An S_N2 reaction mechanism for hydrolysis of siloxane linkages and the implications for dissolution of quartz

Abstract

Density functional theory calculations were performed on an extended silicate molecular cluster to test the hypothesis that inclusion of Cl⁻ to charge-balance protonation of bridging O atoms in siloxane bridges, and H₂O attack via an S_N2 mechanism, could explain the observed activation energy of quartz dissolution under acidic conditions. The results are consistent with this hypothesis and help explain the observed solution pH and salt-dependence of quartz dissolution rates.

Introduction

Dissolution of silica and silicates is a topic that relates to biogeochemical cycles^1,2, soil productivity^3,4, corrosion of man-made materials⁵, and geothermal energy extraction^6–10. Consequently, myriad field, laboratory, and simulation studies have been published on the rates and mechanisms of silica and silicate dissolution. However, field-laboratory^11–14, laboratory^15–20, field-laboratory-simulation studies^21–23, and simulation studies^24–41 result in significant discrepancies in rates.

There is no consensus molecular mechanism for quartz dissolution. The current scientific hypothesis is that hydrolysis of Si-O_br-Si linkages (O_br for “bridging O atom” hereafter) is the rate-limiting step. No other reaction has been posited that can transform SiO₄-tetrahedra with 4 Si- O_br-Si linkages in the solid into the aqueous species Si(OH)₄. This study focuses on modeling the Si- O_br-Si hydrolysis reaction with the goal of determining a reaction pathway consistent with observed activation energies (ΔE_a) of silica dissolution. With such a reaction pathway determined for realistic models of surface SiO₄-tetrahedra, we posit that it will become possible to predict rates of hydrolysis and dissolution over a range of temperatures and solution compositions for use in weathering where rates are slow and subsurface reactions at higher T and ionic strength solutions.

Field and laboratory experiments related to silica weathering or dissolution are too numerous to cover so we reference a recent review that covers the literature on this topic²³. We cite three key papers that provide reliable ΔE_a values, however, as these observables are the main targets of our simulation efforts. Dove (1999) performed dissolution experiments on quartz at pH ≈5.7 over a temperature range of 175 to 295 °C in distilled water and with chloride salts at 0.04 to 0.15 M to derive ΔE_a values of 71 to 75 kJ·mol^{− 1}.¹⁴ Icenhower and Dove (2000) studied fused silica dissolution rates from 25 to 250 °C in distilled and 0.05 M NaCl solutions and determined ΔE_a to be 74.5 ± 1.4 kJ·mol^{− 1}.²⁰ Crundwell (2017) analyzed quartz dissolution data and concluded the ΔE_a = 70 kJ·mol^{− 1}.²² Because simulations do not represent all potential steps in the dissolution process, we consider these values to be upper limits for the acceptable ΔE_a values determined in our DFT calculations.

Quantum mechanical studies of Si-O-Si hydrolysis and silica dissolution began with Casey et al. (1990) who determined that H⁺-transfer to the O_br was not the rate-limiting step because modeled H-D kinetic isotopic effects were significantly different in experiment and calculation³⁰. These types of studies continued for three decades, but the results were either based on models that did not represent silica surface chemistry (e.g., the molecule H₃SiOSiH₃) or resulted in activation energies significantly higher than observation (e.g., Criscenti et al., 2006, where ΔE_a = 112 kJ·mol^{− 1}).³² More recent studies have used machine-learning methods to derive ΔE_avalues from experimental data²¹, and kinetic Monte Carlo techniques based on observed ΔE_a values to predict behavior consistent (i.e., etch pit formation) with experimental dissolution rates⁴⁰. Periodic DFT simulations have focused more on the quartz-water interface as a function of solution composition^25,27,28,41, and these studies have indicated changes in the intrasurface H-bonding (i.e., H-bonding between SiOH and O_br sites) that could explain the observed salt-dependence of silica dissolution rates¹⁴.

Periodic DFT simulations have revealed that association of the Cl⁻ with the quartz surface is probable and that this near approach to the surface can help lower the calculated energy of a H⁺ attached to an O_br. Previous work had not examined this potential ion-pairing of Cl⁻ and H⁺ at the silica surface, but lower dielectric constants at mineral-water interfaces are likely to help stabilize such ion pairing⁴². This paper focuses on such ion-paired structures and the subsequent hydrolysis reaction pathway. In addition, simulations predicted that rather than direct attack of H₂O from solution to form a 5-coordinate surface Si (^[5]Si), H₂O could diffuse just below the silica surface and form the ^[5]Si from an S_N2 mechanism. The hypothesized reaction mechanism is shown in Fig. 1. These two variations from previous modeling studies result in a significant lowering of the calculated ΔE_a which is more consistent with observed values.

Fig. 1.

Hypothesized reaction mechanism. From left to right, a lone pair of electrons from the bridging O bonds to an H⁺, and a lone pair of electrons from H₂O bonds beneath a Q¹ Si to form the middle structure that will lead to a transitions state. Finally, a lone pair of electrons from the bridging O^−[5]Si bond in the middle structure moves to the bridging O to form the products in the right structure. The bonds on the left-most Si in each model show the generic bonds of a Si to the remainder of the silica model.

Results and discussion

Figure 2 shows the quartz cluster model used for this work. The model was chosen to represent a general silicate that exhibited a Q¹ Si, where we hypothesize that dissolution can occur with hydration of that Si from below and breaking of the Si-O_br bond, while a Cl⁻ stabilizes the H⁺ on O_br. After energy minimizing the model, the Si-O bond distances were 1.64 ± 0.02 Å and the Si-O-Si bond angles were 134.4 ± 7.3º, which agree with experimental data for those measurements of 1.61 to 1.62 Å for Si-O bond lengths in α-quartz⁴³, and 144º for Si-O-Si angles in α-quartz⁴⁴. Our Si-O-Si bond angle average is less than that for α-quartz; however, in lower-pressure crystalline silicate minerals and glasses, the Si-O-Si angle can range from 134 to 150º, which would include our calculated results⁴⁵.

Fig. 2.

The labeled and enlarged spheres in this model show the areas of interest discussed in this work. The model above shows a pentavalent Q¹ Si bonded to three OH⁻, one H₂O, and a O_br between the Q¹ Si of interest and a Q⁴ Si. A Cl⁻ is adjacent to the bridging H that is bonded to the O_br. Atom colors are white for H, red for O, yellow for Si, and green for Cl.

Figure 3 shows the relative electronic energy (E) compared with the lowest energy model configuration (Fig. 3a), the change in the distance between O_br and H_br (Fig. 3b), and the distance between the H_br on the O_br and Cl⁻ (Fig. 3c), versus the distance between O_br and the Q¹ Si. The Si(OH)₃(H₂O) – O_br distance was held constant during the energy minimization calculations, except for the first (1.76 Å) and last (3.15 Å) calculations, which were minimized with no constraints.

Fig. 3.

Relative electronic energy (E) (a), O_br-H_br bond length (b), and the H_br-Cl⁻ distance as the bond between O(H)_br and Si(OH)₃(OH₂) of the model increases and breaks (c).

The plot in Fig. 3a shows the trend in the energy change relative to the final model as the Si(OH)₃(H₂O) – O_br bond distance increases, breaks, and dissociates. The initial model had an energy that was 14 kJ·mol^− 1 higher than the lowest energy (final) model. The model with the Si(OH)₃(H₂O) – O_br distance of 2.21 Å had a relative energy of 56 kJ·mol^− 1 higher than the energy of the final model (at 0 kJ·mol^− 1); the higher energy model also had one imaginary (negative) vibrational frequency of −66 cm^− 1 for the Si(OH)₃(H₂O) – O_br vibrational mode, making it the transition state for the reaction and provide the ΔE_a. The ΔE_aresult is consistent with the experimental results from the literature that we cited in the introduction^14,20,22 and does not exceed those results dramatically like prior calculated results from the literature.

In addition, a recent paper fit data obtained from 285 quartz and silica dissolution experiments and found an ΔE_a of 70 kJ·mol^− 1,²² which agrees well with our result, suggesting that dissolution mechanism presented here could be prominent. We posit that the flexibility of the reaction is greater in model clusters compared to real surfaces where lattice constraints could increase the ΔE_a.

Modeling periodic quartz surfaces using the proposed reaction mechanism would be a logical step to test this hypothesis; however, a prior study showed that the cluster structures such as those used here can give similar results to condensed phases used in periodic model calculations⁴⁶. Using a small molecular model, (OH)₃Si-O-Si(OH)₃, Gibbs’ et al., (2006) calculated bond lengths, bond angles, bond critical points, local potentials, and kinetic energy densities that were similar to quartz⁴⁶. (OH)₃Si-O-Si(OH)₃ and quartz also showed close similarity of electron density distribution between their model and quartz and showed that Si-O bond energies were similar between (OH)₃Si-O-Si(OH)₃ (462 kJ·mol^{− 1}) and quartz (465 kJ·mol^{− 1})⁴⁶. Therefore, we agree with those authors’ assertion that these similarities justify the use of small molecular models to act as proxies for quartz polymorphs.

We also ran single-point energy calculations using implicit solvation to test the effect on the calculated activation and reaction energies on the initial, transition state, and final structures. The results show a −5 kJ/mol lower ΔE_a(51 vs. 56 kJ/mol) and the reaction energy difference is +5 (19 vs. 14 kJ/mol). We consider these differences to be insignificant as we do not claim accuracy of better than ± 10 kJ/mol in our calculations. Replicate DFT calculations would give 0 kJ/mol differences, but those results could differ from experiment by as much as 10 kJ/mol. Furthermore, the implicit solvation calculations were performed with a permittivity (ε) of 78.4 for bulk water, but the dielectric constant at the interface is likely to be significantly lower²³, so the effect of long-range solvation on reaction energetics would be less.

Figure 3b shows that as the Si(OH)₃(H₂O) – O_br distance increases, the O_br-H_br bond length decreases, making the latter bond stronger and more stable. The O_br-H_br bond length decreased by nearly 0.2 Å due to Si(OH)₃(H₂O) dissolution.

As the O_br-H_br bond length decreases, the distance between the H_br and Cl⁻ increases (Fig. 3c) by nearly 0.3 Å as the Si(OH)₃(H₂O) group dissociates. This result suggests that Cl⁻ is shielding the H_br from attack by H₂O to form H₃O⁺, as well as stabilizing and lowering the energy of the cluster. As the Si(OH)₃(H₂O) dissociates and the O_br-H_br distance decreases, the Cl⁻ moves away from the H_br.

To evaluate this potential phenomenon, we also calculated the effect of Cl⁻ proximity to the H_br, using the two models shown in Fig. 4. We compared that model in Fig. 4a, which is the same model shown in Fig. 2, with the model shown in Fig. 4b, where we moved the Cl⁻ away from the H_br, so that no interaction could occur between the H_br and Cl⁻. After energy minimizing the structure in Fig. 4b, the E for that model was 196 kJ·mol^− 1 higher in free energy than the minimized model in Fig. 4a. The distance between the H⁺ on the O_br and Cl^- in the model in Fig. 4a is 1.78 Å and 3.11 Å in Fig. 4b model. Therefore, Cl⁻ interacting with the proton on the O_br is lowering the model energy by stabilizing the proton on the O_br. Coupled with the results showing that Cl⁻ moves away from H_br as the Si(OH)₃(H₂O) group dissociates, the results suggest that Cl^- stabilizes the H⁺ on O_br and is necessary for our proposed mechanism.

Fig. 4.

(a) is the same the model in Fig. 1, whereas in (b), the Cl⁻ was moved far from the protonated O_brH_br. The hydrating H₂O molecules were also moved away from O_brH_br, because without the Cl⁻ near the H_br, H₂O deprotonates O_brH_br to form H₃O⁺.

The calculated rate constants, k_r, shown in Table 1 were obtained using Eq. (1) and are compared with a select calculated result³³. We calculated the rate constants at 298.15 K and 1.01 bar and at 468.15 K and 206 bar; the latter were to make the result pertinent to well UO_FU_5832_PT from U.S. Department of Energy (US DOE) Forge (Frontier Observatory for Research in Geothermal Energy) from a sampling depth of 7450 m. Our k_r result at 298.15 K suggests that the reaction we hypothesize would proceed 1000 times slower than that using the calculation from the literature³³.; that work used a model that was smaller and charged (Si₂O₈H₉⁺), whereas our model was larger and neutral (Si₈O₃₁H₃₁Cl). These differences, coupled with the lack of Cl⁻ counter ion for charge balance and stability in the other work, could account for the differences in model results.

Table 1.

Rate constant, k_r, results from this work and calculations performed in reference 44*. The Q results for the reactant model (Q) and transition state model (Q^‡) are from the Gaussian 16 log file results at the temperatures shown. Reference 44 did not report Q results.

Temperature (K)	kr (s− 1)	Q	Q‡
298.15	9.2 × 105	1.98 × 1043	1.86 × 1046
468.15	1.2 × 1010	1.30 × 1055	2.61 × 1058
298*	8.6 × 108*

There is a large discrepancy between our results and model predictions reported in the literature (Table 1)³³; our proposed mechanism where H₂O attacks a Q¹ Si by S_N2 attack coupled with Cl⁻ stabilizing a proton on the adjacent O_br is different that prior mechanism, so we expect our results to differ from those of previous work that used a H₆Si₂O₇ + H₃O⁺ model. Furthermore, the previous work assumed the protonated O_br as the reactant when this state is significantly higher in energy than the non-protonated state. Good agreement between our ΔE_a results and experimental data^14,22 suggest that our results are more realistic.

Using our calculated rate constants shown in Table 1, a surface site density of 8 O_br per nm², and Eq. 2, the rates were calculated at 298.15 and 498.15 K (Table 2). As discussed in the methods section, the % O_br as O_brH site density was based on XPS data⁴⁷.

Table 2.

Rate of dissolution reaction as a function of H⁺ site density percentage (pH).

pH	% Obr as ObrH	T 298 K	T 468 K
		Rate (Obr sites·nm− 2·s− 1))	Rate (Obr sites·nm− 2·s− 1))
1	7.29	5.35 × 105	6.75 × 109
2	5.14	3.78 × 105	4.77 × 109
3	3.93	2.88 × 105	3.64 × 109
4	3.21	2.36 × 105	2.98 × 109
5	2.86	2.10 × 105	2.65 × 109
6	2.14	1.57 × 105	1.99 × 109

The results in Table 2 show that as the % O_br as O_brH decreased (pH increase), the rate of the reaction decreased at both temperatures, as expected for a mechanism that would occur under acidic conditions. The rates are 468.15 K (i.e., geothermal conditions) are much greater than the rates at 298.15 K.

Conclusion

This work has shown that silicate dissolution under acidic conditions could occur at Q¹ Si atoms by S_N2 attack, if H₂O attacks from beneath the Si atom to form a ^[5]Si. Simultaneously, the O_br adjacent to the Q¹ Si would be protonated under acidic conditions, and the H⁺ on the O_br would be stabilized by an anion, such as Cl⁻. The calculated energy of activation of 56 kJ·moL^− 1 for this work agrees more accurately with data from the literature than prior computational study results have. We reported rate constant and rate results at 298.15 and 498.15 K for the calculated E_a result, which could be used to guide experiments and to determine if our proposed mechanism is occurring. Our results show that as pH increases, the rate of dissolution by the proposed S_N2 mechanism decreases at both temperatures, as would be expected for a H⁺-driven mechanism.

Methods

Models were built using Materials Studio 2016 (Biovia, Inc.; San Diego, CA - https://www.3ds.com/products/biovia/materials-studio) and energy minimized using Gaussian 16 (ES64L-G16RevB.01 http://www.gaussian.com⁴⁸. The model stoichiometry was Si₈O₃₁ H₃₁Cl. The initial structure for the cluster was extracted from an unpublished periodic DFT simulation. We added 5 H₂O molecules to our model to hydrate and stabilize the Cl⁻ that was added to stabilize a proton that was added to one of the bridging O atoms (O_br). Although the small silica model that we used might not be able to mimic the chemistry of quartz, earlier authors found that small models can capture quartz chemistry accurately⁴⁶. Our goal for this work is to explore a reaction mechanism using a silica model where H₂O attacks a Q¹ Si from below and breaks away from a protonated O_br to form Si(OH)₃(H₂O). If activation energy results from this work match those from experiment, this method could then be applied to larger silica models and periodic models that better model quartz.

Energy minimization calculations were performed using density functional theory (DFT)^49,50 with the M06-2X functional⁵¹. We chose the M06-2X functional due to its ability predict benchmark level (CCSD/6–31 + G(d)) geometries for the transition states structures of Diels Alder reactions⁵². The 6-311G(d, p) basis set was used for all calculations⁵³. For the energy minimization calculations, a fine CPHF grid⁴⁹, and ultrafine integration grid⁴⁸ were used. The models were first geometry optimized then subjected to frequency calculation to confirm that each model had reached a potential energy minimum and that each model exhibited only real (> 0 cm^{− 1}) vibrational frequencies.

Single-point energy calculations were run with the SMD implicit solvation model on the reactants⁵⁵, transition state and product structures to evaluate the potential effect of long-range solvation on the energies. We note that a permittivity (ε) of 78.4 for bulk water was used although the value of ε is likely to be significantly lower at the quartz-water interface. Hence, these results will represent the maximum potential impact of long-range solvation on our results.

For the stepwise dissolution calculations, the minimized structure in Fig. 1 was modified by incrementally increasing the distance between the Q¹ Si and the O_br in that model in ca. 0.1 Å steps until the Si(OH)₃(OH₂) group was dissociated from the cluster. For each of the stepwise calculations, the Si-O_br distance was held constant by freezing those atoms and the distance between them using the “−1” method in Gaussian 16; all other atoms were minimized without constraints. The end member models of the stepwise calculations were energy minimized without constraining the distance between the O_br and the Q¹ Si. The transition state model underwent a frequency calculation without constraints. This model was used to find the energy of activation (E_a) of the dissolution and had one and only one imaginary vibrational frequency.

For the stepwise calculations, we report electronic energies (E) that were relative to the E of the model having the fully dissociated Si(OH)₃(OH₂); that is, the E of that model was set to 0 kJ·mol^− 1 and E of the other stepwise dissociation models were calculated relative to that model.

Rate constants were calculated using Eq. 1⁵⁶:

1

Here, k_r is the rate constant, k_B is the Boltzmann constant, h is the Planck constant, T is the temperature, and ΔE_a is the change in energy between the minimized reactant (initial) structure and the minimized transition state structure. Q^‡ and Q are the v = 0 total partition functions of the transition state and reactant structures, respectively. obtained from Gaussian 16.

Reaction rates were used by using the rate law in Eq. (2):

2

Here, k_r is the rate constant, O_br site density is estimated as 8 O_br sites per nm²from the quartz (101) model upon which the model in this work was based^25,41, and % O_br as O_brH was estimated using a plot of SiOH₂⁺ surface site density percentage versus pH based on X-ray photoelectron spectroscopy (XPS) data⁴⁷. Previous work showed that the proton affinity of SiOH₂⁺ is approximately equal to that of O_brH surface density^27,56, so we halved the site density present on the plot from the literature (based on SiOH₂⁺) to obtain percentage of H⁺ site density for Eq. 2.

The bond distance for O_br-H_br and the distance between H_br and Cl⁻ were obtained from each of the stepwise, energy-minimized models using the bond length using Materials Studio 2016.

Author contributions

JDK and HDW designed the study. HDW performed the calculations and wrote the first draft of the manuscript Methods and Results and Discussion sections. JDK wrote the first draft of the Introduction. JDK and HDW edited the manuscript.

Funding

This work was funded by the U.S. Department of Energy, Office of Science, Energy Earthshots Initiatives under Award Number DE-SC0024703.

Data availability

Data is provided within the manuscript.

Declarations

Competing interests

The authors declare no competing interests.