Category Archives: Homogeneous mixture model

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

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

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

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

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

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

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

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

The “uphill” diffusion experiment

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

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

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

The “uphill” diffusion phenomenon explained

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

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

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

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

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

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

Conventional through-diffusion

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

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

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

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

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

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

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

“Uphill” diffusion

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

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

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

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

giving the corresponding steady-state flux

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

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

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

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

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

Reexamining the Tertre et al. (2024) statements

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

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

A way forward (if anybody cares)

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

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

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

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

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

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

Footnotes

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

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

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

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

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

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

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

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

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

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

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).

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

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A testable difference

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Comparison with experiment

Van Loon et al. (2007)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Glaus et al. (2010)

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

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

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

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

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

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

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

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

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

Insight from closed-cell tests

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

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

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

Footnotes

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

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

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

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

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

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

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

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

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

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

Sorption part IV: What is K_d?

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Measuring K_d

Researchers traditionally measure sorption on montmorillonite in batch tests, where a small amount of solids is mixed with a tracer-spiked solution (typical solid-to-liquid ratios are \(\sim 1 – 10\) g/l). After equilibration, solids and solution are usually separated by centrifugation and the supernatant is analyzed.

This procedure evidently counts tracer cations that reside in diffuse layers as sorbed. But tracer ions may also sorb due to other mechanisms, in particular due to bonding on specific surface hydroxyl groups, on the edges of individual montmorillonite layers. These different types of “sorption” are in the clay literature usually referred to as “cation exchange” and “surface complexation”, respectively.

The amount of tracer “sorbed” in the ways just described is quantified by the distribution coefficient \(K_d\), defined as

\begin{equation} s = K_d\cdot c_\mathrm{eq} \end{equation}

where \(s\) denotes the amount of tracers “on the solids”, and \(c_\mathrm{eq}\) is the corresponding equilibrium concentration in the aqueous phase. As the amount “on the solids” can be inferred from the amount of tracers that has been removed from the initial solution, we can evaluate \(K_d\) from

\begin{equation} K_d = \frac{\left ( c_\mathrm{init} – c_\mathrm{final} \right ) \cdot V_\mathrm{sol}} {c_\mathrm{final}\cdot m_\mathrm{s}} \end{equation}

where \(c_\mathrm{init}\) is the initial tracer concentration (i.e. before adding the clay), \(c_\mathrm{final}\) is the tracer concentration in the supernatant, \(V_\mathrm{sol}\) is the solution volume, and \(m_s\) is the mass of the solids.

If the purpose of a study is solely to quantify the amount of tracer “on the solids”, it is adequate to define sorption as including both “cation exchange” and “surface complexation”, and to use \(K_d\) as the measure of this sorption. However, if our main concern is to describe transport in compacted bentonite, \(K_d\) is a rather blunt tool, since it quantifies both ions that dominate the transport capacity (“cation exchange”), and ions that are immobile, or at least contribute to an actual delay of diffusive fluxes (“surface complexation”).

A good illustration of this problem is the traditional diffusion-sorption model, which incorrectly assumes that all ions quantified by \(K_d\) are immobilized. In earlier blog posts, we have discussed the consequences of this model assumption, and the empirical evidence against it. A complication when discussing sorption is that researchers often “measure” \(K_d\) by fitting the traditional diffusion-sorption model to data — although the model is not valid for compacted bentonite.

Moreover, when evaluating \(K_d\) in batch tests, or when using this parameter in models, authors assume that the solids are in equilibrium locally with a bulk water phase. But there is no compelling evidence that such a phase exists in compacted water saturated bentonite. On the contrary, several observations strongly suggest that compacted bentonite lacks significant amounts of bulk water. This, in turn, suggests that \(K_d\) actually quantifies the equilibrium between a bentonite sample and an external solution.

Indeed, even in batch tests is the final concentration measured in a solution (the supernatant) separated from the clay (the sediment), as a consequence of the centrifugation, as illustrated here:

This figure also illuminates additional and perhaps more subtle complications when evaluating \(K_d\) from batch tests. Firstly, such values are implicitly assumed independent of “sample” density. There are, however, arguments for that \(K_d\) in general depends on density, as will be explored below. The question is then to what density range we can apply batch test values when modeling compacted systems, or if they can be applied at all. Note that the “sample” that is measured on in a batch test (see figure) has a more or less well-defined density. But sediment densities are, to my knowledge, never investigated in these types of studies.¹

Secondly, it could be questioned if the supernatant have had time to equilibrate with the sediment, i.e whether \(c_\mathrm{final} = c_\mathrm{eq}\). Instead, as far as I know, researchers routinely assume that the equilibrium established prior to centrifugation remains.

In the following, we use the homogeneous mixture model to analyze in more detail the nature of \(K_d\) in compacted bentonite.

K_d in the homogeneous mixture model

As usual when analyzing bentonite with the homogeneous mixture model, we assume an external solution in contact with a homogeneous bentonite domain at a specific density (water-to-solid mass ratio \(w\)). The bentonite and the external solution are separated via a semi-permeable component, which allows for the passage of water and ions, but does not allow for the passage of clay (symbols are explained below):

This model resembles the alternative test set-up for determining \(K_d\) in compacted systems used by Van Loon and Glaus (2008), where the clay is contained in a sample holder, and the tracer is supplied through a filter from an external circulating solution. This approach has the advantages that the state of the clay is controlled throughout the test (which, e.g., allows for investigating how \(K_d\) depends on density), and that the equilibration process is better controlled (avoiding the possible disruptive procedure of centrifugation). The obvious disadvantage is that equilibration — being diffusion controlled — may take a long time.

When applying the homogeneous mixture model in earlier blog posts, we have assumed “simple” ions, which contribute to the ion population of the clay only in terms of the interlayer concentration, \(c^\mathrm{int}\). This concentration quantifies the amount of mobile ions involved in establishing Donnan equilibrium between clay and external solutions. But many “non-simple” ions actually do seem to be immobilized/delayed by also associate with surfaces (\(\mathrm{H}^+\), \(\mathrm{Ni}^{2+}\), \(\mathrm{Zn}^{2+}\), \(\mathrm{Co}^{2+}\), \(\mathrm{P_2O_7^{4-}}, …\)). For a more general description, we therefore extend the homogeneous mixture model with a second contribution to the ion population: \(s^\mathrm{int}\) (ions per unit mass).

Using the traditional terminology, the ions quantified by \(c^\mathrm{int}\) are to be identified as “sorbed by ion exchange”, and those quantified by \(s^\mathrm{int}\) as “sorbed by surface complexation”. But since the ion exchange process does not immobilize ions and primarily should be associated with Donnan equilibrium, we want to avoid referring to them as “sorbed”. Also, with the traditional terminology, all ions in the homogeneous mixture model are described as “sorbed”, which obviously not is very useful.

We therefore introduce different terms, and refer to the ions quantified by \(c^\mathrm{int}\) as aqueous interlayer species, and to the ions quantified by \(s^\mathrm{int}\) as truly sorbed ions. With this terminology, the term “sorption” puts emphasis on ions being immobile.² Moreover, the description now also applies to anions, without having to refer to them as e.g. “sorbed by ion exchange”.

In analogy with the traditional diffusion-sorption model, we assume a linear relation between \(s^\mathrm{int}\) and \(c^\mathrm{int}\)

\begin{equation} s^\mathrm{int} = \Lambda\cdot c^\mathrm{int} \tag{1} \end{equation}

where \(\Lambda\) is a distribution coefficient quantifying the relation between the amount of aqueous species in the interlayer domain and amount of truly sorbed substance.³

The amount of an aqueous species in the homogeneous mixture model is \(V_p\cdot c^\mathrm{int}\), where \(V_p\) is the total pore volume. The total amount of an ion per unit mass is thus \(V_p\cdot c^\mathrm{int}/m_s + s^\mathrm{int}\), where \(m_s\), as before, denotes total solid mass.

To get an expression for \(K_d\) in the homogeneous mixture model, we must associate ions “on the solids” (\(s\)) with the concentration in the external solution. Here we choose the simplest way to do this, and write

\begin{equation} s = \frac{V_p\cdot c^\mathrm{int}}{m_s} + s^\mathrm{int} = K_d\cdot c^\mathrm{ext} \tag{2} \end{equation}

which implies that we define all ions in the bentonite sample to be “on the solids”. To be fully consistent, we should perhaps subtract the contribution expected to be found in the clay if it behaved like a conventional porous system (\(V_p\cdot c^\mathrm{ext}/m_s\)). But, since we are mostly interested in the limit of small \(V_p/m_s\), this contribution can be thought of as becoming arbitrary small, and we therefore don’t bother with including it in the formulas. In any case, this “conventional porewater” contribution would simply give an extra term \(-w/\rho_w\) in the equations we are about to derive, and can be included if desired.

Using eqs. 1 and 2, we get the expression for \(K_d\) in the homogeneous mixture model

\begin{equation} K_d = \frac{w\cdot\Xi }{\rho_w} + \Lambda\cdot \Xi \tag{3} \end{equation}

where we also have used the definition of the ion equilibrium coefficient \(\Xi = c^\mathrm{int}/c^\mathrm{ext}\), and utilized that \(V_p/m_s = w/\rho_w\), where \(\rho_w\) is the density of water.⁴

A full analysis of eq. 3 is a major task, but a few things are immediately clear:

\(K_d\) generally has two contributions: one from Donnan equilibrium (\(w\cdot\Xi/\rho_w\)) and one from true sorption (\(\Lambda\cdot \Xi\)). Using the traditional terminology, these contributions correspond for cations to “sorption by ion exchange” and “sorption by surface complexation”, respectively. But note that eq. 3 is valid also for anions.
For a simple cation (\(\Lambda = 0\)), \(K_d\) merely quantifies the aqueous interlayer concentration.⁵ As we have discussed earlier, \(K_d\) quantifies in this case a type of enhancement of the transport capacity. I think it is unfortunate that a mechanism that dominates the mass transfer capacity traditionally is labeled “sorption”.
For cations with \(\Lambda \neq 0\), \(K_d\) is not a measure of true sorption, because we always expect a significant Donnan contribution. In this case \(K_d\) quantifies a mixture of transport enhancing and transport inhibiting mechanisms. Clearly, it is unsatisfactory to use the term “sorption” for mechanisms that both enhance and reduce transport capacity (at least when the objective is a transport description).
For simple anions, the above expression gives a positive value for \(K_d\). Traditionally, the \(K_d\) concept has not been applied to these types of ions, and e.g. chloride is often described as “non-sorbing”, with \(K_d =0\). Since \(\Xi \rightarrow 0\) as \(w \rightarrow 0\) generally for anions, this result (\(K_d = 0\)) is recovered in this limit.⁶

K_d for simple cations

We end this post by examining expressions for \(K_d\) for simple cations in some specific cases. In the following we consequently assume \(\Lambda = 0\), and this section relies heavily on the ion equilibrium framework in the homogeneous mixture model, with the main relation

\begin{equation} \Xi \equiv \frac{c^\mathrm{int}}{c^\mathrm{ext}} = \Gamma f_D^{-z} \tag{4} \end{equation}

where \(z\) is the charge number of the ion, \(\Gamma \equiv \gamma^\mathrm{ext}/\gamma^\mathrm{int}\) is an activity coefficient ratio, and \(f_D = e^\frac{F\psi^\star}{RT}\) is the so-called Donnan factor, with \(\psi^\star\) (\(<0\)) being the Donnan potential.

Simple cation tracers in a 1:1 system

We assume a bentonite sample at water-to-solid mass ratio \(w\) in equilibrium with an external 1:1 solution (e.g. NaCl) of concentration \(c^\mathrm{bgr}\). The Donnan factor is in this case, in the limit \(c^\mathrm{bgr} \ll c_\mathrm{IL}\)⁷

\begin{equation} f_D = \Gamma_+\frac{c^\mathrm{bgr}}{c_\mathrm{IL}} \end{equation}

where \(\Gamma_+\) is the activity coefficient ratio for the cation of the 1:1 electrolyte, and, as usual

\begin{equation} c_\mathrm{IL} = \frac{CEC\cdot \rho_w}{w\cdot F} \end{equation}

where \(CEC\) is the cation exchange capacity, and \(F\) is the Faraday constant (1 eq/mol). We furthermore assume the presence of a mono-valent cation tracer, which, by definition, does not influence \(f_D\). The ion equilibrium coefficient for this tracer is (from eq. 4)

\begin{equation} \Xi = \Gamma\cdot \Omega_{1:1}\cdot \frac{\rho_w}{w} \end{equation}

where \(\Gamma\) is the activity coefficient ratio for the tracer, and we have defined

\begin{equation} \Omega_{1:1} \equiv \frac{CEC}{F\cdot c^\mathrm{bgr}\cdot\Gamma_+} \end{equation}

\(K_d\) for a simple mono-valent tracer in a 1:1 electrolyte is thus (using eq. 3 with \(\Lambda = 0\))

\begin{equation} K_{d} = \Gamma \cdot \Omega_{1:1} \tag{5} \end{equation} \begin{equation} \text{ (mono-valent simple tracer in 1:1 system)} \end{equation}

For a divalent tracer we instead have

\begin{equation} \Xi = \Gamma \cdot \Omega_{1:1}^2 \cdot \left (\frac{\rho_w}{w} \right )^2 \end{equation}

giving

\begin{equation} K_d = \Gamma \cdot \Omega_{1:1}^2 \cdot \frac{\rho_w} {w} \tag{6} \end{equation} \begin{equation}\text{(di-valent simple tracer in 1:1 system)} \end{equation}

Eqs. 5 and 6 are essentially identical⁸ with the expression for \(K_d\) in a 1:1 system, derived in Glaus et al. (2007), which we used in the analysis of filter influence in cation through-diffusion.

Simple cation tracers in a 2:1 system

In a 2:1 system (e.g \(\mathrm{CaCl_2}\)), the Donnan factor is, in the limit \(c^\mathrm{bgr} \ll c_\mathrm{IL}\)

\begin{equation} f_D = \sqrt{2 \Gamma_{++}\frac{c^\mathrm{bgr}}{c_\mathrm{IL}}} \end{equation}

where index “++” refers to the cation of the 2:1 background electrolyte. Thus, for a mono-valent tracer

\begin{equation} \Xi = \Gamma\cdot \sqrt{\Omega_{2:1}} \cdot \sqrt{\frac{\rho_w}{w}} \end{equation}

where

\begin{equation} \Omega_{2:1} \equiv \frac{CEC}{2F\cdot c^\mathrm{bgr}\cdot\Gamma_{++}} \end{equation}

\(K_d\) for a mono-valent simple tracer in a 2:1 electrolyte is consequently

\begin{equation} K_{d} = \Gamma \cdot \sqrt{\Omega_{2:1}}\cdot\sqrt{\frac{w}{\rho_w}} \tag{7} \end{equation} \begin{equation} \text{(simple mono-valent tracer in 2:1 system)} \end{equation}

For a divalent tracer we instead have

\begin{equation} \Xi = \Gamma \cdot \Omega_{2:1} \cdot \frac{\rho_w}{w} \end{equation}

giving

\begin{equation} K_d = \Gamma \cdot \Omega_{2:1} \tag{8} \end{equation} \begin{equation} \text{(simple di-valent tracer in 2:1 system)} \end{equation}

Density dependence of K_d

Note that \(K_d\) for a mono-valent ion in a 1:1 system does not explicitly depend on density (eq. 5), while \(K_d\) for a di-valent ion diverges as \(w\rightarrow 0\) (eq. 6). In contrast, \(K_d\) in a 2:1 system has no explicit density dependence for di-valent tracers (eq. 8), while \(K_d\) vanishes for a mono-valent tracer in the limit \(w \rightarrow 0\) (eq. 7).

These results imply that we expect \(K_d\) to generally depend on sample density in systems where the charge number of the tracer ions differs from that of the cation of the background electrolyte. It may therefore not be appropriate to use values of \(K_d\) evaluated in batch-type tests for analyzing compacted systems.

Note also that \(K_d\) may have significant density dependence also in cases where the present analysis gives no explicit \(w\)-dependence on \(K_d\). This was demonstrated e.g. by Van Loon and Glaus (2008) for cesium tracers in sodium dominated bentonite. Interpreted in terms of the homogeneous mixture model, their results show that the interlayer activity coefficients vary significantly with density. In particular, the results imply either that the interlayer activity coefficient for cesium becomes small (\(\Gamma_\mathrm{Cs} \gg 1\)), or that the interlayer activity coefficient for sodium becomes large (\(\Gamma_\mathrm{Na} \ll 1\)), in the high density limit.

Footnotes

[1] A sediment density is, reasonably, related to e.g. initial solid-to-water ratio and to the details of the centrifugation procedure.

[2] I am not very happy with this terminology, but we need a way to distinguish this type of sorption from how the term “sorption” is used in the bentonite literature, where it nowadays essentially refers to the process of taking up an ion from a bulk water phase to some other phase. This is the reason for why there are so many quotation marks around the word “sorption” in the text.

[3] I don’t know if this is a valid assumption, but it seems like the natural starting point.

[4] The presence of water density in the formulas reflects the fact that we are using molar units (substance per unit volume), which is natural, as \(K_d\) typically has units of volume per mass. How to associate a density to water in the homogeneous mixture model is a bit subtle, and we don’t focus on that aspect here (it may be the issue of future posts). In the presented formulas \(\rho_w\) can rather be viewed as a unit conversion factor.

[5] When \(\Lambda = 0\), we can rearrange eq. 3 as

\begin{equation} \Xi = \frac{K_d\cdot \rho_w}{w} = \frac{K_d\cdot \rho_d}{\phi} \equiv \kappa \end{equation}

where \(\rho_d\) is dry density, \(\phi\) is porosity, and \(\kappa\) was defined as a scaled, dimensionless version of \(K_d\) by Gimmi and Kosakowski (2011), discussed in a previous blog post. Interpreted using the homogeneous mixture model, \(\kappa\) is thus simply the ion equilibrium coefficient for simple cations.

[6] By including the “conventional porewater” contribution in the definition of \(K_d\), as discussed earlier, we get for these types of anions

\begin{equation} K^\prime_d = \frac{w\cdot \Xi}{\rho_w} – \frac{w}{\rho_w} = \frac{w}{\rho_w} \left ( \Xi – 1 \right) \end{equation}

This is typically a negative quantity, and quantifies anion exclusion, in the Schofield sense of the term. We have, also with this definition, that \(K^\prime_d \rightarrow 0\) as \(w \rightarrow 0\).

[7] We assume \(c^\mathrm{bgr} \ll c_\mathrm{IL}\) in this and all following cases. For compacted bentonite \(c_\mathrm{IL}\) is of the order of several molar, and the derived approximations are thus valid for “typical” background concentrations (\(< 1\) M). Also, for an arbitrary value of \(c^\mathrm{bgr}\), one can in principle always choose a sufficiently low value of \(w\) to satisfy \(c^\mathrm{bgr} \ll c_\mathrm{IL}\).

[8] If the selectivity coefficient is identified with that derived in Birgersson (2017).

Extracting anion equilibrium concentrations from through-diffusion tests

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Recently, we discussed reported equilibrium chloride concentrations in sodium dominated bentonite, and identified a need to assess the individual studies. As most data is obtained from through-diffusion experiments, we here take a general look at how anion equilibrium is a part of the through-diffusion set-up, and how we can use reported model parameters to extract the experimentally accessible equilibrium concentrations.

We define the experimentally accessible concentration of a chemical species in a bentonite sample as

\begin{equation} \bar{c} = \frac{n}{m_\mathrm{w}} \end{equation}

where \(n\) is the total amount of the species,¹ and \(m_{w}\) is the total water mass in the clay.² It should be clear that \(\bar{c}\), which we will refer to as the clay concentration, is accessible without relying on any particular model concept.

An equilibrium concentration is defined as the corresponding clay concentration (i.e. \(\bar{c}\)) of a species when the clay is in equilibrium with an external solution with species concentration \(c^\mathrm{ext}\). A convenient way to express this equilibrium is in terms of the ratio \(\bar{c}/c^\mathrm{ext}\).

The through-diffusion set-up

A through-diffusion set-up consists of a (bentonite) sample sandwiched between a source and a target reservoir, as illustrated schematically here (for some arbitrary time):

The sample length is labeled \(L\), and we assume the sample to be initially empty of the diffusing species. A test is started by adding a suitable amount of the diffusing species to the source reservoir. Diffusion through the bentonite is thereafter monitored by recording the concentration evolution in the target reservoir,³ giving an estimation of the flux out of the sample (\(j^\mathrm{out}\)). The clay concentration for anions is typically lower than the corresponding concentration in the source reservoir.

Although a through-diffusion test is not in full equilibrium (by definition), local equilibrium prevails between clay and external solution⁴ at the interface to the source reservoir (\(x=0\)). Thus, even if the source concentration varies, we expect the ratio \(\bar{c}(0)/c^\mathrm{source}\) to stay constant during the course of the test.⁵

The effective porosity diffusion model

Our primary goal is to extract the concentration ratio \(\bar{c}(0)/c^\mathrm{source}\) from reported through-diffusion parameters. These parameters are in many anion studies specific to the “effective porosity” model, rather than being accessible directly from the experiments. We therefore need to examine this particular model.

The effective porosity model divides the pore space into a bulk water domain and a domain that is assumed inaccessible to anions. The porosity of the bulk water domain is often referred to as the “effective” or the “anion-accessible” porosity, and here we label it \(\epsilon_\mathrm{eff}\).

Anions are assumed to diffuse in the bulk water domain according to Fick’s first law

\begin{equation} \label{eq:Fick1_eff} j = -\epsilon_\mathrm{eff} \cdot D_p \cdot \nabla c^\mathrm{bulk} \tag{1} \end{equation}

where \(D_p\) is the pore diffusivity in the bulk water phase. This relation is alternatively expressed as \(j = -D_e \cdot \nabla c^\mathrm{bulk}\), which defines the effective diffusivity \(D_e = \epsilon_\mathrm{eff} \cdot D_p\).

Diffusion is assumed to be the only mechanism altering the concentration, leading to Fick’s second law

\begin{equation} \label{eq:Fick2_eff} \frac{\partial c^\mathrm{bulk}}{\partial t} = D_p\cdot \nabla^2 c^\mathrm{bulk} \tag{2} \end{equation}

Connection with experimentally accessible quantities

The bulk water concentration in the effective porosity model relates to the experimentally accessible concentration as

\begin{equation} \label{eq:cbar_epsilon} \bar{c} = \frac{\epsilon_\mathrm{eff}}{\phi} c^\mathrm{bulk} \tag{3} \end{equation}

where \(\phi\) is the physical porosity of the sample. Since a bulk water concentration varies continuously across interfaces to external solutions, we have \(c^\mathrm{bulk}(0) = c^\mathrm{source}\) at the source reservoir, giving

\begin{equation} \label{eq:cbar_epsilon0} \frac{\bar{c}(0)}{c^\mathrm{source}} = \frac{\epsilon_\mathrm{eff}} {\phi} \tag{4} \end{equation}

This equation shows that the effective porosity parameter quantifies the anion equilibrium concentration that we want to extract. That is not to say that the model is valid (more on that later), but that we can use eq. 4 to translate reported model parameters to an experimentally accessible quantity.

In principle, we could finish the analysis here, and use eq. eq. 4 as our main result. But most researchers do not evaluate the effective porosity in the direct way suggested by this equation (they may not even measure \(\bar{c}\)). Instead, they evaluate \(\epsilon_\mathrm{eff}\) from a fitting procedure that also includes the diffusivity as a parameter. It is therefore fruitful to also include the transport aspects of the through-diffusion test in our analysis.

From closed-cell diffusion tests, we know that the clay concentration evolves according to Fick’s second law, both for many cations and anions. We will therefore take as an experimental fact that \(\bar{c}\) evolves according to

\begin{equation} \label{eq:Fick2_exp} \frac{\partial \bar{c}}{\partial t} = D_\mathrm{macr.} \nabla^2 \bar{c} \tag{5} \end{equation}

This equation defines the diffusion coefficient \(D_\mathrm{macr.}\), which should be understood as an empirical quantity.

Combining eqs. 3 and 2 shows that \(D_p\) governs the evolution of \(\bar{c}\) in the effective porosity model (if \(\epsilon_\mathrm{eff}/\phi\) can be considered a constant). A successful fit of the effective porosity model to experimental data thus provides an estimate of \(D_\mathrm{macr.}\) (cf. eq. 5), and we may write

\begin{equation} D_p = D_\mathrm{macr.} \tag{6} \end{equation}

With the additional assumption of constant reservoir concentrations, eq. 2 has a relatively simple analytical solution, and the corresponding outflux reads

\begin{equation} \label{eq:flux_analytic} j^\mathrm{out}(t) = j^\mathrm{ss} \left ( 1 + 2\sum_{n=1}^\infty \left (-1 \right)^n e^{-\frac{\pi^2n^2 D_\mathrm{p} t}{L^2}} \right ) \tag{7} \end{equation}

where \(j^\mathrm{ss}\) is the steady-state flux. In steady-state, \(c^\mathrm{bulk}\) is distributed linearly across the sample, and we can express the gradient in eq. 1 using the reservoir concentrations, giving

\begin{equation} j^\mathrm{ss} = \epsilon_\mathrm{eff} \cdot D_\mathrm{p} \cdot \frac{c^\mathrm{source}}{L} \tag{8} \end{equation}

where we have assumed zero target concentration.

Treating \(j^\mathrm{ss}\) as an empirical parameter (it is certainly accessible experimentally), and using eq. 6, we get another expression for \(\epsilon_\mathrm{eff}\) in terms of experimentally accessible quantities

\begin{equation} \epsilon_\mathrm{eff} = \frac{j^\mathrm{ss}\cdot L}{c^\mathrm{source} \cdot D_\mathrm{macr.} } \tag{9} \end{equation}

This relation (together with eqs. 4 and 6) demonstrates that if we fit eq. 7 using \(D_p\) and \(j^\mathrm{ss}\) as fitting parameters, the equilibrium relation we seek is given by

\begin{equation} \label{eq:exp_estimate} \frac{\bar{c}(0)}{c^\mathrm{source}} = \frac{j^\mathrm{ss}\cdot L} {\phi \cdot c^\mathrm{source} \cdot D_\mathrm{macr.} } \tag{10} \end{equation}

This procedure may look almost magical, since any explicit reference to the effective porosity model has now disappeared; eq. 10 can be viewed as a relation involving only experimentally accessible quantities.

But the validity of eq. 10 reflects the empirical fact that the (steady-state) flux can be expressed using the gradient in \(\bar{c}\) and the physical porosity. The effective porosity model can be successfully fitted to anion through-diffusion data simply because it complies with this fact. Consequently, a successful fit does not validate the effective porosity concept, and essentially any description for which the flux can be expressed as \(j = -\phi\cdot D_p \cdot \nabla\bar{c}\) will be able to fit to the data.

We may thus consider a generic model for which eq. 5 is valid and for which a steady-state flux is related to the external concentration difference as

\begin{equation} \label{eq:jss_general} j_\mathrm{ss} = – \beta\cdot D_p \cdot \frac{c^\mathrm{target} – c^\mathrm{source}}{L} \tag{11} \end{equation}

where \(\beta\) is an arbitrary constant. Fitting such a model, using \(\beta\) and \(D_p\) as parameters, will give an estimate of \(\bar{c}(0)/c^\mathrm{source}\) (\(=\beta / \phi\)).

Note that the system does not have to reach steady-state — eq. 11 only states how the model relates a steady-state flux to the reservoir concentrations. Moreover, the model being fitted is generally numerical (analytical solutions are rare), and may account for e.g. possible variation of concentrations in the reservoirs, or transport in the filters connecting the clay and the external solutions.

The effective porosity model emerges from this general description by interpreting \(\beta\) as quantifying the volume of a bulk water phase within the bentonite sample. But \(\beta\) can just as well be interpreted e.g. as an ion equilibrium coefficient (\(\phi\cdot \Xi = \beta\)), showing that this description also complies with the homogeneous mixture model.

Additional comments on the effective porosity model

The effective porosity model can usually be successfully fitted to anion through-diffusion data (that’s why it exists). The reason is not because the data behaves in a manner that is difficult to capture without assuming that anions are exclusively located in a bulk water domain, but simply because this model complies with eqs. 5 and 11. We have seen that also the homogeneous mixture model — which makes the very different choice of having no bulk water at all within the bentonite — will fit the data equally well: the two fitting exercises are equivalent, connected via the parameter identification \(\epsilon_\mathrm{eff} \leftrightarrow \phi\cdot\Xi\).

Given the weak validation of the effective porosity model, I find it concerning that most anion through-diffusion studies are nevertheless reported in a way that not only assumes the anion-accessible porosity concept to be valid, but that treats \(\epsilon_\mathrm{eff}\) basically as an experimentally measured quantity.

Perhaps even more remarkable is that authors frequently treat the effective porosity model as were it some version of the traditional diffusion-sorption model. This is often done by introducing a so-called rock capacity factor \(\alpha\) — which can take on the values \(\alpha = \phi + \rho\cdot K_d\) for cations, and \(\alpha = \epsilon_\mathrm{eff}\) for anions — and write \(D_e = \alpha D_a\), where \(D_a\) is the “apparent” diffusion coefficient. The reasoning seems to go something like this: since the parameter in the governing equation in one model can be written as \(D_e/\epsilon_\mathrm{eff}\), and as \(D_e/(\phi + \rho\cdot K_d)\) in the other, one can view \(\epsilon_\mathrm{eff}\) as being due to negative sorption (\(K_d < 0\)).

But such a mixing of completely different mechanisms (volume restriction vs. sorption) is just a parameter hack that throws most process understanding out the window! In particular, it hides the fact that the effective porosity and diffusion-sorption models are incompatible: their respective bulk water domains have different volumes. Furthermore, this lumping together of models has led to that anion diffusion coefficients routinely are reported as “apparent”, although they are not; the underlying model contains a pore diffusivity (eq. 2). As I have stated before, the term “apparent” is supposed to convey the meaning that what appears as pure diffusion is actually the combined result of diffusion, sorption, and immobilization. Sadly, in the bentonite literature, “apparent diffusivity” often means “actual diffusivity”.

Footnotes

[1] For anions, the total amount is relatively easy to measure by e.g. aqueous extraction. Cations, on the other hand, will stick to the clay, and need to be exchanged with some other type of cation (not initially present). In any case, the total amount of a species (\(n\)) can in principle be obtained experimentally, in an unambiguous manner.

[2] Another reasonable choice would be to divide by the total sample volume.

[3] If the test is designed as to have a significant change of the source concentration, it is a good idea to also measure the concentration evolution in this reservoir.

[4] Here we assume that the transfer resistance of the filter is negligible.

[5] Provided that the rest of the aqueous chemistry remains constant, which is not always the case. For instance, cation exchange may occur during the course of the test, if the set-up involves more than one type of cation, and there may be ongoing mineral dissolution.

The danger of log-log plots — measuring and modeling “apparent” diffusivity

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In a previous blog post, we discussed how the diffusivity of simple cations¹ has a small, or even negligible, dependence on background concentration (or, equivalently, on \(K_d\)), and how this observation motivates modeling compacted bentonite as a homogeneous system, containing only interlayer pores.

Despite the indisputable fact that “\(D_a\)”² for simple ions does not depend much on \(K_d\), the results have seldom been modeled using a homogeneous bentonite model. Instead there are numerous attempts in the bentonite literature to both measure and model a variation of “\(D_a\)” with \(K_d\), usually with a conclusion (or implication) that “\(D_a\)” depends significantly on \(K_d\). In this post we re-examine some of these studies.

The claimed \(K_d\)-dependency is often “supported” by the so-called surface diffusion model. I have previously shown that this model is incorrect.³ Here we don’t concern ourselves with the inconsistencies, but just accept the resulting expression as the model to which authors claim to fit data. This model expression is

\begin{equation} D_a = \frac{D_p + \frac{\rho K_d}{\phi} D_s}{1+\frac{\rho K_d}{\phi}} \tag{1} \end{equation}

where \(D_p\) and \(D_s\) are individual domain diffusivities for bulk water and surface regions, respectively, \(\rho\) is dry density, \(\phi\) porosity, and \(K_d\), of course, is assumed to quantify the distribution of ions between bulk water and surfaces as \(s = K_d\cdot c^\mathrm{bulk}\), where \(s\) is the amount of ions on the surface (per unit dry mass), and \(c^\mathrm{bulk}\) is the corresponding bulk water concentration.

Muurinen et al. (1985)

Muurinen et al. (1985) measured diffusivity in high density bentonite samples at various background concentrations, using a type of closed-cell set-up. They also measured corresponding values of \(K_d\) in batch “sorption” tests. The results for cesium, in samples with density in the range \(1870 \;\mathrm{kg/m^3}\) — \(2030 \;\mathrm{kg/m^3}\), are presented in the article in a figure similar to this:

The markers show experimental data, and the solid curve shows the model (eq. 1) with \(D_p = 1.2 \cdot 10^{-10}\;\mathrm{m^2/s}\)⁴ and \(D_s = 4.3\cdot 10^{-13}\;\mathrm{m^2/s}\).

The published plot may give the impression of a systematic variation of \(D_a\) for cesium, and that this variation is captured by the model. But the data is plotted with a logarithmic y-axis, which certainly is not motivated. Let’s see how the plot looks with a linear y-axis (we keep the logarithmic x-axis, to clearly see the model variation).

Now the impression is quite different: this way of plotting reveals that the experimental data only cover a part where the model does not vary significantly. With the adopted range on the x-axis (as used in the article) we actually don’t see the full variation of the model curve. Extending the x-axis gives the full picture:

With the full model variation exposed, it is evident that the model fits the data only in a most superficial way. The model “fits” only because it has insignificant \(K_d\)-dependency in the covered range, in similarity with the measurements.

The defining feature of the model is that the diffusivity is supposed to transition from one specific value at high \(K_d\), to a significantly different value at low \(K_d\). As no such transition is indicated in the data, the above “fit” does not validate the model.

Muurinen et al. (1985) also measured diffusion of strontium in two samples of density \(1740 \;\mathrm{kg/m^3}\). The figures below show the data and corresponding model curve.

The left diagram is similar to how the data is presented in the article, while the right diagram utilizes a linear y-axis and shows the full model variation. The line shows the surface diffusion model with parameters \(D_p = 1.2 \cdot 10^{-10}\;\mathrm{m^2/s}\) and \(D_s = 8.8 \cdot 10^{-12}\;\mathrm{m^2/s}\). In this case it is clear even from the published plot that the experimental data shows no significant variation.

The only reasonable conclusion to make from the above data is that cesium and strontium diffusivity does not significantly depend on \(K_d\) (which implies a homogeneous system). This is actually also done in the article:

The apparent diffusivities of strontium and cesium do not change much when the salt concentration used for the saturation of the samples is changed and the sorption factors change. The surface diffusion model agrees fairly well with the observed diffusion-sorption behaviour.

I agree with the first sentence but not with the second. In my mind, the two sentences contradict each other. From the above plots, however, it is trivial to see that the surface diffusion model does not agree (in any reasonable sense) with observations.

Eriksen et al. (1999)

Although Muurinen et al. (1985) concluded insignificant \(K_d\)-dependency on the diffusion coefficients for strontium and cesium, researchers have continued throughout the years to fit the surface diffusion model to experimental data on these and other ions.

Eriksen et al. (1999) present old and new diffusion data for strontium and cesium (and sodium), fitted and plotted in the same way as in Muurinen et al. (1985). Here are the evaluated diffusivities for cesium plotted against evaluated \(K_d\), as presented in the article, and re-plotted in different ways with a linear y-scale:

The curve shows the surface diffusion model (eq. 1), with parameters \(D_p = 8 \cdot 10^{-10}\;\mathrm{m^2/s}\) and \(D_s = 6 \cdot 10^{-13}\;\mathrm{m^2/s}\). The points labeled “Eriksen 99” are original data obtained from through-diffusion tests on “MX-80” bentonite at dry density 1800 \(\mathrm{kg/m^3}\).⁵ The source for the data points labeled “Muurinen 94” is the PhD thesis of A. Muurinen.⁶

The upper left plot shows the data as presented in the article; again, a logarithmic y-axis is used. In this case, a zoomed-in view with a linear y-axis (upper right diagram) may still give the impression that the data has a systematic variation that is captured by the model. But viewing the whole range reveals that the model is fitted to data where variation is negligible (bottom diagrams), just as in Muurinen et al. (1985).

Data and model for strontium presented in Eriksen et al. (1999) look like this:

The model (line) has parameters \(D_p = 3 \cdot 10^{-10}\;\mathrm{m^2/s}\) and \(D_s = 1 \cdot 10^{-11}\;\mathrm{m^2/s}\), and the source for the data points labeled “Eriksen 84” is found here.

In this case, not even the diagram presented in the article (left) seems to support the promoted model. This is also confirmed when utilizing a linear y-axis, and showing the full model variation (right diagram).

Eriksen et al. (1999) conclude that strontium diffusivities are basically independent of \(K_d\), but claim, in contrast to Muurinen et al. (1985), that cesium diffusivity depends significantly on \(K_d\):

[I]n the \(K_d\) interval 0.01 to 1 the apparent \(\mathrm{Cs}^+\) diffusivity decreases by approximately one order of magnitude whereas for \(\mathrm{Na}^+\) and \(\mathrm{Sr}^{2+}\) the apparent diffusivity is virtually constant.

They also claim that the surface diffusion model fits the data:

\(D_\mathrm{a}\) curves for \(\mathrm{Cs}^+\) and \(\mathrm{Sr}^{2+}\), calculated using a Eq. (6) [eq. 1 here], are plotted in Fig. 4. As can be seen, good fits to experimental data were obtained […]

Note that the variation in the model for cesium is motivated by three data points with relatively high diffusivity and basically the same \(K_d \sim 0.05\;\mathrm{m^3/kg}\). It seems like the model has been fitted to these points, while the point at \(K_d \sim 0.02\;\mathrm{m^3/kg}\) has been mainly neglected. The resulting model has a huge bulk water diffusivity (\(D_p\)), which is about 7 times larger than in the corresponding fit in Muurinen et al. (1985), and only 2.5 times smaller than the diffusivity for cesium in pure water.

Note that, if you claim that the surface diffusion model fits in this case, you implicitly claim that the observed variation — which still is negligible on the scale of the full model variation — is caused by the influence of this enormous (for a 1800 \(\mathrm{kg/m^3}\) sample) bulk pore water diffusivity; with a more “reasonable” value for \(D_p\), the model no longer fits. There are consequently valid reasons to doubt that the claimed \(K_d\) dependence is real. We will return to this fit in the next section.

Gimmi & Kosakowski (2011)

We have now seen several examples of authors erroneously claiming (or implying) that a surface diffusion model is valid, when the actual data for “\(D_a\)” has no significant \(K_d\)-dependency. For reasons I cannot get my head around, this flawed treatment is still in play.

Rather than identifying the obvious problem with the previously presented fits, Gimmi and Kosakowski (2011) instead extended the idea of expressing the diffusivity as a function of \(K_d\) by using scaled, dimensionless quantities

\begin{equation} D_\mathrm{arw} = \frac{D_\mathrm{a}\tau_w}{D_0} \tag{2} \end{equation}

\begin{equation} \kappa = \frac{\rho K_d}{\phi} \end{equation}

where \(D_0\) is the corresponding diffusivity in pure water and \(\tau_w\) is the “tortuosity factor” for water in the system of interest. This factor is simply the ratio between the water diffusivity in the system of interest and the water diffusivity in pure water (I have written about the problem with factors like this here).

The idea — it seems — is that using \(D_\mathrm{arw}\) and \(\kappa\) as variables should make it possible to directly compare the mobility of a given species in systems differing in density, clay content, etc.

Even though it makes some sense that the diffusivity of a specific species scales with the diffusivity of water in the same system, the above procedure inevitably introduces more variation in the data — both because an additional measured quantity (water diffusivity) is involved when evaluating the scaled diffusivity, but also because water diffusivity may depend differently on density as compared with the diffusivity of the species under study.

Also Gimmi and Kosakowski (2011) use the flawed surface diffusion model for analysis, and their expression for \(D_\mathrm{arw}\) is

\begin{equation} D_\mathrm{arw} = \frac{1+\mu_s\kappa}{1+\kappa} \tag{3} \end{equation}

where \(\mu_s = D_s\tau_w/D_0\) is a “relative surface mobility”. This equation is obtained from eq. 1, by dividing by \(D_p\) and assuming \(D_p = D_0/\tau_w\).

Gimmi and Kosakowski (2011) fit eq. 3 to a large set of collected data, measured in various types of material, including bentonites, clay rocks, and clayey soils. This is their result for cesium⁷ (the model curve is eq. 3 with \(\mu_s = 0.031\)⁸)

Viewed as a whole, this data is more scattered as compared with the previous studies. This is reasonably an effect of the larger diversity of the samples, but also an effect of multiplying the “raw” diffusion coefficient with the factor \(\tau_w\) (eq. 2).

Just as in the previous studies we have looked at, the published plot (similar to the left diagram) may give the impression of a systematic variation of the diffusivity with \(K_d\) (it contains partly the same data). But just as before, a linear y-axis (right diagram) reveals that the model is fitted only to data where variation is negligible.

Note that the three data points that contributed to the majority of the variation in the fitted model in Eriksen et al. (1999) here appear as outliers.⁹ The variation with \(K_d\) for cesium claimed in that study is thus invalidated by this larger data set.

As we have noted already, the only reasonable conclusion to draw from this data is that there is no systematic \(K_d\)-dependency on diffusivity of cesium or strontium, and that it does not — in any reasonable sense — fit the surface diffusion model. Yet, also Gimmi and Kosakowski (2011) imply that the surface diffusion is valid:

The data presented here show a general agreement with a simple surface diffusion model, especially when considering the large errors associated with the \(D_\mathrm{erw}\) and \(D_\mathrm{arw}\).

This paper, however, contains an even worse “fit” to strontium data, as compared to the earlier studies (the left diagram is similar to the how it is presented in the article, the right diagram uses a linear y-axis; the line is eq. 3 with \(\mu_s = 0.24\)⁸):

This data does not suggest a variation in accordance with the adopted model even when plotted in a log-log diagram. With a linear y-axis, the dependence rather seems to be the opposite: \(D_\mathrm{arw}\) appears to increase with \(\kappa\). However, I suspect that this is a not a “real” dependence, but rather an effect of trying to construct a “relative” diffusivity; note that while \(\kappa\) spans four orders of magnitude, \(D_\mathrm{arw}\) scatters only by a factor of 5 or 6. Nevertheless, how this data can be claimed to show “general agreement” with the surface diffusion model is a mystery to me.

The view is similar for sodium (the left diagram is similar to the how it is presented in the article, the right diagram uses a linear y-axis; the line is eq. 3 with \(\mu_s=0.52\)⁸):

Even if the model in this case only displays minor variation, it can hardly be claimed to fit the data: again, the data suggests a diffusivity that increases with \(\kappa\). But a significant amount of these data points have \(D_\mathrm{arw} > 1\), which is not likely to be true, as it indicates that the relative mobility for sodium is larger than for water. Consequently, the major contribution of the variation seen in this data is most probably noise.

Gimmi and Kosakowski (2011) also examined diffusivity for calcium, and the data looks like this (the left diagram is similar to the how it is presented in the article, the right diagram uses a linear y-axis; the line is eq. 3 with \(\mu_s=0.1\)⁸):

Here it looks like the data, to some extent, behaves in accordance with the model also when plotted with linear y-axis covering the full model variation. However, there are significantly less data reported for calcium (as compared with cesium, strontium, and sodium) and the model variation is supported only by a few data points¹⁰. I therefore put my bet on that if calcium diffusivity is studied in more detail, the dependence suggested by the above plot will turn out to be spurious.¹¹

Some thoughts

I am more than convinced that the only reasonable starting point for modeling saturated bentonite is a homogeneous description. I had nevertheless expected to at least have to come up with an argument against the multi-porous view put forward in the considered publications (and in many others). I am therefore quite surprised to find that this argument is already provided by the data in the very same publications (and even by the statements, sometimes): there is nothing in the data here reviewed that seriously suggests that cation diffusion is influenced by a heterogeneous pore structure.

Still, the unsupported idea that cations in compacted bentonite are supposed to diffuse in two (or more) different types of water domains has evidently propagated through the scientific literature for decades, and a multi-porous view is mainstream in modern bentonite research. It is difficult to not feel disheartened when faced with this situation. What would it take for researchers to begin scrutinize their assumptions? Is nobody interested in the topics we are supposed to study?

Footnotes

[1] Unfortunately, a quantity which by many is incorrectly interpreted as an “apparent” diffusivity.

[2] I use quotation marks to indicate that \(D_a\) is a parameter in the traditional diffusion-sorption model, a model not valid for compacted bentonite. Still, this parameter is often reported as if it was a directly measured quantity.

[3] I have also derived a correct version of the surface diffusion model, which does not involve apparent diffusivity.

[4] The article states \(\epsilon D_p = 3.5\cdot 10^{-11}\; \mathrm{m^2/s}\), where \(\epsilon\) is the porosity. \(D_p = 1.2\cdot 10^{-10} \; \mathrm{m^2/s}\) corresponds to \(\epsilon = 0.29\).

[5] In this study, both \(K_d\) and \(D_a\) were evaluated by fitting the traditional diffusion-sorption model to concentration measurements.

[6] I have had no access to this document, and I have not verified e.g. sample density (this data set is different from that presented in the previous section). Instead, I have read these values from the diagram in Eriksen et al. (1999).

[7] They actually divide their cesium data into two categories, which show quite different mobility. The data shown here — which includes bentonite samples — is for systems categorized as being “non-illite” or having Cs concentration above “trace”.

[8] According to the article table, the fitted values for \(\mu_s\) are 0.52 (Na), 0.39 (Sr), 0.087 (Ca), and 0.015 (Cs). The plotted lines, however, appear to instead use what is listed as “mean \(\mu_s\)”. Here, I have used these \(\mu_s\)-values: 0.52 (Na), 0.24 (Sr), 0.1 (Ca), and 0.031 (Cs).

[9] This cluster contains a fourth data point, from Jensen and Radke (1988).

[10] All data for calcium is essentially from only two different sources: Staunton (1990) and Oscarsson (1994).

[11] It would also be more than amazing if it turns out — after it is verified that Cs, Na, and (especially) Sr show no significant \(K_d\) dependence — that Ca diffusivity actually varies in accordance with the flawed surface-diffusion model!

Donnan equilibrium and the homogeneous mixture model

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We can directly apply the homogeneous mixture model for bentonite to isolated systems — e.g. closed-cell diffusion tests — as discussed previously. For systems involving external solutions we must also handle the chemical equilibrium at solution/bentonite interfaces.

I have presented a framework for calculating the chemical equilibrium between an external solution and a bentonite component in the homogeneous mixture model here. In this post I will discuss and illustrate some aspects of that work.

Overview

We assume a homogeneous bentonite domain in contact with an external solution, with the clay particles prevented from crossing the domain interface. For real systems, this corresponds to the frequently encountered set-up with bentonite confined in a sample holder by means of e.g. a metal filter. From the assumptions of the homogeneous model — that all ions are mobile and allowed to cross the domain interface — it follows that the type of equilibrium to consider is the famous Donnan equilibrium. I have discussed the Donnan effect and its relevance for bentonite quite extensively here.

Since the adopted model assumes a homogeneous bentonite domain, the only region where Donnan equilibrium comes into play is at the interface between the bentonite and the external solution. This is quite different from how Donnan equilibrium calculations are implemented in many multi-porous models, where the equilibrium is internal to the clay — between assumed “macro” and “micro” compartments of the pore structure. The need for performing Donnan equilibrium calculations is thus minimized in the homogeneous mixture model (as mentioned, isolated systems require no such calculations). Note also that the semi-permeable mechanism in multi-porous models is required to act on the pore-scale. I have never seen any description or explanation how such a mechanism is supposed to work.¹ In the homogeneous mixture model, on the other hand, the semi-permeable interface corresponds directly to a macroscopic and experimentally well-defined component: the confining filter.

The problem to be solved can be illustrated like this

Schematic illustration of an external solution in contact with a homogeneous bentonite domain

The aim is to relate the set of species concentrations in the external solution (\(\{c_i^\mathrm{ext}\}\)) to those in the clay domain (\(\{c_i^\mathrm{int}\}\)) when the system is in equilibrium. This is done by applying the standard approach to Donnan equilibrium, as found in textbooks on the subject. If there is anything “radical” about this framework, it is thus not in the way Donnan equilibrium is implemented, but rather in treating bentonite as a single phase: this approach is formally equivalent to assuming the bentonite to be an aqueous solution.

Chemical equilibrium

I prefer to formulate the Donnan equilibrium framework in a way that separates effects due to difference in the local chemical environment from effects due to differences in electrostatic potential between the two compartments. An important reason for focusing on this separation is that the local environment affects the chemistry under all circumstances, while the (relative) value of the electrostatic potential only is relevant when bentonite is contacted with an external solution. We therefore express the chemical equilibrium as

\begin{equation} \frac{c_i^\mathrm{int}}{c_i^\mathrm{ext}} = \frac{\gamma_i^\mathrm{ext}}{\gamma_i^\mathrm{int}}\cdot e^{-\frac{z_iF\psi^\star}{RT}} \tag{1} \end{equation}

This formula is achieved by setting the electro-chemical potential equal for each species in the two compartments. Here \(\gamma_i\) denotes the activity coefficient for species \(i\), and \(\psi^*\) is the electrostatic potential difference between the compartments, which we refer to as the Donnan potential.

I find it convenient to rewrite this expression using some fancy Greek letters

\begin{equation} \label{eq:chem_eq2} \Xi_i = \Gamma_i \cdot f_D^{-z_i} \tag{2} \end{equation}

Here I call \(\Xi_i = c_i^\mathrm{int}/c_i^\mathrm{ext}\) the ion equilibrium coefficient for species \(i\). This quantity expresses the essence of ion equilibrium in the homogeneous mixture model, and will appear in many places in the analysis. \(\Xi_i\) has two factors:

\(\Gamma_i = \gamma_i^\mathrm{ext}/\gamma_i^\mathrm{int}\) expresses the chemical aspect of the equilibrium: when \(\Gamma_i\) is large (\(>1\)), the species has a chemical preference for residing in the interlayer pores, and when \(\Gamma_i\) is small (\(<1\)), the species has a preference for the external solution. In general, \(\Gamma_i\) for any specific species \(i\) is a function of all species concentrations in the system.
\(f_D^{-z_i}\), where \(f_D = e^{\frac{F\psi^\star}{RT}}\) is a dimensionless transformation of the Donnan potential (this is basically the Nernst equation), which we here call the Donnan factor. \(f_D\) expresses the electrostatic aspect of the equilibrium, and is the same for all species. The effect on \(\Xi_i\), however, is different for species of different charge number, because of the exponent \(-z_i\) in the full expression.

I want to emphasize that eqs. 1 and 2 express the exact same thing: chemical equilibrium between the two compartments.

Illustrations

To get a feel for the quantity \(\Xi\), here is a hopefully useful animation

Relation beteween internal and external concentration for varying Xi

It may also be helpful to see the influence of \(f_D\) on the equilibrium. Since the Donnan potential is negative, \(f_D\) is less than unity and typical values in relevant bentonite systems is \(f_D \sim\) 0.01 — 0.4. Due to the exponent \(-z_i\) in eq. 2, this influence on the equilibrium looks quite different for species with different valency. For mono- and di-valent cations, the behavior looks like this (here is put \(\Gamma = 1\) for both species)

Variation of internal cation concentrations with varying Donnan factor

The typical behavior for cations is that the internal concentration is much larger than the corresponding external concentration (at \(f_D = 0.01\) in the above animation, the internal concentration for the di-valent cation is enhanced by a factor \(\Xi = 10 000\)!). For anions, the internal concentration is instead lower than the external concentration,² as shown here (\(\Gamma = 1\) for both species)

Variation of internal anion concentration with the Donnan factor

Equation for \(f_D\)

For a complete description, we need an equation for calculating \(f_D\). This is derived by requiring charge neutrality in the two compartments and look like

\begin{equation*} \sum_i z_i\cdot\Gamma_i \cdot c_i^\mathrm{ext} \cdot f_D^{-z_i} – c_{IL} = 0 \tag{3} \end{equation*}

where

\begin{equation*} c_{IL} = \frac{CEC}{F \cdot w} \end{equation*}

is the structural charge present in the clay (i.e. negative montmorillonite layer charge) expressed as a monovalent interlayer concentration. Here \(CEC\) is the cation exchange capacity of the clay component, \(w\) the water-to-solid mass ratio,³ and \(F\) is the Faraday constant.

The way eq. 3 is formulated implies that the external concentrations should be used as input to the calculation. This is typically the case as the external concentrations are under experimental control.

In typical geochemical systems it is required to account for aqueous species with valency at least in the range -2 — +2 (e.g. \(\mathrm{Ca}^{2+}\), \(\mathrm{Na}^{+}\), \(\mathrm{Cl}^{-}\), \(\mathrm{SO_4}^{2-}\)), which implies that the equation for calculating \(f_D\) is generally a polynomial equation of degree four or higher.

An important special case is the 1:1 system — e.g. pure Na-montmorillonite contacted with a NaCl solution — which has an equation for \(f_D\) of only degree two, and thus have a relatively simple analytical solution

\begin{equation*} f_D = \frac{c_{IL}}{2c^\mathrm{ext} \Gamma_\mathrm{Cl}} \left ( \sqrt{1+ \frac{4(c^\mathrm{ext})^2 \Gamma_\mathrm{Na}\Gamma_\mathrm{Cl}} {c_{IL}^2}} – 1 \right ) \end{equation*}

With the machinery in place for calculating the Donnan potential, here is an animation demonstrating the response in internal sodium and chloride concentrations as the external NaCl concentration is varied. In this calculation \(c_{IL} = 2\) M, and \(\Gamma_\mathrm{Na} = \Gamma_\mathrm{Cl} = 1\)

Relation between internal and external Na and Cl concentrations

Comment on through-diffusion

To me, the last illustration makes it absolutely clear that Donnan equilibrium and the homogeneous mixture model provide the correct principal explanation for e.g. the behavior of tracer ions in through-diffusion tests. If you choose to relate the flux in through-diffusion tests to the external concentration difference — which is basically done in all published studies, via the parameter \(D_e\) — you will evaluate large “diffusivities” for cations and small “diffusivities” for anions. These “diffusivities” will, moreover, have the opposite dependence on background concentration: the cation flux diverges in the low background concentration limit,⁴ while the anion flux approaches zero.

But this behavior is seen to be caused by differently induced internal concentration gradients. If fluxes are related to these gradients — which they of course should, if you strive for an actual Fickian description — you find that the diffusivities are no different from what is evaluated in closed-cell tests. Relating the steady-state flux to the external concentration difference in the homogeneous mixture model gives (assuming zero tracer concentration on the outflow side)

\begin{equation*} j_\mathrm{ss} = -\phi\cdot D_c \cdot \nabla c^\mathrm{int} = \phi\cdot D_c \cdot\Xi\cdot \frac{c^\mathrm{source}}{L} \end{equation*}

where \(c^\mathrm{source}\) denotes the tracer concentration in the external solution on the inflow side, \(\phi\) is the porosity, \(D_c\) is the pore diffusivity in the interlayer domain, and \(L\) is the length of the bentonite sample. From the above equation can directly be identified

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

\(D_e\) is thus not a diffusion coefficient, but basically a measure of \(\Xi\).

Note that this explanation for the behavior of \(D_e\) does not invoke any notion of an anion accessible volume, nor any “sorption” concept for cations.⁵

Additional comments

When I first published on Donnan equilibrium in bentonite, I was a bit confused and singled out the term “Donnan equilibrium” to refer to anions only, while calling the corresponding cation equilibrium “ion-exchange equilibrium”. To refer to “both” types of equilibrium we used the term “ion equilibrium”.⁶ Of course, Donnan equilibrium applies to ions of any charge and, being better informed, I should have used a more stringent terminology. In later publications I have tried to make amends by pointing out that the process of cation exchange is part of the establishment of Donnan equilibrium.

Being new to the Donnan equilibrium world, I also invented some of my own nomenclature and symbols: e.g. I named the ratio between internal and external concentration the ion equilibrium coefficient (\(\Xi\)). Conventionally, if I now have understood correctly, this concentration ratio is referred to as the “Donnan ratio”, and is usually labeled \(r\) (although I’ve also seen \(K\)).

But the term “Donnan ratio” seems to be used slightly differently in different contexts, e.g. defined either as \(c^\mathrm{int}/c^\mathrm{ext}\) or as \(c^\mathrm{ext}/c^\mathrm{int}\), and is sometimes related more directly to the Donnan potential (if no distinction is made between activities and concentrations, we can write \(f_D^{-z_i} = c_i^\mathrm{int}/c_i^\mathrm{ext}\)). I therefore will continue to use the term “ion equilibrium coefficient” — with label \(\Xi\) — in the context of bentonite systems. This usage has also been picked up in some other clay publications. The ion equilibrium coefficient should be understood as strictly defined as \(\Xi = c^\mathrm{int}/c^\mathrm{ext}\) for any species, and never to define, or being defined by, the Donnan potential.

To emphasize the difference between effects due to the presence of a Donnan potential and effects due to different local chemical environments, I will refer to \(f_D\) as the Donnan factor. (This term does not seem to be used conventionally for any other quantity, although there are examples where it is used as a synonym for Donnan ratio.)

Finally, as in any other approach, the current framework requires a description for the activity coefficients. For activity coefficients in the external solution, there are quite a number of models already available. For the interlayer, modeling — and measuring! — activities is an open research area (at least I hope that this research area is open).

Footnotes

[1] This is just one of several major “loose ends” in most multi-porous models. I have earlier discussed the lack of treatment of swelling, and the incorrect treatment of fluxes in different domains. Update (220622): The lack of a semi-permeable component in multi-porosity models is further discussed here.

[2] This does not have to be the case in principle, if \(\Gamma\) for the anion is large, at the same time as the external concentration is not too low.

[3] Hence, it is implied that we use concentration units based on water mass (molality).

[4] What actually happens is that the transport resistance in the filters begins to dominate.

[5] Speaking of “sorption”, we have noted before that this term nowadays is used to mean any type of uptake between bulk water and some other domain (where the species may or may not be immobile). In this sense, there is “sorption” in the homogeneous mixture model (for both cations and anions), but only at interfaces to external solutions. It thus translates to a boundary condition, rather than being part of the transport dynamics within the clay (which makes life much simpler from a numeric perspective). Update (220622): The homogeneous mixture model is extended to deal with ions that truly sorbs here.

[6] It turns out Donnan himself actually used this terminology (“ionic equilibria”)

Sorption part II: Letting go of the bulk water

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Disclaimer: The following discussion applies fully to ions that only interact with bentonite by means of being part of an electric double layer. Here such ions are called “simple” ions. Species with more specific chemical interactions will be discussed in separate blog posts.

The “surface diffusion” model is not suitable for compacted bentonite

In the previous post on sorption¹ we derived a correct “surface diffusion” model. The equation describing the concentration evolution in such a model is a real Fick’s second law, meaning that it only contains the actual diffusion coefficient (apart from the concentration itself)

\begin{equation}
\frac{\partial c}{\partial t} = D_\mathrm{sd} \cdot\nabla^2 c \tag{1}
\end{equation}

Note that \(c\) in this equation still denotes the concentration in the presumed bulk water,² while \(D_\mathrm{sd}\) relates to the mobility, on the macroscopic scale, of a diffusing species in a system consisting of both bulk water and surfaces.³

Conceptually, eq. 1 states that there is no sorption in a surface diffusion model, in the sense that species do not get immobilized. Still, the concept of sorption is frequently used in the context of surface diffusion, giving rise to phrases such as “How Mobile Are Sorbed Cations in Clays and Clay Rocks?”. The term “sorption” has evidently shifted from referring to an immobilization process, to only mean the uptake of species from a bulk water domain to some other domain (where the species may or may not be mobile). In turn, the role of the parameter \(K_d\) is completely shifted: in the traditional model it quantifies retardation of the diffusive flux, while in a surface diffusion model it quantifies enhancement of the flux (in a certain sense).

A correct⁴ surface diffusion model resolves several of the inconsistencies experienced when applying the traditional diffusion-sorption model to cation diffusion in bentonite. In particular, the parameter referred to as \(D_e\) may grow indefinitely without violating physics (because it is no longer a real diffusion coefficient), and the insensitivity of \(D_\mathrm{sd}\) to \(K_d\) may be understood because \(D_\mathrm{sd}\) is the real diffusion coefficient (it is not an “apparent” diffusivity, which is expected to be influenced by a varying amount of immobilization).

Still, a surface diffusion model is not a very satisfying description of bentonite, because it assumes the entire pore volume to be bulk water. To me, it seems absurd to base a bentonite model on bulk water, as the most characteristic phenomenon in this material — swelling — relies on it not being in equilibrium with a bulk water solution (at the same pressure). It is also understood that the “surfaces” in a surface diffusion model correspond to montmorillonite interlayer spaces — here defined as the regions where the exchangeable ions reside⁵ — which are known to dominate the pore volume in any relevant system.

Indeed, assuming that diffusion occurs both in bulk water and on surfaces, it is expected that \(D_\mathrm{sd}\) actually should vary significantly with background concentration, because a diffusing ion is then assumed to spend considerably different times in the two domains, depending on the value of \(K_d\).⁶

Using the sodium diffusion data from Tachi and Yotsuji (2014) as an example, \(\rho\cdot K_d\) varies from \(\sim 70\) to \(\sim 1\), when the background concentration (NaCl) is varied from 0.01 M to 0.5 M (at constant dry density \(\rho=800\;\mathrm{kg/m^3}\)). Interpreting this in terms of a surface diffusion model, a tracer is supposed to spend about 1% of the time in the bulk water phase when the background concentration is 0.01 M, and about 41% of the time there when the background concentration is 0.5 M⁷. But the evaluated values of \(D_\mathrm{sd}\) (referred to as “\(D_a\)” in Tachi and Yotsuji (2014)) show a variation less than a factor 2 over the same background concentration range.

Insignificant dependence of \(D_\mathrm{sd}\) on background concentration is found generally in the literature data, as seen here (data sources: 1, 2, 3, 4, 5)

Diffusion coefficients as a function of background concentration for Sr, Cs, and Na.

These plots show the deviation from the average of the macroscopically observed diffusion coefficients (\(D_\mathrm{macr.}\)). These diffusion coefficients are most often reported and interpreted as “\(D_a\)”, but it should be clear from the above discussion that they equally well can be interpreted as \(D_\mathrm{sd}\). The plots thus show the variation of \(D_\mathrm{sd}\), in test series where \(D_\mathrm{sd}\) (reported as “\(D_a\)”) has been evaluated as a function of background concentration.⁸ The variation is seen to be small in all cases, and the data show no systematic dependencies on e.g. type of ion or density (i.e., at this level of accuracy, the variation is to be regarded as scatter).

The fact that \(D_\mathrm{sd}\) basically is independent of background concentration strongly suggests that diffusion only occurs in a single domain, which by necessity must be interlayer pores. This conclusion is also fully in line with the basic observation that interlayer pores dominate in any relevant system.

Diffusion in the homogeneous mixture model

A more conceptually satisfying basis for describing compacted bentonite is thus a model that assigns all pore volume to the surface regions and discards the bulk water domain. I call this the homogeneous mixture model. In its simplest version, diffusive fluxes in the homogeneous mixture model is described by the familiar Fickian expression

\begin{equation} j = -\phi\cdot D_c \cdot \nabla c^\mathrm{int} \tag{2} \end{equation}

where the concentration of the species under consideration, \(c^\mathrm{int}\), is indexed with an “int”, to remind us that it refers to the concentration in interlayer pores. The corresponding diffusion coefficient is labeled \(D_c\). Notice that \(c^\mathrm{int}\) and \(D_c\) refer to macroscopic, averaged quantities; consequently, \(D_c\) should be associated with the empirical quantity \(D_\mathrm{macr.}\) (i.e. what we interpreted as \(D_\mathrm{sd}\) in the previous section, and what many unfortunately interpret as \(D_a\)) — \(D_c\) is not the short scale diffusivity within an interlayer.

For species that only “interact” with the bentonite by means of being part of an electric double layer (“simple” ions), diffusion is the only process that alters concentration, and the continuity equation has the simplest possible form

\begin{equation} \frac{\partial n}{\partial t} + \nabla\cdot j = 0 \end{equation}

Here \(n\) is the total amount of diffusing species per volume porous system, i.e. \(n = \phi c^\mathrm{int}\). Inserting the expression for the flux in the continuity equation we get

\begin{equation} \frac{\partial c^\mathrm{int}}{\partial t} = D_c \cdot \nabla^2 c^\mathrm{int} \tag{3} \end{equation}

Eqs. 2 and 3 describe diffusion, at the Fickian level, in the homogeneous mixture model for “simple” ions. They are identical in form to the Fickian description in a conventional porous system; the only “exotic” aspect of the present description is that it applies to interlayer concentrations (\(c^\mathrm{int}\)), and more work is needed in order to apply it to cases involving external solutions.

But for isolated systems, e.g. closed-cell diffusion tests, eq. 3 can be applied directly. It is also clear that it will reproduce the results of such tests, as the concentration evolution is known to obey an equation of this form (Fick’s second law).

Model comparison

We have now considered three different models — the traditional diffusion-sorption model, the (correct) surface diffusion model, and the homogeneous mixture model — which all can be fitted to closed-cell diffusion data, as exemplified here

three models fitted to diffusion data for Sr from Sato et al. (92)

The experimental data in this plot (from Sato et al. (1992)) represent the typical behavior of simple ions in compacted bentonite. The plot shows the resulting concentration profile in a Na-montmorillonite sample of density 600 \(\mathrm{kg/m^3}\), where an initial planar source of strontium, embedded in the middle of the sample, has diffused for 7 days. Also plotted are the identical results from fitting the three models to the data (the diffusion coefficient and the concentration at 0 mm were used as fitting parameters in all three models).

From the successful fitting of all the models it is clear that bentonite diffusion data alone does not provide much information for discriminating between concepts — any model which provides a governing equation of the form of Fick’s second law will fit the data. Instead, let us describe what a successful fit of each model implies conceptually

The traditional diffusion-sorption model
The entire pore volume is filled with bulk water, in contradiction with the observation that bentonite is dominated by interlayer pores. In the bulk water strontium diffuse at an unphysically high rate. The evolution of the total ion concentration is retarded because most ions sorb onto surface regions (which have zero volume) where they become immobilized.
The “surface diffusion” model
The entire pore volume is filled with bulk water, in contradiction with the observation that bentonite is dominated by interlayer pores. In the bulk water strontium diffuse at a reasonable rate. Most of the strontium, however, is distributed in the surface regions (which have zero volume), where it also diffuse. The overall diffusivity is a complex function of the diffusivities in each separate domain (bulk and surface), and of how the ion distributes between these domains.
The homogeneous mixture model
The entire pore volume consists of interlayers, in line with the observation that bentonite is dominated by such pores. In the interlayers strontium diffuse at a reasonable rate.
From these descriptions it should be obvious that the homogeneous mixture model is the more reasonable one — it is both compatible with simple observations of the pore structure and mathematically considerably less complex as compared with the others.

The following table summarizes the mathematical complexity of the models (\(D_p\), \(D_s\) and \(D_c\) denote single domain diffusivities, \(\rho\) is dry density, and \(\phi\) porosity)

Note that the simplicity of the homogeneous mixture model is achieved by disregarding any bulk water phase: only with bulk water absent is it possible to describe experimental data as pure diffusion in a single domain. This process — pure diffusion in a single domain — is also suggested by the observed insensitivity of diffusivity to background concentration.

These results imply that “sorption” is not a valid concept for simple cations in compacted bentonite, regardless of whether this is supposed to be an immobilization mechanism, or if it is supposed to be a mechanism for uptake of ions from a bulk water to a surface domain. For these types of ions, closed-cell tests measure real (not “apparent”) diffusion coefficients, which should be interpreted as interlayer pore diffusivities (\(D_c\)).

Footnotes

[1] Well, the subject was rather on “sorption” (with quotes), the point being that “sorbed” ions are not immobilized.

[2] Eq. 1 can be transformed to an equation for the “total” concentration by multiplying both sides by \(\left (\phi + \rho\cdot K_d\right)\).

[3] Unfortunately, I called this quantity \(D_\mathrm{macr.}\) in the previous post. As I here compare several different diffusion models, it is important to separate between model parameters and empirical parameters, and the diffusion coefficient in the “surface diffusion” model will henceforth be called \(D_\mathrm{sd}\). \(D_\mathrm{macr.}\) is used to label the empirically observed diffusion parameter. Since the “surface diffusion” model can be successfully fitted to experimental diffusion data, the value of the two parameters will, in the end, be the same. This doesn’t mean that the distinction between the parameters is unimportant. On the contrary, failing to separate between \(D_\mathrm{macr.}\) and the model parameter \(D_\mathrm{a}\) has led large parts of the bentonite research community to assume \(D_\mathrm{a}\) is a measured quantity.

[4] It might seem silly to point out that the model should be “correct”, but the model which actually is referred to as the surface diffusion model in the literature is incorrect, because it assumes that diffusive fluxes in different domains can be added.

[5] There is a common alternative, implicit, and absurd definition of interlayer, based on the stack view, which I intend to discuss in a future blog post. Update (220906): This interlayer definition and stacks are discussed here.

[6] Note that, although \(D_\mathrm{sd}\) is not given simply by a weighted sum of individual domain diffusivities in the surface diffusion model, it is some crazy function of the ion mobilities in the two domains.

[7] With this interpretation, the fraction of bulk water ions is given by \(\frac{\phi}{\phi+\rho K_d}\).

[8] The plot may give the impression that such data is vast, but these are basically all studies found in the bentonite literature, where background concentration has been varied systematically. Several of these use “raw” bentonite (“MX-80”), which contains soluble minerals. Therefore, unless this complication is identified and dealt with (which it isn’t), the background concentration may not reflect the internal chemistry of the samples, i.e. the sample and the external solution may not be in full chemical equilibrium. Also, a majority of the studies concern through-diffusion, where filters are known to interfere at low ionic strength, and consequently increase the uncertainty of the evaluated parameters. The “optimal” tests for investigating the behavior of \(D_\mathrm{macr.}\) with varying background concentrations are closed-cell tests on purified montmorillonite. There are only two such tests reported (Kozaki et al. (2008) and Tachi and Yotsuji (2014)), and both are performed on quite low density samples.

The “uphill” diffusion experiment

The “uphill” diffusion phenomenon explained

Conventional through-diffusion

“Uphill” diffusion

Reexamining the Tertre et al. (2024) statements

A way forward (if anybody cares)

Footnotes

Implied structure of the free water domain

Arbitrary division between diffuse layer and free water

Donnan effect on the microscopic scale?!

Footnotes

A testable difference

Comparison with experiment

Van Loon et al. (2007)

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

Glaus et al. (2010)

Insight from closed-cell tests

Footnotes

Measuring Kd

Kd in the homogeneous mixture model

Kd for simple cations

Simple cation tracers in a 1:1 system

Simple cation tracers in a 2:1 system

Density dependence of Kd

Footnotes

The through-diffusion set-up

The effective porosity diffusion model

Connection with experimentally accessible quantities

Additional comments on the effective porosity model

Footnotes

Muurinen et al. (1985)

Eriksen et al. (1999)

Gimmi & Kosakowski (2011)

Some thoughts

Footnotes

Overview

Chemical equilibrium

Illustrations

Equation for \(f_D\)

Comment on through-diffusion

Additional comments

Footnotes

The “surface diffusion” model is not suitable for compacted bentonite

Diffusion in the homogeneous mixture model

Model comparison

Footnotes

Measuring K_d

K_d in the homogeneous mixture model

K_d for simple cations

Density dependence of K_d