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

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 Effects in Mixed Matrix Membranes with Embedded Carbon Nanotubes

Polymeric membranes with embedded nanoparticles, e.g., nanotubes, show a significant increase in permeability of the target component while maintaining selectivity. However, the question of the reasons for this behavior of the composite membrane has not been unequivocally answered to date. In the present work, based on experimental data on the permeability of polymer membranes based on Poly(vinyl trimethylsilane) (PVTMS) with embedded CNTs, an approach to explain the abnormal behavior of such composite membranes is proposed. The presented model considered the mass transfer of gases and liquids through polymeric membranes with embedded CNTs as a parallel transport of gases through the polymeric matrix and a "percolation" cluster-bound regions around the embedded CNTs. The proposed algorithm for modeling parameters of a percolation cluster of embedded tubular particles takes into account an agglomeration and makes it possible to describe the threshold increase and subsequent decrease permeability with increasing concentration of embedded particles. The numerical simulation of such structures showed: an increase in the particle length leads to a decrease in the percolation concentration in a matrix of finite size, the power of the percolation cluster decreases significantly, but the combination of these effects leads to a decrease in the influence of the introduced particles on the properties of the matrix in the vicinity of the percolation threshold; an increase in the concentration of embedded particles leads to an increase in the probability of the formation of agglomerates and the characteristic size of the elements that make up the percolation cluster, the influence of individual particles decreases and the characteristics of the percolation transition determine the ratio of the sizes of agglomerates and matrix; and an increase in the lateral linear dimensions of the matrix leads to a nonlinear decrease in the proportion of the matrix, which is affected by the introduced particles, and the transport characteristics of such MMMs deteriorate.

Percolation analysis for estimating the maximum size of particles passing through nanosphere membranes

Percolation theory can be used to study the flow-related properties of various porous systems. In particular, recently developed membranes from silica nanoparticles with surface grafted polymer brushes represent a quintessential hard-sphere soft-shell system for which fluid-flow behavior can be illuminated via a percolation framework. However, a critical parameter in membrane design involves the maximum pass-through size of particles. While percolation theory considers path connectedness of a system, little explicit consideration is given to the size of the paths that traverse the space. This paper employs a hard-sphere soft-shell percolation model to investigate maximum particle pass-through size of membranes. A pixelated (as opposed to continuous) representation of the geometry is created, and combined with readily available homology software to analyze percolation behavior. The model is validated against previously published results. For a given sphere volume fraction, the maximum diameter of a percolating path is determined by applying iterative dilations to the spheres until the percolation threshold is reached. A simple approximate relationship between maximum particle size and sphere volume fraction is derived for application to membrane design. Experimental particle cutoff size results for the polymer modified silica nanoparticle membranes were used as a partial verification of the model created in this paper. The presence of a distribution of sphere sizes (naturally created by the manufacturing process) is found to have negligible effect, compared to results for a single sphere size.

Fluid flow through packings of elastic shells

Fluid transport in porous materials is commonly studied in geological samples (soil, sediments, etc.) or idealized systems, but the fluid flow through compacted granular materials, consisting of substantially strained granules, remains relatively unexplored. As a step toward filling this gap, we study a model of liquid transport in packings of deformable elastic shells using finite-element and lattice-Boltzmann methods. We find that the fluid flow abruptly vanishes as the porosity of the material falls below a critical value, and the flow obstruction exhibits features of a percolation transition. We further show that the fluid flow can be captured by a simplified permeability model in which the complex porous material is replaced by a collection of disordered capillaries, which are distributed and shaped by the percolation transition. To that end, we numerically explore the divergence of hydraulic tortuosity τ_{H} and the decrease of a hydraulic radius R_{h} as the percolation threshold is approached. We interpret our results in terms of scaling predictions derived from the percolation theory applied to random packings of spheres.

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.