An analytically solvable reaction-diffusion model for chemical dynamics in solutions

Abstract

• Smoluchowski equation for a harmonic potential with a finite sink of the form, S ( x , t ) = k 0 e kt δ ( x - x c e - kt ) is solved analytically. • The moving of the prepared excited-state PES due to solvation dynamics is implicitly incorporated through S ( x , t ) term. • Exact analytical solutions for the survival probability Q ( t ) and probability distribution P ( x , t ) are obtained and verified, • The features of the considered model and insights into chemical process are derived using the obtained time-domain solutions. • The time-dependent realization of Marcus’ rate prediction about electron transfer reactions is presented. Problems with diffusing probability distribution in the presence of sink traps, where the moving entity can be a particle, an entity, or a relevant coordinate in a reaction can represent a variety of physical processes. So far, the theoretical solutions gave time-domain understanding only if the problem had a translational invariance in potential or a mirror symmetry about the trap or the both. In this paper, we present a time-domain solution for diffusion in the presence of a harmonic potential well ( V ( x ) = 1 2 kx 2 ) with a finite absorbing sink S ( x , t ) . Considering the Smoluchowski equation as the governing equation for the process, a special exit condition given by, S ( x , t ) = k 0 e kt δ ( x - x c e - kt ) can be solved analytically using variable transformations from a simple model. Interesting insights into the diffusion dynamics emerge from the interplay between the initial position of the distribution, diffusivity, position of the sink, etc. The derived survival probability profile gives an understanding of chemical dynamics in condensed phases.