The Geometry behind the Glass Transition and Frictional Jamming in Systems of Two-Dimensional Hard Disks

TL;DR

This paper explores the geometric structures underlying the glass transition and jamming in 2D hard disk systems, linking topological tilings to dynamic arrest phenomena and suggesting broader implications for 3D systems.

Contribution

It introduces a geometric perspective connecting tiling structures to dynamical transitions in 2D and 3D hard particle systems, offering a new understanding of glass and jamming transitions.

Findings

Dynamic arrest occurs at similar area fractions in simulations and experiments.

Identification of a pentagonal tiling (floret) associated with the transition.

Topological mechanisms underlying caging and jamming phenomena.

Abstract

The relation between dynamics and structure in systems of Brownian bidisperse 2D hard disks with arrested dynamics is examined using numerical simulations. Surprisingly, the suspensions show dynamic arrest at an area fraction of {\phi} {\approx} 0.777 over a wide range of disk-size ratios. This is in close agreement with the experimental findings of [Nat. Mater. 18, 1118 (2019)] ({\phi} {\approx} 0.776) for a quasi-2D colloidal suspension of spheres with large-to-small size ratio of approximately 1.4. Intriguingly, this also matches a jamming transition ({\phi} {\approx} 0.773 to 0.777) found experimentally in a 2D bidisperse granular packing of disks for a similar aspect ratio [Powders and Grains (2025)]. Adopting a geometric viewpoint allows for the identification of the floret pentagonal tiling ({\phi} {\approx} 0.777343), which is comprised of congruent (elongated) pentagonal tiles. In analogy to foam and tissue models, it is the presence of this congruent reference state that induces a dynamical transition in the disordered fluid of the disks. That is, the change in dynamics observed both in a uniformly compressed granular medium and a colloidal one at finite temperature are implied to be caused by the same topological mechanism. Extending this reasoning to a honeycomb lattice could also explain the experimentally observed onset of caging dynamics ({\phi} {\approx} 0.6) in a similar bidisperse, quasi-2D system of colloidal spheres [Nature 587, 225 (2020)]. An outlook is provided on how this concept may be applied to 3D: removing one in four particles from a face-centered-cubic arrangement leads to a square bipyramidal honeycomb} with associated volume fraction {\eta} {\approx} 0.55536. The ideas put forward here thus also provide a possible origin for random loose packing of (frictional) hard spheres in 3D and can potentially shed light onto the nature of the glass transition.