Physical meaning of Cahn-Hilliard boundary conditions

Consider the 1D Cahn-Hilliard equation for a two-component mixture, on an interval $$x\in[a,b]$$:

$$\frac{dc}{dt} = -\frac{d}{dx}j(x)$$ where the flux $$j(x) = -D\frac{d}{dx}\left(c^3 - c - \gamma\frac{d^2c}{dx^2}\right)$$

The typical boundary conditions are:

1. $$j(x)=0$$ at $$x=a$$ and $$x=b$$, meaning no flux across the boundaries.

2. $$\frac{dc}{dx}=0$$ at $$x=a$$ and $$x=b$$.

What is the physical rationale for boundary condition 2?

It is tempting to compare boundary condition 2 to the zero-flux boundary condition for the diffusion equation, which has the same mathematical form $$\frac{dc}{dx}=0$$ at $$x=a$$ and $$x=b$$. However, for the diffusion equation, $$j(x) = -D\frac{dc}{dx}$$, so the condition $$\frac{dc}{dx}=0$$ is equivalent to $$j(x)=0$$, which is analogous to boundary condition 1, not 2.

You are absolutely correct that the first condition is the no-flux condition. The second condition is a bit more subtle and actually determines which value of $$c$$ is favored at the boundary! The normal derivative of $$c$$ directly determines the contact angle of droplets at the wall! The standard condition $$\partial_x c=0$$ implies a neutral condition where the contact angle is 90 degrees. This boundary condition can be directly derived by minimizing the total free energy including a contact potential. We discuss this in some detail in one of our recent publications: https://doi.org/10.1063/5.0207761 (see particularly Eqs. 1 and 3).

• Thanks for the interesting reference! The idea that the boundary conditions come out of the free energy minimization is very helpful. Commented Jul 22 at 19:53

However, for the diffusion equation, $$j(x) = -D\frac{dc}{dx}$$, so the condition $$\frac{dc}{dx}=0$$ is equivalent to $$j(x)=0$$, which is analogous to boundary condition 1, not 2.

You are correct if everything was purely diffusion, but your overall flux is not diffusive, so $$\text dc/\text dx$$ does not imply that $$j(x)=0$$ (just plug that in and see for yourself).

You are correct that $$\text dc/\text dx=0$$ means that there is no diffusive flux though.

• The last sentence is only correct for ideal diffusion but not for the non-ideal mixtures described by the Cahn-Hilliard equation. Commented Jul 21 at 18:03
• @DavidZwicker Yep! Which is why the OP is confused Commented Jul 21 at 18:05
• I guess I’m confused by your notion of „diffusion“. For me the flux $j$ in the Cahn-Hilliard equation describes precisely the inter-diffusion of the two species. Commented Jul 21 at 18:25
• I'm confused by your last sentence, because I didn't say that $dc/dx=0$ means flux is zero. Are you suggesting that "diffusive flux" is a physical quantity distinct from $j(x)$? If so, what is diffusive flux? Commented Jul 21 at 22:59
• I said that for the diffusion equation, $dc/dx=0$ is equivalent to $j(x)=0$, but not for the Cahn-Hilliard equation (I thought your last sentence was referring to the Cahn-Hilliard equation). I guess I'm confused how your answer is addressing the question. Do you know why $dc/dx=0$ is a physically reasonable boundary condition for the Cahn-Hilliard equation? Commented Jul 22 at 1:21