A new approach to the theory of Brownian coagulation and diffusion-limited reactions

TL;DR

This paper introduces a new analytical approach to Brownian coagulation and diffusion-limited reactions, addressing limitations of traditional models by accurately describing regimes with small reaction radii and homogeneous particle distribution.

Contribution

The paper develops a generalized analytical framework that interpolates between continuum and free molecular regimes, improving reaction rate calculations for small reaction radii.

Findings

Traditional diffusion approach valid only for large particles

New interpolation expression describes transition between regimes

Reaction rate for small radii deviates from classical models in 2D

Abstract

An overview of the author's papers on the new approach to the Brownian coagulation theory and its generalization to the diffusion-limited reaction rate theory is presented. The traditional diffusion approach of the Smoluchowski theory for coagulation of colloids is critically analyzed and shown to be valid only in the particular case of coalescence of small particles with large ones. It is shown that, owing to rapid diffusion mixing, coalescence of comparable size particles occurs in the kinetic regime, realized under condition of homogeneous spatial distribution of particles, in the two modes, continuum and free molecular. Transition from the continuum to the free molecular mode can be described by the interpolation expression derived within the new analytical approach with fitting parameters that can be specified numerically, avoiding semi-empirical assumptions of the traditional models. A similar restriction arises in the traditional approach to the diffusion-limited reaction rate theory, based on generalization of the Smoluchowski theory for coagulation of colloids. In particular, it is shown that the traditional approach is applicable only to the special case of reactions with a large reaction radius, and becomes inappropriate to calculation of the reaction rate in the case of a relatively small reaction radius. In the latter, more general case particles collisions occur mainly in the kinetic regime (rather than in the diffusion one) characterized by homogeneous (at random) spatial distribution of particles. The calculated reaction rate for a small reaction radius in 3-d formally coincides with the expression derived in the traditional approach for reactions with a large reaction radius, however, notably deviates at large times from the traditional result in the plane (2-d) geometry, that has wide applications also in the membrane biology as well as in some other important areas.