Dynamics of a Polymer Network Modeled by a Fractal Cactus

Abstract

In this paper, we focus on the relaxation dynamics of a polymer network modeled by a fractal cactus. We perform our study in the framework of the generalized Gaussian structure model using both Rouse and Zimm approaches. By performing real-space renormalization transformations, we determine analytically the whole eigenvalue spectrum of the connectivity matrix, thereby rendering possible the analysis of the Rouse-dynamics at very large generations of the structure. The evaluation of the structural and dynamical properties of the fractal network in the Rouse type-approach reveals that they obey scaling and the dynamics is governed by the value of spectral dimension. In the Zimm-type approach, the relaxation quantities show a strong dependence on the strength of the hydrodynamic interaction. For low and medium hydrodynamic interactions, the relaxation quantities do not obey power law behavior, while for slightly larger interactions they do. Under strong hydrodynamic interactions, the storage modulus does not follow power law behavior and the average displacement of the monomer is very low. Remarkably, the theoretical findings with respect to scaling in the intermediate domain of the relaxation quantities are well supported by experimental results from the literature.

Mechanisms of the Complex Thermo-Mechanical Behavior of Polymer Glass Across a Wide Range of Temperature Variations

“…According to the physical meaning of the parameters, it is interesting to see that E 1 and E 2 in this model should correspond to the measured moduli of the overall microstructure and the microstructure after β-relaxation, respectively. Recent theoretical works on the dynamics of polymer network also revealed the relationship between the relaxation-induced microstructure and modulus changes [36,37]. Therefore, this study provides a convenient and physically sound method to determine the parameters.…”

Spectral analysis and consensus problems for a class of fractal network models

“…Furthermore, the spectrum of Laplacian matrix for fractal cactus model can also be used to study the mechanical relaxation dynamics and structural properties of fractal polymer networks [43]. Moreover, it is worth noting that the fractal cactus can be regarded as the interpolation between two non-fractal networks, namely Husimi cactus and triangular Kagome lattice [43,44]. These also illustrate the importance of studying the fractal network model and the properties of Laplacian matrix spectrum on it.…”

“…For example, the chemical synthesis of molecular fractal Sierpinski hexagonal fractal and defect-free Sierpinski triangles have been used to design the synthesis of polymer systems [40][41][42]. Furthermore, the spectrum of Laplacian matrix for fractal cactus model can also be used to study the mechanical relaxation dynamics and structural properties of fractal polymer networks [43]. Moreover, it is worth noting that the fractal cactus can be regarded as the interpolation between two non-fractal networks, namely Husimi cactus and triangular Kagome lattice [43,44].…”

“…The extended fractal cactus model is denoted as G m (t), and its random walk properties have been studied by Junhao Peng and Elena Agliari [44], but the spectral properties of the Laplacian matrix on it have not been solved. Moreover, the previous work has proved that the consensus problems on the fractal network researched by the spectral method has extremely important applications in the field of science and engineering [45][46][47][48], and the fractal cactus network model has also been proved to play an important role in the study of molecular polymer networks and other issues [43], so the study of the spectrum of the Laplacian matrix and consensus problems of this extended network models has potential application value. For example, if a multi-agent system is constructed using the fractal network model, the convergence speed, delay robustness and noise robustness of the system can be analyzed by the spectrum of the Laplacian matrix of the network.…”

“…In this paper, the eigenvalue iteration relation of the Laplacian matrix corresponding to fractal network G m (t) is calculated by using the decimation procedure [49][50][51], and the spectrum corresponding to the matrix after any iteration is obtained. On the basis of obtaining the spectrum of the network G m (t), we further studied the consensus problem and solved the convergence speed, delay robustness, and noise robustness including first-order and second-order noise consensus on the system built according to network model G m (t) which play a key role in the research of traffic network, sensor network, multi-agent system and polymer network [18][19][20][21]43]. It is worth noting that when the coefficient m=3, the network G 3 (t) is the fractal cactus model, so the properties and applications of the Laplacian spectrum of the network G m (t) are also applicable to the fractal cactus model [43].…”