Electrostatics can be quite subtle. The following comment on the
interlayer ion distribution, in
Kjellander et
al. (1988), was an eye-opener for me
The ion concentration profile is determined by the net force acting
on each ion. The electrostatic potential from the uniform surface
charges is constant between the two walls, which means that the
forces due to these charges cancel each other completely. Thus, the
large counter-ion concentration in the electric double layer near the
walls is solely a consequence of the repulsive interactions between
the ions.
Interlayer cations are not attracted to the surfaces, but are pushed
towards them due to repulsion between the ions themselves! My
intuition has been that interlayer counter-ions distribute due to
attraction with the surfaces, but the perspective given in the above
quotation certainly makes a lot of sense. Here I use the word
“perspective” because I don’t fully agree with the statement that
the ion distribution is solely a consequence of repulsion. To
discuss the issue further, let’s flesh out the reasoning in
Kjellander et
al. (1988) and draw some pictures.
Here we discuss an idealized model of an interlayer as a dielectric continuum sandwiched between two parallel infinite planes of uniform surface charge density.1 The system is thus symmetric around the axis normal to the surfaces (the model is one-dimensional).
From this result follows that the electric field vanishes between two
equally negatively charged surfaces. The electrostatic field
configuration for an “empty” interlayer can thus be illustrated like
this
This means that the two interlayer surfaces don’t “care” about the
counter-ions, in the sense that this part of the electrostatic energy
(ion – surfaces) is independent of the counter-ion distribution.
To consider the fate of the counter-ions we continue to explore the axial symmetry. The counter-ion distribution varies only in the direction normal to the surfaces, and we can treat it as a sequence of thin parallel planes of uniform charge. Since the size of the electric field from such planes is independent of distance, the force on a positive test charge (= the electric field) at any position in the interlayer depends only on the difference in total amount of charge on each side of this position, as illustrated here
This, in turn, implies both that the electric field is zero at the mid position, and that the electric field elsewhere is directed towards the closest surface (since symmetry requires equal amount of charge in the two halves of the interlayer2). The counter-ions indeed repel each other towards the surfaces! The charge density must therefore increase towards the surfaces, and we understand that the equilibrium electric field qualitatively must look like this3
However, as far as I see, the “indifference” of the surfaces to the
counter-ions is a matter of perspective. Consider e.g. making the
interlayer distance very large. In this limit, the system is more
naturally conceptualized as two single surfaces. It is then awkward to
describe the ion distribution at one surface as caused by repulsion
from other ions arbitrarily far away, rather than as caused by
attraction to the surface. But for the case most relevant
for compacted bentonite — i.e. interlayers, or what is often
described as “overlapping” electric double layers — the natural
perspective is that counter-ions distribute as a consequence of
repulsion among themselves.
This perspective also implies that anions (co-ions) distribute within the interlayer as a consequence of attraction to counter-ions rather than repulsion from the surfaces! (The above figure applies, with all arrows reversed.) This insight should not be confused with the fact that repulsion between anions and surfaces is not really the mechanism behind “anion exclusion”. Rather, the implication here is that anion-surface repulsion can be viewed as not even existing within an interlayer.
[R]ather than contributing to repulsion, electrostatic interactions
actually reduce the pressure. This is clearly seen from e.g. the
Poisson-Boltzmann solution for two charged surfaces, where the
resulting osmotic pressure corresponds to an ideal solution with a
concentration corresponding to the value at the midpoint (cf. the
quotation from Kjellander et al. (1988) above). But the midpoint
concentration — and hence the osmotic pressure — is lowered as
compared with the average, because of electrostatic attraction
between layers and counter-ions.
But the final sentence should rather be formulated as
But the midpoint concentration — and hence the osmotic pressure — is
lowered as compared with the average, because of electrostatic
repulsion between the counter-ions.
In the original post, I also write
This plot demonstrates the attractive aspect of electrostatic
interactions in these systems. While the NaCl pressure is only
slightly reduced, Na-montmorillonite shows strong non-ideal
behavior. In the “low” concentration regime (< 2 mol/kgw) we
understand the pressure reduction as an effect of counter-ions
electrostatically attracted to the clay surfaces.
The last part is better formulated as
In the “low” concentration regime (< 2 mol/kgw) we understand the
pressure reduction as an effect of electrostatic repulsion among
the counter-ions.
I think the implication here is quite wild: In a sense, electrostatic
repulsion reduces swelling pressure!
Footnotes
[1] The treatment in Kjellander et al. (1988) is more advanced, including effects of image charges and ion-ion correlations, but it does not matter for the present discussion.
[2] Actually, the whole distribution is required to be
symmetric around the interlayer midpoint.
[3] The quantitative picture is of course achieved from solving the Poisson-Boltzmann equation. The picture may be altered when considering more involved mechanisms, such as image charge interactions or ion-ion correlations; Kjellander et al. (1988) show that the effect of image charges may reduce the ion distribution at very short distances, while the effect of ion-ion correlations is to further increase the accumulation towards the surfaces. Note that neither of these effects involve direct interaction with the surface charge.
Mu07 actually report 9 more data points, but these originate from Muurinen et al. (2004) (which we have already assessed). This is not fully acknowledged in Mu07, but below I try to sort out the status of all data presented. We refer to Muurinen et al. (2004) as Mu04.
In similarity to Mu04, Mu07 is an equilibrium study (i.e. not a diffusion study) performed on purified “MX-80” bentonite. One of the main objectives in Mu07 is to investigate possible influence of sample preparation on the chloride equilibrium concentrations. The samples in Mu07 cover a large density range (0.6 — 1.5 g/cm3), but were all equilibrated with a single type of solution: 0.1 M NaCl.
Originally conducted and already reported tests
Mu07 state that the study is a continuation of the investigations
presented in Mu04 and present data on five different sets of samples,
prepared and equilibrated using different methods (labeled A —
E). What is not explicitly stated — but what is obvious if comparing
tables 1 and 2 in Mu04 with table 1 in Mu07 — is that sample sets D
and E are the same as previously reported in Mu04.
I find this quite remarkable, since two of these samples were
dismissed as “not reliable” in Mu04
(In my
assessment, I dismissed all tests in Mu04); here the same results
— which show an increase in equilibrium chloride content with
density — are not only re-reported, but modeled! The authors don’t
even seem aware that they have previously discarded the samples,
writing: “Surprisingly, it seems that the concentrations in the
sample types D and E start to increase at the highest
densities“. Furthermore, one of the (previously reported) data points
of sample set E, which have a clay concentration larger than
the corresponding concentration of the equilibrating solution, is
not included in Mu07. Needless to say, excluding data points
without motivation, or including previously discarded data is not good
scientific practice.
As the sample overview table in Mu07 also has some misprints,1 I here present a (hopefully) correct version that also indicates original publication (for the indicated sample-IDs, see the assessment of Mu04).
Type
Density (g/cm3)
Time d.w. (days)
Time 0.1 M (days)
Clay conc. (mM)
origin/remark
A
0.625
18+35
217
37
Mu07
A
0.812
18+35
217
29
Mu07
A
1.200
18+35
217
20
Mu07
B
0.670
222
107
35
Mu07
B
0.937
222
107
23
Mu07
B
1.389
222
107
15
Mu07
C
0.622
30
36
48
Mu07
C
0.731
30
36
30
Mu07
C
1.113
30
36
17
Mu07
C
1.382
30
36
14
Mu07
C
1.517
30
36
12
Mu07
D
0.754
0
40
65
Mu04 (S2-02)
D
0.855
0
40
39
Mu04 (S2-21)
D
1.273
0
36
22
Mu04 (S2-04)
D
1.636
0
36
24
Mu04 (S2-17; deemed “not reliable” in Mu04)
D
1.764
0
85
48
Mu04 (S2-18; deemed “not reliable” in Mu04)
E
0.750
0
12(?)
109
Mu04 (not included!)
E
0.875
0
12
61
Mu04
E
1.225
0
12
25
Mu04
E
1.516
0
12
12
Mu04
E
1.543
0
12
14
Mu04
In the following, the focus is solely on samples sets A — C (as
mentioned, the others have already been
assessed).
Material
The material appear to be the same as used in Mu04. I therefore refer to the assessment of that study for a detailed discussion. In brief, the material is purified “MX-80” bentonite, with a montmorillonite content above 90% and about 90% sodium as exchangeable cation.
Samples
Samples in the three different sample sets A — C were prepared in different ways. For set A, the clay was initially dispersed in deionized water at quite low density. After an equilibration time of 18 days (which included ultrasound treatment), the dispersion was slowly squeezed to achieve the intended densities. This squeezing phase lasted 35 days, after which the samples were contacted with 0.1 M NaCl and equilibrated for 217 days.
Samples in set B were prepared in the same type of sample holder as
those in set A, but the bentonite powder was directly compacted to the
desired density, and the samples were water saturated by contact with
deionized water for 222 days (!). Thereafter, the samples were
contacted with 0.1 M NaCl and equilibrated for 107 days.
The external solution was not circulated in the preparation of samples
in sets A and B. In contrast, samples in set C were prepared in cells
with external circulation. The bentonite powder was directly
compacted to the desired density, and the samples were water
saturated by contact with (circulating) deionized water for 30
days. The samples were then equilibrated with (circulating) 0.1 M NaCl
solution for 36 days.
Even if the preparation protocols are described quite detailed in Mu07, we are not given any information on sample geometry. We are not even told if the samples have the same geometry! (Given that they were prepared in different types of equipment, different geometries may certainly be the case.) Without knowledge on e.g. the characteristic diffusion lengths, it is impossible to assess e.g. whether the adopted equilibration times are adequate. Reasonably, the size of the samples are on the cm scale, and since the equilibration times are very long, we can guess that they have had time to equilibrate. This is in contrast to the samples in Mu04, which we have reason to suspect have not been completely equilibrated, as discussed in the assessment of that study. (Note that these samples are included in Mu07, as sample sets D and E.)
Mu07 does not provide any information on how sample density was
measured. Since we neither know the dimensions of the samples it is
therefore impossible to estimate any uncertainty of the reported
densities.
Chloride equilibrium concentrations
The following plot summarizes the reported chloride equilibrium
concentrations and corresponding densities in sample sets A — C.
Although the data show some significant scatter (e.g. for the two
lowest densities in sample set C), the main impression is that the
three different ways of preparing and equilibrating samples result in
quite similar values for the chloride equilibrium concentrations. Thus,
even if we know little about the samples, this coherence in the
results indicates that they have been properly equilibrated.
Possible interface excess salt
As we have discussed in
severalpreviousblog
posts, when performing equilibrium tests it is important to handle
the possibility that the samples have an increased salt content in the
interface regions. In the assessment of Mu04, my guess was that the
samples had not been handled specifically to deal with this possible
measuring artifact, and I neither see any reason to believe that this
issue has been addressed in sample sets A — C (we can, however, rule
out that too much salt entered these samples during saturation, since
deionized water was used in this phase).
The possible influence of interface excess depends, apart from general
sample treatment, on e.g. sample thickness and the concentration of
the equilibrating external solution. As noted above, we have no
information on sample thickness, but the external concentration is in
this regard quite low (we showed in an earlier post that the problem
of interface excess salt becomes more severe for thin samples and low
external concentrations). Therefore, we can certainly not exclude the
possibility that the reported equilibrium concentrations are
systematically overestimated due to possible influence of an interface
excess, especially for the denser samples (see
here for details on this).
An argument against that interface excess has significantly influenced
the results is the similar result for the three different sample
sets. Of course, this depends on how similar (or dissimilar) the
samples in the different sets are, of which we have no
information. Under any circumstance, it is very clear that Mu07
provides too little information to fully rely on the reported values.
Summary and verdict
From one perspective, Mu07 is a very straightforward study: samples of
purified bentonite (almost pure Na-montmorillonite) at various density
have been equilibrated with a single type of external solution (0.1 M
NaCl). The results also look reasonably coherent. However, the paper
contains way too little information on e.g. sample geometry and how
density and concentration were measured to fully rely on the
results. In particular, we cannot rule out a systematic overestimation
due to influence of interface excess salt. Furthermore, the main
reason to believe that equilibrium was achieved, is the
similarity between the different test sets.
My decision, however, is to keep these result to use e.g. for possible
qualitative process understanding (specifically, chloride exclusion).
But I will certainly keep in mind the quite extensive lack of
information associated with this data.
To celebrate that I have built myself a tool for solving the Poisson-Boltzmann equation for two parallel charged plates and specified external solution, here is a cosy little animation
The animation shows the anion concentration profile (blue) between the plates as the distance varies, in systems in equilibrium with an external 100 mM 1:1 salt solution. Also plotted is the corresponding internal concentration level as calculated from the ideal Donnan equilibrium formula (orange). The layer charge density in the Poisson-Boltzmann calculation is 0.111 C/m2, and the corresponding cation exchange capacity in the Donnan calculation is 0.891 eq/kg.
As the distance between the plates increases, the Poisson-Boltzmann profile increasingly deviates from the Donnan concentration. At lower density (larger plate distance) it is clear that the Poisson-Boltzmann solution allows for considerably more anions between the plates as compared with the Donnan result. On the other hand, for denser systems, the difference between the two solutions decreases; this is especially true when considering the relative difference — keep in mind that the external concentration is kept constant, at 100 mM.
In fact, in systems relevant for e.g. radioactive waste storage — spanning an effective montmorillonite density range from \(\rho_\mathrm{mmt} =\) 1.60 g/cm3 to \(\rho_\mathrm{mmt} =\) 1.15 g/cm3, say — the difference between the Poisson-Boltzmann and the Donnan results is virtually negligible (it should also be kept in mind that the continuum assumption underlying the Poisson-Boltzmann calculation is not valid in this density range). Here are plotted snapshots of these two limiting cases, together with the Poisson-Boltzmann solution for a single plate (the Gouy-Chapman model)
Vl07 is centered around a set of through-diffusion tests in “KWK” bentonite samples of nominal dry densities 1.3 g/cm3, 1.6 g/cm3, and 1.9 g/cm3. For each density, chloride tracer diffusion tests were conducted with NaCl background concentrations 0.01 M, 0.05 M, 0.1 M, 0.4 M, and 1.0 M. In total, 15 samples were tested. The samples are cylindrical with diameter 2.54 cm and height 1 cm, giving an approximate volume of 5 cm3. We refer to a specific test or sample using the nomenclature “nominal density/external concentration”, e.g. the sample of density 1.6 g/cm3 contacted with 0.1 M is labeled “1.6/0.1”.
After maintaining steady-state, the external solutions were replaced
with tracer-free solutions (with the same background concentration),
and tracers in the samples were allowed to diffuse out. In this way,
the total tracer amount in the samples at steady-state was
estimated. For tests with background concentrations 0.01 M, 0.1 M, and
1.0 M, the outflux was monitored in some detail, giving more
information on the diffusion process. After finalizing the tests, the
samples were sectioned and analyzed for stable (non-tracer)
chloride. In summary, the tests were performed in the following
sequence
Saturation stage
Through-diffusion stage
Transient phase
Steady-state phase
Out-diffusion stage
Sectioning
Uncertainty of samples
The used bentonite material is referred to as “Volclay KWK”. Similar
to
“MX-80”, “KWK” is just a brand name (it seems to be used mainly
in wine and juice
production). In contrast to “MX-80”, “KWK” has been used in
only a fewresearchstudies related to
radioactive waste storage. Of the studies I’m aware, only
Vejsada et
al. (2006) provide some information relevant
here.1
Vl07 state that “KWK” is similar to “MX-80” and present a table with chemical composition and exchangeable cation population of the bulk material. As the chemical composition in this table is identical to what is found in various “technical data sheets”, we conclude that it does not refer to independent measurements on the actual material used (but no references are provided). I have not been able to track down an exact origin of the stated exchangeable cation population, but the article gives no indication that these are original measurements (and gives no reference). I have found a specification of “Volclay bentonite” in this report from 1978(!) that states similar numbers (this document also confirms that “MX-80” and “KWK” are supposed to be the same type of material, the main difference being grain size distribution). We assume that exchangeable cations have not been determined explicitly for the material used in Vl07.
In a second table, Vl07 present a mineral composition of “KWK”, which I assume has been determined as part of the study. But this is not fully clear, as the only comment in the text is that the composition was “determined by XRD-analysis”. The impression I get from the short material description in Vl07 is that they rely on that the material is basically the same as “MX-80” (whatever that is).
Montmorillonite content
Vl07 state a smectite content of about
70%. Vejsada et
al. (2006), on the other hand, state a smectite content of 90%,
which is also stated in the 1978 specification of “Volclay
bentonite”. Note that 70% is lower and 90% is higher than any
reported montmorillonite content in “MX-80”. Regardless whether or
not Vl07 themselves determined the mineral content, I’d say that the
lack of information here must be considered when estimating an
uncertainty on the amount of montmorillonite (“smectite”) in the
used material. If we also consider the claim that “KWK” is similar
to “MX-80”, which has a documented montmorillonite content
in the range 75 — 85%, an uncertainty range for “KWK” of 70 —
90% is perhaps “reasonable”.
Cation population
Vl07 state that the amount exchangeable sodium is in the range 0.60 —
0.65 eq/kg, calcium is in the range 0.1 — 0.3 eq/kg, and magnesium is
in the range 0.05 — 0.2 eq/kg. They also state a cation exchange
capacity in the range 0.76 — 1.2 eq/kg, which seems to have been
obtained from just summing the lower and upper limits, respectively,
for each individual cation. If the material is supposed to be similar
to “MX-80”, however, it should have a cation exchange capacity in
the
lower regions of this range. Also,
Vejsada et
al. (2006) state a cation exchange capacity of 0.81 eq/kg. We
therefore assume a cation exchange capacity in the range 0.76 — 0.81,
with at least 20% exchangeable divalent ions.
Soluble accessory minerals
According to Vl07, “KWK” contains substantial amounts of accessory carbonate minerals (mainly calcite), and Vejsada et al. (2006) also state that the material contains calcite. The large spread in calcium and magnesium content reported for exchangeable cations can furthermore be interpreted as an artifact due to dissolving calcium- and magnesium minerals during the measurement of exchangeable cations (but we have no information on this measurement). Vl07 and Vejsada et al. (2006) do not state any presence of gypsum, which otherwise is well documented in “MX-80”. I do not take this as evidence for “KWK” being gypsum free, but rather as an indication of the uncertainty of the composition (the 1978 specification mentions gypsum).
Sample density
Vl07 don’t report measured sample densities (the samples are ultimately sectioned into small pieces), but estimate density from the water uptake in the saturation stage. The reported average porosity intervals are 0.504 — 0.544 for the 1.3 g/cm3 samples, 0.380 — 0.426 for the 1.6 g/cm3 samples, and 0.281 — 0.321 for the 1.9 g/cm3 samples. Combining these values with the estimated interval for montmorillonite content, we can derive an interval for the effective montmorillonite dry density by combining extreme values. The result is (assuming grain density 2.8 g/cm3, adopted in Vl07).
Sample density (g/cm3)
EMDD interval (g/cm3)
1.3
1.04 — 1.32
1.6
1.36 — 1.67
1.9
1.67 — 1.95
These intervals must not be taken as quantitative estimates, but as giving an idea of the uncertainty.
Uncertainty of external solutions
Samples were water saturated by first contacting them from one side with the appropriate background solution (NaCl). From the picture in the article, we assume that this solution volume is 200 ml. After about one month, the samples were contacted with a second NaCl solution of the same concentration, and the saturation stage was continued for another month. The volume of this second solution is harder to guess: the figure shows a smaller container, while the text in the figure says “200 ml”. The figure shows the set-up during the through-diffusion stage, and it may be that the containers used in the saturation stage not at all correspond to this picture. Anyway, to make some sort of analysis we will assume the two cases that samples were contacted with solutions of either volume 200 ml, or 400 ml (200 ml + 200 ml) during saturation.
The through-diffusion tests were started by replacing the two saturating solutions: on the left side (the source) was placed a new 200 ml NaCl solution, this time spiked with an appropriate amount of 36Cl tracers, and on the right side (the target) was placed a fresh, tracer free NaCl solution of volume 20 ml. The through-diffusion tests appear to have been conducted for about 55 days. During this time, the target solution was frequently replaced in order to keep it at a low tracer concentration. The source solution was not replaced during the through-diffusion test.
As (initially) pure NaCl solutions are contacted with bentonite that contains significant amounts of calcium and magnesium, ion exchange processes are inevitably initiated. Thus, in similarity with some of the earlierassessed studies, we don’t have full information on the cation population during the diffusion stages. As before, we can simulate the process to get an idea of this ion population. In the simulation we assume a bentonite containing only sodium and calcium, with an initial equivalent fraction of calcium of 0.25 (i.e. sodium fraction 0.75). We assume sample volume 5 cm3, cation exchange capacity 0.785 eq/kg, and Ca/Na selectivity coefficient 5.
Below is shown the result of equilibrating an external
solution of either 200 or 400 ml with a sample of density 1.6 cm3/g,
and the corresponding result for density 1.3 cm3/g and external volume
400 ml. As a final case is also displayed the result of first
equilibrating the sample with a 400 ml solution, and then replacing it
with a fresh 200 ml solution (as is the procedure when the
through-diffusion test is started).
Although the results show some spread, these simulations make it relatively clear that the ion population in tests with the lowest background concentration (0.01 M) probably has not changed much from the initial state. In tests with the highest background concentration (1.0 M), on the other hand, significant exchange is expected, and the material is consequently transformed to a more pure sodium bentonite. In fact, the simulations suggest that the mono/divalent cation ratio is significantly different in all tests with different background concentrations.
Note that the simulations do not consider possible dissolution of accessory minerals and therefore may underestimate the amount divalent ions still left in the samples. We saw, for example, that the material used in Muurinen et al. (2004) still contained some calcium and magnesium although efforts were made to convert it to pure sodium form. Note also that the present analysis implies that the mono/divalent cation ratio probably varies somewhat in each individual sample during the course of the diffusion tests.
Direct measurement of clay concentrations
Chloride
clay concentration profiles were measured in all samples after
finishing the diffusion tests, by dispersing sample sections in
deionized water. Unfortunately, Vl07 only present this chloride
inventory in terms of “effective” or
“Cl-accessible porosity”, a concept often encountered in
evaluation of diffusivity. However, “effective porosity” is
not what is measured, but is rather an interpretation of
the evaluated amount of chloride in terms of a certain pore volume
fraction. Vl07 explicitly define effective porosity as
\(V_\mathrm{Cl}/V_\mathrm{1g}\), where \(V_\mathrm{1g}\) is the “volume
of a unit mass of wet bentonite”, and \(V_\mathrm{Cl}\) is the “volume
of the Cl-accessible pores of a unit mass of bentonite”. While
\(V_\mathrm{1g}\) is accessible experimentally, \(V_\mathrm{Cl}\) is
not. Vl07 further “derive” a formula for the effective porosity
(called \(\epsilon_\mathrm{eff}\) hereafter)
where \(n’_\mathrm{Cl}\) is the amount chloride per mass bentonite, \(\rho_\mathrm{Rf}\) is the density of the “wet” bentonite, and \(C_\mathrm{bkg}\) is the background NaCl concentration.2 In contrast to \(V_\mathrm{Cl},\) these three quantities are all accessible experimentally, and the concentration \(n’_\mathrm{Cl}\) is what has actually been measured. For a result independent of how chloride is assumed distributed within the bentonite, we thus multiply the reported values of \(\epsilon_\mathrm{eff}\) by \(C_\mathrm{bkg}\), which basically gives the (experimentally accessible) clay concentration
Here we also have divided by sample porosity, \(\phi\), to relate the clay concentration to water volume rather than total sample volume. Note that eq. 2 is not derived from more fundamental quantities, but allows for “de-deriving” a quantity more directly related to measurements. (I.e., what is reported as an accessible volume is actually a measure of the clay concentration.)
It is, however, impossible (as far as I see) to back-calculate the actual value of \(n’_ \mathrm{Cl}\) from provided formulas and values of \(\epsilon_\mathrm{eff}\), because masses and volumes of the sample sections are not provided. Therefore, we cannot independently assess the procedure used to evaluate \(\epsilon_\mathrm{eff}\), and simply have to assume that it is adequate.3 Here are the reported values of \(\epsilon_\mathrm{eff}\) for each test, and the corresponding evaluation of \(\bar{C}\) using eq. 2 (column 3)
*) The table in Vl07 says 0.076, but the concentration profile diagram says 0.090. **) The table in Vl07 says 0.16, but this must be a typo.
When using eq. 2 we have adopted porosities 0.536, 0.429, and 0.322,
respectively, for densities 1.3 g/cm3, 1.6 g/cm3, and 1.9 g/cm3.
The tabulated \(\epsilon_\mathrm{eff}\) values are evaluated as averages of the clay concentration profiles (presented as effective porosity profiles), which look like this for the samples exposed to background concentrations 0.01 M, 0.1 M and 1.0 M (profiles for 0.05 M and 0.4 M are not presented in Vl07)
The chloride concentration increases near the interfaces in all samples; we have discussed this interface excess effect in previousposts. Vl07 deal with this issue by evaluating the averages only for the inner parts of the samples. I performed a similar evaluation, also presented in the above figures (blue lines). In this evaluation I adopted the criterion to exclude all points situated less than 2 mm from the interfaces (Vl07 seem to have chosen points a bit differently). The clay concentration reevaluated in this way is also listed in the above table (last column). Given that I have only used nominal density for each sample (I don’t have information on the actual density of the sample sections), I’d say that the re-evaluated values agree well with those de-derived from reported \(\epsilon_\mathrm{eff}\). One exception is the sample 1.9/0.01, which is seen to have concentration points all over the place (or maybe detection limit is reached?). While Vl07 choose the lowest three points in their evaluation, here we choose to discard this result altogether. I mean that it is rather clear that this concentration profile cannot be considered to represent equilibrium.
As the reevaluation gives similar values as those reported, and since
we lack information for a full analysis, we will use the values
de-derived from reported \(\epsilon_\mathrm{eff}\) in the continued
assessment (except for sample 1.9/0.01).
Diffusion related estimations
Vl07 determine diffusion parameters by fitting various mathematical expressions to flux data.4 Parameters fitted in this way generally depend on the underlying adopted model, and we have discussed how equilibrium concentrations can be extracted from such parameters in an earlier blog post. In Vl07 it is clear that the adopted mathematical and conceptual model is the effective porosity diffusion model. When first presented in the article, however, it is done so in terms of a sorption distribution coefficient (\(R_d\)) that is claimed to take on negative values for anions. The presented mathematical expressions therefore contain a so-called rock capacity factor, \(\alpha\), which relates to \(R_d\) as \(\alpha = \phi + \rho_d\cdot R_d\). But such use of a rock capacity factor is a mix-up of incompatible models that I have criticized earlier. However, in Vl07 the description involving a sorption coefficient is in words only — \(R_d\) is never brought up again — and all results are reported, interpreted and discussed in terms of effective (or “chloride-accessible”) porosity, labeled \(\epsilon\) or \(\epsilon_\mathrm{Cl}\). We here exclusively use the label \(\epsilon_\mathrm{eff}\) when referring to formulas in Vl07. The mathematics is of course the same regardless if we call the parameter \(\alpha\), \(\epsilon\), \(\epsilon_\mathrm{Cl}\), or \(\epsilon_\mathrm{eff}\).
Mass balance in the out-diffusion stage
Vl07 measured the amount of tracers accumulated in the two reservoirs during the out-diffusion stage. The flux into the left side reservoir, which served as source reservoir during the preceding through-diffusion stage, was completely obscured by significant amounts of tracers present in the confining filter, and will not be considered further (also Vl07 abandon this flux in their analysis). But the total amount of tracers accumulated in the right side reservoir, \(N_\mathrm{right}\),5 can be used to directly estimate the chloride equilibrium concentration.
The initial concentration profile in the out-diffusion stage is linear (it is the steady-state profile), and the total amount of tracers, \(N_\mathrm{tot}\),6 can be expressed
where \(\bar{c}_0\) is the initial clay concentration at the left side interface, and \(V_\mathrm{sample}\) (\(\approx\) 5 cm3) is the sample volume.
A neat feature of the out-diffusion process is that two thirds of the
tracers end up in the left side reservoir, and one third in the right
side reservoir, as illustrated in this simulation
\(\bar{c}_0\) can thus be estimated by using
\(N_\mathrm{tot} = 3\cdot N_\mathrm{right}\) in eq. 3, giving
where \(c_\mathrm{source}\) is the tracer concentration in the left side reservoir in the through-diffusion stage.7 Although eq. 4 depends on a particular solution to the diffusion equation, it is independent of diffusivity (the diffusivity in the above simulation is \(1\cdot 10^{-10}\) m2/s). Eq. 4 can in this sense be said to be a direct estimation of \(\bar{c}_0\) (from measured \(N_\mathrm{right}\)), although maybe not as “direct” as the measurement of stable chloride, discussed previously.
Vl07 state eq. 4 in terms of a “Cl-accessible porosity”, but this is still just an interpretation of the clay concentration; \(\bar{c}_0\) is, in contrast to \(\epsilon_\mathrm{eff}\), directly accessible experimentally in principle. From the reported values of \(\epsilon_\mathrm{eff}\) we may back-calculate \(\bar{c}_0\), using the relation \(\bar{c}_0 / c_\mathrm{source} = \epsilon_\mathrm{eff}/\phi\). Alternatively, we may use eq. 4 directly to evaluate \(\bar{c}_0\) from the reported values of \(N_\mathrm{right}\). Curiously, these two approaches result in slightly different values for \(\bar{c}_0/c_\mathrm{source}\). I don’t understand the cause for this difference, but since \(N_\mathrm{right}\) is what has actually been measured, we use these values to estimate \(\bar{c}_0.\) The resulting equilibrium concentrations are
Test
\(N_\mathrm{right}\) (10-10 mol)
\(\bar{c}_0/c_\mathrm{source}\) (-)
1.3/0.01
4.10
0.038
1.3/0.05
10.2
0.097
1.3/0.1
17.8
0.168
1.3/0.4
41.4
0.395
1.3/1.0
52.4
0.445
1.6/0.01
1.21
0.014
1.6/0.05
3.64
0.043
1.6/0.1
6.15
0.072
1.6/0.4
13.0
0.154
1.6/1.0
21.6
0.225
1.9/0.01
0.41
0.006
1.9/0.05
1.14
0.018
1.9/0.1
1.64
0.025
1.9/0.4
3.19
0.051
1.9/1.0
8.19
0.113
We have now investigated two independent estimations of the chloride equilibrium concentrations: from mass balance of chloride tracers in the out-diffusion stage, and from measured stable chloride content. Here are plots comparing these two estimations
The similarity is quite extraordinary! With the exception of two
samples (1.3/0.4 and 1.9/0.1), the equilibrium chloride concentrations
evaluated in these two very different ways are essentially the
same. This result strongly confirms that the evaluations are adequate.
Steady-state fluxes
Vl07 present the flux evolution in the through-diffusion stage only for a single test (1.6/1.0), and it looks like this (left diagram)
The outflux reaches a relatively stable value after about 7 days,
after which it is meticulously monitored for a quite long time period.
The stable flux is not completely constant, but decreases slightly
during the course of the test. We anyway refer to this part as the
steady-state phase, and to the preceding part as the transient phase.
One reason that the steady-state is not completely stable is, reasonably, that the source reservoir concentration slowly decreases during the course of the test. The estimated drop from this effect, however, is only about one percent,8 while the recorded drop is substantially larger, about 7%. Vl07 do not comment on this perhaps unexpectedly large drop, but it may be caused e.g. by the ongoing conversion of the bentonite to a purer sodium state (see above).
Most of the analysis in Vl07 is based on anyway assigning a single
value to the steady-state flux. Judging from the above plot, Vl07 seem
to adopt the average value during the steady-state phase, and it is
clear that the assigned value is well constrained by the measurements
(the drop is a second order effect). The steady-state flux can
therefore be said to be directly measured in the through-diffusion
stage, rather than being obtained from fitting a certain model to
data.
Vl07 only implicitly consider the steady-state flux, in terms of a fitted “effective diffusivity” parameter, \(D_e\) (more on this in the next section). We can, however, “de-derive” the corresponding steady-state fluxes using \(j_\mathrm{ss} = D_e\cdot c_\mathrm{source}/L\), where \(L\) (= 0.01 m) is sample length. When comparing different tests it is convenient to use the normalized steady state flux \(\widetilde{j}_\mathrm{ss} = j_\mathrm{ss}/c_\mathrm{source}\), which then relates to \(D_e\) as \(\widetilde{j}_\mathrm{ss} = D_e/L\). Indeed, “effective diffusivity” is just a scaled version of the normalized steady-state flux, and it makes more sense to interpret it as such (\(D_e\) is not a diffusion coefficient). From the reported values of \(D_e\) we obtain the following normalized steady-state fluxes (my apologies for a really dull table)
Test
\(D_e\) (10-12 m2/s)
\(\widetilde{j}_\mathrm{ss}\) (10-10 m/s)
1.3/0.01
2.6
2.6
1.3/0.05
7.5
7.5
1.3/0.1
16
16
1.3/0.4
25
25
1.3/1.0
49
49
1.6/0.01
0.39
0.39
1.6/0.05
1.1
1.1
1.6/0.1
2.3
2.3
1.6/0.4
4.6
4.6
1.6/1.0
10
10
1.9/0.01
0.033
0.033
1.9/0.05
0.12
0.12
1.9/0.1
0.24
0.24
1.9/0.4
0.5
0.5
1.9/1.0
1.2
1.2
Plotting \(\widetilde{j}_\mathrm{ss}\) as a function of background concentration gives the following picture
The steady-state flux show a very consistent behavior: for all three
densities, \(\widetilde{j}_\mathrm{ss}\) increases with background
concentration, with a higher slope for the three lowest background
concentrations, and a smaller slope for the two highest background
concentrations. Although we have only been able to investigate the
1.6/1.0 test in detail, this consistency confirms that the
steady-state flux has been reliably determined in all tests.
Transient phase evaluations
So far, we have considered estimations based on more or less direct
measurements: stable chloride concentration profiles, tracer mass
balance in the out-diffusion stage, and steady-state fluxes. A major
part of the analysis in Vl07, however, is based on fitting solutions
of the diffusion equation to the recorded flux.
Vl07 state somewhat different descriptions for the through- and
out-diffusion stages. For out-diffusion they use an expression for the
flux into the right side reservoir (the sample is assumed located
between \(x=0\) and \(x=L\))
where \(j_\mathrm{ss}\) is the steady-state flux,9 \(D_e\) is “effective diffusivity”, and \(\epsilon_\mathrm{eff}\) is the effective porosity parameter (Vl07 also state a similar expression for the diffusion into the left side reservoir, but these results are discarded, as discussed earlier). For through-diffusion, Vl07 instead utilize the expression for the amount tracer accumulated in the right side reservoir
were \(S\) denotes the cross section area of the sample.
It is clear that Vl07 use \(D_e\) and \(\epsilon_\mathrm{eff}\) as fitting parameters, but not exactly how the fitting was conducted. \(D_e\) seems to have been determined solely from the the through-diffusion data, while separate values are evaluated for \(\epsilon_\mathrm{eff}\) from the through- and out-diffusion stages. As already discussed, Vl07 also provide a third estimation of \(\epsilon_\mathrm{eff}\), based on mass-balance in the out-diffusion stage. To me, the study thereby gives the incorrect impression of providing a whole set of independent estimations of \(\epsilon_\mathrm{eff}\). Although eqs. 5 and 6 are fitted to different data, they describe diffusion in one and the same sample, and an adequate fitting procedure should provide a consistent, single set of fitted parameters \((D_e, \epsilon_\mathrm{eff})\). Even more obvious is that the estimation of \(\epsilon_\mathrm{eff}\) from fitting eq. 5 should agree with the estimation from the mass-balance in the out diffusion stage — the accumulated amount in the right side reservoir is, after all, given by the integral of eq. 5. A significant variation of the reported fitting parameters for the same sample would thus signify internal inconsistency (experimental- or modelwise).
In the following reevaluation we streamline the description by solely using fluxes as model expressions,4 and by emphasizing steady-state flux as a parameter, which I think gives particularly neat expressions,10 (“TD” and “OD” denote through- and out-diffusion, respectively)
Here we use the pore diffusivity, \(D_p\), instead of the combination \(D_e/\epsilon_\mathrm{eff}\) in the exponential factors, and \(\widetilde{j} = j/c_\mathrm{source}\) denotes normalized flux. This formulation clearly shows that the time evolution is governed solely by \(D_p\), and that \(\widetilde{j}_\mathrm{ss}\) simply acts as a scaling factor.
In my opinion, using \(\widetilde{j}_\mathrm{ss}\) and \(D_p\) gives a formulation more directly related to measurable quantities; the steady-state flux is directly accessible experimentally, as we just examined, and \(D_p\) is an actual diffusion coefficient (in contrast to \(D_e\)) that can be directly evaluated from clay concentration profiles. Of course, eqs. 7 and 8 provide the same basic description as eqs. 5 and 6, and \(\widetilde{j}_\mathrm{ss}\) and \(D_p\) are related to the parameters reported in Vl07 as
When reevaluating the reported data we focus on the above discussed consistency aspect, i.e. whether or not a single model (a single pair of parameters) can be satisfactory fitted to all available data for the same sample. In this regard, we begin by noting that the fitting parameters are already constrained by the direct estimations. We have already concluded that the recorded steady-state flux basically determines \(\widetilde{j}_\mathrm{ss}\), and if we combine this with the estimated chloride clay concentration, \(D_p\) is determined from \(j_\mathrm{ss} = \phi\cdot D_p\cdot \bar{c}_0/L\), i.e.
Here are plotted values of \(D_p\) evaluated in this manner
Note that these values basically remain constant for samples of similar density (within a factor of 2) as the background concentration is varied by two orders of magnitude. This is the expected behavior of an actual diffusion coefficient,11 and confirms the adequacy of the evaluation; the numerical values also compares rather well with corresponding values for “MX-80” bentonite, measured in closed-cell tests (indicated by dashed lines in the figure).
Using eq. 10, we can also evaluate values of \(D_p\) corresponding to
the various reported fitted parameters \(\epsilon_\mathrm{eff}\). The
result looks like this (compared with the above evaluations from
direct estimations)
As pointed out above, a consistent evaluation requires that the
parameters fitted to the out-diffusion flux (red) are very similar
to those evaluated from considering the mass balance in the same process
(blue). We note that the resemblance is quite reasonable, although
some values — e.g. tests 1.3/1.0 and 1.6/1.0 — deviate in a perhaps
unacceptable way.
\(D_p\) evaluated from reported through-diffusion parameters, on the other hand, shows significant scattering (green). As the rest of the values are considerably more collected, and as the steady-state fluxes show no sign whatsoever that the diffusion coefficient varies in such erratic manner, it is quite clear that this scattering indicates problems with the fitting procedure for the through-diffusion data.
The 1.6/1.0 test
To further investigate the fitting procedures, we take a detailed look at the 1.6/1.0 test, for which flux data is provided. Vl07 report fitted parameters \(D_e = 1.0\cdot 10^{-11}\) m2/s and \(\epsilon_\mathrm{eff} = 0.063\) to the through-diffusion data, corresponding to \(\widetilde{j}_\mathrm{ss} = 1.0\cdot 10^{-9}\) m/s and \(D_p = 1.6\cdot 10^{-10}\) m2/s. We have already concluded that the steady-state flux is well captured by this data, but to see how well fitted \(\epsilon_\mathrm{eff}\) (or \(D_p\)) is, lets zoom in on the transient phase
This diagram also contains models (eq. 7) with different values of \(D_p\), and with a slightly different value of \(j_\mathrm{ss}\).12 It is clear that the model presented in the paper (black) completely misses the transient phase, and that a much better fit is achieved with \(D_p = 9.7\cdot10^{-11}\) m2/s (and \(\widetilde{j}_\mathrm{ss} = 1.06\cdot 10^{-9}\) m/s) (red). This difference cannot be attributed to uncertainty in the parameter \(D_p\) — the reported fit is simply of inferior quality. With that said, we note that all information on the transient phase is contained within the first three or four flux points; the reliability could probably have been improved by measuring more frequently in the initial stage.13
A reason for the inferior fit may be that Vl07 have focused only on the linear part of eq. 6; the paper spends half a paragraph discussing how the approximation of this expression for large \(t\) can be used to extract the fitting parameters using linear regression. Does this mean that only experimental data for large times where used to evaluate \(D_e\) and \(\epsilon_\mathrm{eff}\)? Since we are not told how fitting was performed, we cannot answer this question. Under any circumstance, the evidently low quality of the fit puts in question all the reported \(\epsilon_\mathrm{eff}\) values fitted to through-diffusion data. This is actually good news, as several of the corresponding \(D_p\) values were seen to be incompatible with constraints from direct estimations. We can thus conclude with some confidence that the inconsistency conveyed by the differently evaluated fitting parameters does not indicate experimental shortcomings, but stems from bad fitting of the through-diffusion model. Therefore, we simply dismiss the reported \(\epsilon_\mathrm{eff}\) values evaluated in this way. Note that the re-fitted value for \(D_p\) \((9.7\cdot10^{-11}\) m2/s) is consistent with those evaluated from direct estimations.
We note that when fitting the transient phase, it is appropriate to
use a value of \(\widetilde{j}_\mathrm{ss}\) slightly larger than the
average value adopted by Vl07 (as the model does not account for the
observed slight drop of the steady-state flux). This is only a minor
variation in the \(\widetilde{j}_\mathrm{ss}\) parameter itself (from
\(1.02\cdot10^{-9}\) to \(1.06\cdot10^{-9}\) m/s), but, since this value
sets the overall scale, it indirectly influences the fitted value of
\(D_p\) (model fitting is subtle!).
More questions arise regarding the fitting procedures when also examining the presented out-diffusion stage for the 1.6/1.0 sample. The tabulated fitted value for this stage is \(\epsilon_\mathrm{eff}\) = 0.075, while it is implied that the same value has been used for \(D_e\) as evaluated from the the through-diffusion stage (\(1.0\cdot 10^{-11}\) m2/s). The corresponding pore diffusivity is \(D_p = 1.33\cdot 10^{-10}\) m2/s. The provided plot, however, contains a different model than tabulated, and looks similar to this one (left diagram)
Here the presented model (black dashed line) instead corresponds to
\(D_p = 8.5\cdot 10^{-11}\) m2/s (or \(\epsilon_\mathrm{eff}\) =
0.118). The model corresponding to the tabulated value (orange) do not
fit the data! I guess this error may just be due to a typo in the
table, but it nevertheless gives more reasons to
not trust the reported \(\epsilon_\mathrm{eff}\) values fitted to
diffusion data.
The above diagram also shows the model corresponding to the reported parameters from the through-diffusion stage (black solid line). Not surprisingly, this model does not fit the out-diffusion data, confirming that it does not appropriately describe the current sample. The model we re-fitted in the through-diffusion stage (red), on the other hand, captures the outflux data quite well. By also slightly adjusting \(\widetilde{j}_{ss}\), from from \(1.06\cdot10^{-9}\) to \(0.99\cdot10^{-9}\) m/s, to account for the drop in steady-state flux during the course of the through-diffusion test, and by plotting in a lin-lin rather than a log-log diagram, the picture looks even better! In a lin-lin plot (right diagram), it is easier to note that the model presented in the graph of Vl07 actually misses several of the data points. Could it be that Vl07 used visual inspection of the model in a log-log diagram to assess fitting quality? If so, data points corresponding to very low fluxes are given unreasonably high weight.14 This could be (another) reason for the noted difference between \(D_p\) evaluated from fitted parameters to the out-diffusion flux, and from the total accumulated amount of tracer (which should be equal).
From examining the reported results of sample 1.6/1.0 we have seen that the fitting procedures adopted in Vl07 appear inappropriate, but also that a consistent model can be successfully fitted to all available data (using a single \(D_p\)). Vl07 don’t provide flux data for any other sample, but we must conclude that the reported fitted \(\epsilon_\mathrm{eff}\) parameters cannot be trusted. Luckily, the preformed refitting exercise confirms the results obtained from analysis of stable chloride profiles and accumulated amount of tracers in out-diffusion, and we conclude that these results most probably are reliable. The corresponding value of \(\bar{c}_0/c_\mathrm{source}\) (using eq. 11) for the refitted model is here compared with the estimations from direct measurements
Summary and verdict
Chloride equilibrium concentrations evaluated from mass balance of the tracer in the out-diffusion stage and from stable chloride content show remarkable agreement. On the other hand, the scattering of estimated concentrations increases substantially if they are also evaluated from the reported fitted diffusion parameters. This could indicate underlying experimental problems, as a consistent evaluation should result in a single value for the equilibrium concentration; the various evaluations — stable chloride, out-diffusion mass balance, through-diffusion fitting and out-diffusion fitting — relate, after all, to a single sample.
By reexamining the evaluations we have found, however, that the problem is associated with how the fitting to diffusion data has been conducted (and presented), rather than indicating fundamental experimental issues. In the test that we have been able to examine in detail (1.6/1.0), we found that the reported models do not fit data, but also that it is possible to satisfactorily refit a single model that is also compatible with the direct methods for evaluating the equilibrium concentration. For the rest of the samples, we have also been able to discard the fitted diffusion parameters, as they are not compatible e.g. with how the steady-state flux (very consistently) vary with density and background concentration.
For these reasons, we discard the reported “effective porosity”
parameters evaluated from fitting solutions of the diffusion equation
to flux data, and keep the results from direct measurements of
chloride equilibrium concentrations (from stable chloride profile
analysis and mass-balance in the out-diffusion stage). I judge the
resulting chloride equilibrium concentrations as reliable and that
they can be used for increased qualitative process understanding. I
furthermore judge the directly measured steady-state fluxes as
reliable. This study thus provide adequate values for both chloride
equilibrium concentrations and diffusion coefficients.
However, a frustrating problem is that, although the equilibrium concentrations are well determined, we have little information on the exact state of the samples in which they have been measured. We basically have to rely on that the “KWK” material is “similar” to “MX-80”, keeping in mind that “MX-80” is not really a uniform material (from a scientific point of view). Also, the exchangeable mono/divalent cation ratio is most probably quite different in samples contacted with different background concentrations.
Yet, I judge the present study to provide the best information
available on chloride equilibrium in compacted bentonite, and will use
it e.g. for investigating the salt exclusion mechanism in these
systems (Ialreadyhave). That this information is the best available is, however, also
a strong argument for that more and better constrained data is
urgently needed.
The (reliable) results are presented in the diagram below, which includes “confidence areas”, that takes into account the spread in equilibrium concentrations, in samples where more than a single evaluation were performed, and the estimated uncertainty in effective montmorillonite dry density (the actual points are plotted at nominal density, assuming 80% montmorillonite content)
[1] Vejsada et al. (2006) call their material “KWK 20-80”. In other
contexts, I have also found the versions
“KWK
food grade” and
“KWK krystal klear”. I have given up my attempts at trying to
understand the difference between these “KWK” variants.
[3] This should be relatively straightforward, but I get at bit nervous e.g. about the presence of a rather arbitrary factor 0.85 in the presented formula (eq. 19 in Van Loon et al. (2007)).
[4] As always for these types of diffusion tests, the raw data consists of simultaneously measured values of time (\(\{t_i\}\)) and reservoir concentrations (\(\{c_i\}\)). From these, flux can be evaluated as (\(A\) is sample cross sectional area, and \(V_\mathrm{res}\) is reservoir volume)
\(\bar{j}_i\) is the mean flux in the time interval between \(t_{i-1}\)
and \(t_i\), and should be associated with the average time of the
same interval: \(\bar{t}_i = (t_i + t_{i-1})/2\). The above formula
assumes no solution replacement after the \((i-1)\):th measurement (if
the solution is replaced, \(\left (c_i – c_{i-1} \right )\) should be
replaced with \(c_i\)).
Alternatively one can work with the accumulated amount of substance, which e.g. is \(N(t_i) = \sum_{j=1}^i c_j\cdot V_\mathrm{res}\), in case the solution is replaced after each measurement. I prefer using the flux because eq. * only depends on two consecutive measurements, while \(N(t_i)\) in principle depends on all measurements up to time \(t_i\). Also, I think it is easier to judge how well e.g. a certain model fits or is constrained by data when using fluxes; the steady-state, for example, then corresponds to a constant value.
Van Loon et
al. (2007) seem to have utilized both fluxes and accumulated
amount of substance in their evaluations, as discussed in later
sections.
[8] From total test time, recorded flux, and sample cross sectional area, we estimate that about \(5.8\cdot 10^{-8}\) mol of tracer is transferred from the source reservoir during the course of the test (\(50\) days\(\cdot 2.7\cdot 10^{-11}\) mol/m2/s\(\cdot 0.0005\) m2). This is about 1% of the total amount tracer, \(c_\mathrm{source} \cdot V_\mathrm{source} = 2.65 \cdot 10^{-5}\) M \(\cdot 0.2\) L = \(5.3\cdot 10^{-6}\) mol.
[9] Van Loon et al. (2007) label this parameter \(J_L\), and don’t relate it explicitly to the steady-state flux. From the experimental set-up it is clear, however, that the initial value of the out-diffusion flux (into the right side reservoir) is the same as the previously maintained steady-state flux. Note that the expressions for the fluxes in the out-diffusion stage in Van Loon et al. (2007) has the wrong sign.
[10] The description provided by eqs. 5 and 6 not only mixes expressions for flux and accumulated amount tracer, but also contains three dependent parameters \(D_e\), \(\epsilon_\mathrm{eff}\), and \(j_\mathrm{ss}\) (e.g. \(j_\mathrm{ss} = D_e/(c_\mathrm{source}\cdot L)\)). In this reformulation, the model parameters are strictly only \(\widetilde{j}_\mathrm{ss}\) and \(D_p\). We have also divided out \(c_\mathrm{source}\) to obtain equations for normalized fluxes. Note that the expression for \(\widetilde{j}_{TD}(L,t)\) is essentially the same that we have used in previousassessments of through-diffusion tests. Note also that eqs. 7 and 8 imply the relation \(\widetilde{j}_{OD}(L,t) = \widetilde{j}_{ss} – \widetilde{j}_{TD}(L,t)\), reflecting that the out-diffusion process is essentially the through-diffusion process in reverse.
[11] Note the similarity with that diffusivity also is basically independent of background concentration for simple cations. Note also that there is no reason to expect completely constant \(D_p\) for a given density, because the samples are not identically prepared (being saturated with saline solutions of different concentration).
[12] As we here consider a single sample, we alternate a bit sloppily between steady-state flux (\(j_\mathrm{ss} \)) and normalized steady-state flux (\(\widetilde{j}_\mathrm{ss}\)), but these are simply related by a constant: \(\widetilde{j}_\mathrm{ss} = j_\mathrm{ss} / c_\mathrm{source}\). For the 1.6/1.0 test this constant is (as tabulated) \(c_\mathrm{source} = 2.65\cdot 10^{-2}\) mol/m3.
[13] I think it is a bit amusing that the pattern of data points suggests measurements being performed on Mondays, Wednesdays, and Fridays (with the test started on a Wednesday).
[14] I have warned about the dangers of log-log plots earlier.
“Multi-porosity” models1 — i.e
models that account for both a bulk water phase and one, or several,
other domains within the clay — have become increasingly
popular in bentonite research during the last couple of decades. These
are obviously macroscopic, as is clear e.g. from the benchmark
simulations described in
Alt-Epping et
al. (2015), which are specified to be discretized into 2 mm thick
cells; each cell is consequently assumed to contain billions and
billions individual montmorillonite particles. The macroscopic
character is also relatively clear in their description of two
numerical tools that have implemented multi-porosity
PHREEQC and CrunchFlowMC have implemented a Donnan approach to
describe the electrical potential and species distribution in the
EDL. This approach implies a uniform electrical potential
\(\varphi^\mathrm{EDL}\) in the EDL and an instantaneous equilibrium
distribution of species between the EDL and the free water (i.e.,
between the micro- and macroporosity, respectively). The assumption
of instantaneous equilibrium implies that diffusion between micro-
and macroporosity is not considered explicitly and that at all times
the chemical potentials, \(\mu_i\), of the species are the same in the
two porosities
On an abstract level, we may thus illustrate a multi-porosity approach
something like this (here involving two domains)
The model is represented by one
continuum for the “free water”/”macroporosity” and one for the
“diffuse layer”/”microporosity”,2 which are
postulated to be in equilibrium within each macroscopic cell.
But such an equilibrium (Donnan equilibrium)
requires a
semi-permeable component. I am not aware of any suggestion for such
a component in any publication on multi-porosity
models. Likewise, the co-existence of diffuse layer and free water
domains requires
a mechanism that prevents swelling and maintains the pressure
difference — also the water chemical potential should of course be
the equal in the two “porosities”.3
Note that the questions of what constitutes the semi-permeable
component and what prevents swelling have a clear answer in
the homogeneous mixture model. This answer also corresponds to an
easily identified real-world object: the metal filter (or similar
component) separating the sample from the external solution.
Multi-porosity models, on the other hand, attribute no particular
significance to interfaces between sample and external
solutions. Therefore, a candidate for the semi-permeable component has
to be — but isn’t — sought elsewhere. Donnan equilibrium
calculations are virtually meaningless without identifying this
component.
The partitioning between diffuse layer and free water in
multi-porosity models is, moreover, assumed to be controlled by water
chemistry, usually by means of the
Debye length. E.g. Alt-Epping et al. (2015) write
To determine the volume of the microporosity, the surface area of
montmorillonite, and the Debye length, \(D_L\), which is the distance
from the charged mineral surface to the point where electrical
potential decays by a factor of e, needs to be known. The volume of
the microporosity can then be calculated as
\begin{equation*}
\phi^\mathrm{EDL} = A_\mathrm{clay} D_L,
\end{equation*}
where \(A_\mathrm{clay}\) is the charged surface area of the clay
mineral.
I cannot overstate how strange the multi-porosity description
is. Leaving the abstract representation, here is an attempt to
illustrate the implied clay structure, at the “macropore” scale
The view emerging from the above description is actually even more
peculiar, as the “micro” and “macro” volume fractions are supposed
to vary with the Debye length. A more general illustration of how the
pore structure is supposed to function is shown in this animation
(“I” denotes ionic strength)
What on earth could constitute such magic semi-permeable membranes?!
(Note that they are also supposed to withstand the inevitable pressure
difference.)
Here, the informed reader may object and point out that no researcher
promoting multi-porosity has this magic pore structure in
mind. Indeed, basically all multi-porosity publications instead
vaguely claim that the domain separation occurs on the nanometer scale
and present microscopic illustrations, like this (this is a
simplified version of what is found in
Alt-Epping et
al. (2015))
In the remainder of this post I will discuss how the idea of a domain separation on the microscopic scale is even more preposterous than the magic membranes suggested above. We focus on three aspects:
The implied structure of the free water domain
The arbitrary domain division
Donnan equilibrium on the microscopic scale is not really a valid concept
Implied structure of the free water domain
I’m astonished by how little figures of the microscopic scale are
explained in many publications. For instance, the illustration above
clearly suggests that “free water” is an interface region with
exactly the same surface area as the “double layer”. How can that
make sense? Also, if the above structure is to be taken seriously it
is crucial to specify the extensions of the various water layers. It
is clear that the figure shows a microscopic view, as it depicts an
actual diffuse layer.4 A diffuse layer width varies, say, in the
range 1 – 100 nm,5 but authors seldom reveal if we are
looking at a pore 1 nm wide or several hundred nm wide. Often we are
not even shown a pore — the water film just ends in a void, as in the
above figure.6
The vague nature of these descriptions indicates that they are merely “decorations”, providing a microscopic flavor to what in effect still is a macroscopic model formulation. In practice, most multi-porosity formulations provide some ad hoc mean to calculate the volume of the diffuse layer domain, while the free water porosity is either obtained by subtracting the diffuse layer porosity from total porosity, or by just specifying it. Alt-Epping et al. (2015), for example, simply specifies the “macroporosity”
The total porosity amounts to 47.6 % which is divided into
40.5 % microporosity (EDL) and 7.1 % macroporosity (free
water). From the microporosity and the surface area of
montmorillonite (Table 7), the Debye length of the EDL calculated
from Eq. 11 is 4.97e-10 m.
Clearly, nothing in this description requires or suggests that the
“micro” and “macroporosities” are adjacent waterfilms on the
nm-scale. On the contrary, such an interpretation becomes quite
grotesque, with the “macroporosity” corresponding to half a
monolayer of water molecules! An illustration of an actual pore of
this kind would look something like this
This interpretation becomes even more bizarre, considering that
Alt-Epping et
al. (2015) assume advection to occur only in this half-a-monolayer
of water, and that the diffusivity is here a factor 1000 larger than
in the “microporosity”.
As another example, Appelo
and Wersin (2007) model a cylindrical sample of “Opalinus clay”
of height 0.5 m and radius 0.1 m, with porosity 0.16, by discretizing
the sample volume in 20 sections of width 0.025 m. The void volume of
each section is consequently
\(V_\mathrm{void} = 0.16\cdot\pi\cdot 0.1^2\cdot 0.025\;\mathrm{m^3} =
1.257\cdot10^{-4}\;\mathrm{m^3}\). Half of this volume (“0.062831853”
liter) is specified directly in the input file as the volume of the
free water;7 again, nothing suggests that this water
should be distributed in thin films on the nm-scale. Yet,
Appelo and Wersin (2007)
provide a figure, with no length scale, similar in spirit to that
above, that look very similar to this
They furthermore write about this figure (“Figure 2”)
It should be noted that the model can zoom in on the nm-scale
suggested by Figure 2, but also uses it as the representative form
for the cm-scale or larger.
I’m not sure I can make sense of this statement, but it seems that they imply that the illustration can serve both as an actual microscopic representation of two spatially separated domains and as a representation of two abstract continua on the macroscopic scale. But this is not true!
Interpreted macroscopically, the vertical dimension is fictitious, and
the two continua are in equilibrium in each paired cell. On a
microscopic scale, on the other hand, equilibrium between paired cells
cannot be assumed a priori, and it becomes crucial to specify
both the vertical and horizontal length scales. As
Appelo and Wersin (2007)
formulate their model assuming equilibrium between paired cells, it is
clear that the above figure must be interpreted macroscopically (the
only reference to a vertical length scale is that the “free
solution” is located “at infinite distance” from the surface).
We can again work out the implications of anyway interpreting the model microscopically. Each clay cell is specified to contain a surface area of \(A_\mathrm{surf}=10^5\;\mathrm{m^2}\).8 Assuming a planar geometry, the average pore width is given by (\(\phi\) denotes porosity and \(V_\mathrm{cell}\) total cell volume)
The double layer thickness is furthermore specified to be 0.628 nm.9 A microscopic interpretation of this particular model thus implies that the sample contains a single type of pore (2.51 nm wide) in which the free water is distributed in a thin film of width 1.25 nm — i.e. approximately four molecular layers of water!
Rather than affirming that multi-porosity model formulations are macroscopic at heart, parts of the bentonite research community have instead doubled down on the confusing idea of having free water distributed on the nm-scale. Tournassat and Steefel (2019) suggest dealing with the case of two parallel charged surfaces in terms of a “Dual Continuum” approach, providing a figure similar to this (surface charge is -0.11 C/m2 and external solution is 0.1 M of a 1:1 electrolyte)
Note that here the perpendicular length scale is specified,
and that it is clear from the start that the electrostatic potential
is non-zero everywhere. Yet,
Tournassat and Steefel
(2019) mean that it is a good idea to treat this system as if it
contained a 0.7 nm wide bulk water slice at the center of the
pore. They furthermore express an almost “postmodern” attitude
towards modeling, writing
It should be also noted here that this model refinement does not
imply necessarily that an electroneutral bulk water is present at
the center of the pore in reality. This can be appreciated in Figure
6, which shows that the Poisson–Boltzmann predicts an overlap of the
diffuse layers bordering the two neighboring surfaces, while the
dual continuum model divides the same system into a bulk and a
diffuse layer water volume in order to obtain an average
concentration in the pore that is consistent with the
Poisson–Boltzmann model prediction. Consequently, the pore space
subdivision into free and DL water must be seen as a convenient
representation that makes it possible to calculate accurately the
average concentrations of ions, but it must not be taken as evidence
of the effective presence of bulk water in a nanoporous medium.
I can only interpret this way of writing (“…does not imply
necessarily that…”, “…must not be taken as evidence of…”)
that they mean that in some cases the bulk phase should be
interpreted literally, while in other cases the bulk phase
should be interpreted just as some auxiliary component. It is my
strong opinion that such an attitude towards modeling only contributes
negatively to process understanding (we may e.g. note that later in
the article, Tournassat
and Steefel (2019) assume this perhaps non-existent bulk water to
be solely responsible for advective flow…).
I say it again: no matter how much researchers discuss them in microscopic terms, these models are just macroscopic formulations. Using the terminology of Tournassat and Steefel (2019), they are, at the end of the day, represented as dual continua assumed to be in local equilibrium (in accordance with the first figure of this post). And while researchers put much effort in trying to give these models a microscopic appearance, I am not aware of anyone suggesting a reasonable candidate for what actually could constitute the semi-permeable component necessary for maintaining such an equilibrium.
Arbitrary division between diffuse layer and free water
Another peculiarity in the multi-porosity descriptions showing that they cannot be interpreted microscopically is the arbitrary positioning of the separation between diffuse layer and free water. We saw earlier that Alt-Epping et al. (2015) set this separation at one Debye length from the surface, where the electrostatic potential is claimed to have decayed by a factor of e. What motivates this choice?
Most publications on multi-porosity models define free water as a region where the solution is charge neutral, i.e. where the electrostatic potential is vanishingly small.10 At the point chosen by Alt-Epping et al. (2015), the potential is about 37% of its value at the surface. This cannot be considered vanishingly small under any circumstance, and the region considered as free water is consequently not charge neutral.
The diffuse layer thickness chosen by Appelo and Wersin (2007) instead corresponds to 1.27 Debye lengths. At this position the potential is about 28% of its value at the surface, which neither can be considered vanishingly small. At the mid point of the pore (1.25 nm), the potential is about 8%11 of the value at the surface (corresponding to about 2.5 Debye lengths). I find it hard to accept even this value as vanishingly small.
Note that if the boundary distance used by Appelo and Wersin (2007) (1.27 Debye lengths) was used in the benchmark of Alt-Epping et al. (2015), the diffuse layer volume becomes larger than the total pore volume! In fact, this occurs in all models of this kind for low enough ionic strength, as the Debye length diverges in this limit. Therefore, many multi-porosity model formulations include clunky “if-then-else” clauses,12 where the system is treated conceptually different depending on whether or not the (arbitrarily chosen) diffuse layer domain fills the entire pore volume.13
In the example from Tournassat and Steefel (2019) the extension of the diffuse layer is
1.6 nm, corresponding to about 1.69 Debye lengths. The potential is
here about 19% of the surface value (the value in the midpoint is
12% of the surface
value). Tournassat
and Appelo (2011) uses yet another separation distance — two Debye
lengths — based on
misusing the concept of exclusion volume in the Gouy-Chapman model.
With these examples, I am not trying to say that a better criterion is needed for the partitioning between diffuse layer and bulk. Rather, these examples show that such a partitioning is quite arbitrary on a microscopic scale. Of course, choosing points where the electrostatic potential is significant makes no sense, but even for points that could be considered having zero potential, what would be the criterion? Is two Debye lengths enough? Or perhaps four? Why?
These examples also demonstrate that researchers ultimately do not
have a microscopic view in mind. Rather, the “microscopic”
specifications are subject to the macroscopic constraints.
Alt-Epping et
al. (2015), for example, specifies a priori that the system
contains about 15% free water, from which it follows that the diffuse
layer thickness must be set to about one Debye length (given the
adopted surface area). Likewise,
Appelo and Wersin (2007)
assume from the start that Opalinus clay contains 50% free water, and
set up their model accordingly.14Tournassat and Steefel
(2019) acknowledge their approach to only be a “convenient
representation”, and don’t even relate the diffuse layer
extension to a specific value of the electrostatic
potential.15 Why
the free water domain anyway is considered to be positioned in the
center of the nanopore is a mystery to me (well, I guess because
sometimes this interpretation is supposed to be taken literally…).
Note that none of the free water domains in the considered models are actually charged, even though the electrostatic potential in the microscopic interpretations is implied to be non-zero. This just confirms that such interpretations are not valid, and that the actual model handling is the equilibration of two (or more) macroscopic, abstract, continua. The diffuse layer domain is defined by following some arbitrary procedure that involves microscopic concepts. But just because the diffuse layer domain is quantified by multiplying a surface area by some multiple of the Debye length does not make it a microscopic entity.4
Donnan effect on the microscopic scale?!
Although we have already seen that we cannot interpret multi-porosity models microscopically, we have not yet considered the weirdest description adopted by basically all proponents of these models: they claim to perform Donnan equilibrium calculations between diffuse layer and free water regions on the microscopic scale!
The underlying mechanism for a Donnan effect is the establishment of charge separation, which obviously occur on the scale of the ions, i.e. on the microscopic scale. Indeed, a diffuse layer is the manifestation of this charge separation. Donnan equilibrium can consequently not be established within a diffuse layer region, and discontinuous electrostatic potentials only have meaning in a macroscopic context.
Consider e.g. the interface between bentonite and an external solution
in
the
homogeneous mixture model. Although this model ignores the
microscopic scale, it implies charge separation and a continuously
varying potential on this scale, as illustrated here
The regions where the potential varies are exactly what we categorize
as diffuse layers (exemplified in two ideal microscopic geometries).
The discontinuous potentials encountered in multi-porosity model descriptions (see e.g. the above “Dual Continuum” potential that varies discontinuously on the angstrom scale) can be drawn on paper, but don’t convey any physical meaning.
Here I am not saying that Donnan equilibrium calculations cannot be performed in multi-porosity models. Rather, this is yet another aspect showing that such models only have meaning macroscopically, even though they are persistently presented as if they somehow consider the microscopic scale.
An example of this confusion of scales is found in
Alt-Epping et
al. (2018), who revisit the benchmark problem of
Alt-Epping et
al. (2015) using an alternative approach to Donnan equilibrium:
rather than directly calculating the equilibrium, they model the clay
charge as immobile mono-valent anions, and utilize the
Nernst-Planck
equations. They present “the conceptual model” in a figure very
similar to this one
This illustration simultaneously conveys both a micro- and macroscopic view. For example, a mineral surface is indicated at the bottom, suggesting that we supposedly are looking at an actual interface region, in similarity with the figures we have looked at earlier. Moreover, the figure contains entities that must be interpreted as individual ions, including the immobile “clay-anions”. As in several of the previous examples, no length scale is provided (neither perpendicular to, nor along the “surface”).
On the other hand, the region is divided into cells, similar to the
illustration in Appelo and Wersin (2007). These can hardly have any other meaning
than to indicate the macroscopic discretization in the adopted
transport code (FLOTRAN). Also, as the “Donnan porosity” region
contains the “clay-anions” it can certainly not represent a diffuse
layer extending from a clay surface; the only way to make sense of
such an “immobile-anion” solution is that it represents a
macroscopic homogenized clay domain (a homogeneous mixture!).
Furthermore, if the figure is supposed to show the microscopic scale
there is no Donnan effect, because there is no charge separation!
Taking the depiction of individual ions seriously, the interface
region should rather look something like this in equilibrium
This illustrates the fundamental problem with a Donnan effect between microscopic compartments: the effect requires a charge separation, whose extension is the same as the size of the compartments assumed to be in equilibrium.16
Despite the confusion of the illustration in Alt-Epping et al. (2018), it is clear that a macroscopic model is adopted, as in our previous examples. In this case, the model is explicitly 2-dimensional, and the authors utilize the “trick” to make diffusion much faster in the perpendicular direction compared to the direction along the “surface”. This is achieved either by making the perpendicular diffusivity very high, or by making the perpendicular extension small. In any case, a perpendicular length scale must have been specified in the model, even if it is nowhere stated in the article. The same “trick” for emulating Donnan equilibrium is also used by Jenni et al. (2017), who write
In the present model
set-up, this approach was implemented as two connected domains in
the z dimension: one containing all minerals plus the free porosity
(z=1) and the other containing the Donnan porosity, including the
immobile anions (CEC, z=2, Fig. 2). Reproducing instantaneous
equilibrium between Donnan and free porosities requires a much
faster diffusion between the porosity domains than along the
porosity domains.
Note that although the perpendicular dimension (\(z\)) here is referred
to without unit(!), this representation only makes sense in a
macroscopic context.
Jenni et al. (2017) also provide a statement that I think fairly well sums up the multi-porosity modeling endeavor:17
In a Donnan porosity concept, cation exchange can be seen as
resulting from Donnan equilibrium between the Donnan porosity and
the free porosity, possibly moderated by additional specific
sorption. In CrunchflowMC or PhreeqC (Appelo and Wersin, 2007;
Steefel, 2009; Tournassat and Appelo, 2011; Alt-Epping et al., 2014;
Tournassat and Steefel, 2015), this is implemented by an explicit
partitioning function that distributes aqueous species between the
two pore compartments. Alternatively, this ion partitioning can be
modelled implicitly by diffusion and electrochemical migration
(Fick’s first law and Nernst-Planck equations) between the free
porosity and the Donnan porosity, the latter containing immobile
anions representing the CEC. The resulting ion compositions of the
two equilibrated porosities agree with the concentrations predicted
by the Donnan equilibrium, which can be shown in case studies
(unpublished results, Gimmi and Alt-Epping).
Ultimately, these are models that, using one approach or the other,
simply calculates Donnan equilibrium between two abstract,
macroscopically defined domains (“porosities”,
“continua”). Microscopic interpretations of these models lead — as
we have demonstrated — to multiple absurdities and errors. I am not
aware of any multi-porosity approach that has provided any kind of
suggestion for what constitutes the semi-permeable component required
for maintaining the equilibrium they are supposed to describe.
Alternatively expressed: what, in the previous figure,
prevents the “immobile anions” from occupying the entire clay
volume?
The most favorable interpretation I can make of multi-porosity approaches to bentonite modeling is a dynamically varying “macroporosity”, involving magical membranes (shown above). This, in itself, answers why I cannot take multi-porosity models seriously. And then we haven’t yet mentioned the flawed treatment of diffusive flux.
[1] This category has many other names,
e.g. “dual
porosity” and “dual continuum”, models. Here, I mostly use the term
“multi-porosity” to refer to any model of this kind.
[3] This lack of a full
description is very much related to the incomplete description of
so-called
“stacks” — I am not aware of any reasonable suggestion of a
mechanism for keeping stacks together.
[4] Note the difference between a diffuse layer and a diffuse layer domain. The former is a structure on the nm-scale; the latter is a macroscopic, abstract model component (a continuum).
[5] The scale of an electric double layer is
set by the Debye length, \(\kappa^{-1}\). From the formula for a 1:1
electrolyte, \(\kappa^{-1} = 0.3 \;\mathrm{nm}/\sqrt{I}\), the Debye
length is seen to vary between 0.3 nm and 30 nm when ionic strength
is varied between 1.0 M to 0.0001 M (\(I\) is the numerical value of
the ionic strength expressed in molar units). Independent of the
value of the factor used to multiply \(\kappa^{-1}\) in order to
estimate the double layer extension, I’d say that the estimation 1 –
100 nm is quite reasonable.
[6] Here, the informed reader may perhaps point out that authors don’t really mean that the free water film has exactly the same geometry as the diffuse layer, and that figures like the one above are more abstract representations of a more complex structure. Figures of more complex pore structures are actually found in manymulti-porositypapers. But if it is the case that the free water part is not supposed to be interpreted on the microscopic scale, we are basically back to a magic membrane picture of the structure! Moreover, if the free water is not supposed to be on the microscopic scale, the diffuse layer will always have a negligible volume, and these illustrations don’t provide a mean for calculating the partitioning between “micro” and “macroporosity”.
It seems to me that not specifying the extension of the free water is a way for authors to dodge the question of how it is actually distributed (and, as a consequence, to not state what constitutes the semi-permeable component).
[7] The PHREEQC input files are provided as
supplementary material to Appelo and Wersin (2007). Here I consider the input corresponding
to figure 3c in the article. The free water is specified with
keyword “SOLUTION”.
[8] Keyword
“SURFACE” in the PHREEQC input file for figure 3c in the paper.
[9] Using the identifier “-donnan” for the “SURFACE”
keyword.
[10] We assume a boundary
condition such that the potential is zero in the solution infinitely
far away from any clay component.
[11] Assuming exponential decay, which is only strictly true for a single clay layer of low charge.
[12] For example,
Tournassat and Steefel
(2019) write
(\(f_{DL}\) denotes the volume fraction of the diffuse layer):
In PHREEQC and CrunchClay, the volume of the diffuse layer (\(V_{DL}\) in m3), and hence the \(f_{DL}\) value, can be defined as a multiple of the Debye length in order to capture this effect of ionic strength on \(f_{DL}\): \begin{equation*} V_{DL} = \alpha_{DL}\kappa^{-1}S \tag{22} \end{equation*} \begin{equation*} f_{DL} = V_{DL}/V_{pore} \end{equation*} […] it is obvious that \(f_{DL}\) cannot exceed 1. Equation (22) must then be seen as an approximation, the validity of which may be limited to small variations of ionic strength compared to the conditions at which \(f_{DL}\) is determined experimentally. This can be appreciated by looking at the results obtained with a simple model where: \begin{equation*} \alpha_{DL} = 2\;\mathrm{if}\;4\kappa^{-1} \le V_{pore}/S\;\mathrm{and,} \end{equation*} \begin{equation*} f_{DL} = 1 \;\mathrm{otherwise.} \end{equation*}
[13] Some tools (e.g. PHREEQC) allow to put a maximum size limit on the diffuse layer domain, independent of chemical conditions. This is of course only a way for the code to “work” under all conditions.
[14] As icing on the cake, these estimations of free water in bentonite (15%) and Opalinus clay (50%) appear to be based on the incorrect assumption that “anions” only reside in such compartments. In the present context, this handling is particularly confusing, as a main point with multi-porosity models (I assume?) is to evaluate ion concentrations in other types of compartments.
[16] Donnan equilibrium between microscopic
compartments can be studied in
molecular dynamics simulations, but they require the considered
system to be large enough for the electrostatic potential to reach
zero. The semi-permeable component in such simulations is
implemented by simply imposing constraints on the atoms making up
the clay layer.
In contrast to the
earlierassessed
studies, Mu04 is not a diffusion study, but considers directly the
clay concentration in samples equilibrated with an external
solution. Moreover, Mu04 uses purified “MX-80” bentonite, ion
exchanged to a more pure sodium form.
Mu04 contains data from two quite different types of samples. 15
samples originate from a study on pressure response in montmorillonite
contacted with external NaCl solutions of varying concentration
(Karnland et al., 2005; in the following referred to as Ka05). The
remaining 10 samples were prepared for determining basal distance
using small-angle X-ray scattering (SAXS). We refer to these two sets
of samples as the swelling pressure samples and the SAXS
samples, respectively. A more detailed description of the sample
analysis is given in
Muurinen (2006), in the following referred to as Mu06.
Material
The used material is referred to as “purified MX-80”. Mu06 states that this material was produced by mixing “MX-80” powder and NaCl solutions in bottles, where the solutions were repeatedly replaced. Mu06 also states that “During this process, part of the dissolving accessory minerals was removed as well.” Ka05 more explicitly say that the raw material was “converted into a homo-ionic Na+ state and coarser grains were removed (Muurinen et al., 2002). The montmorillonite content was thereby increased to above 90% of the total material.”1 With no better estimate of the montmorillonite content, we therefore associate the stated densities with effective montmorillonite dry density, i.e. we assume a montmorillonite content of 100%. We should keep in mind the uncertainty of this parameter, and that, reasonably, this choice somewhat overestimates the effective montmorillonite dry density.
The purified material was found to leach sulfate and carbonate,
indicating that it still contains some amount of soluble accessory
minerals. It follows that the montmorillonite is not completely of
pure sodium form, as confirmed by the reported exchangeable ion
population: 0.74 eq/kg sodium, 0.06 eq/kg calcium, and 0.03 eq/kg
magnesium (i.e. a di/mono-valent ratio of about 10/90). It is
interesting that the material still contains a non-negligible amount
of divalent ions, given that quite a lot of effort was put into
producing it. Nevertheless, we can assume that this material contains
considerably more sodium as compared with the “raw” “MX-80”
encountered in the
previouslyassessed
studies.
Samples overview
The swelling pressure samples were originally cylindrical, with diameter 5 cm and length 2 cm, giving a volume of approximately 39 cm3. After termination of the swelling pressure tests, these samples were cut into pieces, to be used for different types of analyses. The samples cover large ranges of density and external NaCl concentration, as listed here
Neither Mu04 nor Mu06 provide much information about preparation and handling of the SAXS samples. It is stated that these are cylindrical with diameter 2.5 cm and length 0.5 cm, giving a total volume of 2.45 cm3. Although not stated, also these sample have reasonably been sub-divided, as e.g. density was determined (and some parts were obviously used for SAXS).
The SAXS samples varies substantially in density, but were only contacted with external NaCl solutions of concentration 0.1 or 0.3 M. The table below identifies each sample by external solution concentration and, presumably, measured density (how density was determined is not reported)
NaCl conc. (M)
\(\rho_d\) (measured?) (g/cm3)
0.1
0.750
0.1
0.875
0.1
1.225
0.1
1.516
0.1
1.543
0.3
0.954
0.3
1.058
0.3
1.206
0.3
1.559
0.3
1.662
In the following, we separately discuss the chloride concentration
evaluations of the swelling pressure samples and the SAXS samples.
Swelling pressure samples
Chloride concentration was
evaluated3 in three separate pieces of each original sample, as
indicated in this figure:4
Chloride content was determined by dispersing each piece, containing about 1 g of clay, in de-ionized water, centrifuging, and analyzing the supernatant. The pieces were located at different heights of the original cylinder (see figure), giving some spatial resolution of the chloride distribution, reported in Mu06. Mu04, however, only report the average value for each sample. For some samples, the value reported in Mu04 does not perfectly match the average calculated from the values listed in Mu06 (cf. the plots below).
Let’s anticipate the “verdict” for these samples: the evaluated clay
concentrations are not useful for quantitative understanding of
ion equilibrium, and I will not use them e.g. for validating anion
exclusion models.
That these samples are not adequately equilibrated is best seen from
looking at the evaluated concentrations in the samples contacted with
0.1 and 0.3 M NaCl, here plotted with spatial resolution
The indicated densities (\(\rho_d\)) are the average of the spatially resolved values reported in Mu06 (these differ a bit from what is reported in Mu04). The profiles show several peculiarities:
The densest samples in both test sets (S2-18 and S2-16, respectively) contain the second highest amount of chloride.
Samples S2-21 and S2-02 have a huge difference in chloride concentration, even though they have quite similar density.
In both test sets, the chloride concentration is very similar in the samples with densities \(\sim\) 1.3 g/cm3 and \(\sim\) 1.6 g/cm3 (S2-04 vs. S2-17, and S2-14 vs. S2-15).
These observations strongly indicate that the samples either have not
been adequately equilibrated, or that they have not been adequately
handled after test termination (or both). Consequently, the results
are of little help for adequate quantitative process
understanding.5 Mu04 acknowledge this shortcoming,
but takes different action
At high dry densities (>1630 kg/m3 ) and low NaCl concentrations, the concentrations in the porewater tend to increase with increasing density. The phenomenon is not seen with the thinner SAXS samples, however. One possible explanation is that during saturation too much chloride is transported into the sample and the equilibration time has been too short to reach the equilibrium. Three such samples marked with (*) in Tables 1 and 3 have been omitted from the treatment of the results.
But one cannot simply omit only the samples that deviate from the
expected qualitative behavior while assuming that the rest of the
results are adequate! This is especially true when the source for the
shortcoming has not been clarified. In fact, we just identified
additional peculiarities in the data. Consequently, not only should
the rest of the samples equilibrated with 0.1 M and 0.3 M NaCl be
omitted, but also those equilibrated with 1.0 M and 3.0 M.
For completeness, here are the chloride concentration profiles for
the tests with high background concentration
Although we discard them, it may be interesting to identify possible
reasons for these flawed results. Previously, we discussed why
equilibrium salt concentrations may be overestimated. Both factors
identified there may apply here: failing to handle possible interface
excess, and issues related to directly saturating samples with a
saline solution.
Saturating with saline solutions
From the different reports it is clear that the samples were saturated directly with the saline solution. It is, however, not fully clear if the saturation was performed from only one end of the sample, or from both. In Mu04 and Mu06, the assumption seems to be that the samples were saturated from one side only, although this is not described in any detail. Mu04 write
The compacted samples were closed in metal tubes and saturated
through a sinter at one end.
In Ka05, however, the statement is
The water solutions were slowly circulated behind the bottom filters
to start with, in order not to trap the original air in the samples.
where the formulation “to start with” suggests that the solution was
eventually also contacted from the top.
The reason why this detail may be important is that with solution at
one end only the effective diffusion distance for salt is doubled as
compared with having solution at both ends. A doubling of diffusion
length, in turn, increases the characteristic diffusion time by a
factor of 4.
We estimate the time needed for excess salt to diffuse out by
considering a model with an initial unit concentration in the entire
domain (domain length \(L\)), and boundary condition of zero
concentration at the end points. The midpoint concentration in such a
model, for various values of \(L\) (0.01 — 0.02 m) and diffusion
coefficients (\(1\cdot 10^{-10}\) — \(1\cdot 10^{-11}\;\mathrm{m^2/s}\)),
evolve like this
We see that, depending on parameter values, the set-up may be such that a possible “overshoot” of salt have not had time to completely diffuse out of the sample during the course of the swelling pressure tests, which were conducted for about a month. In particular, if the effective diffusion length was 2 cm during the major part of the saturation process, it is very plausible that the equilibrium process was not completed for certain samples (this depends of course also on the detailed values of diffusion coefficient and equilibration time).
Supported by this simple analysis, we cannot rule out that the samples
initially took up more salt than dictated by the final state, and that
this salt may not have had time to fully diffuse out again.
Interface excess
Concerning interface excess (a potential problem regardless of whether or not the sample has reached full equilibrium before termination), no detailed information is given on the dismantling procedure. It seems relatively clear, though, that the outer parts of the original samples were not sectioned off. Ka05 write
After reaching pressure equilibrium and a minimum test time of 1
month, the test solutions were disconnected and the samples were
removed and split in order to make detailed analyses of the water
ratio, sample density, pore-water chemistry, water activity and
microstructure.
Mu06 writes (“Figure 4” is similar to the figure above)
The bentonite sample cylinders obtained from the swelling pressure
measurements were cut into smaller pieces according to Figure 4 in
order to provide samples for different analyses and
measurements. Half of the sample piece was used for the porewater
studies while the other half […] was left in Clay Technology AB
for their studies.
My interpretation is that the original sample was cut in half during the dismantling in the swelling pressure study, and that one half was sent off elsewhere for the analysis presented in Mu04 (the upper part of the disc indicated in the above figure). Thus, it seems plausible that the interface regions were not sectioned off during dismantling, and that the samples were stored/transported for an appreciable amount of time. Possible excess salt would consequently had time to even out in the sample before further analysis. This interpretation is in line with the evaluated rather flat chloride profiles: note the contrast between these and the quite pronouncednon-linear profiles observed at the interfaces in studies where samples are sectioned at test termination.
SAXS samples
Mu04 (and Mu06) provide almost no information about how the SAXS samples were handled. Reasonably, also these samples were split, with some part being used for the SAXS measurement and another for determining water content, but we have no information on this. In fact, not even the SAXS results are properly reported for these samples; only evaluated “interlamellar spaces” for the samples equilibrated at 0.3 M are discussed; neither Mu04 nor Mu06 report SAXS data for the 0.1 M samples.
The reports are also somewhat contradictory. In the caption to a table in Mu04 it is stated that the SAXS samples were first saturated with de-ionized water, and thereafter equilibrated with the salt solutions. Mu06, on the other hand, states
The samples were compacted into the cells and saturated through a
filter plate from one side with 0.1 or 0.3 M NaCl solutions for 12
days.
Should the last statement rather be that equilibration was
performed for 12 days, after saturation? Under any
circumstance, the lack of information on handling of the SAXS samples
is a major flaw and must be considered in the assessment.
If I should guess, I believe that possible interface excess on these
samples where not handled, i.e. I believe that the end parts
were not sectioned off when the samples were dismantled (also, with
only 5 mm thick samples, there is not much to section off…). Note
that the SAXS samples are thin (5 mm) and were equilibrated with
solutions of relatively low concentration (0.1 M and 0.3 M). Based on
the
analysis in the previous post, these samples are expected to be very
sensitive to an interface excess effect.
Here is plotted the reported chloride clay concentrations for the SAXS
samples, together with corresponding (average) values for the swelling
pressure samples at the same background concentration6
Note that, although the density dependence on the SAXS sample data appears more reasonable compared with the swelling pressure samples, the SAXS sample data seem to have a scatter of at least a factor of 2 (see e.g. the leftmost SAXS points for 0.3 M and the rightmost points for 0.1 M). Note also that in one sample (0.1 M, 750 kg/cm3), the evaluated clay concentration is larger than the background concentration!
Summary and verdict
I discard the chloride concentrations measured in the swelling pressure samples, based on the reported results: it is clear that the observed scatter and spurious dependencies demonstrate that the samples were not properly equilibrated, in order to use the results for quantitative process understanding. To accept e.g. a result that the equilibrium chloride concentration increases with density I require a considerably more rigorous study. Moreover, I mean that all results must be discarded, not only those that obviously deviate from the expected qualitative behavior.
I also discard the results of the SAXS samples. Although we don’t have
any clear indication that they were incorrectly prepared, I judge the
uncertainties and lack of information to be too large in order to rely
on the results. Almost no information is provided! Furthermore, the
reports do not give any hint that the issue of interface excess is
identified and handled — an effect we can expect to be substantial in
these samples.
I am saddened to have to discard these results, because, in my mind, adequate results from equilibration of homo-ionic samples would be very valuable for increased process understanding. I strongly believe that the bentonite research community should strive for conducting many more of these relatively simple tests on purified clays, rather than complicated through-diffusion tests. In properly conducted equilibrium tests, concentration data is accessed directly and there is no risk for the results to be obscured by issues related to ionic transport.
[3] Muurinen et al. (2004) also report chloride concentrations from
so-called squeezing tests. Squeezing tests are not adequate for
evaluating equilibrium clay concentrations, and I intend to write a
future blog post on the subject. Here we simply ignore the squeezing
results.
[4] The pieces labeled “B” were used
to determine density (water content).
[6] The plots also show the difference in average concentration and density for the swelling pressure samples as reported in Muurinen et al. (2004) and Muurinen (2006); these points should lie on top of each other.
When discussing semi-permeability, we noted that a bentonite sample that is saturated with a saline solution probably contains more salt in the initial stages of the process than what is dictated by the final state Donnan equilibrium. This salt must consequently diffuse out of the sample before equilibrium is reached.
The reason for such a possible “overshoot” of the clay concentration is that an infiltrating solution is not subject to a Donnan effect (between sample and external solution) when it fills out the air-filled voids of an unsaturated sample. Also, even if the region near the interface to the external solution becomes saturated — so that a Donnan effect is active — a sample may still take up more salt than prescribed by the final state, due to hyperfiltration: with a net inflow of water and an active Donnan effect, salt will accumulate at the inlet interface (unless the interface is flushed). This increased concentration, in turn, alters the Donnan equilibrium at the interface, with the effect that more salt diffuses into the clay.
These effects are relevant for our ongoing assessment of studies of chloride equilibrium concentrations. If bentonite samples are saturated with saline solutions, without taking precautions against these effects, evaluated equilibrium concentrations may be overestimated. Note that, even if saturating a sample may be relatively fast, it may take a long time for salt to reach full equilibrium, depending on details of the experimental set-up. In particular, if the set-up is such that the external solution does not flow past the inlet, equilibration may take a very long time, being limited by diffusion in filters and tubing.
Interface excess salt
Another way for evaluated salt concentrations to overestimate the true equilibrium value — which is independent of whether or not the sample has been saturated with a saline solution — is due to excess salt at the sample interfaces.
Suppose that you determine the equilibrium salt concentration in a bentonite sample in the following way. First you prepare the sample in a test cell and contact it with an external salt solution via filters. When the system (bentonite + solution) has reached equilibrium (taking all the precautions against overestimation discussed above), the concentration profile may be conceptualized like this
The aim is to determine \(\bar{c}_\mathrm{clay}\), the
clay concentration of the species of interest
(e.g. chloride), and to relate it to the corresponding concentration in the
external solution (\(c_ \mathrm{ext}\)).
After ensuring the value of \(c_\mathrm{ext}\) (e.g. by sampling or controlling the external solution), you unload the test cell and isolate the bentonite sample. In doing so, we must keep in mind that the sample will begin to swell as soon as the force on it is released, if only water is available. In the present example it is difficult not to imagine that some water is available, e.g. in the filters.1
It is thus plausible that the actual concentration profile look
something like this directly after the sample has been isolated
We will refer to the elevated concentration at the interfaces as the
interface excess. The exact shape of the resulting
concentration profile depends reasonably on the detailed procedure for
isolating the sample.2 If the ion content of the sample is measured
as a whole, and/or if the sample is stored for an appreciable amount
of time before further analysis (so that the profile evens out due to
diffusion), it is clear that the evaluated ion content will be larger
than the actual clay concentration.
To quantify how much the clay concentration may be overestimated due
to the interface excess, we introduce an effective penetration
depth, \(\delta\)
\(\delta\) corresponds to a depth of the external concentration that
gives the same interface excess as the actual distribution. Using this
parameter, it is easy to see that the clay concentration evaluated as
the average over the entire sample is
This expression is quite interesting. We see that the relative
overestimation, reasonably, depends linearly on \(\delta\) and on the
inverse of sample length. But the expression also contains the ratio
\(r \equiv c_\mathrm{ext}/\bar{c}_\mathrm{clay}\), indicating that the effect may
be more severe for systems where the clay concentration is small in
comparison to the external concentration (high density, low
\(c_\mathrm{ext}\)).
An interface excess is more than a theoretical concept, and is frequently observed e.g. in anion through-diffusion studies. We have previously encountered them when assessing the diffusion studies of Muurinen et al. (1988) and Molera et al. (2003).3Van Loon et al. (2007) clearly demonstrate the phenomenon, as they evaluate the distribution of stable chloride (the background electrolyte) in the samples after performing the diffusion tests.4 Here is an example of the chloride distribution in a sample of density 1.6 g/cm3 and background concentration of 0.1 M5
The line labeled \(\bar{c}_\mathrm{clay}\) is evaluated from the average of only the interior sections (0.0066 M), while the line labeled \(\bar{c}_\mathrm{eval}\) is the average of all sections (0.0104 M). Using the full sample to evaluate the chloride clay concentration thus overestimates the value by a factor 1.6. From eq. 1, we see that this corresponds to \(\delta = 0.2\) mm. For a sample of length 5 mm with the same penetration depth, the corresponding overestimation is a factor of 2.1.
Here is plotted the relative overestimation (eq. 1) as a function of \(\delta\) for several systems of varying length and \(r\) (\(= c^\mathrm{ext}/\bar{c}_\mathrm{clay}\))
We see that systems with large \(r\) and/or small \(L\) become
hypersensitive to this effect. Thus, even if it may be expected that
\(\delta\) decreases with increasing \(r\)6, we may still expect an
increased overestimation for such systems.
To avoid this potential overestimation of the clay concentration, I
guess the best practice is to quickly remove the first couple of
millimeters on both sides of a sample after it has been unloaded. In
many through-diffusion tests, this is done as part of the study, as
the concentration profile across the sample often is measured. In
studies where samples are merely equilibrated with an external
solution, however, removing the interface regions may not be
considered.
Summary
We have here discussed some plausible reasons for why an evaluated
equilibrium salt concentration in a clay sample may be overestimated:
If samples are saturated directly with a saline solution. Better practice is to first saturate the sample with pure water (or a dilute solution) and then to equilibrate with respect to salt in a second stage.
If the external solution is not circulated. Diffusion may then occur over very long distances (depending on test design). The reasonable practice is to always circulate external solutions.
If interface excess is not handled. This is an issue even if saturation is done with pure water. The most convenient way to deal with this is to section off the first millimeters on both sides of the samples as quickly as possible after they are unloaded.
Footnotes
[1] One way to minimize this possible effect could be to
empty the filter before unloading the test cell. This may, however,
be difficult unless the filter itself is flushable. Also, you may
run into the problem of beginning to dry the sample.
[2] The only study I’m aware of that has
systematically investigated these types of concentration profiles is
Glaus et
al. (2011). They claim, if I understand correctly, that the
interface excess is not caused by swelling during
dismantling. Rather, they mean that the profile is the result of an
intrinsic density decrease that occurs in interface regions. Still,
they don’t discuss how swelling are supposed to be inhibited,
neither during dismantling, nor in order for the density
inhomogeneity to remain. Under any circumstance, the conclusions in
this blog post are not dependent on the cause for the presence of a
salt interface excess.
[3] In through-diffusion tests, the problem of the
interface excess is usually not that the equilibrium clay
concentration is systematically overestimated, since the detailed
concentration profile often is sampled in the final state. Instead,
the problem becomes how to separate the linear and non-linear parts
of the profile.
where \(\rho_0\) is the air density at sea level, and
\(\alpha = RT/(Mg) \approx 7500\) m is a constant. Integrating
the above formula from sea level to the height of Mount Everest
(\(\approx 9000\) m) gives
More advanced research finds a neat interpretation of this relation: the accessible height for air is 5200 m. Above this limit air is excluded, probably due to repulsion from the bedrock at these altitudes — there are reasons to believe that such rock has significantly different properties compared with rock at sea level (e.g. positive gravitational potential). In fact, both experimental work and theoretical modelling — even at the atomistic level! — have given strong evidence for the air exclusion effect: best fitting to available data is achieved with so-called air-free models.
As an example of the success of this research, one has been able to explain the existence of life in the highest regions of the Tibetan Plateu: air exists in these regions in hidden valleys (also called interpeak volumes) below the 5200 m-level, which consequently have air density \(\rho_0\). Much of present day air exclusion research is actually devoted to quantifying the amount of hidden valleys, given measurements of air density in various regions around the world (valleys that otherwise would be very difficult to discover).
Even if this research field lately has progressed heavily, there is
still a lot of exciting work waiting to be done. Of the many topics
can be mentioned so-called partial air exclusion on the outer borders
to certain high plains, air transport between hidden valleys (which
typically are connected), and the possibility of having different
accessible heights for different types of air.
A future potential application of the air exclusion effect is to build
storage e.g. for food at high altitudes. With no air around, food is
expected to stay fresh forever!
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 systemsother 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).
[…] 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.
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-stateuphill
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.
[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
[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”?
[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.
[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.
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”
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 waythis study is referenced in otherpublications, \(\alpha\) 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 \(\alpha\).
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
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.
[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:
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.