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

3.2. SINGLE MODE POM-POM MODEL 51
γ . [sec -1 ]
0.25
0.2
1.225
0.7
3.0
5.0
0.25
affine
S xy
0.15
0.1
0.05
0
0 5 10 15 20
time [sec]
Figure 3.1: Evolution of S xy for simple shear shear, ˙γ = 1sec −1 . Affine deformation
corresponds to τ b = ∞ . The thin solid line corresponds to the value of τ b for which
the steady state value of S xy is maximised.
value of S xy can be shown to be
τ b(max) =
√
3/2
. (3.3)
˙γ
This is shown as the thin solid line in figure 3.1. The dashed curves in figure 3.1, which
correspond to values of τ b other than τ b(max) , all fall below the τ b(max) curve in steady
state. When τ b < τ b(max) the rapid reptation means that the flow is not fast enough,
on average, to orientate the backbones significantly before they relax. For τ b > τ b(max)
the backbones are orientated through the optimum angle but the flow then continues
to orientate them towards the x-axis before the slow orientation relaxation produces
a dynamic equilibrium.
The contribution of a single backbone to S xy is maximised
when it lies in the x-y plane, making an angle of 45 o with the x-axis. At this angle
the contribution to S xy is 1/2. Note that even in the affine case S xy never reaches this
value since, due to the isotropic initial distribution of backbones, all of the backbones
will never simultaneously point in the same direction.
For nearly exponential shear the full analytic solution for S xy is
(1 + 2 α τ b ) ( e αt − 1 )
S xy =
),
(6 ατ b + 2ατ b e 2αt + 2ατ b e − t
τ b − 4ατ b e (α− 1 )t − τ b + 2e t
τ b − 2e (α− 1 )t τ b + 9 +
3
ατ b
(3.4)
with α as defined in equation 3.1. Figure 3.2 shows plots of S xy for varying backbone
orientation times. The solutions for S xy separate into two distinct regimes. For 1/τ b < α