Contribution of interstitial OH groups to the incorporation of water in forsterite

Abstract

Water incorporation in forsterite samples synthesized under low to medium silica-activity conditions mostly occurs via a substitutional mechanism in which a Si vacancy is compensated by four protons. Corresponding IR absorption spectra display a cluster of narrow and weakly anharmonic OH-stretching bands at wavenumbers above 3,500 cm⁻¹. However, this diagnostic spectrum is often superimposed to one broader absorption band, rarely two, displaying pronounced temperature-dependent properties and tentatively assigned to H atoms in interstitial position (Ingrin et al. in Phys Chem Miner 40:499–510, 2013). Here, we investigate the structural and vibrational properties of selected interstitial H-bearing defects in forsterite using a first-principles modeling approach. We show that the broad bands discussed by Ingrin et al. (Phys Chem Miner 40:499–510, 2013) are most likely related to interstitial OH groups in the vacant octahedral sites alternating with the M2 sites along the c axis of the forsterite structure. The corresponding OH defects lead to the formation of fivefold coordinated Si species. Their peculiar thermal properties stem from the vibrational phase relaxation due to the anharmonic coupling of the high-energy local OH-stretching mode with a low-energy vibrational mode. This “exchange mode” corresponds to the hindered longitudinal translation of the OH group. These results suggest that at high pressure, hydrogen incorporation in forsterite is dominated by coexisting interstitial OH groups and (4H)_Si defects.

Appendix: Computation of anharmonic coupling parameters

We calculate the anharmonic coupling coefficients between the OH-stretching mode and the other vibrational modes of the defective forsterite. This is done by studying the variations of the dynamical matrix coefficients when the atoms are displaced out of their equilibrium position along the normal coordinates of the OH-stretching mode.

Assuming that the exchange mode is localized in the same region as the interstitial OH group, only vibrational properties at the Brillouin zone center are considered. In this case, the coefficients of the dynamical matrix are defined by:

$$D_{ij\alpha \beta } = \frac{1}{{\sqrt {m_{i} } \sqrt {m_{j} } }}\frac{{\partial^{2} E_{\text{tot}} }}{{\partial x_{i\alpha } \partial x_{j\beta } }} ,$$

(6)

where m _i and x _iα stand for the mass of atom i and the displacement of atom i along the Cartesian coordinate α, respectively. Let us call Z _liα the orthonormal eigenvectors of the dynamical matrix. The Z _liα are related to the normal coordinates Q _liα of the vibration mode l by $\tilde{m}_{l}$ Z _liα  = (m _i )^1/2 Q _liα . The eigenvalues ω ² _l of the dynamical matrix can be written as:

$$\omega_{l}^{2} = \left\langle {Z_{l} } \right|D\left| {Z_{l} } \right\rangle ,$$

(7)

where the bra-ket products indicate the sum over the i and α indexes.

They are related to the derivatives of the total energy by:

$$\omega_{l}^{2} = \frac{1}{{\tilde{m}_{l} }}\frac{{\partial^{2} E_{tot} }}{{\partial Q_{l}^{2} }} ,$$

(8)

where $\tilde{m}_{l}$ is the effective mass of mode l. One can then obtain

$$\frac{{4!C_{1122} }}{{\tilde{m}_{\text{ex}} }} = \frac{{\partial^{2} \left\langle {Z_{\text{ex}} } \right|D\left| {Z_{\text{ex}} } \right\rangle }}{{\partial Q_{\text{LVM}}^{2} }} = \left\langle {Z_{\text{ex}} \left| {\frac{{\partial^{2} D}}{{\partial Q_{\text{LVM}}^{2} }}} \right|Z_{\text{ex}} } \right\rangle$$

(9)

Thus, from Eq. (9), the C ₁₁₂₂ coefficients can be obtained from the second-order derivatives of the dynamical matrix with respect to atomic displacements along the normal coordinates of the LVM. To keep the computation time under a reasonable limit, an approximate approach is used to describe the coupling between the OH-stretching LVM and the other vibrational modes of the system. In this approach, it is assumed that the mode–mode dominant coupling terms are related to atoms in a close proximity to the OH defect.

The full dynamical matrix of the OH_ia supercell is computed first, leading to 342 vibrational modes. The corresponding OH-stretching wavenumber is 3,585 cm⁻¹, in excellent agreement with the result of the calculation restricted to the OH group only (3,583 cm⁻¹). A reduced-size model of the defect (30 atoms per unit-cell) is then built using the primitive unit-cell of forsterite. This smaller model preserves the overall geometry of the defect, with the occurrence of a fivefold Si species and the OH group pointing to the neighboring O2 atom. The O_i–H bond length is almost unchanged at 0.977 Å. The Si–O_i and Mg–O_i distances increase by ~1 % to 1.72 and 1.94 Å, respectively, whereas the O_i(H)…O2 distance shortens to 2.70 Å. Consistently, a slight decrease in the OH-stretching wavenumber to 3,540 cm⁻¹ is observed.

The second-order derivatives of the dynamical matrix coefficients with respect to atomic displacements along the LVM normal coordinates are then obtained by calculating the dynamical matrix of the small system for four finite atomic displacements λ Q _OH along the OH-stretching mode normal coordinates, with λ = − 0.01, −0.005, +0.005, +0.01. Under the assumption that the second-order derivatives of the dynamical matrix coefficients can be transferred from the small to the large system, the mode–mode coupling in the large system is described by considering the displacement of these 30 atoms only. In other words, the projection in Eq. (9) is performed using the $\frac{{\partial^{2} D_{ij\alpha \beta } }}{{\partial Q_{\text{LVM}}^{2} }}$ coefficients computed on the small system and the atomic displacements Z _ex corresponding to the full dynamical matrix. The δω parameters are then obtained using Eq. (5).