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

1.6. NON-LINEAR RHEOLOGY 9
1.5.2 Linear continuous shear
Equation 1.10 can also be used to model a constant rate forward deformation provided
that the deformation rate is small in comparison to the longest relaxation time of the
material. For a continuous shear deformation commencing at time t = 0 the model
predicts,
σ xy (t) =
∫ t
0
G(t − t ′ ) ˙γdt ′ . (1.16)
In these shear experiments the transient shear viscosity, η + (t) = σ xy (t)/ ˙γ, is often
plotted as a function of time. In this plot a Newtonian fluid would show a constant
response at all deformation rates (see figure 1.1). For a linear viscoelastic fluid, whose
constitutive equation is given by equation 1.8, the transient shear viscosity, η 0 (t), will
be independent of the applied shear rate. This behaviour is seen in polymeric fluids
at low shear rates (less than the reciprocal of the longest relaxation time) and also at
early times when the applied strain, ˙γt, is small. Under these conditions the material
is described as being in linear response. The master curve can be obtained from the
linear relaxation spectrum. If G(t) is taken to be the sum of independent Maxwell
modes (equation 1.15) then equation 1.16 gives
η 0 (t) = ∑ i
g i τ i [1 − exp(−t/τ i )] . (1.17)
At higher deformation rates the transient curves of polymeric fluids often deviate from
linear response at strains of order one. The form of the deviation can be used to classify
the material’s non-linear response.
A similar transient viscosity, based on the first normal stress difference, can be
defined for extensional flows η + E (t) = N 1/˙ɛ. Using equation 1.8 it can be shown that,
for a linear viscoelastic fluid in uniaxial extension, η + E (t) = 3η 0(t).
1.6 Non-linear rheology
Non-linear rheology refers to flows in which the strain rates and accumulated strains
are large. More specifically, the strain rate must be faster than some characteristic
time of the material, usually the terminal time. Strains must be of order one or larger
to observe non-linear effects. These flows are useful for a number of reasons. Nonlinear
measurements on polymer liquids often show very striking behaviour and can,
consequently, be more sensitive to molecular details than weaker deformations. For
example, the extensional rheology of a polydisperse system can be strongly dependent
on the high molecular weight tail of the distribution. A material’s response to different
deformation geometries is often qualitatively different under non-linear deformations.