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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):

Through diffusion schematics

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:

cesium diffusivivty vs. Kd, model and measurements. From Muurinen et al. (1985)

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!

Swelling pressure, part V: Suction

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There are several “descriptions” of bentonite swelling. While a few of them actually denies any significant role played by the exchangeable cations, most of these descriptions treat the exchangeable ions as part of an osmotic system. I have earlier discussed how the terms “osmotic” or “osmosis” may cause some confusion in different contexts, and discussed the confusion surrounding the treatment of electrostatic forces.

In this blog post I discuss the description of bentonite swelling often adopted in the fields of soil mechanics and geotechnical engineering. In particular, we focus on the concept of suction, which is central in these research fields, while being basically absent in others.

As far as I understand, suction is just the water chemical potential “disguised” as a pressure variable; although I have trouble finding clear-cut definitions, it seems clear that suction is directly inherited from the “water potential” concept, which has been central in soil science for a long time. Applied to bentonite, the geotechnical description is thus not principally different from the osmotic approach that I have presented previously. But the way the suction concept is (and isn’t) applied may cause unnecessary confusion regarding the swelling mechanisms. I think a root for this confusion is that suction involves both osmotic and capillary mechanisms.

Matric suction (capillary suction)

Matric suction is typically associated with capillarity, a fundamental mechanism in many conventional soil materials under so-called unsaturated conditions. A conventional soil with a significant amount of small enough pores shows capillary condensation, i.e. it contains liquid water below the condensation point for ordinary bulk water. Naturally, the equilibrium vapor pressure increases with the amount of water in the soil, as the pores containing liquid water become larger. For conventional soils, it therefore makes sense to speak of the degree of saturation of a sample, and to relate saturation and equilibrium vapor pressure by means of a water retention curve. Underlying this picture is the notion that the solid parts constitute a “soil skeleton” (the matrix), and that the soil can be viewed as a vessel that can be more or less filled with water.

The pressure of the capillary water is lower than that of the surrounding air, and is related to the curvature of the interfaces between the two phases (menisci), as expressed by the Young-Laplace equation. For a spherically symmetric meniscus this equation reads

\begin{equation} \Delta p = p_w – p_a = \frac{2\sigma}{r} \tag{1} \end{equation}

where \(p_a\) and \(p_w\) denote the pressures of air and capillary water, respectively, \(\sigma\) is the surface tension, and \(r\) is the radius of curvature of the interface. The sign of \(r\) depends on whether the interface bulges inwards (“concave”, \(r<0\)) or outwards (“convex”, \(r>0\)). For capillary water, \(r\) is negative and \(\Delta p\) — which is also called the Laplace pressure — is a negative quantity.

As far as I understand, matric suction is simply defined as the negative Laplace pressure, i.e.

\begin{equation} s_m = p_a – p_w \tag{2} \end{equation}

With this definition, suction has a straightforward physical meaning as quantifying the difference in pressure of the two fluids occupying the pore space, and clearly relates to the everyday use of the word.

Suction — in this capillary sense — gives a simple principal explanation for (apparent) cohesion in e.g. unsaturated sand: individual grains are pushed together by the air-water pressure difference, as schematically illustrated here (the yellow stuff is supposed to be two grains of sand, and the blue stuff water)

Net force for two sand grains exposed to matric suction

It is reasonable to assume that the net force transmitted by the soil skeleton — usually quantified using the concept of effective stress — governs several mechanical properties of the soil sample, e.g. shear strength. The above description also makes it reasonable to assume that effective stress depends on suction.

Thus, in unsaturated conventional soil are quantities like degree of saturation, pore size distribution, (matric) suction, effective stress, and shear strength very much associated. Another way of saying this is that there is an optimal combination of water content and particle size distribution for constructing the perfect sand castle.

The chemical potential of the capillary water is related to matric suction. Choosing pure bulk water under pressure \(p_a\)1 as reference, the chemical potential of the liquid phase in the soil is obtained by integrating the Gibbs-Duhem equation from \(p_a\) to \(p_w\)

\begin{equation} \mu = \mu_0 + \int_{p_a}^{p_w}v dP = \mu_0 + v\cdot \left(p_w – p_a \right) = \mu_0 -v\cdot s_m \tag{3} \end{equation}

where \(\mu_0\) is the reference chemical potential, \(v\) is the molar volume of water, and we have assumed incompressibility.

The above expression shows that matric suction in this case directly quantifies the (relative) water chemical potential. Note, however, that eq. 3 does not define matric suction; \(s_m\) is defined as a pressure difference between two phases (eq. 2), and happens to quantify the chemical potential under the present circumstances (pure capillary water).

A chemical potential can generally be expressed in terms of activity (\(a\))

\begin{equation} \mu = \mu_0 + RT \ln a \tag{4} \end{equation}

For our case, water activity is to a very good approximation equal to relative humidity, the ratio between the vapor pressures in the state under consideration and in the reference state, i.e. \(a = p_v/p_{v,0}\). Combining eqs. 3 and 4, we see that the vapor pressure in this case is related to matric suction as

\begin{equation} \frac{p_v}{p_{v,0}} = e^{-v\cdot s_m/RT} \end{equation}

Using the Young-Laplace equation (eq. 1) for \(s_m\) we can also write this as

\begin{equation} \frac{p_v}{p_{v,0}} = e^{\frac{2v\sigma}{RTr}} \end{equation}

This is the so-called Kelvin equation, which relates the equilibrium vapor pressure to the curvature of an air-pure water interface. Note that, since \(r<0\) for capillary water, the vapor pressure is lower than the corresponding bulk value (\(p_v < p_{v,0}\)).

Osmotic suction and total suction

So far, we have discussed suction in a capillary context, and related it to water chemical potential or vapor pressure. Now consider how the picture changes if the pores in our conventional soil contain saline water. Matric suction — i.e. the actual pressure difference between the pore solution and the surrounding air, sticking with eq. 2 as the definition — is in general different from the pure water case, because solutes influence surface tension. Also, water activity (vapor pressure) is different from the pure water case, but there is no longer a direct relation between water activity and matric suction, because water activity is independently altered by the presence of solutes.

The water chemical potential of a saline bulk solution (i.e. with no capillary effects), can be written in terms of the osmotic pressure, \(\pi(c)\)

\begin{equation} \mu(c) = \mu_0 – v\cdot\pi(c) \tag{5} \end{equation}

where we have assumed a salt concentration \(c\), and indicated that the osmotic pressure, and hence the chemical potential, depends on this concentration.

Although eq. 5 is of the same form as eq. 3, matric suction and osmotic pressure are very different quantities. The former is defined under circumstances where an actual pressure difference prevail between the air and water phases. In contrast, there is no pressure difference between the phases in a container containing both a solution and a gas phase. \(\pi(c)\) corresponds to the elevated pressure that must be applied for the solution to be in equilibrium with pure water kept at the reference pressure.

Despite the different natures of matric suction and osmotic pressure, the fields of geotechnical engineering and soil mechanics insist on also referring to \(\pi(c)\) as a suction variable: the osmotic suction. Similarly, total suction is defined as the sum of matric and osmotic suction

\begin{equation} \Psi = s_m + \pi(c) \end{equation}

These definitions seem to have no other purpose than to be able to write the water chemical potential generally as

\begin{equation} \mu = \mu_0 -v\cdot \Psi \tag{6} \end{equation}

Total suction is thus de facto defined simply as the (relative) value of the water chemical potential, expressed as a pressure (I think this is completely analogous to “total water potential” in soil science).

Eq. 6 shows that \(\Psi\) is directly related to water activity, or vapor pressure, and we can write

\begin{equation} \frac{p_v}{p_{v,0}} = e^{-v\cdot \Psi/RT} \tag{7} \end{equation}

This relation is quite often erroneously referred to as the Kelvin equation (or “Kelvin’s law”) in the bentonite literature. But note that the above equation just restates the definition of water activity, because \(v\cdot\Psi\) cannot be reduced to anything more concrete than the relative value of the water chemical potential. The Kelvin equation, on the other hand, expresses something more concrete: the equilibrium vapor pressure for a curved air-water interface. Some clay literature refer to the above relation as the “Psychrometric law”, but that name seems not established in other fields.2

A definition is motivated by its usefulness, and total change in water chemical potential is of course central when considering e.g. moisture movement in soil. My non-geotechnical brain, however, is not fond of extending the “suction” variable in the way outlined above. To start with, there is already a variable to use: the water chemical potential. Also, “total suction” no longer has the direct relation to the everyday use of the word suction: there is no “sucking” going on in a saline bulk solution,3 while in a capillary there is. Furthermore, with a saline pore solution there is no direct relation between (total) suction and e.g. effective stress or shear strength.

Although both matric suction and osmotic pressure under certain circumstances can be measured in a direct way, it seems that (total) suction usually is quantified by measuring/controlling the vapor pressure with which the soil sample is in equilibrium. Actually, one of the more comprehensive definitions of various “suctions” that I have been able to find — in Fredlund et al. (2012) — speaks only of various vapor pressures (although based on the capillary and osmotic concepts):4

Matric or capillary component of free energy: Matric suction is the equivalent suction derived from the measurement of the partial pressure of the water vapor in equilibrium with the soil-water relative to the partial pressure of the water vapor in equilibrium with a solution identical in composition with the soil-water.

Osmotic (or solute) component of free energy: Osmotic suction is the equivalent suction derived from the measurement of the partial pressure of the water vapor in equilibrium with a solution identical in composition with the soil-water relative to the partial pressure of water vapor in equilibrium with free pure water.

Total suction or free energy of soil-water: Total suction is the equivalent suction derived from the measurement of the partial pressure of the water vapor in equilibrium with the soil-water relative to the partial pressure of water vapor in equilibrium with free pure water.

It seems that such operational definitions of suction has made the term synonymous with “vapor pressure depression” in large parts of the bentonite scientific literature.

Suction in bentonite

In the above discussion we had mainly a conventional soil in mind. When applying the suction concepts to bentonite,5 I think there are a few additional pitfalls/sources for confusion. Firstly, note that the definitions discussed previously involve “a solution identical in composition with the soil-water”. But soil-water that contains appreciable amounts of exchangeable ions — as is the case for bentonite — cannot be realized as an external solution.

It seems that this “complication” is treated by assuming that an external solution in equilibrium with a bentonite sample is the soil-water (this is analogous to how many geochemists use the term “porewater” in bentonite contexts). Not surprisingly, this treatment has bizarre consequences. The conclusion for e.g. a salt free bentonite sample — which is in equilibrium with pure water — is that it lacks osmotic suction, and that its lowered vapor pressure (when isolated and unloaded) is completely due to matric suction! I think this is such an odd outcome that it is worth repeating: A system dominated by interlayer pores, containing dissolved cations at very high concentrations, is described as lacking osmotic pressure! It is not uncommon to find descriptions like this one (from Lang et al. (2019))

The total suction of unsaturated soils consists of matric and osmotic suctions (Yong and Warkentin, 1975; Fredlund et al., 2012; Lu and Likos, 2004). In clays, the matric suction is due to surface tension, adsorptive forces and osmotic forces (i.e. the diffuse double layer forces), whereas the osmotic suction is due to the presence of dissolved solutes in the pore water.

We apparently live in a world where “matric suction” consists of “osmotic forces”, while the same “osmotic forces” do not contribute to “osmotic suction”. Except when the clay contains excess ions, in which case we have an arbitrary combination of the two “suctions” (note also that “osmotic suction” and “osmotic swelling” are two quite different things).

Although the above consequence is odd, it is still only a matter of definition: accepting that “matric suction” involves osmotic forces (which I don’t recommend), the description may still be adequate in principle; after all, “total suction” quantifies the reduction of the water chemical potential.

But the focus on “matric suction” also reveals a conceptual view of bentonite structure that I find problematic: it suggests a first order approximation of bentonite as a conventional soil, i.e. as an assemblage of solid grains separated from an aqueous phase (and a gas phase). This “matric” view is fully in line with the idea of “free water” in bentonite, and it is quite clear that this is a prevailing view in the geotechnical, as well as in the geochemical, literature. For instance, with the formulation “the presence of dissolved solutes in the pore water” in the above quotation, the “pore water” the authors have in mind is a charge neutral bulk water solution.

With the “matric” conceptual view, the degree of saturation becomes a central variable in much soil mechanical analyses of bentonite. When dealing with actual unsaturated bentonite samples, I guess this makes sense, but once a sample is saturated this variable has lost much of its meaning.6 Consider e.g. the different expected behaviors if drying e.g. a water saturated metal filter or a saturated bentonite sample.

The different nature of drying a metal filter compared with drying a saturated bentonite sample

The equilibrium vapor pressure of both these systems is lower than the corresponding pure bulk water value. For the metal filter, the lowered water activity is of course due to capillarity, i.e. there is an actual pressure reduction in the water phase (matric suction!). When lowering the external vapor pressure below the equilibrium point (i.e. drying), capillary water migrates out of the filter, while the metal structure itself remains intact. In this case, as the system remains defined in a reasonable way, it is motivated to speak of the saturation state of the filter.

For a drying bentonite sample, the behavior is not as well defined, and depends on how the drying is performed and on initial water content. For a quasi-static process, where the external vapor pressure is lowered in small steps at an arbitrary slow rate, it should be clear that the entire sample will respond simply by shrinking. In this case it does not make much sense to speak of the sample as still being saturated, nor to speak of it as having become unsaturated.

For a more “violent” drying process, e.g. placing the bentonite sample in an oven at 105 °C , it is also clear that — rather than resulting in a neatly shrunken, dense piece of clay — the sample now will suffer from macroscopic cracks and other deformations. Neither in this case does it make much sense to try to define the degree of saturation, in relation to the sample initially put in the oven.

Note also that if we, instead of drying, increase the external vapor pressure from the initial equilibrium value, the metal filter will not respond much at all, while the bentonite sample immediately will begin to swell.

I hope that this example has made it clear, not only that the degree of saturation is in general ill-defined for bentonite, but also that a bentonite sample behaves more as an aqueous solution rather than as a conventional soil: if we alter external vapor pressure, an aqueous solution responds by either “swelling” (taking up water) or “shrinking” (giving off water). A main aspect of this conceptual view of bentonite — which we may call the “osmotic” view — is that water does not form a separate phase7. This was pointed out e.g. by Bolt and Miller (1958) (referring to this type of system as an “ideal clay-water system”)

In contrast to the familiar case described is the ideal clay-water system in which the particles are not in direct contact but are separated by layers of water. Removal of water from such a system does not introduce a third phase but merely causes the particles to move closer to one another with the pores remaining water saturated.

From these considerations it follows that a generally consistent treatment is to relate bentonite water activity to water content, rather than to degree of saturation.

Another consequence of adopting a “matric” view of bentonite (i.e. to include osmotic forces in “matric suction”) is that “matric suction” loses its direct connection with effective stress. This can be illustrated by taking the “osmotic” view: just as the mechanical properties of an aqueous solution (e.g. viscosity) do not depend critically on whether or not it is under (osmotic) pressure, we should not expect e.g. bentonite shear strength to be directly related to swelling pressure.8

Footnotes

[1] Often, the air is at atmospheric pressure, in which case the reference is the ordinary standard state.

[2] The relatively common misspelling “Psychometric law” is kind of funny.

[3] The cautious reader may remark that saline solutions do “suck”, in terms of osmosis. But note the following: 1) Osmosis requires a semi-permeable membrane, separating the solution from an external water source. We have said nothing about the presence of such a component in the present discussion. The way osmotic suction sometimes is described in the literature makes me suspect that some authors are under the impression that the mere presence of a solute causes a pressure reduction in the liquid. 2) In the presence of a semi-permeable membrane, osmosis has no problem occurring without a pressure difference between between the two compartments. 3) For cases when the solution is acted on by an increased hydrostatic pressure, water is transported from lower to higher pressure. It is difficult to say that there is any “sucking” in such a process (I would argue that the establishment of a pressure difference is an effect, rather than a cause, in the case of osmosis) 4) The idea that a solution has a well-defined partial water pressure is wrong.

[4] I’m still not fully satisfied with this definition: It may be noted that the definitions are somewhat circular (“matric suction is the equivalent suction…”), so they still require that we have in mind that “suction” also is defined in terms of a certain vapor pressure ratio (e.g. eq. 7). Note also that the headings speak of “free energy”. Perhaps I am nitpicking, but (free) energy is an extensive quantity, while suction (pressure) is intensive. Thus, “free energy” here really mean “specific free energy” (or “partial free energy”, i.e. chemical potential). I think the soil science literature in general is quite sloppy with making this distinction.

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

[6] If you press bentonite granules to form a cohesive sample you certainly end up with a system having both water filled interlayer pores and air-filled macropores (or perhaps an even more complex pore structure). This blog post mainly concerns saturated bentonite, by which I mean bentonite material which does not contain any gas phase. We can thus speak of saturated bentonite, although a degree of saturation variable is not well defined.

[7] Rather, montmorillonite and water form a homogeneous mixture.

[8] However, bentonite strength relates indirectly to swelling pressure (under specific conditions) because both quantities depends on a third: density.