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

2.5. BRANCHED POLYMERS 41
density polyethylene (LDPE) using the multimode approach described by Inkson et al.
(1999). Commercial LDPE is a polydisperse blend of molecules with multiple irregularly
spaced long chain branches. Each section of this complex molecular architecture
has its own time-scales for orientation and stretch relaxation. Under the multimode
method these sections are modelled by a superposition of pom-pom molecules of differing
relaxation times and arm numbers. The stress contribution of these pom-pom
molecules is then summed to obtain the total stress,
σ
≈
=
n∑ n∑
σ = 3 g i λ 2
≈
i S (2.43)
i ≈ i.
i=1 i=1
The approximation is the decoupling of the modes of a connected molecule. There
ought to be interactions between the separate sections on the same molecule, however
these are neglected. The moduli, g i , and backbone orientation times, τ bi for a particular
melt can be determined from the linear viscoelastic behaviour of the material.
However, the values of τ s and q must be determined from a non-linear flow experiment,
usually uniaxial extension. In the multimode formulation of the pom-pom model
the direct computation of timescales from molecular structure is replaced by fitting to
experimental rheology. This set of variables {τ bi , g i , τ si , q i } can then by used to make
successful predictions for other simple flows, such as shear and planar extension, and for
complex geometries [Lee et al. (2001)]. The method is a generalisation of the practice
of fitting a spectrum of Maxwell modes to linear oscillatory shear data. In this case the
spectrum is extended to include non-linear parameters and the pom-pom model is used
as the underlying theory. It should be noted that the range of phenomena captured by
this approach is significantly larger than that of linear response.
2.5.4 Discussion of multimode pom-pom model
On first inspection the decoupling of different sections of a connected molecule appears
to be severe approximation. It is then, perhaps, surprising that the method works
so well. Blackwell et al. (2001) have investigated this decoupling by deriving a more
rigorous tube theory for symmetric molecules containing multiple layers of branching
including the appropriate coupling through the branch points. They compared their
predictions to those of the multimode pom-pom and managed to identify some discrepancies
which also appear in the comparison of the pom-pom model with real data.
Generally, the errors were not serious, though. It is likely that the errors associated
with the decoupling occur a large strains, particularly in steady state. In this case the
errors will be screened out by the viscous contribution from modes which are in linear
response. This mechanism also explains why the multi-mode pom-pom can be used to
model non-linear shear of branched polymer melts despite the omission of CCR (see