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

2.3. DOI-EDWARDS MODEL OF ENTANGLED POLYMERS 25
The constant of proportionality, G 0 , relating the stress to the fraction of unrelaxed
tube is called the plateau modulus. The concept is generalised from calculation of the
modulus in a cross-linked network. The plateau modulus is related to the molecular
parameters by
G 0 = 4 ρRT
. (2.31)
5 M e
The experimental value of the plateau modulus is often used as a method of determining
M e , or equivalently the tube diameter. This can be problematic since the experimental
plateau in G ′ (ω) is only independent of chain length in the limit of very highly entangled
materials. For example see figure 2.8(b). The plateau can be seen by eye since data
are available on an exceptionally well entangled material (Z ≈ 44). However, it would
be less clear if the data for the higher molecular weight melts were not available. The
factor of 4/5 arises because the entanglements allows longitudinal motion of the chain
along the tube to relax stress. The proportion of stress relaxed in this way is 1/5 of
the total stress and in this sense entanglements are distinct from crosslinks since this
longitudinal motion is not possible in crosslinked systems. This factor has resulted in
wide spread confusion and contradiction. For an explanation and a proposed solution
to this see Likhtman and McLeish (2002). I will adopt their conventions in this thesis.
In particular the definition of the entanglement modulus G e = σ xy (τ e )/γ, the relaxation
modulus a t = τ e after a small step shear is safer. This is directly equivalent to the
cross linked version since no relaxation on length-scales longer than the tube diameter
will have occurred. In terms of the molecular parameters G e is given by
G e = ρRT
M e
. (2.32)
Equation 2.30 demonstrates the strong influence of entanglements on the chain
dynamics, in that it they increase the characteristic relaxation time by a factor of
3Z. The above result can also be derived directly by solving a microscopic Langevin
equation as in the derivation of the Rouse model (see section 2.4.1).
The relaxation spectrum derived above can be used with equation 1.14 to produce
predictions for the complex modulus of a linear polymer fluid under linear oscillatory
shear.
These model predictions, together with some experimental data on a range
of nearly monodisperse polystyrene melts are shown in figure 2.8.
The qualitative
similarities between the theory and data are very encouraging. Both sets show a plateau
region in G ′ , that increases in width with molecular weight. In both cases the plateau
height is independent of molecular weight. The experimental data show an upturn at
high frequencies that is not present in the theory. This can be rectified by taking into
account the free Rouse motion of the chain at length-scales which are shorter than the
tube diameter. Similar qualitative agreement can be seen for G ′′ (ω).