Electrostatics can be quite subtle. The following comment on the interlayer ion distribution, in Kjellander et al. (1988), was an eye-opener for me

The ion concentration profile is determined by the net force acting on each ion. The electrostatic potential from the uniform surface charges is constant between the two walls, which means that the forces due to these charges cancel each other completely. Thus, the large counter-ion concentration in the electric double layer near the walls is solely a consequence of the repulsive interactions between the ions.

Interlayer cations are not attracted to the surfaces, but are pushed towards them due to repulsion between the ions themselves! My intuition has been that interlayer counter-ions distribute due to attraction with the surfaces, but the perspective given in the above quotation certainly makes a lot of sense. Here I use the word “perspective” because I don’t fully agree with the statement that the ion distribution is solely a consequence of repulsion. To discuss the issue further, let’s flesh out the reasoning in Kjellander et al. (1988) and draw some pictures.

Here we discuss an idealized model of an interlayer as a dielectric continuum sandwiched between two parallel infinite planes of uniform surface charge density.¹ The system is thus symmetric around the axis normal to the surfaces (the model is one-dimensional).

From electrostatics we know that the electric field originating from a plane of uniform surface charge has the same size at any distance from the plane (we discussed this fact in the blog post on electrostatics and swelling pressure). We may draw such electric fields like this

From this result follows that the electric field vanishes between two equally negatively charged surfaces. The electrostatic field configuration for an “empty” interlayer can thus be illustrated like this

This means that the two interlayer surfaces don’t “care” about the counter-ions, in the sense that this part of the electrostatic energy (ion – surfaces) is independent of the counter-ion distribution.

To consider the fate of the counter-ions we continue to explore the axial symmetry. The counter-ion distribution varies only in the direction normal to the surfaces, and we can treat it as a sequence of thin parallel planes of uniform charge. Since the size of the electric field from such planes is independent of distance, the force on a positive test charge (= the electric field) at any position in the interlayer depends only on the difference in total amount of charge on each side of this position, as illustrated here

This, in turn, implies both that the electric field is zero at the mid position, and that the electric field elsewhere is directed towards the closest surface (since symmetry requires equal amount of charge in the two halves of the interlayer²). The counter-ions indeed repel each other towards the surfaces! The charge density must therefore increase towards the surfaces, and we understand that the equilibrium electric field qualitatively must look like this³

However, as far as I see, the “indifference” of the surfaces to the counter-ions is a matter of perspective. Consider e.g. making the interlayer distance very large. In this limit, the system is more naturally conceptualized as two single surfaces. It is then awkward to describe the ion distribution at one surface as caused by repulsion from other ions arbitrarily far away, rather than as caused by attraction to the surface. But for the case most relevant for compacted bentonite — i.e. interlayers, or what is often described as “overlapping” electric double layers — the natural perspective is that counter-ions distribute as a consequence of repulsion among themselves.

This perspective also implies that anions (co-ions) distribute within the interlayer as a consequence of attraction to counter-ions rather than repulsion from the surfaces! (The above figure applies, with all arrows reversed.) This insight should not be confused with the fact that repulsion between anions and surfaces is not really the mechanism behind “anion exclusion”. Rather, the implication here is that anion-surface repulsion can be viewed as not even existing within an interlayer.

A couple of corrections

With this (to me) new perspective in mind, I’d like to correct a few formulations in the blog post on electrostatics and swelling. In that post, I write

[R]ather than contributing to repulsion, electrostatic interactions actually reduce the pressure. This is clearly seen from e.g. the Poisson-Boltzmann solution for two charged surfaces, where the resulting osmotic pressure corresponds to an ideal solution with a concentration corresponding to the value at the midpoint (cf. the quotation from Kjellander et al. (1988) above). But the midpoint concentration — and hence the osmotic pressure — is lowered as compared with the average, because of electrostatic attraction between layers and counter-ions.

But the final sentence should rather be formulated as

But the midpoint concentration — and hence the osmotic pressure — is lowered as compared with the average, because of electrostatic repulsion between the counter-ions.

In the original post, I also write

This plot demonstrates the attractive aspect of electrostatic interactions in these systems. While the NaCl pressure is only slightly reduced, Na-montmorillonite shows strong non-ideal behavior. In the “low” concentration regime (< 2 mol/kgw) we understand the pressure reduction as an effect of counter-ions electrostatically attracted to the clay surfaces.

The last part is better formulated as

In the “low” concentration regime (< 2 mol/kgw) we understand the pressure reduction as an effect of electrostatic repulsion among the counter-ions.

I think the implication here is quite wild: In a sense, electrostatic repulsion reduces swelling pressure!

Footnotes

[1] The treatment in Kjellander et al. (1988) is more advanced, including effects of image charges and ion-ion correlations, but it does not matter for the present discussion.

[2] Actually, the whole distribution is required to be symmetric around the interlayer midpoint.

[3] The quantitative picture is of course achieved from solving the Poisson-Boltzmann equation. The picture may be altered when considering more involved mechanisms, such as image charge interactions or ion-ion correlations; Kjellander et al. (1988) show that the effect of image charges may reduce the ion distribution at very short distances, while the effect of ion-ion correlations is to further increase the accumulation towards the surfaces. Note that neither of these effects involve direct interaction with the surface charge.