Stress relaxation of near-critical gels

Abstract

The time-dependent stress relaxation for a Rouse model of a cross-linked polymer melt is completely determined by the spectrum of eigenvalues of the connectivity matrix. The latter has been computed analytically for a mean-field distribution of cross-links. It shows a Lifshitz tail for small eigenvalues and all concentrations below the percolation threshold, giving rise to a stretched exponential decay of the stress relaxation function in the sol phase. At the critical point the density of states is finite for small eigenvalues, resulting in a logarithmic divergence of the viscosity and an algebraic decay of the stress relaxation function. Numerical diagonalization of the connectivity matrix supports the analytical findings and has furthermore been applied to cluster statistics corresponding to random bond percolation in two and three dimensions.

On the Stochastic Dynamics of Disordered Spin Models

“…Finally, in several problems of physical interest, as the gelation process, dilute random matrices play an important role [15]. The methods and results here discussed will be relevant for these problems too.…”

Dynamical scaling behavior of percolation clusters in scale-free networks

“…(4) defines the so-called percolation threshold. It is very useful to extend the present model, by also allowing the strength of each bond to be weighted [19,20] following a normalized coupling strength distribution D(µ). Thus in the corresponding connectivity matrices, each of the nonzero values of C ik can be chosen according to the distribution D(µ).…”

“…We are interested in the dynamic behavior of random graphs with arbitrary degree distributions and hence in the density ρ(λ) of their eigenfrequencies. Following the ideas used in the analysis of gel dynamics [20] and hyperbranched polymers [19], we display an integral equation for ρ(λ) for a special class of random graphs with arbitrary degree distributions [13,31,32,33]. This integral equation allows us to determine ρ(λ) for the classes of scale-free networks discussed in Ref.…”

“…For scale-free networks it was found that the density of the eigenvalues of C has a triangular form with a power-law tail [14,15,16]. On the other hand, many problems ranging from the dynamics of randomly branched polymers [19] and the stress relaxation of near critical gels [20], over random resistor-capacitor networks [21] to glassy relaxation dynamics [22], depend on the Laplacian A; A is connected to C via:…”

“…A whole series of works based on A were devoted to classical deterministic and random graphs, such as Cayleytrees (dendrimers), hyperbranched macromolecules, the Erdös-Rényi (ER) random graph and bond diluted Cayley-trees [17,18,19,20,21,22,23,24,25,26,27,28,29]. Based on the Laplacian, many time and frequencydependent observables can be written as integrals over ρ(λ), the density of eigenvalues of A: The structure of these observables is either…”