Random walks and reactions on dendrimer structures

Abstract

We perform model calculations of individual particle random walks on dendrimer structures, due to the emerging interest and wealth of recent experimental data. We use varying coordination numbers and generation orders. We find that it is practically impossible to verify numerically the customary walk properties with the existing asymptotic formalism, because of finite-size effects, even at short times and large system sizes. Several different sets of boundary conditions are employed, producing large differences in the observed behavior for the number of sites visited at least once, as a function of time, S N . We also investigate the case of reacting (annihilating) particles, using the well-known A+A and A+B models, in which case we find an anomalous scaling for the decay of the particle density, with the scaling exponents being f=0.80±0.03 and 0.50±0.03, respectively. We discuss how diffusion in dendrimers is analogous to biased random walks in one dimension, which demonstrates a breakdown of the usual scaling between the reaction progress (or reactant survival) and S N . An appendix gives results for biased walks in one dimension.