Simulations and electrical conductivity of percolated networks of finite rods with various degrees of axial alignment

We present a three-dimensional simulation and calculation of electrical conductivity above the filler percolation threshold for networks containing finite, conductive cylinders as a function of axial orientation $\left(S\right)$ and aspect ratio $\left(L/D\right)$. At a fixed volume fraction and $L/D$, the simulations exhibit a critical degree of orientation, $S_c$, above which the electrical conductivity decreases dramatically. With increasing filler concentration and aspect ratio, this critical orientation shifts to higher degrees of alignment. Additionally, at a fixed volume fraction and $L/D$, the simulated electrical conductivity displays a maximum at slight uniaxial orientation, which is less pronounced at higher volume fractions and aspect ratios. Our approach can be used as a predictive tool to design the optimal filler concentration and degree of orientation required to maximize electrical conductivity in polymer nanocomposites with conductive cylindrical fillers of finite dimension.

Figure 1

(Color online) (a) Simulated morphologies of finite rods having different amounts of axial alignment. The axial orientation parameter, $S$, is defined as $S=\left(3⟨{\text{cos}}^2\theta ⟩-1\right)/2$ and varies from 0 to 1 corresponding to isotropic and perfectly aligned rods, respectively. The simulation volume is 1 unit long with a cross-sectional area of $0.1{\text{unit}}^2$. These simulated rods have an $L/D=10$ and volume fractions of 0.075, 0.082, and 0.118 for $S$ of 0, 0.11, and 0.58, respectively. The electrical conductivity of these simulated morphologies was calculated using the random resistor network model. (b) The critical volume fraction, ${\varphi }_c$, was determined at fixed $S$ using a power law to relate electrical conductivity and volume fraction for three values of $L/D$. For comparison, ${\varphi }_c$ from the excluded volume model [Eq. (4)] is shown as a line and notable discrepancies are observed at lower $L/D$.

Figure 2

Simulated electrical conductivity ($\mathrm{Siemens}$/unit) vs $S$ for (a) $L/D=10$, (b) 20, and (c) 80. The volume fraction, $\varphi$, is given for each curve.

Figure 3

(Color online) (a) The critical axial orientation parameter, $S_c$, was determined at fixed concentration and $L/D$ using a power law to relate electrical conductivity and the axial orientation parameter, $S$ [Eq. (8)]. (b) $S_c$ as a function of volume fraction of rods for three values of $L/D$. At $S=0$, the points correspond to ${\varphi }_{c,\text{iso}}$ in Table .

Figure 4

(Color online) Experimental results by Du et al. (Ref. 5) from uniaxially aligned nanocomposite of SWCNT in poly(methyl methacrylate). The electrical conductivity has been replotted at fixed loading as a function of $S$. The nanotube bundles in these composites have $L/D\sim 45$ and $S$ was measured using an x-ray scattering method.