Tacticity in chiral phononic crystals

The study of vibrational properties in engineered periodic structures relies on the early intuitions of Haüy and Boscovich, who regarded crystals as ensembles of periodically arranged point masses interacting via attractive and repulsive forces. Contrary to electromagnetism, where mechanical properties do not couple to the wave propagation mechanism, in elasticity this paradigm inevitably leads to low stiffness and high-density materials. Recent works transcend the Haüy-Boscovich perception, proposing shaped atoms with finite size, which relaxes the link between their mass and inertia, to achieve unusual dynamic behavior at lower frequencies, leaving the stiffness unaltered. Here, we introduce the concept of tacticity in spin-spin-coupled chiral phononic crystals. This additional layer of architecture has a remarkable effect on their dispersive behavior and allows to successfully realize material variants with equal mass density and stiffness but radically different dynamic properties.

If we now substitute these quantities into the equation expressing ω * , The bounds for ω * are given by the cases where m 1 ≈ m 2 , leading to ω * → 2 π , or m 2 m 1 leading to ω * → 1 π .
For the sake of completeness, the ω * for the syndiotactic phononic crystal presented in the main text is calculated according to the inline equation shown on page 3 of the manuscript ω * = f BG a ρ C , with f BG = 374 Hz, a = 0.059 m, ρ = 203 kg·m −3 , C = 1 MPa leading to ω * = 0.314 (with 2/π = 0.637 for monoatomic crystals).

Supplementary Note 2
Frequency response of TRO+ and mode shapes. Figure 2 of the main text shows a detailed view of the rigid body modes of the TO and TROs. The presence of the antiresonance in the TRO frequency spectra indicates that an additional resonance can be expected at higher frequency. Indeed, the inspection of a wider range of frequencies allows us to confirm this expectation, as shown in Supplementary Fig. 4.

Supplementary Note 3
Analytical model of the TRO. In this section, we formulate and solve an analytical model for a single TRO with the aim of (i) capturing the basic physics underlying its resonance and antiresonance phenomena and (ii) predicting the (low-)frequency response function (frf) in a closed analytical form.
As presented in the main text, the conceived model of the TRO includes deformable elastic elements such as tapered and twisted beams, tilted with respect to the vertical direction of an angle Ψ, connected to disks (also deformable).
A detailed analytical description of such a multi-structure can be derived using asymptotic techniques 1, 2 . Here, to obtain the (low-)frequency response function (frf) of a TRO in a closed analytical form, we assume that (i) the disks are rigid (i.e., of infinite stiffness), (ii) the elastic properties of the beams isotropic and (iii) their cross-sections of constant circular shape throughout their longitudinal coordinate. This allowed the modeling of the beams according to the Euler-Bernoulli theory. The tilting of the beams with respect to the vertical coordinate has been retained so to point out its effect on the frf of a single TRO, as the one schematically represented in Supplementary Fig. 5, where the three Euler-Bernoulli beams are represented by the dashed lines. Each of them, e.g. the one marked in red, supports a displacement vector of the form: The transverse w and longitudinal v displacements of the beam are governed by two decoupled PDEs 3 : for the longitudinal and flexural fields, respectively. In Eq. such that e θ = R(∆/2, e z )e x and e 3 = R(∆/2, e z )e y , being: We assume that the TRO in Supplementary Fig. 5 vibrates as a result of a time-harmonic prescribed displacement applied to the base of the beams x 1 = 0, The displacement field in Eq. (1) is therefore time-harmonic of frequency ω, i.e.: Moreover, we introduce the scaled variables: where u 0 is the amplitude of the time harmonic external excitation. Using Eqs. (4) and (5), and focusing on the low-frequency regime, we can rewrite the PDEs (2) as ODEs: where we have introduced (·) ≡ ∂/∂x 1 (·), whose general solutions are: The motion of a rigid disk in Fourier space comprises a vertical displacement and a rotation around the e z axis, i.e.: In Eq. (8) we have introduced the normalized vertical displacement amplitude of the disk as U out =Ū out /u 0 and its normalized angular displacement around e z , Θ out =Θ out r/u 0 , being r the radius of the disk.
In a regime of small deformations, the following set of boundary conditions apply to the longitudinal displacement field of the beam (first relation in Eq. (7)): whereas the end conditions for the flexural displacement field (second function in Eqs. (7)) are: The simplified set of boundary conditions for the flexural motion (10) is compatible with those used by Orta and Yilmaz 4 . However, in our model we also take into account the force F (see Supplementary Fig. 5), which results from the longitudinal deformation of the beams, which was on the contrary neglected by Orta and Yilmaz 4 . It is worth mentioning here that although the longitudinal force F plays no role on the dynamics of the disk when the beams are tilted, it significantly contributes to the dynamics of the disk at Ψ = π/2 (i.e., for vertical beams). Therefore in our analytical model we have retained the longitudinal forces due to the elongation of the beams to avoid limitations on Ψ values the model can treat. A related problem was described by Tallarico et al. 5 in the context of a two dimensional mass-truss lattice with its unit cell containing a tilted resonators. In this work, it is shown that the structure is degenerate -i.e. possesses a vanishing torsional frequency -at zero tilting angle and that such degeneracy can be cured introducing flexural ligaments. For these reasons, we believe that accounting for both flexural and longitudinal reaction forces to the supporting beams is of pivotal importance to obtain consistent analytical models of geometrically chiral and tactic structures.
In Eq. (11) we have introduced: with: We refer to the functions (13) as the longitudinal and torsional frf, respectively. In Supple- Analogous conclusions hold for Supplementary Fig. 7 where we show the comparison of the FE torsional frequency response function with its analytical approximation in Eq. (13).
Moreover, the frf of a non-tilted TO (Ψ = π/2 in Eq. (13)) reduces to: perfectly agree with the exponential fit at the same frequencies shown in Supplementary  Fig. 8a,b.
In the isotactic case ( Supplementary Fig. 8c of the edge of the right plate was deemed a sufficiently good proxy. However this approximation is responsible for the peak around 1000 Hz visible in transmission function reported in Fig. 1c of the main text, which is due to the fact that the amplitude measured at the edge of the plate becomes extremely small, due to a local mode of the plate, therefore implying a division of the measured output signal by a small value (causing the artificial peak). 'x=0.302mm