Effect of long-chain branching on the relation between steady-flow and dynamic viscosity of polyethylene melts

The steady-state viscosity η, the dynamic viscosity η′, and the storage modulus G′ of several high-density and low-density polyethylene melts were investigated by using the Instron rheometer and the Weissenberg rheogoniometer. The theoretical relation between the two viscosities as proposed earlier is:\documentclass{article}\pagestyle{empty}\begin{document}$ \eta \left( {\dot \gamma } \right){\rm } = {\rm }\int {H\left( {\ln {\rm }\tau } \right)} {\rm }h\left( \theta \right)g\left( \theta \right)^{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}} \tau {\rm }d{\rm }\ln {\rm }\tau $\end{document}, where \documentclass{article}\pagestyle{empty}\begin{document}$ \theta {\rm } = {\rm }{{\dot \gamma \tau } \mathord{\left/ {\vphantom {{\dot \gamma \tau } 2}} \right. \kern-\nulldelimiterspace} 2} $\end{document}; \documentclass{article}\pagestyle{empty}\begin{document}$ {\dot \gamma } $\end{document} is the shear rate, H is the relaxation spectrum, τ is the relaxation time, \documentclass{article}\pagestyle{empty}\begin{document}$ g\left( \theta \right){\rm } = {\rm }\left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\left {\cot ^{ - 1} \theta {\rm } + {\rm }{\theta \mathord{\left/ {\vphantom {\theta {\left( {1 + \theta ^2 } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + \theta ^2 } \right)}}} \right] $\end{document}, and \documentclass{article}\pagestyle{empty}\begin{document}$ h\left( \theta \right){\rm } = {\rm }\left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\left {\cot ^{ - 1} \theta {\rm } + {\rm }{{\theta \left( {1{\rm } - {\rm }\theta ^2 } \right)} \mathord{\left/ {\vphantom {{\theta \left( {1{\rm } - {\rm }\theta ^2 } \right)} {\left( {1{\rm } + {\rm }\theta ^2 } \right)^2 }}} \right. \kern-\nulldelimiterspace} {\left( {1{\rm } + {\rm }\theta ^2 } \right)^2 }}} \right] $\end{document}. Good agreement between the experimental and calculated values was obtained, without any coordinate shift, for high-density polyethylenes as well as for a low density sample with low n_w, the weight-average number of branch points per molecule. The correlation, however, was poor with low-density samples with large values of the long-chain branching index n_w. This lack of coordination can be related to n_w. The empirical relation of Cox and Merz failed in a similar way.