Lattice model for percolation on a plane of partially aligned sticks with length dispersity.

A lattice-based model for continuum percolation is applied to the case of randomly located, partially aligned sticks with unequal lengths in 2D which are allowed to cross each other. Results are obtained for the critical number of sticks per unit area at the percolation threshold in terms of the distributions over length and orientational angle and are compared with findings from computer simulations. Consistent with findings from computer simulations, our model shows that the percolation threshold is (i) elevated by increasing degrees of alignment for a fixed length distribution, and (ii) lowered by increasing degrees of length dispersity for a fixed orientational distribution.

A lattice model for the impact of volume fraction fluctuations upon percolation by cylinders.

A lattice model for continuum percolation by cylindrical rods is generalized to account for inhomogeneities in the volume fraction that are indicative of particle clustering or aggregation. The percolation threshold is evaluated from a formalism that uses two different categories of occupied sites (denoting particles) with different occupation probabilities that represent large and small local volume fractions. Our modeling framework enables independent variations in (i) the strength of the correlation that adjacent particles experience high (or low) effective volume fractions, (ii) the disparity between the macroscopically averaged volume fraction and (say) the volume fraction characterizing the regions with high effective particle concentrations, and (iii) the overall proportion of particles that are located in regions with either high or low volume fraction.

Bethe lattice model with site and bond correlations for continuum percolation by isotropic systems of monodisperse rods.

A model for connectedness percolation in isotropic systems of monodisperse cylinders is developed that employs a generalization of the tree-like Bethe lattice. The traditional Bethe lattice is generalized by incorporating (within a heuristic, mean-field framework) a pair of correlation parameters that describe (i) the states of occupancy of neighboring sites and (ii) the states of directly adjacent bonds, which are also allowed to be in either of two possible states. Averaging over the fluctuating states of neighboring bonds provides an operational means to modulate the dependence upon volume fraction of the average number of next-nearest-neighbor rod-rod contacts without altering the number of such nearest-neighbor interparticle contacts.

Distribution over pore radii in random and isotropic systems of polydisperse rods with finite aspect ratios.

Excluded-volume arguments are applied toward modeling the pore-size distribution in systems of randomly arranged cylindrical rods with finite and nonuniform aspect ratios. An explicit expression for the pore-size distribution is obtained by way of an analogy to a hypothetical system of fully penetrable objects, through a mapping that is designed to preserve the volume fraction occupied by the particle cores and the specific surface area. Results are presented for the mean value and standard deviation of the pore radius as functions of the rod aspect ratio, volume fraction, and polydispersity (degree of nonuniformity in the aspect ratios of the particles).

A model for the pore size distribution in random and isotropic mixtures of cylindrical fibres and circular disks.

A model for the distribution of pore sizes in systems of randomly distributed cylindrical fibers and circular disks is developed that derives from simple excluded-volume considerations. The dimensions of the particles with finite hard cores are mapped onto those of a hypothetical system of fully penetrable objects in a manner that preserves the overall core-occupied volume fraction and specific surface area. This mapping enables us to obtain estimates for the pore size distribution, and for the first and second moments of the pore sizes, as functions of the particle volume fraction and mixture composition in terms of closed-form, analytical expressions.

Random geometric graph description of connectedness percolation in rod systems.

The problem of continuum percolation in dispersions of rods is reformulated in terms of weighted random geometric graphs. Nodes (or sites or vertices) in the graph represent spatial locations occupied by the centers of the rods. The probability that an edge (or link) connects any randomly selected pair of nodes depends upon the rod volume fraction as well as the distribution over their sizes and shapes, and also upon quantities that characterize their state of dispersion (such as the orientational distribution function).

Connectedness percolation in isotropic systems of monodisperse spherocylinders.

The dependence upon aspect ratio and connectedness range of the percolation threshold for monodisperse spherocylinders as calculated from an analogy between lattice and continuum percolation is compared with findings from: (i) recent Monte Carlo (MC) simulations, and: (ii) a recent elaboration of an integral equation-based approach that uses the connectedness Ornstein-Zernike equation. The results from each of these approaches are found to be in reasonable semi-quantitative agreement, especially for the case of particles that have a finite hard core diameter.

A lattice model for connectedness percolation in mixtures of rods and disks.

Geometric percolation in mixtures of isotropically oriented rods and disks is examined from the perspective of a tree-like lattice model with a distribution over the co-ordination numbers (or vertex degrees). Correlations between the particle locations are described within a mean-field approximation that employs operationally-defined parameters to characterize pairwise interactions. The percolation threshold is studied as a function of the particle size disparity and the strength of inter-particle correlations.

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

The critical path approximation ('CPA') is integrated with a lattice-based approach to percolation to provide a model for conductivity in nanofiber-based composites. Our treatment incorporates a recent estimate for the anisotropy in tunneling-based conductance as a function of the relative angle between the axes of elongated nanoparticles. The conductivity is examined as a function of the volume fraction, degree of clustering, and of the mean value and standard deviation of the orientational order parameter.

Percolation thresholds for polydisperse circular disks: a lattice-based exploration.

The percolation threshold for polydisperse systems of (i) isotropically oriented, and (ii) perfectly aligned, circular disks is calculated within an analogy to a lattice model. Our results are expressed in terms of moments of the distribution function over the disk radii, and they closely resemble findings obtained from an integral equation approach. The threshold is found to be quite sensitive to polydispersity in the disk radii and, for monodisperse systems, to approach a plateau for large values of the aspect ratio (thickness to diameter ratio).

Percolation in polydisperse systems of aligned rods: a lattice-based analysis.

A model is developed for percolation in polydisperse systems of oriented cylinders that integrates excluded volume arguments with an analogy to site percolation on a modified Bethe lattice. Results from this treatment are presented for the volume fraction at the percolation threshold (denoted ϕc) as a function of the degree of polydispersity, mixture composition, and degree of orientational ordering. For monodisperse systems, ϕc is found to be a monotonically increasing function of the traditional orientational order parameter that quantifies degree of alignment.

A percolation-based model for the conductivity of nanofiber composites.

A model is presented that integrates the critical path approximation with percolation theory to describe the dependence of electrical conductivity upon volume fraction in nanofiber-based composites. The theory accounts for clustering and correlation effects that reflect non-randomness in the spatial distribution of the particles. Results from this formalism are compared to experimental measurements performed upon carbon nanotube-based conductive nanocomposites.

Quasiuniversal connectedness percolation of polydisperse rod systems.

The connectedness percolation threshold (η(c)) and critical coordination number (Z(c)) 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) η(c) 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 Z(c) can become smaller than unity, in analogy with observations reported for generalized and complex networks.

Geometric percolation in polydisperse systems of finite-diameter rods: effects due to particle clustering and inter-particle correlations.

The impact of particle clustering and correlation upon the percolation behavior of polydisperse cylinders with finite hard core diameter is examined within an analogy to a lattice percolation problem. Percolation thresholds and percolation and backbone probabilities are explored as functions of the degree of clustering and extent of correlation among the inter-particle contacts. The percolation threshold and volume fractions occupied by the infinite network and by the cluster backbone are shown to be quite sensitive to the formation of inter-connected cliques of particles and to the presence of correlation among particle contacts.

A simple model for the pore size distribution in random fibre networks.

A heuristic approach based upon excluded volume arguments is developed for modelling the distribution of pore sizes in isotropic networks of randomly distributed cylindrical fibres. Our formalism accounts for the finite hard core diameters of the fibres, and leads to compact, analytically tractable expressions that span the complete range of volume fractions. Results are presented for the mean and mean-squared pore radii as functions of the fibre volume fraction, and for the partition coefficient of a spherical tracer particle into such a network under conditions such that steric effects are dominant.

Tracer diffusion in fibre networks: the impact of spatial fluctuations in the fibre distribution.

A mean-field formalism that addresses spatial non-uniformities in fibre networks is combined with the cylindrical cell model to calculate the diffusion constant for a spherical tracer. Deviations from randomness in the fibre distribution are described by an operational distribution over volume fractions that is parametrized by mean values for the pore radii and void space chord lengths. Weight factors for elements with different radii in the cell model are assigned in a manner that enforces agreement with the distribution over pore sizes predicted by our treatment of heterogeneous networks.

Connectedness percolation in monodisperse rod systems: clustering effects.

A model is presented that examines the impact of local clustering upon the percolation behaviour of interpenetrable rod-like particles. The percolation threshold, as well as percolation and backbone probabilities, are evaluated as functions of the particle aspect ratio and degree of clustering by way of an analogy to a lattice site percolation problem. The formation of local, physically connected cliques of particles is shown to raise the percolation threshold whilst reducing the percolation and backbone fractions for a fixed volume fraction of particles.

A simple model for characterizing non-uniform fibre-based composites and networks.

A mean-field model is presented that describes non-uniformities in the spatial distribution of fibres in networks and composites in terms of fluctuations in the local composition. The mean pore radius, specific surface area, lineal path function, and chord length probability density are expressed as functions of the fibre volume fraction within a heuristic formalism. The impact of statistical heterogeneities in the fibre distribution upon the elastic moduli is assessed within the semi-empirical Reuss-Voigt-Hill averaging scheme.

Connectedness percolation in polydisperse rod systems: A modified Bethe lattice approach.

A mean-field theory is presented for the percolation behavior of systems of rodlike particles characterized by length polydispersity. An analogy to the problem of site percolation on a modified Bethe lattice is employed to estimate the percolation threshold, percolation probability, and backbone fraction as functions of the rod volume fraction and polydispersity. Model calculations reveal that the percolation probability and backbone fraction depend sensitively upon the rod length distribution, while the percolation threshold is governed primarily by the weight-averaged rod length.

Elastic moduli of cellulose nanoparticle-reinforced composites: a micromechanical model.

A model for the elastic coefficients of fiber-reinforced materials is applied toward the analysis of the tensile and shear moduli of nanocomposites reinforced by rod-like cellulose nanoparticles. Our formalism integrates results from percolation theory with micromechanical and effective medium approaches. Polydispersity in the fiber length distribution and anisotropies in the stiffness coefficients of cellulose nanoparticles are taken into account explicitly.

Integral equation theory for athermal solutions of linear polymers.

An integral equation model is developed for athermal solutions of flexible linear polymers with particular reference to good solvent conditions. Results from scaling theory are used in formulating form factors for describing the single chain structure, and the impact of solvent quality on the chain fractal dimension is accounted for. Calculations are performed within the stringlike implementation of the polymer reference interaction site model with blobs (as opposed to complete chains) treated as the constituent structural units for semidilute solutions.

Macromolecule-Induced Clustering of Hard Spheres.

The connectivity Ornstein-Zernike formalism, together with the polymer reference interaction site model (PRISM), is employed to describe connectivity and network formation in mixtures of spheres and polymers. Results are presented for the percolation of spheres induced by both flexible coil-like and rigid rod-like linear polymers; the Percus-Yevick (PY) approximation is used throughout. Our results are compared with predictions based on the adhesive hard sphere (AHS) model, and correlations with the polymer-mediated second virial coefficient between spheres are discussed.