Category Archives: Sorption

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)

Summary models

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
\(D_\mathrm{e}^\mathrm{clay}\)(\(10^{-10}\;\mathrm{m^2/s}\))143.70.860.530.38
\(D_\mathrm{e}^\mathrm{filter}\)(\(10^{-10}\;\mathrm{m^2/s}\))0.861.00.860.860.86
Calculated
\(\omega\)(-)9.32.10.60.40.3

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
\(K_d\)(\(10^{-3}\) \(\mathrm{m^3/kg}\))467.31.81.20.74
Calculated
CEC(eq/kg)0.460.730.900.840.74

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)88.538.512.29.37.8
\(s_{out}\)(\(10^{-12}\) mol/kg)76.317.41.41.1\(\sim 0\)
\(c_{source}\)(\(10^{-9}\) \(\mathrm{mol/m^3}\))3.110.58.17.79.4
Assumed
\(K_d\)(\(10^{-3}\) \(\mathrm{m^3/kg}\))888.81.761.260.88
Calculated
\(c_{in}\)(\(10^{-9}\) \(\mathrm{mol/m^3}\))1.04.46.97.48.9
\(c_{out}\)(\(10^{-9}\) \(\mathrm{mol/m^3}\))0.92.00.80.9\(\sim 0\)
\(\omega\)(-)21.363.370.310.19\(\sim 0\)

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

concentration profile across fileters, assuming symmetric or assymetric configurations

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 \(D_\mathrm{e}^\mathrm{clay} = D_\mathrm{a}\left(\phi + \rho K_d\right)\))

ReportedCalculated
Density\(C_\mathrm{bkg}\)\(D_\mathrm{a}\)\(K_d\)\(D_\mathrm{e}^\mathrm{clay}\)\(\omega\)
(\(\mathrm{kg/m^3}\))(\(\mathrm{mol/m^3}\))(\(10^{-10}\;\mathrm{m^2/s}\))(\(10^{-3}\;\mathrm{m^3/kg}\))(\(10^{-10}\;\mathrm{m^2/s}\))(-)
4001000.4311019.36.8
8001000.3515042.214.8
8005000.40155.11.8
1200100.2150012644.2
12001000.1813028.29.9
12005000.25134.01.4
1600100.14120026994.2
16001000.109014.45.1
16005000.20154.91.7
18001000.098013.04.6
18005000.12153.31.1

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

  1. 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).
  2. Ions in the porewater “sorb” on solid surfaces and become immobilized. The amount of “sorbed” ions per unit solid mass, \(s\), is related to \(c\) via a distribution coefficient \(K_d\) \begin{equation} s = K_d\cdot c \tag{2} \end{equation}

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

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

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

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

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

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

with

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

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

The immobilization assumption is not valid

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

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

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

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

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

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

A correct “surface diffusion” model

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

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

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

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

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

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

and eq. 8 reduces to

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

Comparing with eq. 1 directly shows

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

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

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

A complete conceptual change

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

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

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

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

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

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

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

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

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

Footnotes

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

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

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