Complete mathematical theory of the jamming transition: A perspective

TL;DR

This paper presents a comprehensive microscopic mathematical theory of the jamming transition in frictionless particles, deriving key mechanical properties and their critical behavior from first principles, and validating predictions with numerical simulations.

Contribution

The paper develops the first parameter-free microscopic theory of the jamming transition that quantitatively matches numerical results for shear modulus and viscosity.

Findings

Microscopic expressions for shear modulus and viscosity near jamming.

Quantitative agreement with numerical simulations for shear modulus.

First derivation of viscosity divergence from microscopic Hamiltonian.

Abstract

The jamming transition of frictionless athermal particles is a paradigm to understand the mechanics of amorphous materials at the atomic scale. Concepts related to the jamming transition and the mechanical response of jammed packings have cross-fertilized into other areas such as atomistic descriptions of the elasticity and plasticity of glasses. In this perspective article, the microscopic mathematical theory of the jamming transition is reviewed from first-principles. The starting point of the derivation is a microscopically-reversible particle-bath Hamiltonian from which the governing equation of motion for the grains under an external deformation is derived. From this equation of motion, microscopic expressions are obtained for both the shear modulus and the viscosity as a function of the distance from the jamming transition (respectively, above and below the transition). Regarding the vanishing of the shear modulus at the unjamming transition, this theory, as originally demonstrated in [Zaccone & Scossa-Romano, Phys. Rev. B 83, 184205 (2011)], is currently the only quantitative microscopic theory in parameter-free agreement with numerical simulations of [O'Hern et al. Phys. Rev. E 68, 011306 (2003)] for jammed packings. The divergence of the viscosity upon approaching the jamming transition from below is derived here, for the first time, from the same microscopic Hamiltonian. The quantitative microscopic prediction of the diverging viscosity is shown to be in fair agreement with numerical results of sheared 2D soft disks from [Olsson & Teitel, Phys. Rev. Lett. 99, 178001 (2007)].