Rotational Diffusion - Rotational Version of Fick's Law

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} = \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 for which the mathematical identity holds

\frac{1}{\sin\theta} \frac{\partial}{\partial \theta}\left( \sin\theta \frac{\partial Y^{m}_{l}}{\partial \theta} \right) + \frac{1}{\sin^{2} \theta} \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(\theta, \phi, t) = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} C_{lm} Y^{m}_{l}(\theta, \phi) e^{-t/\tau_{l}}

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

\tau_{l} = \frac{1}{D_{\mathrm{rot}}l(l+1)}

Read more about this topic:  Rotational Diffusion

Famous quotes containing the words version and/or law:

    “It is never the thing but the version of the thing:
    The fragrance of the woman not her self,
    Her self in her manner not the solid block,
    The day in its color not perpending time,
    Time in its weather, our most sovereign lord,
    The weather in words and words in sounds of sound.”
    —Wallace Stevens (1879–1955)

    “There’s no law against taking off a spaceship. It’s never been done so they haven’t gotten around to prohibiting it.”
    —Rip Van Ronkel, and Robert A. Heinlein (1907–1988)