Mark–Houwink equation

Applications

The Mark–Houwink equation can be used in size-exclusion chromatography (SEC)/gel permeation chromatography (GPC) 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 (imagine eg. the coiling of comb-like polystyrene vs. linear polystyrene). This means that the molecular weight associated with a given retention time/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, ${\displaystyle [\eta ]M}$, is proportional to the hydrodynamic radius and therefore independent of substance. It follows that

${\displaystyle [\eta {]}_A{M}_A=[\eta {]}_B{M}_B}$

is true at any given retention volume/time. Substitution of ${\displaystyle [\eta ]}$ using the Mark–Houwink equation gives:

${\displaystyle K_A{M}_A^{a_A+1}=K_B{M}_B^{a_B+1}}$

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 ${\displaystyle \log M}$ vs. retention volume ) in eg. toluene at 40 °C. This calibration can then be used to determine the "polystyrene equivalent" molecular weight of eg. a polyethylene sample or any other polymer for which standards might not be available if the Mark–Houwink parameters for both substances are known in this solvent and at this temperature.