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 tot− h[A]2 or to [A]x, whereh=1- ds/2, X=1+2/ds=(h-2)(h-1), anddsis the effective spectral dimension. We note that ford = ds=1, h =1/2 andX = 3, for percolating clustersds = 4/3,h = 1/3 andX = 5/2, while for “dust” ds <1, 1 >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.