The time dependent solution of the Smoluchowski equation: Kinetics of escaping from the well for different dimensionalities

Abstract

A recently developed method of analytical solution of the Smoluchowski equation is applied to the investigation of the kinetics of diffusional escape from a potential well for different space dimensionalities n. The kinetics is described by the time dependence of the well occupation probability Nn(t). The formulas derived are valid for times longer than the time t0=a 2/D of relaxation within the well U(r) [a is the Onsager’s radius determined by U(a)=kT ]. In the absence of absorption (or reaction) within the well the kinetics for low n=1,2 is found to be nonexponential at any time from the very beginning and for any depth of the well. The characteristic escaping time, however, significantly depends on the well depth. At longer times we obtain long time tails Nn(t)∼t−n/2 (n=1,2), typical for corresponding free diffusion processes. A simple general analytical expression for N1(t) is derived. In the limit of high reactivity within the well N1(t) appears to be exponential at moderately short t, but at long times it changes to the asymptotic function ∼t−3/2 (unlike the weak reactivity tail ∼t−1/2 ). In the two-dimensional (2D) case in the high reactivity limit the kinetics is also exponential at short t. The long time asymptotics somewhat differs from the weak reactivity one (∼t−1):N2(t)∼1/t ln2 t. It is shown that the three-dimensional (3D) escaping process is described by the same expression as the one-dimensional (1D) one in the high reactivity limit, i.e., at moderately short t, N3(t)∼exp(−Wt), while at long times N3(t)∼t−3/2. Some possible extensions of the method are discussed.

Geminate recombination of hydroxyl radicals generated in 200 nm photodissociation of aqueous hydrogen peroxide

“…As shown by Shushin [17], the decay kinetics of diffusional escape from such a potential well exhibits two regimes: (i) an exponential decay on a short time scale (with rate constant W=W r +W d , where W r,d are the recombination and dissociation rates of the radicals in the potential well, respectively), and t -1/2 behavior on the longer time scale, due to slow recombination of radicals that escaped beyond the Onsager radius a, at which U(a)=-kT. In this theory, the escape probability p d of the radical partners is equal to W r /W and the effective radius R eff of the reaction in the bulk is given by a product a(1-p d ) [18]. Fig.…”

“…Fig. 1 demonstrates the fit of the experimental kinetics to Shushin's theory expressions [17,18] for the survival probability of a radical pair migrating out of the potential well.…”

“…This suggests a weak interaction between the radicals (in other words, the existence of a solvent cage around the radical partners). Phenomenologically, this interaction can be described in terms of a mean force potential U(r) having a profile of a well [17,18]. As shown by Shushin [17], the decay kinetics of diffusional escape from such a potential well exhibits two regimes: (i) an exponential decay on a short time scale (with rate constant W=W r +W d , where W r,d are the recombination and dissociation rates of the radicals in the potential well, respectively), and t -1/2 behavior on the longer time scale, due to slow recombination of radicals that escaped beyond the Onsager radius a, at which U(a)=-kT.…”

Investigation of geminate recombination of radical ion pairs generated by dissociation of exciplexes in moderately polar solvents using the photoconductivity technique

“…Shushin [56][57][58] has considered kinetics of the diffusion dissociation of interacting molecules for the general shape of the interaction potential possessing a well and a barrier at small distances. It has been shown that the kinetics of the diffusion escape from a well is exponential at short times and follows a power law at long times [56][57][58].…”

External Force Affected Escape of Brownian Particles from a Potential Well

“…The well is assumed to be isotropic and highly localized (short range). Though, detailed analysis shows 16,17 that fairly deep potential well resulted, for example, from the Coulomb interaction (at large r) can also be considered as highly localized in some conditions.…”