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

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 wave numbers, allowing us to identify a structural fractal dimension df. In three dimensions, d(f) approximate to 2.0 for 2.5 <= beta <= 3.8, such that each of the structure factors can be collapsed onto a universal curve. In two dimensions, we instead find 1.0 less than or similar to d(f) <= 1.34 for 2.1 <= beta <= 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.

Automated summary

This study calculates the static structure factors of large-scale, mechanically stable, jammed packings of frictionless spheres and disks with broad size dispersity. The analysis of the power-law behavior of the structure factors reveals that in three dimensions, the structural fractal dimension (df) is approximately 2.0 for -beta values between 2.5 and 3.8, allowing the collapse of structure factors onto a universal curve. In two dimensions, the fractal dimension (df) ranges from 1.0 to 1.34 for -beta values between 2.1 and 2.9. Additionally, the fractal behavior persists even after removing rattler particles, indicating that the long-wavelength structural properties of the packings are controlled by the large particle backbone conferring mechanical rigidity to the system.