Growth, percolation, and correlations in disordered fiber networks

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 mss density correlations in paper sheets.