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

3.2. SINGLE MODE POM-POM MODEL 53
At constant α, decreasing τ b shifts the S xy curve to the right but does not alter its
shape since at lower orientation times it takes longer for the shear rate to reach a
sufficient size to orientate the backbone. Decreasing α broadens the curve because the
time interval between beginning orientation to completely orientating the backbone is
dilated. For intermediate relaxation times, where α ∼ 1/τ b , the solution is complicated
by the discontinuity in shear rate at t=0. This is observed in the rapid rise in degree
of orientation at early times followed by a shallowing of the gradient as the effect of
reptation becomes significant seen most clearly in the τ b = 0.1sec curve. The main
difference in behaviour of S xy between simple and exponential shear is that in the
exponential case the rising shear rate will always become large enough to orientate the
backbone sections and to cause S xy to tend to zero at long times regardless of how
small the backbone orientation time is.
In planar extension with a velocity field v x = ˙ɛx, v y = −˙ɛy and v z = 0, The rate
at which the flow stretches the backbone segments is ˙ɛ(S xx − S yy ). Figure 3.3 shows
the solutions to the pom-pom equations for orientation difference, S xx − S yy , in planar
extension. For ˙ɛ > 1/2τ b the backbone sections become fully aligned along the direction
1
0.8
0.6
S xx - S yy
0.4
0.2
τ b
[sec]
0.03
0.1
0.3
0.4
1.0
3.0
0
0 2 4 6 8
time [sec]
Figure 3.3: Evolution of S xx − S yy for a planar extensional flow, ˙ɛ = 1sec −1 .
of extension so that S xx − S yy approaches 1 at large times. For ˙ɛ < 1/2τ b the strain
rate is not sufficiently fast to align the backbones fully before they relax. The plateau
value of S xx − S yy falls with decreasing orientation time. Rapidly reptating backbones
hardly orientate at all in slow flows.
Figure 3.4 shows the predicted behaviour of S xx − S yy in exponential shear. When
α > 1/τ b the deformation is essentially affine. The resulting curves have some similarities
with those of planar extension. As is the case with extensional flows the molecular
segments align parallel to the x-axis. For α < 1/τ b the initial shear rate is too small to