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]

## Applications to aerosols

The mobility of non-spherical aerosol particles can be described by the hydrodynamic radius. In the continuum limit, where the mean free path of the particle is negligible compared to a characteristic length scale of the particle, the hydrodynamic radius is defined as the radius that gives the same magnitude of the frictional force, ${\textstyle {\boldsymbol {F))_{d))$ as that of a sphere with that radius, i.e.

${\displaystyle {\boldsymbol {F))_{d}=6\pi \mu R_{hyd}{\boldsymbol {v))}$

where ${\textstyle \mu }$ is the viscosity of the surrounding fluid, and ${\textstyle {\boldsymbol {v))}$ is the velocity of the particle. This is analogous to the Stokes' radius, however this is untrue as the mean free path becomes comparable to the characteristic length scale of the particulate - a correction factor is introduced such that the friction is correct over the entire Knudsen regime. As is often the case,[5] the Cunningham correction factor ${\textstyle C}$ is used, where:

${\displaystyle {\boldsymbol {F))_{d}={\frac {6\pi \mu R_{hyd}{\boldsymbol {v))}{C)),\quad {\text{where:))\quad C=1+{\text{Kn))(\alpha +\beta {\text{e))^{\frac {\gamma }{\text{Kn))})}$,

where ${\textstyle \alpha ,\beta ,{\text{ and ))\gamma }$ were found by Millikan[6] to be: 1.234, 0.414, and 0.876 respectively.

## 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 (2): 573–577. Bibcode:1990PNAS...87..573G. doi:10.1073/pnas.87.2.573. PMC 53307. PMID 2153957.
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.
5. ^ Sorensen, C. M. (2011). "The Mobility of Fractal Aggregates: A Review". Aerosol Science and Technology. 45 (7): 765–779. Bibcode:2011AerST..45..765S. doi:10.1080/02786826.2011.560909. ISSN 0278-6826. S2CID 96051438.
6. ^ Millikan, R. A. (1923-07-01). "The General Law of Fall of a Small Spherical Body through a Gas, and its Bearing upon the Nature of Molecular Reflection from Surfaces". Physical Review. 22 (1): 1–23. Bibcode:1923PhRv...22....1M. doi:10.1103/PhysRev.22.1. ISSN 0031-899X.

## References

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