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

TL;DR

This work tackles whether grain-size segregation can emerge in jammed granular matter without external biases. It develops an exact configurational-statistical framework for two-size disks in a narrow channel, modeling jammed microstates as assemblies of 17 statistically interacting quasiparticles built from 16 tiles, with two reference vacua guiding the analysis. A single energy-parameter inequality $\Delta\mathcal V$ partitions the jamming protocols into two ordering regimes—size segregation and size alternation—with a border case $\Delta\mathcal V=0$ yielding persistent size randomness; exact expressions for excess volume and configurational entropy are derived as functions of the small-disk fraction and agitation intensity. To validate robustness, the authors compare a highly symmetric model with a higher-stability variant, showing consistent ordering tendencies and confirming that steric effects alone can drive size-based ordering under jammed conditions. Overall, the paper provides exact, protocol-dependent insights into how steric constraints can cause or inhibit segregation in polydisperse, quasi-one-dimensional granular systems.

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.

Figures (13)

• Figure 1: Population densities (\ref{['eq:27']}) of particles from species $m=1,3,\ldots,17$ versus the fraction $\bar{N}_\mathrm{S}$ of small disks in the limit $\beta=0$. Note the different vertical scales in (a) and (b).
• Figure 2: (a) Entropy $\bar{S}$ and (b) excess volume $\bar{V}$ versus the fraction $\bar{N}_\mathrm{S}$ of small disks for various values of $\beta$.
• Figure 3: Population densities (a) $\bar{N}_1$ at $\bar{N}_\mathrm{S}=0$ and (b) $\bar{N}_{15}$, $\bar{N}_{17}$ at $\bar{N}_\mathrm{S}=1$ plotted versus $\beta^{-1}$. All other $\bar{N}_m$ vanish identically if only large or small disks are present.
• Figure 4: Population densities $\bar{N}_m$ for particles from species (a) $m=1,15,17$, (b) $m=3,5,7,8$ and (c) $m=9,11,13,14$ at $\bar{N}_\mathrm{S}=0.5$ plotted versus $\beta^{-1}$. Note the different vertical scales. In panel (d) the solid curves show $\bar{S}$ versus $\bar{V}$ parametrically for varying $\beta$ at different values of $\bar{N}_\mathrm{S}$. The dashed curves represent $\bar{S}$ versus $\bar{V}$ for vaying $\bar{N}_\mathrm{S}$ at $\beta=0,5,10,20,30$ (top to bottom).
• Figure 5: Population densities $\bar{N}_m$ for particles from species all species for $\bar{N}_\mathrm{S}=0.4$ (left) and $\bar{N}_\mathrm{S}=0.6$ (right). Note the different vertical scales.