Category Archives: Electric double layer

Are interlayer cations not attracted to the surfaces?!

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

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

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

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

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

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

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

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

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

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

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

A couple of corrections

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

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

But the final sentence should rather be formulated as

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

In the original post, I also write

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

The last part is better formulated as

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

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

Footnotes

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

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

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

Solving the Poisson-Boltzmann equation

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

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

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

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

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

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).

Semi-permeability, part I

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Descriptions in bentonite literature

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Chagneau et al. (2015) write

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

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

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

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

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

Bentonite as a non-ideal semi-permeable membrane

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

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

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

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

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

“Semi-permeability” in experiments

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

Tracer through-diffusion

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

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

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

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

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

Measuring hydraulic conductivity

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

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

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

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

Mass transfer in a salt gradient

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

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

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

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

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

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

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

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

Footnotes

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

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

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

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

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

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

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

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

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

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

Molecular dynamics simulations do not support complete anion exclusion

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We have discussed various aspects of “anion exclusion” on this blog. This concept is often used to justify multi-porosity models of compacted bentonite, by reasoning that the exclusion mechanism makes parts of the pore space inaccessible to anions. But we have seen that this reasoning has no theoretical backup: studies making such assumptions usually turn out to refer to conventional electric double layer theory, described e.g. by the Poisson-Boltzmann equation. In the following, we refer to the notion of compartments inaccessible to anions as complete anion exclusion.

In fact, a single, physically reasonable concept underlies basically all descriptions of anion exclusion in the clay literature: charge separation. Although the required mathematics may differ for different systems — may it be using Donnan’s “classical equations”, or the Poisson-Boltzmann equation — the underlying mechanism is the same. In the following we refer to this type of description as traditional theory or Donnan theory. It is important to recognize that traditional theory is incompatible with complete anion exclusion: the Poisson-Boltzmann equation predicts anions everywhere.

In more recent years, however, a different meaning of the term “anion exclusion” has sneaked into the literature. This seems to be related to the dawn of molecular dynamics (MD) simulations of clays. In particular, the study of Rotenberg et al. (2007) — which I think is the first published MD simulation of montmorillonite interlayers in contact with an external compartment — is frequently cited as demonstrating qualitatively different results as compared with the traditional models. E.g. Kosakowski and Berner (2013) write

Very often it is assumed that negatively charged ions are strongly hindered to enter the interlayer space (Kosakowski et al., 2008; Rotenberg et al., 2007), although other authors come to different conclusions (Karnland et al., 2007). Note that we favor the former view with our montmorillonite setup.

Although the terms “assumed” and “conclusions” seem misplaced, it is clear that Kosakowski and Berner (2013) mean that the interlayer space is essentially anion-free, rather than obeying ordinary Donnan equilibrium (the approach used in Karnland et al. (2007)).

A similar citation is found in Tournassat and Steefel (2015)

The interlayer space can be seen as an extreme case where the diffuse layer vanishes leaving only the Stern layer of the adjacent basal surfaces. For this reason, the interlayer space is often considered to be completely free of anions (Tournassat and Appelo 2011), although this hypothesis is still controversial (Rotenberg et al. 2007c; Birgersson and Karnland 2009).

Here Tournassat and Steefel (2015) conceive of the interlayer space as something distinctly different from a diffuse layer¹, and they mean that the MD result stands in contrast to conventional Donnan theory (Birgersson and Karnland, (2009)).

As a third example, Wersin et al. (2016) write

Based upon [results from anion diffusion tests], anion-exclusion models have been formulated, which subdivide the water-filled pore space into interlayer, diffuse (or electric) double layer (DDL) and “free” water porosities (Wersin et al. 2004; Tournassat & Appelo 2011; Appelo 2013). In this formulation, anions are considered to reside in the “free” electrically neutral solution and in the DDL in the external (intergranular) pores, whereas the interlayer (intragranular) space is considered devoid of anions. Support for this model has been given by molecular dynamics simulations (Rotenberg et al. 2007), but this issue remains controversial (Birgersson & Karnland 2009)

The term “anion-exclusion” is here fully transformed to refer to complete exclusion, rather than to the traditional theory from which the term was coined. Note that the picture of bentonite given in this and the previous quotations is basically the contemporary mainstream view, which we discussed in a previous blog post. This description has not emerged from considering MD results that are allegedly in contradiction with traditional Donnan equilibrium theory. Rather, it has resulted from misusing the concept of exclusion-volume. The study of Rotenberg et al. (2007) (Rot07, in the following) supports the contemporary mainstream view only to the extent that it is at odds with the predictions of traditional theory. But is it? Let’s take a look at the relevant MD studies.

Rotenberg et al. (2007)

Rot07 is not primarily a study of the anion equilibrium, but considers more generally the transition of species between an external compartment² and interlayer pores: water, cations (Na and Cs), and anions (Cl). The study only concerns interlayers with two monolayers of water, in the following referred to as a 2WL system. There is of course nothing wrong with exclusively studying the 2WL system, but this study alone cannot be used to support general model assumptions regarding interlayers (which anyway is commonplace, as we saw above). The meaning of the term “interlayer” in modern clay literature is quite confusing, but there is at least full consensus that it includes also states with three monolayers of water (3WL) (we’ll get back to those). Rot07 furthermore consider only a single external concentration, of 0.52 M.

Here is an illustration of the simulated system:

A cell (outlined with dashed lines) containing two montmorillonite layers (yellow) and six chloride ions (green) is repeated infinitely in all directions (the cell depth in the direction normal to the picture is 20.72 Å). While only chloride ions are indicated in this figure, also cations, water atoms, and montmorillonite atoms are explicitly accounted for in the simulation.

Note that the study neither varies density (interlayer distance) nor external concentration (number of chloride ions) — two variables essential for studying anion equilibrium. I don’t mean this as direct criticism, but it should be recognized when the study is used to support assumptions regarding interlayers in other models.

What I do want to criticize, however, is that Rot07 don’t actually compare with Donnan theory. Instead, they seem to be under the impression that traditional theory predicts complete exclusion in their system. Consider this passage in the introduction

Due to the negative charge of clay layers, anions should be repelled by the external surfaces, and excluded from the interlayers. On the contrary, cations are attracted by the surfaces, and may exchange with the natural interlayer counterions.

Here they associate two different terms with the anions: they are repelled by the “external surfaces” and excluded from “interlayers”. I can only interpret this as meaning that anions are completely excluded from interlayers, especially as the wording “on the contrary” is used when describing cations.³

The study comprises both a “plain” MD simulation of the (presumed) equilibrium state, and separate calculations of free energy profiles. In the “plain” MD simulation, anions do not enter the interlayers, and the calculation of the free energy profile gives a barrier of ~9 kT for chloride to enter the interlayer.

These results motivate the authors to conclude that the “thermal fluctuations do not allow anions to overcome the free energy barrier corresponding to their entrance into the interlayer” and that “anions are excluded from the interlayer: the probability for an anion reaching the interface to enter into the interlayer is very small (of the order of e^-9 ~ 10^-4)”

It is important to keep in mind that the authors are under the impression that this result and conclusion are in line with the traditional description of anion exclusion.³ When summarizing their findings they write

All the results are in agreement with the common sense on ionic exchange and anion exclusion.

and

The results confirm the generally admitted ideas of ionic exchange and anion exclusion

The problem is that this “common sense” and these “generally admitted ideas” are based on misconceptions of traditional theory (I also think one should be careful with using terms like these in scientific writing). Consequently, the authors erroneously conclude that their results confirm, rather than contrast, traditional theory. This is opposite to how this study is referred to in later publications, as was exemplified above.

The anion exclusion predicted from Donnan theory for the system in Rot07 is estimated as follows. The adopted montmorillonite unit cell (Na_0.75Si₈Al_3.25Mg_0.75O₂₀OH₄) has structural charge 0.75e, and lateral dimensions 8.97 Å × 5.18 Å. With an interlayer width of 6.1 Å we thus have for the concentration of interlayer charge

\begin{equation} c_{IL} = \frac{0.75/N_A}{8.97\cdot 5.18\cdot 6.1 \mathrm{Å^3}} = 4.39 \;\mathrm{M} \end{equation}

where \(N_A\) is the Avogadro constant. Using this value for \(c_{IL}\) in the expression for internal anion concentration in an ideal 1:1 Donnan system,

\begin{equation} c^\mathrm{int} = \frac{c_{IL}}{2} \left ( \sqrt{1+\frac{4\cdot (c^\mathrm{ext})^2}{c_{IL}^2}} – 1 \right ) \tag{1} \end{equation}

together with \(c^\mathrm{ext}\) = 0.52 M, gives

\begin{equation} c^\mathrm{int} = 0.06 \;\mathrm{M} \end{equation}

This should be the anion interlayer concentration expected from “generally admitted ideas”, and Rot07 should have concluded that their results differ by a factor ~1000 (or more) from traditional theory. This is not to say that the calculations are incorrect (more on that later), but it certainly puts the results in a different light. A discrepancy of this magnitude should reasonably be of interest to investigate further.

Hsiao and Hedström (2015)

Considerably more detailed MD simulations of the 2WL system are provided by Hsiao and Hedström (2015) (Hsi15, hereafter). In contrast to Rot07, Hsi15 specifically focus on the anion equilibrium, and they explicitly compare with both conventional Donnan theory, and the results of Rot07. In these simulations, chloride actually populates the interlayer.

Hsi15 also analyze the convergence behavior, by varying system size and simulation time. This analysis makes it clear both that most of the simulations presented in the paper are properly converged, and that the simulation of Rot07 is not. With external concentration 1.67 M, Hsi15 demonstrate that, during intervals of 20 ns, the interlayer concentration fluctuates between basically zero and 0.13 M (converged value: 0.04 M), in a system with similar size as that of Rot07. Given that the total simulation time of the earlier study is 20 ns, and that it also adopts a considerably lower external concentration, its result of zero chloride concentration in the interlayer is no surprise.

The converged interlayer concentrations in Hsi15 look like this in the direction normal to the basal surfaces (simulation time: 150 ns, layer size: 8 × 4 unit cells, external concentration: 1.67 M)

Note that the simulation contains two interlayer pores (indicated by the dotted lines; cf. the illustration of the simulated system) and that sodium and chloride populate the same central layer, sandwiched by the two water layers (not shown). The nearly identical chloride profiles is a strong confirmation that the simulation is converged.

The chloride interlayer concentrations evaluated in Hsi15 deviate strongly from the predictions of the ideal Donnan formula. With \(c_{IL}\) = 4.23 M (as reported in the article) and \(c^\mathrm{ext}\) = 1.67 M, eq. 1 gives \(c^\mathrm{int}\) = 0.580 M, while the MD results are in the range 0.033 M — 0.045 M, i.e. more than a factor 10 lower (but not a factor 1000).

Hsi15 also calculate the free energy profiles along the coordinate connecting the external compartment and the interlayer, similar to the technique utilized by Rot07 (as far as I understand). For the external concentration of 1.67 M they evaluate a free energy barrier of ~3.84 kT, which corresponds to an interlayer concentration of 0.036 M, and is in good agreement with the directly evaluated concentrations.

Note that Hsi15 — in contrast to Rot07 — conclude significant deviation between the MD results of the 2WL system and ideal traditional theory. Continuing their investigation (again, in contrast to Rot07), Hsi15 found that the contribution from ion hydration to the free energy barrier basically make up for the entire discrepancy with the ideal Donnan formula. Moreover, even though the ideal Donnan formula strongly overestimates the actual values obtained from MD, it still shows the correct dependency on external concentration: when the external concentration is lowered to 0.55 M, the evaluated free energy barrier increases to ~5.16 kT, which corresponds to a reduction of the internal concentration by about a factor of 10. This is in agreement with Donnan theory, which gives for the expected reduction (0.55/1.67)² ≈ 0.11.

From the results of Hsi15 (and Rot07, for that matter), a relatively clear picture emerges: MD simulated 2WL systems function as Donnan systems. Anions are not completely excluded, and the dependency on external concentration is in line with what we expect from a varying Donnan potential across the interface between interlayer and external compartment (Hsi15 even comment on observing the space-charge region!).

The simulated 2WL system is, however, strongly non-ideal, as a consequence of the ions not being optimally hydrated. Hsi15 remark that the simulations probably overestimate this energy cost, e.g. because atoms are treated as non-polarizable. This warning should certainly be seriously considered before using the results of MD simulated 2WL systems to motivate multi-porosity in compacted bentonite. But, concerning assumptions of complete anion exclusion in interlayers, another system must obviously also be considered: 3WL.

Hedström and Karnland (2012)

MD simulations of anion equilibrium in the 3WL system are presented in Hedström and Karnland (2012) (Hed12, in the following). Hed12 consider three different external concentrations, by including either 12, 6, or 4 pairs of excess ions (Cl^– + Na⁺). This study also varies the way the interlayer charge is distributed, by either locating unit charges on specific magnesium atoms in the montmorillonite structure, or by evenly reducing the charge by a minor amount on all the octahedrally coordinated atoms.

Here are the resulting ion concentration profiles across the interlayer, for the simulation containing 12 chloride ions, and evenly distributed interlayer charge (simulation time: 20 ns, layer size: 4 × 4 unit cells)

Chloride mainly resides in the middle of the interlayer also in the 3WL system, but is now separated from sodium, which forms two off-center main layers. The dotted lines indicate the extension of the interlayer.

The main objectives of this study are to simply establish that anions in MD equilibrium simulations do populate interlayers, and to discuss the influence of unavoidable finite-size effects (6 and 12 are, after all, quite far from Avogadro’s number). In doing so, Hed12 demonstrate that the system obeys the principles of Donnan equilibrium, and behaves approximately in accordance with the ideal Donnan formula (eq. 1). The authors acknowledge, however, that full quantitative comparison with Donnan theory would require better convergence of the simulations (the convergence analysis was further developed in Hsi15). If we anyway make such a comparison, it looks like this

#Cl TOT	Layer charge	#Cl IL	\(c^\mathrm{ext}\)	\(c^\mathrm{int}\) (Donnan)	\(c^\mathrm{int}\) (MD)
12	distr.	1.8	1.45	0.62	0.42 (67%)
12	loc.	1.4	1.50	0.66	0.32 (49%)
6	distr.	0.6	0.77	0.20	0.14 (70%)
6	loc.	1.3	0.67	0.15	0.30 (197%)
4	distr.	0.2	0.54	0.10	0.05 (46%)
4	loc.	0.18	0.54	0.10	0.04 (41%)

The first column lists the total number of chloride ions in the simulations, and the second indicates if the layer charge was distributed on all octahedrally coordinated atoms (“distr.”) or localized on specific atoms (“loc.”) The third column lists the average number of chloride ions found in the interlayer in each simulation. \(c^\mathrm{ext}\) denotes the corresponding average molar concentration in the external compartment. The last two columns lists the corresponding average interlayer concentration as evaluated either from the Donnan formula (eq. 1 with \(c_{IL}\) = 2.77 M, and the listed \(c^\mathrm{ext}\)), or from the simulation itself.

The simulated results are indeed within about a factor of 2 from the predictions of ideal Donnan theory, but they also show a certain variation in systems with the same number of total chloride ions,⁴ indicating incomplete convergence (compare with the fully converged result of Hsi15). It is also clear from the analysis in Hed12 and Hsi15 that the simulations with the highest number och chloride ions (12) are closer to being fully converged.⁵ Let’s therefore use the result of those simulations to compare with experimental data.

Comparison with experiments

In an earlier blog post, we looked at the available experimental data on chloride equilibrium concentrations in Na-dominated bentonite. Adding the high concentration chloride equilibrium results from Hed12 and Hsi15 to this data (in terms of \(c^\mathrm{int}/c^\mathrm{ext}\)), gives the following picture⁶ (the 3WL system corresponds to pure montmorillonite of density ~1300 kg/m³, and the 2WL system corresponds to ~1600 kg/m³, as also verified experimentally).

The x-axis shows montmorillonite effective dry density, and applied external concentrations for each data series are color coded, but also listed in the legend. Note that this plot contains mainly all available information for drawing conclusions regarding anion exclusion in interlayers.⁷ To me, the conclusions that can be drawn are to a large extent opposite to those that have been drawn:

The amount chloride in the simulated 3WL system corresponds roughly to measured values. Consequently, MD simulations do not support models that completely exclude anions from interlayers.
The 3WL results instead suggest that interlayers contain the main contribution of chloride. Interlayers must consequently be handled no matter how many additional pore structures a model contains.
For systems corresponding to 2WL interlayers, there is a choice: Either,
1. assume that the discrepancy between simulations and measurements indicates the existence of an additional pore structure, where the majority of chloride resides, or
2. assume that presently available MD simulations of 2WL systems overestimate “anion” exclusion.⁸
If making choice no. 1. above, keep in mind that the additional pore structure cannot be 3WL interlayers (they are virtually non-existent at 1600 kg/m³), and that it should account for approximately 0% of the pore volume.

Tournassat et al. (2016)

Tournassat et al. (2016) (Tou16, in the following) present more MD simulations of interlayer pores in contact with an external compartment, with a fixed amount of excess ions, at three different interlayer distances: 2WL (external concentration ~0.5 M), 3WL (~0.4 M), and 5WL (~0.3 M).

In the 2WL simulations, no anions enter the interlayers. Tou16 do not reflect on the possibility that 2WL simulations may overestimate exclusion, as suggested by Hsi15⁹, but instead use this result to argue that anions are basically completely excluded from 2WL interlayers. They even imply that the result of Rot07 is more adequate than that of Hsi15

In the case of the 2WL hydrate, no Cl^– ion entered the interlayer space during the course of the simulation, in agreement with the modeling results of Rotenberg et al. (2007b), but in disagreement with those of Hsiao and Hedström (2015).

But, as discussed, there is no real “disagreement” between the results of Hsi15 and Rot07. To refute the conclusions of Hsi15, Tou16 are required to demonstrate well converged results, and analyze what is supposedly wrong with the simulations of Hsi15. It is, furthermore, glaringly obvious that most of the anion equilibrium results in Tou16 are not converged.

Regarding convergence, the only “analysis” provided is the following passage

The simulations were carried out at the same temperature (350 K) as the simulations of Hsiao and Hedström (2015) and with similar simulation times (50 ns vs. 100-200 ns) and volumes (27 × 10⁴ Å³ vs. 15 × 10⁴ Å³), thus ensuring roughly equally reliable output statistics. The fact that Cl^– ions did not enter the interlayer space cannot, therefore, be attributed to a lack of convergence in the present simulation, as Hsiao and Hedström have postulated to explain the difference between their results and those of Rotenberg et al. (2007b).

I mean that this is not a suitable procedure in a scientific publication — the authors should of course demonstrate convergence of the simulations actually performed! (Especially after Hsi15 have provided methods for such an analysis.¹⁰)

Anyhow, Tou16 completely miss that Hsi15 demonstrate convergence in simulations with external concentration 1.67 M; for the system relevant here (0.55 M), Hsi15 explicitly write that the same level of convergence requires a 10-fold increase of the simulation time (because the interlayer concentration decreases approximately by a factor of 10, as predicted by — Donnan theory). Thus, the simulation time of Tou16 (53 ns) should be compared with 2000 ns, i.e. it is only a few percent of the time required for proper convergence.

Further confirmation that the simulations in Tou16 are not converged is given by the data for the systems where chloride has entered the interlayers. The ion concentration profiles for the 3WL simulation look like this

The extension of the interlayers is indicated by the dotted lines. Each interlayer was given slightly different (average) surface charge density, which is denoted in the figure. One of the conspicuous features of this plot is the huge difference in chloride content between different interlayers: the concentration in the mid-pore (0.035 M) is more than three times that in left pore (0.010 M). This clearly demonstrates that the simulation is not converged (cf. the converged chloride result of Hsi15). Note further that the larger amount of chloride is located in the interlayer with the highest surface charge, and the least amount is located in the interlayer with the smallest surface charge.¹¹ I think it is a bit embarrassing for Clays and Clay Minerals to have used this plot for the cover page.

As the simulation times (53 ns vs. 40 ns), as well as the external concentrations (~0.5 M vs. ~0.4 M), are similar in the 2WL and and 3WL simulations, it follows from the fact that the 3WL system is not converged, that neither is the 2WL system. In fact, the 2WL system is much less converged, given the considerably lower expected interlayer concentration. This conclusion is fully in line with the above consideration of convergence times in Hsi15.

For chloride in the 3WL (and 5WL) system, Tou16 conclude that “reasonable quantitative agreement was found” between MD and traditional theory, without the slightest mentioning of what that implies.¹² I find this even more troublesome than the lack of convergence. If the authors mean that MD simulations reveal the true nature of anion equilibrium (as they do when discussing 2WL), they here pull the rug out from under the entire mainstream bentonite view! With the 3WL system containing a main contribution, interlayers can of course not be modeled as anion-free, as we discussed above. Yet, not a word is said about this in Tou16.

In this blog post I have tried to show that available MD simulations do not, in any reasonable sense, support the assumption that anions are completely excluded from interlayers. Frankly, I see this way of referencing MD studies mainly as an “afterthought”, in attempts to justify the misuse of the exclusion-volume concept. In this light, I am not surprised that Hed12 and Hsi15 have not gained reasonable attention, while Tou16 nowadays can be found referenced to support claims that anions do not have access to “interlayers”.¹³

Footnotes

[1] I should definitely discuss the “Stern layer” in a future blog post.

[2] The view of bentonite (“clay”) in Rotenberg et al. (2007) is strongly rooted in a “stack” concept. What I refer to as an “external compartment” in their simulation, they actually conceive of as a part of the bentonite structure, calling it a “micropore”.

[3] That Rotenberg et al. (2007) expresses this view of anion exclusion puzzles me somewhat, since several of the same authors published a study just a few years later where Donnan theory was explored in similar systems: Jardat et al. (2009).

[4] Since the number of chloride ions found in the interlayer is not correlated with how layer charge is distributed, we can conclude that the latter parameter is not important for the process.

[5] The small difference in the two simulations with 4 chloride ions is thus a coincidence.

[6] I am in the process of assessing the experimental data, and hope to be able to better sort out which of these data series are more relevant. So far I have only looked at — and discarded — the study by Muurinen et al. (1988). This study is therefore removed from the plot.

[7] There are of course several other results that indirectly demonstrate the presence of anions in interlayers. Anyway, I think that the bentonite research community, by now, should have managed to produce better concentration data than this (both simulated and measured).

[8] As the cation (sodium) may give a major contribution to the hydration energy barrier (this is not resolved in Hsiao and Hedström (2015)), it may be inappropriate to refer to this part as “anion” exclusion (remember that it is salt that is excluded from bentonite). It may be noted that sodium actually appear to have a hydration barrier in e.g. the Na/Cs exchange process, which has been explored both experimentally and in MD simulations.

[9] Tournassat et al. (2016) even refer to Hsiao and Hedström (2015) as presenting a “hypothesis” that “differences in solvation energy play an important role in inhibiting the entry of Cl^– in the interlayer space”, rather than addressing their expressed concern that the hydration energy cost may be overestimated.

[10] Ironically, Tournassat et al. (2016) choose to “rely” on the convergence analysis in Hsiao and Hedström (2015), while simultaneously implying that the study is inadequate.

[11] As the interlayers have different surface charge, they are not expected to have identical chloride content. But the chloride content should reasonably decrease with increasing surface charge, and the difference between interlayers should be relatively small.

[12] Here we have to disregard that the “agreement” is not quantitative. It is not even qualitative: the highest chloride content was recorded in the interlayer pore with highest charge, in both the 3WL and the 5WL system.

[13] There are even examples of Hedström and Karnland (2012) being cited to support complete exclusion!

The mechanism for “anion” exclusion

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Repulsion between surfaces and anions is not really the point

Many publications dealing with “anion” exclusion in compacted bentonite describe the phenomenon as being primarily due to electrostatic repulsion of anions from the negatively charged clay surfaces. This explanation, which may seem plausible both at a first and a second glance, is actually not that satisfactory. There are two major issues to consider:

Although it is popular to use the word “anion” when referring to the phenomenon, it must be remembered that the anions are accompanied by cations, in order to maintain overall charge neutrality; it really is salt that is excluded from the bentonite. This observation shows that the above “explanation” is incomplete: it can be argued with the same logic that salt should accumulate, because the clay surfaces attract the cations of the external salt.
Salt exclusion occurs generally in Donnan systems, also in those that lack surfaces. Its principal explanation can consequently not involve the presence of surfaces. For a simpler system, e.g. potassium ferrocyanide, the “explanation” above translates to claiming that exclusion is caused by “anions” being electrostatically repelled by the ferrocyanide ions. In this case it may be easier to spot the shortcoming of such a claim, and to consider also the potassium ions (which attract anions), as well as the role played by the cations of the excluded salt.

What, then, is the primary cause for salt exclusion? Let us continue with using potassium ferrocyanide as an example of a simple Donnan system, and then translate our findings to the case of compacted bentonite.

Ferrocyanide

Consider a potassium ferrocyanide solution separated from a potassium chloride solution by a membrane permeable to all but the ferrocyanide ions. The ionic configuration near the membrane then looks something like this

Because potassium ions can pass the membrane, and because they have an entropic driving force to migrate out of the ferrocyanide solution, a (microscopic) region is formed in the external solution next to the membrane, with an excess amount of positive charge. Similarly, a region is formed next to the membrane in the ferrocyanide solution with an excess amount of negative charge. Thus, a region of charge separation exists across the membrane — similar to the depletion zone in a p-n junction — over which the electrostatic potential varies. The electric field (= a varying potential) at the interface acts as to pull back potassium ions towards the ferrocyanide solution. The equilibrium width of the space charge region is set when the diffusive flux is balanced by the flux due to the electric field.

With a qualitative understanding of the electrostatic potential configuration we can now give the most plain answer to what causes “anion” exclusion: it is because of the potential difference across the membrane. Chloride ions behave in the opposite way as compared to potassium, with an entropic driving force to enter the ferrocyanide solution, while being pulled back towards the external solution due to the electric field across the membrane.

Here the mindful reader may perhaps object and point out that the electric field restricting the chloride inflow reasonably originates from the ferrocyanide anions. It thus may seem that “anion” exclusion, after all, is caused by repulsion from other negative charges.

Indeed, electrostatic repulsion of anions requires the “push” of some other negatively charged entity. But note that the potential is constant in the interior of the ferrocyanide solution, and only varies near the membrane. The variation of the potential is caused by separation of charge: chloride is as much “pushed” out of the ferrocyanide solution by the ferrocyanide as it is “pulled” out of it, due to electrostatic attraction, by the excess potassium on the other side. Repulsion between charges of equal sign occurs also in the interior of the ferrocyanide solution (or in any ionic solution), but does not in itself lead to salt exclusion.

Bentonite

The above description can be directly transferred to the case of compacted bentonite. Replacing the potassium ferrocyanide with e.g. K-montmorillonite, salt exclusion occurs mainly because potassium can migrate out of the clay region, while montmorillonite particles cannot. Again, we have charge separation with a resulting varying electrostatic potential across the interface.

Admittedly, the general situation is more complicated in bentonite because of the extension of montmorillonite particles; viewed as “anions”, these are irregularly shaped macromolecules with hundreds or thousands of charge centers.

The ion configuration in a bentonite suspension therefore looks quite different from a corresponding ordinary solution, as the montmorillonite charge obviously is constrained to individual particles. Dilute systems thus have charge separation on the particle scale and show salt exclusion even without charge separation at the interface to the external solution. These types of systems (suspensions) have historically been the subject of most studies on “anion” exclusion, and are usually treated theoretically using the Gouy-Chapman model.

With increasing density, however, the effect of a varying potential between montmorillonite particles diminishes, while the effect of charge separation at the interface increases. For dense systems (> 1.2 g/cm³, say), we may therefore approximate the internal potential as constant and only consider the variation across the interface to the external solution using Donnan’s “classical” framework.¹

Here is an illustration of the validity of this approximation:

Internal and external potential in compacted bentonite system

The figure shows the difference between the external (green) and the average internal (orange) potentials in a 1:1 system of density 1.3 g/cm³ and with external concentration 0.1 M, calculated using Donnan’s “classical” equation. Also plotted is the electrostatic potential across the interlayer (blue) as calculated using the Poisson-Boltzmann equation,² in a similar system (interlayer distance 1 nm). It is clear that the variation of the Poisson-Boltzmann potential from the average is small in comparison with the Donnan potential.

Repulsion between chloride and montmorillonite particles of course occurs everywhere in compacted bentonite, whereas the phenomenon mainly responsible for salt exclusion occurs only near the interfaces. Merely stating electrostatic repulsion as the cause for salt exclusion in compacted bentonite does not suffice, just as in the case of ferrocyanide.

To illustrate that the salt exclusion effect depends critically on exchangeable cations being able to diffuse out of the bentonite, consider the following thought experiment.³ Compacted K-montmorillonite is contacted with a NaCl solution. But rather than having a conventional component separating the solution and the clay, we imagine a membrane that does not allow for the passage of neither potassium nor clay, but that allows for the passage of sodium and chloride. Since potassium is not allowed to diffuse out of the bentonite, no charge separation occurs across the membrane. With no space charge region, the electrostatic potential does not vary and NaCl is not excluded! (to the extent that the Donnan approximation is valid)

NaCl + K-montmorillonite with interface only permeable for Na and Cl

A charge neutral perspective

The explanation for “anion” exclusion that we have explored rests on the formation of a potential difference across the interface region between bentonite and external solution. But remember that it is salt — in our example KCl — that is excluded from the bentonite (or the ferrocyanide solution), and that the cation (K) gains energy by being transferred from the external to the internal solution. The electrical work for transferring a unit of KCl is thus zero (which makes sense since KCl is a charge neutral entity). In this light, it may seem unsatisfactory to offer the potential difference as the sole explanation for salt exclusion.

I therefore think that the following kinematic way of reasoning is very helpful. Instead of considering the mass transfer of Cl across the membrane in terms of oppositely directed “electric” and “diffusive” parts, we lump them together with equal amounts of K transfer, giving two equal but oppositely directed fluxes of KCl. Reasonably, the KCl flux into the ferrocyanide solution is proportional to the external ion concentrations

\begin{equation} j^\mathrm{in}_\mathrm{KCl} = A\cdot c_\mathrm{K}^\mathrm{ext}\cdot c_\mathrm{Cl}^\mathrm{ext} \end{equation}

while the outflux is proportional to the internal ion concentrations

\begin{equation} j^\mathrm{out}_\mathrm{KCl} = -A\cdot c_\mathrm{K}^\mathrm{int}\cdot c_\mathrm{Cl}^\mathrm{int} \end{equation}

\(A\) is a coefficient accounting for the transfer resistance across the interface region. Requiring the sum of these fluxes to be zero gives the following relation

\begin{equation} c_\mathrm{K}^\mathrm{ext}\cdot c_\mathrm{Cl}^\mathrm{ext} = c_\mathrm{K}^\mathrm{int}\cdot c_\mathrm{Cl}^\mathrm{int} \end{equation}

which is the (ideal) Donnan equation.

We can therefore interpret KCl exclusion as an effect of potassium in the clay providing a potential for “out-transfer”, as soon as the chance is given, i.e. when chloride enters from the external solution. From this perspective salt exclusion could maybe be said to be a form of cation “rejection”.

Footnotes

[1] Note also that the Gouy-Chapman model is not valid in the high density limit, although it is applied (or alluded to) in this limit in many publications. But e.g. Schofield (1947) states (about the Gouy-Chapman solution):

[T]he equation is applicable to cases in which the distance between opposing surfaces considerably exceeds the distance between neighboring point charges on the surfaces; for there will then be a range of electrolyte concentrations over which the radius of the ionic atmosphere is less than the former and greater than the latter.

This criterion is not met in compacted bentonite, where instead the interlayer distance is comparable to the distance between neighboring charge centers on the surfaces. Invalid application of the Gouy-Chapman model also seems to underlie the flawed but widespread “anion-accessible porosity” concept.

[2] This calculation uses the equations presented in Engström and Wennerström (1978), and assumes no excess ions and a surface charge density of 0.111 \(\mathrm{C/m^2}\). For real consistency this calculation should really be performed with the boundary condition of 0.1 M external concentration. However, since the purpose of the graph is just to demonstrate the sizes of the two potential variations, and since I have yet to acquire a reasonable tool for performing Poisson-Boltzmann calculations with non-zero external concentration, I disregard this inconsistency. Moreover, the continuum assumption of the Poisson-Boltzmann description is anyway beginning to lose its validity at these interlayer distances. Update (220831): Solutions to the Poisson-Boltzmann equation with non-zero external concentration are presented here.

[3] Perhaps this could be done as a Molecular Dynamics simulation?

Stacks make no sense

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At the atomic level, montmorillonite is built up of so-called TOT-layers: covalently bonded sheets of aluminum (“O”) and silica (“T”) oxide (including the right amount of impurities/defects). In my mind, such TOT-layers make up the fundamental particles of a bentonite sample. Reasonably, since montmorillonite TOT-layers vary extensively in size, and since a single cubic centimeter of bentonite contains about ten million billions (\(10^{16}\)), they are generally configured in some crazily complicated manner.

Stack descriptions in the literature

But the idea that the single TOT-layer is the fundamental building block of bentonite is not shared with many of today’s bentonite researchers. Instead, you find descriptions like e.g. this one, from Bacle et al. (2016)

Clay mineral particles consist of stacks of parallel negatively-charged layers separated by interlayer nanopores. Consequently, compacted smectite contains two major classes of pores: interlayer nanopores (located inside the particles) and larger mesopores (located between the particles).

or this one, from Churakov et al. (2014)

In compacted rocks, montmorillonite (Mt) forms aggregates (particles) with 5–20 TOT layers (Segad et al., 2010). A typical radial size of these particles is of the order of 0.01 to 1 \(\mathrm{\mu m}\). The pore space between Mt particles is referred to as interparticle porosity. Depending on the degree of compaction, the interparticle porosity contributes 10 to 30% of the total water accessible pore space in Mt (Holmboe et al., 2012; Kozaki et al., 2001).

Such statements show that researchers have something more complex in mind than individual TOT-layers when speaking of “particles”: they are some sort of assemblages of TOT-layers. The quotation of Bacle et al. (2016), using both the terms “stacks” and “particles”, even hints at an idea of a hierarchy of fundamental structures. Such a hierarchy is expressed explicitly in e.g. Navarro et al. (2017), who provide a figure with the caption “Schematic particle arrangement in highly compacted Na-bentonite” that looks similar to this one:

“Schematic particle arrangement in highly compacted Na-bentonite” from Navarro et al. (2017)

Here it is clear that they differ between “aggregates” (which I’m not sure is the same thing as “particles”), “stacks”, and individual TOT-layers (which I assume are represented by the line-shaped objects). In the following, however, we will use the term “stack” to refer to any kind of suggested fundamental structure built up from individual TOT-layers.

The one-sentence version of this blog post is:

Stacks make no sense as fundamental building blocks in models of water saturated, compacted bentonite.

The easiest argument against stacks is, in my mind, to simply work out the geometrical consequences. But before doing that we will examine some of the references given to support statements about stacks in compacted systems. Often, no references are given at all, but when they are, they usually turn out to be largely irrelevant for the system under study, or even to support an opposite view.

Inadequate referencing

As an example (of many) of inadequate referencing, we use the statement above from Churakov et al. (2014) as a starting point. I think this is a “good” statement, in the sense that it makes rather precise claims about how compacted bentonite is supposed to be structured, and provides references for some key statements, which makes it easier to criticize.

Churakov et al. (2014) reference Segad et al. (2010) for the statement that montmorillonite forms “particles” with 5 – 20 TOT-layers. In turn, Segad et al. (2010) write:

Clay is normally not a homogeneous lamellar material. It might be better described as a disordered structure of stacks of platelets, sometimes called tactoids — a tactoid typically consists of 5-20 platelets.^19-21

Here the terminology is quite different from the previous quotations: TOT-layers are called “platelets”, and “particles” are called “tactoids”. Still, they use the phrase “stacks of platelets”, so I think we can continue with using “stack” as a sort of common term for what is being discussed.¹ We may also note that here is used the word “clay”, rather than “montmorillonite” (as does Bacle et al. (2016)), but it is clear from the context of the article that it really is montmorillonite/bentonite that is discussed.

Anyhow, Segad et al. (2010) do not give much direct information on the claim we investigate, but provide three new references. Two² of these — Banin (1967) and Shalkevich et al. (2007) — are actually studies on montmorillonite suspensions, i.e. as far away as you can get from compacted bentonite in terms of density; the solid mass fraction in these studies is in the range 1 – 4%.

The average distance between individual TOT-layers in this density limit is comparable with, or even larger than, their typical lateral extension (~100 nm). Therefore, much of the behavior of low density montmorillonite depends critically on details of the interaction between layer edges and various other components, and systems in this density limit behave very differently depending on e.g. ionic strength, cation population, preparation protocol, temperature, time, etc. This complex behavior is also connected with the fact that pure Ca-montmorillonite does not form a sol, while the presence of as little as 10 – 20% sodium makes the system sol forming. The behaviors and structures of montmorillonite suspensions, however, say very little about how the TOT-layers are organized in compacted bentonite.

We have thus propagated from a statement in Churakov et al. (2014), and a similar one in Segad et al. (2010), that montmorillonite in general, in “compacted rocks” forms aggregates of 5 – 20 TOT-layers, to studies which essentially concern different types of materials. Moreover, the actual value of “5 – 20 TOT layers” comes from Banin (1967), who writes

Evidence has accumulated showing that when montmorillonite is adsorbed with Ca, stable tactoids, containing 5 to 20 parallel plates, are formed (1). When the mineral is adsorbed with Na, the tactoids are not stable, and the single plates are separated from each other.

This source consequently claims that the single TOT-layers are the fundamental units, i.e. it provides an argument against any stack concept! (It basically states that pure Ca-montmorillonite does not form a sol.) In the same manner, even though Segad et al. (2010) make the above quoted statement in the beginning of the paper, they only conclude that “tactoids” are formed in pure Ca-montmorillonite.

The swelling and sedimentation behavior of Ca-montmorillonite is a very interesting question, that we do not have all the answers to yet. Still, it is basically irrelevant for making statements about the structure in compacted — sodium dominated³ — bentonite.

Churakov et al. (2014) also give two references for the statement that the “interparticle porosity” in montmorillonite is 10 – 30% of the total porosity: Holmboe et al. (2012) and Kozaki et al. (2001). This is a bizarre way of referencing, as these two studies draw incompatible conclusions, and since Holmboe et al. (2012) — which is the more adequately performed study — state that this type of porosity may be absent:

At dry density \(>1.4 \;\mathrm{g/cm^3}\) , the average interparticle porosity for the [natural Na-dominated bentonite and purified Na-montmorillonite] samples used in this study was found to be \(1.5\pm1.5\%\), i.e. \(\le 3\%\) and significantly lower than previously reported in the literature.

Holmboe et al. (2012) address directly the discrepancy with earlier studies, and suggest that these were not properly analyzed

The apparent discrepancy between the basal spacings reported by Kozaki et al. (1998, 2001) using Kunipia-F washed Na-montmorillonite, and by Muurinen et al. (2004), using a Na-montmorillonite originating from Wyoming Bentonite MX-80, and the corresponding average basal spacings of the [Na-montmorillonite originating from Wyoming bentonite MX-80] samples reported in this study may partly be due to water contents and partly to the fact that only apparent \(\mathrm{d_{001}}\) values using Bragg’s law, without any profile fitting, were reported in their studies.

If Kozaki et al. (2001) should be used to support a claim about “interparticle porosity”, it consequently has to be done in opposition to — not in conjunction with — Holmboe et al. (2012). It would then also be appropriate for authors to provide arguments for why they discard the conclusions of Holmboe et al. (2012).⁴

Stacks in compacted bentonite make no geometrical sense

The literature is full of fancy figures of bentonite structure involving stacks. A typical example is found in Wu et al. (2018), and looks similar to this:

stacks illustration from Wu et al. (2018)

This illustration is part of a figure with the caption “Schematic representation of the different porosities in bentonite and the potential diffusion paths.”⁵ The regular rectangles in this picture illustrate stacks that each seems to contain five TOT-layers (I assume this throughout). Conveniently, these groups of five layers have the same length within each stack, while the length varies somewhat between stacks. This is a quite common feature in figures like this, but it is also common that all stacks are given the same length.

Another feature this illustration has in common with others is that the particles are ordered: we are always shown edges of the TOT-layers. I guess this is partly because a picture of a bunch of stacks seen from “the top” would be less interesting, but it also emphasizes the problem of representing the third dimension: figures like these are in practice figures of straight lines oriented in 2D, and the viewer is implicitly required to imagine a 3D-version of this two-dimensional representation.

A “realistic” stack picture

But, even as a 2D-representation, these figures are not representative of what an actual configuration of stacks of TOT-layers looks like. Individual TOT-layers have a distinct thickness of about 1 nm, but varies widely in the other two dimensions. Ploehn and Liu (2006) analyzed the size distribution of Na-montmorillonite (“Cloisite Na+”) using atomic force microscopy, and found an average aspect ratio of 180 (square-root of basal area divided by thickness). A representative single “TOT-line” drawn to scale is consequently quite different from what is illustrated in in most stack-pictures, and look like this (click on the figure to see it in full size)

Representative TOT-layer drawn to scale with water films

In this figure, we have added “water layers” on each side of the TOT-layer (light red), with the water-to-solid volume ratio of 1⁶. Neatly stacking five such units shows that the rectangles in the Wu et al. (2018)-figure should be transformed like this

actual veiw of stacks in Wu et al. (2018)

But this is still not representative of what an assemblage of five randomly picked TOT-layers would look like, because the size distribution has a substantial variance. According to Ploehn and Liu (2006), the aspect ratio follows approximately a log-normal distribution. If we draw five values from this distribution for the length of five “TOT-lines”, and form assemblages, we end up with structures that look like this:⁷

These are the kind of units that should fill the bentonite illustrations. They are quite irregularly shaped and are certainly not identical (this would be even more pronounced when considering the third dimension, and if the stacks contain more layers).

It is easy to see that it is impossible to construct a dense structure with these building blocks, if they are allowed a random orientation. The resulting structure rather looks something like this

Such a structure evidently has very low density, and are reminiscent of the gel structures suggested in e.g. Shalkevich et al. (2007) (see fig. 7 in that paper). This makes some sense, since the idea of stacks of TOT-layers (“tactoids”) originated from studies of low density structures, as discussed above.

Note that the structure in pictures like that in Wu et al. (2018) has a substantial density only because it is constructed with stacks with an unrealistic shape. But even in these types of pictures is the density not very high: with some rudimentary image analysis we conclude that the density in the above picture is only around 800 kg/m³. Also the figure from Navarro et al. (2017) above gives a density below 1000 kg/m³, although there it is explicitly stated that it is a representation of “highly compacted bentonite”.

The only manner in which the “realistic” building blocks can be made to form a dense structure is to keep them in the same orientation. The resulting structures then look e.g. like this

Dense structrue of color coded realistic stacks

where we have color coded each stack, to remind ourselves that these units are supposed to be fundamental.

Just looking at this structure of a “stack of stacks” should make it clear how flawed the idea is of stacks as fundamental structural units in compacted bentonite (note also how unrepresentative the stack-pictures found in the literature are). One of many questions that immediately arises is e.g. why on earth the tiny gaps between stacks (indicated by arrows) should remain. This brings us to the next argument against stacks as fundamental units for compacted water saturated bentonite:

What is supposed to keep stacks together?

Compacted bentonite of interest e.g. for sealing in radioactive waste repositories exerts swelling pressure of several MPa when in contact with external water. This osmotic pressure is a consequence of the presence of the mobile exchangeable cations in montmorillonite. Each “realistic” unit that we have imagined above is thus required to be at a huge elevated pressure, and the individual TOT-layers have a strong driving force to separate. And, unless a mechanism is provided for why such a separation is impossible, this is of course what we expect to happen! As far as I am aware, such a separation inhibiting mechanism has never been suggested in any publication that promotes the stack concept in compacted bentonite. To get a feel for the absurdity of this issue, let’s take a new look at the figure from Navarro et al. (2017)

The pressure configuration in the structure suggested by Navarro et al. (2017)

Assuming that this system is in equilibrium with an external water reservoir at zero pressure (i.e. atmospheric absolute pressure), the pressure in the compartment labeled “intra-aggregate space” is also close to zero. On the other hand, in the “stacks” located just a few nm away, the pressure is certainly above 10 MPa in many places! A structure like this is obviously not in mechanical equilibrium! (To use the term “obvious” here feels like such an understatement.)

Implications

To sum up what we have discussed so far, the following picture emerges. The bentonite literature is packed with descriptions of compacted water saturated bentonite as built up of stacks as fundamental units. These descriptions are so commonplace that they often are not supported by references. But when they are, it seems that the entire notion is based on misconceptions. In particular, structures identified in low density systems (suspensions, gels) have been carried over, without reflection, to descriptions of compacted bentonite. Moreover, all figures illustrating the stack concept are based on inadequate representations of what an arbitrary assemblage of TOT-layers arranged in this way actually would look like. With a “realistic” representation it quickly becomes obvious that it makes little sense to base a fundamental unit in compacted systems on the stack concept.

My impression is that this flawed stack concept underlies the entire contemporary mainstream view of compacted bentonite, as e.g. expressed by Wu et al. (2018):

A widely accepted view is that the total porosity of bentonite consists of \(\epsilon_ {ip}\) and \(\epsilon_ {il}\) (Tachi and Yotsuji, 2014; Tournassat and Appelo, 2011; Van Loon et al., 2007). \(\epsilon_ {ip}\) is a porosity related to the space between the bentonite particles and/or between the other grains of minerals present in bentonite. It can further be subdivided into \(\epsilon_ {ddl}\) and \(\epsilon_ {free}\). The diffuse double layer, which forms in the transition zone from the mineral surface to the free water space, contains water, cations and a minor amount of anions. The charge at the negative outer surface of the montmorillonite is neutralized by an excess of cations. The free water space contains a charge-balanced aqueous solution of cations and anions. \(\epsilon_ {il}\) represents the space between TOT-layers in montmorillonite particles exhibiting negatively charged surfaces. Due to anion exclusion effect, anions are excluded from the interlayer space, but water and cations are present.

This view can be summarized as:

The fundamental building blocks are stacks of TOT-layers (“particles”, “aggregates”, “tactoids”, “grains”…)
Electric double layers are present only on external surfaces of the stacks.
Far away from external surfaces — in the “inter-particle” or “inter-aggregate” pores — the diffuse layers merge with a bulk water solution
Interlayer pores are defined as being internal to the stacks, and are postulated to be fundamentally different from the external diffuse layers; they play by a different set of rules.

I don’t understand how authors can get away with promoting this conceptual view without supplying reasonable arguments for all of its assumptions⁸ — and with such a complex structure, there are a lot of assumptions.

As already discussed, the geometrical implications of the suggested structure do not hold up to scrutiny. Likewise, there are many arguments against the presence of substantial amounts of bulk water in compacted bentonite, including the pressure consideration above. But let’s also take a look at what is stated about “interlayers” and how these are distinguished from electric double layers (I will use quotation marks in the following, and write “interlayers” when specifically referring to pores defined as internal to stacks).

“Interlayers”

“Interlayers” are often postulated to be completely devoid of anions. We discussed this assumption in more depth in a previous blog post, where we discovered that the only references supplied when making this postulate are based on the Poisson-Boltzmann equation. But this is inadequate, since the Poisson-Boltzmann equation does describe diffuse layers, and predicts anions everywhere.

By requiring anion-free “interlayers”, authors actually claim that the physico-chemistry of “interlayers” is somehow qualitatively different from that of “external surfaces”, although these compartments have the exact same constitution (charged TOT-layer surface + ions + water). But an explanation for why this should be the case is never provided, nor is any argument given for why diffuse layer concepts are not supposed to apply to “interlayers”.⁹ This issue becomes even more absurd given the strong empirical evidence for that anions actually do reside in interlayers.

The treatment of anions is not the only ad hoc description of “interlayers”. It also seems close to mandatory to describe them as having a maximum extension, and as having an extension independently parameterized by sample density. E.g. the influential models for Na-bentonite of Bourg et al. (2006) and Tournassat and Appelo (2011) both rely on the idea that “interlayers” swell out to a certain volume that is smaller than the total pore volume, but that still depends on density.

In e.g. Bourg et al. (2006), the fraction of “interlayer” pores remains essentially constant at ~78%, as density decreases from 1.57 g/cm³ to 1.27 g/cm³, while the “interlayers” transform from having 2 monolayers of water (2WL) to having 3 monolayers (3WL). This is a very strange behavior: “interlayers” are acknowledged as having a swelling potential (2WL expands to 3WL), but do, for some reason, not affect 22% of the pore volume! Although such a behavior strongly deviates from what we expect if “interlayers” are treated with conventional diffuse layer concepts, no mechanism is provided.

In contrast, it should be noted that the established explanation for “tactoid” formation in pure Ca-montmorillonite involves no ad hoc assumptions of this sort, but rests on treating all ions as part of diffuse layers.

Another type of macabre consequence of defining “interlayer” pores as internal to stacks is that a completely homogeneous system is described has having no interlayer pores (because it has no stacks). E.g. Tournassat and Appelo (2011) write (\(n_c\) is the number of TOT-layers in a stack)

[…] the number of stacks in the \(c\)-direction has considerable influence on the interlayer porosity, with interlayer porosity increasing with \(n_c\) and reaching the maximum when \(n_c \approx 25\). The interlayer porosity halves with \(n_c\) when \(n_c\) is smaller than 3, and becomes zero for \(n_c = 1\).¹⁰

It is not acceptable that using the term interlayer should require accepting stacks as fundamental units. But the usage of the term as being internal to stacks is so widespread in the contemporary bentonite literature, that I fear it is difficult to even communicate this issue. Nevertheless, I am certain that e.g. Norrish (1954) does not depend on the existence of stacks when using the term like this:

Fig. 7 shows the relationship between interlayer spacing and water content for Na-montmorillonite. There is good agreement between the experimental points and the theoretical line, showing that interlayer swelling accounts for all, or almost all, of physical swelling.

The stack view obstructs real discovery

A severe consequence of the conceptual view just discussed is that “stacking number” — the (average) number of TOT-layers that stacks are supposed to contain — has been established as fitting parameter in models that are clearly over-parameterized. An example of this is Tournassat and Appelo (2011), who write¹¹

Our predictive model excludes anions from the interlayer space and relates the interlayer porosity to the ionic strength and the montmorillonite bulk dry density. This presentation offers a good fit for measured anion accessible porosities in bentonites over a wide range of conditions and is also in agreement with microscopic observations.

But since anions do reside in interlayers,¹² it would be better if the model didn’t fit: an over-parameterized or conceptually flawed model that fits data provides very little useful information.

A similar more recent example is Wu et al. (2018). In this work, a model based on the stack concept is successfully fitted both to data on \(\mathrm{ReO_4^-}\) diffusion in “GMZ” bentonite and to data on \(\mathrm{Cl^-}\) diffusion in “KWK” bentonite, by varying “stacking number” (among other parameters). Again, as the model assumes anion-free “interlayer” pores, a better outcome would be if it was not able to fit the data. Moreover, this paper focuses mainly on the ability of the model, while not at all emphasizing the fact that about ten (!) times more \(\mathrm{ReO_4^-}\) was measured in “GMZ” as compared with \(\mathrm{Cl^-}\) in “KWK”, at similar conditions in certain cases. The latter observation is quite puzzling and is, in my opinion, certainly worth deeper investigation (and I am fully convinced that it is not explained by differences in “stacking number”).

Footnotes

[1] The terminology in the bentonite literature is really all over the place. You may e.g. also find the term “tactoid” used as Navarro et al. (2017) use “aggregate”, or the term “platelet” used for a stack of TOT-layers…

[2] The third reference is an entire book on clays.

[3] Note that “sodium dominated” in this context means ~20% or more.

[4] It may be noticed that Kozaki et al. (2001) see no X-ray diffraction peaks for low density samples:

The basal spacing of water-saturated montmorillonite was determined by the XRD method. […] It was found that a basal spacing of 1.88 nm, corresponding to the three-water layer hydrate state […] was not observed before the dry density reached 1.0 Mg/m³.

My interpretation of this observation is that the diffraction peak has shifted to even lower reflection angles (in agreement with the observations of Holmboe et al. (2012)), not registered by the equipment. The alternative interpretation must otherwise be that “stacks” suddenly cease to exist below 1.0 g/cm³. (Yet, Kozaki et al. (2001) continues to use a certain d-value in their analysis, also for densities below 1.0 g/cm³.)

[5] I have discussed “diffusion paths” in an earlier blog post. This illustration certainly fits that discussion.

[6] A water-to-solid volume ratio of 1 corresponds basically to interlayers of three monolayers of water (3WL).

[7] To construct these units, I made the additional choice of placing each layer randomly in the horizontal direction, with the constraint that all layers should be confined within the range of the longest one in each unit.

[8] By “get away with” I mean “pass peer-review”, and by “don’t understand” I mean “understand”.

[9] This is reminiscent of how certain authors imply that the interlayer is non-diffusive under so-called crystalline swelling.

[10] A mathematical remark: if the interlayer porosity “halves with \(n_c\)” (what does that mean?) when \(n_c = 2\) (“smaller than 3”), it is impossible to simultaneously have zero interlayer porosity for \(n_c = 1\) (unless the interlayer porosity is zero for any \(n_c\)).

[11] I guess the word “presentation” here really should be “representation”?

[12] Note that one of the authors of this paper also claims in a later paper that anions do populate 3-waterlayer interlayers, in accordance with the Poisson-Boltzmann equation:

The agreement between PB calculations and MD simulation predictions was somewhat worse in the case of the \(\mathrm{Cl^-}\) concentration profiles than in the case of the \(\mathrm{Na^+}\) profiles (Figure 3), perhaps reflecting the poorer statistics for interlayer Cl concentrations […] Nevertheless, reasonable quantitative agreement was found (Table 2).

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).

Sorption part III: Donnan equilibrium in compacted bentonite

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Consider this basic experiment: contact a water saturated sample of compacted pure Na-montmorillonite, with dry mass 10 g and cation exchange capacity 1 meq/g, with an external solution of 100 ml 0.1 M KCl. Although such an experiment has never been reported¹, I’m convinced that all agree that the outcome would be similar to what is illustrated in this animation.

Potassium diffuses in, and sodium diffuses out of the sample until equilibrium is established. At equilibrium also a minor amount of chloride is found in the sample. The indicated concentration levels are chosen to correspond roughly to results from from similar type of experiments.²

Although results like these are quite unambiguous, the way they are described and modeled in the bentonite³ literature is, in my opinion, quite a mess. You may find one or several of the following terms used to describe the processes

Cation exchange
Sorption/Desorptioṇ
Anion exclusion
Accessible porosity
Surface complexation
Donnan equilibrium
Donnan exclusion
Donnan porosity/volume
Stern layer
Electric double layer
Diffuse double layer
Triple layer
Poisson-Boltzmann
Gouy-Chapman
Ion equilibrium
…

In this blog post I argue for that the primary mechanism at play is Donnan equilibrium, and that most of the above terms can be interpreted in terms of this type of equilibrium, while some of the others do not apply.

Donnan equilibrium: effect vs. model

In the bentonite literature, the term “Donnan” is quite heavily associated with the modeling of anion equilibrium; e.g. the term “Donnan exclusion” is quite common , and you may find statements that researchers use “Donnan porespace models” as models for “anion exclusion”, or a “Donnan approach” to model “anion porosity”.⁴ Sometimes the term “Donnan effect” is used synonymously with “Salt exclusion”. Also when authors acknowledge cations as being part of “Donnan” equilibrium, the term is still used mainly to label a model or an “approach”.

But I would like to push for that “Donnan equilibrium” primarily should be the name of an observable effect, and that it applies equally to both anions and cations. This effect — which was hypothesized by Gibbs already in the 1870s — relies basically only on two things:

An electrolytic system, i.e. the presence of charged aqueous species (ions).
The presence of a semi-permeable component that is permeable to some of the charges, but does not allow for the passage of at least one type of charge.

In equilibrated systems fulfilling these requirements it is — to use Donnan’s own words — “thermodynamically necessary” that the permeant ions distribute unequally across the semi-permeable component. This phenomenon — unequal ion distributions on the different sides of the semi-permeable component — should, in my opinion, be the central meaning of the term “Donnan equilibrium”.

The first publication of Donnan on the effect actually concerned osmotic pressure response, in systems of Congo Red separated from solutions of sodium chloride and sodium hydroxide. The same year (1911) he also published the ionic equilibrium equations for some specific systems.⁵ In particular he considered the equilibrium of NaCl initially separated from NaR, where R is an impermeant anion (e.g. that of Congo Red), leading to the famous relation (“int” denotes the solution containing R)

\begin{equation} c_\mathrm{Na^+}^\mathrm{ext}\cdot c_\mathrm{Cl^-}^\mathrm{ext} = c_\mathrm{Na^+}^\mathrm{int}\cdot c_\mathrm{Cl^-}^\mathrm{int} \tag{1} \end{equation}

Unfortunately, this relation alone (or relations derived from it) is often what the term “Donnan” is associated with in today’s clay research literature, with the implication that systems not obeying it are not Donnan systems. But the above relation assumes ideal conditions and complete ionization of the salts — issues Donnan persistently seems to have grappled with. In a review on the effect he writes

The exact equations can, however, be stated only in terms of the chemical potentials of Willard Gibbs, or of the ion activities or ionic activity-coefficients of G. N. Lewis. Indeed an accurate experimental study of the equilibria produced by ionically semi-permeable membranes may prove to be of value in the investigation of ionic activity coefficients.

It must therefore be understood that, if in the following pages ionic concentrations and not ionic activities are used, this is done in order to present a simple, though only approximate, statement of the fundamental relationships.

The issue of (the degree of) ionization was explicitly addressed in publications following the 1911 article; Donnan & Allmand (1914) motivated their investigations of the \(\mathrm{KCl/K_4Fe(CN)_6}\) system by that “it was deemed advisable to test the relation when using a better defined, non-dialysable anion than that of Congo-red”, and the study of the Na/K equilibrium in Donnan & Garner (1919) used ferrocyanide solutions on both sides of the membrane in an attempt to overcome the difficulty of the “uncertainty as to the manner of ionisation of potassium ferrocyanide” (and thus for the simplified equations to apply).

I mean that since non-ideality and ion association are general issues when treating salt solutions, it does not make much sense to use the term “Donnan equilibrium” only when some particular equation applies; as long as the mechanism for the observed behavior is that some charges diffuse through a semi-permeable component, while some others don’t, the effect should be termed Donnan equilibrium.

Donnan equilibrium in gels, soils and clays

After Donnan’s original publications in 1911, the effect was soon recognized in colloidal systems. Procter & Wilson (1916) used Donnan’s equations to analyze the swelling of gelatin jelly immersed in hydrochloric acid. In this case chloride is the charge compensating ion, allowed to move between the phases, while the immobile charge is positive charges on the gelatin network. Thus, no semi-permeable membrane is necessary for the effect; alternatively one could say that the gel constitutes its own semi-permeable component. The Donnan equilibrium in protein solutions was further and extensively investigated by Loeb.

As far as I am aware, Mattson was first to identify the Donnan effect in “soil” suspensions,⁶ attributing e.g. “negative adsorption” of chloride as a consequence of Donnan equilibrium, and explicitly referencing the works of Procter and Loeb. Mattson describes the suspension in terms of electric double layers with a diffuse “atmosphere of cations” surrounding the “micelle” (the soil particle), and refers to Donnan equilibrium as the distribution of an electrolyte between the “micellar” and the “inter-micellar” solutions. Oddly,⁷ he uses Donnan’s original framework (e.g. eq. 1) to quantify the equilibrium, although the electrostatic potential and the ion concentrations varies significantly in the investigated systems. A more appropriate treatment would thus be to use e.g. the Gouy-Chapman description for the ion distribution near a charged plane surface (which he refers to!).

Instead, Schofield (1947) analyzed Mattson’s data using this approach. He also comments on its (the Gouy-Chapman model) range of validity

… [T]he equation is applicable to cases in which the distance between opposing surfaces considerably exceeds the distance between neighboring point charges on the surfaces; for there will then be a range of electrolyte concentrations over which the radius of the ionic atmosphere is less than the former and greater than the latter. In Mattson’s measurements on bentonite suspension, these distances are roughly 500 A. and 10 A. respectively, so there is an ample margin.

He continues to comment on the validity of Donnan’s original equations

When the distance ratio has narrowed to unity, it is to be expected that the system will conform to the equation of the Donnan membrane equilibrium. This equation fits closely the measurements of Procter on gelatine swollen in dilute hydrochloric acid. […] In a bentonite suspension the charges are so far from being evenly distributed that the Donnan equation is not even approximately obeyed.

From these statements it should be clear that the general behavior (cation exchange, salt exclusion) of ions in bentonite equilibrated with an external solution is due to the Donnan effect.⁸ The appropriate theoretical treatment of this effect differs, however, depending on details of the investigated system. To argue whether or not e.g. the Gouy-Chapman description should be classified as a “Donnan” approach is purely semantic.

It is also clear that in the case of compacted bentonite the distance ratio is narrowed to unity — the typical interlayer distance is 1 nm, which also is the typical distance between structural charges in the montmorillonite particles. It is thus expected that Donnan’s original treatment may work for such systems (adjusted for non-ideality), while the Gouy-Chapman description is not valid.⁹

The message I am trying to convey is neatly presented in Overbeek (1956) — a text I highly recommend for further information. Overbeek distinguishes between “classical” (Donnan’s original) and “new” (accounting for variations in potential etc.) treatments of Donnan equilibrium, and says the following about dense systems

If the particles come very close together the potential drop between [surface and interlayer midpoint] becomes smaller and smaller as illustrated in Fig. 4. This means that the local concentrations of ions are not very variable and that we are again back at the classical Donnan situation, where distribution of ions, osmotic pressure and Donnan potential are simply given by the elementary equations as treated in section 2. It is remarkable that the new treatment of the Donnan effects may deviate strongly from the classical treatment when the colloid concentration is low, but not when it is high.

It thus seems plausible that Donnan equilibrium in compacted bentonite can be treated using Donnan’s original equations. But — as interlayer pores is a quite extreme chemical environment — substantial non-ideal behavior may be expected. Treating such behavior is a large challenge for chemical modeling of compacted bentonite, but can not be avoided, since interlayers dominate the pore structure.

Cation exchange is Donnan equilibration

The term “Donnan” in modern bentonite literature is, as mentioned, quite heavily associated with the fate of anions interacting with bentonite. In contrast, cations are often described as being “sorbed” onto the “solids”. This sorption is usually separated into two categories: cation exchange and surface complexation.

Surface complexation reactions are typically described using “surface sites”, and are usually written something like this (exemplified with sodium sorption)

\begin{equation} \equiv \mathrm{S^-} + \mathrm{Na^{+}(aq)} \leftrightarrow \equiv \mathrm{SNa} \end{equation}

where the “surface site” is labeled \(\equiv \mathrm{S}^-\)

Cation exchange is also typically written in terms of “sites”, but requires the exchange of ions (duh!), like this (here exemplified for calcium/sodium exchange)

\begin{equation} \mathrm{2XNa} + \mathrm{Ca^{2+}(aq)} \leftrightarrow \mathrm{X_2Ca} + 2\mathrm{Na^+(aq)} \tag{2} \end{equation}

where X represents an “exchange site” in the solid phase.

In the clay literature the distinction between “surface complexation” and “ion exchange” reactions is rather blurred. You can e.g. find statements that “the ion exchange model can be seen as a limiting case of the surface complex model…”, and it is not uncommon that ion exchange is modeled by means of a surface complexation model. It also seems rather common that ion exchange is understood to involve surface complexation.

Underlying these modeling approaches and descriptions is the (sometimes implicit) idea that exchanged ions are immobile, which clearly has motivated e.g. the traditional diffusion-sorption model for bentonite and claystone. This model assumes that ion exchange binds cations to the solid, making them immobile, while diffusion occurs solely in a bulk water phase (which, incredibly, is assumed to fill the entire pore volume).

However, the idea that the exchanged ion is immobile does not agree with descriptions in the more general ion exchange literature, which instead acknowledge the process as an aspect of the Donnan effect.

Indeed, already in 1919, Donnan & Garner reported Na/K exchange equilibrium in a system consisting of two ferrocyanide solutions separated by a membrane impermeable to ferrocyanide, and it is fully clear that the particular distribution of cations in such systems is just as “thermodynamically necessary” as the distribution of chloride in the initial work on Congo Red and ferrocyanide.

Applied to clays, it is clear that cation exchange occurs even without postulating specific “sorption sites” or immobilization. On the contrary, ion exchange occurs in Donnan systems precisely because the ions are mobile.

In his book “Ion exchange”,¹⁰ Freidrich Helfferich describes ion exchange as diffusion, and distinguishes it from “chemical” processes

Occasionally, ion exchange has been referred to as a “chemical” process, in contrast to adsorption as a “physical” process. This distinction, though plausible at first glance, is misleading. Usually, in ion exchange as a redistribution of ions by diffusion, chemical factors are less significant than in adsorption where the solute is held by the sorbent by forces which may not be purely electrostatic.

Furthermore, in describing a general ion exchange system, he states the exact characteristics of a Donnan system, with the crucial point that the exchangeable ion is “free”, albeit subject to the constraint of electroneutrality

Ion exchangers owe their characteristic properties to a peculiar feature of their structure. They consist of a framework which is held together by chemical bonds or lattice energy. This framework carries a positive or negative electric surplus charge which is compensated by ions of opposite sign, the so-called counter ions. The counter ions are free to move within the framework and can be replaced by other ions of the same sign. The framework of a cation exchanger may be regarded as a macromolecular or crystalline polyanion, that of an anion exchanger as a polycation.

To give a very simple picture, the ion exchanger may be compared to a sponge with counter ions floating in the pores. When the sponge is immersed in a solution, the counter ions can leave the pores and float out. However, electroneutrality must be preserved, i.e., the electric surplus charge of the sponge must be compensated at any time by a stoichiometrically equivalent number of counter ions within the pores. Hence a counter ion can leave the sponge only when, simultaneously, another counter ion enters and takes over the task of contributing its share to the compensation of the framework charge.

With this “sponge” model at hand, he argues for that the reaction presented in eq. 2 above should be reformulated

[T]he model shows that ion exchange is essentially a statistical redistribution of counter ions between the pore liquid and the external solution, a process in which neither the framework nor the co-ions take part. Therefore Eqs. (1-1) [eq. 2 above] and (1-2) should be rewritten: \begin{equation} 2\overline{\mathrm{Na^+}} + \mathrm{Ca^{2+}} \leftrightarrow \overline{\mathrm{Ca^{2+}}} + 2\mathrm{Na^{+}} \end{equation} \begin{equation} 2\overline{\mathrm{Cl^-}} + \mathrm{SO_4^{2+}} \leftrightarrow \overline{\mathrm{SO_4^{2-}}} + 2\mathrm{Cl^{-}} \end{equation} Quantities with bars refer to the inside of the ion exchanger.

This “statistical redistribution” is of course nothing but the establishment of Donnan equilibrium between the external solution and the exchanger phase (as in the animation above). Naturally, Donnan equilibrium — using either the “classical” or the “new” equations — is at the heart of many analyses of ion exchange systems.

Unfortunately, this has not been the tradition in the compacted bentonite research field, where a “diffuse layer” approach to cation exchange has only been considered in more recent years, and then usually as a supplement to already existing models and tools. We are therefore in the rather uneasy situation that ion exchange in bentonite nowadays often is explained in terms of both a Donnan effect and as specific surface complexation.

Considering the robust evidence for significant ion mobility in interlayer pores, I strongly doubt surface complexation to be relevant for describing ion exchange in bentonite.¹¹ Instead, I believe that not separating these processes obscures the analysis of species that actually do sorb in these systems. In any event, the exact effects of Donnan equilibrium — a mechanism dependent on nothing but that some charges diffuses through the semi-permeable component, while some others don’t — must first and foremost be worked out.

A demonstration of compacted bentonite as a Donnan system

To demonstrate how well the Donnan effect in compacted bentonite is captured by Donnan’s original description, we use the following relation, derived from eq. 1 (i.e we assume only the presence of a 1:1 salt, apart from the impermeable component)

\begin{equation} \frac{c_\mathrm{Cl^-}^\mathrm{int}}{c_\mathrm{Cl^-}^\mathrm{ext}} = -\frac{1}{2}\frac{z}{c_\mathrm{Cl^-}^\mathrm{ext}} + \sqrt{\frac{1}{4} (\frac{z}{c_\mathrm{Cl^-}^\mathrm{ext}})^2+1} \tag{3} \end{equation}

Here \(z\) denotes the concentration of cations compensating impermeable charge. Eq. 3 quantifies anion exclusion, and is seen to depend only on the ratio \(c_\mathrm{Cl^-}^\mathrm{ext}/z\).

This equation is plotted in the diagram below, together with data of chloride exclusion in sodium dominated bentonite (Van Loon et al., 2007) and in potassium ferrocyanide (Donnan & Allmand, 1914)

Anion exclusion in bentonite and ferrocyanide compared with Donnan's ideal formula

I find this plot amazing. Although some points refer to bentonite at density 1900 \(\mathrm{kg/m^3}\) (corresponding to \(z \approx 5\) M), while others refer to a solution of approximately 25 mM \(\mathrm{K_4Fe(CN)_6}\) (\(z \approx 0.1\) M), the anion exclusion behavior is basically identical! Moreover, it fits the ideal “Donnan model” (eq. 3) quite well!

There is of course a lot more to be said about the detailed behavior of these systems, but I think a few things stand out:

It should be obvious that the basic mechanism for anion exclusion is the same in these two systems. This observed similarity thus invalidates the idea that anion exclusion in compacted bentonite is due to an intricate, ionic strength-dependent partitioning of a complex pore structure into parts which either are, or are not, accessible to chloride. In other words, the above plot is another demonstration that the concept of “accessible anion porosity” is nonsense.
The similarity between compacted bentonite and the simpler ferrocyanide system confirms Overbeek’s statement above, that Donnan’s “elementary” equations apply when the colloid concentration (i.e. density) is high enough.
The slope of the curve at small external concentrations directly reflects the amount of exchangeable cations that contributes to the Donnan effect. The similarity between model and experimental data thus confirms that the major part of the cations are mobile, i.e. not adsorbed by surface complexation. The similarity between the bentonite system and the ferrocyanide system also suggests that non-ideal corrections to the theory is better dealt with by means of e.g. activity coefficients, rather than by singling out a quite different mechanism (surface complexation) in one of the systems.

Footnotes

[1] The only equilibrium study of this kind I am aware of, that involves compacted, purified, homo-ionic clay, is Karnland et al. (2011). This study concerns Na/Ca exchange, and does not investigate the associated chloride equilibrium.

[2] I have assumed a K/Na selectivity coefficient of 2, and 95% salt exclusion.

[3] “Bentonite” is used in the following as an abbreviation for bentonite and claystone, or any clay system with significant cation exchange capacity.

[4] This particular publication states that I am one of the researchers using a “Donnan approach” to model “anion porosity”. Let me state for the record that I never have modeled “anion porosity”, or have any intentions to do so.

[5] This article has an English translation.

[6] In my head, a “soil suspension” and a “soil particle” are not very well defined entities. As I understand, Mattson investigated “Sharkey soil” and “Bentonite”. Sharkey soil is reported to have a cation exchange capacity of around 0.3 eq/kg, and the bentonite appear to be of “Wyoming” type. It is thus reasonably clear that Mattson’s “soil” particles are montmorillonite particles.

[7] Mattson and co-workers published a whole series of papers on “The laws of soil colloidal behavior” during the course of over 15 years, and appear to have caused both awe and confusion in the soil science community. I find it a bit amusing that there is a published paper (Kelley, 1943) which in turn reviews and comments on Mattson’s papers. Some statements in this paper include: “It seems to be generally agreed that some of [Mattsons papers] are difficult to understand.” and “The extensive use by [Mattson and co-workers] of terms either coined by them or used in new settings, the frequent contradictions of statement and inconsistencies in definition, and perhaps most important of all, the use by the authors of theoretical reasoning founded, not on experimentally determined data, but on calculations based on purely hypothetical premises, make it difficult to condense these papers into a form suitable for publication without doing injustice to the authors or sacrificing strict accuracy.”

[8] It may be worth noting that the only works referenced by Schofield — apart from a paper on dye adsorption — are Mattson, Procter and Donnan. Remarkably, Gouy is not referenced!

[9] Of course, one can instead solve the Poisson-Boltzmann equation for “overlapping” double layers.

[10] In its introduction is found the following gem: “A spectacular evolution began in 1935 with the discovery by two English chemists, Adams and Holmes, that crushed phonograph records exhibit ion-exchange properties.” Who wouldn’t want to hear more of that story?!

[11] As a further argument for that the concept of immobile exchangeable ions in bentonite is flawed, one can take a look at the spread in reported values for the fraction of such ions. You can basically find any value between \(>99\%\) and \(\sim 0\%\) for the same type of systems. To me, this indicates overparameterization rather than physical significance.

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.

A couple of corrections

Footnotes

Implied structure of the free water domain

Arbitrary division between diffuse layer and free water

Donnan effect on the microscopic scale?!

Footnotes

Descriptions in bentonite literature

Bentonite as a non-ideal semi-permeable membrane

“Semi-permeability” in experiments

Tracer through-diffusion

Measuring hydraulic conductivity

Mass transfer in a salt gradient

Footnotes

Rotenberg et al. (2007)

Hsiao and Hedström (2015)

Hedström and Karnland (2012)

Comparison with experiments

Tournassat et al. (2016)

Footnotes

Repulsion between surfaces and anions is not really the point

Ferrocyanide

Bentonite

A charge neutral perspective

Footnotes

Stack descriptions in the literature

Inadequate referencing

Stacks in compacted bentonite make no geometrical sense

A “realistic” stack picture

What is supposed to keep stacks together?

Implications

“Interlayers”

The stack view obstructs real discovery

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

Donnan equilibrium: effect vs. model

Donnan equilibrium in gels, soils and clays

Cation exchange is Donnan equilibration

A demonstration of compacted bentonite as a Donnan system

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