Einstein's formula - Knudsen Diffusion (DiffuDict)

[w_creig]

Hi,

I noted in the GeoDict guide that the Knudsen diffusivity is determined by Einstein's formula, which does not include a term for pore diameter: [latex]D=E\frac{(x_t-x_0)(x_t-x_0)^T}{2t}[/latex]

What part of Einstein's formula relates to the pore diameter term (here, d) in the usual formula used?
[latex]D_{KA}=\frac{d}{3}\sqrt{\frac{8RT}{\pi M_A}}[/latex]

Is it the displacement of the molecules?

[Jürgen Becker]

Hi,

the first formula ((16) of DiffuDict 2023 User Guide) is used to compute the 3x3 diffusivity matrix from a set of tracked molecules. For every molecule we have a start point [latex=inline]x_0[/latex] and an end point [latex=inline]x_t[/latex] and we can use this set  to compute the diffusivity by:
[latex]D=\frac{E((x_t-x_0)(x_t-x_0)^T)}{2t} [/latex]
The 3D geometry and the velocity are an input for the simulation in this case.

Your second formula is actually formula (17) of the DiffuDict 2023 User Guide:
[latex]d_0=\frac{1}{3}L\bar{v}[/latex]
where our char. Length L is your d, and the mean thermal velocity is
[latex]\bar{v}=\sqrt{\frac{8RT}{\pi M_A}}[/latex]
(which is also formula (26) of the User Guide). This computes the one-dimensional diffusivity of a cylindrical pore and neglects all 3D effects that a more complex structure may show. However, by assuming that there is some characteristic length L of the 3D structure, one can use the value [latex=inline]d_0[/latex] as a scaling factor for the diffusivity computed with Einsteins formula, and then one gets a dimensionless 3x3 diffusivity [latex=inline]D^{\ast}[/latex] that is independent from the diffusing species.  This is then formula (18) of the User Guide:
[latex]D=d_0\cdot D^{\ast}[/latex]
The [latex=inline]D^{\ast}[/latex] could be understood as the three-dimensional pore shape factor that is missing in your cylindrical pore formula.

However, if you have a look at page 38 of the DiffuDict 2023 User Guide, you can see that if you enter GeoDict with a single cylindrical pore structure, you get exactly the result from your second formula in the pore direction (and 0 in all other directions).

Best regards.