Linear Viscoelasticity in Entangling Polymer Systems

Abstract

Molecular models for polymers, such as the Rouse model, yield steady state shear compliance $J_e$ and terminal relaxation time ${\tau }_{\max }$ of the forms $J_e\mathrm{,\; \propto M\hspace{0.17em}/\hspace{0.17em}cRT}$ and ${\tau }_{\max }\mathrm{\propto \eta M\hspace{0.17em}/\hspace{0.17em}cRT}$ , in which $c$ is the polymer concentration, $M$ is the molecular weight, and $\eta$ is the zero shear viscosity. Recent experiments have shown that in entangling systems these forms change, becoming $J_e{\mathrm{\propto \hspace{0.17em}/\hspace{0.17em}c}}^2\mathrm{RT}$ and ${\tau }_{\max }{\mathrm{\propto \eta \hspace{0.17em}/\hspace{0.17em}c}}^2\mathrm{RT}$ . It is proposed here that the changes are a consequence of the highly uncorrelated nature of entanglement drag interactions, as opposed to the smoothly varying interactions inherent in the Rouse analysis. A new model is proposed to account for this difference. The resulting predictions are consistent with the experimental forms of $\mathrm{G\prime (\omega ),\; G″(\omega )}$ , and $\mathrm{G(t)}$ as well as those of ${\theta }_{\max }$ and $J_e$ . The molecular weight between entanglement points, $M_e$ , was estimated from several different viscoelastic properties of undiluted polystyrene. With the exception of that from $J_e$ , similar values of $M_e$ were obtained in all cases. Calculations of $J_e$ for mixtures of two molecular weights $M_A$ and $M_B$ were made with and without the assumption of an uniformly effective drage coefficient. The former predicted an unrealistically weak dependence on polydispersity; the latter agreed with experimental results for cases in which $M_B{\mathrm{\hspace{0.17em}/\hspace{0.17em}M}}_A$ was in the range of 2, but overestimated the effects of polydispersity when the difference in molecular weights was greater.