Vibrations and diverging length scales near the unjamming transition

TL;DR

This study investigates vibrational properties and diverging length scales in jammed particle packings near the unjamming transition, revealing critical scaling behaviors and characteristic length divergences.

Contribution

It provides a detailed numerical analysis of vibrational states and length scales approaching the unjamming point, highlighting new critical scaling laws.

Findings

Density of vibrational states remains finite at zero frequency near unjamming.

Crossover frequency scales as a power-law with distance from the transition.

Characteristic length scales diverge as power-laws at the unjamming transition.

Abstract

We numerically study the vibrations of jammed packings of particles interacting with finite-range, repulsive potentials at zero temperature. As the packing fraction $\phi$ is lowered towards the onset of unjamming at $\phi_{c}$, the density of vibrational states approaches a non-zero value in the limit of zero frequency. For $\phi>\phi_{c}$, there is a crossover frequency, $\omega^{*}$ below which the density of states drops towards zero. This crossover frequency obeys power-law scaling with $\phi-\phi_{c}$. Characteristic length scales, determined from the dominant wavevector contributing to the eigenmode at $\omega^{*}$, diverge as power-laws at the unjamming transition.