Anisotropy in electrical conductivity of films of aligned intersecting conducting rods

Numerical simulations by means of the Monte Carlo method have been performed to study the electrical properties of a two-dimensional composite filled with rodlike particles. The main goal was to study the effect of the alignment of such rods on the anisotropy of its electrical conductivity. A continuous model was used. In this model, the rods have zero-width (i.e., infinite aspect ratio) and they may intersect each other. To involve both the low conductive host matrix and highly conductive fillers (rods) in the consideration, a discretization algorithm based on the use of a supporting mesh was applied. The discretization is equivalent to the substitution of rods with the polyominoes. Once discretized, the Frank-Lobb algorithm was applied to evaluate the electrical conductivity. Our main findings are (i) the alignment of the rods essentially affects the electrical conductivity and its anisotropy, (ii) the discrete nature of computer simulations is crucial. For slightly disordered system, high electrical anisotropy was observed at small filler content, suggesting a method to enable the production of optically transparent and highly anisotropic conducting films.

Figure 1

Order parameter, $s$, versus the degree of anisotropy, $A$. Model I calculated using Eq. (12), Model II calculated numerically.

Figure 2

Transformation of the continuous problem of rods into a “zoo of lattice animals” (linear polyominoes).

Figure 3

Transformation of the supporting mesh into a resistor network. All possible combinations of the conductivities are indicated.

Figure 4

Comparison of the results obtained within the frameworks of Models I and II. Electrical conductivity, $\sigma$, versus the filling fraction, $p$, for $m=2048$ and $s=0.8$. Here, the filled and open symbols correspond to the longitudinal and transversal directions, respectively. The lines are provided simply as visual guides. Solid lines correspond to Model I, dashed lines correspond to Model II.

Figure 5

Example of scaling analysis for electrical conductivity, $\sigma$ vs filling fraction, $p$, for the different values of $L/l=16,32,64$ but with a fixed value $m/L=16$. (a) Longitudinal conductivity, (b) transversal conductivity. The lines are provided simply as visual guides.

Figure 6

Examples of the size distribution of “lattice animals” (polyominoes) vs. angle for different values of $m$ in the polar diagram. The results were averaged over ${10}^5$ independently located rods. $k^{*}=m/L$. Only each tenth point is shown to allow a clearer view. The curves correspond to Eq. (17). Two gray arcs correspond to the values of the polar radius 1 and $\sqrt{2}$.

Figure 7

Examples of the distribution of “lattice animals” (polyominoes), by size, in an isotropic system ($s=0$) for three different values of $m$ according to Eq. (A9). The solid lines are provided simply as visual guides. The vertical dashed line corresponds to $\sqrt{2}$.

Figure 8

Critical values of the filling ratio, $p_c$ (a), and the number density, $n_c$ (b), versus the order parameter, $s$, at different values of $m$. Here, the filled and open symbols correspond to the longitudinal and transversal directions, respectively. The lines are provided simply as visual guides.

Figure 9

Critical number density, $n_c$, versus the critical filling ratio, $p_c$ at different values of $m$ and the order parameter, $s$. Here, the filled and open symbols correspond to the longitudinal and transversal directions, respectively. The lines are provided simply as visual guides.

Figure 10

Examples of fittings of $1/n_c$ versus $1/m$ with the power equation $1/n_c\left(m\right)=1/n_c\left(\infty \right)+\alpha /m^{\beta }$ used for evaluation of the values of $n_c\left(\infty \right)$. Here, $\alpha$ and $\beta$ are the fitting parameters.

Figure 11

Relative critical number density, $n^{*}=n_c/n_c^i$, ($n_c^i\approx 6.044$) versus order parameter, $s$, for the values obtained in the limit of $m\to \infty$. Here, the filled and open symbols correspond to the longitudinal and transversal directions, respectively. The dashed line was obtained using Eq. (9).

Figure 12

Electrical anisotropy ratio, $\delta$, versus the filling fraction, $p$, (a) for $m=2048$ and different values of order parameter $s$, (b) for fixed value of order parameter, $s=0.8$, and different values of $m$, (c) for fixed value of order parameter, $s=1$, and different values of $m$.

Figure 13

Example of a polygon formed by a line segment on a checkered paper. The size of the polyomino is 8.

Figure 14

Example of a horizontal line segment on checkered paper, $5<l<6$, i.e., $\lfloor l_x\rfloor =5$. The line segment may intersect only 5 or 6 vertical lines.