Scaling of phononic transport with connectivity in amorphous solids

Abstract

The effect of coordination on transport is investigated theoretically using random networks of springs as model systems. An effective medium approximation is made to compute the density of states of the vibrational modes, their energy diffusivity (a spectral measure of transport) and their spatial correlations as the network coordination $z$ is varied. Critical behaviors are obtained as $z\to z_c$ where these networks lose rigidity. A sharp cross-over from a regime where modes are plane-wave-like toward a regime of extended but strongly-scattered modes occurs at some frequency ${\omega }^{\ast }\sim z-z_c$, which does not correspond to the Ioffe-Regel criterion. Above ${\omega }^{\ast }$ both the density of states and the diffusivity are nearly constant. These results agree remarkably with recent numerical observations of repulsive particles near the jamming threshold \cite{ning}. The analysis further predicts that the length scale characterizing the correlation of displacements of the scattered modes decays as $1/\sqrt{\omega }$ with frequency, whereas for $\omega <<{\omega }^{\ast }$ Rayleigh scattering is found with a scattering length $l_s\sim (z-z_c{)}^3/{\omega }^4$. It is argued that this description applies to silica glass where it compares well with thermal conductivity data, and to transverse ultrasound propagation in granular matter.

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