Faster diffusion across an irregular boundary

First-passage times to a fractal boundary: Local persistence exponent and its log-periodic oscillations

We investigate the statistics of the first-passage time (FPT) to a fractal self-similar boundary of the Koch snowflake. When the starting position is fixed near the absorbing boundary, the FPT distribution exhibits an apparent power-law decay over a broad range of timescales, culminated by an exponential cutoff. By extensive Monte Carlo simulations, we compute the local persistence exponent of the survival probability and reveal its log-periodic oscillations in time due to self-similarity of the boundary. The effect of the starting point on this behavior is analyzed in depth. Theoretical bounds on the survival probability are derived from the analysis of diffusion in a circular sector. Physical rationales for the refined structure of the survival probability are presented.

Longevity evaluation of non-pumping reactive wells for control of groundwater contamination: Application of upscaling methods

Non-pumping reactive wells (NPRWs) are subsurface structures used for the passive treatment of contaminated groundwater using wells containing reactive media. In the vicinity of NPRWs, a combination of hydrogeological and chemical processes makes it difficult to predict their longevity. In this study, we evaluated the longevity of NPRWs using the upscaling methods. A horizontal two-dimensional sandbox was constructed to mimic the hydrogeological and chemical processes in a single unit of NPRW (unit NPRW). The groundwater flow and solute transport were simulated numerically to validate the processes of contaminant spreading prevention in the sandbox. Dye tracing and arsenic transport tests showed different performance of NPRW due to induced flow and uneven consumption of reactivity, which is dependent on the pathway length and residence time of the coal waste. Through numerical modeling of the experiments, the fate-related processes of contamination around NPRW were described in detail in both spatial and temporal terms. The stepwise approach of the upscaling methods was used to predict the contamination-blocking performance of the entire facility based on the reactivity of the materials and the contamination removal of the unit NPRW.

Diffusion toward a nanoforest of absorbing pillars

Spiky coatings (also known as nanoforests or Fakir-like surfaces) have found many applications in chemical physics, material sciences, and biotechnology, such as superhydrophobic materials, filtration and sensing systems, and selective protein separation, to name but a few. In this paper, we provide a systematic study of steady-state diffusion toward a periodic array of absorbing cylindrical pillars protruding from a flat base. We approximate a periodic cell of this system by a circular tube containing a single pillar, derive an exact solution of the underlying Laplace equation, and deduce a simple yet exact representation for the total flux of particles onto the pillar. The dependence of this flux on the geometric parameters of the model is thoroughly analyzed. In particular, we investigate several asymptotic regimes, such as a thin pillar limit, a disk-like pillar, and an infinitely long pillar. Our study sheds light onto the trapping efficiency of spiky coatings and reveals the roles of pillar anisotropy and diffusional screening.

Steady-state flux of diffusing particles to a rough boundary formed by absorbing spikes periodically protruding from a reflecting base

We study steady-state flux of particles diffusing on a flat surface and trapped by absorbing spikes of arbitrary length periodically protruding from a reflecting base. It is assumed that the particle concentration, far from this comblike boundary, is kept constant. To find the flux, we use a boundary regularization approach that replaces the initial highly rough and heterogeneous boundary by an effective boundary which is smooth and uniform. After such a replacement, the two-dimensional diffusion problem becomes essentially one-dimensional, and the steady-state flux can be readily found. Our main results are simple analytical expressions determining the position of the smooth effective boundary and its uniform trapping rate as functions of the spike length and interspike distance. It is shown that the steady-state flux to the effective boundary is identical to its counterpart to the initial boundary at large distances from this boundary. Our analytical results are corroborated by Brownian dynamics simulations.

Trapping of diffusive particles by rough absorbing surfaces: boundary smoothing approach

We present analytical results for the first-passage statistics of Brownian particles near a comblike absorbing boundary. Our approach is based on the method of boundary homogenization (or boundary smoothing) when an equivalent flat boundary is introduced to maintain the same diffusion flux as the original rough boundary. By using the conformal invariance of the Laplace equation we derive an analytic expression for the position of an equivalent boundary in terms of its spatial period and amplitude. The main analytical results being initially obtained for the steady state system provide important insights into the statistical characteristics of diffusive transport near rough boundaries (high order moments of the trapping time statistics).