Percolation: Connecting the Dots

This work develops a general framework for the exact determination or systematic approximation of percolation thresholds for systems in thermal equilibrium. We critically review the existing theoretical approaches to continuum percolation, in particular, connectedness percolation theory. Analyzing percolation as a graph problem, we propose a different integral equation, which provides the exact solution to a variety of one-dimensional problems. In higher dimensions, we can systematically close the integral equation, yielding a hierarchy of progressively more accurate lower bounds to the percolation threshold. This allows us to reliably predict the impact of particle shape and polydispersity on the percolation threshold.

We continue with a comprehensive overview of exactly solvable percolation problems. We demonstrate that all these problems can be associated with a tree-like network topology below criticality. Mapping the percolation problems onto a corresponding branching process, when joined with a new characterization of percolation, provides unified and simplified solutions to this entire class of problems.

Finally, we analyze percolation of rigid carbon black aggregates with a fractal structure. Combining simulation results and experiments on carbon black composites, we investigate the impact of the aggregate morphology on the percolation threshold and composite conductivity. Despite their complex appearance, we introduce a simple mapping of carbon black aggregates onto a system of fully penetrable spheres, which accurately predicts the percolation threshold of large aggregates.