On Variational Arguments for Vibrational Modes near Jamming

TL;DR

This paper introduces a new variational approach to accurately predict the scaling of vibrational soft modes near the jamming transition in amorphous solids, improving understanding of their elastic properties.

Contribution

A novel variational argument is developed that correctly predicts the scaling of soft vibrational modes near jamming, overcoming previous limitations.

Findings

Soft modes relate to local strain response in connected networks.

Soft mode volume scales as 1/δz, with δz being excess coordination.

Predictions are confirmed through numerical verification.

Abstract

Amorphous solids tend to present an abundance of soft elastic modes, which diminish their transport properties, generate heterogeneities in their elastic response, and affect non-linear processes like thermal activation of plasticity. This is especially true in packings of particles near their jamming transition, for which effective medium theory and variational arguments can both predict the density of vibrational modes. However, recent numerics support that one hypothesis of the variational argument does not hold. We provide a novel variational argument which overcomes this problem, and correctly predicts the scaling properties of soft modes near the jamming transition. Soft modes are shown to be related to the response to a local strain in more connected networks, and to be characterized by a volume $1/\delta z$, where $\delta z$ is the excess coordination above the Maxwell threshold. These predictions are verified numerically.