On the Rigidity of Amorphous Solids

TL;DR

This paper provides a theoretical framework for understanding the microscopic origins of rigidity in amorphous solids, revealing critical behaviors near jamming and their implications for vibrational properties and material response.

Contribution

It derives scaling laws and characterizes the non-local geometric nature of rigidity in amorphous systems near jamming, applying to various materials like granular matter and glasses.

Findings

Presence of a boson peak related to coordination

Rigidity characterized by a large length scale diverging at jamming

Shear modulus can be much smaller than bulk modulus near jamming

Abstract

We poorly understand the properties of amorphous systems at small length scales, where a continuous elastic description breaks down. This is apparent when one considers their vibrational and transport properties, or the way forces propagate in these solids. Little is known about the microscopic cause of their rigidity. Recently it has been observed numerically that an assembly of elastic particles has a critical behavior near the jamming threshold where the pressure vanishes. At the transition such a system does not behave as a continuous medium at any length scales. When this system is compressed, scaling is observed for the elastic moduli, the coordination number, but also for the density of vibrational modes. In the present work we derive theoretically these results, and show that they apply to various systems such as granular matter and silica, but also to colloidal glasses. In particular we show that: (i) these systems present a large excess of vibrational modes at low frequency in comparison with normal solids, called the "boson peak" in the glass literature. The corresponding modes are very different from plane waves, and their frequency is related to the system coordination; (ii) rigidity is a non-local property of the packing geometry, characterized by a length scale which can be large. For elastic particles this length diverges near the jamming transition; (iii) for repulsive systems the shear modulus can be much smaller than the bulk modulus. We compute the corresponding scaling laws near the jamming threshold. Finally, we discuss the applications of these results to the glass transition, the transport, and the geometry of the random close packing.