An exact mathematical calculation of the relaxation time spectrum of polymeric materials from the dynamic modulus using Mellin transforms

A molecular-level understanding of sub-Tg relaxation processes in industrial resins is of uttermost importance for real-world applications. These processes are characterized by the distribution of relaxation times depicted by the relaxation time spectrum usually computed numerically from experimental data or physics-based molecular models using approximate mathematical relationships. In this Note, I show how to remove these mathematical approximations explicitly by deriving an exact mathematical relationship that allows for the computation of the relaxation time spectrum from the dynamic modulus, thus relegating the inaccuracies to the numerical evaluation of the integrals with the consequent increase in the accuracy of the calculated spectrum.