Figure 4. (a) Incident waveform at the focal plane for different maximum incident intensities ($I_{max}$). The simulation results (solid lines) using the nonlinear acoustic code (Soneson Reference Soneson2009) are compared with the experimental measurements (cross points) at low intensity. (b) Evolution of the minimum and maximum incident pressure (${P}_{{min,max}}^{{inc}}$) and of the minimum local pressure (${P}_{{surface}}^{{position}}$) at a specific position for a surface with the maximum incident intensity ($I_{{max}}$).

Such asymmetry has important implications in the local minimum generated in the interference region. For $r > 0$, the local minima located at $\lambda$, $\lambda /2$ and on the wall are equal to ${P}_{{j}}^{\lambda }={P}_{{j}}^{\lambda /2}={P}_{{j}}^{{wall}}=(1+r_{w,j}) {P}_{{min}}^{{inc}}$. However, for $r<0$, the reflected wave is inverted (with a phase change by ${\rm \pi}$), and the local minimum pressures are ${P}_{{j}}^{3\lambda /4}={P}_{{j}}^{\lambda /4}={P}_{{min}}^{{inc}}-r_{w,j}{P}_{{max}}^{{inc}}$. Figure 4(b) shows the minimum pressures generated by the combination of nonlinear acoustics and the reflection for the different surfaces. As the reflection from SHS 2 at low intensity (figure 2) has the same amplitude as the glass surface, we use $r_{w,shs2}=-0.8$ to better estimate the reflection. Due to the reflection of the strong maximum incident pressure, minimum pressures in SHS 2 experiments reach the lowest level, which explains the biggest cavitation cloud observed. Cavitation with SHS 2 appears at $I=2.5~{\rm kW}\ {\rm cm}^{-2}$, i.e. at ${P}_{{shs2}}^{\lambda /4}\approx - 16$ MPa, which corresponds to the threshold for cloud cavitation in water. Pressure in the glass experiment only reaches ${P}_{{glass}}^{wall}\approx - 15$ MPa at the maximum intensity. This explains why no clouds are observed in water for the glass cases. Cavitation, in this case, occurs only on the glass interface due to the presence of surface nuclei. It should be noted that the nonlinear waveform reflected with a phase shift of ${\rm \pi}$ is unstable and will recover to a sine wave at the transducer, as shown by Tanter et al. (Reference Tanter, Thomas, Coulouvrat and Fink2001).

3.3. Wetting transition

The stability of the SH state is also studied by using the bottom camera and the acoustic reflection at a small amplitude. Figure 5(a) shows the images from the bottom taken after the exposure of SHS 2 to different intense acoustic pulses. Note that the surface is moved between each high impulse experiment, so each trial is performed on a fresh area of the SHS. For intensities smaller than ${\approx }0.1\ {\rm kW}\ {\rm cm}^{-2}$, the air layer remains unchanged, as shown by the brownish colour. Starting at $0.25\ {\rm kW}\ {\rm cm}^{-2}$, a change in colour is observed, and a clear interference colour pattern appears at $0.59\ {\rm W}\ {\rm cm}^{-2}$. These colours arise from the white light interference inside the air layer of the SHS and indicate that the air-layer thickness has decreased at the focal region due to the acoustic pulse. For $I_{max}> 1.1\ {\rm kW}\ {\rm cm}^{-2}$, a dark disk appears in the centre surrounded by interference fringes. The radius of this disk increases with the acoustic intensity, and its evolution with $I_{max}$ is reported in figure 5(c).

Figure 5. (a) Bottom view of SHS 2 after exposition of different high-intensity acoustic pulses showing the appearance of interference fringes and a wetted region in black of radius ($R_w$). (b) Acoustic reflection of the SHS 2 of a small amplitude pulse before (blue) and after (red) the exposure to a high-intensity pulse of $I_{max}=2.5\ {\rm kW}\ {\rm cm}^{-2}$. (c) Evolution of the wetted radius ($R_w$) observed experimentally as a function of the maximum acoustic intensity ($I_{max}$). The solid and dashed lines correspond to the wetted radius predicted by the sliding and touch-down models, respectively.

This black disk might correspond to a local SH breakdown (i.e. wetting transition) since SHS 1, which did not trap an air layer, appears black in figure 2. To verify this, the acoustic reflection signals at low amplitude from the SHS 2 surface before (shown in dashed blue) and after (shown in red) an intense acoustic pulse of $2.5\ {\rm kW}\ {\rm cm}^{-2}$ are compared in figure 5(b). Before, the reflection corresponds to an air–water reflection, as shown in figure 2. The reflection after the high-intensity pulse is drastically changed and is similar to the reflection of SHS 1 of figure 2. This observation confirms that high-intensity pulses trigger a local wetting transition, corresponding to the dark area where water penetrates inside the microstructure. It should be noted that this transition occurs at an amplitude smaller ($I_{max}=1.1 \ {\rm kW}\ {\rm cm}^{-2}$) than that of the appearance of the cavitation cloud ($I_{max}=2.5\ {\rm kW}\ {\rm cm}^{-2}$), ruling out a transition driven by cavitation.

The transition from a Cassie–Baxter to a Wenzel wetting state is usually understood by looking at the local force balance of the air–water interface in contact with the microstructure. A small pressure increase in the liquid phase is balanced by a capillary pressure arising from the bending of the interface between pillars (Quéré Reference Quéré2008). At higher liquid pressure, the interface can either touch the bottom of the SHS (i.e. touch-down scenario) or slide on the pillar when the local contact angle on the pillars reaches the advancing one (i.e. sliding scenario) (Moulinet & Bartolo Reference Moulinet and Bartolo2007; Reyssat et al. Reference Reyssat, Yeomans and Quéré2007). These different scenarios are associated with two critical liquid pressures

(3.2)\begin{equation} p_t^c\approx\frac{2\gamma H}{(\sqrt{2}P-D)^2}, \end{equation}

for the touch down, with $\gamma$ the air–water surface tension, and $\theta _a$ the advancing contact angle; for the sliding (Moulinet & Bartolo Reference Moulinet and Bartolo2007)

(3.3)\begin{equation} p_s^c=\frac{2\varPhi}{1-\varPhi}|\cos{\theta_a}|\frac{2\gamma}{D}. \end{equation}

Previously, we showed that the oscillating acoustic pressure at the SHS 2 interface almost vanishes due to the acoustic boundary condition, and thus the fast-oscillating acoustic pressure (at 1.1 MHz) cannot explain the wetting transition. Nonetheless, acoustic waves are known to apply radiation pressure to interfaces (Borgnis Reference Borgnis1952) and are commonly used to trap objects in acoustofluidics (Bruus Reference Bruus2012). Such radiation forces arise from a change of medium momentum induced by the acoustic scattering or reflection of an interface or an object (Westervelt Reference Westervelt1957; Baresch, Thomas & Marchiano Reference Baresch, Thomas and Marchiano2013). These radiation forces originate from the difference between the time-averaged acoustic energy densities, i.e. radiation pressure, across the interface. It is a cumulative process, and these forces hence occur on a longer time scale than the fast-oscillating acoustic pressure. If we assume that the SHS 2 behaves as a pure water–air interface and an incident angle of 90$^o$, the radiation pressure applied on the interface is given by (Borgnis Reference Borgnis1952)

(3.4)\begin{equation} p_{rad}(r)=\frac{2I(r)}{c}, \end{equation}

with $I$ the incident acoustic intensity, $c$ the sound speed in water and $r$ the radial coordinate on the surface centred on the focal axis. The acoustic intensity, and thus the radiation pressure, depends on $r$ due to the HIFU geometry, as shown in figure 6(b). The wetting transition would occur if $p_{rad} > p_{s,t}^c$, and a critical radius $r_c$ can be extracted from the simulated intensity field whereby $p_{rad}(r_c) = p_{s,t}^c$. The predictions for $r_c$ using the sliding and touch-down models presented above are compared with the experimental data in figure 5(c) for $\theta _a = 110^\circ$.

Figure 6. Evolution of the maximum acoustic pressure in the axial (a) and radial (b) directions at low amplitude. The red lines correspond to the prediction of the acoustic simulation using the nonlinear code (Soneson Reference Soneson2009), while the blue dots represent the experimental points. (c) Evolution of the input pressure of the simulation ($p_{sim}$) fitting the different maximum and minimum measurements as a function of the generator voltage ($V_{gen}$).

This simplified model predicts the beginning of the transition around $I_{max}=900\ {\rm W}\ {\rm cm}^{-2}$ for the sliding scenario, which is in good agreement with the experimental data. Moreover, the wetted radius $R_w$ agrees well with the critical radius predicted in the range $1\unicode{x2013}2.5 \ {\rm kW}\ {\rm cm}^{-2}$ of $I_{max}$. The good agreement indicates that the wetting transition occurs where the acoustic beam is more intense and also validates the wetting transition as being driven by radiation pressure. It should be noted that the interference pattern starts to appear at a lower intensity (see figure 5(a) for $I_{max} = 0.59\ {\rm kw}\ {\rm cm}^{-2}$). This may suggest that the interface starts to slide at lower radiation pressure than predicted but does not reach the microstructure bottom. A more comprehensive analysis involving time dependence, similar to the one done for a free surface in Issenmann et al. (Reference Issenmann, Wunenburger, Chraibi, Gandil and Delville2011), is needed to better understand the interface motion under radiation pressure. In addition to developing such a complex model, a study of the influence of pulse duration and a better acoustic calibration would be needed to fully capture the interface dynamics in the future.

For larger intensity, the water penetration area is underestimated by the sliding model. This deviation coincides with the appearance of an intense cavitation cloud, which can reflect an acoustic wave and also create a strong pressure during its collapse. Therefore, the local acoustic field is expected to greatly differ from the simple combination of an incident and reflected wave, thereby modifying the radiation pressure on the surface. Note that the local acoustic field can be inferred from the imprint left on the SHS. In figure 5(a), concentric grey circles surrounding the wetted area appear and are spaced by ${\approx }\lambda /4$. However, such circles cannot be explained by the secondary lobes of the incident beam, spaced by ${\approx }\lambda$ (see figure 6b), which tends to support a strong modification of the acoustic field by the cavitation cloud.