Phonon gap and localization lengths in floppy materials

TL;DR

This paper investigates the vibrational and elastic properties of floppy materials, revealing a phonon gap and associated length scales that diverge as the system approaches marginal stability, with implications for amorphous solids near unjamming.

Contribution

It introduces a theoretical and numerical analysis of phonon gaps and localization lengths in floppy elastic networks, clarifying their dependence on coordination and frequency.

Findings

Presence of a phonon gap below a critical frequency $^*$

Localization length $l_c$ diverges as the phonon gap vanishes

Two characteristic length scales, $l_c$ and $l^*$, govern elasticity and boundary effects

Abstract

Gels of semi-flexible polymers, network glasses made of low valence elements, softly compressed ellipses and dense suspensions under flow are examples of floppy materials. These systems present collective motions with almost no restoring force. We study theoretically and numerically the frequency-dependence of the response of these materials, and the length scales that characterize their elasticity. We show that isotropic floppy elastic networks present a phonon gap for frequencies smaller than a frequency $\omega^*$ governed by coordination, and that the elastic response is localized on a length scale $l_c\sim 1/\sqrt{\omega^*}$ that diverges as the phonon gap vanishes (with a logarithmic correction in the two dimensional case). $l_c$ also characterizes velocity correlations under shear, whereas another length scale $l^*\sim 1/\omega^*$ characterizes the effect of pinning boundaries on elasticity. We discuss the implications of our findings for suspensions flows, and the correspondence between floppy materials and amorphous solids near unjamming, where $l_c$ and $l^*$ have also been identified but where their roles are not fully understood.