Fractal dimensions of jammed packings with power-law particle size distributions in two and three dimensions

TL;DR

This study analyzes the fractal dimensions of jammed packings with power-law particle size distributions in two and three dimensions, revealing universal behaviors and structural properties related to size dispersity.

Contribution

It introduces a method to compute structure factors for jammed packings with broad size distributions and identifies the fractal dimensions governing their large-scale structure.

Findings

In 3D, fractal dimension d_f ≈ 2.0 for 2.5 ≤ β ≤ 3.8.

In 2D, fractal dimension ranges from 1.0 to 1.34 for 2.1 ≤ β ≤ 2.9.

Fractal behavior persists after removing rattler particles, indicating backbone control.

Abstract

Static structure factors are computed for large-scale, mechanically stable, jammed packings of frictionless spheres (three dimensions) and disks (two dimensions) with broad, power-law size dispersity characterized by the exponent $-\beta$. The static structure factor exhibits diverging power-law behavior for small wavenumbers, allowing us to identify a structural fractal dimension, $d_f$. In three dimensions, $d_f \approx 2.0$ for $2.5 \le \beta \le 3.8 $, such that each of the structure factors can be collapsed onto a universal curve. In two dimensions, we instead find $1.0 \lesssim d_f \lesssim 1.34 $ for $2.1 \le \beta \le 2.9 $. Furthermore, we show that the fractal behavior persists when rattler particles are removed, indicating that the long wavelength structural properties of the packings are controlled by the large particle backbone conferring mechanical rigidity to the system. A numerical scheme for computing structure factors for triclinic unit cells is presented and employed to analyze the jammed packings.