Diffusion and trapping in dendrimer structures

We investigate diffusion on models of large molecules with dendrimer structure. The models we use are topological Cayley trees. We focus on diffusion properties, such as the excursion distance, the mean square displacement of the diffusing particles, and the area probed, as given by the random walk parameter S_N, the number of the distinct sites visited. Finally we look at the trapping kinetics for randomly distributed stationary traps of low concentrations. We use different coordination numbers, z, and different generation orders g of a dendrimer structure. We show that for calculations on dendrimers where we simulate the entire structure the finite size effects dominate making the results worthless. We, therefore, implemented two algorithms that do not retain the underlying structure, and thus effectively the random walk is performed on an equivalent infinite structure. We find a universal linear behavior for all the properties examined, thus differing from the regular lattices. The average displacement R grows linearly with time, implying that R² grows as t² instead of t. The S_N property also grows linearly with time, which resembles the random walk case in three-dimensional lattices, but with a different prefactor which strongly depends on z. Trapping is dependent on S_N, as expected, and follows exactly its behavior. Due to the nature of these structures the random walk is indeed a type of a “biased” walk. We devise a model in which we take out this “bias” factor, and in this model we show that the system becomes diffusive in nature, i.e. the mean square displacement is linear with time. These results could be utilized if one is to design a dendrimer system with specifically desired properties.