Diffusion-Limited Binary Reactions: A Hierarchy of Non-Classical Regimes
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DIFFUSION-LIMITED BINARY REACTIONS: A HIERARCHY OF
NON-CLASSICAL REGIMES PANOS ARGYRAKIS°, RAOUL KOPELMAN°° and KATJA LINDENBERGt
"*Departmentof Chemistry, The University of Michigan, Ann Arbor, MI 48109 and Department of Physics, University of Thessaloniki, GR-54006 Thessaloniki, Greece ** Departments of Chemistry and Physics, The University of Michigan, Ann Arbor, MI 48109 t Department of Chemistry and Institute for Nonlinear Science, University of California at San Diego, La Jolla, CA 92093-0340 ABSTRACT We discuss the various regimes of kinetic behavior of the densities of reactants for the A + B --+ 0 reaction from initial to asymptotic times. Scaling arguments and Monte Carlo simulations demonstrate an unexpectedly rich hierarchy of cross-overs among time exponents ir the decay law p - t-0. For instance, in one dimension possible time domains include classical (a = 1), A + A-type (a = 1/2), Zeldovich (a = 1/4), asymptotic correlated (a = 3/4) and finally nonalgebraic (exponential) finite size regimes. Simulation and theory are consistent with respect to both exponefits and cross-over times for one and two dimensions. INTRODUCTION Diffusion-limited binary reactions in low dimensions lead to the spontaneous formation of spatial structures and to associated "anomalous" rate laws for the global densities p(t) of the reacting species. The irreversible reaction A + A -* 0 under "normal" circumstances is described by the rate law p = -kp
2
whereas the asymptotic rate law for dimensions d < 2 is p = -kp(1+
2
/d).
The higher exponent is a consequence of the spatial distribution of A's. A random or "mixed" distribution of A's has its maximum at zero separation, indicative of the presence of many extremely close nearest neighbor pairs of reactant particles. An anomalous rate law implies many fewer close reactant pairs. Indeed, in dimensions lower than two an initially random distribution quickly settles into a distribution that peaks at a nonzero nearest neighbor separation, leading to an almost crystal-like arrangement of reactants. This non-random distribution arises from the fact that diffusion is not an effective mixing mechanism in low dimensions. Another example of anomalous kinetics is the diffusion-limited irreversible reaction A+ B -- 0. Under normal circumstances the rate laws for the global densities PA and PB are PA = PB = -kpAPB. If PA(t = 0) = PB(t = 0), then the densities of the two species are equal at all times 2 and we can dispense with the subscripts so that once again p = -kp . The actual asymptotic rate law in dimensions d < 4 for an initially random distribution of reactants is instead p = -kp(1+4/d). Here the principal cause of the anomalous behavior is the formation of aggregates of like particles. The spatial regions in which the density of one type of particle is overwhelmingly greater than that of the other grow in time (while the total density within each aggregate of course decreases with time). Since the reaction can essentially only occur at the interfaces between aggregates, the re
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