Percolation in metal-insulator composites of randomly packed spherocylindrical nanoparticles

While classical percolation is well understood, percolation effects in randomly packed or jammed structures are much less explored. Here we investigate both experimentally and theoretically the electrical percolation in a binary composite system of disordered spherocylinders, to identify the relation between structural (percolation) and functional properties of nanocomposites. Experimentally, we determine the percolation threshold $p_c$ and the conductivity critical exponent $t$ for composites of conducting ($\mathrm{Cr}{\mathrm{O}}_2$) and insulating (${\mathrm{Cr}}_2{\mathrm{O}}_3$) rodlike nanoparticles that are nominally geometrically identical, yielding $p_c=0.305\pm 0.026$ and $t=2.52\pm 0.03$ respectively. Simulations and modeling are implemented through a combination of the mechanical contraction method and a variant of random walk (de Gennes ant) approach, in which charge diffusion is correlated with the system conductivity via the Nernst-Einstein relation. The percolation threshold and critical exponents identified through finite-size scaling are in good agreement with the experimental values. Curiously, the calculated percolation threshold for spherocylinders with an aspect ratio of 6.5, $p_c=0.312\pm 0.002$, is very close (within numerical errors) to the one found previously in two other distinct systems of disordered jammed spheres and simple cubic lattice, an intriguing and surprising result.

Figure 1

Sample systems of disordered packing, including closely packed homegrown lemons (top-left panel), a simulation snapshot of randomly packed conducting (green, with $p=0.33$) and insulating (red) spherocylinders near the percolation threshold (top-right panel), and an SEM image of the densely packed $\mathrm{Cr}{\mathrm{O}}_2/{\mathrm{Cr}}_2{\mathrm{O}}_3$ experimental sample at approximately the same fraction $p$ (bottom panel). The average width and length of the $\mathrm{Cr}{\mathrm{O}}_2/{\mathrm{Cr}}_2{\mathrm{O}}_3$ nanoparticles are measured to be approximately 40 nm and 300 nm, respectively.

Figure 2

The XRD intensity vs $2\theta$ plots, with the peaks corresponding to the original $\mathrm{Cr}{\mathrm{O}}_2$ nanoparticles before annealing and ${\mathrm{Cr}}_2{\mathrm{O}}_3$ nanoparticles after annealing.

Figure 3

Sample resistivity as a function of fraction $p$ of $\mathrm{Cr}{\mathrm{O}}_2$ at five different temperatures (a) $T=300\mathrm{K}$, (b) $T=270\mathrm{K}$, (c) $T=250\mathrm{K}$, (d) $T=220\mathrm{K}$, and (e) $T=200\mathrm{K}$. Insets: Sample conductivity as a function of $p-p_c$ in log-log scale to identify the critical exponent $t$ (where the error of $p_c$ determined from a separate nonlinear fitting procedure has been incorporated).

Figure 4

Spanning percolation probability as a function of fraction $p$ of the conducting spherocylinders, for (a) two sample systems with $N=1728$ (open circles) and $N=10648$ (filled squares) particles, and (b) an enlarged portion for all six systems of different $N$. Each set of simulation data (shown as symbols) is fitted to Eq. (5), with results shown as dashed curves. The vertical dashed line in both (a) and (b) indicates the location of $p_c=0.312$.

Figure 5

Width $\mathrm{\Delta }$ of the percolation transition as a function of system size $L$ rescaled by the particle diameter $d$. The error bars are from the fitting of Eq. (5) to obtain $\mathrm{\Delta }$ at each $L$. The dashed line is the power-law fit to Eq. (3), yielding a value of the critical exponent $\nu =0.646\pm 0.005$.

Figure 6

Effective percolation threshold $p_c^{\mathrm{eff}}$ as a function of ${(L/d)}^{-1/\nu }$. The dashed line is the fit to Eq. (4), and its intersection with the $y$ axis gives the percolation threshold $p_c=0.312\pm 0.002$ for the infinite system with $L\to \infty$.

Figure 7

Rescaled conductivity, $\sigma \left(p\right)/\sigma (p=1)$, as a function of $p-p_c$ for systems of $N=10648$ particles. The fitting to the scaling relation $\sigma \propto {(p-p_c)}^{\mu }$ (shown as the dashed line) gives the conductivity critical exponent $\mu =1.62\pm 0.04$.