Perrin friction factors

Perrin friction factors

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 spheroids (i.e., to ellipsoids 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 spheroids, the axial ratio "p > 1" since the axial semiaxis is longer than the equatorial semiaxes. Conversely, in oblate spheroids, the axial ratio "p < 1" since the axial semiaxis is shorter than the equatorial semiaxes. 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.

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{mathrm{def{=} 2 frac{mathrm{atanh} xi}{xi}

where the parameter xi is defined

:xi stackrel{mathrm{def{=} frac{sqrt{left| p^{2} - 1 ight|{p}

Similarly, for "oblate" spheroids (i.e., discus-shaped spheroids with two long axes and one short axis)

:S stackrel{mathrm{def{=} 2 frac{mathrm{atan} xi}{xi}

For spheres, S = 2, as may be shown by taking the limit p ightarrow 1 for the prolate or oblate spheroids.

Translational friction factor

The frictional coefficient of an arbitrary spheroid of volume V equals

:f_{tot} = f_{sphere} f_{P}

where f_{sphere} is the translational friction coefficient of a sphere of equivalent "volume" (Stokes' law)

:f_{sphere} = 6 pi eta R_{eff} = 6pi eta left(frac{3V}{4pi} ight)^{(1/3)}

and f_{P} is the Perrin translational friction factor

:f_{P} stackrel{mathrm{def{=} frac{2p^{2/3{S}

The frictional coefficient is related to the diffusion constant "D" by the Einstein relation

:D = frac{k_{B}T}{f_{tot

Hence, f_{tot} can be measured directly using analytical ultracentrifugation, or indirectly using various methods to determine the diffusion constant (e.g., NMR and dynamic light scattering).

Rotation friction factor

There are two rotational friction factors for a general spheroid, one for a rotation about the axial semiaxis (denoted F_{ax}) and other for a rotation about one of the equatorial semiaxes (denoted F_{eq}).
Perrin showed that

:F_{ax} stackrel{mathrm{def{=} left( frac{4}{3} ight) frac{xi^{2{2 - (S/p^{2})}

:F_{eq} stackrel{mathrm{def{=} left( frac{4}{3} ight) frac{(1/p)^{2} - p^{2{2 - S left [ 2 - (1/p)^{2} ight] }

for both prolate and oblate spheroids. For spheres, F_{ax} = F_{eq} = 1, as may be seen by taking the limit p ightarrow 1.

These formulae may be numerically unstable when p approx 1, since the numerator and denominator both go to zero into the p ightarrow 1 limit. In such cases, it may be better to expand in a series, e.g.,

:frac{1}{F_{ax = 1.0 + left(frac{4}{5} ight) left( frac{xi^{2{1 + xi^{2 ight) + left(frac{4 cdot 6}{5 cdot 7} ight) left( frac{xi^{2{1 + xi^{2 ight)^{2} + left(frac{4 cdot 6 cdot 8}{5 cdot 7 cdot 9} ight) left( frac{xi^{2{1 + xi^{2 ight)^{3} + 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

: au_{ax} = left( frac{1}{k_{B}T} ight) frac{F_{eq{2}

whereas that for the equatorial direction vectors is

: au_{eq} = left( frac{1}{k_{B}T} ight) frac{F_{ax}F_{eq{F_{ax} + F_{eq

These time constants can differ significantly when the axial ratio ho deviates significantly from 1, especially for prolate spheroids. Experimental methods for measuring these time constants include fluorescence anisotropy, NMR, flow birefringence and dielectric spectroscopy.

It may seem paradoxical that au_{ax} involves F_{eq}. 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 au_{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.


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Perrin — General= Perrin may refer to: *Perrin Air Force Base, located in rural Grayson County, Texas *Perrin friction factors, in hydrodynamics *Perrin number, in mathematics *Perrin s Beaked Whale, the newest species of Beaked Whale to be described… …   Wikipedia

  • Oblate spheroidal coordinates — Figure 1: Coordinate isosurfaces for a point P (shown as a black sphere) in oblate spheroidal coordinates (μ, ν, φ). The z axis is vertical, and the foci are at ±2. The red oblate spheroid (flattened sphere) corresponds to μ=1, whereas the blue… …   Wikipedia

  • Rotational diffusion — is a process by which the equilibrium statistical distribution of the overall orientation of particles or molecules is maintained or restored. Rotational diffusion is the counterpart of translational diffusion, which maintains or restores the… …   Wikipedia

  • Flow birefringence — In biochemistry, flow birefringence is a hydrodynamic technique for measuring the rotational diffusion constants (or, equivalently, the rotational drag coefficients). The birefringence of a solution sandwiched between two concentric cylinders is… …   Wikipedia

  • Fluorescence anisotropy — In chemistry, fluorescence anisotropy assays the rotational diffusion of a molecule from the decorrelation of polarization in fluorescence, i.e., between the exciting and emitted (fluorescent) photons. This decorrelation can measure the tumbling… …   Wikipedia

  • Fan death — is a South Korean urban legend which states that an electric fan, if left running overnight in a closed room, can cause the death (by suffocation, poisoning, or hypothermia) of those inside. Fans manufactured and sold in Korea are equipped with a …   Wikipedia

  • Herbert Spencer — Infobox Philosopher region = Western Philosophy era = 19th century philosophy color = #B0C4DE image caption =Herbert Spencer name =Herbert Spencer birth =27 April, 1820 death =8 December, 1903 school tradition = Evolutionism,… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”