Percolation Problems and Corrosion of Disordered Solids

Chemical etching of disordered solids

This study has been inspired to some experimental results about the slow chemical etching of thin (almost 2d) aluminum films immerged in a finite volume of a corrosive solution. The experiments consisted in monitoring the evolution of the corrosion front. One observes that this evolution is very rapid at an early stage and then slow down up to stop in a static situation. In this state the chemical concentration of the etchant in the solution is significative and the final corrosion front is fractal up to a characteristic scale with fractal dimension D~1.33. Our theoretical study consisted in the mathematical and numerical analysis of a dynamical model for this chemical etching of thin films of a disordered solid by a finite volume of a corrosive solution [24][32]. The results of this model agree very well with both the dynamical evolution and the fractal geometrical properties of the final corrosion front observed in the experiments. Furthermore we have shown, through a random field theory approach to this dynamical model, that it belongs to the random percolation universality class and in particular to the one of gradient percolation [28][49]. Finally, we have used this model to study the statistics of the chemical fracture events of these systems. To this aim we have followed a combined approach of percolation theory and probability theory of extremal events finding a good theoretical prediction for the probability law of failure events [37].

Models for the formation of fractal coasts

The problem of the formation of rocky fractal coasts, with an observed fractal dimension D~1.3, is one of the important subjects concerning out of equilibrium fractal dynamics which is not yet been solved. Inspired by the works about the corrosion of disordered solids, we have formulated and studied a model for the formation and self-stabilization through erosion of fractal coasts whose fractal dimension is near to the observed one. This model takes into account both the disordered nature of the rocky coasts and the erosion process implied by the dissipation of a part of the see energy on the coast. In this model the see is modelled as a resonator dissipating energy both on the boundaries (i.e., the coast) and in the bulk (i.e., viscous dissipation). Moreoevr the slow chemical aging, mainly by saltation, of the coastal rocks is also taken into account. The results are very encoraging since this dynamical model reaches an attractive stationary state in which the coast is fractal with fractal dimension D~1.3. A theoretical explanation of this results is given through percolation theory and a dynamical version of gradient percolation.

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