Tree-ansatz percolation of hard spheres

Area of Science:

• Physics
• Materials Science
• Statistical Mechanics

Background:

• Particle suspensions are crucial in various scientific fields.
• Understanding percolation phenomena in disordered systems is complex.
• Hard sphere models provide a fundamental basis for studying such systems.

Purpose of the Study:

• To develop an analytical model for continuum percolation in hard sphere suspensions.
• To investigate the relationship between particle connectivity and percolation thresholds.
• To provide a method for accurately predicting percolation behavior.

Main Methods:

• Modeling particle suspensions as random geometric graphs.
• Analyzing the network structure of connected spheres.
• Exploiting the diminishing prevalence of closed loops for small connectivity ranges.
• Deriving an analytic expression for the percolation threshold.

Main Results:

• A tree-like network structure emerges for small connectivity ranges (δ/D).
• An accurate analytic expression for the percolation threshold was derived.
• The derived formula's accuracy increases as the connectivity range diminishes.
• A rescaling method extends the formula's validity to wider connectivity ranges.

Conclusions:

• The tree-like approximation is effective for understanding percolation in dilute hard sphere systems.
• The derived analytic expression offers a predictive tool for percolation thresholds.
• The rescaling approach enhances the applicability of the model across different connectivity regimes.