- Rotational diffusion
**Rotational diffusion**is a process by which theequilibrium 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 theangular velocity about each of an object'sprincipal axes is inversely proportional to itsmoment of inertia about that axis. Therefore, there should be three independent relaxationtime constant s 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,spheroid al particles have two time constants for rotational diffusion; these time constants may be determined from thePerrin friction factors , in analogy with the Einstein relation of translationaldiffusion .These time constants may be determined experimentally from

fluorescence anisotropy ,flow birefringence ,dielectric spectroscopy , the linewidths of liquid-stateNMR 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 certainvirus es.**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 ormagnetic 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)" inspherical 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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