# How to formulate a PDE for diffusion between two different materials?

Suppose I have two (connected) materials with different diffusion coefficients for which I am modelling diffusion. Consider the one dimensional case. I am not sure what conditions to impose at the point where they are connected. Continuity of concentration makes sense. I am not sure what to impose in terms of the flux.

Perhaps counter to intuition, the concentration is not necessarily continuous. Materials tend to segregate impurities to different degrees, and Nature here requires that matter is conserved, not concentration. When formulating the equations, this means that no matter can disappear: the outgoing flux from the left side must equal the incoming flux from the right side.

Thus, for a small* amount of dissolved C near an interface between A and B, for example, we would have

$$J_C|_{0^-}=-D_{CA}\left.\frac{\partial C_C(x)}{\partial x}\right|_{0^-}=-D_{CB}\left.\frac{\partial C_C(x)}{\partial x}\right|_{0^+}=J_C|_{0^+},$$

where $$J$$ is the flux, $$0^-$$ and $$0^+$$ ($$=0$$) are respectively the left and right sides of the interface, $$D$$ is the diffusivity of C in the host material, and $$C_C$$ is the concentration of C.

At steady state, it may also be useful to work with a segregation coefficient $$K$$ that expresses the equilibrium concentration of C in B, say, given a certain concentration of C in A:

$$KC_C|_{0^-}=C_C|_{0^+}.$$

One could also work with the solubility coefficient $$\sigma_{C}$$ and permeability $$P_{C}=D_{C}\sigma_{C}$$; see, for instance, Crank's Mathematics of Diffusion.

Or, one can work with the chemical potential, a generalized version of concentration that incorporates interactions with the host environment. Matter moves to eliminate chemical potential gradients, so this is an example of a parameter that is equalized across an interface (at equilibrium).

*So that $$D$$ can be assumed to be constant and so that the interface itself doesn't move.

• So the flux condition at the first equation is that flux is continuous at the boundary between AB
– Novo
Commented Jul 14, 2021 at 18:12
• That’s right. Flux continuity is always a good first condition to impose when you’ve got something mobile that can’t be created or destroyed. Commented Jul 14, 2021 at 21:57