## Polymer physics

### Related Topics:

Polymer Viscosity

## 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:

ηrel = η / ηS

ηsp = (η - ηS) / ηS = ηrel - 1

ηred = ηsp / c = (ηrel - 1) / c

where η is the viscosity of the solution, ηS 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 = ηS (1 + 5/2 φp)

or

ηsp =  5/2 φp = 2.5 Np vh / = 2.5 NA c vh / M

where Np / V is the number of particles per unit volume, vh the hydrodynamic volume of a polymer particle and M its molecular weight. The hydrodynamic volume of a particle can be rewritten as follows

vh / M = 4/3 · π · (Rh2 / M)3/2 M1/2

Then the specific viscosity of a very dilute solution reads

[η] = (10 π / 3) NA (Rh,02 / M)3/2 M1/2

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 NA (Rh,02 / M)3/2 M1/2αh3 = Φ (Rh,02 / M)3/2 M1/2αh3

where αh = Rh / Rh,0  is the expansion of a polymer coil in a good solvent over that of one in θ-solvent and Rh,0 is the radius of an unperturbed polymer. The equation is known as the Flory-Fox equation.9 Under θ-conditions it simplifies to

[η]θ  = ΦθRh,023/2 / M

The constant Φθ has a value of about 4.2·1024 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

αh3 = [η] / [η]θ

The values of αh typically vary between unity for a θ-solvent to about three for very good solvents increasing with molecular weight.

Both 〈Rh,02〉 and αh can be expressed as a function of molecular weight M:

〈Rh,021/2 = C1 (M / M0)1/2;  〈Rh21/2 = C1 (M / M0)ν

αh = (vh / vh,0)1/3 = C2 (M / M0)(ν - 1/2)

where C1 and C2 are constants, M0 is the molar mass of a monomer and 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 Ma

K = const M0-3ν

This equation is known as the Mark-Howink or Mark-Howink-Staudinger equation2-4, 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

ηrel = η / ηS = 160 / 100 = 1.6

ηsp = ηrel - 1 = 0.6

ηred = ηsp / c = 0.6 / 0.001 g/ml = 600 ml/g

Assuming the concentration is sufficiently close to zero so that [η] ≈ 6.0 102 ml/g, the molecular weight can be estimated with the the Mark-Houwink relation:

[η] = K Ma

600 = 11.5 10-3 M0.73M =  2.9 106 g/mol

##### References
1. M.D. Lechner, K. Gehrke and E.H. Nordmeier, Makromolekulare Chemie, Birkhaeuser, Basel 1993
2. H. Mark, in R. Saenger, Der feste Koerper, Hirzel, Leipzig, 1938
3. R. Houwink , J. Prakt. Chem., Vol. 157, Issue 1-3, p. 15 (1940)
4. H. Staudinger, Die Hochmolekulare Organischen Verbindungen, Julius Springer, Berlin 1932
5. H.K. Mahabadi, and A. Rudin, Poly. J., Vol. 11, No.2, pp 123-131 (1979)
6. J. Brandup, E.H. Immergut, and E.A. Grulke, Polymer Handbook, 4th ed., Wiley, New York 1999
7. The ratio Rh / Rg is typically in the range 0.65 - 0.75.8
8. L.J. Fetters, J.S. Lindner, and J.W. Mays, J. Phys. Chem. Ref. Data, Vol. 23, No. 4 (1994)
9. T.G. Fox, P.J. Flory, J. Am. Chem. Soc., 73, 1904-1908 (1951)
• ## Summary

### Intrinsic Viscosity

is a measure for the internal friction in polymer solutions at the limit of zero polymer concentration. Thus, this quantity describes the effect of completely separated polymer chains on the solution viscosity.

• ### Mark-Houwink Relation

The viscosity of very dilute polymer solutions can be described with a simple power law, known as the Mark-Houwink relation.

• The Mark-Houwink parameters can be determined from a double logaritmic plot of intrinsic viscosity versus molecular weight which yields straight lines.

• ### Gel permeation chromatography (GPC)

is frequently used to measure the molecular weight distribution of a polymer sample. The retention time of a polymer in a GC column is directly proportional to its size. Thus a GPC seperates dissolved polymers by their hydrodynamic volume.

• A GPC chromatogram provides the molecular weight distribution of a polymer sample because the hydrodynamic volume of a polymer particle is directly related to the molecular weight.

•

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