Rotational diffusion

For rotational diffusion about a single axis, the mean-square angular deviation in time ${\displaystyle t}$ is

${\displaystyle \left\langle \theta ^{2}\right\rangle =2D_{r}t}$,

where ${\displaystyle D_{r}}$ is the rotational diffusion coefficient (in units of radians²/s). The angular drift velocity ${\displaystyle \Omega _{d}=(d\theta /dt)_{\rm {drift}}}$ in response to an external torque ${\displaystyle \Gamma _{\theta }}$ (assuming that the flow stays non-turbulent and that inertial effects can be neglected) is given by

${\displaystyle \Omega _{d}={\frac {\Gamma _{\theta }}{f_{r}}}}$,

where ${\displaystyle f_{r}}$ is the frictional drag coefficient. The relationship between the rotational diffusion coefficient and the rotational frictional drag coefficient is given by the Einstein relation (or Einstein–Smoluchowski relation):

${\displaystyle D_{r}={\frac {k_{\rm {B}}T}{f_{r}}}}$,

where ${\displaystyle k_{\rm {B}}}$ is the Boltzmann constant and ${\displaystyle T}$ is the absolute temperature. These relationships are in complete analogy to translational diffusion.

The rotational frictional drag coefficient for a sphere of radius ${\displaystyle R}$ is

${\displaystyle f_{r,{\textrm {sphere}}}=8\pi \eta R^{3}}$

where ${\displaystyle \eta }$ is the dynamic (or shear) viscosity.[3]

The rotational diffusion of spheres, such as nanoparticles, may deviate from what is expected when in complex environments, such as in polymer solutions or gels. This deviation can be explained by the formation of a depletion layer around the nanoparticle.[4]

A rotational version of Fick's law of diffusion can be defined. Let each rotating molecule be associated with a unit vector ${\displaystyle {\hat {n}}}$; for example, ${\displaystyle {\hat {n}}}$ 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 ${\displaystyle {\hat {n}}}$ at time t. Here, θ and φ represent the spherical angles, with θ being the polar angle between ${\displaystyle {\hat {n}}}$ and the z-axis and φ being the azimuthal angle of ${\displaystyle {\hat {n}}}$ in the x-y plane.

The rotational version of Fick's law states

${\displaystyle {\frac {1}{D_{\mathrm {rot} }}}{\frac {\partial f}{\partial t}}=\nabla ^{2}f={\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \phi ^{2}}}}$.

This partial differential equation (PDE) may be solved by expanding f(θ, φ, t) in spherical harmonics ${\displaystyle Y_{l}^{m}}$ for which the mathematical identity holds

${\displaystyle {\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial Y_{l}^{m}}{\partial \theta }}\right)+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}Y_{l}^{m}}{\partial \phi ^{2}}}=-l(l+1)Y_{l}^{m}(\theta ,\phi )}$.

Thus, the solution of the PDE may be written

${\displaystyle f(\theta ,\phi ,t)=\sum _{l=0}^{\infty }\sum _{m=-l}^{l}C_{lm}Y_{l}^{m}(\theta ,\phi )e^{-t/\tau _{l}}}$,

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

${\displaystyle \tau _{l}={\frac {1}{D_{\mathrm {rot} }l(l+1)}}}$.

• Diffusion equation
• Perrin friction factors
• Rotational correlation time
• False diffusion

• Cantor, CR; Schimmel PR (1980). Biophysical Chemistry. Part II. Techniques for the study of biological structure and function. W. H. Freeman.
• Berg, Howard C. (1993). Random Walks in Biology. Princeton University Press.