Can one hear the shape of an electrode? II. Theoretical study of the Laplacian transfer

Abstract

The flux across resistive irregular interfaces driven by a force deriving from a Laplacian potential is computed on a rigorous basis. The theory permits one to relate the size of the active zone Aact. to the derivative of the spectroscopic impedance Zspect.(r) with respect to the surface resistivity r through:. It is shown that the macroscopic transfer properties through a system of arbitrary shape are determined by the characteristics of a first-passage interface-interface random walk operator. More precisely, it is the distribution of the harmonic measure (or normalized primary current) on the eigenmodes of this linear operator that controls the transfer. In addition, it is also shown that, whatever the dimension, the impedance of a weakly polarizable electrode for any irregular geometry scales under a homothety transformation as L d−1 , L being the size of the system and d its topological dimension. In this new formalism, the question addressed in the title is transformed in a open mathematical question: "Knowing the distribution of the harmonic measure on the eigenmodes of the self-transport operator, can one retrieve the shape of the interface?"

Laplacian eigenfunctions in NMR. I. A numerical tool

“…[10] brings the relaxation length D/K which is the distance a particle should travel near the boundary before surface relaxation effects reduce its expected magnetization. The relaxation length is also called ''unscreened perimeter length'' and it plays an important role in diffusive transport phenomena (10,11,(18)(19)(20)(21)(22)(23)(24). The third dimensionless variable…”

Diffusion-Reorganized Aggregates: Attractors in Diffusion Processes?

“…Halsey first indicated that the response of such systems depends on the probability that a particle starting on the interface comes back to it [5]. Generalizing these ideas, it has been recently shown that the Laplacian transfer across irregular interfaces is controlled by a single linear operatorQ which maps the static surface onto itself through effective "Brownian bridges" [6]. In this context, each surface has an operatorQ, which is symmetric and positive.…”

“…It is also interesting to return to the exact concept of the self-transport operator attached to a given morphology [6]. Extending the notion of diffusive self-transport in this context, it might be that DRA shapes could be considered as "eigenshapes" of these operators.…”

“…This led us to study a dynamical process in which the notion of aQ operator, which maps the surface onto itself through average diffusion, is transformed into a real operation which transforms the surface through discretized diffusion [7]. The evolution mechanism then proceeds in the following steps ( Fig.…”

Laplacian transfer across a rough interface: numerical resolution in the conformal plane

“…More recently, they showed that the frequential behavior of a rough electrode could be obtained by studying the spectral properties of the so-called self-transport operator [5,6]. The latter measures the probability for two given sites of an interface to be linked by random walk through the electrolyte.…”