Jammed disks of two sizes in a channel: segregation driven by steric forces

TL;DR

This paper provides an exact statistical mechanics analysis of jammed two-size disk systems in a channel, revealing how steric forces alone can drive segregation without directional bias.

Contribution

It introduces a novel exact characterization of jammed microstates, including volume and entropy, using quasiparticle species and analyzes segregation driven solely by steric forces.

Findings

Exact formulas for jammed macrostate volume and entropy.

Identification of regimes that promote or suppress size mixing.

Demonstration of segregation driven by steric forces without bias.

Abstract

Disks of two sizes are confined to a long and narrow channel. The axis and the plane of the channel are horizontal. The channel is closed off by pistons that freeze jammed microstates out of loose disk configurations, agitated randomly at calibrated intensity and subject to moderate pressure. Disk sizes and channel width are such that under jamming no disks remain loose and all disks touch one wall. The protocol permits disks to move past each other prior to jamming, which facilitates randomness in the sequence of large and small disks. We present exact results for the characterization of jammed macrostates including volume and entropy for given fractions of small and large disks as functions of energy parameters which depend on the jamming protocol. Our analysis divides the disk sequence of jammed microstates into overlapping tiles out of which we construct 17 species of statistically interacting quasiparticles. Jammed macrostates then depend on the fractions of small and large disks and on a dimensionless control parameter inferred from measures for expansion work against the pistons and intensity of random agitations. Two models are introduced for comparison of key technical aspects: one model emphasizes symmetry and the other mechanical stability. We distinguish regimes for the energy parameters that either enhance or suppress mixing of disk sizes in jammed macrostates. The latter case, if realizable, is a manifestation of grain segregation driven by steric forces alone, without directional bias.