Adjacency matrix formulation of energy flow in dendrimeric polymers

Abstract

Dendrimers are synthetic, highly branched polymers with an unusually high density of chromophores. As a result of their extremely high absorption cross-sections for visible light, they represent some of the most promising new materials for energy harvesting. Although the signature of the bonding structure in dendrimers is an essentially fractal geometry, the three-dimensional molecular folding of most higher generation materials results in a chromophore layout that is more obviously akin to concentric spherical shells. The number of chromophores in each shell is a simple function of the distance from the central core. The energy of throughput optical radiation, on capture by any of the chromophores, passes by a multi-step but highly efficient process to the photoactive core. Modeling this crucial migration process presents a number of challenges. It is far from a simple diffusive random walk; each step is subject to an intricate interplay of geometric and spectroscopic features. In this report, the first results of a new approach to the theory is described, developed and adapted from an adjacency matrix formulation. It is shown how this method offers not only kinetic information but also insights into the typical number of steps and the patterns of internal energy flow.

Energy harvesting: a review of the interplay between structure and mechanism

“…A recently developed operator approach is founded upon an adjacency matrix representation, constructed on the basis of the chemical connectivity between individual chromophores [81][82][83]. A square matrix, whose order is equal to the number of chromophores, represents the propensities (probabilities associated with a specific time interval) for energy migration between the chromophores.…”