Lineal measures of clustering in overlapping particle systems

Chord-length distributions cannot generally be obtained from small-angle scattering

The methods used to extract chord-length distributions from small-angle scattering data assume a structure consisting of spatially uncorrelated and disconnected convex regions. These restrictive conditions are seldom met for a wide variety of materials such as porous materials and semicrystalline or phase-separated copolymers, the structures of which consist of co-continuous phases that interpenetrate each other in a geometrically complex way. The significant errors that would result from applying existing methods to such systems are discussed using three distinct models for which the chord-length distributions are known analytically. The models are a dilute suspension of hollow spheres, the Poisson mosaic and the Boolean model of spheres.

Structure of a tractable stochastic mimic of soft particles

The structure and assembly of soft particles is difficult to characterize because their interpenetrability allows them to be packed at ever higher density albeit with an increasing penalty in energy and/or pressure. Alternatively, the use of impenetrable particles (such as hard spheres) as a reference model for soft particles can fail because the packing densities are limited by the impossibility of complete space filling. We recently introduced the stochastic penetration algorithm (SPA) so as to allow for the computationally efficient integration of hard sphere models while including overlaps seen in soft interactions [Craven et al., J. Chem. Phys., 2013, 138, 244901]. Moving beyond the initial one-dimensional case studied earlier, we now consider the spatial properties of systems of stochastically penetrable spheres in dimensions d≤ 3 through the use of molecular dynamics simulations and analytic methods. The stochastic potential allows spheres to either interpenetrate with a probability δ or collide elastically otherwise. For δ > 0 the particles interpenetrate (overlap), reducing the effective volume occupied by the particles in the system. We find that the occupied volume can be accurately predicted using analytic expressions derived from mean field arguments for the particle overlap probabilities with the exception of an observed clustering regime. This anomalous clustering behavior occurs at high densities and small δ. We find that this regime is coincident with that observed in deterministic penetrable models. The behavior of the stochastic penetrable particles also indicates that soft particles would be characterizable through a single reduced parameter that captures their overlap probability.

Grain reconstruction of porous media: application to a Bentheim sandstone

The two-point correlation measured on a thin section can be used to derive the probability density of the radii of a population of penetrable spheres. The geometrical, transport, and deformation properties of samples derived by this method compare well with the properties of the digitized real sample and of the samples generated by the standard grain reconstruction method.

Grain reconstruction of porous media: application to a low-porosity Fontainebleau sandstone

The fundamental issue of reconstructing a porous medium is examined anew in this paper, thanks to a sample of low-porosity Fontainebleau sandstone that has been analyzed by computed microtomography. Various geometric properties are determined on the experimental sample. A statistical property, namely, the probability density of the covering radius, is determined. This is used in order to reconstruct a porous medium by means of a Poissonian generation of polydisperse spheres. In a second part, the properties of the real experimental sample and of the reconstructed one are compared. The most important success of the present reconstruction technique is the fact that the numerical sample percolates despite its low porosity. Moreover, other geometrical features and conductivity are found to be in good agreement.

Design of inhomogeneous materials with given structural properties

We describe a technique applicable to optimize certain quantities associated with the two-phase structure described by a model of penetrable grains. The solution is given in a form of an intensity surface that controls locations of the grains. Particular examples include maximization of the expected phase 2 volume and design of functionally graded materials with a given density profile.

Discrete approximation of the linear Boolean model of heterogeneous materials

The Boolean model is a random set process in which random shapes are positioned according to outcomes of a Poisson process. Two- and three-dimensional versions of the model characterize structures of certain heterogeneous materials. Linear transects of the Boolean model produce a one-dimensional Boolean model that summarizes some material properties. Two functions from linear transects, clump-length and lineal path distributions, provide information on material phase connectivity. Computation of these distributions is notoriously difficult. We provide a discrete approximation to the one-dimensional convex-grain Boolean model that yields stable, linear-time, recursive algorithms to approximate these functions. Computer simulations demonstrate accuracy and speed.

Chord-distribution functions of three-dimensional random media: approximate first-passage times of Gaussian processes

The main result of this paper is a semianalytic approximation for the chord-distribution functions of three-dimensional models of microstructure derived from Gaussian random fields. In the simplest case the chord functions are equivalent to a standard first-passage time problem, i.e., the probability density governing the time taken by a Gaussian random process to first exceed a threshold. We obtain an approximation based on the assumption that successive chords are independent. The result is a generalization of the independent interval approximation recently used to determine the exponent of persistence time decay in coarsening. The approximation is easily extended to more general models based on the intersection and union sets of models generated from the isosurfaces of random fields. The chord-distribution functions play an important role in the characterization of random composite and porous materials. Our results are compared with experimental data obtained from a three-dimensional image of a porous Fontainebleau sandstone and a two-dimensional image of a tungsten-silver composite alloy.