Multiple Entanglements between Two Chains in Polymer Melts: An Analysis of Primitive Paths Based on Frenet Trihedron

We propose an analysis of primitive paths for entangled polymers based on combining energy minimization with the Frenet trihedron geometric analysis. Using this method, which we abbreviate as FT-PPA, we identify the chain and monomer index associated with each entanglement, classify the different types of entanglements, and determine the fraction of each type under both equilibrium and shear conditions. Our analysis reveals that in entangled polymer melts, a pair of neighboring polymer chains can form multiple (two or more) entanglements (MuEs). The fraction of MuEs follows a very good exponential decay with the number of entanglements between the two entangled chains at equilibrium, with the rate of decay decreasing with increasing chain length. In addition, a significant fraction of entanglements are intervening entanglements (InEs), which are formed by a third chain between two MuEs on a tagged chain. We apply FT-PPA to shear banding in entangled polymer melts under fast shear deformation and show that the minimum in the spatial distribution of MuEs and InEs in the gradient direction, even at equilibrium, is strongly correlated with the center position of the fast band. This result supports our previous inference (ACS Macro Lett., 2021, 10, 1517–1523) that shear banding in entangled polymer melts is rooted in the structural heterogeneity prior to the application of shear.