Category Archives: Diffusion

$D_a$ is not “apparent” (sorption, part I)

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

The traditional diffusion-sorption model

A persistent idea, that has been around since the very start of research on compacted bentonite¹, is that “sorbed” ions are immobilized, and early diffusion models were based on two assumptions

The entire pore volume contains bulk water, where ions diffuse according to \begin{equation} j = -D_e \nabla c \tag{1} \end{equation} where $c$ is the ion concentration and $D_e = \phi D_p$, with $\phi$ being the porosity, and $D_p$ a so-called pore diffusivity (containing information on the ion mobility at the macroscopic scale of the porous system).
Ions in the porewater “sorb” on solid surfaces and become immobilized. The amount of “sorbed” ions per unit solid mass, $s$, is related to $c$ via a distribution coefficient $K_d$ \begin{equation} s = K_d\cdot c \tag{2} \end{equation}

With these assumptions, the total amount of ions in the bentonite (per volume porous medium) is

\begin{equation} n = \phi\cdot c + \rho\cdot s = \left (\phi+\rho K_d \right) c \tag{3} \end{equation}

where $\rho$ is the (dry) density of the clay. Applying the continuity equation, $\partial n/\partial t = -\nabla\cdot j$, gives

\begin{equation} \frac{\partial c}{\partial t} = \frac{D_e}{\phi + \rho K_d}\nabla^2c \tag{4} \end{equation}

This equation is usually written in terms of the so-called apparent diffusion coefficient $D_a$

\begin{equation} \frac{\partial c}{\partial t} = D_a \nabla^2c \tag{5} \end{equation}

with

\begin{equation} D_a = \frac{D_e}{\phi + \rho K_d} \tag{6} \end{equation}

The model defined by eqs. 5 and 6 — which we will refer to as the traditional model — has a distinct physical interpretation: ions diffuse relatively fast in a bulk water phase, while being retarded due to “sorption” onto the solid surfaces. Although eq. 5 has the form of Fick’s second law, it is clear that $D_a$ is not a real diffusion coefficient, but is influenced both by diffusion ($D_e$) and “sorption” ($K_d$).

The immobilization assumption is not valid

The traditional model has a major problem: it is not valid for compacted bentonite. There is — and was — strong evidence that the immobilization assumption does not hold for all types of “sorbing” ions. E.g Jenny and Overstreet (1939) reported transport of $\mathrm{Fe^{2+}}$, $\mathrm{H^+}$, $\mathrm{K^+}$, $\mathrm{Na^+}$, and $\mathrm{Ca^{2+}}$ as samples of H- K- Na- and Ca-bentonite was brought in contact with samples of Fe/H-bentonite (without gel contact there was no transport), and van
Schaik et al. (1966) demonstrated steady-state fluxes of $\mathrm{Na^+}$ through Na-montmorillonite, corresponding to pore diffusivites significantly larger than the $\mathrm{Na^+}$ diffusivity in pure water. Moreover, ion mobility is an essential component of the most well-established theoretical concept used for describing montmorillonite — the electric double layer. The ionic part of the electric double layer is even referred to as the diffuse layer, for crying out loud!²

So it should come as no surprise that the pioneers in radwaste barrier research quickly came to the conclusion that the description underlying eq. 6 does not hold. Several research groups reported values of $K_d$ and $D_a$ that gives unreasonably large values of $D_e$ when evaluated using eq. 6 (even larger than the corresponding diffusivities in pure water, in several cases).

What is more surprising is that these conclusions did not lead to a complete reevaluation of the underlying assumptions, which is the preferred procedure for a model producing nonsense. Instead, the invalidity of the traditional model was only considered a problem for the specific ions for which it yield “unrealistic” parameter values (in particular, strontium and cesium), and an adopted “remedy” was the so-called surface diffusion model. The surface diffusion model keeps the description expressed by eqs. 1, 2, 5, and 6, while replacing the relation $D_e = \phi\cdot D_p$ with the following monster

\begin{equation} D_e = \phi\cdot D_p + \rho\cdot K_d\cdot D_s \tag{7} \end{equation}

where $D_s$ is a so-called surface diffusion coefficient.

This extension of the traditional model brings about several problems, the most glaring one being that eq. 7 is wrong. Therefore, to be able to continue the discussion, we first have to derive a correct “surface diffusion”³ model.

A correct “surface diffusion” model

Eq. 7 is not valid because it relies on the incorrect assumption that diffusive fluxes from different domains are additive. To derive the correct equations, we go back to the only conceptual modification made in the surface diffusion model: that the “sorbed” ions are no longer assumed immobile. Thus, the basic assumptions of the surface diffusion model are:

The entire pore volume contains bulk water, where ions diffuse.
Ions in the porewater “sorb” on the solid surfaces.
“Sorbed” ions also diffuse, along the surfaces.

There are consequently two domains (bulk water and surfaces) where diffusion is assumed to take place, and a correct equation for the macroscopic flux is in this case (for details, see here)

\begin{equation} j = -\phi D_\mathrm{macr.} \frac{\bar{c}}{c}\nabla c \tag{8} \end{equation}

where $D_\mathrm{macr.}$ is the diffusion coefficient on the macroscopic scale, which conveys information on the mobility in both of the involved domains, as well as their geometrical configuration. $\bar{c}$ is the average concentration of all mobile entities; here, this include all ions — those in the porewater as well as those on the surfaces. The bulk water concentration is still labelled $c$. Using eq. 3, we can write

\begin{equation} \phi\bar{c} = n = \left (\phi + \rho K_d \right) c \tag{9} \end{equation}

and eq. 8 reduces to

\begin{equation} j = – D_\mathrm{macr.}\left (\phi + \rho K_d \right) \nabla c \tag{10} \end{equation}

Comparing with eq. 1 directly shows

\begin{equation} D_e = D_\mathrm{macr.}\left (\phi + \rho K_d \right) \tag{11} \end{equation}

Further, when using eq. 10 in the continuity equation, the factor $\left (\phi + \rho K_d \right)$ cancels, giving

\begin{equation} \frac{\partial c}{\partial t} = D_\mathrm{macr.} \nabla^2 c \tag{12} \end{equation}

A complete conceptual change

Eqs. 11 and 12 are the main result of a correctly derived “surface diffusion” model. Comparing with eqs. 5 and 6, we see that, although the equations are deceptively similar, a “surface diffusion” model brings about a complete conceptual change: with the immobilization assumption released, it is actually the real diffusion coefficient that appear in Fick’s second law (eq. 12). In fact, the surface diffusion model does not contain any apparent diffusivity!

Similarly, $D_e$ is not a real diffusion coefficient in the surface diffusion model, as clearly seen by eq. 11. Rather, $D_e$ is the product of the diffusion coefficient and a concentration factor, stemming from the fact that the general flux equation reads $j = -\frac{D}{RT} \bar{c}\cdot \nabla \mu$, where $\mu$ is the chemical potential of the diffusing ion.

In the same vein, $K_d$ does not quantify “sorption” — at least if “sorption” is supposed to refer to a process that retards the diffusive flux. On the contrary, $K_d$ acts as a kind of amplifier of the flux (eq. 10). This implication was already pointed out by Jahnke and Radke (1985):

Thus, $K_d$ values are of extreme importance during the steady release period. Counter to what may have been anticipated, larger $K_d$ values lead to higher release rates.

The conceptual changes of a “surface diffusion” model make a lot of sense when interpreting diffusion data:

$D_e$ is expected to vary with $K_d$ and can basically grow without limit, without becoming unphysical.
“$D_a$” for a whole bunch of cations is universally found to be insensitive to changes in $K_d$, even though $K_d$ may vary orders of magnitudes (by altering the background concentration). This is the expected behavior of a real diffusion coefficient, and it makes all sense to instead interpret this parameter as $D_\mathrm{macr.}$, using eq. 12.
When assuming all ions in the system to be mobile, we expect the governing equation (eq. 12) to be a real Fick’s second law, i.e. that it contains nothing but the actual diffusion coefficient.

Despite this sense-making, I have never seen these points being spelled out in the bentonite literature. Rather, it appears as the implication of releasing the immobilization assumption has not been fully comprehended. To this day, the traditional model is used as the basis for interpretation in basically all publications, to the extent that “$D_a$”, “$D_e$”, and “$K_d$” usually are considered to be experimental quantities. Similarly, most modern model development is focused on extending the traditional model, using flawed concepts regarding diffusivity in multi-porous systems.

I also find it symptomatic that “surface diffusion” not uncommonly is treated as being an optional mechanism, or even being completely discarded, while the dominant contribution of “sorbed” ions to the diffusive flux seems to have to be rediscovered every 25^th year, or so.

Note that the conclusions made here are not based on any model that I favor, but are the consequence of a correct conceptual treatment of what is stated in the contemporary bentonite literature. Although a “surface diffusion” model (correctly treated!) make much more sense as compared with the traditional model in interpreting cation tracer diffusion, it still involves an assumption that I don’t understand how anybody can endorse: it assumes the entire pore volume to consist of bulk water. I will discuss this point in a separate blog post. Update (210225): Letting go of the bulk water is discussed here.

Footnotes

[1] In the following I will write “bentonite”, but I mean any type of clay system with significant ion exchange properties.

[2] I guess a reason for not acknowledging interlayer diffusivity could have been the incorrect notions that interlayer water is crystalline, and that diffuse layers only “form” at higher water content.

[3] Since the established surface diffusion model is that expressed by eq. 7, I use quotation marks when speaking about models that don’t comply with this equation.

Anion-accessible porosity – a brief history

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Genesis

In the beginning there was the Poisson-Boltzmann equation. Solving it for the case of a salt solution in contact with a negatively charged plane surface (a.k.a. the Gouy-Chapman model) gives the concentration of cations and anions in the solution as a function of the distance to the surface, like this¹

Illustration of Gouy-Chapman concentration profiles

Note:

The suppression of the anion concentration near the surface is often referred to as negative adsorption or anion exclusion. The total amount of excluded anions per unit surface area (indicated in green), usually labeled $\Gamma^-$, is obtained by integrating the Poisson-Boltzmann equation.
There are, nevertheless, anions everywhere! This model will give zero anion concentration only for an infinitely negative electrostatic potential (or if $c_0 = 0$, of course).

A clever way to utilize negative adsorption is for estimating the amount of smectite surface area in a soil sample, first suggested by Schofield (1947). This is done by comparing measured values of negative adsorption with the appropriate expression evaluated from the Gouy-Chapman model. When doing the necessary math² for such an analysis you naturally end up with expressions like

\begin{equation} \frac{\Gamma^-}{c_0} \sim \text{const.}\cdot \kappa^{-1} \tag{1} \end{equation}

where $c_0$ is the external anion concentration (i.e. far from the surface), and $\kappa^{-1}$ is the Debye length. This equation, having the dimension of length, can be interpreted as the width, $d_{ex}$, of a region devoid of anions, which gives the same amount of negative adsorption as the full exclusion region, as illustrated here (yellow)

However, note:

This is just an equivalent, fictitious region.
Anions are still everywhere!

Due to its convenience in the analysis, the notion of an equivalent region devoid of anions — often referred to in terms of “volume of exclusion” — became rather popular. At the same time, authors stopped emphasizing that this is a fictitious region. A clear example of such a transition is Edwards and Quirk (1962) who states that $\Gamma^-/c_0$ “can be regarded as the surface depth from which chloride ions are excluded”, while in Edwards et al. (1965) the same quantity (multiplied by area) is referred to as “the volume from which chloride is excluded”. The latter statement is, strictly speaking, wrong: the actual volume from which anions are excluded is the entire region where the concentration deviates from $c_0$, and the exclusion is only partial — there are anions everywhere!

Compacted bentonite

But the idea of an actual region devoid of anions seems to have stuck, and I believe that this influenced the interpretation of diffusion in compacted bentonite³ in terms of “effective porosity” or “anion accessible-porosity”. Concepts which, in turn, have motivated the idea that bentonite contains bulk water (“free water”, “pore water”).

The first example of this usage in studies of compacted bentonite, that I know of, is in Muurinen et al. (1988) reporting chloride through-diffusion in bentonite with various densities and background concentrations.

The tracer concentration of the porewater clearly depends on the compaction of bentonite and on the salt concentration of the circulating water. The effective porosity can even be less than one percent when the salt concentration is low and compaction high. Also, the diffusivities strongly depend on the density of bentonite and on the salt concentration.

The low tracer concentration in bentonite in the diffusion tests […] are indicative of ion-exclusion [5]. Ion-exclusion probably decreases the effective size of the pores, which changes the geometric factor, of bentonite and thus the apparent diffusivity. In addition to the geometric factor, the effective diffusivity takes into account the effective pore volume; thus, the dependence is even stronger.

“Effective porosity” has not been defined earlier in the article, so it is difficult to know precisely what the authors mean by the term. But it is relatively clear⁴ from the second paragraph that they explain the measured fluxes as being a result of a physical variation of the pore volume accessible to anions, rather than as a variation of the tracer concentration in a homogeneous system. This is also supported by their writing in the conclusions section: “The decreased pore size and porosity caused by ion-exclusion could at least qualitatively explain the dependence.”

However, the reference they provide (“[5]”) is Soudek et al. (1984), who calculate anion exclusion by means of — the Poisson-Boltzmann equation! (Which predicts anions everywhere.) In fact, Soudek et al. (1984) calculate what they term “Donnan exclusion” in a homogeneous model of “parallel, equally-spaced platelets”. Thus, the reference supplied by Muurinen et al. (1988) is in direct contradiction with their interpretation that the pore size and porosity is decreasing with the salt concentration.

Soudek et al. (1984) even provide an example of how the average chloride concentration between the platelets depends on the separation distance, when in equilibrium with an external solution of 10 mM, and write

Note the extremely strong rejection of the co-ion. At 50 w% clay ($\sim 25$Å plate separation) almost 90% of the anions are rejected.

which is completely in line with the observation of Muurinen et al. (1988) that “The effective porosity can even be less than one percent when the salt concentration is low and compaction high”, if only “effective porosity” is replaced by “concentration between the plates”.

It makes me a bit tired to discover that the record could have been set straight over 30 years ago regarding which pores anions can access. Instead the bentonite research community, for the most part, doubled down on the idea that anions only have accesses to parts of the pore volume, or that compacted bentonite contains a significant amount of bulk water.

An explicit description of interpreting “chloride through diffusion porosity” as a specific, limited part of the pore volume is given by Bradbury and Baeyens (2003)

In the interlayer spaces and regions where the individual montmorillonite stacks are in close proximity, double layer overlap will occur and anion exclusion effects will take place. Exclusion will probably be so large that it is highly unlikely that anions can move through these regions (Bolt and de Haan, 1982). However, Cl anions do move relatively readily through compacted bentonite since diffusion rates have been measured in ‘‘through-diffusion’’ tests […]

If the Cl anions cannot move through the interlayer and overlapping double layer regions because of anion exclusion effects, then it is reasonable to propose that the ‘‘free water’’ must provide the diffusion pathways (Fig. 1). Therefore, the hypothesis is that the pore volume associated with the transport of chloride (and other anions) is the ‘‘free water’’ volume, and that this is the porewater in a compacted bentonite.

Here they refer to Bolt and De Haan, (1982) ⁵, when arguing for that anions do not have access to interlayers. But the analysis in this reference is based on nothing but — the Poisson-Boltzmann equation! (which predicts anions everywhere)

Another thing to note is the notion of “overlapping” diffuse layers. Studies of negative adsorption to quantify surface area typically look at soil suspensions, with a solid part of a few percent. In such systems it is justified to perform the analysis on a single diffuse layer because the distance between separate montmorillonite particles is large enough. But at higher density there is not enough space between separate clay particles for the ion concentrations to ever reach the “external” value ($c_0$) — the diffuse layers “overlap”.⁶

It has been shown that effects of “overlapping” diffuse layers on the resulting negative adsorption is significant already at a a solid content of 6%. When carrying over the anion exclusion analysis to compacted bentonite — with solid content typically above 70%! — it therefore becomes near impossible to believe that the system should contain regions unaffected by the montmorillonite (“free water”). Yet, the argumentation above, apart from being flawed in the way it refers to the Poisson-Boltzmann equation, relies critically on the existence of such regions.

The mindful reader may remark that compacted bentonite, if it mainly contains “overlapping” diffuse layers, perhaps is devoid of anions after all. But the Poisson-Boltzmann equation predicts anions everywhere also for “overlapping” diffuse layers. Actually, the model by Soudek et al. (1984), discussed above, considers this case.

Despite the improbability that montmorillonite particles in compacted bentonite can be spaced so far apart as to allow for bulk water within the system, the idea of anions only having access to “free” water was nevertheless further pursued by Van Loon et al. (2007). They provide a picture similar to this

The idea here (and elsewhere) is that bentonite consists of “stacks” of individual montmorillonite particles (TOT-layers) interlaced with interlayer water.⁷ The space between “stacks” is assumed large enough for diffuse layers to fully develop, and to merge into a bulk solution (“free water”), whose volume depends on the ionic strength, reminiscent of the excluded volume in eq 1.⁸ Anions are postulated to only have access to this “free” water.

But as references for anion exclusion is once again simply given studies based on the Poisson-Boltzmann equation (in particular, Bolt and De Haan, (1982)). But these — as I hope has been made clear by now — predict anions everywhere, and consequently do not support the suggested model. In this case, the mismatch between model and supporting references stands out, as the term “effective porosity” is used interchangeably with the term “Cl-accessible porosity”; if Gouy-Chapman theory in a convoluted way can be used to define an “effective” porosity (having no other meaning than a fictitious, equivalent volume), there is no possibility whatsoever to use it to support the idea of anions having access to only parts of the pore space. Ironically, “anion-accessible porosity” seems to be the most popular term nowadays for describing effects of anion exclusion in compacted bentonite.

The strongest confirmation that the modern-day concept of anion-accessible porosity is simply a misuse of the exclusion-volume concept is given in Tournassat and Appelo (2011). They provide a quite extensive background for the type of anion exclusion they consider, and it is based on the excluded-volume concept discussed above. They even explicitly calculate the excluded-volume (named “total chloride exclusion distance”) only to directly discard it as not suitable

However, this binary representation (absence or presence of chloride, Fig. 3) is not very representative of the system since the EDL is not completely devoid of anions.

Yet, after making this statement that anions are everywhere (in the diffuse layer) they anyway define anion accessible porosity as an effective, fictitious volume!⁹

Interlayers

Apart from treating the diffuse layer incorrectly, Bradbury and Baeyens (2003), Van Loon et al. (2007) and Tournassat and Appelo (2011) all make the additional unjustified assumption that interlayers — which in these studies are defined as distinctly different from diffuse layers¹⁰ — are completely devoid of anions. Bradbury and Bans (2003) cites conventional Poisson-Boltzmann based studies to incorrectly support this claim (see above). Also Van Loon et al. (2007) use Bolt and De Haan, (1982) as a reference¹¹

Due to the very narrow space, the double layers in the interlayers overlap and the electric potential in the truncated layer becomes large leading to a complete exclusion of anions from the interlayer (Bolt and de Haan, 1982; Pusch et al., 1990; Olin, 1994; Wersin et al., 2004). The interlayer water thus contains exclusively cations that compensate the permanent charges located in the octahedral layer of the clay.

Of the other sources cited, Pusch et al. (1990) mention “Donnan exclusion” as the reason preventing anions from having access to interlayers. But this is incorrect – Donnan equilibrium always gives a non-zero anion concentration in the interlayer (as long as the external concentration is non-zero). Wersin et al. (2004) only claim that anions are “excluded” from interlayers, without further explanation or references. (I haven’t managed to read Olin (1994) .)

Tournassat and Appelo (2011) cites Bourg et al. (2003) to support the claim that anions have no access to interlayers

When the dry density is above $1.8 \;\mathrm{kg/dm^3}$, almost all the porosity resides in the interlayers of Na-montmorillonite. Since anions are excluded from the interlayers, the anion-accessible porosity becomes zero, and anion-diffusion is minimal (Bourg et al., 2003)

But in Bourg et al. (2003) is explicitly stated that anion exclusion from interlayers is only “partial”!

To sum up…

The idea that anions have access only to parts of the pore volume is widespread in today’s compacted bentonite research community. In this blog post I have shown that this idea emerges from misusing the concept of exclusion-volume, and that all references used to support ideas of “complete exclusion” rests on the Poisson-Boltzmann equation. The Poisson-Boltzmann equation, however, predicts anions everywhere! Thus, the concept of an anion-accessible porosity, and the related idea that compacted bentonite contains different “types” of water, have not been provided with any kind of theoretical support.

In contrast, the result that anions have access to the entire pore volume is further supported both by molecular dynamics simulations, as well as by the empirical evidence for salt in interlayers.

Footnotes

[1] This figure is just an illustration, not an actual result. Update (220831): Actual solutions to the Poisson-Boltzmann equation are presented here.

[2] Schofield writes with an enthusiasm seldom seen in modern scientific papers: “I considered that it would be possible to compute the negative adsorption of the repelled ions from the basic assumptions of Gouy’s theory of the diffuse electric double layer, and therefore invited Mr. M. H. Quenouille to tackle the mathematical difficulties involved. Complete solutions have now been obtained for electrolytes in which the ions have valency ratios 1:2, 1:1, and 2:1, and a full account of this work will be submitted for publication shortly.”

[3] “Bentonite” is used in the following as an abbreviation of “Bentonite and claystone”.

[4] I mean that the word “probably” as used here does not belong in a scientific text.

[5] Sciencedirect.com dates this reference to 1979. The book has a second revised edition, however, published in 1982.

[6] I use quotation marks when writing “overlap” because I think this wording gives the wrong impression in compacted clay: with an average distance between montmorillonite particles of around 1 nm, the concept of individual diffuse layers has lost its meaning.

[7] I plan to comment on “stacks” in a future blog post. Update (211027): Stacks make no sense.

[8] The volume is, however, not proportional to the Debye length, but depends exponentially on ionic strength.

[9] The “anion accessible porosity” is defined in this paper as $\epsilon_{an} = \epsilon_{free} + \epsilon_D\cdot c_D/c_{free}$, where $\epsilon_{free}$ is the porosity of a presumed bulk water phase in the bentonite, and $\epsilon_D$ quantifies the volume of an arbitrarily chosen “Donnan volume” which is (Donnan) equilibrated with the “free” solution. $c_D$ is the anion concentration in this “Donnan volume”, and $c_{free}$ is the anion concentration in the bulk water.

[10] In this context, “interlayers” are defined as being parts of “stacks”. I really need to write about “stacks”… Update (211027): Stacks make no sense

[11] Bolt and de Haan (and others) are fond of writing that anions in very narrow confinement are “almost completely excluded” or “virtually completely excluded”, indicating that they may neglect anions in these compartments, but also that they are aware of that the equations they use never give exactly zero anion concentration. When working with soil suspensions of only a few percent solids it may be a valid approximation to neglect anions in nm-wide pores. In compacted bentonite it is not.

What is a “diffusion path” anyway?

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In a previous post I discussed how parts of the bentonite¹ research community unjustifiably explain variation in (effective) diffusion rates as changes in “diffusion paths”. But what do authors really mean when using the term “diffusion path”?

In the geochemical/reactive transport literature, “diffusive pathways” are usually introduced when discussing the (presumably) related concept of tortuosity. For example, Steefel and Maher (2009) present a figure very similar to the one below, with the caption “Tortuous diffusion paths in porous material.”

Tortuous diffusion paths in porous material (Steefel and Maher (2009))

The text explains further that these are paths “the solute […] follow[s] in tortuous media”². Several questions immediately arise. For example:

What is it, exactly, that follows these paths?
Why do the paths have a direction?
Why are these particular paths singled out? What stops e.g. the “Long Path” from taking this obvious shortcut:

Alternative path in Steefel and Maher's illustration

Let’s start with a hopefully obvious statement: Individual molecules or ions do not follow “paths” in a diffusive process, but conduct random motion. So paths as those in the figure above are certainly not trajectories of single particles.³

An answer to what is illustrated may be found in Van Brakel and Heertjes (1974) — a main reference in the bentonite research community when discussing tortuosity etc.⁴ In this work, the system is analyzed in steady-state, and the following description is given for “diffusion paths”

Assume the pore space of the porous medium to be completely filled with what we will call diffusion paths. The main direction of the diffusion paths is the same as that of the concentration gradient. In the way the diffusion paths wind through the pore space they can be compared with the streamlines for laminar flow in porous media.

Here it may sound as if the authors reject Fickian diffusion (which is always parallel to the concentration gradient), but it is rather that they use the term “concentration gradient” for denoting the externally applied constant concentration difference, required to maintain a steady-state flux. To me, this is quite confusing, because the comparison of diffusion paths with streamlines implies that they are directed along a (negative) concentration gradient. Two “gradients” must thus be kept in mind simultaneously: the external concentration difference, and the actual gradient on the pore-scale.

But with this distinction made, it is clear what Van Brakel and Heertjes mean by “diffusion paths”, and that their aim is to reduce a more complex 3D problem to an effective 1D description. It also becomes relatively clear that this is the way the term is used in much of the bentonite literature. It explains e.g. why the “paths” in the above picture (and others) have a direction: they must be thought of as the steady-state flow on the pore scale, with an implied constant concentration difference applied across the sample, making the macroscopic flow effectively 1D.

This implied reduction to steady-state transport in 1D is often found in the literature, e.g. in Shackelford and Moore (2013)

This increased tortuosity reduces the macroscopic concentration gradient (i.e., increases the distance over which the concentration difference is applied) and, therefore, reduces the diffusive mass flux relative to that which would exist in the absence of the porous medium.

or in Altman et al. (2015)

$\tau$ is a geometrical factor ($\le 1$) representing the reduction in the effective concentration gradient (d(Me)/dx) due to the fact that diffusion paths through a porous medium will generally be greater, i.e. more tortuous, than the straight-line distance between the system boundaries, i.e. dx.

I am quite puzzled by this description for several reasons. Firstly, I find it unsatisfying that these definitions require the system to be in steady-state. Information on the influence of geometry is of course contained in the diffusion coefficient itself, independent of any external concentration differences. To associate “tortuosity” with such concentration differences, rather than with the mobility of the diffusing substance, seems inadequate to me. Moreover, the procedure of reducing a “macroscopic” concentration gradient due to path length seems to only work for an isolated path. At least, the procedure must become more involved for a system of connected paths, something I’ve not seen commented on by authors adopting this concept.

Secondly, note that a “diffusion path” — with the steady-state definition — simply indicates net mass transfer of diffusing substance. The absence of a “diffusion path” in a region does not mean that dissolved constituents don’t migrate there, but only that flux contributions in different directions add up to zero net mass transfer. I did a silly random walk simulation to illustrate this point (the concentration of walkers is kept at a constant finite value on the left side of the domain, while it is kept at zero on the right side)

Note that with the definitions here discussed, we must accept that the vertical section in this illustration is not a diffusion path. This situation is quite distinct from laminar advective transport, where — if I’m thinking correctly — the absence of a streamline implies the absence of motion.

Thirdly, if you consider a porous system to be a network of thin cylinders, I guess the steady-state flux will basically resemble the system itself (interconnected 1D-spaghetti configured in 3D). I suspect that this is the view many authors have in mind when speaking of “diffusion paths”. But, if so, why not simply speak of “paths”? Note also that the pore volume of smectitic systems mainly consists of 2D water films configured in 3D (it is lasagna, not spaghetti).

Lastly, what about non-steady-state transport? Concepts like the ones discussed here are also used when describing closed-cell diffusion tests , but are seldom (never?) defined in any other way. How could e.g. “tortuosity” reduce a macroscopically applied (1D) gradient in this case? And what is even meant by “diffusion paths”, if these are defined in steady-state? Since non-steady state is the general case, it would be more satisfying if quantities obtained under such circumstances were applied to the steady-state, rather than the other way around.

To get a feel for how pore geometry influences diffusion in non-steady-state, I conducted some more random walk simulations. In the animation below is compared random walks in an unrestricted 2D plane (blue) with random walks on a square net (red; strip width: 1 length unit, square size: 20 length units)

To quantify the diffusivity, we plot the average of the square of the displacements, $\langle r^2 \rangle$, as a function of time⁵ in the two systems

We see that the diffusivity — which is directly proportional to the slope of these curves — is very close to twice as large in the unrestricted case as compared with diffusion on the net. From such a result it may be tempting to conclude that this reduction by a factor of two is due to longer “diffusion paths” on the net (and relate it to $\sqrt{2}$, which conveniently is the ratio between the side and the diagonal of a square). But note that the diffusional process is isotropic also on the net, as demonstrated by identical slopes of the angle-resolved $\langle r^2 \rangle$-curves. Thus, interpreted in terms of “diffusion paths” on the net — however these should be defined — the conclusion is that the “paths” have the same length in any direction.

But the situation is easily analyzed from the simple model underlying the figures displayed above: in the 2D-plane, the random walk process has a maximized variance, because movements in the $x-$ and $y-$directions are uncorrelated. The net geometry, on the other hand, correlates these variables: if a walker has free passage in the $y$-direction, it is restricted in the $x$-direction, and vice versa. Thus, the diffusivity is not diminished due to longer “paths”, but because the geometrical restrictions reduce the variance of the underlying process. This effect will depend on the relative reduction of dimensions: with line-like pores in a 3D configuration, the reduction factor becomes 1/3 (I guess this is what what is alluded to for a “homogeneous isotropic pore network” in the often-cited work Dykhuzien and Casey (1989) ), but for the case relevant for bentonite — diffusion in 2D-planes configured in 3D — the factor is 2/3 (which I haven’t seen stated anywhere). I furthermore don’t understand why such a factor should be termed “tortuosity”, because there is nothing intrinsically “tortuous” about it (in a sense one could even argue that individual trajectories in the unrestricted 2D-plane are more “tortuous” than the ones on the net).

By making the net non-isotropic, e.g. by replacing the squares by rectangles like this

the correlation between the $x$- and $y$-variables alters (it is now twice as likely for a walker to have no restriction in the $y$-direction as in the $x$-direction), which is directly reflected in the diffusivities

<r2> vs t for square and rectangular net

The diffusivity in the $y$-direction is now about twice as large as the diffusivity in the $x$-direction. Also the diffusivity in the diagonal directions is significantly larger than in the $x$-direction. Following a naive definition of “tortuosity”, this result may seem surprising (is the “solute” in the $x$-direction following a longer path than than in the diagonal directions?). Still, with a correct averaging procedure I guess the diffusivity can be related to “paths” on the net (However, I still don’t understand how to differ “diffusion paths” from geometrical paths).

With these simulations I simply want to argue for that it seems considerably more subtle and complex to relate pore geometry to diffusivity, than how it typically is presented in the bentonite literature. To be frank, I consider most talk about “diffusion paths” in the bentonite literature, as well as most definitions of various geometric factors, to be just that: talk. There is an established “tradition” to mention certain concepts (geometric factors, tortuosity, constrictivity, paths…), but in the end the introduced factors are usually only functioning as fudge factors, leading to unjustified claims about the nature of bentonite. Similarly, discussions on actual values of such factors are in principle always only qualitative.

As an example, Choi and Oscarson (1996) interpret different values of diffusivity measured in Na- and Ca-bentonite directly as a difference in “tortuosity”:

We attribute this to the larger quasicrystal, or particle, size of Ca- compared to Na-bentonite. Hence, Ca-bentonite has a greater proportion of relatively large pores; this was confirmed by Hg intrusion porosimetry. This means the diffusion pathways in Ca-bentonite are less tortuous than those in Na-bentonite.

But they could have been considerably more quantitative than this. In the paper, tortuosity is defined as $\tau = L^2/L_e^2$, where “$L$ is the straight-line macroscopic distance between two points defining the transport path, and $L_e$ is the actual, microscopic or effective distance between the same two points.” Tortuosity is furthermore evaluated from HTO diffusion to $\tau_{\ce{Na}} = 0.062$ in Na-bentonite, and $\tau_{\ce{Ca}} = 0.117$ in Ca-bentonite. Combining these expressions gives $L_{e,\ce{Na}} = 1.37\cdot L_{e,\ce{Ca}}$.

What is implicitly said in this work is thus that the “actual, microscopic or effective distance between two points” is 1.37 times longer in Na-bentonite as compared with Ca-bentonite. I mean that it would be suitable for authors making these kind of (implicit) claims to provide a quantitative idea of how the pore space is modified in order to achieve this particular alteration of distances. To me, it is not even obvious why “larger quasicrystals” implies shorter “diffusion paths” — note that the effect of the “net” geometries above are scale independent.

But rather than making a quantitative discussion, Choi and Oscarson (1996) give the following caveat

In reality, $\tau$ may account for more than just the pore geometry of the clay. Another factor that may be included in $\tau$ is, for instance, the variation in the viscosity of the solution within the pore space (Kemper et al., 1964).

I find this an incredible statement. It is similar to saying that, “in reality”, Earth’s gravity constant ($g$) may include effects of air resistance.

Footnotes

[1] “Bentonite” is used in the following as an abbreviation of “Bentonite and claystone”.

[2] That is at least my interpretation. A fuller quotation reads: “[Tortuosity] is defined as the ratio of the path length the solute would follow in water alone, $L$, relative to the tortuous path length it would follow in porous media, $L_e$” (while the following equation actually contains the square of this ratio).

[3] I am not fully convinced that all authors keep this in mind at all times. How should e.g. the following passage from Charlet et al. (2017) be interpreted: “An important geometric parameter is the tortuosity factor, $\tau$, that quantifies the travelled distance of a dissolved constituent through the pore network compared to actual distance between two points.”? Or this one from Van loon et al. (2018) : “The tortuosity is a measure for path lengthening and takes into account that molecules have to diffuse around grains and thus take a longer way.”?

[4] Which is quite amazing, considering that this paper deals with diffusion in a gas phase in macroporous systems.

[5] Since all walkers start in the same point (the origin) the data show finite-size effects for small times. The presented data is therefore taken after an initiation time, labeled $t^\star$.

The problem with geometric factors

5 Replies

Many papers on diffusion in compacted bentonite and claystone¹ assume a direct connection between diffusivities measured in clay ($D_\mathrm{clay}$), and the geometrical configuration of the pore space. It seems close to mandatory to include
an expression of this type \begin{equation} D_\mathrm{clay} = \frac{1}{G}\cdot D_\mathrm{w}\tag{1} \end{equation}

where $D_\mathrm{w}$ is the corresponding diffusion coefficient in bulk water (usually taken as the limiting value at infinite dilution), and $G$ is a factor postulated to have pure geometric meaning. The geometric multiplier is written in a variety of ways, sometimes in terms of other factors which individually are supposed to convey specific geometric information, e.g. $1/G = \delta/\tau^2$, where $\delta$ is the “constrictivity” and $\tau^2$ is the “tortuosity factor”².

However, the above described procedure is futile:

A geometric factor can not be deduced a priori, because the pore space geometry of any realistic system is way too complicated. So-called tortuosities have only been calculated in some trivial cases, relating e.g. to highly symmetric networks of identical cylinders, which has basically no resemblance to the chaotic smectite-lasagna expected to be found in bentonite. Indeed, many papers on these issues admit that the geometric factor in practice functions as a “fudge factor” (to be “optimized”).
Even if a geometric factor could be deduced from only considering the pore space configuration, it is not expected to relate to $D_\mathrm{clay}$ in accordance with eq. 1. If we imagine removing all influence of pore geometry on the diffusivity in the clay, what remains is diffusion in an environment heavily influenced by the presence of exchangeable cations. Thus, a correct geometric factor should not multiply $D_\mathrm{w}$, but the diffusion coefficient measured on the short time and length scale in the clay. Such measurements exist — at least for water diffusion — and show values different from $D_\mathrm{w}$.

Because it is practically impossible to independently deduce a geometric factor, it is de facto defined by the ratio $D_\mathrm{clay}/D_\mathrm{w}$ in many studies. I dare to say that in any work within the bentonite research community, a geometric factor has no other meaning than $D_\mathrm{clay}/D_\mathrm{w}$ (or some function of this parameter, depending on definitions).

This means — at best — that the use of a geometric factor does nothing for a model description, but only replaces the parameter $D_\mathrm{clay}$ with the parameter $D_\mathrm{clay}/D_\mathrm{w}$; it simply corresponds to the awkward choice of specifying diffusion coefficients in units of $D_\mathrm{w}$.

Often, however, the procedure has worse consequences, because eq. 1 (as used in practice) represents an unjustified claim; it is not correct to use the value of $D_\mathrm{clay}/D_\mathrm{w}$ to make specific statements about the pore space geometry, while ignoring other possible mechanisms contributing to the value of $D_\mathrm{clay}$. If diffusion instead is described without invoking a geometric factor, $D_\mathrm{clay}$ itself functions as as a model parameter, and no implicit claims are made about why it has the value it has.

There are many examples in the literature where this confusion shows up. In e.g. Appelo and Wersin (2007), it is concluded that tortuosity for “anions” is larger than for water, while tortuosity for “cations” is smaller than for water. This conclusion is based on fitting of geometric factors without independent information on the pore space geometry:

[…] the pore may become filled entirely with a diffuse double layer when it narrows sufficiently. This constricts the passage of anions, and since the anions must circumnavigate the obstacle, they have greater tortuosity than tritium. This explains that a model with a tortuosity factor for iodide that is 1.4 times higher than for tritium better matches the data.

[…] in interlayers and pore constrictions the cations pass in relatively larger amounts than in free porewater, and consequently, they have smaller tortuosity than tritium.

Similar descriptions are given in Altmann et al. (2012) (in a section describing”advances” in process understanding with the FUNMIG project)

At low ionic strengths, cations are mostly located in the diffuse layer. This layer is of relatively low volume and connected throughout the porosity — cations need less time to explore this small volume and therefore to pass from one side of the media to the other.[…]

Anions are excluded from the diffuse layer and, therefore,also have a smaller volume to explore than neutral tracers. They however must pass through small pores in order to explore the total porosity, and these pores act as electrostatic throats […], forming an energy barrier to anion movement, reducing the probability of passage (resulting in reduced $D_e$) and increasing the time needed to explore all of the porosity.

And the same type of argument — that different diffusive behavior of cations and anions is caused by differences in “pathways” — is found in Charlet et al. (2017)

The different pathways of anions and cations in these regions lead to different effective diffusion rates (Van Loon et al., 2007; Mazurek et al., 2009)

But effective diffusion rates — which can basically be made to vanish e.g. for chloride, as has been demonstrated in bentonite — do not vary because of changes in “pathways”, but because of changes in boundary conditions, due to the presence of interfaces to external solutions.

To investigate the behavior of $D_\mathrm{clay}/D_\mathrm{w}$ one must of course consider real — not effective — diffusion coefficients (unfortunately, actual diffusion coefficients are usually referred to as “apparent” in the bentonite research community). Real diffusion coefficients reflect the mobility of the diffusing species, and can be measured e.g. in non-steady-state closed-cell tests, which are not influenced by interfaces to external solutions³. A very good appreciation for how such diffusion coefficients behave is given by the vast amount of data measured by Kozaki et al. in Na-montmorillonite (Kunipia-F). Some of this data (sources: 1, 2, 3, 4, 5) is plotted below, in terms of $D_\mathrm{clay}/D_\mathrm{w}$ (using a linear $y$-axis, for a change).

This plot clearly shows that if $D_\mathrm{clay}/D_\mathrm{w}$ is interpreted as having pure geometrical meaning, the conclusion must be that chloride and sodium have basically identical “tortuosity”, while the “tortuosity” of the cations calcium and cesium is considerably larger⁴. Thus, the closed-cell data disproves any claim that “tortuosity” for anions is larger than for cations. Moreover, from this data it is quite a stretch to state that “tortuosity” for anions is larger than for water, and it is plain wrong to state that “tortuosity” for cations is smaller than for for water. (For the record, I’m strongly convinced that both calcium and cesium occupy the same pore volume as sodium in this material.)

I should emphasize that I am not primarily arguing for a better geometrical interpretation of $D_\mathrm{clay}/D_\mathrm{w}$, but rather that this whole procedure (using eq. 1) should be avoided. This is not to say that pore geometry doesn’t influence clay diffusivity — most reasonably it does — but analysis based on eq. 1 will be flawed, because of the unjustified model assumptions.

When it comes to geometrical understanding of diffusion in bentonite, I think it would be more rewarding to focus on the quite large variation of reported diffusivities in seemingly identical systems. As an example, the figure below shows the diffusivity of cesium in very similar, pure Na-montmorillonite, measured in four different closed-cell studies: apart from Kozaki et al.’s data (blue), also Sato et al. (1992) (green), and Tachi and Yotsuji (2014) (orange).

Cs diffusion data from Kozaki et al. Sato et al., and Tachi and Yotsuji

In aggregation, this figure demonstrates almost an order of magnitude variability for the diffusion coefficient of cesium, even in very pure systems. If this kind of variability cannot be explained, it is of course not very relevant to develop models describing (smaller) differences in diffusivity between different types of ions in different types of bentonites.

If you claim to understand the variation seen in this plot, rather than simply finding an appropriate factor to blame (particle size?), here is a challenge: Give a prescription for how to prepare a Kunipia-F/P sample in order to minimize/maximize the resulting diffusivity⁵.

Footnotes

[1] In the following, the term “bentonite” is used as a shorthand for “compacted bentonite and claystone”.

[2] The nomenclature here is a mess, but I leave it to people who really want to use these concepts to actually define them in detail.

[3] They can also be deduced from the transient behavior in “steady-state” tests, if the transport capacity of the confining filters is not limiting the process.

[4] Note that “tortuosity”, as used e.g. by Appelo and Wersin (2007), is related to the inverse of $D_\mathrm{clay}/D_\mathrm{w}$.

[5] The diffusion direction should still be along the axis of sample compaction, of course.

You cannot add fluxes from different domains in multiporosity models

2 Replies

Most models of compacted bentonite and claystone¹ assume that the material contains different domains with distinctly different properties. In particular, it is common to assume that the system contains both an aqueous domain (a bulk water solution) and one or several domains associated with the solid surfaces.² Models of this type is here called multiporosity models.

A “standard” procedure in multiporosity models is to assume that the diffusive flux in any domain can be fully expressed using quantities that relates only to that same domain, while a “total” flux is given by summing the contributions from each domain. In e.g. the case of having one bulk water domain and one surface domain, the diffusive flux is typically written something like

$$ \label{eq1} j = j_{bulk} + j_{surface} = -D_e\cdot\nabla c -D_s \cdot\nabla c_s \tag{1} $$

were $D_e$ and $c$ denote, respectively, the diffusion coefficient and concentration in the bulk water domain, and the corresponding parameters in the surface domain are denoted $D_s$ and $c_s$. It is important to keep in mind that what is referred to as a “total” flux in these descriptions is the experimentally accessible, macroscopic flux (here simply labelled $j$).

This approach is basically adopted by all proponents of multiporosity models, and has been so for a long time, see e.g.:

There is, however, a big problem with this procedure:

Equation 1 does not make sense

The statement of eq. 1 — that the macroscopic, measurable flux has two independent contributions — is only true if the different domains are completely isolated from each other. This can e.g. be seen by making $D_s$ vanishingly small in eq. 1; the flux is still non-zero due to the contribution from the completely unaffected $j_{bulk}$-term. In contrast, for non-isolated domains the expected behavior is that a macroscopic diffusive flux will vanish if the diffusivity of any domain is made to approach zero.

Consequently, a large part of the compacted bentonite research field seems to treat diffusion by assuming the different domains to be isolated “pipes”. But this is of course not how bentonite works! If you postulate multiple diffusive domains in the clay, you must certainly allow for diffusing species to be able to be transferred between them. And here is where it gets mysterious: the default choice is to also assume that the domains are non-isolated. In fact, the domains are typically assumed to be in equilibrium. In e.g. the case of having one bulk water and one surface domain, the equilibrium is usually expressed something like

$$
c_s = K\cdot c
\tag{2}
$$

where $K$ is some sort of partition coefficient.

Thus, the “standard” treatment is to combine eq. 1 — even though it is only valid for isolated domains — with an equilibrium assumption (eq. 2) to arrive at expressions which typically look something like this

$$
j = \left (D_e + K\cdot D_s\right)\cdot \nabla c
\tag{3}
$$

Note how this equation has the appearance of an addition of diffusion coefficients, as the “total” flux now is related to a single concentration gradient (which in practice always is a bulk solution concentration).

Since eq. 3 results from combining two incompatible assumptions (isolation and equilibrium), it should come as no surprise that it is not a valid description for multiporosity models.

I cannot get my head around why this erroneous description has prevailed within the compacted clay research field for so long. The problem of deriving “effective” properties of heterogeneous systems appears in many research fields, and applies to many other quantities than diffusivity, e.g. heat conductivity, electrical conductance, elastic moduli, etc. It should therefore be clear also in the compacted clay world that obtaining the effective diffusivity for a system as a whole is considerably more involved than simply adding diffusivities from each contributing domain.

Some illustrations

To discuss the issue further, let’s make some illustrations. A molecule or ion in a porous system conducts random “jiggling” motion, exploring all regions to which it has access, as illustrated in this figure

This animation shows a simulated random walk at the microscopic scale, in a “made-up” generic porous system, consisting of solids (brown), a bulk water domain (blue), and a surface domain (pink). The green dots show the trace of the random walker; light green when the walker is located in the bulk water domain (with higher mobility), and darker green when it is located in the surface domain (with lower mobility). Note how the walker frequently switches between the domains — which is exactly what is required for the two domains to be in equilibrium.

As diffusivity is fundamentally related to the statistical properties of the random walk,³ it should be crystal clear that the macroscopic diffusion coefficient (reflecting mobility on length scales beyond what is shown in the animation) depends in a complicated way on the mobility of the involved domains, just as it depends on the detailed geometrical configuration. Oddly, the complex geometrical dependency is usually acknowledged within the bentonite research field (by means of e.g. formation and/or tortuosity factors), which makes it even more incomprehensible that the issue of multiple domain diffusivities is treated as it is.

In contrast, the figure below illustrates the microscopic behavior implied by eq. 1.

Here is shown one random walker in the bulk domain (green trace) and one in the surface domain (red trace). Since no exchange occurs between the domains, they are not in equilibrium (the concept does not apply). The lack of exchange between the domains is of course completely non-physical, but it is only under this assumption that the system is described by two independent fluxes (eq. 1). It may also be noted that the two types of walkers experience completely different geometrical configurations (i.e. formation factors), none of which corresponds to the geometrical configuration in the case of non-isolated domains.

A more reasonable formulation

If you insist on having a multiporosity model, the correct way to go about is to assume equilibrium within a so called representative elementary volume (REV), and to formulate the model on the macroscopic — averaged — level. On this level of description, it is legitimate to speak of both a bulk water and a surface concentration at any given point in the model (they represent REV averages), but there is of course only a single flux, as well as a single value of the chemical potential at any given point. The diffusive flux can be written quite generally as

\begin{equation} j = -\phi
D_\mathrm{macr.}\cdot\frac{\bar{c}}{RT}\nabla\mu
\end{equation}

where $\bar{c}$ is the macroscopic, averaged, concentration, $\phi$ is the porosity, and $\mu$ is the (electro-) chemical potential. $D_\mathrm{macr.}$ is the macroscopic diffusion coefficient, containing statistical information on how the diffusing substance traverses the porous system on length scales larger than the REV (in particular, it contains averaged information on the mobility in all involved domains, as well as their detailed geometrical configuration).

The value of $\mu$ can be obtained from considering the microscopic scale. Since the system by assumption is in equilibrium within a REV, $\mu$ has the same value everywhere within this volume. On the other hand, the (electro-) chemical potential may also be expressed e.g. as

\begin{equation}
\mu(\xi ) = \mu_0 + RT\cdot\ln a(\xi ) + zF\psi(\xi ) = \mathrm{const}
\end{equation}

where $\xi$ is a microscopic variable denoting the position within the REV, $\mu_0$ is a reference potential, $a(\xi )$ is the chemical activity and $\psi(\xi )$ the electrostatic potential. In the bulk water domain, the electrostatic potential is zero by definition (ignoring possible couplings between different types of charged species), and $\mu$ can here be evaluated as (where the index “b” is used to denote bulk)

\begin{equation}
\mu_b = \mu_0 + RT\cdot\ln a_b
\tag{4}
\end{equation}

But this is by definition the value of the (electro-)chemical potential everywhere within the REV.

Thus, going back to the macroscopic scale and using eq. 4 for $\mu$, we have for the flux

\begin{equation}
j = -\phi D_\mathrm{macr.}\cdot\frac{\bar{c}}{a_b}\nabla a_b
\end{equation}

Ignoring also possible gradients in activity coefficients, the flux can be simplified as

\begin{equation}
j = -\phi D_\mathrm{macr.}\cdot\frac{\bar{c}}{c}\nabla c
\end{equation}

Where, as before, $c$ denotes a bulk solution concentration. For the case of having one bulk domain, and one surface domain, $\bar{c}$ can be written

\begin{equation}
\bar{c} = \frac{\phi_b}{\phi}c + \frac{\phi_s}{\phi}c_s
\end{equation}

where $\phi_b$ and $\phi_s$ are the porosities for the bulk and surface domains, respectively ($\phi = \phi_b + \phi_s$). Using also eq. 2, the flux can finally be written

\begin{equation}
j = -D_\mathrm{macr.}\left (\phi_b+\phi_s\cdot K\right )\nabla c
\tag{5}
\end{equation}

This expression can be compared to the result from the “standard” procedure, eq. 3. Although eq. 3 is derived using incompatible, non-physical assumptions, it is similar to eq. 5 , but involves two ($D_e$ and $D_s$) rather than a single diffusion parameter ($D_\mathrm{macr.}$). If these two parameters are used merely as fitting parameters (which they basically always are), eq. 3 can always be made to resemble eq. 5. In this sense, eq. 3 is not wrong. Rather, it is not even wrong.

Footnotes

[1] In the following I will only say bentonite, but I mean any type of clay system with significant swelling/ion exchange properties.

[2] I strongly oppose the way bulk water phases are included in such models, but this post is not a critique of that model choice. Here, I instead point out that virtually all published descriptions of multiporosity models handle the flux incorrectly.

[3] Specifically, the average of the square of the displacement of the walker is proportional to time, with the diffusion coefficient as proportionality constant (apart from a dimensional factor).

The Bentonite Report

Reflections on bentonite research

Category Archives: Diffusion

\(D_a\) is not “apparent” (sorption, part I)

The traditional diffusion-sorption model

The immobilization assumption is not valid

A correct “surface diffusion” model

A complete conceptual change

Footnotes

Anion-accessible porosity – a brief history

Genesis

Compacted bentonite

Interlayers

To sum up…

Footnotes

What is a “diffusion path” anyway?

Footnotes

The problem with geometric factors

Footnotes

You cannot add fluxes from different domains in multiporosity models

Equation 1 does not make sense

Some illustrations

A more reasonable formulation

Footnotes