Dynamics of Vicsek fractals, models for hyperbranched polymers

Dynamics of a Polymer Network Modeled by a Fractal Cactus

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.

Scaling of average receiving time on weighted polymer networks with some topological properties

In this paper, a family of the weighted polymer networks is introduced depending on the number of copies f and a weight factor r. The topological properties of weighted polymer networks can be completely analytically characterized in terms of the involved parameters and/or of the fractal dimension. Moreover, assuming that the walker, at each step, starting from its current node, moves to any of its neighbors with probability proportional to the weight of edge linking them, namely weight-dependent walk. Then, we calculate the average receiving time (ART) with weighted-dependent walks, which is the sum of mean first-passage times (MFPTs) for all nodes absorpt at the trap located at the central node as a recursive relation. The obtained remarkable results display that when [Formula: see text], the ART grows sublinearly with the network size; when [Formula: see text], ART grows with increasing size N _g as [Formula: see text]; when [Formula: see text], ART grows with increasing size N _g as ln N _g . In the treelike polymer networks, ART grows with linearly with the network size N _g when r = 1. Thus, the weighted polymer networks are more efficient than treelike polymer networks in receiving information.

Marginally compact hyperbranched polymer trees

Assuming Gaussian chain statistics along the chain contour, we generate by means of a proper fractal generator hyperbranched polymer trees which are marginally compact. Static and dynamical properties, such as the radial intrachain pair density distribution ρ_pair(r) or the shear-stress relaxation modulus G(t), are investigated theoretically and by means of computer simulations. We emphasize that albeit the self-contact density diverges logarithmically with the total mass N, this effect becomes rapidly irrelevant with increasing spacer length S. In addition to this it is seen that the standard Rouse analysis must necessarily become inappropriate for compact objects for which the relaxation time τ_p of mode p must scale as τ_p ∼ (N/p)^5/3 rather than the usual square power law for linear chains.

Relaxation dynamics of a multihierarchical polymer network

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.

Extended Vicsek fractals: Laplacian spectra and their applications

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.

Local orientational mobility in regular hyperbranched polymers

We study the dynamics of local bond orientation in regular hyperbranched polymers modeled by Vicsek fractals. The local dynamics is investigated through the temporal autocorrelation functions of single bonds and the corresponding relaxation forms of the complex dielectric susceptibility. We show that the dynamic behavior of single segments depends on their remoteness from the periphery rather than on the size of the whole macromolecule. Remarkably, the dynamics of the core segments (which are most remote from the periphery) shows a scaling behavior that differs from the dynamics obtained after structural average. We analyze the most relevant processes of single segment motion and provide an analytic approximation for the corresponding relaxation times. Furthermore, we describe an iterative method to calculate the orientational dynamics in the case of very large macromolecular sizes.

Anomalous behavior of trapping in extended dendrimers with a perfect trap

Compact and extended dendrimers are two important classes of dendritic polymers. The impact of the underlying structure of compact dendrimers on dynamical processes has been much studied, yet the relation between the dynamical and structural properties of extended dendrimers remains not well understood. In this paper, we study the trapping problem in extended dendrimers with generation-dependent segment lengths, which is different from that of compact dendrimers where the length of the linear segments is fixed. We first consider a particular case that the deep trap is located at the central node, and derive an exact formula for the average trapping time (ATT) defined as the average of the source-to-trap mean first passage time over all starting points. Then, using the obtained result we deduce a closed-form expression for the ATT to an arbitrary trap node, based on which we further obtain an explicit solution to the ATT corresponding to the trapping issue with the trap uniformly distributed in the polymer systems. We show that the trap location has a substantial influence on the trapping efficiency measured by the ATT, which increases with the shortest distance from the trap to the central node, a phenomenon similar to that for compact dendrimers. In contrast to this resemblance, the leading terms of ATTs for the three trapping problems differ drastically between extended and compact dendrimers, with the trapping processes in the extended dendrimers being less efficient than in compact dendrimers.

Spectrum of walk matrix for Koch network and its application

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.

Eigenvalues for the transition matrix of a small-world scale-free network: Explicit expressions and applications

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.

Dynamics of semiflexible recursive small-world polymer networks

One of the fundamental issues in polymer physics is to reveal the relation between the structures of macromolecules and their various properties. In this report, we study the dynamical properties of a family of deterministically growing semiflexible treelike polymer networks, which are built in an iterative method. From the analysis of the corresponding dynamical matrix we derive the solution for its eigenvalues and their multiplicities, making use of a combined numerical and analytical approach. The eigenvalue spectra allow us to investigate the mechanical relaxation forms in depth for different values of the stiffness parameter. We observe that the dynamics of semiflexible networks is sensitive to the stiffness parameter. Our work paves a way to explore the structures of the highly symmetric polymers and provides a comprehensive understanding of the role of semiflexibility for the regular treelike networks which possess a small-world feature.

Full eigenvalues of the Markov matrix for scale-free polymer networks

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.

Two universality classes for random hyperbranched polymers

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.

Hyperbranched polymer stars with Gaussian chain statistics revisited

Conformational properties of regular dendrimers and more general hyperbranched polymer stars with Gaussian statistics for the spacer chains between branching points are revisited numerically. We investigate the scaling for asymptotically long chains especially for fractal dimensions df = 3 (marginally compact) and df = 2.5 (diffusion limited aggregation). Power-law stars obtained by imposing the number of additional arms per generation are compared to truly self-similar stars. We discuss effects of weak excluded-volume interactions and sketch the regime where the Gaussian approximation should hold in dense solutions and melts for sufficiently large spacer chains.

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

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.

Relaxation dynamics of scale-free polymer networks

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.

Topology and energy transport in networks of interacting photosynthetic complexes

We address the role of topology in the energy transport process that occurs in networks of photosynthetic complexes. We take inspiration from light-harvesting networks present in purple bacteria and simulate an incoherent dissipative energy transport process on more general and abstract networks, considering both regular structures (Cayley trees and hyperbranched fractals) and randomly generated ones. We focus on the the two primary light-harvesting complexes of purple bacteria, i.e., the LH1 and LH2, and we use network-theoretical centrality measures in order to select different LH1 arrangements. We show that different choices cause significant differences in the transport efficiencies, and that for regular networks, centrality measures allow us to identify arrangements that ensure transport efficiencies which are better than those obtained with a random disposition of the complexes. The optimal arrangements strongly depend on the dissipative nature of the dynamics and on the topological properties of the networks considered, and depending on the latter, they are achieved by using global versus local centrality measures. For randomly generated networks, a random arrangement of the complexes already provides efficient transport, and this suggests the process is strong with respect to limited amount of control in the structure design and to the disorder inherent in the construction of randomly assembled structures. Finally, we compare the networks considered with the real biological networks and find that the latter have in general better performances, due to their higher connectivity, but the former with optimal arrangements can mimic the real networks' behavior for a specific range of transport parameters. These results show that the use of network-theoretical concepts can be crucial for the characterization and design of efficient artificial energy transport networks.

Dynamic structure factor of vibrating fractals

Motivated by novel experimental work and the lack of an adequate theory, we study the dynamic structure factor S(k,t) of large vibrating fractal networks at large wave numbers k. We show that the decay of S(k,t) is dominated by the spatially averaged mean square displacement of a network node, which evolves subdiffusively in time, ((u[over →](i)(t)-u[over →](i)(0))(2))∼t(ν), where ν depends on the spectral dimension d(s) and fractal dimension d(f). As a result, S(k,t) decays as a stretched exponential S(k,t)≈S(k)e(-(Γ(k)t)(ν)) with Γ(k)∼k(2/ν). Applications to a variety of fractal-like systems are elucidated.

Dynamic structure factor of vibrating fractals: proteins as a case study

We study the dynamic structure factor S(k,t) of proteins at large wave numbers k, kR(g)≫1, where R(g) is the gyration radius. At this regime measurements are sensitive to internal dynamics, and we focus on vibrational dynamics of folded proteins. Exploiting the analogy between proteins and fractals, we perform a general analytic calculation of the displacement two-point correlation functions, <[u(−>)(i)(t)-u(−>)(j)(0)](2)>. We confront the derived expressions with numerical evaluations that are based on protein data bank (PDB) structures and the Gaussian network model (GNM) for a few proteins and for the Sierpinski gasket as a controlled check. We use these calculations to evaluate S(k,t) with arrested rotational and translational degrees of freedom, and show that the decay of S(k,t) is dominated by the spatially averaged mean-square displacement of an amino acid. The latter has been previously shown to evolve subdiffusively in time, <[u(−>)(i)(t)-u(−>)(i)(0)](2)> ~t(ν), where ν is the anomalous diffusion exponent that depends on the spectral dimension d(s) and fractal dimension d(f). As a result, for wave numbers obeying k(2)<u(−>)(2)>≳1, S(k,t) effectively decays as a stretched exponential S(k,t)≃S(k)e(-(Γ(k)t)(β)) with β≃ν, where the relaxation rate is Γ(k)~(k(B)T/mω(o)(2))(1/β)k(2/β), T is the temperature, and mω(o)(2) the GNM effective spring constant describing the interaction between neighboring amino acids. The static structure factor is dominated by the fractal character of the native fold, S(k)~k(-d(f)), with negligible to marginal influence of vibrations. The analytical expressions are first confronted with numerically based calculations on the Sierpinski gasket, and very good agreement is found between simulations and theory. We then perform PDB-GNM-based numerical calculations for a few proteins, and an effective stretched exponential decay of the dynamic structure factor is found, albeit their relatively small size. However, when rotational and translational diffusion are added, we find that their contribution is never negligible due to finite size effects. While we can still attribute an effective stretching exponent β to the relaxation profile, this exponent is significantly larger than the anomalous diffusion exponent ν. We compare our theory with recent neutron spin-echo studies of myoglobin and hemoglobin and conclude that experiments in which the rotational and translational degrees of freedom are arrested, e.g., by anchoring the proteins to a surface, will improve the detection of internal vibrational dynamics.

Relaxation dynamics of a polymer network modeled by a multihierarchical structure

We numerically analyze the scaling behavior of experimentally accessible dynamical relaxation forms for polymer networks modeled by a finite multihierarchical structure. In the framework of generalized Gaussian structures, by making use of the eigenvalue spectrum of the connectivity matrix, we determine the averaged monomer displacement under local external forces as well as the mechanical relaxation quantities (storage and loss moduli). Hence we generalize the known analysis for both classes of fractals to the case of multihierarchical structure, for which even though we have a mixed growth algorithm, the above cited observables still give information about the two different underlying topologies. For very large lattices, reached via an algebraic procedure that avoids the numerical diagonalizations of the corresponding connectivity matrices, we depict the scaling of both component fractals in the intermediate time (frequency) domain, which manifests two different slopes.

Proteins as fractals: role of the hydrodynamic interaction

Exploiting the fractal nature of folded proteins, we study the effect of the hydrodynamic interaction between amino acids using a Zimm-type model. We compute the time-dependent mean square displacement of an amino acid and the time-dependent autocorrelation function of the distance between two amino acids, and we show that these dynamic quantities evolve anomalously, similar to the Rouse-type behavior, yet with modified dynamic exponents. Good agreement is found with recent neutron spin-echo studies of myoglobin and hemoglobin.

Determining global mean-first-passage time of random walks on Vicsek fractals using eigenvalues of Laplacian matrices

The family of Vicsek fractals is one of the most important and frequently studied regular fractal classes, and it is of considerable interest to understand the dynamical processes on this treelike fractal family. In this paper, we investigate discrete random walks on the Vicsek fractals, with the aim to obtain the exact solutions to the global mean-first-passage time (GMFPT), defined as the average of first-passage time (FPT) between two nodes over the whole family of fractals. Based on the known connections between FPTs, effective resistance, and the eigenvalues of graph Laplacian, we determine implicitly the GMFPT of the Vicsek fractals, which is corroborated by numerical results. The obtained closed-form solution shows that the GMFPT approximately grows as a power-law function with system size (number of all nodes), with the exponent lies between 1 and 2. We then provide both the upper bound and lower bound for GMFPT of general trees, and show that the leading behavior of the upper bound is the square of system size and the dominating scaling of the lower bound varies linearly with system size. We also show that the upper bound can be achieved in linear chains and the lower bound can be reached in star graphs. This study provides a comprehensive understanding of random walks on the Vicsek fractals and general treelike networks.

Continuum diffusion on networks: trees with hyperbranched trunks and fractal branches

The probability that a random walker returns to its origin for large times scales as t;{-d[over ]2} , where d[over ] is the spectral dimension. We calculate d[over ] for a class of tree structures using a renormalization technique on an infinite continued fraction. We consider a wide range of homogeneous networks based on replacing the branches of a self-similar tree with arbitrary fractals and composite fractals. We also consider a new class of inhomogeneous hyperbranched trees.

Spacers’ role in the dynamics of hyperbranched polymers

We investigate hyperbranched polymers (HBPs) and highlight the relation between their architecture and their viscoelastic behavior, while paying special attention to the role of the chainlike spacer segments between branching points. For this we study the dynamics of HBP in solution, based on the generalized Gaussian structure formalism, an extension of the Rouse model, which disregards hydrodynamical and excluded volume effects. For HBP the dynamical effects display, beside the obvious contributions of localized modes on the spacers, also remarkable features, as we highlight based on the exact renormalization procedure recently developed by us in J. Chem. Phys. 123, 034907 (2005). We exemplify these features by analyzing the dynamics of randomly linked star polymers and study the impact both of the length and of the spacers' mobility on the normal modes' spectra. We compute these modes both by numerical diagonalization and also by employing our renormalization procedure; the excellent agreement between these methods allows us to extend the range of investigations to very large HBP.

Dynamics of end-linked star-polymer structures

In this work we focus on the dynamics of macromolecular networks formed by end-linking identical polymer stars. The resulting macromolecular network can then be viewed as consisting of spacers which connect branching points (the cores of the stars). We succeed in analyzing exactly, in the framework of the generalized Gaussian model, the eigenvalue spectrum of such networks. As applications we focus on several topologies, such as regular networks and dendrimers; furthermore, we compare the results to those found for regular hyperbranched structures. In so doing, we also consider situations in which the beads of the cores differ from the beads of the spacers. The analytical procedure which we use involves an exact real-space renormalization, which allows to relate the star network to a (much simpler) network, in which each star is reduced to its core. It turns out that the eigenvalue spectrum of the star-polymer structure consists of two parts: one follows in terms of polynomial equations from the relaxation spectrum of the corresponding renormalized structure, while the second part involves the motion of the spacer chains themselves. Finally, we show exemplarily the situation for copolymeric dendrimers, calculate their spectra, and from them their storage and the loss moduli.

Spacetime structures of continuous-time quantum walks

The propagation by continuous-time quantum walks (CTQWs) on one-dimensional lattices shows structures in the transition probabilities between different sites reminiscent of quantum carpets. For a system with periodic boundary conditions, we calculate the transition probabilities for a CTQW by diagonalizing the transfer matrix and by a Bloch function ansatz. Remarkably, the results obtained for the Bloch function ansatz can be related to results from (discrete) generalized coined quantum walks. Furthermore, we show that here the first revival time turns out to be larger than for quantum carpets.

Slow transport by continuous time quantum walks

Continuous time quantum walks (CTQWs) do not necessarily perform better than their classical counterparts, the continuous time random walks (CTRWs). For one special graph, where a recent analysis showed that in a particular direction of propagation the penetration of the graph is faster by CTQWs than by CTRWs, we demonstrate that in another direction of propagation the opposite is true. In this case a CTQW initially localized at one site displays a slow transport. We furthermore show that when the CTQW's initial condition is a totally symmetric superposition of states of equivalent sites, the transport gets to be much more rapid.

Dynamical scaling behavior of percolation clusters in scale-free networks

In this work we investigate the spectra of Laplacian matrices that determine many dynamic properties of scale-free networks below and at the percolation threshold. We use a replica formalism to develop analytically, based on an integral equation, a systematic way to determine the ensemble averaged eigenvalue spectrum for a general type of treelike networks. Close to the percolation threshold we find characteristic scaling functions for the density of states rho(lambda) of scale-free networks. rho(lambda) shows characteristic power laws rho (lambda) approximately lambda (alpha(1) ) or rho (lambda) approximately lambda (d(2) ) for small lambda, where alpha(1) holds below and alpha(2) at the percolation threshold. In the range where the spectra are accessible from a numerical diagonalization procedure the two methods lead to very similar results.