Dense random packing with a power-law size distribution: the structure factor, mass-radius relation, and pair distribution function

TL;DR

This paper investigates the microstructural properties of dense random packings of disks with a power-law size distribution, revealing fractal characteristics and deriving a structure factor approximation useful for small-angle scattering analysis.

Contribution

It introduces a theoretical and numerical analysis of fractal properties in dense disk packings with power-law size distributions, including an exact 1D toy model and a new structure factor approximation.

Findings

Packings exhibit fractal properties with the power-law exponent matching the fractal dimension.

Derived an approximate structure factor formula applicable in arbitrary dimensions.

Numerical and theoretical evidence supports the fractal nature of such packings.

Abstract

We consider dense random packing of disks with a power-law distribution of radii and investigate their correlation properties. We study the corresponding structure factor, mass-radius relation and pair distribution function of the disk centers. A toy model of dense segments in one dimension (1d) is solved exactly. It is shown theoretically in 1d and numerically in 1d and 2d that such packing exhibits fractal properties. It is found that the exponent of the power-law distribution and the fractal dimension coincide. An approximate relation for the structure factor in arbitrary dimension is derived, which can be used as a fitting formula in small-angle scattering. The findings can be useful for understanding microstructural properties of various systems like ultra-high performance concrete, high-internal-phase ratio emulsions or biological systems.