Anisotropy in electrical conductivity of two-dimensional films containing aligned nonintersecting rodlike particles: Continuous and lattice models

The electrical conductivity of two-dimensional films filled with rodlike particles (rods) was simulated by the Monte Carlo method. The main attention has been paid to the investigation of the effect of the rod alignment on the electrical properties of the films. Both continuous and lattice approaches were used. Intersections of particles were forbidden. Our main findings are (i) both models demonstrate similar behaviors, (ii) at low concentration of rods, both approaches lead to the same dependencies of the electrical conductivity on the concentration of the rods, (iii) the alignment of the rods essentially affects the electrical conductivity, (iv) at some concentrations of partially aligned rods, the films may be conducting only in one direction, and (v) the films may simultaneously be both highly transparent and electrically anisotropic.

Figure 1

Transformation of continuous system of rods into a discrete system of polyominoes.

Figure 2

Example of the packing density $p$ vs the number density $n$ for $s=0,1$ and $m=256,2048$. The continuous model.

Figure 3

Fragment of a square lattice with three deposited 3-mers of different orientations. All possible combinations of the conductivities are indicated.

Figure 4

Continuous model: Longitudinal and transversal effective electrical conductivity $\sigma$ vs order parameter $s$ for (a) $m=128,n=1.4,1.5$ and (b) $m=2048,n=13.56,20$. The dashed line corresponds to the value ${\sigma }_g$. The solid lines are provided simply as visual guides.

Figure 5

Continuous model: Anisotropy of the effective electrical conductivity $\delta$ [Eq. (4)] vs packing density $p$ for different values of the order parameter $s$. (a) $m=256$ ($k^{*}=8$), (b) $m=2048$ ($k^{*}=64$). The results are averaged over ten independent statistical runs, except at $s=1$ (five runs). The lines are provided simply as visual guides.

Figure 6

Continuous model: Examples of the phase diagram in the $(p,s)$ plane. (a) $m=512$ ($k^{*}=16$), (b) $m=2048$ ($k^{*}=64$).

Figure 7

Lattice model: Examples of the effective longitudinal and transversal electrical conductivities vs $s$ for two particular values of the packing density, $p=p^{*}$ (boxes) and $p=p_c$ (circles). (a) $k=4,L={10}^{3}k$. (b) $k=64,L=100k$. The dashed line corresponds to the value ${\sigma }_g$. The solid lines are provided simply as visual guides.

Figure 8

Lattice model: Anisotropy of the effective electrical conductivity $\delta$ [Eq. (4)] vs the packing density $p$ for different values of the order parameter $s$. (a) $k=8$, (b) $k=64$. The results are averaged over ten independent statistical runs. The lines are provided simply as visual guides.

Figure 9

Lattice model: Examples of the phase diagrams in the $(p,s)$ plane. (a) $k=16$, (b) $k=64$. $L=100k$.

Figure 10

Example of the longitudinal and transversal effective electrical conductivities vs packing density $p$, for values of the order parameter $s=0,1$. For the continuous model, $m=2048$ ($k^{*}=64$), the results are averaged over ten independent statistical runs, except at $s=1$ (5 runs). For the lattice model, $k=64,L=100k$, the results are averaged over ten independent statistical runs. The dashed line corresponds to the value ${\sigma }_g$.