DOI:10.1002/asia.200600076 Diffusion-Influenced Reversible Trapping Problem in the Presence of an External Field Soohyung Park and Kook Joe Shin* [a] Introduction The diffusion-influenced trapping problem has been exten- sively studied theoretically and experimentally during the last few decades. [1–21] Usually, the trapping problem has been treated in the irreversible case, whether the traps are perfect or imperfect. For the irreversible perfect traps, the exact so- lution is known only for one dimension (1D). [1–5] However, the asymptotic behavior is known not only for 1D but also for higher dimensions. In the absence of an external field, it is known that the long-time asymptotic behavior of the sur- vival probability follows exp const  t d=ðdþ2Þ   in d dimen- sions. [6–13] The slower decay relative to that of the classical exponential decay results from the existence of rare but large trap-free regions, where the lifetime of the particle is extremely long. This asymptotic behavior was proved rigor- ously by Donsker and Varadhan. [7] Grassberger and Procaccia [14] found that, in the presence of an external field, the survival probability decays exponen- tially expðkt Þ (k is an effective rate constant depending on the field strength a). Moreover, it was found that there exists a critical field strength that separates two different re- gimes. [14–16] For the external field strength below its critical value, k is proportional to the square of the field strength, whereas it is proportional to the field strength above its crit- ical value. This transition can be explained qualitatively as follows. At low field strengths, the surviving particles spend all the time in large trap-free regions resisting the drift im- parted by the applied external field (“localized behavior”), whereas at high field strengths, they go with the drift (“delo- calized behavior”). Eisele and Lang proved this heuristic analysis rigorously. [15] The trapping problem has been studied intensively in 1D as the effects of field strength and many-body correlations [17] are much more conspicuous than in higher dimensions. The experimental studies of the transfer of the excitation or charge carrier in many organic and nonorganic crystals [17, 18] for high anisotropy (quasi-1D behavior) is an additional source of interest to the 1D case. Movaghar et al. [3] consid- ered the hopping motion of a charge carrier on a 1D lattice with deep traps in the presence of an external field. They obtained an expression for the survival probability, which is exact over the whole time regime and arbitrary field strength for the low concentration of traps. Studies of the re- laxation of photoconductivity in polydiacetylene crystals demonstrated that the predictions of their theory are in good agreement with the photocurrent kinetics observed over a wide range of fields, temperature, and times. [17] Glass- er and Agmon [19] considered the trapping problem with Abstract: We investigate the field effect on the diffusion-influenced re- versible trapping problem in one di- mension. The exact Green function for a particle undergoing diffusive motion between two static reversible traps with a constant external field is ob- tained. From the Green function, we derive the various survival probabili- ties. Two types of trap distribution for the many-body problem are consid- ered, the periodic and random distribu- tions. The mean survival probability is obtained for the crossing-forbidden case for the two types of trap distribu- tion. For the periodic distribution it decays exponentially. For the random trap distribution, similar to the irrever- sible case, there exists a critical field strength at which the long time asymp- totic behavior undergoes a kinetic tran- sition from the power law to exponen- tial behaviors. The difference between equilibrium concentrations for the two types of trap distribution due to the fluctuation effect of trap concentration vanishes as the field strength increases. Keywords: diffusion · external fields · kinetic transition · survival probability [a] S. Park, Prof. K. J. Shin School of Chemistry Seoul National University Seoul 151-747, Korea Fax: (+ 82) 2-889-1568 E-mail : shinkj@snu.ac.kr 216 # 2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Chem. Asian. J. 2006, 1–2, 216 – 223 FULL PAPERS