Random packing of spheres in Menger sponge

Area of Science:

• Physics
• Materials Science
• Statistical Mechanics

Background:

• Understanding granular materials and their packing is crucial in various scientific fields.
• Fractal geometry offers a framework to describe complex structures found in nature and engineered systems.
• Previous studies have explored packing in Euclidean spaces, but fractal collectors present unique challenges.

Purpose of the Study:

• To numerically investigate random packing of spheres within fractal collectors (2 < d < 3).
• To measure the random packing saturation limit and analyze scaling properties of density autocorrelations.
• To test phenomenological relations between saturation density and collector dimension.

Main Methods:

• Numerical simulations using the Random Sequential Adsorption (RSA) algorithm.
• Analysis of density-autocorrelation functions to determine scaling properties.
• Measurement of RSA kinetics coefficients.

Main Results:

• The random packing saturation limit was successfully measured for fractal collectors.
• Scaling properties of density autocorrelations were analyzed, revealing insights into packing structure.
• Phenomenological relations between saturation density and collector dimension were tested and validated.

Conclusions:

• The study confirms that fractal systems with non-integer dimensions (d < 3) generally adhere to established dimensional relations for random packing.
• The RSA algorithm is effective for simulating sphere packing in complex fractal geometries.
• Findings contribute to a deeper understanding of granular packing in non-Euclidean spaces.