Diffusion-limited reaction rates on self-affine fractals

Abstract

The diffusion-limited reaction rate is determined on an approximately self-affine corrugated (random) surface fractal. We obtain the exact result for the low roughness and the asymptotic results (in three time regions) for the arbitrary and large roughness surfaces. These results show the anomalous time dependence for the mean flux and the mean excess flux for the large and small roughness surfaces, respectively. The intermediate time behavior of the reaction flux for the small roughness interface has the form 〈J〉∼ t^-1/2 + const t^-3/2+H, but for the large roughness interfaces it has same form as predicted earlier, 〈J〉 ∼ t^-1+H/2, where H is Hurst's exponent. This nonuniversality and dependence of intermediate time behavior on the strength of fractality of the interface is not conceived by earlier works. We also show the localization of the active zones in the presence of roughness. Finally, these results unravel the connection between the total reaction flux and the crossover times to the roughness characteristics like fractal dimension, lower and upper fractal cutoff lengths, and the amplitude of the fluctuations (strength) of the fractal.