Dimensional Analysis of Diffusive Association Rate Equations

Diffusive adsorption/association is a fundamental step in almost all chemical reactions in diluted solutions, such as organic synthesis, polymerization, self-assembly, biomolecular interactions, electrode dynamics, catalysis, chromatography, air and water environmental dynamics, and social and market dynamics. However, predicting the rate of such a reaction is challenging using the equations established over 100 years ago. From dimensional analysis, we can guess a series of equations including historical ones and new ones that have the correct final unit, the number of molecules associated per second, which is constructed from concentration, diffusion coefficient, and size of the molecules only. These equations are roughly divided into two groups, continuous models and discrete models. The continuous models integrate Fick’s concentration gradient in the solution near the target to calculate the association rate, and are the historical solutions reported. The discrete models integrate the probability density function of each probe that is discretely distributed in the solution. I have introduced a key concept of the nearest neighbor diffusion time to stabilize these time-dependent solutions. These models are applied to analyze a set of experimental results for comparison to achieve consistency and clarify assumptions among models.