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

2.4. CHAIN STRETCH AND CONSTRAINT RELEASE 35
Here c/N is the polymer chain concentration.
obtained from
S(q) =
∫ Z ∫ Z
0
0
⎛
exp ⎝− ∑ α,β
q α q β
2
∫ s ′ ∫ s ′
s
s
〈 ∂Rα (s 1 )
∂s
The single chain structure factor is
⎞
〉
∂R β (s 2 )
∂s ′ ds 1 ds 2
⎠ dsds ′ . (2.40)
〈 〉
Equations 2.39 and 2.40 show that a knowledge of the function f αβ(s,s ′ ) = ∂Rα(s) ∂R β (s ′ )
∂s ∂s ′
is sufficient to evaluate both the stress and the single chain structure factor. By taking
suitable averages of equation 2.36 a deterministic PDE for f
≈
(s, s ′ ) can be obtained. In
deriving an expression for f αβ(s,s ′ ) the following closure approximation is necessary.
〈R α (s, t)R β (s ′ , t)λ(t)〉 ≈ λ(t)〈R α (s, t)R β (s ′ , t)〉. (2.41)
Milner et al. (2001) verified this approximation by direct stochastic simulation of equation
2.36 and found it to be valid whenever strong CCR is present. The resulting PDE
for f (s, s ′ ) is then solved by converting to a Fourier sine series.
≈
2.4.2 Comments on the MMcL model
The MMcL model is significant in that it demonstrated that a full Rouse-like treatment
of constraint release is sufficient to produce a monotonically increasing steady
state shear stress as a function of shear rate. This was achieved without needing to
include chain stretch, something which had not been managed previously with simpler
implementations of CCR. If c ν > 0.07 this theory predicts that all linear polymers
have no shear banding instability regardless of the number of entanglements. However,
a quantitative comparison of this theory with non-linear shear experiments is difficult.
In experimental studies, the large build up of normal stresses under shear limits
the range of shear rates. The usual resolution is to perform experiments on entangled
solutions which have lower absolute stresses in comparison with equivalent melts.
To achieve entangled solutions very high molecular weight polymers are used. The
maximum molecular weight which can be used is limited by the ability to synthesise
monodisperse samples in sufficient quantities. Thus the number of entanglements in
the measured systems is not huge. For examples of these studies see Menezes (1980),
Menezes and Graessley (1982) or Bercea et al. (1993). In this case the window of shear
rates which are simultaneously non-linear and non-stretching is very small. The effect
of CLF and thermal constraint release make the range smaller still. This indicates the
need to add chain stretch to the theory.
Another significant assumption of the model is that the system can be described
by a single constraint release time. This is valid for the window of flow rates to which
the MMcL theory pertains since the simulations of Milner et al. (2001) confirm that all