# Predicting the ratio of translational diffusion coefficients for a sphere and a sheet

Imagine that I have a spherical particle of molecular weight $M$, volume $V$, and some experimentally observed center-of-mobility translational diffusion coefficient $D_{sphere}$ in water. I take this sphere and compress it into a sheet with thickness $L$, keeping the $M$ and $V$ constant, and experimentally measure a new translational diffusion coefficient $D_{sheet}$.

Can I ab initio predict the ratio $\frac{D_{sheet}}{D_{sphere}}$? Does the rigidity of the sheet matter?

-
There will not be one ratio. The translational diffusivity becomes anisotropic and gets coupled to a rotational diffusivity. –  Johannes Jan 18 '11 at 1:35
Johannes, right, one needs to consider the three mobility tensors. However, I imagine we should still be able to predict an (experimentally observable) translational diffusion coefficient at the center of mobility. –  MorningCoffee1998 Jan 18 '11 at 1:58
On second thought, in the limit of very long times, the disk orientations randomize and the displacement tensor regains an isotropic form. So in that sense a simple ratio will emerge. One needs to solve a not-so-easy hydrodynamic (Stokes flow) problem though. –  Johannes Jan 18 '11 at 3:44
Right, if the displacement tensor does not become isotropic, one would be able to create a Feynman brownian ratchet. –  MorningCoffee1998 Jan 18 '11 at 20:51
@Johannes You should be able to get this from the ratio of the mobilities of the disk and the sphere instead, shouldn't you? Stokes' drag. Clearly this depends on the orientation of the cylinder, so it's a tensor as opposed to a scalar. –  Chay Paterson Jul 7 '13 at 21:22