Numerical and analytical studies of the onset of percolation in high-aspect-ratio fiber fiber systems such as nanotube reinforced polymers available in the literature have consistently modeled fibers as penetrable, straight, capped cylinders, also referred to as spherocylinders. In reality, however, fibers of very high-aspect ratio embedded in a polymer do not come into direct physical contact with each other, let alone exhibit any degree of penetrability. Further, embedded fibers of very high-aspect ratio are often actually wavy, rather than straight. In this two-part paper we address these critical differences between known physical systems, and the presently used spherocylinder percolation model. In Paper I we evaluate the effect of allowing penetration of the model fibers on simulation results by comparing the soft-core and the hard-core approaches to modeling percolation onset. We use Monte Carlo simulations to investigate the relationship between percolation threshold and excluded volume for both modeling approaches. Our results show that the generally accepted inverse proportionality between percolation threshold and excluded volume holds for both models. We further demonstrate that the error introduced by allowing the fibers to intersect is non-negligible, and is a function of both aspect ratio and tunneling distance. Thus while the results of both the soft-core model and hard-core assumptions can be matched to select experimental results, the hard-core model is more appropriate for modeling percolation in nanotubes-reinforced composites. The hard-core model can also potentially be used as a tool in calculating the tunneling distance in composite materials, given the fiber morphology and experimentally derived electrical percolation threshold. In Paper II we investigate the effect of the waviness of the fibers on the onset of percolation in fiber reinforced composites.

Nanotube sheets, or ‘‘bucky papers,’ ’ have been proposed for use in actuating, structural and electrochemical systems, based in part on their potential mechanical properties. Here, we present results of detailed simulations of networks of nanotubes/ropes, with special emphasis on the effect of joint morphology. We perform detailed simulations of three-dimensional joint deformation, and use the results to inform simulations of two-dimensional �2D � networks with intertube connections represented by torsion springs. Upper bounds are established on moduli of nanotube sheets, using the 2D Euler beam-network simulations. Comparisons of experimental and simulated response for HiPco-nanotube and laser-ablated nanotube sheets, indicate that �2–30-fold increases in moduli may be achievable in these materials. Increasing the numbers of interrope connections appears to be the best target for improving nanotube sheet stiffnesses in materials containing straight segments. © 2004 American Institute of Physics. �DOI: 10.1063/1.1687995� I

The onset of electrical percolation in nanotube-reinforced composites is often modeled by considering the geometric percolation of a system of penetrable, straight, rigid, capped cylinders, or spherocylinders, despite the fact that embedded nanotubes are not straight and do not penetrate one another. In Part I of this work we investigated the applicability of the soft-core model to the present problem, and concluded that the hard-core approach is more appropriate for modeling electrical percolation onset in nanotube-reinforced composites and other high-aspect-ratio fiber systems. In Part II, we investigate the effect of fiber waviness on percolation onset. Previously, we studied extensively the effect of joint morphology and waviness in two-dimensional nanotube networks. In this work, we present the results of Monte Carlo simulations studying the effect of waviness on the percolation threshold of randomly oriented fibers in three dimensions. The excluded volumes of fibers were found numerically, and relationships between these and percolation thresholds for two different fiber morphologies were found. We build on the work of Part I, and extend the results of our soft-core, wavy fiber simulations to develop an analytical solution using the more relevant hard-core model. Our results show that for high- aspect-ratio fibers, the generally accepted inverse proportionality between percolation threshold and excluded volume holds, independent of fiber waviness. This suggests that, given an expression for excluded volume, an analytical solution can be derived to identify the percolation threshold of a system of high-aspect-ratio fibers, including nanotube-reinforced composites. Further, we show that for high aspect ratios, the percolation threshold of the wavy fiber networks is directly proportional to the analytical straight fiber solution and that the constant of proportionality is a function of the nanotube waviness only. Thus the onset of percolation can be adequately modeled by applying a factor based on fiber geometry to the analytical straight fiber solution.

Waviness alters both geometric and mechanical properties of stochastic fibrous networks and significantly affects overall mechanical response, but few results are available in the literature on the subject. In this work, we explore the importance of the dimension of constituent fibers ͑1D vs 2D͒ in determination of percolation thresholds, and other fundamental statistical properties of fibers having geometric waviness, in adaptation of classical theories on random lattices to practical applications, including analysis of nanotube ropes and collagen bundles. Although the so-called ''curl ratio'' clearly affects the statistical properties, as evaluated by Kallmes and Corte a few decades ago, we have found some results in this classic work to be inaccurate for systems containing fibers of moderate waviness. Our main findings include the independence of the mean number of crossings with regard to waviness, as well as the nonlinear dependence of probability of intersection on waviness. Our investigation of percolation in wavy fiber networks reveals that the percolation threshold is significantly increased, with increasing curl ratio. In addition, several nontrivial results related to network properties of infinite straight lines are also described, some of which are believed to have wide applications in practice.

Using a micromechanics approach, we recently investigated the theoretical limits on achievable moduli in nanotube mats by stiffening of bonds. However, the waviness intrinsic to many manufacturing processes also clearly plays an important role in stiffness of these materials. To study the effect of waviness on mechanical properties, we modeled fiber segments as sinusoids, generated networks comprised of these fibers, and performed simulations of deformations of the networks. In contradiction of classical work by Kallmes and Corte ͓Tappi J. 43, 737 ͑1960͔͒, we found the number of fiber crossings in these networks to be independent of fiber waviness, leading to identification of the number of fiber crossings as a necessary and sufficient parameter to specify network geometry, for either wavy or straight fibers. Our mechanical modeling results suggest that reducing the waviness of nanotube ropes would significantly improve Young's moduli in these materials. However, reduction of waviness would not produce the improvements achievable with higher bond density; for random sheets, assuring connections among all intersecting ropes appears to be the most direct route toward improving the overall sheet properties. There remains a persistent discrepancy between statistically predicted bond densities and physical bond densities, based on moduli of these materials.