Rate processes on fractals: Theory, simulations, and experiments

Heterogeneous kinetics are shown to differ drastically from homogeneous kinetics. For the elementary reaction A + A → products we show that the diffusion-limited reaction rate is proportional to t − h[A] 2 or to [A] x , where h=1- d s /2, X=1+ 2/d s =(h-2)(h-1 ), and d s is the effective spectral dimension. We note that for d = d s =1, h =1/2 and X = 3 , for percolating clusters d s = 4/3, h = 1/3 and X = 5/2 , while for “dust” d s h > 1/2 and ∞ > X > 3. Scaling arguments, supercomputer simulations and experiments give a consistent picture. The interplay of energetic and geometric heterogeneity results in fractal-like kinetics and is relevant to excitation fusion experiments in porous membranes, films, and polymeric glasses. However, in isotopic mixed crystals, the geometric fractal nature (percolation clusters) dominates.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45150/1/10955_2005_Article_BF01010846.pd

Single and multiple random walks on random lattices: Excitation trapping and annihilation simulations

Random walk simulations of exciton trapping and annihilation on binary and ternary lattices are presented. Single walker visitation efficiencies for ordered and random binary lattices are compared. Interacting multiple random walkers on binary and ternary random lattices are presented in terms of trapping and annihilation efficiencies that are related to experimental observables. A master equation approach, based on Monte Carlo cluster distributions, results in a nonclassical power relationship between the exciton annihilation rate and the exciton density.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45145/1/10955_2005_Article_BF01012307.pd

Fractal chemical kinetics: Reacting random walkers

Computer simulations on binary reactions of random walkers ( A + A → A ) on fractal spaces bear out a recent conjecture: ( ρ −1 − ρ 0 −1 ) ∞ t f , where ρ is the instantaneous walker density and ρ 0 the initial one, and f = d s /2, where d s is the spectral dimension. For the Sierpinski gaskets: d =2, 2 f =1.38 ( d s =1.365); d =3, 2 f =1.56 ( d s =1.547); biased initial random distributions are compared to unbiased ones. For site percolation: d = 2, p =0.60, 2 f = 1.35 ( d s =1.35); d=3, p =0.32, 2 f =1.37 ( d s =1.4); fractal-to-Euclidean crossovers are also observed. For energetically disordered lattices, the effective 2 f (from reacting walkers) and d s (from single walkers) are in good agreement, in both two and three dimensions.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45149/1/10955_2005_Article_BF01012924.pd

Single random walker on disordered lattices

Random walks on square lattice percolating clusters were followed for up to 2×105 steps. The mean number of distinct sites visited 〈 (S N ⊃> gives a spectral dimension of d s = 1.30±0.03 consistent with superuniversality ( d s =4J3) but closer to the alternative d s = 182/139, based on the low dimensionality correction. Simulations are also given for walkers on an energetically disordered lattice, with a jump probability that depends on the local energy mismatch and the temperature. An apparent fractal behavior is observed for a low enough reduced temperature. Above this temperature, the walker exhibits a “crossover” from fractal-to-Euclidean behavior. Walks on two- and three-dimensional lattices are similar, except that those in three dimensions are more efficient.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45148/1/10955_2005_Article_BF01012923.pd

Percolation and cluster distribution. III. Algorithms for the site-bond problem

Algorithms for estimating the percolation probabilities and cluster size distribution are given in the framework of a Monte Carlo simulation for disordered lattices for the generalized site-bond problem. The site-bond approach is useful when a percolation process cannot be exclusively described in the context of pure site or pure bond percolation. An extended multiple labeling technique (ECMLT) is introduced for the generalized problem. The ECMLT is applied to the site-bond percolation problem for square and triangular lattices. Numerical data are given for lattices containing up to 16 million sites. An application to polymer gelation is suggested.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45140/1/10955_2005_Article_BF01011170.pd

ANOMALOUS DIFFUSION BY TUNNELLING ON A SQUARE LATTICE - TRAPPING AND FUSION OF TRIPLET EXCITONS IN MIXED-CRYSTALS OF NAPHTHALENE

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