J. Phys. I France
Volume 3, Numéro 8, August 1993
Page(s) 1801 - 1818
DOI: 10.1051/jp1:1993216
J. Phys. I France 3 (1993) 1801-1818

Active surface and adaptability of fractal membranes and electrodes

Ricardo 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 $r_{\rm s}$ is finite there exists a crossover length $L_{\rm c}$ : In pores of sizes smaller than $L_{\rm c}$. the current is homogeneously distributed. In pores of sizes larger than $L_{\rm c}$, the same behavior as in the case $r_{\rm s}$ = 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 $r_{\rm s}$. We then introduce a coarse-graining procedure which maps the problem of non-null $r_{\rm s}$ into that of $r_{\rm s}$ = 0. This permits us to obtain the dependence of the admittance and of the active surface on $r_{\rm s}$. 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.

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