Hong Life CalculatorsUnderstanding the Mark-Houwink intrinsic viscosity calculator — concept and principle
Intrinsic viscosity [η] measures how much a single polymer chain thickens a dilute solution, and it grows with chain length. The Mark-Houwink (Mark-Houwink-Sakurada) equation links the two, so dilute-solution viscometry can be used to estimate molar mass, and gel permeation chromatography (GPC) can use it for universal calibration. This calculator gives [η] for a polymer of known molar mass, solvent and temperature.
How it is calculated
- [η] = K × Mᵃ
[η] is in mL/g (the same units as K), M is the molar mass in g/mol, and K and a are constants for a specific polymer, solvent and temperature, taken from tables or measured. The defaults are of the order reported for polystyrene in tetrahydrofuran (THF) near room temperature. To go the other way, solve M = ([η] ÷ K)^(1/a).
The principle
A flexible chain in solution is a loose, solvent-filled coil. Its contribution to viscosity depends on the volume the coil occupies per unit mass. The coil size grows with M, but more slowly than its mass in a poor solvent and faster in a good solvent, where the chain swells. The exponent a captures this: a ≈ 0 for compact spheres, a = 0.5 for a random coil in a theta solvent, typically about 0.6 to 0.8 for flexible coils in good solvents, and above 1 for stiff, rod-like molecules.
Worked example
- Defaults: K = 0.0114 mL/g, a = 0.716, M = 200,000 g/mol.
- Mᵃ = 200,000^0.716 = 6,245.07.
- [η] = 0.0114 × 6,245.07 = 71.19 mL/g.
- Check: (71.194 ÷ 0.0114)^(1/0.716) returns 200,000 g/mol.
| a | Intrinsic viscosity [η] (mL/g) |
|---|---|
| 0.179 | 0.1013419416 |
| 0.358 | 0.9008937828 |
| 0.537 | 8.008625009 |
| 0.716 | 71.19382525 |
| 0.895 | 632.8877613 |
| 1.07 | 5,626.146889 |
| 1.43 | 444,610.5924 |
How to read the result
Because a is below 1, doubling M does not double [η]: here it multiplies [η] by 2^0.716 ≈ 1.64, to about 117 mL/g. A larger [η] at the same molar mass indicates a more expanded coil, from a better solvent or a stiffer chain. When the equation is used to estimate M from viscometry of a polydisperse sample, the result is the viscosity-average molar mass Mv, which lies between Mn and Mw and is usually closer to Mw.
Common mistakes
- Mixing units: many tables give K for [η] in dL/g; 1 dL/g = 100 mL/g, so K must be converted accordingly.
- Using K and a for a different solvent or temperature; both change with conditions.
- Extrapolating outside the molar-mass range in which the constants were measured.
Frequently asked questions
What is the difference between intrinsic and relative viscosity?
Relative viscosity is solution viscosity divided by solvent viscosity at one concentration; intrinsic viscosity is the limit of (relative − 1)/concentration as concentration goes to zero.
What is universal calibration in GPC?
The product [η]M is proportional to the hydrodynamic volume, so polymers with the same [η]M elute together, allowing calibration across different polymers.
Where do K and a come from?
From measurements on fractions of known molar mass; large compilations such as the Polymer Handbook list values.
Sources: Flory, P. J. Principles of Polymer Chemistry. Cornell University Press, 1953. · Brandrup, J., Immergut, E. H. & Grulke, E. A. (eds.). Polymer Handbook. Wiley.