View - Martin Kröger - ETH Zürich

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
Kröger 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,