# Analytic derivation of the Cox–Merz rule using the MLD “toy” model for polydisperse linear polymers

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## Abstract

A general constitutive formalism, the “naïve” polydisperse MLD model, has been developed by Mead et al. (Macromolecules 31:7895–7914, 1998) and Mead (Rheol Acta 46:369–395, 2007) at both the tube coordinate level and the mathematically simplified “toy” level independent of the tube coordinate. The model includes constraint release generated by convection-driven chain retraction (which is equivalent to “convective constraint release” (CCR)), reptation, and tube contour length fluctuations. The properties of the mathematically simplified naïve polydisperse “toy” MLD model are explored in linear and nonlinear steady shear flows where we analytically derive the Cox–Merz rule relating the steady shear viscosity to the modulus of the linear viscoelastic dynamic viscosity. The Cox–Merz rule relating the linear viscoelastic material properties and the nonlinear material properties is shown to be a direct consequence of convective constraint release. The specific feature of CCR that leads to this result is that the relaxation rate due to convective constraint release is proportional to the shear rate, \(\dot{{\gamma }}\), independent of molecular weight. The viability of this well-known empirical relationship is a direct consequence of a coincidence in the mathematical structure of the linear viscoelastic material properties and convective constraint release. There is no physical analogy or relationship between the molecular relaxation mechanisms operative in linear (diffusive relaxation) and nonlinear (convective relaxation) flow regimes. The polydisperse MLD model predictions of the individual molecular weight component contributions to the flow curve, and interpretations thereof, are effectively identical to those first postulated by Bersted (J Appl Polym Sci 19:2167–2177, 1975, J Appl Polym Sci 20:2705–2714, 1976). Following the theoretical developments, a limited experimental study is executed with a commercial polydisperse polystyrene melt. Nearly quantitative agreement between the polydisperse MLD theory and experimental measurements of steady-shear viscosity and dynamic moduli is achieved over a wide range of shear rates.

## Keywords

Cox–Merz rule Complex modulus Modeling## Notes

### Acknowledgements

DWM acknowledges stimulating discussions with Professor Masao Doi of Tokyo University, Japan. DWM gratefully acknowledges Professor L.J. Fetters of Cornell University (then Exxon Research & Engineering Co.) for performing the triple detector GPC analysis of the commercial Fina polystyrene sample. DWM also gratefully acknowledges Dr. M.K. Lyon of Exxon-Mobil for stimulating discussions and assistance with software for the simulations. This project was partially supported by NSF-GOALI #DMR-9807262 in collaboration with Rheometric Scientific Inc. (now TA Instruments). Final manuscript preparation was done at the Benjamin Levich Institute for Physio-Chemical Hydrodynamics, City College of New York, CUNY, with financial support from NSF Grant #0625072.

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