On the rigidity of amorphous solids

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 implications of these results for the glass transition, the transport, and the geometry of the random close packing.

The jamming scenario—an introduction and outlook

“…Once floppy modes are deformed to generate anomalous modes, they gain a longitudinal component of order ∆Z as follows from the variational argument, whereas the transverse component remains of order one [17,25]. Relative corrections in the anomalous mode energies is thus of order −e/∆Z 2 .…”

“…The answer is related to the fact that at the jamming threshold, packings of frictionless discs and spheres are isostatic, i.e., they are marginal solids that can just maintain their stability [5,[12][13][14][15][16][17]. The origin of this is the following.…”

“…1.7 are consistent with this, as for a Hertzian interaction, G/B ∼ p 1/3 and ∆Z ∼ |∆φ| 0.5 ∼ p 1/3 . The result that the shear modulus generally scales as G ∼ k ∆Z, can be understood [17]: independent of the force law, the ratio G/B should scale as G/B ∼ ∆Z. Empirical support for this behavior in grains is reviewed in [29].…”

“…The difference between the jamming transition and generic rigidity percolation is that in the former case configurations must be such that all the contact forces must be repulsive. This imposes a geometric constraint on the network that is not enforced in standard rigidity percolation models [17]. We refer to [31] for further details.…”

“…[39,40] show that the diffusivity is nearly flat for frequencies from ω * up to the Debye frequency, where it drops to zero. Thus, there is a crossover from weakly-scattered plane waves to strong-scattering, diffusive behavior at ω * ; this can be understood from the heterogeneous nature of the anomalous modes [17,40], or by assuming the continuity of the length scale characterizing plane waves and anomalous modes at the crossover frequency, in conjunction with the scaling laws for the shear and bulk moduli discussed in section 1.2.5 [39,40].…”