The bimolecular irreversible reaction, A + B --> C, between species A and B which diffuse in space is a paradigmatic example to study the impact of incomplete mixing on chemical kinetics. The classical approach to describe well mixed chemical system is through the mass action law and the associated ordinary differential equations. This description is valid in the mean-field limit of an infinite system size, that is, for an infinite number of particles of A and B. In the presence of spatial heterogeneities (which can lead to incomplete mixing on a local scale) and for finite numbers of particles of A and B, this behaviour can change. In natural systems, the number of constituents is finite.
The well-mixed thermodynamic limit effectively prevents the formation of zones where only one reactant is present, and where reactions stop, because neither species A nor species B can completely deplete. Due to the creation of such a zones, or islands of non-reactive particles, the system is no longer well mixed and diffusion effects start to play an important role and will dominate the kinetics of the system. While diffusion attenuates initial concentration contrasts, chemical reaction can amplify them by depleting the species concentrations wherever they are in contact. This leads to segregation, the formation of islands of the respective species.
In the video below is shown the temporal evolution of the local concentration of two chemical s A and B originally well mixed with small heterogeneous fluctuations in a unidimensional system. At early times the concentrations of both reactants decrease due to reactions. As the reactions locally deplete one of the two reactants the formation of islands of single chemical rises. The reaction become diffusion-limited since only new mass transfer will allow A and B to meet and react again.
* (pdf), Journal of Chemical Physics, 2011