Modeling percolation in high-aspect-ratio fiber systems. I. Soft-core versus hard-core models.

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.

Modeling percolation in high-aspect-ratio fiber systems. II. The effect of waviness on the percolation onset.

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.

Direct stochastic simulation of Ca2+ motion in Xenopus eggs.

The release of important intracellular ions has been widely modeled using two approaches, namely, (1) Fickian diffusion, in which sometimes tensorial diffusion coefficients are used to fit observed temporally varying concentrations of calcium, and (2) cellular automata, which produce a set of localized finite difference equations that result in complex global behavior. Here, we take a different approach, employing some assumed, a priori, distribution of ion-binding proteins in the cell, and some assumed biochemical capture and release characteristics to explain ionic motion, and ultimately, distribution. We study several scenarios for ion distribution, based on differences in binder action and distribution. The numbers and strengths of ion binders, spatial variation in inositol 1,4,5-triphosphate concentration, together with the escalating distribution of ionic diffusion speed, are found to be key factors leading to concavity in the Ca2+ wave shape. We also offer an explanation for geometrical effects on previously observed ion diffusion speeds in the cellular cortex of the Xenopus laevis egg during fertilization, based on an angle-of-view correction.

Differences between collagen morphologies, properties and distribution in diabetic and normal biobreeding and Sprague-Dawley rat sciatic nerves.

Both structural and functional differences between normal and diabetic nerve have been observed, in human patients and animal models. We hypothesize that these structural differences are quantifiable, morphologically and mechanically, with the ultimate aim of understanding the contribution of these differences to permanent nerve damage. The outer collagenous epineurial and perineurial tissues of mammalian peripheral nerves mechanically and chemically shield the conducting axons. We have quantified differences in these collagens, using whole-nerve uniaxial testing, and immunohistochemistry of collagens type I, III, and IV in diabetic and normal nerves. We present results of two studies, on normal and diabetic BioBreeding (BB), and normal, diabetic and weight-controlled Sprague-Dawley (SD) rats, respectively. Overall, we measured slightly higher uniaxial moduli (e.g. 5.9 MPa vs. 3.5 MPa, BB; 10.7 MPa vs. 10.0 MPa, SD at 40% strain) in whole nerves as well as higher peak stresses (e.g. 0.99 MPa vs. 0.74 MPa, BB; 2.16 MPa vs. 1.94 MPa, SD at 40% strain) in the diabetics of both animal models. We measured increased concentrations of types III and IV collagens in the diabetics of both models and mixed upregulation results were observed in type I protein levels. We detected small differences in mechanical properties at the tissue scale, though we found significant structural and morphometric differences at the fibril scale. These findings suggest that whole-tissue mechanical testing is not a sufficient assay for collagen glycation, and that fibrillar and molecular scale assays are needed to detect the earliest stages of diabetic protein glycation.

A mechanical model for collagen fibril load sharing in peripheral nerve of diabetic and nondiabetic rats.

Peripheral neuropathy affects approximately 50% of the 15 million Americans with diabetes. It has been suggested that mechanical effects related to collagen glycation are related to the permanence of neuropathy. In the present paper, we develop a model for load transfer in a whole nerve, using a simple pressure vessel approximation, in order to assess the significant of stiffening of the collagenous nerve sheath on endoneurial fluid pressure. We also develop a fibril-scale mechanics model for the nerve, to model the straightening of wavy fibrils, producing the toe region observed in nerve tissue, and also to interrogate the effects of interfibrillar crosslinks on the overall properties of the tissue. Such collagen crosslinking has been implicated in complications in diabetic tissues. Our fibril-scale model uses a two-parameter Weibull model for fibril strength, in combination with statistical parameters describing fibril modulus, angle, wave-amplitude, and volume fraction to capture both toe region and failure region behavior of whole rat sciatic nerve. The extrema of equal and local load-sharing assumptions are used to map potential differences in diabetic and nondiabetic tissues. This work may ultimately be useful in differentiating between the responses of normal and heavily crosslinked tissue.

Nerve collagens from diabetic and nondiabetic Sprague-Dawley and biobreeding rats: an atomic force microscopy study.

BACKGROUND: Alterations in rat's nerve collagens due to diabetes may be related to the permanence of damage due to diabetic neuropathy. We (1) provide a methodology for determining the diameters of collagen fibers accounting for atomic force microscope (AFM) imaging artifacts, (2) present data on structural differences in sciatic nerve endoneurial, epineurial and tail tendon collagens of control and diabetic Sprague-Dawley and BioBreeding rats, and (3) compare results with literature values. METHODS: We measured collagen diameters and band spacing on endoneurial and epineurial sciatic nerve tissue, and tail tendon, in control and diabetic rats (STZ-induced 12-week diabetic SD and 16-week spontaneously diabetic BB rats). We also developed a model to interpret the raw AFM data. RESULTS: All types of fibrillar collagen diameters studied became larger for diabetic versus control animals. Values for diabetic and control collagen fiber diameters in SD rats were 78 nm and 72 nm for SN epineurium, and 49 nm and 43 nm for SN endoneurium. For diabetic and control BB rats, these values were 83 nm and 77 nm (SN epineurium) and 49 nm and 43 nm (SN endoneurium). Values of 161 nm and 125 nm were found for diabetic and control tail tendon of BB rats. No significant changes were observed in any of the five comparisons made in D-band spacings that ranged from 63 to 69 nm. CONCLUSIONS: The best means we have found to reduce raw AFM data is to measure several diameters with a single scan, using valley-to-valley measurements. Structural, fibrillar collagens of the nerve and tendon become larger in rats exposed to prolonged diabetes.

Analytical approximation of the two-dimensional percolation threshold for fields of overlapping ellipses.

Percolation of particle arrays is of high interest in microstructural design of materials. While there have been numerous contributions to theoretical modeling of percolation in particulate systems, no analytical approximation for the generalized problem of variable aspect-ratio ellipses has been reported. In the present work, we (1) derive, and verify through simulation, an analytical percolation approach capable of identifying the percolation point in two-phase materials containing generalized ellipses of uniform shape and size; and (2) explore the dependence of percolation on the particle aspect ratio. We validate our technique with simulations tracking both cluster sizes and percolation status, in networks of elliptical and circular particles. We also outline the steps needed to extend our approach to three-dimensional particles (ellipsoids). For biological materials, we ultimately aim to provide direct insight into the contribution of each single phase in multiphase tissues to mechanical or conductive properties. For engineered materials, we aim to provide insight into the minimum amount of a particular phase needed to strongly influence properties.