Application of fractional calculus to the modeling of the complex rheological behavior of polymers: From the glass transition to flow behavior. I. The theoretical model

Abstract

Abstract A model based on the concept of fractional calculus is proposed for the description of the dynamic elastic modulus ( E * = E ′ + iE ″, where E * is the complex modulus, E ′ is the storage modulus, or real part of the complex modulus, and E ″ is the loss modulus, or imaginary part of the complex modulus) under isothermal and isochronal conditions for amorphous polymers, including both the glass transition process and the flow behavior. The differential equations obtained for this model, which we call the extended fractional Zener model (EFZM), have differential and/or integral operators of fractional order between 0 and 1. The application of the Fourier transform to the fractional equation of the EFZM, the association of their parameters to the relaxation times of the cooperative or noncooperative molecular movements of polymer chains, and the isothermal and isochronal diagrams of E ′ and tan δ = E ″/ E ′ were evaluated. These theoretical diagrams were typical curves that clearly showed the glass‐transition (α‐relaxation) and flow behavior. The EFZM will enable us to analyze the complex rheological behavior of amorphous polymers. © 2008 Wiley Periodicals, Inc. J Appl Polym Sci, 2008