Rate of diffusion-limited reactions for a fractal aggregate of reactive spheres

Abstract

We study the reaction rate for a fractal cluster of perfectly absorbing, stationary spherical sinks in a medium containing a mobile reactant. The effectiveness factor η, which is defined as the ratio of the total reaction rate of the cluster to that without diffusional interactions, is calculated. The scaling behavior of η is derived for arbitrary fractal dimension based on the Kirkwood–Riseman approximation. The asymptotic as well as the finite size scaling of η are confirmed numerically by the method of multipole expansion, which has been proven to be an excellent approximation. The fractal assembly is made of N spheres with its dimension varying from D<1 to D=3. The number of sinks can be as high as N∼O(104). The asymptotic scaling behavior of the effectiveness factor is η∼N1/D−1 for D>1, η∼(ln N)−1 for D=1, and η∼N0 for D<1. The crossover behavior indicates that while in the regime of D>1 the screening effect of diffusive interactions grows with the size, for D<1 it is limited in a finite range and decays with decreasing D. The conclusion is also applicable to transport phenomena like dissolution, heat conduction, and sedimentation.