View - Martin Kröger - ETH Zürich

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.