Viscoelasticity of networks consisting of crosslinked or entangled macromolecules. I. normal modes and mechanical spectra

A molecular theory is developed to describe quantitatively the mechanical behavior of entanglement networks of linear, randomly coiling molecules. The theory is based on the model of Rouse for a single molecule and is a generalization of the theory of Duiser and Staverman for chemically crosslinked networks. A new model for an "entanglement point" is suggested, which accounts for the frictional force due to the velocity difference between two entangled molecules at that point. This leads to a modification of the diffusion equation in the Rouse theory. It turns out that the relaxation spectrum of a network corresponds to that of an assembly of "decoupled" equivalent molecules in which the ends of some submolecules have a mobility which is smaller by a slip factor S. This parameter may have values from zero (crosslinked molecules) to unity (free molecules in dilute solution). The relaxation spectra of the decoupled equivalent molecules can be obtained by applying a property of the Sturm sequence in a computer program, without actually solving for the eigenvalues of the modified matrix in the Rouse diffusion equation. Application of the theory to experimental data yields the number of elastically effective network chains of an entanglement network by integration of the area under the curve in the long-time region of the relaxation spectrum