Monte Carlo renormalization group study of the percolation problem of discs with a distribution of radii

Area covered by disks in small-bounded continuum percolating systems: An application to the string percolation model

In string percolation model, the study of colliding systems at high energies is based on a continuum percolation theory in two dimensions, where the number of strings distributed in the surface of interest is strongly determined by the size and energy of the colliding particles. It is also expected that the surface where the disks are lying be finite, defining a system without periodic boundary conditions. In this work, we report modifications to the fraction of the area covered by disks in continuum percolating systems due to a finite number of disks and bounded by different geometries: circle, ellipse, triangle, square, and pentagon, which correspond to the first Fourier modes of the shape fluctuation of the initial state after the particle collision. We find that the deviation of the fraction of area covered by disks from its corresponding value in the thermodynamic limit satisfies a universal behavior, where the free parameters depend on the density profile, number of disks, and shape of the boundary. Consequently, it is also found that the color suppression factor of the string percolation model is modified by a damping function related to the small-bounded effects. Corrections to the temperature and the speed of sound defined in string systems are also shown for small and elliptically bounded systems.

Percolation of particles on recursive lattices using a nanoscale approach. II. The effect of size and shape disparities

The preparation of many composites requires the intermixing of several macromolecular fluids along with the addition of rigid filler particles. These fillers are usually polydisperse and there is extensive experimental evidence that their size and shape profoundly affect the properties of the resulting material. In particular, it is generally found that the percolation threshold decreases as the size disparity between the different particles present in a system increases, and that the threshold decreases with increasing aspect ratio of the particles. Here, a recursive approach that we have recently introduced is applied to the study of the percolation of particles of different sizes and shapes, without the presence of a polymer matrix, on a lattice in various phases including metastable states. 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 are easily taken into account. In the previous paper of this series, we introduced the recursive approach and showed how correlations among particles of the same size can affect percolation. Before considering the complete system made of particles of various sizes and shapes embedded in a polymer matrix, in the third paper of the series, we describe here the properties of systems made of particles without any matrix. The approach appears to be extremely successful since it is able to capture most of the important features observed in experiments.

Monte Carlo results for continuum percolation in low and high dimensions

We report on Monte Carlo simulations of continuum percolation thresholds, by implementing highly efficient algorithms for very large samples. Our work, which includes percolation of hyperspheres, hypercubes, and boxes, in various dimensions, sizes, and shapes, has confirmed the expected dependence of the threshold on Vex, the total excluded volume, and on Bc, the average number of bonds per site. We have further confirmed that Vex=Bc, and that Bc is dependent on the objects shape, for which we offer a possible explanation. In particular we find that, counterintuitively, one can have Bc<1, as we have found for hyperspheres of dimension > or =12. From our many results for differently sized hyperspheres, we were also able to derive the correlation length exponent v solely from the behavior of the thresholds using finite-size scaling.