shear-thinning:
the fatal failure of reptation
Small
Angle Neutron Scattering (SANS) studies on polymer melts
under steady-state flow provide
in situ information on how the flow field is transmitted to the melt
at a molecular scale. Such experiments, called "Rheo-SANS, are
difficult to set up and require special equipment but their results
are fundamental to
test experimentally the admitted claim that the shear-thinning of
entangled polymer chains is due to significant orientation of the
entanglement segments under the shear flow.
I
refer below to two significant Rheo-SANS studies, one by Watanabe et
al in Japan published in 2007 and the other one by Noirez et al in
France published in 2009.
1.
Hiroshi Watanabe, Toshiji Kanaya, and Yoshiaki Takahashi "Rheo-SANS
behavior of Entangled Polymer Chains with Local Label Under Fast
Shear Flow".
Activity
Report on Neutron Scattering Research: Experimental Reports 14
(2007)
Report
Number: 146.
2.
L.
Noirez, H. Mendil-Jakani, P. Baroni Macromol.
Rapid Commun. 2009,
30,
1709–1714 ""New
Light on Old Wisdoms on Molten Polymers: Conformation, Slippage and
Shear Banding in Sheared Entangled and Unentangled Melts".
Both
studies conclude that
the chains
remain largely undeformed under steady-state shear flow conditions
for which extensive shear-thinning was present.
This
means that there is no change of the rms end-to-end distance of the
chain as the flow crosses from Newtonian to non- Newtonian, in
contradiction with the concept of the deformation of single chains
during shear-thinning. These results represent a formidable challenge
to the reptation model of melt deformation.
Classical
visco-elasticity theory considers the rheological deformation of
polymer melts as resulting from the behavior of singular chains
embedded in a sea of interactions from other chains. In
the existing theories of macromolecular physics, the accent is put on
determining the shape of the individual macromolecules, often called
their configuration. The presence of neighboring and interpenetrating
macromolecules is perceived as a disturbance to the ideal
configuration of the chain. In the traditional texts, the field of
interaction responsible for the disturbance is homogeneous. One can,
therefore, describe the behavior of the melt by describing what
happens to a single chain after it has been established how to
incorporate the effect of the interactions between the chains.
Thermodynamically
speaking, "molecular dynamics" deals with the physics of
systems that are single chains This is the case for the popular
reptation model introduced by de Gennes and fined tuned by Doi and
Edwards, Marrucci, Wagner, Mc Leish and many others. Macromolecules
are able to rearrange their multiple configurations with the change
of the thermal or mechanical energy input. In the case of shear
deformation, the Newtonian viscosity is classically considered to
describe the internal friction between the bonds of interacting
macromolecules which assume a stable thermodynamics state, the
equilibrium state at a given temperature and pressure.
The
non-Newtonian
behavior, shear-thinning,
is due to a modification by the flow of the dimensions of the
macromolecules, i.e. of their configuration, which can be calculated
from the effect of the shear rate on the rms end to end distance of
the macromolecule and the amount of slippage (disentanglement)
occurring. Theoretical models predict that for a shear rate strong
enough to overpower the ability of the chain to relax, -and this
happens at the reptation time-, shear-thinning starts to be observed,
corresponding to an increase of the rms end to end distance (chain
orientation). In
the classical formula that describe the non-Newtonian dependence of
viscosity with shear rate, the amount of shear-thinning is only
function of two parameters (in addition to the strain rate, of
course): the Newtonian viscosity and the value of the reptation time.
But these two parameters can be correlated to each other and to the
dimensions and interactions between the chains, which simplifies the
description of the flow deformation process to the description of the
dependence of the reptation time with temperature, pressure, and
chain length (the interactions between the macromolecules, defined by
"their entanglement", is already incorporated in the
definition of the reptation time).
In
summary, the effect of strain rate, temperature, molecular weight,
could all be resumed to a simple explanation: the deformation and
relaxation of single macromolecular chains confined to move within
the boundaries of a tube, the entanglement tube, whose lifetime was
the reptation time. The whole process would continuously be
happening, from very low strain rate to high shear-thinning producing
strain rate. Additionally, the reptation model provided a new
understanding of "entanglement" by quantifying the
dimensions of the tube and correlating it to the reptation time. The
interactions between the macromolecules could be described
topologically, the tube serving as the new topological description
of the environment of the bonds.
This
was the beauty of the reptation model of de Gennes which
had succeeded in scaling the effect of all variables into the
description of a single parameter, the reptation time. Of course,
this extraordinary tour de force had to be refined over the years to
account for a better description of reality, in particular the
molecular weight dependence of the reptation time, which did not
follow the predicted M3 variation.
Yet,
despite of all its elegance, apparent success and sophistication,
the reptation model is not correctly describing the
reality of the interactions between the macromolecules
(Rheo-SANS experiments of Watanebe et al and Noirez et al) and
should be abandoned.
If
something as simple as shear-thinning (see the Fig. at the top) cannot be explained by the
current theoretical model, I am sorry to say, there is no other
solution than abandon it.
The
reason for this radical proposition is that the
dynamics of the interactions defining the melt properties should not
be defined by thermodynamic systems which are the single
macromolecules.
I
propose to re-consider the mechanism of deformation of polymer
melts giving rise to shear-thinning, correctly describing the
Rheo-SANS experimental results, and to re-consider the concept of
entanglement, the corner stone of polymer physics.
