# 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?

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 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