Demges

For other similar tests, see Demges, Atm. Chun anal., 1917, 22, 193. 

OELEPINE Arrane synthesis 91 DE MAYO Cycloaddition 92 DEMJANOV Rearrangement 93 Demges 219 Dennstedt 64... 

Chain Stretch the Doi-Edwards-Marrucci-Grizzuti (DEMG) Theory... 

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. 

In fast shearing flows, with shear rates greater than I/Tj, the DEMG theory shows overshoots in both shear stress <7 and first normal stress dilference Nj as functions of time after start-up of steady shearing [21 ], in agreement with experiments (as will be presented in Section 11.5.1.1). These overshoots in both <7 and are an improvement over DE theory, which shows only the... 

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).

Note the good agreement between predictions and experimental data for both the steady-state shear stress and first normal stress difference in Fig. 11.7. These predictions substantially improve on those of the Doi-Edwards and DEMG theories, especially in removing, or nearly removing, the maximum in shear stress as a function of shear rate. They also show that with... 

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. 

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].

In steady-state uniaxial extensional flows, the MLD and DEMG equations predict three regions of flow see Fig. 11.4. For monodisperse polymers, these regions are defined by the extension rate e relative to the two relaxation times Tj and T. For e < 1/, the extension rate is too... 

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).

Equation 11.45 is the equation for the orientation tensor S. This equation is similar to that for the Doi-Edwards or DEMG theory see Eq. 11.8. Equation 11.46 is the stretch equation. It is also similar to its counterpart in the DEMG theory see Eq. 11.9. The main difference in the theory for the pom-pom is that the stretch X is limited to be equal to or less than q, the number of arms on each end of the backbone. If the stretch attains the value q, the arms start to be pulled into the backbone tube. The length of arm pulled into the backbone tube is defined by S, which is measured in units of numbers of entanglements. Thus, 1 - c = the fraction of each arm that has been pulled into the backbone tube. The evolution equation... 

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. 

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