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.