Simulation study of the electrical tunneling network conductivity of suspensions of hard spherocylinders

Using Monte Carlo simulations, we investigate the electrical conductivity of networks of hard rods with aspect ratios 10 and 20 as a function of the volume fraction for two tunneling conductance models. For a simple, orientationally independent tunneling model, we observe nonmonotonic behavior of the bulk conductivity as a function of volume fraction at the isotropic-nematic transition. However, this effect is lost if one allows for anisotropic tunneling. The relative conductivity enhancement increases exponentially with volume fraction in the nematic phase. Moreover, we observe that the orientational ordering of the rods in the nematic phase induces an anisotropy in the conductivity, i.e., enhanced values in the direction of the nematic director field. We also compute the mesh number of the Kirchhoff network, which turns out to be a simple alternative to the computationally expensive conductivity of large systems in order to get a qualitative estimate.

Figure 1

Snapshots of percolating clusters at different volume fractions.

Figure 2

Conductivity (in ${log}_{10}$) for the normal tunneling model Eq. (2) as a function of volume fraction for two different contact shell diameters, $\lambda =1.1D$ (blue symbols, mean value as a dashed line) and $\lambda =1.2D$ (green symbols, solid line). In addition, on the second $y$ axis the mean nematic order parameter ($S_2,$ red, dotted line) values of the entire system (not just the connected clusters) is plotted, and the vertical solid (brown) line indicates the coexistence volume fraction in the nematic phase for $L/D=20.$

Figure 3

(a) The average degree of a rod as a function of volume fraction for $L/D=20$ and $\lambda =1.2D.$ The average degree is simply calculated as twice the size of the edge set divided by the number of nodes in a given network. (b) The mean distance to a rod's nearest neighbor in the largest cluster as a function of volume fraction. In both plots, the error bars are obtained by averaging over all independent realisations of largest clusters per volume fraction.

Figure 4

Comparison of the dependence of $\sigma$ (in ${log}_{10}$) on $\eta$ for the two tunneling conductance models as given in Eq. (2) (labelled Iso.) and Eq. (3) (labelled Aniso.), and for $L/D=20.$ (a) $\lambda =1.1D$, (b) $\lambda =1.2D.$ The nematic order parameter $S_2$ is shown on the second $y$ axis in each case.

Figure 5

Conductivity (in ${log}_{10}$) for the anisotropic tunnel model Eq. (3) as a function of volume fraction, for $L/D=10$ and $\lambda =1.1D,$ which is above the percolation threshold for the entire of range of chosen volume fractions here. The mean nematic order parameter ($S_2$) is plotted on the second $y$ axis (red dashed).

Figure 6

The relative conductivity enhancement in percentages for the anisotropic tunneling model as a function of volume fraction, which is calculated based on the data shown in Fig. 4. ${\sigma }_{\mathrm{co}}$ denotes $\sigma$ in the conductive state, i.e., in the presence of percolating clusters ($\lambda =1.2D$), and ${\sigma }_{\mathrm{ins}}$ denotes the mean conductivity value in the absence of percolating clusters ($\lambda =1.1D$). The inset plot shows $ln{\sigma }_{\mathrm{co}}/{\sigma }_{\mathrm{ins}}$ as a function of $\eta$ restricted to the nematic phase, with the linear fit shown in dashed.

Figure 7

Conductivity (in ${log}_{10}$) in the nematic phase for $L/D=20,\lambda =1.2D,$ using the model Eq. (3), as a function of volume fraction. $\sigma$ measured along the director $\vec{n}$ of the rods (orange, triangular), compared to $\sigma$ measured in the orthogonal direction to $\vec{n}$ (blue, round).

Figure 8

Mean mesh numbers (red, star and dotted) and conductivity in ${log}_{10}$ (second $y$ axis, solid and dashed curves) as a function of $\eta ,$ for $L/D=20$ and $\lambda =1.2D.$