Many papers on diffusion in compacted bentonite and claystone¹ assume a direct connection between diffusivities measured in clay (\(D_\mathrm{clay}\)), and the geometrical configuration of the pore space. It seems close to mandatory to include
an expression of this type \begin{equation} D_\mathrm{clay} = \frac{1}{G}\cdot D_\mathrm{w}\tag{1} \end{equation}

where \(D_\mathrm{w}\) is the corresponding diffusion coefficient in bulk water (usually taken as the limiting value at infinite dilution), and \(G\) is a factor postulated to have pure geometric meaning. The geometric multiplier is written in a variety of ways, sometimes in terms of other factors which individually are supposed to convey specific geometric information, e.g. \(1/G = \delta/\tau^2\), where \(\delta\) is the “constrictivity” and \(\tau^2\) is the “tortuosity factor”².

However, the above described procedure is futile:

1. A geometric factor can not be deduced a priori, because the pore space geometry of any realistic system is way too complicated. So-called tortuosities have only been calculated in some trivial cases, relating e.g. to highly symmetric networks of identical cylinders, which has basically no resemblance to the chaotic smectite-lasagna expected to be found in bentonite. Indeed, many papers on these issues admit that the geometric factor in practice functions as a “fudge factor” (to be “optimized”).
2. Even if a geometric factor could be deduced from only considering the pore space configuration, it is not expected to relate to \(D_\mathrm{clay}\) in accordance with eq. 1. If we imagine removing all influence of pore geometry on the diffusivity in the clay, what remains is diffusion in an environment heavily influenced by the presence of exchangeable cations. Thus, a correct geometric factor should not multiply \(D_\mathrm{w}\), but the diffusion coefficient measured on the short time and length scale in the clay. Such measurements exist — at least for water diffusion — and show values different from \(D_\mathrm{w}\).

Because it is practically impossible to independently deduce a geometric factor, it is de facto defined by the ratio \(D_\mathrm{clay}/D_\mathrm{w}\) in many studies. I dare to say that in any work within the bentonite research community, a geometric factor has no other meaning than \(D_\mathrm{clay}/D_\mathrm{w}\) (or some function of this parameter, depending on definitions).

This means — at best — that the use of a geometric factor does nothing for a model description, but only replaces the parameter \(D_\mathrm{clay}\) with the parameter \(D_\mathrm{clay}/D_\mathrm{w}\); it simply corresponds to the awkward choice of specifying diffusion coefficients in units of \(D_\mathrm{w}\).

Often, however, the procedure has worse consequences, because eq. 1 (as used in practice) represents an unjustified claim; it is not correct to use the value of \(D_\mathrm{clay}/D_\mathrm{w}\) to make specific statements about the pore space geometry, while ignoring other possible mechanisms contributing to the value of \(D_\mathrm{clay}\). If diffusion instead is described without invoking a geometric factor, \(D_\mathrm{clay}\) itself functions as as a model parameter, and no implicit claims are made about why it has the value it has.

There are many examples in the literature where this confusion shows up. In e.g. Appelo and Wersin (2007), it is concluded that tortuosity for “anions” is larger than for water, while tortuosity for “cations” is smaller than for water. This conclusion is based on fitting of geometric factors without independent information on the pore space geometry:

[…] the pore may become filled entirely with a diffuse double layer when it narrows sufficiently. This constricts the passage of anions, and since the anions must circumnavigate the obstacle, they have greater tortuosity than tritium. This explains that a model with a tortuosity factor for iodide that is 1.4 times higher than for tritium better matches the data.

[…] in interlayers and pore constrictions the cations pass in relatively larger amounts than in free porewater, and consequently, they have smaller tortuosity than tritium.

Similar descriptions are given in Altmann et al. (2012) (in a section describing”advances” in process understanding with the FUNMIG project)

At low ionic strengths, cations are mostly located in the diffuse layer. This layer is of relatively low volume and connected throughout the porosity — cations need less time to explore this small volume and therefore to pass from one side of the media to the other.[…]

Anions are excluded from the diffuse layer and, therefore,also have a smaller volume to explore than neutral tracers. They however must pass through small pores in order to explore the total porosity, and these pores act as electrostatic throats […], forming an energy barrier to anion movement, reducing the probability of passage (resulting in reduced \(D_e\)) and increasing the time needed to explore all of the porosity.

And the same type of argument — that different diffusive behavior of cations and anions is caused by differences in “pathways” — is found in Charlet et al. (2017)

The different pathways of anions and cations in these regions lead to different effective diffusion rates (Van Loon et al., 2007; Mazurek et al., 2009)

But effective diffusion rates — which can basically be made to vanish e.g. for chloride, as has been demonstrated in bentonite — do not vary because of changes in “pathways”, but because of changes in boundary conditions, due to the presence of interfaces to external solutions.

To investigate the behavior of \(D_\mathrm{clay}/D_\mathrm{w}\) one must of course consider real — not effective — diffusion coefficients (unfortunately, actual diffusion coefficients are usually referred to as “apparent” in the bentonite research community). Real diffusion coefficients reflect the mobility of the diffusing species, and can be measured e.g. in non-steady-state closed-cell tests, which are not influenced by interfaces to external solutions³. A very good appreciation for how such diffusion coefficients behave is given by the vast amount of data measured by Kozaki et al. in Na-montmorillonite (Kunipia-F). Some of this data (sources: 1, 2, 3, 4, 5) is plotted below, in terms of \(D_\mathrm{clay}/D_\mathrm{w}\) (using a linear \(y\)-axis, for a change).

This plot clearly shows that if \(D_\mathrm{clay}/D_\mathrm{w}\) is interpreted as having pure geometrical meaning, the conclusion must be that chloride and sodium have basically identical “tortuosity”, while the “tortuosity” of the cations calcium and cesium is considerably larger⁴. Thus, the closed-cell data disproves any claim that “tortuosity” for anions is larger than for cations. Moreover, from this data it is quite a stretch to state that “tortuosity” for anions is larger than for water, and it is plain wrong to state that “tortuosity” for cations is smaller than for for water. (For the record, I’m strongly convinced that both calcium and cesium occupy the same pore volume as sodium in this material.)

I should emphasize that I am not primarily arguing for a better geometrical interpretation of \(D_\mathrm{clay}/D_\mathrm{w}\), but rather that this whole procedure (using eq. 1) should be avoided. This is not to say that pore geometry doesn’t influence clay diffusivity — most reasonably it does — but analysis based on eq. 1 will be flawed, because of the unjustified model assumptions.

When it comes to geometrical understanding of diffusion in bentonite, I think it would be more rewarding to focus on the quite large variation of reported diffusivities in seemingly identical systems. As an example, the figure below shows the diffusivity of cesium in very similar, pure Na-montmorillonite, measured in four different closed-cell studies: apart from Kozaki et al.’s data (blue), also Sato et al. (1992) (green), and Tachi and Yotsuji (2014) (orange).

In aggregation, this figure demonstrates almost an order of magnitude variability for the diffusion coefficient of cesium, even in very pure systems. If this kind of variability cannot be explained, it is of course not very relevant to develop models describing (smaller) differences in diffusivity between different types of ions in different types of bentonites.

If you claim to understand the variation seen in this plot, rather than simply finding an appropriate factor to blame (particle size?), here is a challenge: Give a prescription for how to prepare a Kunipia-F/P sample in order to minimize/maximize the resulting diffusivity⁵.

Footnotes

[1] In the following, the term “bentonite” is used as a shorthand for “compacted bentonite and claystone”.

[2] The nomenclature here is a mess, but I leave it to people who really want to use these concepts to actually define them in detail.

[3] They can also be deduced from the transient behavior in “steady-state” tests, if the transport capacity of the confining filters is not limiting the process.

[4] Note that “tortuosity”, as used e.g. by Appelo and Wersin (2007), is related to the inverse of \(D_\mathrm{clay}/D_\mathrm{w}\).

[5] The diffusion direction should still be along the axis of sample compaction, of course.