LAPLACIAN TRANSPORT TOWARDS PARTIALLY PASSIVATED 2D IRREGULAR INTERFACES: A CONJECTURAL EXTENSION OF THE MAKAROV THEOREM

Abstract

In several phenomena of practical interest, such as catalyst deactivation, fouling in heat transfer and other systems of technological and scientific relevance, an irregular surface accessed by diffusion can be progressively passivated. In a diffusion limited situation, an interface that works unevenly due to Laplacian screening is simultaneously and unevenly passivated. To study this phenomenon, we describe a process in which the regions of the surface that are initially working, are transformed into passive, reflecting zones. As a consequence, at each step, a new part of the interface becomes active. In turn, this new active zone is passivated, and so on. It is found that the length of the successive active zones remains approximately constant for a prefractal interface. The concept of active zone in Laplacian transport can then be successfully extended to elucidate this self-limiting behavior of the passivation process. A conjecture is then proposed which states that, in D=2, the information dimension of the harmonic measure on a fractal supporting a "passivated or reflecting subfractal" (of smaller dimension) is equal to 1. This constitutes an extension of Makarov theorem. From our results, fractal geometry is revealed as a potential candidate to engineer substrate morphologies that are robust to Laplacian passivation.