J. Phys. I France 3 (1993) 1801-1818
Active surface and adaptability of fractal membranes and electrodesRicardo Gutfraind and Bernard Sapoval
Laboratoire de Physique de la Matière Condensée , C.N.R.S., Ecole Polytechnique, 91128 Palaiseau, France
(Received 4 March 1993, accepted in final form 5 April 1993)
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 is finite there exists a crossover length : In pores of sizes smaller than . the current is homogeneously distributed. In pores of sizes larger than , the same behavior as in the case = 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 . We then introduce a coarse-graining procedure which maps the problem of non-null into that of = 0. This permits us to obtain the dependence of the admittance and of the active surface on . 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.
© Les Editions de Physique 1993