Category Archives: Sorption

Sorption part IV: What is K_d?

Measuring K_d

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

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

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

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

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

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

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

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

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

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

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

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

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

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

K_d in the homogeneous mixture model

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

K_d for simple cations

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

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

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

Simple cation tracers in a 1:1 system

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

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

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

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

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

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

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

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

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

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

For a divalent tracer we instead have

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

giving

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

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

Simple cation tracers in a 2:1 system

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

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

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

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

where

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

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

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

For a divalent tracer we instead have

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

giving

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

Density dependence of K_d

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

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

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

Footnotes

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

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

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

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

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

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

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

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

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

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

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

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

Extracting anion equilibrium concentrations from through-diffusion tests

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

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

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

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

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

The through-diffusion set-up

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

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

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

The effective porosity diffusion model

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

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

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

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

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

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

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

Connection with experimentally accessible quantities

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

where we have assumed zero target concentration.

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

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

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

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

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

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

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

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

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

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

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

Additional comments on the effective porosity model

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

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

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

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

Footnotes

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

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

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

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

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

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

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

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

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

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

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

Muurinen et al. (1985)

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

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

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

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

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

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

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

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

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

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

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

Eriksen et al. (1999)

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

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

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

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

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

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

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

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

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

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

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

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

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

Gimmi & Kosakowski (2011)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Some thoughts

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

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

Footnotes

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

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

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

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

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

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

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

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

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

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

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

Donnan equilibrium and the homogeneous mixture model

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

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

Overview

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

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

The problem to be solved can be illustrated like this

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

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

Chemical equilibrium

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

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

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

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

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

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

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

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

Illustrations

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

Relation beteween internal and external concentration for varying Xi

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

Variation of internal cation concentrations with varying Donnan factor

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

Variation of internal anion concentration with the Donnan factor

Equation for \(f_D\)

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

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

where

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

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

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

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

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

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

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

Relation between internal and external Na and Cl concentrations

Comment on through-diffusion

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

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

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

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

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

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

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

Additional comments

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

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

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

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

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

Footnotes

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

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

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

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

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

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

Sorption part III: Donnan equilibrium in compacted bentonite

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

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

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

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

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

Donnan equilibrium: effect vs. model

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

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

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

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

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

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

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

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

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

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

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

Donnan equilibrium in gels, soils and clays

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

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

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

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

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

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

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

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

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

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

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

Cation exchange is Donnan equilibration

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A demonstration of compacted bentonite as a Donnan system

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

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

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

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

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

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

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

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

Footnotes

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

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

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

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

[5] This article has an English translation.

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

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

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

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

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

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

Sorption part II: Letting go of the bulk water

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Diffusion in the homogeneous mixture model

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

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

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

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

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

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

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

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

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

Model comparison

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

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

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

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

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

From these descriptions it should be obvious that the homogeneous mixture model is the more reasonable one — it is both compatible with simple observations of the pore structure and mathematically considerably less complex as compared with the others.

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

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

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

Footnotes

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

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

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

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

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

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

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

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

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

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

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

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

where \(\omega\) is the relative filter resistance

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

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

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

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

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

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

i.e. independent of any clay parameters.

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

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

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

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

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

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

which approaches zero when \(\omega\) becomes large.

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

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

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

Sodium tracer profiles from Glaus et al. (2007)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Effective diffusivities for strontium in Molera and Eriksen (2002)

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

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

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

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

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

calcium outflux in Tinnacher et al. (2016)

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

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

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

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

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

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

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

Footnotes

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

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

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

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

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

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

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

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

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

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

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

Appendix: Derivation of eqs. 1 and 2

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

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

where \(R\) is the transfer resistance.

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

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

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

Solving for \(c_\mathrm{in}\) and \(c_\mathrm{out}\) gives

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

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

Defining the relative filter resistance as

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

we can express the concentration drop across the clay as

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

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

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

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

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

which is eq. 1 above.

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

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

which is eq. 2 above.

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

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

The traditional diffusion-sorption model

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

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

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

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

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

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

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

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

with

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

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

The immobilization assumption is not valid

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

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

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

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

where \(D_s\) is a so-called surface diffusion coefficient.

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

A correct “surface diffusion” model

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

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

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

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

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

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

and eq. 8 reduces to

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

Comparing with eq. 1 directly shows

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

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

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

A complete conceptual change

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

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

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

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

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

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

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

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

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

Footnotes

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

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

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

Measuring Kd

Kd in the homogeneous mixture model

Kd for simple cations

Simple cation tracers in a 1:1 system

Simple cation tracers in a 2:1 system

Density dependence of Kd

Footnotes

The through-diffusion set-up

The effective porosity diffusion model

Connection with experimentally accessible quantities

Additional comments on the effective porosity model

Footnotes

Muurinen et al. (1985)

Eriksen et al. (1999)

Gimmi & Kosakowski (2011)

Some thoughts

Footnotes

Overview

Chemical equilibrium

Illustrations

Equation for \(f_D\)

Comment on through-diffusion

Additional comments

Footnotes

Donnan equilibrium: effect vs. model

Donnan equilibrium in gels, soils and clays

Cation exchange is Donnan equilibration

A demonstration of compacted bentonite as a Donnan system

Footnotes

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

Diffusion in the homogeneous mixture model

Model comparison

Footnotes

Footnotes

Appendix: Derivation of eqs. 1 and 2

The traditional diffusion-sorption model

The immobilization assumption is not valid

A correct “surface diffusion” model

A complete conceptual change

Footnotes

Measuring K_d

K_d in the homogeneous mixture model

K_d for simple cations

Density dependence of K_d