the angle between two particles, $a=l/d$ is the aspect ratio, and $\langle \cdot\rangle$ corresponds to the number-averaged value. For random orientations of cylinders, $\langle \sin\theta \rangle=\pi/4$, and, in the limit of $a\rightarrow \infty$ (slender-rod limit), we have $V_{ex}=(\pi/2)d l^2$.

In the slender-rod limit, the excluded volume rule for capped cylinders was confirmed by cluster expansion approach [@Bug1985] and MC simulations [@Neda1999]. However, for a finite aspect ratio, important deviations from this rule were observed [@Neda1999; @Florian2001]. From the numerical data of simulations [@Berhan2007] the following general expression for $n_c$ was derived $$\label{eq:3dnc}
n_c = \left(1+ c_1 a^{-c_2}\right)V^{-1}_{ex},$$ where $c_1=3.526$ and $c_2=0.569$.

The effects of partial penetration on percolation behavior have also been analyzed [@Balberg1987; @DeBondt1992; @Berhan2007]. The degree of penetration was defined by the ratio of the diameter of the core to the outer diameter of the soft shell, $\gamma$ [@Berhan2007]. For the soft-core limit $\gamma=0$ and for the hard-core $\gamma=1$. An inverse proportionality rule (Eq. ) was also observed for this hard core/soft shell model. For the slender-rod limit ($a\rightarrow \infty$), the constant of proportionality was shown to be dominated by the aspect ratio $a$, with there being little dependence on the $\gamma$ value. The effects of size dispersity 