Crossover from anomalous to normal diffusion in porous media

Adsorption of Diffusing Tracers, Apparent Tortuosity, and Application to Mesoporous Silica

The apparent tortuosity due to adsorption of diffusing tracers in a porous material is determined by a scaling approach and is used to analyze recent data on LiCl and alkane diffusion in mesoporous silica. The slope of the adsorption isotherm at small loadings is written as β = qA/qG, where qA is the adsorption-desorption ratio and qG = ϵ/(as) - 1 is a geometrical factor depending on the range a of the tracer-wall interaction, the porosity ϵ, and the specific surface area s. The adsorption leads to a decrease of effective diffusion coefficient, which is quantified by multiplying the geometrical tortuosity factor τgeom by an apparent tortuosity factor τapp. In wide pores or when the adsorption barrier is high, τapp = β + 1, as obtained in previous works, but in narrow pores there is an additional contribution from frequent adsorption-desorption transitions. These results are obtained in media with parallel pores of constant cross sections, where the ratio between the effective pore width ϵ/s and the actual width is ≈0.25. Applications to mesoporous silica samples are justified by the small deviations from this ideal ratio. In the analysis of alkane self-diffusion data, the fractions of adsorbed molecules predicted in a recent theoretical work are used to estimate τgeom of the silica samples, which is ≫1 only in the sample with the narrowest pores (nominal 3 nm). The application of the model to Li+ ion diffusion leads to similar values of τgeom and to a difference of energy barriers of desorption and adsorption for those ions of ∼0.06 eV. Comparatively, alkane self-diffusion provides the correct order of magnitude of τgeom, with adsorption playing a less important role, whereas adsorption effects on Li+ diffusion are much more important.

Characterizing network topology using first-passage analysis

Understanding the topological characteristics of complex networks and how they affect navigability is one of the most important goals in science today, as it plays a central role in various economic, biological, ecological, and social systems. Here we apply first-passage analysis tools to investigate the properties and characteristics of random walkers in networks with different topology. Starting with the simplest two-dimensional square lattice, we modify its topology incrementally by randomly reconnecting links between sites. We characterize these networks by first-passage time from a significant number of random walkers without interaction, varying the departure and arrival locations. We also apply the concept of first-passage simultaneity, which measures the likelihood of two walkers reaching their destination together. These measures, together with the site occupancy statistics during the processes, allowed us to differentiate the studied networks, especially the random networks from the scale-free networks, by their navigability. We also show that small-world features can also be highlighted with the proposed technique.

Interplay of adsorption and surface mobility in tracer diffusion in porous media

We model the diffusion of a tracer that interacts with the internal surface of a porous medium formed by a packing of solid spheres. The tracer executes a lattice random walk in which hops from surface to bulk sites and hops on the surface have small probabilities compared to hops from bulk sites; those probabilities are related to bulk and surface diffusion coefficients and to a desorption rate. A scaling approach distinguishes three regimes of steady state diffusion, which are confirmed by numerical simulations. If the product of desorption rate and sphere diameter is large, dominant bulk residence is observed and the diffusion coefficient is close to the bulk value. If that product is small and the surface mobility is low, the tracers are adsorbed most of the time but most hops are executed in the bulk. However, for high surface mobility, there is a nontrivial regime of dominant surface displacement, since the connectivity of solid walls allows the tracers to migrate to long distances while they are adsorbed. In this regime, we observe rounded tracer paths on the sphere walls, which are qualitatively similar to those of a recent experiment on polystyrene particle diffusion. The calculated average residence times are proportional to the bulk and surface densities of an equilibrium ensemble of noninteracting tracers, and the relation between those densities sets the adsorption isotherm. Simulations performed with initially uniform (nonequilibrium) distribution of tracers in the pores show other nontrivial results in cases of dominant surface residence: slow increase of the mean-square displacement at short times, since the tracer has not explored a homogeneous medium, and a remarkable slowdown between the first encounter with a solid wall and the first hop from that point. Relations between our results and other models of diffusion and adsorption in porous media are discussed.

Ubiquity of anomalous transport in porous media: Numerical evidence, continuous time random walk modelling, and hydrodynamic interpretation

Anomalous transport in porous media is commonly believed to be induced by the highly complex pore space geometry. However, this phenomenon is also observed in porous media with rather simple pore structure. In order to answer how ubiquitous can anomalous transport be in porous media, we in this work systematically investigate the solute transport process in a simple porous medium model with minimal structural randomness. The porosities we consider range widely from 0.30 up to 0.85, and we find by lattice Boltzmann simulations that the solute transport process can be anomalous in all cases at high Péclet numbers. We use the continuous time random walk theory to quantitatively explain the observed scaling relations of the process. A plausible hydrodynamic origin of anomalous transport in simple porous media is proposed as a complement to its widely accepted geometric origin in complex porous media. Our results, together with previous findings, provide evidence that anomalous transport is indeed ubiquitous in porous media. Consequently, attentions should be paid when modelling solute transport by the classical advection-diffusion equation, which could lead to systematic error.

Scaling relations in the diffusive infiltration in fractals

In a recent work on fluid infiltration in a Hele-Shaw cell with the pore-block geometry of Sierpinski carpets (SCs), the area filled by the invading fluid was shown to scale as F∼t^{n}, with n<1/2, thus providing a macroscopic realization of anomalous diffusion [Filipovitch et al., Water Resour. Res. 52, 5167 (2016)WRERAQ0043-139710.1002/2016WR018667]. The results agree with simulations of a diffusion equation with constant pressure at one of the borders of those fractals, but the exponent n is very different from the anomalous exponent ν=1/D_{W} of single-particle diffusion in the same fractals (D_{W} is the random-walk dimension). Here we use a scaling approach to show that those exponents are related as n=ν(D_{F}-D_{B}), where D_{F} and D_{B} are the fractal dimensions of the bulk and the border from which diffusing particles come, respectively. This relation is supported by accurate numerical estimates in two SCs and in two generalized Menger sponges (MSs), in which we performed simulations of single-particle random walks (RWs) with a rigid impermeable border and of a diffusive infiltration model in which that border is permanently filled with diffusing particles. This study includes one MS whose external border is also fractal. The exponent relation is also consistent with the recent simulational and experimental results on fluid infiltration in SCs, and explains the approximate quadratic dependence of n on D_{F} in these fractals. We also show that the mean-square displacement of single-particle RWs has log-periodic oscillations, whose periods are similar for fractals with the same scaling factor in the generator (even with different embedding dimensions), which is consistent with the discrete scale invariance scenario. The roughness of a diffusion front defined in the infiltration problem also shows this type of oscillation, which is enhanced in fractals with narrow channels between large lacunas.

Nanoparticle diffusion in crowded and confined media

We identify distinct mechanisms controlling slowing of nanoparticle diffusion through complex media featuring both rigid geometrical confinement and soft mobile crowders. Towards this end, we use confocal microscopy and single particle tracking to probe the diffusion of 400 nm nanoparticles suspended in Newtonian water, in a Newtonian glycerol/water mixture, or in a non-Newtonian polymer solution through a model porous medium, a packed bed of microscale glass beads. The mobility of nanoparticles, as quantified by the long-time diffusion coefficient extracted from the particle mean-squared displacement, slows as the average pore size of the packed bed media decreases for both Newtonian and non-Newtonian solutions. The distribution of particle displacements is non-Gaussian, consistent with the spatial heterogeneity of the geometrical confinement imposed by the packed bed. The slowing of nanoparticle mobility in all solutions follows the predictions of models that describe hydrodynamic interactions with the packed bed. In non-Newtonian solutions, depletion interactions due to the polymers near the glass beads result in temporary adsorption of particles onto the bead surface, as indicated by a stretched-exponential distribution of residence times. Our results therefore suggest that the confined diffusive dynamics of nanoparticles in polymer solutions is controlled by two competing mechanisms: hydrodynamic interactions between particles and spatial obstacles, which dictate the long-time slowing of diffusion, and depletion interactions between particles and confining walls due to the macromolecules, which control transient adsorption and hence alter the statistics of the short-time motion.