## Flow Properties of Polymers

Time-independent Fluids

Polymer solutions, dispersions, and melts are usually non-Newtonian liquids. This means their
*apparent viscosity* (*η*)^{1} depends on the applied shear rate
and increases rapidly with increasing molecular weight (number of repeat units).
Thus, the viscosity of a polymer melt is always larger than that of the corresponding
monomer. This is due to entanglement and intermolecular forces between polymer molecules.

The shear rate (*γ*) - shear stress (*τ*) relationship of time independent
non-Newtonian fluids can be described by the general equation

or graphically by a curve of shear stress as a function of shear rate. The four basic types of time-independent fluids are shown in the figures below.

It must be emphazised that these types are an idealization of the
real flow behavior of fluids. Most polymer solutions and melts
exhibit shear thinning, that is, they belong to the class of
*pseudoplastic* materials, whereas shear-thickening or *
dilatant* behavior is rarely
observed. Some common examples of shear-thickening fluids are
cornstarch in water and nanoparticles dispersed in a (polymer)
solution.

The observed shear thinning of polymer melts and
solutions is caused by disentanglement of polymer chains during
flow. Polymers with a sufficiently high molecular weight are always
entangled (like spagetti) and randomly oriented when at rest. When
sheared, however, they begin to disentangle and to allign which
causes the viscosity to drop. The degree of disentanglement will
depend on the shear rate. At sufficiently high shear rates the
polymers will be completely disentangled and fully alligned. In this
regime, the viscosity of the polymer melt or solution will be
independent of the shear rate, i.e. the polymer will behave like a
Newtonian liquid again.^{2} The same is true for very low shear rates;
the polymer chains move so slowly that entanglement does not impede
the shear flow. The viscosity at infinite slow shear is
called *zero shear rate viscosity* (*η _{0}*). The typical behavior is ilustrated in the figure
below that shows the dependence of the apparent viscosity, η, of a polymeric melt on shear rate.

The behavior of fluids in the shear-thinning regime can be described with the power-law equation of Oswald and de Waele:

This equation may be written in logarithmic form,

This means, a log-log plot of shear stress (*τ*) versus shear strain (d*γ*/d*t*)
should yield a straight line if the polymer solution or melt behaves like a
pseudoplastic liquid. Usually a straight line can be drawn over one
to two decades of shear rate, but over a wider range deviations from
the Oswald law can be expected.

The apparent viscosity is defined by

If we combine this expression with the Oswald equation, we obtain a second power-law equation for the apparent viscosity:

A power law can also be used to describe the behavior of a dilatant (shear-thickening)
liquid. In this case, the value of the exponent *n* will be greater than one.
Again, noticeable deviations can be expected when the Oswald
equation is applied over a
wider range of shear rates.

Some other fluids require a threshold shear stress before they
start to flow. This kind of fluid is called a *plastic fluid*
and if the flowing liquid has a constant viscosity it is called a *
Bingham liquid*. However, such behavior is not observed in
ordinary polymer melts and solutions. Typical examples for plastic
flow behavior are polymer/silica micro- and nanocomposites. The
solid-like behavior at low shear stress can be explained by the
formation of a silica network structure arising from attractive
particle-particle interactions due to hydrogen bonding between
silanol groups. Once the particle network breaks down upon
application of a critical yield stress (*τ _{y}*), the polymer shows normal flow behavior.

The flow behavior of plastic fluids having a constant viscosity *η _{p}*
above the yield stress can be described with the Bingham equation:

whereas non-Newtonian (shear-thinning) behavior of a plastic fluid can be described with the Herschel-Bulkley model:

Using the standard definition for viscosity: *η* = *τ */* γ*, the apparent
viscosity of an Bingham viscoplastic material can be written as

Thus, the apparent viscosity of a Bingham fluid decreases with increasing shear rate and reaches at very high shear rates the constant limit *η _{p}*.

^{1}The
apparent viscosity is often given the symbol *η* instead of *μ* to distinguish it from the Newtonian viscosity.

^{2}The second plateau is rarley observed for polymer melts because it requires extremely high shear rates which might also cause the polymer chains to break
(shear-induced degradation).