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The hydrodynamic radius of a macromolecule or colloid particle is R_. The macromolecule or colloid particle is a collection of N subparticles. This is done most commonly for
polymer A polymer (; Greek '' poly-'', "many" + ''-mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
s; the subparticles would then be the units of the polymer. R_ is defined by : \frac \ \stackrel\ \frac \left\langle \sum_ \frac \right\rangle where r_ is the distance between subparticles i and j, and where the angular brackets \langle \ldots \rangle represent an
ensemble average In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents ...
. The theoretical hydrodynamic radius R_ was originally an estimate by
John Gamble Kirkwood John "Jack" Gamble Kirkwood (May 30, 1907, Gotebo, Oklahoma – August 9, 1959, New Haven, Connecticut) was a noted chemist and physicist, holding faculty positions at Cornell University, the University of Chicago, California Institute of Technol ...
of the
Stokes radius The Stokes radius or Stokes–Einstein radius of a solute is the radius of a hard sphere that diffuses at the same rate as that solute. Named after George Gabriel Stokes, it is closely related to solute mobility, factoring in not only size but also ...
of a polymer, and some sources still use ''hydrodynamic radius'' as a synonym for the Stokes radius. Note that in
biophysics Biophysics is an interdisciplinary science that applies approaches and methods traditionally used in physics to study biological phenomena. Biophysics covers all scales of biological organization, from molecular to organismic and populations. ...
, hydrodynamic radius refers to the Stokes radius, or commonly to the apparent Stokes radius obtained from
size exclusion chromatography Size-exclusion chromatography (SEC), also known as molecular sieve chromatography, is a chromatographic method in which molecules in solution are separated by their size, and in some cases molecular weight. It is usually applied to large molecules ...
. The theoretical hydrodynamic radius R_ arises in the study of the dynamic properties of polymers moving in a
solvent A solvent (s) (from the Latin '' solvō'', "loosen, untie, solve") is a substance that dissolves a solute, resulting in a solution. A solvent is usually a liquid but can also be a solid, a gas, or a supercritical fluid. Water is a solvent for ...
. It is often similar in magnitude to the
radius of gyration ''Radius of gyration'' or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentr ...
.


Applications to aerosols

The mobility of non-spherical aerosol particles can be described by the hydrodynamic radius. In the
continuum limit In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approximate real-world processe ...
, where the
mean free path In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...
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, \boldsymbol_d as that of a sphere with that radius, i.e. :\boldsymbol_d = 6\pi\mu R_\boldsymbol where \mu is the viscosity of the surrounding fluid, and \boldsymbol 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, the Cunningham correction factor C is used, where: :\boldsymbol_d = \frac, \quad \text \quad C = 1+\text(\alpha + \beta \text^), where \alpha, \beta, \text \gamma were found by Millikan to be: 1.234, 0.414, and 0.876 respectively.


Notes


References

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