Development of the “binary interaction” theory for entangled polydisperse linear polymers

Based on the theoretical principles previously described in the literature, the development of the “naïve” binary interaction model is detailed in this paper. The new theory is effectively a sweeping generalization of the “Double Reptation” model. The “switch function” has been shown to be an essential feature of any constraint release model for Doi–Edwards type molecular models that invoke the concept of a discrete slip-link tube and is used in our formulation. Using the assumption of a constant entanglement density, a slip link linear density evolution equation is derived to rigorously count matrix entanglements. This function has no counterpart in the conventional Doi–Edwards theory, or its derivatives, and is absolutely required to properly generalize the “Double Reptation” model so that nonlinear flows can be modeled. The binary interaction polydispersity model is complex mathematically but can be rigorously and justifiably simplified by suppressing the tube coordinate dependence using a boundary layer analysis. The simplification process can be continued to the continuum level to create a hierarchy of approximate binary interaction models, thereby making large-scale numerical simulations of complex flows viable, indeed straightforward.