## Monday, June 28, 2010

### Anomalous Diffusion

Following up on the last post, the model of “anomalous diffusion” represents the opposite case – one where a “new” model is introduced when the old model works just fine.

The standard model of molecular diffusion is that developed by Fick, namely

$\bigg. J = - D \frac{\partial \phi}{\partial x} \bigg.$

where J is the diffusion flux with dimensions of [moles length−2 time−1], $\, D$ is the diffusion coefficient with dimensions of [length2 time−1], and $\, \phi$ is the concentration with dimensions of [moles length−3]. And if you like time-dependent situations, the good doctor Fick has an equation for you too:

$\frac{\partial \phi}{\partial t} = D\,\frac{\partial^2 \phi}{\partial x^2}\,\!$

These equations work quite well for a wide range of systems. The only challenge is knowing "D", the coefficient. While there are various efforts made to predict D from first principles or such, the best situation of course is to simply measure it using any of the various tests developed over the last century.

For systems with low molecular weight solvents at a constant temperature, D is found to be constant. Which is great news. It really is. Except for the people who then think that it has to be constant. And when it isn't constant, they fall apart and create a whole new field of research. And give the field a new name: anomolous diffusion.

This happens most often with materials diffusing through polymer matrices - as amount of diffusant increases, the diffusivity coefficient changes and the diffusion rates are different ("anomalous") from the case of a constant diffusitivity coefficient.

The reality is there is nothing anomalous about it all except the thought patterns behind the research.