Molecular modelling of entangled polymer fluids under flow The ...

2.3. DOI-EDWARDS MODEL OF ENTANGLED POLYMERS 27
Cloizeaux (1990), Milner and McLeish (1998b)], however each of these approaches is
either an incomplete solution or is based around uncontrolled assumptions. Recently
Likhtman and McLeish (2002) obtained a solution via a combined stochastic and analytic
method. In this approach analytic arguments are used to derive expressions for
the relaxation but with unknown coefficients. Numerical values for these coefficients
are found by fitting the results to stochastic simulations of the first passage problem of
a full Rouse chain in a tube. If thermal constraint release (see section 2.4) is included as
well, this method produces good agreement with the viscosity measurements of Colby
et al. (1987), which cover a particularly wide range of molecular weights spanning either
side of M c . This theory has also been shown to produce excellent predictions for the
shape of G ′ (ω) and G ′′ (ω) of monodisperse linear polymer melts.
2.3.3 Non-linear rheology
The DE model can be generalised to non-linear flows. This has been achieved, with some
success, for a range of flow situations. For example continuous forward deformations
can be modelled by assuming the deformation rate to be small with respect to the chain
Rouse time. This still allows a window of non-linear rates since the reptation time and
the Rouse time are fairly well separated in an entangled melt (τ d /τ R = 3Z). In this
case the chain retraction can be assumed to be instantaneous. The model thus consists
of three processes: affine deformation of the tube due to convection, instantaneous
retraction along the tube and reptation, as described above. This model has been
solved using a variety of different pre-averaging approximations. The most widely used
of these is the independent alignment approximation (IAA) which disregards the fact
that, as the chain retractions, orientation information is passed from one tube segment
to the next. This considerably simplifies the solution of the model. The model predicts
a number of qualitative features that are in agreement with experimental data. These
were considerable improvements over preceding models. In extension the model predicts
extension thinning in steady state, in agreement with data on monodisperse samples.
In non-linear shear the model predicts a transient overshoot in shear stress, but not
in normal stress, and strong shear thinning at high shear rates. Qualitatively, all of
these features are observed experimentally. The issue of shear thinning is particularly
significant. The Rouse model, outlined in section 2.2.4, has a constant shear viscosity,
and hence the shear thinning phenomenon can be attributed to the influence of chain
entanglements. However, the DE model predicts too much shear thinning which itself
can be problematic (see section 2.4).
An even greater degree of success was achieved in the prediction of stress relaxation
after a large step shear deformation. The model assumes a perfectly affine deformation
followed by two relaxation processes. The chain relaxes its primitive path length back