Active surface and adaptability of fractal membranes and electrodes

Abstract

We study the properties of a Laplacian potential around an irregular object of finite surface resistance. This can describe the electrical potential in an irregular electrochemical cell as well as the concentration in a problem of diffusion towards an irregular membrane of finite permeability. We show that using a simple fractal generator one can approximately predict the localization of the active zones of a deterministic fractal electrode of zero resistance. When the surface resistance r s is finite there exists a crossover length L c : In pores of sizes smaller than L c the current is homogeneously distributed. In pores of sizes larger than L c the same behavior as in the case r s =0 is observed, namely the current concentrates at the entrance of the pore. From this consideration one can predict the active surface localization in the case of finite r s . We then introduce a coarse-graining procedure which maps the problem of non-null r s into that of r s =0. This permits us to obtain the dependence of the admittance and of the active surface on r s . Finally, we show that the fractal geometry can be the most efficient for a membrane or electrode that has to work under very variable conditions