Category Archives: Semi-permeability

Multi-porosity models cannot be taken seriously (Semi-permeability, part II)

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“Multi-porosity” models1 — i.e models that account for both a bulk water phase and one, or several, other domains within the clay — have become increasingly popular in bentonite research during the last couple of decades. These are obviously macroscopic, as is clear e.g. from the benchmark simulations described in Alt-Epping et al. (2015), which are specified to be discretized into 2 mm thick cells; each cell is consequently assumed to contain billions and billions individual montmorillonite particles. The macroscopic character is also relatively clear in their description of two numerical tools that have implemented multi-porosity

PHREEQC and CrunchFlowMC have implemented a Donnan approach to describe the electrical potential and species distribution in the EDL. This approach implies a uniform electrical potential \(\varphi^\mathrm{EDL}\) in the EDL and an instantaneous equilibrium distribution of species between the EDL and the free water (i.e., between the micro- and macroporosity, respectively). The assumption of instantaneous equilibrium implies that diffusion between micro- and macroporosity is not considered explicitly and that at all times the chemical potentials, \(\mu_i\), of the species are the same in the two porosities

On an abstract level, we may thus illustrate a multi-porosity approach something like this (here involving two domains)

The model is represented by one continuum for the “free water”/”macroporosity” and one for the “diffuse layer”/”microporosity”,2 which are postulated to be in equilibrium within each macroscopic cell.

But such an equilibrium (Donnan equilibrium) requires a semi-permeable component. I am not aware of any suggestion for such a component in any publication on multi-porosity models. Likewise, the co-existence of diffuse layer and free water domains requires a mechanism that prevents swelling and maintains the pressure difference — also the water chemical potential should of course be the equal in the two “porosities”.3

Note that the questions of what constitutes the semi-permeable component and what prevents swelling have a clear answer in the homogeneous mixture model. This answer also corresponds to an easily identified real-world object: the metal filter (or similar component) separating the sample from the external solution. Multi-porosity models, on the other hand, attribute no particular significance to interfaces between sample and external solutions. Therefore, a candidate for the semi-permeable component has to be — but isn’t — sought elsewhere. Donnan equilibrium calculations are virtually meaningless without identifying this component.

The partitioning between diffuse layer and free water in multi-porosity models is, moreover, assumed to be controlled by water chemistry, usually by means of the Debye length. E.g. Alt-Epping et al. (2015) write

To determine the volume of the microporosity, the surface area of montmorillonite, and the Debye length, \(D_L\), which is the distance from the charged mineral surface to the point where electrical potential decays by a factor of e, needs to be known. The volume of the microporosity can then be calculated as \begin{equation*} \phi^\mathrm{EDL} = A_\mathrm{clay} D_L, \end{equation*} where \(A_\mathrm{clay}\) is the charged surface area of the clay mineral.

I cannot overstate how strange the multi-porosity description is. Leaving the abstract representation, here is an attempt to illustrate the implied clay structure, at the “macropore” scale

The view emerging from the above description is actually even more peculiar, as the “micro” and “macro” volume fractions are supposed to vary with the Debye length. A more general illustration of how the pore structure is supposed to function is shown in this animation (“I” denotes ionic strength)

What on earth could constitute such magic semi-permeable membranes?! (Note that they are also supposed to withstand the inevitable pressure difference.)

Here, the informed reader may object and point out that no researcher promoting multi-porosity has this magic pore structure in mind. Indeed, basically all multi-porosity publications instead vaguely claim that the domain separation occurs on the nanometer scale and present microscopic illustrations, like this (this is a simplified version of what is found in Alt-Epping et al. (2015))

In the remainder of this post I will discuss how the idea of a domain separation on the microscopic scale is even more preposterous than the magic membranes suggested above. We focus on three aspects:

  • The implied structure of the free water domain
  • The arbitrary domain division
  • Donnan equilibrium on the microscopic scale is not really a valid concept

Implied structure of the free water domain

I’m astonished by how little figures of the microscopic scale are explained in many publications. For instance, the illustration above clearly suggests that “free water” is an interface region with exactly the same surface area as the “double layer”. How can that make sense? Also, if the above structure is to be taken seriously it is crucial to specify the extensions of the various water layers. It is clear that the figure shows a microscopic view, as it depicts an actual diffuse layer.4 A diffuse layer width varies, say, in the range 1 – 100 nm,5 but authors seldom reveal if we are looking at a pore 1 nm wide or several hundred nm wide. Often we are not even shown a pore — the water film just ends in a void, as in the above figure.6

The vague nature of these descriptions indicates that they are merely “decorations”, providing a microscopic flavor to what in effect still is a macroscopic model formulation. In practice, most multi-porosity formulations provide some ad hoc mean to calculate the volume of the diffuse layer domain, while the free water porosity is either obtained by subtracting the diffuse layer porosity from total porosity, or by just specifying it. Alt-Epping et al. (2015), for example, simply specifies the “macroporosity”

The total porosity amounts to 47.6 % which is divided into 40.5 % microporosity (EDL) and 7.1 % macroporosity (free water). From the microporosity and the surface area of montmorillonite (Table 7), the Debye length of the EDL calculated from Eq. 11 is 4.97e-10 m.

Clearly, nothing in this description requires or suggests that the “micro” and “macroporosities” are adjacent waterfilms on the nm-scale. On the contrary, such an interpretation becomes quite grotesque, with the “macroporosity” corresponding to half a monolayer of water molecules! An illustration of an actual pore of this kind would look something like this

This interpretation becomes even more bizarre, considering that Alt-Epping et al. (2015) assume advection to occur only in this half-a-monolayer of water, and that the diffusivity is here a factor 1000 larger than in the “microporosity”.

As another example, Appelo and Wersin (2007) model a cylindrical sample of “Opalinus clay” of height 0.5 m and radius 0.1 m, with porosity 0.16, by discretizing the sample volume in 20 sections of width 0.025 m. The void volume of each section is consequently \(V_\mathrm{void} = 0.16\cdot\pi\cdot 0.1^2\cdot 0.025\;\mathrm{m^3} = 1.257\cdot10^{-4}\;\mathrm{m^3}\). Half of this volume (“0.062831853” liter) is specified directly in the input file as the volume of the free water;7 again, nothing suggests that this water should be distributed in thin films on the nm-scale. Yet, Appelo and Wersin (2007) provide a figure, with no length scale, similar in spirit to that above, that look very similar to this

They furthermore write about this figure (“Figure 2”)

It should be noted that the model can zoom in on the nm-scale suggested by Figure 2, but also uses it as the representative form for the cm-scale or larger.

I’m not sure I can make sense of this statement, but it seems that they imply that the illustration can serve both as an actual microscopic representation of two spatially separated domains and as a representation of two abstract continua on the macroscopic scale. But this is not true!

Interpreted macroscopically, the vertical dimension is fictitious, and the two continua are in equilibrium in each paired cell. On a microscopic scale, on the other hand, equilibrium between paired cells cannot be assumed a priori, and it becomes crucial to specify both the vertical and horizontal length scales. As Appelo and Wersin (2007) formulate their model assuming equilibrium between paired cells, it is clear that the above figure must be interpreted macroscopically (the only reference to a vertical length scale is that the “free solution” is located “at infinite distance” from the surface).

We can again work out the implications of anyway interpreting the model microscopically. Each clay cell is specified to contain a surface area of \(A_\mathrm{surf}=10^5\;\mathrm{m^2}\).8 Assuming a planar geometry, the average pore width is given by (\(\phi\) denotes porosity and \(V_\mathrm{cell}\) total cell volume)

\begin{equation} d = 2\cdot \phi \cdot \frac{V_\mathrm{cell}}{A_\mathrm{surf}} = 2\cdot \frac{V_\mathrm{void}}{A_\mathrm{surf}} = 2\cdot \frac{1.26\cdot 10^{-4}\;\mathrm{m^3}}{10^{5}\;\mathrm{m^2}} = 2.51\;\mathrm{nm} \end{equation}

The double layer thickness is furthermore specified to be 0.628 nm.9 A microscopic interpretation of this particular model thus implies that the sample contains a single type of pore (2.51 nm wide) in which the free water is distributed in a thin film of width 1.25 nm — i.e. approximately four molecular layers of water!

Rather than affirming that multi-porosity model formulations are macroscopic at heart, parts of the bentonite research community have instead doubled down on the confusing idea of having free water distributed on the nm-scale. Tournassat and Steefel (2019) suggest dealing with the case of two parallel charged surfaces in terms of a “Dual Continuum” approach, providing a figure similar to this (surface charge is -0.11 C/m2 and external solution is 0.1 M of a 1:1 electrolyte)

Note that here the perpendicular length scale is specified, and that it is clear from the start that the electrostatic potential is non-zero everywhere. Yet, Tournassat and Steefel (2019) mean that it is a good idea to treat this system as if it contained a 0.7 nm wide bulk water slice at the center of the pore. They furthermore express an almost “postmodern” attitude towards modeling, writing

It should be also noted here that this model refinement does not imply necessarily that an electroneutral bulk water is present at the center of the pore in reality. This can be appreciated in Figure 6, which shows that the Poisson–Boltzmann predicts an overlap of the diffuse layers bordering the two neighboring surfaces, while the dual continuum model divides the same system into a bulk and a diffuse layer water volume in order to obtain an average concentration in the pore that is consistent with the Poisson–Boltzmann model prediction. Consequently, the pore space subdivision into free and DL water must be seen as a convenient representation that makes it possible to calculate accurately the average concentrations of ions, but it must not be taken as evidence of the effective presence of bulk water in a nanoporous medium.

I can only interpret this way of writing (“…does not imply necessarily that…”, “…must not be taken as evidence of…”) that they mean that in some cases the bulk phase should be interpreted literally, while in other cases the bulk phase should be interpreted just as some auxiliary component. It is my strong opinion that such an attitude towards modeling only contributes negatively to process understanding (we may e.g. note that later in the article, Tournassat and Steefel (2019) assume this perhaps non-existent bulk water to be solely responsible for advective flow…).

I say it again: no matter how much researchers discuss them in microscopic terms, these models are just macroscopic formulations. Using the terminology of Tournassat and Steefel (2019), they are, at the end of the day, represented as dual continua assumed to be in local equilibrium (in accordance with the first figure of this post). And while researchers put much effort in trying to give these models a microscopic appearance, I am not aware of anyone suggesting a reasonable candidate for what actually could constitute the semi-permeable component necessary for maintaining such an equilibrium.

Arbitrary division between diffuse layer and free water

Another peculiarity in the multi-porosity descriptions showing that they cannot be interpreted microscopically is the arbitrary positioning of the separation between diffuse layer and free water. We saw earlier that Alt-Epping et al. (2015) set this separation at one Debye length from the surface, where the electrostatic potential is claimed to have decayed by a factor of e. What motivates this choice?

Most publications on multi-porosity models define free water as a region where the solution is charge neutral, i.e. where the electrostatic potential is vanishingly small.10 At the point chosen by Alt-Epping et al. (2015), the potential is about 37% of its value at the surface. This cannot be considered vanishingly small under any circumstance, and the region considered as free water is consequently not charge neutral.

The diffuse layer thickness chosen by Appelo and Wersin (2007) instead corresponds to 1.27 Debye lengths. At this position the potential is about 28% of its value at the surface, which neither can be considered vanishingly small. At the mid point of the pore (1.25 nm), the potential is about 8%11 of the value at the surface (corresponding to about 2.5 Debye lengths). I find it hard to accept even this value as vanishingly small.

Note that if the boundary distance used by Appelo and Wersin (2007) (1.27 Debye lengths) was used in the benchmark of Alt-Epping et al. (2015), the diffuse layer volume becomes larger than the total pore volume! In fact, this occurs in all models of this kind for low enough ionic strength, as the Debye length diverges in this limit. Therefore, many multi-porosity model formulations include clunky “if-then-else” clauses,12 where the system is treated conceptually different depending on whether or not the (arbitrarily chosen) diffuse layer domain fills the entire pore volume.13

In the example from Tournassat and Steefel (2019) the extension of the diffuse layer is 1.6 nm, corresponding to about 1.69 Debye lengths. The potential is here about 19% of the surface value (the value in the midpoint is 12% of the surface value). Tournassat and Appelo (2011) uses yet another separation distance — two Debye lengths — based on misusing the concept of exclusion volume in the Gouy-Chapman model.

With these examples, I am not trying to say that a better criterion is needed for the partitioning between diffuse layer and bulk. Rather, these examples show that such a partitioning is quite arbitrary on a microscopic scale. Of course, choosing points where the electrostatic potential is significant makes no sense, but even for points that could be considered having zero potential, what would be the criterion? Is two Debye lengths enough? Or perhaps four? Why?

These examples also demonstrate that researchers ultimately do not have a microscopic view in mind. Rather, the “microscopic” specifications are subject to the macroscopic constraints. Alt-Epping et al. (2015), for example, specifies a priori that the system contains about 15% free water, from which it follows that the diffuse layer thickness must be set to about one Debye length (given the adopted surface area). Likewise, Appelo and Wersin (2007) assume from the start that Opalinus clay contains 50% free water, and set up their model accordingly.14 Tournassat and Steefel (2019) acknowledge their approach to only be a “convenient representation”, and don’t even relate the diffuse layer extension to a specific value of the electrostatic potential.15 Why the free water domain anyway is considered to be positioned in the center of the nanopore is a mystery to me (well, I guess because sometimes this interpretation is supposed to be taken literally…).

Note that none of the free water domains in the considered models are actually charged, even though the electrostatic potential in the microscopic interpretations is implied to be non-zero. This just confirms that such interpretations are not valid, and that the actual model handling is the equilibration of two (or more) macroscopic, abstract, continua. The diffuse layer domain is defined by following some arbitrary procedure that involves microscopic concepts. But just because the diffuse layer domain is quantified by multiplying a surface area by some multiple of the Debye length does not make it a microscopic entity.4

Donnan effect on the microscopic scale?!

Although we have already seen that we cannot interpret multi-porosity models microscopically, we have not yet considered the weirdest description adopted by basically all proponents of these models: they claim to perform Donnan equilibrium calculations between diffuse layer and free water regions on the microscopic scale!

The underlying mechanism for a Donnan effect is the establishment of charge separation, which obviously occur on the scale of the ions, i.e. on the microscopic scale. Indeed, a diffuse layer is the manifestation of this charge separation. Donnan equilibrium can consequently not be established within a diffuse layer region, and discontinuous electrostatic potentials only have meaning in a macroscopic context.

Consider e.g. the interface between bentonite and an external solution in the homogeneous mixture model. Although this model ignores the microscopic scale, it implies charge separation and a continuously varying potential on this scale, as illustrated here

The regions where the potential varies are exactly what we categorize as diffuse layers (exemplified in two ideal microscopic geometries).

The discontinuous potentials encountered in multi-porosity model descriptions (see e.g. the above “Dual Continuum” potential that varies discontinuously on the angstrom scale) can be drawn on paper, but don’t convey any physical meaning.

Here I am not saying that Donnan equilibrium calculations cannot be performed in multi-porosity models. Rather, this is yet another aspect showing that such models only have meaning macroscopically, even though they are persistently presented as if they somehow consider the microscopic scale.

An example of this confusion of scales is found in Alt-Epping et al. (2018), who revisit the benchmark problem of Alt-Epping et al. (2015) using an alternative approach to Donnan equilibrium: rather than directly calculating the equilibrium, they model the clay charge as immobile mono-valent anions, and utilize the Nernst-Planck equations. They present “the conceptual model” in a figure very similar to this one

This illustration simultaneously conveys both a micro- and macroscopic view. For example, a mineral surface is indicated at the bottom, suggesting that we supposedly are looking at an actual interface region, in similarity with the figures we have looked at earlier. Moreover, the figure contains entities that must be interpreted as individual ions, including the immobile “clay-anions”. As in several of the previous examples, no length scale is provided (neither perpendicular to, nor along the “surface”).

On the other hand, the region is divided into cells, similar to the illustration in Appelo and Wersin (2007). These can hardly have any other meaning than to indicate the macroscopic discretization in the adopted transport code (FLOTRAN). Also, as the “Donnan porosity” region contains the “clay-anions” it can certainly not represent a diffuse layer extending from a clay surface; the only way to make sense of such an “immobile-anion” solution is that it represents a macroscopic homogenized clay domain (a homogeneous mixture!).

Furthermore, if the figure is supposed to show the microscopic scale there is no Donnan effect, because there is no charge separation! Taking the depiction of individual ions seriously, the interface region should rather look something like this in equilibrium

This illustrates the fundamental problem with a Donnan effect between microscopic compartments: the effect requires a charge separation, whose extension is the same as the size of the compartments assumed to be in equilibrium.16

Despite the confusion of the illustration in Alt-Epping et al. (2018), it is clear that a macroscopic model is adopted, as in our previous examples. In this case, the model is explicitly 2-dimensional, and the authors utilize the “trick” to make diffusion much faster in the perpendicular direction compared to the direction along the “surface”. This is achieved either by making the perpendicular diffusivity very high, or by making the perpendicular extension small. In any case, a perpendicular length scale must have been specified in the model, even if it is nowhere stated in the article. The same “trick” for emulating Donnan equilibrium is also used by Jenni et al. (2017), who write

In the present model set-up, this approach was implemented as two connected domains in the z dimension: one containing all minerals plus the free porosity (z=1) and the other containing the Donnan porosity, including the immobile anions (CEC, z=2, Fig. 2). Reproducing instantaneous equilibrium between Donnan and free porosities requires a much faster diffusion between the porosity domains than along the porosity domains.

Note that although the perpendicular dimension (\(z\)) here is referred to without unit(!), this representation only makes sense in a macroscopic context.

Jenni et al. (2017) also provide a statement that I think fairly well sums up the multi-porosity modeling endeavor:17

In a Donnan porosity concept, cation exchange can be seen as resulting from Donnan equilibrium between the Donnan porosity and the free porosity, possibly moderated by additional specific sorption. In CrunchflowMC or PhreeqC (Appelo and Wersin, 2007; Steefel, 2009; Tournassat and Appelo, 2011; Alt-Epping et al., 2014; Tournassat and Steefel, 2015), this is implemented by an explicit partitioning function that distributes aqueous species between the two pore compartments. Alternatively, this ion partitioning can be modelled implicitly by diffusion and electrochemical migration (Fick’s first law and Nernst-Planck equations) between the free porosity and the Donnan porosity, the latter containing immobile anions representing the CEC. The resulting ion compositions of the two equilibrated porosities agree with the concentrations predicted by the Donnan equilibrium, which can be shown in case studies (unpublished results, Gimmi and Alt-Epping).

Ultimately, these are models that, using one approach or the other, simply calculates Donnan equilibrium between two abstract, macroscopically defined domains (“porosities”, “continua”). Microscopic interpretations of these models lead — as we have demonstrated — to multiple absurdities and errors. I am not aware of any multi-porosity approach that has provided any kind of suggestion for what constitutes the semi-permeable component required for maintaining the equilibrium they are supposed to describe. Alternatively expressed: what, in the previous figure, prevents the “immobile anions” from occupying the entire clay volume?

The most favorable interpretation I can make of multi-porosity approaches to bentonite modeling is a dynamically varying “macroporosity”, involving magical membranes (shown above). This, in itself, answers why I cannot take multi-porosity models seriously. And then we haven’t yet mentioned the flawed treatment of diffusive flux.

Footnotes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Semi-permeability, part I

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Descriptions in bentonite literature

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Chagneau et al. (2015) write

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

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

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

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

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

Bentonite as a non-ideal semi-permeable membrane

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

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

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

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

“Semi-permeability” in experiments

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

Tracer through-diffusion

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

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

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

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

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

Measuring hydraulic conductivity

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

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

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

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

Mass transfer in a salt gradient

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

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

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

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

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

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.