Shear rate dependent relaxation spectrum for polyethylene melts

Abstract

Many of the familiar relations in the theory of linear viscoelasticity which relate the relaxation spectrum H(τ) to the experimentally measured quantities were extended to embrace the concept of shear dependent relaxation spectrum H(τ, \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^ $\end{document}). The concept is based on the relation H(τ, \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^ $\end{document}) = H(τ)h(θ)g(θ)3/2, where θ = c \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^ $\end{document} τ and c = 0.5. Here \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^ $\end{document} is the shear rate and τ is the relaxation time. The replacement of H(τ) by H(τ',\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^ $\end{document}) in the integral relations of the linear theory produced the corresponding nonlinear parameters, e.g., the shear rate dependent viscosity η(\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^ $\end{document}), the shear stress relaxation following cessation of steady‐state flow σ(\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^ $\end{document},t) and the dynamic parameters under parallel superimposed rotation η'(ω) and G'(ω, \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^ $\end{document}). The concept was used successfully for several high‐density polyethylene melts in interconversion between η(\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^ $\end{document}), c(\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^ $\end{document},t) and the dynamic viscosity η'(ω) without involving coordinate shifts or empirical parameters. The failure of this approach for interconversion between η(ω) and η(\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^ $\end{document}) for the highly branched polyethylenes was ascribed to the level of long‐chain branching. For the high‐density resins, the agreement between measured η'(ω,\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^ $\end{document}) and G'(ω,\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^ $\end{document}) and those calculated using H(τ,\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^ $\end{document}) was not quantitative since the parallel superimposed experiments require an additional term in ∂H(τ,\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^ $\end{document})/∂ \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \gamma \limits^ ^2 $\end{document} inside the integral.