Relaxation dynamics of perturbed regular hyperbranched fractals

Abstract

We focus on perturbed regular hyperbranched fractals (pRHF), which are RHF whose f coordinated centers (f CC) are traps. We compute the mechanical properties (storage and loss modulus) and the average displacement in the framework of generalized Gaussian structures, by making use of the eigenvalue spectrum of the connectivity matrix. We generalize the analysis to the case of a connectivity matrix perturbed by a diagonal and pure imaginary operator. Although the above-cited observables in this new situation lose their original meaning, they still give important information about the underlying structures and they could help to analyze other phenomena where complex operators are involved. We obtain analytically the eigenvalue spectrum for pRHF. A drastic change was observed in the behavior of the studied quantities even for a very small perturbation strength. However, it is still possible to depict the scaling of the fractals in the intermediate time (frequency) domains.

Analytical model for the dynamics of semiflexible dendritic polymers

“…IV, SD; scaling behavior is, on the other hand, the rule for linear chains and for fractals. [28][29][30][31] In fact, for FD the corresponding curves in the region of intermediate frequencies can be approximated through logarithmic forms. 8 The broadening of the spectra with growing q, as discussed above, manifests itself through a broadening of the [G (ω)]-shapes.…”

Laplacian spectra of recursive treelike small-world polymer networks: Analytical solutions and applications

“…It is worth stressing that this result is also different from that for Vicsek fractals. [47][48][49][50] For the loss modulus G (ω), we plot in a double scale the results in Fig. 7.…”

“…In the intermediate region, no power-law behavior is observed, which is in marked contrast to that corresponding to Vicsek fractals. [48][49][50] It is also important to notice that in the intermediate region, G (ω) and G (ω) display different behaviors for the smallworld structure.…”

“…As the name suggests, Viscek fractals are fractals, their relaxation patterns are determined by the fractal dimension and spectral dimension. [47][48][49][50] For the small-world treelike polymer networks, they are non-fractal, and thus exhibit different relaxation patterns.…”

Relaxation dynamics of a polymer network modeled by a multihierarchical structure

“…In the last few decades a large and still open issue in polymer physics is the relationship between the geometry of macromolecules and their dynamics. While the first works started from linear polymeric systems [1,2] and their segmental dynamics, [3][4][5] in recent years attention turned to more and more complex topologies such as star polymers [6][7][8][9][10], dendrimers [6,[8][9][10][11][12][13][14][15][16], hyperbranched polymers [11,13,14,[17][18][19][20][21][22], or small-world networks [23][24][25][26][27]. The concept of fractals introduced by Mandelbrot [28] has turned out to be a useful tool in a large variety of scientific domains, such as disordered systems, growth phenomena, chemical reactions controlled by diffusion, and energy transfer.…”