DEMG model

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 constitutive equations for the DEMG model, Eqs. 11.8 to 11.10, can also be generalized to account for a molecular weight distribution. When this is done, the resulting equations are similar to the MED model except without the reptative and convective constraint release terms (the last terms of Eqs. 11.32 and 11.34 ... 

In the linear viscoelastic limit, the DEMG model reduces to the original reptation model of Doi and Edwards, with only the slowest relaxation mode retained. Since the DEMG given by Eqs. 11.38 through 11.40 does not contain reptative constraint release, we have set the relaxation... 

Figure 11.8 Comparison of the predictions of the MLD modei (soiid iines) and DEMG model (broken iines) with experimentai data (symbois) for the viscosity, shear stress and first normal stress difference of a 7 wt% soiution of neariy monodisperse polystyrene of molecular weight 2.89 million in tricresylphosphate at 40 °C.The open circles are the dynamic viscosity if oi. The parameter values for the MLD theory are G 5 = 3000 dyn/cm Tj, = 3.06 s, and = 0.13 s. From Pattamaprom and Larson [36].

Figure 11.11 Comparison of the predictions of the DEMG model (solid line) and MLD model (dashed and dotted lines) to experimental data (symbols) for the uniaxial extensional viscosity rif (e) versus extension rate f. The data are for a 6% solution of 10.2 million molecular weight polystyrene In diethyl phthalate at 21 °C.The parameters used In the MLD and DEMG theories are G 5 = 294 Pa for both models = 21 s, and Tj = 0.51 s for the DEMG theory, and = 83.4 s, and Tj = 1.08 s for the "Mllner-McLeish" method of obtaining the time constants for the MLD model (dashed line), and Tj = 123 s, and Tj = 1.58 s for the "Doi-Kuzuu" method (dotted line). From Bhattacharjeeeta/. [49).

Thus, one of the differences between the predictions of the pom-pom model and the DEMG model for linear molecules is the inequality in Eq. 11.46. In steady imiaxial extension, this criterion produces a saturation value of the stress and a local maximum in the extensional viscosity, while in the DEMG model, the stress has no saturation value. (If finite extensibility is included in the DEMG model, then there is a saturation in viscosity, but no viscosity maximum.) Experimental data of McLeish et al. [97] for a melt of polyisoprene H molecules confirm the predicted decrease in extensional viscosity at high extension rate see Fig. 11.25. 

Figure 11.4 Predictions of the DEMG theory for %/J7o as a function of Weissenberg number, for Z = 20 and 100 entanglements per chain.The chain is taken to be infinitely extensible,/.e, k X) = 1. The prediction of the Doi-Edwards model is also shown. Erom Marrucci and Grizzuti (20).

In Eq. 11.18, 5 is the unit tensor. Equation 11.18 is a differential equation for the orientation tensor S, rather than a history integral equation like that shown by Eq. 11.2. Differential equations are much easier to handle numerically than history integral equations, which motivated the development of Eq. 11.18 as an approximation to a history integral. Equation 11.18 captures much of the behavior predicted by the integral DE equation, Eq. 11.2, except it predicts a zero second normal stress difference, rather than the value predicted by the regular DE model, Eq. 11.2, which is -2/7 times the first normal stress difference at low shear rates. The differential equation Eq. 11.18 for S can also be used as a replacement for the integral expression in the DEMG theory. 

---