Acoustics of bubbles trapped in microgrooves: From isolated subwavelength resonators to superhydrophobic metasurfaces

We study the acoustic response of flat-meniscus bubbles trapped in the grooves of a microstructured hydrophobic substrate immersed in water. In the first part of the paper, we consider a single bubble subjected to a normally incident plane wave. We use the method of matched asymptotic expansions, based on the smallness of the gas-to-liquid density ratio, to describe the near field of the groove, where the compressibility of the liquid can be neglected, and an acoustic region, on the scale of the wavelength, which is much larger than the groove opening in the resonance regime of interest. We find that bubbles trapped in grooves support multiple subwavelength resonances, which are damped—radiatively—even in the absence of dissipation. Beyond the fundamental resonance, at which the pinned meniscus is approximately parabolic, we find a sequence of higher-order antiresonance and resonance pairs; at the antiresonances (whose frequencies are independent of the gas properties and groove size), the gas is idle and the scattering vanishes, while the liquid pressure is in balance with capillary forces.

In the second part of the paper, we develop a multiple-scattering theory for dilute arrays of trapped bubbles, where the frequency response of a single bubble enters via a scattering coefficient. For an infinite array and subwavelength spacing between the bubbles, the resonances are suppressed by an interference effect associated with the strong logarithmic interactions between quasistatic line sources; the antiresonances are robust. In contrast, for finite arrays, however large, we find strong and highly oscillatory deviations from the frequency response of an infinite array in a sequence of intervals about the resonance frequencies of a single bubble; these deviations are shown to be associated with edge excitation, in the finite case, of surface “spoof plasmon” waves, which exist in the infinite case precisely in the said frequency intervals; the resonant peaks in these intervals correspond to the formation of standing surface waves in the finite array.

Figure 1

Schematic of the dimensional problem.

Figure 2

Schematic of the dimensionless problem.

Figure 3

Variation of $\beta$ with $\lambda =K{\chi }^2{\mathrm{\Omega }}^2$: solid line: exact solution; dashed line: strong interfacial-tension approximation (3.27); thin vertical lines: antiresonances

Figure 4

The magnitude and phase of $A$ as functions of $\mathrm{\Omega }$ for $K=1$. The solid curves are obtained using the exact variation of $\beta$ with $K{\chi }^2{\mathrm{\Omega }}^2$, while the dashed curves are obtained using the small-$K{\chi }^2{\mathrm{\Omega }}^2$ approximation (3.27). The vertical lines indicate the values of the resonance frequencies, obtained using (4.1), and antiresonance frequencies, obtained using (B18) and (C1).

Figure 5

Loci in the $(\mathrm{\Omega },K)$ plane of the resonances (solid lines) and antiresonances (dashed lines) of a single trapped bubble. The dash-dotted line shows approximation (4.4) for the fundamental resonance.

Figure 6

Contour maps of the imaginary part of $q(x,y)$ at resonance for $K=1$ (cf. Fig. 4). (a) Zeroth-order resonance, $\mathrm{\Omega }=0.26$. (b) First-order resonance, $\mathrm{\Omega }=2.02$. Also shown is the imaginary part of $\eta$. The small rectangles, of unity area, indicate the gas cavity; their color corresponds to the (imaginary part of) gas-domain pressure value, $Im\overline{q}$.

Figure 7

Contour maps of the real part of $q(x,y)$ at the first-order antiresonance for $K=1$. Also shown is the real part of $\eta$. The small rectangle indicates the corresponding gas-domain pressure value $Re\left(\overline{q}\right)$.

Figure 8

Dimensionless schematic of a dilute array of trapped bubbles.

Figure 9

Infinite array of grooves: $\left|w\right|,argw$, and $argR$ as functions of $l$ (or $\mathrm{\Omega })$ for $L/a=10$ and $K=1$. The vertical lines indicate the corresponding single-groove resonance frequencies, obtained using (4.1), and antiresonance frequencies, obtained using (B18) and (C1).

Figure 10

(Top) Comparison between the near-field response for an infinite array and a finite array consisting of $2N+1=41$ grooves. $(L/a=10$ and $K=1$ for both arrays.) (Bottom) Dispersion curves for surface waves guided by the infinite array; the sound line, representing propagation in the bulk, is also shown.

Figure 11

Same as Fig. 10 but focused on a small-$l$ interval. The added horizontal lines mark the $\mathcal{K}l$ values for which a whole number of surface-wave wavelengths fit the finite array: these are provided by (5.18) with $M=1,2,...,20$. The added vertical lines mark the corresponding $l$ values, showing approximate agreement with the position of the resonant peaks of the finite array.

Figure 12

Plane-wave scattering from a finite array of $2N+1=41$ grooves $(L/a=10$ and $K=1)$. For $X<0\left(X>0\right)$, we plot $ImQ$ for $l$ just below (above) the cutoff of the first surface-wave branch (cf. Figs. 10 and 11). The inset zooms upon the array region.