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:

    “Truth cannot be defined or tested by agreement with ‘the world’; for not only do truths differ for different worlds but the nature of agreement between a world apart from it is notoriously nebulous. Rather—speaking loosely and without trying to answer either Pilate’s question or Tarski’s—a version is to be taken to be true when it offends no unyielding beliefs and none of its own precepts.”
    —Nelson Goodman (b. 1906)

    “Faith, I have been a truant in the law,
    And never yet could frame my will to it,
    And therefore frame the law unto my will.”
    —William Shakespeare (1564–1616)