3d Monte Carlo simulation of site-bond continuum percolation of spheres

We present off-lattice Monte Carlo simulations of site-bond percolation of semi-penetrable spheres or, equivalently, of hard spheres with a finite bond range. We will show that the crucial parameter is the effective volume fraction ($\phi_{\rm e}$), i.e. the volume that is occupied or within the bond range of at least one particle. For the equivalent system of semi-penetrable spheres $1-\phi_{\rm e}$ is the porosity. The bond percolation threshold ($p_{\rm b}$) can be described in terms of $\phi_{\rm e}$ by a simple analytical expression: $\log(\phi_{\rm e})/\log(\phi_{\rm ec})+\log(p_{\rm b})/\log(p_{\rm bc})=1$, with $p_{\rm bc}=0.12$ independent of the bond range and $\phi_{\rm ec}$ a constant that decreases with increasing bond range