Trapping in dendrimers and regular hyperbranched polymers

Abstract

Dendrimers and regular hyperbranched polymers are two classic families of macromolecules, which can be modeled by Cayley trees and Vicsek fractals, respectively. In this paper, we study the trapping problem in Cayley trees and Vicsek fractals with different underlying geometries, focusing on a particular case with a perfect trap located at the central node. For both networks, we derive the exact analytic formulas in terms of the network size for the average trapping time (ATT)---the average of node-to-trap mean first-passage time over the whole networks. The obtained closed-form solutions show that for both Cayley trees and Vicsek fractals, the ATT display quite different scalings with various system sizes, which implies that the underlying structure plays a key role on the efficiency of trapping in polymer networks. Moreover, the dissimilar scalings of ATT may allow to differentiate readily between dendrimers and hyperbranched polymers.

Definitive version accepted for publication in The Journal of Chemical Physics

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• Random walks on weighted networks
• Random walks in weighted networks with a perfect trap: An application of Laplacian spectra
• Eigenvalues of normalized Laplacian matrices of fractal trees and dendrimers: Analytical results and applications
• Laplacian spectra of recursive treelike small-world polymer networks: Analytical solutions and applications
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