Relaxation time spectrum molecular weight distribution relationships

Single exponential decay \(\exp(-t/\tau)\) relationships, which define the molecular weight distribution (MWD) of a polymer as a function of the polymer’s relaxation time spectrum (RTS), have been derived by Wu (Polym Eng Sci 28:538–543, 1988) and Thimm et al. (J Rheol 43:1663–1672, 1999). Experimental validation studies with monodisperse polymers, with quite precisely known MWDs, have been used to test their reliability. It has been established that neither formula is always able to accurately recover the MWDs of monodisperse polymers from their experimentally determined RTS. In this paper, different and more general relationships, based on theoretical results of Anderssen and Loy (Bull Aust Math Soc 65:449–460, 2002a) for decays of the form \(\exp(-\theta(t)/\tau)\), where the derivative of θ(t) is a completely monotone function, are derived, analyzed, and applied. It is shown how to transform these general relationships to equivalent single exponential decay relationships for which Laplace transform solutions are derived. In order to illustrate the interrelationship between an RTS and its corresponding MWD, an explicit analytic solution is given. The paper concludes with a discussion of the rheological implications for the BSW model.