The Doi-Edwards Constitutive Equation

Doi and Edwards (1978a, 1979, 1986) developed a constitutive equation for entangled polymeric fluids that combines the linear viscoelastic response predicted by de Gennes 

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Figure 3.33 The solid curve is the dimensionless shear stress versus dimensionless shear rate yrd predicted by the Doi-Edwards constitutive equation, Eq. (3-71). The dashed curve adds a speculated contribution to the stress from Rouse modes. (From Doi and Edwards 1979, reproduced by permission of The Royal Society of Chemistry.)...

We noted in Section 11.3.1 that the prediction of a maximum in steady-state shear stress as a function of shear rate was a basic failure of the Doi-Edwards constitutive equation. Such a prediction implies that the same shear stress is produced by more than one shear rate. In such a situation, a uniform shearing flow at shear rates near or above the inverse of the reptation time Tj, would become unstable to a so-called constitutive instability, leading to spontaneous... 

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

P.KCurrie, Constitutive equations for polymer melts predicted by the Doi-Edwards and Curtiss-Bird kinetic theory models, J. of Non-Newt. Fluid Mech. 11 (1982), 53-68. 

Problem 3.11 A theory incorporating convective constraint release and chain stretch into the Doi-Edwards model gives the constitutive equations below (Larson et al. 1998) ... 

In this chapter, we have presented the fundamentals of molecular theory for the viscoelasticity of flexible homogeneous polymers, namely the Rouse/Zimm theory for dilute polymer solutions and unentangled polymer melts, and the Doi-Edwards theory for concentrated polymer solutions and entangled polymer melts. In doing so, we have shown how the constitutive equations from each theory have been derived and then have compared theoretical prediction with experiment. The material presented in this chapter is very important for understanding how the molecular parameters of polymers are related to the rheological properties of homopolymers. 

A primitive model of nonlinear behavior can be obtained by simply replacing the infinitesimal strain tensor in Eq. 10.3 by a tensor that can describe finite strain. However, there is no unique way to do this, because there are a number of tensors that can describe the configuration of a material element at one time relative to that at another time. In this book we will make use of the Finger and Cauchy tensors, B and C, respectively, which have been found to be most useful in describing nonlinear viscoelasticity. We note that the Finger tensor is the inverse of the Cauchy tensor, i.e., B = C. A strain tensor that appears in constitutive equations derived from tube models is the Doi-Edwards tensor Q, which is defined below and used in Chapter 11. The definitions of these tensors and their components for shear and uniaxial extension are given in Appendix B. 

Currie, P. K. Constitutive equations forpolymer melts Predictions by the Doi-Edwards andCurtiss-Bird kinetic theory models. /. Non-Newt FI. Mech. (1982) 11, pp. 53-68... 

Doi and Edwards noted that since % = 2 is expected to be much smaller than the reptation time Tj, then for flows that are fast compared to the rate of reptation 1/Tj, but slow compared to the rate of retraction 1 /t, one can assume that the chains remain completely retracted during flow i.e., there is no chain stretch. Under this assumption, Doi and Edwards, in a seminal series of papers [12-15] derived the famous constitutive equation that bears their name. The Doi-Edwards (DE) constitutive equation, introduced in Section 10.3.4, is written as ... 

Eq shown for (a) an HDPE (HDPE i) with = 104,000 and = 18,900 at 150 °C,and (b) a poiypropyiene (PP 2) with = 586,600 and = 61,750 at 180 °C, into which long-chain branches were introduced by electron-beam radiation.The lines are predictions of the Doi-Edwards theories and phenomenological constitutive equations of Wagner etal. Erom Wagner efo/. [87]. 

The simplest nonlinear tube model is the classical Doi-Edwards (DE) constitutive equation for linear polymers, which accounts for reptation and affine rotation of tube segments. The Doi-Edwards equation predicts thinning in both shear and extension, because it accounts for orientation of tube segments, but it is unable to predict extension thickening because it neglects the stretching of tube segments. Inclusion of tube stretch leads to the Doi-Edwards-Marrucci-... 

M. Doi and S. F. Edwards, Dynamics of concentrated polymer systems. Part I. Brownian motion in the equilibrium state, J. Chem. Soc. Faraday Trans. II, 74, 1789 (1978) M. Doi and S. F. Edwards, Dynamics of concentrated polymer systems. Part 2. Molecular motion under flow, J. Chem. Soc. Faraday Trans.II, 74, 1802 (1978) M. Doi and S. F. Edwards, Dynamics of concentrated polymer systems. Part 3. The constitutive equation, J. Chem. Soc. Faraday Trans. II, 74,1818 (1978) M. Doi and S. F. Edwards, Dynamics of concentrated polymer systems. Part 4. Rheological properties, J. Chem. Soc. Faraday Trans. II, 75,38 (1979). 

M.Doi, S.F.Edwards, Dynamics of concentrated polymer systems- Part 3 The constitutive equation, J. Chem. Soc, Faraday Trans II 2A (1978), 1818-1832. 

Curtiss and Bird introduce reptation in a maimer which does not involve the tube concept, at least not in an explicit way. Their model leads to a constitutive equation in which the stress is the sum of two contributions. On contribution is exactly 1/3 the expression obtained by Doi and Edwards when those workers invoke the independent alignment approximation, i.e., that contribution is a special case of the BKZ relative strain equation. The othCT contribution depends on strain rate and is proportional to a link tension coefficent c (0 < e < 1) which diaracterizes the forces along the chain arising from the continued displacements of chain relative to surroundings. 

Doi first proposed the generalized dynamic equations for the concentrated solution of rod-like polymers. Such constitutive equations can be derived from the molecular theory developed by Doi and Edwards (1986). The basis for the molecular theory is the Smoluchowski equation or Fokker-Planck equation in thermodynamics with the mean field approximation of molecular interaction. 

Without derivation, note that Doi and Edwards developed expressions for the probability distribution fimction for the chain as well as the relationship between the stress and the chain orientations to arrive at an evolution equation for the stress to the chain orientation process, which is a function of the macroscopic deformations. The resulting constitutive equation is... 

M. Doi and S. F. Edwards, Dynamics of Concentrated Poljrmer Systems Part 3.-The Constitutive Equation tZ Chem. Soc., Faraday Trans. 2 74,1818—1832 (1978). 

The modern approach to constitutive equations for melts and concentrated solutions has evolved from the reptation idea popularized by de Gennes and Doi and Edwards. In this picture, the mobility of a polymer chain in any direction except along the backbone is visualized as constrained to an imaginary tubelike region formed by the entanglements with the neighboring chains, which serve to restrict... 

In 1978, Doi and Edwards adopted this view and developed the tube theory for polymer melts. The main objective of this work was to derive the constitutive equations for polymer melts from a molecular level model. It was in this work that the tube concept and reptational motion were combined for a concise mathematical description of polymer melt dynamics. 

To describe the polymer stress in this equation, one can probe any of rheological constitutive models proposed for the long-chain branched polymers the partially extending convection (PEC) model of R. Larson, [117], the molecular stress function (MSP) theory of M. Wagner et al. [118,119], the modified extended pom-pom (mXPP) model of M.H. Verbeeten et al. [120], etc. Here, the PEC model has been chosen as it can be easily tuned to describe the overshoot position for a wide class of polymers by changing a value of the non-linear parameter Thus, = 1 in the case of linear polymer (Doi-Edwards limit), 0 < < 1 in the case of branched polymers, and = 0 in the... 

This derivation of the viscosity has been made much more rigorous (but still reaches the same conclusion) by Doi and Edwards (1978). They derive a constitutive relationship from this reptation molecular picture (Larson, 1988). Their result can be expressed as a complete constitutive equation in the form... 

Much research in the last few decades focused on the simulation of LCPs for various processes. Suitable rheological constitutive equations are required for this simulation. Leslie-Ericksen (LE) theory describes the flow behaviour and molecular orientation of many LCPs. LE model is limited to low shear rates and weak molecular distortions. But at high shear rate, the rate of molecular distortions is too fast. Doi and Edwards developed their model to describe the complex dynamics of macromolecules at high shear rate (Doi and Edwards 1978). Doi theory is applicable for lyotropic LCPs of small and moderate concentrations. Due to the complex nature of Doi theory, it is also challenging for simulation. Leonov s continuum theory of weak viscoelastic nematodynamics, developed on the basis of thermodynamics and constitutive relations, consider the nematic viscoelasticity, deformation of molecules as well as evolution of director. 

The Rouse model gives the following integral-type constitutive equation (Doi and Edwards 1986)... 

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