The fractionation of isotopes is widely used in geology to interpret global processes such as CO2 cycle and weathering. The calculation of the isotopic fractionation factor at the equilibrium in minerals is now faithfully performed (, ) and gives insights on the mechanisms of fractionation.
However, in natural systems, most of equilibriums involve a liquid phase. But the fast growth of measurement methods (in complicated systems -biotic, liquid) has not been catched up by calculations. This is challenging because, for consistency, solutions and liquids have to be treated with the same method. The harmonic approx- imation that is commonly accepted to treat minerals was straightforwardly extended to liquids. Results on iron isotopes for the equilibrium of two species in a solution are promising but the values of the fractionation at the equilibrium between minerals and solution are unfortunately in disagreement with the experiment. It suggests that the error (due to the anharmonicity in liquids) cancels out for similar phases but can not be neglected for mineral/solution equilibriums.
The objective of my PhD project was to calculate realistic and consistent fractionation factors, which requires the use of advanced ab initio methods performed on HPC centers. Two approaches were chosen, depending on the kind of equilibrium. Concerning species in liquids, following the approach proposed in , we investigated within the harmonic approximation the eﬀect of the conﬁgurational disorder in liquids on the fractionation factor estimation . Then, we studied solution/mineral equilibriums with a consistent method that takes into account the eﬀect of anharmonicity (Path Integral Molecular Dynamics coupled with Thermodynamics Integration). Eﬀorts have been done to reduce the important computational resources required by this method.
Conﬁgurational disorder in solutions The fractionation factor can be computed as an average over indepen- dent conﬁgurations extracted from an ergodic trajectory. In former work on the fractionation factor, only one conﬁguration was considered. In this study, we investigated the viability of this approximation and proposed a more eﬃcient sampling that is based on the study of the liquids structural properties and on its impact of the fractionation properties.
Trajectories of H4SiO4, H3SiO− 4 or H2SiO2− 4 in water were simulated within the framework of Car-Parrinello MD (with the CPMD package with Troullier-Martins pseudopotentials and the BLYP functional). Analysis codes were written to study the structural liquid properties.
The method that was developed is based on the work of Rustad and Bylaska: Several conﬁgurations are relaxed (with the pw-scf code Quantum Espresso); The vibrational spectra is computed using Density Func- tional Perturbation Theory (for liquids the Γ-point approximation has been tested); Finally the fractionation factor is calculated at the temperature of the dynamics as detailed in Méheut et al..
A ﬁne study of the vibrational properties of diﬀerent conﬁgurations has been done in order to point out the modes that contribute the most to the fractionation properties. We also investigated the correlation between the structural properties of liquids and the calculated value of the fractionation factor with analytical in-house codes (i.e. statistics on the number of water molecules in the ﬁrst solvation layer, bond-length distribution or size of the water rings around the silicon atom).
Apart from liquids, we also computed the Quartz and Kaolinite minerals within the harmonic approxima- tion with the same pseudopotentials as for the dynamics. Quartz is used as a reference to compare the results to experiments and to other calculations performed with the PBE functional.
Two publications on this study will be submitted for publication soon: One regarding the employed method- ology and the second one on the geological implications of the obtained results.
Anharmonicity eﬀects in solution
Since the fractionation factor between a solution and a mineral is impor- tant in geology (i.e. for precipitations, dissolutions and adsorptions), our objective was to consider a consistent method for both phases which takes into account quantum anharmonic eﬀects. A method of choice is the Path Integral Molecular Dynamics coupled with Thermodynamics Integration. The fractionation factor between Li+ in water and Li2O was studied. For comparison with experiments, clays have also been computed within the harmonic approximation.
For this study, we closely collaborated with the group of research of Mark. E. Tuckerman, which is a spe- cialist of PIMD, in order to develop this method and optimize our calculations on geological systems. This collaboration has resulted in the publication of the developed methodology in J. Chem. Theory Comput. .
The test of this method on ”real” systems has ﬁrst been done with empirical potentials in order to have an estimation of the best parameters for the path-integrals dynamics and to observe the diﬀerences between anharmonic (performed with PINY package) and harmonic calculations (with Gulp and PINY). At the same time, we studied the reliability of the potentials. We compared the vibrational properties, which drive the fractionation, with experimental data and with ab initio calculations. We observed an important eﬀect due to anharmonicity in solution whereas it is negligible in Li2O, and the obtained fractionation factors are in very good agreement with the experiments.
The ongoing part of this study consists in ab initio calculations on similar systems (with anharmonic methods) and on more complex Li-containing minerals (with harmonic methods) in order to compare the results more precisely with experiments. The objective is to ﬁnally have a consistent and realistic calculation of the fraction- ation factor between a mineral and a solution which would be a world ﬁrst.
This work will result in the publication of two additional articles which are currently in preparation.
During this project, I worked with specialists of several methods for applications to realistic systems in the ﬁeld of geology. Some of these methods are widely used in large ﬁelds such as physics or biology (namely DFT, MD and CPMD) and PIMD is an attractive method that allows the calculation of very ﬁne quantum eﬀects. The main objective of my PhD thesis, which was to develop a method to compute the fractionation factor in complicated systems involving liquid phases, was successfully achieved. The results were presented at the Goldschmidt conference in 2013 and grew interest for geologists, for instance in order to understand the fractionation properties of silicon at extreme pH. More developments are now foreseen to improve the methods in order to have even more realistic results.
 Opfergelt S. and Delmelle P. (2012) silicon isotopes and continental weathering processes: Assessing controls on Si transfer to the ocean. C R Geosci. 344, 723–738.
 Schauble E. A. (2011) First-principles estimates of equilibrium magnesium isotope fractionation in silicate, oxide, carbonate and hexaaquamagnesium(2+) crystals. Geochim. Cosmochim. Acta 75, 844 – 869.
 M´eheut M., Lazzeri M., Balan E. and Mauri F. (2007) Equilibrium isotopic fractionation in the kaolinite, quartz, water system: Prediction from ﬁrst-principles density-functional theory. Geochim. Cosmochim. Acta 71, 3170 – 3181.
 Beard B. L., Handler R. M., Scherer M. M., Wu L., Czaja A. D., Heimann A. and Johnson C. M. (2010) Iron isotope fractionation between aqueous ferrous iron and goethite. Earth Planet. Sci. Lett. 295, 241 – 250.
 Rustad J. R. and Bylaska E. J. (2007) Ab Initio Calculation of Isotopic Fractionation in B(OH)3(aq) and BOH4-(aq). J. Am. Chem. Soc. 129, 2222–2223.
 Ceriotti M. and Markland T. E. (2013) Eﬃcient methods and practical guidelines for simulating isotope eﬀects. J. Chem. Phys. 138, 014112
 Vigier N., Decarreau A., Millot R., Carignan J., Petit S. and France-Lanord C. (2008) Quantifying Li isotope fractionation during smectite formation and implications for the Li cycle. Geochim. Cosmochim. Acta 72, 780 – 792.
 Marsalek O., Chen P-Y, Dupuis R., Benoit M., M´eheut M., Baˇci´c Z. and Mark E. Tuckerman (2014) Eﬃcient Calculation of Free Energy Diﬀerences Associated with Isotopic Substitution Using Path-Integral Molecular Dynamics. J. Chem. Theory Comput. 10 (4), 1440-1453
 Prabhatasree G., Choudhury N. and Chaplot S. L. (2004) Lattice dynamics of lithium oxide. Pramana journal of physics 63, 409-412
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