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 ${\displaystyle [\eta ]}$ and molecular weight ${\displaystyle M}$:^[1][2]

${\displaystyle [\eta ]=KM^{a}}$

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

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, ${\displaystyle [\eta ]M}$, 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. 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 ${\displaystyle logM}$ 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