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 {Hleft( {ln {rm }tau } right)} {rm }hleft( theta right)gleft( theta right)^{{raise0.7exhbox{$3$} !mathord{left/ {vphantom {3 2}}right.kern-nulldelimiterspace}!lower0.7exhbox{$2$}}} 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 } $end{document} is the shear rate, H is the relaxation spectrum, τ is the relaxation time, documentclass{article}pagestyle{empty}begin{document}$ gleft( 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] $end{document}, and documentclass{article}pagestyle{empty}begin{document}$ hleft( 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.