# Diffusion in porous media - interface flow

i have been working on a problem lately where i think i m missing some basic understanding:

I consider the diffusion of a macromolecule in porous media, which i see as Brownian motion through the empty spaces. As i found, it makes sense to then state that the diffusivity of the macromolecule $$D$$ is reduced as a function of the fraction of filled space. Now if i say that my porous media is bounded by bulk solvent where the molecule moves freely with $$D_0$$, since diffusion is lower in the media, will there be an accumulation of molecules inside of it?

As an example, consider a disk shaped medium surrounded by bulk liquid. If i now have a certain concentration of molecules on the outside of the medium, will the concentration inside be higher, lower, or the same?

FIRST IDEAS: i think it has something to do with how i would write down Fick's law, i.e. if the spatially varying diffusivity is inside the derivative or not. What would the correct Langevin equation be for the description, given that noise is multiplicative?

Any help is welcome!

Cheers, M

If the 'empty spaces' inside the porous medium are still much, much larger than the macromolecules then Fick's Law:

$$J=-D\frac{\rm{d}\varphi}{\rm{d}x}$$

holds as per normal. So macromolecules would still diffuse from higher concentrations in the bulk solution into the lower concentration 'empty spaces'.

But if the 'empty' spaces were of the same size as the macromolecules then the latter would have difficulty diffusing into porous medium.

• Yes! but what with the case of small empty spaces... Then i expect to see that a meso/macroscopic diffusion coefficient is lower in the media, and i am wondering if the concentration in there is higher, compared to the outside. Since the whole system (inside + outside) is at the same temperature, maybe it is the same? Feb 8, 2021 at 16:48
• As I wrote: with empty space of the same or small size than the macromolecules, how then are the latter to move into the former? It's physically impossible.
– Gert
Feb 8, 2021 at 17:08
• Assuming that the spaces are somewhat larger than the molecules, they should get through, but hindered, which would lead to a smaller diffusivity Feb 8, 2021 at 17:15
• You're not wrong but you seem to be answering your own question! LOL
– Gert
Feb 8, 2021 at 17:17
• ^^ I guess my question is: if the empty spaces are somewhat larger than the molecules, and we expect the diffusivity to be smaller in the media, to what will the system equilibrate? in terms of the concentration ratio inside/outside? Feb 8, 2021 at 17:22