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

TL;DR

The paper investigates mixing properties of bi-disperse ellipsoid packings in granular matter using X-ray tomography and Voronoi-based analyses of local packing fractions $\Phi_l$. It demonstrates that bi-disperse packings behave as uncorrelated mixtures of two mono-disperse packings, with no long-range correlations beyond the first neighbor shell, and that the width of the local packing fraction distribution follows a linear master curve $\sigma(\langle \Phi_l \rangle)$ while a derived mixture-variance model accurately describes the data. The key contribution is a simple mean-field-like mixing framework for aspherical binary packings, quantified by weights $w_b=0.4$, $w_s=0.6$ and a nearly constant $\Delta \Phi \approx 0.035$, which aligns with random-network/jamming theories. This framework suggests broader applicability to other bi-disperse non-spherical systems and provides a benchmark for testing the generality of mean-field descriptions in granular matter. The work is parameter-specific to a fixed size and shape ratio, inviting future studies to map the parameter space and extend to other shapes and friction regimes.

Abstract

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.

Figures (5)

• Figure 1: The top row shows photographs of the two types of pharmaceutical placebo pills with two different aspect ratios used in this study and of a bi-disperse packing of ellipsoids created from a mixture of such particles. The bottom image shows a computer-generated illustration of a small section of a larger bi-disperse ellipsoid packing, together with the Voronoi cell of two specific particles. The illustration is generated from position data of ellipsoid particles extracted from a tomography image. Voronoi cells are obtained by the Set-Voronoi algorithm Schaller2013PhilMagpomelo which, as is visible in the picture, can lead to curved faces and edges.
• Figure 2: Local packing fraction distribution of a bi-disperse packing ($\langle\Phi_l\rangle = 0.693$) and the mono-disperse components when only small/big particles are considered ($\langle\Phi_l^s\rangle = 0.679$, $\langle\Phi_l^b\rangle = 0.714$). Inset: Difference of the mean local packing fraction of the big and small particles in all our bi-disperse packings is approximately constant ($\Delta \Phi = \langle\Phi_l^b\rangle - \langle\Phi_l^s\rangle \approx 0.035$) .
• Figure 3: Width $\sigma$ of the local packing fraction distributions of monodisperse and bidisperse packings. The open triangle and circle points are the widths of the distributions of bidisperse packings if only small or big particles are considered. The width of these distributions matches the ones of monodisperse packings. In addition, packings from Ref Schaller2015EPL are included (solid squares). The illustration of particles merely facilitates color identification in the plots and is not to scale.
• Figure 4: Local packing fraction correlation. The two arrows mark the minimal and maximal touching distance of two ellipsoids in the packing.
• Figure 5: Comparison of data from our experiments for normalised free volumes $v_f$ and $A$ with the data presented by Yuan et alyuan2020.