Category Archives: Diffusion

Assessment of chloride equilibrium concentrations: Muurinen et al. (1988)

In the ongoing assessment of chloride equilibrium concentrations in bentonite, we here take a closer look at the study by Muurinen et al. (1988), in the following referred to as Mu88.¹ In the quite messy plot containing all reported chloride equilibrium concentrations, we thus investigate the twelve points indicated here

Mu88 points highlighted in plot with all chloride equilibrium data

Mu88 performed both chloride and uranium through-diffusion tests on “MX-80” bentonite, as well as sorption tests. Here we focus solely on the chloride diffusion. We also disregard one diffusion test series that does not vary external concentration (it was conducted with an unspecified “artificial groundwater” and varied sample density).

Left are two test series performed with nominal sample densities 1.2 g/cm³ and 1.8 g/cm³, respectively. For each of these densities, chloride through-diffusion tests were performed with external NaCl concentrations of 0.01 M, 0.1 M, and 1.0 M, respectively. The samples were cylindrical with a diameter of 3.0 cm, and a length of 1.5 cm, giving a volume of 10.6 cm³. To refer to a specific test or sample, we use the nomenclature “nominal density/external concentration”, e.g. the test performed at nominal density 1.2 g/cm³ and external solution 0.1 M is referred to as “1.2/0.1”.

Uncertainty of bentonite samples

“MX-80” is not the name of some specific standardized material, but simply a product name.² It is quite peculiar that that “MX-80” nevertheless is a de facto standard in the research field for clay buffers in radwaste repositories. But, being a de facto standard, several batches of bentonite with this name have been investigated and reported throughout the years. We consequently have some appreciation for its constitution, and the associated variation.

In Mu88, the material used is only mentioned by name, and it is only mentioned once (in the abstract!). We therefore can’t tell which of the studies that is more appropriate to refer to. Instead, let’s take a look at how “MX-80” has been reported generally.

Report	Batch year	Mmt content	CEC	Na-content
		(%)	(eq/kg)	(%)
TR-06-30 (“WySt”)	1980	82.5	0.76	83
NTR 83-12	< 1983	75.5	0.76	86
TR-06-30 (“WyL1”)	1995	79.5	0.77	–
TR-06-30 (“WyL2”)	1999	79.8	0.75	71
TR-06-30 (“WyR1”)	2001	82.7	0.75	75
TR-06-30 (“WyR2”)	2001	80.0	0.71	71
NTB 01-08	< 2002	–	0.79^*	85
WR 2004-02³	< 2004	80 — 85	0.84^*	65

*) These values were derived from summing the exchangeable ions, and are probably overestimations.

Montmorillonite content

Reported montmorillonite content varies in the range 75 — 85%. For the present context, this primarily gives an uncertainty in adopted effective montmorillonite dry density, which, in turn, is important for making relevant comparison between bentonite materials with different montmorillonite content. For the “MX-80” used in Mu88 we here assume a montmorillonite content of 80%. In the table below is listed the corresponding effective montmorillonite densities when varying the montmorillonite content in the range $x =$ 0.75 — 0.85, for the two nominal dry densities.

Dry density	EMDD ($x$=0.75)	EMDD ($x$=0.80)	EMDD ($x$=0.85)
(g/cm³)	(g/cm³)	(g/cm³)	(g/cm³)
1.2	1.01	1.05	1.09
1.8	1.61	1.66	1.70

The uncertainty in montmorillonite content thus translates to an uncertainty in effective montmorillonite dry density on the order of 0.1 g/cm³.

Cation population

While reported values of the cation exchange capacity of “MX-80” are relatively constant, of around 0.75 eq/kg,⁴ the reported fraction of sodium ions is seen to vary, in the range 70 — 85 %. The remaining population is mainly di-valent rare-earth metal ions (calcium and magnesium). This does not only mean that different studies on “MX-80” may give results for quite different types of systems, as the mono- to di-valent ion ratio may vary, but also that samples within the study may represent quite different systems. We examine this uncertainty below, when discussing the external solutions.

Soluble calcium minerals

The uncertainty of how much divalent cations are available is in fact larger than just discussed. “MX-80” is reported to contain a certain amount of soluble calcium minerals, in particular gypsum. These provide additional sources for divalent ions, which certainly will be involved in the chemical equilibration as the samples are water saturated. Reported values of gypsum content in “MX-80” are on the order of 1%. With a molar mass of 0.172 kg/mol, this contributes to the calcium content by $2\cdot 0.01/0.172$ eq/kg $\approx 0.12$ eq/kg, or about 16% of the cation exchange capacity.

Sample density

The samples in Mu88 that we focus on have nominal dry density of 1.2 and 1.8 g/cm³. The paper also reports measured porosities on each individual sample, listed in the below table together with corresponding values of dry density⁵

Test	$\phi$	$\rho_d$
	(-)	(g/cm³)
1.2/0.01	0.54	1.27
1.2/0.1	0.52	1.32
1.2/1.0	0.49	1.40
1.8/0.01	0.37	1.73
1.8/0.1	0.31	1.89
1.8/1.0	0.34	1.81

We note a substantial variation in measured density for samples with the same nominal density: for the 1.2 g/cm³ samples, the standard deviation is 0.06 g/cm³, and for the 1.8 g/cm³ samples it is 0.07 g/cm³. Moreover, while the mean value for the 1.8 g/cm³ samples is close to the nominal value (1.81 g/cm³), that for the 1.2 g/cm³ samples is substantially higher (1.33 g/cm³).

It is impossible to know from the information provided in Mu88 if this uncertainty is intrinsic to the procedure of preparing the samples, or if it is more related to the procedure of measuring the density at test termination.⁶

Uncertainty of external solutions

Mu88 do not describe how the external solutions were prepared. We assume here, however, that preparing pure NaCl solutions gives no significant uncertainty.

Further, the paper contains no information on how the samples were water saturated, nor on the external solution volumes. Since samples with an appreciable amount of di-valent cations are contacted with pure sodium solutions, it is unavoidable that an ion exchange process is initiated. As we don’t know any detail of the preparation process, this introduces an uncertainty of the exact aqueous chemistry during the course of a test.

To illustrate this problem, here are the results from calculating the exchange equilibrium between a sample initially containing 30% exchangeable charge in form of calcium (70% sodium), and external NaCl solutions of various concentrations and volumes

calcium remaining in the bentonite as a function of inital external NaCl concentration for various volumes

In these calculations we assume a sample of density 1.8 g/cm³ with the same volume as in Mu88 (10.6 cm³), a cation exchange capacity of 0.75 eq/kg, and a Ca/Na selectivity coefficient of 5.

In a main series, we varied the external volume between 50 and 1000 ml (solid lines). While the solution volume naturally has a significant influence on the process, it is seen that the initial calcium content essentially remain for the lowest concentration (0.01 M). In contrast, for a 1.0 M solution, a significant amount of calcium is exchanged for all the solution volumes.

The figure also shows a case for sample density 1.2 g/cm³ (dashed line), and a scenario where equilibrium has been obtained twice, with a replacement of the first solution (to a once again pure NaCl solution) (dot-dashed line).

The main lesson from these simulations is that the actual amount of di-valent ions present during a diffusion test depends on many details: the way samples were saturated, volume of external solutions, if and how often solutions were replaced, time, etc. It is therefore impossible to state the exact ion population in any of the tests in Mu88. But, guided by the simulations, it seems very probable that the tests performed at 0.01 M contain a substantial amount of di-valent ions, while those performed at 1.0 M probably resemble more pure sodium systems.

The only information on external solutions in Mu88 is that the “solution on the low concentration side was changed regularly” during the course of a test. This implies that the amount of di-valent cations may not even be constant during the tests.

Uncertainty of diffusion parameters

The diffusion parameters explicitly listed in Mu88 are $D_e$ and “$D_a$”, while it is implicitly understood that they have been obtained by fitting the effective porosity model to outflux data and the measured clay concentration profile in the final state. “$D_a$” is thus really the pore diffusivity $D_p$,⁷ and relates to $D_e$ as $D_e = \epsilon_\mathrm{eff} D_p$, where $\epsilon_\mathrm{eff}$ is the so-called “effective porosity”. In a previous blog post, we discussed in detail how anion equilibrium concentrations can be extracted from through-diffusion tests, and the results derived there is used extensively in this section.

Rather than fitting the model to the full set of data (i.e. outflux evolution and final state concentration profile), diffusion parameters in Mu88 have been extracted in various limits.

Evaluation of $D_e$ in Mu88

The effective diffusivity was obtained by estimating the steady-state flux, dividing by external concentration difference of the tracer, and multiplying by sample length \begin{equation} D_e = \frac{j^\mathrm{ss}\cdot L}{c^\mathrm{source}}\tag{1} \end{equation}

Here it is assumed that the target reservoir tracer concentration can be neglected (we assume this throughout). Eq. 1 is basically eq. 1 in Mu88 (and eq. 8 in the earlier blog post), from which we can evaluate the values of the steady-state flux that was used for the reported values of $D_e$ ($A \approx 7.1$ cm² denotes sample cross sectional area)

Test	$D_e$	$A\cdot j^\mathrm{ss}/c^\mathrm{source}$
	($\mathrm{m^2/s}$)	(ml/day)
1.2/0.01	$7.7\cdot 10^{-12}$	0.031
1.2/0.1	$2.9\cdot 10^{-11}$	0.118
1.2/1.0	$1.2\cdot 10^{-10}$	0.489
1.8/0.01	$3.3\cdot 10^{-13}$	0.001
1.8/0.1	$4.8\cdot 10^{-13}$	0.002
1.8/1.0	$4.0\cdot 10^{-12}$	0.016

The figure below compares the evaluated values of the steady-state flux with the flux evaluated from the measured target concentration evolution,⁸ for samples with nominal dry density 1.8 g/cm³ (no concentration data was reported for the 1.2 g/cm³ samples)

outflux vs. time for 1.8 g/cm3 samples in Muurinen et al. (1988)

These plots clearly show that the transition to steady-state is only resolved properly for the test with highest background concentration (1.0 M). It follows that the uncertainty of the evaluated steady-state — and, consequently, of the evaluated $D_e$ values — increases dramatically with decreasing background concentration for these samples.

Evaluation of $D_p$ in Mu88

Pore diffusivities were obtained in two different ways. One method was to relate the steady-state flux to the clay concentration profile at the end of the test, giving \begin{equation} D_{p,c} = \frac{j^\mathrm{ss}\cdot L}{\phi\cdot\bar{c}(0)} \tag{2} \end{equation}

where $\bar{c}(0)$ denotes the chloride clay concentration at the interface to the source reservoir. The quantity in eq. 2 is called “$D_{ac}$”⁷ in Mu88, and this equation is essentially the same as eq. 2 in Mu88⁹ (and eq. 10 in the previous blog post). Using the steady-state fluxes, we can back-calculate the values of $\bar{c}(0)$ used for this evaluation of $D_{p,c}$

Test	$D_{p,c}$	$A\cdot j^\mathrm{ss}/c^\mathrm{source}$	$\phi$	$\bar{c}(0)/c^\mathrm{source}$
	($\mathrm{m^2/s}$)	(ml/day)	(-)	(-)
1.2/0.01	$7.0\cdot 10^{-11}$	0.031	0.54	0.204
1.2/0.1	$2.8\cdot 10^{-10}$	0.118	0.52	0.199
1.2/1.0	$5.1\cdot 10^{-10}$	0.489	0.49	0.480
1.8/0.01	$2.0\cdot 10^{-11}$	0.001	0.37	0.045
1.8/0.1	$3.1\cdot 10^{-11}$	0.002	0.31	0.050
1.8/1.0	$5.2\cdot 10^{-11}$	0.016	0.34	0.226

Note that, although we did some calculations to obtain them, the values for $\bar{c}(0)/c^\mathrm{source}$ in this table are closer to the actual measured raw data (concentrations). We made the calculation above to “de-derive” these values from the reported diffusion coefficients (combining eqs. 1 and 2 shows that $\bar{c}(0)$ is obtained from the reported parameters as $\bar{c}(0)/c^\mathrm{source} = D_e/(\phi D_{p,c})$).

Here are compared the measured concentration profiles for the samples of nominal density 1.8 g/cm³ and the corresponding slopes used to evaluate $D_{p,c}$ (profiles for the 1.2 g/cm³ samples are not provided in Mu88)

Final state concentration profiles for 1.8 g/cm3 samples in Muurinen et al. (1988)

For background concentrations 1.0 M and 0.1 M, the evaluated slope corresponds quite well to the raw data. For the 0.01 M sample, however, the match not very satisfactory. I suspect that a detection limit may have been reached for the analysis of the profile of this sample. Needless to say, the evaluated value of $\bar{c}(0)$ is very uncertain for the 0.01 M sample.

It may also be noted that all measured concentration profiles deviates from linearity near the interface to the source reservoir. This is a general behavior in through-diffusion tests, which I am quite convinced of is related to sample swelling during dismantling, but there are also other suggested explanations. Here we neglect this effect and relate diffusion quantities to the linear parts of profiles, but this issue should certainly be treated in a separate discussion. Update (220407): non-linear profiles are discussed here.

$D_p$ was also evaluated in a different way in Mu88, by measuring what we here will call the breakthrough time, $t_\mathrm{bt}$ (Mu88 call it “time-lag”). This quantity is fairly abstract, and relates to the asymptotic behavior of the analytical expression for the outflux that apply for constant boundary concentrations (we here assume them to be $c^\mathrm{source}$ and 0, respectively). This expression is displayed in eq. 7 in the previous blog post.

Multiplying the outflux by the sample cross sectional area $A$ and integrating, gives the accumulated amount of diffused tracers. In the limit of long times, this quantity is, not surprisingly, linear in $t$ \begin{equation} A\cdot j^\mathrm{ss} \cdot \left(t – \frac{L^2}{6\cdot D_p} \right ) \end{equation}

$t_\mathrm{bt}$ is defined as the time for which this asymptotic expression is zero. Determining $t_\mathrm{bt}$ from the measured outflux evolution consequently allows for an estimation of $D_p$ as \begin{equation} D_{p,t} = \frac{L^2}{6t_\mathrm{bt}} \tag{3} \end{equation}

This quantity is called “$D_{at}$” in Mu88⁷ (eq. 3 is eq. 3 in Mu88). With another back calculation we can extract the values of $t_\mathrm{bt}$ determined from the raw data

Test	$D_{p,t}$	$t_\mathrm{bt}$
	($\mathrm{m^2/s}$)	(days)
1.2/0.01	$1.4\cdot 10^{-10}$	3.1
1.2/0.1	$2.0\cdot 10^{-10}$	2.2
1.2/1.0	$3.2\cdot 10^{-10}$	1.4
1.8/0.01	$5.0\cdot 10^{-11}$	8.7
1.8/0.1	$5.4\cdot 10^{-11}$	8.0
1.8/1.0	$7.7\cdot 10^{-11}$	5.6

These evaluated breakthrough times are indicated in the flux plots above for samples of nominal dry density 1.8 g/cm³. For the 0.1 M and 0.01 M samples it is obvious that this value is very uncertain — without a certain steady-state flux it is impossible to achieve a certain breakthrough time. The breakthrough time for the 1.8/1.0 test, on the other hand, simply appears to be incorrectly evaluated: in terms of outflux vs. time, the breakthrough time should be the time where the flux has reached 62% of the steady-state value.¹⁰

As no raw data is reported for the 1.2 g/cm³ tests, the quality of the evaluated breakthrough times cannot be checked for them. It may be noted, however, that the evaluated breakthrough times are significantly shorter in this case as compared with the 1.8 g/cm³ tests. Consequently, while the sampling frequency is high enough to properly resolve the transient stage of the outflux evolution for the 1.8g/cm³ tests, it must be substantially higher in order to resolve this stage in the 1.2g/cm³ tests (I guess a rule of thumb is that sampling frequency must be at least higher than $1/t_{bt}$).

With the pore diffusivities evaluated from $t_\mathrm{bt}$ we get a second estimation of $\bar{c}(0)/c^\mathrm{source}$, using eq. 2. These values are listed in the table below and compared with the direct evaluation from the steady-state concentration profiles.

Test	$\bar{c}(0)/c^\mathrm{source}$	$\bar{c}(0)/c^\mathrm{source}$
	(breakthrough)	(profile)
1.2/0.01	0.102	0.204
1.2/0.1	0.279	0.199
1.2/1.0	0.765	0.480
1.8/0.01	0.018	0.045
1.8/0.1	0.029	0.050
1.8/1.0	0.153	0.226

In a well conducted study these estimates should be similar; $D_{p,c}$ and $D_{p,t}$ are, after all, estimations of the same quantity: the pore diffusivity $D_p$.⁷ But here we note a discrepancy of approximately a factor 2 between several values of $\bar{c}(0)$.

It is difficult to judge generally which of the estimations are more accurate, but we have seen that for the 1.8/0.1 and 1.8/0.01 tests, the flux data is not very well resolved, giving a corresponding uncertainty on the equilibrium concentration estimated from the breakthrough time. On the other hand, also the concentration profile is poorly resolved in the case of 0.01 M at 1.8 g/cm³.

However, in cases where the value of $\bar{c}(0)/c^\mathrm{source}$ is substantial (as for the 1.8/1.0 test and, reasonably, for all tests at 1.2 g/cm³), we expect the estimation directly from the concentration profile to be accurate and robust (as for the 1.8 g/cm³ test at high NaCl concentration). For the 1.2 g/cm³ samples we cannot say much more than this, since Mu88 don’t provide the concentration raw data. For the 1.8/1.0 test, however, we can continue the analysis by fitting the model to all available data.

Re-evaluation by fitting to the full data set

Note that all evaluations in Mu88 are based on making an initial estimation of the steady-state flux, giving $D_e$ (eq. 1). This value of $D_e$ (or $j^{ss}$) is thereafter fixed in the subsequent estimation of $D_{p,c}$ (eq. 2). Likewise, an estimation of the steady-state flux is required for estimating the breakthrough time. Here is an animation showing the variation of the model when transitioning from the value of the pore diffusivity estimated from breakthrough time ($7.7\cdot 10^{-11}$ m²/s), to the value estimated from concentration profile ($5.2\cdot 10^{-11}$ m²/s) for the 1.8/1.0 test, keeping the steady-state flux fixed at the initial estimation

Note that the axes for the flux is on top (time) and to the right (accumulation rate). This animation confirms that the diffusivity evaluated from breakthrough time in Mu88 gives a way too fast process: the slope of the steady-state concentration profile is too small, and the outflux evolution has a too short transient stage. On the other hand, using the diffusivity estimated from the concentration profiles still doesn’t give a flux that fit very well. The problem is that this fitting is performed with a fixed value of the steady-state flux. By instead keeping the slope fixed at the experimental values, while varying diffusivity (and thus steady state flux), we get the following variation

This animation shows that the model can be fitted well to all data (at least for the 1.8/1.0 test). The problem with the evaluation in Mu88 is that it assumes the steady-state to be fully reached at the later stages of the test. As the above fitting procedure shows, this is only barely true. The experiments could thus have been designed better by conducting them longer, in order to better sample the steady-state phase (and the steady-state flux should have been fitted to the entire data set). Nevertheless, for this sample, the steady-state flux obtained by allowing for this parameter to vary is only slightly different from that used in Mu88 (17.5 rather than 16.3 $\mathrm{\mu}$l/day, corresponding to a change of $D_p$ from $5.2\cdot10^{-11}$ to $5.6\cdot10^{-11}$ m²/s). Moreover, this consideration should not be a problem for the 1.2 g/cm³ tests, if they were conducted for as long time as the 1.8 g/cm³ tests, because steady-state is reached much faster (in those tests, sampling frequency may instead be a problem, as discussed above).

As we were able to fit the full model to all data, we conclude that the value of $\bar{c}(0)/c^\mathrm{source}$ obtained from $D_{p,c}$ is probably the more robust estimation¹¹, and that there appears to be a problem with how the breakthrough times have been determined. For the 1.8 g/cm³ samples we have demonstrated that this is the case, for the 1.2 g/cm³ we can only make an educated guess that this is the case.

Summary and verdict

We have seen that the results on chloride diffusion in Mu88 suffer from uncertainty from several sources:

The “MX-80” material is not that well defined
Densities vary substantially for samples at the same nominal density
Without knowledge of e.g water saturation procedures and solution volumes, it is impossible to estimate the proper ion population during the course of a test
It is, however, highly likely that tests performed at low NaCl concentrations contain substantial amounts of di-valent ions, while those at high NaCl concentration are closer to being pure sodium systems.
The reported diffusivities give a corresponding uncertainty in the chloride equilibrium concentrations of about a factor of 2. While some tests essentially have a too high noise level to give certain estimations, the problem for the others seems to stem from the estimation of breakthrough times.

Here is an attempt to encapsulate the above information in an updated plot for the chloride equilibrium data in Mu88

Uncertainty estimations for chloride equilibrum concnetrations in Muurinen et al. (1988)

The colored squares represent “confidence areas” based on the variation within each nominal density (horizontally), and on the variation of $\bar{c}(0)/c^\mathrm{source}$ from the two reported values on pore diffusivity⁷ (vertically). The limits of these rectangles are simply the 95% confidence interval, based on these variations, and assuming a normal distribution.

Data points put within parentheses are estimations judged to be improper (based on either re-evaluation of the raw data, or informed guesses).

From the present analysis my decision is to not use the data from Mu88 to e.g. validate models for anion exclusion. Although there seems to be nothing fundamentally wrong with how these test were conducted, they suffer from so many uncertainties of various sources that I judge the data to not contribute to quantitative process understanding.

Footnotes

[1] This work is referred to as “Muurinen et al. (1989)” by several authors.

[2] MX-80 is not only a brand name, but also a band name.

[3] This report is “Bentonite Mineralogy” by L. Carlson (Posiva WR 2004-02), but it appears to not be included in the INIS database. It can, however, be found with some elementary web searching.

[4] It’s interesting to note that the cation exchange capacity of “MX-80” remains more or less constant, while the montmorillonite content has some variation. This implies that the montmorillonite layer charge varies (and is negatively correlated with montmorillonite content). Could it be that the manufacturer has a specified cation exchange capacity as requirement for this product?

[5] To convert porosity to dry density, I used $\rho_d = \rho_s\cdot(1-\phi)$, with solid grain density $\rho_s = 2.75$ g/cm³.

[6] A speculation is that the uncertainty stems from the measurement procedure, as this was done on smaller sections of the full samples. It is not specified in Mu88 what the reported porosity represent, but it is reasonable to assume that it is the average of all sections of a sample.

[7] At the risk of losing some clarity, I refuse to use the term “apparent diffusivity” for something which actually is a real pore diffusivity.

[8] These values were not tabulated, but I have read them off from the graphs in Mu88.

[9] Mu88 use the concentration based on the total volume in their expression, while $\bar{c}$ is defined in terms of water volume (water mass, strictly). Eq.2 therefore contains the physical porosity. In their concentration profile plots, however, Mu88 use $\bar{c}$ as variable (called $c_{pw}$ — the “concentration in the pore water”)

[10] Plugging the breakthrough time $L^2/6D_p$ into the expression for the flux gives

\begin{equation} j^\mathrm{out}(t_\mathrm{bt}) = j^{ss}\cdot\left ( 1 + 2 \sum (-1)^n e^\frac{-\pi^2 n^2}{6} \right ) \approx 0.616725\cdot j^{ss} \end{equation}

I find it amusing that this value is close to the reciprocal golden ratio (0.618033…). Finding the breakthrough time from a flux vs. time plot thus corresponds (approximately) to splitting the y-axis according to the golden ratio.

[11] Note that the actual evaluated values of $D_{p,c}$ in Mu88 still may be uncertain, because they also depend on the values of the steady-state flux, which we have seen were not optimally evaluated.

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

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

Test principles

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

Schematics of samples in Kahr et al. (1985(

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

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

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

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

Distribution of water in the samples

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

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

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

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

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

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

Results

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

Mobility of interlayer cations confirmed

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

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

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

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

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

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

Anions enter interlayers (and are mobile)

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

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

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

Dependence of diffusivity on water content and density

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

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

Interlayer chemistry

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

Schematics of KI dissolution in interlayer water

This study seems to have made no impact at all

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

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

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

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

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

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

By regarding the clay-gel as a concentrated electrolytic system Marinsky has calculated (30) distribution coefficients for Sr²⁺ and Cs⁺ ions in good agreement with experimentally determined K_d-values. The low anionic exchange capacity and hence the low anion concentration in the pore solution caused by Donnan exclusion also explain the low concentrations of anionic tracers within the clay-gel
[…]
For simple cations the ion-exchange process is dominating and there is, as also pointed out by Marinsky (30), no need to suppose that the counterions are immobilized. It ought to be emphasized that for the compacted bentonite used in the diffusion experiments discussed in this report the water content corresponds roughly to 2-4 water molecule layers (31). There is therefore really no “free water” and the measured diffusivity $\bar{D}$ can be regarded as corresponding approximately to the diffusivity within the adsorbed phase […]

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

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

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

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

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

Footnotes

[1] “Ion diffusion in highly compacted bentonite”

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

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

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

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

The mechanism for “anion” exclusion

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Repulsion between surfaces and anions is not really the point

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

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

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

Ferrocyanide

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

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

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

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

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

Bentonite

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

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

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

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

Here is an illustration of the validity of this approximation:

Internal and external potential in compacted bentonite system

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

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

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

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

A charge neutral perspective

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

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

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

while the outflux is proportional to the internal ion concentrations

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

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

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

which is the (ideal) Donnan equation.

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

Footnotes

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

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

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

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

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

Stacks make no sense

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At the atomic level, montmorillonite is built up of so-called TOT-layers: covalently bonded sheets of aluminum (“O”) and silica (“T”) oxide (including the right amount of impurities/defects). In my mind, such TOT-layers make up the fundamental particles of a bentonite sample. Reasonably, since montmorillonite TOT-layers vary extensively in size, and since a single cubic centimeter of bentonite contains about ten million billions ($10^{16}$), they are generally configured in some crazily complicated manner.

Stack descriptions in the literature

But the idea that the single TOT-layer is the fundamental building block of bentonite is not shared with many of today’s bentonite researchers. Instead, you find descriptions like e.g. this one, from Bacle et al. (2016)

Clay mineral particles consist of stacks of parallel negatively-charged layers separated by interlayer nanopores. Consequently, compacted smectite contains two major classes of pores: interlayer nanopores (located inside the particles) and larger mesopores (located between the particles).

or this one, from Churakov et al. (2014)

In compacted rocks, montmorillonite (Mt) forms aggregates (particles) with 5–20 TOT layers (Segad et al., 2010). A typical radial size of these particles is of the order of 0.01 to 1 $\mathrm{\mu m}$. The pore space between Mt particles is referred to as interparticle porosity. Depending on the degree of compaction, the interparticle porosity contributes 10 to 30% of the total water accessible pore space in Mt (Holmboe et al., 2012; Kozaki et al., 2001).

Such statements show that researchers have something more complex in mind than individual TOT-layers when speaking of “particles”: they are some sort of assemblages of TOT-layers. The quotation of Bacle et al. (2016), using both the terms “stacks” and “particles”, even hints at an idea of a hierarchy of fundamental structures. Such a hierarchy is expressed explicitly in e.g. Navarro et al. (2017), who provide a figure with the caption “Schematic particle arrangement in highly compacted Na-bentonite” that looks similar to this one:

“Schematic particle arrangement in highly compacted Na-bentonite” from Navarro et al. (2017)

Here it is clear that they differ between “aggregates” (which I’m not sure is the same thing as “particles”), “stacks”, and individual TOT-layers (which I assume are represented by the line-shaped objects). In the following, however, we will use the term “stack” to refer to any kind of suggested fundamental structure built up from individual TOT-layers.

The one-sentence version of this blog post is:

Stacks make no sense as fundamental building blocks in models of water saturated, compacted bentonite.

The easiest argument against stacks is, in my mind, to simply work out the geometrical consequences. But before doing that we will examine some of the references given to support statements about stacks in compacted systems. Often, no references are given at all, but when they are, they usually turn out to be largely irrelevant for the system under study, or even to support an opposite view.

Inadequate referencing

As an example (of many) of inadequate referencing, we use the statement above from Churakov et al. (2014) as a starting point. I think this is a “good” statement, in the sense that it makes rather precise claims about how compacted bentonite is supposed to be structured, and provides references for some key statements, which makes it easier to criticize.

Churakov et al. (2014) reference Segad et al. (2010) for the statement that montmorillonite forms “particles” with 5 – 20 TOT-layers. In turn, Segad et al. (2010) write:

Clay is normally not a homogeneous lamellar material. It might be better described as a disordered structure of stacks of platelets, sometimes called tactoids — a tactoid typically consists of 5-20 platelets.^19-21

Here the terminology is quite different from the previous quotations: TOT-layers are called “platelets”, and “particles” are called “tactoids”. Still, they use the phrase “stacks of platelets”, so I think we can continue with using “stack” as a sort of common term for what is being discussed.¹ We may also note that here is used the word “clay”, rather than “montmorillonite” (as does Bacle et al. (2016)), but it is clear from the context of the article that it really is montmorillonite/bentonite that is discussed.

Anyhow, Segad et al. (2010) do not give much direct information on the claim we investigate, but provide three new references. Two² of these — Banin (1967) and Shalkevich et al. (2007) — are actually studies on montmorillonite suspensions, i.e. as far away as you can get from compacted bentonite in terms of density; the solid mass fraction in these studies is in the range 1 – 4%.

The average distance between individual TOT-layers in this density limit is comparable with, or even larger than, their typical lateral extension (~100 nm). Therefore, much of the behavior of low density montmorillonite depends critically on details of the interaction between layer edges and various other components, and systems in this density limit behave very differently depending on e.g. ionic strength, cation population, preparation protocol, temperature, time, etc. This complex behavior is also connected with the fact that pure Ca-montmorillonite does not form a sol, while the presence of as little as 10 – 20% sodium makes the system sol forming. The behaviors and structures of montmorillonite suspensions, however, say very little about how the TOT-layers are organized in compacted bentonite.

We have thus propagated from a statement in Churakov et al. (2014), and a similar one in Segad et al. (2010), that montmorillonite in general, in “compacted rocks” forms aggregates of 5 – 20 TOT-layers, to studies which essentially concern different types of materials. Moreover, the actual value of “5 – 20 TOT layers” comes from Banin (1967), who writes

Evidence has accumulated showing that when montmorillonite is adsorbed with Ca, stable tactoids, containing 5 to 20 parallel plates, are formed (1). When the mineral is adsorbed with Na, the tactoids are not stable, and the single plates are separated from each other.

This source consequently claims that the single TOT-layers are the fundamental units, i.e. it provides an argument against any stack concept! (It basically states that pure Ca-montmorillonite does not form a sol.) In the same manner, even though Segad et al. (2010) make the above quoted statement in the beginning of the paper, they only conclude that “tactoids” are formed in pure Ca-montmorillonite.

The swelling and sedimentation behavior of Ca-montmorillonite is a very interesting question, that we do not have all the answers to yet. Still, it is basically irrelevant for making statements about the structure in compacted — sodium dominated³ — bentonite.

Churakov et al. (2014) also give two references for the statement that the “interparticle porosity” in montmorillonite is 10 – 30% of the total porosity: Holmboe et al. (2012) and Kozaki et al. (2001). This is a bizarre way of referencing, as these two studies draw incompatible conclusions, and since Holmboe et al. (2012) — which is the more adequately performed study — state that this type of porosity may be absent:

At dry density $>1.4 \;\mathrm{g/cm^3}$ , the average interparticle porosity for the [natural Na-dominated bentonite and purified Na-montmorillonite] samples used in this study was found to be $1.5\pm1.5\%$, i.e. $\le 3\%$ and significantly lower than previously reported in the literature.

Holmboe et al. (2012) address directly the discrepancy with earlier studies, and suggest that these were not properly analyzed

The apparent discrepancy between the basal spacings reported by Kozaki et al. (1998, 2001) using Kunipia-F washed Na-montmorillonite, and by Muurinen et al. (2004), using a Na-montmorillonite originating from Wyoming Bentonite MX-80, and the corresponding average basal spacings of the [Na-montmorillonite originating from Wyoming bentonite MX-80] samples reported in this study may partly be due to water contents and partly to the fact that only apparent $\mathrm{d_{001}}$ values using Bragg’s law, without any profile fitting, were reported in their studies.

If Kozaki et al. (2001) should be used to support a claim about “interparticle porosity”, it consequently has to be done in opposition to — not in conjunction with — Holmboe et al. (2012). It would then also be appropriate for authors to provide arguments for why they discard the conclusions of Holmboe et al. (2012).⁴

Stacks in compacted bentonite make no geometrical sense

The literature is full of fancy figures of bentonite structure involving stacks. A typical example is found in Wu et al. (2018), and looks similar to this:

stacks illustration from Wu et al. (2018)

This illustration is part of a figure with the caption “Schematic representation of the different porosities in bentonite and the potential diffusion paths.”⁵ The regular rectangles in this picture illustrate stacks that each seems to contain five TOT-layers (I assume this throughout). Conveniently, these groups of five layers have the same length within each stack, while the length varies somewhat between stacks. This is a quite common feature in figures like this, but it is also common that all stacks are given the same length.

Another feature this illustration has in common with others is that the particles are ordered: we are always shown edges of the TOT-layers. I guess this is partly because a picture of a bunch of stacks seen from “the top” would be less interesting, but it also emphasizes the problem of representing the third dimension: figures like these are in practice figures of straight lines oriented in 2D, and the viewer is implicitly required to imagine a 3D-version of this two-dimensional representation.

A “realistic” stack picture

But, even as a 2D-representation, these figures are not representative of what an actual configuration of stacks of TOT-layers looks like. Individual TOT-layers have a distinct thickness of about 1 nm, but varies widely in the other two dimensions. Ploehn and Liu (2006) analyzed the size distribution of Na-montmorillonite (“Cloisite Na+”) using atomic force microscopy, and found an average aspect ratio of 180 (square-root of basal area divided by thickness). A representative single “TOT-line” drawn to scale is consequently quite different from what is illustrated in in most stack-pictures, and look like this (click on the figure to see it in full size)

Representative TOT-layer drawn to scale with water films

In this figure, we have added “water layers” on each side of the TOT-layer (light red), with the water-to-solid volume ratio of 1⁶. Neatly stacking five such units shows that the rectangles in the Wu et al. (2018)-figure should be transformed like this

actual veiw of stacks in Wu et al. (2018)

But this is still not representative of what an assemblage of five randomly picked TOT-layers would look like, because the size distribution has a substantial variance. According to Ploehn and Liu (2006), the aspect ratio follows approximately a log-normal distribution. If we draw five values from this distribution for the length of five “TOT-lines”, and form assemblages, we end up with structures that look like this:⁷

These are the kind of units that should fill the bentonite illustrations. They are quite irregularly shaped and are certainly not identical (this would be even more pronounced when considering the third dimension, and if the stacks contain more layers).

It is easy to see that it is impossible to construct a dense structure with these building blocks, if they are allowed a random orientation. The resulting structure rather looks something like this

Such a structure evidently has very low density, and are reminiscent of the gel structures suggested in e.g. Shalkevich et al. (2007) (see fig. 7 in that paper). This makes some sense, since the idea of stacks of TOT-layers (“tactoids”) originated from studies of low density structures, as discussed above.

Note that the structure in pictures like that in Wu et al. (2018) has a substantial density only because it is constructed with stacks with an unrealistic shape. But even in these types of pictures is the density not very high: with some rudimentary image analysis we conclude that the density in the above picture is only around 800 kg/m³. Also the figure from Navarro et al. (2017) above gives a density below 1000 kg/m³, although there it is explicitly stated that it is a representation of “highly compacted bentonite”.

The only manner in which the “realistic” building blocks can be made to form a dense structure is to keep them in the same orientation. The resulting structures then look e.g. like this

Dense structrue of color coded realistic stacks

where we have color coded each stack, to remind ourselves that these units are supposed to be fundamental.

Just looking at this structure of a “stack of stacks” should make it clear how flawed the idea is of stacks as fundamental structural units in compacted bentonite (note also how unrepresentative the stack-pictures found in the literature are). One of many questions that immediately arises is e.g. why on earth the tiny gaps between stacks (indicated by arrows) should remain. This brings us to the next argument against stacks as fundamental units for compacted water saturated bentonite:

What is supposed to keep stacks together?

Compacted bentonite of interest e.g. for sealing in radioactive waste repositories exerts swelling pressure of several MPa when in contact with external water. This osmotic pressure is a consequence of the presence of the mobile exchangeable cations in montmorillonite. Each “realistic” unit that we have imagined above is thus required to be at a huge elevated pressure, and the individual TOT-layers have a strong driving force to separate. And, unless a mechanism is provided for why such a separation is impossible, this is of course what we expect to happen! As far as I am aware, such a separation inhibiting mechanism has never been suggested in any publication that promotes the stack concept in compacted bentonite. To get a feel for the absurdity of this issue, let’s take a new look at the figure from Navarro et al. (2017)

The pressure configuration in the structure suggested by Navarro et al. (2017)

Assuming that this system is in equilibrium with an external water reservoir at zero pressure (i.e. atmospheric absolute pressure), the pressure in the compartment labeled “intra-aggregate space” is also close to zero. On the other hand, in the “stacks” located just a few nm away, the pressure is certainly above 10 MPa in many places! A structure like this is obviously not in mechanical equilibrium! (To use the term “obvious” here feels like such an understatement.)

Implications

To sum up what we have discussed so far, the following picture emerges. The bentonite literature is packed with descriptions of compacted water saturated bentonite as built up of stacks as fundamental units. These descriptions are so commonplace that they often are not supported by references. But when they are, it seems that the entire notion is based on misconceptions. In particular, structures identified in low density systems (suspensions, gels) have been carried over, without reflection, to descriptions of compacted bentonite. Moreover, all figures illustrating the stack concept are based on inadequate representations of what an arbitrary assemblage of TOT-layers arranged in this way actually would look like. With a “realistic” representation it quickly becomes obvious that it makes little sense to base a fundamental unit in compacted systems on the stack concept.

My impression is that this flawed stack concept underlies the entire contemporary mainstream view of compacted bentonite, as e.g. expressed by Wu et al. (2018):

A widely accepted view is that the total porosity of bentonite consists of $\epsilon_ {ip}$ and $\epsilon_ {il}$ (Tachi and Yotsuji, 2014; Tournassat and Appelo, 2011; Van Loon et al., 2007). $\epsilon_ {ip}$ is a porosity related to the space between the bentonite particles and/or between the other grains of minerals present in bentonite. It can further be subdivided into $\epsilon_ {ddl}$ and $\epsilon_ {free}$. The diffuse double layer, which forms in the transition zone from the mineral surface to the free water space, contains water, cations and a minor amount of anions. The charge at the negative outer surface of the montmorillonite is neutralized by an excess of cations. The free water space contains a charge-balanced aqueous solution of cations and anions. $\epsilon_ {il}$ represents the space between TOT-layers in montmorillonite particles exhibiting negatively charged surfaces. Due to anion exclusion effect, anions are excluded from the interlayer space, but water and cations are present.

This view can be summarized as:

The fundamental building blocks are stacks of TOT-layers (“particles”, “aggregates”, “tactoids”, “grains”…)
Electric double layers are present only on external surfaces of the stacks.
Far away from external surfaces — in the “inter-particle” or “inter-aggregate” pores — the diffuse layers merge with a bulk water solution
Interlayer pores are defined as being internal to the stacks, and are postulated to be fundamentally different from the external diffuse layers; they play by a different set of rules.

I don’t understand how authors can get away with promoting this conceptual view without supplying reasonable arguments for all of its assumptions⁸ — and with such a complex structure, there are a lot of assumptions.

As already discussed, the geometrical implications of the suggested structure do not hold up to scrutiny. Likewise, there are many arguments against the presence of substantial amounts of bulk water in compacted bentonite, including the pressure consideration above. But let’s also take a look at what is stated about “interlayers” and how these are distinguished from electric double layers (I will use quotation marks in the following, and write “interlayers” when specifically referring to pores defined as internal to stacks).

“Interlayers”

“Interlayers” are often postulated to be completely devoid of anions. We discussed this assumption in more depth in a previous blog post, where we discovered that the only references supplied when making this postulate are based on the Poisson-Boltzmann equation. But this is inadequate, since the Poisson-Boltzmann equation does describe diffuse layers, and predicts anions everywhere.

By requiring anion-free “interlayers”, authors actually claim that the physico-chemistry of “interlayers” is somehow qualitatively different from that of “external surfaces”, although these compartments have the exact same constitution (charged TOT-layer surface + ions + water). But an explanation for why this should be the case is never provided, nor is any argument given for why diffuse layer concepts are not supposed to apply to “interlayers”.⁹ This issue becomes even more absurd given the strong empirical evidence for that anions actually do reside in interlayers.

The treatment of anions is not the only ad hoc description of “interlayers”. It also seems close to mandatory to describe them as having a maximum extension, and as having an extension independently parameterized by sample density. E.g. the influential models for Na-bentonite of Bourg et al. (2006) and Tournassat and Appelo (2011) both rely on the idea that “interlayers” swell out to a certain volume that is smaller than the total pore volume, but that still depends on density.

In e.g. Bourg et al. (2006), the fraction of “interlayer” pores remains essentially constant at ~78%, as density decreases from 1.57 g/cm³ to 1.27 g/cm³, while the “interlayers” transform from having 2 monolayers of water (2WL) to having 3 monolayers (3WL). This is a very strange behavior: “interlayers” are acknowledged as having a swelling potential (2WL expands to 3WL), but do, for some reason, not affect 22% of the pore volume! Although such a behavior strongly deviates from what we expect if “interlayers” are treated with conventional diffuse layer concepts, no mechanism is provided.

In contrast, it should be noted that the established explanation for “tactoid” formation in pure Ca-montmorillonite involves no ad hoc assumptions of this sort, but rests on treating all ions as part of diffuse layers.

Another type of macabre consequence of defining “interlayer” pores as internal to stacks is that a completely homogeneous system is described has having no interlayer pores (because it has no stacks). E.g. Tournassat and Appelo (2011) write ($n_c$ is the number of TOT-layers in a stack)

[…] the number of stacks in the $c$-direction has considerable influence on the interlayer porosity, with interlayer porosity increasing with $n_c$ and reaching the maximum when $n_c \approx 25$. The interlayer porosity halves with $n_c$ when $n_c$ is smaller than 3, and becomes zero for $n_c = 1$.¹⁰

It is not acceptable that using the term interlayer should require accepting stacks as fundamental units. But the usage of the term as being internal to stacks is so widespread in the contemporary bentonite literature, that I fear it is difficult to even communicate this issue. Nevertheless, I am certain that e.g. Norrish (1954) does not depend on the existence of stacks when using the term like this:

Fig. 7 shows the relationship between interlayer spacing and water content for Na-montmorillonite. There is good agreement between the experimental points and the theoretical line, showing that interlayer swelling accounts for all, or almost all, of physical swelling.

The stack view obstructs real discovery

A severe consequence of the conceptual view just discussed is that “stacking number” — the (average) number of TOT-layers that stacks are supposed to contain — has been established as fitting parameter in models that are clearly over-parameterized. An example of this is Tournassat and Appelo (2011), who write¹¹

Our predictive model excludes anions from the interlayer space and relates the interlayer porosity to the ionic strength and the montmorillonite bulk dry density. This presentation offers a good fit for measured anion accessible porosities in bentonites over a wide range of conditions and is also in agreement with microscopic observations.

But since anions do reside in interlayers,¹² it would be better if the model didn’t fit: an over-parameterized or conceptually flawed model that fits data provides very little useful information.

A similar more recent example is Wu et al. (2018). In this work, a model based on the stack concept is successfully fitted both to data on $\mathrm{ReO_4^-}$ diffusion in “GMZ” bentonite and to data on $\mathrm{Cl^-}$ diffusion in “KWK” bentonite, by varying “stacking number” (among other parameters). Again, as the model assumes anion-free “interlayer” pores, a better outcome would be if it was not able to fit the data. Moreover, this paper focuses mainly on the ability of the model, while not at all emphasizing the fact that about ten (!) times more $\mathrm{ReO_4^-}$ was measured in “GMZ” as compared with $\mathrm{Cl^-}$ in “KWK”, at similar conditions in certain cases. The latter observation is quite puzzling and is, in my opinion, certainly worth deeper investigation (and I am fully convinced that it is not explained by differences in “stacking number”).

Footnotes

[1] The terminology in the bentonite literature is really all over the place. You may e.g. also find the term “tactoid” used as Navarro et al. (2017) use “aggregate”, or the term “platelet” used for a stack of TOT-layers…

[2] The third reference is an entire book on clays.

[3] Note that “sodium dominated” in this context means ~20% or more.

[4] It may be noticed that Kozaki et al. (2001) see no X-ray diffraction peaks for low density samples:

The basal spacing of water-saturated montmorillonite was determined by the XRD method. […] It was found that a basal spacing of 1.88 nm, corresponding to the three-water layer hydrate state […] was not observed before the dry density reached 1.0 Mg/m³.

My interpretation of this observation is that the diffraction peak has shifted to even lower reflection angles (in agreement with the observations of Holmboe et al. (2012)), not registered by the equipment. The alternative interpretation must otherwise be that “stacks” suddenly cease to exist below 1.0 g/cm³. (Yet, Kozaki et al. (2001) continues to use a certain d-value in their analysis, also for densities below 1.0 g/cm³.)

[5] I have discussed “diffusion paths” in an earlier blog post. This illustration certainly fits that discussion.

[6] A water-to-solid volume ratio of 1 corresponds basically to interlayers of three monolayers of water (3WL).

[7] To construct these units, I made the additional choice of placing each layer randomly in the horizontal direction, with the constraint that all layers should be confined within the range of the longest one in each unit.

[8] By “get away with” I mean “pass peer-review”, and by “don’t understand” I mean “understand”.

[9] This is reminiscent of how certain authors imply that the interlayer is non-diffusive under so-called crystalline swelling.

[10] A mathematical remark: if the interlayer porosity “halves with $n_c$” (what does that mean?) when $n_c = 2$ (“smaller than 3”), it is impossible to simultaneously have zero interlayer porosity for $n_c = 1$ (unless the interlayer porosity is zero for any $n_c$).

[11] I guess the word “presentation” here really should be “representation”?

[12] Note that one of the authors of this paper also claims in a later paper that anions do populate 3-waterlayer interlayers, in accordance with the Poisson-Boltzmann equation:

The agreement between PB calculations and MD simulation predictions was somewhat worse in the case of the $\mathrm{Cl^-}$ concentration profiles than in the case of the $\mathrm{Na^+}$ profiles (Figure 3), perhaps reflecting the poorer statistics for interlayer Cl concentrations […] Nevertheless, reasonable quantitative agreement was found (Table 2).

Extracting anion equilibrium concentrations from through-diffusion tests

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

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

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

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

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

The through-diffusion set-up

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

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

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

The effective porosity diffusion model

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

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

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

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

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

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

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

Connection with experimentally accessible quantities

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

where we have assumed zero target concentration.

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

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

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

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

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

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

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

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

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

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

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

Additional comments on the effective porosity model

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

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

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

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

Footnotes

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

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

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

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

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

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

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

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

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

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

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

Muurinen et al. (1985)

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

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

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

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

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

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

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

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

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

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

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

Eriksen et al. (1999)

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

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

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

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

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

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

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

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

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

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

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

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

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

Gimmi & Kosakowski (2011)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Some thoughts

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

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

Footnotes

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

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

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

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

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

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

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

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

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

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

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

Donnan equilibrium and the homogeneous mixture model

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

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

Overview

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

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

The problem to be solved can be illustrated like this

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

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

Chemical equilibrium

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

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

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

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

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

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

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

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

Illustrations

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

Relation beteween internal and external concentration for varying Xi

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

Variation of internal cation concentrations with varying Donnan factor

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

Variation of internal anion concentration with the Donnan factor

Equation for $f_D$

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

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

where

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

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

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

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

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

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

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

Relation between internal and external Na and Cl concentrations

Comment on through-diffusion

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

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

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

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

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

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

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

Additional comments

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

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

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

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

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

Footnotes

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

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

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

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

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

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

Sorption part III: Donnan equilibrium in compacted bentonite

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

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

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

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

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

Donnan equilibrium: effect vs. model

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

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

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

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

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

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

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

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

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

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

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

Donnan equilibrium in gels, soils and clays

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

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

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

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

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

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

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

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

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

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

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

Cation exchange is Donnan equilibration

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A demonstration of compacted bentonite as a Donnan system

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

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

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

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

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

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

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

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

Footnotes

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

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

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

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

[5] This article has an English translation.

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

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

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

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

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

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

Sorption part II: Letting go of the bulk water

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Diffusion in the homogeneous mixture model

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

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

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

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

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

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

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

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

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

Model comparison

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

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

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

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

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

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

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

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

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

Footnotes

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

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

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

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

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

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

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

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

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

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

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

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

where $\omega$ is the relative filter resistance

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

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

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

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

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

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

i.e. independent of any clay parameters.

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

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

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

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

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

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

which approaches zero when $\omega$ becomes large.

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

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

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

Sodium tracer profiles from Glaus et al. (2007)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Effective diffusivities for strontium in Molera and Eriksen (2002)

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

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

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

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

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

calcium outflux in Tinnacher et al. (2016)

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

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

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

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

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

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

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

Footnotes

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

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

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

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

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

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

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

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

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

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

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

Appendix: Derivation of eqs. 1 and 2

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

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

where $R$ is the transfer resistance.

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

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

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

Solving for $c_\mathrm{in}$ and $c_\mathrm{out}$ gives

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

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

Defining the relative filter resistance as

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

we can express the concentration drop across the clay as

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

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

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

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

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

which is eq. 1 above.

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

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

which is eq. 2 above.

Dry density	EMDD (\(x\)=0.75)	EMDD (\(x\)=0.80)	EMDD (\(x\)=0.85)
(g/cm³)	(g/cm³)	(g/cm³)	(g/cm³)
1.2	1.01	1.05	1.09
1.8	1.61	1.66	1.70

Test	\(\phi\)	\(\rho_d\)
	(-)	(g/cm³)
1.2/0.01	0.54	1.27
1.2/0.1	0.52	1.32
1.2/1.0	0.49	1.40
1.8/0.01	0.37	1.73
1.8/0.1	0.31	1.89
1.8/1.0	0.34	1.81

Test	\(D_e\)	\(A\cdot j^\mathrm{ss}/c^\mathrm{source}\)
	(\(\mathrm{m^2/s}\))	(ml/day)
1.2/0.01	\(7.7\cdot 10^{-12}\)	0.031
1.2/0.1	\(2.9\cdot 10^{-11}\)	0.118
1.2/1.0	\(1.2\cdot 10^{-10}\)	0.489
1.8/0.01	\(3.3\cdot 10^{-13}\)	0.001
1.8/0.1	\(4.8\cdot 10^{-13}\)	0.002
1.8/1.0	\(4.0\cdot 10^{-12}\)	0.016

Test	\(D_{p,c}\)	\(A\cdot j^\mathrm{ss}/c^\mathrm{source}\)	\(\phi\)	\(\bar{c}(0)/c^\mathrm{source}\)
	(\(\mathrm{m^2/s}\))	(ml/day)	(-)	(-)
1.2/0.01	\(7.0\cdot 10^{-11}\)	0.031	0.54	0.204
1.2/0.1	\(2.8\cdot 10^{-10}\)	0.118	0.52	0.199
1.2/1.0	\(5.1\cdot 10^{-10}\)	0.489	0.49	0.480
1.8/0.01	\(2.0\cdot 10^{-11}\)	0.001	0.37	0.045
1.8/0.1	\(3.1\cdot 10^{-11}\)	0.002	0.31	0.050
1.8/1.0	\(5.2\cdot 10^{-11}\)	0.016	0.34	0.226

Test	\(D_{p,t}\)	\(t_\mathrm{bt}\)
	(\(\mathrm{m^2/s}\))	(days)
1.2/0.01	\(1.4\cdot 10^{-10}\)	3.1
1.2/0.1	\(2.0\cdot 10^{-10}\)	2.2
1.2/1.0	\(3.2\cdot 10^{-10}\)	1.4
1.8/0.01	\(5.0\cdot 10^{-11}\)	8.7
1.8/0.1	\(5.4\cdot 10^{-11}\)	8.0
1.8/1.0	\(7.7\cdot 10^{-11}\)	5.6

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

Uncertainty of bentonite samples

Montmorillonite content

Cation population

Soluble calcium minerals

Sample density

Uncertainty of external solutions

Uncertainty of diffusion parameters

Evaluation of \(D_e\) in Mu88

Evaluation of \(D_p\) in Mu88

Re-evaluation by fitting to the full data set

Summary and verdict

Footnotes

Test principles

Distribution of water in the samples

Results

Mobility of interlayer cations confirmed

Anions enter interlayers (and are mobile)

Dependence of diffusivity on water content and density

Interlayer chemistry

This study seems to have made no impact at all

Footnotes

Repulsion between surfaces and anions is not really the point

Ferrocyanide

Bentonite

A charge neutral perspective

Footnotes

Stack descriptions in the literature

Inadequate referencing

Stacks in compacted bentonite make no geometrical sense

A “realistic” stack picture

What is supposed to keep stacks together?

Implications

“Interlayers”

The stack view obstructs real discovery

Footnotes

The through-diffusion set-up

The effective porosity diffusion model

Connection with experimentally accessible quantities

Additional comments on the effective porosity model

Footnotes

Muurinen et al. (1985)

Eriksen et al. (1999)

Gimmi & Kosakowski (2011)

Some thoughts

Footnotes

Overview

Chemical equilibrium

Illustrations

Equation for \(f_D\)

Comment on through-diffusion

Additional comments

Footnotes

Donnan equilibrium: effect vs. model

Donnan equilibrium in gels, soils and clays

Cation exchange is Donnan equilibration

A demonstration of compacted bentonite as a Donnan system

Footnotes

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

Diffusion in the homogeneous mixture model

Model comparison

Footnotes

Footnotes

Appendix: Derivation of eqs. 1 and 2