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 test 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?

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)

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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.

Are interlayer cations not attracted to the surfaces?!

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

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

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

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

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

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

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

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

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

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

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

A couple of corrections

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

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

But the final sentence should rather be formulated as

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

In the original post, I also write

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

The last part is better formulated as

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

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

Footnotes

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

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

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

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

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Introduction

In the ongoing assessment of chloride equilibrium concentrations in bentonite, we here take a closer look at the study by Muurinen et al. (2007), in the following referred to as Mu07. We thus assess the 11 points indicated here

Mu07 actually report 9 more data points, but these originate from Muurinen et al. (2004) (which we have already assessed). This is not fully acknowledged in Mu07, but below I try to sort out the status of all data presented. We refer to Muurinen et al. (2004) as Mu04.

In similarity to Mu04, Mu07 is an equilibrium study (i.e. not a diffusion study) performed on purified “MX-80” bentonite. One of the main objectives in Mu07 is to investigate possible influence of sample preparation on the chloride equilibrium concentrations. The samples in Mu07 cover a large density range (0.6 — 1.5 g/cm³), but were all equilibrated with a single type of solution: 0.1 M NaCl.

Originally conducted and already reported tests

Mu07 state that the study is a continuation of the investigations presented in Mu04 and present data on five different sets of samples, prepared and equilibrated using different methods (labeled A — E). What is not explicitly stated — but what is obvious if comparing tables 1 and 2 in Mu04 with table 1 in Mu07 — is that sample sets D and E are the same as previously reported in Mu04.

I find this quite remarkable, since two of these samples were dismissed as “not reliable” in Mu04 (In my assessment, I dismissed all tests in Mu04); here the same results — which show an increase in equilibrium chloride content with density — are not only re-reported, but modeled! The authors don’t even seem aware that they have previously discarded the samples, writing: “Surprisingly, it seems that the concentrations in the sample types D and E start to increase at the highest densities“. Furthermore, one of the (previously reported) data points of sample set E, which have a clay concentration larger than the corresponding concentration of the equilibrating solution, is not included in Mu07. Needless to say, excluding data points without motivation, or including previously discarded data is not good scientific practice.

As the sample overview table in Mu07 also has some misprints,¹ I here present a (hopefully) correct version that also indicates original publication (for the indicated sample-IDs, see the assessment of Mu04).

Type	Density (g/cm3)	Time d.w. (days)	Time 0.1 M (days)	Clay conc. (mM)	origin/remark
A	0.625	18+35	217	37	Mu07
A	0.812	18+35	217	29	Mu07
A	1.200	18+35	217	20	Mu07
B	0.670	222	107	35	Mu07
B	0.937	222	107	23	Mu07
B	1.389	222	107	15	Mu07
C	0.622	30	36	48	Mu07
C	0.731	30	36	30	Mu07
C	1.113	30	36	17	Mu07
C	1.382	30	36	14	Mu07
C	1.517	30	36	12	Mu07
D	0.754	0	40	65	Mu04 (S2-02)
D	0.855	0	40	39	Mu04 (S2-21)
D	1.273	0	36	22	Mu04 (S2-04)
D	1.636	0	36	24	Mu04 (S2-17; deemed “not reliable” in Mu04)
D	1.764	0	85	48	Mu04 (S2-18; deemed “not reliable” in Mu04)
E	0.750	0	12(?)	109	Mu04 (not included!)
E	0.875	0	12	61	Mu04
E	1.225	0	12	25	Mu04
E	1.516	0	12	12	Mu04
E	1.543	0	12	14	Mu04

In the following, the focus is solely on samples sets A — C (as mentioned, the others have already been assessed).

Material

The material appear to be the same as used in Mu04. I therefore refer to the assessment of that study for a detailed discussion. In brief, the material is purified “MX-80” bentonite, with a montmorillonite content above 90% and about 90% sodium as exchangeable cation.

Samples

Samples in the three different sample sets A — C were prepared in different ways. For set A, the clay was initially dispersed in deionized water at quite low density. After an equilibration time of 18 days (which included ultrasound treatment), the dispersion was slowly squeezed to achieve the intended densities. This squeezing phase lasted 35 days, after which the samples were contacted with 0.1 M NaCl and equilibrated for 217 days.

Samples in set B were prepared in the same type of sample holder as those in set A, but the bentonite powder was directly compacted to the desired density, and the samples were water saturated by contact with deionized water for 222 days (!). Thereafter, the samples were contacted with 0.1 M NaCl and equilibrated for 107 days.

The external solution was not circulated in the preparation of samples in sets A and B. In contrast, samples in set C were prepared in cells with external circulation. The bentonite powder was directly compacted to the desired density, and the samples were water saturated by contact with (circulating) deionized water for 30 days. The samples were then equilibrated with (circulating) 0.1 M NaCl solution for 36 days.

Even if the preparation protocols are described quite detailed in Mu07, we are not given any information on sample geometry. We are not even told if the samples have the same geometry! (Given that they were prepared in different types of equipment, different geometries may certainly be the case.) Without knowledge on e.g. the characteristic diffusion lengths, it is impossible to assess e.g. whether the adopted equilibration times are adequate. Reasonably, the size of the samples are on the cm scale, and since the equilibration times are very long, we can guess that they have had time to equilibrate. This is in contrast to the samples in Mu04, which we have reason to suspect have not been completely equilibrated, as discussed in the assessment of that study. (Note that these samples are included in Mu07, as sample sets D and E.)

Mu07 does not provide any information on how sample density was measured. Since we neither know the dimensions of the samples it is therefore impossible to estimate any uncertainty of the reported densities.

Chloride equilibrium concentrations

The following plot summarizes the reported chloride equilibrium concentrations and corresponding densities in sample sets A — C.

Although the data show some significant scatter (e.g. for the two lowest densities in sample set C), the main impression is that the three different ways of preparing and equilibrating samples result in quite similar values for the chloride equilibrium concentrations. Thus, even if we know little about the samples, this coherence in the results indicates that they have been properly equilibrated.

Possible interface excess salt

As we have discussed in several previous blog posts, when performing equilibrium tests it is important to handle the possibility that the samples have an increased salt content in the interface regions. In the assessment of Mu04, my guess was that the samples had not been handled specifically to deal with this possible measuring artifact, and I neither see any reason to believe that this issue has been addressed in sample sets A — C (we can, however, rule out that too much salt entered these samples during saturation, since deionized water was used in this phase).

The possible influence of interface excess depends, apart from general sample treatment, on e.g. sample thickness and the concentration of the equilibrating external solution. As noted above, we have no information on sample thickness, but the external concentration is in this regard quite low (we showed in an earlier post that the problem of interface excess salt becomes more severe for thin samples and low external concentrations). Therefore, we can certainly not exclude the possibility that the reported equilibrium concentrations are systematically overestimated due to possible influence of an interface excess, especially for the denser samples (see here for details on this).

An argument against that interface excess has significantly influenced the results is the similar result for the three different sample sets. Of course, this depends on how similar (or dissimilar) the samples in the different sets are, of which we have no information. Under any circumstance, it is very clear that Mu07 provides too little information to fully rely on the reported values.

Summary and verdict

From one perspective, Mu07 is a very straightforward study: samples of purified bentonite (almost pure Na-montmorillonite) at various density have been equilibrated with a single type of external solution (0.1 M NaCl). The results also look reasonably coherent. However, the paper contains way too little information on e.g. sample geometry and how density and concentration were measured to fully rely on the results. In particular, we cannot rule out a systematic overestimation due to influence of interface excess salt. Furthermore, the main reason to believe that equilibrium was achieved, is the similarity between the different test sets.

My decision, however, is to keep these result to use e.g. for possible qualitative process understanding (specifically, chloride exclusion). But I will certainly keep in mind the quite extensive lack of information associated with this data.

Footnotes

[1] The presented diagram in Mu07 (fig. 5) don’t suffer from these typos.

Solving the Poisson-Boltzmann equation

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To celebrate that I have built myself a tool for solving the Poisson-Boltzmann equation for two parallel charged plates and specified external solution, here is a cosy little animation

The animation shows the anion concentration profile (blue) between the plates as the distance varies, in systems in equilibrium with an external 100 mM 1:1 salt solution. Also plotted is the corresponding internal concentration level as calculated from the ideal Donnan equilibrium formula (orange). The layer charge density in the Poisson-Boltzmann calculation is 0.111 C/m², and the corresponding cation exchange capacity in the Donnan calculation is 0.891 eq/kg.

As the distance between the plates increases, the Poisson-Boltzmann profile increasingly deviates from the Donnan concentration. At lower density (larger plate distance) it is clear that the Poisson-Boltzmann solution allows for considerably more anions between the plates as compared with the Donnan result. On the other hand, for denser systems, the difference between the two solutions decreases; this is especially true when considering the relative difference — keep in mind that the external concentration is kept constant, at 100 mM.

In fact, in systems relevant for e.g. radioactive waste storage — spanning an effective montmorillonite density range from \(\rho_\mathrm{mmt} =\) 1.60 g/cm³ to \(\rho_\mathrm{mmt} =\) 1.15 g/cm³, say — the difference between the Poisson-Boltzmann and the Donnan results is virtually negligible (it should also be kept in mind that the continuum assumption underlying the Poisson-Boltzmann calculation is not valid in this density range). Here are plotted snapshots of these two limiting cases, together with the Poisson-Boltzmann solution for a single plate (the Gouy-Chapman model)

This figure clearly shows that the Gouy-Chapman model is not at all valid in any relevant system, unless you postulate larger voids in the bentonite. But why would you do that?

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

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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

Saturation stage
Through-diffusion stage
- Transient phase
- Steady-state phase
Out-diffusion stage
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
1.6	1.36 — 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)
1.3/0.01	0.034	0.06	0.051
1.3/0.05	0.045	0.08	–
1.3/0.1	0.090^*	0.17	0.162
1.3/0.4	0.140	0.26	–
1.3/1.0	0.220	0.41	0.400

1.6/0.01	0.009	0.02	0.019
1.6/0.05	0.016^**	0.04	–
1.6/0.1	0.029	0.07	0.066
1.6/0.4	0.060	0.14	–
1.6/1.0	0.110	0.26	0.239

1.9/0.01	0.009	0.03	discarded
1.9/0.05	0.007	0.02	–
1.9/0.1	0.015	0.05	0.044
1.9/0.4	0.017	0.05	–
1.9/1.0	0.044	0.14	0.128

*) 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)	\(\bar{c}_0/c_\mathrm{source}\) (-)
1.3/0.01	4.10	0.038
1.3/0.05	10.2	0.097
1.3/0.1	17.8	0.168
1.3/0.4	41.4	0.395
1.3/1.0	52.4	0.445

1.6/0.01	1.21	0.014
1.6/0.05	3.64	0.043
1.6/0.1	6.15	0.072
1.6/0.4	13.0	0.154
1.6/1.0	21.6	0.225

1.9/0.01	0.41	0.006
1.9/0.05	1.14	0.018
1.9/0.1	1.64	0.025
1.9/0.4	3.19	0.051
1.9/1.0	8.19	0.113

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)
1.3/0.01	2.6	2.6
1.3/0.05	7.5	7.5
1.3/0.1	16	16
1.3/0.4	25	25
1.3/1.0	49	49

1.6/0.01	0.39	0.39
1.6/0.05	1.1	1.1
1.6/0.1	2.3	2.3
1.6/0.4	4.6	4.6
1.6/1.0	10	10

1.9/0.01	0.033	0.033
1.9/0.05	0.12	0.12
1.9/0.1	0.24	0.24
1.9/0.4	0.5	0.5
1.9/1.0	1.2	1.2

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)

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“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.

Semi-permeability, part I

Footnotes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Assessment of chloride equilibrium concentrations: Muurinen et al. (2004)

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In the ongoing assessment of chloride equilibrium concentrations in bentonite, we here take a closer look at the study by Muurinen et al. (2004), in the following referred to as Mu04. We thus assess the 25 points indicated here

In contrast to the earlier assessed studies, Mu04 is not a diffusion study, but considers directly the clay concentration in samples equilibrated with an external solution. Moreover, Mu04 uses purified “MX-80” bentonite, ion exchanged to a more pure sodium form.

Mu04 contains data from two quite different types of samples. 15 samples originate from a study on pressure response in montmorillonite contacted with external NaCl solutions of varying concentration (Karnland et al., 2005; in the following referred to as Ka05). The remaining 10 samples were prepared for determining basal distance using small-angle X-ray scattering (SAXS). We refer to these two sets of samples as the swelling pressure samples and the SAXS samples, respectively. A more detailed description of the sample analysis is given in Muurinen (2006), in the following referred to as Mu06.

Material

The used material is referred to as “purified MX-80”. Mu06 states that this material was produced by mixing “MX-80” powder and NaCl solutions in bottles, where the solutions were repeatedly replaced. Mu06 also states that “During this process, part of the dissolving accessory minerals was removed as well.” Ka05 more explicitly say that the raw material was “converted into a homo-ionic Na⁺ state and coarser grains were removed (Muurinen et al., 2002). The montmorillonite content was thereby increased to above 90% of the total material.”¹ With no better estimate of the montmorillonite content, we therefore associate the stated densities with effective montmorillonite dry density, i.e. we assume a montmorillonite content of 100%. We should keep in mind the uncertainty of this parameter, and that, reasonably, this choice somewhat overestimates the effective montmorillonite dry density.

The purified material was found to leach sulfate and carbonate, indicating that it still contains some amount of soluble accessory minerals. It follows that the montmorillonite is not completely of pure sodium form, as confirmed by the reported exchangeable ion population: 0.74 eq/kg sodium, 0.06 eq/kg calcium, and 0.03 eq/kg magnesium (i.e. a di/mono-valent ratio of about 10/90). It is interesting that the material still contains a non-negligible amount of divalent ions, given that quite a lot of effort was put into producing it. Nevertheless, we can assume that this material contains considerably more sodium as compared with the “raw” “MX-80” encountered in the previously assessed studies.

Samples overview

The swelling pressure samples were originally cylindrical, with diameter 5 cm and length 2 cm, giving a volume of approximately 39 cm³. After termination of the swelling pressure tests, these samples were cut into pieces, to be used for different types of analyses. The samples cover large ranges of density and external NaCl concentration, as listed here

Sample-ID²	NaCl conc. (M)	\(\rho_d\) (nominal) (g/cm³)
S2-21	0.1	0.786
S2-02	0.1	0.786
S2-04	0.1	1.257
S2-17	0.1	1.571
S2-18	0.1	1.729
S2-13	0.3	0.786
S2-14	0.3	1.257
S2-15	0.3	1.571
S2-16	0.3	1.729
S2-05	1.0	1.257
S2-08	1.0	1.571
S2-11	1.0	1.729
S2-06	3.0	1.257
S2-09	3.0	1.571
S2-12	3.0	1.729

Neither Mu04 nor Mu06 provide much information about preparation and handling of the SAXS samples. It is stated that these are cylindrical with diameter 2.5 cm and length 0.5 cm, giving a total volume of 2.45 cm³. Although not stated, also these sample have reasonably been sub-divided, as e.g. density was determined (and some parts were obviously used for SAXS).

The SAXS samples varies substantially in density, but were only contacted with external NaCl solutions of concentration 0.1 or 0.3 M. The table below identifies each sample by external solution concentration and, presumably, measured density (how density was determined is not reported)

NaCl conc. (M)	\(\rho_d\) (measured?) (g/cm³)
0.1	0.750
0.1	0.875
0.1	1.225
0.1	1.516
0.1	1.543
0.3	0.954
0.3	1.058
0.3	1.206
0.3	1.559
0.3	1.662

In the following, we separately discuss the chloride concentration evaluations of the swelling pressure samples and the SAXS samples.

Swelling pressure samples

Chloride concentration was evaluated³ in three separate pieces of each original sample, as indicated in this figure:⁴

Chloride content was determined by dispersing each piece, containing about 1 g of clay, in de-ionized water, centrifuging, and analyzing the supernatant. The pieces were located at different heights of the original cylinder (see figure), giving some spatial resolution of the chloride distribution, reported in Mu06. Mu04, however, only report the average value for each sample. For some samples, the value reported in Mu04 does not perfectly match the average calculated from the values listed in Mu06 (cf. the plots below).

Let’s anticipate the “verdict” for these samples: the evaluated clay concentrations are not useful for quantitative understanding of ion equilibrium, and I will not use them e.g. for validating anion exclusion models.

That these samples are not adequately equilibrated is best seen from looking at the evaluated concentrations in the samples contacted with 0.1 and 0.3 M NaCl, here plotted with spatial resolution

The indicated densities (\(\rho_d\)) are the average of the spatially resolved values reported in Mu06 (these differ a bit from what is reported in Mu04). The profiles show several peculiarities:

The densest samples in both test sets (S2-18 and S2-16, respectively) contain the second highest amount of chloride.
Samples S2-21 and S2-02 have a huge difference in chloride concentration, even though they have quite similar density.
In both test sets, the chloride concentration is very similar in the samples with densities \(\sim\) 1.3 g/cm³ and \(\sim\) 1.6 g/cm³ (S2-04 vs. S2-17, and S2-14 vs. S2-15).

These observations strongly indicate that the samples either have not been adequately equilibrated, or that they have not been adequately handled after test termination (or both). Consequently, the results are of little help for adequate quantitative process understanding.⁵ Mu04 acknowledge this shortcoming, but takes different action

At high dry densities (>1630 kg/m³ ) and low NaCl concentrations, the concentrations in the porewater tend to increase with increasing density. The phenomenon is not seen with the thinner SAXS samples, however. One possible explanation is that during saturation too much chloride is transported into the sample and the equilibration time has been too short to reach the equilibrium. Three such samples marked with (*) in Tables 1 and 3 have been omitted from the treatment of the results.

But one cannot simply omit only the samples that deviate from the expected qualitative behavior while assuming that the rest of the results are adequate! This is especially true when the source for the shortcoming has not been clarified. In fact, we just identified additional peculiarities in the data. Consequently, not only should the rest of the samples equilibrated with 0.1 M and 0.3 M NaCl be omitted, but also those equilibrated with 1.0 M and 3.0 M.

For completeness, here are the chloride concentration profiles for the tests with high background concentration

Although we discard them, it may be interesting to identify possible reasons for these flawed results. Previously, we discussed why equilibrium salt concentrations may be overestimated. Both factors identified there may apply here: failing to handle possible interface excess, and issues related to directly saturating samples with a saline solution.

Saturating with saline solutions

From the different reports it is clear that the samples were saturated directly with the saline solution. It is, however, not fully clear if the saturation was performed from only one end of the sample, or from both. In Mu04 and Mu06, the assumption seems to be that the samples were saturated from one side only, although this is not described in any detail. Mu04 write

The compacted samples were closed in metal tubes and saturated through a sinter at one end.

In Ka05, however, the statement is

The water solutions were slowly circulated behind the bottom filters to start with, in order not to trap the original air in the samples.

where the formulation “to start with” suggests that the solution was eventually also contacted from the top.

The reason why this detail may be important is that with solution at one end only the effective diffusion distance for salt is doubled as compared with having solution at both ends. A doubling of diffusion length, in turn, increases the characteristic diffusion time by a factor of 4.

We estimate the time needed for excess salt to diffuse out by considering a model with an initial unit concentration in the entire domain (domain length \(L\)), and boundary condition of zero concentration at the end points. The midpoint concentration in such a model, for various values of \(L\) (0.01 — 0.02 m) and diffusion coefficients (\(1\cdot 10^{-10}\) — \(1\cdot 10^{-11}\;\mathrm{m^2/s}\)), evolve like this

We see that, depending on parameter values, the set-up may be such that a possible “overshoot” of salt have not had time to completely diffuse out of the sample during the course of the swelling pressure tests, which were conducted for about a month. In particular, if the effective diffusion length was 2 cm during the major part of the saturation process, it is very plausible that the equilibrium process was not completed for certain samples (this depends of course also on the detailed values of diffusion coefficient and equilibration time).

Supported by this simple analysis, we cannot rule out that the samples initially took up more salt than dictated by the final state, and that this salt may not have had time to fully diffuse out again.

Interface excess

Concerning interface excess (a potential problem regardless of whether or not the sample has reached full equilibrium before termination), no detailed information is given on the dismantling procedure. It seems relatively clear, though, that the outer parts of the original samples were not sectioned off. Ka05 write

After reaching pressure equilibrium and a minimum test time of 1 month, the test solutions were disconnected and the samples were removed and split in order to make detailed analyses of the water ratio, sample density, pore-water chemistry, water activity and microstructure.

Mu06 writes (“Figure 4” is similar to the figure above)

The bentonite sample cylinders obtained from the swelling pressure measurements were cut into smaller pieces according to Figure 4 in order to provide samples for different analyses and measurements. Half of the sample piece was used for the porewater studies while the other half […] was left in Clay Technology AB for their studies.

My interpretation is that the original sample was cut in half during the dismantling in the swelling pressure study, and that one half was sent off elsewhere for the analysis presented in Mu04 (the upper part of the disc indicated in the above figure). Thus, it seems plausible that the interface regions were not sectioned off during dismantling, and that the samples were stored/transported for an appreciable amount of time. Possible excess salt would consequently had time to even out in the sample before further analysis. This interpretation is in line with the evaluated rather flat chloride profiles: note the contrast between these and the quite pronounced non-linear profiles observed at the interfaces in studies where samples are sectioned at test termination.

SAXS samples

Mu04 (and Mu06) provide almost no information about how the SAXS samples were handled. Reasonably, also these samples were split, with some part being used for the SAXS measurement and another for determining water content, but we have no information on this. In fact, not even the SAXS results are properly reported for these samples; only evaluated “interlamellar spaces” for the samples equilibrated at 0.3 M are discussed; neither Mu04 nor Mu06 report SAXS data for the 0.1 M samples.

The reports are also somewhat contradictory. In the caption to a table in Mu04 it is stated that the SAXS samples were first saturated with de-ionized water, and thereafter equilibrated with the salt solutions. Mu06, on the other hand, states

The samples were compacted into the cells and saturated through a filter plate from one side with 0.1 or 0.3 M NaCl solutions for 12 days.

Should the last statement rather be that equilibration was performed for 12 days, after saturation? Under any circumstance, the lack of information on handling of the SAXS samples is a major flaw and must be considered in the assessment.

If I should guess, I believe that possible interface excess on these samples where not handled, i.e. I believe that the end parts were not sectioned off when the samples were dismantled (also, with only 5 mm thick samples, there is not much to section off…). Note that the SAXS samples are thin (5 mm) and were equilibrated with solutions of relatively low concentration (0.1 M and 0.3 M). Based on the analysis in the previous post, these samples are expected to be very sensitive to an interface excess effect.

Here is plotted the reported chloride clay concentrations for the SAXS samples, together with corresponding (average) values for the swelling pressure samples at the same background concentration⁶

Note that, although the density dependence on the SAXS sample data appears more reasonable compared with the swelling pressure samples, the SAXS sample data seem to have a scatter of at least a factor of 2 (see e.g. the leftmost SAXS points for 0.3 M and the rightmost points for 0.1 M). Note also that in one sample (0.1 M, 750 kg/cm³), the evaluated clay concentration is larger than the background concentration!

Summary and verdict

I discard the chloride concentrations measured in the swelling pressure samples, based on the reported results: it is clear that the observed scatter and spurious dependencies demonstrate that the samples were not properly equilibrated, in order to use the results for quantitative process understanding. To accept e.g. a result that the equilibrium chloride concentration increases with density I require a considerably more rigorous study. Moreover, I mean that all results must be discarded, not only those that obviously deviate from the expected qualitative behavior.

I also discard the results of the SAXS samples. Although we don’t have any clear indication that they were incorrectly prepared, I judge the uncertainties and lack of information to be too large in order to rely on the results. Almost no information is provided! Furthermore, the reports do not give any hint that the issue of interface excess is identified and handled — an effect we can expect to be substantial in these samples.

I am saddened to have to discard these results, because, in my mind, adequate results from equilibration of homo-ionic samples would be very valuable for increased process understanding. I strongly believe that the bentonite research community should strive for conducting many more of these relatively simple tests on purified clays, rather than complicated through-diffusion tests. In properly conducted equilibrium tests, concentration data is accessed directly and there is no risk for the results to be obscured by issues related to ionic transport.

Footnotes

[1] The reference “(Muurinen et al., 2002)” appears to be what eventually became Muurinen et al. (2004); the 2002-reference is stated to be submitted to Applied Clay Science, but has the same title as the 2004-reference (published in Physics and Chemistry of the Earth).

[2] These IDs are used in Karnland et al. (2005) and Muurinen (2006), but not in Muurinen et al. (2004). Sample S2-21 is incorrectly specified in table 3 in Muurinen (2006) (but not in the actual tables of results). Karnland et al. (2005) report an additional sample — S2-22 — that is not included in Muurinen (2006) or Muurinen et al. (2004) (background concentration 1.0 M, density 0.786 g/cm³).

[3] Muurinen et al. (2004) also report chloride concentrations from so-called squeezing tests. Squeezing tests are not adequate for evaluating equilibrium clay concentrations, and I intend to write a future blog post on the subject. Here we simply ignore the squeezing results.

[4] The pieces labeled “B” were used to determine density (water content).

[5] This does not in any way alter the fact that the study has contributed greatly to qualitative process understanding: as we have discussed several times on this blog, Karnland et al. (2005) proves that anions have access to montmorillonite interlayers.

[6] The plots also show the difference in average concentration and density for the swelling pressure samples as reported in Muurinen et al. (2004) and Muurinen (2006); these points should lie on top of each other.

How salt equilibrium concentrations may be overestimated

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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.

Atmosphere exclusion

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By applying simple physical principles we can obtain an expression for the density of Earth’s atmosphere at a certain altitude \(h\)

\begin{equation} \rho(h) = \rho_0 e^{-h/\alpha} \end{equation}

where \(\rho_0\) is the air density at sea level, and \(\alpha = RT/(Mg) \approx 7500\) m is a constant. Integrating the above formula from sea level to the height of Mount Everest (\(\approx 9000\) m) gives

\begin{equation} \int_0^{9000}\rho_0 e^{-\frac{h}{\alpha}} dh = \rho_0\alpha\left(1-e^{-9000/7500}\right) \approx \rho_0\cdot 5200\;\mathrm{m} \end{equation}

More advanced research finds a neat interpretation of this relation: the accessible height for air is 5200 m. Above this limit air is excluded, probably due to repulsion from the bedrock at these altitudes — there are reasons to believe that such rock has significantly different properties compared with rock at sea level (e.g. positive gravitational potential). In fact, both experimental work and theoretical modelling — even at the atomistic level! — have given strong evidence for the air exclusion effect: best fitting to available data is achieved with so-called air-free models.

As an example of the success of this research, one has been able to explain the existence of life in the highest regions of the Tibetan Plateu: air exists in these regions in hidden valleys (also called interpeak volumes) below the 5200 m-level, which consequently have air density \(\rho_0\). Much of present day air exclusion research is actually devoted to quantifying the amount of hidden valleys, given measurements of air density in various regions around the world (valleys that otherwise would be very difficult to discover).

Even if this research field lately has progressed heavily, there is still a lot of exciting work waiting to be done. Of the many topics can be mentioned so-called partial air exclusion on the outer borders to certain high plains, air transport between hidden valleys (which typically are connected), and the possibility of having different accessible heights for different types of air.

A future potential application of the air exclusion effect is to build storage e.g. for food at high altitudes. With no air around, food is expected to stay fresh forever!