# Listening to immersed superhydrophobic surfaces: Acoustic inspection of air plastron layers

Pierre-Brice Bintein, Pierre-Yves Passaggia, Nicolas Mazellier, Adrien Bussonnière

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Pierre-Brice Bintein, Pierre-Yves Passaggia, Nicolas Mazellier, Adrien Bussonnière. Listening to immersed superhydrophobic surfaces: Acoustic inspection of air plastron layers. Physical Review Fluids, 2025, 10 (11), pp.114905. 10.1103/ghsv-hy11. hal-05382671

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# Listening to immersed superhydrophobic surfaces: acoustic inspection of air plastron layers

Pierre-Brice Bintein

Matière et Systèmes Complexes, Univ. Paris Cité, CNRS, UMR 7057, 75013 Paris, France

Pierre-Yves Passaggia and Nicolas Mazellier

PRISME, Univ. Orléans, INSA CVL, UR 4229, France

Adrien Bussonnière*

Matière et Systèmes Complexes, Univ. Paris Cité, CNRS, UMR 7057, 75013 Paris, France

(Dated: November 26, 2025)

Super-hydrophobic coatings are used to separate surfaces from the surrounding liquid by trapping an air layer called plastron in its microstructures. The durability and therefore the performance of these surfaces lie in the capacity to retain this plastron, which might be destabilized by excessive stresses or dissolution. Plastron monitoring is therefore essential to quantify the amount of gas shielding the surface from the liquid and to determine its temporal evolution. Such air layers, which may blend multiple length scales ranging from nm up to cm, are challenging to probe. In this paper, we develop a new versatile technique based on the acoustic resonance of the plastron, which allows passive probing of the total volume of air trapped in the surface roughness. Validation of this technique is performed by comparing the mean acoustic thickness against direct optical observations and shows very good agreement. As this acoustic method uses portable transducers and is sensitive to the air layer's volume regardless of the super-hydrophobic surface, we foresee that it can pave the way toward plastron monitoring under more realistic conditions such as complex geometries in different flow regimes. Additionally, this new technique is shown to efficiently monitor plastron on timescales of a few milliseconds, enabling real-time characterization necessary to better assess destabilization mechanisms leading to the collapse of a plastron.

## I. INTRODUCTION

Super-hydrophobic (SH) surfaces are becoming increasingly popular in industrial applications as they repel water, therefore delaying undesirable phenomena such as corrosion, bio-fouling, icing, and friction drag, among others [1, 2]. Such surfaces are also common in nature, for instance in plants to provide self-cleaning properties, and are also used by some animals to breathe underwater [3–7]. When submerged in water, these bio-inspired surfaces entrap a thin gas layer often called "plastron," which shields the surface from its environment. The endurance of this resulting Cassie-Baxter state, and thus the gas plastron, essentially relies on the size of the surface roughness and on the contact angle between the solid surface, the surrounding liquid, and the trapped gas [8–10], two properties of the surface tailored for a specific application and associated flow regime. To ensure the stability of the air pocket, the spatial scales of the largest roughness elements range between a few tenths of a millimetre down to sub-micrometer scales depending on the characteristics of the surface and its operating conditions. Nonetheless, under extreme circumstances, for instance when the capillary forces no longer balance the pressure and shear stresses at the interface, or when the air layer diffuses over long periods of time, the plastron depletes and may vanish, leading to the Wenzel state [11–14]. Conversely, gas bubbles can be used to replenish the plastron (inducing a life cycle of the thickness) [15–17]. Therefore, the actual performance of underwater SH surfaces depends on the amount of trapped gas within their structures and underlines the need for dynamic monitoring of the plastron volume.

At such scales, measurements of the plastron thickness are challenging, even in well-controlled laboratory experiments. Direct optical imaging with long-range microscopes provides access to the interface position and displacement, which are directly linked to plastron thickness for SH surfaces with well-defined geometries [13], but fails for random roughness with cavities and micropores [14]. In addition, such imaging techniques can have high temporal resolution, yet low spatial resolution due to spherical aberrations and wavelength limitations. A compromise may, however, be found with laser-induced fluorescence combined with confocal microscopy [11, 12, 18]. Other optical methods include reflectance spectroscopy [19, 20] or interference microscopy [9, 21]. These techniques, although extremely sensitive, require special surfaces and preparations, together with costly and non-portable optical equipment; moreover, they

<footnote>* Contact author: adrien.bussonniere@cnrs.fr</footnote>
only provide a very local view of the samples and thereby prevent measurements at the industrial scale over large surfaces.

Here we show that a robust alternative using acoustic waves can be used to probe the global volume of the plastron and span a wide range of spatial scales at low cost. Similar to bubbles in liquids, the air layer trapped in the microstructure is expected to strongly interact with acoustic waves. For bubbles, the large compressibility of air coupled with the liquid inertia results in a strong acoustic resonator [22, 23], which efficiently scatters acoustic waves [24]. This remarkable property of bubbles has been widely used in applications such as medical ultrasound, where micro-bubble contrast agents are employed [25]. In the case of a plastron, although the air volume differs from a bubble, it is anticipated that a similar monopolar resonance [22] exists and that the air volume can be measured based on the plastron natural resonant frequency. It should be noted that acoustic waves have already been used to detect and collapse plastrons [26] or to generate "plastronic" (capillary) waves [27]. However, these studies used the acoustic radiation pressure generated by high-frequency (MHz) focused ultrasounds to interact locally with the air layer and did not consider the global plastron resonance.

In this work, we combine acoustic resonance and optical microscopy to perform a side-by-side comparison between independent quantitative measurements of plastron thicknesses for spheres with different radii and three different sizes of roughness elements, coated with a commercially available super-hydrophobic paint. In Section II, we present the theoretical basis of the new acoustic method for air layer monitoring by deriving a model predicting the resonance of a plastron; it allows the extraction of the air volume or the equivalent thickness from the measured natural frequency. To validate this model, we first independently measure the air layer thickness in Section III by combining confocal microscopy on dry planar surfaces and microscopy imaging on immersed spheres to detect the air-water interface position. In Section IV, the experimental acoustic method is presented and used to measure the air volume around the SH spheres; comparison with previous optical results shows good agreement, validating the acoustic technique while highlighting its advantages. The underlying acoustic model is further validated by varying the liquid pressure. In Section V, the robustness and versatility of this new method are demonstrated by applying it to the temporal monitoring of plastrons. Finally, conclusions are drawn, and the limitations of the technique are discussed in Section VI.

## II. THEORETICAL MODEL OF PLASTRON ACOUSTIC RESONANCE

Bubbles of gas immersed in liquids are known to resonate acoustically [22, 28]. We expect plastrons surrounding SH spheres to behave similarly in the long-wavelength limit; their resonance will then be associated with the breathing mode of the air layer, corresponding to an isotropic (radial) compression-dilatation of the plastron thickness. Such oscillation involves the compressibility of the air layer, which acts as a spring, and the inertia of the water around the object, which acts as a mass. The resulting resonant pulsation for a bubble while neglecting surface tension is called the Minnaert angular frequency [22] and is given by $\omega_M = \sqrt{3\kappa p_{g0}/(\rho R_0^2)}$, where $p_{g0}$ is the gas pressure, $\kappa$ is the polytropic coefficient, $\rho$ is the liquid density, and $R_0$ is the bubble radius.

Around the spherical plastrons studied in this work, the radial flow is identical to the one associated with spherical bubble oscillations; the mass involved in plastron oscillation is the same. The main difference between both oscillations lies in the oscillator spring associated with the gas compressibility. For the same external radius, the amount of gas in the plastron is drastically smaller, which translates into a higher stiffness. Indeed, the gas behavior is governed by the ideal gas law, and $pV^\kappa$ is constant, where $V$ is the gas volume. For bubbles, one gets $p_g = p_{g0}(R_0/R)^{3\kappa}$, which, once linearized for small radial oscillations of amplitude $x$ ($R = R_0 + x$, $x \ll R_0$), leads to a restoring pressure of the form $p_g = p_{g0}(1 - 3x\kappa/R_0)$. To describe the plastron, we assume that the steel interior remains still and thus does not participate in the observed resonance; this is supported by the large acoustic impedance of steel compared to that of air and water. The plastron is modelled as a spherical air shell with uniform mean thickness $h_0$ at rest, and with constant radius $R_0$ approximated by the steel ball radius $D/2$, where $D$ is the ball diameter without coating (as the typical roughness height and $h_0$ will both be of the order of tens to hundreds of micrometers, negligible with respect to the diameters $D$ of several millimetres). The instantaneous air volume is then given by $V = 4\pi R_0^2 h$ for an oscillation of the plastron thickness $h$ with small amplitude $x$ ($h = h_0 + x$, $x \ll h_0$), so that the gas's pressure change is now $p_g = p_{g0}(1 - x\kappa/h_0)$. By analogy, the plastron resonant frequency can therefore be obtained by replacing the bubble restoring pressure ($3p_{g0}\kappa/R_0$) with the one of the plastron ($p_{g0}\kappa/h_0$) in the Minnaert frequency, leading to:

$$f_0 = \frac{1}{2\pi} \sqrt{\frac{\kappa p_{g0}}{\rho R_0 h_0}} \quad (1)$$

The polytropic coefficient $\kappa$ appearing in eq. (1) depends on the gas-temperature evolution during the compression-dilatation process [29]. Due to their high specific-heat capacities, water and steel (interior of the sphere) can be
considered as heat reservoirs and impose a constant temperature at both the solid-gas and the gas-liquid interfaces. The gas temperature therefore depends on the rate at which the heat diffuses out of the plastron. The air thermal diffusion layer is given by $\delta_{th} = \sqrt{2D_{th}/\omega}$, with $D_{th}$ the thermal gas diffusivity of the order of $20 \times 10^{-6} \text{ m}^2/\text{s}$, which is to be compared to the air-layer thickness $h_0$. If $2\delta_{th} > h_0$ (as there are two interfaces), the heat leaks out of the plastron and the oscillation is isothermal, while if $2\delta_{th} < h_0$, the gas compression-dilatation is adiabatic. $h_0$ is, however, unknown and is the aim of the acoustic measurement. In the following, we assume an isothermal behaviour ($\kappa = 1$) and verify it a posteriori. This hypothesis is motivated by the fact that the plastron is not a uniform air layer but is made of interconnected gas pockets (Cassie-Baxter state) trapped in the large surface roughness: we can therefore expect the solid inclusions to thermalize the air layer faster. Note that a critical frequency $f_i$ above which a plastron follows an isothermal behaviour can be constructed based on the criterion $2\delta_{th} > h_0$ and using equation (1) as $f_i = \sqrt[3]{p_0^2/(\rho^2 D_{th} R_0^2)}/4\pi$.

We recall that this model is valid in the long-wavelength limit, i.e. when pressure excitation is uniform over the object ($R_0 \ll \lambda_{water}$). Moreover, the resonance described here relies on the assumption that the air layer vibrates as a monopole. This requires that the air has time to redistribute over the object to accommodate plastron thickness inhomogeneities, which is justified if the inhomogeneities expand over a size smaller than the acoustic wavelength in air ($c_{air}/f_0$). We verified a posteriori that these two conditions were fulfilled for the spheres used in this study.

The formula (1) shows that $f_0$ is expected to depend on the size of the SH spheres, on the pressure of the trapped gas, and, more importantly, on the plastron thickness allowing for potential acoustic plastron monitoring. These different dependencies are tested in Section IV to validate the model, while in Section III we independently measure the air-layer thickness.

### III. FABRICATION AND OPTICAL CHARACTERIZATION OF SH SPHERES

#### A. SH spheres manufacturing

Super-hydrophobic spheres were manufactured as follows: steel ball bearings with diameters [5, 10, 20] mm were spray-coated with a bottom layer of Ultra-Ever Dry (UED), then manually covered with calibrated carbon-based powders in order to attach particles of specific sizes to vary the surface roughness. After drying, a second layer of UED bottom coating is applied and dried, after which an additional layer of nanometric roughness elements is made with a top layer of UED coating. Three different types of roughness were obtained in this way: the first one, referred to as "P80", used a powder with ISO grade P80 corresponding to particles of 200 µm mean size; the second one, denoted as "P220", used an ISO grade P220 powder of 68 µm particles; and the third one, "NAR," was prepared without added particles. The same spheres were used and described in more detail in the work of Castagna et al. [30].

#### B. Confocal measurement of surface roughness

Flat samples of each type of roughness were prepared for visual inspection in a clean room with a Keyence VHX-5000 digital microscope, allowing the acquisition of 3D profiles with a large depth of field (102 µm) and a high spatial resolution in the direction normal to the surface (down to 5 µm). An objective VH-Z500T enabled magnification up to 5000× at a 4.4mm distance of observation. The scan was stopped when a 3D model covering a $1.5 \times 2.0 \text{ mm}^2$ portion was obtained, a size enabling the statistical convergence of the porosity measurements. Results are shown in Fig. 1(a-c) for each type of roughness element used in this study.

#### C. Visualisation of plastron interface

We now turn to immersed spheres in order to localize the plastron interface and measure its altitude above the solid surface. Direct side-view measurements are not possible as roughness elements prevent observation of the bottom surface of the spheres. In addition, the reflective nature of the plastron hinders the detection of a reference altitude. Therefore, only indirect measurements by comparison of situations with and without plastron can be made. As shown thereafter, the plastron interface appears to have a complicated shape linked to the random underlying roughness, so that a close view with high magnification is necessary to precisely determine its position; the SH sphere must then be kept at the very same location in order to scan the same area of its surface with and without plastron. Due to the thin magnetic needle attaching system used here to minimize external perturbation of the plastron, the sphere is mobile and slightly moves during its immersion, so that blocking it while removing water is illusory. Here we chose to
<figure><img src="image_2.png" /><figcaption>FIG. 1. 3D confocal microscopy images of a portion (1.5 × 2.0 mm²) of a flat plate covered with super-hydrophobic coatings. (a) NAR, (b) P220, (c) P80 (see the main text for details on coatings). The colour scales indicate the local height (z) in each case.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>FIG. 2. Experimental method to optically measure the position of the plastron's interface with sketches of (a) the set-up used to modify the pressure inside a vacuum tank while visualizing the plastron and (b) the typical plastron evolution associated with variations of the pressure *p*. The roughness of the SH sphere (sketched in grey) is omitted. The blue line represents the interface and shows the main deformation due to the volume change of the plastron. Turquoise levels reflect the air density. The number of the different steps (1-5) corresponds to the images shown in Fig. 3.</figcaption></figure>

remove the air instead, which turned out to maintain the sphere firmly still while maintaining the same optical index: quick and repeatable measurements are then possible.

The experimental setup is sketched in Fig. 2(a): the tested SH sphere is immersed in a transparent cell placed inside a vacuum tank and filled with previously degassed water in order to avoid bubble formation on the cell walls during the course of the experiment. The transparent acrylic vacuum tank with dimensions 200 × 158 × 218 mm³ and 3 cm thick walls is connected to a vacuum pump with a manometer and a release valve, allowing variation of the pressure. To precisely visualize the interface position, we use a long-range microscope (Navitar) 12X zoom lens mounted on a 5.5 Mpx Zyla Camera (Andor) focused on an equatorial edge of the sphere, while the sample is illuminated by an overhead projector. Acquisition is performed at a frequency of 10 Hz; the best resolution of 0.475 µm/px cannot focus all the plastron details at the pixel size, but resolutions within a couple of µm can still be achieved.

The evolution of the plastron during a typical pressure cycle is illustrated in Fig. 2(b) with corresponding pictures given in the example of Fig. 3(a); the associated long-range microscopic pictures showing the magnified surface are given in Fig. 3(b), where the interface is identified and highlighted by continuous coloured lines at different instants during the experiment, indexed from 1 to 5. The pressure cycle consists of a decrease in pressure to inflate the air layer up to the detachment of a bubble (1 to 3). The pressure is then restored to the atmospheric pressure while the plastron enters the microstructures due to its smaller volume (3 to 5). This allows accurate measurement of the plastron interface position relative to the roughness ridge line.

This pressure cycle is optically monitored as shown in Fig. 3(a-b). At atmospheric pressure above the cell (1), the interface highlighted in yellow in Fig. 3(b1) (and retained in the following pictures to show the interface evolution) appears distorted at such a small scale, covering most of the roughness elements but leaving some peaks emerging, with associated anchored triple lines. This complex shape results from contact-line motion during plastron formation
<figure><img src="image_4.png" /><figcaption>FIG. 3. (a) Pictures of an immersed SH sphere (with $D = 10$ mm and P80 roughness treatment) placed in a vacuum tank with varying the ambient pressure $p$, as denoted in (b). (b) Long-range microscope images taken at the equator of the immersed sphere showing the plastron at different steps of the pressure cycle (1-5). The yellow line shows the interface position at room pressure, the orange line shows it for a partially removed plastron and the red one corresponds to the wet crest line (ridge line) when the plastron moved inside the roughness. (c) Sketch of the two lengths to add for estimating the mean altitude of the plastron: $\Delta z_{Nav}$ (measured with long-range microscope) is the mean plastron elevation above the mean ridge height $z_{Ridge}$ (measured with confocal microscope). Colours hold for the lines drawn in (b), and their average positions. (d) Evolution of the elevation $\Delta z_{Nav}$ with respect to the sphere diameter $D$.</figcaption></figure>

(which may be pinned at specific locations), together with small interface deformations during the sphere's descent. Indeed, the global curvature of the interface ensures pressure balance at the depth of immersion through the Laplace capillary pressure: the plastron can withstand small pressure variations by interface deflections and motions.

When decreasing the pressure (2-3), the volume increases as the air pocket follows the isothermal ideal gas law. The air accumulates at the top of the sphere because of buoyancy (see Fig. 3(a2)), while the interface at the equator remains at the same location (Fig. 3(b2)). The pressure drop is stopped at 0.1 bar once the bubble on top detaches, while the plastron position at the equator remains unchanged (Fig. 3(a3-b3)). Then, as the total plastron volume has decreased, the plastron shrinks while restoring the pressure (4-5); the interface bends towards the sphere centre (Fig. 3(b4)) and ultimately moves inside the texture, becoming invisible and revealing the highest part of the roughness elements, i.e. the ridge line (Fig. 3(b5)). This process occurs globally, as the entire sphere loses its shiny aspect, as seen in Fig. 3(a5).

## D. Estimation of plastron thickness

The sharp crest line highlighted in red in Fig. 3(b5) is denoted here as the ridge line; it is a projection of the real crest lines over the width of the focal plane, i.e. the depth of field of the long-range microscope. The average elevation of this ridge line relative to the roughness bottom is denoted $z_{Ridge}$ in the following, as sketched in Fig. 3(c).
<figure><img src="image_5.png" /><figcaption>FIG. 4. Porosity $\phi$ as a function of the height $z$ for the surface with roughness P80, derived from the formula (2) applied to the scanned surface of Fig. 1(c). Dashed lines coloured as in Fig. 3 indicate the values of $z_{Ridge}$ and $\Delta z_{Nav}$ measured with confocal and long-range microscopes, respectively. Their sum is the plastron altitude so that the gray area under $\phi$ up to this altitude is the plastron thickness optically deduced $h_{opt}$ according to eq. (3).</figcaption></figure>

From the Navitar measurements, we deduce the mean elevation of the plastron above the ridge line as the difference between the mean plastron position at atmospheric pressure and the mean ridge altitude, i.e. the average distance between the yellow and red lines in Fig. 3(c), denoted as $\Delta z_{Nav}$. The value of $\Delta z_{Nav}$ is reported in Fig. 3(d) as a function of the sphere diameter for the three types of roughness. No clear dependence on the diameter is found, while $\Delta z_{Nav}$ depends on the surface roughness, indicating that the interface position depends solely on the roughness type. Note that $\Delta z_{Nav}$ does not represent the air-layer thickness, as most of the air is trapped inside the microstructure and is hidden in the immersed images.

To measure this hidden volume, we now turn to the confocal measurements. To consistently estimate $z_{Ridge}$ in both optical measurements, it is calculated as the mean projection of the altitudes in the $y$ direction (see Fig. 1) over a thickness chosen as the depth of field of the long-range microscope, ranging from 10 to 20 $\mu$m depending on the zoom used. A standard deviation of this height is computed by repeating the averaging procedure for different $y$ positions, leading to deviations of 3 to 8 $\mu$m in the worst case, for P80 and $D = 5$ mm. An example of such an average projected height is drawn with a thick red line in Fig. 3(c) for the specific P80 roughness. Once $z_{Ridge}$ is determined, the air-liquid interface can then be positioned in the confocal measurement by adding $\Delta z_{Nav}$ to $z_{Ridge}$.

Then, the amount of air trapped in the roughness below the interface is determined using the 2D porosity $\phi(z)$, which represents the area of air normalized by the total area at a specific height $z$:

$$ \phi(z) = \frac{1}{S} \int_{S} 1(x, y, z) \, dS, \qquad (2) $$

where 1 is the indicator function equal to 1 for air or 0 for solid, and $S$ is the surface scanned with the confocal microscope. The porosity evolution with height for the P80 sample is shown in Fig. 4. It starts from 0 at the bottom (no air) and reaches 1 (no solid) at the roughness maximum height ($z \approx 0.6$ mm). Finally, the total air volume, or equivalently the mean air thickness $h_{opt}$ of the plastron, can be calculated by integrating the 2D porosity from the bottom ($z = 0$) up to the mean air-water interface position ($z = z_{Ridge} + \Delta z_{Nav}$):

$$ h_{opt} = \int_{0}^{z_{Ridge} + \Delta z_{Nav}} \phi(z) \, dz. \qquad (3) $$

This optical height is illustrated in Fig. 4 as the gray area beneath the porosity curve. Consistent with the average nature of $z_{Ridge}$ and $\Delta z_{Nav}$, $h_{opt}$ is the uniform thickness of a hypothetical equivalent plastron with the same volume as the real one but in the shape of a spherical shell. It is thus possible—though tedious—to determine the plastron volume or equivalent thickness by combining optical measurements that require non-portable equipment. The results for the three types of roughness will be used in the following to validate the acoustic model.
<figure><img src="image_6.png" /><figcaption>FIG. 5. (a) Schematic of the acoustic experimental setup. A SH sphere of diameter $D$ that has trapped an air layer of thickness $h_0$ is held between an acoustic projector and a hydrophone to measure its plastron resonance by frequency sweeping (bottom). A reference signal is acquired without the sphere (top). (b) Typical signals recorded: the black signal is the reference signal and the blue one is obtained for the SH sphere of diameter 10 mm and roughness P80, at an exciting frequency $f = 2.2$ kHz close to its resonance. (c) Amplitude and phase of the normalized scattered pressure as a function of the tested frequency. The red line is a fit using eq. (5).</figcaption></figure>

# IV. ACOUSTIC MEASUREMENTS

## A. Experimental set-up and procedure

For the acoustic characterisation, each sphere is held again with a magnetic needle and immersed in a tank made of glass walls covered with absorbent material (Aptile SF5048, Precision Acoustics) to limit the influence of the boundaries and filled with tap water. The immersion is now controlled by a motorized linear stage (Thorlabs LTS300C/M) mounted over the cell and moving at a low speed (0.5 mm/s) to ensure the reproducibility of the plastron formation. This velocity is indeed smaller than the critical one above which dynamical effects can increase plastron thickness as reported in [31]. An acoustic projector (DNH Aqua 30) is used to excite the plastron, and a hydrophone amplified by a Nexus conditioner (Brüel & Kjaer 8103 & 2692, respectively) records the acoustic response, as sketched in Fig. 5(a). The excitation and hydrophone signals are generated/recorded by a USB oscilloscope (Tiepie HS5-540). The entire experiment is controlled using an in-house Matlab code based on the LibTiepie library. Several tank sizes and sphere/source positions were tested and were found to have only a small influence on the acoustic results.

To precisely characterize the acoustic resonance, we probe the plastron response as described for bubbles in [28]. A single-tone sine-wave voltage at a given frequency $f$ is used to excite the projector. The hydrophone voltage $s(t)$ first exhibits some amplitude growth, then its envelope stabilizes at a certain amplitude after 6 ms (see Fig. 5 (b)); to ensure being in the forced oscillation regime, we only analyse the signal between 10 and 30 ms. The hydrophone signal is recorded, then another sine wave is used at a different frequency, and so on. The frequencies probed here typically range from 0.5 to 10 kHz. For each frequency, we used synchronous detection to robustly extract the forced oscillation response $\bar{S}(\omega) = \langle s(t) \cos(\omega t) \rangle + i \langle s(t) \sin(\omega t) \rangle$, where $\bar{.}$ and $\langle \cdot \rangle$ denote the Fourier transform and time average, respectively.

The measured signal $s(t)$ is the convolution of the pressure received by the hydrophone $p(t)$ with the unknown hydrophone response $R_h(t)$, so that $\bar{S}(\omega) = \bar{R}_h(\omega)\bar{p}(\omega)$. To extract the SH-sphere response, we first note that the pressure measured by the hydrophone is the sum of two contributions: $p = p_{inc} + p_{sc}$, with $p_{inc}$ the incident pressure generated by the projector and $p_{sc}$ the pressure scattered by the sphere. Measuring the response with the sphere ($\bar{S}$) and without it ($\bar{S}_0$), the SH-sphere response function $\bar{R}_S(\omega)$ can be extracted as:

$$ \bar{R}_S = \frac{\bar{S} - \bar{S}_0}{\bar{S}_0} = \frac{\bar{R}_h(\bar{p}_{inc} + \bar{p}_{sc}) - \bar{R}_h\bar{p}_{inc}}{\bar{R}_h\bar{p}_{inc}} = \frac{\bar{p}_{sc}}{\bar{p}_{inc}}. \quad (4) $$

Figure 5(c) shows the plastron response of a SH sphere of diameter 10 mm and roughness P80. A resonance is clearly visible at 2.2 kHz, where a maximum amplitude can be observed together with a $\pi/2$ phase shift. Note that measurements with non-coated spheres were also performed and showed little to no difference from the reference signal obtained without the sphere, indicating that the observed resonance arises solely from the air layer.

To characterize the measured resonances, we refer again to the similar acoustic resonance of a bubble, which results
<figure><img src="image_7.png" /><figcaption>FIG. 6. (a) Natural resonance frequency of plastrons around SH spheres as a function of the sphere diameter for the 3 roughnesses tested. Each symbol corresponds to one sphere and is the average frequency over at least 3 acoustic measurements, with error bars showing their standard deviations. Solid lines are least-square fits of (1) where the fitting parameter $h_0$ is assumed to be constant for each of the 3 roughness families. (b) Same as in (a), in a log-log scale. The green solid line corresponds to the critical frequency $f_i$ above which an isothermal behaviour is ensured. (c) Acoustic thickness given by eq. (1) for each SH sphere as a function of the sphere diameter, in a logarithmic-linear scale. The average thicknesses (solid lines) inferred from the fits with eq. (1) plotted in (a) are again reported for each family, together with their confidence interval (dashed lines).</figcaption></figure>

in a strong scattered field modelled as a damped oscillator in [28]:

$$ \frac{\bar{p}_{sc}}{\bar{p}_{inc}} = \frac{R_0}{r} \frac{\omega^2}{\omega_0^2 - \omega^2 + i\Gamma\omega}, \qquad (5) $$

where $R_0$ is the sphere radius, $r$ the distance between the SH-sphere centre and the hydrophone, $\omega_0$ the resonant pulsation, and $\Gamma$ the damping coefficient (assumed here to be frequency-independent). The natural plastron frequency is estimated by fitting the experimental function $\bar{R}_S$ given by eq. (4) with the theoretical prediction of eq. (5) using a least-squares fit. An example is shown in red in Fig. 5(c). The factor $R_0/r$ is left free to improve the fit with the experiments but remains in a reasonable range (around unity, consistent with the hydrophone-sphere distance). Furthermore, the parameter $\Gamma$ could be related to physical dissipation mechanisms [32], but such processes are tedious to model in a confined environment [28]. Thus, only $\omega_0$ from the fit is used to derive the thickness of the plastron from experiments, thanks to eq. (1).

## B. Variations with the SH spheres

To test the theoretical prediction of equation (1), spheres with different diameters and three different roughnesses were tested. All spheres showed a resonance from which the natural frequency was extracted. Fig. 6(a) shows this frequency $f_0$ as a function of the sphere diameter $D$ and of its roughness, each associated with a specific colour. Each symbol is the average of at least three independent acoustic measurements of $f_0$ for a single SH sphere. The error bars correspond to the standard deviations of the measured frequencies and are smaller than the symbol size for most spheres, indicating a remarkable reproducibility of the acoustic measurement.

Natural frequencies of the different spheres cluster according to the surface roughness, implying that each surface treatment is associated with a characteristic air-layer thickness. Nonetheless, significant scattering is observed for the P220 and P80 cases, which suggests dispersion induced by the complex fabrication of each sphere with added roughness (probably due to inhomogeneous particle sticking). Despite this dispersion, the resonant frequency decreases with sphere diameter and follows the predicted $f_0 \propto D^{-1/2}$ trend, as shown in Fig. 6(b).

Assuming eq. (1) is valid, the plastron thickness of each sphere can be extracted, and the results are reported in Fig. 6(c). No clear trend with the sphere diameter is observed. An average thickness with a standard deviation can be extracted for each roughness; these values are plotted with continuous and dashed lines in Fig. 6(c). As anticipated, the air-layer thickness increases with the roughness; again, substantial variability arises for spheres with added powder. We recall that an isothermal behaviour was assumed to extract these thicknesses, which requires that $h < 2\delta_{th}$, i.e. $f_0 > f_i$. This condition is fulfilled for the NAR and P220 roughness as the corresponding natural frequencies are higher than $f_i$ (see Fig. 6(b)). For the P80 roughness, $h \approx 2\delta_{th}$ and the coefficient $\kappa$ might differ slightly from 1 [29]. However, as previously suggested, these plastrons do not consist of uniform air layers but cover
<figure><img src="image_8.png" /><figcaption>FIG. 7. Plastron thickness measured from the acoustic method $h_{acou}$ plotted against the thickness measured using the optical method $h_{opt}$, averaged over 3 diameters for each roughness element. The dashed line shows the 1:1 comparison between the two methods.</figcaption></figure>

<table><caption>TABLE I. Characteristics and thicknesses of the samples reported in this study.</caption><thead><tr><th>Type</th><th>$z_{Ridge}$ (μm)</th><th>$\Delta z_{Nav}$ (μm)</th><th>$h_{opt}$ (μm)</th><th>$h_{acou}$ (μm)</th></tr></thead><tbody><tr><td>NAR</td><td>20</td><td>8</td><td>28</td><td>14</td></tr><tr><td>P220</td><td>120</td><td>15</td><td>49</td><td>50</td></tr><tr><td>P80</td><td>440</td><td>28</td><td>153</td><td>143</td></tr></tbody></table>

solid inclusions, which will most probably thermalize the air faster, making an isothermal behaviour very likely even in the P80 case.

Note that in more realistic applications, plastron thicknesses are likely to be of the same order of magnitude as here while covering a larger surface area. Such plastrons have lower natural frequencies, which increase the thermal diffusion layer, thereby ensuring an isothermal behaviour. The theoretical curves corresponding to each roughness shown in Fig. 6 indicate that a relative agreement is found between the experiments and the model provided in eq. (1). Although encouraging, this is not sufficient to fully validate the acoustic method, essentially because of the variability of the results. In the following, we compare these acoustic measurements side by side with the previous optical results on the same sphere to avoid variability arising from the fabrication.

### C. Comparison with optical measurements

As shown previously, the optical method allowed the estimation of the plastron equivalent thickness $h_{opt}$, defined as the thickness of a uniform air layer, i.e. the air volume divided by the sphere's surface. This equivalent thickness is precisely the one involved in the acoustic resonance. To validate the acoustic model, we extract the equivalent acoustic thickness from the measured resonant pulsations: $h_{acou} = p_0/(\omega_0^2\rho R_0)$, still with equation (1), assuming an isothermal behaviour ($\kappa = 1$).

As suggested by the acoustic results, variability seems to arise from the fabrication of the spheres; to avoid this potential source of error, we compare the optical and acoustic results solely on the spheres that were tested both optically and acoustically (3 spheres per roughness). Fig. 7 shows the equivalent thickness measured from the acoustic method as a function of that measured with the optical method. The agreement between the two measurements is excellent, as the experimental points follow the 1:1 dashed line, which fully validates the acoustic method for probing plastron thickness.

It is worth emphasizing again that the acoustic error bars are mainly induced by variability among the different spheres tested; the resonant frequencies measured on one sphere are highly reproducible and lead to a thickness error of only a few micrometers. For the optical measurements, the error bar mainly comes from the uncertainty in determining $z_{Ridge}$, which depends on the depth of field of the long-range microscope and the random nature of the roughness, i.e. the number of peaks and valleys in the image. The results are summarized in Table I.
<figure><img src="image_9.png" /><figcaption>FIG. 8. Evolution of the acoustic resonance frequency f with the pressure p for the sphere D = 10 mm and P80 roughness. Blue data (full squares) are measurements at atmospheric pressure p₀ and indicate the typical data spread around the mean frequency f₀. These values at atmospheric conditions are used to normalize the fitted resonance frequencies obtained when varying the pressure (empty red squares). The dashed line has a unity slope.</figcaption></figure>

### D. Variations with the pressure

The last parameter of the acoustic model is the gas pressure of the plastron in eq. (1). The acoustic setup is placed in the same vacuum tank described in section III C, but connected either to a vacuum pump or to a compressor in order to impose pressures lower or higher than the atmospheric one. External pressure p has two effects on a plastron: changing its internal pressure p_g = p as well as its volume V = 4πR₀²h, with h the thickness at a pressure p, owing to air compressibility. The trapped air follows the isothermal ideal-gas law, so that p_gV, and hence p_gh, remain constant. Therefore, increasing or decreasing the pressure shrinks or expands the plastron, respectively. Note that we use smaller decompression than in section III C so that the air bubble forming on top remains small and connected to the plastron.

According to eq. (1), the plastron frequency when the external pressure is imposed at p is given by:

$$f = \frac{1}{2\pi} \sqrt{\frac{\kappa p_g}{\rho R_0 h}} = \frac{1}{2\pi} \sqrt{\frac{\kappa p_g^2}{\rho R_0 h_0 p_0}} = f_0 \frac{p}{p_0}, \quad (6)$$

as p_g = p and with f₀ the frequency when the external pressure is the atmospheric one p₀.

Fig. 8 shows the results of such acoustic measurements in normalized form. The experimental data indeed lie on a straight line with slope one, which further validates the acoustic model eq. (1) together with the ideal-gas law evolution. Note that the interface curvature could induce a pressure difference between the liquid and the gas (i.e. given by the Laplace pressure); nonetheless, such a deviation remains small and is negligible here. The presence of a bubble connected to the plastron at low pressures (as seen in Fig. 3) does not significantly modify its resonance, which confirms that the acoustic method monitors the air volume regardless of its distribution. Deviations from the identity line at lower pressure originate from parasitic cell acoustic resonances that distort the plastron signal.

## V. APPLICATION TO PLASTRON MONITORING

### A. Quality control

Our previous results obtained in Fig. 6(c), where several spheres were tested, show an important dispersion of thicknesses obtained for spheres with the same expected roughness. We can now safely attribute this variability to the fabrication method. The acoustic characterization therefore constitutes an easy and reliable method to control the quality of a super-hydrophobic coating. Here, we can conclude that the addition of glued particles is a critical step in the fabrication of SH spheres (as NAR spheres without additional powder show a smaller dispersion), and special care must be taken in order to have reproducible plastron thickness. Nonetheless, let us recall that the acoustic technique
<figure><img src="image_10.png" /><figcaption>FIG. 9. (a) Temporal evolution of acoustically measured height of plastron around a sphere with $D = 10$ mm and roughness P80, decreasing by dissolution. Colours hold for three successive experiments. (b) Acoustic signals for experiments in blue in (a) after 48 h of immersion: amplitude (up) and phase (bottom) of the normalized scattered pressure as the function of the tested frequency. These resonance signals remain nearly the same during the last 24 h of the experiment, revealing the presence of non-dissolving residual gas.</figcaption></figure>

probes an air volume, from which an equivalent thickness can be deduced, but a defect cannot be localized using the current procedure: only its overall effect on the air volume is detectable.

## B. Plastron dissolution

The technique is now tested dynamically by letting the immersed plastron dissolve in water. A SH sphere with $D = 10$ mm and roughness P80 is acoustically monitored over time. Figure 9(a) shows the evolution of the acoustically detected thickness, which represents the first quantitative tracking of such a dissolution for a plastron on random roughness. A good reproducibility is obtained between the 3 different experiments. Note that the typical thickness reduction that can be optically detected (by direct visualization) corresponds to its retraction down to the crest lines and is of the order of $\Delta z_{Nav} \approx 30$ µm (as seen in Fig. 3(d)). The plastron could thus be optically followed only for less than a couple of hours while the acoustic method reveals the dissolution dynamics over more than 15 hours.

The plastron dissolution can be modelled using the Epstein-Plesset diffusion model proposed for a spherical bubble [33]. The mass rate adapted for a plastron is then given by:

$$ \frac{dm}{dt} = 4\pi R_0^2 D_g \Delta c \left( \frac{1}{R_0} + \frac{1}{\sqrt{\pi D_g t}} \right), \qquad (7) $$

where $D_g$ is the diffusion coefficient of dissolved air in water ($D_g \approx 2.10^{-9}$ m²/s) and $\Delta c = c_0 - c_s$ is the dissolved gas-mass concentration difference between the surface assumed to be at saturation ($c_s \approx 22.8$ g/m³) and the one far away ($c_0 \approx 20.4$ g/m³ measured with an oxygen probe pyroscience OXROB10). After one hour, the mass diffusion layer is larger than $R_0$ and the second term on the right-hand side becomes negligible. Combining the expression for the air mass $m = \rho_g 4\pi R_0^2 h$ and eq. (7), the plastron thinning rate is:

$$ \frac{dh}{dt} = \frac{D_g \Delta c}{\rho_g R_0}. \qquad (8) $$

This is in agreement with the experimental observation of a linear decrease. Moreover, the thinning rate measured at 3 nm/s is close to the one predicted by diffusion (1 nm/s). This discrepancy suggests an additional convective effect, but further refinements are out of the scope of this study.

After approximately 20 hours, the thinning slows down and seems to reach a stable thickness as shown in Fig. 9(a). One experiment ran for a total duration of 48 hours, and the acoustic response indeed no longer evolved after 20 hours, remaining the same until the end of the experiment; the final amplitude and phase of the acoustic sphere response are plotted in Fig. 9(b). This suggests that a stable thickness is reached at long time. Similar thin air layers remaining for days have been observed in previous studies of plastrons covering natural or artificial SH surfaces
<figure><img src="image_11.png" /><figcaption>FIG. 10. Time evolution (a) and its Fourier transform (b) of the reference impulsion $s_0$ used to probe a broadband spectrum. (c) Amplitude of the acoustic response of a SH sphere (D=10 mm and P80) measured using the impulsion method, in blue, and the succession of monochromatic excitations in red.</figcaption></figure>

with hierarchical roughness elements [11, 20, 34]. The stability of such layers can be explained as follows: as the interface curves towards the gas phase, it significantly decreases the gas pressure inside the air (through the Laplace pressure), which in turn cancels out the gas concentration difference between the interface and the liquid through Henry's law. The signal's amplitude is broader than at the beginning of the experiment (as seen by comparison with Fig. 5(c)). Yet, it still has a discernible maximum associated with a phase jump of $\pi/2$ around a frequency of 10 kHz, corresponding to an equivalent thickness of 5 µm. However, the spread of both signals suggests the presence of inhomogeneous air layers, and possibly disconnected air pockets with a broad distribution of sizes rather than a connected homogeneous thin layer. Despite the local complexity of such small interfaces that would hinder other means of detection, the global acoustic method still provides access to the total residual gas volume, nearly 1.6 µL in the present case.

### C. Towards high-frequency monitoring

The acoustic method presented up to now is relatively slow as it consists of a succession of monochromatic acoustic excitation each lasting 30 ms, resulting in a total acquisition time of around 10 s (depending on the number of frequencies probed). To drastically improve the acquisition speed, acoustic pulses can be used to generate a broad spectrum, allowing multiple frequencies to be scanned at once. A cardinal-sinus-like pulse is used here to demonstrate the potential of this technique as shown in Fig. 10(a). The pulse is modified to compensate for the different transfer functions of the equipment without the sphere (projector, glass tank and hydrophone) in order to obtain a relatively constant spectral excitation between 50 Hz and 3 kHz and 0 for other frequencies (see Fig. 10(b)).

Similar to the monochromatic method, the SH sphere's response is calculated using $\bar{R}_s = \bar{s}/\bar{s}_0 - 1$ where $\bar{s}$ is the Fourier transform, while $s$ and $s_0$ are the signals with and without the sphere respectively. The main difference is that Fourier transforms are used here, while a synchronous detection was used for the monochromatic method, which reduced the measurement noise. The comparison between the two methods, shown in Fig. 10(c), shows an excellent agreement, demonstrating the potential of acoustics to monitor plastron dynamics at high frequencies (here 50 Hz). Faster monitoring could be achieved by tuning the pulse, but the rate will be limited to a fraction of the natural frequency of the probed plastron, as at least a few periods need to be recorded. This improvement of the method opens new possibilities to probe plastron dynamics at short times and might, for instance, allow to unravel plastron destabilisation mechanisms.

## VI. CONCLUSION & PERSPECTIVES

We propose a novel acoustic method to monitor the air volume of plastrons covering immersed super-hydrophobic spheres. These isolating air layers are essential for the applications of immersed SH surfaces, yet represent a weakness as they are sensitive to external perturbations such as shear or pressure variations, as well as dissolution, which
induces their long-term shrinking. Such effects can lead to the appearance of wet areas, or even to the destabilisation and full collapse of the plastrons. Thus, their characterization and monitoring are crucial.

Validated against usual tedious optical measurements, the innovative technique presented here is based on acoustic resonance and allows to measure the full volume of a plastron, or its equivalent mean-air thickness, for spheres of several millimetres with random roughness elements, and over time scales ranging from a few milliseconds to several days. Such high spatial and temporal resolutions enhance the prospects of applications of acoustic monitoring; it is especially adapted for surfaces with random roughness having cavities and micropores, which prevent direct visualization and the use of advanced optical techniques.

In addition, this quick, low-cost and versatile method makes use of a simple, adaptable setup; its only intrusive part is two small and portable hydrophones immersed in water, which could be displaced or replicated along larger or multiple surfaces with more complex geometries. We therefore expect such acoustic measurements to help unravelling the destabilisation mechanisms of a plastron and to serve for potential applications of SH surfaces. Nonetheless, it should be pointed out that the monopolar resonance used here is pertinent solely in the long-wavelength limit, i.e. for $c/\omega_0 > R_0$ with the c the sound speed in water and $\omega_0 = 2\pi f_0$. Using eq. (1), this provides the critical size below which this technique is usable : $R_c < h_0c^2\rho/p_0$ which is of the order of 2 m for a 100 µm thick plastron. Although this criterion proscribes a proper upscaling of the method as it, we believe that this technique is still of interesting use at the lab scale or to study plastrons used by plants and small animals. At larger scales, we expect the plastron response to be governed by acoustic diffusion of higher vibrations modes and will require more dedicated studies.

In this study, plastrons were assumed to form a continuous layer of air around the sphere which is confirmed by the good agreement between the optical and acoustic layer. For larger surfaces, plastron might be composed of a collection of disconnected air layers. The acoustic response of such plastrons is expected to behave differently as each layers possesses its own resonant frequency. Nonetheless, these air patches will be located close to each other and will therefore be coupled which might result in a collective low-frequency resonance, similar to the one studied here [35]. Future works remain necessary to clarify the acoustic behaviour of such peculiar coupled oscillators.

Finally, interface deformation at resonance $\xi_0$ induced by the acoustic excitation $p_{ac}$ is given by [28] and is $\xi_0 = Qp_{ac}/(\rho R_0 \omega_0^2)$ with $Q = \omega_0/\Gamma$ the oscillator quality factor typically ranging from 1 to 10 in our experiments. Using the definition of $\omega_0$ we find $\xi_0/h_0 = Qp_{ac}/p_0$. The proposed technique remains non-invasive as long as $\xi_0/h_0 \le 10^{-2}$ corresponding to $p_{ac} \le 100$ Pa, which is the case for classical underwater sources. From a different perspective, more powerful sources could be used to probe the critical pressure required for plastron collapse or to force interface motion which could be of interest to interact with the hydrodynamic boundary layer. In the same vein, in turbulent conditions, flow pressure fluctuations might excite the resonance studied here and trigger interface motion.

## ACKNOWLEDGMENTS

We would like to thank Valentin Leroy for useful discussions. This work was funded by the French National Research Agency (ANR) through the IDEFHYX Project (ANR-22-CE51-0021). For the purpose of Open Access, a CC-BY public copyright licence has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.

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