Effect of nanotube alignment on percolation conductivity in carbon nanotube/polymer composites

Percolation conductivity of a stick network depends on alignment as well as concentration. We show that both dependences exhibit critical (power-law) behavior, and study the alignment threshold in detail. The highest conductivity occurs for slightly aligned, rather than isotropic, sticks. Experiments on single wall carbon nanotube composites are supported by Monte Carlo simulations. These results should be broadly applicable to percolating networks of anisotropic conductors.

Figure 1

(a) Electrical conductivity of isotropic SWNT/PMMA composites as a function of loading. Inset: a log-log plot of conductivity vs reduced mass fraction determines the critical composition, $m_c$. (b) Electrical conductivity of the $2\mathrm{wt}\%$ SWNT/PMMA composite along the alignment direction as a function of alignment. Inset: a log-log plot of conductivity vs reduced FWHM determines the critical alignment, ${\mathrm{FWHW}}_c$. (c) Electrical conductivity along the alignment direction as a function of SWNT alignment for composites with various loadings. ▵: $3\mathrm{wt}\%$; ▴: $2\mathrm{wt}\%$; ◻: $1\mathrm{wt}\%$; ∎: $0.5\mathrm{wt}\%$; ◆: $0.38\mathrm{wt}\%$.

Figure 2

2D stick-in-unit-square simulations with stick alignment determined by a cutoff angle ${\theta }_{\mu }$. (a) 50 sticks with ${\theta }_{\mu }=90°$ (randomly distributed) do not exhibit a percolation path; (b) 100 sticks with ${\theta }_{\mu }=5°$ (highly aligned) do not exhibit a percolation path; (c) 100 sticks with ${\theta }_{\mu }=90°$ (randomly distributed) exhibit a percolation path (the gray area). All sticks have $L=0.255$ relative to the unit square.

Figure 3

Percolation probability, $P$, along the stick alignment direction from 1000 simulations as a function of the stick alignment $\left({\theta }_{\mu }\right)$ for various numbers of sticks (420–900) as indicated. The critical angles, ${\theta }_c$ are ▵: 11°; ▴: 21°; ◻: 31°; ∎: 33°; ◆: 37°. In addition, $P$ was found perpendicular to the alignment direction for 600 sticks (+) and exhibits ${\theta }_c=31°$. All sticks have $L=0.108$ relative to the unit square.