Mixing properties of bi-disperse ellipsoid assemblies: Mean-field behaviour in a granular matter experiment

The structure and spatial statistical properties of amorphous ellipsoid assemblies have profound scientific and industrial significance in many systems, from cell assays to granular materials. This paper uses a fundamental theoretical relationship for mixture distributions to explain the observations of an extensive X-ray computed tomography study of granular ellipsoidal packings. We study a size-bi-disperse mixture of two types of ellipsoids of revolutions that have the same aspect ratio of alpha approximately equal to 0.57 and differ in size, by about 10% in linear dimension, and compare these to mono-disperse systems of ellipsoids with the same aspect ratio. Jammed configurations with a range of packing densities are achieved by employing different tapping protocols. We numerically interrogate the final packing configurations by analyses of the local packing fraction distributions calculated from the Voronoi diagrams. Our main finding is that the bi-disperse ellipsoidal packings studied here can be interpreted as a mixture of two uncorrelated mono-disperse packings, insensitive to the compaction protocol. Our results are consolidated by showing that the local packing fraction shows no correlation beyond their first shell of neighbours in the binary mixtures. We propose a model of uncorrelated binary mixture distribution that describes the observed experimental data with high accuracy. This analysis framework will enable future studies to test whether the observed mean-field behaviour is specific to the particular granular system or the specific parameter values studied here or if it is observed more broadly in other bi-disperse non-spherical particle systems.

💡 Research Summary

This paper investigates the structural mixing properties of a binary mixture of ellipsoidal particles with identical aspect ratio (α≈0.57) but a modest size disparity of roughly 10 % in linear dimension. The authors use pharmaceutical placebo pills as model particles: small ellipsoids with semi‑axis a = 4.45 mm and large ellipsoids with a = 5.10 mm. Apart from size, the two species share the same material density and low‑friction surface coating, ensuring that shape and surface properties do not confound the size effect. The mixture ratio is fixed at 2 : 3 (large : small), which yields equal total volume for each species in the container.

Samples are prepared in a cylindrical vessel, initially poured loosely, then compacted by vertical tapping with a variety of protocols (different tap strengths and numbers). Each tap produces a jammed configuration; after a series of taps the system reaches a range of global packing fractions Φ_g between about 0.60 and 0.76. The final configurations are imaged using high‑resolution cone‑beam X‑ray micro‑CT (voxel size 56 µm). Particle positions and orientations are extracted via a watershed‑based algorithm, and a Set‑Voronoi construction is applied to obtain the Voronoi cell for each particle. The local packing fraction for particle i is defined as Φ_l,i = V_particle,i / V_Voronoi,i.

The central observable is the distribution of Φ_l across all particles. For each packing the authors compute the mean ⟨Φ_l⟩ and the standard deviation σ. They also separate the data into the two species, obtaining ⟨Φ_l⟩_small, σ_small and ⟨Φ_l⟩_large, σ_large. In monodisperse packings (either all small or all large particles) the σ versus ⟨Φ_l⟩ relationship collapses onto a single linear trend, σ(⟨Φ_l⟩)=0.122−0.147⟨Φ_l⟩, which agrees with earlier results for ellipsoids of other aspect ratios.

When the binary mixture is considered as a whole, its σ is larger than that of a monodisperse system with the same ⟨Φ_l⟩, as expected for a broader distribution. Remarkably, however, the σ values of the two subsystems (small‑only and large‑only) lie exactly on the same linear curve as the monodisperse data. This observation suggests that the binary packing can be regarded, to first order, as a statistical mixture of two independent monodisperse packings.

To formalise this, the authors adopt the standard mixture‑distribution formula for the second moment: σ²_mix = Σ_i w_i