The hydrodynamic radius of a macromolecule or colloid particle is ${\displaystyle R_{\rm {hyd}}}$. The macromolecule or colloid particle is a collection of ${\displaystyle N}$ subparticles. This is done most commonly for polymers; the subparticles would then be the units of the polymer. ${\displaystyle R_{\rm {hyd}}}$ is defined by

${\displaystyle {\frac {1}{R_{\rm {hyd}}}}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{2N^{2}}}\left\langle \sum _{i\neq j}{\frac {1}{r_{ij}}}\right\rangle }$

where ${\displaystyle r_{ij}}$ is the distance between subparticles ${\displaystyle i}$ and ${\displaystyle j}$, and where the angular brackets ${\displaystyle \langle \ldots \rangle }$ represent an ensemble average.[1] The theoretical hydrodynamic radius ${\displaystyle R_{\rm {hyd}}}$ was originally an estimate by John Gamble Kirkwood of the Stokes radius of a polymer, and some sources still use hydrodynamic radius as a synonym for the Stokes radius.

Note that in biophysics, hydrodynamic radius refers to the Stokes radius,[2] or commonly to the apparent Stokes radius obtained from size exclusion chromatography.[3]

The theoretical hydrodynamic radius ${\displaystyle R_{\rm {hyd}}}$ arises in the study of the dynamic properties of polymers moving in a solvent. It is often similar in magnitude to the radius of gyration.[4]

## Notes

1. ^ J. Des Cloizeaux and G. Jannink (1990). Polymers in Solution Their Modelling and Structure. Clarendon Press. ISBN 0-19-852036-0. Chapter 10, Section 7.4, pages 415-417.
2. ^ Harding, Stephen (1999). "Chapter 7: Protein Hydrodynamics" (PDF). Protein: A comprehensive treatise. JAI Press Inc. pp. 271–305. ISBN 1-55938-672-X.
3. ^ Goto, Yuji; Calciano, Linda; Fink, Anthony (1990). "Acid-induced unfolding of proteins". Proc. Natl. Acad. Sci. USA. 87: 573–577. doi:10.1073/pnas.87.2.573. PMC 53307.
4. ^ Gert R. Strobl (1996). The Physics of Polymers Concepts for Understanding Their Structures and Behavior. Springer-Verlag. ISBN 3-540-60768-4. Section 6.4 page 290.

## References

Grosberg AY and Khokhlov AR. (1994) Statistical Physics of Macromolecules (translated by Atanov YA), AIP Press. ISBN 1-56396-071-0