Doi-Edwards equation

At the high polymer concentration used in plasticized systems the viscosity of amorphous polymer is given by the modified Rouse theory at low molecular weight, M - 2Mr [from equation (47)] and by the modified Doi-Edwards equation at high molecular weight. In the first case... 

In terms of network analogy, the damping function may be viewed as the expression of the retraction of the strands as compared to the continuum. The Lodge model thus corresponds to no retraction (affine deformation, a=0 in equation (30) ), the Doi-Edwards equation corresponds to complete retraction (a=0.2), whereas incomplete retraction makes the damping function more softly decreasing (0 < a < 0.2). In the later cases, the deformation is non-affine since there is a difference between that of the continuum and that of the network strands. Wagner [33] showed that the Doi Edwards strain function... 

The Doi—Edwards equation is a special case of a separable K—BKZ equation, for which... 

This latter approximation shows that the strain dependence of the Doi-Edwards equation is softer than that of the temporary network model roughly by the factor 1 -p (7i — 3)/5, There is also a differential approximation to the Doi-Edwards equation (Marmcci 1984 Larson 1984b) ... 

The Doi— Edwards equation predicts that the ratio 2 is —2/7 = —0.29 at low shear rates. This changes to 4 2/ l i = —1/7 = —0.14 when the independent alignment approximation is dropped (Osaki et al. 1981). With or without the independent alignment approximation, the ratio —is predicted to decrease towards zero as the shear rate increases. The prediction of fof entangled solutions contrasts with that predicted... 

The Doi-Edwards equation predicts an overshoot in shear stress as a function of time after inception of steady shearing, but no overshoot in the first normal stress difference (Doi and Edwards 1978a). Typical overshoots in these quantities for a polydisperse melt are shown in Fig. 1-10. For monodisperse melts, the Doi-Edwards model predicts that the shear-stress maximum should occur at a shear strain yt = Yp, of about 2, roughly independently of... 

Problems 3.7 through 3.11 test your ability to work with reptation ideas and the Doi-Edwards equation. 

Problem 3.8 Numerically solve Eq. (3-78), the differential approximation to the Doi-Edwards equation for entangled linear melts, in a steady-state shearing flow. Plot the dimensionless shear stress ayijG against Weissenberg number W/ = )>r for Wi between 0.1 and 100. 

Problem 3,9 Integrate the Doi-Edwards equation (3-71) using the Currie expression for the Q tensor, Eq. (3-75), for steady-state shearing, for yr = 0.1, 0.3, 1.0, 3.0, and 10.0, using only one relaxation time in the spectrum. Plot the values of dimensionless shear stress OnjG versus yr on the same plot as in Problem 3.8. How close is the prediction of the approximate differential model to that of the exact integral model ... 

The Doi-Edwards equation is a K-BKZ model, since the scalar functions and are derivatives of a strain energy function and depend on the first and second invariants of the Finger tensor, which are defined by Eqs. 10.8 and 10.9. While these two functions cannot be written in a closed form, Currie [13] has shown that they can be approximated by the following analytical expressions. 

Once S and A have been obtained by solving Eqs. 11.8 and 11.9, andk5(A)ffomEq. 11.11, the stress tensor trcan be obtained from Eq. 11.10. Fortunately, the set of Eqs. 11.8 through 11.11, which defines the toy version of the DEMG theory, is nearly identical in its predictions to the fiill DEMG model. The DEMG model, in full or toy form, improves some aspects of the Doi-Edwards equation but not others. 

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-... 

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