Transfer across Random versus Deterministic Fractal Interfaces

A numerical study of the transfer across random fractal surfaces shows that their responses are very close to the response of deterministic model geometries with the same fractal dimension. The simulations of several interfaces with prefractal geometries show that, within very good approximation, the flux depends only on a few characteristic features of the interface geometry: the lower and higher cut-offs and the fractal dimension. Although the active zones are different for different geometries, the electrode reponses are very nearly the same. In that sense, the fractal dimension is the essential "universal" exponent which determines the net transfer.Comment: 4 pages, 6 figure

Chemical fracture and distribution of extreme values

When a corrosive solution reaches the limits of a solid sample, a chemical fracture occurs. An analytical theory for the probability of this chemical fracture is proposed and confirmed by extensive numerical experiments on a two dimensional model. This theory follows from the general probability theory of extreme events given by Gumbel. The analytic law differs from the Weibull law commonly used to describe mechanical failures for brittle materials. However a three parameters fit with the Weibull law gives good results, confirming the empirical value of this kind of analysis.Comment: 7 pages, 5 figures, to appear in Europhysics Letter

Explicit Construction of the Brownian Self-Transport Operator

Applying the technique of characteristic functions developped for one-dimensional regular surfaces (curves) with compact support, we obtain the distribution of hitting probabilities for a wide class of finite membranes on square lattice. Then we generalize it to multi-dimensional finite membranes on hypercubic lattice. Basing on these distributions, we explicitly construct the Brownian self-transport operator which governs the Laplacian transfer. In order to verify the accuracy of the distribution of hitting probabilities, numerical analysis is carried out for some particular membranes.Comment: 30 pages, 9 figures, 1 tabl

Etching of random solids: hardening dynamics and self-organized fractality

When a finite volume of an etching solution comes in contact with a disordered solid, a complex dynamics of the solid-solution interface develops. Since only the weak parts are corroded, the solid surface hardens progressively. If the etchant is consumed in the chemical reaction, the corrosion dynamics slows down and stops spontaneously leaving a fractal solid surface, which reveals the latent percolation criticality hidden in any random system. Here we introduce and study, both analytically and numerically, a simple model for this phenomenon. In this way we obtain a detailed description of the process in terms of percolation theory. In particular we explain the mechanism of hardening of the surface and connect it to Gradient Percolation.Comment: Latex, aipproc, 6 pages, 3 figures, Proceedings of 6th Granada Seminar on Computational Physic

Transition from Knudsen to molecular diffusion in activity of absorbing irregular interfaces

We investigate through molecular dynamics the transition from Knudsen to molecular diffusion transport towards 2d absorbing interfaces with irregular geometry. Our results indicate that the length of the active zone decreases continuously with density from the Knudsen to the molecular diffusion regime. In the limit where molecular diffusion dominates, we find that this length approaches a constant value of the order of the system size, in agreement with theoretical predictions for Laplacian transport in irregular geometries. Finally, we show that all these features can be qualitatively described in terms of a simple random-walk model of the diffusion process.Comment: 4 pages, 4 figure

Percolation-dependent Reaction Rates in the Etching of Disordered Solids

A prototype statistical model for the etching of a random solid is investigated in order to assess the influence of disorder and temperature on the dissolution kinetics. At low temperature, the kinetics is dominated by percolation phenomena, and the percolation threshold determines the global reaction time. At high temperature, the fluctuations of the reaction rate are Gaussian, whereas at low temperature they exhibit a power law tail due to chemical avalanches. This is an example where microscopic disorder directly induces non-classical chemical kinetics.Comment: Revtex, 4 pages, 5 figure

Surface Hardening and Self-Organized Fractality Through Etching of Random Solids

When a finite volume of etching solution is in contact with a disordered solid, complex dynamics of the solid-solution interface develop. If the etchant is consumed in the chemical reaction, the dynamics stop spontaneously on a self-similar fractal surface. As only the weakest sites are corroded, the solid surface gets progressively harder and harder. At the same time it becomes rougher and rougher uncovering the critical spatial correlations typical of percolation. From this, the chemical process reveals the latent percolation criticality hidden in any random system. Recently, a simple minimal model has been introduced by Sapoval et al. to describe this phenomenon. Through analytic and numerical study, we obtain a detailed description of the process. The time evolution of the solution corroding power and of the distribution of resistance of surface sites is studied in detail. This study explains the progressive hardening of the solid surface. Finally, this dynamical model appears to belong to the universality class of Gra dient Percolation.Comment: 14 pages, 15 figures (1457 Kb