Abstract
Chemical reactions generically require that particles come into contact. In practice, reaction is often imperfect and can necessitate multiple random encounters between reactants. In confined geometries, despite notable recent advances, there is to date no general analytical treatment of such imperfect transport-limited reaction kinetics. Here, we determine the kinetics of imperfect reactions in confining domains for any diffusive or anomalously diffusive Markovian transport process, and for different models of imperfect reactivity. We show that the full distribution of reaction times is obtained in the large confining volume limit from the knowledge of the mean reaction time only, which we determine explicitly. This distribution for imperfect reactions is found to be identical to that of perfect reactions upon an appropriate rescaling of parameters, which highlights the robustness of our results. Strikingly, this holds true even in the regime of low reactivity where the mean reaction time is independent of the transport process, and can lead to large fluctuations of the reaction time - even in simple reaction schemes. We illustrate our results for normal diffusion in domains of generic shape, and for anomalous diffusion in complex environments, where our predictions are confirmed by numerical simulations.
Highlights
Chemical reactions generically require that particles come into contact
This observable is involved in various areas of biological and soft matter physics and is relevant in the context of reaction kinetics, because two reactants have to meet before any reaction can occur[11,12,13]
This allows us to answer the following questions: (i) Is reaction limited by transport or reactivity? (ii) What is the magnitude of the fluctuations of the reaction time? In particular, is the first moment sufficient to fully determine reaction kinetics? (iii) Do reaction kinetics depend on the choice of model of imperfect reactivity– namely partially reflecting (Robin) conditions[23,35,36]
Summary
The probability that a reaction happens after exactly n visits to the target is given by p(1 − p)n−1, in which case Tr(p) is the sum of the first passage time (starting from r) and of n − 1 independently distributed first return times (see Fig. 1). These results indicate that our approximations (i.e. the use of the O’Shaughnessy-Procaccia operator, the hypothesis that scaling of all moments hold up to r = 1, large volume limit) lead to accurate predictions for the mean reaction time and its full distribution. Our above results for discrete models show that the full distribution of reaction times (a)
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