Dynamic theory of die swell for entangled polymeric liquids in tube extrusions: New set of equations for the growth and ultimate extrudate swelling ratios under the free states

A new dynamic theory of die swell for entangled polymeric liquids in a steady simple shear flow is proposed which can be used to predict the correlation of the time-dependent and time-independent extrudate swelling behaviors to the molecular parameters of polymers and the operational variables. The theory is based on the O-W-F constitutive equation and the free recovery from Poioeuille flow with different ratios. Experiments show that the magnitudes of the simple shear in the steady simple shear flow may be resolved into the free recoil resulting from the recoverable elastic strains and the viscous treating resulting from the unrecoverable viscoelastic strains. For distinguishing the recoil from the viscous treating, a partition function and two exponential fractions of conformation for the recoil and the viscous heating were defined. Thus the instantaneous, delayed and ultimate recoverable strain, arid recoil in the free recovery were correlated to the partition function, the fraction of recoverable conformation, the molecular parameters, and the operational variables. Also the dynamics of the growth equation on the delayed viscoelastic strain arid the delayed recoil in free state were deduced. After introducing the condition of uniform two dimensional extensions, the definition of swell ratio and the operational variables into the above correlation expressions and growth equations, then the correlations of tire delayed extrudate swelling effect and the ultimate extrudate swelling effects to the molecular parameters arid the operational variables were derived. Finally, two new sets of equations on the growth variables and ultimate extrudate swelling ratios under the dynamic and equilibrium states were also deduced from this dynamic theory of die swell. The first set of equations on the ultimate extrudate swelling ratio under the free arid equilibrium states was verified by HDPE experimental data at two temperatures and different operational variables. The second set of equations on the growth extrudate swelling ratios under free arid dynamic states was verified by PBD experimental data with different molecular weights and different operational variables. An excellent agreement is obtained, which shows that the two sets of equations for the growth and ultimate extrudate swelling ratios can be used directly to predict,the correlation of extrudate swelling ratios to the molecular parameters and the operational variables.