# Effective diffusion coefficient of a species in a heterogeneous mixture

I want to calculate the effective diffusion coefficient of hydrogen radical in a mixed-phase consisting of metal oxide, metal hydride and metal. I consider the interfaces to be diffuse.

How can I calculate the effective diffusivity or effective diffusion coefficient of a species (ion or radical) in a heterogeneous mixture consisting of 3 different components? Can I weight average the diffusivities or is it more complicated than that?

• I’m general, with great difficulty. – Jon Custer Oct 18 '18 at 14:42

## 1 Answer

This greatly depends on macro structure. If the phases were say arranged in a sheet-like structure (like a diffusion front) then the low diffusivity phase would be the dominating determinate of the diffusion rate. For this macrostructure, if we assume steady state with no change in phase or length from diffusion or time, then mass transport of the bulk can be approximated by:

$$\frac{\mathrm dn}{\mathrm dt} = \frac{A \times \nabla C}{\sum\frac{L_i}{D_i}}$$

If the high diffusivity phase is continuous then that could be the dominating determinate of the diffusion rate. That does not mean it is easily calculable, just noting that dominating factors can change. If no phase is continuous or other phases are continuous then this also becomes convoluted. For non-sheet-like structures calculating diffusion requires a computer model of the grain structure and finite element analysis to resolve an approximate solution.