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.

Sudden, “Step” Electron Capture by Conjugated Polymers

“…Unlike the small and basically spherical solutes that were tested above with eq , these molecules are extended in one dimension in solution. Reactions with such molecules have been shown to be accommodated by theoretical models that express R eff for diffusion of point particles to extended polymers described as ellipsoids or lines of spheres. ,,− For reactions having large R eff , capture greatly increases in rate at early times and remains significant into the nanosecond regime for long extended polymers of the type used in this work. In previous pulse radiolysis work, , it was noted that another source of prompt or step capture just began to be apparent at modest concentrations of the polymer, possibly related to presolvated or “dry” electron capture.…”

Operator algebraic methods in the theory ofdiffusion‐influencedreaction kinetics

“…In this section, we will confine ourselves to the description of the effects of high reactant concentration on the diffusion‐influenced kinetics of irreversible and reversible bimolecular reactions, mainly based on the reduced distribution function (RDF) formalism 10,93–105 . Another subject of the many‐particle effects that is not treated in this review is the reaction involving a reactant with many active sites 106–113 or a group of clustered reactant molecules 5,9,114–128 …”

Kinetic Response of Surfaces Defined by Finite Fractals

“…[24][25][26] We therefore reformulate/transform the diffusion dynamics described by Eq. (1) by using Cantor-like basis sets 13,15,27 to allow analytical/numerical solution of the problem (Fig. 1), while preserving the scale invariance properties of the original structure.…”

“…We therefore reformulate/transform the diffusion dynamics described by Eq. (1) by using Cantor-like basis sets [13][14][15] to allow analytical/numerical solution of the problem and show that the amount of particles captured on a fractal surface, V (t), is still related to its D F through a simple scaling law,…”