On the intermediate asymptote of diffusion-limited reactions in a fractal porous catalyst

One of the main numerical results of studies of reaction and diffusion in p ore-fractal geometries is the existence of an intermediate low-slope asympt ote in the plot of log (rate) vs, log k, which separates the known asymptot es of kinetics- and diffusion-controlled. rates. Moreover, comparison of th e rates in a fractal catalyst with those in a uniform-pore object, showed t hat the former is superior in the k-insensitive domain. We derive here anal ytical solutions to the reaction and diffusion process in three pore-fracta l geometries: A simple pore-tree with a clear hierarchy exhibits an interme diate asymptote, when all pore-generations are diffusion limited; this asym ptote depends on geometric parameters only and its domain of existence is l arger with trees of a large number of generations. A pore-tree with mixed h ierarchies do not admit such an asymptote but its rate dependence on k does admit three domains with a weak dependence in the intermediate domain. An ordered pore-fractal 'catalyst', like the Sierpinsky gasket, for which nume rical results are available, is more complex than the two previous structur es as it contains closed-loops: At low ii's it can be modeled as self-nesti ng catalytic squares while with large k's the loops are not important and t he object can be viewed as a combination of pore trees with mixed hierarchi es. (C) 1998 Elsevier Science Ltd. All rights reserved.