Modelling hydrogen mobility in forsterite as diffusion coupled to inter-site reaction

Abstract

Whilst the diffusivity of hydrogen (H) in forsterite (Mg₂SiO₄) has been extensively studied, there still remain some puzzling observations. Firstly, experiments measuring ostensibly the same process have provided different results in terms of diffusion coefficients. Secondly, despite H diffusion in forsterite generally being associated with diffusion of M-vacancies charge compensated by 2H⁺, a plethora of H-bearing point defects have been observed, including those associated with Si vacancies, trivalent cations and tetravalent cations in the form of so-called ‘clinohumite-type’ point defects. This has been tentatively associated with some form of inter-site reaction, such as one in which a M-site vacancy associated with 2H⁺ reacts with tetrahedrally coordinated Ti⁴⁺. Equivalent reactions can be constructed to form all point defects mentioned above. Here, we present a series of numerical models in which these processes are simulated. In the models, the mobility of H is described using a diffusion coefficient (D*) for the hydrogenated M-site vacancies and an equilibrium constant (K) for the relevant inter-site reaction(s). Reevaluation of published data shows that the extracted D* and K values are consistent between some different datasets, even in situations where the phenomenological (chemical) diffusion coefficient $\tilde{D}$, extracted simply using solutions of Fick’s second law, did not agree. The ‘true’ mobility (D*) of the M-site vacancy associated with 2H⁺ must be between 1 and 2 orders of magnitude greater than previously determined $\tilde{D}$ in order to form measurable profiles of all point defects observed by vibrational spectroscopy. Density functional theory calculations of the K of each of the inter-site reactions implemented in our model show good agreement (within an order of magnitude) with those determined experimentally for the reactions forming ‘clinohumite-type’ defects, but considerable disagreement (~ 3 orders of magnitude) for the defects involving trivalent cations, potentially due to assumptions related to binding of different components within individual defects. Overall, the first-order implication is that H diffusion profiles that we observe in natural and experimental samples are unlikely to be formed by simple diffusion alone. These models provide a new methodological framework for further understanding of complex ionic diffusion mechanisms in olivine and the other nominally anhydrous minerals.