Monte Carlo studies of diffusion-limited trapping reactions in restricted geometries.

Abstract

The particle distributions and macroscopic reaction rate laws of the diffusion-limited trapping reaction $(A + T \to T)$ in low and fractal dimensions was investigated by means of computer simulations. Herein, we re-examine the Smoluchowski model (1917) of one trap sitting in a swarm of particles but broaden the scope to study the effects of relative reactant mobility, microscopic reaction probability and lattice geometry. Diffusion processes were modeled on random walks on a lattice: A ring was used for one-dimensional space while a Sierpinski gasket was used to approximate fractal and/or disordered systems. Particle configurations were analyzed primarily using nearest-neighbor distance distributions and concentration profiles. Non-equilibrium transient reaction rate laws were obtained by measuring the particle density decay and comparing to theoretical predictions. Population distributions were utilized to explain the irregularities of the rate laws. Our simulation results agree with most of the theoretical predictions. However, these simulations have extended the current model to new parameter regimes and, likewise, have stimulated new vistas of theoretical work. For systems far from equilibrium, a self-organization of reacting species was observed, resulting in anomalous reaction rate laws. In particular, the anomalous results were found to be related to a growing non-Hertzian distribution of partices as time progresses. A difference in the nearest-neighbor distance distribution exists for different mobilities of the trap and A species. It is found, however, that the concentration profile shows no dependence upon the reactant mobility. For reactions where the reaction probability is less than unity, the concentration profile changes dramatically from the diffusion-limited to the reaction-limited regime. The average nearest-neighbor distance, however, shows no dependence upon the reaction probability as time approaches infinity. For any finite time, however, there exists a corrective dependence on the reaction probability. Calculations of the heterogeneity exponent h for the trapping problems show an interesting dependence on reaction probability and on time. These dependencies manifest themselves in a cline plot which represents a crossover between textbook reaction kinetics and nonclassical behavior.