Invariant Distribution of Voronoi volumes

In 1989 Sam Edwards and coauthors have proposed that the complexity of static granular systems could be understood by a statistical mechanics approach reducing the description of the system state to a few control parameters only. An essential part of Edwards' idea is that in static granular media volume plays the role held by energy in usual thermodynamics. Therefore, an understanding of the volume distribution function is the key to connect microscopic details of the system with macroscopic state variables.

Tomography setup: X-ray source with shutter(left), fluidized bed on rotational stage (center) and X-ray camera (right).

We analyze the structural properties of static granular packings produced in 18 different experiments, 6 with acrylic spheres in air and 12 with glass beads prepared in water-fluidized beds. The packing fractions range from 0.56 to 0.64. Three-dimensional density maps have been obtained for these systems using X-ray Computed Tomography. Coordinates of the bead centers have been calculated for more than two million spheres with a precision better than 1% of their diameters, which is better than the uncertainty arising from polydispersity (5% for the glass particles and 2% for the acrylic particles).

Results

(Top) Distributions of the Voronoi cell volumes plotted versus V/d³ (d is the particle diameter) for 18 different experiments.

(Bottom) All the distributions collapse onto a universal curve when plotted vs. (V-V_min)/(⟨V⟩- V_min). ⟨V⟩ is the average Voronoi volume. V_min = 0.694 d³ is the smallest Voronoi cell that can be built in a equal-spheres packing. The theoretical line is a Gamma distribution with a shape parameter of 12.

Conclusion

We have shown that the local volume distributions of granular packings of monodisperse spherical grains are described by a Gamma distribution. This universal distribution function was derived using a statistical mechanics approach and the assumption that the volumes are composed of a set of elementary cells. Granular samples have been prepared by water fluidization pulses, tapping, and pouring. The agreement between theory and experiment indicates that the local volume distribution is not sensitive to different ways of sampling the granular phase space. This suggests some generic ergodicity, which is an encouraging road sign on the way to a statistical mechanics of static granular media.

Publication

Tomaso Aste, Tiziana Di Matteo, Mohammad Saadatfar, Tim J. Senden, Matthias Schröter, and Harry L. Swinney
An invariant distribution in static granular media
Europhysics Letters, 79, 24003 (2007).
[Journal URL], [pdf], [arxiv]