Category Archives: Donnan equilibrium

Semi-permeability, part I

Descriptions in bentonite literature

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Chagneau et al. (2015) write

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

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

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

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

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

Bentonite as a non-ideal semi-permeable membrane

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

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

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

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

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

“Semi-permeability” in experiments

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

Tracer through-diffusion

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

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

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

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

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

Measuring hydraulic conductivity

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

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

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

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

Mass transfer in a salt gradient

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

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

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

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

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

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

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

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

Footnotes

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

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

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

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

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

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

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

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

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

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

Molecular dynamics simulations do not support complete anion exclusion

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

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

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

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

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

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

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

Here Tournassat and Steefel (2015) conceive of the interlayer space as something distinctly different from a diffuse layer,¹ and they mean that the MD result stands in contrast to conventional Donnan theory (Birgersson and Karnland, (2009)).

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

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

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

Rotenberg et al. (2007)

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

Here is an illustration of the simulated system:

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

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

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

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

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

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

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

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

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

and

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

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

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

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

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

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

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

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

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

Hsiao and Hedström (2015)

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

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

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

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

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

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

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

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

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

Hedström and Karnland (2012)

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

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

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

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

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

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

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

Comparison with experiments

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

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

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

Tournassat et al. (2016)

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

In the 2WL simulations, no anions enter the interlayers. Tou16 do not reflect on the possibility that 2WL simulations may overestimate exclusion, as suggested by Hsi15,⁹ but instead use this result to argue that anions are basically completely excluded from 2WL interlayers. They even imply that the result of Rot07 is more adequate than that of Hsi15

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

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

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

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

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

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

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

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

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

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

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

Footnotes

[1] I should definitely discuss the “Stern layer” in a future blog post. Update (250113): Stern layers are discussed here.

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

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

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

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

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

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

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

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

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

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

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

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

Kahr et al. (1985) — the diffusion study that could have changed everything

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On the surface, “Ionendiffusion in Hochverdichtetem Bentonit”¹ by G. Kahr, R. Hasenpatt, and M. Müller-Vonmoos, published by NAGRA in March 1985, looks like an ordinary mundane 37-page technical report. But it contains experimental results that could have completely changed the history of model development for compacted clay.

Test principles

The tests were conducted in a quite original manner. By compacting granules or powder, the investigators obtained samples that schematically look like this

Schematics of samples in Kahr et al. (1985(

The bentonite material — which was either Na-dominated “MX-80”, or Ca-dominated “Montigel” — was conditioned to a specific water-to-solid mass ratio \(w\). At one of the faces, the bentonite was mixed with a salt (in solid form) to form a thin source for diffusing ions. This is essentially the full test set-up! Diffusion begins as soon as the samples are prepared, and a test was terminated after some prescribed amount of time, depending on diffusing ion and water content. At termination, the samples were sectioned and analyzed. In this way, the investigators obtained final state ion distributions, which in turn were related to the initial states by a model, giving the diffusion coefficients of interest.

Note that the experiments were conducted without exposing samples to a liquid (external) solution; the samples were “unsaturated” to various degree, and the diffusing ions dissolve within the bentonite. The samples were not even confined in a test cell, but “free-standing”, and consequently not under pressure. They were, however, stored in closed vessels during the course of the tests, to avoid changes in water content.

With this test principle a huge set of diffusion tests were performed, with systematic variation of the following variables:

Bentonite material (“MX-80” or “Montigel”)
Water-to-solid mass ratio (7% — 33%)
Dry density (1.3 g/m³ — 2.1 g/m³ )
Diffusing salt (SrCl₂, SrI₂, CsCl, CsI, UO₂(NO₃)₂, Th(NO₃)₄, KCl, KI, KNO₃, K₂SO₄, K₂CO₃, KF)

Distribution of water in the samples

From e.g. X-ray diffraction (XRD) we know that bentonite water at low water content is distributed in distinct, sub-nm thin films. For simplicity we will refer to all water in the samples as interlayer water, although some of it, reasonably, forms interfaces with air. The relevant point is that the samples contain no bulk water phase, but only interfacial (interlayer) water.

I argue extensively on this blog for that interlayer water is the only relevant water phase also in saturated samples under pressure. In the present case, however, it is easier to prove that this is the case, as the samples are merely pressed bentonite powder at a certain water content; the bentonite water is not pressurized, the samples are not exposed to liquid bulk water, nor are they in equilibrium with liquid bulk water. Since the water in the samples obviously is mobile — as vapor, but most reasonably also in interconnected interlayers — it is a thermodynamic consequence that it distributes as to minimize the chemical potential.

There is a ton of literature on how the montmorillonite basal spacing varies with water content. Here, we use the neat result from Holmboe et al. (2012) that the average interlayer distance varies basically linearly² with water content, like this

average basal distance vs. water content from Holmboe et al. (2012)

XRD-studies also show that bentonite water is distributed in rather distinct hydration states, corresponding to 0, 1, 2, or 3 monolayers of water.³ We label these states 0WL, 1WL, 2WL, and 3WL, respectively. In the figure is indicated the approximate basal distances for pure 1WL (12.4 Å), 2WL (15.7 Å), and 3WL (19.0 Å), which correspond roughly to water-to-solid mass ratios of 0.1, 0.2, and 0.3, respectively.

From the above plot, we estimate roughly that the driest samples in Kahr et al. (1985) (\(w \sim 0.1\)) are in pure 1WL states, then transitions to a mixture of 1WL and 2WL states (\(w\sim 0.1 – 0.2\)), to pure 2WL states (\(w \sim 0.2\)), to a mixture of 2WL and 3WL states (\(w\sim 0.2 – 0.3\)), and finally to pure 3WL states (\(w\sim 0.3\)).

Results

With the knowledge of how water is distributed in the samples, let’s take a look at the results of Kahr et al. (1985).

Mobility of interlayer cations confirmed

The most remarkable results are of qualitative character. It is, for instance, demonstrated that several cations diffuse far into the samples. Since the samples only contain interlayer water, this is a direct proof of ion mobility in the interlayers!

Also, cations are demonstrated to be mobile even when the water content is as low as 7 or 10 %! As such samples are dominated by 1WL states, this is consequently evidence for ion mobility in 1WL states.

A more quantitative assessment furthermore shows that the cation diffusivities varies with water content in an almost step-wise manner, corresponding neatly to the transitions between various hydration states. Here is the data for potassium and strontium

De vs. water content for potassium and strontium from Kahr et al. (1985)

This behavior further confirms that the ions diffuse in interlayers, with an increasing diffusivity as the interlayers widen.

It should also be noted that the evaluated values of the diffusivities are comparable to — or even larger⁴ — than corresponding results from saturated, pressurized tests. This strongly suggests that interlayer diffusivity dominates also in the latter types of tests, which also has been confirmed in more recent years. The larger implication is that interlayer diffusion is the only relevant type of diffusion in general in compacted bentonite.

Anions enter interlayers (and are mobile)

The results also clearly demonstrate that anions (iodide) diffuse in systems with water-to-solid mass ratio as low as 7%! With no other water around, this demonstrates that anions diffuse in — and consequently have access to — interlayers. This finding is strongly confirmed by comparing the \(w\)-dependence of diffusivity for anions and cations. Here is plotted the data for iodide and potassium (with the potassium diffusivity indicated on the right y-axis)

De vs. water content for iodide and potassium from Kahr et al. (1985)

The iodide mobility increases as the system transitions from 1WL to 2WL, in a very similar way as for potassium (and strontium). If this is not a proof that the anion diffuse in the same domain as the cation I don’t know what is! Also for iodide the value of the diffusivity is comparable to what is evaluated in water saturated systems under pressure, which implies that interlayer diffusivity dominates generally in compacted bentonite, also for anions.

Dependence of diffusivity on water content and density

A conclusion made in Kahr et al. (1985), that I am not sure I fully agree with, is that diffusivity mainly depends on water content rather than density. As seen in the diagrams above, the spread in diffusivity is quite substantial for a given value of \(w\). There is actually some systematic variation here: for constant \(w\), diffusivity tend to increase with dry density.

Although using unsaturated samples introduces additional variation, the present study provides a convenient procedure to study diffusion in systems with very low water content. A more conventional set-up in this density limit has to deal with enormous pressures (on the order of 100 MPa).

Interlayer chemistry

An additional result is not acknowledged in the report, but is a direct consequence of the observations: the tests demonstrate that interlayers are chemically active. The initially solid salt evidently dissolves before being able to diffuse. Since these samples are not even close to containing a bulk water phase (as discussed above), the dissolution process must occur in an interlayer. More precisely, the salt must dissolve in interface water between the salt mineral and individual montmorillonite layers, as illustrated here

Schematics of KI dissolution in interlayer water

This study seems to have made no impact at all

In the beginning of 1985, the research community devoted to radioactive waste barriers seems to have been on its way to correctly identify diffusion in interlayers as the main transport mechanism, and to recognize how ion diffusion in bentonite is influenced by equilibrium with external solutions.

Already in 1981, Torstenfelt et al. (1981) concluded that the traditional diffusion-sorption model is not valid, for e.g. diffusion of Sr and Cs, in compacted bentonite. They also noted, seemingly without realizing the full importance, that these ions diffused even in unsaturated samples with as low water-to-solid mass ratio as 10%.

A significant diffusion was observed for Sr in dry clay, although slower than for water saturated clay, Figure 4, while Cs was almost immobile in the dry clay.

A year later also Eriksen and Jacobsson (1982) concluded that the traditional diffusion model is not valid. They furthermore pointed out the subtleties involved when interpreting through-diffusion experiments, due to ion equilibrium effects

One difficulty in correlating the diffusivities obtained from profile analysis to the diffusivities calculated from steady state transport data is the lack of knowledge of the tracer concentration at the solution-bentonite interface. This concentration is generally higher for sorbing species like positive ions (counterions to the bentonite) and lower for negative ions (coions to the bentonite) as shown schematically in figure 11. The equilibrium concentration of any ion in the bentonite and solution respectively is a function of the ionic charge, the ionic strength of the solution and the overall exchanger composition and thereby not readily calculated

In Eriksen and Jacobsson (1984) the picture is fully clear

By regarding the clay-gel as a concentrated electrolytic system Marinsky has calculated (30) distribution coefficients for Sr²⁺ and Cs⁺ ions in good agreement with experimentally determined K_d-values. The low anionic exchange capacity and hence the low anion concentration in the pore solution caused by Donnan exclusion also explain the low concentrations of anionic tracers within the clay-gel

[…]

For simple cations the ion-exchange process is dominating and there is, as also pointed out by Marinsky (30), no need to suppose that the counterions are immobilized. It ought to be emphasized that for the compacted bentonite used in the diffusion experiments discussed in this report the water content corresponds roughly to 2-4 water molecule layers (31). There is therefore really no “free water” and the measured diffusivity \(\bar{D}\) can be regarded as corresponding approximately to the diffusivity within the adsorbed phase […]

Furthermore, also Soudek et al. (1984) had discarded the traditional diffusion-sorption model, identified the exchangeable cations as giving a dominating contribution to mass transfer, and used Donnan equilibrium calculations to account for the suppressed internal chloride concentration.

In light of this state of the research front, the contribution of Kahr et al. (1985) cannot be described as anything but optimal. In contrast to basically all earlier studies, this work provides systematic variation of several variables (most notably, the water-to-solid ratio). As a consequence, the results provide a profound confirmation of the view described by Eriksen and Jacobsson (1984) above, i.e. that interlayer pores essentially govern all physico-chemical behavior in compacted bentonite. A similar description was later given by Bucher and Müller-Vonmoos (1989) (though I don’t agree with all the detailed statements here)

There is no free pore water in highly compacted bentonite. The water in the interlayer space of montmorillonite has properties that are quite different from those of free pore water; this explains the extremely high swelling pressures that are generated. The water molecules in the interlayer space are less mobile than their free counterparts, and their dielectric constant is lower. The water and the exchangeable cations in the interlayer space can be compared to a concentrated salt solution. The sodium content of the interlayer water, at a water content of 25%, corresponds approximately to a 3-n salt solution, or six times the concentration in natural seawater. This more or less ordered water is fundamentally different from that which engineers usually take into account; in the latter case, pore water in a saturated soil is considered as a freely flowing fluid. References to the porosity in highly compacted bentonite are therefore misleading. Highly compacted bentonite is an unfamiliar material to the engineer.

Given this state of the research field in the mid-80s, I find it remarkable that history took a different turn. It appears as the results of Kahr et al. (1985) made no impact at all (it may be noticed that they themselves analyzed the results in terms of the traditional diffusion-sorption model). And rather than that researchers began identifying that transport in interlayers is the only relevant contribution, the so-called surface diffusion model gained popularity (it was already promoted by e.g. Soudek et al. (1984) and Neretnieks and Rasmuson (1983)). Although this model emphasizes mobility of the exchangeable cations, it is still centered around the idea that compacted bentonite contains bulk water.⁵ Most modern bentonite models suffer from similar flaws: they are formulated in terms of bulk water, while many effects related to interlayers are treated as irrelevant or optional.

For the case of anion diffusion the historical evolution is maybe even more disheartening. In 1985 the notions of “effective” or “anion-accessible” porosities seem to not have been that widely spread, and here was clear-cut evidence of anions occupying interlayer pores. But just a few years later the idea began to grow that the pore space in compacted bentonite should be divided into regions which are either accessible or inaccessible to anions. As far as I am aware, the first use of the term “effective porosity” in this context was by Muurinen et al. (1988), who, ironically, seem to have misinterpreted the Donnan equilibrium approach presented by Soudek et al. (1984). To this day, this flawed concept is central in many descriptions of compacted clay.

Footnotes

[1] “Ion diffusion in highly compacted bentonite”

[2] Incidentally, the slope of this line corresponds to a water “density” of 1.0 g/cm³.

[3] This is the region of swelling often referred to as “crystalline”.

[4] I’m not sure the evaluation in Kahr et al. (1985) is fully correct. They use the solution to the diffusion equation for an impulse source (a Gaussian), but, to my mind, the source is rather one of constant concentration (set by the solubility of the salt). Unless I have misunderstood, the mathematical expression to be fitted to data should then be an erfc-function, rather than a Gaussian. Although this modification would change the numerical values of the evaluated diffusion coefficients somewhat, it does not at all influence the qualitative insights provided by the study.

[5] I have discussed the surface diffusion model in some detail in previous blog posts.

The mechanism for “anion” exclusion

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Repulsion between surfaces and anions is not really the point

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

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

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

Ferrocyanide

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

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

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

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

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

Bentonite

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

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

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

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

Here is an illustration of the validity of this approximation:

Internal and external potential in compacted bentonite system

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

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

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

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

A charge neutral perspective

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

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

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

while the outflux is proportional to the internal ion concentrations

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

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

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

which is the (ideal) Donnan equation.

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

Footnotes

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

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

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

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

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

Sorption part IV: What is K_d?

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Measuring K_d

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

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

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

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

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

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

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

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

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

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

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

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

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

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

K_d in the homogeneous mixture model

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

K_d for simple cations

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

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

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

Simple cation tracers in a 1:1 system

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

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

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

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

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

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

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

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

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

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

For a divalent tracer we instead have

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

giving

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

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

Simple cation tracers in a 2:1 system

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

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

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

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

where

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

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

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

For a divalent tracer we instead have

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

giving

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

Density dependence of K_d

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

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

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

Footnotes

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

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

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

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

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

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

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

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

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

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

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

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

Donnan equilibrium and the homogeneous mixture model

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

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

Overview

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

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

The problem to be solved can be illustrated like this

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

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

Chemical equilibrium

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

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

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

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

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

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

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

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

Illustrations

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

Relation beteween internal and external concentration for varying Xi

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

Variation of internal cation concentrations with varying Donnan factor

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

Variation of internal anion concentration with the Donnan factor

Equation for \(f_D\)

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

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

where

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

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

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

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

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

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

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

Relation between internal and external Na and Cl concentrations

Comment on through-diffusion

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

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

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

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

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

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

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

Additional comments

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

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

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

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

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

Footnotes

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

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

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

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

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

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

Sorption part III: Donnan equilibrium in compacted bentonite

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

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

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

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

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

Donnan equilibrium: effect vs. model

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

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

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

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

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

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

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

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

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

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

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

Donnan equilibrium in gels, soils and clays

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

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

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

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

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

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

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

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

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

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

It thus seems plausible that Donnan equilibrium in compacted bentonite can be treated using Donnan’s original equations. But — as interlayer pores 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.

Descriptions in bentonite literature

Bentonite as a non-ideal semi-permeable membrane

“Semi-permeability” in experiments

Tracer through-diffusion

Measuring hydraulic conductivity

Mass transfer in a salt gradient

Footnotes

Rotenberg et al. (2007)

Hsiao and Hedström (2015)

Hedström and Karnland (2012)

Comparison with experiments

Tournassat et al. (2016)

Footnotes

Test principles

Distribution of water in the samples

Results

Mobility of interlayer cations confirmed

Anions enter interlayers (and are mobile)

Dependence of diffusivity on water content and density

Interlayer chemistry

This study seems to have made no impact at all

Footnotes

Repulsion between surfaces and anions is not really the point

Ferrocyanide

Bentonite

A charge neutral perspective

Footnotes

Measuring Kd

Kd in the homogeneous mixture model

Kd for simple cations

Simple cation tracers in a 1:1 system

Simple cation tracers in a 2:1 system

Density dependence of Kd

Footnotes

Overview

Chemical equilibrium

Illustrations

Equation for \(f_D\)

Comment on through-diffusion

Additional comments

Footnotes

Donnan equilibrium: effect vs. model

Donnan equilibrium in gels, soils and clays

Cation exchange is Donnan equilibration

A demonstration of compacted bentonite as a Donnan system

Footnotes

Measuring K_d

K_d in the homogeneous mixture model

K_d for simple cations

Density dependence of K_d