Theoretical models of Scale-free Polymer Networks - Ribeiro, Marcus Vinicius Alves (2023)
Scale-free Networks (SFNs) are structures built with nodes that show a degree distribution that follows a power law. SFNs are used with great success in several real networks. In this work, the networks are modeled from an algorithm that constructs scale-free networks without loops by changing the minimum, Kmin and the maximum, K max , allowed degrees, γ, which measures the density of links and N , total number of monomers. In this work, we will study the theoretical polymers dynamics models focused on Generalized Scale-Free Networks (GSFNs) on arbitrary tree-like polymers. For the Rouse Model, we monitor the influence of each of the parameters K min , Kmax , γ, and N . In the Semiflexible Model, which fixes the angles between the bonds between the nearest neighbors, we add one more parameter: the stiffness parameter q. In the Copolymer Model, we consider of the parameters: η = NA/NB, the ratio between the number of monomers of type A and B, and σ = ζ A /ζ B , where ζ A and ζ B are friction constants of monomers of the A and B, respectively. In all the cases, we will analyze the eigenvalue (λ) spectra of the connectivity matrix A and the dynamical behavior of these networks, focusing on the Complex Dynamic Modulus, with its two parts: the Storage Modulus (G‘) and the Loss Modulus (G”) and on the average displacement << Y (t) >>. For eigenvalues, we can notice the influence of the parameters of the Rouse and Flexibility model in terms of the degeneracy of λ = 1. Differently, in the Copolymer Model, we have two eigenvalues with high degeneracy: = 1 and λ = σ. In all models are encountered two situations: connectivity-independent behaviors at very small and very large ω, namely for very small ω one has G’(ω) ∼ ω 2 and G ‘’ (ω) ∼ ω 1 , for very large ω has G 0’(ω) ∼ ω 0 and G ‘’(ω) ∼ ω −1 .