Category Archives: Misconceptions

Sorption part III: Donnan equilibrium in compacted bentonite

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 are 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.

Bentonite swelling pressure, part IV: electrostatics

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Swelling is not due to electrostatic repulsion between montmorillonite particles

Few things confuse me more than how the role of electrostatics in clay swelling is described in the scientific literature. Consider e.g. this statement from Bratko et al. (1986)

The interaction between charged aggregates in solution is generally interpreted in terms of electrostatic repulsion between double layers surrounding the aggregates.

But in the same paper we learn that the main contribution to the force between two charged surfaces in solution is the entropy of mixing of counter-ions, and that electrostatic interactions actually may result in an attractive force between the surfaces.

Nevertheless, I think Bratko et al. (1986) are right: swelling is, for some reason, often “interpreted” in terms of electrostatic repulsion between electric double layers. It is easy to find statements that e.g. the expression for the osmotic pressure in the Gouy-Chapman model describes “the electrostatic force per unit area”, or that lamellar phases are “electrostatically swollen”, with an osmotic pressure “mainly of electrostatic origin”. Segad (2013) writes

The interactions between the negatively charged platelets lead to a repulsive long-ranged electrostatic force promoting swelling.

and Tester et al. (2016) write

The DLVO theory describes the interaction between two colloidal particles as a balance between electrostatic repulsion, in this case between two negatively charged clay layers, and vdW attraction.

Laird (2006) claims that electrostatics cause both repulsion and (strong) attraction between clay layers

A balance between strong electrostatic-attraction and hydration-repulsion forces controls crystalline swelling. The extent of crystalline swelling decreases with increasing layer charge. Double-layer swelling occurs between quasicrystals. An electrostatic repulsion force develops when the positively charged diffuse portions of double layers from two quasicrystals overlap in an aqueous suspension. Layer charge has little or no direct effect on double-layer swelling.

Although many authors reasonably understand the actual mechanisms of double layer repulsion, I think it is very unfortunate that this language is established and contributes to unnecessary confusion.

To gain some intuition for that clay swelling is not primarily due to electrostatic repulsion between montmorillonite particles, let us consider the Poisson-Boltzmann equation. This is, after all, the description usually referred to when authors speak of “electrostatic repulsion” between clay layers. The Poisson-Boltzmann equation may be used to describe the electrostatic potential, and the corresponding counter-ion equilibrium distribution, between two equally charged surfaces, and a typical result looks like this¹

Here we assume two negatively charged parallel surfaces with uniform charge density, and the counter-ions are represented by a continuous charge density. The system is assumed infinitely extended in the x- (in/out of the page) and y- (up/down) directions, and thus rotationally symmetric around the z-axis.

With a lot of equal charges “facing” each other, the illustration may indeed give the impression that there somehow is an electrostatic repulsion between the surfaces. That this is not the case, however, may be seen from the symmetry of the potential. In fact, replacing one of the negatively charged surfaces by a neutral surface at half the distance does not change the solution to the Poisson-Boltzmann equation! A charged and a neutral surface thus experience the same repulsion as two charged surfaces, if only placed at half the distance.²

charged-neutral and charged-chared diffuse layer

With one surface being uncharged, “interpreting” the force as an electrostatic repulsion between the particles makes little sense.

A related way to convince yourself that there is no electrostatic repulsion between the two charged surfaces is to consider the electric field generated by one “half” of the original system. This field vanishes on the outside of the considered “half”-system.

This means that removing a “half”-system would not be “noticed” by the other “half”-system, in the sense that the electric field configuration remains the same (and corresponds to having a neutral particle at half the distance).

It may be helpful to also remember from electrostatics that the electric field outside a plate capacitor vanishes. Thus, configuring two plate capacitors as shown below, there is no electric field between the positively charged surfaces, regardless of how close they are³.

This “plate capacitor” configuration is actually reminiscent of the charge distribution in an interlayer at low water content (where the continuum assumption of the Poisson-Boltzmann equation is not valid).

A squeezed ion cloud

Having established that there is no direct electrostatic repulsion between clay particles, the obvious question is: what is the main cause for the repulsion? What the two configurations above have in common — with either two charged surfaces or one charged and one neutral surface — is that they restrict the counter ions to a certain volume. Hence, there is an entropic driving force for transporting more water into the region between the surfaces, thereby pushing them apart. Nelson
(2013) describes this quite well⁴

One may be tempted to say, “Obviously two negatively charged surfaces will repel.” But wait: Each surface, together with its counterion cloud, is an electrically neutral object! Indeed, if we could turn off thermal motion the mobile ions would collapse down to the surfaces, rendering them neutral. Thus the repulsion between like-charged surfaces can only arise as an entropic effect. As the surfaces get closer than twice their Gouy–Chapman length, their diffuse counterion clouds get squeezed; they then resist with an osmotic pressure.

Notice that the presence of this osmotic pressure requires contact with an “external” solution. The existence of a repulsive force between clay layers thus requires that water is available to be transported into the interlayer region. This seems to often be “forgotten” about in many descriptions of clay swelling. But let Kjellander et al. (1988) remind us

The PB pressure between two planar surfaces with equal surface charge equals $P_\mathrm{ionic} = k_BT\sum_i n_i(0)$, where $n_i(0)$ is the ion density at the midplane between the surfaces. Due to symmetry there is no electrostatic force between the two halves of the system (the electrostatic fluctuation forces due to ion-ion correlations are neglected). To obtain the net pressure when the system is surrounded by a bulk electrolyte solution, it is necessary to subtract the external pressure calculated in the same approximation; this is given by the ideal gas contribution $P_\mathrm{bulk} = k_BT \sum_i n_i^\mathrm{bulk}$.

There is no repulsive force of this kind in an isolated, internally equilibrated, clay.

Moreover, the force is usually conceived of as repulsive because the water chemical potential of the surrounding (“external”) solution is typically larger than in the clay. But from an osmotic viewpoint there is nothing fundamentally different going on when the external phase is, say, vapor of low pressure (set e.g. by a saturated salt solution), causing the clay to lose water, i.e to shrink. Thus, if swelling is “interpreted” as electrostatic repulsion between montmorillonite particles, then drying should be “interpreted” as electrostatic attraction between the same particles.

The fact that there is repulsion — in the osmotic sense — between a montmorillonite particle and a neutral surface has huge implications for how to handle interfaces between montmorillonite and other phases in compacted bentonite components. Rather than to simply assume these to contain bulk water (“free water”), there is every reason to believe that the physico-chemical conditions at such interfaces are similar to “ordinary” interlayer pores. Since any type of mineral alteration occurs at such interfaces, there is no escape from understanding interlayer chemistry if a satisfying geochemical description of bentonite is desired.

The role of electrostatics in clay swelling

Although swelling is not primarily due to direct electrostatic repulsion between clay particles, electrostatics is of course essential to consider when calculating the osmotic pressure. And rather 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.

Moreover, a treatment of the electrostatic problem beyond the mean-field (i.e. beyond the Poisson-Boltzmann description) shows that ion-ion correlation cause an explicit attraction between equally charged surfaces (similar to a van der Waals force). In systems with divalent counter ions, this attraction is large enough to prevent swelling beyond a certain limit — a prediction in qualitative agreement with observation. Electrostatics could thus be claimed to contribute to prevent clay swelling.

I think comparison with a simple salt solution can be useful. Nobody (?) would come up with the idea that the primary reason for the osmotic pressure of a NaCl solution is due to electrostatic repulsion between, say, chloride ions. In fact, the electrostatic interactions in such a solution reduce the osmotic pressure compared with a corresponding ideal solution.

Below is plotted the swelling pressure of Na-montmorillonite as a function of the average concentration of counter-ions (data from here). For comparison, the osmotic pressures of a NaCl solution and an ideal solution are also plotted (data from here), as a function of the total amount of ions (i.e. two times the NaCl concentration)⁵

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 dramatic increase of swelling pressure in the high concentration limit is reasonably an effect of hydration of ions and surfaces; it should be kept in mind that an average ion concentration of 3 mol/kgw in Na-montmorillonite roughly corresponds to a water-to-solid-mass ratio of only 0.3, and an average interlayer width below 1 nm.

Even though there seems to be quite some confusion regarding clay swelling pressure in the bentonite literature, the message here is not that everything about it is in reality understood. On the contrary, there are quite a number of behaviors that, as far as I’m aware, lack fully satisfactory explanations. For example, at room temperature the basal spacing in Ca-montmorillonite is never observed to be larger than $\sim 19$ Å⁶, corresponding to a (dry) density of approximately $1300 \;\mathrm{kg/m^3}$; yet, this material systematically exerts swelling pressure at considerably lower density ($\sim 700 \;\mathrm{kg/m^3}$). But in order to tackle issues like these, it is essential to be clear about the swelling mechanisms that we actually do understand.

Update (221018): A correction to this blog post is discussed here.

Footnotes

[1] This particular calculation uses the formulas presented in Engström and Wennerström (1978), and assumes mono-valent counter-ions at room temperature, a charge density of $-0.1 \;\mathrm{C/m^2}$, and a surface-surface distance of 2 nm.

[2] Here is only considered the Poisson-Boltzmann pressure. If e.g. van der Waals attraction between the surfaces is included, the resulting forces are not necessarily equal. The point here, however, concerns the repulsion due to the presence of diffuse layers.

[3] Having strictly zero field is of course an ideal result, corresponding to an infinitely extended capacitor.

[4] The quotation is taken from a draft version of this book.

[5] The graph denoted “Ideal solution” is simply the van’t Hoff relation $\Pi = RT c$, which strictly is only valid in the low concentration limit. It is nevertheless here extended to the whole concentration range. In the same way, the NaCl-curve is simply $\Pi = \varphi RT c$, where $\varphi$ is the osmotic coefficient for NaCl. Sorry about that.

[6] Upon cooling, Svensson and Hansen (2010) actually observed a basal spacing of 21.6 Å in pure Ca-montmorillonite.

Filter influence: why cation through-diffusion tests at low ionic strength should be avoided

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In through-diffusion tests, diffusion is monitored from an external source reservoir, through a clay sample, into an external target reservoir. As the sample typically is sandwiched between two confining filters, the full set-up can be abstracted as transport across three conductive components, coupled in series (filter-clay-filter).

Solving this problem — which is not specifically related to diffusion in clay, and applies equally well to e.g. electric currents or laminar fluid flow — the steady-state flux can be written as (for details, see appendix)

\begin{equation} j = D_\mathrm{e}^\mathrm{clay}\frac{1}{1+\omega} \frac{c_\mathrm{source}}{L_\mathrm{clay}} \tag{1} \end{equation}

where $\omega$ is the relative filter resistance

\begin{equation} \omega =D_\mathrm{e}^\mathrm{clay}\left (\frac{1}{D_\mathrm{e}^\mathrm{filter1}} + \frac{1}{D_\mathrm{e}^\mathrm{filter2}} \right ) \frac{L_\mathrm{filter}}{L_\mathrm{clay}} \tag{2} \end{equation}

Here $D_e$ denotes the effective diffusivity for the different components¹, $c_\mathrm{source}$ is the constant source reservoir concentration, $L_\mathrm{clay}$ is the length of the clay sample, and $L_\mathrm{filter}$ is the length of the filters (we assume that the two filters have the same length).

Eq. 1 shows that in the limit $\omega \ll 1$, the flux is expressed solely in terms of clay parameters²

\begin{equation} j = D_\mathrm{e}^\mathrm{clay}\cdot \frac{c_\mathrm{source}} {L_\mathrm{clay}} \end{equation}

In the other limit ($\omega \gg 1$), the flux is instead completely controlled by the filters,

\begin{equation} j = \frac{D_\mathrm{e}^\mathrm{filter1} \cdot D_\mathrm{e}^\mathrm{filter2}} { D_\mathrm{e}^\mathrm{filter1}+D_\mathrm{e}^\mathrm{filter2}} \cdot \frac{c_\mathrm{source}}{L_\mathrm{filter}} \end{equation}

i.e. independent of any clay parameters.

It is thus clear that through-diffusion tests should be designed to have $\omega$ reasonably small; preferably, one should have $\omega \ll 1$, which allows for leaving the filters out of the analysis.

While filter parameters in practice are limited to a quite small range for a given ion³, $D_\mathrm{e}^\mathrm{clay}$ is known to grow indefinitely for many cations, as the background concentration tends to zero¹. Consequently, for such ions, there always exists a background concentration limit, beyond which the filters completely control the resulting flux (i.e. where $\omega \gg 1$).

Even though the effect of filters in through-diffusion tests has been identified for a long time, there are numerous examples in the bentonite literature where filter influence is ignored, or not fully identified, leading to erroneous interpretations. For example, when describing through-diffusion tests with strontium in Boom Clay, Altmann et al. (2012) write

The resulting $\alpha$ value of $\sim 440$ corresponds to a $K_d$ value similar to that measured on dispersed and intact Boom Clay. However, if this $\alpha$ is used to estimate the corresponding $D_e$ value via $D_e = \alpha\cdot D_a$, the value obtained is $\sim 45$ times higher than $D_e(\mathrm{HTO})$, which is an unrealistically large difference. This is probably because the necessary conditions for calculating $D_e$ by Fick’s law were not satisfied as indicated by the fact that the concentration profiles measured at the end of the through-diffusion experiment were unexpectedly ‘flat’, i.e. did not vary in a linear fashion between the surfaces in contact with the source and sink solutions. The reason for this behaviour is not yet known.

But a “flat” concentration profile is a key signature of filters limiting the flux, as the (external) concentration difference across the clay is (see appendix)

\begin{equation} c_\mathrm{in} – c_\mathrm{out} = \frac{c_\mathrm{source}}{1+\omega} \end{equation}

which approaches zero when $\omega$ becomes large.

Consequently, the reported behavior strongly indicates that Boom Clay has a very high transport capacity for strontium under the right conditions (the test was performed with a sodium background concentration of approximately 0.02 M), leading to the filters limiting the flux. This, in turn, implies that the value for $D_e$ in the clay is underestimated, rather than being “unrealistically large”.

What is demonstrated in this test — but not concluded — is that the principal diffusion mechanism in Boom Clay is the same as in compacted bentonite: ions assumed “sorbed” contribute to — and probably dominate — the diffusive flux. The traditional sorption-diffusion model is not valid for Boom Clay.

Glaus et al. (2007) clearly demonstrate filter influence on Na diffusion in Na-montmorillonite, performed over a large range of $\mathrm{NaClO_4}$ background concentrations. The concentration profiles across the samples at the time of termination look like this⁴

Sodium tracer profiles from Glaus et al. (2007)

The profiles become increasingly “flat” with decreasing background concentration, demonstrating an increasing transport capacity of the clay (demonstrating this transport capacity was the main purpose of the study). The tests in Glaus et al. (2007) are analyzed assuming a filter-clay-filter configuration, with identical diffusivities for the two filters (for a given test). The clay component is described using the traditional sorption-diffusion model⁵. From the reported fitted model parameters, we can calculate the corresponding relative filter resistances, using eq. 2. The result is as follows (all these samples have $L_\mathrm{clay}=5.4$ mm and $L_\mathrm{filter}=1.55$ mm.)

$C_\mathrm{bkg}$	($\mathrm{mol/m^3}$)	10	100	500	700	1000
			Reported
$D_\mathrm{e}^\mathrm{clay}$	($10^{-10}\;\mathrm{m^2/s}$)	14	3.7	0.86	0.53	0.38
$D_\mathrm{e}^\mathrm{filter}$	($10^{-10}\;\mathrm{m^2/s}$)	0.86	1.0	0.86	0.86	0.86
			Calculated
$\omega$	(-)	9.3	2.1	0.6	0.4	0.3

Indeed, $\omega \gg 1$ for the test performed at 10 mM $\mathrm{NaClO_4}$, and filters fully control the flux. Filter influence is also significant in the test at 100 mM ($\omega = 2.1$), while the effect is less important in the tests at higher background concentration. These results fully reflect the appearance of the concentration profiles above.

The filter influence is also clearly seen in the behavior of the outfluxes at the different background concentrations (dotted graphs)

sodium tracer outflux and source concentration evolution in Glaus et al. (2007)

For the tests at high background concentration (i.e. small $\omega$), steady-state⁶ is reached in about 8 – 10 days. In the 10 mM-test, on the other hand, the system is far from steady-state even after 45 days⁷ — the outflux is still increasing, even though the source concentration (dash-dotted graphs) has dropped significantly. A prolonged transient state is thus another key signature of filters limiting the flux.

This prolonged transient appears because the clay has to be “filled up” with ions before a steady-state can be established. It is important not to confuse this effect with that of retardation due to increased “sorption”: here, it is the filters that cannot “fill up” the clay fast enough, while the diffusive transport capacity of the clay actually increases with increasing “sorption”. Note that this increased transport capacity is not due to increased diffusivity, but exactly because the clay accommodates an increasing amount of tracers as the background concentration decreases.

For the most part, Glaus et al. (2007) treat the filter influence adequately, allowing them to draw correct conclusions regarding diffusion in compacted bentonite. Going into detail, however, I think there is some inconsistency in the parameters, demonstrating the inherent difficulties with handling cation through-diffusion at low ionic strength. $K_d$ has, as far as I see, been used as a free fitting parameter in the modeling of the tests.⁸ But for the specific case of sodium tracers diffusing in pure sodium montmorillonite, this parameter is constrained by the simple relation (which also is derived in the article)

\begin{equation} K_d = \frac{\mathrm{CEC}} {1\;\mathrm{eq/mol}} \cdot \frac{1} {C_\mathrm{bkg}} \tag{3} \end{equation}

where $C_{bkg}$ denotes the background concentration, and CEC is the cation exchange capacity. The reported $K_d$ values, thus corresponds to these CEC values

$C_\mathrm{bkg}$	($\mathrm{mol/m^3}$)	10	100	500	700	1000
			Reported
$K_d$	($10^{-3}$ $\mathrm{m^3/kg}$)	46	7.3	1.8	1.2	0.74
			Calculated
CEC	(eq/kg)	0.46	0.73	0.90	0.84	0.74

As the documented CEC for the used material (purified “Milos” montmorillonite) is $\sim 0.88$ eq/kg, this evaluation indicates that the fitted $K_d$ is significantly underestimated for the test performed at 10 mM.

The reason for this underestimation can be further investigated by using the end values of the recorded clay concentration profile, and assuming the CEC value (i.e. assuming $K_d$, using eq. 3). From the definition of $K_d$ we can thereby calculate $c_\mathrm{in}$ and $c_\mathrm{out}$.

$C_\mathrm{bkg}$	($\mathrm{mol/m^3}$)	10	100	500	700	1000
			Reported
$s_{in}$	($10^{-12}$ mol/kg)	88.5	38.5	12.2	9.3	7.8
$s_{out}$	($10^{-12}$ mol/kg)	76.3	17.4	1.4	1.1	$\sim 0$
$c_{source}$	($10^{-9}$ $\mathrm{mol/m^3}$)	3.1	10.5	8.1	7.7	9.4
			Assumed
$K_d$	($10^{-3}$ $\mathrm{m^3/kg}$)	88	8.8	1.76	1.26	0.88
			Calculated
$c_{in}$	($10^{-9}$ $\mathrm{mol/m^3}$)	1.0	4.4	6.9	7.4	8.9
$c_{out}$	($10^{-9}$ $\mathrm{mol/m^3}$)	0.9	2.0	0.8	0.9	$\sim 0$
$\omega$	(-)	21.36	3.37	0.31	0.19	$\sim 0$

This calculation gives a concentration drop across the inlet filter ($c_\mathrm{source} – c_\mathrm{in}$) that is considerably larger than half the value of the concentration in the source reservoir ($c_\mathrm{source}$), for the tests made at 10 mM and 100 mM. Such a behavior is impossible if the diffusivities of the two filters are identical! This reevaluation thus suggests that it is not strictly valid to assume identical filter diffusivities when evaluating these kinds of tests. Of course, if the tests are performed under conditions with small $\omega$, this assumption will make little difference, because the filter influence is anyway small. But under conditions with $\omega \gg 1$, the exact values of both filter diffusivities will significantly influence the analysis. The concentration profiles across the filters in the 10 mM case can be illustrated like this

The main achievement in Glaus et al. (2007) is that they, despite filter transport complications, manage to verify that the effective diffusivity in the clay, both for sodium and strontium tracers, scale with sodium background concentration as

\begin{equation} D_\mathrm{e}^\mathrm{clay} \propto \frac{1}{C_\mathrm{bkg}^Z} \tag{4} \end{equation}

where $Z$ is the valency of the tracer (i.e., $Z = 1$ for sodium, and $Z = 2$ for strontium). Not only is this relation crucial for understanding bentonite diffusion at a deeper level, it also allows for assessing filter influence on evaluated diffusion parameters in general. Eq. 4 implies a dramatic effect of the background concentration on the relative filter resistance for strontium (note from eq. 2 that also $\omega$ will scale as $C_\mathrm{bkg}^{-Z}$): lowering the background concentration e.g. from 0.5 M to 0.1 M, increases $\omega$ by a factor of 25; lowering it from 0.5 M to 0.01 M gives a factor of 2500! (I don’t think it is a coincidence that the strontium tests in Glaus et al. (2007) are restricted to $C_\mathrm{bkg}\ge 0.5\;\mathrm{M}$.)

Molera and Eriksen (2002) report diffusion parameters evaluated for strontium in “MX-80” bentonite of various densities and in the background concentration ($\mathrm{NaClO_4}$) range 0.5 M – 0.01 M. The tests were evaluated using the traditional sorption-diffusion model for the clay, and by taking the filters into account. The filter diffusivities were, however, assumed identical in the two filters, and kept constant (for a given ion) in all models. From the reported fitted parameters, we can evaluate $\omega$, using eq. 2 (they used “$D_\mathrm{a}$” as fitting parameter rather than $D_\mathrm{e}^\mathrm{clay}$, but these are related via $D_\mathrm{e}^\mathrm{clay} = D_\mathrm{a}\left(\phi + \rho K_d\right)$)

		Reported		Calculated
Density	$C_\mathrm{bkg}$	$D_\mathrm{a}$	$K_d$	$D_\mathrm{e}^\mathrm{clay}$	$\omega$
($\mathrm{kg/m^3}$)	($\mathrm{mol/m^3}$)	($10^{-10}\;\mathrm{m^2/s}$)	($10^{-3}\;\mathrm{m^3/kg}$)	($10^{-10}\;\mathrm{m^2/s}$)	(-)
400	100	0.43	110	19.3	6.8
800	100	0.35	150	42.2	14.8
800	500	0.40	15	5.1	1.8
1200	10	0.21	500	126	44.2
1200	100	0.18	130	28.2	9.9
1200	500	0.25	13	4.0	1.4
1600	10	0.14	1200	269	94.2
1600	100	0.10	90	14.4	5.1
1600	500	0.20	15	4.9	1.7
1800	100	0.09	80	13.0	4.6
1800	500	0.12	15	3.3	1.1

In this evaluation is used $D_\mathrm{e}^\mathrm{filter} = 0.925 \cdot 10^{-10}\;\mathrm{m^2/s}$, $L_\mathrm{filter} = 0.81$ mm, and $L_\mathrm{clay} = 5.0$ mm for all tests.

Filter transport dominates ($\omega \gg 1$) in all but the tests performed at 500 mM (and even in these tests, there is significant filter influence). It can therefore be questioned whether the parameters have been adequately evaluated. That the fitted parameter values ($K_d$ and/or $D_\mathrm{a}$) are not adequate is seen when plotting $D_\mathrm{e}^\mathrm{clay}$ against background concentration (the “expected dependency” line assumes the $D_\mathrm{e}^\mathrm{clay}$ value of the 1200 $\mathrm{kg/m^3}$ sample at 500 mM background concentration).

Effective diffusivities for strontium in Molera and Eriksen (2002)

The $D_\mathrm{e}^\mathrm{clay}$ values do not obey Glaus’ relation, which they are expected to do, as “MX-80” is a sodium dominated clay. Note that the above plot suggests that $D_\mathrm{e}^\mathrm{clay}$ in Molera and Eriksen (2002) at background 0.01 M may be underestimated by roughly two orders of magnitude! Nevertheless, the actual clay diffusivity estimated in this study (unfortunately interpreted as “apparent” diffusivity) compares relatively well with other measurements, e.g. Kim et al. (1993), indicating that the underestimation of $D_\mathrm{e}^\mathrm{clay}$ is rooted in a similar underestimation of $K_d$.

Results like those of Glaus et al. (2007) and Molera and Eriksen (2002) show that cation through-diffusion tests at low background concentration should be avoided if possible: Both studies explicitly take into account filters when evaluating model parameters, yet the evaluations can be demonstrated to be inconsistent in the low background concentration limit. Although experimental design — as well as corresponding modeling — can be of various quality, the low concentration limit is fundamental: no matter how rigorous the analysis, the results will still be uncertain, simply because the experiment itself conveys less and less information on transport parameters in the clay.

Thus, unless the explicit purpose is to explore the low background concentration limit, it is better to stay away from it, thereby reducing the risk of drawing incorrect conclusions. An example of using data influenced by filter resistance to draw far-reaching conclusions regarding bentonite structure is found in Tinnacher et al. (2016).

This study uses the result from a single through-diffusion test in pure Na-montmorillonite (prepared from SWy-2) at 800 $\mathrm{kg/m^3}$⁹ to review “single porosity models”, and to argue for that this system is dominated by bulk water ($>70\%$) — a rather bizarre conclusion, in my opinion.

The test was done with a background electrolyte of 0.1 M NaCl, by adding a small amount of $\mathrm{CaBr_2}$ (1 mM) to the source reservoir, and monitoring the accumulation of calcium and bromide in the target reservoir (which was kept virtually tracer free by frequent replacement). The recorded normalized outflux of calcium looks like this¹⁰

calcium outflux in Tinnacher et al. (2016)

As this test concerns diffusion of a di-valent cation at relatively low ionic strength, there are strong reasons to suspect that filter resistance influences the flux evolution. If I understand correctly, this test was actually performed using the exact same equipment as used in the study by Molera and Eriksen (2002), where we evaluated a value $\omega = 14.8$ for strontium at the same conditions, albeit in a different clay material (see above).

But using the reported model parameters in Tinnacher et al. (2016) gives $\omega = 0.77$ ($D_\mathrm{e}^\mathrm{clay} = 2.06\cdot 10^{-10}\; \mathrm{m^2/s}$, $D_\mathrm{e}^\mathrm{filter} = 0.85\cdot 10^{-10}\; \mathrm{m^2/s}$, $L_\mathrm{filter} = 0.79$ mm, and $L_\mathrm{clay} = 5$ mm). This result — indicating only moderate filter influence — is a bit surprising, given the results from Molera and Eriksen (2002), and given that calcium appears to diffuse faster than strontium in Na-montmorillonite.

However, these model parameters are not consistent with the recorded steady-state flux. The normalized flux ($j/c_\mathrm{source}$) can be calculated from eq. 1

\begin{equation} \frac{j}{c_\mathrm{source}} = \frac{1}{\frac{L_\mathrm{clay}}{D_\mathrm{e}^\mathrm{clay}} + \frac{2\cdot L_\mathrm{filter}}{D_\mathrm{e}^\mathrm{filter}}} = 2.33\cdot 10^{-8} \;\mathrm{m/s} \end{equation}

which is significantly smaller than the observed flux of $3.5 \cdot 10^{-8} \;\mathrm{m/s}$. In order to match the observed flux instead requires $D_\mathrm{e}^\mathrm{clay} = 5.0\cdot10^{-10}\; \mathrm{m^2/s}$, indicating significant filter influence after all ($\omega = 1.86$).

Of course, the calculated flux could match the observed flux by instead altering the filter diffusivity (or by altering both the filter and clay diffusivities). But matching the fluxes by only altering the filter diffusivity requires $D_\mathrm{e}^\mathrm{filter} = 3.67\cdot 10^{-10}\; \mathrm{m^2/s}$, which is unrealistically large (it corresponds to a geometric factor of unity and porosity 0.46).

This analysis shows that the evaluated value for $D_\mathrm{e}^\mathrm{clay}$ for calcium in Tinnacher et al. (2016) is conditioned on the adopted value for filter diffusivity, and that the experiment most probably is significantly influenced by filter limitations. It is consequently not suited for reviewing “single porosity models”.¹¹

Footnotes

[1] Note that for bentonite, $D_\mathrm{e}^\mathrm{clay}$ is not a real diffusion coefficient! But, since it is the parameter that quantifies the steady-state flux given the external concentration difference ($c_\mathrm{in} – c_\mathrm{out}$), it is precisely what is required in this analysis.

[2] Except for $c_\mathrm{source}$, of course; without a source concentration there wouldn’t be much flux.

[3] Typically, $L_\mathrm{filter} \sim 1$ mm and $D_\mathrm{e}^\mathrm{filter} \sim 0.1\cdot D_0$, where $D_0$ is the corresponding diffusivity in pure bulk water.

[4] The data underlying these plots are found in the supporting information to Glaus et al. (2007). There it is, however, presented as “normalized” concentrations, without a full description of how this normalization has been performed. I have used the concentration values as plotted, but scaled them spatially to the proper sample length (5.4 mm).

[5] In contrast to basically any other diffusion study, the traditional model is (in a sense) concluded invalid in Glaus et al. (2007). For this reason, the quantity usually labeled $D_\mathrm{e}$ is in this paper labeled $^cD$, where “c” is short for “conditional”. Here, we continue to label this quantity $D_\mathrm{e}^\mathrm{clay}$, in order to relate it to other studies.

[6] These tests were performed with a changing source reservoir concentration (also plotted), and the system is never strictly in steady-state, as reflected in a weak decay of the flux at long times. Still, there is a distinct difference between this “quasi”-steady-state and the initial transient state, and the presented theoretical analysis is still useful to apply.

[7] The supporting information unfortunately leaves out the data between days 45 and 100.

[8] This quantity is referred to as $R_d$ in Glaus et al. (2007).

[9] Tinnacher et al. (2016) states the density as both 800 $\mathrm{kg/m^3}$ and 790 $\mathrm{kg/m^3}$. I have used the former value.

[10] Oddly, the “flux” data is presented in Tinnacher et al. (2016) without correcting for a certain amount of “dead” volume that is not being exchanged during the target concentration measurements. Consequently, what is called “flux” in the article is strictly not the real flux, and all model curves look like a hedgehog’s back. In the plot presented here, this correction has been performed, and it does not exactly resemble the published plot. In practice, these differences are not important for the point I’m trying to make: the steady-state flux is still the same.

[11] I mean that diffusion studies in general are not very useful on their own for drawing conclusions on e.g. the presence of bulk water in bentonite. But that’s a separate discussion.

Appendix: Derivation of eqs. 1 and 2

We assume that the steady-state flux in any of the conductive units is linearly dependent on the concentration difference applied across it ($\Delta c$)

\begin{equation} j = -\frac{1}{R}\Delta c \end{equation}

where $R$ is the transfer resistance.

With constant source and target concentrations, the steady-state flux in the system under consideration (filter-clay-filter) can be expressed using any of the involved units (the flux is the same everywhere)

\begin{equation} j = \frac{1}{R_\mathrm{filter1}} \cdot \left(c_\mathrm{source} – c_\mathrm{in}\right) = \frac{1}{R_\mathrm{clay}} \cdot \left(c_\mathrm{in} – c_\mathrm{out} \right) = \frac{1}{R_\mathrm{filter2}}\cdot c_\mathrm{out} \tag{A1} \end{equation}

Here is also assumed, without loss of generality, that the target reservoir concentration is zero.

Solving for $c_\mathrm{in}$ and $c_\mathrm{out}$ gives

\begin{equation} c_\mathrm{in} = \frac{R_\mathrm{clay} + R_\mathrm{filter2}} {R_\mathrm{filter1} + R_\mathrm{clay}+ R_\mathrm{filter2} } c_\mathrm{source} \end{equation}

\begin{equation} c_\mathrm{out} = \frac{R_\mathrm{filter2} }{R_\mathrm{filter1} + R_\mathrm{clay} + R_\mathrm{filter2} } c_\mathrm{source} \end{equation}

Defining the relative filter resistance as

\begin{equation} \omega = \frac{R_\mathrm{filter1} + R_\mathrm{filter2}} {R_\mathrm{clay}} \end{equation}

we can express the concentration drop across the clay as

\begin{equation} c_\mathrm{in} – c_\mathrm{out} = \frac{R_\mathrm{clay}} {R_\mathrm{filter1} + R_\mathrm{filter2} + R_\mathrm{clay}} c_\mathrm{source} = \frac{c_\mathrm{source}}{1+\omega} \end{equation}

Specializing to the case of Fickian diffusion, the resistances may be expressed as

\begin{equation} R = \frac{L}{D_\mathrm{e}} \tag{A2} \end{equation}

and the steady-state may be written (using the middle expression in eq. A1)

\begin{equation} j = D_\mathrm{e}^\mathrm{clay}\frac{1}{1+\omega} \frac{c_\mathrm{source}} {L_\mathrm{clay}} \end{equation}

which is eq. 1 above.

Using eq. A2, the relative filter resistance becomes (assuming equal filter lengths)

\begin{equation} \omega = D_\mathrm{e}^\mathrm{clay} \left ( \frac{1}{D_\mathrm{e}^\mathrm{filter1}} + \frac{1}{D_\mathrm{e}^\mathrm{filter2}} \right ) \frac{L_\mathrm{filter}}{L_\mathrm{clay}} \end{equation}

which is eq. 2 above.

$D_a$ is not “apparent” (sorption, part I)

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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. Species with more specific chemical interactions will be discussed in separate blog posts.

The traditional diffusion-sorption model

A persistent idea, that has been around since the very start of research on compacted bentonite¹, is that “sorbed” ions are immobilized, and early diffusion models were based on two assumptions

The entire pore volume contains bulk water, where ions diffuse according to \begin{equation} j = -D_e \nabla c \tag{1} \end{equation} where $c$ is the ion concentration and $D_e = \phi D_p$, with $\phi$ being the porosity, and $D_p$ a so-called pore diffusivity (containing information on the ion mobility at the macroscopic scale of the porous system).
Ions in the porewater “sorb” on solid surfaces and become immobilized. The amount of “sorbed” ions per unit solid mass, $s$, is related to $c$ via a distribution coefficient $K_d$ \begin{equation} s = K_d\cdot c \tag{2} \end{equation}

With these assumptions, the total amount of ions in the bentonite (per volume porous medium) is

\begin{equation} n = \phi\cdot c + \rho\cdot s = \left (\phi+\rho K_d \right) c \tag{3} \end{equation}

where $\rho$ is the (dry) density of the clay. Applying the continuity equation, $\partial n/\partial t = -\nabla\cdot j$, gives

\begin{equation} \frac{\partial c}{\partial t} = \frac{D_e}{\phi + \rho K_d}\nabla^2c \tag{4} \end{equation}

This equation is usually written in terms of the so-called apparent diffusion coefficient $D_a$

\begin{equation} \frac{\partial c}{\partial t} = D_a \nabla^2c \tag{5} \end{equation}

with

\begin{equation} D_a = \frac{D_e}{\phi + \rho K_d} \tag{6} \end{equation}

The model defined by eqs. 5 and 6 — which we will refer to as the traditional model — has a distinct physical interpretation: ions diffuse relatively fast in a bulk water phase, while being retarded due to “sorption” onto the solid surfaces. Although eq. 5 has the form of Fick’s second law, it is clear that $D_a$ is not a real diffusion coefficient, but is influenced both by diffusion ($D_e$) and “sorption” ($K_d$).

The immobilization assumption is not valid

The traditional model has a major problem: it is not valid for compacted bentonite. There is — and was — strong evidence that the immobilization assumption does not hold for all types of “sorbing” ions. E.g Jenny and Overstreet (1939) reported transport of $\mathrm{Fe^{2+}}$, $\mathrm{H^+}$, $\mathrm{K^+}$, $\mathrm{Na^+}$, and $\mathrm{Ca^{2+}}$ as samples of H- K- Na- and Ca-bentonite was brought in contact with samples of Fe/H-bentonite (without gel contact there was no transport), and van
Schaik et al. (1966) demonstrated steady-state fluxes of $\mathrm{Na^+}$ through Na-montmorillonite, corresponding to pore diffusivites significantly larger than the $\mathrm{Na^+}$ diffusivity in pure water. Moreover, ion mobility is an essential component of the most well-established theoretical concept used for describing montmorillonite — the electric double layer. The ionic part of the electric double layer is even referred to as the diffuse layer, for crying out loud!²

So it should come as no surprise that the pioneers in radwaste barrier research quickly came to the conclusion that the description underlying eq. 6 does not hold. Several research groups reported values of $K_d$ and $D_a$ that gives unreasonably large values of $D_e$ when evaluated using eq. 6 (even larger than the corresponding diffusivities in pure water, in several cases).

What is more surprising is that these conclusions did not lead to a complete reevaluation of the underlying assumptions, which is the preferred procedure for a model producing nonsense. Instead, the invalidity of the traditional model was only considered a problem for the specific ions for which it yield “unrealistic” parameter values (in particular, strontium and cesium), and an adopted “remedy” was the so-called surface diffusion model. The surface diffusion model keeps the description expressed by eqs. 1, 2, 5, and 6, while replacing the relation $D_e = \phi\cdot D_p$ with the following monster

\begin{equation} D_e = \phi\cdot D_p + \rho\cdot K_d\cdot D_s \tag{7} \end{equation}

where $D_s$ is a so-called surface diffusion coefficient.

This extension of the traditional model brings about several problems, the most glaring one being that eq. 7 is wrong. Therefore, to be able to continue the discussion, we first have to derive a correct “surface diffusion”³ model.

A correct “surface diffusion” model

Eq. 7 is not valid because it relies on the incorrect assumption that diffusive fluxes from different domains are additive. To derive the correct equations, we go back to the only conceptual modification made in the surface diffusion model: that the “sorbed” ions are no longer assumed immobile. Thus, the basic assumptions of the surface diffusion model are:

The entire pore volume contains bulk water, where ions diffuse.
Ions in the porewater “sorb” on the solid surfaces.
“Sorbed” ions also diffuse, along the surfaces.

There are consequently two domains (bulk water and surfaces) where diffusion is assumed to take place, and a correct equation for the macroscopic flux is in this case (for details, see here)

\begin{equation} j = -\phi D_\mathrm{macr.} \frac{\bar{c}}{c}\nabla c \tag{8} \end{equation}

where $D_\mathrm{macr.}$ is the diffusion coefficient on the macroscopic scale, which conveys information on the mobility in both of the involved domains, as well as their geometrical configuration. $\bar{c}$ is the average concentration of all mobile entities; here, this include all ions — those in the porewater as well as those on the surfaces. The bulk water concentration is still labelled $c$. Using eq. 3, we can write

\begin{equation} \phi\bar{c} = n = \left (\phi + \rho K_d \right) c \tag{9} \end{equation}

and eq. 8 reduces to

\begin{equation} j = – D_\mathrm{macr.}\left (\phi + \rho K_d \right) \nabla c \tag{10} \end{equation}

Comparing with eq. 1 directly shows

\begin{equation} D_e = D_\mathrm{macr.}\left (\phi + \rho K_d \right) \tag{11} \end{equation}

Further, when using eq. 10 in the continuity equation, the factor $\left (\phi + \rho K_d \right)$ cancels, giving

\begin{equation} \frac{\partial c}{\partial t} = D_\mathrm{macr.} \nabla^2 c \tag{12} \end{equation}

A complete conceptual change

Eqs. 11 and 12 are the main result of a correctly derived “surface diffusion” model. Comparing with eqs. 5 and 6, we see that, although the equations are deceptively similar, a “surface diffusion” model brings about a complete conceptual change: with the immobilization assumption released, it is actually the real diffusion coefficient that appear in Fick’s second law (eq. 12). In fact, the surface diffusion model does not contain any apparent diffusivity!

Similarly, $D_e$ is not a real diffusion coefficient in the surface diffusion model, as clearly seen by eq. 11. Rather, $D_e$ is the product of the diffusion coefficient and a concentration factor, stemming from the fact that the general flux equation reads $j = -\frac{D}{RT} \bar{c}\cdot \nabla \mu$, where $\mu$ is the chemical potential of the diffusing ion.

In the same vein, $K_d$ does not quantify “sorption” — at least if “sorption” is supposed to refer to a process that retards the diffusive flux. On the contrary, $K_d$ acts as a kind of amplifier of the flux (eq. 10). This implication was already pointed out by Jahnke and Radke (1985):

Thus, $K_d$ values are of extreme importance during the steady release period. Counter to what may have been anticipated, larger $K_d$ values lead to higher release rates.

The conceptual changes of a “surface diffusion” model make a lot of sense when interpreting diffusion data:

$D_e$ is expected to vary with $K_d$ and can basically grow without limit, without becoming unphysical.
“$D_a$” for a whole bunch of cations is universally found to be insensitive to changes in $K_d$, even though $K_d$ may vary orders of magnitudes (by altering the background concentration). This is the expected behavior of a real diffusion coefficient, and it makes all sense to instead interpret this parameter as $D_\mathrm{macr.}$, using eq. 12.
When assuming all ions in the system to be mobile, we expect the governing equation (eq. 12) to be a real Fick’s second law, i.e. that it contains nothing but the actual diffusion coefficient.

Despite this sense-making, I have never seen these points being spelled out in the bentonite literature. Rather, it appears as the implication of releasing the immobilization assumption has not been fully comprehended. To this day, the traditional model is used as the basis for interpretation in basically all publications, to the extent that “$D_a$”, “$D_e$”, and “$K_d$” usually are considered to be experimental quantities. Similarly, most modern model development is focused on extending the traditional model, using flawed concepts regarding diffusivity in multi-porous systems.

I also find it symptomatic that “surface diffusion” not uncommonly is treated as being an optional mechanism, or even being completely discarded, while the dominant contribution of “sorbed” ions to the diffusive flux seems to have to be rediscovered every 25^th year, or so.

Note that the conclusions made here are not based on any model that I favor, but are the consequence of a correct conceptual treatment of what is stated in the contemporary bentonite literature. Although a “surface diffusion” model (correctly treated!) make much more sense as compared with the traditional model in interpreting cation tracer diffusion, it still involves an assumption that I don’t understand how anybody can endorse: it assumes the entire pore volume to consist of bulk water. I will discuss this point in a separate blog post. Update (210225): Letting go of the bulk water is discussed here.

Footnotes

[1] In the following I will write “bentonite”, but I mean any type of clay system with significant ion exchange properties.

[2] I guess a reason for not acknowledging interlayer diffusivity could have been the incorrect notions that interlayer water is crystalline, and that diffuse layers only “form” at higher water content.

[3] Since the established surface diffusion model is that expressed by eq. 7, I use quotation marks when speaking about models that don’t comply with this equation.

Anion-accessible porosity – a brief history

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Genesis

In the beginning there was the Poisson-Boltzmann equation. Solving it for the case of a salt solution in contact with a negatively charged plane surface (a.k.a. the Gouy-Chapman model) gives the concentration of cations and anions in the solution as a function of the distance to the surface, like this¹

Illustration of Gouy-Chapman concentration profiles

Note:

The suppression of the anion concentration near the surface is often referred to as negative adsorption or anion exclusion. The total amount of excluded anions per unit surface area (indicated in green), usually labeled $\Gamma^-$, is obtained by integrating the Poisson-Boltzmann equation.
There are, nevertheless, anions everywhere! This model will give zero anion concentration only for an infinitely negative electrostatic potential (or if $c_0 = 0$, of course).

A clever way to utilize negative adsorption is for estimating the amount of smectite surface area in a soil sample, first suggested by Schofield (1947). This is done by comparing measured values of negative adsorption with the appropriate expression evaluated from the Gouy-Chapman model. When doing the necessary math² for such an analysis you naturally end up with expressions like

\begin{equation} \frac{\Gamma^-}{c_0} \sim \text{const.}\cdot \kappa^{-1} \tag{1} \end{equation}

where $c_0$ is the external anion concentration (i.e. far from the surface), and $\kappa^{-1}$ is the Debye length. This equation, having the dimension of length, can be interpreted as the width, $d_{ex}$, of a region devoid of anions, which gives the same amount of negative adsorption as the full exclusion region, as illustrated here (yellow)

However, note:

This is just an equivalent, fictitious region.
Anions are still everywhere!

Due to its convenience in the analysis, the notion of an equivalent region devoid of anions — often referred to in terms of “volume of exclusion” — became rather popular. At the same time, authors stopped emphasizing that this is a fictitious region. A clear example of such a transition is Edwards and Quirk (1962) who states that $\Gamma^-/c_0$ “can be regarded as the surface depth from which chloride ions are excluded”, while in Edwards et al. (1965) the same quantity (multiplied by area) is referred to as “the volume from which chloride is excluded”. The latter statement is, strictly speaking, wrong: the actual volume from which anions are excluded is the entire region where the concentration deviates from $c_0$, and the exclusion is only partial — there are anions everywhere!

Compacted bentonite

But the idea of an actual region devoid of anions seems to have stuck, and I believe that this influenced the interpretation of diffusion in compacted bentonite³ in terms of “effective porosity” or “anion accessible-porosity”. Concepts which, in turn, have motivated the idea that bentonite contains bulk water (“free water”, “pore water”).

The first example of this usage in studies of compacted bentonite, that I know of, is in Muurinen et al. (1988) reporting chloride through-diffusion in bentonite with various densities and background concentrations.

The tracer concentration of the porewater clearly depends on the compaction of bentonite and on the salt concentration of the circulating water. The effective porosity can even be less than one percent when the salt concentration is low and compaction high. Also, the diffusivities strongly depend on the density of bentonite and on the salt concentration.

The low tracer concentration in bentonite in the diffusion tests […] are indicative of ion-exclusion [5]. Ion-exclusion probably decreases the effective size of the pores, which changes the geometric factor, of bentonite and thus the apparent diffusivity. In addition to the geometric factor, the effective diffusivity takes into account the effective pore volume; thus, the dependence is even stronger.

“Effective porosity” has not been defined earlier in the article, so it is difficult to know precisely what the authors mean by the term. But it is relatively clear⁴ from the second paragraph that they explain the measured fluxes as being a result of a physical variation of the pore volume accessible to anions, rather than as a variation of the tracer concentration in a homogeneous system. This is also supported by their writing in the conclusions section: “The decreased pore size and porosity caused by ion-exclusion could at least qualitatively explain the dependence.”

However, the reference they provide (“[5]”) is Soudek et al. (1984), who calculate anion exclusion by means of — the Poisson-Boltzmann equation! (Which predicts anions everywhere.) In fact, Soudek et al. (1984) calculate what they term “Donnan exclusion” in a homogeneous model of “parallel, equally-spaced platelets”. Thus, the reference supplied by Muurinen et al. (1988) is in direct contradiction with their interpretation that the pore size and porosity is decreasing with the salt concentration.

Soudek et al. (1984) even provide an example of how the average chloride concentration between the platelets depends on the separation distance, when in equilibrium with an external solution of 10 mM, and write

Note the extremely strong rejection of the co-ion. At 50 w% clay ($\sim 25$Å plate separation) almost 90% of the anions are rejected.

which is completely in line with the observation of Muurinen et al. (1988) that “The effective porosity can even be less than one percent when the salt concentration is low and compaction high”, if only “effective porosity” is replaced by “concentration between the plates”.

It makes me a bit tired to discover that the record could have been set straight over 30 years ago regarding which pores anions can access. Instead the bentonite research community, for the most part, doubled down on the idea that anions only have accesses to parts of the pore volume, or that compacted bentonite contains a significant amount of bulk water.

An explicit description of interpreting “chloride through diffusion porosity” as a specific, limited part of the pore volume is given by Bradbury and Baeyens (2003)

In the interlayer spaces and regions where the individual montmorillonite stacks are in close proximity, double layer overlap will occur and anion exclusion effects will take place. Exclusion will probably be so large that it is highly unlikely that anions can move through these regions (Bolt and de Haan, 1982). However, Cl anions do move relatively readily through compacted bentonite since diffusion rates have been measured in ‘‘through-diffusion’’ tests […]

If the Cl anions cannot move through the interlayer and overlapping double layer regions because of anion exclusion effects, then it is reasonable to propose that the ‘‘free water’’ must provide the diffusion pathways (Fig. 1). Therefore, the hypothesis is that the pore volume associated with the transport of chloride (and other anions) is the ‘‘free water’’ volume, and that this is the porewater in a compacted bentonite.

Here they refer to Bolt and De Haan, (1982) ⁵, when arguing for that anions do not have access to interlayers. But the analysis in this reference is based on nothing but — the Poisson-Boltzmann equation! (which predicts anions everywhere)

Another thing to note is the notion of “overlapping” diffuse layers. Studies of negative adsorption to quantify surface area typically look at soil suspensions, with a solid part of a few percent. In such systems it is justified to perform the analysis on a single diffuse layer because the distance between separate montmorillonite particles is large enough. But at higher density there is not enough space between separate clay particles for the ion concentrations to ever reach the “external” value ($c_0$) — the diffuse layers “overlap”.⁶

It has been shown that effects of “overlapping” diffuse layers on the resulting negative adsorption is significant already at a a solid content of 6%. When carrying over the anion exclusion analysis to compacted bentonite — with solid content typically above 70%! — it therefore becomes near impossible to believe that the system should contain regions unaffected by the montmorillonite (“free water”). Yet, the argumentation above, apart from being flawed in the way it refers to the Poisson-Boltzmann equation, relies critically on the existence of such regions.

The mindful reader may remark that compacted bentonite, if it mainly contains “overlapping” diffuse layers, perhaps is devoid of anions after all. But the Poisson-Boltzmann equation predicts anions everywhere also for “overlapping” diffuse layers. Actually, the model by Soudek et al. (1984), discussed above, considers this case.

Despite the improbability that montmorillonite particles in compacted bentonite can be spaced so far apart as to allow for bulk water within the system, the idea of anions only having access to “free” water was nevertheless further pursued by Van Loon et al. (2007). They provide a picture similar to this

The idea here (and elsewhere) is that bentonite consists of “stacks” of individual montmorillonite particles (TOT-layers) interlaced with interlayer water.⁷ The space between “stacks” is assumed large enough for diffuse layers to fully develop, and to merge into a bulk solution (“free water”), whose volume depends on the ionic strength, reminiscent of the excluded volume in eq 1.⁸ Anions are postulated to only have access to this “free” water.

But as references for anion exclusion is once again simply given studies based on the Poisson-Boltzmann equation (in particular, Bolt and De Haan, (1982)). But these — as I hope has been made clear by now — predict anions everywhere, and consequently do not support the suggested model. In this case, the mismatch between model and supporting references stands out, as the term “effective porosity” is used interchangeably with the term “Cl-accessible porosity”; if Gouy-Chapman theory in a convoluted way can be used to define an “effective” porosity (having no other meaning than a fictitious, equivalent volume), there is no possibility whatsoever to use it to support the idea of anions having access to only parts of the pore space. Ironically, “anion-accessible porosity” seems to be the most popular term nowadays for describing effects of anion exclusion in compacted bentonite.

The strongest confirmation that the modern-day concept of anion-accessible porosity is simply a misuse of the exclusion-volume concept is given in Tournassat and Appelo (2011). They provide a quite extensive background for the type of anion exclusion they consider, and it is based on the excluded-volume concept discussed above. They even explicitly calculate the excluded-volume (named “total chloride exclusion distance”) only to directly discard it as not suitable

However, this binary representation (absence or presence of chloride, Fig. 3) is not very representative of the system since the EDL is not completely devoid of anions.

Yet, after making this statement that anions are everywhere (in the diffuse layer) they anyway define anion accessible porosity as an effective, fictitious volume!⁹

Interlayers

Apart from treating the diffuse layer incorrectly, Bradbury and Baeyens (2003), Van Loon et al. (2007) and Tournassat and Appelo (2011) all make the additional unjustified assumption that interlayers — which in these studies are defined as distinctly different from diffuse layers¹⁰ — are completely devoid of anions. Bradbury and Bans (2003) cites conventional Poisson-Boltzmann based studies to incorrectly support this claim (see above). Also Van Loon et al. (2007) use Bolt and De Haan, (1982) as a reference¹¹

Due to the very narrow space, the double layers in the interlayers overlap and the electric potential in the truncated layer becomes large leading to a complete exclusion of anions from the interlayer (Bolt and de Haan, 1982; Pusch et al., 1990; Olin, 1994; Wersin et al., 2004). The interlayer water thus contains exclusively cations that compensate the permanent charges located in the octahedral layer of the clay.

Of the other sources cited, Pusch et al. (1990) mention “Donnan exclusion” as the reason preventing anions from having access to interlayers. But this is incorrect – Donnan equilibrium always gives a non-zero anion concentration in the interlayer (as long as the external concentration is non-zero). Wersin et al. (2004) only claim that anions are “excluded” from interlayers, without further explanation or references. (I haven’t managed to read Olin (1994) .)

Tournassat and Appelo (2011) cites Bourg et al. (2003) to support the claim that anions have no access to interlayers

When the dry density is above $1.8 \;\mathrm{kg/dm^3}$, almost all the porosity resides in the interlayers of Na-montmorillonite. Since anions are excluded from the interlayers, the anion-accessible porosity becomes zero, and anion-diffusion is minimal (Bourg et al., 2003)

But in Bourg et al. (2003) is explicitly stated that anion exclusion from interlayers is only “partial”!

To sum up…

The idea that anions have access only to parts of the pore volume is widespread in today’s compacted bentonite research community. In this blog post I have shown that this idea emerges from misusing the concept of exclusion-volume, and that all references used to support ideas of “complete exclusion” rests on the Poisson-Boltzmann equation. The Poisson-Boltzmann equation, however, predicts anions everywhere! Thus, the concept of an anion-accessible porosity, and the related idea that compacted bentonite contains different “types” of water, have not been provided with any kind of theoretical support.

In contrast, the result that anions have access to the entire pore volume is further supported both by molecular dynamics simulations, as well as by the empirical evidence for salt in interlayers.

Footnotes

[1] This figure is just an illustration, not an actual result. Update (220831): Actual solutions to the Poisson-Boltzmann equation are presented here.

[2] Schofield writes with an enthusiasm seldom seen in modern scientific papers: “I considered that it would be possible to compute the negative adsorption of the repelled ions from the basic assumptions of Gouy’s theory of the diffuse electric double layer, and therefore invited Mr. M. H. Quenouille to tackle the mathematical difficulties involved. Complete solutions have now been obtained for electrolytes in which the ions have valency ratios 1:2, 1:1, and 2:1, and a full account of this work will be submitted for publication shortly.”

[3] “Bentonite” is used in the following as an abbreviation of “Bentonite and claystone”.

[4] I mean that the word “probably” as used here does not belong in a scientific text.

[5] Sciencedirect.com dates this reference to 1979. The book has a second revised edition, however, published in 1982.

[6] I use quotation marks when writing “overlap” because I think this wording gives the wrong impression in compacted clay: with an average distance between montmorillonite particles of around 1 nm, the concept of individual diffuse layers has lost its meaning.

[7] I plan to comment on “stacks” in a future blog post. Update (211027): Stacks make no sense.

[8] The volume is, however, not proportional to the Debye length, but depends exponentially on ionic strength.

[9] The “anion accessible porosity” is defined in this paper as $\epsilon_{an} = \epsilon_{free} + \epsilon_D\cdot c_D/c_{free}$, where $\epsilon_{free}$ is the porosity of a presumed bulk water phase in the bentonite, and $\epsilon_D$ quantifies the volume of an arbitrarily chosen “Donnan volume” which is (Donnan) equilibrated with the “free” solution. $c_D$ is the anion concentration in this “Donnan volume”, and $c_{free}$ is the anion concentration in the bulk water.

[10] In this context, “interlayers” are defined as being parts of “stacks”. I really need to write about “stacks”… Update (211027): Stacks make no sense

[11] Bolt and de Haan (and others) are fond of writing that anions in very narrow confinement are “almost completely excluded” or “virtually completely excluded”, indicating that they may neglect anions in these compartments, but also that they are aware of that the equations they use never give exactly zero anion concentration. When working with soil suspensions of only a few percent solids it may be a valid approximation to neglect anions in nm-wide pores. In compacted bentonite it is not.

Evidence for anions in montmorillonite interlayers (swelling pressure, part II)

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It is easy to find models assuming montmorillonite interlayers devoid of “anions” . Here I will present empirical evidence that such an assumption is incorrect. Before doing so, just a quick remark on the term “anions” in this context. If anions reside in interlayers, they certainly do so accompanied by excess cations, in order to maintain overall charge neutrality. Thus, when speaking of “anions” in the interlayer we really mean “salt” (= anion(s) + cation(s)). In the following I will use the term “salt” because it better reflects the overall charge neutral character of the process (we are not interested in pushing a handful of negative charge into an interlayer).

The nature of bentonite swelling

The evidence for salt having access to interlayers follows directly from the observed swelling pressure response to changes in external salinity. It is therefore important to first understand the thermodynamic basis for swelling pressure, which I wrote about in an earlier post (the same nomenclature is adopted here). In essence, swelling is a consequence of balancing the water chemical potential¹ in the clay with that in the external solution², and swelling pressure quantifies the difference in chemical potential between the external solution and the non-pressurized bentonite sample, as illustrated here

chemical potentials in non-pressurizied bentoniote and in external solution

Since the chemical potential in the external solution depends on the salt content, we generally expect a response in swelling pressure when altering external salinity.

Labeling the salt concentration $c^{ext}$, we write the chemical potential of the external solution in terms of an osmotic pressure³

\begin{equation}
\mu_w^{ext} = \mu_0 – P_{osm.}^{ext}(c^{ext})\cdot v_w
\tag{1}
\end{equation}

where $v_w$ is the partial molar volume of water. $P_{osm}^{ext}$ is not the pressure in the external solution, but the pressure that would be required to keep the solution in equilibrium with pure water. The actual pressure in this compartment is the same as for the reference state: $P_0$. It may seem confusing to use a “pressure” to specify the chemical potential, but we will see that it has its benefits. Experimentally we have full control of $P_{osm}^{ext}$ by choosing an appropriate $c^{ext}$.

Response in an indifferent clay

With salt in the external solution, the big question is what happens to the chemical potential of the clay. We will start by assuming (incorrectly) that external salt cannot access the interlayers. This means that the chemical potential of the (non-pressurized) bentonite does not change when the external salinity changes. We refer to this hypothetical bentonite as indifferent. In analogy with the external solution, we write the chemical potential of the indifferent non-pressurized bentonite as⁴ \begin{equation} \mu_w^{int}(P_0) = \mu_0 -P_s^0\cdot v_w \;\; \;\; \;\; \text{(indifferent clay)} \end{equation}

were $P_s^0$ is the swelling pressure in case of pure water as external solution. By assumption, $\mu_w^{int}(P_0)$ does not depend on the external salinity (it is independent of $P_{osm}^{ext}$). The chemical potential in the indifferent clay at an elevated pressure $P$ is

\begin{equation}
\mu_w^{int}(P) = \mu_0 – P_s^0\cdot v_w +(P-P_0)\cdot v_w \;\;
\;\; \;\; \text{(indifferent clay)}
\tag{2}
\end{equation}

The swelling pressure (defined as the difference in pressure between bentonite and external solution, when the two are in equilibrium: $P_s \equiv P_{eq} – P_0$) in an indifferent clay is given by equating eqs. 1 and 2, giving the neat formula

\begin{equation}
P_s(c^{ext}) = P_s^0 – P_{osm}^{ext}(c^{ext}) \;\; \;\; \;\;
\text{(indifferent clay)}
\end{equation}

Note the following:

Although an indifferent clay is not affected by salt, it certainly has a swelling pressure response, demonstrating that swelling pressure depends as much on the external solution as it does on the clay.
Since swelling pressure in this case decreases linearly with the osmotic pressure of the external solution, it is predicted to vanish when the osmotic pressure equals $P_s^0$.
External osmotic pressures larger than $P_s^0$ implies “drying” of the clay (water transport from the clay into the external compartment)

If the above derivation feels a bit messy, with all the different types of pressure quantities to keep track of, here is a hopefully helpful animation

Animation swelling pressure response without anions in interlayer

Real swelling pressure response

Equipped with the swelling pressure response of an indifferent clay, let’s compare with the real response: The swelling pressure response in real bentonite deviates strongly from the indifferent clay response. This is seen e.g. here for Na-montmorillonite equilibrated in sequence with NaCl solutions of increasing concentration⁵ (data from Karnland et al., 2005 )

Swelling pressure response to salinities mid range densities

Swelling pressure indeed drops with increased concentration, but the drop is not linear in $P^{ext}_{osm}$, and is weaker as compared with the indifferent clay response (shown by dashed lines). All samples in the diagram above exert swelling pressure when $P^{ext}_{osm} \gg P_s^0$, i.e. far beyond the point where the swelling pressure in an indifferent clay is lost.

The deviation of the observed response from that of an indifferent clay directly demonstrates that the clay is affected by salt, i.e. that the chemical potential of the non-pressurized clay depends on the external salt concentration. The only reasonable way for salt to influence the chemical potential in the bentonite is of course to reside in the interlayer pores. Consequently, the observed swelling pressure response proves that salt from the external solution enters the interlayer pores.

Here is an illustration of how the chemical potentials relate to the swelling pressure in real bentonite

Swelling pressure repsonse real bentonite

Although the observed swelling pressure response in itself is sufficient to dismiss the idea that salt does not have access to interlayers, the study by Karnland et al., (2005) provides a much broader verification of the thermodynamic nature of swelling pressure. In particular, the chemical potential was measured (by means of vapor pressure) separately in the same samples as used for swelling pressure tests, after they had been isolated and unloaded. The terms in the relation $P_s = \left(\mu_w^{ext} – \mu_w^{int}(P_0) \right)/v_w$ were thus checked independently, as indicated here

Measurements performed in Karnland et al. (2005)

A striking confirmation of salt residing in interlayers is given by the observation that the chemical potential in the non-pressurized samples is lower than that in the corresponding external solution, as well as that in non-pressurized samples of similar density, but equilibrated with pure water.

Another interesting observation is that the sample with the highest density behaves qualitatively similar to the others: although the external osmotic pressure never exceeded $P_s^0$ ($\approx$56 MPa), the response strongly deviates from that of an indifferent clay

swelling pressure response to salinity high density

Because the pore space of samples this dense ($2.02\;\mathrm{g/cm^3}$) mainly consists of mono- and bihydrated interlayers, this similarity in response shows that salt has access also to such pores.

Implications

The issue of whether “anions” have access to montmorillonite interlayers has — for some reason — been a “hot” topic within the bentonite research community for a long time, and a majority of contemporary models rest on some version of the assumption that “anions” does not have access to the full pore volume. But, as far as I can see, this whole idea is based on misconceptions. I guess that saying so may sound quite grandiose, but note that swelling pressure is not at all considered in most chemical models of bentonite. And if it is, it is usually treated incorrectly. As an example, here is what Bradbury and Baeyens (2003) writes in a very influential publication

One of the main premises in the approach proposed here is that 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. Thus, to a first approximation, the composition of the external saturating aqueous phase should be a second-order effect which has little influence on the initial compacted bentonite porewater composition.

If the composition of the re-saturating water were to play an important role in determining the porewater composition, then it should also have a significant influence on swelling (Bolt, 1979). Dixon (2000) recently reviewed the role of salinity on the development of swelling pressure in bentonite buffer and backfill materials. He concluded that provided the initial dry densities were greater than 900 $\mathrm{kg\;m^{-3}}$, the swelling pressures developed are unaffected for groundwater salinities $< 75 \;\mathrm{g\;l^{-1}}$. Even brines appear to have little or no influence for initial dry densities $>1500 \;\mathrm{kg\;m^{-3}}$.

But, as we just have learned, a system with a weak swelling pressure response necessarily has a significant contribution to its water chemical potential due to externally provided salt. In contrast, the approximation discussed in the first paragraph of the quotation — which is basically that of an indifferent clay — maximizes the swelling pressure response. Thus, the discussed “main premise” does not hold, and the provided empirical “support” is actually an argument for the opposite (i.e. that salt has access to the clay).

Footnotes

[1] In the following I will write only “chemical potential” — it is always the chemical potential of water that is referred to.

[2] This is just a complicated way of saying that swelling is (an effect of) osmosis.

[3] Some may say that $P_{osm}^{ext}$ is simply the “suction” of the solution, but I am not a fan of using that concept in this context. I will comment on “suction” in a later blog post. Update 210816: “suction” is discussed here.

[4] The density dependence of the chemical potential in the bentonite is not explicitly stated here, in order to keep the formulas readable, but we assume throughout that the bentonite has some specific water-to-solid mass ratio $w$.

[5] The NaCl concentrations are 0.0 M, 0.1 M, 0.3 M, 1.0 M, and 3.0 M.

The problem with geometric factors

5 Replies

Many papers on diffusion in compacted bentonite and claystone¹ assume a direct connection between diffusivities measured in clay ($D_\mathrm{clay}$), and the geometrical configuration of the pore space. It seems close to mandatory to include
an expression of this type \begin{equation} D_\mathrm{clay} = \frac{1}{G}\cdot D_\mathrm{w}\tag{1} \end{equation}

where $D_\mathrm{w}$ is the corresponding diffusion coefficient in bulk water (usually taken as the limiting value at infinite dilution), and $G$ is a factor postulated to have pure geometric meaning. The geometric multiplier is written in a variety of ways, sometimes in terms of other factors which individually are supposed to convey specific geometric information, e.g. $1/G = \delta/\tau^2$, where $\delta$ is the “constrictivity” and $\tau^2$ is the “tortuosity factor”².

However, the above described procedure is futile:

A geometric factor can not be deduced a priori, because the pore space geometry of any realistic system is way too complicated. So-called tortuosities have only been calculated in some trivial cases, relating e.g. to highly symmetric networks of identical cylinders, which has basically no resemblance to the chaotic smectite-lasagna expected to be found in bentonite. Indeed, many papers on these issues admit that the geometric factor in practice functions as a “fudge factor” (to be “optimized”).
Even if a geometric factor could be deduced from only considering the pore space configuration, it is not expected to relate to $D_\mathrm{clay}$ in accordance with eq. 1. If we imagine removing all influence of pore geometry on the diffusivity in the clay, what remains is diffusion in an environment heavily influenced by the presence of exchangeable cations. Thus, a correct geometric factor should not multiply $D_\mathrm{w}$, but the diffusion coefficient measured on the short time and length scale in the clay. Such measurements exist — at least for water diffusion — and show values different from $D_\mathrm{w}$.

Because it is practically impossible to independently deduce a geometric factor, it is de facto defined by the ratio $D_\mathrm{clay}/D_\mathrm{w}$ in many studies. I dare to say that in any work within the bentonite research community, a geometric factor has no other meaning than $D_\mathrm{clay}/D_\mathrm{w}$ (or some function of this parameter, depending on definitions).

This means — at best — that the use of a geometric factor does nothing for a model description, but only replaces the parameter $D_\mathrm{clay}$ with the parameter $D_\mathrm{clay}/D_\mathrm{w}$; it simply corresponds to the awkward choice of specifying diffusion coefficients in units of $D_\mathrm{w}$.

Often, however, the procedure has worse consequences, because eq. 1 (as used in practice) represents an unjustified claim; it is not correct to use the value of $D_\mathrm{clay}/D_\mathrm{w}$ to make specific statements about the pore space geometry, while ignoring other possible mechanisms contributing to the value of $D_\mathrm{clay}$. If diffusion instead is described without invoking a geometric factor, $D_\mathrm{clay}$ itself functions as as a model parameter, and no implicit claims are made about why it has the value it has.

There are many examples in the literature where this confusion shows up. In e.g. Appelo and Wersin (2007), it is concluded that tortuosity for “anions” is larger than for water, while tortuosity for “cations” is smaller than for water. This conclusion is based on fitting of geometric factors without independent information on the pore space geometry:

[…] the pore may become filled entirely with a diffuse double layer when it narrows sufficiently. This constricts the passage of anions, and since the anions must circumnavigate the obstacle, they have greater tortuosity than tritium. This explains that a model with a tortuosity factor for iodide that is 1.4 times higher than for tritium better matches the data.

[…] in interlayers and pore constrictions the cations pass in relatively larger amounts than in free porewater, and consequently, they have smaller tortuosity than tritium.

Similar descriptions are given in Altmann et al. (2012) (in a section describing”advances” in process understanding with the FUNMIG project)

At low ionic strengths, cations are mostly located in the diffuse layer. This layer is of relatively low volume and connected throughout the porosity — cations need less time to explore this small volume and therefore to pass from one side of the media to the other.[…]

Anions are excluded from the diffuse layer and, therefore,also have a smaller volume to explore than neutral tracers. They however must pass through small pores in order to explore the total porosity, and these pores act as electrostatic throats […], forming an energy barrier to anion movement, reducing the probability of passage (resulting in reduced $D_e$) and increasing the time needed to explore all of the porosity.

And the same type of argument — that different diffusive behavior of cations and anions is caused by differences in “pathways” — is found in Charlet et al. (2017)

The different pathways of anions and cations in these regions lead to different effective diffusion rates (Van Loon et al., 2007; Mazurek et al., 2009)

But effective diffusion rates — which can basically be made to vanish e.g. for chloride, as has been demonstrated in bentonite — do not vary because of changes in “pathways”, but because of changes in boundary conditions, due to the presence of interfaces to external solutions.

To investigate the behavior of $D_\mathrm{clay}/D_\mathrm{w}$ one must of course consider real — not effective — diffusion coefficients (unfortunately, actual diffusion coefficients are usually referred to as “apparent” in the bentonite research community). Real diffusion coefficients reflect the mobility of the diffusing species, and can be measured e.g. in non-steady-state closed-cell tests, which are not influenced by interfaces to external solutions³. A very good appreciation for how such diffusion coefficients behave is given by the vast amount of data measured by Kozaki et al. in Na-montmorillonite (Kunipia-F). Some of this data (sources: 1, 2, 3, 4, 5) is plotted below, in terms of $D_\mathrm{clay}/D_\mathrm{w}$ (using a linear $y$-axis, for a change).

This plot clearly shows that if $D_\mathrm{clay}/D_\mathrm{w}$ is interpreted as having pure geometrical meaning, the conclusion must be that chloride and sodium have basically identical “tortuosity”, while the “tortuosity” of the cations calcium and cesium is considerably larger⁴. Thus, the closed-cell data disproves any claim that “tortuosity” for anions is larger than for cations. Moreover, from this data it is quite a stretch to state that “tortuosity” for anions is larger than for water, and it is plain wrong to state that “tortuosity” for cations is smaller than for for water. (For the record, I’m strongly convinced that both calcium and cesium occupy the same pore volume as sodium in this material.)

I should emphasize that I am not primarily arguing for a better geometrical interpretation of $D_\mathrm{clay}/D_\mathrm{w}$, but rather that this whole procedure (using eq. 1) should be avoided. This is not to say that pore geometry doesn’t influence clay diffusivity — most reasonably it does — but analysis based on eq. 1 will be flawed, because of the unjustified model assumptions.

When it comes to geometrical understanding of diffusion in bentonite, I think it would be more rewarding to focus on the quite large variation of reported diffusivities in seemingly identical systems. As an example, the figure below shows the diffusivity of cesium in very similar, pure Na-montmorillonite, measured in four different closed-cell studies: apart from Kozaki et al.’s data (blue), also Sato et al. (1992) (green), and Tachi and Yotsuji (2014) (orange).

Cs diffusion data from Kozaki et al. Sato et al., and Tachi and Yotsuji

In aggregation, this figure demonstrates almost an order of magnitude variability for the diffusion coefficient of cesium, even in very pure systems. If this kind of variability cannot be explained, it is of course not very relevant to develop models describing (smaller) differences in diffusivity between different types of ions in different types of bentonites.

If you claim to understand the variation seen in this plot, rather than simply finding an appropriate factor to blame (particle size?), here is a challenge: Give a prescription for how to prepare a Kunipia-F/P sample in order to minimize/maximize the resulting diffusivity⁵.

Footnotes

[1] In the following, the term “bentonite” is used as a shorthand for “compacted bentonite and claystone”.

[2] The nomenclature here is a mess, but I leave it to people who really want to use these concepts to actually define them in detail.

[3] They can also be deduced from the transient behavior in “steady-state” tests, if the transport capacity of the confining filters is not limiting the process.

[4] Note that “tortuosity”, as used e.g. by Appelo and Wersin (2007), is related to the inverse of $D_\mathrm{clay}/D_\mathrm{w}$.

[5] The diffusion direction should still be along the axis of sample compaction, of course.

You cannot add fluxes from different domains in multiporosity models

2 Replies

Most models of compacted bentonite and claystone¹ assume that the material contains different domains with distinctly different properties. In particular, it is common to assume that the system contains both an aqueous domain (a bulk water solution) and one or several domains associated with the solid surfaces.² Models of this type is here called multiporosity models.

A “standard” procedure in multiporosity models is to assume that the diffusive flux in any domain can be fully expressed using quantities that relates only to that same domain, while a “total” flux is given by summing the contributions from each domain. In e.g. the case of having one bulk water domain and one surface domain, the diffusive flux is typically written something like

$$ \label{eq1} j = j_{bulk} + j_{surface} = -D_e\cdot\nabla c -D_s \cdot\nabla c_s \tag{1} $$

were $D_e$ and $c$ denote, respectively, the diffusion coefficient and concentration in the bulk water domain, and the corresponding parameters in the surface domain are denoted $D_s$ and $c_s$. It is important to keep in mind that what is referred to as a “total” flux in these descriptions is the experimentally accessible, macroscopic flux (here simply labelled $j$).

This approach is basically adopted by all proponents of multiporosity models, and has been so for a long time, see e.g.:

There is, however, a big problem with this procedure:

Equation 1 does not make sense

The statement of eq. 1 — that the macroscopic, measurable flux has two independent contributions — is only true if the different domains are completely isolated from each other. This can e.g. be seen by making $D_s$ vanishingly small in eq. 1; the flux is still non-zero due to the contribution from the completely unaffected $j_{bulk}$-term. In contrast, for non-isolated domains the expected behavior is that a macroscopic diffusive flux will vanish if the diffusivity of any domain is made to approach zero.

Consequently, a large part of the compacted bentonite research field seems to treat diffusion by assuming the different domains to be isolated “pipes”. But this is of course not how bentonite works! If you postulate multiple diffusive domains in the clay, you must certainly allow for diffusing species to be able to be transferred between them. And here is where it gets mysterious: the default choice is to also assume that the domains are non-isolated. In fact, the domains are typically assumed to be in equilibrium. In e.g. the case of having one bulk water and one surface domain, the equilibrium is usually expressed something like

$$
c_s = K\cdot c
\tag{2}
$$

where $K$ is some sort of partition coefficient.

Thus, the “standard” treatment is to combine eq. 1 — even though it is only valid for isolated domains — with an equilibrium assumption (eq. 2) to arrive at expressions which typically look something like this

$$
j = \left (D_e + K\cdot D_s\right)\cdot \nabla c
\tag{3}
$$

Note how this equation has the appearance of an addition of diffusion coefficients, as the “total” flux now is related to a single concentration gradient (which in practice always is a bulk solution concentration).

Since eq. 3 results from combining two incompatible assumptions (isolation and equilibrium), it should come as no surprise that it is not a valid description for multiporosity models.

I cannot get my head around why this erroneous description has prevailed within the compacted clay research field for so long. The problem of deriving “effective” properties of heterogeneous systems appears in many research fields, and applies to many other quantities than diffusivity, e.g. heat conductivity, electrical conductance, elastic moduli, etc. It should therefore be clear also in the compacted clay world that obtaining the effective diffusivity for a system as a whole is considerably more involved than simply adding diffusivities from each contributing domain.

Some illustrations

To discuss the issue further, let’s make some illustrations. A molecule or ion in a porous system conducts random “jiggling” motion, exploring all regions to which it has access, as illustrated in this figure

This animation shows a simulated random walk at the microscopic scale, in a “made-up” generic porous system, consisting of solids (brown), a bulk water domain (blue), and a surface domain (pink). The green dots show the trace of the random walker; light green when the walker is located in the bulk water domain (with higher mobility), and darker green when it is located in the surface domain (with lower mobility). Note how the walker frequently switches between the domains — which is exactly what is required for the two domains to be in equilibrium.

As diffusivity is fundamentally related to the statistical properties of the random walk,³ it should be crystal clear that the macroscopic diffusion coefficient (reflecting mobility on length scales beyond what is shown in the animation) depends in a complicated way on the mobility of the involved domains, just as it depends on the detailed geometrical configuration. Oddly, the complex geometrical dependency is usually acknowledged within the bentonite research field (by means of e.g. formation and/or tortuosity factors), which makes it even more incomprehensible that the issue of multiple domain diffusivities is treated as it is.

In contrast, the figure below illustrates the microscopic behavior implied by eq. 1.

Here is shown one random walker in the bulk domain (green trace) and one in the surface domain (red trace). Since no exchange occurs between the domains, they are not in equilibrium (the concept does not apply). The lack of exchange between the domains is of course completely non-physical, but it is only under this assumption that the system is described by two independent fluxes (eq. 1). It may also be noted that the two types of walkers experience completely different geometrical configurations (i.e. formation factors), none of which corresponds to the geometrical configuration in the case of non-isolated domains.

A more reasonable formulation

If you insist on having a multiporosity model, the correct way to go about is to assume equilibrium within a so called representative elementary volume (REV), and to formulate the model on the macroscopic — averaged — level. On this level of description, it is legitimate to speak of both a bulk water and a surface concentration at any given point in the model (they represent REV averages), but there is of course only a single flux, as well as a single value of the chemical potential at any given point. The diffusive flux can be written quite generally as

\begin{equation} j = -\phi
D_\mathrm{macr.}\cdot\frac{\bar{c}}{RT}\nabla\mu
\end{equation}

where $\bar{c}$ is the macroscopic, averaged, concentration, $\phi$ is the porosity, and $\mu$ is the (electro-) chemical potential. $D_\mathrm{macr.}$ is the macroscopic diffusion coefficient, containing statistical information on how the diffusing substance traverses the porous system on length scales larger than the REV (in particular, it contains averaged information on the mobility in all involved domains, as well as their detailed geometrical configuration).

The value of $\mu$ can be obtained from considering the microscopic scale. Since the system by assumption is in equilibrium within a REV, $\mu$ has the same value everywhere within this volume. On the other hand, the (electro-) chemical potential may also be expressed e.g. as

\begin{equation}
\mu(\xi ) = \mu_0 + RT\cdot\ln a(\xi ) + zF\psi(\xi ) = \mathrm{const}
\end{equation}

where $\xi$ is a microscopic variable denoting the position within the REV, $\mu_0$ is a reference potential, $a(\xi )$ is the chemical activity and $\psi(\xi )$ the electrostatic potential. In the bulk water domain, the electrostatic potential is zero by definition (ignoring possible couplings between different types of charged species), and $\mu$ can here be evaluated as (where the index “b” is used to denote bulk)

\begin{equation}
\mu_b = \mu_0 + RT\cdot\ln a_b
\tag{4}
\end{equation}

But this is by definition the value of the (electro-)chemical potential everywhere within the REV.

Thus, going back to the macroscopic scale and using eq. 4 for $\mu$, we have for the flux

\begin{equation}
j = -\phi D_\mathrm{macr.}\cdot\frac{\bar{c}}{a_b}\nabla a_b
\end{equation}

Ignoring also possible gradients in activity coefficients, the flux can be simplified as

\begin{equation}
j = -\phi D_\mathrm{macr.}\cdot\frac{\bar{c}}{c}\nabla c
\end{equation}

Where, as before, $c$ denotes a bulk solution concentration. For the case of having one bulk domain, and one surface domain, $\bar{c}$ can be written

\begin{equation}
\bar{c} = \frac{\phi_b}{\phi}c + \frac{\phi_s}{\phi}c_s
\end{equation}

where $\phi_b$ and $\phi_s$ are the porosities for the bulk and surface domains, respectively ($\phi = \phi_b + \phi_s$). Using also eq. 2, the flux can finally be written

\begin{equation}
j = -D_\mathrm{macr.}\left (\phi_b+\phi_s\cdot K\right )\nabla c
\tag{5}
\end{equation}

This expression can be compared to the result from the “standard” procedure, eq. 3. Although eq. 3 is derived using incompatible, non-physical assumptions, it is similar to eq. 5 , but involves two ($D_e$ and $D_s$) rather than a single diffusion parameter ($D_\mathrm{macr.}$). If these two parameters are used merely as fitting parameters (which they basically always are), eq. 3 can always be made to resemble eq. 5. In this sense, eq. 3 is not wrong. Rather, it is not even wrong.

Footnotes

[1] In the following I will only say bentonite, but I mean any type of clay system with significant swelling/ion exchange properties.

[2] I strongly oppose the way bulk water phases are included in such models, but this post is not a critique of that model choice. Here, I instead point out that virtually all published descriptions of multiporosity models handle the flux incorrectly.

[3] Specifically, the average of the square of the displacement of the walker is proportional to time, with the diffusion coefficient as proportionality constant (apart from a dimensional factor).

\(C_\mathrm{bkg}\)	(\(\mathrm{mol/m^3}\))	10	100	500	700	1000
			Reported
\(D_\mathrm{e}^\mathrm{clay}\)	(\(10^{-10}\;\mathrm{m^2/s}\))	14	3.7	0.86	0.53	0.38
\(D_\mathrm{e}^\mathrm{filter}\)	(\(10^{-10}\;\mathrm{m^2/s}\))	0.86	1.0	0.86	0.86	0.86
			Calculated
\(\omega\)	(-)	9.3	2.1	0.6	0.4	0.3