## Viscosity of Polymer Solutions

Part I: Intrinsic Viscosity of Dilute
Solutions

High molecular weight polymers greatly increase the viscosity of liquids in which they are dissolved. The increase in viscosity is caused by strong internal friction between the randomly coiled and swollen macromolecules and the surrounding solvent molecules. How much a polymer increases the viscosity of a solvent will depend on both the nature of the polymer and solvent.

Three important quantities frequently
encountered in the field of polymer solution rheology are the *relative viscosity*, the
*specific* and *reduced specific viscosity*. These
quantities are defined as follows:

*η _{rel}* =

*η*/

*η*

_{S}*η _{sp}* = (

*η*-

*η*) /

_{S}*η*=

_{S}*η*- 1

_{rel} *η _{red}* =

*η*/

_{sp}*c*= (

*η*- 1) /

_{rel}*c*

where *η* is the viscosity of the solution, *η _{S}*
that of the solvent and

*c*is the polymer concentration, usually expressed in grams per 100 cm³ or in grams per cm³. Another important quantity in very dilute solutions at vanishing shear rate is the

*intrinsic viscosity*(also called limiting viscosity number) which is defined as

[*η*]
describes the increase in viscosity of individual polymer chains.
Assuming the polymers are spherical impenetrable particles, the increase in
viscosity can be calculated with Einstein's viscosity relationship:

*η* = *η _{S}* (1 + 5/2

*φ*)

_{p}or

*η _{sp}* = 5/2

*φ*= 2.5

_{p}*N*/

_{p}v_{h}*V*= 2.5

*N*

_{A}c

*v*/

_{h}*M*

where *N _{p}* /

*V*is the number of particles per unit volume,

*v*the hydrodynamic volume of a polymer particle and

_{h}*M*its molecular weight. The hydrodynamic volume of a particle can be rewritten as follows

*v _{h}* /

*M*= 4/3

*· π*· (

*R*

_{h}^{2 }/

*M*)

^{3/2}

*M*

^{1/2}

Then the specific viscosity of a very dilute solution reads

[*η*] = (10 *π* / 3) *N*_{A}
(*R _{h,0}*

^{2 }/

*M*)

^{3/2}

*M*

^{1/2}

This and similar expressions for other particle geometries can be
found in many text books.^{1} In the case of dissolved, soft
polymer particles, Einsteins relation has to be modifed. It has been
shown that the equation is still applicable to dissolved polymers if the hard
sphere radius is replaced by the hydrodynamic radius of the polymer
coil. Then the equation can be rewritten in the form

[*η*] = *k N _{A}*
(⟨

*R*

_{h,0}^{2}⟩

^{ }/

*M*)

^{3/2}

*M*

^{1/2}

*α*

_{h}^{3}= Φ (⟨

*R*

_{h,0}^{2}⟩

^{ }/

*M*)

^{3/2}

*M*

^{1/2}

*α*

_{h}^{3}

where *α _{h}* =

*R*/

_{h}*R*is the expansion of a polymer coil in a good solvent over that of one in

_{h,0 }*θ-*solvent and

*R*is the radius of an unperturbed polymer. The equation is known as the Flory-Fox equation.

_{h,0}^{9}Under

*θ*-conditions it simplifies to

[*η*]* _{θ}* =
Φ

_{θ}⟨

*R*

_{h,0}^{2}⟩

^{3/2}/

*M*

The constant Φ_{θ} has a value of about
4.2·10^{24} for rigid spherical particles if [*η*] is expressed as
a function of the radius of
gyration.^{1,7} If the intrinsic viscosity is measured in
both a very dilute *θ-*solvent and in a "good" solvent, the expansion
can be directly estimated:^{5}

*α _{h}*

^{3}= [

*η*] / [

*η*]

_{θ }The values of *α _{h} *typically vary between unity for a

*θ-*solvent to about three for very good solvents increasing with molecular weight.

Both 〈*R _{h,0}*

^{2}〉 and

*α*can be expressed as a function of molecular weight

_{h}*M*:

〈*R _{h,0}*

^{2}⟩

^{1/2}=

*C*(

_{1}*M*/

*M*)

_{0}^{1/2}; 〈

*R*

_{h}^{2}⟩

^{1/2}=

*C*(

_{1}*M*/

*M*)

_{0}^{ν}

*α _{h}* = (

*v*

_{h}/

*v*

_{h,0})

^{1/3}=

*C*(

_{2}*M*/

*M*)

_{0}^{(ν - 1/2)}

where *C _{1}* and

*C*are constants,

_{2}*M*is the molar mass of a monomer and

_{0}*ν*is a scaling exponent. The value of

*ν*depends on the solvent-polymer system and its temperature. For example, under

*θ-*conditions the scaling exponent has the value

*ν*= 1/2 and in a good solvent

*ν*= 3/5. With these expressions, the equation for the intrinsic viscosity can be written in the form

[*η*] =
*K* *M*^{(3ν - 1)} = *K* *M*^{a}

*K* = const *M _{0}*

^{-3ν}

This equation is known as the *Mark-Howink* or Mark-Howink-Staudinger equation^{2-4}, where
*K*
has the dimensions cm³/g x (g/mol)^{a}.
Mark-Houwink parameters have been tabulated for a large number of polymer-solvent systems
in standard references.^{6} These parameters are
usually determined from a double logaritmic plot of intrinsic
viscosity versus molecular weight:

ln [*η*] =
ln
*K* + a ln *M*

### Intrinsic Viscosity versus Molecular Weight

**Example**:

0.1 g of atactic polystyrene of unknown molecular weight is
dissolved in 100 ml benzene. The Mark-Houwink parameters of this system
are *a* = 0.73 and *K* = 11.5 10^{-3}. To estimate the molecular weight, the viscosity of both the solvent and the solution have to be measured. A measurement with an Ubbelohde capillary viscometer yields
following results:

Pure benzene: 100 sec.

Polystyrene solution: 160 sec.

The viscosity is given by

*η _{rel}* =

*η*/

*η*= 160 / 100 = 1.6

_{S}*η _{sp}* =

*η*- 1 = 0.6

_{rel} *η _{red}* =

*η*/

_{sp}*c*= 0.6 / 0.001 g/ml = 600 ml/g

Assuming the concentration is sufficiently
close to zero so that [*η*] ≈ 6.0 10^{2} ml/g, the molecular weight
can be estimated with the the Mark-Houwink relation:

[*η*] = *K* *M*^{a}

600 = 11.5 10^{-3} *M*^{0.73} ⇒ *M *
= 2.9 10^{6} g/mol

##### References

- M.D. Lechner, K. Gehrke and E.H. Nordmeier,
*Makromolekulare Chemie*, Birkhaeuser, Basel 1993 - H. Mark, in R. Saenger,
*Der feste Koerper*, Hirzel, Leipzig, 1938 - R. Houwink ,
*J. Prakt. Chem.*, Vol. 157, Issue 1-3, p. 15 (1940) - H. Staudinger,
*Die Hochmolekulare Organischen Verbindungen*, Julius Springer, Berlin 1932 - H.K. Mahabadi, and A. Rudin,
*Poly. J.*, Vol. 11, No.2, pp 123-131 (1979) - J. Brandup, E.H. Immergut, and E.A. Grulke, Polymer Handbook, 4th ed., Wiley, New York 1999
- The ratio
*R*_{h}/*R*_{g}is typically in the range 0.65 - 0.75.^{8} - L.J. Fetters, J.S. Lindner, and J.W. Mays,
*J. Phys. Chem. Ref. Data*, Vol. 23, No. 4 (1994) - T.G. Fox, P.J. Flory,
*J. Am. Chem. Soc.*, 73, 1904-1908 (1951)