Hydrodynamic radius

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. For polymers in solution, ${\displaystyle R_{\rm {hyd}}}$ 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]