  Evolving structure and rheology across Kuhn scale interfaces in entangled
polymer fluids under flow play a prominent role in processing of manufactured
plastics, and have numerous other applications. Quantitative tracking of chain
conformation statistics on the Kuhn scale is essential for developing
computational models of such phenomena. For this purpose, we formulate here a
two-scale/two-mode model of entangled polymer chains under flow. Each chain is
partitioned by successive entanglements into strands that are in one of two
modes: entangled or dangling. On the strand scale, conformation statistics of
ideal (non-interacting) strands follows a differential evolution equation for
the second moment of its end-to-end distance. The latter regulates persistent
random walks sampling conformation statistics of ideal entangled strands on the
Kuhn scale, as follows from a generalized Green-Kubo relation and the Maximum
Entropy Principle. We test it numerically for a range of deformation rates at
the start-up of simple elongational and shear flows. A self-consistent
potential, representing segmental interactions, modifies strand conformation
statistics on the Kuhn scale, as it renormalizes the parameters controlling the
persistent random walk. The generalized Green-Kubo relation is then inverted to
determine how the second moment of the strand end-to-end distance is changed by
the self-consistent potential. This allows us to devise a two-scale propagation
scheme for the statistical weights of subchains of the entangled chain. The
latter is used to calculate local volume fractions for each chemical type of
Kuhn segments in entangled chains, thus determining the self-consistent
potential.
