Relaxation dynamics of scale-free polymer networks

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.

Exploring the applications of fractional calculus: Hierarchically built semiflexible polymers

“…Now, for several classes of polymers the role of their topology on the dynamics in external fields has been studied using the standard GGS formalism. This classes include stars and dendritic polymers [3], scale-free polymer networks [4,5], hyperbranched polymers modeled by Vicsek fractals [6,7], Sierpinski gaskets [8], and also multihierarchical fractals [9]. In this respect regular fractal structures are of much interest since their dynamical properties may display scaling.…”

Dynamics of Dual Scale-Free Polymer Networks

“…[13]. Being the dual structures [14] of treelike scale-free networks, which have a power-law distribution for their degrees [4,9,15,16], our networks contain dual units with their sizes following the same power-law decay. The limiting topologies that one can get as a function of this power-law exponent, γ, are networks made of huge dual units for very low values of γ and linear chains for very high values of γ.…”

“…In particular the concept of scale-free networks was applied with great success to World Wide Web [1,2], metabolic networks in biological organisms [3], reaction-diffusion processes [4], financial networks [5], and transport networks [6,7], to name only a few, but also to model hyperbranched polymers [8,9]. Inspired by recent experimental techniques allowing to make chemical transformations from hyperbranched polymers to functional core-shell nanogel systems [10] as well as to being interested in the fundamental role of the presence of loops in the polymer networks (e.g.…”

“…In this article we construct the scale-free networks by following the algorithm developed elsewhere [9], but differently from this reference here we consider their dual structures. This means that instead of adding a node with functionality k we construct a ring or a ring with additional bonds (à la small-world network [21,27,28]) consisting of k nodes.…”

“…In this way, we stick for p = 0 to the ring limit and for p = 1 we obtain complete graphs. The relaxation dynamics of these networks is studied in the framework of generalized Gaussian structures (GGS) [9,[17][18][19][20][21][22][23][24][25], which concentrates on the role of connectivity of the structures. In the model, the monomers are visualized as beads experiencing viscous friction connected only to their nearest neighbours by means of elastic springs.…”