Influence of geometry on light harvesting in dendrimeric systems. II. nth-nearest neighbor effects and the onset of percolation

We explore the effect of imposing different constraints (biases, boundary conditions) on the mean time to trapping (or mean walklength) for a particle (excitation) migrating on a finite dendrimer lattice with a centrally positioned trap. By mobilizing the theory of finite Markov processes, we are able to obtain exact analytic expressions for site-specific walklengths as well as the overall walklength for both nearest-neighbor and second-nearest-neighbor displacements. This allows the comparison with and generalization of earlier results [A. Bar-Haim, J. Klafter, J. Phys. Chem. B 102 (1998) 1662; A. Bar-Haim, J. Klafter, J. Lumin. 76, 77 (1998) 197; O. Flomenbom, R.J. Amir, D. Shabat, J. Klafter, J. Lumin. 111 (2005) 315; J.L. Bentz, F.N. Hosseini, J.J. Kozak, Chem. Phys. Lett. 370 (2003) 319]. A novel feature of this work is the establishment of a connection between the random walk models studied here and percolation theory. The full dynamical behavior was also determined via solution of the stochastic master equation, and the results obtained compared with recent spectroscopic experiments.