Theoretical derivation of molecular weight scaling for rheological parameters

Recently, a method has been established to determine moments (functionals) of the molecular weight distribution (MWD) of a given polymer directly from measurements of the linear viscoelastic relaxation modulus of that polymer. In part, the need to compute such quantities (functionals) is motivated by the experimentally observed scaling of rheological properties of polymers with respect to moments of their MWD. Although various authors have advanced different ad hoc arguments to derive various molecular weight scaling results for a variety of rheological parameters, such as the zero-shear viscosity, no formal procedure for deriving molecular weight scaling for rheological parameters has been proposed. In this paper, a natural parametric generalization of the reptation based mixing rules is introduced which includes single and double reptation as special cases. For this generalization, it is shown, by invoking the mean value theorem for integrals, how to formalize the derivation of molecular weight scaling for rheological parameters. In particular, from the point of view of choosing practical mixing rules, this paper establishes that when the relaxation function is characterized by a single time constant, the molecular weight scaling is independent of the standard linear mixing rules. (C) 1998 Elsevier Science B.V. All rights reserved.