Entangled polymers: Constraint release, mean paths, and tube bending energy

Constitutive equations for entangled polymer melts and solutions are often derived from single chain tube models. Instead of keeping the chain coordinates. these models operate with a coarse-grained description in terms of positions of the tube segments. The dynamics of the tube is then imposed by constraint release, tube renewal at the ends, and deformation by the flow. However, the step of coarse-graining is rarely discussed, and the tube free energy and tube statistics are not derived. Moreover, the microscopic definition of the tube is rarely specified. In this paper we propose to define the tube as a mean path. i.e., a line connecting positions of each monomer averaged over entanglement relaxation time tau(e). We propose one simple model where such a coarse-graining step can be performed exactly, resulting in a free energy containing the usual Gaussian chain term and an additional bending energy term. This free energy leads to a path in space which is locally smooth and differentiable but has random walk statistics at length scales larger than the tube diameter. This eliminates several problems in previous tube models which use derivatives over Contour variables. We then proceed to modify the constitutive equation of Graham et al. (2003) to include the bending energy in constraint release terms. The resulting theory does not contain uncertainties of the original theory and has a clearer and better defined microscopic origin.