Mark–Houwink equation

The Mark–Houwink equation, also known as the Mark–Houwink–Sakurada equation or the Kuhn–Mark–Houwink–Sakurada equation or the Landau–Kuhn–Mark–Houwink–Sakurada equation or the Mark-Chrystian equation gives a relation between intrinsic viscosity $[\eta ]$ and molecular weight $M$ :^[1]^[2]

[\eta ]=KM^{a}

From this equation the molecular weight of a polymer can be determined from data on the intrinsic viscosity and vice versa.

The values of the Mark–Houwink parameters, $a$ and $K$ , depend on the particular polymer-solvent system as well as temperature. For solvents, a value of $a=0.5$ is indicative of a theta solvent. A value of $a=0.8$ is typical for good solvents. For most flexible polymers, $0.5\leq a\leq 0.8$ . For semi-flexible polymers, $a\geq 0.8$ . For polymers with an absolute rigid rod, such as Tobacco mosaic virus, $a=2.0$ .

It is named after Herman F. Mark and Roelof Houwink.

Applications

The Mark-Houwink equation is used in size-exclusion chromatography (SEC) to construct the so called universal calibration curve which can be used to determine the molecular weight of a polymer A using a calibration done with polymer B.

In SEC molecules are separated based on hydrodynamic volume, i.e. the size of the coil a given polymer forms in solution. The hydrodynamic volume, however, cannot simply be related to molecular weight (compare comb-like polystyrene vs. linear polystyrene). This means that the molecular weight associated with a given retention volume is substance specific and that in order to determine the molecular weight of a given polymer a molecular-weight size marker of the same substance must be available. However, the product of the intrinsic viscosity and the molecular weight, $[\eta ]M$ , is proportional to the hydrodynamic radius and therefore independent of substance. It follows that

[\eta ]_{A}M_{A}=[\eta ]_{B}M_{B}

is true at any given retention volume. Substitution of $[\eta ]$ using the Mark-Houwink equation gives:

K_{A}M_{A}^{a_{A}+1}=K_{B}M_{B}^{a_{B}+1}

which can be used to relate the molecular weight of any two polymers using their Mark-Houwink constants (i.e. "universally" applicable for calibration).

For example, if narrow molar mass distribution standards are available for polystyrene, these can be used to construct a calibration curve (typically $logM$ vs. retention volume ) in eg. toluene at 40 °C. This calibration can then be used to determine the "polystyrene equivalent" molecular weight of a polyethylene sample if the Mark-Houwink parameters for both substances are known in this solvent at this temperature.^[3]

References

^ Hiemenz, Paul C., and Lodge, Timothy P.. Polymer Chemistry. Second ed. Boca Raton: CRC P, 2007. 336, 338–339.
^ Rubinstein, Michael, and Colby, Ralph H.. Polymer Physics. Oxford University Press, 2003.
^ Mori, Sadao, and Barth, Howard G.. Size Exclusion Chromatography. First ed. Springer-Verlag Berlin Heidelberg New York, 1999. 107-110.

[1] Hiemenz, Paul C., and Lodge, Timothy P.. Polymer Chemistry. Second ed. Boca Raton: CRC P, 2007. 336, 338–339.

[2] Rubinstein, Michael, and Colby, Ralph H.. Polymer Physics. Oxford University Press, 2003.

[3] Mori, Sadao, and Barth, Howard G.. Size Exclusion Chromatography. First ed. Springer-Verlag Berlin Heidelberg New York, 1999. 107-110.

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