Void percolation and conduction of overlapping ellipsoids

Percolation through voids around toroidal inclusions

In the case of media comprised of impermeable particles, fluid flows through voids around impenetrable grains. For sufficiently low concentrations of the latter, spaces around grains join to allow transport on macroscopic scales, whereas greater impenetrable inclusion densities disrupt void networks and block macroscopic fluid flow. A critical grain concentration ρ_{c} marks the percolation transition or phase boundary separating these two regimes. With a dynamical infiltration technique in which virtual tracer particles explore void spaces, we calculate critical grain concentrations for randomly placed interpenetrating impermeable toroidal inclusions; the latter consist of surfaces of revolution with circular and square cross sections. In this manner, we study continuum percolation transitions involving nonconvex grains. As the radius of revolution increases relative to the length scale of the torus cross section, the tori develop a central hole, a topological transition accompanied by a cusp in the critical porosity fraction for percolation. With a further increase in the radius of revolution, as constituent grains become more ringlike in appearance, we find that the critical porosity fraction converges to that of high-aspect-ratio cylindrical counterparts only for randomly oriented grains.

Percolation phase transition by removal of k^{2}-mers from fully occupied lattices

Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse site percolation by the removal of k×k square tiles (k^{2}-mers) from square lattices. The process starts with an initial configuration, where all lattice sites are occupied and, obviously, the opposite sides of the lattice are connected by occupied sites. Then the system is diluted by removing k^{2}-mers of occupied sites from the lattice following a random sequential adsorption mechanism. The process finishes when the jamming state is reached and no more objects can be removed due to the absence of occupied sites clusters of appropriate size and shape. The central idea of this paper is based on finding the maximum concentration of occupied sites, p_{c,k}, for which the connectivity disappears. This particular value of the concentration is called the inverse percolation threshold and determines a well-defined geometrical phase transition in the system. The results obtained for p_{c,k} show that the inverse percolation threshold is a decreasing function of k in the range 1≤k≤4. For k≥5, all jammed configurations are percolating states, and consequently, there is no nonpercolating phase. In other words, the lattice remains connected even when the highest allowed concentration of removed sites is reached. The jamming exponent ν_{j} was measured, being ν_{j}=1 regardless of the size k considered. In addition, the accurate determination of the critical exponents ν, β, and γ reveals that the percolation phase transition involved in the system, which occurs for k varying between one and four, has the same universality class as the standard percolation problem.

Void distributions reveal structural link between jammed packings and protein cores

Dense packing of hydrophobic residues in the cores of globular proteins determines their stability. Recently, we have shown that protein cores possess packing fraction ϕ≈0.56, which is the same as dense, random packing of amino-acid-shaped particles. In this article, we compare the structural properties of protein cores and jammed packings of amino-acid-shaped particles in much greater depth by measuring their local and connected void regions. We find that the distributions of surface Voronoi cell volumes and local porosities obey similar statistics in both systems. We also measure the probability that accessible, connected void regions percolate as a function of the size of a spherical probe particle and show that both systems possess the same critical probe size. We measure the critical exponent τ that characterizes the size distribution of connected void clusters at the onset of percolation. We find that the cluster size statistics are similar for void percolation in packings of amino-acid-shaped particles and randomly placed spheres, but different from that for void percolation in jammed sphere packings. We propose that the connected void regions are a defining structural feature of proteins and can be used to differentiate experimentally observed proteins from decoy structures that are generated using computational protein design software. This work emphasizes that jammed packings of amino-acid-shaped particles can serve as structural and mechanical analogs of protein cores, and could therefore be useful in modeling the response of protein cores to cavity-expanding and -reducing mutations.

Percolation through Voids around Randomly Oriented Polyhedra and Axially Symmetric Grains

Porous materials made up of impermeable grains constrain fluid flow to voids around the impenetrable inclusions. A percolation transition marks the boundary between densities of grains permitting bulk transport and concentrations blocking traversal on macroscopic scales. With dynamical infiltration of void spaces using virtual tracer particles, we treat inclusion geometries exactly. We calculate the critical number density per volume ρ_{c} for a variety of axially symmetric shapes and faceted solids with the former including cylinders, ellipsoids, cones, and tablet shaped grains from highly oblate (platelike) to highly prolate (needlelike) extremes. For the latter, results suggest a common asymptotic value identical to the counterpart for aligned cylindrical grains. We find percolation thresholds for each of the five platonic solids (i.e., tetrahedra, cubes, octahedra, dodecahedra, and icosahedra) as well as truncated icosahedra. For each polyhedron type, we consider aligned and randomly oriented grains, finding distinct percolation thresholds for the former versus the latter only for cubes. The anomalous diffusion exponents we find differ from those of the universality class for discrete models on 3D lattices.

Modeling Oil Recovery for Mixed Macro- and Micro-Pore Carbonate Grainstones

In general, modeling oil-recovery is a challenging problem involving detailed fluid flow calculations with required structural details that challenge current experimental resolution. Recent laboratory experiments on mixed micro- and macro-pore suggest that there is a systematic relationship between remaining oil saturation (ROS) and the fraction of micro-pores. Working with experimental measurements of the pores obtained from X-ray tomography and mercury intrusion capillary pressure porosimetry, we define a digital rock model exemplifying the key structural elements of these carbonate grainstones. We then test two fluid-flow models: invasion percolation model and effective medium model. Although invasion percolation identifies the important impact of macro-pore percolation on permeability, it does not describe the dependence of ROS on micro-pore percentage. We thus modified the effective medium model by introducing a single-parameter descriptor, r_eff. Oil from pores r ≥ r_eff is fully removed, while for the remaining pores with r < r_eff, their contribution is scaled by (r/r_eff)². Applying this straightforward physics to pore size distributions for the mixed-pore grainstones reproduces the experimental ROS dependence.

Percolation through voids around overlapping spheres: a dynamically based finite-size scaling analysis

The percolation threshold for flow or conduction through voids surrounding randomly placed spheres is calculated. With large-scale Monte Carlo simulations, we give a rigorous continuum treatment to the geometry of the impenetrable spheres and the spaces between them. To properly exploit finite-size scaling, we examine multiple systems of differing sizes, with suitable averaging over disorder, and extrapolate to the thermodynamic limit. An order parameter based on the statistical sampling of stochastically driven dynamical excursions and amenable to finite-size scaling analysis is defined, calculated for various system sizes, and used to determine the critical volume fraction ϕc=0.0317±0.0004 and the correlation length exponent ν=0.92±0.05.

Diffusion amid random overlapping obstacles: similarities, invariants, approximations

Efficient and accurate numerical techniques are used to examine similarities of effective diffusion in a void between random overlapping obstacles: essential invariance of effective diffusion coefficients (D(eff)) with respect to obstacle shapes and applicability of a two-parameter power law over nearly entire range of excluded volume fractions (φ), except for a small vicinity of a percolation threshold. It is shown that while neither of the properties is exact, deviations from them are remarkably small. This allows for quick estimation of void percolation thresholds and approximate reconstruction of D(eff) (φ) for obstacles of any given shape. In 3D, the similarities of effective diffusion yield a simple multiplication "rule" that provides a fast means of estimating D(eff) for a mixture of overlapping obstacles of different shapes with comparable sizes.

Diffusion in cytoplasm: effects of excluded volume due to internal membranes and cytoskeletal structures

The intricate geometry of cytoskeletal networks and internal membranes causes the space available for diffusion in cytoplasm to be convoluted, thereby affecting macromolecule diffusivity. We present a first systematic computational study of this effect by approximating intracellular structures as mixtures of random overlapping obstacles of various shapes. Effective diffusion coefficients are computed using a fast homogenization technique. It is found that a simple two-parameter power law provides a remarkably accurate description of effective diffusion over the entire range of volume fractions and for any given composition of structures. This universality allows for fast computation of diffusion coefficients, once the obstacle shapes and volume fractions are specified. We demonstrate that the excluded volume effect alone can account for a four-to-sixfold reduction in diffusive transport in cells, relative to diffusion in vitro. The study lays the foundation for an accurate coarse-grain formulation that would account for cytoplasm heterogeneity on a micron scale and binding of tracers to intracellular structures.

Geometric percolation thresholds of interpenetrating plates in three-dimensional space

The geometric percolation thresholds for circular, elliptical, square, and triangular plates in the three-dimensional space are determined precisely by Monte Carlo simulations. These geometries represent oblate particles in the limit of zero thickness. The normalized percolation points, which are estimated by extrapolating the data to zero radius, are etac=0.961 4+/-0.000 5, 0.864 7+/-0.000 6, and 0.729 5+/-0.000 6 for circles, squares, and equilateral triangles, respectively. These results show that the noncircular shapes and corner angles in the plate geometry tend to increase the interparticle connectivity and therefore reduce the percolation point. For elliptical plates, the percolation threshold is found to decrease moderately, when the aspect ratio epsilon is between 1 and 1.5, but decrease rapidly for epsilon greater than 1.5. For the binary dispersion of circular plates with two different radii, etac is consistently larger than that of equisized plates, with the maximum value located at around r1/r2=0.5.

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).