Rotational diffusion

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 equilibrium statistical distribution of particles' position in space.

The random re-orientation of molecules (or larger systems) is an important process for many biophysical probes. Due to the equipartition theorem, larger molecules re-orient more slowly than do smaller objects and, hence, measurements of the re-orientation times can give insight into the overall mass and distribution of mass within an object. Quantitatively, the mean square of the angular velocity about each of an object's principal axes is inversely proportional to its moment of inertia about that axis. Therefore, there should be three independent relaxation time constants for re-orientation, corresponding to each of the three principal axes; however, two or even three of these time constants may be the same if the object is symmetrical in its principal axes. For example, spheroidal particles have two time constants for rotational diffusion; these time constants may be determined from the Perrin friction factors, in analogy with the Einstein relation of translational diffusion.

These time constants may be determined experimentally from fluorescence anisotropy, flow birefringence, dielectric spectroscopy, the linewidths of liquid-state NMR peaks and other related biophysical methods. However, it is very difficult to discern the three different time constants; usually, only one is possible. Two time constants can sometimes be measured when there is a great difference between them, e.g., for very long, thin ellipsoids such as certain viruses.

Rotational version of Fick's law

A rotational version of Fick's law of diffusion can be defined. Let each rotating molecule be associated with a vector n of unit length n·n=1; for example, n might represent the orientation of an electric or magnetic dipole moment. Let "f(θ, φ, t)" represent the probability density distribution for the orientation of n at time "t". Here, θ and φ represent the spherical angles, with θ being the polar angle between n and the "z"-axis and φ being the azimuthal angle of n in the "x-y" plane. The rotational version of Fick's law states

:$frac\{1\}\{D\_\{mathrm\{rot\}\; frac\{partial\; f\}\{partial\; t\}\; =\; abla^\{2\}\; f\; =\; frac\{1\}\{sin\; heta\}\; frac\{partial\}\{partial\; heta\}left(\; sin\; heta\; frac\{partial\; f\}\{partial\; heta\}\; ight)\; +\; frac\{1\}\{sin^\{2\}\; heta\}\; frac\{partial^\{2\}\; f\}\{partial\; phi^\{2$

This partial differential equation (PDE) may be solved by expanding "f(θ, φ, t)" in spherical harmonics for which the mathematical identity holds

:$frac\{1\}\{sin\; heta\}\; frac\{partial\}\{partial\; heta\}left(\; sin\; heta\; frac\{partial\; Y^\{m\}\_\{l\{partial\; heta\}\; ight)\; +\; frac\{1\}\{sin^\{2\}\; heta\}\; frac\{partial^\{2\}\; Y^\{m\}\_\{l\{partial\; phi^\{2\; =\; -l(l+1)\; Y^\{m\}\_\{l\}$

Thus, the solution of the PDE may be written

:$f(\; heta,\; phi,\; t)\; =\; sum\_\{l=0\}^\{infty\}\; sum\_\{m=-l\}^\{l\}\; C\_\{lm\}\; Y^\{m\}\_\{l\}(\; heta,\; phi)\; e^\{-t/\; au\_\{l$

where "C_lm" are constants fitted to the initial distribution and the time constants equal

:$au\_\{l\}\; =\; frac\{1\}\{D\_\{mathrm\{rotl(l+1)\}$

ee also

* Perrin friction factors

References

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