Assigning macromolecular meaning to nonlinear continuum rheology

The Oldroyd 8-constant continuum framework has yielded elegant analytical solutions for many polymer processing flows. However, continuum frameworks are silent on macromolecular structure. We can assign macromolecular meaning to the continuum constants by bridging continuum frameworks to the macromolecular theory of polymeric liquid dynamics. When the Oldroyd 8-constant framework has been bridged to rigid dumbbell theory (two-step), no higher order rheology was predicted (ν1=ν2=0). By higher order, we mean the nonlinear rheology. This troubled Bird (1972), motivating his modified Oldroyd 8-constant continuum framework, which does predict higher order rheology, to which meaning in rigid dumbbell theory is assigned. By two-step, we mean we get the three Jeffreys model constants from the macromolecular expression for the complex viscosity, and then solve five equations simultaneously for the five remaining constants. In this paper, in three steps, we bridge the Bird 8-constant framework to the more versatile rotarance theory (general rigid bead-rod theory). By three-step, we mean we get the three Jeffreys model constants from the macromolecular expression for the complex viscosity, and then solve three equations simultaneously for the next three, and finally solving two equation simultaneously for the remaining two higher order constants. By versatile, we mean accommodating any axisymmetric macromolecular structure (including the rigid dumbbell). We find the constants in the Bird 8-constant framework to be explicit functions of just one dimensionless macromolecular attribute: the ratio of the moment of inertia about the molecular axis, to the moment about either transverse axis. We thus assign macromolecular meaning to the higher order rheology. In passing, we also discover a new bridge to the Oldroyd 8-constant framework (three-step), which also assigns macromolecular meaning to the higher order rheology.