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Intrinsic viscosity

## Summary

Intrinsic viscosity ${\displaystyle \left[\eta \right]}$ is a measure of a solute's contribution to the viscosity ${\displaystyle \eta }$ of a solution. It should not be confused with inherent viscosity, which is the ratio of the natural logarithm of the relative viscosity to the mass concentration of the polymer.

Intrinsic viscosity is defined as

${\displaystyle \left[\eta \right]=\lim _{\phi \rightarrow 0}{\frac {\eta -\eta _{0}}{\eta _{0}\phi }}}$

where ${\displaystyle \eta _{0}}$ is the viscosity in the absence of the solute, ${\displaystyle \eta }$ is (dynamic or kinematic) viscosity of the solution and ${\displaystyle \phi }$ is the volume fraction of the solute in the solution. As defined here, the intrinsic viscosity ${\displaystyle \left[\eta \right]}$ is a dimensionless number. When the solute particles are rigid spheres at infinite dilution, the intrinsic viscosity equals ${\displaystyle {\frac {5}{2}}}$, as shown first by Albert Einstein.

In practical settings, ${\displaystyle \phi }$ is usually solute mass concentration (c, g/dL), and the units of intrinsic viscosity ${\displaystyle \left[\eta \right]}$ are deciliters per gram (dL/g), otherwise known as inverse concentration.

## Formulae for rigid spheroids

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

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

## General ellipsoidal formulae

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

## 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, G. B. (1922). "The motion of ellipsoidal particles immersed in a viscous fluid". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. The Royal Society. 102 (715): 161–179. Bibcode:1922RSPSA.102..161J. doi:10.1098/rspa.1922.0078. ISSN 0950-1207.
• Simha, R. (1940). "The Influence of Brownian Movement on the Viscosity of Solutions". The Journal of Physical Chemistry. American Chemical Society (ACS). 44 (1): 25–34. doi:10.1021/j150397a004. ISSN 0092-7325.
• Mehl, J. W.; Oncley, J. L.; Simha, R. (1940-08-09). "Viscosity and the Shape of Protein Molecules". Science. American Association for the Advancement of Science (AAAS). 92 (2380): 132–133. Bibcode:1940Sci....92..132M. doi:10.1126/science.92.2380.132. ISSN 0036-8075. PMID 17730219.
• Saitô, Nobuhiko (1951-09-15). "The Effect of the Brownian Motion on the Viscosity of Solutions of Macromolecules, I. Ellipsoid of Revolution". Journal of the Physical Society of Japan. Physical Society of Japan. 6 (5): 297–301. Bibcode:1951JPSJ....6..297S. doi:10.1143/jpsj.6.297. ISSN 0031-9015.
• Scheraga, Harold A. (1955). "Non‐Newtonian Viscosity of Solutions of Ellipsoidal Particles". The Journal of Chemical Physics. AIP Publishing. 23 (8): 1526–1532. Bibcode:1955JChPh..23.1526S. doi:10.1063/1.1742341. ISSN 0021-9606.