19 - Finite reaction rates

Summary

When the rate of coalescence is finite the typical time for reaction competes with the typical diffusion time, and a crossover between the reaction-limited regime and the diffusion-limited regime is observed. The model cannot be solved exactly, but it can be approached through an approximation based on the IPDF method. The kinetic crossover is well captured by this approximation.

A model for finite coalescence rates

Until now we have dealt with infinite coalescence rates: when a particle hops into an occupied site the coalescence reaction is immediate. Thus, the typical reaction time is zero and the process is diffusion-limited. We want now to discuss the case in which the coalescence rate (and the typical time for the coalescence reaction) is finite. In this case a competition arises between the typical transport time and the typical reaction time.

The model we have in mind is the following: when a particle attempts to hop onto a site that is already occupied, the move is allowed and coalescence takes place with probability k (0 ≤ k ≤ 1). The attempt is rejected, and the state of the system remains unchanged, with probability (1―k). The case of k = 1 corresponds to an infinite coalescence rate, which we have studied so far. The opposite limit, of k-0, describes diffusion of the particles with hard-core repulsion, but no reactions take place.

Suppose that 0 < k « 1. Reactions then require a large number of collisions, and the kinetics is dominated by the long reaction times (the reaction-limited regime).