Finite-size effects in nanocomposite thin films and fibers

Monte Carlo simulations of finite-size effects for continuum percolation in three-dimensional, rectangular sample spaces filled with spherical particles were performed. For samples with any dimension less than 10–20 times the particle diameter, finite-size effects were observed. For thin films in the finite-size regime, percolation across the thin direction of the film gave critical volume fraction ($p_c$) values that differed from those along the plane of the film. Simulations perpendicular to the film for very thin samples resulted in $p_c$ values lower than the classical limit of ∼29% (for spheres in a three-dimensional matrix) which increased with film thickness. For percolation along thin films, while holding film thickness constant, $p_c$ increased with increasing sample size, which is a modification of the finite-sized scaling effect for cubic samples. For samples with a large aspect ratio (fibers) and a finite-sized cross-sectional area, the critical volume fraction increased with sample length, as the sample became quasi-one-dimensional. The results are discussed in the context of adding volume along or perpendicular to the percolation direction. From an experimental perspective, these findings indicate that sample shape, as well as relative size, influences percolation in the finite-size regime.

Figure 1

Percolation probability curves for cubic samples doped with spheres. Fit lines are error function. Sample dimensions are relative to sphere diameter. The dashed horizontal line indicates 50% percolation probability.

Figure 2

Thin film to cubic to fiberlike transition for five different cross-sectional areas. Sample dimensions are relative to sphere diameter and percolation is tested for along $L$ the percolation length. $A$ is the cross-sectional area perpendicular to $L$. Fit lines are from the standard $p_c$ finite-sized scaling form (taking the absolute value of the left-hand side) with $p_c^{\infty }$ and $\nu$ values as follows: $A$ = 1 × 1, $1.003\pm 0.004$, $2.11\pm 0.08$; $A$ = 5 × 5, $0.495\pm 0.003$, $2.23\pm 0.08$; $A$ = 10 × 10, $0.376\pm 0.001$, $1.72\pm 0.05$; $A$ = 50 × 50, $0.2995\pm 0.0006$, $0.88\pm 0.05$; $A$ = 100 × 100, $0.299\pm 0.003$, $0.94\pm 0.05$. As expected, the smallest $L$ values do not scale; more low $L$ values had to be excluded as cross-sectional area increased. An elevated $\nu$ indicates a slower approach to the infinite value as $L$ is increased. We also scaled samples with the same aspect ratio (AR) versus $L$. For a small number of samples with AR = 2, $p_c^{\infty }$ = 0.289 and $\nu$ = 0.88; $\nu$ increased as AR varied from 1 (e.g., AR = 1/5, $\nu$ = 2.2; AR = 15,$\nu =1.1$). Again, small $L$ values do not scale.

Figure 3

Critical volume fraction in two directions ($x$ and $y$ as indicated above) for cubic to filmlike samples. The $x$ and $z$ dimensions are always the same (=$d$), so percolation occurs perpendicular to or along a filmlike sample. Dimensions are in units of the sphere diameter. Arrows point in the direction of larger samples. For comparison, the values in Fig. 2 are in the $y$ direction.