Continuum percolation with discs having a distribution of radii

Percolation of binary disk systems: Modeling and theory

The dispersion and connectivity of particles with a high degree of polydispersity is relevant to problems involving composite material properties and reaction decomposition prediction and has been the subject of much study in the literature. This work utilizes Monte Carlo models to predict percolation thresholds for a two-dimensional systems containing disks of two different radii. Monte Carlo simulations and spanning probability are used to extend prior models into regions of higher polydispersity than those previously considered. A correlation to predict the percolation threshold for binary disk systems is proposed based on the extended dataset presented in this work and compared to previously published correlations. A set of boundary conditions necessary for a good fit is presented, and a condition for maximizing percolation threshold for binary disk systems is suggested.

Continuum percolation of overlapping disks with a distribution of radii having a power-law tail

We study the continuum percolation problem of overlapping disks with a distribution of radii having a power-law tail; the probability that a given disk has a radius between R and R+dR is proportional to R(-(a+1)), where a>2. We show that in the low-density nonpercolating phase, the two-point function shows a power-law decay with distance, even at arbitrarily low densities of the disks, unlike the exponential decay in the usual percolation problem. As in the problem of fluids with long-range interaction, we argue that in our problem, the critical exponents take their short-range values for a>3-η(sr) whereas they depend on a for a<3-η(sr) where η(sr) is the anomalous dimension for the usual percolation problem. The mean-field regime obtained in the fluid problem corresponds to the fully covered regime, a≤2, in the percolation problem. We propose an approximate renormalization scheme to determine the correlation length exponent ν and the percolation threshold. We carry out Monte Carlo simulations and determine the exponent ν as a function of a. The determined values of ν show that it is independent of the parameter a for a>3-η(sr) and is equal to that for the lattice percolation problem, whereas ν varies with a for 2<a<3-η(sr). We also determine the percolation threshold of the system as a function of the parameter a.

Cortical free-association dynamics: distinct phases of a latching network

A Potts associative memory network has been proposed as a simplified model of macroscopic cortical dynamics, in which each Potts unit stands for a patch of cortex, which can be activated in one of S local attractor states. The internal neuronal dynamics of the patch is not described by the model, rather it is subsumed into an effective description in terms of graded Potts units, with adaptation effects both specific to each attractor state and generic to the patch. If each unit, or patch, receives effective (tensor) connections from C other units, the network has been shown to be able to store a large number p of global patterns, or network attractors, each with a fraction a of the units active, where the critical load p_{c} scales roughly like p_{c}≈CS^{2}/aln(1/a) (if the patterns are randomly correlated). Interestingly, after retrieving an externally cued attractor, the network can continue jumping, or latching, from attractor to attractor, driven by adaptation effects. The occurrence and duration of latching dynamics is found through simulations to depend critically on the strength of local attractor states, expressed in the Potts model by a parameter w. Here we describe with simulations and then analytically the boundaries between distinct phases of no latching, of transient and sustained latching, deriving a phase diagram in the plane w-T, where T parametrizes thermal noise effects. Implications for real cortical dynamics are briefly reviewed in the conclusions.

Percolation of diffusionally evolved two-phase systems

Percolation thresholds and critical exponents for universal scaling laws are computed for microstructures that derive from phase-transformation processes in two dimensions. The computed percolation threshold for nucleation and growth processes, p(c)≈0.6612, is similar to those obtained by random placement of disks and greater than that of spinodal decomposition, p(c)≈0.4987. Three critical exponents for scaling behavior were computed and do not differ significantly from universal values. The time evolution of a characteristic microstructural length was also computed: For spinodal decomposition, this length grows according to a power law after a short incubation period; for nucleation and growth, there are several transitions in the nature of the growth law. We speculate that the transitions in nucleation and growth derive from competing effects of coalescence at short times and then subsequent coarsening. Short-range order is present, but different, for both classes of microstructural evolution.

Asymmetry in the percolation thresholds of fully penetrable disks with two different radii

We perform simulations of continuum gradient percolation to measure the percolation threshold of systems of fully penetrable disks of two different radii. As a function of volumetric proportion v, the percolation threshold phic(v,lambda) is approximately symmetric about v=0.5 for a fixed ratio lambda of the radii. However, the difference from symmetry is strongly statistically significant. We also improve, by an order of magnitude, the measurement of the percolation threshold for disks of equal radius: phic*=0.676 347 5(6).

Percolation of particles on recursive lattices using a nanoscale approach. I. Theoretical foundation at the atomic level

Powdered materials of sizes ranging from nanometers to micrometers are widely used in materials science and are carefully selected to enhance the performance of a matrix. Fillers have been used in order to improve properties, such as mechanical, rheological, electrical, magnetic, thermal, etc., of the host material. Changes in the shape and size of the filler particles are known to affect and, in some cases, magnify such enhancement. This effect is usually associated with an increased probability of formation of a percolating cluster of filler particles in the matrix. In this series of papers, we will consider lattice models. Previous model calculations of percolation in polymeric systems generally did not take into account the possible difference between the size and shape of monomers and filler particles and usually neglected interactions or accounted for them in a crude fashion. In our approach, the original lattice is replaced by a recursive structure on which calculations are done exactly and interactions as well as size and shape disparities can be easily taken into account. Here, we introduce the recursive approach, describe how to derive the percolation threshold as a function of the various parameters of the problem, and apply the approach to the analysis of the effect of correlations among monodisperse particles on the percolation threshold of a system. In the second paper of the series, we tackle the issue of the effect of size and shape disparities of the particles on their percolation properties. In the last paper, we describe the effects due to the presence of a polymer matrix.

Measurement of the percolation threshold for fully penetrable disks of different radii

We perform simulations of gradient percolation to study the percolation threshold for systems of homogeneous fully penetrable disks of variable radii. We find that, if the radii follow a uniform distribution, the percolation threshold is 0.686 610+/-0.000 007. We also investigate binary dispersions, studying the influence of constitutive parameters on the percolation threshold and suggesting an empirical formula for the threshold. We find that, with the appropriate parameters, a percolation threshold of approximately 0.76 can be achieved. The minimal threshold of 0.676 339+/-0.000 004 is achieved by disks of equal radius.