A Constitutive Model for Entangled Polydisperse Linear Flexible Polymers with Entanglement Dynamics and a Configuration Dependent Friction Coefficient. Part I: Model Derivation

A new polydisperse toy constitutive model is derived and developed from fundamental principles and ideas governing the nonlinear rheology of linear flexible polymers [Mead et al., J. Rheol. 59, 335-363 (2015)]. Specifically, the new model is comprised of four fundamental pieces. First, the model contains a simple differential description of the entanglement dynamics of discrete entanglement pairs. Second, the model contains a differential description of the ij entanglement pair orientation tensor dynamics. Third, following a similar development by Mead and Mishler [J. Non-Newtonian Fluid Mech. 197, 61-79 and 80-90 (2013).], a diluted stretch tube is constructed to describe the relative stretch of each component in the molecular weight distribution (MWD). Fourth, a description of configuration dependent friction coefficients is generated by generalizing the monodisperse formulation of Ianniruberto et al. [Macromolecules 45, 8058-8066 (2012)]. The polydisperse stress calculator is developed from the orientation, stretch and entanglement density and is fundamentally different from other molecular models that assume a constant entanglement density. The resulting model is comprised of three differential evolution equations and is simple to code and fast to execute. The model can simulate arbitrary fast nonlinear flows of arbitrary MWD\u27s. In the slow flow linear viscoelastic limit, the model collapses to the double reptation model. This welcome result has positive implications with respect to our model parameter determination [Ye et al., J. Rheol. 47, 443-468 (2003); Ye and Sridhar, Macromolecules 38, 3442-3449 (2005)] for making quantitative calculations