Relaxation dynamics of Sierpinski hexagon fractal polymer: Exact analytical results in the Rouse-type approach and numerical results in the Zimm-type approach

Abstract

In this paper, we focus on the relaxation dynamics of Sierpinski hexagon fractal polymer. The relaxation dynamics of this fractal polymer is investigated in the framework of the generalized Gaussian structure model using both Rouse and Zimm approaches. In the Rouse-type approach, by performing real-space renormalization transformations, we determine analytically the complete eigenvalue spectrum of the connectivity matrix. Based on the eigenvalues obtained through iterative algebraic relations we calculate the averaged monomer displacement and the mechanical relaxation moduli (storage modulus and loss modulus). The evaluation of the dynamical properties in the Rouse-type approach reveals that they obey scaling in the intermediate time/frequency domain. In the Zimm-type approach, which includes the hydrodynamic interactions, the relaxation quantities do not show scaling. The theoretical findings with respect to scaling in the intermediate domain of the relaxation quantities are well supported by experimental results.

Dynamics of a Polymer Network Modeled by a Fractal Cactus

“…The two dashed reference lines exhibit the that slope equals 1. In addition, we find that, under HI, the intermediate time range gets smaller (as was also established in several previous works [17,21,25,39]). As discussed before, typical for the structure under investigation is the intermediate regime.…”

“…The relaxation dynamics of some regular fractal structures has been intensively investigated in several previous studies [17,21,25,39]. For the dual Sierpinski gaskets and for the Sierpinski hexagon, the conclusion drawn was that the relevant physical quantities that describe the dynamics (averaged monomer displacement and mechanical moduli) do not obey power laws in the Zimm approaches.…”

“…The problem of relating the dynamical features of macromolecules with their structure has a long standing history, starting from the landmark works of Rouse [1] and Zimm [2] who focused on the investigation of dilute solutions of linear polymers. With the continuous advancement in polymer synthesis and analysis, the attention turned to macromolecules with more and more complex architectures like star polymers [3,4,5], dendrimers [5,6,7,8,9,10,11,12,13], hyperbranched polymers [5,14,15,16,17,18,19,20,21,22], and polymer networks [23,24,25,26,27,28,29,30]. Nowadays, available experimental techniques in supramolecular chemistry allow for synthesizing a large variety of polymers with precisely controlled molecular structures such as the spherical and cylindrical supramolecular dendrimers [31], the gel-like supramolecular networks [32], and the honeycomb lattices [33], culminating with molecular fractals [34,35,36,37,38].…”

“…Previous works have shown that, in the Zimm model, the dynamical quantities do not obey power-laws in their intermediate time/frequency domain for fractal networks containing loops on loops (as dual Sierpinski gaskets [39] and Sierpinski hexagon [25]), whereas, for loopless fractals (as Vicsek fractal [17,21]), they do obey power laws. In these works [17,21,25,39], the HI parameter ranged from 0.05 to 0.25. The lost of scaling was attributed to the presence of loops and chiefly loops on loops that produce a position-dependent shortening of the effective bond lengths inside the structure, and thus impair scaling.…”

Relaxation dynamics of generalized scale-free polymer networks

“…Due to the continuous advancement in polymer synthesis and analysis, new macromolecules or supramolecules with very complex architectures and tunable properties have been synthesized. Paralleling the advancement on polymers synthesis, the Rouse-type approach has been successfully applied in the theoretical works devoted to the study of polymers with more complex architecture, such as dendrimers and their derivatives 26 , 34 – 37 , star polymers 34 , hyperbranched or fractal polymer networks 31 , 38 – 40 , multilayered or multihierarchical polymer networks 41 – 43 , small-world polymer networks 44 – 46 and scale-free polymer networks 8 . The present work extents all these studies by considering the treelike generalized scale-free polymer networks and we systematically monitor the influence of the modularity parameters on the topology and dynamics of these complex networks.…”