P Bonnand
email: pierre.bonnand@uca.fr

E Bruand

A K Matzen

M Jerram

F Schiavi

B J Wood

M Boyet

A N Halliday

Redox control on chromium isotope behaviour in silicate melts in contact with magnesiochromite

Keywords: Cr isotopes, Redox conditions, Magnesiochromite, partial melting, fractional crystallisation

Transition metal isotopes are particularly useful for understanding the conditions under which magmatic processes occur. Moreover, those with more than one oxidation state (e.g. Cr 2+ , Cr 3+ and Cr 6+ ) may also provide powerful constraints on the evolution of the redox state of the mantle over time. By investigating the Cr isotopic compositions in both magnesiochromite and silicate melts during experiments performed at 1300 °C and under controlled redox conditions (-12 < logfO2 < -4), this study presents the first experimental petro-isotopic investigation of Cr isotope fractionation and reveals clear systematics between Cr concentration, isotopic compositions and fO2. Two series of experiments were performed to study (a) the dissolution of a natural magnesiochromite into Cr-free silicate melts (series A) and (ii) the crystallisation of magnesiochromite from Cr-doped silicate melts (series B). In agreement with previous studies, the Cr solubility in the silicate melts at equilibrium with magnesiochromite is strongly controlled by oxygen fugacity. Melts produced at low fO2 are enriched in Cr compared to more oxidised melts. In series A experiments, the Cr isotopic composition of silicate melts are lighter than the initial chromite starting material. The experiments, performed at -12 < logfO2 < -6, reveal that Cr isotopic compositions of the silicate melts are correlated with fO2. This demonstrates that, as for the Cr solubility, Cr isotopes are sensitive to fO2 and could be used to track changes in redox conditions in hightemperature processes. Furthermore, the Cr isotopic compositions of silicate melts that are reacted under more oxidising conditions (logfO2 > -6) are much lighter than those of melts equilibrated with magnesiochromite at lower oxygen fugacity. The observed variations can be explained by changes in bonding environment for Cr under oxidised conditions in the silicate melts and/or in the magnesiochromite grains. Similarly, the second set of experiments designed to study fractional crystallisation (series B) suggest that Cr isotope fractionation is larger under oxidising conditions.

Introduction

Transition metals are useful for understanding the conditions of differentiation processes such as core formation (e.g. [START_REF] Bourdon | Isotope tracers of core formation[END_REF]. Elements with more than one oxidation state can also provide powerful constraints on the evolution of the redox conditions in the mantle through time (e.g. [START_REF] Williams | Iron isotope fractionation and the oxygen fugacity of the mantle[END_REF]). The ability to determine isotopic fractionations associated with such differentiation processes has provided new tools whereby early Earth conditions can be explored. Chromium (Cr) is of particular interest because it can be present in two main oxidation states in silicate melts (Cr 2+ and Cr 3+ ), although during quenching Cr 2+ is oxidised by Fe 3+ and only Cr 3+ has been measured in room-temperature silicate glasses [START_REF] Berry | A XANES determination of the oxidation state of chromium in silicate glasses[END_REF]. Chromium isotopic compositions have been measured in a number of terrestrial and extra-terrestrial samples (e.g. [START_REF] Schoenberg | The stable Cr isotope inventory of solid Earth reservoirs determined by double spike MC-ICP-MS[END_REF][START_REF] Moynier | Isotopic Evidence of Cr Partitioning into Earth's Core[END_REF][START_REF] Farkas | Chromium isotope variations (delta Cr-53/52) in mantle-derived sources and their weathering products: Implications for environmental studies and the evolution of delta Cr-53/52 in the Earth's mantle over geologic time[END_REF][START_REF] Li | Precise measurement of Cr isotope ratios using a highly sensitive Nb2O5 emitter by thermal ionization mass spectrometry and an improved procedure for separating Cr from geological materials[END_REF], Bonnand et al. 2016a, Bonnand et al. 2016b[START_REF] Schoenberg | The stable Cr isotopic compositions of chondrites and silicate planetary reservoirs[END_REF][START_REF] Xia | Chromium isotope heterogeneity in the mantle[END_REF][START_REF] Bonnand | Oxidized conditions in iron meteorite parent bodies[END_REF][START_REF] Shen | Hightemperature inter-mineral Cr isotope fractionation: A comparison of ionic model predictions and experimental investigations of mantle xenoliths from the North China Craton[END_REF][START_REF] Shen | Stable chromium isotope fractionation during magmatic differentiation: Insights from Hawaiian basalts and implications for planetary redox conditions[END_REF][START_REF] Bonnand | The influence of igneous processes on the chromium isotopic compositions of terrestrial basalts[END_REF]). However, very little experimental work has been done to independently verify the isotopic effects found in nature. The aim of this study is to investigate the impact of oxygen fugacity on the behaviour of Cr and its isotopic compositions during partial melting and fractional crystallisation in a basaltic system. To this end, the Cr isotopic compositions of silicate melts and chromite have been measured in the run products of 1-atm experiments performed under controlled oxygen fugacity.

Background

Chromium in the mantle and in silicate melts

The Cr concentrations in terrestrial basalts are highly variable and can be up to 2000 µg g -1 (e.g. [START_REF] Maclennan | Geochemical variability in a single flow from northern Iceland[END_REF]. The Cr concentration in mantle peridotites is also variable ranging from 1000 µg g -1 to about ~4000 µg g -1 (e.g. Bodinier and Godard 2013 and references therein). In basaltic and peridotitic systems, Cr-bearing minerals are spinel, garnet, orthopyroxene and clinopyroxene. During partial melting, Cr is a compatible element meaning that Cr concentrations in the melt are lower than that of the solid residue. The solubility of Cr in the silicate melt is mainly controlled by the partition coefficient between spinel and/or garnet and the silicate melts. The crystal-melt partition coefficient (Dcrystal-melt) of Cr into spinel (~ 170, Liu and O'Neill 1994) is higher than that of garnet (~ 5.5, McKenzie and O'Nions 1995) and this suggest that melts produced at higher pressure should be enriched in Cr compared to silicate melts produced at low pressure (e.g. [START_REF] Bonnand | The influence of igneous processes on the chromium isotopic compositions of terrestrial basalts[END_REF]. During fractional crystallisation in basaltic systems, Cr is mainly hosted in spinel and clinopyroxene depending on the chemical composition of the silicate melt and the temperature of crystallisation.

The presence of Cr 2+ in silicate melts has been demonstrated by in-situ XANES measurements, performed by [START_REF] Berry | The effect of composition on Cr2+/Cr3+ in silicate melts[END_REF]. They also show that the ratio Cr 2+ /CrTOT varies with oxygen fugacity and the chemical composition of the silicate melts. The solubility of chromium in silicate melts is dependent on temperature, oxygen fugacity and the chemical composition of the silicate melt [START_REF] Roeder | Crystallisation of chromite and chromium solubility in basaltic melts[END_REF][START_REF] Hanson | The systematics of Cr3+ and Cr2+ partitioning between olivine and liquid in the presence of spinel[END_REF][START_REF] Huang | Mobility of chromium in high temperature crustal and upper mantle fluids[END_REF]. A number of experimental studies have demonstrated that Cr 2+ is more soluble than Cr 3+ (e.g. [START_REF] Murck | The effects of temperature, oxygen fugacity and melt composition on the behaviour of chromium in basic and ultrabasic melts[END_REF]. The Cr solubility in silicate melts increases under reduced conditions (from 0.072 Cr2O3 wt % at logfO2 = -5.1 to 1.43 wt. % at logfO2 = -12.78, [START_REF] Roeder | Crystallisation of chromite and chromium solubility in basaltic melts[END_REF]. This observation is consistent with high Cr concentration in the lunar basalts, compared with their terrestrial counterparts. Chromium solubility also increases with temperature (at logfO2 = -8, from 0.085 wt % at 1200 ˚C to 0.162 wt. % at 1300 ˚C, [START_REF] Roeder | Crystallisation of chromite and chromium solubility in basaltic melts[END_REF]Reynolds 1991, Hanson and[START_REF] Hanson | The systematics of Cr3+ and Cr2+ partitioning between olivine and liquid in the presence of spinel[END_REF]. In silicate melts, Cr 3+ is octahedrally coordinated whereas Cr 2+ is located in a distorted octahedral site [START_REF] Keppler | Crystal field spectra and geochemistry of transition metal ions in silicate melts and glasses[END_REF]).

Chromite

In ultramafic and mafic magmatic reservoirs, the main Cr-bearing phases are spinel and pyroxene. There are two types of spinel minerals: 2-3 spinels such as MgAl2O4 and MgFe2 3+ O4 and 2-4 spinels such as Fe2TiO4 and the high pressure form of Mg2SiO4 [START_REF] Biagioni | The systematics of the spinel-type minerals: An overview Am[END_REF]. Chromite is a 2-3 spinel-structured oxide, common in a variety of rocks such as mantle xenoliths and basalts but also in metamorphic and sedimentary rocks [START_REF] Dick | Chromian spinel as a petrogenetic indicator in abyssal and alpinetype peridotites and spatially associated lavas[END_REF]. The 2-3 spinel structure A 2+ B 3+

2O4 is based on a nearly ideal cubic close-packed array of oxygen atoms with tetrahedral and octahedral sites. The cations in the tetrahedral sites (A, IV fold coordinated) are usually Mg 2+ , Fe 2+ , Zn 2+ , Mn 2+ . The cations in the octahedral sites (B, VI fold coordinated) are Al 3+ , Fe 3+ and Cr 3+ . The spinel structure can be modified to accommodate the presence of high Fe 3+ concentration (e.g. Lenaz and Lughi 2013). In this case the Fe 3+ can move to the A site and this structure is referred to as inverse spinel. It is commonly assumed however, that magnesiochromite has a normal spinel structure. The normal and inverse structures are two end-members which form a continuum and depending on the chemical composition and the physical conditions, 3+ cations can migrate to the A site.

It is important to note, however, that the spinel structure is such that it can host a large number of trace elements including transition metals and high-field-strength elements (HFSE) [START_REF] Nielsen | Major-and trace-element magnetitemelt equilibria[END_REF]. With such high partition coefficients, spinel plays an important role in the trace element budget in most natural igneous rocks, despite relatively low modal proportions. For example, spinel controls the behaviour of many transition metals during partial melting and fractional crystallisation (e.g. [START_REF] Li | Subsolidus Phase Relations in the System MgO-SiO 2-Ci^O in Equilibrium with Metallic Cr, and their Significance for the Petrochemistry of Chromium Subsolidus phase relations have been determined in the systems[END_REF]. Indeed, chromite is often associated with economically important platinum group element (PGE) deposits [START_REF] Naldrett | PGE tenor and metal ratios within and below the Merensky Reef, Bushveld Complex: Implications for its genesis[END_REF].

The variations in the chemical compositions of natural chromite has attracted a lot of attention and several parameters are commonly used to describe the observed compositions:

Cr# = Cr 3+ / (Cr 3+ + Al 3+ ) Fe 3+ # = Fe 3+ / (Cr 3+ + Al 3+ + Fe 3+ ) Mg# = Mg 2+ / (Mg 2+ + Fe 2+ )
where Cr 3+ , Al 3+ , Fe 3+ , Fe 2+ and Mg 2+ are atomic abundances calculated assuming 4 oxygen atoms in the spinel structure and that Cr is only present as Cr 3+ . In natural spinel, there are large compositional variations between several endmembers and these variations have been linked to the chemical compositions of basalts and/or peridotite, crystallisation temperature, and pressure (e.g. [START_REF] Warren | Global variations in abyssal peridotite compositions[END_REF]). A number of previous studies have shown that partitioning of transition metals and/or HFSE between spinel/chromite and melts depends on temperature, pressure and the chemical composition of the crystal and melt [START_REF] Nielsen | Major-and trace-element magnetitemelt equilibria[END_REF][START_REF] Horn | Experimentally determined partitioning of high field strength-and selected transition elements between spinel and basaltic melt[END_REF][START_REF] Nielsen | Magnetitemelt HFSE partitioning[END_REF][START_REF] Blundy | Partitioning of trace elements between crystals and melts[END_REF], Righter et al. 2006[START_REF] Davis | Revisiting the electron microprobe method of spinel-olivine-orthopyroxene oxybarometry applied to spinel peridotites[END_REF]. Redox state also plays a major role in controlling the chemical composition of natural chromite and many experimental studies have calibrated these variations (e.g. Wood et al. 1990, Frost and[START_REF] Frost | The Redox State of Earth's Mantle[END_REF]. Although chromite plays a major role in controlling the Cr behaviour in magmatic system, no experimental work has been conducted on the impact of chromite on the behaviour of Cr isotopes.

Chromium isotopes

The variations in Cr isotopes are reported using the notation: Most recent analyses on mantle xenoliths and mineral separates have shown that Cr isotopes are fractionated during partial melting [START_REF] Xia | Chromium isotope heterogeneity in the mantle[END_REF][START_REF] Farkas | Chromium isotope variations (delta Cr-53/52) in mantle-derived sources and their weathering products: Implications for environmental studies and the evolution of delta Cr-53/52 in the Earth's mantle over geologic time[END_REF][START_REF] Bai | Diffusion-driven chromium isotope fractionation in ultramafic cumulate minerals: Elemental and isotopic evidence from the Stillwater Complex[END_REF].

The residues become heavier while partial melting proceeds and the melts are therefore believed to be lighter than the residues [START_REF] Xia | Chromium isotope heterogeneity in the mantle[END_REF][START_REF] Shen | Hightemperature inter-mineral Cr isotope fractionation: A comparison of ionic model predictions and experimental investigations of mantle xenoliths from the North China Craton[END_REF]. Furthermore, data from mineral separates reveal that, at equilibrium, the Cr isotopic compositions become heavier in the order olivine ≤ cpx = opx < spinel [START_REF] Shen | Hightemperature inter-mineral Cr isotope fractionation: A comparison of ionic model predictions and experimental investigations of mantle xenoliths from the North China Craton[END_REF]. It has also been demonstrated that Cr isotopes are fractionated during magmatic differentiation on Earth and on the Moon [START_REF] Bonnand | The influence of igneous processes on the chromium isotopic compositions of terrestrial basalts[END_REF][START_REF] Shen | Stable chromium isotope fractionation during magmatic differentiation: Insights from Hawaiian basalts and implications for planetary redox conditions[END_REF], Bonnand et al. 2016b). Recently, it has been proposed that the isotopic fractionation occurring during fractional crystallisation depends on the the Cr 3+ /CrTOT ratios in both the silicate melts and the crystallising phases [START_REF] Farkas | Chromium isotope variations (delta Cr-53/52) in mantle-derived sources and their weathering products: Implications for environmental studies and the evolution of delta Cr-53/52 in the Earth's mantle over geologic time[END_REF][START_REF] Schoenberg | The stable Cr isotopic compositions of chondrites and silicate planetary reservoirs[END_REF][START_REF] Bonnand | The influence of igneous processes on the chromium isotopic compositions of terrestrial basalts[END_REF][START_REF] Shen | Stable chromium isotope fractionation during magmatic differentiation: Insights from Hawaiian basalts and implications for planetary redox conditions[END_REF]. The Cr isotopic variations in HED (Howardite-Eucrite-Diogenite) have also been linked to magmatic processes as well as Cr volatility during planetary accretion [START_REF] Zhu | Tracking the volatile and magmatic history of Vesta from chromium stable isotope variations in eucrite and diogenite meteorites[END_REF]. Comparing Cr isotope signatures in samples from different planetary bodies is complicated further by the fact that processes active during planet formation may have affected the isotopic compositions of the major reservoirs. For example, the observed difference in Cr isotopes (δ 53 Cr) between the Moon and the Earth has been attributed to the volatility of Cr during the late stages of the Moon forming Giant Impact [START_REF] Sossi | Volatile loss following cooling and accretion of the Moon revealed by chromium isotopes[END_REF]. In several studies, the variations observed in Cr isotopes in silicate melts and mantle xenoliths have been linked to the isotopic composition of chromite [START_REF] Farkas | Chromium isotope variations (delta Cr-53/52) in mantle-derived sources and their weathering products: Implications for environmental studies and the evolution of delta Cr-53/52 in the Earth's mantle over geologic time[END_REF][START_REF] Xia | Chromium isotope heterogeneity in the mantle[END_REF][START_REF] Shen | Hightemperature inter-mineral Cr isotope fractionation: A comparison of ionic model predictions and experimental investigations of mantle xenoliths from the North China Craton[END_REF], Bonnand et al. 2016[START_REF] Bonnand | The influence of igneous processes on the chromium isotopic compositions of terrestrial basalts[END_REF][START_REF] Shen | Stable chromium isotope fractionation during magmatic differentiation: Insights from Hawaiian basalts and implications for planetary redox conditions[END_REF]. Our ability to estimate the Cr isotopic composition of silicate reservoirs is hampered by our limited understanding of the fractionation factors in high-temperature processes.

Analytical methods

Experiments

The starting compositions used in this study are listed in Table 1. These compositions were synthesised using reagent-grade oxide powders, which were mixed by grinding under ethanol in an agate mortar and pestle.

Two series of experiments were conducted during the course of this study. The first (series A) was designed to study the dissolution of chromite into a Cr-free silicate melt. The second (series B) was designed to study the fractional crystallisation of chromite out of the silicate melts. The only chemical difference between the two silicate melts used in this study is the Cr concentration (see Table 1). The bulk compositions of the melts were chosen such that they were completely molten at 1300°C. To avoid Fe loss from the silicates, loops were presaturated by performing multiple saturation experiments (e.g. [START_REF] Hanson | The systematics of Cr3+ and Cr2+ partitioning between olivine and liquid in the presence of spinel[END_REF]. We performed three saturation experiments (72 h per experiment) for each loop at the required temperature and fO2.

For each series A experiment, 100 mg of starting material A and 5 mg of natural chromite grains (50-200 µm) were combined and pressed into a small pellet (~4 mm diameter). For each series B experiment, 100 mg of starting material B was weighed and pressed into a small pellet. Experimental conditions (e.g. oxygen fugacity, loop material) for each experiment are given in Table 2. Using the series A experimental method, we performed a time series, using a single Pt loop. The loop was used to perform 5 experiments (A3a, A3b, A3c, A3d and A3)

with run durations from 2 h to 168 h (Table 2). Between each run, the silicate material was removed and the loop was cleaned in a cold bath of concentrated HF. In addition to the time series, six experiments were performed using method A (A1 to A7), and three experiments were performed using method B (B1 to B3). For each series, the experiment number is given with increasing oxygen fugacity (i.e. A1 is the experiment performed under the more reducing conditions for the series A experiments).

Experiments were performed at the Department of Earth Sciences of the University of Oxford. A schematic is presented in Fig. 1. The experiments were performed by holding the powdered starting material in the hot spots of a gas-mixing furnace (Fig. S1). During the experiments, the temperature was controlled to within ± 5 °C of the desired temperature (1300 °C), and monitored by a thermocouple adjacent to the sample. Oxygen fugacity was controlled using flowing CO and CO2. Experiments were performed from two log units below to six log units above the Iron-Wüstite (IW) buffer. An O2 sensor, made of yttria-stabilized zirconia, was used to monitor the oxygen fugacity during the duration of each experiment.

The measured fO2 is thought to be accurate to within 0.1 (1 s.d.) log units. Depending on oxygen fugacity, starting materials were suspended using Pt or Re-loops. The samples for series A experiments were slowly placed into the furnace at 800°C, the temperature was then increased to 1300°C where it was held for 168 h. The experimental procedure was similar for the series B experiments except, in order to facilitate the crystallization of chromite, temperature cycling was employed at the beginning of the experiment. After insertion into the furnace the temperature was increased to 1550 °C, followed by cooling to 1300 °C at 5 °C h -1 ;

the temperature was increased again to 1450 °C followed by cooling to 1300 °C at 5°C h -1 after which the temperature was kept constant (at 1300 °C) for 168 h. At the end of all experiments, samples were quenched by dropping them into water.

Previous studies have reported Cr loss during experiments performed both at high and low fO2 [START_REF] Hanson | The systematics of Cr3+ and Cr2+ partitioning between olivine and liquid in the presence of spinel[END_REF]. While Cr loss at low fO2 is likely due to Cr entering the loop, Cr volatility was observed in experiments performed in air (logfO2 = -0.68, [START_REF] Hanson | The systematics of Cr3+ and Cr2+ partitioning between olivine and liquid in the presence of spinel[END_REF], an fO2 that is higher than those used in this study (logfO2 range from -12 to -4). More recently, [START_REF] Sossi | Evaporation of moderately volatile elements from silicate melts: experiments and theory[END_REF] proposed that Cr is volatile at 1-atm under oxidizing conditions (logfO2 > -7) at 1300°C. Specifically, they lost almost 80% of their starting Cr after 15 minutes. In contrast, the experiments of Norris and Wood [START_REF] Norris | Earth's volatile contents established by melting and vaporization[END_REF], also conducted at 1300°C and an logfO2 range of -13 to -7, provide evidence that Cr is not volatile. Our experiments show no clear evidence of Cr loss, either to the loop, or by volatility.

Sample preparation

Sample preparation for isotope measurements required a different approach between the series A and B experiments. For the A series, the samples retrieved after the experiments were lightly crushed in an agate mortar and pestle and the large chromite grains (>200 µm) were picked from the silicate glass using a binocular stereoscope. Chromite grains were dissolved in a mixture of acids (HNO3-HCl), using Parr bombs. Despite our best efforts to separate the two phases, a small amount of silicate melt often adhered to the chromite grains. As we show below, this represents a negligible change to the Cr budget of the sample. The silicate melt fractions were first dissolved in cold HF-HNO3 (3:1) for 24h and then centrifuged to make sure that no chromite grains were present. The samples were then dissolved in 6M HCl to remove the fluorides. For the series B experiments, after crushing in an agate pestle and mortar, the silicate melts were picked under a binocular microscope. The small size of the chromite grains (< 15 µm) meant that it was not possible to effectively separate the chromite grains from the silicate melts by hand. In order to achieve this separation, we performed a sequential leaching of chromite rich sample splits. To this end, the samples were dissolved in cold diluted (3M) HF for 24h. Following this, the samples were then centrifuged for 10 min at 3000 rpm and the supernatant was removed and discarded. The samples were then rinsed with de-ionised water and centrifuged again. The isolated chromite grains were dissolved in acid mixtures (HNO3-HCl).

For microprobe and Raman spectroscopy analyses, small pieces of each experiment, containing both silicate melts and chromite grains, were picked, mounted in epoxy and polished.

Microprobe analyses

A Cameca SX5 tactis microprobe and a Cameca SX100 microprobe at the Laboratoire Magmas et Volcans (LMV) were used to determine major-elements composition of the chromites and silicate melts. A 15 kV accelerating voltage, a 15 nA beam current, and a 1 µm electron beam were used. Standards used to calibrate the instrument were Al on Al2O3, Mg on MgO, Na on albite, Ti and Mn on MnTiO3, Ca and Si on wollastonite, Fe on Fe2O3 and Cr on Cr2O3. Cr#, Fe 3+ # and Mg# presented in this study have been calculated from EPMA analyses by assuming ideal stoichiometry of the spinel phase. Fe 3+ # and Mg# were calculated by assigning Fe cations either as ferric or ferrous ions to balance negative charge [START_REF] Stormer | The effects of recalculation on estimates of temperature and oxygen fugacity from analyses of multicomponent iron-titanium oxides[END_REF].

Previous studies have shown that Fe 3+ /FeTOT determination using EPMA analyses only, can be affected by large analytical uncertainties (e.g. [START_REF] Wood | Upper mantle oxidation state: Ferric iron contents of Iherzolite spinels by 57Fe Mössbauer spectroscopy and resultant oxygen fugacities[END_REF]. [START_REF] Wood | Upper mantle oxidation state: Ferric iron contents of Iherzolite spinels by 57Fe Mössbauer spectroscopy and resultant oxygen fugacities[END_REF] provided a robust methodology to correct for this effect using Mössbauer calibrated standards. This technique has been widely used to correct EPMA analyses (e.g. [START_REF] Parkinson | Peridotites from the Izu-Bonin-Mariana forearc (ODP Leg 125): evidence for mantle melting and melt-mantle interaction in a supra-subduction zone setting[END_REF]. Recently, [START_REF] Davis | Revisiting the electron microprobe method of spinel-olivine-orthopyroxene oxybarometry applied to spinel peridotites[END_REF] confirmed that the correction improves accuracy and precision of Fe 3+ /FeTOT values if the analysed chromite lies on the mantellic spinel MgO-Cr# correlation. In this study, we have analysed Mössbauer calibrated chromite standards to assess the accuracy of our EMPA methodology (supplementary material). We found that the Fe 3+ /FeTOT ratios obtained with our EPMA measurements are in agreement with Mössbauer Fe 3+ /FeTOT ratios obtained for the same samples (Wood andVirgo, 1989, Ionov and[START_REF] Ionov | The oxidation state of subcontinental mantle: oxygen thermobarometry of mantle xenoliths from central Asia[END_REF], which suggests that no correction is needed. Using the standard data, we have determined the analytical uncertainty (2 s.d.) associated with Mg#, Cr# and Fe 3+ # calculations (± 0.013, ± 0.010 and ± 0.005, respectively).

Chromium isotope measurements

Protocols for the separation of Cr from silicate matrices have been previously published (e.g. [START_REF] Trinquier | High-precision analysis of chromium isotopes in terrestrial and meteorite samples by thermal ionization mass spectrometry[END_REF]Bonnand et al. 2016a). Only a brief summary is given here:

Aliquots of ~2 µg of chromium of the solutions were spiked with the requisite amount of 50 Cr-54 Cr double spike [START_REF] Bonnand | Accurate and precise determination of stable Cr isotope compositions in carbonates by double spike MC-ICP-MS[END_REF]). The solutions were dried down and taken up in 6M HCl to ensure isotopic equilibrium between the samples and the spike. The protocol used to separate the Cr fraction from the matrix is a two-column procedure (Bonnand et al. 2016a).

The first column is designed to remove the major cations while the second column removes the few remaining isobaric interferences, such as Ti. The isotopic measurements were performed on a ThermoFisher Triton TIMS at the University of Oxford and at the LMV.

Typical Cr isotope measurements consisted of 54 blocks of 10 cycles. The gains were measured daily and the baselines (30 s) were measured before each block. The amplifier rotation was used to cancel out the gain differences between each Faraday cup. The reproducibility obtained for the JP-1 reference material over the course of this study is δ 53 Cr = -0.108 ± 0.014 ‰ (2 s.d., n = 8) in agreement with previous measurements (Bonnand et al. 2016a[START_REF] Li | Precise measurement of Cr isotope ratios using a highly sensitive Nb2O5 emitter by thermal ionization mass spectrometry and an improved procedure for separating Cr from geological materials[END_REF][START_REF] Zhu | An improved method of Cr purification for high precision measurement of Cr isotopes by double spike MC-ICP-MS[END_REF].

Results

Times series

The chemical compositions of the silicate melts in the time series experiments do not vary except for Cr. The Cr concentration evolves from 0.02 ± 0.03 wt. % after 2 hours to 0.18 ±0.02 wt. % after 120 and 168 hours (Fig. 2). and A7 (logfO2 -5.8, and -3.9, respectively). Their chromite grains are chemically homogeneous in Mg# but heterogeneous in Fe 3+ # with the rims enriched in Fe 3+ and depleted in Cr compared to the cores (Table 4). Interestingly, A6 and A7 chromites are enriched in Mg relative to Fe 2+ compared to the starting material compositions (Fig. 4). Overall, all series-A experiments show covariation between log fO2 and Cr# (and Fe 3+ #) in the rims of the chromite grains (Fig. 5).

Series B

In this experimental series, chromite grains are euhedral and up to 20 microns in size (Fig. S4). The chemical composition of chromite grains crystallised in the series B experiments are highly heterogeneous (Table 4 and Fig. 6). The three experiments have Mg# values ranging from 0.67 to 0.79. The Fe 3+ # and Cr# vary from 0 to 0.09 and 0.47 to 0.84, respectively. The core of the bigger chromite grains have generally lower Mg# and Fe 3+ # and higher Cr# than their rims or the small chromite grains. Differences between B experiments can also be observed, for example, the small (i.e. rims) chromite grains in experiments B1 have systematically lower Fe 3+ # and Mg# compared to B3 experiment (Fig. 5).

Chromium isotopic compositions

The Cr isotopic compositions of all silicate melts and chromite grains are given in Table 5. The results presented in this study show that the chemical and isotopic compositions of the silicate melts in our experiments vary through time and depend on oxygen fugacity.

During the time series experiments the silicate melt compositions are invariant for all the major elements but Cr concentrations vary (Fig. 2). In these experiments, the initial Cr concentration in the melt is ~0 and it gradually increases from ~50 µg g -1 after 2 h to 1200 µg g -1 after 120 h, as expected from dissolution of the chromite grains into the silicate melts. The fact that the Cr concentration in the melt reaches a plateau after 120 h suggests that the Cr saturation concentration in the melt has been reached, which at 1300 ºC and logfO2 = -8.6, is approximately 1200 µg g -1 .

The Cr isotopic compositions of the silicate melts also evolve with time. After 2 hours, the δ 53 Cr value in the melt is -0.251 ‰ and it rapidly evolves to heavier values after 24 h (-0.167 ‰), reaching a plateau after 120 h (-0.143 ‰, Fig. 2). The dissolution of chromite into the silicate melts seems to be driven by a kinetic reaction. Indeed, the first Cr going into the melts is isotopically light compared to the chromite starting material and it evolves toward heavier values with time (Fig. 2). As with the Cr concentrations, it seems that the isotopic composition of the melt reaches steady state after 120-168 h. Due to the slow diffusion rate of Cr 3+ in the chromite and the resulting zoning observed in chromite of the series A experiments, it is impossible to argue for full isotopic equilibrium in the experimental charge.

However, the plateau observed in the time series experiments suggests that the Cr isotopic compositions of the silicate melts are at local equilibrium with the rims of the chromite and we argue that the silicate melts approach Cr isotopic composition equilibrium after 120 h. We therefore decided to run our other experiments for 168 h in order to reach the Cr saturation concentration in the silicate melts and to approach isotopic equilibrium.

Chemical evolution of the silicate melt compositions

The chemical composition of the silicate melts in individual experiments and between the experiments of the series A and B are homogeneous, except for Cr. In Figure 3, the Cr concentration in both series of experiments are plotted against oxygen fugacity. The Cr concentration in the melt increases with decreasing oxygen fugacity (Fig. 3). The two series of experiments lie on the same solubility curve highlighting that, in both cases, the Cr concentration of the melt has closely approached equilibrium with the chromite. Importantly, the increase in Cr solubility in the silicate melts with decreasing fO2 is in agreement with previous studies that had already shown that Cr is more soluble in silicate melts under reducing conditions (Fig. 2, [START_REF] Roeder | Crystallisation of chromite and chromium solubility in basaltic melts[END_REF][START_REF] Hanson | The systematics of Cr3+ and Cr2+ partitioning between olivine and liquid in the presence of spinel[END_REF].

In a silicate melt, chromium occupies two principal oxidation states, Cr 2+ and Cr 3+ . Cr 2+ is absent from room-temperature silicate glasses due to the oxidation of Cr (Fe 3+ + Cr 2+ → Fe 2+ + Cr 3+ ) during quenching [START_REF] Berry | The effect of composition on Cr2+/Cr3+ in silicate melts[END_REF]). This makes it impossible to measure the Cr 2+ /Cr 3+ ratio in our silicate glasses. [START_REF] Hanson | The systematics of Cr3+ and Cr2+ partitioning between olivine and liquid in the presence of spinel[END_REF] have proposed a method to calculate the Cr 2+ /Cr 3+ ratios in silicate melts in the presence of spinel. The main assumption in this method is that i) at the Ni-NiO oxygen buffer, Cr is only present in Cr 3+ , and ii) that the Cr 3+ concentration in the melt is independent of fO2 [START_REF] Hanson | The systematics of Cr3+ and Cr2+ partitioning between olivine and liquid in the presence of spinel[END_REF]. The latter assumption is based on the fact that, in the presence of spinel, the activity of Cr 3+ (aCrO1.5) remains constant and does not vary with fO2. In order to calculate the Cr 2+ /Cr 3+ ratio in our silicate melts, we used the following equation:

Cr 2+ /Cr 3+ = (Crmelt -Croxy) / Croxy
where Crmelt and Croxy are the Cr concentrations in a given experiment and in the most oxidised experiment, respectively. The similarity in composition and experimental conditions between the A and B series means that for the Cr 2+ /Cr 3+ calculation, we assume that the Cr 3+ concentration in our experiments is the average Cr concentration in the experiments performed above the Ni-NiO buffer (A6, A7 and B3). In Figure 7, we report the log(Cr 2+ /Cr 3+ ) versus the log of oxygen fugacity for our series of experiments. We obtain a slope of -0.25 ± 0.01 (series A) and -0.27 ± 0.01 (series B) close to the theoretical slope of -0.25 expected for a redox reaction involving a one electron exchange. This is strong evidence that the increase in Cr solubility in our experiments under reducing conditions is controlled by the difference in solubility between Cr 3+ and Cr 2+ in silicate melts. This conclusion is in agreement with previous studies on this topic (e.g. [START_REF] Hanson | The systematics of Cr3+ and Cr2+ partitioning between olivine and liquid in the presence of spinel[END_REF].

Together with Cr, Fe is another redox sensitive element and the Fe 3+ /FeTOT in the silicate melt is also expected to vary with fO2. We have calculated the Fe 3+ /FeTOT ratio in our silicate melts using the calibration given in O' Neill et al. (2018). The results of this calculation are shown in Figure 7b. As expected, given the temperature and the redox conditions in which we performed our experiments the Fe 3+ /FeTOT ratio varies greatly in our silicate melts from 0.02 to 0.45 (Table 3). The amount of Fe 3+ in the melt varies less than the Cr 3+ /CrTOT ratio calculated above because the oxidation reaction for Fe happens at higher fO2 than that of Cr.

Chromite chemical evolution

The chromite grains analysed in this study are heterogeneous. The large variations in chemical composition, illustrated by the zoning observed in the chromite from the series A experiments, clearly indicate that the cores of the chromite grains are not at equilibrium with the silicate melts. This can be easily explained by the slow diffusion rate of Cr 3+ in chromite [START_REF] Suzuki | Cr and Al diffusion in chromite spinel: Experimental determination and its implication for diffusion creep[END_REF]). However, several systematics in the chromite zoning can be identified (Fig. 4 andS1). For the two most oxidised experiments (exp A6 and A7), the Mg# is homogeneous and higher than the Mg# in the starting material. This observation indicates that, under oxidising conditions, Mg 2+ and Fe 2+ diffuse relatively fast and have sufficient time to equilibrate with the melt. In the same experiments, the Fe 3+ # is higher in the rim than in the cores (Fig. 4). This suggests that the original chromite grain was depleted in Fe 3+ compared to the chromite at equilibrium with the silicate melts. The trends observed can be explained by diffusion of Fe 3+ from the melt together with diffusion of Al and Cr from the chromite to the melt.

In the more reduced experiments, the chromite grains are zoned in Mg#, Fe 3+ # and Cr#.

The Mg# is higher in the rims than in the cores (Fig. 4), indicating that the starting material was Fe 2+ rich compared to the chromite at equilibrium with the silicate melts at different fO2.

This indicates that Fe 2+ diffuses out of the chromite grain even at low fO2 but the diffusion rate of Mg 2+ and Fe 2+ in the chromite is slower under more reducing conditions. The Mg# in the rims does not correlate with fO2. The Fe 3+ # and Cr# are both lower in the rims than in the cores which suggests that the chromite grains are losing Fe and Cr to the melt and/or gaining Al from the melt. This is consistent with a decrease in Fe 3+ /FeTOT and Cr 3+ /CrTOT in the silicate melts described in the section above. As for the most oxidised experiments, the cores of chromite are not at equilibrium with the silicate melts.

In Figure 4, all chromite data are reported and general trends can be easily seen. First, there is a clear fO2 control on the rate of diffusion of Fe 2+ in the chromite grains. Indeed, although all rims tend to have higher Mg# values compared to the starting material, it is clearly the most oxidised experiments that have the highest Mg#. Considering Fe 3+ # and Cr#, the rims values are also extremely variable and, overall, the oxidised experiments have higher Fe 3+ # values compared to the more reduced experiments. The rim chemical compositions are strongly controlled by fO2 and this is shown in Figure 5. There is a positive correlation between Cr# and Fe 3+ # in the rim compositions and fO2. The most reduced experiments have rim Cr# and Fe 3+ # compositions of 0.5 and 0, respectively. The more oxidised experiments have rim compositions of 0.7 and 0.12. This clearly shows that the composition of the chromite grains in our experiments is controlled by fO2. Under reducing conditions, Fe 3+ and Cr 3+ are lost from the initial chromite. Under more oxidised conditions, Fe 3+ becomes more stable in the chromite structure and Cr is also favoured. Assuming the rim compositions are in equilibrium with the silicate melts then, this indicates that the partition coefficient of Cr and Fe between chromite and a silicate melt is strongly fO2 dependent. The DCr (DCr = Crchromite/Crmelt) calculated between the rims and the silicate melts in the experiments series A are presented in Figure 8. The DCr values vary from 55 to 570 and there is a good correlation between DCr and fO2. We also present DCr3+ (DCr3+ = Crchromite/Cr 3+ melt) using the calculations presented in section 5.1. In the series A, the DCr3+ increases with fO2 (from logfO2 = -12 to logfO2 = -6) and then there is a break with smaller DCr3+ values at logfO2 > -6 (Fig. 8). This suggests that Cr 3+ becomes less compatible in chromite under oxidising conditions.

In the series B experiments, the chromite grains are also extremely variable. In this study, we have systematically analysed the large and small chromite grains (Fig. 6). Both core and rims were analysed in the large chromite (> 15 µm). The data are presented in Figure 6. The core compositions in all experiments have high Cr2O3 concentrations and low Mg#. The rim and small chromites compositions have lower Cr2O3 concentrations and higher Mg# (Table 4). This suggests that during fractional crystallisation the chemical composition of the chromite strongly depends on the silicate melt chemical composition. Indeed, the decrease in Cr2O3 content in the chromite is controlled by the concentration in the silicate melts while crystallisation proceeds. Interestingly, the Fe 3+ content in the chromite also evolves with fractional crystallisation but is strongly dependent on the redox conditions of the experiments. This is also true for Cr content but the variations are smaller. Our experiments show that the redox conditions during the experiments control the chemical composition of the crystallising phases. This is best observed in Figure 5, where the Fe 3+ # and Cr# of the small chromite grains are plotted versus fO2. The Fe 3+ content of the chromite increases with increasing fO2

as observed for the series A chromite. The fact that the chromite grains are chemically heterogeneous indicates that they are not all in equilibrium with the silicate melts. The slow diffusion of Fe 3+ and Cr in the chromite lattice means that there is no chemical reequilibration after the chromites have crystallised out of the silicate melts. This observation provides evidence that the small chromites and rims of the larger grains are more likely to represent the chemical composition of the chromite at equilibrium with the silicate melts. As with the series A experiments, the Cr contents of the small chromite grains were used to calculate partition coefficients (DCr) which vary from 150 to 390 and are positively correlated with fO2 (Fig. 8). The partition coefficient values are very similar to the values obtained with the series A experiments. The DCr3+ for the series B experiments are slightly negatively correlated with fO2 which confirms that under oxidising conditions Cr 3+ becomes less compatible in chromite. The Fe 3+ # in the small chromite grains most likely at equilibrium with the silicate melts is correlated with fO2 (Figure 5). This correlation is very similar to the correlation shown above for the series A experiments. The similarity in Cr concentration, DCr and Fe 3+ # of small chromite obtained for the series B experiments and the rims of series A chromite, suggests that although there is not chemical equilibrium between the silicate melts and the entire chromite grains, the rims of the chromite grains are at local equilibrium with the silicate melts.

Chromium isotope variations

Chromites

The Cr isotopic compositions of the chromites in the series A experiments show little variation. This can be explained by the lack of equilibration between the cores of the chromite grains and the silicate melts, due in large part to the grain size. In this study, the chromites were handpicked from experimental facilitating analysis of single grains. The fact that the reaction rims in the chromite grain are small (relative to the size of the grain) means that Cr isotope measurements of chromite is dominated by the composition of the core. Thus, it is unsurprising that the Cr isotopic compositions of the chromite grains are within error of that of the starting material. This suggests that these compositions are not at equilibrium with the silicate melts and are not discussed further in this manuscript.

Series A

The Cr isotopic compositions of the silicate melts in the series A experiments are lighter than the Cr isotopic composition of the initial chromite added. This suggests that chromite is isotopically heavier than silicate melts, an observation that agrees with both ab-initio calculation and inferences based on lunar and terrestrial samples [START_REF] Schoenberg | The stable Cr isotope inventory of solid Earth reservoirs determined by double spike MC-ICP-MS[END_REF][START_REF] Moynier | Isotopic Evidence of Cr Partitioning into Earth's Core[END_REF][START_REF] Shen | Hightemperature inter-mineral Cr isotope fractionation: A comparison of ionic model predictions and experimental investigations of mantle xenoliths from the North China Craton[END_REF], Bonnand et al. 2016[START_REF] Bonnand | The influence of igneous processes on the chromium isotopic compositions of terrestrial basalts[END_REF][START_REF] Farkas | Chromium isotope variations (delta Cr-53/52) in mantle-derived sources and their weathering products: Implications for environmental studies and the evolution of delta Cr-53/52 in the Earth's mantle over geologic time[END_REF]). In the series A experiments, the Cr isotopic compositions of the silicate melts can be divided into two domains: experiments at log fO2 above -6, silicate melts have δ 53 Cr of approximately -0.45 ‰, while experiments run at logfO2 below -6, silicate melts have δ 53 Cr of approximately -0.15 ‰ (Fig. 9). Additionally, for those experiments run at logfO2 lower than -6, the Cr isotopic composition of the silicate melts is negatively correlated with the redox conditions and the Cr 3+ /CrTOT of the silicate melts (Fig. 10). In these experiments, the Cr isotopic composition in the melts decreases from δ 53 Cr=-0.12‰ to δ 53 Cr=-0.15‰ as logfO2 increases from -12 to -7.

It is possible that the observed variations are controlled by the redox conditions. In this case, the change in isotopic composition could be due to the fact that, at equilibrium, Cr 2+ and Cr 3+ in the melts have different isotopic compositions. If this is the case, then we would expect to see δ 53 Cr varying with the Cr 3+ /CrTOT ratio (and fO2). It is important to note, however, that the δ 53 Cr of silicate melts is lighter with increasing Cr 3+ /CrTOT, implying that Cr 3+ is lighter than Cr 2+ . Ab-initio calculations suggest the opposite; Cr 3+ is assumed to be isotopically heavy compared to Cr 2+ (ΔCr 3+ -Cr 2+ = 0.35*10 6 /T 2 , [START_REF] Moynier | Isotopic Evidence of Cr Partitioning into Earth's Core[END_REF]. In summary, the Cr isotopic composition of silicate melts in contact with magnesiochromite depends on oxygen fugacity. Under reducing conditions, the Cr isotopic composition of the silicate melt is isotopically heavier than silicate melts produced at higher fO2. This suggests that Cr isotopes could be a powerful tool to study redox conditions in high temperature environments.

Importantly, the two most oxidised experiments have very light Cr isotopic compositions, which could reflect diverse causes. The small Cr concentration in the silicate melts indicates that the amount of chromite dissolving into the melt in oxidised conditions is small and could potentially limit the attainment of local equilibrium between chromite and the silicate melts. It is important to note, however, that the chromite grains in those experiments are the least zoned (although different from the starting composition) which suggests that they have reached equilibrium more rapidly than those under reducing conditions. It is surprising that in the most oxidised conditions, where all Cr is believed to be Cr 3+ , that the silicate melts exhibit the largest isotopic shifts. Although this observation is consistent with the observation made for more reduced experiments (melts lighter than chromite), the shift in isotopic composition is large, and seemingly abrupt.

Three main mechanisms can be put forward to explain these light values. The Cr isotopic compositions in the silicate melts are controlled by (i) the chemical composition and the structure of the silicate melt, (ii) the chemical composition and the structure of the chromite grains and (iii) the isotopic fractionation during volatilisation of Cr under high fO2 conditions.

As explained above, depending on the Fe 3+ /FeTOT ratio in silicate melt, the Fe 3+ coordination in the silicate melts vary. In this study, the two most oxidised experiments have Fe 3+ /FeTOT ratio > 0.3 and are also characterised by Raman spectra that could indicate a change in coordination number for Fe. There is a possibility that this change in the melt structure might also affect the coordination of Cr 3+ in the melt. The Cr isotopic shift observed at about logfO2 = -6 could be the result of a change in Cr 3+ coordination in the melt. In this case, a change of Cr 3+ coordination in the melt from VI fold to IV fold could explain the lighter Cr isotopic composition in the silicate melts in oxidising conditions. This suggestion is supported by the changes in the Raman spectra of both experiments performed under oxidizing conditions (Fig. S5). However, it is generally thought that lower coordination number favours heavy isotopes [START_REF] Young | High-temperature equilibrium isotope fractionation of non-traditional stable isotopes: Experiments, theory, and applications[END_REF]. Alternatively, a change in Cr neighbours in the silicate melt structure could also explain the shift in Cr isotopic composition. A lower force constant in the Cr-O bond could indeed results in lighter Cr isotopic compositions. It is also possible that Cr may be present under Cr 6+ in the silicate melts above logfO2 = -6 and this could be responsible for the observed isotopic shifts. Given the fact that Cr undergoes redox changes during quenching, it is very difficult to assess a change in oxidation state and in coordination number.

The second hypothesis is that the chemical composition of the chromite grain controls the observed isotopic shift. In this case, the hypothesis is that under oxidising conditions, Fe 3+ becomes more compatible in magnesiochromite resulting in a competition between Cr 3+ and Fe 3+ . This is supported by the decrease in DCr3+ with fO2 observed in our experiments (550 to 450 for experiments above and below logfO2 = -6, respectively). This would suggest that the result of this competition would be a change in the equilibrium fractionation factor between chromite and the silicate melt. In this case, the fractionation factor between chromite and silicate melts is higher under oxidising conditions and would be inversely correlated with DCr.

Finally, it had been recently argued that Cr is volatile under oxidised conditions (Sossi et al. 2019). The variations in the most oxidised experiments may be caused by loss of isotopically heavy Cr during volatilisation. Our experiments are not designed to quantify the volatility of Cr because the samples after the experiments are not 100% recovered which inhibits mass balance calculation. It is thus difficult to assess whether volatile loss has occurred. The constant flow of gas employed in our experiments means that any evaporative loss will not be at isotopic equilibrium with the silicate melt. [START_REF] Sossi | Evaporation of moderately volatile elements from silicate melts: experiments and theory[END_REF] reported that Cr concentration decreased by ~40% at logfO2 = -5.6 in 15 minutes at 1300 °C. If volatilisation were the main process controlling the Cr budget in our silicate melts, we would have observed severe depletion in Cr concentration in our experiment performed at logfO2 = -3.9 and for ~10,000 minutes. Since such depletion did not occur, we consider that volatile loss of Cr is not a significant influence on our results.

Series B

In the series B experiments, the Cr isotopic measurements of the chromites were made on multi-grain samples, due to the small size (< 10 µm) of the chromites precipitated from the silicate melt. The HCl leaching method used in this study did not selectively sample the small chromite grains but can be regarded as an average composition of chromite in each experiment and not the Cr isotopic composition at equilibrium with the silicate melts.

Nevertheless, the Cr isotopic compositions of the chromite in the series B experiments are heavy (by ~0.3 ‰) compared to the Cr isotopic composition of the starting material. This suggests that during fractional crystallisation the chromites are isotopically heavy compared to the silicate melts. This observation is in agreement with the Cr isotopic variations observed in lunar basalt and terrestrial basalts (Bonnand et al. 2016[START_REF] Bonnand | The influence of igneous processes on the chromium isotopic compositions of terrestrial basalts[END_REF][START_REF] Shen | Stable chromium isotope fractionation during magmatic differentiation: Insights from Hawaiian basalts and implications for planetary redox conditions[END_REF]) and also with ab-initio calculations [START_REF] Moynier | Isotopic Evidence of Cr Partitioning into Earth's Core[END_REF]) and mineral separates data where chromite is the heaviest of Cr-bearing phases [START_REF] Shen | Hightemperature inter-mineral Cr isotope fractionation: A comparison of ionic model predictions and experimental investigations of mantle xenoliths from the North China Craton[END_REF].

The Cr isotopic compositions of the silicate melts in the series B experiments are isotopically light compared to the starting material. Like the series A experiments, the Cr isotopic composition of the melt in the more reduced experiment in series B is also heavier than the isotopic composition of the silicate melts at higher fO2 (-0.24 and -0.44 at logfO2 of -10 and -6, respectively; see Fig. 9). Unfortunately, we do not have access to both the melts and the chromite at complete equilibrium. In this series of experiments, the main reaction is fractional crystallisation. We have modelled the Cr isotopic composition of the silicate melts during fractional crystallisation using the initial Cr concentration and isotopic composition of the starting silicate melts. In Figure 11, we show two fractional crystallisation models, one

Rayleigh fractionation model and one equilibrium fractionation model. Using this conceptual framework, the isotopic fractionation measured in our experiments represents an average over the entire crystallisation sequence. Two cases can be considered, one in which the isotopic fractionation factor is constant during the entire crystallisation sequence; the second in which it evolves during crystallisation proceeds. In the models presented in Figure 11, the Cr isotopic compositions of the crystallising chromite is lighter than that measured for the chromite in the series B experiments (Fig. 11). This provides evidence that the fractionation factors vary during crystallisation and are averages from the start of the crystallising sequence to the end. The variation could be linked to the change in the silicate melts chemical composition but also to the change in temperature during the experiments. The results of these models show that experiments B1 and B2 can be explained by a similar ε factor, whereas the most oxidised experiment B3 clearly requires a larger isotopic fractionation between crystallising chromite and the melt. This is in agreement with the observation that under more oxidised conditions the melt is more depleted in heavy isotopes than the melts produced at lower fO2. This shift in isotopic composition could be linked to the fact that the chemical composition of the chromite grains is controlled by fO2. The fractionation factor is therefore linked to fO2 and it is surprising that fractionation factors between chromite and Cr 3+ in the melt are larger than chromite and Cr 2+ in the melt. This is difficult to explain with ab-initio calculations because changes in valence state are supposed to lead to higher fractionation factor than changes in bonding environment [START_REF] Schauble | Applying Stable Isotope Fractionation Theory to New Systems[END_REF]. In this case, however, it seems that bonding environment in chromite under oxidising conditions strongly favours heavy isotopes. As previously proposed for the series A experiments, the Cr isotopic shift at high fO2 could be explained by (i) a change in bonding environment in the melts, (ii) a change in bonding environment in the crystallising chromites and (iii) Cr isotopic fractionation during Cr volatilisation under oxidising conditions.

Implications for Cr isotopes at high temperature

The results of this study can be used to make general comments on Cr isotope behaviour during igneous processes such as partial melting. Our measurements predict that during partial melting, the silicate melts will be enriched in light isotopes, and the residues will be enriched in heavy isotopes. Recent measurements of ocean island basalts [START_REF] Shen | Stable chromium isotope fractionation during magmatic differentiation: Insights from Hawaiian basalts and implications for planetary redox conditions[END_REF][START_REF] Bonnand | The influence of igneous processes on the chromium isotopic compositions of terrestrial basalts[END_REF]) and in mantle peridotites [START_REF] Xia | Chromium isotope heterogeneity in the mantle[END_REF][START_REF] Shen | Hightemperature inter-mineral Cr isotope fractionation: A comparison of ionic model predictions and experimental investigations of mantle xenoliths from the North China Craton[END_REF][START_REF] Bai | Diffusion-driven chromium isotope fractionation in ultramafic cumulate minerals: Elemental and isotopic evidence from the Stillwater Complex[END_REF] show that residues tend to be 0.05 ‰ heavier than products. Furthermore, the results on the time series experiments demonstrate that in non-equilibrium partial melting reactions, Cr isotope fractionations are driven by kinetic reaction and the melts are enriched in light isotopes. Nonequilibrium melting reactions in natural systems would results in enrichment in heavy isotopes in the residues as suggested by [START_REF] Xia | Chromium isotope heterogeneity in the mantle[END_REF]. Importantly, our measurements were performed at 1 atmosphere and so the results of these experiments are not directly transferable to natural systems. The impact of pressure on the Cr isotope behaviour should be studied in order to evaluate this.

The isotopic variations observed in the experiments performed in this study clearly highlight for the first time, the role played by oxygen fugacity in the behaviour of Cr and its isotopes. It seems that fractionations between chromite and silicate melts are larger under oxidising conditions and this possibly could be useful for studying redox reactions in natural systems. This observation, however, is in disagreement with recent data suggesting that during fractional crystallisation, the Cr isotopic fractionation is smaller under oxidising conditions [START_REF] Shen | Stable chromium isotope fractionation during magmatic differentiation: Insights from Hawaiian basalts and implications for planetary redox conditions[END_REF][START_REF] Bonnand | The influence of igneous processes on the chromium isotopic compositions of terrestrial basalts[END_REF]. Indeed, the observed variations in terrestrial samples for some OIBs can be modelled with fractionation factors (Δ 53 Crcrystal-melts) ranging from -0.005 to -0.02 ‰ [START_REF] Shen | Stable chromium isotope fractionation during magmatic differentiation: Insights from Hawaiian basalts and implications for planetary redox conditions[END_REF][START_REF] Bonnand | The influence of igneous processes on the chromium isotopic compositions of terrestrial basalts[END_REF] As also shown in the series A experiments, it seems that the fractionation factor is higher under oxidised conditions. Our work also shows that it is likely that the fractionation factor during fractional crystallisation varies with the chemical composition of the chromites, the chemical composition of the silicate melts and temperature. These factors, in addition to the fO2, likely play a major role in controlling Cr isotope behaviour in igneous processes.

For

  the silicate melts in series A and B experiments, the δ 53 Cr values range from -0.449 ± 0.014 to -0.117 ± 0.014 ‰ and from -0.437 ± 0.014 to -0.240 ± 0.014 ‰, respectively. The chromites of the series A experiments have an average δ 53 Cr value of -0.108 ± 0.014 ‰ (2 s.d., n = 7) within error of the chromite grain added to the starting material (δ 53 Cr = -0.105 ± 0.014 ‰). The starting material used in the series B experiments has a δ 53 Cr value -0.060 ± 0.014 ‰. The chromites from the series B experiments are more heterogeneous with δ 53 Cr values ranging from 0.063 ± 0.014 to 0.150 ± 0.014 ‰. 5. Discussion 5.1. Chemical and isotopic evolution during the times series experiments

Figure 1 :

 1 Figure captions:

Figure 2 :

 2 Figure 2: Chromium concentrations in the silicate melts versus time. The experiments shown in this

Figure 3 :

 3 Figure 3: Cr chemical compositions in the silicate melts of series A and B experiments. The grey and

Figure 4 :

 4 Figure 4: Cr# (a) and Fe 3+ # (b) for the chromite in the series A experiments. The checkered square is

Figure 5 :

 5 Figure 5: Cr number (green) and Fe 3+ # (blue) in the rim (or small) chromite versus oxygen fugacity. 919

Figure 6 :

 6 Figure 6: Fe 3+ # (a) and Cr# (b) for the chromite in the series B experiments. 921

Figure 7 :

 7 Figure 7: Log(Cr 2+ /Cr 3+ ) (a) and Cr 3+ /CrTOT (b) in the silicate melts versus log of oxygen fugacity. In (b), in blue, Fe 3+ /FeTOT in the silicate melts calculated with O'Neill (2018).

Figure 8 :

 8 Figure 8: Partition coefficients DCrchromite-melt (dark green) and DCr 3+ chromite-melt (light green) versus

Figure 9 :

 9 Figure 9: Cr isotopic compositions versus logfO2 in series A (a) and series B (b) experiments. The red

Figure 10 :

 10 Figure 10: Chromium isotopic composition versus the Cr 3+ /CrTOT ratio in the series A experiments

Figure 11 :

 11 Figure 11: Chromium isotopic composition of melts and chromite versus Cr concentration for the

Table 3 .

 3 The Cr isotopic compositions (expressed as δ53 Cr, relative to NIST SRM 979) of the silicate melts of the time series experiments evolve There is no variation within the three experiments for all major elements except FeO The chromite grains are chemically heterogeneous in Mg#, Fe 3+ # and Cr#. Experiments A1 and A2 (logfO2 -11.9, and -9.6, respectively) are strongly zoned with clear dissolutionreprecipitation textures (Fig.S3). The newly crystallised material at the rim of the crystal is up to 50 microns thick. Experiment A3 (logfO2 -8.3) has a thin dark rim characterised by a depletion in Fe and enrichments in Cr and Al without obvious dissolution-reprecipitation textures. For the chromites of the first group of samples (A1 to A5), the rims have higher Mg# values and lower Fe 3+ # and Cr# values compared to the core compositions. The increase in Mg# and decrease in Fe 3+ # and Cr# values, across experiments, are correlated with oxygen fugacity. As fO2 decreases, Fe 3+ # and Cr# decrease.The second group of experiments is composed of the two most oxidised experiments A6

	from -0.167 ± 0.012 ‰ after 24 h to -0.138 ± 0.012 ‰ after 168 h (Fig. 2).
	4.2. Silicate melt chemical compositions
	4.2.1. Series A

The chemical compositions of the silicate melts of the series A experiments are reported in Table

3

. Overall, the variations in major element composition are minor with the exception of FeO in the more reduced experiment (exp A1). The SiO2, Al2O3, MgO and CaO concentrations in the silicate melts are 47.2, 15.4, 11.7 and 17.6 wt. %, respectively. The Cr concentration (Cr2O3) in the silicate melts vary from 0.81 to 0.09 wt. %. The Cr concentrations in the silicate melts increase with decreasing oxygen fugacity (Fig.

3

).

4.2.2. Series B

The chemical compositions of the silicate melts of the series B experiments are reported in extremely variable

(Figs. 4 and S2)

. Based on chemical compositions, the series A experiments can be divided into two groups of experiments with similar behaviour. The first group of experiments is composed of exp. A1 to A5 and were performed at logfO2 < -6.

  . Under more reducing conditions, the fractionation factor used to model the lunar basalts variations is -0.07 ‰. The dichotomy between two recent datasets of natural OIB samples and our experiments suggest that either chromite is not controlling the Cr budget during fractional crystallisation of OIB samples or that other physical parameters play a role in controlling Cr isotope fractionation. Our study clearly shows that a better understanding of Cr isotopes behaviour in high temperature systems is needed. DCr is positively correlated with fO2 and DCr 3+ is weakly negatively correlated. This indicates that Cr 3+ becomes less compatible in chromite under oxidising conditions because of the competition with Fe 3+ . In the series B experiments, the chromite grains are heterogeneous and several processes play a role in the observed chemical evolution: the redox conditions, the temperature and the chemical composition of the evolving silicate melts. Raman spectroscopy was used to analyse both silicate melts (series A and B) and in chromite (series B only). In summary, the Raman spectra are strongly influenced by the presence of Fe 3+ in both the melts and the chromite grains. The changes in the Raman spectra do not allow us to definitively show a change in Fe 3+ coordination in the silicate melts. However, the differences in Raman spectra clearly indicate that the experiments performed under oxidising conditions are structurally distinct from the experiments performed under reducing conditions.The Cr isotopic compositions measured in the silicate melts in both series A and B experiments are strongly influenced by oxygen fugacity. The Cr isotopic composition of the silicate melts in the experiments from the series A group performed at logfO2 < -6 are

	6. Conclusions
	We have investigated the variations in chemical composition and Cr isotopic compositions
	in both magnesiochromite and silicate melts during experiments performed under controlled
	redox conditions. Two series of experiments were performed to study both dissolution (series
	A) and crystallisation (series B) reactions.
	Overall, the chemical composition of the silicate melts is relatively constant with the
	exception of Cr. As expected, the Cr solubility in silicate melts in our experiments is
	controlled by temperature and oxygen fugacity. We confirm that chromium becomes more
	soluble under reducing conditions. The chemical compositions of the chromite in the
	dissolution experiments (series A) are extremely variable as demonstrated by their zoning in
	each experiment. The Fe

3+ # in the rims correlates with fO2 and this indicates that the chemical composition of the chromites are controlled by fO2. The partition coefficient of Cr between chromite and silicate melts vary with fO2; correlated with fO2. This suggests that Cr isotopes is a powerful tool to study changes in redox conditions in high temperature processes. The Cr isotopic composition of the silicate performed under more oxidising conditions are isotopically much lighter. Two main factors (not mutually exclusive) are proposed to explain such variations: (i) a change in Cr bonding environment in the silicate melt (ii) a change in Cr bonding environment in the chromite. More work is needed to definitively determine the factors that control the isotopic behaviour of Cr in silicate melts. Finally, the Cr isotopic composition for the series B experiments indicate that Cr isotopes are fractionated during fractional crystallisation of magnesiochromite from silicate melts.

Table 1 :

 1 Average chemical composition of the starting silicate melts in series A and B experiments.

	Wt. %	Series A	2 s.d.	Series B	2 s.d.
	MgO	11.78	0.17	11.74	0.25
	Al2O3	15.59	0.15	15.39	0.29
	SiO2	47.22	0.31	45.78	0.43
	Cr2O3	n.d.	n.d.	1.97	0.04
	FeO	7.36	0.11	7.41	0.21
	CaO	17.58	0.16	17.43	0.45
	Total	99.53	0.38	99.72	0.51

Table 5 :

 5 Cr isotopic composition for the silicate melts and the chromite for the series A and B experiments.

	series A	logfO2	δ 53 Crmelt	2 s.e.	δ 53 Crchromite	2 s.e.
	A1	-11.9	-0.117	0.006	-0.110	0.006
	A2	-9.6	-0.137	0.005	n.d.	n.d.
	A3	-8.4	-0.138	0.005	-0.115	0.004
	A4	-7.5	-0.149	0.005	n.d.	n.d.
	A5	-6.6	-0.153	0.004	n.d.	n.d.
	A6	-5.8	-0.449	0.005	-0.102	0.009
	A7	-3.9	-0.431	0.005	-0.108	0.009
	Time series
	A3a	-8.4	-0.251	0.018	n.d.	n.d.
	A3b	-8.4	-0.167	0.007	n.d.	n.d.
	A3c	-8.4	-0.147	0.005	n.d.	n.d.
	A3d	-8.4	-0.143	0.005	n.d.	n.d.
	A3	-8.4	-0.138	0.005	-0.115	0.004
	series B
	B1	-10.2	-0.240	0.006	0.063	0.010
	B2	-7.8	-0.274	0.006	0.088	0.011
	B3	-5.5	-0.437	0.008	0.150	0.008
	starting material		-0.060	0.007
	JP-1	N/A	-0.108	0.014*
	* 2 s.d. n=8

Acknowledgments:

This research was supported by STFC grant (ST/M001318/1) to ANH and BJW. M.B. acknowledge support from the European Union's Horizon 2020 research and innovation programme (grant agreement no 682778). We thank J.-L. Devidal for assistance with microprobe analyses and D. Smythe for numerous discussions. We would like to thank Ronny Schoenberg and two anonymous reviewers for their helpful comments.

Declaration of competing interest:

We have no competing financial interests or personal relationship that could have appeared to influence the work reported in this paper.