Modeling percolation in high-aspect-ratio fiber systems. I. Soft-core versus hard-core models

Numerical and analytical studies of the onset of percolation in high-aspect-ratio fiber fiber systems such as nanotube reinforced polymers available in the literature have consistently modeled fibers as penetrable, straight, capped cylinders, also referred to as spherocylinders. In reality, however, fibers of very high-aspect ratio embedded in a polymer do not come into direct physical contact with each other, let alone exhibit any degree of penetrability. Further, embedded fibers of very high-aspect ratio are often actually wavy, rather than straight. In this two-part paper we address these critical differences between known physical systems, and the presently used spherocylinder percolation model. In Paper I we evaluate the effect of allowing penetration of the model fibers on simulation results by comparing the soft-core and the hard-core approaches to modeling percolation onset. We use Monte Carlo simulations to investigate the relationship between percolation threshold and excluded volume for both modeling approaches. Our results show that the generally accepted inverse proportionality between percolation threshold and excluded volume holds for both models. We further demonstrate that the error introduced by allowing the fibers to intersect is non-negligible, and is a function of both aspect ratio and tunneling distance. Thus while the results of both the soft-core model and hard-core assumptions can be matched to select experimental results, the hard-core model is more appropriate for modeling percolation in nanotubes-reinforced composites. The hard-core model can also potentially be used as a tool in calculating the tunneling distance in composite materials, given the fiber morphology and experimentally derived electrical percolation threshold. In Paper II we investigate the effect of the waviness of the fibers on the onset of percolation in fiber reinforced composites.

Figure 1

(Color online) (a) Soft-core model and (b) hard-core model with soft shell of radius $R$ for a spherocylinder of length $L$ and radius $r$.

Figure 2

$\ln \left(s\right)$ vs $\ln (R∕L)$ for straight fibers $(L=0.15)$.

Figure 3

$(1+s)$ vs $L∕R$.

Figure 4

(Color online) Schematic of (a) a bundle of seven nanotubes and (b) the equivalent spherocylinder used in calculation shown encasing the bundle

Figure 5

Hard-core analytical solution for percolation threshold vs $t$ for nanotube bundles.