Isotropic Chiral Acoustic Phonons in 3D Quasicrystalline Metamaterials

The elastic properties of three-dimensional (3D) crystalline mechanical metamaterials, unlike those of amorphous structures, are generally strongly anisotropic—even in the long-wavelength limit and for highly symmetric crystals. Aiming at isotropic linear elastic wave propagation, we therefore study 3D periodic approximants of 3D icosahedral quasicrystalline mechanical metamaterials consisting of uniaxial chiral metarods. Considering the increasing order of the approximants, we approach nearly isotropic effective speeds of sound and isotropic acoustical activity. The latter is directly connected to circularly polarized 3D metamaterial chiral acoustic phonons—for all propagation directions in three dimensions.

Figure 1

(a) Illustration of the $q/p=3/2$ approximant of a 3D icosahedral quasicrystalline metamaterial. The black points result from the cut-and-project method. The enlarged area highlights three types of segments between these points, i.e., the edges (red), the face diagonals (blue dotted) of the two types of rhombohedra, and the shortest body diagonals (gray dashed) of the thinner rhombohedra. An animated version of the $3/2$ approximant is shown in Supplemental Material, Video 1 [39]. If we replace each segment by either an achiral homogeneous cylindrical rod or by a homogeneous uniaxial chiral rod, we approach an achiral or a chiral 3D quasicrystal in the limit of large approximant order, respectively. (b) Next, we approximate a homogeneous uniaxial chiral rod with full rotational symmetry by the depicted chiral metarod with threefold rotational symmetry. The metarod is composed of ordinary achiral elastic rods. This approximation is justified for wavelengths large compared to the length $L$. (c) Resulting chiral architecture corresponding to the enlarged region in (a). Throughout this Letter, we consider the following normalized parameters: $d/L=0.3$, $R/L=0.2$, $h/L=0.035$, $r/a_{6\mathrm{D}}=0.01$ (leading to the same rod radius for all rods in the architecture for fixed $a_{6\mathrm{D}}$), and fixed $a_{6\mathrm{D}}=100\mu \mathrm{m}$, which is amenable to state-of-the-art 3D laser printing of polymers [18]. Therefore, we use typical polymer parameters for all rods: Young’s modulus $E=4.18\mathrm{GPa}$, Poisson’s ratio $\nu =0.4$, and mass density $\rho =1.15\times {10}^3{\mathrm{kg}/\mathrm{m}}^3$.

Figure 2

Calculated phonon dispersion relations for approximant architectures as illustrated in Fig. 1 for propagation along a cubic face diagonal ($\mathrm{\Gamma }\mathrm{K}$ direction, left) and a principal cubic direction ($\mathrm{\Gamma }\mathrm{M}$ direction, right). (a) $q/p=1/1$, (b) $q/p=2/1$, (c) $q/p=3/2$, and (d) $q/p=5/3$. The lowest two transverse bands and their backfolded parts are colored in red, the other less important bands are plotted in gray. These data are shown in magnified form in Fig. S6 [39]. (e) and (f) exhibit the relative frequency splitting, $2(f_2-f_1)/(f_2+f_1)$, between the two transverse bands. The colors for the four approximants are indicated in the legend. (e) $\mathrm{\Gamma }\mathrm{K}$ direction. (f) $\mathrm{\Gamma }\mathrm{M}$ direction. The results shown in Figs. 3 and 4 have been evaluated for the wave number indicated by the dashed gray straight lines.

Figure 3

Calculated results for increasing order (from left to right) of 3D quasicrystal approximants for $|\mathbf{k}|=1/9(\pi /a_{6\mathrm{D}})$ [cf. dashed gray lines in Figs. 2 and 2]. (a) $q/p=1/1$, (b) $q/p=2/1$, (c) $q/p=3/2$, and (d) $q/p=5/3$. In the first two rows, we depict the direction dependence of the two transverse phonon eigenfrequencies, $f_1$ and $f_2$ on a false-color scale. In addition, the length of the vector from the origin to the surface of the plot is proportional to the eigenfrequency for this propagation direction. In the third row, we depict the relative frequency splitting between the two transverse eigenfrequencies, $2(f_2-f_1)/(f_2+f_1)$, in the same manner.

Figure 4

Calculated results for increasing order (from left to right) of 3D quasicrystal approximants for $|\mathbf{k}|=1/9(\pi /a_{6\mathrm{D}})$ [cf. dashed gray lines in Figs. 2 and 2]. (a) $q/p=1/1$, (b) $q/p=2/1$, (c) $q/p=3/2$, and (d) $q/p=5/3$. The first row exhibits the polarization degree, $\zeta$, on a false-color scale (cf. Fig. 3). $\zeta =0$ corresponds to linear polarization and $\zeta =1$ to circular polarization. The second row shows real-space trajectories of the phonon displacement vector versus time (mean over one periodic cell), i.e., a circle corresponds to circular phonon polarization, an ellipse to elliptical polarization, and a line to linear polarization. The center of each miniature on the sphere indicates the corresponding wave vector direction in 3D. The behavior of the two transverse phonon bands, with indices $i=1$ and $i=2$, is shown in red and black, respectively. For the $5/3$ approximant, chiral phonons are obtained for all phonon propagation directions in 3D.