M Roskosz

E Deloule

J Ingrin

C Depecker

D Laporte

S Merkel

L Remusat

H Leroux

Kinetic D/H fractionation during hydration and dehydration of silicate glasses, melts and nominally anhydrous minerals

The distribution of hydrogen isotopes during diffusion-driven aqueous processes in

silicate glasses, melts and crystals was investigated. Hydration/dehydration experiments were performed on silica glasses at 1000°C and 1 bar total pressure. Dehydration triggered by decompression-driven bubble nucleation and growth was performed on rhyolitic melts at 800°C and a few hundred MPa. Hydrogen extraction from a nominally anhydrous mineral (grossular) single crystal was carried out at 800°C and ambient pressure. After these three series of experiments, pronounced water (sensu lato) concentration profiles were observed in all recovered samples. In the grossular single-crystal, a large spatial variation in H isotopes (D variation > 550‰) was measured across the sample. This isotopic distribution correlates with the hydrogen extraction profile. The fit to the data suggests an extreme decoupling between hydrogen and deuterium diffusion coefficients (D H and D D respectively), akin to the decoupling expected in a dilute ideal gas (D H /D D 1.41). Conversely, no measurable spatially-and time-resolved isotopic variations were measured in silicate glasses and melts. This contrasted behavior of hydrogen isotopes likely stands in the different water speciation and solution mechanisms in the three different materials. Glasses and melts contain essentially hydroxyl and molecular water groups but the mobile species is molecular water in both cases. Protonated defects make up most of the water accommodated in grossular and other nominally anhydrous minerals (NAM). These defects are also the mobile species that diffuse against polarons. These results are crucial to accurately model the degassing behavior of terrestrial and lunar magmas and to derive the initial D/H of water trapped in fluid inclusions commonly analyzed in mantle NAMs, which suffered complex geological histories.

Introduction:

Water is one of the most ubiquitous volatile compounds in the solar system. It is found in primitive meteorites, it is detected in Earth-like planetary bodies including Mars and the Moon, and it makes up the surface of a number of asteroids. Water in these rocks has been preserved over time even when intense thermal processing of the planetary materials occurred. Water chemically reacts with silicates and oxides to form protonated point defects forming OH-bonds or strongly bound molecular water layers deeply stabilized in the silicate structure. However, despite the common occurrence of hydrous phases in planetary materials, the origin of water, its budget and timing of delivery are not fully understood. In particular, the formation and persistence of hydrous silicate dusts within the protoplanetary disk is very controversial [START_REF] Prinn | Origin and Evolution of Planetary and Satellite Atmospheres[END_REF][START_REF] Bose | Quartz-coesite transition revisited: reversed experimental determination at 500-1200°C and retrieved thermochemical properties[END_REF][START_REF] Drake | Origin of water in terrestrial planets[END_REF]. The budget and the carrier of the water delivered to planets is also debated (e.g. [START_REF] Lunine | The origin of water on Mars[END_REF][START_REF] Albarède | Volatile accretion history of the terrestrial planets and dynamic implications[END_REF][START_REF] Hartogh | Ocean-like water in the Jupiter-family comet 103P/ Hartley 2[END_REF][START_REF] Sarafian | Early accretion of water in the inner solar system from a carbonaceous chondrite-like source[END_REF][START_REF] Hallis | Evidence for primordial water in Earth's deep mantle[END_REF][START_REF] Altwegg | 67P/Churyumov-Gerasimenko, a Jupiter family comet with a high D/H ratio[END_REF][START_REF] Hauri | Water in the Moon's interior: Truth and consequences[END_REF][START_REF] Barnes | An asteroidal origin for water in the Moon[END_REF]. Clues to understanding the origin of water lie essentially in its hydrogen isotopic composition. However, interpretations of this powerful proxy can remain biased because these isotope variations do not only probe geochemical reservoirs. Chemical reactions driven by thermodynamic equilibrium and disequilibrium processes also affect the isotopic signature of silicate materials. Therefore to provide a reliable interpretation of the D/H ratios measured in these different materials (i.e.

to make mass balance calculation or to model degassing), the assessment of equilibrium/kinetic isotope fractionation is critical.

In the past decades, a large body of literature has been devoted to the determination of equilibrium fractionation factors of coexisting H-bearing phases in gas, liquid and solid.

Conversely, little is known on the effect of disequilibrium and kinetically-controlled reactions. First order modeling of degassing, assuming an ideal behavior for dissolved water molecules (sensu lato, including protons, OH and H 2 O groups) is commonly carried out and provides an upper limit to the magnitude of kinetic fractionation (e.g. [START_REF] Tartèse | The abundance, distribution, and isotopic composition of hydrogen in the Moon as revealed by basaltic lunar samples: Implications for the volatile inventory of the Moon[END_REF][START_REF] Barnes | An asteroidal origin for water in the Moon[END_REF]). Yet no experimental quantifications are available to disentangle reservoir, equilibrium and partial disequilibrium effects except on a few hydrous minerals (e.g. [START_REF] Pilorgé | D/H diffusion in serpentine[END_REF].

In this experimental work, we investigate the isotopic consequences of partial hydration/dehydration/degassing of silicates. Experimental conditions and materials were selected to mimic different processes at play in various rock-forming environments. Firstly, the hydration/dehydration of a silica glass in contact with water vapor was performed to document the possible interactions between vapor and the amorphous silicate components detected in most of primitive planetary disks and repeatedly observed in the matrix of well preserved chondrites [START_REF] Abreu | Early solar system processes recorded in the matrices of two highly pristine CR3 carbonaceous chondrites, MET00426 and QUE99177[END_REF]. Although silica is a very simplified analogue, it is an excellent proxy because the speciation of water and its dynamics have been extensively studied (e.g. [START_REF] Doremus | Diffusion of water in silica glass[END_REF]Davis and Tomozawa, 1996;Davis et al., 1995;[START_REF] Behrens | Ar, CO 2 and H 2 O diffusion in silica glasses at 2 kbar pressure[END_REF]. Secondly, magma degassing was modeled by the controlled decompression of a presaturated rhyolitic melt, leading to diffusion-controlled nucleation and growth of H 2 O bubbles within the melt. Thirdly, the isotopic effect of the loss of water stored in nominally anhydrous minerals (NAMs) was studied by high temperature, ambient pressure, hydrogen extraction experiments in a single grossular crystal. With these results in hand, the relationships between water speciation and isotopic effects are discussed and the parameters required to quantitatively account for the kinetic fractionation of hydrogen isotopes are provided.

Experimental and analytical methods

Sample preparation

2.1.1. Silica glass Silica samples were prepared from Infrasil 301 and Herasil 102 (Heraeus Co.) for hydration and dehydration experiments, respectively. The nominal water content of Infrasil 301 is less than 8 ppm whereas Herasil contains less than 150 ppm. In both cases, the parent rods (1 cm in diameter) were cut as disks (approximately 1-2 mm thick) in ethanol and double-sided polished to an optical quality (using SiC papers and 1/4m diamond suspension on polishing clothes and ethanol as the polishing fluid). Annealing was performed in a tubular furnace (Nabertherm Co.) at the university of Lille (UMET) at 1000°C (3°C). Run durations extended to 30 days. Water was introduced into the furnace as vapor. It was generated by bubbling pure Ar in a large volume (500 mL) of ultrapure (Milli-Q) water at 20°C (1°C) before injection into the tubular furnace. In these conditions, the f H2O was close to 20x10 -3 bar (the saturation pressure at 1 bar total pressure and 20°C). Presaturation of the alumina tube was performed during a week before running the experiments. The experimental setup was designed to limit the causes of hydrogen isotope fractionation to the vapor production and potentially to its adsorption on the sample. In this respect, we note that at 20°C, the vapor should be lighter than the liquid reservoir by about 81‰ [START_REF] Horita | Liquid-vapor fractionation of oxygen and hydrogen isotopes of water from the freezing to the critical temperature[END_REF]. Furthermore, the negligible amount of water vaporized at 20°C during the runs implies that the isotopic composition of the liquid reservoir (and so the vapor reservoir) can be considered as constant over the entire duration of experiments.

After quench, samples were sliced perpendicular to the surface of disks (slices were 131, 165 and 186 m in thickness for the sample annealed for 3, 8 and 30 days respectively). Again, samples were polished to optical quality without water and stored in a glovebox under inert atmosphere (Ar) before analyses of profiles by FTIR and SIMS. Dehydration experiments were performed in the same furnace but without injection of water vapor. Only dry argon flowed during the experiments. The furnace was heated to 1600°C for 2 days before the run in order to extract as much water as possible from the alumina tube. The thicknesses of the samples analyzed by FTIR were 193 and 147 m in thickness for the reference and the sample annealed for 3 hours, respectively.

Rhyolitic melts

The rhyolitic samples were described in a previous study [START_REF] Mourtada-Bonnefoi | Kinetics of bubble nucleation in a rhyolitic melt: an experimental study of the effect of ascent rate[END_REF]. The starting material was an obsidian from Güney Dagi, Turkey (76.51% SiO 2 , 12.56% Al 2 O 3 , 0.70% FeO, 0.01% MgO, 0.25% CaO, 4.47% Na 2 O, 4.24% K 2 O, 0.03% TiO 2 , 0.07% MnO in wt%). Samples were presaturated in water in an internally heated pressure vessel and were isobarically quenched to prevent bubble nucleation at 250 MPa and 1000°C.

The saturated glasses initially contained about 6.6 ± 0.2 wt% H 2 O as determined by FTIR [START_REF] Hamada | Simulating bubble number density of rhyolitic pumices from Plinian eruptions: constraints from fast decompression experiments[END_REF]. Glass cylinders were prepared from this starting material and welded in a new platinum capsule (with no water added). The capsule was placed in an externally heated pressure vessel fitted with a rapid quench extension. It was first cold-pressurized to 230-240 MPa and then heated to 800°C in 60-90 min. Oxygen fugacity was not buffered during the decompression experiments, but was assumed to remain close to the Ni-NiO buffer (as in the saturation experiments) because of the short run durations involved. All experiments began by a 15 min relaxation stage at 260 MPa, 800°C. Pressure was then decreased at a rate equal to 27.8 or 167 kPa.s -1 down to a final value P f ranging from 119 to 59 MPa. As soon as pressure P f was reached, the experiment was quenched isobarically at an initial rate of 200°C.s -1 . We focused on two samples decompressed at 27.8 kPa.s -1 (samples C20 and C19 quenched at P f =88.6 and 85.1 MPa, respectively; Mourtada-Bonnefoi and Laporte, 2004), and one sample decompressed at 167 kPa.s -1 (sample B14 quenched at P f =70.7 MPa). In these three experiments, the homogeneous bubble nucleation occurred at 90 ± 2 MPa, and yielded a core with a relatively small number of large bubbles: about 10 bubbles per mm 3 in the experiments at 27.8 kPa.s -1 , with a mean diameter of 77 m (C20) to 187 m (C19); 350 bubbles per mm 3 in the experiment at 167 kPa.s -1 , with a mean diameter equal to 77 m (B14). The SIMS analyses were performed in the cores of these three samples.

Grossular crystal

Grossular (Ca 3 Al 2 Si 3 O 12 ) is known to incorporate greater concentrations of hydrogen than any other natural member of the garnet group. The single crystal studied (38.16% SiO 2 , 16.15% Al 2 O 3 , 1.11% FeO, 7.45% Fe 2 O 3 , 0.24% MgO, 34.93% CaO, 0.02% Na 2 O, 0.42% TiO 2 , 0.43% MnO in wt%) originate from a vesuvianite-bearing fissure in a relatively large rodingite body (approximate dimension 10 m x 2 m x 5 m), located in the serpentinite massif of Rotkopf in the Zillertaler Alps, Austria. Some parts of the material were without fractures, free of inclusions of other minerals, and of gem quality. A double-sided polished parallelepiped sample (4740x2890x1140 µm) was annealed in dry air, at 800°C (5°C) and at ambient pressure to extract hydrogen. The mechanisms involved in this dehydration are discussed below. A high-temperature furnace equipped with a horizontal alumina tube was used for this purpose. After the extraction experiment, a slice was cut from the central part perpendicular to the larger thickness of the annealed sample and polished on both sides (591x2890x1140 µm; see detail of the method in [START_REF] Ingrin | Diffusion of Hydrogen in Minerals[END_REF]. The SIMS analyses (D/H and water content) were performed across the sample approximately perpendicular to the initial free surface of the sample. In addition, an infrared absorption profile was then measured along the SIMS profile.

Analytical protocols

Fourier-transform Infrared Spectroscopy

Fourier-transform Infrared Spectroscopy (FTIR) analyses were performed on two different instruments (LASIR, Université Lille 1). For silica samples, the FTIR absorbance spectra were recorded by a Perkin Elmer FTIR Spotlight 300 equipped with the FPA technology (Focal Plan Area) and a MCT (Mercury Cadmium Telluride) detector. Spectra were recorded in the range of wave number from 700 cm -1 to 5000 cm -1 with a spectral resolution of 4 cm -1 and 32 scans. They were recorded in point mode with a step of 10 µm and a slit of 10x100 µm 2 . This geometry was chosen to improve the signal without modifying the depth spatial resolution (the length of the samples was 5 to 10 times longer than their thickness).

Turning to the grossular crystal, the OH diffusion profiles were analyzed with a Bruker Hyperion 3000 FTIR-microscope coupled to a Bruker Vertex 70 spectrometer equipped with a liquid-N 2 -cooled MCT detector in the laboratory of Infrared and Raman Spectrochemistry.

The whole system was continuously purged with dry air to minimize atmospheric contamination. The sample was placed on a 2 mm thick CaF 2 disc. The infrared beam was focused on the surface of the doubly polished samples with a cassegrainian objective (15x for a spot of 60 x 60 µm and 36x for a spot of 30 x 30 µm). A profile was measured across the whole sample with the larger spot size (60 x 60 µm every 40 µm) while a shorter profile was measured with the smaller spot size (30 x 30 µm every 30 µm). Measurements were also tested in focusing the beam into the middle of the 951 µm thick sample. No major difference was observed. A total of 256 scans were accumulated with a resolution of 4 cm -1 for each measurement. The background spectrum was measured before the profile measurement on the CaF 2 disc.

Secondary Ion Mass Spectrometry

The D/H were acquired using the Cameca IMS 1270 and IMS 1280 ion probes at CRPG-CNRS-Nancy during 3 sessions. An additional session using the Cameca IMS 7f at the Institut Jean Lamour (Nancy) was dedicated to the measurement of the SiH/Si close to the bubbles formed in the decompressed rhyolitic melts. In order to minimize surface hydrogen contamination, the D/H and H 2 O concentrations were measured on standards and samples dried for 48h at 70°C before gold coating and then stored several hours before analysis in the SIMS storage chamber under ultra-high vacuum (2-310 -8 torr).

On the IMS 7f, a 10 nA O -primary ion beam was focused on a 20 µm diameter spot.

The 30 Si and 29 SiH secondary ions were measured with a mass resolution of 3000. The SiH/Si were then considered as representative of the water content in the sample. No standards were used during this session. The absolute measured values of this ratio are therefore certainly affected by matrix effects and cannot be taken as face values. Therefore, we only consider in the following the relative variations of the SiH + /Si + ratio along concentration profiles close to the bubbles and from a sample to another.

Isotope ratios and water concentrations were determined on both IMS 1270 and 1280. During the first session, a 6 nA O -primary ion beam was focused on a 10 µm diameter spot. The H + and D + secondary ions were measured with a mass resolution of 2000, by peak switching in monocollection mode and EM ion counting (H + for 4 s, D + for 20 s, and background for 4 s at mass 0.98). During the second and third sessions, a 2 nA Cs + primary ion beam was focused on a 10 µm diameter area. The electron gun was used for charge compensation. The EGUN emission was adjusted to limit the H -emission below 1000 cps with the primary beam off. The H -and D -secondary ions were measured with a high mass resolution of 2000, by peak switching in monocollection mode (H -on FC for 4 s, D -on EM for 20 s and background for 4 s at mass 0.98 for FC and for 4 s at the mass 1.98 for EM). For each measurement, the sample was pre-sputtered for 2 minutes. The beam position relative to the field aperture and the magnetic field were checked automatically and 20 to 24 cycles acquired. The D/H instrumental fractionation and the H useful yield were determined by using the MPI DING reference glasses [START_REF] Jochum | MPI-DING reference glasses for in situ microanalysis: New reference values for element concentrations and isotope ratios[END_REF]. Finally, during the third session, the water content was also investigated by measuring the OH -/O -and OH -/Si -ratios.

Measurements were made with the same primary Cs + beam and EGUN setting as in the second session. The negative secondary ions were collected by peak switching in monocollection mode at a mass resolution of 7000 to ensure the full separation of 17 O and 16 OH. The acquisition cycles included 16 O on FC, 17 O and 16 OH on EM, 28 Si on FC and 30 Si on EM. The FC and EM backgrounds were measured at the masses 15.8 and 16.8 respectively.

For each measurement, the sample was pre-sputtered for 2 minutes, the beam position and the magnetic field were checked automatically and 12 cycles were acquired.

All isotopic data collected were internally standardized. Data collected on silica samples are expressed as the deviation (D in ‰) from the isotopic compositions of the starting materials measured during the same analytical session. Data collected on rhyolite glasses are expressed as the deviation (D in ‰) from the isotopic compositions measured on the same samples at long distances from bubbles (>200 m), where the water content was found identical to the initial water content (see below). In both cases, several analyses (n=4-10) were performed and did not reveal measurable variations of the starting isotopic composition of these glasses. Data collected on the grossular crystal were processed in a more advanced way, fully described in the result section. Data are presented as the deviation (D in ‰) from the mean composition of the inner part of the sample, where FTIR and SIMS data suggest no significant water loss (see below for details). The D notations in this article are therefore not expressed relative to VSMOW (Vienna Standard Mean Ocean Water) with the notable exception of Fig. 5b (see below for more details). This study thus describes the amplitude of the fractionation produced by partial hydration/dehydration of planetary materials and does not attempt to reproduce quantitatively the absolute D/H values measured on natural samples.

Results

Hydration of silica glass during vapor/solid interactions

Water uptake during gas-glass interaction at 1000°C only affects a few hundreds of microns below the sample surface (Table 1, Fig. 1a). The speciation of water in our samples is unambiguously dominated by hydroxyl groups as repeatedly observed in similar glasses hydrated at temperatures above 700°C (Davis and Tomozawa, 1996;[START_REF] Behrens | Ar, CO 2 and H 2 O diffusion in silica glasses at 2 kbar pressure[END_REF]. For this reason, OH quantification was performed by considering the extinction coefficient provided by Davis et al. (1996) for the OH band at 3673 cm -1 . At the gas-glass interface, the OH concentration is close to the equilibrium water solubility predicted from literature data extrapolated to 1000°C and to a vapor pressure of 20x10 -3 bar [START_REF] Moulson | Water in silica glass[END_REF].

The measured concentrations (134 ± 4 ppm) are in excellent agreement with predictions (124 ppm) for long run experiments (8 and 30 days). Conversely, the concentration measured at the surface of the sample annealed for 3 days (160 ppm) is slightly higher than the equilibrium value. The reason for this slight overshoot is unclear but a time-dependent variability of the surface water concentration has been reported for silica glasses and was attributed to the competing effects of structural relaxation and water diffusion [START_REF] Tomozawa | Time dependent diffusion coefficient of water into silica glass at low temperature[END_REF].

Another typical feature of such hydration experiments on silica glass is that the diffusion coefficient of water is a strong function of the water content. It increases with the water content as depolymerization of the silica network proceeds (e.g. [START_REF] Doremus | Diffusion of water in silica glass[END_REF][START_REF] Behrens | Ar, CO 2 and H 2 O diffusion in silica glasses at 2 kbar pressure[END_REF]. A direct consequence is that concentration profiles present an unusual shape with a water-saturated rim at the surface, which grows over time and is followed, in depth, by an abrupt drop (Fig. 1a). In this context, a constant diffusion coefficient cannot be used for long-run experiments (e.g. [START_REF] Behrens | Ar, CO 2 and H 2 O diffusion in silica glasses at 2 kbar pressure[END_REF]. The primary goal of our study is not to provide a determination of the concentration-dependence of the water diffusion coefficient in our samples. As a consequence, we did not model our dataset with such concentrationdependence. We note however that it can be satisfactorily modeled with D H2O = 10 -14 m 2 .s -1 , which is a reasonable average value for our experimental conditions (Moulson and Roberts, 1961;[START_REF] Behrens | Ar, CO 2 and H 2 O diffusion in silica glasses at 2 kbar pressure[END_REF].

Finally, turning to the isotopic composition of the water dissolved in these silica samples, the dataset clearly points to an absence of detectable and systematic variation at the level of ~20-30‰ (1s.d.). This holds true for all analyzed samples (Table 2, Fig. 1b).

Therefore our results suggest that no measurable diffusion-driven fractionation affects the D/H signature of water during the hydration of silicate glasses at high temperature.

Dehydration of silica glass during vapor/solid interactions

Dehydration of OH-bearing silica glass does not proceed by the exact reverse reaction as hydration experiments (Moulson and Roberts, 1961). After only 3 hours at 1000°C, the bulk sample has uniformly lost a large fraction of the initial water content (Table 1, Fig. 2a).

Hence, dehydration occurs within a much shorter timescale than hydration of initially anhydrous silica glass samples. This result combined to the very limited extent of the concentration gradient at the vapor-solid interface suggests that dehydration is limited by water surface desorption and not by its bulk outward diffusion.

Turning to the isotopic composition of the remaining water dissolved in the sample after the experiment, analytical results show a potentially small constant difference (D =-2322 ‰ at 1sd) across the sample relative to the starting material (Table 2, Fig. 2b). This deviation is not significant and the isotopic composition of the dehydrated sample is taken as homogeneous at the level of ~20-30 ‰ (1 s.d.). As discussed below, any kinetic process would let the solid residue preferentially enriched in the heavy isotope (deuterium). The small difference measured here is opposite.

Rhyolitic magma degassing during decompression

During the decompression of the melt, the water content of the molten silicate is assumed to be minimum close to the bubbles and to increase with increasing distance from the bubbles as a result of diffusive bubble growth. For each sample selected for this study, microprobe analyses of the glass were previously made as far as possible from the bubbles visible in the polished section [START_REF] Mourtada-Bonnefoi | Kinetics of bubble nucleation in a rhyolitic melt: an experimental study of the effect of ascent rate[END_REF]. These previous results provide strong arguments for disequilibrium degassing since the water contents are similar to the initial water content: about 6.6 wt% and much higher than the equilibrium values, at P f , of 3.9 -3.4 wt% [START_REF] Zhang | H 2 O in rhyolitic glasses and melts: measurement, speciation, solubility, and diffusion[END_REF][START_REF] Tamic | The solubility of H 2 O and CO 2 in rhyolitic melts in equilibrium with a mixed CO 2 /H 2 O fluid phase[END_REF]. The SiH/Si measured in this study by IMS 7f confirms and extends previous conclusions. Firstly, the SiH/Si signal is constant and almost identical for all samples at distances as close as 200 microns from the bubbles (Table 3, Fig. 3a), and decreases closer to the bubbles. Secondly, for the sample with the largest bubbles (sample C19 decompressed at 27.8 kPa.s -1 and quenched at P f = 85.1 MPa), the SiH/Si at the interface with the bubble drops down to 60% (extrapolated from available data) of the initial signal. If the measured ratios are proportional to the water content in the concentration range of interest (from 7 to 3 wt%), then it is possible to estimate the water content at the interface to be about 4 wt%. This value is close to the water solubility at equilibrium [START_REF] Zhang | H 2 O in rhyolitic glasses and melts: measurement, speciation, solubility, and diffusion[END_REF][START_REF] Tamic | The solubility of H 2 O and CO 2 in rhyolitic melts in equilibrium with a mixed CO 2 /H 2 O fluid phase[END_REF]. For the two other samples, exhibiting smaller bubbles, the water content at the interface is higher than the equilibrium values (3.8-3.9 wt% at P f = 88.6 MPa and 3.4-3.5 wt% at P f = 70.7 MPa): about 6 wt% in sample C20 decompressed at 27.8 kPa.s -1 and quenched just below the nucleation pressure (P f = 88.6 MPa); about 5.6 wt% in sample B14 decompressed at 167 kPa.s -1 and quenched at P f = 70.7 MPa. Therefore, for all samples disequilibrium degassing occurred during decompression.

Turning to the isotopic composition of the water dissolved in the rhyolitic glass close to the bubbles, the results clearly point to a lack of detectable variation at the level of 20‰ (1s.d.). This holds true for all analyzed areas and for the three samples (Table 3, Fig. 3b). The decompression rate and the final pressure have no effect on the D/H of the remaining water.

Here again, our results suggest that no significant diffusion-driven fractionation affects the D/H signature of water during magma degassing.

Grossular dehydration

Based on the position of the main OH bands [START_REF] Rossman | The hydrous components in garnets: grossularhydrogrossular[END_REF] identified seven different classes of FTIR spectrum in natural grossular. This grossular belongs to class 5 spectra. It means that they are dominated by a band at 3611 cm -1 and two side bands at about 3664 cm -1 and 3562 cm -1 . This grossular is a typical single crystal from cavities in meta-rodingites found in alpine type serpentinites. The integrated absorbance in the region 3700-3400 cm -1 was determined to calculate the OH content of the samples (Fig. 4). The absorption coefficient proposed by [START_REF] Rossman | The hydrous components in garnets: grossularhydrogrossular[END_REF] was used. This grossular crystal initially contained 0.14 (± 0.03) wt% H 2 O and the OH content was homogenous within experimental and statistical errors. It is of note that the correct absorption coefficients for garnets have been a matter of debate [START_REF] Maldener | Hydrogen in some natural garnets studied by nuclear reaction analysis and vibrational spectroscopy[END_REF]), but it is of minor consequence here as we only need information on the relative variation of the OH-content during experiments.

After annealing in air for 10h at 800°C, the hydrogen extraction is pronounced (Fig. 5a). A large and quasi-symmetric OH concentration profile was measured by FTIR spectroscopy across the sample with two spot sizes (30x30 microns and 60x60 microns). The maximum absorbance measured was then used to normalize the OH contents determined locally. This maximum absorbance was in excellent agreement with the average initial absorbance measured before annealing. Consequently, the crystal can reasonably be considered as a semi-infinite medium, a simplifying assumption for processing the dataset in order to extract diffusion coefficients. These FTIR profiles are also fully consistent with the OH -/O -and OH -/Si -profiles measured by SIMS (Table 4, Fig. 5a). In details, the FTIR profile collected over spots of 30x30 microns is slightly more pronounced. This discrepancy may result from the more local nature of the analysis. It is therefore possible to derive, from these four separate datasets, a fairly accurate diffusion coefficient describing the hydrogen extraction in these experimental conditions. Within the assumption of a semi-infinite medium, the full profile across the sample can be modeled analytically with the help of the function proposed by Crank ( 1956):

(1) where C(x,t) is the local concentration at coordinates (x,t), where the origin is located at the mid-point of the slab, C 0 is the initial homogeneous concentration at t=0, t is time (s) , x is distance (m), D is the diffusion coefficient (m 2 .s -1 ) and L is the half-length of the sample (m).

In this experiment, the boundary conditions are C=0 for x=-L, L for any t and C=C 0 for -L<x<L at t=0. With this equation in hands, the diffusion coefficient for hydrogen extraction at T = 800°C derived from the four different profiles are very consistent: 4 (±1) x 10 -13 m 2 .s -1 , 3 (±1)

x 10 -13 m 2 .s -1 , 2.8 (±1.1) x 10 -13 m 2 .s -1 , 2.7 (±1) x 10 -13 m 2 .s -1 for FTIR (30x30), FTIR (60x60), OH -/O -and OH -/Si -respectively (Fig. 5a).

The isotope distribution within the grossular crystal drastically contrasts with the results collected on glasses and melts. For this nominally anhydrous mineral (NAM) single crystal, a large isotopic gradient was measured. From the center of the sample to the edges, the measured D, without standard corrections and calculated against VSMOW varies by more than 550 ‰ (Table 4, Fig. 5b). Such a profile is expected in the case of a diffusiondriven fractionation during hydrogen extraction. Across the central 600 µm exhibiting a flat profile of water content (FTIR, OH -/O -and OH -/Si -ratios) and where no D/H redistribution is expected, the raw D measured (without standardization nor any corrections, but calculated against VSMOW) only varies within a 70‰ range with a typical spot to spot reproducibility of ±30‰ (Fig. 5b). This variation is significant but much lower than the overall measured variation (covering more than 550‰). Moreover, across this inner part of the sample, the measured D is also reproducible from a session to another. It is independent of the direction of the measurements and shows a continuous trend (Fig. 5b). The reason for this non-flat profile is unclear but may be a geometrical artifact resulting from the cutting and processing of the sample after water extraction experiments. It may also represent the remnant of a continuous variation of the D/H across the single crystal from which the sample was prepared. This artifact marginally affects the quality of the fits and their quantitative analysis (See Appendix).

Nonetheless, we propose a simple linear correction for this artifact. It is based on the fact that after the experiment, no D/H change can be expected where no measurable water extraction profile can be resolved. We therefore used the continuous linear trend measured on the sample to propose an internal correction of the data. We first defined the inner part of the sample by the presence of a flat plateau in the FTIR, OH -/O -and OH -/Si -ratios for C/C 0 >0.95 (Fig. 5a). A linear regression is then applied to the D dataset (Fig. 5b) and this contribution is subtracted to the entire data set (Fig. 5c). We therefore assumed that a continuous drift affected the entire sample and used the linear regression (D =-377(±19)+0.1(±0.3)x) to internally standardize the whole dataset. The consequence of this procedure is to set the inner part of the samples at a D0‰ (the D reported on Fig. 5c are therefore relative to the homogeneous inner part of the sample). Moreover, without additional processing, a quasi-symmetrical isotopic profile is also obtained. This correction does not affect the quantitative analysis and the discussion since the same fits can be adjusted to the raw data as shown in the appendix.

Discussion

Water speciation in silica glass, rhyolitic melt and grossular

The speciation of water in minerals, liquids and melts controls its solubility and its mobility. In this study, the nature of the hydrogen carrier is also pivotal to model our data and propose applications to natural samples. The speciation of water in silicate glasses and melts has been a matter of concern for several decades. It is now widely accepted that both molecular water and hydroxyl groups generally coexist, their relative proportions depending on the silicate composition, temperature, pressure and water partial pressure (e.g. [START_REF] Stolper | The speciation of water in silicate melts[END_REF][START_REF] Schmidt | Water diffusion in phonolite melts[END_REF][START_REF] Fanara | Water diffusion in potassium-rich phonolitic and trachytic melts[END_REF]. The water speciation may also change as melt is quenched into glass [START_REF] Dingwell | Relaxation in silicate melts[END_REF][START_REF] Zhang | Kinetics of the reaction H2O+O=2OH in rhyolitic and albitic glasses: preliminary results[END_REF]. One notable exception however is the case for silica glasses and melts. In these materials, when the total water content is lower than 0.3 wt% and at temperature higher than 650°C, water is essentially dissolved as hydroxyl groups (Davis and Tomozawa, 1996). The speciation of water in our silica samples is thus dominated by OH groups and the amount of molecular water is negligible. The situation is different for hydrous rhyolites [START_REF] Zhang | Kinetics of the reaction H 2 O+O=2OH in rhyolitic glasses upon cooling : geospeedometry and comparison with glass transition[END_REF][START_REF] Ihinger | The speciation of dissolved water in rhyolitic melt[END_REF][START_REF] Hui | Pressure dependence of the speciation of dissolved water in rhyolitic melts[END_REF]. The proportion of OH groups increases with melting temperature and pressure but decreases as the total amount of water increases. Here, samples were saturated below 1 GPa at 800°C and contained about 7 wt% water. In these conditions, proportions of OH and molecular H 2 O are anticipated to be similar [START_REF] Ihinger | The speciation of dissolved water in rhyolitic melt[END_REF]. There is therefore a clear expected difference between the speciation of water in silica and in water-saturated rhyolitic samples.

Turning to the water speciation in garnet minerals (and in other NAMs), H 2 O molecules are not detected. Instead, hydrogen is bound to an oxygen-atom, forming a hydroxyl bond, which in turns induces local charge excess usually compensated by a cation vacancy or a cation substitution. The dominant incorporation mechanism in grossular is the "hydrogarnet" substitution, where four OH -molecules are associated with a silicon vacancy, replacing a SiO 4 tetrahedron [START_REF] Cohen-Addad | Etude de la substitution du groupement SiO 4 par (OH) 4 dans les composés Al 2 Ca 3 (OH) 12 et Al 2 Ca 3 (SiO 4 ) 2 . 16 (OH) 3.36 de type grenat[END_REF]. Hydroxyl groups might also be associated with an octahedral (Al 3+ ) vacancy, which would, for instance require 3 OH -molecules for local charge balance (Geiger et al., 1991). Finally, a mechanism involving titanium has been suggested (Johnson, 2003) and can possibly occur in titanium-bearing grossular including the crystal used in this study.

Microscopic mechanisms controlling water diffusion in silica glass, rhyolitic melt and grossular

Microscopic mechanisms are likely at the origin of the contrasted behavior of hydrogen isotopes reported in this work. The three different materials exhibit very different diffusion mechanisms, which in turns probably explains the different isotopic evolutions observed in glasses, melts and crystals. In amorphous silica, it is generally accepted that water diffuses as molecular water (e.g. [START_REF] Doremus | Diffusion of water in silica glass[END_REF]. These molecules eventually react with the glass network to form immobile hydroxyls (or silanol) in favourable sites following a reaction such as:

H 2 O+SiOSiSiOH + HOSi (2)
Even below 1000°C, this reaction is fast compared to the OH intrinsic mobility and reaches the local equilibrium with most of the immobile water present as hydroxyl [START_REF] Doremus | Reactivity in solids[END_REF][START_REF] Horita | Liquid-vapor fractionation of oxygen and hydrogen isotopes of water from the freezing to the critical temperature[END_REF][START_REF] Doremus | Diffusion of water in silica glass[END_REF]. Our results are also in agreement with pioneering results of Moulson and Roberts (1961) suggesting that water diffusivities are always lower during hydration than during dehydration experiments (Figs. 1 and2).

The diffusion of water in more complex and geologically relevant melts such as rhyolites, dacites, andesites and basalts has been extensively studied in the past (e.g. [START_REF] Zhang | Diffusion of H, C, and O components in silicate melts[END_REF]. For some of them, it has been suggested that hydroxyl groups may contribute significantly to the diffusive transport of water. This is the case for dacite, possibly basalts [START_REF] Behrens | H 2 O diffusion in dacitic and andesitic melts[END_REF] and haploandesite when the total amount of dissolved water is low [START_REF] Ni | Hydroxyl and molecular H2O in a haploandesitic melt[END_REF]. In the latter case, the contribution of hydroxyl to the overall water diffusion can be up to 10-20%. Nonetheless, there is no report of such significant contribution of hydroxyl to water diffusion in rhyolitic melts. Therefore, it is reasonable here to assume that the behavior of water during bubble formation in rhyolites during decompression does not differ from its behavior in silica. In both cases, molecular water is the mobile species while OH groups have a much lower mobility [START_REF] Zhang | Diffusion of H, C, and O components in silicate melts[END_REF].

Hydrogen extraction from a garnet crystal is a more complicated process because local charge balance must be conserved. One of the fastest mechanisms of hydrogen release in several iron-bearing minerals was identified as the redox reaction:

Fe 3+ +O 2-+ 1/2H 2 <-> Fe 2+ +OH -.
(3)

This mechanism was ubiquitously proposed for other nominally anhydrous minerals such as diopside [START_REF] Ingrin | Hydrogen in diopside: Diffusion, kinetics of extractionincorporation, and solubility[END_REF][START_REF] Woods | Hydrogen in diopside: Diffusion profiles[END_REF] and olivine [START_REF] Mackwell | Diffusion of Hydrogen in Olivine: Implications for Water in the Mantle[END_REF]. It also likely operates for iron-rich grossular. Now, the possible mechanisms controlling hydrogen migration through the garnet structure is less well understood than for other NAMs. A few studies suggest nonetheless that processes operating for garnets are similar to other NAMs [START_REF] Blanchard | Kinetics of deuteration in pyrope[END_REF][START_REF] Kurka | Kinetics of hydrogen extraction and deuteration in grossular[END_REF][START_REF] Zhang | Kinetics of deuteration in andradite[END_REF]. In ferromagnesian olivines, hydrogen diffusion is charge compensated either by a fast flux of polarons or by a slower coupling to metal vacancies diffusion [START_REF] Mackwell | Diffusion of Hydrogen in Olivine: Implications for Water in the Mantle[END_REF]Kohlstedt and Mackwell, 1998). Concerning iron-poor grossulars, [START_REF] Kurka | Kinetics of hydrogen extraction and deuteration in grossular[END_REF] showed that migration of hydrogen is likely coupled to the slow migration of metal vacancies. Nonetheless, in iron-bearing natural minerals the extraction mechanism likely involves counter-fluxes of polarons [START_REF] Mackwell | Diffusion of Hydrogen in Olivine: Implications for Water in the Mantle[END_REF][START_REF] Ingrin | Hydrogen in diopside: Diffusion, kinetics of extractionincorporation, and solubility[END_REF][START_REF] Demouchy | Water Diffusion in Synthetic Iron-Free Forsterite[END_REF]. In other words, the extraction of hydrogen from this mineral may be described as the oriented jump of protons outward the crystal structure and not as the diffusion of free molecular water as for silicate melts and glasses. In minerals with low-iron content (typically lower than 7% in diopside) the diffusion rate of polarons, which is lower than the mobility of hydrogen measured by H-D exchange control the rate of hydrogen extraction [START_REF] Ingrin | Diffusion of Hydrogen in Minerals[END_REF]. For higher concentration of iron the diffusion of polaron is faster than the diffusion of hydrogen, in this case it is the mobility of hydrogen that controls the extraction [START_REF] Ingrin | Diffusion of Hydrogen in Minerals[END_REF]. The grossular presently studied contains more than 8 wt% FeO; it is thus likely that the migration of hydrogen during extraction balanced by a counter-flux of polarons is controlled by the mobility of hydrogen.

The control of extraction by the mobility of protons in interstitial positions and not water is likely the origin of the contrasted behavior of hydrogen isotopes between glasses and melts and the grossular crystal.

Modelling diffusion-driven D/H fractionation in glasses, melts and crystals

The same model of D/H fractionation in glasses and melts can be used to explain our experimental data collected on partially hydrated Infrasil 301 preforms (containing initially about 8 ppm water) as well as for decompressed rhyolitic melts. Since the natural abundance of deuterium is very low (0.0156%), the diffusion coefficient derived from these experiments can be taken as the diffusion coefficient of the pure H-bearing mobile species (here H or H 2 O). Then, D profiles are easily modeled provided that the concentration profiles of the corresponding D-bearing molecules can be calculated. We note at this point that this modeling strategy relies on the determination of the effective diffusion coefficient of water in our samples. As this coefficient is sensitive to the local water content, the quantitative modeling of isotopic profiles may be difficult. However, the lack of any measurable isotopic variation along diffusion profiles makes it unnecessary and impossible to quantitatively address this difficulty. As a case study, we focus below on the dataset collected on the silica glass. We assumed a constant diffusion coefficient of D H2O = 10 -14 m 2 .s - 1 , which is a reasonable average value for our experimental conditions (Moulson and Roberts, 1961;[START_REF] Behrens | Ar, CO 2 and H 2 O diffusion in silica glasses at 2 kbar pressure[END_REF].

In this framework, the classical empirical relationship relating the ratio of diffusion coefficients of two isotopes to their inverse mass ratio was used [START_REF] Richter | Isotope fractionation by diffusion in molten silicates[END_REF]:

D i / D j =(m j / m i )  , (4)
where D i and D j are diffusion coefficients for isotopes i and j, m i and m j are their respective masses and the -exponent is estimated by adjusting Eq. 4 to the experimental isotopic profiles. This formalism was successfully applied to an increasingly large number of isotopic systems from Li to Fe isotopes in natural samples (e.g. [START_REF] Sio | Discerning crystal growth from diffusion profiles in zoned olivine by in situ Mg-Fe isotopic analyses[END_REF][START_REF] Oeser | Processes and time scales of magmatic evolution as revealed by Fe-Mg chemical and isotopic zoning in natural olivines[END_REF][START_REF] Richter | Reassessing the cooling rate and geologic setting of Martian meteorites MIL 03346 and NWA 817[END_REF] and experimental charges (e.g. [START_REF] Richter | Isotope fractionation by diffusion in molten silicates[END_REF][START_REF] Roskosz | Experimental quantification of the fractionation of Fe isotopes during metal segregation from a silicate melt[END_REF].

It was rarely applied to the D-H system (e.g. [START_REF] Saal | Hydrogen isotopes in lunar volcanic glasses and melt inclusions reveal a carbonaceous chondrite heritage[END_REF]. So far, no direct experimental study were specifically dedicated to this system, while, as an empirical law, the -coefficient has to be experimentally (or computationally) determined. In principle, kinetic theory applied to a perfect gas provides an upper limit of 0.5 for the -value. For condensed matter it is thus expected that  should take values between 0 and 0.5. With Eq. 4 in hands, the D D and the diffusion profile of deuterated mobile species were derived as a function of .

As discussed before, molecular water is the dominant mobile species involved in conventional hydration/dehydration reactions in glasses and melts. Nevertheless, molecular hydrogen diffusion also occurs in more exotic environments, namely i) in very reducing conditions, ii) when the hydrogen fugacity (f H2 ) is high, iii) when radiolysis occurs in situ, for instance during ionizing irradiation of silicates. The last process was recently found efficient to separate D and H of water (and hydroxyl) groups dissolved in amorphous and crystalline silicates [START_REF] Roskosz | Experimental investigation of irradiation-driven hydrogen isotope fractionation in analogues of protoplanetary hydrous silicate dust[END_REF]. For this reason, we provide models describing the spatial redistribution of H and D, in an amorphous silicate, caused by the diffusion of either molecular water or hydrogen atoms (Fig. 6 a and b). If water molecules are the mobile species (diffusion of H 2 O and HDO, mass ratio=18/19), the lack of any measurable D profiles can be explained by any values for  from 0 to 0.5 (Fig. 6 a andb). In other words,  is unconstrained here. More importantly the moderate mass difference between H 2 O and HDO makes the diffusion-driven fractionation of H and D very limited (on the order of 30‰). The situation would be different however in environments where the mobility of hydrogen atoms is favored (instead of molecular water). In this case, a significant fractionation would be expected (Fig. 6 a and b). The amplitude of the fractionation for =0.5 is of several hundreds of ‰. Such a large diffusion-driven fractionation was indeed recently produced experimentally by electron irradiation of hydrous analogues of the dust present in the protoplanetary disk [START_REF] Roskosz | Experimental investigation of irradiation-driven hydrogen isotope fractionation in analogues of protoplanetary hydrous silicate dust[END_REF].

The formalism may, in principle, be somewhat different in the case of the grossular single-crystal. The classical theory of the isotopic mass-dependence of diffusion rates in crystalline solids was developed in the late 1950s. A comprehensive review of this theory can be found in [START_REF] Van Orman | theoretical constraints on the isotope effect for diffusion in minerals[END_REF]. In summary, the amplitude of the diffusiondriven isotope fractionation diminishes in proportion to the degree to which the diffusion process deviates from that of a purely random walk [START_REF] Schoen | Correlation and the isotope effect for diffusion in crystalline solids[END_REF][START_REF] Tharmalingam | Isotope effect in vacancy diffusion[END_REF]. The basic equation that relates the diffusivities of two isotopes is then (LeClaire, 1966):

(5)

where and represent the frequency of a jump of two isotopes i and j to an adjacent lattice site (vacancy or interstitial) and f is the correlation coefficient. In the simplest case of a random walk, with no correlation between successive jumps, (i.e. when the motion of the atom during a jump is fully decoupled from the vibrations of the surrounding atoms), f = 1.

Furthermore, when there is no such coupling between the vibrations of the jumping atom and of the rest of the crystal, the ratio of jump frequencies of the two isotopes is given by [START_REF] Leclaire | Some comments on the mass effect in diffusion[END_REF].

(6)

Combining Eq. 5 and 6, allow us write Eq. 4 with =0.5 when f=1. There are often correlations between jumps, and generally, f < 1. Nonetheless, there are a few cases where f is close to unity. Notably, diffusion of a trace element by a simple interstitial mechanism is uncorrelated because all adjacent interstitial sites of the same type are equivalent. For this reason, diffusion-driven isotope fractionation is expected to be large, close to a square root of mass dependence, when an interstitial mechanism is involved.

The diffusion of protons in NAMs likely obeys this description. Though microscopic mechanisms may differ from one mineral to the other and as a function of their chemical composition, diffusion of trace amount of H-defects is always faster than any vacancy in the crystal structure. Hydrogen diffusion is thus essentially interstitial in olivine and other NAMs in general. Therefore, the same procedure as for glasses was applied to the D profile. The model only considers the mobility of hydrogen and deuterium atoms. A good fit to the data is obtained for  of 0.5 (Fig. 5c andAppendix). The same data were also plotted against the remaining fraction of hydrogen in the sample (Fig. 6c andAppendix). In this frame, the whole dataset can be satisfactorily described by a Rayleigh distillation equation:

where F is the remaining fraction of hydrogen in the solid residue and  factor. This factor can describe either an equilibrium fractionation between coexisting Hbearing species, or a purely kinetic process. In the latter case, assuming that desorbed hydrogen molecules behave as an ideal gas, the fractionation factor should be close to . Our best fit is obtained for =1.460.04, which is in excellent agreement with the expected behavior of an ideal gas. These two different approaches therefore firmly suggest that the diffusion of protons and deuterons within the structure of grossular occurs at a rate that only depends on their masses. In other words and following the description of the isotope effects of diffusion in crystalline solids, there is no significant interaction between the lattice vibrations of mineral and those of protons and deuterons.

Geochemical implications

The source of water in planetary bodies is investigated by measuring the D/H ratio because different regions of the solar system show very distinct D/H signatures [START_REF] Hartogh | Ocean-like water in the Jupiter-family comet 103P/ Hartley 2[END_REF][START_REF] Altwegg | 67P/Churyumov-Gerasimenko, a Jupiter family comet with a high D/H ratio[END_REF]. The hydrogen isotope fingerprint is thus a major proxy to study the provenance of water in bodies throughout the solar system, including Earth, the Moon, Mars, and the asteroid belt (e.g. Alexander et al. 2012;[START_REF] Saal | Hydrogen isotopes in lunar volcanic glasses and melt inclusions reveal a carbonaceous chondrite heritage[END_REF][START_REF] Sarafian | Early accretion of water in the inner solar system from a carbonaceous chondrite-like source[END_REF][START_REF] Hallis | Evidence for primordial water in Earth's deep mantle[END_REF][START_REF] Barnes | An asteroidal origin for water in the Moon[END_REF]. However, several episodes of hydration/dehydration/evaporation/degassing have affected these planetary bodies overtime (see for instance [START_REF] Saal | Volatile content of lunar volcanic glasses and the presence of water in the Moon's interior[END_REF] for the Moon). A reliable modeling of the isotopic modifications induced by the successive events that may have affected the measured samples is thus required to assess the origin of water in these bodies. [START_REF] Saal | Hydrogen isotopes in lunar volcanic glasses and melt inclusions reveal a carbonaceous chondrite heritage[END_REF] measured the isotopic composition of water accommodated in lunar glasses and inclusions.

They performed a detailed analysis of their data with a particular emphasis on the kinetic isotopic effect of dehydration. They considered the possible contribution of molecular water, hydroxyl and molecular hydrogen to this kinetic fractionation. An undisputable conclusion of their models is that the typical -value for molecular hydrogen has to be lower than 0.06. Our experimental results collected on silica glasses are fully consistent with this conclusion (Fig. 6). This implies that as far as degassing of a melt is concerned the kinetic fractionation caused by degassing is very limited, at a few ‰ level and will be difficult to measure except if extensive distillation episodes are involved.

A drastically different conclusion is drawn when phases of interest are NAMs instead of glasses. In this case, the migration of protons through the crystalline network appears to be particularly efficient to separate D and H. To our knowledge no experiments have been carried out on other minerals of this family (e.g. olivines and pyroxenes). Concerning hydrous minerals, [START_REF] King | The role of dehydrogenation on the hydrogen contents and dD of kaersutites[END_REF] reported diffusion-driven D/H fractionation during the partial dehydration of an amphibole (kaersutite). They also observed marked isotopic profiles spatially correlated to water concentration profiles. These profiles were understood as the consequence of the diffusion of an interstitial species (protons/deuterons) and the dataset is satisfactorily described by Eq. 4 with a -value close to 0.5. Obviously, our results urge for the quantification of this -coefficient for major upper mantle minerals (i.e.

pyroxenes and olivines). Grossular and amphibole crystals, however, exhibit very comparable results, with  ≈ 0.5. This suggests that  could also be very close to 0.5 for a large array of minerals. If so, the determination of the source reservoir of planetary water inferred from inclusions hosted in such minerals should be carefully corrected for the possible multi-stage hydration/dehydration (e.g. [START_REF] Hallis | Evidence for primordial water in Earth's deep mantle[END_REF]. If diffusion within the inclusion itself is probably inefficient to modify significantly the isotopic composition of dissolved water, the transport through the host mineral could modify this isotopic composition at the level of several 100‰.

Conclusion

Several (see text for more details). The fit to the data presented here is for =0.5.

Figure captions :

 captions Figure captions:

Figure 2 :

 2 Figure 2: a) Hydroxyl concentration profiles measured by FTIR after the partial dehydration of nominally water-rich (~150 ppm) silica samples at 1000°C. b) Associated D relative to the mean isotopic composition of the inner part of the sample and measured by SIMS. The grey bar represent the typical 1sd error envelop.

Figure 3 :

 3 Figure 3: a) Hydroxyl concentration profiles measured around bubbles formed by magma decompression in a rhyolitic melt (data measured by SIMS and normalized to the initial water concentration measured far from the bubbles, see text for more details. b) Associated D relative to the mean isotopic composition measured by SIMS far from the bubbles. The grey bar represent the typical 1sd error envelop.

Figure 4 :

 4 Figure 4: Transmission -FTIR spectra (in the region 3000-4000 cm -1 ) collected across the grossular sample after hydrogen extraction experiment. The spectra exhibiting large absorptions were collected at the center of the sample.

Figure 5 :

 5 Figure 5: a) Hydroxyl concentration profiles measured across a grossular single crystal after partial water extraction (Data measured by SIMS and FTIR and normalized to the initial water content measured at the center of the sample, see text for more details). b) Associated raw D/H measured by SIMS along 4 different profiles and collected during different analytical sessions. These data are given without matrix effect corrections against VSMOW. The best linear fit to the data collected at the center of the sample is also shown (see text for more details). c) D/H measured along hydroxyl concentration profiles internally corrected for matrix effect based on the linear fit shown on panel b

Figure 6 :

 6 Figure 6: Models for D/H redistibution during diffusion at 1000°C of water molecules and protons in a silica glass after a) 8 days and b) 30 days (see text for more details). c) Rayleigh distillation plot describing the D/H redistibution during water extraction from a grossular single crystal.

  Figure1

  experimental set-ups and silicate materials (glasses, melts, minerals) were used in order to mimic water-silicate interactions in different geo/cosmochemical environments and investigate associated H-isotope fractionation. Controlled hydration/dehydration experiments were performed. When molecular water is the mobile species (in hydrous phases and hydrated glasses and melts), no significant kinetic isotopic effect could be measured (within ±20‰); conversely when hydrogen is the dominant mobile species (i.e. in nominally anhydrous minerals) within the silicate framework, an extreme kinetic isotopic effect is observed. The geochemical and geophysical contexts (e.g. presence of water vapor vs. liquid; bubble degassing vs. hydration; diffusion in solids vs. diffusion in liquids, high vs. low pressure environments…) are not the dominant factors controlling the extent of the kinetically driven isotope fractionation. The speciation of water within the silicate and the microscopic mechanism involved in the diffusion process are, instead, the dominant factors. They may obviously be influenced by extrinsic intensive parameters, but is it essentially controlled by the crystal chemistry of the silicates of interest. Our results can

	thus probably be applied to a much broader range of geological and astrophysical
	environments provided that the cristallochemistry of 'water' accommodated into the
	minerals of interest are known.

Table 1 :

 1 Hydroxyl concentration profiles measured after the partial hydration of nominally water-poor (8 ppm) silica samples, and the partial dehydration of nominally water-rich (~150 ppm) silica samples at 1000°C.

		Hydration at 1000°C.		Dehydration at 1000°C.
	Hydroxyl concentration (in ppm)	Hydroxyl concentration (in ppm)
	Distance (μm)	30 days	8 days	3 days	Distance (μm)	Starting	3 hours
						material
	0	136	132	160	0	0	0
	10	135	130	148	20	1	1
	20	133	120	132	40	6	1
	30	135	110	110	60	19	1
	40	134	101	88	80	22	1
	50	132	94	64	100	28	5
	60	127	83	39	120	43	5
	70	121	75	25	140	42	10
	80	114	68	16	160	55	12
	90	107	60	13	180	61	11
	100	100	52	11	200	64	13
	110	94	44	10	220	66	17
	120	87	37	9	240	66	19
	130	82	30	9	260	67	19
	140	76	24	9	280	65	19
	150	70	17	9	300	66	20
	160	64	12	9	320	66	19
	170	57	9	9	340	66	20
	180	52	7	10	360	68	20
	190	47	7	9	380	67	20
	200	41	8	9	400	66	22
	210	36	8	9	420	67	21
	220	30	9	9	440	67	22
	230	25	9	9	460	67	21
	240	20	9	9	480	67	22
	250	15	9	9	500	68	22
	260	13	8	9	520	67	22
	270	10	8	8	540	68	22
	280	7	8	9	560	72	22
	290	7	8	9	580	67	22
	300	8	8	9	600	71	23
	310	8	8	8	620	68	23
	320	9	8	8	640	67	23
	330	9	9	8	660	67	22
	340	9	8	9	680	67	23
	350	10	9	10	700	67	23
	360	10	10	11	720	69	23
	370	11	11	12	740	73	22
	380	10	11	11	760	69	23
	390	10	11	10	780	68	23
	400	10	11	10	800	68	23
	410	10	12	9	820	68	23
	420	10	11	11	840	68	23
	430	10	10	11	860	71	23
	440	10	9	12	880	71	23
	450	9	11	12	900	69	23
	460	9	10	10	920	68	22
	470	9	10	10	940	71	23
	480	8	10	9	960	71	23
	490	8	8	8	980	73	23
					1000	67	23
					1020	71	24
					1040	67	25
					1060	68	25
					1080	71	24
					1110	67	22
					1120	70	23

Table 3 :

 3 Hydrogen concentration and isotope composition profiles measured by SIMS around bubbles formed by magma decompression in a rhyolitic melt (H/Si data are normalized to the initial water concentration measured far from the bubbles, see text for more details).

	Normalized H/Si concentration profiles		Hydrogen isotopic compositions
	Sample	Distance (μm)	H/Si (normalized)	27.8 kPa/s to 85 Mpa Distance D (‰. 1 (μm) s.d.)	27.8 kPa/s to 88 Mpa 167 kPa/s to 70 Mpa Distance D (‰. 1 Distance D (‰. (μm) s.d.) (μm) 1 s.d.)
		25	0.87	368 -9 ± 3	757 -15 ± 6	25	5 ± 6
	Mpa 70 to kPa/s 167	56 89 128 163 194 226 258	0.88 0.93 0.94 1.00 1.01 1.01 0.99	337 -16 ± 4 306 -9 ± 5 280 0 ± 3 257 2 ± 5 228 -3 ± 4 206 -17 ± 5 186 33 ± 4	479 -41 ± 6 465 18 ± 4 450 -13 ± 6 425 -4 ± 7 473 28 ± 5 434 23 ± 5 425 23 ± 6	55 16 ± 6 85 -8 ± 6 115 -3 ± 6 145 2 ± 5 175 -3 ± 5 205 -10 ± 5 25 29 ± 6
		285	0.99	128	7 ± 5	253 -37 ± 7	45 -4 ± 8
		25	0.68	73 12 ± 8	25 -34 ± 6	70	2 ± 8
	Mpa 85 to kPa/s 27.8	55 87 120 158 192 228 264	0.79 0.85 0.90 0.95 0.98 0.99 1.02	111 -28 ± 15 147 1 ± 18 80 15 ± 18 52 1 ± 10 26 10 ± 18 20 1 ± 15 20 17 ± 16	59 83 32 ± 6 2 ± 7 107 35 ± 7 132 30 ± 5 157 25 ± 6 182 15 ± 3 206 -4 ± 4	89 114 134 -6 ± 7 9 ± 8 1 ± 7 159 -8 ± 7 179 -9 ± 7 204 -7 ± 9 224 -6 ± 8
		301	0.99	50 -23 ± 8	231 -14 ± 3	25 -30 ± 7
	Mpa	25 53	0.94 0.96	80 -11 ± 11 110 23 ± 8	256 281 -8 ± 4 4 ± 3	53 -8 ± 8 84 8 ± 6
	88	90	0.96	140 -13 ± 11	306 -8 ± 3	111	4 ± 5
	to	130	0.97	20	7 ± 17	30 -26 ± 4	141	3 ± 9
	kPa/s	177 215	1.00 1.00			55 12 ± 4 80 1 ± 3	168 198	9 ± 7 5 ± 9
	27.8	249 285	1.00 1.00			105 -3 ± 3 317 3 ± 3	232 -3 ± 6 261 13 ± 7
						314	0 ± 2
						38 -10 ± 4
						63 -9 ± 2
						269 31 ± 3
						113 30 ± 4
						138 26 ± 4
						25 -1 ± 3
						49 -11 ± 10
						75 -8 ± 9
						101 -5 ± 11
						125 -32 ± 13
						149 -18 ± 12
						175 -17 ± 11
						201 -18 ± 11

Table 4 :

 4 Hydroxyl concentration profiles measured across the grossular single crystal after partial water extraction (Data measured by SIMS and FTIR). They are normalized to the initial water content measured at the center of the sample, see text for more details). Associated hydrogen isotope compositions (relative to VSMOW and without matrix effect correction) measured by SIMS along 4 different profiles and collected during different analytical sessions.

	Normalized hydroxyl concentration profiles	Hydrogen isotope compositions (‰. 1 s.d.)

Distance µm FTIR (60 µm) FTIR (30 µm) OH/Si (SIMS) OH/O (SIMS) Distance µm

						Profil #1	Profil #2	Profil #3	Profil #4
	20	48			3		363 ± 40
	25		68	67	3		321 ± 49
	30		34		23		513 ± 14
	50		69	68	23		510 ± 20
	60	60	48		43		451 ± 10
	75		75	72	43		479 ± 13
	90		60		63		290 ± 14
	100	70			63		320 ±	7
	100		82	81	73		233 ± 10
	120		68		73		245 ± 11
	125		85	82	93		108 ± 11
	140	79			93		131 ±	8
	150		74		113		75 ± 15
	150		88	89	113		105 ± 10
	175		90	88	133		20 ± 17
	180	86	77		133		51 ± 11
	200		92	91	153		-1 ± 16
	210		82		153		24 ± 14
	220	91			173		14 ± 11
	240		86		173		36 ± 12
	250		94	93	193		-16 ± 15
	260	94			193		8 ± 14
	270		90		213		-11 ± 10
	300	96	93		213		-7 ±	8
	300		98	96	228 23 ± 13
	330		95		233		7 ± 14
	340	97			253 28 ±	7
	350		98	97	253		-1 ± 10
	360		97		273		-12 ± 12
	380	98			275	3 ±	7
	390		98		300	-2 ±	7
	400		99	98	323		-22 ± 12
	420	99	99		325 -32 ±	9
	450		100		350 25 ±	7
	450		100	99	373		-13 ± 11
	460	100			375	-1 ±	7
	500	100			400 -12 ±	5
	500		100	100	423		-15 ±	6
	540	100			425 -12 ±	7
	550		99	98	450 -12 ±	5
	580	100			473		3 ±	8
	600		99	99	475 20 ±	5
	620	100			500 -13 ±	5
	650		98	99	523		-22 ±	7
	660	100			525 -23 ±	6
	700	99			550	8 ±	7
	700		97	98	573		5 ±	7
	740	99			575 -23 ±	5
	750		96	96	600	-7 ±	5
	775		94	91	623		-13 ±	6
	780	98			625	-7 ±	5
	800		93	91	650	2 ±	6
	820	96			673		-7 ±	5
	825		91	89	675	-8 ± 10
	850		91	91	700 19 ±	5
	860	92			723		11 ±	7
	875		86	85	725	-4 ±	8
	900	88			750	6 ±	6
	900		85	84	773		8 ±	5
	925		82	79	775 -13 ±	6
	940	80			800 12 ±	5
	950		79	75	823		-3 ±	4
	975		71	67	825	4 ±	6
	980	71			850 15 ±	5
	1000		67	61	873		24 ±	5
	1020	66			875	5 ±	7
	1025		68	60	923		58 ±	6
	1060	50			973		68 ±	6
					1023		94 ±	7
					1063		534 ± 11

Acknowledgment

This work was supported by the French ANR Program (2011JS56 004 01, FrIHIDDA) and by the Region Nord-Pas de Calais to M.R. The FTIR apparatus are supported by the Chevreul Institute, the Conseil Regional de Nord-Pas de Calais, the ERDF, and the Institut National des Sciences de l'Univers (INSU, CNRS).

Appendix:

In this appendix, we present fit to the raw isotopic data (spatially resolved in panel a and Rayleigh distillation plot of the same dataset on panel b) measured across a grossular single crystal after partial water extraction. These figures are the counterparts of Fig. 5c and Fig. 6c respectively. The presented data here were not processed. Instead, the data set was only internally standardized for the instrumental mass fractionation, considering the average isotopic composition of the inner part of the sample as the mean isotopic composition of the bulk starting grossular crystal (see text for the precise definition of the inner part of the sample). Fit to the data are identical to those presented in Fig. 5c and6c


Giorgio Audrito
email: giorgio.audrito@unito.it

Mirko Viroli
email: mirko.viroli@unibo.it

Ferruccio Damiani
email: ferruccio.damiani@unito.it

Jacob Beal
email: jakebeal@ieee.org

On a Higher-order Calculus of Computational Fields

Keywords: Distributed computing, Core calculus, Type system, Denotational semantics, Operational semantics, Computational adequacy

Computational fields have been proposed as an effective abstraction to fill the gap between the macro-level of distributed systems (specifying a system's collective behaviour) and the micro-level (individual devices' actions of computation and interaction to implement that collective specification), thereby providing a basis to better facilitate the engineering of collective APIs and complex systems at higher levels of abstraction. This approach is particularly suited to complex large-scale distributed systems, like the Internet-of-Things and Cyber-Physical Systems, where new mechanisms are needed to address composability and reusability of collective adaptive behaviour. This work introduces a full formal foundation for field computations, in terms of a core calculus equipped with typing, denotational, and operational semantics. Critically, we apply techniques for formal programming languages to collective adaptive systems: we provide formal establishment of a link between the micro-and macro-levels of collective adaptive systems, via a result of computational adequacy and abstraction for the (aggregate) denotational semantics with respect to the (per-device) operational semantics.

Background

Aggregate computing [START_REF] Beal | Aggregate programming for the Internet of Things[END_REF] is a paradigm aiming to address the complexity of largescale distributed systems, by means of the notion of computational field [START_REF] Mamei | Programming pervasive and mobile computing applications: The TOTA approach[END_REF] (or simply field ): this is a collective, distributed map from computational events (when and where a device executes a computational action, also called a round ) to computational objects (data values of any sort, including higher-order objects such as functions and processes) representing the result of computation at that event. Computing with fields means computing such global data structures, and defining a reusable block of behaviour means to define a reusable computation from fields to fields. This functional view holds at any level of abstraction, from low-level mechanisms of the language up to whole applications, which ultimately work by getting input fields from sensors and processing them to produce output fields for actuators. Most importantly, computing with fields is functional and hence declarative: (i) the designer focusses on the intended global goal of system behaviour, while the dynamics of interactions is left to the underlying platform (i.e., semantics); and (ii) one can scale with complexity by relying on functional composition: libraries of reusable building blocks can be constructed, and successive layering can be used to bottom-up derive whole applications.

The field calculus [START_REF] Damiani | A type-sound calculus of computational fields[END_REF] is a tiny functional language providing basic constructs to work with fields. 4 It provides a unifying approach to understanding and analysing the wide range of approaches to distributed systems engineering that make use of computational fields [START_REF] Beal | Organizing the aggregate: Languages for spatial computing[END_REF][START_REF] Viroli | From fieldbased coordination to aggregate computing[END_REF]. The operational semantics of field calculus [START_REF] Damiani | A type-sound calculus of computational fields[END_REF] can act as a blueprint for actual implementations where myriad devices interact via proximity-based broadcasts. More recently, the field calculus has been used to investigate formal properties of resiliency to environment changes [START_REF] Nishiwaki | Digamma-calculus: A universal programming language of selfstabilizing computational fields[END_REF][START_REF] Viroli | Engineering resilient collective adaptive systems by self-stabilisation[END_REF] and to device distribution [START_REF] Beal | Self-adaptation to device distribution in the internet of things[END_REF]. Its expressiveness has been investigated by introducing the cone Turing Machine [START_REF] Audrito | Space-time universality of field calculus[END_REF].

The higher-order field calculus [12] combines self-organisation and code mobility by extending the field calculus with a semantics for distributed first-class functions. It allows self-organisation code to be naturally handled like any other data, e.g., dynamically constructed, compared, spread across devices, and executed in safely encapsulated distributed scopes. Ultimately, this calculus provides programmers with a novel first class abstraction, a "distributed function field". This is a dynamically evolving map from a network of devices to a set of executing distributed processes: in each space-time region where the proces is the same, devices form a coalition collectively carrying on that process in isolation.

Contributions of [3]

This paper presents syntax and operational semantics of the higher-order field calculus together with new contributions: a type system for the higher-order version of the calculus, a denotational semantics, and associated properties. The new, enhanced syntax is parametric in the set of the modeled data values (in [12] Booleans, numbers, and pairs were explicitly modeled). Moreover, the if construct has been removed by encapsulating its branching capability into function calls, which now take the form of a function field applied to arguments, implicitly enacting branching. Then, a novel key insight and technical result of this paper is that the notoriously difficult problem of reconciling local and global behaviour in a complex adaptive system [START_REF] Viroli | Engineering resilient collective adaptive systems by self-stabilisation[END_REF] can be connected to a well-known problem in programming languages: correspondence between denotational and operational semantics. On the one hand, denotational semantics can be used to characterise computations in terms of their collective effect across space (available devices) and time (device computation events)-i.e., the macro level. On the other hand, operational semantics gives a transition system dictating each device's individual and local computing/interactive behaviour-i.e., the micro level. Correspondence between the two, formally proved in this paper via computational adequacy and a form of abstraction (c.f. [START_REF] Curien | Definability and full abstraction[END_REF][START_REF] Stoughton | Fully abstract models of programming languages[END_REF]) that we call computational abstraction, thus provides a formal micro-macro connection: one designs a system considering the denotational semantics of programming constructs, and an underlying platform running the distributed interpreter defined by the operational semantics guarantees a consistent execution. This is a significant step towards effective methods for the engineering of self-adaptive systems, achieved thanks to the standard theory and framework of programming languages.

Conclusions, Related and Future Work

The work presented in this paper builds on a sizable body of prior work, for which the field calculus can somewhat act as a lingua franca: foundational approaches to group interaction (ambients [START_REF] Cardelli | Mobile ambients[END_REF], shared-spaces [START_REF] Viroli | Spatial coordination of pervasive services through chemical-inspired tuple spaces[END_REF]), device abstraction languages (TOTA [START_REF] Mamei | Programming pervasive and mobile computing applications: The TOTA approach[END_REF], Hood [START_REF] Whitehouse | Hood: a neighborhood abstraction for sensor networks[END_REF]), pattern languages [START_REF] Nagpal | Programmable Self-Assembly: Constructing Global Shape using Biologically-inspired Local Interactions and Origami Mathematics[END_REF], information movement languages [START_REF] Newton | Region streams: Functional macroprogramming for sensor networks[END_REF], and spatial computing languages (MGS [START_REF] Giavitto | Computational models for integrative and developmental biology[END_REF] and Proto [START_REF] Bachrach | Composable continuous space programs for robotic swarms[END_REF]). Accordingly, future plans include consolidation of this work to investigate variants of the field calculus [2], to support an analytical methodology and a practical toolchain for system development, and to isolate fragments of the calculus that satisfy behavioural properties such as self-stabilisation developed in [START_REF] Viroli | Engineering resilient collective adaptive systems by self-stabilisation[END_REF].

Much as λ-calculus[START_REF] Church | A set of postulates for the foundation of logic[END_REF] captures the essence of functional computation and FJ[START_REF] Igarashi | Featherweight Java: A minimal core calculus for Java and GJ[END_REF] the essence of class-based object-oriented programming.