Dense Random Packing of Disks With a Power-Law Size Distribution in Thermodynamic Limit: Fractal-like Properties

TL;DR

This paper investigates the fractal-like properties of densely packed disks with a power-law size distribution in the thermodynamic limit, analyzing their structure factor and finite-size effects to understand scattering behavior.

Contribution

The study extends previous work by analyzing the structure factor in reciprocal space for large systems, revealing fractal decay and finite-size effects in dense disk packings with power-law sizes.

Findings

Fractal power-law decay of the structure factor is observed in reciprocal space.

The fractal range shrinks with decreasing packing fraction.

Finite-size effects show a parabolic structure factor at very low momenta.

Abstract

The correlation properties of a random system of densely packed disks, obeying a power-law size distribution, are analyzed in reciprocal space in the thermodynamic limit. This limit assumes that the total number of disks increases infinitely, while the mean density of the disk centers and the range of the size distribution are kept constant. We investigate the structure factor dependence on momentum transfer across various number of disks and extrapolate these findings to the thermodynamic limit. The fractal power-law decay of the structure factor is recovered in reciprocal space within the fractal range, which corresponds to the range of the size distribution in real space. The fractal exponent coincides with the exponent of the power-law size distribution as was shown previously by the authors [A. Yu. Cherny, E. M. Anitas, V. A. Osipov, J. Chem. Phys. 158, 044114 (2023)]. The dependence of the structure factor on density is examined. As is found, the power-law exponent remains unchanged but the fractal range shrinks when the packing fraction decreases. Additionally, the finite-size effects are studied at extremely low momenta of the order of the inverse system size. We show that the structure factor is parabolic in this region and calculate the prefactor analytically. The obtained results reveal fractal-like properties of the packing and can be used to analyze small-angle scattering from such systems.