Category Archives: Semi-permeability

Post-publication review: Tournassat and Steefel (2015), part II

“Electrostatic properties, high surface area, and anion exclusion”

As stated in the first part, I find it remarkable that the authors use general terms such as “clay minerals” when the actual subject matter is specifically systems with a significant cation exchange capacity, and montmorillonite in particular. I will continue to refer to these systems as “bentonite” in the following, disregarding the constant references to “clay minerals” in TS15.

Stacks

After having established that montmorillonite and illite have structural negative charge, it begins:

Clay mineral particles are made of layer stacks and the space between two adjacent layers is named the interlayer space (Fig. 5).

This is the first mention of clay “particles” in the article, and they are introduced as if this is a most well-established concept in bentonite science (incredibly, it is also the first occurrence of the term “interlayer”). We will refer to “clay mineral particle” constructs as “stacks” in the following. I have written a detailed post on why stacks make little sense, where I demonstrate their geometrical impossibility and show that most references given to support the concept are studies on suspensions that often imply that montmorillonite do not form stacks. Sure enough, this is also the case in TS15

The number of layers per montmorillonite particle depends on the water chemical potential and on the nature and external concentration of the layer charge compensating cation (Banin and Lahav 1968; Shainberg and Otoh 1968; Schramm and Kwak 1982a; Saiyouri et al. 2000)

Banin and Lahav (1968), Shainberg and Otoh (1968), and Schramm and Kwak (1982) all report studies on montmorillonite suspensions. The abstract of Shainberg and Otoh (1968) even states “The breakdown of the tactoids occurred when the equivalent fraction of Na increased from 0.2 to 0.5. Montmorillonite clay saturated with 50% calcium (and less) exists as single platelets.”, and the abstract of Schramm and Kwak (1982) states “Upon exchange of Ca-counterions for Li-, Na-, or K-counterions, a sharp initial decrease in tactoid size was observed over approximately the first 30% of cation exchange.”. These are just different ways of saying that sodium dominated montmorillonite is sol forming.

I want to stress the absurdity of the description given in TS15. A pure fantasy is stated about how compacted bentonite is structured. As “support” for the claim are given references to studies on “dilute suspensions”. It should be clear that the way TOT-layers interact in such suspensions essentially says nothing about how they are organized at high density. But even if we pretend that these results are applicable, the given references say that most of the relevant systems (montmorillonite with about 30% sodium or more) do not form stacks.

Disregarding the references, note also how bizarre the above statement is that the number of layers in a “particle” depends on “the water chemical potential and on the nature and external concentration of the layer charge compensating cation”: stacks are supposed to be fundamental structural units, yet the number of layers in a stack is supposed to depend on the entire water chemistry?! (It makes sense, of course, for stacks in actual suspensions.) Also, for montmorillonite an actual number of layers is nowhere stated in TS15.

TS15 further complicate things by lumping together montmorillonite and illite. In contrast to Na-montmorillonite, illite has by definition a mechanism for keeping adjacent TOT-layers together: its layer charge density is higher and compensated by potassium, which doesn’t hydrate that well, leading to collapsed interlayers. As far as I understand, one characterizing feature of illite is that the collapsed interlayers are manifested as a “10-angstrom peak” in X-ray diffraction measurements.

To treat montmorillonite and illite on equal footing (in a laid-back single sentence) again shows how nonsensical this description is. Stacking in montmorillonite suspensions occurs as a consequence of an increased ion-ion correlation effect when the fraction of e.g. calcium becomes large (> 70-80%). This process requires the ions to be diffusive and is distinctly different from the interlayer collapse in illite.

I actually have a hard time understanding what exactly is meant by the term “illite” here. In clay science it is clear that what is referred to by this name are systems that may have a quite considerable cation exchange capacity.¹ Reasonably, such systems contain other types of cations besides potassium² (as they are exchangeable), and must contain compartments where such ions can diffuse (as they are exchangeable). To increase the complexity, there are also “illite-smectite interstratified clay minerals”, which typically are in “smectite-to-illite” transitional states. For these, it seems reasonable to assume that the remaining smectite layers provide both diffusable interlayer pores and the cation exchange capacity. I don’t know if such “smectite layers” provides the cation exchange capacity in general in systems that researchers call illite. Neither do I understand how researchers can accept and use this, in my view, vague definition of “illite”. Anyway, it is the task of TS15 to sort out what they mean by the term. This is not done, and instead we get the following sentence

Illite particles typically consist of 5 to 20 stacked TOT layers (Sayed Hassan et al. 2006).

This study (Sayed Hassan et al., 2006) concerns one particular material (illite from “the Le Puy ore body”) that has been heavily processed as part of the study.³ I mean that such a specific study cannot be used as a single reference for the general nature of “illite particles”. Moreover, the stated stack size (5 — 20 layers) is nowhere stated in Sayed Hassan et al. (2006)!⁴

In their laid-back sentence, TS15 also implicitly define “interlayer space” as being internal to stacks. I criticized this way of redefining already established terms in the stack blog post, and TS15 serves as a good illustration of the problem: are we not supposed to be able to use the term “interlayer” without accepting the fantasy concept of stacks? To be clear, “interlayer spaces” in the context of montmorillonite simply means, and must continue to mean, spaces between adjacent TOT basal surfaces. It drives me half mad that the “stack-internal” definition is so common in contemporary bentonite scientific literature that this point seems almost impossible to communicate.

The provided illustration (“Fig 5”) explicitly shows how TS15 differ between “interlayers” that are assumed internal, and “outer basal surfaces” that are assumed external to the stack.

This illustration misrepresents the actual result of assembling a set of TOT-layers, just like any other “stack” picture found in the literature. The figure shows five identical TOT-layers that can be estimated to be smaller than 20 nm in lateral extension (while the text “conveniently” states that they should be 50 — 200 nm). Compared with “realistic” stacks, formed by randomly drawing TOT-layer sizes from an actual distribution, the depicted stack in TS15 looks like this⁵ (see here for details)

Besides the fact that “realistic” stack units cannot be used to form the structure of compacted bentonite, it should also be clear from this picture that “outer basal surfaces” and “interlayers” (in the sense of being internal to the stack) are not well defined. Note further that in actual compacted systems (above 1.2 g/cm³, say) such “realistic” stacks would be pushed together, something like this

In this picture, why should e.g. the interface between the green and the red stack be defined as an interface between two “outer surfaces” rather than an interlayer? Also, is this interface supposed to change nature and become an “interlayer”, as the water chemical potential or the external ion content changes? Like all other proponents of stack descriptions that I have encountered, TS15 do not in any way explain how “interlayers” and “outer surfaces” are supposed to function fundamentally differently. Similarly, they do not describe how the number of layers in a stack depends on water chemistry, nor do they provide a mechanism for why (sodium dominated) montmorillonite stacks of are supposed to keep together.

I want to emphasize that I do not favor any construction with “realistic” stacks, but only use them to illustrate the absurd consequences of taking a stack description seriously, and to demonstrate that all such descriptions in the bentonite literature are essentially pure fantasies, including the one given in TS15. I’m also quite baffled as to why TS15 (and others) provide such completely nonsensical descriptions, and how these can end up in review articles. I believe a hint is given in this formulation

[T]he number of stacked TOT layers in montmorillonite particles dictates the distribution of water in two distinct types of porosity: the interlayer porosity […] and the inter-particle porosity.

The only way I can make sense of this whole description is as an embarrassing attempt to motivate the introduction of models with several “distinct types of porosity”: the outcome is simply a macroscopic multi-porosity model (which will also be evident in later sections).

I’ve written a detailed blog post on why multi-porosity models cannot be taken seriously. There I point out that basically all authors promoting multi-porosity for some reason attempt to dress it up in terms of microscopic concepts, while the models obviously are macroscopic. Moreover, no one has ever suggested a mechanism for how equilibrium is supposed to be maintained between the different types of “porosities”.

Anion exclusion

After hallucinating about the structure of compacted bentonite, TS15 change gear and begin an “explanation” of anion exclusion. Let’s go through the description in detail.

The negative charge of the clay layers is responsible for the presence of a negative electrostatic potential field at the clay mineral basal surface–water interface.

I cannot really make sense of the term “negative electrostatic potential field”, although I think I understand what the authors are trying to say here. What is true is that the electrostatic potential near a montmorillonite basal surface is lowered compared to a point farther away. But whether or not the value of the potential is negative is irrelevant, as we are free to choose the reference level. If the zero level is chosen at a point very far from the surface, which often is done, it is true that the potential is negative at the surface. But the key principle is that the potential decreases towards the surface.⁶ A varying electrostatic potential signifies an electric field, which in this case is directed towards the surface (\(E = -d\phi/dx\)).

Furthermore, the electric field is not present merely because of the presence of negative charge, but because this charge is constrained to be positioned in the atomic structure of the clay. Remember that the structural clay charge is compensated by counter-ions, and that the system as a whole is charge neutral. The reason for the presence of an electric field near the surface is due to charge separation. And the reason for the potential decreasing (i.e. the electric field pointing towards the surface) is because it is the negative charge that is unable to be completely freely distributed.⁷

The concentrations of ions in the vicinity of basal planar surfaces of clay minerals depend on the distance from the surface considered. In a region known as the electrical double layer (EDL), concentrations of cations increase with proximity to the surface, while concentrations of anions decrease.

Having established that the electrostatic potential varies in the vicinity of the surface, it follows trivially that the ion concentrations also vary. I also find it peculiar to label the regions where the concentrations varies as the EDL. An electric double layer is a structure that includes both the surface charge and the counter-ions (hence the word “double”). What is described here should preferably be called a diffuse layer. Note, moreover, that the way an electric double layer here is introduced implies that TS15 consider a single interface, i.e. some variant of the Gouy-Chapman model (this becomes clear below). But this model is not applicable to compacted bentonite.

At infinite distance from the surface, the solution is neutral and is commonly described as bulk or free solution (or water).

Here I think it becomes obvious that the authors try to motivate the presence of bulk water within the clay structure. As described in the blog post on “Anion accessible porosity”, it is only reasonable to assume that diffuse layers merge with a bulk solution in systems that are very sparse — i.e. in suspensions.⁸ This is how e.g. Schofield (1947) utilized the Gouy-Chapman model to estimate surface area. But how is the solution next to a basal surface in compacted bentonite supposed to merge with a bulk solution? Even if we use the authors’ own fantasy stack constructs, the typical structure of compacted bentonite must be envisioned something like this (I have color coded different stacks to be able to understand where they begin and end).

The regions where basal surfaces of different stacks face each other (labelled A) are way too small in order to merge with a bulk solution (and, as asked earlier, how are these regions even different from “interlayers”?). Furthermore, regions adjacent to external edge-surfaces of these imaginary stack units (B, C) are not at all considered by applying a Gouy-Chapman model. The only way to make “sense” out of the present description is to imagine larger voids in the clay structure, something like this

But even if such voids would exist (in equilibrated water-saturated bentonite under reasonable conditions, they do not) they would only constitute an exotic exception to the typical pore structure. By focusing on this type of possible “anion” exclusion, TS15 completely miss the point.

This spatial distribution of anions and cations gives rise to the anion exclusion process that is observed in diffusion experiments.

Now I’m lost. I don’t understand how ion distributions are supposed to cause a process. I think the authors here allude to Schofield’s approach to estimating surface area in montmorillonite suspensions. As discussed in detail in the blog post on anion-accessible porosity, if the suspension is so dilute that we can consider each clay layer independently, and if we equilibrate it with an external solution, we can measure its salt content, and use the Gouy-Chapman model to e.g. estimate surface area from the amount of excluded salt (as compared with the external solution).

But, as also discussed in the blog post on salt exclusion, the “Schofield type” of exclusion is not what we expect to be dominating in a dense system. Rather, in denser systems (and in Donnan systems generally — no surfaces need to be involved), salt exclusion occurs mainly because of charge separation at interfaces with the external solution. I find it revealing that TS15 so far in the article has not at all mentioned such interfaces.

Moreover, in the above sentence TS15 causally states that the anion exclusion process is “observed in diffusion experiments”, without further clarification. Given that the previous section treated diffusion, a reader would expect to have been introduced to the anion exclusion process and how it is observed in diffusion experiments. But this subsection is the first time in TS15 where the term “anion exclusion” is used! In the section on diffusion, “anion accessible porosity” was briefly mentioned, and I suppose a reader is here presumed to connect the dots. But the presence of an exclusion process certainly does not imply an “anion accessible porosity”! Furthermore, anion exclusion is not necessarily observed in diffusion experiments. A more correct statement is that we observe effects of salt exclusion in experiments where a bentonite sample is contacted with an external solution via a semi-permeable component (which typically is a filter that keeps the clay in place). The effect is most conveniently studied in equilibrium rather than diffusion tests, and salt exclusion is not present in e.g. closed-cell diffusion tests. Note that exclusion effects are always related to an external solution.

As the ionic strength increases, the EDL thickness decreases, with the result that the anion accessible porosity increases as well.

Here it is fully clear that TS15 conflate “anion accessible porosity” and “anion exclusion”. If we consider the “Schofield type” of salt exclusion, it is true that the so-called “exclusion volume” changes with the ionic strength. However, an exclusion volume is not a physical space, but an effective, equivalent quantity. It is derived from the Gouy-Chapman model, which always has anions present everywhere.

Even more importantly, the “Schofield type” of exclusion is not really of interest in dense systems (nor is the Gouy-Chapman model valid in such systems). As discussed above, one must instead consider salt exclusion stemming from charge separation at interfaces with the external solution. For this case it does not even make sense to define an exclusion volume.

I can only interpret this entire paragraph as another fruitless attempt to motivate a multi-porous modeling approach. In this subsection we have so far been told that “two distinct types of porosity” can be defined (they cannot), and we have vaguely been hinted that “bulk or free solution” also is relevant for modelling compacted bentonite. And with the last quoted sentence it is relatively clear that TS15 try to establish that the relative sizes of various “porosities” are controlled by a simple parameter (ionic strength).

The final paragraph of this subsection contain several statements that makes my jaw drop.

An equivalent anion accessible porosity can be estimated from the integration of the anion concentration profile (Fig. 6) from the surface to the bulk water (Sposito 2004)

Here the authors suddenly use the phrase “equivalent”! They are thus obviously aware of that “anion accessible porosity” is a spurious concept?! ?!?! I really don’t know what to say. Their own graph (“Fig. 6”) even show that the Gouy-Chapman model has anions (salt) everywhere! Note that this statement also implies that “the bulk water” is assumed to exist within the clay.

In compacted clay material, the pore sizes may be small as compared to the EDL size. In that case, it is necessary to take into account the EDLs overlap between two neighbouring surfaces.

I think this is a very revealing passage. The conditions of compacted bentonite are treated as an exception: pore sizes “may” be smaller than the EDL, and “in that case” it is necessary to account for overlapping diffuse layers. But for compacted bentonite, this is the only relevant situation to consider! Without “overlapping” diffuse layers there is no swelling and no sealing properties. An entire page has been devoted to discussing a model only relevant for suspensions (Gouy-Chapman), while “compacted clay material” here is commented in two sentences…

Clay mineral particles are, however, often segregated into aggregates delimiting inter-aggregate spaces whose size is usually larger than inter-particle spaces inside the aggregates.

All of a sudden — in the middle of a paragraph — we are introduced to a new structural component! “Aggregates” have not been mentioned earlier in the article and is here introduced without any references. It is my strong opinion that this way of writing is not appropriate for a scientific publication, especially not for a review article. I’m not sure what type of system the authors have in mind here, but “aggregates” are typically not present in actual water saturated bentonite. I have commented more on this in the blog post on stacks.

Conclusion

At the end of the previous section (on diffusion), we were promised that this section should qualitatively link “fundamental properties of the clay minerals” to the diffusional behavior of compacted bentonite. Instead, we are given a fictional description of the structure (conflated with structures of other “clay minerals”), along with a confused explanation of anion exclusion that is irrelevant for such systems. Not a single word is said about the equilibrium that must be considered, namely that at interfaces between bentonite and external solutions. Rather, the idea of “overlapping” diffuse layers — which is the ultimate cause for bentonite swelling — is treated as an exception and only commented on in passing (and nothing is said about how to handle such systems). Although nothing is fully spelled out, I can only interpret this entire part as a (failed) attempt to motivate a multi-porous approach to modeling bentonite. And multi-porosity models cannot be taken seriously.

I admit that scrutinizing studies and pointing out flaws can be fun. However, considering that the descriptions in TS15 are the rule rather than the exception in contemporary bentonite research, I mostly feel weary and resigned. I don’t mean that every clay researcher must agree with me that a homogeneous model is the only reasonable starting point for describing compacted bentonite, and I could only wish that this blog was more influential. But I feel almost dizzy thinking about how this research sector is so hermetically sealed that one can spend entire careers in it without ever having to worry about understanding the nature of swelling and swelling pressure.

Footnotes

[1] The Wikipedia article on illite, for example, states that the cation exchange capacity is typically 0.2 — 0.3 eq/kg. Is a significant cation exchange capacity required for classifying something as illite?

[2] E.g. (Poinssot et al., 1999), that TS15 reference as a source on illite, work with sodium exchanged “illite du Puy”, i.e. “Na-illite”.

[3] The material was dispersed by diluting it in alkaline solution and sonicating it. It was thereafter dropped as a suspension on a glass slide and dried.

[4] We may note that the number 5 — 20 TOT-layers in a stack actually showed up when we investigated how this concept is (mis)used in descriptions of bentonite. There it turned out to be a complete misunderstanding of the behavior of suspensions of Ca-montmorillonite.

[5] I am not capable to produce anything reasonable in 3D, but I think a 2D representation still conveys the message.

[6] Perhaps this criticism can be regarded as nitpicking. I have a nagging feeling, though, that electrostatics is quite poorly understood in certain parts of the bentonite research field. Take the phrase “negative electrostatic potential field”, for example. Although it can be understood at face value (a scalar field with negative values), it also appear to mix together stuff related to charges (“negative”), electric fields (“field”), and potentials. It certainly is important to separate these concepts. There are many examples in the clay literature when this is not done. E.g. Madsen and Müller-Vonmoos (1989) mean that two “potential fields” can repel each other (and also misunderstand swelling)

A high negative potential exists directly at the surface of the clay layer. […] When two such negative potential fields overlap, they repel each other, and cause the observed swelling in clay.

Horseman et al. (1996) claim that a potential repels charges:

[…] the net negative electrical potential between closely spaced clay particles repel anions attempting to migrate through the narrow aqueous films of a compact clay […]

And Shackleford and Moore (2013) mean that “overlapping” potentials repel charges

In this case, when the clay is compressed […] to the extent that the electrostatic (diffuse double) layers surrounding the particles overlap, the overlapping negative potentials repel invading anions such that the pore becomes excluded to the anion.

[7] An isolated layer of negative charge of course also has an electric field directed towards it, but this is not the relevant system to consider here. (Such a system will actually have an electric field strength that is independent of the distance to the surface, as long as the layer can be regarded as infinitely extended.)

[8] See also this comment, and this quotation.

Multi-porosity models cannot be taken seriously (Semi-permeability, part II)

Implied structure of the free water domain

I’m astonished by how little figures of the microscopic scale are explained in many publications. For instance, the illustration above clearly suggests that “free water” is an interface region with exactly the same surface area as the “double layer”. How can that make sense? Also, if the above structure is to be taken seriously it is crucial to specify the extensions of the various water layers. It is clear that the figure shows a microscopic view, as it depicts an actual diffuse layer.⁴ A diffuse layer width varies, say, in the range 1 – 100 nm,⁵ but authors seldom reveal if we are looking at a pore 1 nm wide or several hundred nm wide. Often we are not even shown a pore — the water film just ends in a void, as in the above figure.⁶

The vague nature of these descriptions indicates that they are merely “decorations”, providing a microscopic flavor to what in effect still is a macroscopic model formulation. In practice, most multi-porosity formulations provide some ad hoc mean to calculate the volume of the diffuse layer domain, while the free water porosity is either obtained by subtracting the diffuse layer porosity from total porosity, or by just specifying it. Alt-Epping et al. (2015), for example, simply specifies the “macroporosity”

The total porosity amounts to 47.6 % which is divided into 40.5 % microporosity (EDL) and 7.1 % macroporosity (free water). From the microporosity and the surface area of montmorillonite (Table 7), the Debye length of the EDL calculated from Eq. 11 is 4.97e-10 m.

Clearly, nothing in this description requires or suggests that the “micro” and “macroporosities” are adjacent waterfilms on the nm-scale. On the contrary, such an interpretation becomes quite grotesque, with the “macroporosity” corresponding to half a monolayer of water molecules! An illustration of an actual pore of this kind would look something like this

This interpretation becomes even more bizarre, considering that Alt-Epping et al. (2015) assume advection to occur only in this half-a-monolayer of water, and that the diffusivity is here a factor 1000 larger than in the “microporosity”.

As another example, Appelo and Wersin (2007) model a cylindrical sample of “Opalinus clay” of height 0.5 m and radius 0.1 m, with porosity 0.16, by discretizing the sample volume in 20 sections of width 0.025 m. The void volume of each section is consequently \(V_\mathrm{void} = 0.16\cdot\pi\cdot 0.1^2\cdot 0.025\;\mathrm{m^3} = 1.257\cdot10^{-4}\;\mathrm{m^3}\). Half of this volume (“0.062831853” liter) is specified directly in the input file as the volume of the free water;⁷ again, nothing suggests that this water should be distributed in thin films on the nm-scale. Yet, Appelo and Wersin (2007) provide a figure, with no length scale, similar in spirit to that above, that look very similar to this

They furthermore write about this figure (“Figure 2”)

It should be noted that the model can zoom in on the nm-scale suggested by Figure 2, but also uses it as the representative form for the cm-scale or larger.

I’m not sure I can make sense of this statement, but it seems that they imply that the illustration can serve both as an actual microscopic representation of two spatially separated domains and as a representation of two abstract continua on the macroscopic scale. But this is not true!

Interpreted macroscopically, the vertical dimension is fictitious, and the two continua are in equilibrium in each paired cell. On a microscopic scale, on the other hand, equilibrium between paired cells cannot be assumed a priori, and it becomes crucial to specify both the vertical and horizontal length scales. As Appelo and Wersin (2007) formulate their model assuming equilibrium between paired cells, it is clear that the above figure must be interpreted macroscopically (the only reference to a vertical length scale is that the “free solution” is located “at infinite distance” from the surface).

We can again work out the implications of anyway interpreting the model microscopically. Each clay cell is specified to contain a surface area of \(A_\mathrm{surf}=10^5\;\mathrm{m^2}\).⁸ Assuming a planar geometry, the average pore width is given by (\(\phi\) denotes porosity and \(V_\mathrm{cell}\) total cell volume)

\begin{equation} d = 2\cdot \phi \cdot \frac{V_\mathrm{cell}}{A_\mathrm{surf}} = 2\cdot \frac{V_\mathrm{void}}{A_\mathrm{surf}} = 2\cdot \frac{1.26\cdot 10^{-4}\;\mathrm{m^3}}{10^{5}\;\mathrm{m^2}} = 2.51\;\mathrm{nm} \end{equation}

The double layer thickness is furthermore specified to be 0.628 nm.⁹ A microscopic interpretation of this particular model thus implies that the sample contains a single type of pore (2.51 nm wide) in which the free water is distributed in a thin film of width 1.25 nm — i.e. approximately four molecular layers of water!

Rather than affirming that multi-porosity model formulations are macroscopic at heart, parts of the bentonite research community have instead doubled down on the confusing idea of having free water distributed on the nm-scale. Tournassat and Steefel (2019) suggest dealing with the case of two parallel charged surfaces in terms of a “Dual Continuum” approach, providing a figure similar to this (surface charge is -0.11 C/m² and external solution is 0.1 M of a 1:1 electrolyte)

Note that here the perpendicular length scale is specified, and that it is clear from the start that the electrostatic potential is non-zero everywhere. Yet, Tournassat and Steefel (2019) mean that it is a good idea to treat this system as if it contained a 0.7 nm wide bulk water slice at the center of the pore. They furthermore express an almost “postmodern” attitude towards modeling, writing

It should be also noted here that this model refinement does not imply necessarily that an electroneutral bulk water is present at the center of the pore in reality. This can be appreciated in Figure 6, which shows that the Poisson–Boltzmann predicts an overlap of the diffuse layers bordering the two neighboring surfaces, while the dual continuum model divides the same system into a bulk and a diffuse layer water volume in order to obtain an average concentration in the pore that is consistent with the Poisson–Boltzmann model prediction. Consequently, the pore space subdivision into free and DL water must be seen as a convenient representation that makes it possible to calculate accurately the average concentrations of ions, but it must not be taken as evidence of the effective presence of bulk water in a nanoporous medium.

I can only interpret this way of writing (“…does not imply necessarily that…”, “…must not be taken as evidence of…”) that they mean that in some cases the bulk phase should be interpreted literally, while in other cases the bulk phase should be interpreted just as some auxiliary component. It is my strong opinion that such an attitude towards modeling only contributes negatively to process understanding (we may e.g. note that later in the article, Tournassat and Steefel (2019) assume this perhaps non-existent bulk water to be solely responsible for advective flow…).

I say it again: no matter how much researchers discuss them in microscopic terms, these models are just macroscopic formulations. Using the terminology of Tournassat and Steefel (2019), they are, at the end of the day, represented as dual continua assumed to be in local equilibrium (in accordance with the first figure of this post). And while researchers put much effort in trying to give these models a microscopic appearance, I am not aware of anyone suggesting a reasonable candidate for what actually could constitute the semi-permeable component necessary for maintaining such an equilibrium.

Arbitrary division between diffuse layer and free water

Another peculiarity in the multi-porosity descriptions showing that they cannot be interpreted microscopically is the arbitrary positioning of the separation between diffuse layer and free water. We saw earlier that Alt-Epping et al. (2015) set this separation at one Debye length from the surface, where the electrostatic potential is claimed to have decayed by a factor of e. What motivates this choice?

Most publications on multi-porosity models define free water as a region where the solution is charge neutral, i.e. where the electrostatic potential is vanishingly small.¹⁰ At the point chosen by Alt-Epping et al. (2015), the potential is about 37% of its value at the surface. This cannot be considered vanishingly small under any circumstance, and the region considered as free water is consequently not charge neutral.

The diffuse layer thickness chosen by Appelo and Wersin (2007) instead corresponds to 1.27 Debye lengths. At this position the potential is about 28% of its value at the surface, which neither can be considered vanishingly small. At the mid point of the pore (1.25 nm), the potential is about 8%¹¹ of the value at the surface (corresponding to about 2.5 Debye lengths). I find it hard to accept even this value as vanishingly small.

Note that if the boundary distance used by Appelo and Wersin (2007) (1.27 Debye lengths) was used in the benchmark of Alt-Epping et al. (2015), the diffuse layer volume becomes larger than the total pore volume! In fact, this occurs in all models of this kind for low enough ionic strength, as the Debye length diverges in this limit. Therefore, many multi-porosity model formulations include clunky “if-then-else” clauses,¹² where the system is treated conceptually different depending on whether or not the (arbitrarily chosen) diffuse layer domain fills the entire pore volume.¹³

In the example from Tournassat and Steefel (2019) the extension of the diffuse layer is 1.6 nm, corresponding to about 1.69 Debye lengths. The potential is here about 19% of the surface value (the value in the midpoint is 12% of the surface value). Tournassat and Appelo (2011) uses yet another separation distance — two Debye lengths — based on misusing the concept of exclusion volume in the Gouy-Chapman model.

With these examples, I am not trying to say that a better criterion is needed for the partitioning between diffuse layer and bulk. Rather, these examples show that such a partitioning is quite arbitrary on a microscopic scale. Of course, choosing points where the electrostatic potential is significant makes no sense, but even for points that could be considered having zero potential, what would be the criterion? Is two Debye lengths enough? Or perhaps four? Why?

These examples also demonstrate that researchers ultimately do not have a microscopic view in mind. Rather, the “microscopic” specifications are subject to the macroscopic constraints. Alt-Epping et al. (2015), for example, specifies a priori that the system contains about 15% free water, from which it follows that the diffuse layer thickness must be set to about one Debye length (given the adopted surface area). Likewise, Appelo and Wersin (2007) assume from the start that Opalinus clay contains 50% free water, and set up their model accordingly.¹⁴ Tournassat and Steefel (2019) acknowledge their approach to only be a “convenient representation”, and don’t even relate the diffuse layer extension to a specific value of the electrostatic potential.¹⁵ Why the free water domain anyway is considered to be positioned in the center of the nanopore is a mystery to me (well, I guess because sometimes this interpretation is supposed to be taken literally…).

Note that none of the free water domains in the considered models are actually charged, even though the electrostatic potential in the microscopic interpretations is implied to be non-zero. This just confirms that such interpretations are not valid, and that the actual model handling is the equilibration of two (or more) macroscopic, abstract, continua. The diffuse layer domain is defined by following some arbitrary procedure that involves microscopic concepts. But just because the diffuse layer domain is quantified by multiplying a surface area by some multiple of the Debye length does not make it a microscopic entity.⁴

Donnan effect on the microscopic scale?!

Although we have already seen that we cannot interpret multi-porosity models microscopically, we have not yet considered the weirdest description adopted by basically all proponents of these models: they claim to perform Donnan equilibrium calculations between diffuse layer and free water regions on the microscopic scale!

The underlying mechanism for a Donnan effect is the establishment of charge separation, which obviously occur on the scale of the ions, i.e. on the microscopic scale. Indeed, a diffuse layer is the manifestation of this charge separation. Donnan equilibrium can consequently not be established within a diffuse layer region, and discontinuous electrostatic potentials only have meaning in a macroscopic context.

Consider e.g. the interface between bentonite and an external solution in the homogeneous mixture model. Although this model ignores the microscopic scale, it implies charge separation and a continuously varying potential on this scale, as illustrated here

The regions where the potential varies are exactly what we categorize as diffuse layers (exemplified in two ideal microscopic geometries).

The discontinuous potentials encountered in multi-porosity model descriptions (see e.g. the above “Dual Continuum” potential that varies discontinuously on the angstrom scale) can be drawn on paper, but don’t convey any physical meaning.

Here I am not saying that Donnan equilibrium calculations cannot be performed in multi-porosity models. Rather, this is yet another aspect showing that such models only have meaning macroscopically, even though they are persistently presented as if they somehow consider the microscopic scale.

An example of this confusion of scales is found in Alt-Epping et al. (2018), who revisit the benchmark problem of Alt-Epping et al. (2015) using an alternative approach to Donnan equilibrium: rather than directly calculating the equilibrium, they model the clay charge as immobile mono-valent anions, and utilize the Nernst-Planck equations. They present “the conceptual model” in a figure very similar to this one

This illustration simultaneously conveys both a micro- and macroscopic view. For example, a mineral surface is indicated at the bottom, suggesting that we supposedly are looking at an actual interface region, in similarity with the figures we have looked at earlier. Moreover, the figure contains entities that must be interpreted as individual ions, including the immobile “clay-anions”. As in several of the previous examples, no length scale is provided (neither perpendicular to, nor along the “surface”).

On the other hand, the region is divided into cells, similar to the illustration in Appelo and Wersin (2007). These can hardly have any other meaning than to indicate the macroscopic discretization in the adopted transport code (FLOTRAN). Also, as the “Donnan porosity” region contains the “clay-anions” it can certainly not represent a diffuse layer extending from a clay surface; the only way to make sense of such an “immobile-anion” solution is that it represents a macroscopic homogenized clay domain (a homogeneous mixture!).

Furthermore, if the figure is supposed to show the microscopic scale there is no Donnan effect, because there is no charge separation! Taking the depiction of individual ions seriously, the interface region should rather look something like this in equilibrium

This illustrates the fundamental problem with a Donnan effect between microscopic compartments: the effect requires a charge separation, whose extension is the same as the size of the compartments assumed to be in equilibrium.¹⁶

Despite the confusion of the illustration in Alt-Epping et al. (2018), it is clear that a macroscopic model is adopted, as in our previous examples. In this case, the model is explicitly 2-dimensional, and the authors utilize the “trick” to make diffusion much faster in the perpendicular direction compared to the direction along the “surface”. This is achieved either by making the perpendicular diffusivity very high, or by making the perpendicular extension small. In any case, a perpendicular length scale must have been specified in the model, even if it is nowhere stated in the article. The same “trick” for emulating Donnan equilibrium is also used by Jenni et al. (2017), who write

In the present model set-up, this approach was implemented as two connected domains in the z dimension: one containing all minerals plus the free porosity (z=1) and the other containing the Donnan porosity, including the immobile anions (CEC, z=2, Fig. 2). Reproducing instantaneous equilibrium between Donnan and free porosities requires a much faster diffusion between the porosity domains than along the porosity domains.

Note that although the perpendicular dimension (\(z\)) here is referred to without unit(!), this representation only makes sense in a macroscopic context.

Jenni et al. (2017) also provide a statement that I think fairly well sums up the multi-porosity modeling endeavor:¹⁷

In a Donnan porosity concept, cation exchange can be seen as resulting from Donnan equilibrium between the Donnan porosity and the free porosity, possibly moderated by additional specific sorption. In CrunchflowMC or PhreeqC (Appelo and Wersin, 2007; Steefel, 2009; Tournassat and Appelo, 2011; Alt-Epping et al., 2014; Tournassat and Steefel, 2015), this is implemented by an explicit partitioning function that distributes aqueous species between the two pore compartments. Alternatively, this ion partitioning can be modelled implicitly by diffusion and electrochemical migration (Fick’s first law and Nernst-Planck equations) between the free porosity and the Donnan porosity, the latter containing immobile anions representing the CEC. The resulting ion compositions of the two equilibrated porosities agree with the concentrations predicted by the Donnan equilibrium, which can be shown in case studies (unpublished results, Gimmi and Alt-Epping).

Ultimately, these are models that, using one approach or the other, simply calculates Donnan equilibrium between two abstract, macroscopically defined domains (“porosities”, “continua”). Microscopic interpretations of these models lead — as we have demonstrated — to multiple absurdities and errors. I am not aware of any multi-porosity approach that has provided any kind of suggestion for what constitutes the semi-permeable component required for maintaining the equilibrium they are supposed to describe. Alternatively expressed: what, in the previous figure, prevents the “immobile anions” from occupying the entire clay volume?

The most favorable interpretation I can make of multi-porosity approaches to bentonite modeling is a dynamically varying “macroporosity”, involving magical membranes (shown above). This, in itself, answers why I cannot take multi-porosity models seriously. And then we haven’t yet mentioned the flawed treatment of diffusive flux.

Semi-permeability, part I

Footnotes

[1] This category has many other names, e.g. “dual porosity” and “dual continuum”, models. Here, I mostly use the term “multi-porosity” to refer to any model of this kind.

[2] These compartments have many names in different publications. The “diffuse layer” domain is also called e.g. “electrical double layer (EDL)”, “diffuse double layer (DDL)”, “microporosity”, or “Donnan porosity”, and the “free water” is also called e.g. “macroporosity”, “bulk water”, “charge-free” (!), or “charge-neutral” porewater. Here I will mostly stick to using the terms “diffuse layer” and “free water”.

[3] This lack of a full description is very much related to the incomplete description of so-called “stacks” — I am not aware of any reasonable suggestion of a mechanism for keeping stacks together.

[4] Note the difference between a diffuse layer and a diffuse layer domain. The former is a structure on the nm-scale; the latter is a macroscopic, abstract model component (a continuum).

[5] The scale of an electric double layer is set by the Debye length, \(\kappa^{-1}\). From the formula for a 1:1 electrolyte, \(\kappa^{-1} = 0.3 \;\mathrm{nm}/\sqrt{I}\), the Debye length is seen to vary between 0.3 nm and 30 nm when ionic strength is varied between 1.0 M to 0.0001 M (\(I\) is the numerical value of the ionic strength expressed in molar units). Independent of the value of the factor used to multiply \(\kappa^{-1}\) in order to estimate the double layer extension, I’d say that the estimation 1 – 100 nm is quite reasonable.

[6] Here, the informed reader may perhaps point out that authors don’t really mean that the free water film has exactly the same geometry as the diffuse layer, and that figures like the one above are more abstract representations of a more complex structure. Figures of more complex pore structures are actually found in many multi-porosity papers. But if it is the case that the free water part is not supposed to be interpreted on the microscopic scale, we are basically back to a magic membrane picture of the structure! Moreover, if the free water is not supposed to be on the microscopic scale, the diffuse layer will always have a negligible volume, and these illustrations don’t provide a mean for calculating the partitioning between “micro” and “macroporosity”.

It seems to me that not specifying the extension of the free water is a way for authors to dodge the question of how it is actually distributed (and, as a consequence, to not state what constitutes the semi-permeable component).

[7] The PHREEQC input files are provided as supplementary material to Appelo and Wersin (2007). Here I consider the input corresponding to figure 3c in the article. The free water is specified with keyword “SOLUTION”.

[8] Keyword “SURFACE” in the PHREEQC input file for figure 3c in the paper.

[9] Using the identifier “-donnan” for the “SURFACE” keyword.

[10] We assume a boundary condition such that the potential is zero in the solution infinitely far away from any clay component.

[11] Assuming exponential decay, which is only strictly true for a single clay layer of low charge.

[12] For example, Tournassat and Steefel (2019) write (\(f_{DL}\) denotes the volume fraction of the diffuse layer):

In PHREEQC and CrunchClay, the volume of the diffuse layer (\(V_{DL}\) in m³), and hence the \(f_{DL}\) value, can be defined as a multiple of the Debye length in order to capture this effect of ionic strength on \(f_{DL}\): \begin{equation*} V_{DL} = \alpha_{DL}\kappa^{-1}S \tag{22} \end{equation*} \begin{equation*} f_{DL} = V_{DL}/V_{pore} \end{equation*} […] it is obvious that \(f_{DL}\) cannot exceed 1. Equation (22) must then be seen as an approximation, the validity of which may be limited to small variations of ionic strength compared to the conditions at which \(f_{DL}\) is determined experimentally. This can be appreciated by looking at the results obtained with a simple model where: \begin{equation*} \alpha_{DL} = 2\;\mathrm{if}\;4\kappa^{-1} \le V_{pore}/S\;\mathrm{and,} \end{equation*} \begin{equation*} f_{DL} = 1 \;\mathrm{otherwise.} \end{equation*}

[13] Some tools (e.g. PHREEQC) allow to put a maximum size limit on the diffuse layer domain, independent of chemical conditions. This is of course only a way for the code to “work” under all conditions.

[14] As icing on the cake, these estimations of free water in bentonite (15%) and Opalinus clay (50%) appear to be based on the incorrect assumption that “anions” only reside in such compartments. In the present context, this handling is particularly confusing, as a main point with multi-porosity models (I assume?) is to evaluate ion concentrations in other types of compartments.

[15] Yet, Tournassat and Steefel (2019) sometimes seem to favor the choice of two Debye lengths (see footnote 12), for unclear reasons.

[16] Donnan equilibrium between microscopic compartments can be studied in molecular dynamics simulations, but they require the considered system to be large enough for the electrostatic potential to reach zero. The semi-permeable component in such simulations is implemented by simply imposing constraints on the atoms making up the clay layer.

[17] I believe the referred unpublished results now are published: Gimmi and Alt-Epping (2018).

Semi-permeability, part I

Descriptions in bentonite literature

What do authors mean when they say that bentonite has semi-permeable properties? Take for example this statement, from Bradbury and Baeyens (2003)¹

[…] highly compacted bentonite can function as an efficient semi-permeable membrane (Horseman et al., 1996). This implies that the re-saturation of compacted bentonite involves predominantly the movement of water molecules and not solute molecules.

Judging from the reference to Horseman et al. (1996) — which we look at below — it is relatively clear that Bradbury and Baeyens (2003) allude to the concept of salt exclusion when speaking of “semi-permeability” (although writing “solute molecules”). But a lowered equilibrium salt concentration does not automatically say that salt is less transferable.

A crucial question is what the salt is supposed to permeate. Note that a semi-permeable component is required for defining both swelling pressure and salt exclusion. In case of bentonite, this component is impermeable to the clay particles, while it is fully permeable to ions and water (in a lab setting, it is typically a metal filter). But Bradbury and Baeyens (2003) seem to mean that in the process of transferring aqueous species between an external reservoir and bentonite, salt is somehow effectively hindered to be transferred. This does not make much sense.

Consider e.g. the process mentioned in the quotation, i.e. to saturate a bentonite sample with a salt solution. With unsaturated bentonite, most bets are off regarding Donnan equilibrium, and how salt is transferred depends on the details of the saturation procedure; we only know that the external and internal salt concentrations should comply with the rules for salt exclusion once the process is finalized.

Imagine, for instance, an unsaturated sample containing bentonite pellets on the cm-scale that very quickly is flushed with the saturating solution, as illustrated in this state-of-the-art, cutting-edge animation

The evolution of the salt concentration in the sample will look something like this

Initially, as the saturating solution flushes the sample, the concentration will be similar to that of the external concentration (\(c_\mathrm{ext}\)). As the sample reaches saturation, it contains more salt than what is dictated by Donnan equilibrium (\(c_\mathrm{eq.}\)), and salt will diffuse out.

In a process like this it should be obvious that the bentonite not in any way is effectively impermeable to the salt. Note also that, although this example is somewhat extreme, the equilibrium salt concentration is probably reached “from above” in most processes where the clay is saturated with a saline solution: too much salt initially enters the sample (when a “microstructure” actually exists) and is later expelled.

Also for mass transfer between an external solution and an already saturated sample does it not make sense to speak of “semi-permeability” in the way here discussed. Consider e.g. a bentonite sample initially in equilibrium with an external 0.3 M NaCl solution, where the solution suddenly is switched to 1.0 M. Salt will then start to diffuse into the sample until a new (Donnan) equilibrium state is reached. Simultaneously (a minute amount of) water is transported out of the clay, in order for the sample to adapt to the new equilibrium pressure.²

There is nothing very “semi-permeabilic” going on here — NaCl is obviously free to pass into the clay. That the equilibrium clay concentration in the final state happens to be lower than in the external concentration is irrelevant for how how difficult it is to transfer the salt.

But it seems that many authors somehow equate “semi-permeability” with salt exclusion, and also mean that this “semi-permeability” is caused by reduced mobility for ions within the clay. E.g. Horseman et al. (1996) write (in a section entitled “Clays as semi-permeable membranes”)

[…] the net negative electrical potential between closely spaced clay particles repel anions attempting to migrate through the narrow aqueous films of a compact clay, a phenomenon known as negative adsorption or Donnan exclusion. In order to maintain electrical neutrality in the external solution, cations will tend to remain
with their counter-ions and their movement through the clay will also be restricted (Fritz, 1986). The overall effect is that charged chemical species do not move readily through a compact clay and neutral water molecules may be able to pass more freely.

It must be remembered that Donnan exclusion occurs in many systems other than “compact clay”. By instead considering e.g. a ferrocyanide solution, it becomes clear that salt exclusion has nothing to do with how hindered the ions are to move in the system (as long as they move). KCl is, of course, not excluded from a potassium ferrocyanide system because ferrocyanide repels chloride, nor does such interactions imply restricted mobility (repulsion occurs in all salt solutions). Similarly, salt is not excluded from bentonite because of repulsion between anions and surfaces (also, a negative potential does not repel anything — charge does).

In the above quotation it is easy to spot the flaw in the argument by switching roles of anions and cations; you may equally incorrectly say that cations are attracted, and that anions tag along in order to maintain charge neutrality.

The idea that “semi-permeability” (and “anion” exclusion) is caused by mobility restrictions for the ions within the bentonite, while water can “pass more freely” is found in many places in the bentonite literature. E.g. Shackelford and Moore (2013) write (where, again, potentials are described as repelling)

In [the case of bentonite], when the clay is compressed to a sufficiently high density such that the pore spaces between adjacent clay particles are minimized to the extent that the electrostatic (diffuse double) layers surrounding the particles overlap, the
overlapping negative potentials repel invading anions such that the pore becomes excluded to the anion. Cations also may be excluded to the extent that electrical neutrality in solution is required (e.g., Robinson and Stokes, 1959).

This phenomenon of anion exclusion also is responsible for the existence of semipermeable membrane behavior, which refers to the ability of a porous medium to restrict the migration of solutes, while allowing passage of the solvent (e.g., Shackelford, 2012).

Chagneau et al. (2015) write

[…] TOT layers bear a negative structural charge that is compensated by cation accumulation and anion depletion near their surfaces in a region known as the electrical double layer (EDL). This property gives clay materials their semipermeable
membrane properties: ion transport in the clay material is hindered by electrostatic repulsion of anions from the EDL porosity, while water is freely admitted to the membrane.

and Tournassat and Steefel (2019) write (where, again, we can switch roles of “co-” and “counter-ions”, to spot one of the flaws)

The presence of overlapping diffuse layers in charged nanoporous media is responsible for a partial or total repulsion of co-ions from the porosity. In the presence of a gradient of bulk electrolyte concentration, co-ion migration through the pores is hindered, as well as the migration of their counter-ion counterparts because of the electro-neutrality constraint. This explains the salt-exclusionary properties of these materials. These properties confer these media with a semi-permeable membrane behavior: neutral aqueous species and water are freely admitted through the membrane while ions are not, giving rise to coupled transport processes.

I am quite puzzled by these statements being so commonplace.³ It does not surprise me that all the quotations basically state some version of the incorrect notion that salt exclusion is caused by electrostatic repulsion between anions and surfaces — this is, for some reason, an established “explanation” within the clay literature.⁴ But all quotations also state (more or less explicitly) that ions (or even “solutes”) are restricted, while water can move freely in the clay. Given that one of the main features of compacted bentonite components is to restrict water transport, with hydraulic conductivities often below 10^-13 m/s, I don’t really know what to say.

Furthermore, one of the most investigated areas in bentonite research is the (relatively) high cation transport capacity that can be achieved under the right conditions. In this light, I find it peculiar to claim that bentonite generally impedes ion transport in relation to water transport.

Bentonite as a non-ideal semi-permeable membrane

As far as I see, authors seem to confuse transport between external solutions and clay with processes that occur between two external solutions separated by a bentonite component. Here is an example of the latter set-up

The difference in concentration between the two solutions implies water transport — i.e. osmosis — from the reservoir with lower salt concentration to the reservoir with higher concentration. In this process, the bentonite component as a whole functions as the membrane.

The bentonite component has this function because in this process it is more permeable to water than to salt (which has a driving force to be transported from the high concentration to the low concentration reservoir). This is the sense in which bentonite can be said to be semi-permeable with respect to water/salt. Note:

Salt is still transported through the bentonite. Thus, the bentonite component functions fundamentally only as a non-ideal membrane.
Zooming in on the bentonite component in the above set-up, we note that the non-ideal semi-permeable functionality emerges from the presence of two ideal semi-permeable components. As discussed above, the ideal semi-permeable components (metal filters) keep the clay particles in place.

The non-ideal semi-permeability is a consequence of salt exclusion. But these are certainly not the same thing! Rather, the implication is: Ideal semi-permeable components (impermeable to clay) \(\rightarrow\) Donnan effect \(\rightarrow\) Non-ideal semi-permeable membrane functionality (for salt)
The non-ideal functionality means that it is only relevant during non-equilibrium. E.g., a possible (osmotic) pressure increase in the right compartment in the illustration above will only last until the salt has had time to even out in the two reservoirs; left to itself, the above system will eventually end up with identical conditions in the two reservoirs. This is in contrast to the effect of an ideal membrane, where it makes sense to speak of an equilibrium osmotic pressure.
None of the above points depend critically on the membrane material being bentonite. The same principal functionality is achieved with any type of Donnan system. One could thus imagine replacing the bentonite and the metal filters with e.g. a ferrocyanide solution and appropriate ideal semi-permeable membranes. I don’t know if this particular system ever has been realized, but e.g. membranes based on polyamide rather than bentonite seems more commonplace in filtration applications (we have now opened the door to the gigantic fields of membrane and filtration technology). From this consideration it follows that “semi-permeability” cannot be attributed to anything bentonite specific (such as “overlapping double layers”, or direct interaction with charged surfaces).
I think it is important to remember that, even if bentonite is semi-permeable in the sense discussed, the transfer of any substance across a compacted bentonite sample is significantly reduced (which is why we are interested in using it e.g. for confining waste). This is true for both water and solutes (perhaps with the exception of some cations under certain conditions).

“Semi-permeability” in experiments

Even if bentonite is not semi-permeable in the sense described in many places in the literature, its actual non-ideal semi-preamble functionality must often be considered in compacted clay research. Let’s have look at some relevant cases where a bentonite sample is separated by two external solution reservoirs.

Tracer through-diffusion

The simplest set-up of this kind is the traditional tracer through-diffusion experiment. Quite a lot of such tests have been published, and we have discussed various aspects of this research in earlier blog posts.

The traditional tracer through-diffusion test maintains identical conditions in the two reservoirs (the same chemical compositions and pressures) while adding a trace amount of the diffusing substance to the source reservoir. The induced tracer flux is monitored by measuring the amount of tracer entering the target reservoir.

In this case the chemical potential is identical in the two reservoirs for all components other than the tracer, and no additional transport processes are induced. Yet, it should be kept in mind that both the pressure and the electrostatic potential is different in the bentonite as compared with the reservoirs. The difference in electrostatic potential is the fundamental reason for the distinctly different diffusional behavior of cations and anions observed in these types of tests: as the background concentration is lowered, cation fluxes increase indefinitely (for constant external tracer concentration) while anion fluxes virtually vanish.

Tracer through-diffusion is often quantified using the parameter \(D_e\), defined as the ratio between steady-state flux and the external concentration gradient.⁵ \(D_e\) is thus a type of ion permeability coefficient, rather than a diffusion coefficient, which it nevertheless often is assumed to be.

Typically we have that \(D_e^\mathrm{cation} > D_e^\mathrm{water} > D_e^\mathrm{anion}\) (where \(D_e^\mathrm{cation}\) in principle may become arbitrary large). This behavior both demonstrates the underlying coupling to electrostatics, and that “charged chemical species” under these conditions hardly can be said to move less readily through the clay as compared with water molecules.

Measuring hydraulic conductivity

A second type of experiment where only a single component is transported across the clay is when the reservoirs contain pure water at different pressures. This is the typical set-up for measuring the so-called hydraulic conductivity of a clay component.⁶

Even if no other transport processes are induced (there is nothing else present to be transported), the situation is here more complex than for the traditional tracer through-diffusion test. The difference in water chemical potential between the two reservoirs implies a mechanical coupling to the clay, and a corresponding response in density distribution. An inhomogeneous density, in turn, implies the presence of an electric field. Water flow through bentonite is thus fundamentally coupled to both mechanical and electrical processes.

In analogy with \(D_e\), hydraulic conductivity is defined as the ratio between steady-state flow and the external pressure gradient. Consequently, hydraulic conductivity is an effective mass transfer coefficient that don’t directly relate to the fundamental processes in the clay.

An indication that water flow through bentonite is more subtle than what it may seem is the mere observation that the hydraulic conductivity of e.g. pure Na-montmorillonite at a porosity of 0.41 is only 8·10^-15 m/s. This system thus contains more than 40% water volume-wise, but has a conductivity below that of unfractioned metamorphic and igneous rocks! At the same time, increasing the porosity by a factor 1.75 (to 0.72), the hydraulic conductivity increases by a factor of 75! (to 6·10^-13 m/s⁷)

Mass transfer in a salt gradient

Let’s now consider the more general case with different chemical compositions in the two reservoirs, as well as a possible pressure difference (to begin with, we assume equal pressures).

Even with identical hydrostatic pressures in the reservoirs, this configuration will induce a pressure response, and consequently a density redistribution, in the bentonite. There will moreover be both an osmotic water flow from the right to the left reservoir, as well as a diffusive solute flux in the opposite direction. This general configuration thus necessarily couples hydraulic, mechanical, electrical, and chemical processes.

This type of configuration is considered e.g. in the study of osmotic effects in geological settings, where a clay or shale formation may act as a membrane.⁸ But although this configuration is highly relevant for engineered clay barrier systems, I cannot think of very many studies focused on these couplings (perhaps I should look better).

For example, most through-diffusion studies are of the tracer type discussed above, although evaluated parameters are often used in models with more general configurations (e.g. with salt or pressure gradients). Also, I am not aware of any measurements of hydraulic conductivity in case of a salt gradient (but the same hydrostatic pressure), and I am even less aware of such values being compared with those evaluated in conventional tests (discussed previously).

A quite spectacular demonstration that mass transfer may occur very differently in this general configuration is the seeming steady-state uphill diffusion effect: adding an equal concentration of a cation tracer to the reservoirs in a set-up with a maintained difference in background concentration, a tracer concentration difference spontaneously develops. \(D_e\) for the tracer can thus equal infinity,⁹ or be negative (definitely proving that this parameter is not a diffusion coefficient). I leave it as an exercise to the reader to work out how “semi-permeable” the clay is in this case. Update (240822): The “uphill” diffusion effect is further discussed here.

A process of practical importance for engineered clay barrier systems is hyperfiltration of salts. This process will occur when a sufficient pressure difference is applied over a bentonite sample contacted with saline solutions. Water and salt will then be transferred in the same direction, but, due to exclusion, salt will accumulate on the inlet side. A steady-state concentration profile for such a process may look like this

The local salt concentration at the sample interface on the inlet side may thus be larger than the concentration of the injected solution. This may have consequences e.g. when evaluating hydraulic conductivity using saline solutions.

Hyperfiltration may also influence the way a sample becomes saturated, if saturated with a saline solution. If the region near the inlet is virtually saturated, while regions farther into the sample still are unsaturated, hyperfiltration could occur. In such a scenario the clay could in a sense be said to be semi-permeable (letting through water and filtrating salts), but note that the net effect is to transfer more salt into the sample than what is dictated by Donnan equilibrium with the injected solution (which has concentration \(c_1\), if we stick with the figure above). Salt will then have to diffuse out again, in later stages of the process, before full equilibrium is reached. This is in similarity with the saturation process that we considered earlier.

Footnotes

[1] We have considered this study before, when discussing the empirical evidence for salt in interlayers.

[2] This is more than a thought-experiment; a test just like this was conducted by Karnland et al. (2005). Here is the recorded pressure response of a Na-montmorillonite sample (dry density 1.4 g/cm³) as it is contacted with NaCl solutions of increasing concentration

We have considered this study earlier, as it proves that salt enters interlayers.

[3] As a side note, is the region near the surface supposed to be called “diffuse layer”, “electrical double layer”, or “electrostatic (diffuse double) layer”?

[4] Also Fritz (1986), referenced in the quotation by Horseman et al. (1996), states a version of this “explanation”.

[5] This is not a gradient in the mathematical sense, but is defined as \( \left (c_\mathrm{target} – c_\mathrm{source} \right)/L\), where \(L\) is sample length.

[6] Hydraulic conductivity is often also measured using a saline solution, which is commented on below.

[7] Which still is an a amazingly small hydraulic conductivity, considering the the water content.

[8] The study of Neuzil (2000) also provides clear examples of water moving out of the clay, and salt moving in, in similarity with the process considered above.

[9] Mathematically, the statement “equal infinity” is mostly nonsense, but I am trying to convey that a there is a tracer flux even without any external tracer concentration difference.

The Bentonite Report

Reflections on bentonite research

Category Archives: Semi-permeability

Post-publication review: Tournassat and Steefel (2015), part II

“Electrostatic properties, high surface area, and anion exclusion”

Stacks

Anion exclusion

Conclusion

Footnotes

Multi-porosity models cannot be taken seriously (Semi-permeability, part II)

Implied structure of the free water domain

Arbitrary division between diffuse layer and free water

Donnan effect on the microscopic scale?!

Footnotes

Semi-permeability, part I

Descriptions in bentonite literature

Bentonite as a non-ideal semi-permeable membrane

“Semi-permeability” in experiments

Tracer through-diffusion

Measuring hydraulic conductivity

Mass transfer in a salt gradient

Footnotes