On the scaling of molecular weight distribution functionals

When formulating a constitutive equation model or a mixing rule for some sy nthetic or biological polymer, one is essentially solving an inverse proble m. However, the data will not only include the results obtained from simple step strain, oscillatory shear, elongational, and other experiments, but a lso information about the molecular weight scaling of key theological param eters (i.e., molecular weight distribution functionals) such as zero-sheat viscosity, steady-state compliance, and the normal stress differences. In t erms of incorporating such scaling information into the formulation of mode ls, there is a need to understand the relationship between various models a nd their molecular weight scaling, since such information identifies the wa ys in which molecular weight scaling constrains the choice of possible mode ls. In Anderssen and Mead (1998) it was established formally that the membe rs of a quite general class of reptation mixing rules all had the same mole cular weight scaling. The purpose of this paper is to first introduce the c oncept of a generalized reptation mixing rule, which greatly extends the cl ass examined by Anderssen and Mead, and then show that all such rules have the same molecular weight scaling. The proof is similar to that given by An derssen and Mead, but uses the implicit function theorem to establish the u niqueness of the mean values which arise when invoking various integral mea n-value representations for the molecular weight distribution functionals c onsidered. The theological significance of the new generalized two-paramete r mixing rule, proposed in this paper, is examined in some detail in the co nclusions. In particular, it is used to established how one must construct a mixing rule for a general polydispersed polymer where the molecular dynam ics involves some single, some double and some higher levels of multiple re ptation. The work of Maier et al. (1998) and Thimm et al. (2000) is then ut ilized to illustrate and validate this proposal. (C) 2001 The Society of Rh eology.