Figure 4A compares the calculated binding energies between the aluminosilicate dimer (i.e., [(OH)₃-Si-O-Al-(OH)₃]⁻¹) and monovalent cations/clusters, where we see negative binding energies for all cases, suggesting that their interactions are all energetically favorable. This favorability is expected because of the attractive Coulomb interaction between a negatively and a positively charged species. Figure 4B shows that there is a general increase in binding energy (i.e., becomes more negative) as the effective ionic radius of the corresponding monovalent cation decreases. Similar trends have been observed for the interactions of monovalent cations with different chemical species, including (i) guanine and 6-thioguanine tetrads (Deepa et al., 2011), (ii) glutathione (Liu et al., 2013), (iii) tetraoxa[8]circulene sheet (Karaush et al., 2015), (iv) cubane, cyclohexane and adamantane (Gopalsamy and Subramanian, 2014), and (v) methanol (Vayssilov et al., 2000), where the binding energy (also obtained from DFT calculations in referred studies) increases (i.e., becomes more negative) in the order of K⁺ < Na⁺ < Li⁺. Although these chemical species are very different from the aluminosilicate dimer/trimer, their interaction energies with monovalent cations are highly correlated with each other, as illustrated in Supplementary Figure S3 of Supporting Information. Supplementary Figure S3 shows that R ² values of 0.94–1.00 are achieved for linear regressions between binding energies obtained here and those reported in the literature for other chemical species (Vayssilov et al., 2000; Deepa et al., 2011; Liu et al., 2013; Gopalsamy and Subramanian, 2014; Karaush et al., 2015). The trend observed here (increasing interaction energy in the order of K⁺ < Na⁺ < Li⁺) can be attributed to the increase of bond strength (evidenced by bond order) in the order of K-O < Na-O < Li-O, as shown for alkali silicate-based glasses using ab initio calculations (Baral et al., 2017).

However, we do see considerable differences in binding energy between Li⁺ and Cu⁺, although their effective ionic radius is similar. This discrepancy means that the calculated binding energy of alkali with the aluminosilicate dimer cannot be solely explained by ionic radius. This discrepancy may be related to the difference in the absolute hardness of the two cations (35.1 and 6.3 eV for Li⁺ and Cu⁺, respectively) (Parr and Pearson, 1983), which is a measure of the resistance of the metal cation to lose electrons.

The higher binding energy of Cu⁺ with the aluminosilicate dimer compared with the other three alkalis M⁺ (i.e., Na⁺, K⁺, and Li⁺) could be used to explain the observed trend of Cu⁺⇌M⁺ exchange in alkali aluminosilicate glasses (i.e., 20M₂O-10Al₂O₃-70SiO₂) in an early study (Yoko et al., 1991), where the extent of Cu⁺⇌R⁺ exchange is seen to increase in the order of Li⁺ < Na⁺ < K⁺. The occurrence of this Cu⁺⇌R⁺ exchange can be partially attributed to the higher binding energy of Cu⁺ (or bond strength) with the negatively charged aluminosilicate network than the other three alkalis, which promotes the exchange. The trend in the extent of Cu⁺⇌R⁺ exchange (Li⁺ < Na⁺ < K⁺) can be attributed to the increasing difference in binding energy between Cu⁺ and R⁺ in the order of Li⁺ < Na⁺ < K⁺, which gives an exchange driving force in the order of Li⁺ < Na⁺ < K⁺.

Another observation from Figure 4A is that the binding energy of H₃O⁺ with the aluminosilicate dimer is seen to be higher than K⁺ and Na⁺, yet lower than Li⁺, which can be used to explain the observations in a recent experimental study on alkali aluminosilicate glasses (30M₂O-10Al₂O₃-60SiO₂) (Guo et al., 2014). Corrosion experiments in this study (Guo et al., 2014) showed no obvious H⁺ ⇌ Li⁺ exchange for the lithium aluminosilicate glass, suggesting that this glass is resistant to moisture attack, likely due to the higher binding energy of Li⁺ with the aluminosilicate network than H₃O⁺. In contrast, obvious H⁺ ⇌ Na⁺ and H⁺ ⇌ K⁺ exchange was observed in the sodium- and potassium-containing glasses to a hydrogen penetration depth of 0.4 and 3 μm, respectively, within the same experimental timeframe. This observation is likely due to the lower binding energies of Na⁺ and K⁺ (especially the latter, where the binding energy is the lowest among all monovalent cations considered here) with the aluminosilicate network compared to H₃O⁺, as illustrated in Figure 4A.

The binding energies of different divalent cations with the aluminosilicate dimer (i.e., [(OH)₃-Si-O-Al-(OH)₃]⁻¹) and trimer (i.e., [(OH)₃-Al-O-(OH)₂-Si-O-Al-(OH)₃]⁻²) are compared in Figure 5A, which shows that the interaction energy with the trimer is consistently higher than those with the dimer by ∼7.6–8.2 eV. This is expected given that the trimer has a higher negative charge (−2) than the dimer (−1) and hence exhibits stronger attractive Coulomb interactions with the positively charged divalent cations. A comparison of the energy values in Figures 4A and 5A shows that the divalent cations exhibit higher binding energies (∼14.5–22.5 eV) with the dimer than the monovalent cations (∼5.3–8.1 eV). This observation is consistent with previous DFT calculations on other chemical complexes (Deepa et al., 2011; Gatti et al., 2012; Liu et al., 2013; Gopalsamy and Subramanian, 2014; Karaush et al., 2015), where divalent cations (e.g., Ca²⁺ and Mg²⁺) are seen to exhibit higher binding energies than monovalent cations (e.g., K⁺, Na⁺, and Li⁺). Again, the energy values from this study are highly correlated with those reported in refs. (Deepa et al., 2011; Gatti et al., 2012; Liu et al., 2013; Gopalsamy and Subramanian, 2014; Karaush et al., 2015), regardless of the type of chemical complex that the metal cations interact with, as seen by the high R ² values (0.94–1.00) achieved with linear regressions in Supplementary Figure S3 of Supporting Information. These high degrees of correlation among binding energies obtained with different chemical complexes suggest that the differences in cationic binding energies for any given complex are mainly controlled by the inherent properties of the cations (e.g., field strength and ionic potential, as will be shown in the subsequent sections).

Similar to Figure 4B, we have examined the correlation between the binding energy values (Figure 5A) and effective ionic radii of the divalent cations in Figure 5B, where we see approximate inverse correlations. Nevertheless, the R ² values achieved with linear regressions (0.66–0.70, Figure 5B) are lower than those in Figure 4B for the monovalent cations (0.88). It is seen in Figure 5B that the calculated binding energy can be noticeably different even when the effective ionic radius is the same (e.g., Ni²⁺, Co²⁺, and Fe²⁺), similar to the case of Li⁺ and Cu⁺ in Figure 4B. Again, this discrepancy may be related to the difference in the absolute hardness of these divalent cations (e.g., 8.5 and 7.3 eV for Ni²⁺, and Fe²⁺, respectively) (Parr and Pearson, 1983), or the acid softness index defined by Xu et al. (e.g., −10.9, −13 and −21.87 kcal/mol for Ni²⁺, Co²⁺, and Fe²⁺, respectively) (Xu et al., 2017).

The binding energy results in Figure 5 can also be used to explain observations in many literature investigations. For example, MD simulations on ionic transport in C-A-S-H gel show that the diffusion coefficient of Ca²⁺ is over twenty times smaller than Na⁺ (Duque-Redondo et al., 2021). The slower diffusion of Ca²⁺, in this case, is likely due to its higher binding strength with the aluminosilicate network (as shown in Figures 4 and 5), which hinders the diffusion of Ca²⁺ in C-A-S-H gel channel (compared with Na⁺). Furthermore, recent investigations on the chemical durability of alkali/alkaline earth aluminoborate glasses show that the initial glass dissolution rate increases in the order of Mg²⁺ < Ca²⁺ < Li⁺ (Oey et al., 2019) and Mg²⁺ < Li⁺ < Na⁺ (Mascaraque et al., 2019). These observations can be attributed to the opposite trend in the ability of these cations to stabilize the negatively charged network and to hinder network dissolution/destruction in the order of Mg²⁺ > Ca²⁺ > Li⁺ > Na⁺ (as suggested by the binding energies in Figures 4 and 5). More recently, the addition of Mg²⁺ (Gevaudan et al., 2021) and Cu²⁺, and Co²⁺ (Gevaudan et al., 2019) in AAMs has been shown to reduce AAM leaching (especially leaching of Al species) in sulfuric acid. This observed reduction in dealumination in cation-doped AAMs in refs. (Gevaudan et al., 2019; Gevaudan et al., 2021) may be partially attributed to the higher binding strength of the doped Mg²⁺, Cu²⁺ and Co²⁺ cations that help better stabilize Al compared with Na⁺ and Ca²⁺ in reference samples (see the binding energy differences in Figures 4 and 5). This stronger Al stabilization effect of Mg²⁺, Cu²⁺ and Co²⁺ (than Na⁺ and Ca²⁺) may have two underlying mechanisms: (i) on the one hand, due to their higher binding strength, the doped divalent cations enhance the stability of the negatively charged aluminosilicate network in AAMs (i.e., the main binding gel in the investigated AAMs) upon acid attack, rending it more resistant to destruction; (ii) on the other hand, the doped cations may better stabilize a passivation layer (rich in Si and Al according to SEM-EDX analysis (Gevaudan et al., 2019)) at the reaction front that slows down diffusion of chemical species.

Figure 6 (and Supplementary Figure S4 of Supporting Information) compares (i) the binding energy of the aluminosilicate dimer (and trimer) with six divalent cations (M²⁺ = Ca²⁺, Mg²⁺, Zn²⁺, Fe²⁺, Co²⁺, and Ni²⁺) and (ii) the measured log dissolution rate of the corresponding M₂SiO₄ and MCO₃ minerals at different pHs and the log water exchange rate from the hydration sphere of the corresponding dissolved cation to the surrounding solvent (data extracted from refs. (Casey and Westrich, 1992; Pokrovsky and Schott, 2002)). Although the R ² values achieved using linear regressions are not high (∼0.43–0.56), they appear to be inversely correlated, with higher binding energy associated with generally lower dissolution and water exchange rates (Figure 6; Supplementary Figure S4). These inverse correlations may be due to the same underlying mechanism: a higher M-O bond strength leads to a higher binding energy in the case of the current study and a higher resistance to M-O bond-breaking for M₂SiO₄ and MCO₃ mineral dissolution and water exchange. Nevertheless, we note that the mineral dissolution process is highly complex, and measured dissolution rates may be influenced by other factors (Brantley et al., 2008) in addition to M-O bond strength.

The calculated binding energy values for the high valent cations (i.e., Co³⁺, Cr³⁺, Fe³⁺, Ti⁴⁺ and Cr⁶⁺) are presented in Figure 7, which shows higher interaction energies with the trimer than with the dimer, consistent with the divalent cations in Figure 5A. Furthermore, these binding energy values are considerably higher (i.e., more negative) than those for the monovalent and divalent cations given in Figures 4A and 5A, respectively. For example, the binding energies of Fe³⁺ (39.6 and 53.0 eV with the dimer and trimer, respectively) are considerably higher than those of Fe²⁺ (20.0 and 27.9 eV, Figure 5A). This may explain the higher strength of Fe³⁺-O bonds compared to Fe²⁺-O bonds (Gong and Olivetti, 2022) and the lower dissolution rate of Fe-containing minerals under oxidative conditions compared with reductive conditions, as has been reported in the literature (Brantley et al., 2008).

In fact, as seen in Figure 8A, there appears to be a general trend of increasing binding energy with both the aluminosilicate dimer and trimer as the charge of the cation/cluster increases (all the cations and cationic clusters containing OH⁻¹ have been included in the figure). These binding energy values are plotted in Figure 8B against the effective ionic radius of the corresponding cations, which shows that they are generally inversely correlated, as already seen in Figures 4B and 5B for monovalent and divalent cations, respectively. Due to these opposite correlations seen in Figures 8A, B, we have plotted in Figure 9A the binding energy values as a function of cation charge/ionic radii, which is defined as the ionic potential (IP) of the cation introduced by Cartledge (Cartledge, 1928) to describe to what extent the cations are electrostatically attracted by oppositely charged ions. Here, we used the effective ionic radii tabulated by Shannon (Shannon, 1976) to calculate the ionic potential of each studied cation (see the values in Table 2). It is clear from Figure 9A that the cationic binding energies with the aluminosilicate dimer/trimer are positively correlated with the IP of the cations, and their correlations can be accurately captured using 2nd order polynomial functions, as evidenced by the high R ² values achieved for regressions (0.99–1.00). Previous studies have also attempted to connect the IP of cations to their binding energies with organic species (e.g., glutathione (Liu et al., 2013) and calix[2]furano[2]pyrrole (Teixeira dos Santos and Magalhães, 2010)), as well as to other material properties (e.g., formation enthalpies (Wu et al., 2013; Sun et al., 2016) and cation discharge potential (Cartledge, 1928)).

In the glass community, one important term introduced by Dietzel (Dietzel, 1942) to characterize the effect of a single cation on oxide glasses is the cationic field strength ( F ) parameter, which is given by Eq. 2 (Vogel, 2012): F = Z C r C + r O 2 − 2 (2) where Z C is the charge of the cation; r C and r O 2 − are the ionic radii of the cation and oxygen anion, respectively. This field strength parameter has been widely used to classify ions as network modifiers ( F ≈ 0.1–0.4), network formers ( F ≈ 1.3–2.1), and intermediates ( F ≈ 0.5–1) (Vogel, 2012). Again, here we have used the effective ionic radii from Shannon (Shannon, 1976) to calculate the field strength of each cation (see the values in Table 2), where the results are presented in Figure 9B as a function of the calculated binding energies with the aluminosilicate dimer and trimer. The binding energy values are seen to be positively correlated with the cationic field strength, and the correlations can be well captured by 2nd order polynomial functions (R ² values of 0.99–1.00), similar to the case of the ionic potential (Figure 9A). In fact, as shown in Supplementary Figure S5, the ionic potential and field strength of the cations are positively and linearly correlated with an R ² value of 0.99 for a linear regression. In the glass literature, many studies have attempted to use cationic field strength to draw connections with the properties of silicate-based glasses, including effective ionic diffusion (Karlsson et al., 2017), glass transition temperature (Januchta et al., 2017; Atila et al., 2020; Lv et al., 2022), and network connectivity (e.g., Al and B coordination) (Januchta et al., 2017; Atila et al., 2020; Lv et al., 2022).

With these simple polynomial functions in Figure 9 (also given in Eqs. 3, 4) obtained using regression, we can now provide approximate estimates of the binding energies of different cations with the aluminosilicate dimer/trimer by using either the ionic potential or the field strength of the cations, where the ionic potential and field strength of the cations can be readily estimated from well-tabulated ionic radii available in the literature for a wide range of cations beyond those studied here (Shannon, 1976). For example, we can estimate binding energies of different cations with the aluminosilicate dimer, including cations beyond those included in the DFT calculations here (e.g., Rb⁺, Cs⁺, Be²⁺, Ba²⁺, Sr²⁺, Cd²⁺, Pb²⁺, and Al³⁺), by using Eqs. 3, 4. − E b = 0.66 × I P 2 + 4.95 × I P + 1.09 (3) − E b = 53.75 × F 2 + 9.83 × F + 4.11 (4)

The calculated cationic binding energies with the aluminosilicate dimer (values given in Supplementary Table S2 of Supporting Information) are plotted in Figure 10 against the corresponding cationic binding energies with different organic species reported in the literature (calix[2]furano[2]pyrrole (C₂₀N₂O₂H₉) (Teixeira dos Santos and Magalhães, 2010), β-cyclodextrin (C₄₂H₇₀O₃₅) (Stachowicz et al., 2011) and cucurbit[6]uril (C₃₆H₃₆N₂₄O₁₂) (Sinha and Sundararajan, 2014)), where the binding energies have been obtained using DFT calculations. It is clear from Figure 10 that the cationic binding energies estimated using Eqs. 3, 4 for the aluminosilicate dimer are positively and linearly correlated with the DFT-derived binding energies for all three types of organic species. In spite of the large difference in the types of interacting chemical complexes (see the atomic structures of the three organic species in refs. (Teixeira dos Santos and Magalhães, 2010; Stachowicz et al., 2011; Sinha and Sundararajan, 2014)), these linear correlations (R ² values of 0.95–0.99 for linear regressions) in Figure 10 suggest that the relative binding energy values estimated using Eqs. 3, 4 are reasonable, with the overall trend for a range of cations being correctly captured. Nevertheless, we also see that the estimated values from Eqs. 3, 4 are generally more than 30% higher than those from DFT calculations. This discrepancy may be partially attributed to the difference in the charge of the organic complex (neutral) compared with the Al-Si dimer (−1). This is supported by Figure 9, where the interaction energies with the more negatively charged trimer (−2) are consistently higher (∼30–50% higher) than the same cation with the dimer. The linear correlations in Figure 10 and Supplementary Figure S3 also demonstrate the governing effect that the ionic potential and field strength of cations have on the binding energies with a given chemical species.

Finally, we have reported the number of unpaired electrons (alpha spin–beta spin) in the DFT-optimized clusters (the explored configurations with the lowest energy) involving transition metal cations based on our Dmol³ calculations. The results, along with the dipole moments, are summarized in Supplementary Table S3 of Supporting Information for the reference of future studies.