Random geometric graph description of connectedness percolation in rod systems.

Area of Science:

• Statistical physics
• Materials science
• Network theory

Background:

• Continuum percolation describes how connected clusters form in systems of particles.
• Dispersions of rods present unique challenges due to their anisotropic shapes and orientations.
• Existing models often rely on approximations that may not fully capture complex interactions.

Purpose of the Study:

• To reformulate the problem of continuum percolation in rod dispersions.
• To develop a new theoretical framework using weighted random geometric graphs.
• To establish a method for calculating percolation thresholds that accounts for various physical parameters.

Main Methods:

• Representing rod centers as nodes in a random geometric graph.
• Defining edge probabilities based on rod volume fraction, size, shape, and orientation.
• Utilizing the negligible contribution of closed loops for large aspect ratios.
• Equating results to the second-virial approximation of the Ornstein-Zernike equation.

Main Results:

• The reformulated model provides a comprehensive framework for continuum percolation.
• Percolation thresholds derived are equivalent to those from the second-virial approximation.
• The model successfully incorporates rod volume fraction, size, shape, and orientational distributions.
• The formulation allows for the inclusion of inter-rod interactions and many-body effects.

Conclusions:

• Weighted random geometric graphs offer a powerful tool for studying continuum percolation in anisotropic systems.
• The developed method provides accurate percolation thresholds, especially for high aspect ratio rods.
• This approach enhances the understanding of phase transitions and connectivity in complex materials.