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Phonon transport properties of particulate physical gels

Particulate physical gels are sparse, low-density amorphous materials in which clusters of glasses are connected to form a heterogeneous network structure. This structure is characterized by two length scales, $ξ_s$ and $ξ_G$: $ξ_s$ measures the length of heterogeneities in the network structure, and $ξ_G$ is the size of glassy clusters. Accordingly, the vibrational states of such a material also exhibit a multiscale nature with two characteristic frequencies, $ω_\ast$ and $ω_G$, which are associated with $ξ_s$ and $ξ_G$, respectively: (i) phonon-like vibrations in the homogeneous medium at $ω< ω_\ast$, (ii) phonon-like vibrations in the heterogeneous medium at $ω_\ast < ω< ω_G$, and (iii) disordered vibrations in the glassy clusters at $ω> ω_G$. Here, we demonstrate that the multiscale characteristics seen in the static structures and vibrational states also extend to the phonon transport properties. Phonon transport exhibits two distinct crossovers at the frequencies $ω_\ast$ and $ω_G$~(or at wavenumbers of $\sim ξ_s^{-1}$ and $\sim ξ_G^{-1}$). In particular, both transverse and longitudinal phonons cross over between Rayleigh scattering at $ω< ω_\ast$ and diffusive damping at $ω>ω_\ast$. Remarkably, the Ioffe--Regel limit is located at the very low frequency of $ω_\ast$. Thus, phonon transport is localized above $ω_\ast$, even where phonon-like vibrational states persist. This markedly strong scattering behavior is caused by the sparse, porous structure of the gel.