Viscometry method to determine molar masses of polymer

Staudinger proposed this method. Accurate measurement of absolute viscosity being difficult, it is convenient to measure relative viscosity, η_rel, defined as,

Relative Viscosity

Relative viscosity is the ratio between the viscosity of the solution and the viscosity of the pure solvent.

\[\displaystyle {{\eta }_{{rel}}}=\frac{\eta }{{{{\eta }_{0}}}}---(1)\]

η = viscosity of the solution and η₀ = viscosity of the solvent

Specific Viscosity

Relative viscosity reduces unity and is called specific viscosity.

\[\displaystyle {{\eta }_{{sp}}}={{\eta }_{{rel}}}-1---(2)\]

Reduced Viscosity

The ratio between specific viscosity and concentration of the polymer is called reduced viscosity.

\[\displaystyle {{\eta }_{{red}}}=\frac{{{{\eta }_{{sp}}}}}{c}---(3)\]

Where c = concentration of the polymer

Intrinsic Viscosity

Intrinsic viscosity is a limit of reduced viscosity.

\[\displaystyle \left[ \eta \right]=c\overset{{\lim }}{\longrightarrow}0\left( {\frac{{{{\eta }_{{sp}}}}}{c}} \right)---(4)\]

In 1906, Einstein derived the following relation between the viscosity of a dilute suspension of complex spherical molecules and the volume fraction,Φ , of the solute molecules:

\[\displaystyle \eta ={{\eta }_{0}}\left( {1+2.5\phi } \right)---(5)\]

The rearrangement of equation (5) gives,

\[\displaystyle \frac{\eta }{{{{\eta }_{0}}}}-1={{\eta }_{{sp}}}=2.5\phi ---(6)\]

From equations (4) and (5),

\[\displaystyle \left[ \eta \right]=c\overset{{\lim }}{l\longrightarrow}0\left( {\frac{{{{\eta }_{{sp}}}}}{c}} \right)=\frac{{2.5\phi }}{c}---(7)\]

It is tough to measure , the volume fraction of the polymer molecules in solution. The plot of η_sp/c and ln η_red/c → c gives straight lines, which conform to the following equation:

Concentration vs. Reduce and intrinsic-viscosity

\[\displaystyle \text{Huggin,s equation: }\frac{{{{\eta }_{{sp}}}}}{c}=\left[ \eta \right]+{{k}^{'}}{{\left[ \eta \right]}^{2}}c---(8)\]

\[\displaystyle \text{Cramer,s Equation: }\ln \frac{{{{\eta }_{{red}}}}}{c}=\left[ \eta \right]-{{k}^{{''}}}{{\left[ \eta \right]}^{2}}c---(9)\]

Both equations are applicable only in dilute solutions. For, large number of polymers, k’ = 0.4 ± 0.1 and k’’= 0.50 ± 0.05

Staudinger found, in 1950, that for a series of samples of the same polymer in a given solvent and at a constant temperature, the intrinsic viscosity is related to the molar mass of the polymer by the following equation, known as, Mark-Kuhn-Houwink-Sakurada equation, called Staudinger equation:

\[\displaystyle \left[ \eta \right]=k{{\left( {\overline{{{{M}_{{visc}}}}}} \right)}^{a}}---(10)\]

Here, K and a are constant. The value of the exponent a depends upon the geometry or the shape of the macromolecules.

Taking logs on both sides of equation 10, we get,

\[\displaystyle \ln \left[ \eta \right]=\ln k+a\ln \overline{{{{M}_{{visc}}}}}----(11)\]

η vs. molecular viscosity

Plot a graph of [η] → _visc gives a straight line with a slope equal to a and an intercept equal to lnk. The constants a and K can be easily determined.