## Molecules Weight Dependence of Melt Viscosity

The flow behavior of low molecular weight polymers differs significantly from that of high
molecular weight polymers. As one might expect, melts and solutions of
(very) low
molecular weight behave like Newtonian liquids,
that is, the viscosity is shear rate independent. This is not
necessarily the case for polymers of high molecular weight where entanglement of polymer molecules takes place,
that is, entangled polymers show
a much stronger resistance to flow, and since polymers tend to disentangle when subjected to shear
above a critical value *ý _{c}*, the viscosity depends on the shear rate

*ý*whereas below

*ý*the viscosity is more or less constant. The shear rate dependence (shear thinning) is usually expressed in a power law:

_{c} *η * = const. for
*ý* ≤ *ý _{c}*

*η * =* Φ ý*^{n-1} for *ý* ≥ *ý _{c}*

where *ý _{c }*
is the critical shear rate for the transition from Newtonian to
shear thinning behavior,

*Φ*is an interaction constant which depends on the polymer structure, and

*n*is the power-law coefficient describing the shear thinning behavior.

^{6}

To be
able to compare the rheology of polymers with different MW, the
dependence on shear rate has to be eliminated. This can be
achieved by using the zero shear-rate viscosity, *η _{0}*, which is the viscosity in the limit of infinite small shear rate. As the name suggest, this viscosity can not be measured but has to
be extrapolated. At this shear rate, Newtonian or quasi Newtonian behavior
is observed, and when the molecular weight drops below a certain threshold where entanglements
between chains become unlikely or even impossible, the viscosity is
directly proportional to the molecular weight

*M*. The critical molecular weight for entanglement coupling is about two times the entanglement weight,

*M*= 2

_{c}*M*. Above this value, the zero shear rate viscosity can be described by a simple power law:

_{e}^{1,2,5}

*η _{0}* =

*k*·

_{l}*M*,

*M*<

*M*

_{c}*η _{0}* =

*k*·

_{m}*M*

^{α},

*M*

*> M*

_{c}The power-law coefficient *α* of polymer melts has a value of about 3.4 ± 0.2.

As has been shown by Nichetti eta al. (1998)^{3}, the critical shear rate *ý _{c}* is a
function of the molecular weight. At

*ý*=

*ý*the zero shear rate viscosity

_{c}*η*has to be equal to the power law viscosity:

_{0} *η _{0} *=

*k*

_{m}*M*

^{α }=

*Φ*

*ý*

_{c}^{n-1}

or

*ý _{c}* = (

*Φ*/

*k*

_{m}*M*

^{α})

^{1/(n-1)}

The parameter *α* is the inverse of *n*: n = *α*^{-1}.
This is not surprising since *n* describes the
disentanglement (shear-thinning) an *α* the entanglement state
at zero shear rate.

The critical shear rate *ý _{c}* can also be
derived from the characteristic disentanglement or relaxation time

*τ*which is inverse proportional to the critical shear rate:

_{R}^{3,4}

*ý _{c} *=

*τ*

_{R}^{-1}

The relaxation time depends on the polymer concentration and type
of solvent (if present). For example, in a melt *τ _{R}* is proportional to

*M*

^{3.4}whereas for polymers in a dilute theta-solvent

*τ*∼

_{R}*M*

^{3/2}and in a bad solvent

*τ*∼

_{R}*M*.

##### References and Notes

- T.G. Fox, P.J. Flory,
*J. Am. Chem. Soc.*, 70, 2384-2395 (1948) - T.G. Fox, P.J. Flory,
*J. Polym. Sci.*, 14, 315-319 (1954) - D. Nichetti, I. Manas-Zloczowera,
*J. Rheol.*, 42 (4), 951-969 (1998) - M. Bouldin, W.-M. Kulicke, H. Kehler,
*Coll. and Poly. Sci.*, Vol. 266, 9, 793–805 (1988) - Lewis J. Fetters, David J. Lohse, and Scott T. Milner,
*Macromolecules*, 32, 6847-6851 (1999) For

*n*= 1, the viscosity is shear rate independent. Newtonian behavior is observed at both very high and very low shear rates. The first Newtonian limit is often denoted as*η*and the second as_{0 }*η*_{∞}.