Perrin Friction Factors
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In hydrodynamics, the Perrin friction factors are multiplicative adjustments to the translational and rotational friction of a rigid spheroid, relative to the corresponding frictions in spheres of the same volume. These friction factors were first calculated by Jean-Baptiste Perrin. These factors pertain to
spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ...
s (i.e., to
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as the ...
s of revolution), which are characterized by the axial ratio ''p = (a/b)'', defined here as the axial semiaxis ''a'' (i.e., the semiaxis along the axis of revolution) divided by the equatorial semiaxis ''b''. In
prolate spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has c ...
s, the axial ratio ''p > 1'' since the axial semiaxis is longer than the equatorial semiaxes. Conversely, in
oblate spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ...
s, the axial ratio ''p < 1'' since the axial semiaxis is shorter than the equatorial semiaxes. Finally, in
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
s, the axial ratio ''p = 1'', since all three semiaxes are equal in length. The formulae presented below assume "stick" (not "slip") boundary conditions, i.e., it is assumed that the velocity of the fluid is zero at the surface of the spheroid.


Perrin S factor

For brevity in the equations below, we define the Perrin S factor. For ''prolate'' spheroids (i.e., cigar-shaped spheroids with two short axes and one long axis) : S \ \stackrel\ 2 \frac where the parameter \xi is defined : \xi \ \stackrel\ \frac Similarly, for ''oblate'' spheroids (i.e., discus-shaped spheroids with two long axes and one short axis) : S \ \stackrel\ 2 \frac For spheres, S = 2, as may be shown by taking the limit p \rightarrow 1 for the prolate or oblate spheroids.


Translational friction factor

The frictional coefficient of an arbitrary spheroid of volume V equals : f_ = f_ \ f_ where f_ is the translational friction coefficient of a sphere of equivalent ''volume'' ( Stokes' law) : f_ = 6 \pi \eta R_ = 6\pi \eta \left(\frac\right)^ and f_ is the Perrin translational friction factor : f_ \ \stackrel\ \frac The frictional coefficient is related to the diffusion constant ''D'' by the Einstein relation : D = \frac Hence, f_ can be measured directly using
analytical ultracentrifugation Analytical ultracentrifugation is an analytical technique which combines an ultracentrifuge with optical monitoring systems. In an analytical ultracentrifuge (commonly abbreviated as AUC), a sample’s sedimentation profile is monitored in real tim ...
, or indirectly using various methods to determine the diffusion constant (e.g., NMR and
dynamic light scattering Dynamic light scattering (DLS) is a technique in physics that can be used to determine the size distribution profile of small particles in suspension or polymers in solution. In the scope of DLS, temporal fluctuations are usually analyzed usin ...
).


Rotation friction factor

There are two rotational friction factors for a general spheroid, one for a rotation about the axial semiaxis (denoted F_) and other for a rotation about one of the equatorial semiaxes (denoted F_). Perrin showed that : F_ \ \stackrel\ \left( \frac \right) \frac : F_ \ \stackrel\ \left( \frac \right) \frac for both prolate and oblate spheroids. For spheres, F_ = F_ = 1, as may be seen by taking the limit p \rightarrow 1. These formulae may be numerically unstable when p \approx 1, since the numerator and denominator both go to zero into the p \rightarrow 1 limit. In such cases, it may be better to expand in a series, e.g., : \frac = 1.0 + \left(\frac\right) \left( \frac\right) + \left(\frac\right) \left( \frac\right)^ + \left(\frac\right) \left( \frac\right)^ + \ldots for oblate spheroids.


Time constants for rotational relaxation

The rotational friction factors are rarely observed directly. Rather, one measures the exponential rotational relaxation(s) in response to an orienting force (such as flow, applied electric field, etc.). The time constant for relaxation of the axial direction vector is : \tau_ = \left( \frac \right) \frac whereas that for the equatorial direction vectors is : \tau_ = \left( \frac \right) \frac These time constants can differ significantly when the axial ratio \rho deviates significantly from 1, especially for prolate spheroids. Experimental methods for measuring these time constants include
fluorescence anisotropy Fluorescence anisotropy or fluorescence polarization is the phenomenon where the light emitted by a fluorophore has unequal intensities along different axes of polarization. Early pioneers in the field include Aleksander Jablonski, Gregorio Webe ...
, NMR, flow birefringence and
dielectric spectroscopy Dielectric spectroscopy (which falls in a subcategory of impedance spectroscopy) measures the dielectric properties of a medium as a function of frequency.Kremer F., Schonhals A., Luck W. Broadband Dielectric Spectroscopy. – Springer-Verlag, 200 ...
. It may seem paradoxical that \tau_ involves F_. This arises because re-orientations of the axial direction vector occur through rotations about the ''perpendicular'' axes, i.e., about the equatorial axes. Similar reasoning pertains to \tau_{eq}.


References

* Cantor CR and Schimmel PR. (1980) ''Biophysical Chemistry. Part II. Techniques for the study of biological structure and function'', W. H. Freeman, p. 561-562. * Koenig SH. (1975) "Brownian Motion of an Ellipsoid. A Correction to Perrin's Results." Biopolymers 14: 2421–2423. Fluid dynamics