Growth, Percolation, and Correlations in Disordered Fiber Networks

TL;DR

This study introduces a 2D continuum deposition model for disordered fiber networks, analyzing growth, percolation, and correlations, with theoretical and simulation results that match experimental observations.

Contribution

The paper develops a new model for fiber network growth with variable clustering parameter p, deriving growth laws, percolation thresholds, and correlation functions, supported by simulations and mean-field theory.

Findings

Percolation threshold depends on clustering parameter p.

Nontrivial density correlations are observed for p<1.

The model's correlation functions agree qualitatively with experimental data.

Abstract

This paper studies growth, percolation, and correlations in disordered fiber networks. We start by introducing a 2D continuum deposition model with effective fiber-fiber interactions represented by a parameter $p$ which controls the degree of clustering. For $p=1$, the deposited network is uniformly random, while for $p=0$ only a single connected cluster can grow. For $p=0$, we first derive the growth law for the average size of the cluster as well as a formula for its mass density profile. For $p>0$, we carry out extensive simulations on fibers, and also needles and disks to study the dependence of the percolation threshold on $p$. We also derive a mean-field theory for the threshold near $p=0$ and $p=1$ and find good qualitative agreement with the simulations. The fiber networks produced by the model display nontrivial density correlations for $p<1$. We study these by deriving an approximate expression for the pair distribution function of the model that reduces to the exactly known case of a uniformly random network. We also show that the two-point mass density correlation function of the model has a nontrivial form, and discuss our results in view of recent experimental data on mass density correlations in paper sheets.