Intrinsic viscosity

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

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

where $η 0$ 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 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.

Fluid dynamics

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Intrinsic_viscosity". A list of authors is available in Wikipedia.