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# Intrinsic viscosity

Intrinsic viscosity $\left[ \eta \right]$ is a measure of a solute's contribution to the viscosity η of a solution. It is defined as

$\left[ \eta \right] = \lim_{\varphi \rightarrow 0} \frac{\eta - \eta_{0}}{\eta_{0}\varphi}$

where η0 is the viscosity in the absence of the solute and φ is the volume fraction of the solute in the solution. As defined here, the intrinsic viscosity $\left[ \eta \right]$ is a dimensionless number. When the solute particles are rigid spheres, the intrinsic viscosity equals 2.5, as shown first by Albert Einstein.

In practical settings, φ is usually solute mass concentration, and the units of intrinsic viscosity $\left[ \eta \right]$ are inverse concentration (deciliters per gram).

## Formulae for rigid spheroids

Generalizing from spheres to spheroids with an axial semiaxis a (i.e., the semiaxis of revolution) and equatorial semiaxes b, the intrinsic viscosity can be written

$\left[ \eta \right] = \left( \frac{4}{15} \right) (J + K - L) + \left( \frac{2}{3} \right) L + \left( \frac{1}{3} \right) M + \left( \frac{1}{15} \right) R N$

where the constants are defined

$M \ \stackrel{\mathrm{def}}{=}\ \frac{1}{a b^{4}} \frac{1}{J_{\alpha}^{\prime}}$
$K \ \stackrel{\mathrm{def}}{=}\ \frac{M}{2}$
$J \ \stackrel{\mathrm{def}}{=}\ K \frac{J_{\alpha}^{\prime\prime}}{J_{\beta}^{\prime\prime}}$
$L \ \stackrel{\mathrm{def}}{=}\ \frac{2}{a b^{2} \left( a^{2} + b^{2} \right)} \frac{1}{J_{\beta}^{\prime}}$
$N \ \stackrel{\mathrm{def}}{=}\ \frac{6}{a b^{2}} \frac{\left( a^{2} - b^{2} \right)}{a^{2} J_{\alpha} + b^{2} J_{\beta}}$

The J coefficients are the Jeffery functions

$J_{\alpha} = \int_{0}^{\infty} dx \frac{dx}{\left( x + b^{2} \right) \sqrt{\left( x + a^{2} \right)^{3}}}$
$J_{\beta} = \int_{0}^{\infty} dx \frac{dx}{\left( x + b^{2} \right)^{2} \sqrt{\left( x + a^{2} \right)}}$
$J_{\alpha}^{\prime} = \int_{0}^{\infty} dx \frac{dx}{\left( x + b^{2} \right)^{3} \sqrt{\left( x + a^{2} \right)}}$
$J_{\beta}^{\prime} = \int_{0}^{\infty} dx \frac{dx}{\left( x + b^{2} \right)^{2} \sqrt{\left( x + a^{2} \right)^{3}}}$
$J_{\alpha}^{\prime\prime} = \int_{0}^{\infty} dx \frac{x\ dx}{\left( x + b^{2} \right)^{3} \sqrt{\left( x + a^{2} \right)}}$
$J_{\beta}^{\prime\prime} = \int_{0}^{\infty} dx \frac{x\ dx}{\left( x + b^{2} \right)^{2} \sqrt{\left( x + a^{2} \right)^{3}}}$

## General ellipsoidal formulae

It is possible to generalize the intrinsic viscosity formula from spheroids to arbitrary ellipsoids with semiaxes a, b and c.

## Frequency dependence

The intrinsic viscosity formula may also be generalized to include a frequency dependence.

## Applications

The intrinsic viscosity is very sensitive to the axial ratio of spheroids, especially of prolate spheroids. For example, the intrinsic viscosity can provide rough estimates of the number of subunits in a protein fiber composed of a helical array of proteins such as tubulin. More generally, intrinsic viscosity can be used to assay quaternary structure. In polymer chemistry intrinsic viscosity is related to molar mass through the Mark-Houwink equation.A practical method for the determination of intrinsic viscosity is with a Ubbelohde viscometer.

## References

• Jeffery GB. (1922) "The Motion of Ellipsoidal Particles Immersed in a Viscous Fluid", Proc. Roy. Soc., A102, 161-179.
• Simha R. (1940) "The Influence of Brownian Movement on the Viscosity of Solutions", J. Phys. Chem., 44, 25-34.
• Mehl JW, Oncley JL, Simha R. (1940) "Viscosity and the Shape of Protein Molecules", Science, 92, 132-133.
• Saito N. (1951) J. Phys. Soc. Japan, 6, 297.
• Scheraga HA. (1955) "Non-Newtonian Viscosity of Solutions of Ellipsoidal Particles", J. Chem. Phys., 23, 1526-1531.