Monthly Archives: March 2022

Semi-permeability, part I

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Descriptions in bentonite literature

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Chagneau et al. (2015) write

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

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

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

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

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

Bentonite as a non-ideal semi-permeable membrane

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

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

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

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

“Semi-permeability” in experiments

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

Tracer through-diffusion

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

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

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

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

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

Measuring hydraulic conductivity

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

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

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

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

Mass transfer in a salt gradient

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

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

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

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

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

A process of practical importance for engineered clay barrier systems is hyperfiltration of salts. This process will occur when a sufficient pressure difference is applied over a bentonite sample contacted with saline solutions. Water and salt will then be transferred in the same direction, but, due to exclusion, salt will accumulate on the inlet side. A steady-state concentration profile for such a process may look like this

The local salt concentration at the sample interface on the inlet side may thus be larger than the concentration of the injected solution. This may have consequences e.g. when evaluating hydraulic conductivity using saline solutions.

Hyperfiltration may also influence the way a sample becomes saturated, if saturated with a saline solution. If the region near the inlet is virtually saturated, while regions farther into the sample still are unsaturated, hyperfiltration could occur. In such a scenario the clay could in a sense be said to be semi-permeable (letting through water and filtrating salts), but note that the net effect is to transfer more salt into the sample than what is dictated by Donnan equilibrium with the injected solution (which has concentration \(c_1\), if we stick with the figure above). Salt will then have to diffuse out again, in later stages of the process, before full equilibrium is reached. This is in similarity with the saturation process that we considered earlier.

Footnotes

[1] We have considered this study before, when discussing the empirical evidence for salt in interlayers.

[2] This is more than a thought-experiment; a test just like this was conducted by Karnland et al. (2005). Here is the recorded pressure response of a Na-montmorillonite sample (dry density 1.4 g/cm3) as it is contacted with NaCl solutions of increasing concentration

We have considered this study earlier, as it proves that salt enters interlayers.

[3] As a side note, is the region near the surface supposed to be called “diffuse layer”, “electrical double layer”, or “electrostatic (diffuse double) layer”?

[4] Also Fritz (1986), referenced in the quotation by Horseman et al. (1996), states a version of this “explanation”.

[5] This is not a gradient in the mathematical sense, but is defined as \( \left (c_\mathrm{target} – c_\mathrm{source} \right)/L\), where \(L\) is sample length.

[6] Hydraulic conductivity is often also measured using a saline solution, which is commented on below.

[7] Which still is an a amazingly small hydraulic conductivity, considering the the water content.

[8] The study of Neuzil (2000) also provides clear examples of water moving out of the clay, and salt moving in, in similarity with the process considered above.

[9] Mathematically, the statement “equal infinity” is mostly nonsense, but I am trying to convey that a there is a tracer flux even without any external tracer concentration difference.

Assessment of chloride equilibrium concentrations: Molera et al. (2003)

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In the ongoing assessment of chloride equilibrium concentrations in bentonite, we here take a closer look at the study by Molera et al. (2003), in the following referred to as Mo03. We thus assess the 13 points indicated here

Mo03 performed both chloride and iodide through-diffusion tests on “MX-80” bentonite, but here we focus on the chloride results. However, since the only example in the paper of an outflux evolution and corresponding concentration profile is for iodide, this particular result will also be investigated. The tests were performed at background concentrations of 0.01 M or 0.1 M NaClO4, and nominal sample densities of 0.4, 0.8, 1.2, 1.6, and 1.8 g/cm3. We refer to a single test by stating “nominal density/background concentration”, e.g. a test performed at nominal density 1.6 and background concentration 0.1 M is referred to as “1.6/0.1”.

Uncertainty of samples

The material used is discussed only briefly, and the only reference given for its properties is (Müller-Von Moos and Kahr, 1983). I don’t find any reason to believe that the “MX-80” batch used in this study actually is the one investigated in this reference, and have to assume the same type of uncertainty regarding the material as we did in the assessment of Muurinen et al (1988). I therefore refer to that blog post for a discussion on uncertainty in montmorillonite content, cation population, and soluble calcium minerals.

Density

The samples in Mo03 are cylindrical with radius 0.5 cm and length 0.5 cm, giving a volume of 0.39 cm3. This is quite small, and corresponds e.g. only to about 4% of the sample size used in Muurinen et al (1988). With such a small volume, the samples are at the limit for being considered as a homogeneous material, especially for the lowest densities: the samples of density 0.4 g/cm3 contain 0.157 g dry substance in total, while a single 1 mm3 accessory grain weighs about 0.002 — 0.003 g.

Furthermore, as the samples are sectioned after termination, the amount substance in each piece may be very small. This could cause additional problems, e.g. enhancing the effect of drying. The reported profile (1.6/0.1, iodide diffusion) has 10 sections in the first 2 mm. As the total mass dry substance in this sample is 0.628 g, these sections have about 0.025 g dry substance each (corresponding to the mass of about ten 1 mm3 grains). For the lowest density, a similar sectioning corresponds to slices of dry mass 0.006 g (the paper does not give any information on how the low density samples were sectioned).

Mo03 only report nominal densities for the samples, but from the above considerations it is clear that a substantial (but unknown) variation may be expected in densities and concentrations.

A common feature of many through-diffusion studies is that the sample density appears to decrease in the first few millimeters near the confining filters. We saw this effect in the profiles of Muurinen et al (1988), and it has been the topic of some studies, including Mo03. Here, we don’t consider any possible cause, but simply note that the samples seem to show this feature quite generally (below we discuss how Mo03 handle this). Since the samples of Mo03 are only of length 5 mm, we may expect that the major part of them are affected by this effect. Of course, this increases the uncertainty of the actual density of the used samples.

Uncertainty of external solutions

Mo03 do not describe how the external solutions were prepared, other than that they used high grade chemicals. We assume here that the preparation did not introduce any significant uncertainty.

Since “MX-80” contains a substantial amount of divalent ions, connecting this material with (initially) pure sodium solutions inevitably initiates cation exchange processes. The extent of this exchange depends on details such as solution concentrations, reservoir volumes, number of solution replacements, time, etc…

Very little information is given on the volume of the external solution reservoirs. It is only hinted that the outlet reservoir may be 25 ml, and for the inlet reservoir the only information is

The volume of the inlet reservoir was sufficient to keep the concentration nearly constant (within a few percent) throughout the experiments.

Consequently, we do not have enough information to assess the exact ion population during the course of the tests. We can, however, simulate this process of “unintentional exchange” to get some appreciation for the amount of divalent ions still left in the sample, as we did in the assessment of Muurinen et al. (1988). 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 NaClO4 solutions of various concentrations and volumes

In these calculations we assume a sample of density 1.6 g/cm3 (except when indicated), a volume of 0.39 cm3, a cation exchange capacity of 0.75 eq/kg, and a Ca/Na selectivity coefficient of 5.

These simulations make it clear that the tests performed at 0.01 M most probably contain most of the divalent ions initially present in the “MX-80” material: even with an external solution volume of 1000 ml, or with density 0.4 g/cm3, exchange is quite limited. For the tests performed at 0.1 M we expect some exchange of the divalent ions, but we really can’t tell to what extent, as the exact value strongly depends on handling (solution volumes, if solutions were replaced, etc.). That the exact ion population is unknown, and that the divalent/monovalent ratio probably is different for different samples, are obviously major problems of the study (the same problems were identified in Muurinen et al (1988)).

Uncertainty of diffusion parameters

Diffusion model

Mo03 determine diffusion parameters by fitting a model to all available data, i.e the outflux evolution and the concentration profile across the sample at termination. The model is solved by a numerical code (“ANADIFF”) that takes into account transport both in clay samples and filters. The fitted parameters are an apparent diffusivity, \(D_a\), and a so-called “capacity factor”, \(\alpha\). \(\alpha\) is vaguely interpreted as being the combination of a porosity factor \(\epsilon\), and a sorption distribution coefficient \(K_d\), described as “a generic term devoid of mechanism”

\begin{equation} \alpha = \epsilon + \rho\cdot K_d \end{equation}

It is claimed that for anions, \(K_d\) can be treated as negative, giving \(\alpha < \epsilon\). I have criticized this mixing of what actually are incompatible models in an earlier blog post. Strictly, this use of a “generic term devoid of mechanism” means that the evaluated \(\alpha\) should not be interpreted in any particular way. Nevertheless, the way this study is referenced in other publications, \(\alpha\) is interpreted as an effective porosity. It should be noticed, however, that this study is performed with a background electrolyte of NaClO4. The only chloride (or iodide) present is therefore at trace level, and it cannot be excluded that a mechanism of true sorption influences the results (there are indications that this is the case in other studies).

For the present assessment we anyway assume that \(\alpha\) directly quantifies the anion equilibrium between clay and the external solution (i.e. equivalent to the incorrect way of assuming that \(\alpha\) quantifies a volume accessible to chloride). It should be kept in mind, though, that effects of anion equilibrium and potential true sorption is not resolved by the single parameter \(\alpha\).

In practice, then, the model is

\begin{equation} \frac{\partial c}{\partial t} = D_p\frac{\partial^2 c}{\partial x^2} \tag{1} \end{equation}

where \(c\) is the concentration in the clay of the isotope under consideration, and the diffusion coefficient is written \(D_p\) to acknowledge that it is a pore diffusivity (when referring to models and parameter evaluations in Mo03 we will use the notation “\(D_a\)”). The boundary conditions are

\begin{equation} c(0,t) = \alpha \cdot C_0 \;\;\;\; c(L,t) = 0 \tag{2} \end{equation}

where \(C_0\) is the concentration in the source reservoir,1 and \(L\) is the sample length.

This model — that we have discussed before — has a relatively simple analytical solution, and the outflux can be written

\begin{equation} j^\mathrm{out}(t) = j^\mathrm{ss}\left (1 + 2 \sum_{n=1}^\infty \left (-1 \right )^n e^{-\frac{\pi^2n^2D_pt}{L^2}} \right) \end{equation}

where \(j^\mathrm{ss}\) is the corresponding steady-state flux. Here, the steady-state flux is related to the other parameters as

\begin{equation} j^\mathrm{ss} = \alpha\cdot D_p \frac{C_0}{L} \tag{3} \end{equation}

“Fast” and “slow” processes

Oddly, Mo03 model the system as if two independent diffusion processes are simultaneously active. They refer to these as the “fast” and the “slow” processes, and hypothesize that they relate to diffusion in interlayer water2 and “interparticle water”,3 respectively.

The “fast” process is the “ordinary” process that is assumed to reach steady state during the course of the test, and that is the focus of other through-diffusion studies. The “slow” process, on the other hand, is introduced to account for the frequent observation that measured tracer profiles are usually significantly non-linear near the interface to the source reservoir (discussed briefly above). I guess that the reason for this concentration variation is due to swelling when the sample is unloaded. But even if the reason is not fully clear, it can be directly ruled out that it is the effect of a second, independent, diffusion process — because this is not how diffusion works!

If anions move both in interlayers and “interparticle water”, they reasonably transfer back and forth between these domains, resulting in a single diffusion process (the diffusivity of such a process depends on the diffusivity of the individual domains and their geometrical configuration). To instead treat diffusion in each domain as independent means that these processes are assumed to occur without transfer between the domains, i.e. that the bentonite is supposed to contain isolated “interlayer pipes”, and “interparticle pipes”, that don’t interact. It should be obvious that this is not a reasonable assumption. Incidentally, this is how all multi-porous models assume diffusion to occur (while simultaneously assuming that the domains are in local equilibrium…).

We will thus focus on the “fast” process in this assessment, although we also use the information provided by the parameters for the “slow” process. Mo03 report the fitted values for \(D_a\) and \(\alpha\) in a table (and diagrams), and only show a comparison between model and measured data in a single case: for iodide diffusion at 0.1 M background concentration and density 1.6 g/cm3. To make any kind of assessment of the quality of these estimations we therefore have to focus on this experiment (the article states that these results are “typical high clay density data”).

Outflux

The first thing to note is that the modeled accumulated diffusive substance does not correspond to the analytical solution for the diffusion process. Here is a figure of the experimental data and the reported model (as presented in the article), that also include the solution to eqs. 1 and 2.

In fact, the model presented in Mo03 has an incorrect time dependency in the early stages. Here is a comparison between the presented model and analytical solutions in the transient stage

With the given boundary conditions, the solutions to the diffusion equation inevitably has zero slope at \(t = 0\),4 reflecting that it takes a finite amount of time for any substance to reach the outflux boundary. The models presented in Mo03, on the other hand, has a non-zero slope in this limit. I cannot understand the reason for this (is it an underlying problem with the model, or just a graphical error?), but it certainly puts all reported parameter values in doubt.

The preferred way to evaluate diffusion data is, in my opinion, to look at the flux evolution rather than the evolution of the accumulated amount of diffused substance. Converting the reported data to flux, gives the following picture.5

From a flux evolution it is easier to establish the steady-state, as it reaches a constant. It furthermore gives a better understanding for how well constrained the model is by the data. As is seen from the figure, the model is not at all very well constrained, as the experimental data almost completely miss the transient stage. (And, again, it is seen that the model in the paper with \(D_a= 9\cdot 10^{-11}\) m/s2 does not correspond to the analytical solution.)

The short transient stage is a consequence of using thin samples (0.5 cm). Compared e.g. to Muurinen et al (1988), who used three times as long samples, the breakthrough time is here expected to be \(3^2 = 9\) times shorter. As Muurinen et al. (1988) evaluated breakthrough times in the range 1 — 9 days, we here expect very short times. Here are the breakthrough times for all chloride diffusion tests, evaluated from the reported diffusion coefficients (“fast” process) using the formula \(t_\mathrm{bt} = L^2/(6D_a)\).

Test\(D_a\)\(t_\mathrm{bt}\)
(m2/s) (days)
0.4/0.01\(8\cdot 10^{-10}\)0.06
0.4/0.1 \(9\cdot 10^{-10}\) 0.05
0.4/0.1 \(8\cdot 10^{-10}\) 0.06
0.8/0.01 \(3.5\cdot 10^{-10}\) 0.14
0.8/0.1 \(3.5\cdot 10^{-10}\) 0.14
0.8/0.1 \(3.7\cdot 10^{-10}\) 0.13
1.2/0.01 \(1.4\cdot 10^{-10}\)0.34
1.2/0.1 \(2.3\cdot 10^{-10}\) 0.21
1.2/0.1 \(2.0\cdot 10^{-10}\) 0.24
1.6/0.1 \(1.0\cdot 10^{-10}\) 0.48
1.8/0.01 \(2\cdot 10^{-11}\) 2.41
1.8/0.1 \(5\cdot 10^{-11}\) 0.96
1.8/0.1 \(5.5\cdot 10^{-11}\) 0.88

The breakthrough time is much shorter than a day in almost all tests! To sample the transient stage properly requires a sampling frequency higher than \(1/t_{bt}\). As seen from the provided example of a outflux evolution, this is not the case: The second measurement is done after about 1 day, while the breakthrough time is about 0.5 days (moreover, the first measurement appears as an outlier). We have no information on sampling frequency in the other tests, but note that to properly sample e.g. the tests at 0.8 g/cm3 requires measurements at least every third hour or so. For 0.4 g/cm3, the required sample frequency is once an hour! This design choice puts more doubt on the quality of the evaluated parameters.

Concentration profile

The measured concentration profile across the 1.6/0.1 iodide sample, and corresponding model results are presented in Mo03 in a figure very similar to this

Here the two models correspond to the “slow” and “fast” process discussed above (a division, remember, that don’t make sense). Zooming in on the “linear” part of the profile, we can compare the “fast” process with analytical solutions (eqs. 1 and 2)

The analytical solutions correspond directly to the outflux curves presented above. We note that the analytical solution with \(D_p = 9\cdot 10^{-11}\) m/s2 corresponds almost exactly to the model presented by Mo03. As this model basically has the same steady state flux and diffusion coefficient, we expect this similarity. It is, however, still a bit surprising, since the corresponding outflux curve of the model in Mo03 was seen to not correspond to the analytical solution. This continues to cast doubt on the model used for evaluating the parameters.

We furthermore note that the evolution of the activity of the source reservoir is not reported. Once in the text is mentioned that the “carrier concentration” is \(10^{-6}\) M, but since we don’t know how much of this concentration corresponds to the radioactive isotope, we can not directly compare with reported concentration profile across the sample (whose concentration unit is counts per minute per cm3). By extrapolating the above model curve with \(\alpha = 0.15\), we can however deduce that the corresponding source activity for this particular sample is \(C_0 = 1.26\cdot 10^5/0.15\) cpu/cm3 \(= 8.40\cdot 10^5\) cpu/cm3. But it is unsatisfying that we cannot check this independently. Also, we can of course not assume that this value of \(C_0\) is the same in any other of the tests (in particular those involving chloride). We thus lack vital information (\(C_0\)) to be able to make a full assessment of the model fitting.

It should furthermore be noticed that the experimental concentration profile does not constrain the models very well. Indeed, the adopted model (diffusivity \(9\cdot 10^{-11}\) m/s2) misses the two rightmost concentration points (which corresponds to half the sample!). A model that fits this part of the profile has a considerable higher diffusivity, and a correspondingly lower \(\alpha\) (note that the product \(D_p\cdot \alpha\) is constrained by the steady-state flux, eq. 3).

More peculiarities of the modeling is found if looking at the “slow” process (remember that this is not a real diffusion process!). Zooming in on the interface part of the profile and comparing with analytical solutions gives this picture

Here we note that an analytical solution coincides with the model presented in Mo03 with parameters \(D_a = 6\cdot 10^{-14}\) m2/s and \(\alpha = 1.12\) only if it is propagated for about 15 days! Given that no outflux measurements seem to have been performed after about 4 days (see above), I don’t now what to make of this. Was the test actually conducted for 15 days? If so, why is not more of the outflux measured/reported? (And why were the samples then designed to give a breakthrough time of only a few hours?)

Without knowledge of for how long the tests were conducted, the reported diffusion parameters becomes rather arbitrary, especially for the low density samples. For e.g. the samples of density 0.4 g/cm3, even the “slow” process has a diffusivity high enough to reach steady-state within a few days. Simulating the processes with the reported parameters gives the following profiles if evaluated after 1 and 4 days, respectively

The line denoted “total” is what should resemble the measured (unreported) data. It should be clear from these plots that the division of the profile into two separate parts is quite arbitrary. It follows that the evaluated diffusion parameters for the process of which we are interested (“fast”) has little value.

Summary and verdict

We have seen that the reported model fitting leaves a lot of unanswered questions: some of the model curves don’t correspond to the analytical solutions, information on evolution times and source concentrations is missing, and the modeled profiles are divided quite arbitrary into two separate contributions (which are not two independent diffusion process).

Moreover, the ion population (divalent vs. monovalent cations) of the samples are not known, but there are strong reasons to believe that the 0.01 M tests contain a significant amount of divalent ions, while the 0.1 M samples are partly converted to a more pure sodium state.

Also, the small size of the samples contributes to more uncertainty, both in terms of density, but also for the flux evolution because the breakthrough times becomes very short.

Based on all of these uncertainties, I mean that the results of Mo03 does not contribute to quantitative process understanding and my decision is to not to use the study for e.g. validating models of anion exclusion.

A confirmation of the uncertainty in this study is given by considering the density dependence on the chloride equilibrium concentrations for constant background concentration, evaluated from the reported diffusion parameters (\(\alpha\) for the “fast” process).

If these results should be taken at face value, we have to accept a very intricate density dependence: for 0.1 M background, the equilibrium concentration is mainly constant between densities 0.3 g/cm3 and 0.7 g/cm3, and increases between densities 1.0 g/cm3 and 1.45 g/cm3 (or, at least, does not decrease). For 0.01 M background, the equilibrium concentration instead falls quite dramatically between between densities 0.3 g/cm3 and 0.7 g/cm3, and thereafter displays only a minor density dependence.

To accept such dependencies, I require a considerably more rigorous experimental procedure and evaluation. In this case, I rather view the above plot as a confirmation of large uncertainties in parameter evaluation and sample properties.

Footnotes

[1] Strictly, \(c(0,t)\) relates to the concentration in the endpoint of the inlet filter. But we ignore filter resistance in this assessment, which is valid for the 1.6/0.1 sample. Moreover, the filter diffusivities are not reported in Mo03.

[2] Mo03 refer to interlayer pores as “intralayer” pores, which may cause some confusion.

[3] Apparently, the authors assume an underlying stack view of the material.

[4] It may be objected that the analytical solution do not include the filter resistance. But note that filter resistance only will increase the delay. Moreover, the transport capacity of the sample in this test is so low that filters have no significant influence.

[5] The model by Mo03 looks noisy because I have read off values of accumulated concentration from the published graph. The “noise” occurs because the flux is evaluated from the concentration data by the difference formula:

\begin{equation} \bar{j}(\bar{t}_i) =\frac{1}{A} \frac{a(t_{i+1})-a(t_i)}{t_{i+1}-t_{i}} \end{equation}

where \(t_i\) and \(t_{i+1}\) are the time coordinates for two consequitive data points, \(a(t)\) is the accumulated amount diffused substance at time \(t\), \(A\) is the cross sectional area of the sample, \(\bar{t}_i = (t_{i+1} + t_i)/2\) is the average time of the considered time interval, and \(\bar{j}\) denotes the average flux during this time interval.