Polymer System Solutions 585 New York Manta Jim Carlin

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_rel - 1

η_red = η_sp / c = (η_rel - 1) / 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_p = 2.5 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_h the hydrodynamic volume of a polymer particle and 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_h,0  is the expansion of a polymer coil in a good solvent over that of one in θ-solvent and R_h,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   = Φ_θ ⟨R_h,0 ²⟩^3/2 / M

The constant Φ_θ 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

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_h can be expressed as a function of molecular weight M:

 〈R_h,0 ²⟩^1/2 = C₁ (M / M₀ )^1/2;  〈R_h ²⟩^1/2 = C₁ (M / M₀ ) ^ν

α_h = (v _h / v _h,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 M ^a

K = const M₀ ^-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_S = 160 / 100 = 1.6

η_sp = η_rel - 1 = 0.6

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] ≈ 6.0 10² 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 =  2.9 10⁶ 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 R _h / R _g 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)

Polymer System Solutions 585 New York Manta Jim Carlin

Source: https://polymerdatabase.com/polymer%20physics/Solution_Viscosity.html

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