Persistent effects of inertia on diffusion-influenced reactions: Theoretical methods and applications

Abstract

The Cattaneo–Vernotte model has been widely studied to take momentum relaxation into account in transport equations. Yet, the effect of reactions on the Cattaneo–Vernotte model has not been fully elucidated. At present, it is unclear how the current density associated with reactions can be expressed in the Cattaneo–Vernotte model. Herein, we derive a modified Cattaneo–Vernotte model by applying the projection operator method to the Fokker–Planck–Kramers equation with a reaction sink. The same modified Cattaneo–Vernotte model can be derived by a Grad procedure. We show that the inertial effect influences the reaction rate coefficient differently depending on whether the intrinsic reaction rate constant in the reaction sink term depends on the solute relative velocity or not. The momentum relaxation effect can be expressed by a modified Smoluchowski equation including a memory kernel using the Cattaneo–Vernotte model. When the intrinsic reaction rate constant is independent of the reactant velocity and is localized, the modified Smoluchowski equation should be generalized to include a reaction term without a memory kernel. When the intrinsic reaction rate constant depends on the relative velocity of reactants, an additional reaction term with a memory kernel is required because of competition between the current density associated with the reaction and the diffusive flux during momentum relaxation. The competition effect influences even the long-time reaction rate coefficient.