Disordered surface vibrations in jammed sphere packings

Abstract

We study the vibrational properties near a free surface of disordered spring networks derived from jammed sphere packings. In bulk systems, without surfaces, it is well understood that such systems have a plateau in the density of vibrational modes extending down to a frequency scale ${\omega }^{\ast }$. This frequency is controlled by $\Delta Z=⟨Z⟩-2d$, the difference between the average coordination of the spheres and twice the spatial dimension, $d$, of the system, which vanishes at the jamming transition. In the presence of a free surface we find that there is a density of disordered vibrational modes associated with the surface that extends far below ${\omega }^{\ast }$. The total number of these low-frequency surface modes is controlled by $\Delta Z$, and the profile of their decay into the bulk has two characteristic length scales, which diverge as $\Delta Z^{-1/2}$ and $\Delta Z^{-1}$ as the jamming transition is approached.

7 pages, 5 figures