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).
• Although the notion of an EDL captures essential parts of the observed swelling pressure behavior, it is nevertheless quite easy to find claims that e.g. DLVO theory has serious flaws , or even that the fundamental mechanism for clay swelling is not associated with the exchangeable cations¹. While many of these alternative descriptions may be easily (or not so easily) rejected, I fear that they have influenced a way of thinking about other aspects of bentonite, in particular the persistent idea of exchangeable cations as non-diffusive.
• Modern geotechnical descriptions of bentonite often combine capillary concepts (surface tension, saturation,”suction” etc.) with a diffuse double layer mechanism. This seems conceptually incompatible, as a rigid structure — which is required to withstand attractive capillary forces — can hardly exist in a system that, under certain circumstances, is able to swell indefinitely.
• In most geochemical treatments of bentonite, swelling pressure is simply ignored. This is remarkable as most modern reactive-transport models postulate different domains with different “types” of water, making the issue of water equilibrium critical.

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.

Footnotes

[1] You can even find a statement saying that clay swelling has been proved to be “due to long-range interaction between particle surfaces and the water” (I don’t agree).

[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.