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

3.1. INTRODUCTION 49
principle extension ratios to planar extension (see section 1.4.2). In both flows one
direction extends exponentially in time, a second is unchanged and, to conserve volume,
the final direction compresses exponentially. For example
λ 1 (t) = e αt ,
λ 2 (t) = 1,
λ 3 (t) = e −αt . (3.2)
The principle extension ratios are found by taking the square root of the eigenvalues
of the finger tensor, C −1 . Under some flow classification schemes this exponential
≈
growth of one of the principle stretch ratios classifies a flow as “strong” [Tanner and
Huilgol (1975)]. A flow is classified as weak if it evolves in any other way. Hence under
this and most other classification schemes extensional flow and exponential shear flows
are regarded as strong. Simple shear is deemed to be weak since its largest principle
stretch ratio grows linearly with time [Doshi and Dealy (1987)]. These schemes are
based around the idea that a strong flow is likely to stretch the polymer chains of a
melt. Venerus (2000) presents a more detailed summary of flow classification.
A flow classification scheme can be given credibility if it is found to be consistent
with rheological data. For example transient first normal stress growth coefficient data
for branched polymer melts consistently rise above the predictions of linear viscoelastic
theory at large strains. This is the so called strain hardening phenomenon. Conversely
transient shear stress growth coefficient data typically fall below linear viscoelastic predictions
at large strains and this is known as strain softening. In these cases the above
flow classification scheme is consistent, in that the strong flows strain harden and weak
flows soften. Dealy (1990) notes that there is some ambiguity in this use of linear viscoelastic
theory as a reference curve for strain hardening since the phenomenon usually
occurs at large strains where the theory is no longer valid. He suggests alternative
reference curves, such as finite linear viscoelastic theory (FLV) 1 , but observes that all
known flows thin relative to these curves. Continuing, he also points out further problems
in choosing a suitable reference curve for exponential shear, particularly for first
normal stress difference since linear viscoelastic theory predicts a zero value for this
quantity. However, by careful choice of material function, Venerus (2000) has shown
that exponential shear stress data fall below even linear viscoelastic theory and so has
classified it as a weak flow.
In this chapter I will solve the pom-pom equations as derived by McLeish and
Larson (1998) for exponential shear and compare the solutions to those for simple
shear and planar extension in section 3.2. The model will then be tested quantitatively
1 Otherwise known as first-order linear viscoelastic theory or the Lodge rubber-like model