Diffusion in Jammed Particle Packs

Abstract

Using random walk simulations we explore diffusive transport through monodisperse sphere packings over a range of packing fractions, φ, in the vicinity of the jamming transition at φ c .Various diffusion properties are computed over several orders of magnitude in both time and packing pressure. Two well-separated regimes of normal, "Fickian" diffusion, where the mean squared displacement is linear in time, are observed. The first corresponds to diffusion inside individual spheres, while the latter is the long-time bulk diffusion. The intermediate anomalous diffusion regime and the long-time value of the diffusion coefficient are both shown to be controlled by particle contacts, which in turn depend on proximity to φ c . The time required to recover normal diffusion t * scales as (φ − φ c ) −0.5 and the long-time diffusivityIt is shown that the distribution of mean first passage times associated with the escape of random walkers between neighboring particles controls both t * and D ∞ in the limit φ → φ c .

Geometric universality and anomalous diffusion in frictional fingers

“…By contrast, in complex geometries or under flow, the diffusion exponent α may depart from unity, being subdiffusive 0<α<1 or superdiffusive 1<α<2. This type of anomalous transport is typical in complex systems [11][12][13][14][15][16][17][18]. The most famous example of subdiffusion is perhaps de Gennes 'la fourmi dans le labyrinthe' (the ant in the labyrinth), referring to a random walker on a 2D critical percolation cluster [9].…”

Random walks on jammed networks: Spectral properties

“…We consider conduction through isotropically compressed, monodisperse, frictionless jammed spheres created via Discrete Element Method (DEM) simulations [4,5] with an established (de)compression protocol [6]. Multiple disordered packings of jammed mono-sized spheres were generated for a range of packing fractions φ above the jamming transition [7]. The contact networks from packings of N = 10 3 , 10 4 , and 10 5 particles form the objects of our study.…”

“…In this sense the particles are viewed as macroscopic objects composed on internal degrees of freedom which account for the conductive processes. An off-lattice random walker approach [8] can then be used to model the internal degrees of freedom and simulate conduction within and between contacting particles as was done in previous work [7]. Here, however, we take a different approach by effectively coarsegraining the off-lattice random walk within particles to a continuous-time Markov process on the contact network of the particle pack leading to a random walk-type model that is nonetheless analogous to solid-state lattice approaches [9].…”

“…Here, however, we take a different approach by effectively coarsegraining the off-lattice random walk within particles to a continuous-time Markov process on the contact network of the particle pack leading to a random walk-type model that is nonetheless analogous to solid-state lattice approaches [9]. To accomplish this, two simplifying assumptions made in [7] are noted: (i) walkers are constrained to remain within the particles meaning, e.g., no heat flux from particle surfaces due to radiation; and (ii) particle contacts are ideal and provide no barrier for walkers to pass through, e.g. no contact resistance.…”

“…Note, the distribution of z i is one of the topological features that distinguish the networks considered here from classical homogeneous lattice or regular graph random walks where z i is constant. Previous work [7], for packs near the jamming point, found that the bulk, effective conductivity, equivalent to the long-time diffusivity of random walkers constrained to walk within the particles, scales as the inverse of a characteristic time, D ∞ ∼ 1/t * . Accordingly, t * was identified with the median value, t m ∼ t * , of the distribution over all particles of their total volumeaveraged MFPT, t i = (Σ zi j=1 1/τ ij ) −1 , see Fig.…”