Quasiuniversal Connectedness Percolation of Polydisperse Rod Systems

Abstract

The connectedness percolation threshold (ηc) and critical coordination number (Zc) of systems of penetrable spherocylinders characterized by a length polydispersity are studied by way of Monte Carlo simulations for several aspect ratio distributions. We find that (i) ηc is a nearly universal function of the weight-averaged aspect ratio, with an approximate inverse dependence that extends to aspect ratios that are well below the slender rod limit and (ii) that percolation of impenetrable spherocylinders displays a similar quasiuniversal behavior. For systems with a sufficiently high degree of polydispersity, we find that Zc can become smaller than unity, in analogy with observations reported for generalized and complex networks.PACS numbers: 64.60.ah, 61.46.Fg, 82.70.Dd Idealized elongated objects such as perfectly rigid cylinders, spherocylinders and prolate spheroids are prototypical models for a wide array of technologically relevant systems that include liquid crystals, nanocomposites based on filamentous fillers as well as fiber-reinforced materials. Percolation phenomena involving dramatic increases in, e.g., structural rigidity and electrical and thermal conductivities of composites with increasing filler loading are currently of particular interest [1]. These increases are caused by the formation of an infinite cluster of in some sense connected particles at the critical loading, i.e., the percolation threshold.It has been established by analytical [1-3] and numerical [4][5][6][7][8][9][10][11] studies that for dispersions of sufficiently elongated objects of identical size and shape, i.e, "monodisperse" objects, the geometric percolation threshold expressed in terms of the critical volume fraction of particles is inversely proportional to the aspect ratio of the filler particles. This property is exploited in the fabrication of conducting polymeric composites with very low conducting filler contents. Depending on the production processes of the composites, however, the filler particles almost invariably exhibit a pronounced polydispersity in both size and shape [12,13]. Although it represents a possible factor behind huge quantitative discrepancies between theory and experiments [14,15], such polydispersity has received relatively little attention in terms of theoretical modeling until fairly recently [1,16,17], Achievement of a theoretical understanding of how the continuum percolation of fibrous fillers is affected by polydispersity is thus key to the controlled design of a large class of composite materials for practical particle size and shape distributions.Recent analytical results obtained from integral equation methods [16] and from an heuristic mapping onto a generalized Bethe lattice [17] predict that in the slender rod limit, where the particles have asymptotically large values of the aspect ratio, the volume fraction at the percolation threshold is inversely proportional to the weight average L w = L 2 / L of the rod lengths, where the brackets imply number averages over the distribution of...

A percolation-based model for the conductivity of nanofiber composites

“…Excluded volume arguments, 15 integral equation theories, 16,17 analogy to site percolation on a modified Bethe lattice, 18 and Monte Carlo (MC) simulations 19,20 reveal that for polydisperse cylindrical particles that are randomly a) E-mail: apchatte@esf.edu. Fax: 315-470-6856. distributed and oriented, φ c is expected to depend approximately inversely upon the weight-averaged aspect ratio.…”

Depletion-interaction effects on the tunneling conductivity of nanorod suspensions

“…We calculate δ c following the method described in Refs. [27,28,34,35], which consists of constructing the percolation probability P perc (δ) from the values of the percolation distance δ calculated for all realizations of the system with a given combination of rod and sphere concentrations. From the condition P perc (δ c ) = 1/2, which provides a robust estimate of the critical distance [28], we find that the effect of the depletant particles is to lower δ c compared to the case without depletants (φ s = 0), as shown in Figs.…”

Tunneling conductivity in anisotropic nanofiber composites: a percolation-based model

“…Analytical theory based upon both lattice [3] and continuum representations [4][5][6][7][8] as well as computer simulation studies [9][10][11][12] reveal that the volume fraction occupied by the particles at the percolation threshold, denoted c  , is strongly dependent upon the particle shape. For the case of elongated, rod-like, electrically conductive nanoparticles that can be approximately modeled as cylinders, theory and experiment suggest that the conductivity of a composite based upon such filler species also depends upon the degree of orientational alignment of the particles [13][14][15][16].…”