1
|
Jurjiu A, Gomes Maia Júnior D, Galiceanu M. Relaxation dynamics of generalized scale-free polymer networks. Sci Rep 2018; 8:3731. [PMID: 29487316 PMCID: PMC5829225 DOI: 10.1038/s41598-018-21968-9] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/09/2017] [Accepted: 02/13/2018] [Indexed: 11/25/2022] Open
Abstract
We focus on treelike generalized scale-free polymer networks, whose geometries depend on a parameter, γ, that controls their connectivity and on two modularity parameters: the minimum allowed degree, Kmin, and the maximum allowed degree, Kmax. We monitor the influence of these parameters on the static and dynamic properties of the achieved generalized scale-free polymer networks. The relaxation dynamics is studied in the framework of generalized Gaussian structures model by employing the Rouse-type approach. The dynamical quantities on which we focus are the average monomer displacement under external forces and the mechanical relaxation moduli (storage and loss modulus), while for the static and structure properties of these networks we concentrate on the eigenvalue spectrum, diameter, and degree correlations. Depending on the values of network’s parameters we were able to switch between distinct hyperbranched structures: networks with more linearlike segments or with a predominant star or dendrimerlike topology. We have observed a stronger influence on Kmin than on Kmax. In the intermediate time (frequency) domain, all physical quantities obey power-laws for polymer networks with γ = 2.5 and Kmin = 2 and we prove additionally that for networks with γ ≥ 2.5 new regions with constant slope emerge by a proper choice of Kmin. Remarkably, we show that for certain values of the parameter set one may obtain self-similar networks.
Collapse
Affiliation(s)
- Aurel Jurjiu
- Department of Condensed Matter Physics and Advanced Technologies, Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084, Cluj-Napoca, Romania.
| | | | - Mircea Galiceanu
- Departamento de Física, Universidade Federal do Amazonas, 69077-000, Manaus, Brazil.
| |
Collapse
|
2
|
Jurjiu A, Turcu F, Galiceanu M. Dynamics of a Complex Multilayer Polymer Network: Mechanical Relaxation and Energy Transfer. Polymers (Basel) 2018; 10:E164. [PMID: 30966200 PMCID: PMC6415159 DOI: 10.3390/polym10020164] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/25/2017] [Revised: 02/04/2018] [Accepted: 02/06/2018] [Indexed: 01/12/2023] Open
Abstract
In this paper, we focus on the mechanical relaxation of a multilayer polymer network built by connecting identical layers that have, as underlying topologies, the dual Sierpinski gasket and the regular dendrimer. Additionally, we analyze the dynamics of dipolar energy transfer over a system of chromophores arranged in the form of a multilayer network. Both dynamical processes are studied in the framework of the generalized Gaussian structure (GSS) model. We develop a method whereby the whole eigenvalue spectrum of the connectivity matrix of the multilayer network can be determined iteratively, thereby rendering possible the analysis of the dynamics of networks consisting of a large number of layers. This fact allows us to study in detail the crossover from layer-like behavior to chain-like behavior. Remarkably, we highlight the existence of two bulk-like behaviors. The theoretical findings with respect to the decomposition of the intermediate domain of the relaxation quantities, as well as the chain-like behavior, are well supported by experimental results.
Collapse
Affiliation(s)
- Aurel Jurjiu
- Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania.
| | - Flaviu Turcu
- Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania.
| | - Mircea Galiceanu
- Department of Physics, Federal University of Amazonas, 69077-000 Manaus, Brazil.
| |
Collapse
|
3
|
Jurjiu A, Biter TL, Turcu F. Dynamics of a Polymer Network Based on Dual Sierpinski Gasket and Dendrimer: A Theoretical Approach. Polymers (Basel) 2017; 9:E245. [PMID: 30970922 PMCID: PMC6432022 DOI: 10.3390/polym9070245] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/30/2017] [Revised: 06/17/2017] [Accepted: 06/21/2017] [Indexed: 11/16/2022] Open
Abstract
In this paper we focus on the relaxation dynamics of a multihierarchical polymer network built through the replication of the dual Sierpinski gasket in the form of a regular dendrimer. The relaxation dynamics of this multihierarchical structure is investigated in the framework of the generalized Gaussian structure model using both Rouse and Zimm approaches. In the Rouse-type approach, we show a method whereby the whole eigenvalue spectrum of the connectivity matrix of the multihierarchical structure can be determined iteratively, thereby rendering possible the analysis of the Rouse-dynamics at very large generations. Remarkably, the general picture that emerges from both approaches, even though we have a mixed growth algorithm and the monomers interactions are taken into account specifically to the adopted approach, is that the multihierarchical structure preserves the individual relaxation behaviors of its constituent components. The theoretical findings with respect to the splitting of the intermediate domain of the relaxation quantities are well supported by experimental results.
Collapse
Affiliation(s)
- Aurel Jurjiu
- Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania.
| | - Teodor-Lucian Biter
- Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania.
| | - Flaviu Turcu
- Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania.
| |
Collapse
|
4
|
Jurjiu A, Biter TL, Turcu F. Relaxation dynamics of a multihierarchical polymer network. J Chem Phys 2017; 146:034902. [PMID: 28109236 DOI: 10.1063/1.4973936] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/02/2023] Open
Abstract
In this work, we study the relaxation dynamics of a multihierarchical polymer network built by replicating the Vicsek fractal in dendrimer shape. The relaxation dynamics is investigated in the framework of the generalized Gaussian structure model by employing both Rouse and Zimm approaches. In the Rouse-type approach, we show the iterative procedure whereby the whole eigenvalue spectrum of the connectivity matrix of the multihierarchical structure can be obtained. Remarkably, the general picture that emerges from both approaches, even though we have a mixed growth algorithm, is that the obtained multihierarchical structure preserves the individual relaxation behaviors of its components. The theoretical findings with respect to the splitting of the intermediate domain of the relaxation quantities are well supported by experimental results.
Collapse
Affiliation(s)
- Aurel Jurjiu
- Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania
| | - Teodor Lucian Biter
- Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania
| | - Flaviu Turcu
- Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania
| |
Collapse
|
5
|
Jurjiu A, Galiceanu M, Farcasanu A, Chiriac L, Turcu F. Relaxation dynamics of Sierpinski hexagon fractal polymer: Exact analytical results in the Rouse-type approach and numerical results in the Zimm-type approach. J Chem Phys 2016; 145:214901. [PMID: 28799361 DOI: 10.1063/1.4968209] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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.
Collapse
Affiliation(s)
- Aurel Jurjiu
- Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania
| | - Mircea Galiceanu
- Departamento de Fisica, Universidade Federal do Amazonas, 69077-000 Manaus, Brazil
| | - Alexandru Farcasanu
- Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania
| | - Liviu Chiriac
- Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania
| | - Flaviu Turcu
- Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania
| |
Collapse
|
6
|
Dolgushev M, Liu H, Zhang Z. Extended Vicsek fractals: Laplacian spectra and their applications. Phys Rev E 2016; 94:052501. [PMID: 27967151 DOI: 10.1103/physreve.94.052501] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2016] [Indexed: 06/06/2023]
Abstract
Extended Vicsek fractals (EVF) are the structures constructed by introducing linear spacers into traditional Vicsek fractals. Here we study the Laplacian spectra of the EVF. In particularly, the recurrence relations for the Laplacian spectra allow us to obtain an analytic expression for the sum of all inverse nonvanishing Laplacian eigenvalues. This quantity characterizes the large-scale properties, such as the gyration radius of the polymeric structures, or the global mean-first passage time for the random walk processes. Introduction of the linear spacers leads to local heterogeneities, which reveal themselves, for example, in the dynamics of EVF under external forces.
Collapse
Affiliation(s)
- Maxim Dolgushev
- Institute of Physics, University of Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany
- Institut Charles Sadron, Université de Strasbourg and CNRS, 23 rue du Loess, 67034 Strasbourg Cedex, France
| | - Hongxiao Liu
- School of Computer Science, Fudan University, Shanghai 200433, China
- Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
| | - Zhongzhi Zhang
- School of Computer Science, Fudan University, Shanghai 200433, China
- Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
| |
Collapse
|
7
|
Galiceanu M, Jurjiu A. Relaxation dynamics of multilayer triangular Husimi cacti. J Chem Phys 2016; 145:104901. [DOI: 10.1063/1.4962196] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
8
|
Liu H, Lin Y, Dolgushev M, Zhang Z. Dynamics of comb-of-comb networks. Phys Rev E 2016; 93:032502. [PMID: 27078400 DOI: 10.1103/physreve.93.032502] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/28/2015] [Indexed: 11/07/2022]
Abstract
The dynamics of complex networks, a current hot topic in many scientific fields, is often coded through the corresponding Laplacian matrix. The spectrum of this matrix carries the main features of the networks' dynamics. Here we consider the deterministic networks which can be viewed as "comb-of-comb" iterative structures. For their Laplacian spectra we find analytical equations involving Chebyshev polynomials whose properties allow one to analyze the spectra in deep. Here, in particular, we find that in the infinite size limit the corresponding spectral dimension goes as d(s) → 2. The d(s) leaves its fingerprint on many dynamical processes, as we exemplarily show by considering the dynamical properties of polymer networks, including single monomer displacement under a constant force, mechanical relaxation, and fluorescence depolarization.
Collapse
Affiliation(s)
- Hongxiao Liu
- School of Computer Science, Fudan University, Shanghai 200433, China.,Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
| | - Yuan Lin
- School of Computer Science, Fudan University, Shanghai 200433, China.,Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
| | - Maxim Dolgushev
- Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany.,Institut Charles Sadron, Université de Strasbourg and CNRS, 23 rue du Loess, 67034 Strasbourg Cedex, France
| | - Zhongzhi Zhang
- School of Computer Science, Fudan University, Shanghai 200433, China.,Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
| |
Collapse
|
9
|
Xie P, Lin Y, Zhang Z. Spectrum of walk matrix for Koch network and its application. J Chem Phys 2015; 142:224106. [PMID: 26071700 DOI: 10.1063/1.4922265] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Various structural and dynamical properties of a network are encoded in the eigenvalues of walk matrix describing random walks on the network. In this paper, we study the spectra of walk matrix of the Koch network, which displays the prominent scale-free and small-world features. Utilizing the particular architecture of the network, we obtain all the eigenvalues and their corresponding multiplicities. Based on the link between the eigenvalues of walk matrix and random target access time defined as the expected time for a walker going from an arbitrary node to another one selected randomly according to the steady-state distribution, we then derive an explicit solution to the random target access time for random walks on the Koch network. Finally, we corroborate our computation for the eigenvalues by enumerating spanning trees in the Koch network, using the connection governing eigenvalues and spanning trees, where a spanning tree of a network is a subgraph of the network, that is, a tree containing all the nodes.
Collapse
Affiliation(s)
- Pinchen Xie
- School of Computer Science, Fudan University, Shanghai 200433, China
| | - Yuan Lin
- School of Computer Science, Fudan University, Shanghai 200433, China
| | - Zhongzhi Zhang
- School of Computer Science, Fudan University, Shanghai 200433, China
| |
Collapse
|
10
|
Zhang Z, Lin Y, Guo X. Eigenvalues for the transition matrix of a small-world scale-free network: Explicit expressions and applications. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:062808. [PMID: 26172755 DOI: 10.1103/physreve.91.062808] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/30/2014] [Indexed: 06/04/2023]
Abstract
The eigenvalues of the transition matrix for random walks on a network play a significant role in the structural and dynamical aspects of the network. Nevertheless, it is still not well understood how the eigenvalues behave in small-world and scale-free networks, which describe a large variety of real systems. In this paper, we study the eigenvalues for the transition matrix of a network that is simultaneously scale-free, small-world, and clustered. We derive explicit simple expressions for all eigenvalues and their multiplicities, with the spectral density exhibiting a power-law form. We then apply the obtained eigenvalues to determine the mixing time and random target access time for random walks, both of which exhibit unusual behaviors compared with those for other networks, signaling discernible effects of topologies on spectral features. Finally, we use the eigenvalues to count spanning trees in the network.
Collapse
Affiliation(s)
- Zhongzhi Zhang
- School of Computer Science, Fudan University, Shanghai 200433, China and Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
| | - Yuan Lin
- School of Computer Science, Fudan University, Shanghai 200433, China and Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
| | - Xiaoye Guo
- School of Computer Science, Fudan University, Shanghai 200433, China and Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
| |
Collapse
|
11
|
Fürstenberg F, Gurtovenko AA, Dolgushev M, Blumen A. Molecular Dynamics Simulations of Hyperbranched PAMAM Vicsek Fractals. MACROMOL THEOR SIMUL 2014. [DOI: 10.1002/mats.201400063] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/16/2023]
Affiliation(s)
- Florian Fürstenberg
- Theoretical Polymer Physics; University of Freiburg; Hermann-Herder-Str. 3 D-79104 Freiburg Germany
| | - Andrey A. Gurtovenko
- Institute of Macromolecular Compounds; Russian Academy of Sciences; Bolshoi pr. V.O. 31 St.Petersburg 199004 Russia
- Faculty of Physics; St.Petersburg State University; Ul'yanovskaya ul. 1 Petrodvorets St.Petersburg 198504 Russia
| | - Maxim Dolgushev
- Theoretical Polymer Physics; University of Freiburg; Hermann-Herder-Str. 3 D-79104 Freiburg Germany
| | - Alexander Blumen
- Theoretical Polymer Physics; University of Freiburg; Hermann-Herder-Str. 3 D-79104 Freiburg Germany
| |
Collapse
|
12
|
Galiceanu M, Reis AS, Dolgushev M. Dynamics of semiflexible scale-free polymer networks. J Chem Phys 2014; 141:144902. [DOI: 10.1063/1.4897563] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
13
|
Zhang Z, Guo X, Lin Y. Full eigenvalues of the Markov matrix for scale-free polymer networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:022816. [PMID: 25215790 DOI: 10.1103/physreve.90.022816] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/06/2014] [Indexed: 06/03/2023]
Abstract
Much important information about the structural and dynamical properties of complex systems can be extracted from the eigenvalues and eigenvectors of a Markov matrix associated with random walks performed on these systems, and spectral methods have become an indispensable tool in the complex system analysis. In this paper, we study the Markov matrix of a class of scale-free polymer networks. We present an exact analytical expression for all the eigenvalues and determine explicitly their multiplicities. We then use the obtained eigenvalues to derive an explicit formula for the random target access time for random walks on the studied networks. Furthermore, based on the link between the eigenvalues of the Markov matrix and the number of spanning trees, we confirm the validity of the obtained eigenvalues and their corresponding degeneracies.
Collapse
Affiliation(s)
- Zhongzhi Zhang
- School of Computer Science and Shanghai Key Lab of Intelligent Information Processing, Fudan University, Shanghai 200433, China
| | - Xiaoye Guo
- School of Computer Science and Shanghai Key Lab of Intelligent Information Processing, Fudan University, Shanghai 200433, China
| | - Yuan Lin
- School of Computer Science and Shanghai Key Lab of Intelligent Information Processing, Fudan University, Shanghai 200433, China
| |
Collapse
|
14
|
Jurjiu A, Dockhorn R, Mironova O, Sommer JU. Two universality classes for random hyperbranched polymers. SOFT MATTER 2014; 10:4935-4946. [PMID: 24882064 DOI: 10.1039/c4sm00711e] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
We grow AB2 random hyperbranched polymer structures in different ways and using different simulation methods. In particular we use a method of ad hoc construction of the connectivity matrix and the bond fluctuation model on a 3D lattice. We show that hyperbranched polymers split into two universality classes depending on the growth process. For a "slow growth" (SG) process where monomers are added sequentially to an existing molecule which strictly avoids cluster-cluster aggregation the resulting structures share all characteristic features with regular dendrimers. For a "quick growth" (QG) process which allows for cluster-cluster aggregation we obtain structures which can be identified as random fractals. Without excluded volume interactions the SG model displays a logarithmic growth of the radius of gyration with respect to the degree of polymerization while the QG model displays a power law behavior with an exponent of 1/4. By analyzing the spectral properties of the connectivity matrix we confirm the behavior of dendritic structures for the SG model and the corresponding fractal properties in the QG case. A mean field model is developed which explains the extension of the hyperbranched polymers in an athermal solvent for both cases. While the radius of gyration of the QG model shows a power-law behavior with the exponent value close to 4/5, the corresponding result for the SG model is a mixed logarithmic-power-law behavior. These different behaviors are confirmed by simulations using the bond fluctuation model. Our studies indicate that random sequential growth according to our SG model can be an alternative to the synthesis of perfect dendrimers.
Collapse
Affiliation(s)
- A Jurjiu
- Leibniz Institut für Polymerforschung Dresden e.V., Hohe Strasse 6, D-01069, Dresden, Germany.
| | | | | | | |
Collapse
|
15
|
Galiceanu M. Hydrodynamic effects on scale-free polymer networks in external fields. J Chem Phys 2014; 140:034901. [DOI: 10.1063/1.4861218] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
16
|
Liu H, Zhang Z. Laplacian spectra of recursive treelike small-world polymer networks: analytical solutions and applications. J Chem Phys 2013; 138:114904. [PMID: 23534659 DOI: 10.1063/1.4794921] [Citation(s) in RCA: 43] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A central issue in the study of polymer physics is to understand the relation between the geometrical properties of macromolecules and various dynamics, most of which are encoded in the Laplacian spectra of a related graph describing the macrostructural structure. In this paper, we introduce a family of treelike polymer networks with a parameter, which has the same size as the Vicsek fractals modeling regular hyperbranched polymers. We study some relevant properties of the networks and show that they have an exponentially decaying degree distribution and exhibit the small-world behavior. We then study the Laplacian eigenvalues and their corresponding eigenvectors of the networks under consideration, with both quantities being determined through the recursive relations deduced from the network structure. Using the obtained recursive relations we can find all the eigenvalues and eigenvectors for the networks with any size. Finally, as some applications, we use the eigenvalues to study analytically or semi-analytically three dynamical processes occurring in the networks, including random walks, relaxation dynamics in the framework of generalized Gaussian structure, as well as the fluorescence depolarization under quasiresonant energy transfer. Moreover, we compare the results with those corresponding to Vicsek fractals, and show that the dynamics differ greatly for the two network families, which thus enables us to distinguish between them.
Collapse
Affiliation(s)
- Hongxiao Liu
- School of Computer Science, Fudan University, Shanghai 200433, China
| | | |
Collapse
|
17
|
Wu B, Zhang Z. Controlling the efficiency of trapping in treelike fractals. J Chem Phys 2013; 139:024106. [DOI: 10.1063/1.4812690] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
18
|
Julaiti A, Wu B, Zhang Z. Eigenvalues of normalized Laplacian matrices of fractal trees and dendrimers: Analytical results and applications. J Chem Phys 2013; 138:204116. [DOI: 10.1063/1.4807589] [Citation(s) in RCA: 49] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
|
19
|
Lin Y, Zhang Z. Influence of trap location on the efficiency of trapping in dendrimers and regular hyperbranched polymers. J Chem Phys 2013; 138:094905. [DOI: 10.1063/1.4793309] [Citation(s) in RCA: 37] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
20
|
Fürstenberg F, Dolgushev M, Blumen A. Dynamics of semiflexible regular hyperbranched polymers. J Chem Phys 2013; 138:034904. [DOI: 10.1063/1.4775584] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
|
21
|
Galiceanu M. Relaxation dynamics of scale-free polymer networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:041803. [PMID: 23214606 DOI: 10.1103/physreve.86.041803] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/29/2012] [Revised: 09/10/2012] [Indexed: 06/01/2023]
Abstract
We focus on polymer networks with a scale-free topology. In the framework of generalized Gaussian structures, by making use of the eigenvalue spectrum of the connectivity matrix, we determined numerically the averaged monomer displacement under external forces and the mechanical relaxation moduli (storage and loss modulus). First, we monitor these physical quantities and additionally the eigenvalue spectrum for structures of different sizes, but with the same γ, which is a parameter that measures the connectivity of the structure. Second, we vary the parameter γ, and we keep constant the size of the structures. This allows us to study in detail the crossover behavior from a simple linear chain to a starlike structure. As expected we encounter a more chainlike behavior for high values of γ, while for small values of γ a more starlike behavior is observed. In the intermediate time (frequency) domain, we encounter regions of constant slope for some intermediate values of γ.
Collapse
Affiliation(s)
- M Galiceanu
- Departamento de Física, Universidade Federal do Amazonas, 69077-000 Manaus, Brazil.
| |
Collapse
|
22
|
Wu B, Lin Y, Zhang Z, Chen G. Trapping in dendrimers and regular hyperbranched polymers. J Chem Phys 2012; 137:044903. [DOI: 10.1063/1.4737635] [Citation(s) in RCA: 51] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
23
|
Fürstenberg F, Dolgushev M, Blumen A. Analytical model for the dynamics of semiflexible dendritic polymers. J Chem Phys 2012; 136:154904. [PMID: 22519347 DOI: 10.1063/1.3703757] [Citation(s) in RCA: 36] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Florian Fürstenberg
- Theoretical Polymer Physics, University of Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany.
| | | | | |
Collapse
|