A unified quantifier of mechanical disorder in solids

TL;DR

This paper introduces a unified, measurable quantifier of mechanical disorder in solids by deriving a scaling relation between two existing dimensionless measures, supported by simulations and relevant to vibrational spectra and material properties.

Contribution

The authors derive a fundamental scaling relation linking two different measures of mechanical disorder, unifying them into a single quantifier supported by theoretical and simulation data.

Findings

The two disorder quantifiers are intrinsically related through a basic scaling law.

The unified quantifier applies across different physical systems and relates to vibrational properties.

Results have implications for understanding the unjamming transition and vibrational spectra in disordered solids.

Abstract

Mechanical disorder in solids, which is generated by a broad range of physical processes and controls various material properties, appears in a wide variety of forms. Defining unified and measurable dimensionless quantifiers, allowing quantitative comparison of mechanical disorder across widely different physical systems, is therefore an important goal. Two such coarse-grained dimensionless quantifiers (among others) appear in the literature, one is related to the spectral broadening of discrete phononic bands in finite-size systems (accessible through computer simulations) and the other is related the spatial fluctuations of the shear modulus in macroscopically large systems. The latter has been recently shown to determine the amplitude of wave attenuation rates in the low-frequency limit (accessible through laboratory experiments). Here, using two alternative and complementary theoretical approaches linked to the vibrational spectra of solids, we derive a basic scaling relation between the two dimensionless quantifiers. This scaling relation, which is supported by simulational data, shows that the two apparently distinct quantifiers are in fact intrinsically related, giving rise to a unified quantifier of mechanical disorder in solids. We further discuss the obtained results in the context of the unjamming transition taking place in soft sphere packings at low confining pressures, in addition to their implications for our understanding of the low-frequency vibrational spectra of disordered solids in general, and in particular those of glassy systems.