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.
Highlights
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 perform our calculations in the framework of the generalized Gaussian structures (GGSs) model which represents the extension of the Rouse model[13], developed for linear polymer chains, to polymer systems of
The rest of the paper is structured as follows: In Methods we describe our algorithm that creates the networks from a scale-free degree distribution with two additional modularity parameters and we briefly remind the formalism of GGS and solve the system of Langevin differential equations that governs the relaxation dynamics
Summary
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 introduce a new treelike structure that is able to map the transition from a predominant starlike architecture to a linear or dendrimerlike topology This transition is realized for a treelike scale-free polymer network by alternating two modularity parameters, namely the minimum and the maximum allowed degree. If the minimum and the maximum permitted degrees are varied a greater amount of possible architectures are encountered, ranging from starlike to dendrimerlike or linear topology In this way the control of the parameters, which is nothing else than the control of branch points and of the maximum permitted connections of a monomer, allows us to predict the type of the obtained structure; it is a long linear chain with small side chains attached or it is irregular (modified) dendrimer. The connectivity matrix, being the discrete version of the Laplacian operator, is greatly used in many areas of science; for instance, in graph theory applied to biological systems[27], reaction-diffusion systems[28], in the study of fluorescence depolarization under dipolar quasiresonant energy transfer[29,30], the dielectric relaxation functions[31], and the NMR relaxation functions[32,33]
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