Dimensional Crossover in the Growth of Depletion Zone in a Rectangular Capillary: Experiments and Monte Carlo Simulation
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Dimensional Crossover in the Growth of Depletion Zone in a Rectangular Capillary: Experiments and Monte Carlo Simulations Sung Hyun Park, Hailin Peng, Panos Argyrakis1, Haim Taitelbaum2, and Raoul Kopelman Department of Chemistry, University of Michigan, Ann Arbor, MI 48109-1055, U.S.A. 1 Department of Physics, University of Thessaloniki, 54124 Thessaloniki, Greece 2 Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel ABSTRACT The diffusion-limited kinetics of the growth of depletion zone around a static point trap in a thin, long stripe geometry was studied using a laser photobleaching experiment of fluorescein dye inside a rectangular capillary. The dynamics of the depletion zone was monitored by the θdistance, defined as the distance from the trap to the point where the reactant concentration has been depleted to the specific fraction of its initial bulk value. A dimensional crossover from two dimensions to one dimension, due to the finite width of the reaction zone, was observed. The crossover seems to occur for all θ values concurrently when the depletion zone touches the boundary for the first time, suggesting that the boundary information spreads faster than diffusion. Monte Carlo simulations were performed to support the experimental results. The crossover time (τc) is found to scale with the width (L) of the rectangular reaction zone as τc ~ L2, as expected from the Einstein’s diffusion law. INTRODUCTION It has been well established that the kinetic laws for the reactions in a diffusion-limited environment are considerably different from the conventional rate laws, due to the spatial correlations of the reactants, originating from the inefficiency of the diffusive mixing.[1-10] One of the simple cases of the diffusion-limited reaction is the trapping reaction A+T→T, where T is a static trap and A is a diffusing species that is annihilated upon collision with the trap. The occurrence of A-T reactions creates a zone of depletion around the trap, which is a form of selfsegregation of reactants. The growth of the depletion zone in the trapping reaction leads to anomalous kinetics for a variety of dynamic quantities in low dimensions. For example, the nearest neighbor distance was found to increase asymptotically as t1/4 in one dimension (1D) and (ln t)1/2 in two dimensions (2D),[4,5] while the trapping reaction global rate was shown to decrease asymptotically as t-1/2 in 1D and (ln t)-1 in 2D.[4,6,7] A quantity relevant for the description of the depletion zone is the so called θ-distance (rθ), which is the distance from the trap to the point where the concentration of A particles, c(r, t), reaches the specific fraction θ (0 ≤ θ ≤ 1) of its value in the bulk c0 [7], i.e., c(rθ, t) = θ c0.
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The θ-distance has been shown, by theory [7] and experiment [8], to increase asymptotically as t1/2 in 1D. The case in 2D produces the anomalous behavior of non-universality for the θdistance, which has been shown, by theory and experiment, to grow asymptotically as tθ/2. [7, 9] In this study, we
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