Category Archives: Diffusion

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

This is the second part of the review of “Ionic Transport in Nano-Porous Clays with Consideration of Electrostatic Effects” (Tournassat and Steefel (2015) (referred to as TS15 in the following). For background and context please check the first part. That part covered the introduction and the section “Classical Fickan Diffusion Theory”. The next section is titled “Clay mineral surfaces and related properties”, and is further partitioned into two subsections. Here we exclusively deal with the first one of these subsections: “Electrostatic properties, high surface area, and anion exclusion”. It only covers three and a half journal pages, but since the article here goes completely off the rails, there is much to comment on.

“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.

Sorption, part V: A case against Stern layers

It should go without saying that modelers and model developers must justify every feature, mechanism, or component that they use. Failing to do so strongly increases the risk of being fooled by overparameterization rather than gaining insight. The bentonite scientific literature is nonetheless full of incorrect or unjustified model assumptions, several of which have been discussed previously on the blog. Examples include assuming the presence of bulk water, assuming “stack” structures, and assuming that diffusive fluxes from separate domains are additive. Here we discuss yet another unjustified common model component: Stern layers on montmorillonite basal surfaces.

In the bentonite literature, a Stern layer essentially means a layer of “specifically” sorbed ions on the basal surface, as e.g. illustrated here, in a a figure very similar to what is found in Leroy et al. (2006). Illustrations like this are ubiquitous in the literature.

If montmorillonite basal surfaces function roughly as uniform planes of charge we expect the counter-ions to form a diffuse layer, as e.g. described by the Gouy-Chapman model. By introducing a Stern layer, however, many bentonite researchers mean that exchangeable cations in general also interact with basal surfaces by forming immobile surface complexes. Such interactions necessarily involve mechanisms more “chemical” than the pure electrostatic interaction with uniform planes of charge, and the typical description postulates localized “sorption sites”, as illustrated above.

This blog post treats three different main arguments against Stern layers, presented in different sections

I want to make clear that this criticism concerns one particular type of surface: the montmorillonite basal surface. Stern layer models are found in many research fields dealing with solid interfaces, and although they have been criticized more generally, here we have no intention of doing so. Likewise, the process of surface complexation is certainly important generally — even in bentonite, e.g. on edge surfaces of montmorillonite particles.1

To better be able to criticize the use of Stern layers on basal surfaces in bentonite modeling, we begin by discussing the origin of the Stern’s model.

Origins of the Stern Layer model

The Stern layer concepts were introduced by Otto Stern2 as an extension of the Gouy-Chapman model. Stern’s main concern was metallic electrodes in electrochemical applications. In such systems, the surface (electrode) potential is externally controlled, and can typically be on the order of 1 volt. It is easily seen that the Gouy-Chapman model predicts nonsense for such surface potentials. For e.g. a 1:1 system, the counter-ion concentration at the surface is enhanced by a factor of the order of \(10^{17}\)(!), as seen directly from the Boltzmann distribution \(c^\mathrm{surf} = c^\mathrm{ext}\cdot e^{e\psi^0/kT}\), where \(\psi^0\) is the surface potential, \(e\) is the elementary charge, \(kT\) the thermal energy, and \(c^\mathrm{ext}\) is the concentration far away from the surface. The main problem is that the Gouy-Chapman model does not account for the finite size of ions, and therefore can accumulate an arbitrary amount of charge at the surface. To remedy this flaw, Stern suggested to divide the interface region into a “compact” layer and a “diffuse” part, with the division located an ionic radius from the electrode surface (sometimes referred to as the outer Helmholtz plane).

In the simplest version of Stern’s model the compact layer is free of charges but act as a plate capacitor with a prescribed capacitance (per unit area) \(K_0\). In the original paper Stern shows that, with electrode potential \(\psi^0 = 1\) V and external 1:1 solution concentration \(c^\mathrm{ext} = 1\) M, such a capacitive layer reduces the potential where the diffuse layer begins to \(\psi^1 = 0.08\) V; lowering the external concentration to \(c^\mathrm{ext} = 0.01\) M gives \(\psi^1 = 0.18\) V. For these calculations, Stern uses a value of \(K_0 = 0.29\;\mathrm{F/m^2}\), adopted from measurements on mercury electrodes. This version of the Stern model is essentially a way to take into account that ions cannot get arbitrarily close to the surface.

Stern also presented more elaborate versions of the model that include adsorption in the compact layer (as a Langmuir adsorption model). It is such mechanisms that is universally referred to as a Stern layer in the bentonite scientific literature. Clearly, such versions are substantially more conceptually complex; rather than to just account for a finite ion size at the first molecular layer, we must now also consider additional chemical interactions that typically are different for different types of ions. We also need to have an idea about the adsorption capacity.

Lack of a coherent description of “specific sorption” on montmorillonite basal surfaces

When using Stern layers for describing montmorillonite basal surfaces, a first thing to note is that the surface potential is not independently controlled for these systems. In contrast to metallic electrodes, montmorillonite is characterized by a fixed surface charge and the problem of accumulating unrealistically large amounts of ions at the interface is significantly mitigated. As pointed by e.g. Norrish and Bolt already in the 1950s, even if we put all counter-ions within the first nanometer adjacent to the surface, the corresponding ion concentration is not larger than approximately 3 M. Here is a illustration of the montmorillonite basal surface on the nm scale, with a representative number of monovalent counter-ions (top layer oxygen atoms are red and the counter-ions blue).3

Clearly, there is room to accommodate all ions without running into the problems that was initially addressed by Stern’s model. Of course, solely accounting for the finite size of the ions — as is done in the simplest version of Stern’s model — is always well justified and will in principle improve the description. In particular, a pure diffuse layer model overestimates the capacitance of the surface. As shown in the table below, the introduction of an empty Stern layer “fixes” this problem.

Here \(c^\mathrm{ext}\) is the concentration of the 1:1 salt far away from the surface, \(\psi^1\) and \(c^1\) denote the electrostatic potential and the counter-ion concentration, respectively, at the point where the diffuse layer begins (i.e. at the interface to the compact layer), and \(\psi^0\) is the electrostatic potential at the surface. \(K_\mathrm{Stern}\) denotes the corresponding capacitance as calculated from the Stern model, with the choice \(K_0 = 0.29 \;\mathrm{F/m^2}\). \(K_\mathrm{DL}\) is instead the capacitance as calculated from a pure diffuse layer model (in which case the surface potential has the value of \(\psi^1\)). In the calculations are assumed a montmorillonite surface charge of 0.111 \(\mathrm{C}/\mathrm{m ^2}\).

But to simply account for the finite size of the ions by means of an empty compact layer is not how the term Stern layer is used in the bentonite scientific literature. As mentioned, most bentonite researchers mean that parts of the rather sparse collection of ions on the surface interacts chemically (“specific sorption”, “chemisorption”). The question of whether Stern layers on montmorillonite basal surfaces are well motivated thus reduces to what arguments there are for more elaborate chemical mechanisms being active on these surfaces. And descriptions of specific sorption on basal surfaces are really all over the place.

Deshpande and Marshall (1959, 1961) claim that counter-ions are partitioned between (i) chemisorbed ions, which do not contribute to conductivity or activity, (ii) physisorbed ions in a Stern layer, which do not contribute to activity or D.C. conductivity, and (iii) diffuse layer ions, which contribute fully to activity and conductivity. If I interpret their numbers correctly, they state that about 75% — 80% of the counter-ions in pure K-montmorillonite are immobilized. Note that these authors mean that ions in the Stern layer are “physically” adsorbed, while the surface also have “chemically” adsorbed species. Thus, they use the term Stern layer for certain types of physisorption, while stating that ions also bond covalently to the surface.

Shainberg and Kemper (1966, 1967), on the other hand, model ions as either mobile in a diffuse layer or immobile in a Stern layer. They argue that covalently bound ions are “extremely unlikely”, and mean that ions in the Stern layer form “ion pairs” with the surface, as suggested earlier by Heald et al. (1964). They use this idea as a starting point for analyzing differences in exchange selectivity for different monovalent cations in montmorillonite. They claim that “about 20 to 50% of sodium are specifically adsorbed”.

Note that Shainberg and Kemper, just like Deshpande and Marshall, assume Stern layer ions to be “physically” bonded to the surface (i.e. non-covalently), while having a completely different opinion on the presence of chemisorbed ions. Shainberg and Kemper (1966) provide a picture showing the conceptual difference between an ion in the Stern layer (“unhydrated”) and ions in the diffuse layer (“hydrated”) that looks very similar to this

In this context it may be worth to also mention the work of Low and co-workers. Low argued consistently that swelling pressure is not primarily related to the exchangeable ions — something that I strongly disagree with and that I commented briefly on in a previous blog post.4 Directly related to this view, these authors claim that the exchangeable ions for the most part do not dissociate from the surfaces, and in later papers they refer to such ions as being part of a Stern layer.

To me, all the above descriptions seem like little more than speculation. None of these authors discuss how or why e.g. sodium ions (!) are supposed to from ion pairs with a charge center buried far inside the montmorillonite layer, nor how or why they bond covalently with the basal surface. Nevertheless, both Low as well as Shainberg and Kemper seem to have influenced the writings of Sposito, who, in turn, has had quite a huge impact on contemporary descriptions of bentonite. In e.g. Sposito (1992), which specifically discusses montmorillonite (“smectite”), he writes

Despite the long history of continual investigation of the surface and colloid chemistry of smectites (van Olphen, 1977; Sposito, 1984), the structure of the electrical double layer at smectite surfaces and its influence on the rheological properties of smectite suspensions remain topics of lively controversy. One of the most contentious issues is the partitioning of adsorbed monovalent cations among the three possible surface species on the basal planes of smectite particles, such as montmorillonite (see, e.g., Low, 1981, 1987). […] [A] monovalent cation can be adsorbed on the basal planes by three different mechanisms: inner-sphere surface complexes, in which the cation desolvates and is captured by a ditrigonal cavity; outer-sphere surface complexes, in which the cation remains solvated but still is captured by a ditrigonal cavity and immobilized; and the diffuse-ion swarm, in which the cation is attracted to the basal plane, but remains fully dissociated from the smectite surface (Sposito, 1989a, Chap. 7).

The view conveyed here is that exchangeable ions do not interact with montmorillonite basal surfaces as if these, to a first approximation, are planes of uniform charge. Such interaction is only supposed to govern an outer diffuse layer (called a “diffuse-ion swarm” for unclear reasons), and ions are also supposed to interact with the surface by no less than two other “mechanisms”, related to the “ditrigonal cavities”.

Note that while Sposito acknowledges an ongoing “lively controversy” regarding how to describe montmorillonite basal surfaces, he specifies that this debate is limited to how to distribute the exchangeable ions among “three possible surface species.” But, as we will explore below, there is certainly no consensus within colloid chemistry that exchangeable ions are involved in complexation chemistry on the basal surfaces! (I therefore find this way of formulating the “controversy” quite dishonest, to be honest.) For reasons I can’t get my head around, descriptions of “inner-” and “outer-sphere complexes” on montmorillonite basal surfaces are anyway ubiquitous in modern bentonite literature. Let’s therefore take a closer look at how these are introduced.

“Inner-” and “outer-sphere” surface complexes

A description that hardly enlightens me is given in Sposito (1989)5

[The inner- and outer-sphere complexes] constitute the Stern layer on an adsorbent. […] The diffuse-ion swarm and the outer-sphere surface complex mechanisms of adsorption involve almost exclusively electrostatic bonding, whereas inner-sphere complex mechanisms are likely to involve ionic as well as covalent bonding. Because covalent bonding depends significantly on the particular electron configurations of both the surface group and the complexed ion, it is appropriate to consider inner-sphere surface complexation as the molecular basis of the term specific adsorption. Correspondingly, diffuse-ion screening and outer-sphere surface complexation are the molecular basis for the term nonspecific adsorption. Nonspecific refers to the weak dependence on the detailed electron configurations of the surface functional group and adsorbed ion that is to be expected for the interactions of solvated species.

Here, Sposito means that exchangeable ions bond covalently with the montmorillonite basal surface,6 in agreement with Deshpende and Marschall, and in disagreement with Shainberg and Kemper (we have one “extremely unlikely” and one “likely” for covalent bonding…). In contrast to Deshpende and Marschall, however, Sposito means that these “inner-sphere” complexes are part of the Stern layer. To confuse matters even more, Shainberg and Kemper assume their “unhydrated” construct (which corresponds structurally to an “inner-sphere” complex, see above figure) as being part of the Stern layer, but not part of any covalent bonding. Moreover, Shainberg and Kemper assume their “hydrated” construct (corresponding structurally to an “outer-sphere” complex, see above figure) to be part of the diffuse layer, while Sposito wants his “outer-sphere” complexes to be immobile and part of the Stern layer…

Given the above description (and others) it is hard to understand what the difference is supposed to be between an “outer-sphere” complex and an ion in the “diffuse-ion swarm”, other than that the former is simply claimed to be immobilized; both ions are said to interact with “exclusively electrostatic bonding”,7 both are classified as “nonspecific adsorption”, and both are fully hydrated. In my head, this is simply a recipe for achieving an overparameterized model description.

Sposito’s description also makes implicit statements about the montmorillonite basal surface: it contains “surface functional groups” whose specific electron configuration significantly influence covalent bonding, while being insensitive for the formation of “outer-sphere” complexes and the “diffuse ion-swarm”. In Sposito (1984) he suggests that the “functional groups” are groups of oxygen atoms on the surface of the montmorillonite particle (“ditrigonal cavities”) that qualifies as Lewis bases. The presence of atomic substitutions in the octahedral layer is supposed to enhance this Lewis base character

If isomorphic substitution of \(\mathrm{Al}^{3+}\) by \(\mathrm{Fe}^{2+}\) or \(\mathrm{Mg}^{2+}\) occurs in the octahedral sheet, the resulting excess negative charge can distribute itself principally over the 10 surface oxygen atoms of the four silica tetrahedra that are associated through their apexes with a single octahedron in the layer. This distribution of negative charge enhances the Lewis base character of the ditrigonal cavity and makes it possible to form complexes with cations as well as with dipolar molecules.

Note how completely different this whole description is compared to the original Gouy-Chapman conceptual view. Here is implied that montmorillonite basal planes8 cannot be described as a passive layer of charge, but that it is a fully reactive system, including covalent bonding mechanisms.

Frankly, I dismiss the above description of the montmorillonite surface as a Lewis base as pure speculation. I will gladly admit that I am a physicist rather than a chemist, and perhaps I am missing something obvious, but I really don’t see any argumentation behind this description. I am also under the impression that montmorillonite basal planes are relatively chemically stable — that is why they form in the first place, and that is also one reason for why we are interested in using bentonite for e.g. long-term geological waste storage. Furthermore, a meta-argument for dismissing this description is that in later publications we find statements like this, in Sposito (2004):

The \(\mathrm{Na}^+\) that are counterions for the negative structural charge developed as a result of isomorphic substitutions within the clay mineral layer tend to adsorb as solvated species on the basal plane (a plane of hexagonal rings of oxygen ions known as a siloxane surface) near deficits of negative charge originating in the octahedral sheet from substitution of a bivalent cation for \(\mathrm{Al}^{3+}\). This mode of adsorption occurs as a result of the strong solvating characteristics of Na and the physical impediment to direct contact between Na and the site of negative charge posed by the layer structure itself. The way in which this negative charge is distributed on the siloxane surface is not well known, but if the charge tends to be delocalized there, that would also lend itself to outer-sphere surface complexation.

So, 20 years after the surface chemistry of montmorillonite was described as if it was completely understood (Sposito, 1984), the way the negative charge distributes is now described instead as “not well known”… Furthermore, in contrast to earlier statements, the formation of an “outer-sphere complex” is here associated mainly with the hydration properties of sodium. But if the idea of a “surface functional group” is discarded — or at least downplayed — why should a hydrated ion near the surface be described as a surface complex at all?

We note that Sposito (2004) still seem to imply that the “outer-sphere surface complex” is localized and immobile (“adsorbed near deficits of negative charge”) But the evidence is vast that sodium, and several other ions, are quite mobile even in monohydrates (see below).

Deviations from the Gouy-Chapman model do not imply surface complexation

Authors that promote Stern layers on montmorillonite basal surfaces usually rely on the Gouy-Chapman model for describing the diffuse layer part. Lyklema, writing generally on colloid science, explicitly “defends” such an approach

In the following our discussion will be based on the rather pragmatic, though somewhat artificial, subdivision of the solution side of the double layer into two parts: an inner part, or Stern layer where all complications regarding finite ion size, specific adsorption, discrete charges, surface heterogeneity, etc., reside and an outer, Gouy or diffuse layer, that is by definition ideal, i.e. it obeys Poisson-Boltzmann statistics. This model is due to Stern following older ideas of Helmholtz and has over the decades since its inception rendered excellent services, especially in dealing with experimental systems.

Dzombak and Hudson (1995) express a similar attitude

Bolt and co-workers […] investigated in detail the application of the Gouy–Chapman diffuse-layer theory to ion-exchange processes. Their work demonstrated that consideration of electrostatic sorption alone is not sufficient to explain ion-exchange data and that chemisorption (or “specific” sorption) needs to be included in ion-exchange models.

It is not logically consistent to conclude that deviations from the Gouy-Chapman model implies that specific sorption “needs to be included”.9 On the contrary, introducing specific sorption to compensate for a certain model rather than for surface chemical reasons may, in my mind, be a recipe for an overparameterized disaster. I don’t get reassured by statements like this, also from Dzombak and Hudson (1995)

Surface complexation models can be extended to include diffuse-layer sorption. This approach permits their application in modeling the sorption of ions (such as monovalent electrolyte ions) that exhibit weak specific sorption. The generality of such an extended surface complexation approach together with the mathematical power of modern chemical speciation models offers the potential for accurate physicochemical modeling of ion exchange

Reasonably, a complex system may require complex models, but it is certainly dangerous in a modeling context to rely too heavily on “mathematical power” (I guess “numerical power” is the preferred phrase).10

Note that very different attitudes towards the Stern layer concept is found in the colloid science literature, where e.g. Evans and Wennerström (1999) describe it as an “intellectual cul de sac”.11

One way of dealing with these difficulties is to say that the solution layer closest to a charged surface has properties so different from the bulk that it should be treated as a separate entity. This device was introduced in the 1930s by the German electrochemist Stern and the surface layer is commonly referred to as the Stern layer, whose properties are specified by a number of empirical parameters. It is the opinion of the authors of this book that the Stern layer concept is an intellectual cul de sac for the description of electrostatics in colloidal systems. One reason for this point of view is that from modern spectroscopic measurements we know molecular properties are not dramatically changed for a liquid close to a charged surface.

I find it quite perplexing that so many authors in the bentonite scientific community attribute any deviation from the Gouy-Chapman model solely to surface-related mechanisms. The Gouy-Chapman model treats both ions (point particles) and water (a dielectric continuum) in a very simplified manner, and it is clear that “specific ion” effects are ubiquitous, also in systems that lack surfaces. Addressing differences in e.g. selectivity coefficients without considering ion polarizability and hydration, while postulating the existence of localized sorption “sites”, can, to my mind, only lead to incorrect descriptions.

The Poisson-Boltzmann equation is a mean field approximation

Note also that the Poisson-Boltzmann equation — which underlies the Gouy-Chapman model — is only approximate. It is derived by assuming that the electrostatic potential experienced by any ion is the average potential from all other ions (and surfaces). More accurately, the ion distribution around a given ion deviates from the average, as a direct consequence of the presence of the central ion.

Including these ion-ion correlation effects makes the mathematical description considerably more complex. But with the advent of sufficiently powerful computers and algorithms, the electric double layer has been solved basically “exactly”. The “exact” solution may differ strongly from the Poisson-Boltzmann solution, with increasing concentrations towards the surfaces (and consequently a lowering of interlayer midpoint concentrations), and an explicit attractive electrostatic force between the two halves of an interlayer. Using Monte Carlo simulations, Guldbrand et al. (1984) demonstrated that with divalent counter-ions these effects are so large that the system becomes net attractive at a certain interlayer distance, in qualitative disagreement with the Poisson-Boltzmann solution. This effect, which has been thoroughly studied since the 80s, and which we have discussed in several posts on this blog, is the prevailing explanation e.g. for the limited swelling of Ca-montmorillonite.

The lesson here is that observed deviations from predictions of the Poisson-Boltzmann equation not automatically can be taken as evidence for additional active system components, and certainly not as evidence for specific sorption. Note that limited swelling in divalent montmorillonite is explained by the ions being diffusive, not that they are sorbed and immobilized. I cannot overstate the importance of this insight.

It boggles my mind that the entire research area on ion-ion correlations in colloidal systems seem to have made no significant impact on parts of the bentonite scientific community; I seldom find references to works on ion-ion correlation, and when I do it’s quite confused. E.g. Sposito (1992) means that the formation of “quasicrystals”12 is due to “outer-sphere” complexes

The best known example of a montmorillonite quasicrystal is that comprising stacks of four to seven layers. \(\mathrm{Ca}^{2+}\) ions, solvated by six water molecules (outer-sphere surface complex), serve as molecular “cross-links” to help bind the clay layers together through electrostatic forces.

Sure, the ion-ion correlation effect that prevents Ca-montmorillonite from exfoliating is of electrostatic origin, but it is not related to “cross-links” or surface complexes. Sposito furthermore continues by claiming that “even […] Na-montmorillonite” forms “quasicrystals”. Such claim cannot be supported by ion-ion correlation — on the contrary, ion-ion correlation explains why Ca-montmorillonite forms “quasicrystals”, while Na-montmorillonite does not. It is thus relatively clear that Sposito do not refer to ion-ion correlation in the above statement. At the same time, later in the same publication he cites Kjellander et al. (1988) on going beyond the mean-field treatment of the Poisson-Boltzmann equation, and even claims that the Gouy-Chapman model is “completely inaccurate” for systems containing divalent ions. I can only conclude that this is a quite confused description.

Finite-size effects of water molecules

With focus on the first molecular layers at a solid interface, it is clear that finite-size effects of water molecules — which are not treated in the Gouy-Chapman model — reasonably influences the resulting ion distribution. This influence is manifested both as a steric effect — there can only be a discrete number of water molecules between the ion and the surface — and as an effect of how strongly a certain ion is hydrated.

Water molecules are treated explicitly e.g. in molecular dynamics (MD) simulations of montmorillonite/water interfaces, and here are results from simulating a three water-layer interlayer of Na-montmorillonite, from Hedström and Karnland (2012)13

Sodium is seen to accumulate “between” the water layers; in the above illustration we have also included schematic illustrations of the molecular configurations, as conceived by Shainberg and Kemper (1966) (shown earlier). As stated earlier, Shainberg and Kemper (1966) refer to these as “hydrated” and “unhydrated”, but they are clearly the same type of configurations that e.g. Sposito (1992) and Dzombak and Hudson (1995) call “outer-” and “inner-sphere” complexes.

While the above mentioned authors mean that these “complexes” involve specific interactions between ions and surface, the MD simulation suggests that such structures are mainly a consequence of the finite-size of the molecules and ions. In particular, the MD results do not support the idea that these structures depend critically on a specific, non-electrostatic, ion–surface interaction. Indeed, the simulations explicitly treat also the atoms of the montmorillonite layer, which could make it difficult to judge whether the appearance of “complexes” mainly is related to water–ion or ion–surface interactions. But note that Hedström and Karnland (2012) simulate two different systems: one where the montmorillonite charge is put on specific atoms in the octahedral sheet (Mg for Al substitutions), and one where it is distributed on all Al atoms (as a fraction of the elementary charge). Both systems have essentially an identical atomic configuration in the interlayer, which strongly suggest that no critical ion–surface interaction is involved in forming “outer-” and “inner-sphere complexes” (i.e. they really are not surface complexes). I am not aware of any published simulation where the basal surface is represented as a uniform sheet of charge while water molecules are treated explicitly, but I am convinced that “outer-” and “inner-sphere complexes” would appear also in such a simulation.

Regarding MD simulations of montmorillonite interlayers, you can also simply observe them to convince yourself that the counter-ions are not in any reasonable sense immobilized. These types of simulations are routinely used to calculate (quite significant) interlayer diffusion coefficients, for crying out loud!

Experimental evidence of counter-ion mobility

A final argument for why Stern layers on montmorillonite basal surfaces are unjustified is the vast amount of empirical evidence of counter-ion mobility. We have discussed several diffusion studies in earlier blog posts that show that many ions (Na, Cl, K, Sr, I, Cs, Ca,…) have a significant mobility even in very dense systems, dominated by bi- or monohydrated interlayers. In the previous post, we brought up the following result

This figure shows the resulting concentration profiles in two diffusion experiments where sodium and chloride tracers, respectively, have diffused from an initial planar source for the same amount of time (23.7 h), in samples of pure Na-montmorillonite of dry density 1.8 \(\mathrm{g/cm^3}\), equilibrated with deionized water. This result was used previously to dismiss the ludicrous idea that these two ions are supposed to migrate in separate parts of the pore volume, exposed to completely different mechanisms. In the same vein, this result can be used to dismiss the idea of a Stern layer on basal surfaces.

Sodium, which is universally acknowledged to reside in the interlayers, is here demonstrated to diffuse just fine in bi- and monohydrated interlayers. As chloride, which also resides in the interlayers (despite all talk of “anion-accessible porosity”), behave essentially identical, it is quite far-fetched to assume any significant surface complexation mechanism. And anyone who argues for that these tracers actually do not diffuse in the interlayers should be reminded of the seeming “uphill” diffusion experiment,14 which is performed at even higher density, and where the “uphill” diffusion direction once and for all proves that the transport occurs in interlayers.

Strangely, many authors nowadays seem to promote both Stern layers and interlayer mobility in bentonite. Various simulation codes has been modified for this possibility, and there are several examples of researchers pointing out a possibility of “Stern layer diffusion”. I think these authors should carefully examine their chain of assumptions: Surface complexation in a Stern layer (i.e. sorption) is initially suggested to explain e.g. why breakthrough times in cation through-diffusion tests are relatively long as compared with the steady-state flux (i.e. why “\(D_e\)” can be considerably larger than”\(D_a\)”). With evidence for that the “sorbed” ions actually dominate the mass transfer, the sorption mechanism is not reconsidered, but yet another mechanism is suggested: Stern layer mobility… Reasonably, such an approach is not adequate for developing models; researchers employing it should critically consider the intended purpose of a Stern layer component.

Counter-ion mobility is also related to swelling pressure. Bentonite swelling pressure is difficult to describe generally, and I have written a whole series of blog posts on the subject, but it is clear that measured swelling pressures in e.g. moderately dense Na- and Li-montmorillonites is quite well described by the Poisson-Boltzmann equation. As this set of conditions (not too dense clay, simple monovalent ions) are exactly those for which we expect the Poisson-Boltzmann equation to be adequate, this is a strong indication that all counter-ions contribute to the pressure.15 Also, the limited swelling in e.g. Ca-montmorillonite, as previously discussed, is explained by ion-ion correlation effects where all ions are included in the diffuse layer.

Finally, we can take a look at salt exclusion from compacted bentonite. The magnitude of salt exclusion is directly related to the amount of mobile counter-ions. Thus, if most of the counter-ions were immobilized in a Stern layer, bentonite should show small exclusion effects. In contrast, the empirical results for e.g. chloride exclusion in sodium dominated bentonite indicate, again, that all counter-ions are part of a diffuse layer.

This diagram shows the relative amount of chloride in the bentonite as a function of \(c^\mathrm{ext}/c_\mathrm{IL}\), where \(c^\mathrm{ext}\) is the external salt concentration and \(c_\mathrm{IL}\) is the amount exchangeable cations, expressed as a monovalent interlayer concentration. The experimental data is from Van Loon et al. (2007), which we reevaluated and examined in detail in a previous blog post. The lines are the result from applying the “ideal” Donnan formula with various amounts of the counter-ions assumed diffusive. For details on Donnan theory, see this blog post.

Although the experimental data show considerable scatter, there is nothing in this plot that suggests that a fraction of the counter-ions are immobilized. And the quality of this data is certainly good enough to directly dismiss models that assume that the major part of ions are immobilized in a Stern layer.

Footnotes

[1] I find it quite frustrating that many descriptions in the literature only refer abstractly to “mineral surfaces” rather than specifically addressing montmorillonite. At the same time it is often clear from the context that statements regarding “mineral surfaces” should be understood as applicable to montmorillonite basal surfaces. I would much appreciate if researchers promoting Stern layers on basal surfaces would provide descriptions for specific systems, e.g. pure Na-Ca-montmorillonite.

[2] Otto Stern is a fascinating character in the history of science, most famous for the Stern-Gerlasch experiment, that helped pave the way for quantum mechanics. I highly recommend this lecture by the late Sandip Pakvasa. An example of its contents:

A note on Stern’s style of working: He always had a cigar in one hand, and he left actual work with hands to others, as he did not trust his own manual dexterity! […] He described the beneficial effects of a large wooden hammer that he kept in his lab and used it to threaten the apparatus if it did not behave! (apparently it worked!)

[3] Note that di-valent counter-ions would be even more sparsely distributed than this.

[4] It may be worth discussing my objections against the work of Low and co-workers in more detail in a future blog post.

[5] This is a general discussion on sorption on mineral surfaces, and is cited from the second edition of the book (2008). There is also a “Thired” edition.

[6] This particular description is general (for “adsorbents”), but since Sposito, as well as a large part of the contemporary bentonite scientific community, claim that “inner-sphere” complexes are present on montmorillonite basal surfaces, we can conclude that they mean that covalent bonding occur on such surfaces.

[7] Sure, the full quotation is “almost exclusively electrostatic bonding”, but what is a reader supposed to do with that? Such vague and sloppy scientific writing annoys me.

[8] Again, the discussion is on general “mineral surfaces”, but from other writings it is clear that this is supposed to apply to the montmorillonite basal surface.

[9] I furthermore don’t believe that “Bolt and co-workers” concluded that specific sorption “needs” to be included, but that this rather is an interpretation made by Dzombak and Hudson themselves. Bolt considered and downplayed the Stern Layer already in the mid 50s, and although he indeed has expressed positive attitudes (seriously, these guys just write too much!), he continued to downplay its significance in e.g. Bolt (1979), writing “In conclusion it appears justified to assume that for homoionic clays saturated with common ions, if hydrated, the Stern layer will be an “empty” Stern layer according to the terminology of Grahame (1947).”

[10] Note also that the perspective in this quotation is that specific sorption models can be complemented with diffuse layer features — i.e. the existence of sorption “sites” is assumed a priori. But Dzombak and Hudson (1995) never really discuss the nature of such “sites” on montmorillonite basal surfaces, but relies on Sposito’s speculations about “inner-” and “outer-surface” complexes.

[11] Note that Stern’s original paper actually is from 1924. I also suspect that Stern would object to being labeled an “electrochemist”.

[12] The terminology here is quite messy, and other authors may use other terms such as “tactoids”, see this post for a further discussion.

[13] This study was thoroughly discussed in a blog post on MD simulations and anion exclusion.

[14] Anyone making this argument should also provide a plausible suggestion for where a significant non-interlayer pore structure is located at these extreme densities.

[15] The pressure in these types of calculations can be related to the interlayer midpoint concentration. But this does not mean that not all counter-ions are involved in the process.

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

Here’s an opinion: The compacted bentonite research field is currently in a terrible state.

After a period away, I’ve recently begun catching up on newly published research in this field. With a fresh perspective, yet still influenced by writing over 30 long-reads over the past years, I can’t help but wonder: what is the problem? Why are a majority of researchers stuck with a view of bentonite1 that essentially makes no sense? And why has this view been the mainstream for decades now?

I get how this might come across: a solitary man ranting on a blog, criticizing an entire research field in less-than-perfect English. I probably smell bad and have some wild ideas about why General Relativity is wrong as well. But what I’m aiming for with this blog is simply a platform to present an alternative to the mainstream, primarily because it annoys me as a science-minded person how absurd this view is.2 I understand that I will likely struggle to convince anyone who is already invested in this view, but I’m trying to put myself in the shoes of e.g. someone entering this field for the first time.

For these reasons, I will try something a little new here: reviewing already published papers. I have touched on this in various forms before, but then usually with a broader topic in mind. Now I intend to critically assess specific publications from the outset. As a first publication to review in this way, I have chosen “Ionic Transport in Nano-Porous Clays with Consideration of Electrostatic Effects” (Tournassat and Steefel, 2015), for the following reasons

  • It is published in “Reviews in Mineralogy & Geochemistry”, which claims that “The content of each volume consists of fully developed text which can be used for self-study, research, or as a text-book for graduate-level courses.” If anyone aims to learn about ion transport in bentonite from this publication, I would certainly recommend to also consider this review.
  • It is a quite comprehensive source for many of the claims of the contemporary mainstream view that I have described in earlier blog posts. I guess it makes sense for a publication in “Reviews in Mineralogy & Geochemistry” to reflect the typical view of a research field.
  • It considers the seeming uphill diffusion effect that I recently commented on. The effect is as misunderstood in this publication as it is in Tertre et al. (2024).
  • It is published as open access. The article is thus accessible to anyone who wants to check the details.

I will use the abbreviation TS15 in following to refer to this publication.

Overview

The article covers 38 journal pages (+ references) and includes quite a lot of topics. At the highest level of headings, the outline look like this

  • Introduction (p. 1 — 2)
  • Classical Fickian Diffusion Theory (p. 2 — 9)
  • Clay mineral surfaces and related properties (p. 9 — 17)
  • Constitutive equations for diffusion in bulk, diffuse layer, and interlayer water (p. 17 — 23)
  • Relative contributions of concentration, activity coefficient and diffusion potential gradients to total flux (p. 24 — 28)
  • From diffusive flux to diffusive transport equations (p. 28 — 33)
  • Applications (p. 33 — 37)
  • Summary and Perspectives (p. 37 — 38)

Given the quite large scope of TS15, I will present this review in parts, with this first part focusing on the introduction and the section titled “Classical Fickian Diffusion Theory”.

“Introduction”

I find it remarkable that the authors use terms like “clays” and “clay minerals” when speaking of properties such as “low permeability”, “high adsorption capacity” and “swelling behavior”, and of applications such as nuclear waste storage. I mean that using such general terms here is too broad, as the article focuses solely on systems with swelling/sealing ability. Such an ability is generally connected to a significant cation exchange capacity. Here, I will refer to such systems as “bentonite”, although I am aware that I use the term quite sloppily. But I think this is better than to refer to the components as general “clay minerals” — I don’t think anyone consider it a good idea to e.g. use talc or kaolinite as buffer materials in nuclear waste repositories. Moreover, most of the examples considered in the article are systems that can be described as bentonite. Given the title of the article I also expect a definition of “nano-porous clays”. It is not given here, and the term is actually not used at all in the entire text! (Except one time at the very end.)

After providing a brief overview of the application of (sealing) clay materials, the introduction takes, in my opinion, a rather drastic turn (it happens without even changing paragraphs!).

Clay transport properties are however not simple to model, as they deviate in many cases from predictions made with models developed previously for “conventional” porous media such as permeable aquifers (e.g., sandstone). […] In this respect, a significant advantage of modern reactive transport models is their ability to handle complex geometries and chemistry, heterogeneities and transient conditions (Steefel et al. 2014). Indeed, numerical calculations have become one of the principal means by which the gaps between current process knowledge and defensible predictions in the environmental sciences can be bridged (Miller et al. 2010).

I think the first sentence is too subjective and general. Given the above discussion, here the term “clay transport properties” can cover a million things, if read at face value. Are all of them difficult to model? Also, something does not have to be more difficult just because it deviates from the “convention”. I would argue that several aspects of bentonite actually make it easier to model than, say, sandstone. Advective processes, for example, can often be neglected in compacted bentonite.

I find the statement regarding the advantage of reactive transport models highly problematic. Not only does it read more like an advertisement for the authors’ own tools than “fully developed text for self-study”, but the authors also seem ignorant of issues like the dangers of overparameterization (a theme that will recur).

“Classical Fickian Diffusion Theory”

As the title of the next section is “Classical Fickian Diffusion Theory,” a reader expects a discussion focused solely on diffusive process, especially when the immediate subtitle reads “Diffusion Basics.” I therefore find it peculiar that this section actually presents the traditional diffusion-sorption model, which describes a combination of diffusion and sorption processes. The model is summarized in eq. 10 in TS15

\begin{equation} \frac{\partial c}{\partial t} = \frac{D_e} {\phi + \rho_dK_D} \nabla ^ 2 c \end{equation}

where \(c\) is the “pore water” concentration of the considered species, \(D_e\) its “effective diffusivity”, \(K_D\) the sorption partition coefficient, \(\rho_d\) dry density, and \(\phi\) porosity.3 For later considerations we also note that TS15 define the denominator on the right hand side as the “rock capacity factor”, \(\alpha = \phi + \rho_dK_D\).

I find it particularly odd that two of the fundamental assumptions of this specific model are essentially left uncommented, namely that sorbed ions are immobilized and that the pores contain bulk water. Instead, the authors appear to question the assumption of Fickian diffusion in the context of clay systems, i.e. that diffusive fluxes are assumed proportional to corresponding aqueous concentration gradients.

This section aims, as far as I can see, to point out shortcomings in the description of diffusion in bentonite, and to motivate further model development. But it should be clear from the outset that using the traditional diffusion-sorption model as the basis for such an endeavor is doomed to fail. The reason for this failure is not due to assuming Fickian diffusion, but due to the other two model assumptions; it has long been demonstrated that exchangeable ions are mobile, and the notion that compacted bentonite contains mainly bulk water is absurd.

After the traditional diffusion-sorption model has been presented, it is evaluated by investigating how it can be fitted to tracer through-diffusion data (this is restatement of original work of Tachi and Yotsjui (2014)). Not surprisingly, it turns out that fitted diffusion coefficients may be unrealistically large. This is of course a direct consequence of the incorrect assumption of immobility in the traditional diffusion-sorption model. TS15 also appear to dismiss the model, saying

This result […] is not physically correct and points out the inconsistency of the classic Fickian diffusion theory for modeling diffusion processes in clay media.

I am bothered, though, that they keep using the phrase “classic Fickian diffusion theory”, which inevitably focuses on the Fickian aspect rather than on the obviously incorrect assumptions of the chosen model. Also, rather than simply concluding that the model is incorrect, TS15 continues4

[T]he large changes of \(\mathrm{Cs}^+\) diffusion parameters as a function of chemical conditions (\(D_{e,\mathrm{Cs}^+}\) decreases when the ionic strength increases […]) highlight the need to couple the chemical reactivity of clay materials to their transport properties in order to build reliable and predictive diffusion models.

There is no rationale for such a conclusion. I don’t even completely understand what “couple the chemical reactivity of clay materials to their transport properties” mean. Isn’t that what the traditional diffusion-sorption model attempts? What unrealistic \(D_e\) values actually highlights is simply that one should not use a model that assumes immobilization of “sorbed” ions.

To make things worse, TS15 describe the seeming uphill diffusion test and comment

However, the experimental observations were completely different: \(^{22}\mathrm{Na}^+\) accumulated in the high NaCl concentration reservoir as it was depleted in the low NaCl concentration reservoir, evidencing non-Fickian diffusion processes.

This is plain wrong. As explained in detail in an earlier post, the diffusion process in the “uphill” test is certainly Fickian. What the test demonstrates is, again, that “sorbed” ions are not immobile.

TS15 also comment on the results of fitting the model to anion tracer through-diffusion data. Here, as is well known, the fitted “rock capacity factor” \(\alpha = \phi + \rho_dK_D\) becomes significantly lower than the porosity \(\phi\). From the perspective of the traditional diffusion-sorption model, this is completely infeasible, as it implies a negative \(K_D\). But rather than simply dismissing the model, TS15 state

The lower \(\alpha\) values for anions than for water indicate that anions do not have access to all of the porosity.

Also this is incorrect. The porosity5 is an input parameter rather than a fitting parameter in the traditional diffusion-sorption model. When claiming that a small value of \(\alpha\) indicates a decreased porosity, TS15 reinterpret the parameter, on the fly, in terms of a completely different model: the effective porosity model. This model has not been mentioned at all earlier in the article.6

As has been discussed earlier on the blog, the effective porosity model can be fitted to anion tracer through-diffusion data, but now we need to keep track of two different models in the evaluation (something that TS15 do not). Moreover, these two models (the traditional diffusion-sorption and the effective porosity models) are incompatible. But TS15 continue by saying

This result is a first direct evidence of the limitation of the classic Fickian diffusion theory when applied to clay porous media: it is not possible to model the diffusion of water and anions with the same single porosity model. The observation of a lower \(\alpha\) value for anions than for water led to the development of the important concept of anion accessible porosity […]

This is a terrible passage. To begin with, the “Fickian” aspect is also here implied as the problem. But the reason for why the traditional diffusion-sorption model cannot be fitted to anion tracer through-diffusion data is of course because this model assumes the entire pore space to be filled with bulk water. Further, it’s hardly comprehensible what the authors mean by “it is not possible to model the diffusion of water and anions with the same single porosity model”. I think they simply mean that for water you must choose \(\alpha = \phi\), while for anion through-diffusion you instead must “choose” \(\alpha < \phi\). But the result \(\alpha < \phi\) should only lead to the conclusion that the traditional diffusion-sorption model cannot in any reasonable sense be fitted. A favorable reading of this passage is to assume that the authors actually mean that the effective porosity model can only be fitted to anion and water tracer through-diffusion data by using different values of the (effective) porosity, and that any “rock capacity factor” should not appear in this discussion.

Finally, the last sentence gives me headache. Rather than being an “important concept”, I mean that the idea of an “anion accessible porosity” has caused tremendous damage to the development of the bentonite research field for several decades now. We have earlier discussed on the blog that the whole idea of “anion accessible porosity” is based on misunderstandings. We have also demonstrated that the effective porosity model is not valid, even though it can be fitted to anion tracer through-diffusion data. A simple way to see this is to consider closed-cell diffusion data rather than through-diffusion data. Closed-cell tests are simpler than through-diffusion tests, as they don’t involve interfaces between clay and external solutions. We can e.g. take a look at the vast amount of diffusion coefficients for chloride in montmorillonite, presented in Kozaki et al. (1998).

There are in total 55(!) values, corresponding to 55(!) separate tests. These have been systematically varied with respect to density and temperature, but all of them were performed on montmorillonite equilibrated with distilled water. From the perspective of the effective porosity model, the effective porosity in such a system should be minute, perhaps even strictly zero; effective porosities evaluated from chloride through-diffusion tests are well below 1% even at a background concentration as large as 10 mM. Thus, if the idea of “anion accessible porosity” was reasonable, we’d expect extremely low values of the chloride diffusion coefficient in the above plot.7 We’d perhaps also expect a threshold behavior, where chloride diffusivity basically vanishes above a certain density. But this is not at all the behavior: chloride is seen to diffuse just fine in all 55(!) tests, with temperature- and density dependencies that seems reasonable for a homogeneous system. Moreover, chloride behaves very similarly to e.g. sodium, as seen here

Here the sodium data is from Kozaki et al. (1998),8 and it has also been measured in montmorillonite equilibrated with distilled water.

The effective porosity model and the notion of “anion accessible porosity” can consequently be dismissed directly, by comparing with simpler tests than what is done in TS15. The reason that the effective porosity model can be fitted to anion through-diffusion data must be attributed to a misinterpretation of such tests, as they involve also interfaces to external solutions. At least to me it is completely clear that what many researchers interpret as an effective porosity is actually effects of interface equilibrium.

If TS15 were serious about evaluating bentonite diffusion processes in this section I think they should have done the following:

  • Discuss the assumptions of ion immobility of sorbed ions and bulk pore water when presenting the traditional diffusion-sorption model. Moreover, they should not call this “Classical Fickian Diffusion Theory”.
  • Also present and discuss the effective porosity model, as they obviously use it in their evaluations. They actually even seem to promote it! And it is as “Fickian” as the traditional diffusion-sorption model.
  • Evaluate the models using closed-cell data to avoid misinterpretations arising from complications at bentonite/external solution interfaces.
  • Conclude that the traditional diffusion-sorption model is not valid for bentonite, and that this is because of the assumptions of immobility of sorbed ions and bulk pore water.
  • Conclude that the effective porosity model is not valid for bentonite, and that the notion of “anion accessible porosity” is flawed.

Instead, we get a quite confused and incomplete description, mixed with entirely inaccurate statements. In the end, it is difficult to understand what the takeaway message of this section really is. A reader is left with an impression that there is some problem with the “Fickian” aspect of diffusion, but nothing is spelled out. We have also been hinted that “anion accessible porosity” is important, without really having been introduced to the concept/model.

The section ends with the following passage

The limitations of the classic Fickian diffusion theory must find their origin in the fundamental properties of the clay minerals. In the next section, these fundamental properties are linked qualitatively to some of the observations described above.

If “classic Fickian diffusion theory” here is interpreted as “the traditional diffusion-sorption model” (which is literally what has been presented), the first sentence is both incorrect and trivial at the same time. The traditional diffusion-sorption model does not have “limitations” — it is fundamentally incorrect as a model for bentonite. The reason for this is that exchangeable ions are not immobile and that bentonite does not contain significant amounts of bulk water. Both of these reasons can be linked to “fundamental properties” of some specific clay minerals.

But it is clear that TS15 also have vaguely promoted the concept of “anion accessible porosity” and the effective porosity model. Are these not included in “the classic Fickian diffusion theory”? If not, why then is a model that assumes sorption and immobilization?

How can it not be immediately obvious to everyone that the diffusion process is much simpler than the contemporary descriptions?

As we have brought up the data from Kozaki et al. (1998), I would like to end this blog post by further considering actual profiles of chloride and sodium diffusing in montmorillonite.

This figure shows the corresponding normalized concentration profiles after 23.7 hours in closed-cell tests performed at \(50\;^\circ\mathrm{C}\) in Na-montmorillonite at dry density \(1.8 \;\mathrm{g/cm^3}\) that has been equilibrated with distilled water. In the case of sodium, both the profile evaluated from Fick’s second law (orange line) and measured values (circles) are plotted. In the case of chloride, no measured values are available, but the value of the diffusion coefficient is the result of fitting Fick’s second law (green line) to such data.

From the perspective of the traditional diffusion-sorption model, the sodium profile is supposed to represent the combined result of ions diffusing in bulk water, at a rate many orders of magnitude larger than in pure water, while being strongly retarded due to sorption onto “the solid” (where the ions are immobile). This is clearly nonsense, and something that I think TS15 actually tries to communicate.

From the perspective of the effective porosity model, on the other hand, the chloride profile is supposed to be the result of the ion diffusing in an essentially infinitesimal fraction of the pore volume, which magically is perfectly interconnected in all samples on which such tests are conducted. This is of course just as nonsensical as the above interpretation of the sodium profile, but in this case TS15 appear to promote the model (the “important concept of anion accessible porosity”).

Note that these two simple ions, at the end of the day, diffuse very similarly (please stop reading for a moment and contemplate the above plot). If sodium and chloride actually migrate in completely different domains and are subject to completely different physico-chemical processes, this “coincident” would be more than a little weird. Especially given that the two ions show similar diffusive behavior across a wide range of densities. To me, this simple observation makes is evident that ion diffusion in bentonite at the basic level is much simpler than what is suggested by the contemporary mainstream view. I mean that it is completely obvious that all ions in bentonite diffuse in the same type of quite homogeneous domain. And since it cannot be argued that the pore volume is dominated by anything other than interlayers at 1.8 g/cm³, this homogeneous domain is the interlayer domain at any relevant density. The evidence has been available for at least 25 years (in fact much longer than that). How can this be difficult to grasp?

Update (250213): Part II of this review is found here.

Footnotes

[1] By “bentonite” I here mean any type of smectite-rich system with a significant cation exchange capacity.

[2] The irony is that the “alternative” in a broader perspective is more mainstream than the “mainstream” view. I basically propose to obey the laws of thermodynamics.

[3] I have simplified the notation here somewhat compared with how it is written in TS15. As many others, TS15 call this equation “Fick’s second law” (via their eq. 4), which is not correct. Fick’s laws refer strictly to pure diffusion processes. However, the equation has the same form as Fick’s second law, if \(D_e/(\phi + \rho_d K_D)\) is treated as a single constant (often referred to as the apparent diffusivity).

[4] This behavior is of course not unique for cesium; I don’t know why TS15 focus so hard on that ion here.

[5] “Porosity” is a volume ratio. I’m not a fan of that the word has also begun to mean “pore space” in the bentonite scientific literature.

[6] In fact, \(\alpha\) has earlier in the article been unambiguously related to sorption:

If the species \(i\) is also adsorbed on or incorporated into the solid phase, then it is possible to define a rock capacity factor \(\alpha_i\) that relates the concentration in the porous media to the concentration in solution

[7] That the diffusivity is much too large for an effective porosity interpretation to make sense can also be seen from invoking Archie’s law, which is quite popular in bentonite scientific papers.

\begin{equation} D_e = \epsilon_\mathrm{eff}^n D_0\end{equation}

Here \(D_0\) is the diffusivity in pure bulk water, which is about \(2\cdot 10^{-9} \;\mathrm{m^2/s}\) for chloride. Using the popular choice \(n \approx 2\) and choosing e.g. \(\epsilon_\mathrm{eff} = 0.001\) (most probably an overestimation when using distilled water), we get

\begin{equation} D_0 = (5.1\cdot 10^{-11} \;\mathrm{m^2/s})/0.001 = 5.1\cdot 10^{-8} \;\mathrm{m^2/s}\end{equation}

This is more than twenty times the actual value for \(D_0\). (\(D_e = 5.1\cdot 10^{-14} \;\mathrm{m^2/s}\) is evaluated from Kozaki’s data at \(1.4 \;\mathrm{g/cm^3}\) and \(25\;^\circ\mathrm{C}\))

[8] Note! This publication is different from the chloride study.

“Uphill” diffusion in bentonite — a comment on Tertre et al. (2024)

The vast majority of published tests on ion diffusion in bentonite deal with chemically uniform systems, and in a previous blog post I addressed the lack of studies where actual chemical gradients are maintained. But recently such a study was published: “Influence of salinity gradients on the diffusion of water and ionic species in dual porosity clay samples” (Tertre et al., 2024). Although I’m pleased to see these types of experiments being reported, I must admit that the paper as a whole leaves me quite disappointed.

The paper follows a structure recognizable from several others that we have considered previously on the blog: It starts off with an introduction section containing several incorrect or unfounded statements1 regarding bentonite.2 It then presents some experimental results that makes it evident that no real progress has been made for a long time regarding e.g. experimental design.3 The major part of the paper is devoted to a “results and discussion” section with several incorrect statements and inferences, speculation, and irrelevant modeling.

Here I would like to focus on how the study “Seeming steady-state uphill diffusion of \(^{22}\mathrm{Na}^+\) in compacted montmorillonite” (Glaus et al., 2013) is referenced:

[I]nfluence of a background electrolyte concentration gradient on the diffusion of anionic and cationic species at trace concentrations has […] been rarely investigated. Notable exceptions are the DR-A in situ diffusion experiment conducted at the Mont-Terri laboratory (Soler et al., 2019), and an “uphill” diffusion experiment of a \(^{22}\mathrm{Na}^+\) tracer in a compacted sodium montmorillonite (Glaus et al., 2013). These two studies demonstrated the marked influence of background electrolyte concentration gradient on tracer diffusion, and thus the necessity to understand the couplings between diffusion of several charged species present at contrasting concentrations and experiencing different concentration gradients. The experiment from Glaus et al. (2013) also demonstrated the importance of considering diffusion processes occurring in the porosity next to the charged surface of clay minerals (i.e., the porosity associated to the EDL of particles).

This quotation contains two statements relating to Glaus et al. (2013), both of which I think are problematic4

  • It basically claims that the “uphill” phenomenon is due to diffusive couplings between several types of ions. Of course, ion diffusion always involves couplings between different types of ions, due to the requirement of electroneutrality. But it is clear that Tertre et al. (2024) mean that the “uphill” effect is caused by additional couplings that are not present in chemically homogeneous systems.
  • It says that Glaus et al. (2013) demonstrates the importance to consider diffuse layers. I agree with this, but it is written in a way that implies that there also are other relevant “porosities”, and that there are other types of tests where ion diffusion in bentonite is not significantly influenced by the presence of diffuse layers.

As one of the authors of the “uphill” study, I would here like to argue for why I think the above statements are problematic and give some background context.

The “uphill” diffusion experiment

The “uphill” study actually originated from a prediction presented by me in a conference poster session. This poster discussed the role of the quantity \(D_e\), using the exact same theory that we had previously used to explain the diffusive behavior of tracer ions in compacted bentonite as an effect of Donnan equilibrium in a homogeneous system. In particular, it pointed out that \(D_e\) — although universally referred to as the (effective) “diffusion coefficient” — is not a diffusion coefficient in the context of compacted bentonite. I have continued this discussion in later papers, and in several posts on this blog.

In the poster, we suggested the “uphill” experiment as a demonstration of the shortcoming of \(D_e\). If the two reservoirs in a through-diffusion test are maintained at different background concentrations, the theory predicts a non-zero tracer flux for a vanishing external tracer concentration difference, i.e. an “infinite” value of \(D_e\). The suggestion caught the interest of an experimental group, and after a successful collaboration we could present the results of an actual “uphill” experiment. Without making too much of an exaggeration, I would say that the results of this experiment were basically exactly as predicted.

Given this background, it should be clear that the tests in Glaus et al. (2013) follow exactly the same rules as tests in chemically homogeneous systems, rather than demonstrating “the necessity to understand the couplings between diffusion of several charged species present at contrasting concentrations”. Although it is quite clearly stated already in the abstract in Glaus et al. (2013), there is apparently still a need to communicate this explanation. Let me therefore try that here.

The “uphill” diffusion phenomenon explained

Consider an ordinary aqueous solution containing radioactive \(^{22}\mathrm{Na}\) and stable \(^{23}\mathrm{Na}\). The fraction of \(^{22}\mathrm{Na}\) ions can be written \(c_\mathrm{ext}/C_\mathrm{bkg}\), where \(c_\mathrm{ext}\) is the \(^{22}\mathrm{Na}\) concentration, and \(C_\mathrm{bkg}\) is the total sodium concentration (the “tracer” and “background” concentrations, respectively).

Since \(^{23}\mathrm{Na}\) and \(^{22}\mathrm{Na}\) are basically chemically indistinguishable, the same \(^{22}\mathrm{Na}\)-fraction will be maintained in any system with which this solution is in equilibrium. In particular, if the solution is in equilibrium with a montmorillonite interlayer solution, we can write

\begin{equation*} \frac{c_\mathrm{int}}{C_\mathrm{int}} = \frac{c_\mathrm{ext}}{C_\mathrm{bkg}} \tag{1} \end{equation*}

where \(c_\mathrm{int}\) and \(C_\mathrm{int}\) are the \(^{22}\mathrm{Na}\) and total interlayer concentrations, respectively. The total interlayer cation concentration (\(C_\mathrm{int}\)) can be handled in different ways, but it is important to note that this is a substantial number under all conditions, relating to the cation exchange capacity.5 Rearranging eq. 1 gives

\begin{equation*} c_\mathrm{int} = \frac{C_\mathrm{int}}{C_\mathrm{bkg}}\cdot c_\mathrm{ext} \end{equation*}

Since the interlayer cation concentration is always larger than the corresponding background concentration, the above equation tells us that the corresponding interlayer tracer concentration becomes enhanced, by the factor \(C_\mathrm{int}/C_\mathrm{bkg}\).

Conventional through-diffusion

This enhancement mechanism causes the diffusional behavior of \(^{22}\mathrm{Na}\) in conventional through-diffusion experiments in bentonite. In such experiments, the tracer concentration in the target reservoir is usually kept near zero, and the actual steady-state concentration gradient in the interlayers is

\begin{equation*} \frac{\partial c_\mathrm{int}}{\partial x} = \frac{0- C_\mathrm{int}/C_\mathrm{bkg}\cdot c_\mathrm{ext}^{(1)}} {L} = -\frac{C_\mathrm{int}}{C_\mathrm{bkg}}\cdot \frac{ c_\mathrm{ext}^{(1)} }{ L } \end{equation*}

where we have indexed the tracer concentration in the source reservoir with “\((1)\)”, labeled the sample length \(L\), and assumed that ions diffuse in the \(x\)-direction. The corresponding flux is thus (Fick’s law)

\begin{equation*} j_\mathrm{steady-state} = – \phi D_c\frac{\partial c_\mathrm{int}}{\partial x} = \phi D_c\cdot \frac{C_\mathrm{int}}{C_\mathrm{bkg}}\cdot \frac{c_\mathrm{ext}^{(1)} } {L} \tag{2} \end{equation*}

where \(D_c\) denotes the (macroscopic) diffusivity in the interlayers, and \(\phi\) is porosity. Keeping \(c_\mathrm{ext}^{(1)}\) constant, eq. 2 shows that the \(^{22}\mathrm{Na}\) steady-state flux increases indefinitely as the background concentration is made small, in full agreement with experimental observation.6

The picture below illustrates the concentration conditions in an conventional through-diffusion test.

Here we have chosen \(C_\mathrm{int}=\) 4.0 M, the background concentration in the two reservoirs (blue) is put equal to 0.1 M, and the tracer concentration (orange) is put to 0.1 mM in reservoir 1 (and zero i reservoir 2). The corresponding internal tracer gradient is plotted in the right side diagram, and the resulting diffusive flux is indicated by the arrow.

“Uphill” diffusion

To explain the “uphill” effect the only modifications needed in the above derivation is to allow for different background concentrations in the external reservoirs, and to recognize that the tracer concentration in the clay on the “target” side (indexed “\((2)\)”) no longer is zero. Considering the tracer concentration enhancement at both interfaces, the steady-state interlayer concentration gradient then reads

\begin{equation*} \frac{\partial c_\mathrm{int}}{\partial x} = \frac{ C_\mathrm{int}/C_\mathrm{bkg}^{(2)}\cdot c_\mathrm{ext}^{(2)} -C_\mathrm{int}/C_\mathrm{bkg}^{(1)}\cdot c_\mathrm{ext}^{(1)}} {L} \end{equation*}

To be more concrete, let’s assume that \(C_\mathrm{bkg}^{(2)} = 5\cdot C_\mathrm{bkg}^{(1)}\), which is the same ratio as in Glaus et al. (2013). We then have

\begin{equation*} \frac{\partial c_\mathrm{int}}{\partial x} = \frac{C_\mathrm{int}}{C_\mathrm{bkg}^{(1)}} \cdot \frac{ c_\mathrm{ext}^{(2)}/5 – c_\mathrm{ext}^{(1)}} {L} \end{equation*}

giving the corresponding steady-state flux

\begin{equation*} j_\mathrm{steady-state} = \phi D_c\cdot \frac{C_\mathrm{int}}{C_\mathrm{bkg}^{(1)}} \cdot \frac{ c_\mathrm{ext}^{(1)} – c_\mathrm{ext}^{(2)}/5} {L} \end{equation*}

Note that we recover the conventional through-diffusion result (eq. 2) from this expression, if we put \(c_\mathrm{ext}^{(2)}= 0\). But if we e.g. set the tracer concentration equal in both reservoirs, we still have a flux from side \((1)\) to side \((2)\), of size \(j = 4/5 \cdot \phi D_c\cdot C_\mathrm{int}/C_\mathrm{bkg}^{(1)}\cdot c_\mathrm{ext}^{(1)}\). And even if we make \(c_\mathrm{ext}^{(2)}\) larger than \(c_\mathrm{ext}^{(1)}\) — as long as \(c_\mathrm{ext}^{(1)}< c_\mathrm{ext}^{(2)} < 5\cdot c_\mathrm{ext}^{(1)}\) — we still have a diffusive flux from side \((1)\) to side \((2)\), i.e seeming “uphill” diffusion.

Below is illustrated the concentration conditions in an “uphill” configuration.

In contrast to the above illustration for conventional through-diffusion, the background concentration in reservoir 2 is here raised to 0.5 M and the tracer concentration in reservoir 2 is put equal to 0.2 mM. We see that, although tracers are transported to the reservoir with higher concentration, the process is still ordinary Fickian diffusion, as the internal tracer gradient has the same direction as in the conventional case.

We can now conclude what was stated above: The “uphill” diffusion effect is caused by exactly the same mechanism that cause the behavior of cation diffusion in conventional bentonite through-diffusion tests. This mechanism is ion equilibrium between clay and external solutions at the two interfaces. In this particular case, with sodium tracers diffusing in a sodium background, we don’t need to invoke the full ion equilibrium framework in order to quantify the fluxes, but can rely on the very robust result that any two systems in equilibrium have the same tracer fraction (eq. 1).

Reexamining the Tertre et al. (2024) statements

With the explanation for the “uphill” effect established, let’s re-examine the problematic statements in Tertre et al. (2024) identified above

  • Glaus et al. (2013) cannot be used to support a claim of “marked influence” of additional diffusional couplings. The opposite is true: Glaus et al. (2013) found no significant influence from mechanisms beyond those in chemically homogeneous conditions.
  • The “uphill” effect was predicted from taking the idea seriously that diffusion in compacted bentonite is fully governed by interlayer properties. Singling out Glaus et al. (2013) as the study that demonstrates the importance of diffuse layers7 therefore gives the wrong impression. Rather, what Glaus et al. (2013) demonstrates, in conjunction with corresponding conventional through-diffusion results, is that compacted bentonite contains insignificant amounts of bulk water (what Tertre et al. (2024) call “interparticle water”).

A way forward (if anybody cares)

After the uphill study was published I was for a while under the illusion that things would begin to change within the compacted bentonite research field. Not only did the study, to my mind, deal a fatal blow to any bentonite model that relies on the presence of a bulk water phase in the clay. It also opened up a whole new area of interesting studies to conduct. Now, some 11 years later, I can disappointingly conclude that not a single additional study has been presented that explore the ideas here discussed.8 And, regarding bentonite models, bulk water is apparently alive and kicking, as has been discussed ad nauseum on this blog.

Experimentally, there are a number of interesting questions looking for answers. In particular, we actually do expect additional mechanisms to play a role in chemically inhomogeneous systems, e.g. osmosis, and other effects due to presence of salt concentration gradients and electrostatic potential differences. It may be argued for why such effects are not significant in Glaus et al. (2013), but it is of course both of fundamental and practical interest to understand under which conditions they are. The original “uphill” study is e.g. performed at quite extreme density (\(1900\;\mathrm{m^3/kg}\)). How would the result differ at \(1600\;\mathrm{m^3/kg}\) or \(1300\;\mathrm{m^3/kg}\)? Also, how would the results change with other choices of the reservoir concentrations, and how would the results differ if one of the cations is not at trace level (e.g. a system with comparable amounts of sodium and potassium)?

Even under the conditions of the original study, there are several predictions left to verify. If e.g. \(c^{(1)}_\mathrm{ext} = c^{(2)}_\mathrm{ext}/5\), the theory predicts zero flux (implying \(D_e = 0\)). The theory also implies that when performing “conventional” through-diffusion, the actual level of the background concentration in the target reservoir is irrelevant, as long as the tracer concentration is kept at zero.

In fact, one can imagine making a whole cycle of through-diffusion tests to explore the ideas here discussed, as illustrated in this animation

The resulting steady-state flux for various external conditions is indicated by the arrow. Here, the full ion equilibrium framework was used to calculate the internal concentrations (giving an internal gradient also in \(C_\mathrm{int}\)). Background concentrations and total interlayer concentration is chosen to be comparable with Glaus et al. (2013), while the choice for tracer concentration is arbitrary.9

With the risk of sounding hubristic, the number of experiments suggested in the above animation could have given enough material for several Ph.D. theses. But here we are, in the year 2024, without even a replication of the “uphill” effect. Instead, a basically entire research field has been stuck for decades with the ludicrous idea that models of compacted bentonite should be based on a bulk water description. I find this both hilarious and horrific.

Footnotes

[1] For example (follow links to discussions on these issues):

  • It states the traditional diffusion-sorption model as being relevant in these systems. It is not.
  • It somehow manages to combine the traditional diffusion-sorption model with the effective porosity model for anion tracer diffusion, although these two models are incompatible.
  • Related to using the traditional diffusion-sorption model, it assumes \(D_e\) to be a real diffusion coefficient, which it is not. I find this particularly remarkable in a paper that deals with the presence of “saline gradients”. A motivation behind e.g. the “uphill” test is to point out the shortcomings of \(D_e\), as discussed in the rest of this blog post.
  • It claims that “anionic and cationic tracers do not experience the same overall accessible porosity”, which is unjustified.
  • It claims that “diffusion rates” of anions are decreased and “diffusion rates” of cations are increased, compared to “neutral species”, due to different interactions with diffuse layers. But this is not true generally.
  • It implicitly simply assumes a “stack”-view of these clay systems. But stacks don’t make much sense.

[2] I use the word “bentonite” here quite loosely. Tertre et al. (2024) use wordings such as “clayey samples”, “argillaceous rocks” and “clayey formation”, but it is clear that the presented material is supposed to apply to actual bentonite.

[3] I’m specifically thinking about that cation tracer through-diffusion tests at low background concentration is not a good idea, and that it is completely clear from the results presented in Tertre et al. (2024) that some of these are mainly controlled by diffusion in the confining filters. Estimating a “rock capacity factor” larger than 750 for sodium tracers in a sodium-clay (at 20 mM background concentration) should have set off all alarm bells.

[4] Regarding Soler et al. (2019), I think that whole study is problematic, which I might argue for in a separate blog post.

[5] Glaus et al. (2013) invoke the “exchange site” activity \([\mathrm{NaX}]\) to discuss this quantity. I personally prefer relating it to the quantity \(c_\mathrm{IL}\) that is defined within the homogeneous mixture model.

[6] This agreement has been shown to be quantitative, see e.g. Glaus et al. (2007), Birgersson and Karnland (2009) and Birgersson (2017). Note that this result is quite independent on how many “porosities” you choose to include in a model; it’s merely a consequence of treating the dominating pores (interlayers) adequately. Further, note that measuring the diverging fluxes in the limit of low background concentration becomes increasingly difficult, as the confining filters becomes rate limiting.

[7] In the present context, I presume the terms “diffuse layer” and “interlayer” to be more or less equivalent. Other authors instead make an unjustified distinction, that I have addressed here.

[8] There are a few examples of published studies where effects of the kind discussed here are present, but where the authors don’t seem to be aware of it.

[9] Tracer concentrations in Glaus et al. (2013) is much smaller, but this value does not affect any behavior, as long as it is small in comparison with total concentration.

Assessment of chloride equilibrium concentrations: Van Loon et al. (2007)

In the ongoing assessment of chloride equilibrium concentrations in bentonite, we here take a closer look at the study by Van Loon et al. (2007), in the following referred to as Vl07. We thus assess the 54 points indicated here (click on figures to enlarge)

Vl07 is centered around a set of through-diffusion tests in “KWK” bentonite samples of nominal dry densities 1.3 g/cm3, 1.6 g/cm3, and 1.9 g/cm3. For each density, chloride tracer diffusion tests were conducted with NaCl background concentrations 0.01 M, 0.05 M, 0.1 M, 0.4 M, and 1.0 M. In total, 15 samples were tested. The samples are cylindrical with diameter 2.54 cm and height 1 cm, giving an approximate volume of 5 cm3. We refer to a specific test or sample using the nomenclature “nominal density/external concentration”, e.g. the sample of density 1.6 g/cm3 contacted with 0.1 M is labeled “1.6/0.1”.

After maintaining steady-state, the external solutions were replaced with tracer-free solutions (with the same background concentration), and tracers in the samples were allowed to diffuse out. In this way, the total tracer amount in the samples at steady-state was estimated. For tests with background concentrations 0.01 M, 0.1 M, and 1.0 M, the outflux was monitored in some detail, giving more information on the diffusion process. After finalizing the tests, the samples were sectioned and analyzed for stable (non-tracer) chloride. In summary, the tests were performed in the following sequence

  1. Saturation stage
  2. Through-diffusion stage
    • Transient phase
    • Steady-state phase
  3. Out-diffusion stage
  4. Sectioning

Uncertainty of samples

The used bentonite material is referred to as “Volclay KWK”. Similar to “MX-80”, “KWK” is just a brand name (it seems to be used mainly in wine and juice production). In contrast to “MX-80”, “KWK” has been used in only a few research studies related to radioactive waste storage. Of the studies I’m aware, only Vejsada et al. (2006) provide some information relevant here.1

Vl07 state that “KWK” is similar to “MX-80” and present a table with chemical composition and exchangeable cation population of the bulk material. As the chemical composition in this table is identical to what is found in various “technical data sheets”, we conclude that it does not refer to independent measurements on the actual material used (but no references are provided). I have not been able to track down an exact origin of the stated exchangeable cation population, but the article gives no indication that these are original measurements (and gives no reference). I have found a specification of “Volclay bentonite” in this report from 1978(!) that states similar numbers (this document also confirms that “MX-80” and “KWK” are supposed to be the same type of material, the main difference being grain size distribution). We assume that exchangeable cations have not been determined explicitly for the material used in Vl07.

In a second table, Vl07 present a mineral composition of “KWK”, which I assume has been determined as part of the study. But this is not fully clear, as the only comment in the text is that the composition was “determined by XRD-analysis”. The impression I get from the short material description in Vl07 is that they rely on that the material is basically the same as “MX-80” (whatever that is).

Montmorillonite content

Vl07 state a smectite content of about 70%. Vejsada et al. (2006), on the other hand, state a smectite content of 90%, which is also stated in the 1978 specification of “Volclay bentonite”. Note that 70% is lower and 90% is higher than any reported montmorillonite content in “MX-80”. Regardless whether or not Vl07 themselves determined the mineral content, I’d say that the lack of information here must be considered when estimating an uncertainty on the amount of montmorillonite (“smectite”) in the used material. If we also consider the claim that “KWK” is similar to “MX-80”, which has a documented montmorillonite content in the range 75 — 85%, an uncertainty range for “KWK” of 70 — 90% is perhaps “reasonable”.

Cation population

Vl07 state that the amount exchangeable sodium is in the range 0.60 — 0.65 eq/kg, calcium is in the range 0.1 — 0.3 eq/kg, and magnesium is in the range 0.05 — 0.2 eq/kg. They also state a cation exchange capacity in the range 0.76 — 1.2 eq/kg, which seems to have been obtained from just summing the lower and upper limits, respectively, for each individual cation. If the material is supposed to be similar to “MX-80”, however, it should have a cation exchange capacity in the lower regions of this range. Also, Vejsada et al. (2006) state a cation exchange capacity of 0.81 eq/kg. We therefore assume a cation exchange capacity in the range 0.76 — 0.81, with at least 20% exchangeable divalent ions.

Soluble accessory minerals

According to Vl07, “KWK” contains substantial amounts of accessory carbonate minerals (mainly calcite), and Vejsada et al. (2006) also state that the material contains calcite. The large spread in calcium and magnesium content reported for exchangeable cations can furthermore be interpreted as an artifact due to dissolving calcium- and magnesium minerals during the measurement of exchangeable cations (but we have no information on this measurement). Vl07 and Vejsada et al. (2006) do not state any presence of gypsum, which otherwise is well documented in “MX-80”. I do not take this as evidence for “KWK” being gypsum free, but rather as an indication of the uncertainty of the composition (the 1978 specification mentions gypsum).

Sample density

Vl07 don’t report measured sample densities (the samples are ultimately sectioned into small pieces), but estimate density from the water uptake in the saturation stage. The reported average porosity intervals are 0.504 — 0.544 for the 1.3 g/cm3 samples, 0.380 — 0.426 for the 1.6 g/cm3 samples, and 0.281 — 0.321 for the 1.9 g/cm3 samples. Combining these values with the estimated interval for montmorillonite content, we can derive an interval for the effective montmorillonite dry density by combining extreme values. The result is (assuming grain density 2.8 g/cm3, adopted in Vl07).

Sample density
(g/cm3)
EMDD interval
(g/cm3)
1.3 1.04 — 1.32
1.61.36 — 1.67
1.9 1.67 — 1.95

These intervals must not be taken as quantitative estimates, but as giving an idea of the uncertainty.

Uncertainty of external solutions

Samples were water saturated by first contacting them from one side with the appropriate background solution (NaCl). From the picture in the article, we assume that this solution volume is 200 ml. After about one month, the samples were contacted with a second NaCl solution of the same concentration, and the saturation stage was continued for another month. The volume of this second solution is harder to guess: the figure shows a smaller container, while the text in the figure says “200 ml”. The figure shows the set-up during the through-diffusion stage, and it may be that the containers used in the saturation stage not at all correspond to this picture. Anyway, to make some sort of analysis we will assume the two cases that samples were contacted with solutions of either volume 200 ml, or 400 ml (200 ml + 200 ml) during saturation.

The through-diffusion tests were started by replacing the two saturating solutions: on the left side (the source) was placed a new 200 ml NaCl solution, this time spiked with an appropriate amount of 36Cl tracers, and on the right side (the target) was placed a fresh, tracer free NaCl solution of volume 20 ml. The through-diffusion tests appear to have been conducted for about 55 days. During this time, the target solution was frequently replaced in order to keep it at a low tracer concentration. The source solution was not replaced during the through-diffusion test.

As (initially) pure NaCl solutions are contacted with bentonite that contains significant amounts of calcium and magnesium, ion exchange processes are inevitably initiated. Thus, in similarity with some of the earlier assessed studies, we don’t have full information on the cation population during the diffusion stages. As before, we can simulate the process to get an idea of this ion population. In the simulation we assume a bentonite containing only sodium and calcium, with an initial equivalent fraction of calcium of 0.25 (i.e. sodium fraction 0.75). We assume sample volume 5 cm3, cation exchange capacity 0.785 eq/kg, and Ca/Na selectivity coefficient 5.

Below is shown the result of equilibrating an external solution of either 200 or 400 ml with a sample of density 1.6 cm3/g, and the corresponding result for density 1.3 cm3/g and external volume 400 ml. As a final case is also displayed the result of first equilibrating the sample with a 400 ml solution, and then replacing it with a fresh 200 ml solution (as is the procedure when the through-diffusion test is started).

Although the results show some spread, these simulations make it relatively clear that the ion population in tests with the lowest background concentration (0.01 M) probably has not changed much from the initial state. In tests with the highest background concentration (1.0 M), on the other hand, significant exchange is expected, and the material is consequently transformed to a more pure sodium bentonite. In fact, the simulations suggest that the mono/divalent cation ratio is significantly different in all tests with different background concentrations.

Note that the simulations do not consider possible dissolution of accessory minerals and therefore may underestimate the amount divalent ions still left in the samples. We saw, for example, that the material used in Muurinen et al. (2004) still contained some calcium and magnesium although efforts were made to convert it to pure sodium form. Note also that the present analysis implies that the mono/divalent cation ratio probably varies somewhat in each individual sample during the course of the diffusion tests.

Direct measurement of clay concentrations

Chloride clay concentration profiles were measured in all samples after finishing the diffusion tests, by dispersing sample sections in deionized water. Unfortunately, Vl07 only present this chloride inventory in terms of “effective” or “Cl-accessible porosity”, a concept often encountered in evaluation of diffusivity. However, “effective porosity” is not what is measured, but is rather an interpretation of the evaluated amount of chloride in terms of a certain pore volume fraction. Vl07 explicitly define effective porosity as \(V_\mathrm{Cl}/V_\mathrm{1g}\), where \(V_\mathrm{1g}\) is the “volume of a unit mass of wet bentonite”, and \(V_\mathrm{Cl}\) is the “volume of the Cl-accessible pores of a unit mass of bentonite”. While \(V_\mathrm{1g}\) is accessible experimentally, \(V_\mathrm{Cl}\) is not. Vl07 further “derive” a formula for the effective porosity (called \(\epsilon_\mathrm{eff}\) hereafter)

\begin{equation} \epsilon_\mathrm{eff} = \frac{n’_\mathrm{Cl}\cdot \rho_\mathrm{Rf}}{C_\mathrm{bkg}} \tag{1} \end{equation}

where \(n’_\mathrm{Cl}\) is the amount chloride per mass bentonite, \(\rho_\mathrm{Rf}\) is the density of the “wet” bentonite, and \(C_\mathrm{bkg}\) is the background NaCl concentration.2 In contrast to \(V_\mathrm{Cl},\) these three quantities are all accessible experimentally, and the concentration \(n’_\mathrm{Cl}\) is what has actually been measured. For a result independent of how chloride is assumed distributed within the bentonite, we thus multiply the reported values of \(\epsilon_\mathrm{eff}\) by \(C_\mathrm{bkg}\), which basically gives the (experimentally accessible) clay concentration

\begin{equation} \bar{C} = \frac{\epsilon_\mathrm{eff} \cdot C_\mathrm{bkg}}{\phi} \tag{2} \end{equation}

Here we also have divided by sample porosity, \(\phi\), to relate the clay concentration to water volume rather than total sample volume. Note that eq. 2 is not derived from more fundamental quantities, but allows for “de-deriving” a quantity more directly related to measurements. (I.e., what is reported as an accessible volume is actually a measure of the clay concentration.)

It is, however, impossible (as far as I see) to back-calculate the actual value of \(n’_ \mathrm{Cl}\) from provided formulas and values of \(\epsilon_\mathrm{eff}\), because masses and volumes of the sample sections are not provided. Therefore, we cannot independently assess the procedure used to evaluate \(\epsilon_\mathrm{eff}\), and simply have to assume that it is adequate.3 Here are the reported values of \(\epsilon_\mathrm{eff}\) for each test, and the corresponding evaluation of \(\bar{C}\) using eq. 2 (column 3)

Test
\(\epsilon_\mathrm{eff}\)
(reported)
\(\bar{C}/C_\mathrm{bkg}\)
(from \(\epsilon_\mathrm{eff}\))
\(\bar{C}/C_\mathrm{bkg}\)
(re-evaluated)
1.3/0.010.0340.060.051
1.3/0.050.0450.08
1.3/0.10.090*0.170.162
1.3/0.40.1400.26
1.3/1.00.2200.410.400
1.6/0.010.0090.020.019
1.6/0.050.016**0.04
1.6/0.10.0290.070.066
1.6/0.40.0600.14
1.6/1.00.1100.260.239
1.9/0.010.0090.03discarded
1.9/0.050.0070.02
1.9/0.10.0150.050.044
1.9/0.40.0170.05
1.9/1.00.0440.140.128
*) The table in Vl07 says 0.076, but the concentration profile diagram says 0.090.
**) The table in Vl07 says 0.16, but this must be a typo.

When using eq. 2 we have adopted porosities 0.536, 0.429, and 0.322, respectively, for densities 1.3 g/cm3, 1.6 g/cm3, and 1.9 g/cm3.

The tabulated \(\epsilon_\mathrm{eff}\) values are evaluated as averages of the clay concentration profiles (presented as effective porosity profiles), which look like this for the samples exposed to background concentrations 0.01 M, 0.1 M and 1.0 M (profiles for 0.05 M and 0.4 M are not presented in Vl07)

The chloride concentration increases near the interfaces in all samples; we have discussed this interface excess effect in previous posts. Vl07 deal with this issue by evaluating the averages only for the inner parts of the samples. I performed a similar evaluation, also presented in the above figures (blue lines). In this evaluation I adopted the criterion to exclude all points situated less than 2 mm from the interfaces (Vl07 seem to have chosen points a bit differently). The clay concentration reevaluated in this way is also listed in the above table (last column). Given that I have only used nominal density for each sample (I don’t have information on the actual density of the sample sections), I’d say that the re-evaluated values agree well with those de-derived from reported \(\epsilon_\mathrm{eff}\). One exception is the sample 1.9/0.01, which is seen to have concentration points all over the place (or maybe detection limit is reached?). While Vl07 choose the lowest three points in their evaluation, here we choose to discard this result altogether. I mean that it is rather clear that this concentration profile cannot be considered to represent equilibrium.

As the reevaluation gives similar values as those reported, and since we lack information for a full analysis, we will use the values de-derived from reported \(\epsilon_\mathrm{eff}\) in the continued assessment (except for sample 1.9/0.01).

Diffusion related estimations

Vl07 determine diffusion parameters by fitting various mathematical expressions to flux data.4 Parameters fitted in this way generally depend on the underlying adopted model, and we have discussed how equilibrium concentrations can be extracted from such parameters in an earlier blog post. In Vl07 it is clear that the adopted mathematical and conceptual model is the effective porosity diffusion model. When first presented in the article, however, it is done so in terms of a sorption distribution coefficient (\(R_d\)) that is claimed to take on negative values for anions. The presented mathematical expressions therefore contain a so-called rock capacity factor, \(\alpha\), which relates to \(R_d\) as \(\alpha = \phi + \rho_d\cdot R_d\). But such use of a rock capacity factor is a mix-up of incompatible models that I have criticized earlier. However, in Vl07 the description involving a sorption coefficient is in words only — \(R_d\) is never brought up again — and all results are reported, interpreted and discussed in terms of effective (or “chloride-accessible”) porosity, labeled \(\epsilon\) or \(\epsilon_\mathrm{Cl}\). We here exclusively use the label \(\epsilon_\mathrm{eff}\) when referring to formulas in Vl07. The mathematics is of course the same regardless if we call the parameter \(\alpha\), \(\epsilon\), \(\epsilon_\mathrm{Cl}\), or \(\epsilon_\mathrm{eff}\).

Mass balance in the out-diffusion stage

Vl07 measured the amount of tracers accumulated in the two reservoirs during the out-diffusion stage. The flux into the left side reservoir, which served as source reservoir during the preceding through-diffusion stage, was completely obscured by significant amounts of tracers present in the confining filter, and will not be considered further (also Vl07 abandon this flux in their analysis). But the total amount of tracers accumulated in the right side reservoir, \(N_\mathrm{right}\),5 can be used to directly estimate the chloride equilibrium concentration.

The initial concentration profile in the out-diffusion stage is linear (it is the steady-state profile), and the total amount of tracers, \(N_\mathrm{tot}\),6 can be expressed

\begin{equation} N_\mathrm{tot} = \frac{\phi\cdot \bar{c}_0\cdot V_\mathrm{sample}} {2} \tag{3} \end{equation}

where \(\bar{c}_0\) is the initial clay concentration at the left side interface, and \(V_\mathrm{sample}\) (\(\approx\) 5 cm3) is the sample volume.

A neat feature of the out-diffusion process is that two thirds of the tracers end up in the left side reservoir, and one third in the right side reservoir, as illustrated in this simulation

\(\bar{c}_0\) can thus be estimated by using \(N_\mathrm{tot} = 3\cdot N_\mathrm{right}\) in eq. 3, giving

\begin{equation} \frac{\bar{c}_0}{c_\mathrm{source}} = \frac{6 \cdot N_\mathrm{right}} {\phi \cdot V_\mathrm{sample}\cdot c_\mathrm{source}} \tag{4} \end{equation}

where \(c_\mathrm{source}\) is the tracer concentration in the left side reservoir in the through-diffusion stage.7 Although eq. 4 depends on a particular solution to the diffusion equation, it is independent of diffusivity (the diffusivity in the above simulation is \(1\cdot 10^{-10}\) m2/s). Eq. 4 can in this sense be said to be a direct estimation of \(\bar{c}_0\) (from measured \(N_\mathrm{right}\)), although maybe not as “direct” as the measurement of stable chloride, discussed previously.

Vl07 state eq. 4 in terms of a “Cl-accessible porosity”, but this is still just an interpretation of the clay concentration; \(\bar{c}_0\) is, in contrast to \(\epsilon_\mathrm{eff}\), directly accessible experimentally in principle. From the reported values of \(\epsilon_\mathrm{eff}\) we may back-calculate \(\bar{c}_0\), using the relation \(\bar{c}_0 / c_\mathrm{source} = \epsilon_\mathrm{eff}/\phi\). Alternatively, we may use eq. 4 directly to evaluate \(\bar{c}_0\) from the reported values of \(N_\mathrm{right}\). Curiously, these two approaches result in slightly different values for \(\bar{c}_0/c_\mathrm{source}\). I don’t understand the cause for this difference, but since \(N_\mathrm{right}\) is what has actually been measured, we use these values to estimate \(\bar{c}_0.\) The resulting equilibrium concentrations are

Test
\(N_\mathrm{right}\)
(10-10 mol)
\(\bar{c}_0/c_\mathrm{source}\)
(-)
1.3/0.014.100.038
1.3/0.0510.20.097
1.3/0.117.80.168
1.3/0.441.40.395
1.3/1.052.40.445
1.6/0.011.210.014
1.6/0.053.640.043
1.6/0.16.150.072
1.6/0.413.00.154
1.6/1.021.60.225
1.9/0.010.410.006
1.9/0.051.140.018
1.9/0.11.640.025
1.9/0.43.190.051
1.9/1.08.190.113

We have now investigated two independent estimations of the chloride equilibrium concentrations: from mass balance of chloride tracers in the out-diffusion stage, and from measured stable chloride content. Here are plots comparing these two estimations

The similarity is quite extraordinary! With the exception of two samples (1.3/0.4 and 1.9/0.1), the equilibrium chloride concentrations evaluated in these two very different ways are essentially the same. This result strongly confirms that the evaluations are adequate.

Steady-state fluxes

Vl07 present the flux evolution in the through-diffusion stage only for a single test (1.6/1.0), and it looks like this (left diagram)

The outflux reaches a relatively stable value after about 7 days, after which it is meticulously monitored for a quite long time period. The stable flux is not completely constant, but decreases slightly during the course of the test. We anyway refer to this part as the steady-state phase, and to the preceding part as the transient phase.

One reason that the steady-state is not completely stable is, reasonably, that the source reservoir concentration slowly decreases during the course of the test. The estimated drop from this effect, however, is only about one percent,8 while the recorded drop is substantially larger, about 7%. Vl07 do not comment on this perhaps unexpectedly large drop, but it may be caused e.g. by the ongoing conversion of the bentonite to a purer sodium state (see above).

Most of the analysis in Vl07 is based on anyway assigning a single value to the steady-state flux. Judging from the above plot, Vl07 seem to adopt the average value during the steady-state phase, and it is clear that the assigned value is well constrained by the measurements (the drop is a second order effect). The steady-state flux can therefore be said to be directly measured in the through-diffusion stage, rather than being obtained from fitting a certain model to data.

Vl07 only implicitly consider the steady-state flux, in terms of a fitted “effective diffusivity” parameter, \(D_e\) (more on this in the next section). We can, however, “de-derive” the corresponding steady-state fluxes using \(j_\mathrm{ss} = D_e\cdot c_\mathrm{source}/L\), where \(L\) (= 0.01 m) is sample length. When comparing different tests it is convenient to use the normalized steady state flux \(\widetilde{j}_\mathrm{ss} = j_\mathrm{ss}/c_\mathrm{source}\), which then relates to \(D_e\) as \(\widetilde{j}_\mathrm{ss} = D_e/L\). Indeed, “effective diffusivity” is just a scaled version of the normalized steady-state flux, and it makes more sense to interpret it as such (\(D_e\) is not a diffusion coefficient). From the reported values of \(D_e\) we obtain the following normalized steady-state fluxes (my apologies for a really dull table)

Test
\(D_e\)
(10-12 m2/s)
\(\widetilde{j}_\mathrm{ss}\)
(10-10 m/s)
1.3/0.012.62.6
1.3/0.057.57.5
1.3/0.11616
1.3/0.42525
1.3/1.04949
1.6/0.010.390.39
1.6/0.051.11.1
1.6/0.12.32.3
1.6/0.44.64.6
1.6/1.01010
1.9/0.010.0330.033
1.9/0.050.120.12
1.9/0.10.240.24
1.9/0.40.50.5
1.9/1.01.21.2

Plotting \(\widetilde{j}_\mathrm{ss}\) as a function of background concentration gives the following picture

The steady-state flux show a very consistent behavior: for all three densities, \(\widetilde{j}_\mathrm{ss}\) increases with background concentration, with a higher slope for the three lowest background concentrations, and a smaller slope for the two highest background concentrations. Although we have only been able to investigate the 1.6/1.0 test in detail, this consistency confirms that the steady-state flux has been reliably determined in all tests.

Transient phase evaluations

So far, we have considered estimations based on more or less direct measurements: stable chloride concentration profiles, tracer mass balance in the out-diffusion stage, and steady-state fluxes. A major part of the analysis in Vl07, however, is based on fitting solutions of the diffusion equation to the recorded flux.

Vl07 state somewhat different descriptions for the through- and out-diffusion stages. For out-diffusion they use an expression for the flux into the right side reservoir (the sample is assumed located between \(x=0\) and \(x=L\))

\begin{equation} j(L,t) = -2\cdot j_\mathrm{ss} \sum_{n = 1}^\infty \left(-1\right )^n\cdot e^{-\frac{D_e\cdot n^2\cdot \pi^2\cdot t} {L^2\cdot \epsilon_\mathrm{eff}}} \tag{5} \end{equation}

where \(j_\mathrm{ss}\) is the steady-state flux,9 \(D_e\) is “effective diffusivity”, and \(\epsilon_\mathrm{eff}\) is the effective porosity parameter (Vl07 also state a similar expression for the diffusion into the left side reservoir, but these results are discarded, as discussed earlier). For through-diffusion, Vl07 instead utilize the expression for the amount tracer accumulated in the right side reservoir

\begin{equation} A(L,t) = S\cdot L \cdot c_\mathrm{source} \left ( \frac{D_e\cdot t}{L^2} – \frac{\epsilon_\mathrm{eff}} {6} – \frac{2\cdot\epsilon_\mathrm{eff}}{\pi^2} \sum_{n = 1}^\infty \frac{\left(-1\right )^n}{n^2} \cdot e^{-\frac{D_e\cdot n^2\cdot \pi^2\cdot t} {L^2\cdot \epsilon_\mathrm{eff}} }\right ) \tag{6} \end{equation}

were \(S\) denotes the cross section area of the sample.

It is clear that Vl07 use \(D_e\) and \(\epsilon_\mathrm{eff}\) as fitting parameters, but not exactly how the fitting was conducted. \(D_e\) seems to have been determined solely from the the through-diffusion data, while separate values are evaluated for \(\epsilon_\mathrm{eff}\) from the through- and out-diffusion stages. As already discussed, Vl07 also provide a third estimation of \(\epsilon_\mathrm{eff}\), based on mass-balance in the out-diffusion stage. To me, the study thereby gives the incorrect impression of providing a whole set of independent estimations of \(\epsilon_\mathrm{eff}\). Although eqs. 5 and 6 are fitted to different data, they describe diffusion in one and the same sample, and an adequate fitting procedure should provide a consistent, single set of fitted parameters \((D_e, \epsilon_\mathrm{eff})\). Even more obvious is that the estimation of \(\epsilon_\mathrm{eff}\) from fitting eq. 5 should agree with the estimation from the mass-balance in the out diffusion stage — the accumulated amount in the right side reservoir is, after all, given by the integral of eq. 5. A significant variation of the reported fitting parameters for the same sample would thus signify internal inconsistency (experimental- or modelwise).

In the following reevaluation we streamline the description by solely using fluxes as model expressions,4 and by emphasizing steady-state flux as a parameter, which I think gives particularly neat expressions,10 (“TD” and “OD” denote through- and out-diffusion, respectively)

\begin{equation} \widetilde{j}_{TD}(L,t) = \widetilde{j}_\mathrm{ss} \left ( 1 + 2 \sum_{n = 1}^\infty \left(-1\right )^n \cdot e^{-\frac{D_p\cdot n^2\cdot \pi^2\cdot t} {L^2} }\right ) \tag{7} \end{equation}

\begin{equation} \widetilde{j}_{OD}(L,t) = -2\cdot \widetilde{j}_\mathrm{ss} \sum_{n = 1}^\infty \left( -1 \right )^n \cdot e^{-\frac{D_p\cdot n^2\cdot \pi^2\cdot t} {L^2}} \tag{8} \end{equation}

Here we use the pore diffusivity, \(D_p\), instead of the combination \(D_e/\epsilon_\mathrm{eff}\) in the exponential factors, and \(\widetilde{j} = j/c_\mathrm{source}\) denotes normalized flux. This formulation clearly shows that the time evolution is governed solely by \(D_p\), and that \(\widetilde{j}_\mathrm{ss}\) simply acts as a scaling factor.

In my opinion, using \(\widetilde{j}_\mathrm{ss}\) and \(D_p\) gives a formulation more directly related to measurable quantities; the steady-state flux is directly accessible experimentally, as we just examined, and \(D_p\) is an actual diffusion coefficient (in contrast to \(D_e\)) that can be directly evaluated from clay concentration profiles. Of course, eqs. 7 and 8 provide the same basic description as eqs. 5 and 6, and \(\widetilde{j}_\mathrm{ss}\) and \(D_p\) are related to the parameters reported in Vl07 as

\begin{equation} \widetilde{j}_\mathrm{ss} = \frac{D_e}{L} \tag{9} \end{equation}

\begin{equation} D_p = \frac{D_e}{\epsilon_\mathrm{eff}} \tag{10} \end{equation}

When reevaluating the reported data we focus on the above discussed consistency aspect, i.e. whether or not a single model (a single pair of parameters) can be satisfactory fitted to all available data for the same sample. In this regard, we begin by noting that the fitting parameters are already constrained by the direct estimations. We have already concluded that the recorded steady-state flux basically determines \(\widetilde{j}_\mathrm{ss}\), and if we combine this with the estimated chloride clay concentration, \(D_p\) is determined from \(j_\mathrm{ss} = \phi\cdot D_p\cdot \bar{c}_0/L\), i.e.

\begin{equation} D_p = \frac{\widetilde{j}_\mathrm{ss}\cdot L} {\phi\cdot \left (\bar{c}_0 / c_\mathrm{source}\right )} \tag{11} \end{equation}

Here are plotted values of \(D_p\) evaluated in this manner

Note that these values basically remain constant for samples of similar density (within a factor of 2) as the background concentration is varied by two orders of magnitude. This is the expected behavior of an actual diffusion coefficient,11 and confirms the adequacy of the evaluation; the numerical values also compares rather well with corresponding values for “MX-80” bentonite, measured in closed-cell tests (indicated by dashed lines in the figure).

Using eq. 10, we can also evaluate values of \(D_p\) corresponding to the various reported fitted parameters \(\epsilon_\mathrm{eff}\). The result looks like this (compared with the above evaluations from direct estimations)

As pointed out above, a consistent evaluation requires that the parameters fitted to the out-diffusion flux (red) are very similar to those evaluated from considering the mass balance in the same process (blue). We note that the resemblance is quite reasonable, although some values — e.g. tests 1.3/1.0 and 1.6/1.0 — deviate in a perhaps unacceptable way.

\(D_p\) evaluated from reported through-diffusion parameters, on the other hand, shows significant scattering (green). As the rest of the values are considerably more collected, and as the steady-state fluxes show no sign whatsoever that the diffusion coefficient varies in such erratic manner, it is quite clear that this scattering indicates problems with the fitting procedure for the through-diffusion data.

The 1.6/1.0 test

To further investigate the fitting procedures, we take a detailed look at the 1.6/1.0 test, for which flux data is provided. Vl07 report fitted parameters \(D_e = 1.0\cdot 10^{-11}\) m2/s and \(\epsilon_\mathrm{eff} = 0.063\) to the through-diffusion data, corresponding to \(\widetilde{j}_\mathrm{ss} = 1.0\cdot 10^{-9}\) m/s and \(D_p = 1.6\cdot 10^{-10}\) m2/s. We have already concluded that the steady-state flux is well captured by this data, but to see how well fitted \(\epsilon_\mathrm{eff}\) (or \(D_p\)) is, lets zoom in on the transient phase

This diagram also contains models (eq. 7) with different values of \(D_p\), and with a slightly different value of \(j_\mathrm{ss}\).12 It is clear that the model presented in the paper (black) completely misses the transient phase, and that a much better fit is achieved with \(D_p = 9.7\cdot10^{-11}\) m2/s (and \(\widetilde{j}_\mathrm{ss} = 1.06\cdot 10^{-9}\) m/s) (red). This difference cannot be attributed to uncertainty in the parameter \(D_p\) — the reported fit is simply of inferior quality. With that said, we note that all information on the transient phase is contained within the first three or four flux points; the reliability could probably have been improved by measuring more frequently in the initial stage.13

A reason for the inferior fit may be that Vl07 have focused only on the linear part of eq. 6; the paper spends half a paragraph discussing how the approximation of this expression for large \(t\) can be used to extract the fitting parameters using linear regression. Does this mean that only experimental data for large times where used to evaluate \(D_e\) and \(\epsilon_\mathrm{eff}\)? Since we are not told how fitting was performed, we cannot answer this question. Under any circumstance, the evidently low quality of the fit puts in question all the reported \(\epsilon_\mathrm{eff}\) values fitted to through-diffusion data. This is actually good news, as several of the corresponding \(D_p\) values were seen to be incompatible with constraints from direct estimations. We can thus conclude with some confidence that the inconsistency conveyed by the differently evaluated fitting parameters does not indicate experimental shortcomings, but stems from bad fitting of the through-diffusion model. Therefore, we simply dismiss the reported \(\epsilon_\mathrm{eff}\) values evaluated in this way. Note that the re-fitted value for \(D_p\) \((9.7\cdot10^{-11}\) m2/s) is consistent with those evaluated from direct estimations.

We note that when fitting the transient phase, it is appropriate to use a value of \(\widetilde{j}_\mathrm{ss}\) slightly larger than the average value adopted by Vl07 (as the model does not account for the observed slight drop of the steady-state flux). This is only a minor variation in the \(\widetilde{j}_\mathrm{ss}\) parameter itself (from \(1.02\cdot10^{-9}\) to \(1.06\cdot10^{-9}\) m/s), but, since this value sets the overall scale, it indirectly influences the fitted value of \(D_p\) (model fitting is subtle!).

More questions arise regarding the fitting procedures when also examining the presented out-diffusion stage for the 1.6/1.0 sample. The tabulated fitted value for this stage is \(\epsilon_\mathrm{eff}\) = 0.075, while it is implied that the same value has been used for \(D_e\) as evaluated from the the through-diffusion stage (\(1.0\cdot 10^{-11}\) m2/s). The corresponding pore diffusivity is \(D_p = 1.33\cdot 10^{-10}\) m2/s. The provided plot, however, contains a different model than tabulated, and looks similar to this one (left diagram)

Here the presented model (black dashed line) instead corresponds to \(D_p = 8.5\cdot 10^{-11}\) m2/s (or \(\epsilon_\mathrm{eff}\) = 0.118). The model corresponding to the tabulated value (orange) does not fit the data! I guess this error may just be due to a typo in the table, but it nevertheless gives more reasons to not trust the reported \(\epsilon_\mathrm{eff}\) values fitted to diffusion data.

The above diagram also shows the model corresponding to the reported parameters from the through-diffusion stage (black solid line). Not surprisingly, this model does not fit the out-diffusion data, confirming that it does not appropriately describe the current sample. The model we re-fitted in the through-diffusion stage (red), on the other hand, captures the outflux data quite well. By also slightly adjusting \(\widetilde{j}_{ss}\), from from \(1.06\cdot10^{-9}\) to \(0.99\cdot10^{-9}\) m/s, to account for the drop in steady-state flux during the course of the through-diffusion test, and by plotting in a lin-lin rather than a log-log diagram, the picture looks even better! In a lin-lin plot (right diagram), it is easier to note that the model presented in the graph of Vl07 actually misses several of the data points. Could it be that Vl07 used visual inspection of the model in a log-log diagram to assess fitting quality? If so, data points corresponding to very low fluxes are given unreasonably high weight.14 This could be (another) reason for the noted difference between \(D_p\) evaluated from fitted parameters to the out-diffusion flux, and from the total accumulated amount of tracer (which should be equal).

From examining the reported results of sample 1.6/1.0 we have seen that the fitting procedures adopted in Vl07 appear inappropriate, but also that a consistent model can be successfully fitted to all available data (using a single \(D_p\)). Vl07 don’t provide flux data for any other sample, but we must conclude that the reported fitted \(\epsilon_\mathrm{eff}\) parameters cannot be trusted. Luckily, the preformed refitting exercise confirms the results obtained from analysis of stable chloride profiles and accumulated amount of tracers in out-diffusion, and we conclude that these results most probably are reliable. The corresponding value of \(\bar{c}_0/c_\mathrm{source}\) (using eq. 11) for the refitted model is here compared with the estimations from direct measurements

Summary and verdict

Chloride equilibrium concentrations evaluated from mass balance of the tracer in the out-diffusion stage and from stable chloride content show remarkable agreement. On the other hand, the scattering of estimated concentrations increases substantially if they are also evaluated from the reported fitted diffusion parameters. This could indicate underlying experimental problems, as a consistent evaluation should result in a single value for the equilibrium concentration; the various evaluations — stable chloride, out-diffusion mass balance, through-diffusion fitting and out-diffusion fitting — relate, after all, to a single sample.

By reexamining the evaluations we have found, however, that the problem is associated with how the fitting to diffusion data has been conducted (and presented), rather than indicating fundamental experimental issues. In the test that we have been able to examine in detail (1.6/1.0), we found that the reported models do not fit data, but also that it is possible to satisfactorily refit a single model that is also compatible with the direct methods for evaluating the equilibrium concentration. For the rest of the samples, we have also been able to discard the fitted diffusion parameters, as they are not compatible e.g. with how the steady-state flux (very consistently) vary with density and background concentration.

For these reasons, we discard the reported “effective porosity” parameters evaluated from fitting solutions of the diffusion equation to flux data, and keep the results from direct measurements of chloride equilibrium concentrations (from stable chloride profile analysis and mass-balance in the out-diffusion stage). I judge the resulting chloride equilibrium concentrations as reliable and that they can be used for increased qualitative process understanding. I furthermore judge the directly measured steady-state fluxes as reliable. This study thus provide adequate values for both chloride equilibrium concentrations and diffusion coefficients.

However, a frustrating problem is that, although the equilibrium concentrations are well determined, we have little information on the exact state of the samples in which they have been measured. We basically have to rely on that the “KWK” material is “similar” to “MX-80”, keeping in mind that “MX-80” is not really a uniform material (from a scientific point of view). Also, the exchangeable mono/divalent cation ratio is most probably quite different in samples contacted with different background concentrations.

Yet, I judge the present study to provide the best information available on chloride equilibrium in compacted bentonite, and will use it e.g. for investigating the salt exclusion mechanism in these systems (I already have). That this information is the best available is, however, also a strong argument for that more and better constrained data is urgently needed.

The (reliable) results are presented in the diagram below, which includes “confidence areas”, that takes into account the spread in equilibrium concentrations, in samples where more than a single evaluation were performed, and the estimated uncertainty in effective montmorillonite dry density (the actual points are plotted at nominal density, assuming 80% montmorillonite content)

Footnotes

[1] Vejsada et al. (2006) call their material “KWK 20-80”. In other contexts, I have also found the versions “KWK food grade” and “KWK krystal klear”. I have given up my attempts at trying to understand the difference between these “KWK” variants.

[2] Van Loon et al. (2007) label the background concentration \([\mathrm{Cl}]_0\).

[3] This should be relatively straightforward, but I get at bit nervous e.g. about the presence of a rather arbitrary factor 0.85 in the presented formula (eq. 19 in Van Loon et al. (2007)).

[4] As always for these types of diffusion tests, the raw data consists of simultaneously measured values of time (\(\{t_i\}\)) and reservoir concentrations (\(\{c_i\}\)). From these, flux can be evaluated as (\(A\) is sample cross sectional area, and \(V_\mathrm{res}\) is reservoir volume)

\begin{equation} \bar{j}_i = \frac{1}{A} \frac{ \left (c_i – c_{i-1} \right ) \cdot V_\mathrm{res}} {t_i – t_{i-i}} \tag{*} \end{equation}

\(\bar{j}_i\) is the mean flux in the time interval between \(t_{i-1}\) and \(t_i\), and should be associated with the average time of the same interval: \(\bar{t}_i = (t_i + t_{i-1})/2\). The above formula assumes no solution replacement after the \((i-1)\):th measurement (if the solution is replaced, \(\left (c_i – c_{i-1} \right )\) should be replaced with \(c_i\)).

Alternatively one can work with the accumulated amount of substance, which e.g. is \(N(t_i) = \sum_{j=1}^i c_j\cdot V_\mathrm{res}\), in case the solution is replaced after each measurement. I prefer using the flux because eq. * only depends on two consecutive measurements, while \(N(t_i)\) in principle depends on all measurements up to time \(t_i\). Also, I think it is easier to judge how well e.g. a certain model fits or is constrained by data when using fluxes; the steady-state, for example, then corresponds to a constant value.

Van Loon et al. (2007) seem to have utilized both fluxes and accumulated amount of substance in their evaluations, as discussed in later sections.

[5] Van Loon et al. (2007) denote this quantity \(A(L)\).

[6] Van Loon et al. (2007) denote this quantity \(A_w\), \(A_\mathrm{out}\), and \(A_\mathrm{tot}\).

[7] Van Loon et al. (2007) denote this quantity \(C_0\).

[8] From total test time, recorded flux, and sample cross sectional area, we estimate that about \(5.8\cdot 10^{-8}\) mol of tracer is transferred from the source reservoir during the course of the test (\(50\) days\(\cdot 2.7\cdot 10^{-11}\) mol/m2/s\(\cdot 0.0005\) m2). This is about 1% of the total amount tracer, \(c_\mathrm{source} \cdot V_\mathrm{source} = 2.65 \cdot 10^{-5}\) M \(\cdot 0.2\) L = \(5.3\cdot 10^{-6}\) mol.

[9] Van Loon et al. (2007) label this parameter \(J_L\), and don’t relate it explicitly to the steady-state flux. From the experimental set-up it is clear, however, that the initial value of the out-diffusion flux (into the right side reservoir) is the same as the previously maintained steady-state flux. Note that the expressions for the fluxes in the out-diffusion stage in Van Loon et al. (2007) has the wrong sign.

[10] The description provided by eqs. 5 and 6 not only mixes expressions for flux and accumulated amount tracer, but also contains three dependent parameters \(D_e\), \(\epsilon_\mathrm{eff}\), and \(j_\mathrm{ss}\) (e.g. \(j_\mathrm{ss} = D_e/(c_\mathrm{source}\cdot L)\)). In this reformulation, the model parameters are strictly only \(\widetilde{j}_\mathrm{ss}\) and \(D_p\). We have also divided out \(c_\mathrm{source}\) to obtain equations for normalized fluxes. Note that the expression for \(\widetilde{j}_{TD}(L,t)\) is essentially the same that we have used in previous assessments of through-diffusion tests. Note also that eqs. 7 and 8 imply the relation \(\widetilde{j}_{OD}(L,t) = \widetilde{j}_{ss} – \widetilde{j}_{TD}(L,t)\), reflecting that the out-diffusion process is essentially the through-diffusion process in reverse.

[11] Note the similarity with that diffusivity also is basically independent of background concentration for simple cations. Note also that there is no reason to expect completely constant \(D_p\) for a given density, because the samples are not identically prepared (being saturated with saline solutions of different concentration).

[12] As we here consider a single sample, we alternate a bit sloppily between steady-state flux (\(j_\mathrm{ss} \)) and normalized steady-state flux (\(\widetilde{j}_\mathrm{ss}\)), but these are simply related by a constant: \(\widetilde{j}_\mathrm{ss} = j_\mathrm{ss} / c_\mathrm{source}\). For the 1.6/1.0 test this constant is (as tabulated) \(c_\mathrm{source} = 2.65\cdot 10^{-2}\) mol/m3.

[13] I think it is a bit amusing that the pattern of data points suggests measurements being performed on Mondays, Wednesdays, and Fridays (with the test started on a Wednesday).

[14] I have warned about the dangers of log-log plots earlier.

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

“Multi-porosity” models1 — i.e models that account for both a bulk water phase and one, or several, other domains within the clay — have become increasingly popular in bentonite research during the last couple of decades. These are obviously macroscopic, as is clear e.g. from the benchmark simulations described in Alt-Epping et al. (2015), which are specified to be discretized into 2 mm thick cells; each cell is consequently assumed to contain billions and billions individual montmorillonite particles. The macroscopic character is also relatively clear in their description of two numerical tools that have implemented multi-porosity

PHREEQC and CrunchFlowMC have implemented a Donnan approach to describe the electrical potential and species distribution in the EDL. This approach implies a uniform electrical potential \(\varphi^\mathrm{EDL}\) in the EDL and an instantaneous equilibrium distribution of species between the EDL and the free water (i.e., between the micro- and macroporosity, respectively). The assumption of instantaneous equilibrium implies that diffusion between micro- and macroporosity is not considered explicitly and that at all times the chemical potentials, \(\mu_i\), of the species are the same in the two porosities

On an abstract level, we may thus illustrate a multi-porosity approach something like this (here involving two domains)

The model is represented by one continuum for the “free water”/”macroporosity” and one for the “diffuse layer”/”microporosity”,2 which are postulated to be in equilibrium within each macroscopic cell.

But such an equilibrium (Donnan equilibrium) requires a semi-permeable component. I am not aware of any suggestion for such a component in any publication on multi-porosity models. Likewise, the co-existence of diffuse layer and free water domains requires a mechanism that prevents swelling and maintains the pressure difference — also the water chemical potential should of course be the equal in the two “porosities”.3

Note that the questions of what constitutes the semi-permeable component and what prevents swelling have a clear answer in the homogeneous mixture model. This answer also corresponds to an easily identified real-world object: the metal filter (or similar component) separating the sample from the external solution. Multi-porosity models, on the other hand, attribute no particular significance to interfaces between sample and external solutions. Therefore, a candidate for the semi-permeable component has to be — but isn’t — sought elsewhere. Donnan equilibrium calculations are virtually meaningless without identifying this component.

The partitioning between diffuse layer and free water in multi-porosity models is, moreover, assumed to be controlled by water chemistry, usually by means of the Debye length. E.g. Alt-Epping et al. (2015) write

To determine the volume of the microporosity, the surface area of montmorillonite, and the Debye length, \(D_L\), which is the distance from the charged mineral surface to the point where electrical potential decays by a factor of e, needs to be known. The volume of the microporosity can then be calculated as \begin{equation*} \phi^\mathrm{EDL} = A_\mathrm{clay} D_L, \end{equation*} where \(A_\mathrm{clay}\) is the charged surface area of the clay mineral.

I cannot overstate how strange the multi-porosity description is. Leaving the abstract representation, here is an attempt to illustrate the implied clay structure, at the “macropore” scale

The view emerging from the above description is actually even more peculiar, as the “micro” and “macro” volume fractions are supposed to vary with the Debye length. A more general illustration of how the pore structure is supposed to function is shown in this animation (“I” denotes ionic strength)

What on earth could constitute such magic semi-permeable membranes?! (Note that they are also supposed to withstand the inevitable pressure difference.)

Here, the informed reader may object and point out that no researcher promoting multi-porosity has this magic pore structure in mind. Indeed, basically all multi-porosity publications instead vaguely claim that the domain separation occurs on the nanometer scale and present microscopic illustrations, like this (this is a simplified version of what is found in Alt-Epping et al. (2015))

In the remainder of this post I will discuss how the idea of a domain separation on the microscopic scale is even more preposterous than the magic membranes suggested above. We focus on three aspects:

  • The implied structure of the free water domain
  • The arbitrary domain division
  • Donnan equilibrium on the microscopic scale is not really a valid concept

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.

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).

How salt equilibrium concentrations may be overestimated

Saturating with saline solution

When discussing semi-permeability, we noted that a bentonite sample that is saturated with a saline solution probably contains more salt in the initial stages of the process than what is dictated by the final state Donnan equilibrium. This salt must consequently diffuse out of the sample before equilibrium is reached.

The reason for such a possible “overshoot” of the clay concentration is that an infiltrating solution is not subject to a Donnan effect (between sample and external solution) when it fills out the air-filled voids of an unsaturated sample. Also, even if the region near the interface to the external solution becomes saturated — so that a Donnan effect is active — a sample may still take up more salt than prescribed by the final state, due to hyperfiltration: with a net inflow of water and an active Donnan effect, salt will accumulate at the inlet interface (unless the interface is flushed). This increased concentration, in turn, alters the Donnan equilibrium at the interface, with the effect that more salt diffuses into the clay.

These effects are relevant for our ongoing assessment of studies of chloride equilibrium concentrations. If bentonite samples are saturated with saline solutions, without taking precautions against these effects, evaluated equilibrium concentrations may be overestimated. Note that, even if saturating a sample may be relatively fast, it may take a long time for salt to reach full equilibrium, depending on details of the experimental set-up. In particular, if the set-up is such that the external solution does not flow past the inlet, equilibration may take a very long time, being limited by diffusion in filters and tubing.

Interface excess salt

Another way for evaluated salt concentrations to overestimate the true equilibrium value — which is independent of whether or not the sample has been saturated with a saline solution — is due to excess salt at the sample interfaces.

Suppose that you determine the equilibrium salt concentration in a bentonite sample in the following way. First you prepare the sample in a test cell and contact it with an external salt solution via filters. When the system (bentonite + solution) has reached equilibrium (taking all the precautions against overestimation discussed above), the concentration profile may be conceptualized like this

The aim is to determine \(\bar{c}_\mathrm{clay}\), the clay concentration of the species of interest (e.g. chloride), and to relate it to the corresponding concentration in the external solution (\(c_ \mathrm{ext}\)).

After ensuring the value of \(c_\mathrm{ext}\) (e.g. by sampling or controlling the external solution), you unload the test cell and isolate the bentonite sample. In doing so, we must keep in mind that the sample will begin to swell as soon as the force on it is released, if only water is available. In the present example it is difficult not to imagine that some water is available, e.g. in the filters.1

It is thus plausible that the actual concentration profile look something like this directly after the sample has been isolated

We will refer to the elevated concentration at the interfaces as the interface excess. The exact shape of the resulting concentration profile depends reasonably on the detailed procedure for isolating the sample.2 If the ion content of the sample is measured as a whole, and/or if the sample is stored for an appreciable amount of time before further analysis (so that the profile evens out due to diffusion), it is clear that the evaluated ion content will be larger than the actual clay concentration.

To quantify how much the clay concentration may be overestimated due to the interface excess, we introduce an effective penetration depth, \(\delta\)

\(\delta\) corresponds to a depth of the external concentration that gives the same interface excess as the actual distribution. Using this parameter, it is easy to see that the clay concentration evaluated as the average over the entire sample is

\begin{equation} \bar{c}_\mathrm{eval} = \bar{c}_\mathrm{clay}+\frac{2\cdot\delta} {L} \cdot \left (c_\mathrm{ext} – \bar{c}_\mathrm{clay} \right ) \end{equation}

By dividing by the actual value \(\bar{c}_\mathrm{clay}\), we get an expression for the relative overestimation

\begin{equation} \frac{\bar{c}_\mathrm{eval}}{\bar{c}_\mathrm{clay}} = 1 + \frac{2\cdot\delta} {L} \cdot \left (\frac{c_\mathrm{ext}}{\bar{c}_\mathrm{clay}} – 1 \right ) \tag{1} \end{equation}

This expression is quite interesting. We see that the relative overestimation, reasonably, depends linearly on \(\delta\) and on the inverse of sample length. But the expression also contains the ratio \(r \equiv c_\mathrm{ext}/\bar{c}_\mathrm{clay}\), indicating that the effect may be more severe for systems where the clay concentration is small in comparison to the external concentration (high density, low \(c_\mathrm{ext}\)).

An interface excess is more than a theoretical concept, and is frequently observed e.g. in anion through-diffusion studies. We have previously encountered them when assessing the diffusion studies of Muurinen et al. (1988) and Molera et al. (2003).3 Van Loon et al. (2007) clearly demonstrate the phenomenon, as they evaluate the distribution of stable chloride (the background electrolyte) in the samples after performing the diffusion tests.4 Here is an example of the chloride distribution in a sample of density 1.6 g/cm3 and background concentration of 0.1 M5

The line labeled \(\bar{c}_\mathrm{clay}\) is evaluated from the average of only the interior sections (0.0066 M), while the line labeled \(\bar{c}_\mathrm{eval}\) is the average of all sections (0.0104 M). Using the full sample to evaluate the chloride clay concentration thus overestimates the value by a factor 1.6. From eq. 1, we see that this corresponds to \(\delta = 0.2\) mm. For a sample of length 5 mm with the same penetration depth, the corresponding overestimation is a factor of 2.1.

Here is plotted the relative overestimation (eq. 1) as a function of \(\delta\) for several systems of varying length and \(r\) (\(= c^\mathrm{ext}/\bar{c}_\mathrm{clay}\))

We see that systems with large \(r\) and/or small \(L\) become hypersensitive to this effect. Thus, even if it may be expected that \(\delta\) decreases with increasing \(r\)6, we may still expect an increased overestimation for such systems.

To avoid this potential overestimation of the clay concentration, I guess the best practice is to quickly remove the first couple of millimeters on both sides of a sample after it has been unloaded. In many through-diffusion tests, this is done as part of the study, as the concentration profile across the sample often is measured. In studies where samples are merely equilibrated with an external solution, however, removing the interface regions may not be considered.

Summary

We have here discussed some plausible reasons for why an evaluated equilibrium salt concentration in a clay sample may be overestimated:

  • If samples are saturated directly with a saline solution. Better practice is to first saturate the sample with pure water (or a dilute solution) and then to equilibrate with respect to salt in a second stage.
  • If the external solution is not circulated. Diffusion may then occur over very long distances (depending on test design). The reasonable practice is to always circulate external solutions.
  • If interface excess is not handled. This is an issue even if saturation is done with pure water. The most convenient way to deal with this is to section off the first millimeters on both sides of the samples as quickly as possible after they are unloaded.

Footnotes

[1] One way to minimize this possible effect could be to empty the filter before unloading the test cell. This may, however, be difficult unless the filter itself is flushable. Also, you may run into the problem of beginning to dry the sample.

[2] The only study I’m aware of that has systematically investigated these types of concentration profiles is Glaus et al. (2011). They claim, if I understand correctly, that the interface excess is not caused by swelling during dismantling. Rather, they mean that the profile is the result of an intrinsic density decrease that occurs in interface regions. Still, they don’t discuss how swelling are supposed to be inhibited, neither during dismantling, nor in order for the density inhomogeneity to remain. Under any circumstance, the conclusions in this blog post are not dependent on the cause for the presence of a salt interface excess.

[3] In through-diffusion tests, the problem of the interface excess is usually not that the equilibrium clay concentration is systematically overestimated, since the detailed concentration profile often is sampled in the final state. Instead, the problem becomes how to separate the linear and non-linear parts of the profile.

[4] Van Loon et al. (2007) will be assessed regarding evaluated chloride equilibrium concentrations in a future blog post. However, the study was considered in the post on the failure of Archie’s law in bentonite. Update (220721): Van Loon et al. (2007) is assessed in detail here.

[5] Van Loon et al. (2007) reports evaluated values of “effective porosity”, \(\epsilon_\mathrm{eff}\). I have calculated the clay concentration from these as \(\bar{c}_\mathrm{clay} = c_\mathrm{ext}\cdot \epsilon_\mathrm{eff}/\phi\), where \(\phi\) is the physical porosity. Note that \(\bar{c}_\mathrm{clay}\) is a model independent parameter, while \(\epsilon_\mathrm{eff}\) certainly is not.

[6] Because \(r\) and \(\delta\) may co-vary with density.

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.

Assessment of chloride equilibrium concentrations: Molera et al. (2003)

In the ongoing assessment of chloride equilibrium concentrations in bentonite, we here take a closer look at the study by Molera et al. (2003), in the following referred to as Mo03. We thus assess the 13 points indicated here

Mo03 performed both chloride and iodide through-diffusion tests on “MX-80” bentonite, but here we focus on the chloride results. However, since the only example in the paper of an outflux evolution and corresponding concentration profile is for iodide, this particular result will also be investigated. The tests were performed at background concentrations of 0.01 M or 0.1 M NaClO4, and nominal sample densities of 0.4, 0.8, 1.2, 1.6, and 1.8 g/cm3. We refer to a single test by stating “nominal density/background concentration”, e.g. a test performed at nominal density 1.6 and background concentration 0.1 M is referred to as “1.6/0.1”.

Uncertainty of samples

The material used is discussed only briefly, and the only reference given for its properties is (Müller-Von Moos and Kahr, 1983). I don’t find any reason to believe that the “MX-80” batch used in this study actually is the one investigated in this reference, and have to assume the same type of uncertainty regarding the material as we did in the assessment of Muurinen et al (1988). I therefore refer to that blog post for a discussion on uncertainty in montmorillonite content, cation population, and soluble calcium minerals.

Density

The samples in Mo03 are cylindrical with radius 0.5 cm and length 0.5 cm, giving a volume of 0.39 cm3. This is quite small, and corresponds e.g. only to about 4% of the sample size used in Muurinen et al (1988). With such a small volume, the samples are at the limit for being considered as a homogeneous material, especially for the lowest densities: the samples of density 0.4 g/cm3 contain 0.157 g dry substance in total, while a single 1 mm3 accessory grain weighs about 0.002 — 0.003 g.

Furthermore, as the samples are sectioned after termination, the amount substance in each piece may be very small. This could cause additional problems, e.g. enhancing the effect of drying. The reported profile (1.6/0.1, iodide diffusion) has 10 sections in the first 2 mm. As the total mass dry substance in this sample is 0.628 g, these sections have about 0.025 g dry substance each (corresponding to the mass of about ten 1 mm3 grains). For the lowest density, a similar sectioning corresponds to slices of dry mass 0.006 g (the paper does not give any information on how the low density samples were sectioned).

Mo03 only report nominal densities for the samples, but from the above considerations it is clear that a substantial (but unknown) variation may be expected in densities and concentrations.

A common feature of many through-diffusion studies is that the sample density appears to decrease in the first few millimeters near the confining filters. We saw this effect in the profiles of Muurinen et al (1988), and it has been the topic of some studies, including Mo03. Here, we don’t consider any possible cause, but simply note that the samples seem to show this feature quite generally (below we discuss how Mo03 handle this). Since the samples of Mo03 are only of length 5 mm, we may expect that the major part of them are affected by this effect. Of course, this increases the uncertainty of the actual density of the used samples.

Uncertainty of external solutions

Mo03 do not describe how the external solutions were prepared, other than that they used high grade chemicals. We assume here that the preparation did not introduce any significant uncertainty.

Since “MX-80” contains a substantial amount of divalent ions, connecting this material with (initially) pure sodium solutions inevitably initiates cation exchange processes. The extent of this exchange depends on details such as solution concentrations, reservoir volumes, number of solution replacements, time, etc…

Very little information is given on the volume of the external solution reservoirs. It is only hinted that the outlet reservoir may be 25 ml, and for the inlet reservoir the only information is

The volume of the inlet reservoir was sufficient to keep the concentration nearly constant (within a few percent) throughout the experiments.

Consequently, we do not have enough information to assess the exact ion population during the course of the tests. We can, however, simulate this process of “unintentional exchange” to get some appreciation for the amount of divalent ions still left in the sample, as we did in the assessment of Muurinen et al. (1988). Here are the results from calculating the exchange equilibrium between a sample initially containing 30% exchangeable charge in form of calcium (70% sodium), and external NaClO4 solutions of various concentrations and volumes

In these calculations we assume a sample of density 1.6 g/cm3 (except when indicated), a volume of 0.39 cm3, a cation exchange capacity of 0.75 eq/kg, and a Ca/Na selectivity coefficient of 5.

These simulations make it clear that the tests performed at 0.01 M most probably contain most of the divalent ions initially present in the “MX-80” material: even with an external solution volume of 1000 ml, or with density 0.4 g/cm3, exchange is quite limited. For the tests performed at 0.1 M we expect some exchange of the divalent ions, but we really can’t tell to what extent, as the exact value strongly depends on handling (solution volumes, if solutions were replaced, etc.). That the exact ion population is unknown, and that the divalent/monovalent ratio probably is different for different samples, are obviously major problems of the study (the same problems were identified in Muurinen et al (1988)).

Uncertainty of diffusion parameters

Diffusion model

Mo03 determine diffusion parameters by fitting a model to all available data, i.e the outflux evolution and the concentration profile across the sample at termination. The model is solved by a numerical code (“ANADIFF”) that takes into account transport both in clay samples and filters. The fitted parameters are an apparent diffusivity, \(D_a\), and a so-called “capacity factor”, \(\alpha\). \(\alpha\) is vaguely interpreted as being the combination of a porosity factor \(\epsilon\), and a sorption distribution coefficient \(K_d\), described as “a generic term devoid of mechanism”

\begin{equation} \alpha = \epsilon + \rho\cdot K_d \end{equation}

It is claimed that for anions, \(K_d\) can be treated as negative, giving \(\alpha < \epsilon\). I have criticized this mixing of what actually are incompatible models in an earlier blog post. Strictly, this use of a “generic term devoid of mechanism” means that the evaluated \(\alpha\) should not be interpreted in any particular way. Nevertheless, the way this study is referenced in other publications, \(\alpha\) is interpreted as an effective porosity. It should be noticed, however, that this study is performed with a background electrolyte of NaClO4. The only chloride (or iodide) present is therefore at trace level, and it cannot be excluded that a mechanism of true sorption influences the results (there are indications that this is the case in other studies).

For the present assessment we anyway assume that \(\alpha\) directly quantifies the anion equilibrium between clay and the external solution (i.e. equivalent to the incorrect way of assuming that \(\alpha\) quantifies a volume accessible to chloride). It should be kept in mind, though, that effects of anion equilibrium and potential true sorption is not resolved by the single parameter \(\alpha\).

In practice, then, the model is

\begin{equation} \frac{\partial c}{\partial t} = D_p\frac{\partial^2 c}{\partial x^2} \tag{1} \end{equation}

where \(c\) is the concentration in the clay of the isotope under consideration, and the diffusion coefficient is written \(D_p\) to acknowledge that it is a pore diffusivity (when referring to models and parameter evaluations in Mo03 we will use the notation “\(D_a\)”). The boundary conditions are

\begin{equation} c(0,t) = \alpha \cdot C_0 \;\;\;\; c(L,t) = 0 \tag{2} \end{equation}

where \(C_0\) is the concentration in the source reservoir,1 and \(L\) is the sample length.

This model — that we have discussed before — has a relatively simple analytical solution, and the outflux can be written

\begin{equation} j^\mathrm{out}(t) = j^\mathrm{ss}\left (1 + 2 \sum_{n=1}^\infty \left (-1 \right )^n e^{-\frac{\pi^2n^2D_pt}{L^2}} \right) \end{equation}

where \(j^\mathrm{ss}\) is the corresponding steady-state flux. Here, the steady-state flux is related to the other parameters as

\begin{equation} j^\mathrm{ss} = \alpha\cdot D_p \frac{C_0}{L} \tag{3} \end{equation}

“Fast” and “slow” processes

Oddly, Mo03 model the system as if two independent diffusion processes are simultaneously active. They refer to these as the “fast” and the “slow” processes, and hypothesize that they relate to diffusion in interlayer water2 and “interparticle water”,3 respectively.

The “fast” process is the “ordinary” process that is assumed to reach steady state during the course of the test, and that is the focus of other through-diffusion studies. The “slow” process, on the other hand, is introduced to account for the frequent observation that measured tracer profiles are usually significantly non-linear near the interface to the source reservoir (discussed briefly above). I guess that the reason for this concentration variation is due to swelling when the sample is unloaded. But even if the reason is not fully clear, it can be directly ruled out that it is the effect of a second, independent, diffusion process — because this is not how diffusion works!

If anions move both in interlayers and “interparticle water”, they reasonably transfer back and forth between these domains, resulting in a single diffusion process (the diffusivity of such a process depends on the diffusivity of the individual domains and their geometrical configuration). To instead treat diffusion in each domain as independent means that these processes are assumed to occur without transfer between the domains, i.e. that the bentonite is supposed to contain isolated “interlayer pipes”, and “interparticle pipes”, that don’t interact. It should be obvious that this is not a reasonable assumption. Incidentally, this is how all multi-porous models assume diffusion to occur (while simultaneously assuming that the domains are in local equilibrium…).

We will thus focus on the “fast” process in this assessment, although we also use the information provided by the parameters for the “slow” process. Mo03 report the fitted values for \(D_a\) and \(\alpha\) in a table (and diagrams), and only show a comparison between model and measured data in a single case: for iodide diffusion at 0.1 M background concentration and density 1.6 g/cm3. To make any kind of assessment of the quality of these estimations we therefore have to focus on this experiment (the article states that these results are “typical high clay density data”).

Outflux

The first thing to note is that the modeled accumulated diffusive substance does not correspond to the analytical solution for the diffusion process. Here is a figure of the experimental data and the reported model (as presented in the article), that also include the solution to eqs. 1 and 2.

In fact, the model presented in Mo03 has an incorrect time dependency in the early stages. Here is a comparison between the presented model and analytical solutions in the transient stage

With the given boundary conditions, the solutions to the diffusion equation inevitably has zero slope at \(t = 0\),4 reflecting that it takes a finite amount of time for any substance to reach the outflux boundary. The models presented in Mo03, on the other hand, has a non-zero slope in this limit. I cannot understand the reason for this (is it an underlying problem with the model, or just a graphical error?), but it certainly puts all reported parameter values in doubt.

The preferred way to evaluate diffusion data is, in my opinion, to look at the flux evolution rather than the evolution of the accumulated amount of diffused substance. Converting the reported data to flux, gives the following picture.5

From a flux evolution it is easier to establish the steady-state, as it reaches a constant. It furthermore gives a better understanding for how well constrained the model is by the data. As is seen from the figure, the model is not at all very well constrained, as the experimental data almost completely miss the transient stage. (And, again, it is seen that the model in the paper with \(D_a= 9\cdot 10^{-11}\) m/s2 does not correspond to the analytical solution.)

The short transient stage is a consequence of using thin samples (0.5 cm). Compared e.g. to Muurinen et al (1988), who used three times as long samples, the breakthrough time is here expected to be \(3^2 = 9\) times shorter. As Muurinen et al. (1988) evaluated breakthrough times in the range 1 — 9 days, we here expect very short times. Here are the breakthrough times for all chloride diffusion tests, evaluated from the reported diffusion coefficients (“fast” process) using the formula \(t_\mathrm{bt} = L^2/(6D_a)\).

Test\(D_a\)\(t_\mathrm{bt}\)
(m2/s) (days)
0.4/0.01\(8\cdot 10^{-10}\)0.06
0.4/0.1 \(9\cdot 10^{-10}\) 0.05
0.4/0.1 \(8\cdot 10^{-10}\) 0.06
0.8/0.01 \(3.5\cdot 10^{-10}\) 0.14
0.8/0.1 \(3.5\cdot 10^{-10}\) 0.14
0.8/0.1 \(3.7\cdot 10^{-10}\) 0.13
1.2/0.01 \(1.4\cdot 10^{-10}\)0.34
1.2/0.1 \(2.3\cdot 10^{-10}\) 0.21
1.2/0.1 \(2.0\cdot 10^{-10}\) 0.24
1.6/0.1 \(1.0\cdot 10^{-10}\) 0.48
1.8/0.01 \(2\cdot 10^{-11}\) 2.41
1.8/0.1 \(5\cdot 10^{-11}\) 0.96
1.8/0.1 \(5.5\cdot 10^{-11}\) 0.88

The breakthrough time is much shorter than a day in almost all tests! To sample the transient stage properly requires a sampling frequency higher than \(1/t_{bt}\). As seen from the provided example of a outflux evolution, this is not the case: The second measurement is done after about 1 day, while the breakthrough time is about 0.5 days (moreover, the first measurement appears as an outlier). We have no information on sampling frequency in the other tests, but note that to properly sample e.g. the tests at 0.8 g/cm3 requires measurements at least every third hour or so. For 0.4 g/cm3, the required sample frequency is once an hour! This design choice puts more doubt on the quality of the evaluated parameters.

Concentration profile

The measured concentration profile across the 1.6/0.1 iodide sample, and corresponding model results are presented in Mo03 in a figure very similar to this

Here the two models correspond to the “slow” and “fast” process discussed above (a division, remember, that don’t make sense). Zooming in on the “linear” part of the profile, we can compare the “fast” process with analytical solutions (eqs. 1 and 2)

The analytical solutions correspond directly to the outflux curves presented above. We note that the analytical solution with \(D_p = 9\cdot 10^{-11}\) m/s2 corresponds almost exactly to the model presented by Mo03. As this model basically has the same steady state flux and diffusion coefficient, we expect this similarity. It is, however, still a bit surprising, since the corresponding outflux curve of the model in Mo03 was seen to not correspond to the analytical solution. This continues to cast doubt on the model used for evaluating the parameters.

We furthermore note that the evolution of the activity of the source reservoir is not reported. Once in the text is mentioned that the “carrier concentration” is \(10^{-6}\) M, but since we don’t know how much of this concentration corresponds to the radioactive isotope, we can not directly compare with reported concentration profile across the sample (whose concentration unit is counts per minute per cm3). By extrapolating the above model curve with \(\alpha = 0.15\), we can however deduce that the corresponding source activity for this particular sample is \(C_0 = 1.26\cdot 10^5/0.15\) cpu/cm3 \(= 8.40\cdot 10^5\) cpu/cm3. But it is unsatisfying that we cannot check this independently. Also, we can of course not assume that this value of \(C_0\) is the same in any other of the tests (in particular those involving chloride). We thus lack vital information (\(C_0\)) to be able to make a full assessment of the model fitting.

It should furthermore be noticed that the experimental concentration profile does not constrain the models very well. Indeed, the adopted model (diffusivity \(9\cdot 10^{-11}\) m/s2) misses the two rightmost concentration points (which corresponds to half the sample!). A model that fits this part of the profile has a considerable higher diffusivity, and a correspondingly lower \(\alpha\) (note that the product \(D_p\cdot \alpha\) is constrained by the steady-state flux, eq. 3).

More peculiarities of the modeling is found if looking at the “slow” process (remember that this is not a real diffusion process!). Zooming in on the interface part of the profile and comparing with analytical solutions gives this picture

Here we note that an analytical solution coincides with the model presented in Mo03 with parameters \(D_a = 6\cdot 10^{-14}\) m2/s and \(\alpha = 1.12\) only if it is propagated for about 15 days! Given that no outflux measurements seem to have been performed after about 4 days (see above), I don’t now what to make of this. Was the test actually conducted for 15 days? If so, why is not more of the outflux measured/reported? (And why were the samples then designed to give a breakthrough time of only a few hours?)

Without knowledge of for how long the tests were conducted, the reported diffusion parameters becomes rather arbitrary, especially for the low density samples. For e.g. the samples of density 0.4 g/cm3, even the “slow” process has a diffusivity high enough to reach steady-state within a few days. Simulating the processes with the reported parameters gives the following profiles if evaluated after 1 and 4 days, respectively

The line denoted “total” is what should resemble the measured (unreported) data. It should be clear from these plots that the division of the profile into two separate parts is quite arbitrary. It follows that the evaluated diffusion parameters for the process of which we are interested (“fast”) has little value.

Summary and verdict

We have seen that the reported model fitting leaves a lot of unanswered questions: some of the model curves don’t correspond to the analytical solutions, information on evolution times and source concentrations is missing, and the modeled profiles are divided quite arbitrary into two separate contributions (which are not two independent diffusion process).

Moreover, the ion population (divalent vs. monovalent cations) of the samples are not known, but there are strong reasons to believe that the 0.01 M tests contain a significant amount of divalent ions, while the 0.1 M samples are partly converted to a more pure sodium state.

Also, the small size of the samples contributes to more uncertainty, both in terms of density, but also for the flux evolution because the breakthrough times becomes very short.

Based on all of these uncertainties, I mean that the results of Mo03 does not contribute to quantitative process understanding and my decision is to not to use the study for e.g. validating models of anion exclusion.

A confirmation of the uncertainty in this study is given by considering the density dependence on the chloride equilibrium concentrations for constant background concentration, evaluated from the reported diffusion parameters (\(\alpha\) for the “fast” process).

If these results should be taken at face value, we have to accept a very intricate density dependence: for 0.1 M background, the equilibrium concentration is mainly constant between densities 0.3 g/cm3 and 0.7 g/cm3, and increases between densities 1.0 g/cm3 and 1.45 g/cm3 (or, at least, does not decrease). For 0.01 M background, the equilibrium concentration instead falls quite dramatically between between densities 0.3 g/cm3 and 0.7 g/cm3, and thereafter displays only a minor density dependence.

To accept such dependencies, I require a considerably more rigorous experimental procedure and evaluation. In this case, I rather view the above plot as a confirmation of large uncertainties in parameter evaluation and sample properties.

Footnotes

[1] Strictly, \(c(0,t)\) relates to the concentration in the endpoint of the inlet filter. But we ignore filter resistance in this assessment, which is valid for the 1.6/0.1 sample. Moreover, the filter diffusivities are not reported in Mo03.

[2] Mo03 refer to interlayer pores as “intralayer” pores, which may cause some confusion.

[3] Apparently, the authors assume an underlying stack view of the material.

[4] It may be objected that the analytical solution do not include the filter resistance. But note that filter resistance only will increase the delay. Moreover, the transport capacity of the sample in this test is so low that filters have no significant influence.

[5] The model by Mo03 looks noisy because I have read off values of accumulated concentration from the published graph. The “noise” occurs because the flux is evaluated from the concentration data by the difference formula:

\begin{equation} \bar{j}(\bar{t}_i) =\frac{1}{A} \frac{a(t_{i+1})-a(t_i)}{t_{i+1}-t_{i}} \end{equation}

where \(t_i\) and \(t_{i+1}\) are the time coordinates for two consequitive data points, \(a(t)\) is the accumulated amount diffused substance at time \(t\), \(A\) is the cross sectional area of the sample, \(\bar{t}_i = (t_{i+1} + t_i)/2\) is the average time of the considered time interval, and \(\bar{j}\) denotes the average flux during this time interval.

The failure of Archie’s law validates the homogeneous mixture model

A testable difference

In the homogeneous mixture model, the effective diffusion coefficient for an ion in bentonite is evaluated as

\begin{equation} D_e = \phi \cdot \Xi \cdot D_c \tag{1} \end{equation}

where \(\phi\) is the porosity of the sample, \(D_c\) is the macroscopic pore diffusivity of the presumed interlayer domain, and \(\Xi\) is the ion equilibrium coefficient. \(\Xi\) quantifies the ratio between internal and external concentrations of the ion under consideration, when the two compartments are in equilibrium.

In the effective porosity model, \(D_e\) is instead defined as

\begin{equation} D_e = \epsilon_\mathrm{eff}\cdot D_p \tag{2} \end{equation}

where \(\epsilon_\mathrm{eff}\) is the porosity of a presumed bulk water domain where anions are assumed to reside exclusively, and \(D_p\) is the corresponding pore diffusivity of this bulk water domain.

We have discussed earlier how the homogeneous mixture and the effective porosity models can be equally well fitted to a specific set of anion through-diffusion data. The parameter “translation” is simply \(\phi\cdot \Xi \leftrightarrow \epsilon_\mathrm{eff}\) and \(D_c \leftrightarrow D_p\). It may appear from this equivalency that diffusion data alone cannot be used to discriminate between the two models.

But note that the interpretation of how \(D_e\) varies with background concentration is very different in the two models.

  • In the homogeneous mixture model, \(D_c\) is not expected to vary with background concentration to any greater extent, because the diffusing domain remains essentially the same. \(D_e\) varies in this model primarily because \(\Xi\) varies with background concentration, as a consequence of an altered Donnan potential.
  • In the effective porosity model, \(D_p\) is expected to vary, because the volume of the bulk water domain, and hence the entire domain configuration (the “microstructure”), is postulated to vary with background concentration. \(D_e\) thus varies in this model both because \(D_p\) and \(\epsilon_\mathrm{eff}\) varies.

A simple way of taking into account a varying domain configuration (as in the effective porosity model) is to assume that \(D_p\) is proportional to \(\epsilon_\mathrm{eff}\) raised to some power \(n – 1\), where \(n > 1\). Eq. 2 can then be written

\begin{equation} D_e = \epsilon_\mathrm{eff}^n\cdot D_0 \tag{3} \end{equation} \begin{equation}\text{ (effective porosity model)} \end{equation}

where \(D_0\) is the tracer diffusivity in pure bulk water. Eq. 3 is in the bentonite literature often referred to as “Archie’s law”, in analogy with a similar evaluation in more conventional porous systems. Note that with \(D_0\) appearing in eq. 3, this expression has the correct asymptotic behavior: in the limit of unit porosity, the effective diffusivity reduces to that of a pure bulk water domain.

Eq. 3 shows that \(D_e\) in the effective porosity model is expected to depend non-linearly on background concentration for constant sample density. In contrast, since \(D_c\) is not expected to vary significantly with background concentration, we expect a linear dependence of \(D_e\) in the homogeneous mixture model. Keeping in mind the parameter “translation” \(\phi\cdot\Xi \leftrightarrow \epsilon_\mathrm{eff}\), the prediction of the homogeneous mixture model (eq. 1) can be expressed1

\begin{equation} D_e = \epsilon_\mathrm{eff}\cdot D_c \tag{4} \end{equation} \begin{equation} \text{ (homogeneous mixture model)} \end{equation}

We have thus managed to establish a testable difference between the effective porosity and the homogeneous mixture model (eqs. 3 and 4). This is is great! Making this comparison gives us a chance to increase our process understanding.

Comparison with experiment

Van Loon et al. (2007)

It turns out that the chloride diffusion measurements performed by Van Loon et al. (2007) are accurate enough to resolve whether \(D_e\) depends on “\(\epsilon_\mathrm{eff}\)” according to eqs. 3 or 4. As will be seen below, this data shows that \(D_e\) varies in accordance with the homogeneous mixture model (eq. 4). But, since Van Loon et al. (2007) themselves conclude that \(D_e\) obeys Archie’s law, and hence complies with the effective porosity model, it may be appropriate to begin with some background information.

Van Loon et al. (2007) report three different series of diffusion tests, performed on bentonite samples of density 1300, 1600, and 1900 kg/m3, respectively. For each density, tests were performed at five different NaCl background concentrations: 0.01 M, 0.05 M, 0.1 M, 0.4 M, and 1.0 M. The tests were evaluated by fitting the effective porosity model, giving the effective diffusion coefficient \(D_e\) and corresponding “effective porosity” \(\epsilon_\mathrm{eff}\) (it is worth repeating that the latter parameter equally well can be interpreted in terms of an ion equilibrium coefficient).

Van Loon et al. (2007) conclude that their data complies with eq. 3, with \(n = 1.9\), and provide a figure very similar to this one

Effective diffusivity vs. "effective porsity" for a bunch of studies (fig 8 in Van Loon et al. (2007))

Here are compared evaluated values of effective diffusivity and “effective porosity” in various tests. The test series conducted by Van Loon et al. (2007) themselves are labeled with the corresponding sample density, and the literature data is from García-Gutiérrez et al. (2006)2 (“Garcia 2006”) and the PhD thesis of A. Muurinen (“Muurinen 1994”). Also plotted is Archie’s law with \(n\) =1.9. The resemblance between data and model may seem convincing, but let’s take a further look.

Rather than lumping together a whole bunch of data sets, let’s focus on the three test series from Van Loon et al. (2007) themselves, as these have been conducted with constant density, while only varying background concentration. This data is thus ideal for the comparison we are interested in (we’ll get back to commenting on the other studies).

It may also be noted that the published plot contains more data points (for these specific test series) than are reported in the rest of the article. Let’s therefore instead plot only the tabulated data.3 The result looks like this

Effective diffusivity vs. "effective porosity" as evaluated in Van Loon et al. (2007) compared with Archie's law (n=1.9) and the homogenous mixter model predictions.

Here we have also added the predictions from the homogeneous mixture model (eq. 4), where \(D_c\) has been fitted to each series of constant density.

The impression of this plot is quite different from the previous one: it should be clear that the data of Van Loon et al. (2007) agrees fairly well with the homogeneous mixture model, rather than obeying Archie’s law. Consequently, in contrast to what is stated in it, this study refutes the effective porosity model.

The way the data is plotted in the article is reminiscent of Simpson’s paradox: mixing different types of dependencies of \(D_e\) gives the illusion of a model dependence that really isn’t there. Reasonably, this incorrect inference is reinforced by using a log-log diagram (I have warned about log-log plots earlier). With linear axes, the plots give the following impression

Effective diffusivity vs. "effective porosity" as evaluated in Van Loon et al. (2007) compared with Archie's law (n=1.9) and the homogenous mixter model predictions. Linear diagram axes.

This and the previous figure show that \(D_e\) depends approximately linearly on “\(\epsilon_\mathrm{eff}\)”, with a slope dependent on sample density. With this insight, we may go back and comment on the other data points in the original diagram.

García-Gutiérrez et al. (2006) and Muurinen et al. (1988)

The tests by García-Gutiérrez et al. (2006) don’t vary the background concentration (it is not fully clear what the background concentration even is4), and each data point corresponds to a different density. This data therefore does not provide a test for discriminating between the models here discussed.

I have had no access to Muurinen (1994), but by examining the data, it is clear that it originates from Muurinen et al. (1988), which was assessed in detail in a previous blog post. This study provides two estimations of “\(\epsilon_\mathrm{eff}\)”, based on either breakthrough time or on the actual measurement of the final state concentration profile. In the above figure is plotted the average of these two estimations.5

One of the test series in Muurinen et al. (1988) considers variation of density while keeping background concentration fixed, and does not provide a test for the models here discussed. The data for the other two test series is re-plotted here, with linear axis scales, and with both estimations for “\(\epsilon_\mathrm{eff}\)”, rather than the average6

Effective diffusivity vs. "effective porosity" as evaluated in Muurinen et al. (1988) compared with Archie's law (n=1.9) and the homogenous mixter model predictions. Linear diagram axes.

As discussed in the assessment of this study, I judge this data to be too uncertain to provide any qualitative support for hypothesis testing. I think this plot confirms this judgment.

Glaus et al. (2010)

The measurements by Van Loon et al. (2007) are enough to convince me that the dependence of \(D_e\) for chloride on background concentration is further evidence for that a homogeneous view of compacted bentonite is principally correct. However, after the publication of this study, the same authors (partly) published more data on chloride equilibrium, in pure Na-montmorillonite and “Na-illite”,7 in Glaus et al. (2010).

This data certainly shows a non-linear relation between \(D_e\) and “\(\epsilon_\mathrm{eff}\)” for Na-montmorillonite, and Glaus et al. (2010) continue with an interpretation using “Archie’s law”. Here I write “Archie’s law” with quotation marks, because they managed to fit the expression to data only by also varying the prefactor. The expression called “Archie’s law” in Glaus et al. (2010) is

\begin{equation} D_e = A\cdot\epsilon_\mathrm{eff}^n \tag{5} \end{equation}

where \(A\) is now a fitting parameter. With \(A\) deviating from \(D_0\), this expression no longer has the correct asymptotic behavior as expected when interpreting \(\epsilon_\mathrm{eff}\) as quantifying a bulk water domain (see eq. 3). Nevertheless, Glaus et al. (2010) fit this expression to their measurements, and the results look like this (with linear axes)

Effective diffusivity vs. "effective porosity" as evaluated in Glaus al. (2010) compared with "Archie's law" (n=1.9, fitted A) and the homogenous mixter model predictions. Linear diagram axes.

Here is also plotted the prediction of the homogeneous mixture model (eq. 4). For the montmorillonite data, the dependence is clearly non-linear, while for the “Na-illite” I would say that the jury is still out.

Although the data for montmorillonite in Glaus et al. (2010) is non-linear, there are several strong arguments for why this is not an indication that the effective porosity model is correct:

  • Remember that this result is not a confirmation of the measurements in Van Loon et al. (2007). As demonstrated above, those measurements complies with the homogeneous mixture model. But even if accepting the conclusion made in that publication (that Archie’s law is valid), the Glaus et al. (2010) results do not obey Archie’s law (but “Archie’s law”).
  • The four data points correspond to background concentrations of 0.1 M, 0.5 M, 1.0 M, and 2.0 M. If “\(\epsilon_\mathrm{eff}\)” represented the volume of a bulk water phase, it is expected that this value should level off, e.g. as the Debye screening length becomes small (Van Loon et al. (2007) argue for this). Here “\(\epsilon_\mathrm{eff}\)” is seen to grow significantly, also in the transition between 1.0 M and 2.0 M background concentration.
  • These are Na-montmorillonite samples of dry density 1.9 g/cm3. With an “effective porosity” of 0.067 (the 2.0 M value), we have to accept more than 20% “free water” in these very dense systems! This is not even accepted by other proponents of bulk water in compacted bentonite.

Furthermore, these tests were performed with a background of \(\mathrm{NaClO_4}\), in contrast to Van Loon et al. (2007), who used chloride also for the background. The only chloride around is thus at trace level, and I put my bet on that the observed non-linearity stems from sorption of chloride on some system component.

Insight from closed-cell tests

Note that the issue whether or not \(D_e\) varies linearly with “\(\epsilon_\mathrm{eff}\)” at constant sample density is equivalent to whether or not \(D_p\) (or \(D_c\)) depends on background concentration. This is similar to how presumed concentration dependencies of the pore diffusivity for simple cations (“apparent” diffusivities) have been used to argue for multi-porosity in compacted bentonite. For cations, a closer look shows that no such dependency is found in the literature. For anions, it is a bit frustrating that the literature data is not accurate or relevant enough to fully settle this issue (the data of Van Loon et al. (2007) is, in my opinion, the best available).

However, to discard the conceptual view underlying the effective porosity model, we can simply use results from closed-cell diffusion studies. In Na-montmorillonite equilibrated with deionized water, Kozaki et al. (1998) measured a chloride diffusivity of \(1.8\cdot 10^{-11}\) m2/s at dry density 1.8 g/cm3.8 If the effective porosity hypothesis was true, we’d expect a minimal value for the diffusion coefficient9 in this system, since \(\epsilon_\mathrm{eff}\) approaches zero in the limit of vanishing ionic strength. Instead, this value is comparable to what we can evaluate from e.g. Glaus et al. (2010) at 1.9 cm3/g, and 2.0 M background electrolyte: \(D_e/\epsilon_\mathrm{eff} = 7.2\cdot 10^{-13}/0.067\) m2/s = \(1.1\cdot 10^{-11}\) m2/s.

That chloride diffuses just fine in dense montmorillonite equilibrated with pure water is really the only argument needed to debunk the effective porosity hypothesis.

Footnotes

[1] Note that \(\epsilon_\mathrm{eff}\) is not a parameter in the homogeneous mixture model, so eq. 4 looks a bit odd. But it expresses \(D_e\) if \(\phi\cdot \Xi\) is interpreted as an effective porosity.

[2] This paper appears to not have a digital object identifier, nor have I been able to find it in any online database. The reference is, however, Journal of Iberian Geology 32 (2006) 37 — 53.

[3] This choice is not critical for the conclusions made in this blog post, but it seems appropriate to only include the data points that are fully described and reported in the article.

[4] García-Gutiérrez et al. (2004) (which is the study compiled in García-Gutiérrez et al. (2006)) state that the samples were saturated with deionized water, and that the electric conductivity in the external solution were in the range 1 — 3 mS/cm.

[5] The data point labeled with a “?” seems to have been obtained by making this average on the numbers 0.5 and 0.08, rather than the correctly reported values 0.05 and 0.08 (for the test at nominal density 1.8 g/cm3 and background concentration 1.0 M).

[6] Admittedly, also the data we have plotted from the original tests in Van Loon et al. (2007) represents averages of several estimations of “\(\epsilon_\mathrm{eff}\)”. We will get back to the quality of this data in a future blog post when assessing this study in detail, but it is quite clear that the estimation based on the direct measurement of stable chloride is the more robust (it is independent of transport aspects). Using these values for “\(\epsilon_\mathrm{eff}\)”, the corresponding plot looks like this

Effective diffusivity vs. "effective porosity" as evaluated in Van Loon et al. (2007) compared with Archie's law (n=1.9) and the homogenous mixter model predictions. Linear diagram axes. The data for "effective porosity" evaluated solely from measurements of stable chloride measurements.

Update (220721): Van Loon et al. (2007) is assessed in detail here.

[7] To my mind, it is a misnomer to describe something as illite in sodium form. Although “illite” seems to be a bit vaguely defined, it is clear that it is supposed to only contain potassium as counter-ion (and that these ions are non-exchangeable; the basal spacing is \(\sim\)10 Å independent of water conditions). The material used in Glaus et al. (2010) (and several other studies) has a stated cation exchange capacity of 0.22 eq/kg, which in a sense is comparable to the montmorillonite material (a factor 1/4). Shouldn’t it be more appropriate to call this material e.g. “mixed-layer”?

[8] This value is the average from two tests performed at 25 °C. The data from this study is better compiled in Kozaki et al. (2001).

[9] Here we refer of course to the empirically defined diffusion coefficient, which I have named \(D_\mathrm{macr.}\) in earlier posts. This quantity is model independent, but it is clear that it should be be associated with the pore diffusivities in the two models here discussed (i.e. with \(D_c\) in the homogeneous mixture model, and with \(D_p\) in the effective porosity model).