Paper Number
SM46
Session
Polymers Solutions, Melts and Blends
Title
Why the Cox–Merz rule and Gleissle mirror relation work: A quantitative analysis using the Wagner integral framework with a fractional Maxwell kernel
Presentation Date and Time
October 13, 2022 (Thursday) 8:45
Track / Room
Track 6 / Sheraton 3
Authors
- McKinley, Gareth H. (Massachusetts Institute of Technology, Mechanical Engineering)
- John Rathinaraj, Joshua D. (MIT, Mechanical Engineering)
- Keshavarz, Bavand (Massachusetts Institute of Technology, Mechanical Engineering)
Author and Affiliation Lines
Gareth H. McKinley, Joshua D. John Rathinaraj and Bavand Keshavarz
Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02142
Speaker / Presenter
McKinley, Gareth H.
Keywords
experimental methods; theoretical methods; food rheology; polymer blends; polymer melts
Text of Abstract
We derive mathematically a set of conditions for which empirical rheometric relations such as the Cox–Merz rule and Gleissle mirror relationship are satisfied. We consider the Wagner integral constitutive framework and derive analytical expressions for the complex viscosity, the steady shear viscosity, and the transient stress coefficient in the start-up of steady shear. To describe the viscoelastic response we use a fractional Maxwell liquid model as the memory kernel within the non-linear integral constitutive framework. This formulation is especially well-suited for describing polymer melts and solutions that exhibit a broad relaxation spectrum and can also be readily reduced to the canonical Maxwell model for describing idealized viscoelastic liquids that exhibit a single dominant relaxation time. To incorporate the nonlinearities that always become important in real complex fluids at large strain amplitudes, we consider both an exponential damping function as well as the more general Soskey-Winter form. By evaluating analytical expressions for small amplitude oscillatory shear, steady shear, and the start-up of steady shear using these different damping functions, we show that neither the Cox–Merz rule nor the Gleissle mirror relation can be satisfied for materials with a single relaxation mode or narrow relaxation spectrum. We then evaluate the same expressions using asymptotic analysis and direct numerical integration for more representative systems having a wide range of relaxation times. We show that for materials with broad relaxation spectra and sufficiently strong strain-dependent damping the empirical Cox–Merz rule and the Gleissle mirror relations are satisfied either exactly, or to within a constant numerical factor of order unity. By contrast, these relationships are not satisfied in other classes of highly viscoelastic materials such as gluten gels that exhibit only weak strain-dependent damping or strain softening.