Limited Range Fractality of Randomly Adsorbed Rods

TL;DR

This study analyzes the apparent fractal behavior of randomly adsorbed rods in two dimensions, revealing how their orientation, density, and internal structure influence observed fractal dimensions across different length scales.

Contribution

It demonstrates that the apparent fractal dimension depends on rod orientation and coverage, and distinguishes between internal structure and distribution effects at different scales.

Findings

Apparent fractal behavior occurs between average rod length and inter-rod distance.

Fractal dimension decreases with increased anisotropy of rod orientations.

Internal rod structure influences behavior at scales smaller than average rod length.

Abstract

Multiple resolution analysis of two dimensional structures composed of randomly adsorbed penetrable rods, for densities below the percolation threshold, has been carried out using box-counting functions. It is found that at relevant resolutions, for box-sizes, $r$, between cutoffs given by the average rod length $<\ell>$ and the average inter-rod distance $r_1$, these systems exhibit apparent fractal behavior. It is shown that unlike the case of randomly distributed isotropic objects, the upper cutoff $r_1$ is not only a function of the coverage but also depends on the excluded volume, averaged over the orientational distribution. Moreover, the apparent fractal dimension also depends on the orientational distributions of the rods and decreases as it becomes more anisotropic. For box sizes smaller than $<\ell>$ the box counting function is determined by the internal structure of the rods, whether simple or itself fractal. Two examples are considered - one of regular rods of one dimensional structure and rods which are trimmed into a Cantor set structure which are fractals themselves. The models examined are relevant to adsorption of linear molecules and fibers, liquid crystals, stress induced fractures and edge imperfections in metal catalysts. We thus obtain a distinction between two ranges of length scales: $r < <\ell>$ where the internal structure of the adsorbed objects is probed, and $<\ell> < r < r_1$ where their distribution is probed, both of which may exhibit fractal behavior. This distinction is relevant to the large class of systems which exhibit aggregation of a finite density of fractal-like clusters, which includes surface growth in molecular beam epitaxy and diffusion-limited-cluster-cluster-aggregation models.