Intrinsic viscosity

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

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

where the constants are defined

${\displaystyle M\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{ab^{4}}}{\frac {1}{J_{\alpha }^{\prime }}}}$

${\displaystyle K\ {\stackrel {\mathrm {def} }{=}}\ {\frac {M}{2}}}$

${\displaystyle J\ {\stackrel {\mathrm {def} }{=}}\ K{\frac {J_{\alpha }^{\prime \prime }}{J_{\beta }^{\prime \prime }}}}$

${\displaystyle L\ {\stackrel {\mathrm {def} }{=}}\ {\frac {2}{ab^{2}\left(a^{2}+b^{2}\right)}}{\frac {1}{J_{\beta }^{\prime }}}}$

${\displaystyle N\ {\stackrel {\mathrm {def} }{=}}\ {\frac {6}{ab^{2}}}{\frac {\left(a^{2}-b^{2}\right)}{a^{2}J_{\alpha }+b^{2}J_{\beta }}}}$

The ${\displaystyle J}$ coefficients are the Jeffery functions

${\displaystyle J_{\alpha }=\int _{0}^{\infty }{\frac {dx}{\left(x+b^{2}\right){\sqrt {\left(x+a^{2}\right)^{3}}}}}}$

${\displaystyle J_{\beta }=\int _{0}^{\infty }{\frac {dx}{\left(x+b^{2}\right)^{2}{\sqrt {\left(x+a^{2}\right)}}}}}$

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

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

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

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

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

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

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 or with a RheoSense VROC viscometer.