These friction factors were first calculated by Jean-Baptiste Perrin.
Finally, in spheres, 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.
For brevity in the equations below, we define the Perrin S factor.
is defined Similarly, for oblate spheroids (i.e., discus-shaped spheroids with two long axes and one short axis) For spheres,
The frictional coefficient of an arbitrary spheroid of volume
is the translational friction coefficient of a sphere of equivalent volume (Stokes' law) and
is the Perrin translational friction factor The frictional coefficient is related to the diffusion constant D by the Einstein relation Hence,
can be measured directly using analytical ultracentrifugation, or indirectly using various methods to determine the diffusion constant (e.g., NMR and dynamic light scattering).
Perrin showed that for both prolate and oblate spheroids.
In such cases, it may be better to expand in a series, e.g., for oblate spheroids.
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 whereas that for the equatorial direction vectors is These time constants can differ significantly when the axial ratio
Experimental methods for measuring these time constants include fluorescence anisotropy, NMR, flow birefringence and dielectric spectroscopy.