β Scribed by B. Sapoval
No coin nor oath required. For personal study only.
We describe a new and simple way to compute the linear and non-linear response of sell similar electrodes. This method applies in principle to arbitrary irregular geometry and it permits to predict generally that the slope of the Tafel plot is divided by the fractal dimension.
We present here a new way to consider and compute the impedance of self-similar electrodes. In order to find the response of an irregular electrode one have to solve the Laplace equation AV = 0 which governs the electric field distribution in an electrochemical cell with one irregular "working" electrode. Such a cell is represented for instance in Fig. (1). Laplace equation governs the field in the bulk of the electrolyte but one must use the boundary condition that reflects the electrochemical processes at the surface electrode, This surface presents a surface impedance and consequently what is known about the properties of Laplacian fields on irregular surfaces which are totally absorbing cannot be applied directly. This situation is unfortunate.since the Laplacian problem with the boundary condition V=O (the situation of an irregular capacitor) has been thoroughly studied, at least in d=2, and a theorem of important signifance, namely the Makarov theorem which describes the properties of the charge distribution on a irregular (possibly fractal) capacitor has been demonstrated.(1) This theorem states that the information dimension of the harmonic measure (the distribution of electrostatic charge for the capacitor case) on a singly connected object in d = 2 Is exactly equal to 1. This very special property of the Laplacian fields can be imaged in the following manner. If one considers a capacitor with an irregular 511
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