Phononic crystals of poroelastic spheres

An extension of the layer-multiple-scattering method to phononic crystals of poroelastic spheres immersed in a fluid medium is developed. The applicability of the method is demonstrated on specific examples of close-packed fcc crystals of submerged water-saturated meso- and macroporous silica microspheres. It is shown that, by varying the pore size and/or the porosity, the transmission, reflection, and absorption spectra of finite slabs of these crystals are significantly altered. Strong absorption, driven by the slow waves in the poroelastic material and enhanced by multiple scattering, leads to negligible transmittance over an extended frequency range, which might be useful for practical applications in broadband acoustic shielding. The results are analyzed by reference to relevant phononic dispersion diagrams in the viscous and inertial coupling limits, and a consistent interpretation of the underlying physics is provided.

Figure 1

Transmittance and absorbance of an acoustic plane wave incident normally on a slab consisting of eight fcc $\left(111\right)$ planes of submerged water-saturated close-packed porous silica spheres of radius $S=2.5\mu \mathrm{m}$, with porosity $f=10\%$, for different pore sizes: (a) $R_p=10$ nm, (b) $R_p=30$ nm, (c) $R_p=100$ nm, and (d) $R_p=500$ nm. The dotted lines in (a) and (d) display the corresponding transmission spectra for crystals made of homogeneous spheres with elastic parameters calculated using self-consistent effective-medium theory for (lossless) composite elastic media [30] and of water-saturated porous spheres neglecting viscous losses ($\eta =0$), respectively.

Figure 2

The same as in Fig. 1 for spheres of porosity $f=10\%$ but radius $S=2.5$ mm and pore radius $R_p=500\mu \mathrm{m}$.

Figure 3

(a) Change in the density of states of the acoustic field induced by a submerged water-saturated porous silica sphere with porosity $f=10\%$, treated as a lossless homogeneous sphere with elastic parameters calculated using self-consistent effective-medium theory for composite elastic media [30]. (b) The phononic band structure of an fcc crystal of such close-packed homogeneous spheres, in water, along the $\left[111\right]$ direction. Solid (dotted) lines refer to nondegenerate (doubly degenerate) bands of ${\mathrm{\Lambda }}_1$ (${\mathrm{\Lambda }}_3$) symmetry. The shaded areas mark the Bragg (B) and hybridization (H) gaps. (c) Transmittance of an acoustic plane wave incident normally on a $\left(111\right)$ slab of this crystal, eight layers thick.

Figure 4

The same as Fig. 3 but for submerged water-saturated porous silica spheres described by Biot's theory, ignoring viscous losses ($\eta =0$). The peaks in (a) correspond to resonances of given multipole order $\ell$ denoted in the diagram.

Figure 5

Transmittance, absorbance, and reflectance of an acoustic plane wave incident normally on a slab consisting of eight fcc $\left(111\right)$ planes of submerged water-saturated close-packed porous silica spheres of radius $S=2.5\mu \mathrm{m}$ versus the porosity $f$ for different pore sizes. Pore radius (from top to bottom): $R_p=10$, 30, and 100 nm.

Figure 6

Transmittance, absorbance, and reflectance of an acoustic plane wave incident on a slab consisting of eight fcc $\left(111\right)$ planes of submerged water-saturated close-packed porous silica spheres ($S=2.5\mu \mathrm{m}, f=10\%$, and $R_p=100$ nm) with ${\mathbf{k}}_{\parallel }$ along high-symmetry lines of the fcc $\left(111\right)$ surface Brillouin zone, shown in the inset. $\overline{\mathrm{\Gamma }}$: ${\mathbf{k}}_{\parallel }=\frac{2\pi }{a_0}(0,0), \overline{\mathrm{K}}$: ${\mathbf{k}}_{\parallel }=\frac{2\pi }{a_0}\left(\frac{2}{3},0\right), \overline{\mathrm{M}}$: ${\mathbf{k}}_{\parallel }=\frac{2\pi }{a_0}\left(\frac{1}{2},\frac{\sqrt{3}}{6}\right)$.