Scattering Signatures of Invasion Percolation

TL;DR

This paper investigates the scattering properties of three-dimensional percolation clusters, revealing how fractal structures influence scattering at different scales and proposing analytical expressions to model these effects.

Contribution

It introduces a detailed analysis of scattering signatures of invasion percolation clusters, including new models for structure factors across various length scales and conditions.

Findings

Fractal structure persists up to the cluster extent scale.

Scattering behavior depends on three length scales: $\xi$, $d_g$, and $Q^{-1}$.

Approximate expressions for structure factors are proposed.

Abstract

Motivated by recent experiments, we investigate the scattering properties of percolation clusters generated by numerical simulations on a three dimensional cubic lattice. Individual clusters of given size are shown to present a fractal structure up to a scale of order their extent, even far away from the percolation threshold $p_c$. The influence of inter-cluster correlations on the structure factor of assemblies of clusters selected by an invasion phenomenon is studied in detail. For invasion from bulk germs, we show that the scattering properties are determined by three length scales, the correlation length $\xi$, the average distance between germs $d_g$, and the spatial scale probed by scattering, set by the inverse of the scattering wavevector $Q$. At small scales, we find that the fractal structure of individual clusters is retained, the structure factor decaying as $Q^{-d_f}$. At large scales, the structure factor tends to a limit, set by the smaller of $\xi$ and $d_g$, both below and above $p_c$. We propose approximate expressions reproducing the simulated structure factor for arbitrary $\xi$, $d_g$, and $Q$, and illustrate how they can be used to avoid to resort to costly numerical simulations. For invasion from surfaces, we find that, at $p_c$, the structure factor behaves as $Q^{-d_f}$ at all $Q$, i.e. the fractal structure is retained at arbitrarily large scales. Results away from $p_c$ are compared to the case of bulk germs. Our results can be applied to discuss light or neutrons scattering experiments on percolating systems. This is illustrated in the context of evaporation from porous materials.