J. Quintanilla

Efficient measurement of the percolation threshold for fully penetrable discs

We study the percolation threshold for fully penetrable discs by measuring the average location of the frontier for a statistically inhomogeneous distribution of fully penetrable discs. We use two different algorithms to efficiently simulate the frontier, including the continuum analogue of an algorithm previously used for gradient percolation on a square lattice. We find that φc = 0.676 339 ± 0.000 004, thus providing an extra significant digit of accuracy to this constant. (Some figures in this article appear in colour in the electronic version; see www.iop.org)

Local volume fraction fluctuations in random media

Although the volume fraction is a constant for a statistically homogeneous random medium, on a spatially local level it fluctuates. We study the full distribution of volume fraction within an observation window of finite size for models of random media. A formula due to Lu and Torquato for the standard deviation or “coarseness’’ associated with the local volume fraction ξ is extended for the nth moment of ξ for any n. The distribution function FL of the local volume fraction of five different model microstructures is evaluated using analytical and computer-simulation methods for a wide range of window sizes and overall volume fractions. On the line, we examine a system of fully penetrable rods and a system of totally impenetrable rods formed by random sequential addition (RSA). In the plane, we study RSA totally impenetrable disks and fully penetrable aligned squares. In three dimensions, we study fully penetrable aligned cubes. In the case of fully penetrable rods, we will also simplify and numerically i...

Crack Formation and Propagation in Molecular Dynamics Simulations of Polymer Liquid Crystals

In recent papers we have used statistical mechanics to predict multiple phase formation in polymer liquid crystals (PLCs), (1,2) Now we have performed molecular dynamics simulations of PLC copolymers as materials consisting of LC islands in flexible matrices. A method for creating such materials on a computer is described. The overall concentration of the LC units, island size, and spatial distribution of the islands (random, in rows, and evenly distributed throughout the material) were varied. Crack formation and propagation as a function of these parameters were investigated. The local concentration of LC units in each chain has been defined. We found that the probability of a crack initiation site goew symbiotically with the local LC concentration. The first small crack is sometimes a part of the path through which the material breaks, however, although several small cracks may evolve at first, some of these never evolve into larger cracks since crack arrest occurs. The results can be used for creation of real materials with improved mechanical properties.

CLUSTERING IN A CONTINUUM PERCOLATION MODEL

We study properties of the clusters of a system of fully penetrable balls, a model formed by centering equal-sized balls on the points of a Poisson process. We develop a formal expression for the density of connected clusters of k balls (called k-mers) in the system, rst rigorously derived by Penrose 15]. Our integral expressions are free of inherent redundancies, making them more tractable for numerical evaluation. We also derive and evaluate an integral expression for the average volume of k-mers.

New bounds on the elastic moduli of suspensions of spheres

We derive rigorous three‐point upper and lower bounds on the effective bulk and shear moduli of a two‐phase material composed of equisized spheres randomly distributed throughout a matrix. Our approach is analogous to previously derived three‐point cluster bounds on the effective conductivity of suspensions of spheres. Our bounds on the effective elastic moduli are then compared to other known three‐point bounds for statistically homogeneous and isotropic random materials. For the case of totally impenetrable spheres, the bulk modulus bounds are shown to be equivalent to the Beran–Molyneux bounds, and the shear modulus bounds are compared to the McCoy and Milton–Phan‐Thien bounds. For the case of fully penetrable spheres, our bounds are shown to be simple analytical expressions, in contrast to the numerical quadratures required to evaluate the other three‐point bounds.

Gaussian random field models of aerogels

A model capable of predicting pore characteristics and rendering representative images of porous materials is described. A long-term goal is to discriminate between open and closed porosities. Aerogels are modeled by intersecting excursion sets of two independent Gaussian random fields. The parameters of these fields are obtained by matching small-angle neutron scattering data with the scattering function for the model. The chord-length probability density functions are then computed for the model, which contain partial clustering information for the aerogels. Visualizations of this model are performed and compared to electron microscopy images and gas adsorption pore size distributions.

Percolation for a model of statistically inhomogeneous random media

We study clustering and percolation phenomena for a model of statistically inhomogeneous two-phase random media, including functionally graded materials. This model consists of inhomogeneous fully penetrable (Poisson distributed) disks and can be constructed for any specified variation of volume fraction. We quantify the transition zone in the model, defined by the frontier of the cluster of disks which are connected to the disk-covered portion of the model, by defining the coastline function and correlation functions for the coastline. We find that the behavior of these functions becomes largely independent of the specific choice of grade in volume fraction as the separation of length scales becomes large. We also show that the correlation function behaves in a manner similar to that of fractal Brownian motion. Finally, we study fractal characteristics of the frontier itself and compare to similar properties for two-dimensional percolation on a lattice. In particular, we show that the average location of...

Versatility and robustness of Gaussian random fields for modelling random media

One of the authors (JAQ) has recently introduced a method of modelling random materials using excursion sets of Gaussian random fields. This method uses convex quadratic programming to find the optimal admissible field autocorrelation function, providing both theoretical and computational advantages over other techniques such as simulated annealing. In this paper, we discuss the application of this algorithm to model various aerogel systems given small-angle neutron scattering data. We also present new results concerning the robustness of this method.

Local volume fraction fluctuations in periodic heterogeneous media

Although the volume fraction is a constant for statistically homogeneous media, on a spatially local level it fluctuates and depends on the observation window size. In this article, we develop exact analytical expressions for the full local-volume fraction distribution function of periodic arrangements of rods, rectangles, and cubes in a matrix. These formulas depend on the inclusion density and window size.

Deriving the Regression Line with Algebra.

Exploration with spreadsheets and reliance on previous skills can lead students to determine the line of best fit.