Mineral Stability and Cation Exchange Capacity

To simplify the ion-exchange reactions involved, the montmorillonite was Na-saturated to produce a well-controlled Na-montmorillonite (NaMnt) adsorbent. The pH of the supernatant solution was between 1.2 and 2.2 when the SWy-3 montmorillonite was washed with 0.01 mol/L HCl solution. The low pH could, potentially, dissolve the clay mineral and change its physiochemical properties (Amram and Ganor Reference Amram and Ganor2005; Rozalén et al. Reference Rozalén, Huertas, Brady, Cama, García-Palma and Linares2008). Accordingly, the oriented SWy-3 and NaMnt samples were analyzed with XRD to examine the clay mineral structure. The XRD results showed no significant change in the XRD peak positions (Fig. 1), suggesting that the mineral composition of NaMnt remained similar to that of SWy-3.

Fig. 1 Oriented XRD patterns of dry, untreated (SWy-3) and Na-saturated (NaMnt) montmorillonite samples

The total removal of some cations from clay minerals was suggested to be very difficult (Baeyens and Bradbury Reference Baeyens and Bradbury1997), and small cations (e.g. Mg²⁺) were presumably held very strongly in the octahedral vacancies in montmorillonite (Chorom and Rengasamy Reference Chorom and Rengasamy1996). The cations that are difficult to remove would not influence ion-exchange reactions. Table 3 lists exchangeable cations in SWy-3 and NaMnt samples. The total exchangeable cation concentrations included adsorbed cations plus various salts in the samples. These cations would participate in ion-exchange reactions and result in a complicated matrix. However, the NaMnt had very low concentrations of K⁺, Ca²⁺, and Mg²⁺ cations. The exchangeable K⁺, Ca²⁺, and Mg²⁺ cations in the NaMnt sample were not considered in this study due to their low content. Accordingly, Na⁺ was the only dominant exchangeable cation in NaMnt. The formula was (Na_0.58)[Al_3.01Fe(III)_0.41Mn_0.01Mg_0.54Ti_0.02][Si_7.98Al_0.02]O₂₀(OH)₄ adapted from The Clay Minerals Society (2020) to ensure charge neutrality in the clay unit, with a molecular weight (MW) of 744.75 g/mol.

Table 3. Total exchangeable cations in untreated (SWy-3) and Na-saturated montmorillonite (NaMnt). Average ± standard deviations were based on N = 3

Based on the chemical formula of the NaMnt noted above, the theoretically calculated CEC from isomorphic substitution was 778.78 μmol/g (= 0.58/MW×10⁶), which was very close to the reported CEC (764 μmol/g) of SWy-3 from The Clay Minerals Society (2020). The experimentally measured maximum amount of alkali cations adsorbed above pH 7 from different adsorption envelopes came to a similar value of 710 μmol/g (Fig. 2). This value was also very close to the total Na⁺ concentration extracted from the NaMnt (712.8 μmol/g, Table 3). The decrease in the measured values relative to the published or calculated values probably resulted from two contributing factors. One was the loss of small-size clay particles that were decanted with the supernatant solutions during the Na-saturation procedure, which tend to have a larger CEC (Stul and Van Leemput Reference Stul and Van Leemput1982). Another was the presence of feldspar, illite, and quartz impurities (Fig. 1). The CEC of feldspar (2.8–20 μmol/g, Nash and Marshall Reference Nash and Marshall1956), illite (75–240 μmol/g, Baeyens and Bradbury Reference Baeyens and Bradbury2004), and quartz (0 μmol/g) are much lower than montmorillonite (764 μmol/g, The Clay Minerals Society 2020). The primary mineral contributing to the CEC of NaMnt was montmorillonite because of the low concentrations and CEC of the other minerals present (e.g. illite, quartz, and feldspar). Accordingly, the model development discussed in the next section will treat the NaMnt as pure montmorillonite.

In the cation adsorption envelopes, the amount of adsorbed cations was constant at high pH (>7) but decreased at low pH (<7) (Fig. 2). The decrease in cation adsorption at very low pH (<3) could have been caused either by the dissolution of clay minerals or by the competition of H⁺ for adsorption sites (Baeyens and Bradbury Reference Baeyens and Bradbury1997; Poinssot et al. Reference Poinssot, Baeyens and Bradbury1999). The aqueous Al³⁺ and Fe³⁺ concentrations were analyzed at low pH to elucidate the major reason for the decrease in cation adsorption (Fig. 3). The aqueous Al³⁺ and Fe³⁺ concentrations increased rapidly as the pH decreased. The total Al and Fe contents in NaMnt were 4068.45 and 550.52 μmol/g, respectively, based on the chemical formula of NaMnt. Accordingly, the aqueous Al³⁺ and Fe³⁺ concentrations estimated the highest dissolution rates of NaMnt to be 0.1% and 1.2%, respectively. In closed containers, a very small amount of dissolution of Al³⁺ and Fe³⁺ cations will suffice to saturate the supernatant liquid. Conversely, the cation adsorption decreased by >42% of the total CEC at pH < 3 (~300 μmol/g, Fig. 2), which means that the large decrease in cation adsorption at low pH was mainly due to H⁺ competition rather than mineral dissolution.

Fig. 2 Sum of alkali cations adsorbed on NaMnt for various conditions

Fig. 3 Aqueous a Al³⁺ and b Fe³⁺ concentrations and clay dissolution at low pH for various conditions

With increasing Al³⁺ solubility at low pH, competitive adsorption of Al³⁺ cations is possible. The impact of mineral dissolution on the adsorption of other cations is a complex topic to resolve in adsorption modeling exercises. For example, Coulter and Talibudeen (Reference Coulter and Talibudeen1968), Coulter (Reference Coulter1969), and Bloom et al. (Reference Bloom, McBride and Chadbourne1977) observed an increase in CEC with adsorption of Al³⁺ on montmorillonite. In those studies, the Al³⁺ came from allochthonous sources; i.e. the Al³⁺ cations were added. In the present study, the Al³⁺ was from an autochthonous source, coming from the mineral sample itself. As the Al³⁺ ions in the clay mineral dissolved, a concurrent shift may have occurred in the clay’s net charge balance which, in turn, would impact the clay’s net adsorption behavior. As the Al³⁺ re-adsorbs on the same clay from whence it came, another re-shifting of the clay’s net charge balance may occur. The net effect of this Al desorption and re-adsorption could be a local reconfiguration of the clay mineral structure. The influence of Al³⁺ cations on cation adsorption was not addressed in this study because of its low aqueous concentration (at most 0.03 mmol/L at low pH), the low estimates on total solubility (<1.2%), and the similarity in the XRD patterns of Mnt samples at low and high pH values (discussed below).

Model Development

The CEC of clay minerals comes from isomorphic substitution and the protonation/deprotonation of broken edges. However, the model development presented here will focus only on interlayer sites for two reasons: (1) the adsorption capacity of interlayer sites is much larger than the edge sites; and (2) the constrained environments of interlayer sites have a greater retention strength than the edge sites.

First, the charge from isomorphic substitution accounts for the majority of the CEC on montmorillonite (Sposito et al. Reference Sposito, Skipper, Sutton, Park, Soper and Greathouse1999). Published values of the percentage of edge sites on montmorillonite varied from <1% to 15% of the total CEC (McKinley et al. Reference McKinley, Zachara, Smith and Turner1995; Schulthess and Huang Reference Schulthess and Huang1990; Baeyens and Bradbury Reference Baeyens and Bradbury1997; Tournassat et al. Reference Tournassat, Neaman, Villiéras, Bosbach and Charlet2003; Missana et al. Reference Missana, Benedicto, García-Gutiérrez and Alonso2014). The contributions of isomorphic substitution and interlayer adsorption sites were based traditionally on pH-independent adsorption data, while the edge sites are based on pH-dependent adsorption data or potentiometric titration curves (Bradbury and Baeyens Reference Bradbury and Baeyens1997; Fernandes and Baeyens Reference Fernandes and Baeyens2019). However, if cations (e.g. Na⁺, K⁺) are able to adsorb and exchange on interlayer sites, the H⁺ (or hydronium H₃O⁺) cations should be able to do the same. Accordingly, pH-dependent reactions on interlayer sites should be included in ion-adsorption models (Barbier et al. Reference Barbier, Duc and Petit-Ramel2000; Bradbury and Baeyens Reference Bradbury and Baeyens2005; Missana and García-Gutiérrez Reference Missana and García-Gutiérrez2007). That is, some of the pH-dependent CEC data used to estimate the concentrations of edge sites are potentially interlayer sites instead.

Second, the cation adsorption strength in constrained environments (i.e. interlayer sites) can be much greater than that in non-constrained environments (i.e. edge and planar sites). The Cs⁺ cation was found by Dzene et al. (Reference Dzene, Tertre, Hubert and Ferrage2015) to be adsorbed more strongly in the interlayer sites of vermiculite than in the external (edge and basal) sites based on cation adsorption and desorption experiments. The Nanopore Inner-Sphere Enhancement effect (NISE effect; Ferreira and Schulthess Reference Ferreira and Schulthess2011; Schulthess et al. Reference Schulthess, Taylor and Ferreira2011) stated that a cation can be partially or fully dehydrated and adsorb strongly inside nanopore channels when the size of the pore is smaller than the hydrated ionic diameter. The interlayer space of montmorillonite ranges from 0 to 1 nm based on the water content and interlayer cations present (Ferrage et al. Reference Ferrage, Lanson, Sakharov and Drits2005; Iijima et al. Reference Iijima, Tomura and Shoji2010; Ohkubo et al. Reference Ohkubo, Okamoto, Kawamura, Guégan, Deguchi, Ohki, Shimizu, Tachi and Iwadate2018), which is in the same size range as most dehydrated and hydrated cations. Accordingly, the retention of cations tends to be stronger on interlayer sites than on edge sites.

As noted in the Introduction, three types of interlayer adsorption sites on montmorillonite from a previous DFT simulation (Li et al. Reference Li, Schulthess, Co, Sahoo and Alpay2020) were used for the development of the ion-exchange model. All three sites were interlayer sites. The NaMnt montmorillonite has a very similar chemical composition to those used in the DFT study, where, assuming a random distribution of octahedral cations in the montmorillonite, the percentages of three groups of structures with different octahedral Fe-Mg distances (j) were approximately 43%, 43%, and 14%, respectively (Li et al. Reference Li, Schulthess, Co, Sahoo and Alpay2020). As discussed earlier, the NaMnt was treated as pure montmorillonite, particularly in terms of the types of adsorption sites present. The CEC might be smaller than expected due to impurities, but the distribution of sites should remain similar to those in montmorillonite. The maximum amount of cations adsorbed on NaMnt was 710 μmol/g (Fig. 2); accordingly, the CEC of each adsorption site was 305, 305, and 100 μmol/g for sites with j = 1, 2, and 3, respectively. These sites were labelled S _a , S _b , and S _c for j = 1, 2, and 3, respectively.

The H⁺ cation exists in all aqueous solutions, and it is an inevitable competitor in ion-exchange reactions. Therefore, the H⁺ cation was considered as the reference cation for the presentation of all cation-adsorption equilibrium constants. On each site S _i (i = a, b, or c), the Na⁺-H⁺ ion-exchange reaction was:

(1) $S_{i}H+{\mathrm{Na}}^{+}\rightleftharpoons S_i\mathrm{Na}+H^{+}$

The unknown value of K for each site is given by the ratio of total activities of products over reactants:

(2) $K_i=\frac{\left(S_{i}Na\right)\left(H^{+}\right)}{\left(S_{i}H\right)\left({Na}^{+}\right)}$

where () and {} indicated the ionic activities of cations in solution or adsorbed on the solid surface. The known adsorption maxima (Γ_max) for each site was the sum of adsorbed H⁺ and Na⁺ cations:

(3) ${\Gamma }_{max,Si}=S_{i}H+S_i\mathrm{Na}$

Thus, the only unknown parameters were the K constants of Na⁺ on each site, which were optimized based on the least square of errors using IExFit software. The optimization of the negative logarithm of K values (pK, Table 4) showed that the three-site ion-exchange model was able to fit the Na⁺ adsorption data very well for both low and high Na⁺ concentrations (Fig. 4). The pK values increased with sites S _a < S _b < S _c , which means that the Na⁺ adsorption strength decreased relative to H⁺ adsorption in the order S _a > S _b > S _c .

Table 4. The negative logarithm of ion-exchange equilibrium constants (pK) optimized using an ion-exchange model for sites S_a, S_b, and S_c. The CEC for each site was obtained according to Li et al. (Reference Li, Schulthess, Co, Sahoo and Alpay2020) and a maximum retention value of 710 μmol/g (Fig. 2)

Fig. 4 Adsorption envelopes of Na⁺ on NaMnt. The initial conditions are shown in Table 2. Lines are the predicted relationship using the ion-exchange model. The goodness-of-fit is Efron’s pseudo-R ²

When a coexisting cation M ⁺ (namely Cs⁺, K⁺, Li⁺) was present, additional ion-exchange reactions were involved:

(4) $S_{i}H+M^{+}\rightleftharpoons S_{i}M+H^{+}$

with corresponding unknown equilibrium constants:

(5) $K_i=\frac{\left(S_{i}M\right)\left(H^{+}\right)}{\left(S_{i}H\right)\left(M^{+}\right)}$

The nature (e.g. location and CEC) of adsorption sites and the value of K for each cation are characteristic for each clay mineral, and ideally this should hold true regardless of the presence of other minerals and competing cations. Therefore, the number of sites, CEC of each site, and the adsorption equilibrium constant of Na⁺ were kept constant in the presence of other cations. The known adsorption maxima for each site was defined as:

(6) ${\Gamma }_{max,Si}=S_{i}H+S_i\mathrm{Na}+S_{i}M$

Thus, in these competitive ion-exchange reactions, the only unknown parameters were the pK values of the coexisting cations.

The optimization of the pK values showed that the competitive ion-exchange model was able to predict the retention of all the cations very well (Fig. 5, Table 4). The optimized pK values for different cations increased with sites S _a < S _b < S _c , which means that the cation adsorption strength decreased relative to H⁺ in the opposite order. The pK values increased in the order Cs⁺ < K⁺ < Na⁺ < Li⁺ for sites S _a and S _b , but stayed the same for Cs⁺ and K⁺, and Na⁺ and Li⁺ on site S _c , which indicated that the cation adsorption strength decreased in the opposite order on sites S _a and S _b .

Fig. 5 Cation adsorption data of Na⁺-H⁺-M⁺ (M⁺: Li⁺, K⁺, or Cs⁺) on NaMnt as a function of pH. The initial conditions are shown in Table 2. Lines are predicted adsorption data using the ion-exchange model. The goodness-of-fit is Efron’s pseudo-R ²

The alkali cation adsorption strength has been studied widely on montmorillonite, and the hydration energy was suggested to be the determining factor for the isovalent cation adsorption selectivity on clay minerals (Teppen and Miller Reference Teppen and Miller2006; Rotenberg et al. Reference Rotenberg, Morel, Marry, Turq and Morel-Desrosiers2009). In the interlayer space of clay minerals, cations with lower hydration energy tend to dehydrate partially or fully and adsorb strongly (Teppen and Miller Reference Teppen and Miller2006; Salles et al. Reference Salles, Bildstein, Douillard, Jullien and Van Damme2007). The hydration energy of alkali cations in the interlayer space decreases in the order Li⁺ > Na⁺ > K⁺ > Cs⁺ (Salles et al. Reference Salles, Bildstein, Douillard, Jullien and Van Damme2007); thus, the K⁺ and Cs⁺ cations are more easily dehydrated, and are strongly adsorbed on these sites (Teppen and Miller Reference Teppen and Miller2006; Salles et al. Reference Salles, Bildstein, Douillard, Jullien and Van Damme2007). The pK values obtained in the present study (Table 4) were consistent with the proposed adsorption mechanisms for these alkali cations (Teppen and Miller Reference Teppen and Miller2006; Rotenberg et al. Reference Rotenberg, Morel, Marry, Turq and Morel-Desrosiers2009).

Model Testing

Using all the known adsorption parameters (i.e. the number of sites, CEC of each site, and equilibrium K constants), the cation adsorption in various chemical environments can be predicted by constructing their corresponding competitive ion-exchange models. First, the model was tested for competitive cation adsorption as a function of pH. When two types of coexisting cations M ⁺₁ and M ⁺₂ (namely Cs⁺, K⁺, Li⁺) were present, additional competitive ion-exchange reactions occurred. The cation adsorption maxima on site S _i (i = a, b, or c) was:

(7) ${\Gamma }_{max,Si}=S_{i}H+S_i\mathrm{Na}+S_i{M}_1+S_i{M}_2$

Using the same known pK values from Table 4 and Γ_max,Si values based on the previous DFT study (Li et al. Reference Li, Schulthess, Co, Sahoo and Alpay2020), the three-site competitive ion-exchange model showed that the predicted values agreed with the experimental data very well (Fig. 6). The consistency of the pK values in various reactions suggested that cation adsorption followed an ion-exchange reaction, and that the pK values were not influenced by the presence of co-existing cations.

Fig. 6 Cation adsorption data of Na⁺-H⁺-M ⁺₁-M ⁺₂ (M⁺_1,2: Li⁺, K⁺, or Cs⁺) on NaMnt as a function of pH. The initial conditions are shown in Table 2. Lines are predicted adsorption data using the ion-exchange model. The goodness-of-fit is Efron’s pseudo-R ²

Second, the model was tested with Cs⁺ adsorption isotherm data at pH 3.5, 5.5, and 8.3 (Fig. 7). The predicted Cs⁺ and Na⁺ adsorptions were obtained for various initial Cs⁺ concentrations using IExFit software. The model was able to predict the experimental Cs⁺ adsorption isotherm data very well. Thus, the competitive ion-exchange model was able to predict cation adsorption on NaMnt over a wide range of pH, Cs⁺ concentrations, and in the presence of competing cations with the same CEC and pK values. Moreover, the prior determination of the number of sites and CEC of each site on NaMnt reduced the number of unknown variables in the model, which increased the certainty and confidence in the model optimization results.

Fig. 7 Experimental data and predictions from the ion-exchange model and Langmuir equation of cation adsorption on NaMnt at pH a 3.5, b 5.5, and c 8.3. ▲ and ● represent the adsorbed Cs⁺ and Na⁺ experimental data, respectively. The goodness-of-fit is Efron’s pseudo-R ²

Comparison of Ion-Exchange Model with Langmuir Isotherm

The Langmuir isotherm is one of the most widely used adsorption models (Table 5). The Langmuir isotherm is based on an adsorption reaction on one type of site: S + A ⇌ SA. For Cs⁺ adsorption, the Langmuir isotherm predicts the amount adsorbed (q) in μmol/g as

(8) $q=\frac{{\Gamma }_{\max }{K}_{L}c}{1+K_{L}c}$

where c is the equilibrium Cs⁺ concentration (mmol/L), Γ_max is the maximum amount of cation adsorbed (μmol/g), and K_L is the Langmuir constant. The Langmuir isotherm for Cs⁺ adsorption on NaMnt was optimized using LMMpro software which will optimize both K_L and Γ_max values, as well as optimize the K_L value using a fixed Γ_max value (710 μmol/g in this study). Both results are shown in Table 5 and Fig. 7.

Table 5. Recently published Langmuir models for Cs⁺ adsorption on montmorillonite. R ² is the coefficient of determination, and K_L is the Langmuir constant

n.s.: not specified

The cation adsorption isotherm was predicted very well with both the ion-exchange model and the Langmuir equation. Compared to the Langmuir equation, however, the ion-exchange model has several advantages. First, the Langmuir equation does not consider the adsorption of H⁺ cations. The influence of the H⁺ cation on Cs⁺ adsorption was demonstrated by the increase in the CEC and equilibrium K_L constant (from 0.86 to 1.27 and 1.54 L/mmol) with increasing pH in the Langmuir equation (Table 5). Ideally, the K_L constant at equilibrium and total CEC should be the same for the same cation and solid surface regardless of pH changes. Conversely, K did not change in the ion-exchange model at different pH conditions. Second, the Langmuir equation can predict only the adsorption of one cation (namely Cs⁺ in this study), while the ion-exchange model predicts all the competing cations in the system. For example, Fig. 7 showed a good prediction of Na⁺ desorption by the ion-exchange model, which cannot be done by the Langmuir model. Third, various studies have shown that multiple types of adsorption sites exist on montmorillonite (Nolin Reference Nolin1997; Tertre et al. Reference Tertre, Prêt and Ferrage2011; Missana et al. Reference Missana, Benedicto, García-Gutiérrez and Alonso2014; Siroux et al. Reference Siroux, Beaucaire, Tabarant, Benedetti and Reiller2017; Martin et al. Reference Martin, Wissocq, Benedetti and Latrille2018; Wissocq et al. Reference Wissocq, Beaucaire and Latrille2018). The Langmuir equation is based on a single type of site, which does not elucidate the real nature and property of adsorption sites on montmorillonite. Conversely, combining the DFT study (Li et al. Reference Li, Schulthess, Co, Sahoo and Alpay2020) with the ion-exchange model presented in this study provided information not only about the number, location, and chemical structure of each adsorption site (based on the octahedral Fe-Mg distance), but also the CEC and retention strength of each site.

Klika et al. (Reference Klika, Kraus and Vopálka2007) also compared the Langmuir equation with a one-site ion-exchange model that did not include H⁺ cations in the exchange reaction. Both models were able to predict Cs⁺ adsorption on bentonite as a function of the bentonite/water ratio and Cs⁺ concentration very well. However, the ion-exchange model was suggested to be more convenient because of its ability to be used over a wider range of conditions with a single set of model parameter values, while the parameters of the Langmuir model varied with experimental conditions.

The wide application of the Langmuir isotherm in the literature (Table 5) suggests that this equation works very well for the prediction of adsorption data, even though it may not properly clarify real adsorption mechanisms on solid surfaces. The validity of the Langmuir isotherm to predict adsorption data comes from its mathematical similarities to the ion-exchange model. The Cs⁺ adsorption isotherm on NaMnt involved a three-way ion-exchange competition (H⁺-Na⁺-Cs⁺). The total adsorbed Cs⁺ was the sum of the Cs⁺ adsorbed on three types of sites. According to the ion-exchange reactions, the total amount of adsorbed Cs⁺ on NaMnt (q, μmol/g) was

(9) $q=\frac{q_1{K}_2\left({Cs}^{+}\right)}{\left(H^{+}\right)+K_1\left({Na}^{+}\right)+K_2\left({Cs}^{+}\right)}+\frac{q_2{K}_4\left({Cs}^{+}\right)}{\left(H^{+}\right)+K_3\left({Na}^{+}\right)+K_4\left({Cs}^{+}\right)}+\frac{q_3{K}_6\left({Cs}^{+}\right)}{\left(H^{+}\right)+K_5\left({Na}^{+}\right)+K_6\left({Cs}^{+}\right)}$

where q ₁, q ₂, and q ₃ were the maximum adsorption capacity for each site on NaMnt (μmol/g); K ₁, K ₃, and K ₅ were the adsorption equilibrium constants of Na⁺ on sites S _a , S _b , and S _c in NaMnt; K ₂, K ₄, and K ₆ were the adsorption equilibrium constants of Cs⁺ on sites S _a , S _b , and S _c in NaMnt; and (Na⁺) and (Cs⁺) were the aqueous equilibrium Na⁺ and Cs⁺ concentrations. In Eq. 9, the H⁺ competition was constant because the equilibrium aqueous pH was fixed for each Cs⁺ isotherm. The q ₁ to q ₃ and K ₁ to K ₆ values were obtained directly from Table 4. Equation 9 can be modified into three components (one for each site):

(10) $q=\frac{q_1{K}_x{c}_o}{1+\frac{\left(H^{+}\right)}{K_1\left({Na}^{+}\right)}+K_x{c}_o}+\frac{q_2{K}_y{c}_o}{1+\frac{\left(H^{+}\right)}{K_3\left({Na}^{+}\right)}+K_y{c}_o}+\frac{q_3{K}_z{c}_o}{1+\frac{\left(H^{+}\right)}{K_5\left({Na}^{+}\right)}+K_z{c}_o}$

where K _x = $\frac{K_2}{K_1}$ , K _y = $\frac{K_4}{K_3}$ , K _z= $\frac{K_6}{K_5}$ , and c _o = $\frac{\left({Cs}^{+}\right)}{\left({Na}^{+}\right)}$ . While the Cs⁺ concentration is changed in the experiment, the Na⁺ concentration changes as a direct result of the Cs-Na exchange reactions. In Eq. 10, the $\frac{\left({Cs}^{+}\right)}{\left({Na}^{+}\right)}$ ratio is the independent variable. Whenever each of the component values in Eq. 10 is significant, the corresponding $\frac{\left(H^{+}\right)}{K_1\left({Na}^{+}\right)}$ , $\frac{\left(H^{+}\right)}{K_3\left({Na}^{+}\right)}$ , and $\frac{\left(H^{+}\right)}{K_5\left({Na}^{+}\right)}$ values become insignificant and can be ignored. Accordingly, Eq. 10 simplifies to:

(11) $q=\frac{q_1{K}_x{c}_o}{1+K_x{c}_o}+\frac{q_2{K}_y{c}_o}{1+K_y{c}_o}+\frac{q_3{K}_z{c}_o}{1+K_z{c}_o}$

Equation 11 has the same format as the Langmuir isotherm (Eq. 8), which explains why the ion-exchange model yields similar curves as the Langmuir equation.

Numerous studies have been conducted to predict ion adsorption on 2:1 clay mineral surfaces using the Langmuir isotherm (Table 5), which assumes implicitly that the reaction is an adsorption reaction (S + A ⇌ SA) rather than an ion-exchange reaction (SA+B ⇌ SB+A) on a single type of surface site. Even though the Langmuir isotherm was able to predict the cation adsorption data very well, this was nothing more than a mathematical coincidence with the ion-exchange model predictions for specific experimental conditions. That is, the Langmuir isotherm was validated (as in ‘shown to be useful’) for a narrow range of conditions, but was not verified (as in ‘shown to be truthful’) when applied to a wide range of conditions. The competitive multi-site ion-exchange model proposed here was verified by DFT simulations (Li et al. Reference Li, Schulthess, Co, Sahoo and Alpay2020) for the three sites of the model as well as the relative CEC of each site. The model was tested using various adsorption data over a wide range of pH, Cs⁺ concentrations, and multiple competing cations, which validated the proposed model. This also suggests that many published studies that used Langmuir isotherms should be revisited and reevaluated using ion-exchange models. Furthermore, if cations or anions are involved in the models, these models should also include competitive H⁺ or OH^– ions in their exchange reactions.

Relationship Between Interlayer Structure and pK Values

The reported pK values are a measure of the retention strength of cations relative to H⁺ cations. The retention strength of cations is essentially a result of the total energies of two competing environments: total energy of adsorption on the surface sites versus total energy of hydration in the aqueous phase. A high (or more positive) pK value denotes a weakly adsorbing cation, probably outer-sphere adsorption, and a low (or more negative) pK value denotes a strongly adsorbing cation, probably inner-sphere adsorption. An adsorbing cation is not strictly either inner-sphere or outer-sphere adsorbed, but rather it is a combination of the two with a wide range of possibilities (Ferreira et al. Reference Ferreira, Schulthess and Giotto2012; Salles et al. Reference Salles, Douillard, Bildstein, El Ghazi, Prélot, Zajac and Van Damme2015), and this results in a wide range of pK values for different ions. Not surprisingly, the adsorption strength of monovalent ions is often shown to be closely associated with their hydration energies (Teppen and Miller Reference Teppen and Miller2006; Salles et al. Reference Salles, Bildstein, Douillard, Jullien and Van Damme2007). If the adsorption pK values are influenced by the relative degree of hydration of an adsorbing ion averaged across its total retention time, then it should follow that these pK values will also influence the interlayer dimensions of 2:1 clays.

Montmorillonite is expansive, and the interlayer cation has an influence on its interlayer dimension (Fig. 8). The samples were analyzed while wet to ensure the in situ hydration condition of the interlayer cations using XRD. The XRD analysis of cation-saturated montmorillonite showed that the d ₀₀₁ value of HMnt (15.2 Å) was similar to those of LiMnt (15.9 Å) and NaMnt (15.6 Å), which suggested the interlayer thickness of the hydrated H⁺ cation (such as the hydronium cation, H₃O⁺) was similar to the interlayer thickness of hydrated Li⁺ and Na⁺ cations. The d ₀₀₁ values decreased progressively for LiMnt and NaMnt, followed by a broad peak for KMnt (13.6 Å) and a broad peak for CsMnt (12.5 Å). The interlayer dimension follows the hydration energies of these alkali cations, and similar results were observed in the literature for montmorillonite samples (Ferrage et al. Reference Ferrage, Lanson, Sakharov and Drits2005; Iijima et al. Reference Iijima, Tomura and Shoji2010; Ohkubo et al. Reference Ohkubo, Okamoto, Kawamura, Guégan, Deguchi, Ohki, Shimizu, Tachi and Iwadate2018).

Fig. 8 XRD patterns of wet montmorillonite saturated with Li⁺ (LiMnt), Na⁺ (NaMnt), K⁺ (KMnt), Cs⁺ (CsMnt), and H⁺ (HMnt) cations. The d ₀₀₁ values (Å) are also shown. The pH values were between 6 and 7 for alkali cation-saturated montmorillonite, and <1 for the HMnt

The hydration properties of SWy-1 montmorillonite were investigated by Ferrage et al. (Reference Ferrage, Lanson, Sakharov and Drits2005) by modeling experimental XRD patterns; those authors found that the d ₀₀₁ values at 80% relative humidity were 15.49, 15.28, and 12.04 Å when the interlayer cations were Li⁺, Na⁺, and K⁺ cations, respectively. The hydration states of these interlayer cations were suggested to be heterogeneous, with hydrated Li⁺ and Na⁺ consisting mostly of two layers of water molecules, and hydrated K⁺ consisting mostly of one layer of water. A similar study by Morodome and Kawamura (Reference Morodome and Kawamura2011) showed that hydrated Cs⁺ contained one layer of water when the d ₀₀₁ was 12.2 Å.

Collapse of the interlayer region of montmorillonite has been observed widely in other Cs⁺-adsorption experiments (Iijima et al. Reference Iijima, Tomura and Shoji2010; Ohkubo et al. Reference Ohkubo, Okamoto, Kawamura, Guégan, Deguchi, Ohki, Shimizu, Tachi and Iwadate2018). The collapse was suggested to be related to the low hydration energy of the Cs⁺ cation (Cornell Reference Cornell1993; Salles et al. Reference Salles, Bildstein, Douillard, Jullien and Van Damme2007; Iijima et al. Reference Iijima, Tomura and Shoji2010; Ohkubo et al. Reference Ohkubo, Okamoto, Kawamura, Guégan, Deguchi, Ohki, Shimizu, Tachi and Iwadate2018). The interlayer Cs⁺ was shown to have a greater affinity toward the clay surface than toward the water molecules using computational molecular mechanics, which resulted in the dehydration of interlayer Cs⁺ cations and the collapse of the interlayer (Teppen and Miller Reference Teppen and Miller2006).

The XRD pattern of NaMnt changed as a function of the Cs⁺ competitive adsorption at pH 8.3 (Fig. 9). The d ₀₀₁ value of NaMnt did not change when 5% Cs⁺ was adsorbed. When 15% of the Cs⁺ was adsorbed, the peak at 15.6 Å became very small, and a new peak at 12.5 Å appeared. As the Cs⁺ concentration increased, the Cs⁺ cations displaced the interlayer Na⁺ cations, which decreased the number of stacked montmorillonite layers dominated with interlayer Na⁺ and resulted in a broader XRD peak at 15.6 Å. When the adsorbed Cs⁺ reached 33%, the peak at 15.6 Å disappeared, which suggested that the number of stacked montmorillonite layers dominated with interlayer Na⁺ was too low to create an XRD peak. The new broad peak at 12.5 Å indicated the formation of stacked montmorillonite layers with Cs⁺.

Fig. 9 XRD pattern of wet NaMnt samples from Cs⁺ adsorption isotherms at pH 8.3 with different percentages of adsorbed Cs⁺ from Fig. 7c. The NaMnt and CsMnt labels indicate the Na⁺- and Cs⁺-saturated clay samples, respectively. The d ₀₀₁ values (Å) are also shown

Iijima et al. (Reference Iijima, Tomura and Shoji2010) also observed that the interlayer space decreased with increasing exchange of Cs⁺ for Na⁺ cations on montmorillonite. Multiple peaks appeared in their XRD pattern when the aqueous Cs⁺ concentration was ~1–10 mmol/L, which suggested that multiple interlayer spaces exist in the transition phase. Similarly, Ohkubo et al. (Reference Ohkubo, Okamoto, Kawamura, Guégan, Deguchi, Ohki, Shimizu, Tachi and Iwadate2018) observed the coexistence of several d ₀₀₁ peaks in montmorillonite with 36% and 59% Cs⁺ exchange fractions in their XRD patterns, which suggested that Cs⁺ and Na⁺ competed for the interlayer space. These authors proposed that the competition between Cs⁺ and Na⁺ cations in the interlayer space resulted in a non-flat and geometrically heterogeneous interlayer space. However, a non-flat geometry will not yield any distinct peaks based on kinematical scattering theory (Moore and Reynolds Reference Moore and Reynolds1989). Instead, the presence of two peaks suggested the presence of heterogeneous interlayer spaces (a mixture of different interlayer spaces in the clay particle), with a variable distribution of dominance of Cs⁺ or Na⁺ ions in the interlayers. This alternate interpretation of the two coexisting XRD peaks is consistent with Pauling’s rule #5, where the number of different kinds of constituents in a crystal environment tends to be small. This alternate interpretation is also consistent with the observation that the pK values in Table 4 did not need to change when competitive cations were introduced in the adsorption envelope experiments (Figs. 5 and 6). According to the NISE effect (Ferreira and Schulthess Reference Ferreira and Schulthess2011; Schulthess et al. Reference Schulthess, Taylor and Ferreira2011), shrinkage of the interlayer space past specific threshold dimensions should impact the retention strength and pK values of the adsorbing ions. However, as using constant pK values resulted in good data fit in different chemical environments, the retention strength of cations appeared to be independent of the presence of other ions. This, in combination with the XRD data, suggests that some segregation of the interlayer cations existed with minimal ion-ion interactions and influence.

The peak position in the XRD patterns of montmorillonite did not change with pH for samples from the same cation adsorption envelope (Fig. 10). This indicated that the clay interlayer structure was consistent as a function of pH. The XRD patterns of montmorillonite with K-Li-Na, Cs-Li-Na, and Cs-K-Na ion-exchange reactions showed broadening of the 12.5 Å peak toward smaller d ₀₀₁ values (Fig. 11). This suggested that the interlayer structures of three competing cations (Fig. 11) were more heterogeneous than the samples of two competing cations (Figs. 8, 9, and 10).

Fig. 10 XRD pattern of wet NaMnt samples of a Li⁺, b high Na⁺, c K⁺, and d Cs⁺ adsorption from Figs 4 and 5 at various pH values, as indicated. The d ₀₀₁ values (Å) are also shown

Fig. 11 XRD pattern of wet NaMnt samples from K,Li-NaMnt, Cs,Li-NaMnt, and Cs,K-NaMnt adsorption reactions at pH 7 from Fig. 6. The d ₀₀₁ values (Å) are also shown

Using complex impedance spectroscopy and water adsorption isotherms on montmorillonite, Salles et al. (Reference Salles, Douillard, Bildstein, El Ghazi, Prélot, Zajac and Van Damme2015) observed the diffusion coefficients of Li⁺ and Na⁺ increased with relative humidity (RH), and reached a plateau at high RH (around 80%) that were similar to that in bulk water. Conversely, the diffusion coefficients of K⁺ and Cs⁺ reached a maximum at low RH (20% and 10%, respectively), and then remained constant as RH increased. The change in diffusion coefficients suggested that Li⁺ and Na⁺ were easily hydrated in the interlayer space at high RH and adsorbed via outer-sphere adsorption, but K⁺ was hydrated with one water layer and adsorbed via inner-sphere adsorption, and Cs⁺ was barely hydrated and adsorbed via inner-sphere adsorption independent of RH (Salles et al. Reference Salles, Douillard, Bildstein, El Ghazi, Prélot, Zajac and Van Damme2015). The pK values obtained in this study (Table 4) and the XRD patterns in Figs 8 to 11 were consistent with the adsorption mechanisms of outer- and inner-sphere retention for these alkali cations proposed by Salles et al. (Reference Salles, Douillard, Bildstein, El Ghazi, Prélot, Zajac and Van Damme2015).