View - Martin Kröger - ETH Zürich

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1308 FANG, KRÖGER, AND ÖTTINGER FIG. 9. Steady-state shear viscosity vs dimensionless shear rate which here is proportional to a physical shear rate rather than expressed in terms of a relaxation time which depends on molecular weight, for Z 20 and Z 40 with the other parameters 1 1/, 2 1/, and max 10. stretching alone yields the onset of an upturn of the shear stress at high shear rates, but the prediction of the power-law index is worse. When double reptation is considered, the situation is improved. After CCR is added and adjusted to proper intensity, the maximum in the steady-state shear stress disappears and the model predicts an almost constant stress over a wide range of shear rates, which is also observed in experiments of a highly entangled polymer solution Bercea et al. 1993. We thus confirm the previous conclusion that the CCR effect is critical for predicting the right shape of the shear stress curve. In Fig. 9 we plot the steady-state shear viscosity for Z 20 and Z 40 in order to resolve the effect of molecular weight. As a basis for comparison, we assume that the reptation time scales like d M 3 . It turns out that the steady-state viscosity of two different molecular weights coincides in the power-law region, in agreement with the behavior observed in experiments Stratton 1966 and, e.g., molecular dynamics simulations <strong>Kröger</strong> et al. 1993. Similar results are found by the FCS and MLDS models. The effect of molecular weight on N 1 is reported in Fig. 10, the model prediction is in agreement with the second part of observation 2 in Sec. I. D. Steady and transient extinction angle The extinction angle is given by (1/2)arctan(2 xy /N 1 ). In Fig. 11 we show the steady-state extinction angle as a function of shear rate. Obviously, in our model, CCR and double reptation prevent the chain segments from aligning with flow dramatically at high steady shear rates, and thus solve the overestimated steady shear orientation problem of the original DE model. The predictions of our model compare very favorably with the experimental results, while the predictions of the FCS and MLDS models go back to the DE model predictions at very high shear rates in steady flow. Not shown here are the predictions of our model with the parameters 1 2 exp((1)), which also coincide with the predictions of the DE model at high rates. It can be concluded from the above observations that the exponential switch function tunes down the constraint release effect on orientation too strongly, and hence is not appropriate. The reorientation algorithm in the FCS model for constraint release is successful at low and intermediate rates,
THERMODYNAMICALLY ADMISSIBLE REPTATION MODEL 1309 FIG. 10. Steady-state first-normal-stress difference vs dimensionless shear rate for Z 20 and Z 40 with the other parameter values as for Fig. 9. but may profit from an adjustment if a plateau for the steady time-averaged extinction angle at high rates caused by unsteady rotation of chains would be experimentally confirmed. In Fig. 12 we show the transient extinction angle predicted in the startup of shear flow followed by a step down in shear rate. In both predictions by our model and the FCS model, the extinction angle shows an undershoot at the startup of the first shear rate and an immediate undershoot at the inception of the lower shear rate before reaching a steady-state value. These predictions are in agreement with the experimental data Mead 1996; Oberhauser et al. 1998. However, the second undershoot predicted by our model is too small compared with both the experiment and the prediction of the FCS FIG. 11. Steady-state extinction angle as a function of shear rate predicted by the models and experiment.
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View - Martin Kröger - ETH Zürich

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