Disordered surface vibrations in jammed sphere packings

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 $ω^*$. This frequency is controlled by $ΔZ = \langle Z \rangle - 2d$, 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 $ω^*$. The total number of these low-frequency surface modes is controlled by $ΔZ$, and the profile of their decay into the bulk has two characteristic length scales, which diverge as $ΔZ^{-1/2}$ and $ΔZ^{-1}$ as the jamming transition is approached.

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