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.
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.
At the atomic level, montmorillonite is built up of so-called TOT-layers: covalently bonded sheets of aluminum (“O”) and silica (“T”) oxide (including the right amount of impurities/defects). In my mind, such TOT-layers make up the fundamental particles of a bentonite sample. Reasonably, since montmorillonite TOT-layers vary extensively in size, and since a single cubic centimeter of bentonite contains about ten million billions (\(10^{16}\)), they are generally configured in some crazily complicated manner.
Stack descriptions in the literature
But the idea that the single TOT-layer is the fundamental building
block of bentonite is not shared with many of today’s bentonite
researchers. Instead, you find descriptions like e.g. this one, from
Bacle et al. (2016)
Clay mineral particles consist of stacks of parallel
negatively-charged layers separated by interlayer
nanopores. Consequently, compacted smectite contains two major
classes of pores: interlayer nanopores (located inside the
particles) and larger mesopores (located between the particles).
In compacted rocks, montmorillonite (Mt) forms aggregates
(particles) with 5–20 TOT layers (Segad et al., 2010). A typical
radial size of these particles is of the order of 0.01 to 1
\(\mathrm{\mu m}\). The pore space between Mt particles is referred to
as interparticle porosity. Depending on the degree of compaction,
the interparticle porosity contributes 10 to 30% of the total water
accessible pore space in Mt (Holmboe et al., 2012; Kozaki et al.,
2001).
Such statements show that researchers have something more complex in mind than individual TOT-layers when speaking of “particles”: they are some sort of assemblages of TOT-layers. The quotation of Bacle et al. (2016), using both the terms “stacks” and “particles”, even hints at an idea of a hierarchy of fundamental structures. Such a hierarchy is expressed explicitly in e.g. Navarro et al. (2017), who provide a figure with the caption “Schematic particle arrangement in highly compacted Na-bentonite” that looks similar to this one:
Here it is clear that they differ between “aggregates” (which I’m
not sure is the same thing as “particles”), “stacks”, and
individual TOT-layers (which I assume are represented by the
line-shaped objects). In the following, however, we will use the term
“stack” to refer to any kind of suggested fundamental structure
built up from individual TOT-layers.
The one-sentence version of this blog post is:
Stacks make no sense as fundamental building blocks in models of water saturated, compacted bentonite.
The easiest argument against stacks is, in my mind, to simply work out
the geometrical consequences. But before doing that we will examine
some of the references given to support statements about stacks in
compacted systems. Often, no references are given at all, but when
they are, they usually turn out to be largely irrelevant for the
system under study, or even to support an opposite view.
Inadequate referencing
As an example (of many) of inadequate referencing, we
use the statement above from Churakov et al. (2014) as
a starting point. I think this is a “good” statement, in the sense
that it makes rather precise claims about how compacted bentonite is
supposed to be structured, and provides references for some key
statements, which makes it easier to criticize.
Clay is normally not a homogeneous lamellar material. It might be
better described as a disordered structure of stacks of platelets,
sometimes called tactoids — a tactoid typically consists of 5-20
platelets.19-21
Here the terminology is quite different from the previous quotations: TOT-layers are called “platelets”, and “particles” are called “tactoids”. Still, they use the phrase “stacks of platelets”, so I think we can continue with using “stack” as a sort of common term for what is being discussed.1 We may also note that here is used the word “clay”, rather than “montmorillonite” (as does Bacle et al. (2016)), but it is clear from the context of the article that it really is montmorillonite/bentonite that is discussed.
Anyhow, Segad et al. (2010) do not give much direct information on the claim we investigate, but provide three new references. Two2 of these — Banin (1967) and Shalkevich et al. (2007) — are actually studies on montmorillonite suspensions, i.e. as far away as you can get from compacted bentonite in terms of density; the solid mass fraction in these studies is in the range 1 – 4%.
The average distance between individual TOT-layers in this density limit is comparable with, or even larger than, their typical lateral extension (~100 nm). Therefore, much of the behavior of low density montmorillonite depends critically on details of the interaction between layer edges and various other components, and systems in this density limit behave very differently depending on e.g. ionic strength, cation population, preparation protocol, temperature, time, etc. This complex behavior is also connected with the fact that pure Ca-montmorillonite does not form a sol, while the presence of as little as 10 – 20% sodium makes the system sol forming. The behaviors and structures of montmorillonite suspensions, however, say very little about how the TOT-layers are organized in compacted bentonite.
We have thus propagated from a statement in Churakov et al. (2014), and a similar one in Segad et al. (2010), that montmorillonite in general, in “compacted rocks” forms aggregates of 5 – 20 TOT-layers, to studies which essentially concern different types of materials. Moreover, the actual value of “5 – 20 TOT layers” comes from Banin (1967), who writes
Evidence has accumulated showing that when montmorillonite is
adsorbed with Ca, stable tactoids, containing 5 to 20 parallel
plates, are formed (1). When the mineral is adsorbed with Na, the
tactoids are not stable, and the single plates are separated from
each other.
This source consequently claims that the single TOT-layers are the fundamental units, i.e. it provides an argument against any stack concept! (It basically states that pure Ca-montmorillonite does not form a sol.) In the same manner, even though Segad et al. (2010) make the above quoted statement in the beginning of the paper, they only conclude that “tactoids” are formed in pure Ca-montmorillonite.
The swelling and sedimentation behavior of Ca-montmorillonite is a very interesting question, that we do not have all the answers to yet. Still, it is basically irrelevant for making statements about the structure in compacted — sodium dominated3 — bentonite.
Churakov et al. (2014) also give two references for the statement that the “interparticle porosity” in montmorillonite is 10 – 30% of the total porosity: Holmboe et al. (2012) and Kozaki et al. (2001). This is a bizarre way of referencing, as these two studies draw incompatible conclusions, and since Holmboe et al. (2012) — which is the more adequately performed study — state that this type of porosity may be absent:
At dry density \(>1.4 \;\mathrm{g/cm^3}\) , the average interparticle
porosity for the [natural Na-dominated bentonite and purified
Na-montmorillonite] samples used in this study was found to be
\(1.5\pm1.5\%\), i.e. \(\le 3\%\) and significantly lower than
previously reported in the literature.
Holmboe et al. (2012)
address directly the discrepancy with earlier studies, and suggest
that these were not properly analyzed
The apparent discrepancy between the basal spacings reported by Kozaki et al. (1998, 2001) using Kunipia-F washed Na-montmorillonite, and by Muurinen et al. (2004), using a Na-montmorillonite originating from Wyoming Bentonite MX-80, and the corresponding average basal spacings of the [Na-montmorillonite originating from Wyoming bentonite MX-80] samples reported in this study may partly be due to water contents and partly to the fact that only apparent \(\mathrm{d_{001}}\) values using Bragg’s law, without any profile fitting, were reported in their studies.
If
Kozaki et al. (2001)
should be used to support a claim about “interparticle porosity”, it
consequently has to be done in opposition to — not in conjunction
with —
Holmboe et al. (2012).
It would then also be appropriate for authors to provide arguments for
why they discard the conclusions of
Holmboe et al. (2012).4
Stacks in compacted bentonite make no geometrical sense
The literature is full of fancy figures of bentonite structure involving stacks. A typical example is found in Wu et al. (2018), and looks similar to this:
This illustration is part of a figure with the caption “Schematic representation of the different porosities in bentonite and the potential diffusion paths.”5 The regular rectangles in this picture illustrate stacks that each seems to contain five TOT-layers (I assume this throughout). Conveniently, these groups of five layers have the same length within each stack, while the length varies somewhat between stacks. This is a quite common feature in figures like this, but it is also common that all stacks are given the same length.
Another feature this illustration has in common with others is that the particles are ordered: we are always shown edges of the TOT-layers. I guess this is partly because a picture of a bunch of stacks seen from “the top” would be less interesting, but it also emphasizes the problem of representing the third dimension: figures like these are in practice figures of straight lines oriented in 2D, and the viewer is implicitly required to imagine a 3D-version of this two-dimensional representation.
A “realistic” stack picture
But, even as a 2D-representation, these figures are not representative
of what an actual configuration of stacks of TOT-layers looks like.
Individual TOT-layers have a distinct thickness of about 1 nm, but
varies widely in the other two dimensions.
Ploehn and Liu (2006)
analyzed the size distribution of Na-montmorillonite (“Cloisite
Na+”) using atomic force microscopy, and found an average aspect
ratio of 180 (square-root of basal area divided by thickness). A
representative single “TOT-line” drawn to scale is consequently
quite different from what is illustrated in in most stack-pictures,
and look like this (click on the figure to see it in full size)
In this figure, we have added “water layers” on each side of the
TOT-layer (light red), with the water-to-solid volume ratio of
16. Neatly stacking
five such units shows that the rectangles in the
Wu et
al. (2018)-figure should be transformed like this
But this is still not representative of what an assemblage of five
randomly picked TOT-layers would look like, because the size
distribution has a substantial variance. According to
Ploehn and Liu (2006), the
aspect ratio follows approximately a log-normal distribution. If we
draw five values from this distribution for the length of five
“TOT-lines”, and form assemblages, we end up with structures that
look like this:7
These are the kind of units that should fill the bentonite illustrations. They are quite irregularly shaped and are certainly not identical (this would be even more pronounced when considering the third dimension, and if the stacks contain more layers).
It is easy to see that it is impossible to construct a dense structure
with these building blocks, if they are allowed a random
orientation. The resulting structure rather looks something like this
Such a structure evidently has very low density, and are reminiscent of the gel structures suggested in e.g. Shalkevich et al. (2007) (see fig. 7 in that paper). This makes some sense, since the idea of stacks of TOT-layers (“tactoids”) originated from studies of low density structures, as discussed above.
Note that the structure in pictures like that in Wu et al. (2018) has a substantial density only because it is constructed with stacks with an unrealistic shape. But even in these types of pictures is the density not very high: with some rudimentary image analysis we conclude that the density in the above picture is only around 800 kg/m3. Also the figure from Navarro et al. (2017) above gives a density below 1000 kg/m3, although there it is explicitly stated that it is a representation of “highly compacted bentonite”.
The only manner in which the “realistic” building blocks can be
made to form a dense structure is to keep them in the same
orientation. The resulting structures then look e.g. like this
where we have color coded each stack, to remind ourselves that these
units are supposed to be fundamental.
Just looking at this structure of a “stack of stacks” should make it clear how flawed the idea is of stacks as fundamental structural units in compacted bentonite (note also how unrepresentative the stack-pictures found in the literature are). One of many questions that immediately arises is e.g. why on earth the tiny gaps between stacks (indicated by arrows) should remain. This brings us to the next argument against stacks as fundamental units for compacted water saturated bentonite:
What is supposed to keep stacks together?
Compacted bentonite of interest e.g. for sealing in radioactive waste
repositories exerts swelling pressure of several MPa when in contact
with external water. This osmotic pressure is a
consequence
of the presence of the mobile exchangeable cations in
montmorillonite. Each “realistic” unit that we have imagined above is thus required to be at a
huge elevated pressure, and the individual TOT-layers have a strong
driving force to separate. And, unless a mechanism is provided for
why such a separation is impossible, this is of course what we
expect to happen! As far as I am aware, such a separation inhibiting
mechanism has never been suggested in any publication that promotes
the stack concept in compacted bentonite. To get a feel for the
absurdity of this issue, let’s take a new look at the figure
from
Navarro et
al. (2017)
Assuming that this system is in equilibrium with an external water
reservoir at zero pressure (i.e. atmospheric absolute pressure), the
pressure in the compartment labeled “intra-aggregate space” is also
close to zero. On the other hand, in the “stacks” located just a few
nm away, the pressure is certainly above 10 MPa in many places! A
structure like this is obviously not in mechanical equilibrium! (To use
the term “obvious” here feels like such an understatement.)
Implications
To sum up what we have discussed so far, the following picture
emerges. The bentonite literature is packed with descriptions of
compacted water saturated bentonite as built up of stacks as
fundamental units. These descriptions are so commonplace that they
often are not supported by references. But when they are, it seems
that the entire notion is based on misconceptions. In particular,
structures identified in low density systems (suspensions, gels) have
been carried over, without reflection, to descriptions of compacted
bentonite. Moreover, all figures illustrating the stack concept are
based on inadequate representations of what an arbitrary assemblage of
TOT-layers arranged in this way actually would look like. With a
“realistic” representation it quickly becomes obvious that it makes
little sense to base a fundamental unit in compacted systems on the
stack concept.
My impression is that this flawed stack concept underlies the entire
contemporary mainstream view of compacted bentonite, as e.g. expressed
by
Wu et al. (2018):
A widely accepted view is that the total porosity of bentonite
consists of \(\epsilon_ {ip}\) and \(\epsilon_ {il}\) (Tachi and
Yotsuji, 2014; Tournassat and Appelo, 2011; Van Loon et al., 2007).
\(\epsilon_ {ip}\) is a porosity related to the space between the
bentonite particles and/or between the other grains of minerals
present in bentonite. It can further be subdivided into
\(\epsilon_ {ddl}\) and \(\epsilon_ {free}\). The diffuse double layer,
which forms in the transition zone from the mineral surface to the
free water space, contains water, cations and a minor amount of
anions. The charge at the negative outer surface of the
montmorillonite is neutralized by an excess of cations. The free
water space contains a charge-balanced aqueous solution of cations
and anions. \(\epsilon_ {il}\) represents the space between TOT-layers
in montmorillonite particles exhibiting negatively charged
surfaces. Due to anion exclusion effect, anions are excluded from
the interlayer space, but water and cations are present.
This view can be summarized as:
The fundamental building blocks are stacks of TOT-layers
(“particles”, “aggregates”, “tactoids”, “grains”…)
Electric double layers are present only on external
surfaces of the stacks.
Far away from external surfaces — in the “inter-particle” or
“inter-aggregate” pores — the diffuse layers merge with a bulk
water solution
Interlayer pores are defined as being internal to the stacks,
and are postulated to be fundamentally different from the external
diffuse layers; they play by a different set of rules.
I don’t understand how authors can get away with promoting this
conceptual view without supplying reasonable arguments for all of its
assumptions8 — and with such a
complex structure, there are a lot of assumptions.
As already discussed, the geometrical implications of the suggested structure do not hold up to scrutiny. Likewise, there are many argumentsagainst the presence of substantial amounts of bulk water in compacted bentonite, including the pressure consideration above. But let’s also take a look at what is stated about “interlayers” and how these are distinguished from electric double layers (I will use quotation marks in the following, and write “interlayers” when specifically referring to pores defined as internal to stacks).
“Interlayers”
“Interlayers” are often postulated to be completely devoid of anions. We discussed this assumption in more depth in a previous blog post, where we discovered that the only references supplied when making this postulate are based on the Poisson-Boltzmann equation. But this is inadequate, since the Poisson-Boltzmann equation does describe diffuse layers, and predicts anions everywhere.
By requiring anion-free “interlayers”, authors actually claim that the physico-chemistry of “interlayers” is somehow qualitatively different from that of “external surfaces”, although these compartments have the exact same constitution (charged TOT-layer surface + ions + water). But an explanation for why this should be the case is never provided, nor is any argument given for why diffuse layer concepts are not supposed to apply to “interlayers”.9 This issue becomes even more absurd given the strong empirical evidence for that anions actually do reside in interlayers.
The treatment of anions is not the only ad hoc description of
“interlayers”. It also seems close to mandatory to describe them as
having a maximum extension, and as having an extension independently
parameterized by sample density. E.g. the influential models for
Na-bentonite of Bourg
et al. (2006) and
Tournassat and Appelo (2011)
both rely on the idea that “interlayers” swell out to a certain
volume that is smaller than the total pore volume, but that still
depends on density.
In e.g. Bourg et
al. (2006), the fraction of “interlayer” pores remains essentially
constant at ~78%, as density decreases from 1.57 g/cm3 to
1.27 g/cm3, while the “interlayers” transform from having 2
monolayers of water (2WL) to having 3 monolayers (3WL). This is a very
strange behavior: “interlayers” are acknowledged as having a swelling
potential (2WL expands to 3WL), but do, for some reason, not affect
22% of the pore volume! Although such a behavior strongly deviates
from what we expect if “interlayers” are treated with conventional
diffuse layer concepts, no mechanism is provided.
Another type of macabre consequence of defining “interlayer” pores
as internal to stacks is that a completely homogeneous system is
described has having no interlayer pores (because it has no
stacks). E.g.
Tournassat and
Appelo (2011)
write (\(n_c\) is the number of TOT-layers in a stack)
[…] the number of stacks in the \(c\)-direction has considerable influence on the interlayer porosity, with interlayer porosity increasing with \(n_c\) and reaching the maximum when \(n_c \approx 25\). The interlayer porosity halves with \(n_c\) when \(n_c\) is smaller than 3, and becomes zero for \(n_c = 1\).10
It is not acceptable that using the term interlayer should require
accepting stacks as fundamental units. But the usage of the term as
being internal to stacks is so widespread in the contemporary
bentonite literature, that I fear it is difficult to even communicate
this issue. Nevertheless, I am certain that e.g.
Norrish (1954) does not
depend on the existence of stacks when using the term like this:
Fig. 7 shows the relationship between interlayer spacing and water
content for Na-montmorillonite. There is good agreement between the
experimental points and the theoretical line, showing that
interlayer swelling accounts for all, or almost all, of physical
swelling.
The stack view obstructs real discovery
A severe consequence of the conceptual view just discussed is that “stacking number” — the (average) number of TOT-layers that stacks are supposed to contain — has been established as fitting parameter in models that are clearly over-parameterized. An example of this is Tournassat and Appelo (2011), who write11
Our predictive model excludes anions from the interlayer space and
relates the interlayer porosity to the ionic strength and the
montmorillonite bulk dry density. This presentation offers a good
fit for measured anion accessible porosities in bentonites over a
wide range of conditions and is also in agreement with microscopic
observations.
But since anions do reside in interlayers,12 it would be better if the model didn’t fit: an over-parameterized or conceptually flawed model that fits data provides very little useful information.
A similar more recent example is Wu et al. (2018). In this work, a model based on the stack concept is successfully fitted both to data on \(\mathrm{ReO_4^-}\) diffusion in “GMZ” bentonite and to data on \(\mathrm{Cl^-}\) diffusion in “KWK” bentonite, by varying “stacking number” (among other parameters). Again, as the model assumes anion-free “interlayer” pores, a better outcome would be if it was not able to fit the data. Moreover, this paper focuses mainly on the ability of the model, while not at all emphasizing the fact that about ten (!) times more \(\mathrm{ReO_4^-}\) was measured in “GMZ” as compared with \(\mathrm{Cl^-}\) in “KWK”, at similar conditions in certain cases. The latter observation is quite puzzling and is, in my opinion, certainly worth deeper investigation (and I am fully convinced that it is not explained by differences in “stacking number”).
[3] Note that “sodium
dominated” in this context means ~20% or more.
[4] It may be noticed that
Kozaki et
al. (2001) see no X-ray diffraction peaks for low
density samples:
The basal spacing of water-saturated
montmorillonite was determined by the XRD method. […] It was found
that a basal spacing of 1.88 nm, corresponding to the three-water
layer hydrate state […] was not observed before the dry density
reached 1.0 Mg/m3.
My interpretation of this observation is that the diffraction peak has
shifted to even lower reflection angles (in agreement with the
observations
of Holmboe
et al. (2012)), not registered by the equipment. The alternative
interpretation must otherwise be that “stacks” suddenly cease to
exist below 1.0 g/cm3. (Yet,
Kozaki et al. (2001)
continues to use a certain d-value in their analysis, also for densities
below 1.0 g/cm3.)
[5] I have discussed “diffusion
paths” in an
earlier blog post.
This illustration certainly fits that discussion.
[6] A water-to-solid volume ratio of 1 corresponds basically to
interlayers of three monolayers of water (3WL).
[7] To construct these units, I made the additional choice of placing each layer randomly in the horizontal direction, with the constraint that all layers should be confined within the range of the longest one in each unit.
[8] By “get away with” I mean “pass peer-review”, and by “don’t understand” I mean “understand”.
[10] A mathematical remark: if the interlayer porosity “halves with \(n_c\)” (what does that mean?) when \(n_c = 2\) (“smaller than 3”), it is impossible to simultaneously have zero interlayer porosity for \(n_c = 1\) (unless the interlayer porosity is zero for any \(n_c\)).
[11] I guess the word “presentation” here really should be “representation”?
[12] Note that one of the authors of this paper also claims in a later paper that anions do populate 3-waterlayer interlayers, in accordance with the Poisson-Boltzmann equation:
The agreement
between PB calculations and MD simulation predictions was somewhat
worse in the case of the \(\mathrm{Cl^-}\) concentration profiles than
in the case of the \(\mathrm{Na^+}\) profiles (Figure 3), perhaps
reflecting the poorer statistics for interlayer Cl concentrations
[…] Nevertheless, reasonable quantitative agreement was found
(Table 2).
In this blog post I discuss the description of bentonite swelling
often adopted in the fields of soil mechanics and geotechnical
engineering. In particular, we focus on the concept of suction,
which is central in these research fields, while being basically
absent in others.
As far as I understand, suction is just the water chemical potential “disguised” as a pressure variable; although I have trouble finding clear-cut definitions, it seems clear that suction is directly inherited from the “water potential” concept, which has been central in soil science for a long time. Applied to bentonite, the geotechnical description is thus not principally different from the osmotic approach that I have presented previously. But the way the suction concept is (and isn’t) applied may cause unnecessary confusion regarding the swelling mechanisms. I think a root for this confusion is that suction involves both osmotic and capillary mechanisms.
Matric suction (capillary suction)
Matric suction is typically associated with capillarity, a fundamental mechanism in many conventional soil materials under so-called unsaturated conditions. A conventional soil with a significant amount of small enough pores shows capillary condensation, i.e. it contains liquid water below the condensation point for ordinary bulk water. Naturally, the equilibrium vapor pressure increases with the amount of water in the soil, as the pores containing liquid water become larger. For conventional soils, it therefore makes sense to speak of the degree of saturation of a sample, and to relate saturation and equilibrium vapor pressure by means of a water retention curve. Underlying this picture is the notion that the solid parts constitute a “soil skeleton” (the matrix), and that the soil can be viewed as a vessel that can be more or less filled with water.
The pressure of the capillary water is lower than that of the surrounding air, and is related to the curvature of the interfaces between the two phases (menisci), as expressed by the Young-Laplace equation. For a spherically symmetric meniscus this equation reads
where \(p_a\) and \(p_w\) denote the pressures of air and capillary water,
respectively, \(\sigma\) is the surface tension, and \(r\) is the radius
of curvature of the interface. The sign of \(r\) depends on whether the
interface bulges inwards (“concave”, \(r<0\)) or outwards (“convex”,
\(r>0\)). For capillary water, \(r\) is negative and \(\Delta p\) — which
is also called the
Laplace
pressure — is a negative quantity.
As far as I understand, matric suction is simply defined as the negative Laplace pressure, i.e.
With this definition, suction has a straightforward physical meaning
as quantifying the difference in pressure of the two fluids occupying
the pore space, and clearly relates to the everyday use of the word.
Suction — in this capillary sense — gives a simple principal
explanation for (apparent) cohesion in e.g. unsaturated sand:
individual grains are pushed together by the air-water pressure
difference, as schematically illustrated here (the yellow stuff is
supposed to be two grains of sand, and the blue stuff water)
It is reasonable to assume that the net force transmitted by the soil skeleton — usually quantified using the concept of effective stress — governs several mechanical properties of the soil sample, e.g. shear strength. The above description also makes it reasonable to assume that effective stress depends on suction.
Thus, in unsaturated conventional soil are quantities like degree of saturation, pore size distribution, (matric) suction, effective stress, and shear strength very much associated. Another way of saying this is that there is an optimal combination of water content and particle size distribution for constructing the perfect sand castle.
The chemical potential of the capillary water is related to matric suction. Choosing pure bulk water under pressure \(p_a\)1 as reference, the chemical potential of the liquid phase in the soil is obtained by integrating the Gibbs-Duhem equation from \(p_a\) to \(p_w\)
where \(\mu_0\) is the reference chemical potential, \(v\) is the molar volume of water, and we have assumed incompressibility.
The above expression shows that matric suction in this case directly quantifies the (relative) water chemical potential. Note, however, that eq. 3 does not define matric suction; \(s_m\) is defined as a pressure difference between two phases (eq. 2), and happens to quantify the chemical potential under the present circumstances (pure capillary water).
A chemical potential can generally be expressed in terms of activity (\(a\))
\begin{equation} \mu = \mu_0 + RT \ln a \tag{4} \end{equation}
For our case, water activity is to a very good approximation equal to relative humidity, the ratio between the vapor pressures in the state under consideration and in the reference state, i.e. \(a = p_v/p_{v,0}\). Combining eqs. 3 and 4, we see that the vapor pressure in this case is related to matric suction as
This is the so-called
Kelvin
equation, which relates the equilibrium vapor pressure to the
curvature of an air-pure water interface. Note that, since \(r<0\) for
capillary water, the vapor pressure is lower than the corresponding
bulk value (\(p_v < p_{v,0}\)).
Osmotic suction and total suction
So far, we have discussed suction in a capillary context, and related it to water chemical potential or vapor pressure. Now consider how the picture changes if the pores in our conventional soil contain saline water. Matric suction — i.e. the actual pressure difference between the pore solution and the surrounding air, sticking with eq. 2 as the definition — is in general different from the pure water case, because solutes influence surface tension. Also, water activity (vapor pressure) is different from the pure water case, but there is no longer a direct relation between water activity and matric suction, because water activity is independently altered by the presence of solutes.
where we have assumed a salt concentration \(c\), and indicated that the
osmotic pressure, and hence the chemical potential, depends on this
concentration.
Although eq. 5 is of the same form as eq. 3, matric suction and osmotic pressure are very different quantities. The former is defined under circumstances where an actual pressure difference prevail between the air and water phases. In contrast, there is no pressure difference between the phases in a container containing both a solution and a gas phase. \(\pi(c)\) corresponds to the elevated pressure that must be applied for the solution to be in equilibrium with pure water kept at the reference pressure.
Despite the different natures of matric suction and osmotic pressure, the fields of geotechnical engineering and soil mechanics insist on also referring to \(\pi(c)\) as a suction variable: the osmotic suction. Similarly, total suction is defined as the sum of matric and osmotic suction
Total suction is thus de facto defined simply as the (relative)
value of the water chemical potential, expressed as a pressure (I
think this is completely analogous to “total water potential” in
soil science).
Eq. 6 shows that \(\Psi\) is directly related to water activity, or vapor pressure, and we can write
This relation is quite often erroneouslyreferred to as the Kelvin equation (or “Kelvin’s law”) in the bentonite literature. But note that the above equation just restates the definition of water activity, because \(v\cdot\Psi\) cannot be reduced to anything more concrete than the relative value of the water chemical potential. The Kelvin equation, on the other hand, expresses something more concrete: the equilibrium vapor pressure for a curved air-water interface. Some clay literature refer to the above relation as the “Psychrometric law”, but that name seems not established in other fields.2
A definition is motivated by its usefulness, and total change in water chemical potential is of course central when considering e.g. moisture movement in soil. My non-geotechnical brain, however, is not fond of extending the “suction” variable in the way outlined above. To start with, there is already a variable to use: the water chemical potential. Also, “total suction” no longer has the direct relation to the everyday use of the word suction: there is no “sucking” going on in a saline bulk solution,3 while in a capillary there is. Furthermore, with a saline pore solution there is no direct relation between (total) suction and e.g. effective stress or shear strength.
Although both matric suction and osmotic pressure under certain circumstances can be measured in a direct way, it seems that (total) suction usually is quantified by measuring/controlling the vapor pressure with which the soil sample is in equilibrium. Actually, one of the more comprehensive definitions of various “suctions” that I have been able to find — in Fredlund et al. (2012) — speaks only of various vapor pressures (although based on the capillary and osmotic concepts):4
Matric or capillary component of free energy: Matric suction is the equivalent suction derived from the measurement of the partial pressure of the water vapor in equilibrium with the soil-water relative to the partial pressure of the water vapor in equilibrium with a solution identical in composition with the soil-water.
Osmotic (or solute) component of free energy: Osmotic suction is the equivalent suction derived from the measurement of the partial pressure of the water vapor in equilibrium with a solution identical in composition with the soil-water relative to the partial pressure of water vapor in equilibrium with free pure water.
Total suction or free energy of soil-water: Total suction is the equivalent suction derived from the measurement of the partial pressure of the water vapor in equilibrium with the soil-water relative to the partial pressure of water vapor in equilibrium with free pure water.
It seems that such operational definitions of suction has made
the term synonymous with “vapor pressure depression” in large parts
of the bentonitescientificliterature.
Suction in bentonite
In the above discussion we had mainly a conventional soil in mind. When applying the suction concepts to bentonite,5 I think there are a few additional pitfalls/sources for confusion. Firstly, note that the definitions discussed previously involve “a solution identical in composition with the soil-water”. But soil-water that contains appreciable amounts of exchangeable ions — as is the case for bentonite — cannot be realized as an external solution.
It seems that this “complication” is treated by assuming that an external solution in equilibrium with a bentonite sample is the soil-water (this is analogous to how many geochemists use the term “porewater” in bentonite contexts). Not surprisingly, this treatment has bizarre consequences. The conclusion for e.g. a salt free bentonite sample — which is in equilibrium with pure water — is that it lacks osmotic suction, and that its lowered vapor pressure (when isolated and unloaded) is completely due to matric suction! I think this is such an odd outcome that it is worth repeating: A system dominated by interlayer pores, containing dissolved cations at very high concentrations, is described as lacking osmotic pressure! It is not uncommon to find descriptions like this one (from Lang et al. (2019))
The total suction of unsaturated soils consists of matric and
osmotic suctions (Yong and Warkentin, 1975; Fredlund et al., 2012;
Lu and Likos, 2004). In clays, the matric suction is due to surface
tension, adsorptive forces and osmotic forces (i.e. the diffuse
double layer forces), whereas the osmotic suction is due to the
presence of dissolved solutes in the pore water.
We apparently live in a world where “matric suction” consists of “osmotic forces”, while the same “osmotic forces” do not contribute to “osmotic suction”. Except when the clay contains excess ions, in which case we have an arbitrary combination of the two “suctions” (note also that “osmotic suction” and “osmotic swelling” are two quite different things).
Although the above consequence is odd, it is still only a matter of definition: accepting that “matric suction” involves osmotic forces (which I don’t recommend), the description may still be adequate in principle; after all, “total suction” quantifies the reduction of the water chemical potential.
But the focus on “matric suction” also reveals a conceptual view of bentonite structure that I find problematic: it suggests a first order approximation of bentonite as a conventional soil, i.e. as an assemblage of solid grains separated from an aqueous phase (and a gas phase). This “matric” view is fully in line with the idea of “free water” in bentonite, and it is quite clear that this is a prevailing view in the geotechnical, as well as in the geochemical, literature. For instance, with the formulation “the presence of dissolved solutes in the pore water” in the above quotation, the “pore water” the authors have in mind is a charge neutral bulk water solution.
With the “matric” conceptual view, the degree of saturation becomes a central variable in much soil mechanical analyses of bentonite. When dealing with actual unsaturated bentonite samples, I guess this makes sense, but once a sample is saturated this variable has lost much of its meaning.6 Consider e.g. the different expected behaviors if drying e.g. a water saturated metal filter or a saturated bentonite sample.
The equilibrium vapor pressure of both these systems is lower than the
corresponding pure bulk water value. For the metal filter, the lowered
water activity is of course due to capillarity, i.e. there is an
actual pressure reduction in the water phase (matric suction!). When
lowering the external vapor pressure below the equilibrium point
(i.e. drying), capillary water migrates out of the filter, while the
metal structure itself remains intact. In this case, as the system
remains defined in a reasonable way, it is motivated to speak of the
saturation state of the filter.
For a drying bentonite sample, the behavior is not as well defined,
and depends on how the drying is performed and on initial water
content. For a quasi-static process, where the external vapor pressure
is lowered in small steps at an arbitrary slow rate, it should be
clear that the entire sample will respond simply by shrinking. In
this case it does not make much sense to speak of the sample as still
being saturated, nor to speak of it as having become unsaturated.
For a more “violent” drying process, e.g. placing the bentonite sample in an oven at 105 °C , it is also clear that — rather than resulting in a neatly shrunken, dense piece of clay — the sample now will suffer from macroscopic cracks and other deformations. Neither in this case does it make much sense to try to define the degree of saturation, in relation to the sample initially put in the oven.
Note also that if we, instead of drying, increase the external
vapor pressure from the initial equilibrium value, the metal filter
will not respond much at all, while the bentonite sample immediately
will begin to swell.
I hope that this example has made it clear, not only that the degree of saturation is in general ill-defined for bentonite, but also that a bentonite sample behaves more as an aqueous solution rather than as a conventional soil: if we alter external vapor pressure, an aqueous solution responds by either “swelling” (taking up water) or “shrinking” (giving off water). A main aspect of this conceptual view of bentonite — which we may call the “osmotic” view — is that water does not form a separate phase7. This was pointed out e.g. by Bolt and Miller (1958) (referring to this type of system as an “ideal clay-water system”)
In contrast to the familiar case described is the ideal clay-water
system in which the particles are not in direct contact but are
separated by layers of water. Removal of water from such a system
does not introduce a third phase but merely causes the particles to
move closer to one another with the pores remaining water saturated.
From these considerations it follows that a generally consistent treatment is to relate bentonite water activity to water content, rather than to degree of saturation.
Another consequence of adopting a “matric” view of bentonite (i.e. to include osmotic forces in “matric suction”) is that “matric suction” loses its direct connection with effective stress. This can be illustrated by taking the “osmotic” view: just as the mechanical properties of an aqueous solution (e.g. viscosity) do not depend critically on whether or not it is under (osmotic) pressure, we should not expect e.g. bentonite shear strength to be directly related to swelling pressure.8
[1] Often, the air is at atmospheric pressure, in which case the reference is the ordinary standard state.
[2] The relatively common misspelling “Psychometric law” is kind of funny.
[3] The cautious reader may remark that saline solutions do “suck”, in terms of osmosis. But note the following: 1) Osmosis requires a semi-permeable membrane, separating the solution from an external water source. We have said nothing about the presence of such a component in the present discussion. The way osmotic suction sometimes is described in the literature makes me suspect that some authors are under the impression that the mere presence of a solute causes a pressure reduction in the liquid. 2) In the presence of a semi-permeable membrane, osmosis has no problem occurring without a pressure difference between between the two compartments. 3) For cases when the solution is acted on by an increased hydrostatic pressure, water is transported from lower to higher pressure. It is difficult to say that there is any “sucking” in such a process (I would argue that the establishment of a pressure difference is an effect, rather than a cause, in the case of osmosis) 4) The idea that a solution has a well-defined partial water pressure is wrong.
[4] I’m still not fully satisfied with this definition: It may be noted that the definitions are somewhat circular (“matric suction is the equivalent suction…”), so they still require that we have in mind that “suction” also is defined in terms of a certain vapor pressure ratio (e.g. eq. 7). Note also that the headings speak of “free energy”. Perhaps I am nitpicking, but (free) energy is an extensive quantity, while suction (pressure) is intensive. Thus, “free energy” here really mean “specific free energy” (or “partial free energy”, i.e. chemical potential). I think the soil science literature in general is quite sloppy with making this distinction.
[5] “Bentonite” is used in the following as an abbreviation for bentonite and claystone, or any clay system with significant cation exchange capacity.
[6] If you press bentonite granules to form a cohesive sample you certainly end up with a system having both water filled interlayer pores and air-filled macropores (or perhaps an even more complex pore structure). This blog post mainly concerns saturated bentonite, by which I mean bentonite material which does not contain any gas phase. We can thus speak of saturated bentonite, although a degree of saturation variable is not well defined.
[8] However, bentonite strength relates indirectly to swelling pressure (under specific conditions) because both quantities depends on a third: density.
Swelling is not due to electrostatic repulsion between
montmorillonite particles
Few things confuse me more than how the role of electrostatics in
clay swelling is described in the scientific literature. Consider
e.g. this statement from
Bratko et
al. (1986)
The interaction between charged aggregates in solution is generally
interpreted in terms of electrostatic repulsion between double
layers surrounding the aggregates.
But in the same paper we learn that the main contribution to the force
between two charged surfaces in solution is the entropy of mixing of
counter-ions, and that electrostatic interactions actually may result
in an attractive force between the surfaces.
Nevertheless, I think
Bratko et
al. (1986) are right: swelling is, for some reason, often
“interpreted” in terms of electrostatic repulsion between electric
double layers. It is easy to find statements that e.g. the expression
for the osmotic pressure in the Gouy-Chapman model describes
“the
electrostatic force per unit area”, or that lamellar phases are
“electrostatically
swollen”, with an osmotic pressure “mainly of electrostatic
origin”. Segad
(2013) writes
The interactions between the negatively charged platelets lead to a
repulsive long-ranged electrostatic force promoting swelling.
The DLVO theory describes the interaction between two colloidal
particles as a balance between electrostatic repulsion, in this case
between two negatively charged clay layers, and vdW attraction.
Laird (2006) claims
that electrostatics cause both repulsion and (strong)
attraction between clay layers
A balance between strong electrostatic-attraction and
hydration-repulsion forces controls crystalline swelling. The extent
of crystalline swelling decreases with increasing layer
charge. Double-layer swelling occurs between quasicrystals. An
electrostatic repulsion force develops when the positively charged
diffuse portions of double layers from two quasicrystals overlap in
an aqueous suspension. Layer charge has little or no direct effect
on double-layer swelling.
Although many authors reasonably understand the actual mechanisms of
double layer repulsion, I think it is very unfortunate that this
language is established and contributes to unnecessary confusion.
To gain some intuition for that clay swelling is not primarily due to
electrostatic repulsion between montmorillonite particles, let us
consider the
Poisson-Boltzmann equation. This is, after all, the description
usually referred to when authors speak of “electrostatic repulsion”
between clay layers. The Poisson-Boltzmann equation may be used to
describe the electrostatic potential, and the corresponding
counter-ion equilibrium distribution, between two equally charged
surfaces, and a typical result looks like this1
Here we assume two negatively charged parallel surfaces with uniform
charge density, and the counter-ions are represented by a continuous
charge density. The system is assumed infinitely extended in the x-
(in/out of the page) and y- (up/down) directions, and thus
rotationally symmetric around the z-axis.
With a lot of equal charges “facing” each other, the illustration
may indeed give the impression that there somehow is an electrostatic
repulsion between the surfaces. That this is not the case, however,
may be seen from the symmetry of the potential. In fact, replacing one
of the negatively charged surfaces by a neutral surface at half
the distance does not change the solution to the Poisson-Boltzmann
equation! A charged and a neutral surface thus experience the same
repulsion as two charged surfaces, if only placed at half the
distance.2
With one surface being uncharged, “interpreting” the force as an
electrostatic repulsion between the particles makes little
sense.
A related way to convince yourself that there is no electrostatic
repulsion between the two charged surfaces is to consider the electric
field generated by one “half” of the original system. This field
vanishes on the outside of the considered “half”-system.
This means that removing a “half”-system would not be “noticed” by the
other “half”-system, in the sense that the electric field
configuration remains the same (and corresponds to having a neutral
particle at half the distance).
It may be helpful to also remember from electrostatics that
the
electric field outside a plate capacitor vanishes. Thus,
configuring two plate capacitors as shown below, there is no electric
field between the positively charged surfaces, regardless of how close
they are3.
Having established that there is no direct electrostatic repulsion
between clay particles, the obvious question is: what is the
main cause for the repulsion? What the two configurations above have
in common — with either two charged surfaces or one charged and one
neutral surface — is that they restrict the counter ions to a certain
volume. Hence, there is an entropic driving force for transporting
more water into the region between the surfaces, thereby pushing them
apart. Nelson
(2013) describes this quite well 4
One may be tempted to say, “Obviously two negatively charged
surfaces will repel.” But wait: Each surface, together with its
counterion cloud, is an electrically neutral object! Indeed,
if we could turn off thermal motion the mobile ions would collapse
down to the surfaces, rendering them neutral. Thus the repulsion
between like-charged surfaces can only arise as an entropic
effect. As the surfaces get closer than twice their Gouy–Chapman
length, their diffuse counterion clouds get squeezed; they then
resist with an osmotic pressure.
Notice that the presence of this osmotic pressure requires contact with an “external” solution. The existence of a repulsive force between clay layers thus requires that water is available to be transported into the interlayer region. This seems to often be “forgotten” about in many descriptions of clay swelling. But let Kjellander et al. (1988) remind us
The PB pressure between two planar surfaces with equal surface
charge equals \(P_\mathrm{ionic} = k_BT\sum_i n_i(0)\), where \(n_i(0)\)
is the ion density at the midplane between the surfaces. Due to
symmetry there is no electrostatic force between the two halves of
the system (the electrostatic fluctuation forces due to ion-ion
correlations are neglected). To obtain the net pressure when the
system is surrounded by a bulk electrolyte solution, it is necessary
to subtract the external pressure calculated in the same
approximation; this is given by the ideal gas contribution
\(P_\mathrm{bulk} = k_BT \sum_i n_i^\mathrm{bulk}\).
There is no repulsive force of this kind in an isolated, internally
equilibrated, clay.
Moreover, the force is usually conceived of as repulsive because the water chemical potential of the surrounding (“external”) solution is typically larger than in the clay. But from an osmotic viewpoint there is nothing fundamentally different going on when the external phase is, say, vapor of low pressure (set e.g. by a saturated salt solution), causing the clay to lose water, i.e to shrink. Thus, if swelling is “interpreted” as electrostatic repulsion between montmorillonite particles, then drying should be “interpreted” as electrostatic attraction between the same particles.
Although swelling is not primarily due to direct electrostatic
repulsion between clay particles, electrostatics is of course
essential to consider when calculating the osmotic pressure. And
rather 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.
Moreover, a treatment of the electrostatic problem beyond the
mean-field (i.e. beyond the Poisson-Boltzmann description) shows that
ion-ion correlation cause an
explicit attraction between equally charged surfaces (similar to a
van der Waals force). In systems with divalent counter ions, this
attraction is large enough to prevent swelling beyond a certain limit
— a prediction in qualitative agreement with
observation. Electrostatics could thus be claimed to contribute to
prevent clay swelling.
I think comparison with a simple salt solution can be useful. Nobody (?) would come up with the idea that the primary reason for the osmotic pressure of a NaCl solution is due to electrostatic repulsion between, say, chloride ions. In fact, the electrostatic interactions in such a solution reduce the osmotic pressure compared with a corresponding ideal solution.
Below is plotted the swelling pressure of Na-montmorillonite as a
function of the average concentration of counter-ions (data from
here). For comparison, the osmotic pressures of a NaCl solution and
an ideal solution are also plotted (data from
here), as
a function of the total amount of ions (i.e. two times the NaCl
concentration)5
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 dramatic increase of swelling pressure in the high concentration limit is reasonably an effect of hydration of ions and surfaces; it should be kept in mind that an average ion concentration of 3 mol/kgw in Na-montmorillonite roughly corresponds to a water-to-solid-mass ratio of only 0.3, and an average interlayer width below 1 nm.
Even though there seems to be quite some confusion regarding clay swelling pressure in the bentonite literature, the message here is not that everything about it is in reality understood. On the contrary, there are quite a number of behaviors that, as far as I’m aware, lack fully satisfactory explanations. For example, at room temperature the basal spacing in Ca-montmorillonite is never observed to be larger than \(\sim 19\) Å6, corresponding to a (dry) density of approximately \(1300 \;\mathrm{kg/m^3}\); yet, this material systematically exerts swelling pressure at considerably lower density (\(\sim 700 \;\mathrm{kg/m^3}\)). But in order to tackle issues like these, it is essential to be clear about the swelling mechanisms that we actually do understand.
Update (221018): A correction to this blog post is discussed here.
[1] This
particular calculation uses the formulas presented in
Engström and
Wennerström (1978), and assumes mono-valent counter-ions at
room temperature, a charge density of \(-0.1 \;\mathrm{C/m^2}\), and a
surface-surface distance of 2 nm.
[2] Here is only considered the Poisson-Boltzmann
pressure. If e.g. van der Waals attraction between the surfaces is
included, the resulting forces are not necessarily equal. The point
here, however, concerns the repulsion due to the presence of diffuse
layers.
[3] Having strictly zero field is of course an ideal
result, corresponding to an infinitely extended capacitor.
[4] The quotation is taken
from a draft version of this book.
[5] The graph denoted “Ideal solution” is simply
the van’t Hoff relation \(\Pi = RT c\), which strictly is only valid
in the low concentration limit. It is nevertheless here extended to
the whole concentration range. In the same way, the NaCl-curve is
simply \(\Pi = \varphi RT c\), where \(\varphi\) is the osmotic
coefficient for NaCl. Sorry about that.
[6] Upon cooling,
Svensson and
Hansen (2010) actually observed a basal spacing of 21.6 Å in
pure Ca-montmorillonite.
An established procedure in clay research is to differ between
regions of
“crystalline” and “osmotic” swelling.
Although this distinction makes sense in many ways, I think it is
unfortunate that one of the regions has been named “osmotic”, as it
may suggest that bentonite1 swelling is only partly osmotic, or that it is only
osmotic in certain density ranges.
In this post I argue for that bentonite swelling pressure should be understood as an osmotic pressure under all conditions, and discuss the distinction between “crystalline” and “osmotic” swelling in some detail.
Bentonite swelling pressure is an osmotic pressure, under all conditions
A macroscopic definition of osmosis and osmotic pressure cannot depend
on specific microscopic aspects. Here we take the description from
Atkins’
Physical Chemistry2 as a starting point
The phenomenon of osmosis is the spontaneous passage of a pure solvent into a solution separated from it by a semipermeable membrane , a membrane permeable to the solvent but not to the solute. The osmotic pressure , \(\Pi\), is the pressure that must be applied to the solution to stop the influx of solvent.
These definitions are written with simple aqueous solutions in mind,3 but can easily be generalized to include bentonite lab samples. For such a case the role of the “solution” is taken by the bentonite sample, and the “solutes” are the exchangeable cations and other dissolved species, as well as the individual clay particles. The semipermeable membrane in a bentonite set-up is typically filters confining the sample. Note that such filters are impermeable only to the clay particles, while e.g. the exchangeable ions can freely move across them. That the exchangeable ions anyway are located in the sample is because of the electrostatic coupling between them and the clay particles; the filters keep the clay particles in place, and the requirement of charge neutrality forces, in turn, the exchangeable ions to stay in place. Finally, in a bentonite set-up the external water source is in general itself an aqueous solution (often a salt solution). But even if the above description assumes a source of pure solvent it is clear that the mechanism (passage of solvent) is active also if the external source contains several components.
With these remarks it should be clear that water uptake in a
laboratory bentonite sample is an osmotic effect and that swelling
pressure is an osmotic pressure: swelling pressure is the pressure
(difference) that must be applied to prevent further spontaneous
inflow of water from the external source.
Note that the definition of osmotic pressure says nothing about the specific microscopic conditions — it would be rather bizarre if it did. That would imply that the poor lab worker must have knowledge, e.g. of whether a certain interlayer distance is realized in the sample, in order to judge whether or not the measured swelling pressure is an osmotic pressure.
What qualifies swelling pressure as an osmotic pressure is summarized
in the relation
which in earlier blog posts was shown to be generally valid in bentonite. Here \(\Delta \mu_w\) is the difference in water chemical potential between the non-pressurized bentonite and the external solution, and \(v\) is the partial molar volume of water. The presence of \(\Delta \mu_w\) in eq. 1 expresses the “spontaneous” character of the phenomenon: “spontaneous” in this context means movement of water from higher to lower chemical potential. \(\Delta \mu_w\) may have contributions both from entropy and energy, which can be expressed (a bit sloppy) as
where \(\Delta h_w\) and \(\Delta s_w\) are the differences in (partial) molar enthalpy and entropy, respectively, and \(T\) is the absolute temperature.
When only mixing entropy contributes to \(\Delta \mu_w\), and in the
limit of a dilute solution, eq. 1 reduces
to
van
‘t Hoff’s formula \(\Pi = RTc\), where \(c\) is the solute
concentration. Thus, rather than defining osmotic pressure, van ‘t
Hoff’s formula is a limit of the the general relation expressed in
eq. 1.
“Crystalline” vs. “osmotic” swelling
Although a division between “crystalline” and “osmotic” swelling regions can be found in the literature as far back as the 1930s, there doesn’t seem to be fully coherent definitions of these terms.
Note that any of these definitions complies with swelling pressure
being an osmotic pressure of the form discussed above; the release of
heat, or effects of “hydration”, is accommodated by a non-zero
enthalpy contribution (\(\Delta h_w\)) in eq. 2.
Unlike innercrystalline swelling, which acts over small distances
(up to 1 nm), osmotic swelling, which is based on the repulsion
between electric double layers, can act over much larger
distances. In sodium montmorillonite it can result in the complete
separation of the layers. […] The driving force for the osmotic
swelling is the large difference in concentration between the ions
electrostatically held close to the clay surface and the ions in the
pore water of the rock.
Leaving aside what is exactly meant by the term “pore water”, there are several issues here. Firstly, it appears that the authors have in mind a text book version of osmosis — basically van ‘t Hoff’s formula — when writing that the driving force is due to “differences in concentration”. But the actual driving force is differences in water chemical potential, which only under certain circumstances can be translated to differences in solute concentration. Note that also in the case of “crystalline” swelling is water transported from regions of low to regions of (really) high ion concentration. So, with the same logic you can also claim that the driving force for “crystalline” swelling is “large differences in concentration”.
Secondly, the electric double layer is an example of a system where there is no simple relation between ion concentration differences and transport driving forces — the diffuse layer displays an ion concentration gradient in equilibrium, and very weakly overlapping diffuse layers can be conceived of, where the driving force for in-transport of water is minimal, even though the ion concentration closest to the surfaces is large. To arrive at a van ‘t Hoff-like equation for the osmotic pressure of an overlapping diffuse layer, you first have to solve an electrostatic problem (the Poisson-Boltzmann equation, or something worse). With that analysis made, the (approximate4) osmotic pressure can be related to the midpoint concentration in the interlayer space. Madsen and Müller-Vonmoos (1989) present some electrostatic treatment, but, as far as I can see, don’t reflect over the amount of energetics involved in evaluating the osmotic pressure.
Lastly, the way these and many other authors single out the “diffusive” nature of the exchangeable cations when defining “osmotic” swelling implies that they do not consider ions to be diffusive in “crystalline” swelling states. Norrish (1954) states this quite explicitly (writing about the “crystalline” swelling region)
Nor can the interaction of diffuse double layers produce a repulsive force since in this region diffuse double layers are not formed. The repulsive forces of ion hydration and surface adsorption are probably the initial repulsive forces for many other colloids. They can cause surface separations of \(\sim 10\) Å, where the ions could begin to form diffuse double layers.
Even though I cannot find any explicit statements in Norrish (1954), writing like this makes me fear that authors of this era were under the impression that the initial interlayer hydration states consist of actual crystalline (non-liquid) water; I note that e.g. Grim (1953) has a several pages long section entitled “Evidence for the Crystalline State of the Initially Adsorbed Water”. Could it be that the original use of the term “crystalline” swelling was influenced by this belief?
Anyway, nowadays we have vast amount of evidence that interlayer water — at least down to the bihydrate — is liquid-like, and that ions in such states certainly diffuse. It follows that the osmotic pressure in such states has a contribution from mixing entropy.5 It should also be pointed out that the prevailing qualitative explanation for limited swelling in Ca-montmorillonite — which often is described as only displaying “crystalline” swelling — is due to ion-ion correlations in a diffusive system (“overlapping” diffuse layers).
Despite the evidence for interlayer diffusivity, it is very common to find descriptions in the bentonite literature that diffuse layers “develop” or “form” as the interlayers distances (or some other presumed pore) becomes large enough. This is usually claimed without giving a mechanism of how such a “development” or “formation” occurs. I genuinely wonder what authors using such descriptions believe the ions are doing when they have not “formed” a diffuse layer…
My message here is not that a division between “crystalline” and “osmotic” swelling should be discarded — for certain issues it makes a lot of sense to make a distinction, especially as the transition between these regions is not fully understood. But I think authors can do a better job in defining what exactly they mean by terms such as “osmotic”, “crystalline”, “diffusive”, etc. I furthermore wish that another name could be established for the “osmotic” swelling region (Norrish (1954) actually used “Region 1” and “Region 2”), although that seems rather unlikely. Until then we have to live with that bentonite swelling is described as “osmotic” only in a certain density range, while — if reasonable definitions are adopted — bentonite swelling pressure actually is an osmotic pressure under all conditions.
[1] In the following I usually mean bentonite when writing “bentonite”, even though the main points of the blog post also apply to claystone with swelling properties.
[2] The quotation is taken from the 8th
edition.
[3] Note how this description does not refer to any microscopic concepts, nor to differences in concentrations. There seems to be a whole academic field devoted to sorting out misconceptions about osmosis. For further reading, I can recommend e.g. (Kramer and Mayer, 2012) and (Bowler, 2017).
[4] There may be additional significant activity corrections. I guess a solution of the Gouy-Chapman model could be compared to using the Debye-Hückel equation for a conventional aqueous salt solution.
[5] I am not arguing for that swelling is driven by
entropy in these states —
the entropy
contribution is actually negative. But the entropy reasonably has
both a positive (mixing) and a negative (hydration) part.
It is easy to find
models assuming montmorillonite
interlayers devoid
of “anions” . Here I will present empirical evidence that such an
assumption is incorrect. Before doing so, just a quick remark on the
term “anions” in this context. If anions reside in interlayers,
they certainly do so accompanied by excess cations, in order to
maintain overall charge neutrality. Thus, when speaking of “anions”
in the interlayer we really mean “salt” (= anion(s) + cation(s)). In
the following I will use the term “salt” because it better reflects
the overall charge neutral character of the process (we are not
interested in pushing a handful of negative charge into an
interlayer).
The nature of bentonite swelling
The evidence for salt having access to interlayers follows directly
from the observed swelling pressure response to changes in external
salinity. It is therefore important to first understand the
thermodynamic basis for swelling pressure, which I wrote about in
an earlier post (the same nomenclature is adopted here). In essence,
swelling is a consequence of balancing the water chemical
potential1 in the clay with that in the external
solution2, and swelling pressure
quantifies the difference in chemical potential between the external
solution and the non-pressurized bentonite sample, as illustrated here
Since the chemical potential in the external solution depends on the
salt content, we generally expect a response in swelling pressure when
altering external salinity.
Labeling the salt concentration \(c^{ext}\), we write the
chemical potential of the external solution in terms of an osmotic
pressure3
where \(v_w\) is the partial molar volume of water. \(P_{osm}^{ext}\) is not the pressure in the external solution, but the pressure that would be required to keep the solution in equilibrium with pure water. The actual pressure in this compartment is the same as for the reference state: \(P_0\). It may seem confusing to use a “pressure” to specify the chemical potential, but we will see that it has its benefits. Experimentally we have full control of \(P_{osm}^{ext}\) by choosing an appropriate \(c^{ext}\).
Response in an indifferent clay
With salt in the external solution, the big question is what happens
to the chemical potential of the clay. We will start by assuming
(incorrectly) that external salt cannot access the interlayers. This
means that the chemical potential of the (non-pressurized)
bentonite does not change when the external salinity
changes. We refer to this hypothetical bentonite as
indifferent. In analogy with the external solution, we write
the chemical potential of the indifferent non-pressurized bentonite
as4
\begin{equation}
\mu_w^{int}(P_0) = \mu_0 -P_s^0\cdot v_w \;\; \;\; \;\;
\text{(indifferent clay)}
\end{equation}
were \(P_s^0\) is the swelling pressure in case of pure water as
external solution. By assumption, \(\mu_w^{int}(P_0)\) does not depend on the external salinity (it is independent of \(P_{osm}^{ext}\)). The chemical potential in the indifferent clay at an elevated pressure \(P\) is
The swelling pressure (defined as the difference in pressure between
bentonite and external solution, when the two are in equilibrium:
\(P_s \equiv P_{eq} – P_0\)) in an indifferent clay is given by equating eqs. 1and 2, giving the neat formula
Although an indifferent clay is not affected by salt, it
certainly has a swelling pressure response, demonstrating that
swelling pressure depends as much on the external solution as it
does on the clay.
Since swelling pressure in this case decreases linearly with the
osmotic pressure of the external solution, it is predicted to vanish
when the osmotic pressure equals \(P_s^0\).
External osmotic pressures larger than \(P_s^0\) implies “drying” of
the clay (water transport from the clay into the external compartment)
If the above derivation feels a bit messy, with all the different
types of pressure quantities to keep track of, here is a hopefully
helpful animation
Real swelling pressure response
Equipped with the swelling pressure response of an indifferent clay, let’s compare with the real response: The swelling pressure response in real bentonite deviates strongly from the indifferent clay response. This is seen e.g. here for Na-montmorillonite equilibrated in sequence with NaCl solutions of increasing concentration5 (data from Karnland et al., 2005 )
Swelling pressure indeed drops with increased concentration, but the
drop is not linear in \(P^{ext}_{osm}\), and is weaker as compared with
the indifferent clay response (shown by dashed lines). All samples in
the diagram above exert swelling pressure when
\(P^{ext}_{osm} \gg P_s^0\), i.e. far beyond the point where the
swelling pressure in an indifferent clay is lost.
The deviation of the observed response from that of an indifferent
clay directly demonstrates that the clay is affected by salt,
i.e. that the chemical potential of the non-pressurized clay
depends on the external salt concentration. The only reasonable way
for salt to influence the chemical potential in the bentonite is
of course to reside in the interlayer pores. Consequently, the
observed swelling pressure response proves that salt from the
external solution enters the interlayer pores.
Here is an illustration of how the chemical potentials relate to the
swelling pressure in real bentonite
Although the observed swelling pressure response in itself is sufficient to dismiss the idea that salt does not have access to interlayers, the study by Karnland et al., (2005) provides a much broader verification of the thermodynamic nature of swelling pressure. In particular, the chemical potential was measured (by means of vapor pressure) separately in the same samples as used for swelling pressure tests, after they had been isolated and unloaded. The terms in the relation \(P_s = \left(\mu_w^{ext} – \mu_w^{int}(P_0) \right)/v_w\) were thus checked independently, as indicated here
A striking confirmation of salt residing in interlayers is given by the observation that the chemical potential in the non-pressurized samples is lower than that in the corresponding external solution, as well as that in non-pressurized samples of similar density, but equilibrated with pure water.
Another interesting observation is that the sample with the highest density behaves qualitatively similar to the others: although the external osmotic pressure never exceeded \(P_s^0\) (\(\approx\)56 MPa), the response strongly deviates from that of an indifferent clay
Because the pore space of samples this dense (\(2.02\;\mathrm{g/cm^3}\)) mainly consists of mono- and bihydrated interlayers, this similarity in response shows that salt has access also to such pores.
Implications
The issue of whether “anions” have access to montmorillonite interlayers has — for some reason — been a “hot” topic within the bentonite research community for a long time, and a majority of contemporary models rest on some version of the assumption that “anions” does not have access to the full pore volume. But, as far as I can see, this whole idea is based on misconceptions. I guess that saying so may sound quite grandiose, but note that swelling pressure is not at all considered in most chemical models of bentonite. And if it is, it is usually treated incorrectly. As an example, here is what Bradbury and Baeyens (2003) writes in a very influential publication
One of the main premises in the approach proposed here is that 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. Thus, to a first approximation, the composition of the external saturating aqueous phase should be a second-order effect which has little influence on the initial compacted bentonite porewater composition.
If the composition of the re-saturating water were to play an important role in determining the porewater composition, then it should also have a significant influence on swelling (Bolt, 1979). Dixon (2000) recently reviewed the role of salinity on the development of swelling pressure in bentonite buffer and backfill materials. He concluded that provided the initial dry densities were greater than 900 \(\mathrm{kg\;m^{-3}}\), the swelling pressures developed are unaffected for groundwater salinities \(< 75 \;\mathrm{g\;l^{-1}}\). Even brines appear to have little or no influence for initial dry densities \(>1500 \;\mathrm{kg\;m^{-3}}\).
But, as we just have learned, a system with a weak swelling pressure response necessarily has a significant contribution to its water chemical potential due to externally provided salt. In contrast, the approximation discussed in the first paragraph of the quotation — which is basically that of an indifferent clay — maximizes the swelling pressure response. Thus, the discussed “main premise” does not hold, and the provided empirical “support” is actually an argument for the opposite (i.e. that salt has access to the clay).
Footnotes
[1] In the following I will write only “chemical potential” — it is always the chemical potential of water that is referred to.
[2] This is just a complicated way of saying that
swelling is (an effect of) osmosis.
[3] Some may say that \(P_{osm}^{ext}\) is simply the
“suction” of the solution, but I am not a fan of using that
concept in this context. I will comment on “suction” in a later
blog post.
[4] The density dependence of the chemical potential in
the bentonite is not explicitly stated here, in order to keep the
formulas readable, but we assume throughout that the bentonite has
some specific water-to-solid mass ratio \(w\).
[5] The NaCl concentrations are 0.0 M, 0.1 M, 0.3
M, 1.0 M, and 3.0 M.
I am puzzled by how bentonite swelling pressure is presented in present day academic works.
In soil science, the thermodynamic description of the
phenomenon has been around since at least the
1940s. Still, pure thermodynamic approaches to swelling
pressure are not fashionable in modern day research. I
think this is a pity, because for many issues this is the
preferred approach.
Naturally, the notion of
the Electric
Double Layer (EDL) is central in many descriptions of
bentonite swelling pressure,
and EDL models seem to fit the bill very well
for e.g. Li- and Na-montmorillonite at intermediate
densities (at high density, the resulting pressure becomes
increasingly sensitive to variations in model parameters,
such as ionic radii). Models based on the EDL concept also
give a
satisfying qualitative explanation for the limited
swelling of Ca- and Mg-montmorillonite, in terms of
ion-ion correlation. But common EDL approaches — as far
as I’m aware — fail to reproduce the observation that
swelling pressure is significantly reduced in
montmorillonites with heavier monovalent cations
(e.g. K-montmorillonite).
Here, I would like to revisit the pure thermodynamic
description of swelling pressure, which I think may help in
resolving several misconceptions about swelling pressure.
Of course, thermodynamics cannot answer what the microscopic
mechanism of swelling is, but puts focus on other — often
relevant — aspects of the phenomenon. We thus take as input
that, at the same pressure and temperature, the water
chemical potential2 is
lowered in compacted bentonite as compared with pure water,
and we ignore the (microscopic) reason for why this is the
case. We write the chemical potential in
non-pressurized3
bentonite as
\begin{equation}
\mu_w(w,P_0) = \mu_0 + \Delta \mu(w,P_0)
\end{equation}
where \(\mu_0\) is a reference potential of pure bulk water at pressure \(P_0\) (isothermal conditions are assumed, and temperature will be left out of this discussion), and \(w\) is the water-to-solid mass ratio. Note that \(\Delta \mu(w,P_0)\) is a negative quantity.
The chemical potential in a pressurized system is given by
integrating \(d\mu_w = v_wdP\), where \(v_w\) is the partial
molar volume of water,
giving4
\begin{equation}
\mu_w(w,P) = \mu_0 + \Delta \mu(w,P_0) + v_w\cdot (P-P_0)
\end{equation}
In order to define swelling pressure, we require that the bentonite is confined to a certain volume while still having access to externally supplied water, i.e. that it is separated from an external water source by a semi-permeable component. This may sound abstract, but is in fact how any type of swelling pressure test is set up: water is supplied to the sample via e.g. sintered metal filters.
With this boundary condition, a relation between swelling
pressure and the chemical potential is easily obtained by
invoking the condition that, at equilibrium, the chemical
potential is the same everywhere. Assuming an external
reservoir of pure water at pressure \(P_0\), its chemical
potential is \(\mu_0\), and the equilibrium condition reads
\begin{equation}
\mu_w(w,P_{eq}) = \mu_0 + \Delta \mu(w,P_0) + v_w\cdot
(P_{eq}-P_0) = \mu_0
\end{equation}
where \(P_{eq}\) is the pressure in the bentonite at
thermodynamic equilibrium.
Defining the swelling pressure as \(P_s = P_{eq}-P_0\) we get
the desired relation5
\begin{equation}
P_s = -\frac{\Delta \mu(w,P_0)}{v_w}
\tag{4}
\end{equation}
Alternatively this relation can be expressed in terms of activity
(related to the chemical potential as \(\mu = \mu_0 +RT\ln a\))
\begin{equation}
P_s = -\frac{RT}{v_w}\ln a (w,P_0)
\tag{5}
\end{equation}
or, if the activity is expressed in terms of the vapor
pressure, \(P_v\), in equilibrium with the sample,
\begin{equation}
P_s = -\frac{RT}{v_w}\ln \frac{P_v}{P_{v0}}
\tag{6}
\end{equation}
where \(P_{v0}\) is the corresponding vapor pressure of pure bulk water.
The above relation has been presented in the literature for a long time. But, as far as I am aware, direct interpretation of experimental data using eq. 4 is more scarce. Spostio (72) compares swelling pressures in Na-montmorillonite (reported by Warkentin et al 57) with water activities measured in the materials (reported by Klute and Richards 62) and concludes a “quite satisfactory” agreement of eq. 4 (the highest pressures were on the order of 1 MPa). He moreover comments
Future measurements of \(P_S\) and \(\Delta \mu_w\) for pure
clays and soils as a function of water content would do much
to help assess the merit of equation (11)
[eq. 4 here].
Such “future” measurements were indeed presented
by Bucher
et al (1989), for “natural” bentonites in a density
range including very high pressures (\(\sim 40\) Mpa). For
“MX-80” the data looks like this
Here the value of \(v_w\) was set equal to the molar
volume of bulk water when
applying eq. 6. It is interesting to note
that this value, which is necessarily correct in the limit
of low density, appears to be valid for densities as large
as \(2\;\mathrm{g/cm^3}\).
The clearest demonstration of the validity of eq. 4 is in my opinion the study by Karnland et al. (2005), where swelling pressure and vapor pressure were measured on the same samples. The result for Na-montmorillonite is shown below (again, the value of bulk water molar volume was used for \(v_w\)).
The above plots make it clear that the description
underlying eq. 4
(or eq. 5, or eq. 6)
is valid for bentonite, at any density. An important
consequence of this insight — and something I think is
often not emphasized enough — is that swelling pressure
depends as much on the external solution as it does on the
bentonite.
Measuring the response in swelling pressure to changes in the external solution is therefore a powerful method for exploring the physico-chemical behavior of bentonite. I will return to this point in later blog posts, in particular when discussing the “controversial” issue whether “anions” have access to montmorillonite interlayers.
The animation below summarizes the thermodynamic view of the development of swelling pressure: the external reservoir fixes the value of the water chemical potential, and in order for the bentonite sample to attain this level, its pressure increases.
[2] In the following I will simply write
“chemical potential”. Here the water chemical potential is the
only one involved.
[3] Here “non-pressurized” means being at the reference pressure \(P_0\). In practice \(P_0\) is usually atmospheric absolute pressure.
[4] Here it is assumed that \(v_w\) is independent of pressure. Also, using \(w\) as thermodynamic variable implies that the water chemical potential is measured in units of energy per mass, which requires this volume factor to be the partial specific volume of water. Here we assume that the chemical potential is measured in units energy per mol, but use \(w\) for quantifying the amount of water in the clay, since it is the more commonly used variable in the bentonite world. The amount of moles of water is of course in strict one-to-one correspondence with the water mass.
[5] What is said here is that swelling
pressure generally is identified as an osmotic pressure. I
will expand on this in a future blog post.
