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1314 FANG, KRÖGER, AND ÖTTINGER tively in most cases, all the nonlinear properties observed for shear flows summarized in Sec. I from item 2 to item 9 item 1 has already been checked in Part I. For extensional flows, the model exhibits unusual predictions. In order to confirm these, further measurements of the stress for ideal samples linear, monodisperse, entangled polymers in various extensional flows are required. There are difficulties in performing this kind of experiment: a monodisperse polymer is usually sold in powder form, which is often more difficult to shape into bubble free samples than granules; the polymer breaks at much lower Hencky strains and is less homogeneous than an industrial melt Schweizer 1999. Comparisons with two recently formulated models, FCS Hua and Schieber 1998 and MLDS Mead et al. 1998, respectively, in shear flows are also presented. It turns out that all three models are competitive with each other; in many cases, they show similar behaviors. Very detailed comparisons are given in the previous section, e.g., for the MLDS and our model it is shown that a weak switch function proportional to the inverse of relative stretch 1/ predicts more realistic results than obtained by applying a stronger one. In addition to the plateau modulus G N 0 and the reptation time d , our model has two basic parameters, namely Z, the number of entanglements per chain which is proportional to the molecular weight and can be expressed in terms of the Rouse and reptation time, and the maximum stretch max , being equal to the square root of the number of Kuhn steps per entanglement segment. The remaining two model parameters 1 , 2 have been introduced but actually assigned the values 1 2 1/. A ‘‘stable’’ steady-state shear stress curve is obtained by setting 2 2/. Predictions of the steady shear stress at very high shear rates can be also improved by considering the stretching time s to be a decaying function in , to account for the effect of contour length shortening by constraint release in situations where highly stretched conformations occur. Notably, our model is a thermodynamically consistent single-segment theory which has only four degrees of freedom. In the sense of considering the time and memory requirements for the computation, it is very suitable to utilize the model to simulate complex flows by using the CONNFFESSIT idea Laso and Öttinger 1993. To our knowledge, variance reduction techniques Hulsen et al. 1997; Öttinger et al. 1997; van Heel et al. 1999; Gallez et al. 1999; and Bonvin and Picasso 1999 may not be directly applied to the model. The dynamics for the position label—and in particular its jump events upon touching the chain ends—depend on the local value for the macroscopic velocity gradient such that a synchronization of trajectories by cancellation of fluctuations is prevented at first glance. However, it should be a straightforward exercise to extend the presented model to account for polydispersity effects by a mean-field approach, such that the properties of surrounding chains enter into the dynamics of the test chain. The resulting model will necessarily locate beyond simple superposition models. ACKNOWLEDGMENTS The authors would like to thank Professor D. C. Venerus for providing the experimental data, Professor R. G. Larson for providing the simulation program of the MLDS model, J. Neergaard for providing calculations of the FCS model, and Professor J. D. Schieber for extremely helpful comments. J. Fang is grateful for the financial support from the European Community Program BRITE-EuRam.
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