Fractal Structure of Equipotential Curves on a Continuum Percolation Model
This work provides insights into potential distribution in disordered conductive media, relevant for understanding transport in composite materials.
The study numerically investigates electric potential distribution in a 2D continuum percolation model, revealing quasi-equipotential clusters with fractal structure. The fractal dimension of equipotential curves ranges from 1.00 to 1.257, peaking at the percolation threshold.
We numerically investigate the electric potential distribution over a two-dimensional continuum percolation model between the electrodes. The model consists of overlapped conductive particles on the background with an infinitesimal conductivity. Using the finite difference method, we solve the generalized Laplace equation and show that in the potential distribution, there appear the {\it{quasi-equipotential clusters}} which approximately and locally have the same values like steps and stairs. Since the quasi-equipotential clusters has the fractal structure, we compute the fractal dimension of equipotential curves and its dependence on the volume fraction over $[0,1]$. The fractal dimension in [1.00, 1.257] has a peak at the percolation threshold $p_c$.