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

David W. Mead

doi:10.1007/s00397-011-0550-5

Rheologica Acta

October 2011, Volume 50, Issue 9–10, pp 837–866 | Cite as

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

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David W. MeadEmail author

Original Contribution

First Online: 12 May 2011

Received: 18 January 2011

Revised: 22 March 2011

Accepted: 24 March 2011

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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

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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.

Appendix: Interpretation and rationalization of the Cox–Merz Rule for monodisperse linear polymers

The analysis presented in Section “Analytic derivation of the Cox–Merz rule from the polydisperse MLD “toy” model” analytically derives the Cox–Merz rule from the polydisperse MLD toy model with only three justifiable approximations that we have numerically verified in Section “Numerical validation of the polydisperse MLD “toy” model analysis of the Cox–Merz rule”. Although the result is justified within the explicit approximations of the analysis, the underlying physical reasons, if there are indeed any, for the validity of the Cox–Merz rule are not readily apparent. For example, the modulus of the complex dynamic viscosity contains both dissipative and elastic terms while the steady shear viscosity is strictly dissipative. Hence, it is difficult to understand how there can be any physically based relationship between these two material functions. Our objective in this Appendix is to discern whether there are indeed any fundamental physical relationships between the linear and nonlinear dynamical behavior of polymers or whether the widely applicable Cox–Merz relationship is merely a “suspicious coincidence” with no underlying physical basis. In the discussion below, we explore the fundamental causes of the Cox–Merz rule in the restricted and more easily understood case of a monodisperse system.

In slow steady shear flow for a monodisperse polymer, the longest relaxation time is the tube disengagement time, τ _d . For slow flows, the material response is controlled by reptation and agreement with Cox–Merz is assured by virtue of the linear response. It is only the nonlinear portion of the flow curve that requires rationalization with respect to the Cox–Merz rule. In the nonlinear flow regime, as the shear rate is increased, convective constraint release becomes increasingly more important. For a monodisperse system, convective constraint release will be the dominant relaxation mechanism when \(\dot{{\gamma }}\tau_d >1\). The transition from reptation controlled (diffusive) relaxation to CCR is sharp and occurs when \(\dot{{\gamma }}\tau_d \approx 1\). The longest relaxation time in fast flows is the time scale for convective constraint release and is proportional to \(\dot{{\gamma }}^{-1}\) independent of molecular weight. The asymptotic nonlinear viscosity in fast flows can be calculated using the monodisperse special case results from Section “Analytic derivation of the Cox–Merz rule from the polydisperse MLD “toy” model” as \(\eta \left( {\dot{{\gamma }}} \right)=5G_N^\circ \frac{S^\ast }{\dot{{\gamma }}}\approx 0.615\frac{G_N^\circ}{\dot{{\gamma }}}\), independent of molecular weight. This result assumes no stretch is occurring. Any stretch present at high shear rates will raise the viscosity level accordingly.

From the dynamic viscosity, we can develop an expression for the nonlinear viscosity using the Cox–Merz rule. At high frequencies, the storage modulus is much larger than the loss modulus and is approximately equal to the plateau modulus, \(G_N^\circ\) (ignoring high frequency Rouse modes). The high shear rate viscosity of a monodisperse polymer in fast flows according to the Cox–Merz rule (6) is, therefore, \(\eta \left( {\dot{{\gamma }}} \right)\approx \left. {\frac{G'\left( \infty \right)}{\omega }} \right|_{\omega =\dot{{\gamma }}} \approx \frac{G_N^\circ}{\dot{{\gamma }}}\) in approximate agreement with our analysis of steady shear flow derived above (Ianniruberto and Marrucci 1996). The \(\dot{{\gamma }}^{-1}\) scaling of the high shear rate viscosity is approximately consistent with experimental data on monodisperse polymers in fast flow (Kulicke and Kniewske 1984). There is a small discrepancy in the slope of the flow curve at high shear rates. Theoretically, the above simple analysis of the Cox–Merz rule predicts an asymptotic slope of −1. However, experimentally this value is never realized. The slope of the experimental flow curve is less steep, typically −0.85. The cause of this departure is probably related to tube contour length fluctuations, which can never be completely ignored even in convection dominated fast flows. The tube contour length fluctuations incrementally increase the shear viscosity in fast flows by relaxing the tips of the chains which then present a more significant hydrodynamic profile to the flow and incrementally raise the viscosity (Marrucci 1988). These effects are fully discussed in the more detailed version of the MLD theory at the tube coordinate level previously presented (Mead 2007).

The Cox–Merz rule thus has the correct high and low shear rate asymptotes to fit the viscosity flow curve for a monodisperse polymer. For both the complex dynamic viscosity and the steady shear viscosity, the transition from linear (diffusive control) to nonlinear (convection control) behavior is sharp and occurs at a dimensionless frequency/shear rate of unity. The key features leading to the Cox–Merz relationship is the sharp transition in rheological behavior and the \(\dot{{\gamma }}^{-1}\) scaling of the CCR relaxation time. Thus, the modulus of the complex dynamic viscosity has the correct mathematical structure to fit the flow curve. However, it should be clearly noted that there are no physical similarities or analogies between the polymer dynamics or molecular relaxation mechanisms in these two situations! The CCR process is intrinsically a nonlinear effect and there is no corresponding physical phenomenon in small deformation dynamic rheometry. The validity of the Cox–Merz rule is intimately connected to the CCR relaxation process. The relationship of the flow curve to linear viscoelasticity is a consequence of unphysical coincidences in the structure of the mathematical relationships, specifically,

1.

Identical dimensionless transition time scale from linear to nonlinear rheological behavior of \(\dot{{\gamma }}\tau_d \approx 1\) and ωτ _d ≈ 1 signaling a sharp change in the dynamical behavior
2.

The CCR relaxation time scales as \(\dot{{\gamma }}^{-1}\) independent of molecular weight

The ideas discussed above are illustrated diagrammatically in Fig. 16. For polydisperse systems the dynamics are more complicated. However, the essential features of the monodisperse case outlined above also apply. In a polydisperse polymer system, there is also a sharp transition between reptation controlled relaxation and CCR-controlled relaxation at a critical shear rate for each component of the molecular weight distribution, Fig. 2.

Fig. 16
Qualitative sketch of the key features leading to the Cox–Merz rule for a monodisperse system. The relationship between the linear viscoelastic material properties and the nonlinear flow curve is based on coincidental, identical scaling for high dimensionless shear rates/frequencies and sharp transitions between relaxation mechanisms controlling the material response in the linear and nonlinear flow regions. There is no underlying physical basis for the validity of the Cox–Merz rule or any of its derivatives, it is an *unphysical* *coincidence* based on identical dimensionless shear rate/frequency transitions and identical high α asymptotic scaling properties, Fig. 2

The above analysis clearly demonstrates that CCR is the nonlinear relaxation mechanism responsible for the Cox–Merz rule. The \(\dot{{\gamma }}^{-1}\) scaling of the longest relaxation time and the fact that CCR has no molecular weight dependence are the key features responsible for its validity. With the molecular mechanism responsible for the Cox–Merz rule established, we can extrapolate the Cox–Merz rule to other steady deformation fields. For example, since CCR is responsible for the veracity of the Cox–Merz rule, we anticipate that the Cox–Merz rule will also be valid in steady extensional flows, but only when chain stretching is not significant. When chain stretching contributes significantly to the stress, the analytic arguments presented in Section “Analytic derivation of the Cox–Merz rule from the polydisperse MLD “toy” model” will break down. However, there do not currently exist sufficient reliable experimental data on steady-state extensional viscosity to make a firm statement on this point.

Although we have only discussed the CCR relaxation mechanism in relation to linear polymers, similar phenomena certainly occur in long-chain branched polymers. The magnitudes of the relevant relaxation times will change; however, physical arguments identical to those presented above for linear polymers will also apply to long-chain branched polymers. Experimentally, it is well established that the Cox–Merz rule applies to star polymers and, at least approximately, to LDPE (Yasuda et al. 1981; Venkatraman and Okano 1990; Utracki and Gendron 1984; Ferri and Lomellini 1999). The crucial result is that the CCR relaxation rate is independent of molecular architecture and scales as \(\dot{{\gamma }}\), the frequency at which the constraint matrix, i.e. tube, dissolves and renews itself (see Fig. 14) (Marrucci 1996; Marrucci and Ianniruberto 1997). We shall analytically derive the Cox–Merz rule for the MLD model of monodisperse and polydisperse star polymers in a forthcoming publication that details these ideas quantitatively (Chen et al. 2000; Chen and Mead 2011).

Summarizing, the polydisperse MLD toy model, and its relatives (Mead 2007), are all consistent with the well-known empirical Cox–Merz rule. This provides substantial support for the MLD polydispersity theory underpinnings and also provides insight into the true origins of the Cox–Merz relationship. We conclude that there are no fundamental physical connections between the dynamics of polymers in linear deformation fields and nonlinear deformation fields. The Cox–Merz rule is effectively a “suspicious coincidence” based on identical dimensionless frequency transitions and correct asymptotic scaling behavior rather than anything predicated on fundamental polymer physics.

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David W. Mead
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1.Center for the Molecular Study of Condensed Soft MatterIllinois Institute of TechnologyChicagoUSA

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

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