## Viscosity of Polymer Solutions

Part II: Viscosity of Concentrated
Solutions

The viscosity of concetrated polymer solutions and melts has been studied for many decades and a large amount of data can be found in the literature.

The relationship between viscosity and concentration depends on
the polymer structure, its molecular weight and solvent quality, and
becomes much more complicated when the polymer
concentration increases, because the polymer chains start to interact
with each other, first through long range hydrodynamic interaction, and then by
actual contacts, entanglements, aggregates, and finally by a networks of physical crosslinks.
Thus, different relationships have been proposed for dilute and
concentrated polymer solutions. Two common equations that are frequently used to estimate the
reduced specific viscosity of polymer solutions as a function of
concentration are the Huggin's equation for very dilute solutions (> 1%) and the Martin's equation for moderately concentrated solutions,^{1,2}

*η _{sp}*/c = [

*η*] +

*K*[

_{H}*η*]

^{2}c

*η _{sp}*/c = [

*η*] exp(

*K*c [

_{H}*η*]) ⇒ ln

*η*/c = ln[

_{sp}*η*] +

*K*c [

_{H}*η*]

where *K _{H}* is the Huggins parameter, [

*η*] and

*η*the intrinsic viscosity and the specific viscosity respectively, and

_{sp }*c*is the polymer concentration, usually expressed in grams per 100 cm³ or in grams per cm. Both [

*η*] and

*K*are assumed to be independent of concentration but depend on the polymer-solvent system. Thus, if

_{H}*K*is known, the intrinsic viscosity [

_{H}*η*] can be calculated. However, both equations cannot be explicitly solved, meaning approximation methods have to be employed.

^{3}

The Huggins equation has been applied to a large number of systems and has given reasonable accurate estimates for the viscosity of dilute and moderately concentrated polymer solutions. For concentrated polymer solutions, Martin's equation is often expanded into a power series

*η _{sp}*/c ≈ [

*η*] {1 +

*K*c[

_{H}*η*] + (

*K*c[

_{H}*η*])

^{2}/2! + (

*K*c[

_{H}*η*])

^{3}/3! + ...}

According to Matsuoka^{4}, the higher terms describe the viscosity
contributions resulting from the interactions (entanglements) between two,
three, four etc. polymer coils.

### Reduced Specific Viscosity versus Concentration

The Martin and Huggins equation are a good approximation for dilute to medium
concentrations but are known to deviate from the experimental data at high c[*η*] values
whereas the power series provides a much better approximation when truncated after
the third term^{4} (see Figure above).

Many other relationships have been developed for the viscosity of dilute and concentrated polymer solutions. One of the oldest is the Baker equation (1913):

*η _{sp}* =
(1 + [

*η*]

*c*/

*n*)

^{n}- 1

The parameter n depends on the polymer-solvent system but
is assumed to be independent of concentration. As has been shown by Simha et al., the Baker equation can be applied over a
restricted range of concentration.^{2}

##### References

- M.L. Huggins,
*J. Am. Chem. Soc.,*64 (11), 2716 - 2718 (1942) - S.G. Weissberg, R. Simha and S. Rothman,
*J. Res. Natl. Bur. Stand*., 47, 4, 2257 (1951) - M.V.S. Rao,
*Polymer*, Vol. 34, No. 3, pp. 592-596 (1993) - S. Matsuoka, M.K. Cowman,
*Polymer*, 43, 3447 - 3453 (2002)