Zhengbin Deng
email: zhengbin.deng@sund.ku.dk

Marc Chaussidon

Denton S Ebel

Johan Villeneuve

Julien Moureau

Frédéric Moynier

Simultaneous determination of mass-dependent Mg isotopic variations and radiogenic 26 Mg by laser ablation-MC-ICP-MS and implications for the formation of chondrules

Improvements in our understanding of the formation of chondrules requires a better knowledge of the thermal histories and the nature of their solid precursors. We present an in situ nanosecond laser ablation multi-collector inductively-coupled-plasma mass-spectrometry (LA-MC-ICP-MS) technique to measure simultaneously mass-dependent Mg isotopic fractionations and radiogenic 26 Mg in chondritic components, thus allowing us to investigate

within a chronological framework the thermal processes redistributing Mg in chondrules and their precursors. The internal 26 Al-26 Mg isochrons provide initial 26 Al/ 27 Al ratios from 5.46 (± 0.38) × 10 -5 to 6.14 (± 0.92) × 10 -5 for amoeboid olivine aggregates (AOAs) and Ca-, Al-rich inclusions (CAIs), and from 0.16 (± 0.08) × 10 -5 to 1.87 (± 0.92) × 10 -5 for chondrules from Allende and Leoville chondrites, which are consistent with the previously reported values. The combination of these values with up to 2.5‰ variation of the 25 Mg/ 24 Mg ratio within the studied chondrules shows that: (i) AOAs and the precursors of chondrules were likely formed via condensation of rapid-cooling gas reservoirs, and (ii) Mg stable isotopes are probably at disequilibrium between olivines and mesostases in some chondrules, likely due to Mg loss by vaporization during chondrule formation. We use these new observations to propose that Mg isotopes can likely serve as a tracer for the thermal histories of chondrules. We present here a scenario taking into account Mg loss by vaporization from chondrule melt and Mg gain into the melt by olivine dissolution. The existing Mg isotopic observations in chondrule melts and olivines can be explained in a scenario with a homogeneous distribution of Mg isotopes and initial 26 Al in the accretion disk, provided that chondrule precursors have been heated up to sufficiently high peak temperatures (up to 2123 K) and stayed above 1800 K for several tens of minutes to allow for significant Mg evaporation. These conditions are most consistent with a shock wave model for the origin of chondrules.

Introduction

Chondrules are near-spherical objects forming a large fraction of primitive chondrites.

Ordinary chondrules are hundreds of microns to several centimeters in size [START_REF] Friedrich | Chondrule size and related physical properties: A compilation and evaluation of current data across all meteorite groups[END_REF], and are composed of olivine, pyroxene and glassy silicates (also referred as mesostasis phases), indicating that they were once partially or totally molten before being quenched (e.g., [START_REF] Jones | Petrology and thermal history of type IA chondrules in the Semarkona (LL3. 0) chondrite[END_REF][START_REF] Connolly | The formation of chondrules: Petrologic tests of the shock wave model[END_REF][START_REF] Hewins | Chondrule textures and precursor grain size: an experimental study[END_REF][START_REF] Krot | Origin and chronology of chondritic components: A review[END_REF][START_REF] Ebel | Vapor-melt exchange-Constraints on chondrule formation conditions and processes[END_REF][START_REF] Russell | Chondrules: Records of Protoplanetary Disk Processes[END_REF]. The origin of chondrules is a long-standing question.

Various sources of heating have been proposed to be responsible for chondrule formation, e.g., nebula lightning in the disk, solar flares, accretion shock waves, planetesimal bow shocks, and collisions between planetesimals (e.g., [START_REF] Asphaug | Chondrule formation during planetesimal accretion[END_REF][START_REF] Desch | The importance of experiments: Constraints on chondrule formation models[END_REF][START_REF] Connolly | Chondrules: The canonical and non-canonical view[END_REF][START_REF] Russell | Chondrules: Records of Protoplanetary Disk Processes[END_REF][START_REF] Morris | Formation of chondrules by shock waves[END_REF]. Whether these heating processes can or cannot produce the thermal histories observed for chondrules, is a key criterion to discriminate between these different models [START_REF] Desch | The importance of experiments: Constraints on chondrule formation models[END_REF][START_REF] Connolly | Chondrules: The canonical and non-canonical view[END_REF]. This is one of the reasons why a multitude of experiments have been carried out on chondrule analogs to reproduce either textures or chemical features of chondrules (e.g., [START_REF] Hewins | Temperature conditions for chondrule formation[END_REF][START_REF] Connolly | The formation of chondrules: Petrologic tests of the shock wave model[END_REF][START_REF] Connolly | Chondrules: The canonical and non-canonical view[END_REF]and references therein).

However, the thermal histories proposed for chondrules in literature remain quite diverse with cooling rates spanning from several to thousands of degrees per hour and peak temperatures varying between < 1800 K and ≈ 2200 K [START_REF] Desch | The importance of experiments: Constraints on chondrule formation models[END_REF]and references therein). To trace the heat sources at the origin of chondrules. it would be necessary to reconstruct as well as possible the thermal histories of chondrules.

In addition to the thermal history, the nature of the precursors of chondrules is also yet largely unknown. Based on in situ secondary ion mass spectrometry (SIMS) oxygen isotopic analyses, the presence in type I chondrules of relict Mg-rich olivines that are enriched in 16 O relative to other coexisting olivines, pyroxene and mesostases in chondrules and therefore did not equilibrate with the melt have been discovered [START_REF] Weinbruch | Oxygen isotopic composition of individual olivine grains from the Allende meteorite[END_REF][START_REF] Jones | Oxygen isotope heterogeneity in chondrules from the Mokoia CV3 carbonaceous chondrite[END_REF]Pack et al., 2004;[START_REF] Chaussidon | Oxygen isotopic constraints on the origin of magnesian chondrules and on the gaseous reservoirs in the early Solar System[END_REF][START_REF] Rudraswami | Oxygen isotope systematics of chondrules in the Allende CV3 chondrite: high precision ion microprobe studies[END_REF][START_REF] Libourel | Oxygen isotopic constraints on the origin of Mg-rich olivines from chondritic meteorites[END_REF][START_REF] Tenner | Oxygen isotope systematics of chondrule phenocrysts from the CO3.0 chondrite Yamato 81020: evidence for two distinct oxygen isotope reservoirs[END_REF]Marrocchi et al., 2019a). All these imply that chondrules have formed via reworking of pre-existing solid precursors. Refractory inclusions, e.g., amoeboid olivine aggregates (AOAs) and Ca-and Al-rich inclusions (CAIs), were also present among chondrule precursors [START_REF] Krot | Amoeboid olivine aggregates with low-Ca pyroxenes: A genetic link between refractory inclusions and chondrules? Geochim[END_REF][START_REF] Krot | Remelting of refractory inclusions in the chondrule-forming regions: Evidence from chondrule-bearing type C calcium-aluminum-rich inclusions from Allende[END_REF][START_REF] Krot | Calcium-aluminumrich inclusions recycled during formation of porphyritic chondrules from CH carbonaceous chondrites[END_REF][START_REF] Russell | The genetic relationship between refractory inclusions and chondrules[END_REF][START_REF] Nagashima | An amoeboid olivine aggregate surrounded by an igneous ferroan olivine-rich rim from CO3.0 chondrite DOM 08006[END_REF][START_REF] Nagashima | Oxygen isotope systematics in a compound amoeboid olivine aggregate -chondrule object from Acfer 094 meteorite: implications to O-isotope exchange between melt and gas during chondrule formation[END_REF], but their abundance remains an open topic. Finding out the nature of chondrule precursors is key to track material transport in the accretion disk prior to or at the periods of chondrule formation.

The mass-dependent isotopic variations of Mg can complement the existing chemical and O isotopic observations to trace not only the nature of precursors but also the thermal histories of chondrules, and therefore their formation mechanism. As a moderately refractory element, Mg in silicate melts can be significantly vaporized as Mg or MgO at temperatures above 1800 K under nebular conditions [START_REF] Floss | Elemental and isotopic fractionations produced through evaporation of the Allende CV chondrilte: Implications for the origin of HAL-type hibonite inclusions[END_REF][START_REF] Hashimoto | Evaporation kinetics of forsterite and implications for the early solar nebula[END_REF][START_REF] Davis | Isotope mass fractionation during evaporation of Mg 2 SiO 4[END_REF][START_REF] Mendybaev | Experimental evaporation of Mg-and Si-rich melts: Implications for the origin and evolution of FUN CAIs[END_REF]. In vacuum this evaporation can be instantaneous at > 2173 K [START_REF] Hashimoto | Evaporation metamorphism in the early solar nebula-Evaporation experiments on the melt FeO-MgO-SiO 2 -CaO-Al 2 O 3 and chemical fractionations of primitive materials[END_REF]. While free vaporization can lead to enrichments in the heavier isotopes of Mg (up to 40‰/amu in some CAIs), condensation of a Mg-oversaturated gas reservoir can oppositely lead to various scales of enrichments in lighter isotopes of Mg in the condensates relative to the residual gas [START_REF] Richter | Timescales determining the degree of kinetic isotope fractionation by evaporation and condensation[END_REF][START_REF] Simon | A short timescale for changing oxygen fugacity in the solar nebula revealed by high-resolution 26 Al-26 Mg dating of CAI rims[END_REF]. This might be the case for AOAs and some CAIs that have a condensate origin [START_REF] Krot | Amoeboid olivine aggregates with low-Ca pyroxenes: A genetic link between refractory inclusions and chondrules? Geochim[END_REF]Marrocchi et al., 2019b). It is possible that early condensation and evaporation processes in the protosolar nebula have produced distinct mass-dependent Mg isotopic variations in different types of refractory inclusions (e.g., [START_REF] Niederer | Ca isotopes in refractory inclusions[END_REF][START_REF] Clayton | Isotopic variations in the rock-forming elements in meteorites[END_REF]Simon et al., 2005;[START_REF] Davis | Isotope mass fractionation during evaporation of Mg 2 SiO 4[END_REF][START_REF] Mendybaev | Experimental evaporation of Mg-and Si-rich melts: Implications for the origin and evolution of FUN CAIs[END_REF], therefore leaving the fingerprints for discriminating the contribution of these refractory inclusions in chondrules. Moreover, the mass-dependent Mg isotopic fractionations of olivines and mesostases in chondrules can be used to test whether chondrule melts have been subjected to Mg evaporation during the last melting events, which is crucial for deciphering thermal histories of chondrules, i.e., the peak temperatures and the cooling rates. Such evaporation-induced Mg isotopic signatures can be blurred by the effect of olivine dilution when measuring bulk chondrules, therefore calling for an in situ high-precision mass-dependent Mg isotopic study.

In addition to the mass-dependent variations of the concentrations of the three Mg stable isotopes ( 24 Mg, 25 Mg, 26 Mg), a fraction of radiogenic 26 Mg from the decay of 26 Al (T 1/2 = 0.717 Myr; [START_REF] Lee | Demonstration of 26 Mg excesses in Allende and evidence for 26 Al[END_REF]) can be added to chondritic components. Before the extinction of 26 Al, the various phases inside either chondrules or refractory inclusions would develop with time various degrees of enrichments in radiogenic 26 Mg that are proportional to the 27 Al/ 24 Mg ratios, thus forming 26 Al-26 Mg isochrons. Measuring simultaneously the mass-dependent and non mass-dependent Mg isotopic variations of individual phases of chondrules and refractory inclusions in chondrites can be used in theory for resolving the chronology of the thermal processes producing the observed mass-dependent Mg isotopic fractionations (Young et al., 2002a[START_REF] Young | Supracanonical 26 Al/ 27 Al and the residence time of CAIs in the solar protoplanetary disk[END_REF][START_REF] Simon | A short timescale for changing oxygen fugacity in the solar nebula revealed by high-resolution 26 Al-26 Mg dating of CAI rims[END_REF].

Such an in situ determination of mass-dependent Mg isotopic fractionation is presently limited. The main in situ tool to analyze radiogenic 26 Mg variations, SIMS, cannot provide sufficient precision and accuracy for the mass-dependent Mg isotopic measurements due to the strong matrix effects. For instance, a large set of natural or synthetic olivine and komatiitic, basaltic to andesitic glass standards show 8‰/amu variations caused by matrix effects [START_REF] Chaussidon | In situ analysis of non-traditional isotopes by SIMS and LA-MC-ICP-MS: Key aspects and the example of Mg isotopes in olivines and silicate glasses[END_REF], and at fixed MgO contents, minor elements in olivines (e.g., Cr 2 O 3 ) can lead to positive biases of 2‰/amu on the measured Mg isotopic compositions [START_REF] Fukuda | Magnesium isotope analysis of olivine and pyroxene by SIMS: Evaluation of matrix effects[END_REF][START_REF] Nagashima | Investigation of instrumental fractionation in SIMS analyses of magnesium, silicon, and oxygen isotopes in silicates and oxides[END_REF]. It is also noteworthy that the mesostasis phases in chondrules have been notoriously known for their highly heterogeneous chemical compositions (e.g., [START_REF] Jones | Petrology and thermal history of type IA chondrules in the Semarkona (LL3. 0) chondrite[END_REF][START_REF] Alexander | Alkali elemental and potassium isotopic compositions of Semarkona chondrules[END_REF][START_REF] Libourel | Role of gas-melt interaction during chondrule formation[END_REF][START_REF] Olsen | Magnesium and 54 Cr isotope compositions of carbonaceous chondrite chondrules-Insights into early disk processes[END_REF], making the determination of mass-dependent Mg isotopic variations by SIMS for both olivine/pyroxene and mesostasis phases challenging. To circumvent these limitations of SIMS measurement and to investigate the formation of chondrules, we have developed an in situ nanosecond laser ablation multi-collector inductively-coupled-plasma mass-spectrometry (LA-MC-ICP-MS) method to measure with high precision and accuracy simultaneously the 25 Mg/ 24 Mg, 26 Mg/ 24 Mg, 27 Al/ 24 Mg and 44 Ca/ 24 Mg ratios of the major silicate phases of chondrules and refractory inclusions. The results show that matrix-induced Mg isotopic fractionations among various mineral and glass samples can be reduced relative to SIMS by using LA-MC-ICP-MS (1.5-2 ‰/amu variation), which is however still not negligible as suggested earlier (Young et al., 2002a[START_REF] Young | Supracanonical 26 Al/ 27 Al and the residence time of CAIs in the solar protoplanetary disk[END_REF][START_REF] Simon | A short timescale for changing oxygen fugacity in the solar nebula revealed by high-resolution 26 Al-26 Mg dating of CAI rims[END_REF]. We further demonstrate that after proper calibration, LA-MC-ICP-MS can likely provide at the same time the 26 Al-26 Mg isochrons and accurate measurements of 25 Mg/ 24 Mg variations. Two of the studied chondrules were measured by SIMS following a more "classical" in situ approach. All these data are used to put new constraints on the nature of precursors and cooling histories of the present chondrules.

Samples and methods

Mineral and glass standards

Three sets of well-characterized mineral and glass standards were used throughout the development of the techniques for data quality control, including six synthetic olivines (Ol synth Watson #01, #02, #03, #04, #05 and #6) with various MgO contents of 32.83-49.33 wt% [START_REF] Chaussidon | In situ analysis of non-traditional isotopes by SIMS and LA-MC-ICP-MS: Key aspects and the example of Mg isotopes in olivines and silicate glasses[END_REF], eleven komatiitic/basaltic/andesitic glasses (StHs6/80-G, BCR-2G, TB-1G, T1-G, ML3B-G, BHVO-2G, KL2-G, MORB-CLDR01RV, BIR-1G, GOR132-G and GOR128-G) [START_REF] Chaussidon | In situ analysis of non-traditional isotopes by SIMS and LA-MC-ICP-MS: Key aspects and the example of Mg isotopes in olivines and silicate glasses[END_REF], and four CaO-MgO-Al 2 O 3 -SiO 2 (CMAS) glasses (CMAS-synth-I, -II, -III and -IV). We consider that the komatiitic, basaltic and andesitic glasses, and the CMAS glasses can represent two extreme analogs of refractory inclusions and the mesostasis phases in chondrules. The chemical and Mg isotopic compositions of these standards have been provided in Table 1 with the extended details in Table S1. One diopside and one crystalline MORB have also been included to scrutinize the applicability of the method to the samples with different matrices.

Chondrules and refractory inclusions from Allende and Leoville CV3 chondrites

A thick section of Allende CV3 chondrite was mapped for petrography and chemical compositions by scanning electron microscope (SEM) at the Institut de Physique du Globe de Paris (IPGP) and CRPG-CNRS (Nancy), France. Another Leoville CV3 thick section was characterized by electron microprobe (EMP) at the American Museum of Natural History (AMNH), United States, following the protocol described in [START_REF] Ebel | Abundance, major element composition and size of components and matrix in CV, CO and Acfer 094 chondrites[END_REF]. Three Allende chondrules including one type IA (B-Ch5), one Al-rich (C-Ch1) and one type IIAB (B-Ch1), as well three type IA chondrules from Leoville and Leo-4337-t2-ps2B-c7) were selected for LA-MC-ICPMS analyses. The diameter of these chondrules typically ranges between 1 mm and 2 mm (Figs. S1 andS2). These chondrules are porphyritic with significant volumes of mesostases, allowing for a determination of the phase relation for Mg isotopes. Also measured in the same sessions were the refractory inclusions, including one Allende melilite sample (Mel-1) that is zoned with an åkermanite-rich core and a gehlenite-rich rim, one Leoville AOA (i.e., Leo-4337-t2-ps2B-aoa1), and three fine-grained CAIs (i.e., from Leoville (see Fig. S1 and S2 for the compositional maps of these samples).

Laser system and mass spectrometry

In this study, a nanosecond 193 nm ultra-short ATLex 300si excimer laser ablation system was coupled with a Thermo-Scientific Neptune MC-ICP-MS at IPGP for in situ LA-MC-ICP-MS Mg isotopic analysis. The typical duration of the laser pulses is around 4 ns. For the analytical sessions with Allende and Leoville chondrites, sample bracketing with San Carlos olivines (i.e., analysis of San Carlos olivines after each of one or two spots on mineral and glass standards, and the components in chondrites) was used to correct for instrumental fractionation. Duplicate analyses of synthetic olivines and glass standards were conducted over each session to monitor the stability of matrix-induced effects on the measured elemental and Mg isotopic compositions. The washout time was around 10 s after ceasing ablation, and the pause time between two analyses was set to be 60-90 s. Analyses were carried out with a spot size of 40 μm, a repetition rate of 3 Hz, and a gas flow rate of 1.0-1.2 L/min. Each spot contains 400 laser burst shots, and typical ablation depths are 20-25 μm. To investigate the chemistrycontrolled matrix effects, laser fluence was fixed in each session to avoid further complication (3.37 J/cm 2 for the Allende session, Session 1, and 4.23 J/cm 2 for the Leoville session, Session 2) (Table S2), the criteria being that the intensity on 24 Mg + must be similar (± 10%) for olivines with a similar Fo content. On MC-ICP-MS, a medium mass resolution mode (M/ΔM ≈ 5000) was used, which theoretically can resolve all the isobaric interferences from 48 Ti 2+ and 48 Ca 2+ on 24 Mg + , and those from 52 Cr 2+ and 12 C 14 N + on 26 Mg + . Nonetheless, a dilemma is that the masses of doubly charged ions such as 48 Ti 2+ and 48 Ca 2+ are lower than that of 24 Mg + , whereas the mass of molecular ions 12 C 14 N + is higher than that of 26 Mg + . Because interferences can only be avoided by making measurements on one flank of the 24 Mg + or 26 Mg + peak, it is thus not possible to avoid at the same time 48 Ti 2+ or 48 Ca 2+ and 12 C 14 N + . In this study, the priority was to avoid the uncorrectable molecular interference from 12 C 14 N + by measuring the ion intensities at the lower masses of 24, 25 and 26. It was therefore necessary to monitor the intensity of another doubly charged ion of Ca (e.g., 44 Ca 2+ ) to correct for the interference from 48 Ca 2+ on 24 Mg + , which also allows a further estimate on the interferences from 48 Ti 2+ and 52 Cr 2+ (see below).

To monitor the intensities of the masses 22 to 27, we used a dynamic mode with two cup configurations. The Faraday cup positions and the magnitude of the magnetic field were adjusted to align the 24 Mg + (L3-F), 25 Mg + (C-F), 26 Mg + (H2-F) and 27 Al + (H4-F) peaks (Table S2), and the measurements were conducted on the left shoulder of the peaks to avoid 12 C 14 N + isobaric interference (main cup configuration). For every five cycles, the central mass was moved from 25 to 23 to monitor simultaneously the intensities of 44 Ca 2+ (mass 22, L4-F) and 24 Mg + (mass 24, H2-F). Note that two additional sessions on the mineral and glass standards were carried out with either a large spot size of 110 μm or a high laser fluence of 11.47 J/cm 2 to investigate the potential effects from sample loading or sample melting.

Data reduction and notation

Each laser spot contains 18-19 cycles for the main cup configuration and 4-5 cycles for the sub cup configuration. Data reduction was implemented through integrating these cycles except for the transient cycles at the beginning or the end of ablation (intensity varying more than 25% from the plateau intensity). The background of each ablation spot was subtracted with a polynomial fit to the background cycles. The Mg isotopic compositions and variations observed in this study are given in per mil notation that can be defined as:

(1)

 i Mg ‰    i Mg / 24 Mg   sample i Mg / 24 Mg   DSM3  1          1000
where i Mg represents 25 Mg or 26 Mg, and δ i Mg stands for the isotopic difference in per mil relative to the Dead Sea magnesium (DSM-3) standard [START_REF] Galy | Magnesium isotope heterogeneity of the isotope standard SRM980 and new reference materials for magnesium-isotope-ratio measurements[END_REF]. The massindependent or non-mass dependent Mg isotopic fractionations (Δ 26 Mg) can then be provided as:

(2)

 26 Mg   26 Mg   25 Mg  nat
where β nat represents a chosen mass fractionation law for Mg isotopic variations in the natural samples. The Δ 26 Mg value has been usually referred to as δ 26 Mg* in the case of radiogenic 26 Mg excesses or depletions in meteorite samples. Throughout this study, we adopt a β value of 0.517 for both Mg isotopic fractionations in natural samples and those caused by matrix effects, which is in between the kinetic and equilibrium mass fractionation laws, and is close to a Rayleigh process (Young et al., 2002b;[START_REF] Davis | Isotopic mass fractionation laws for magnesium and their effects on 26 Al-26 Mg systematics in solar system materials[END_REF].

SIMS measurements

After LA-MC-ICP-MS measurements, the Allende thick section was cut with diamond wire and mounted into epoxy, and polished to remove the laser ablation pits and gain a new pristine surface for SIMS measurements. Magnesium isotopes and 27 Al/ 24 Mg ratios of two Allende chondrules (i.e., B-Ch5 and C-Ch1) were measured with a CAMECA large radius ims 1280HR2 secondary ion mass spectrometry (SIMS) housed at CRPG-CNRS (Nancy).

Measurements were conducted following the protocols described in [START_REF] Villeneuve | Homogeneous distribution of 26 Al in the solar system from the Mg isotopic composition of chondrules[END_REF] and [START_REF] Luu | High precision Mg isotope measurements of meteoritic samples by secondary ion mass spectrometry[END_REF] except that a Hyperion ion source was used. A primary O -ion beam of 25 nA O -was accelerated at 13 kV and focused to achieve spots of 8-10 μm diameter. The secondary ions of Al and Mg isotopes were measured under multi-collection mode (L'2, C, H1 and H'2 for 24 Mg + , 25 Mg + , 26 Mg + and 27 Al + , respectively) at a mass resolving power of ≈ 2500.

To minimize the isobaric interference of 24 MgH + on 25 Mg + , the vacuum was kept below 4 × [START_REF] Chaussidon | In situ analysis of non-traditional isotopes by SIMS and LA-MC-ICP-MS: Key aspects and the example of Mg isotopes in olivines and silicate glasses[END_REF][START_REF] Fukuda | Magnesium isotope analysis of olivine and pyroxene by SIMS: Evaluation of matrix effects[END_REF][START_REF] Nagashima | Investigation of instrumental fractionation in SIMS analyses of magnesium, silicon, and oxygen isotopes in silicates and oxides[END_REF], measuring precisely and accurately the δ 25 Mg values by SIMS for chondrules requires a strict calibration of the matrix-induced Mg isotopic effects with a large set of standard materials of various matrices, which is beyond the scope of this study.

LA-MC-ICP-MS method for Mg isotopic measurements

Doubly charged ion interferences on Mg isotopes

Arising from the high energy input in the ICP sources (particularly for dry plasma setting by LA-MC-ICP-MS measurements; [START_REF] Janney | Matrix effects in the analysis of Mg and Si isotope ratios in natural and synthetic glasses by laser ablation-multicollector ICPMS: A comparison of single-and double-focusing mass spectrometers[END_REF], various types of doubly charged ions, such as 48 Ca 2+ and 48 Ti 2+ on 24 Mg + , and 52 Cr 2+ on 26 Mg + , can be produced and create mass interferences with the ions of Mg isotopes. It is therefore necessary to quantify the production of these doubly-charged ions during the measurements. This can be approached using three types of mineral and glass standards (Table 1 andS1 CaO. All these standards are terrestrial or synthetized from terrestrial materials, so that they contain no radiogenic 26 Mg excess and form a baseline to detect the interferences resulting in apparent non-mass dependent effects on the measured Mg isotopic ratios. This provides a way to scrutinize the reliability of the protocol chosen for interference correction.

As addressed above, Mg isotopic measurements on the lower flanks of the 24, 25 and 26 peaks are subjected to isobaric interferences from doubly charged ions of Ca, Ti and Cr. As such, the measured Mg isotopic ratios would be biased as following:

(3) In this study, the intensities of 48 Ti 2+ and 52 Cr 2+ were not measured directly. Nonetheless, the production of these two doubly charged ions can be quantified by comparing Δ 26 Mg Ca+Ti+Cr with the measured 44 Ca 2+ / 24 Mg + ratios (equation 6 and Fig. 1). The results from the present mineral and glass standards show that the Δ 26 Mg Ca+Ti+Cr values always define nice linear correlations against the measured 44 Ca 2+ / 24 Mg + ratios with intercepts that are very close to zero (Fig. 1). Considering that the various mineral and glass standards have distinct CaO, TiO 2 and Cr 2 O 3 contents, the linear correlations can support that the doubly charged ion interferences on Mg isotopes in the MC-ICP-MS are predominantly 48 Ca 2+ with very few 48 Ti 2+ and 52 Cr 2+ .

Assuming that the singly charged ionization rate of Mg is close to unity and 48 Ca 2+ , 48 Ti 2+ and 52 Cr 2+ ions have the same transmission rate as 24 Mg + ions in the MC-ICP-MS, the doubly charged ionization rate of Ca in the ICP source (i.e., the production rate of Ca 2+ ) should vary between 16% and 24% between different sessions based on the correlations in Fig. 2.

According to the correlations in Fig. 1, the instrumental 48 Ca 2+ / 44 Ca 2+ ratio within each analytical session can be determined with an uncertainty better than ±5%, thus allowing a sufficient correction of 48 Ca 2+ interferences on 24 Mg + using equation 3 to 5. After correction of 48 Ca 2+ interferences, even the basaltic glass standard with the highest 48 Ti/ 24 Mg ratio of 0.2955 (BCR-2G) turns out to have an apparent non-mass dependent Mg isotopic effect (Δ 26 Mg) within ±0.05‰ identical to those for synthetic Ti-free CMAS glasses (Tables S5 andS6). This suggests that the production rate of doubly charged Ti and Cr ions, in total, should be ≤ 0.02%, and that the present measurements of non-mass dependent Mg isotopic variations in natural samples with 48 Ti/ 24 Mg ratios ≤ 0.3 by LA-MC-ICP-MS were not significantly affected by 48 Ti 2+ and 52 Cr 2+ interferences.

Matrix effects for mass-dependent Mg isotopic measurements

After correction of 48 Ca 2+ interferences, the measured Mg isotopic compositions of the standards may differ from their true isotopic compositions, as the result of matrix effects (i.e.

different chemistry between samples and the bracketing standard). It is notable that such matrix-caused Mg isotopic fractionations ([δ 25 Mg measured -δ 25 Mg true ]) on the mineral and glass standards (excluding the CMAS glasses) are negatively correlated with the Mg Intensity Normalized to the Bracketing Standard (MINBS) in a linear or polynomial manner (Fig. 3):

(7)  25 Mg measured   25 Mg true      a  b  MINBS    c  MINBS   2
The MINBS values here reflect, to first order, the MgO contents of the samples, relative to that of the bracketing standard. This suggests that, at least for olivine, pyroxene and komatiitic/basaltic/andesitic glasses, matrix-induced Mg isotopic effects by LA-MC-ICP-MS are mainly controlled by the differences in major element abundances. A similar systematics for matrix-induced isotopic effects has also been observed in the case of Mg and Fe isotopic analyses of olivines by LA-MC-ICP-MS in [START_REF] Sio | Discerning crystal growth from diffusion profiles in zoned olivine by in situ Mg-Fe isotopic analyses[END_REF]. Since the systematics can vary between sessions, calibration is required for each analytical session. Using these empirical calibrations, a correction can be carried out to eliminate Mg isotopic fractionations induced by matrix effects.

Effects from ablation spot size and laser fluence for CMAS-like samples

It is noteworthy that using a large spot size of 110 μm, the CMAS glass standards form different calibrations for 44 Ca/ 24 Mg and 27 Al/ 24 Mg ratio measurements from those of olivine and komatiitic/basaltic/andesitic glass standards (Fig. 2). We speculate that loading large amounts of sample (i.e. in case of large spot size) may have led to inconsistent ionization rates or inconsistent ion transmission rates for Mg, Al and Ca between standards of different matrices, even within the same analytical session. This calls for cautions to match the matrices between standards and natural samples when using LA-MC-ICP-MS to investigate the 26 Al- 26 Mg system in meteorites. In detail, with a large spot size of ≥ 100 μm, the calibration line defined for 27 Al/ 24 Mg ratio measurements from olivines, and komatiitic to andesitic glass standards cannot be used for CMAS-like samples (e.g., some CAI samples). In fact, it could lead to lower the 27 Al/ 24 Mg ratios, which would result in higher initial 26 of 11.47 J/cm 2 (Fig. 3B). Such different matrix-induced Mg isotopic fractionations are likely due to the inconsistent melting behaviors of the CMAS glasses relative to other standards during their interaction with the nanosecond laser pulses, which can likely produce aerosols with distinct particle size distribution and therefore complicate ionization behaviors in the ICP source. This is likely corroborated by the significant changes in Mg ion intensity for the CMAS glasses relative to other komatiitic to andesitic glass standards at varying laser fluences (Fig. 3B). One may expect to identify an optimal laser fluence for a nanosecond laser system to minimize the matrix effects on the measured δ 25 Mg values for the samples of different matrices.

However, potential inconsistent melting behavior between standards and samples at various laser settings remains a major issue when using a nanosecond laser ablation system to measure in situ mass-dependent Mg isotopic fractionations. This can possibly be overcome by applying a femtosecond laser ablation system, which provides much shorter laser pulses and may avoid melting of samples.

3.4. Applicability of the olivine and komatiitic/basaltic/andesitic glass standards to correct for matrix-induced Mg isotopic fractionations on chondrule phases

The mesostasis phases in chondrules are notoriously known for the complicated chemical compositions, and it is thus unrealistic to prepare standards matching exactly the matrices. 1). Along with the Mg isotopic compositions, our LA-MC-ICP-MS technique is able to obtain the 27 Al/ 24 Mg and 44 Ca/ 24 Mg ratios for the same ablation spots (Fig. 2), which further show that the used olivine and komatiitic to andesitic glass standards can serve as nice analogs for the measured chondrules phases in Allende and Leoville chondrites (Fig. 4). As such, we consider that the matrix-induced Mg isotopic fractionations on the chondrule phases by LA-MC-ICP-MS can be sufficiently tracked via measurements of olivine and komatiitic to andesitic glass standards within the same analytical session.

It is noteworthy that the Allende melilite phases are characterized by the 27 Al/ 24 Mg and 44 Ca/ 24 Mg ratios distinct from those of the komatiitic/basaltic/andesitic glasses but similar to those of the CMAS glasses (Fig. 4), which can at least partially account for the inconsistent matrix-induced Mg isotopic effects on the CMAS glasses relative to the other mineral and glass standards (Fig. 3B). For the data from the Allende and Leoville sessions, we adopted the calibration from olivine and komatiitic/basaltic/andesitic glass standards to correct for the matrix-induced Mg isotopic fractionations on the various phases of the studied chondrules.

Considering that the Allende and Leoville sessions were with low laser fluences of 3.37 or 4.23 J/cm 2 , such a protocol of matrix effect correction may biasedly lead to sub per mil lower δ 25 Mg values for the melilite phases in Allende, and possibly also other refractory inclusions if their compositions have evolved partly to the CMAS compositions. After correction for 48 Ca 2+ interferences, all the present mineral and glass standards show δ 26 Mg* ≈ 0 whatever their 27 Al/ 24 Mg ratios, consistent with the fact that terrestrial samples have no radiogenic 26 Mg excesses (Fig. 5). Using the same interference correction protocol, the crystalline MORB provides a δ 26 Mg* value of -0.02 ± 0.06‰ (2SE, n = 10), confirming that the method also works for crystalline samples. For the measurements of δ 25 Mg values, the olivines and komatiitic to andesitic glass standards all provide δ 25 Mg values identical within uncertainty to the δ 25 Mg values from solution MC-ICP-MS measurements (Fig. 6). However, using the calibration from olivine and komatiitic/basaltic/andesitic glass standards to correct for matrix effects would artificially produce lower δ 25 Mg values for CMAS glasses at low laser fluence, but higher δ 25 Mg values at high laser fluence (Fig. 6).

Results

δ 26 Mg* results

The Allende melilite grain, and the Leoville AOA and CAI samples are characterized by significant enrichments in 26 Mg relative to terrestrial standards, i.e. positive δ 26 Mg* values, forming well-defined 26 Al-26 Mg isochrons (Fig. 7A-C) yielding ( 26 Al/ 27 Al) initial ratios from 5.46 (± 0.38) × 10 -5 to 6.14 (± 0.92) × 10 -5 according to:

(8)  26 Mg*   26 Mg *   initial  1000  26 Al / 27 Al   initial  27 Al / 24 Mg   sample       / 0.13932
where 0.13932 is the reference 26 Mg/ 24 Mg ratio from [START_REF] Catanzaro | Absolute isotopic abundance ratios and atomic weight of magnesium[END_REF]. These initial 26 Al/ 27 Al ratios are consistent with the previous data for refractory inclusions, which were measured by thermal ionization mass spectrometry (TIMS), SIMS, MC-ICP-MS or LA-MC-ICP-MS [START_REF] Lee | Demonstration of 26 Mg excesses in Allende and evidence for 26 Al[END_REF][START_REF] Bizzarro | Mg isotope evidence for contemporaneous formation of chondrules and refractory inclusions[END_REF][START_REF] Young | Supracanonical 26 Al/ 27 Al and the residence time of CAIs in the solar protoplanetary disk[END_REF][START_REF] Simon | A short timescale for changing oxygen fugacity in the solar nebula revealed by high-resolution 26 Al-26 Mg dating of CAI rims[END_REF][START_REF] Jacobsen | 26 Al-26 Mg and 207 Pb-206 Pb systematics of Allende CAIs: Canonical solar initial 26 Al/ 27 Al ratio reinstated[END_REF][START_REF] Larsen | Evidence for magnesium isotope heterogeneity in the solar protoplanetary disk[END_REF][START_REF] Wasserburg | Mg isotopic heterogeneity, Al-Mg isochrons, and canonical 26 Al/ 27 Al in the early solar system[END_REF][START_REF] Kita | Internal 26 Al-26 Mg isotope systematics of a Type B CAI: remelting of refractory precursor solids[END_REF][START_REF] Mishra | Timing and extent of Mg and Al isotopic homogenization in the early inner Solar System[END_REF]. For Allende chondrule C-Ch1, the LA-MC-ICP-MS and SIMS data align along a single isochron which, together with the results of AOA and CAIs, support the robustness of the LA-MC-ICP-MS method to monitor radiogenic 26 Mg in meteorites.

Compared to refractory inclusions, the present chondrules show smaller but resolvable 26 Mg excesses, with 26 Al-26 Mg isochrons having ( 26 Al/ 27 Al) initial ratios from 0.16 (± 0.08) × 10 -5 to 1.87 (± 0.92) × 10 -5 (Fig. 7D-H). These ( 26 Al/ 27 Al) initial ratios are within uncertainty identical to those published for chondrules from carbonaceous and ordinary chondrites (e.g., [START_REF] Kita | A short duration of chondrule formation in the solar nebula: Evidence from 26 Al in Semarkona ferromagnesian chondrules[END_REF][START_REF] Luu | Short time interval for condensation of high-temperature silicates in the solar accretion disk[END_REF][START_REF] Villeneuve | Homogeneous distribution of 26 Al in the solar system from the Mg isotopic composition of chondrules[END_REF] the early Solar System. Note that the low initial 26 Al/ 27 Al ratio of 0.16 (± 0.08) × 10 -5 for the Allende B-CH5 chondrule is likely due to the fact that the linear regression is controlled by the plagioclase grains that are not the major Mg-bearing phases in the mesostases but have very high 27 Al/ 24 Mg ratios of 22.98-51.78 and are thus easily disturbed for Mg isotopes (Fig. 7D).

Including only the data from Mg-rich phases returns to a higher initial 26 Al/ 27 Al ratio of 1.85 (± 1.02) × 10 -5 for the Allende B-CH5 chondrule. In the Leoville chondrule Leo-4337-t2-psB-c7, two Ca-rich pyroxenes and several olivines plot close to the bulk CAI 26 Al-26 Mg isochron [START_REF] Jacobsen | 26 Al-26 Mg and 207 Pb-206 Pb systematics of Allende CAIs: Canonical solar initial 26 Al/ 27 Al ratio reinstated[END_REF][START_REF] Larsen | Evidence for magnesium isotope heterogeneity in the solar protoplanetary disk[END_REF], while the other olivines, Ca-poor pyroxenes and mesostases form an isochron giving a much lower initial 26 Al/ 27 Al ratio of ≈ 0.8 × 10 -5 (Fig. 7I).

δ 25 Mg results

Within the Allende and Leoville chondrules, the Mg-rich olivines show heterogeneous δ 25 Mg values from -1.22 ± 0.24‰ to -0.05 ± 0.24‰, with two out of four type I chondrules (B-Ch5 and Leo-4337-t2-ps2B-c9) containing olivines with δ 25 Mg values as low as -1.22 ± 0.24‰ (Fig. 8). In chondrules B-Ch1, Leo-4337-t2-ps2B-c2 and Leo-4337-t2-ps2B-c7, the olivines have δ 25 Mg values identical within uncertainty to the glassy mesostases or the clinopyroxene and plagioclase (Fig. 8), while, for the other three chondrules (B-Ch5, C-Ch1 and Leo-4337-t2-ps2b-c9), the glassy mesostases or the clinopyroxene and plagioclase are in average 0.68-0.94‰ higher in δ 25 Mg relative to the olivines (Fig. 8). These inter-phase differences are robust since, as explained in Section 3.4, the used olivines and komatiitic to andesitic glasses resemble to the measured chondrule phases in chemical composition and are able to serve as standards to correct for matrix effects for the measurements on chondrules.

Note that previous SIMS data in chondrules from the Acfer 094 C3 chondrite showed that their olivines, at variance with the present results for Allende and Leoville, have high δ 25 Mg values up to +1-2‰ [START_REF] Ushikubo | Contemporaneous formation of chondrules in distinct oxygen isotope reservoirs[END_REF]. Nonetheless, these high δ 25 Mg values in chondrule olivines in [START_REF] Ushikubo | Contemporaneous formation of chondrules in distinct oxygen isotope reservoirs[END_REF] have recently been suggested by [START_REF] Fukuda | Magnesium isotope analysis of olivine and pyroxene by SIMS: Evaluation of matrix effects[END_REF] to be, most probably, the results of an incomplete correction of SIMS matrix effects due to minor elements in olivines, which can be up to +2‰/amu for Cr 2 O 3 contents of 0.8 wt.%. We stress that reliable "bulk" δ 25 Mg values cannot be obtained by averaging the LA-MC-ICP-MS data since most of the ablation spots in olivines have been located in the olivine-core areas to avoid ablation of mesostasis phases.

The most negative δ 25 Mg values in the Leoville AOA are in the olivines (from -1.63 ± 0.24‰ to -1.12 ± 0.25‰), while more variable δ 25 Mg values (-1.40 ± 0.28‰ to -0.03 ± 0.25‰) are present in other Al-and Ca-rich phases (Fig. 8). Such low δ 25 Mg values in AOAs are consistent with previous data by solution MC-ICP-MS for AOAs from the CV chondrite Efremovka [START_REF] Larsen | Evidence for magnesium isotope heterogeneity in the solar protoplanetary disk[END_REF], with 7 out of 8 for these olivines having bulk δ 25 Mg values from -2.465 ± 0.009‰ to -1.148 ± 0.018‰ (Fig. 8), as well as the results of olivines in the AOAs from Kaba (CV3.1) and DOM 08006 (CO3.1) carbonaceous chondrites (δ 25 Mg of -3.8 ± 0.5‰ to -0.2 ± 0.3‰; [START_REF] Fukuda | Correlated isotopic and chemical evidence for condensation origins of olivine in comet 81P/Wild 2 and in AOAs from CV and CO chondrites[END_REF]. The fine-grained CAI samples from Leoville exhibit both negative and positive δ 25 Mg values from -1.12 ± 0.28‰ to +0.35 ± 0.32‰. In contrast to the AOA and the fine-grained CAIs in Leoville, the melilite sample in Allende is showing positive and highly varying δ 25 Mg values of +1.32 ± 0.45‰ to +8.38 ± 0.38‰. These positive δ 25 Mg values are consistent with the commonly observed enrichment in heavy Mg isotopes in type B CAIs due to partial evaporation processes (e.g., [START_REF] Lee | Mg isotopic anomalies in the Allende meteorite and correlation with O and Sr effects[END_REF][START_REF] Lee | Demonstration of 26 Mg excesses in Allende and evidence for 26 Al[END_REF][START_REF] Clayton | Isotopic variations in the rock-forming elements in meteorites[END_REF][START_REF] Davis | Isotope mass fractionation during evaporation of Mg 2 SiO 4[END_REF][START_REF] Mendybaev | Experimental evaporation of Mg-and Si-rich melts: Implications for the origin and evolution of FUN CAIs[END_REF][START_REF] Simon | A short timescale for changing oxygen fugacity in the solar nebula revealed by high-resolution 26 Al-26 Mg dating of CAI rims[END_REF]. Nonetheless, considering the complicated matrix-induced Mg isotopic fractionations by LA-MC-ICP-MS for the CMAS glass standards, a part of the δ 25 Mg variations (up to 2.5‰/amu effects; Fig. 3B) in the studied CAI samples with CMAS-like compositions may have come from the insufficient correction of matrix effects, and examination by solution MC-ICP-MS method would be necessary before speaking on their implications for CAI formation.

The range of δ 25 Mg variations in chondrules by LA-MC-ICP-MS, overall, are consistent with previous data for bulk chondrules by solution MC-ICP-MS (variations up to 1.5‰/amu; Fig. 8; [START_REF] Galy | The formation of chondrules at high gas pressures in the solar nebula[END_REF][START_REF] Bizzarro | Mg isotope evidence for contemporaneous formation of chondrules and refractory inclusions[END_REF][START_REF] Bouvier | Magnesium isotopic fractionation in chondrules from the Murchison and Murray CM2 carbonaceous chondrites[END_REF][START_REF] Luu | Short time interval for condensation of high-temperature silicates in the solar accretion disk[END_REF]Olsen et al., 2016;[START_REF] Van Kooten | Isotopic evidence for primordial molecular cloud material in metal-rich carbonaceous chondrites[END_REF][START_REF] Chen | Chronology of formation of early solar system solids from bulk Mg isotope analyses of CV3 chondrules[END_REF]. However, the present in situ data show that there are Mg-rich olivines in chondrules that trend to δ 25 Mg values as low as those in AOAs reported here and in [START_REF] Larsen | Evidence for magnesium isotope heterogeneity in the solar protoplanetary disk[END_REF] (Fig. 8).

Discussion

Super-cooling gas condensates among chondrule precursors

An important result from the LA-MC-ICP-MS study is that two of the four studied type I chondrules have Mg-rich olivines with low δ 25 Mg values between -1.22 ± 0.24‰ and -1.00 ± 0.29‰ (10 ablation spots) that are overlapping with the δ 25 Mg range of the Leoville and Efremovka AOAs (Fig. 8). There is a wealth of in situ SIMS O isotopic studies showing the presence (often in type I chondrules of carbonaceous chondrites) of relict olivines formed in the nebular reservoirs (or from precursors) that were more enriched in 16 O than the ambient gases where the last chondrule melting events took place (e.g., [START_REF] Weinbruch | Oxygen isotopic composition of individual olivine grains from the Allende meteorite[END_REF][START_REF] Jones | Oxygen isotope heterogeneity in chondrules from the Mokoia CV3 carbonaceous chondrite[END_REF]Pack et al., 2004;[START_REF] Chaussidon | Oxygen isotopic constraints on the origin of magnesian chondrules and on the gaseous reservoirs in the early Solar System[END_REF][START_REF] Rudraswami | Oxygen isotope systematics of chondrules in the Allende CV3 chondrite: high precision ion microprobe studies[END_REF][START_REF] Libourel | Oxygen isotopic constraints on the origin of Mg-rich olivines from chondritic meteorites[END_REF][START_REF] Tenner | Oxygen isotope systematics of chondrule phenocrysts from the CO3.0 chondrite Yamato 81020: evidence for two distinct oxygen isotope reservoirs[END_REF]Marrocchi et al., 2019a). Nonetheless, these 16 O enrichments are generally much smaller than those in AOAs [START_REF] Clayton | Oxygen isotopes in meteorites[END_REF][START_REF] Yurimoto | Oxygen isotopes of chondritic components[END_REF]; and references therein) except in rare cases (Marrocchi et al., 2019a). Thus, the present δ 25 Mg similarity between some chondrule olivines and AOAs does not mean necessarily that the precursors of chondrules formed in the same nebular reservoirs than AOAs, but could instead suggest a formation from a similar rapid-cooling condensation process.

For a cooling gas reservoir, kinetic Mg isotopic effects can arise when Mg oversaturation (P Mg ≥ P Mg,sat ) occurs due to cooling taking place faster than condensation [START_REF] Richter | Timescales determining the degree of kinetic isotope fractionation by evaporation and condensation[END_REF]. The isotope fractionation factors on 25 Mg/ 24 Mg ratio for the condensates are related to the degree of oversaturation [START_REF] Richter | Timescales determining the degree of kinetic isotope fractionation by evaporation and condensation[END_REF], which can be approximated as:

 cond 25/24  1 m 24 Mg m 25 Mg  1          1 P Mg,sat P Mg       (9)
where m represents the atomic mass of isotope 24 Mg or 25 Mg. As such, the δ 25 Mg values of the instant condensates would be:

 25 Mg cond   25 Mg gas  1000   cond 25/24  1   (10)
For a gas reservoir with a chondritic Mg isotopic composition (δ 25 Mg = -0.15 ± 0.04‰, 2SD, n = 38; [START_REF] Teng | Magnesium isotopic composition of the Earth and chondrites[END_REF], an equilibrium condensation at slow cooling (P Mg = P Mg,sat ) would exhibit no kinetic effect. While currently there are no data in the literature on the equilibrium Mg isotopic fractionation between solids and gases, we temporarily assume that such effects are negligible, i.e., the δ 25 Mg value of condensates from equilibrium condensation should be -0.15‰. However, a condensation during fast cooling would lead to various degrees of kinetic Mg isotopic effects depending on P Mg,sat /P Mg . Note that  cond 25/24 can approach 24 / 25 when the cooling is extremely fast relative to the condensation timescale (in other words, P Mg ≫ P Mg,sat ). As modeled in Richter (2004), non-equilibrium condensation at   0.01 (with  a parameter comparing the condensation timescale relative to the cooling timescale,   0.1 implies fast cooling and slow cooling) can lead to δ 25 Mg values in the accumulated

 ≪1
condensates at 1250 K 2‰ lower than in the initial gas.

Thus, the low δ 25 Mg values in the olivines of the present chondrules and AOAs likely suggest a rapid-cooling condensation process (P Mg > P Mg,sat ) for the formation of chondrule precursors and AOAs. Such short timescales for condensation of AOAs' olivines have also been proposed on the basis of their light Si isotopic compositions (Marrocchi et al., 2019b).

The occurrences of rapid-cooling condensation in multiple nebular regions suggest that supercooling of gases may be common across the early solar nebula.

Potential Mg isotopic disequilibrium between olivines and mesostases in chondrules

The LA-MC-ICP-MS results from this study show that in some chondrules, olivines and mesostases have significantly heterogeneous δ 25 Mg values, with the glassy mesostases or the clinopyroxene and plagioclase having δ 25 Mg values higher than those of olivines (Fig. 8). Such inter-phase δ 25 Mg differences in chondrules observed by LA-MC-ICP-MS technique can be further scrutinized in the future via either solution MC-ICP-MS method after careful microscale sampling or femtosecond laser ablation MC-ICP-MS method.

Carbonaceous chondrites have experienced some degrees of late aqueous alteration in the parent bodies and hydration of the mesostases (e.g., [START_REF] Clayton | Oxygen isotope studies of carbonaceous chondrites[END_REF]

Very few δ 25 Mg variations can be expected between the various phases in chondrules if

Mg isotopic equilibrium was established between them at the time of quenching during chondrule formation. In fact, studies on terrestrial mafic rocks have shown that stable Mg isotopic fractionation between silicate minerals like olivines and melts is almost negligible at temperatures above 1400 K (≤ 0.03‰ on δ 25 Mg; [START_REF] Teng | Investigation of magnesium isotope fractionation during basalt differentiation: implications for a chondritic composition of the terrestrial mantle[END_REF]. Thus, the intra-chondrule δ 25 Mg variations suggest the presence of relict isotopic heterogeneities from precursors and also some thermal processes that could affect specifically the Mg isotopic composition of the melted part of chondrules.

Effects of olivine dissolution and melt vaporization on Mg budgets of chondrule melts

Vaporization of Mg can be a possible source of Mg isotopic fractionation in chondrule melts, depending on the peak temperatures that chondrules reached during the heating events.

In vacuum, a melt spherule can lose by evaporation ≈ 0.2% of its Mg per minute at 2000 K and ≈ 3% at 2200 K [START_REF] Hashimoto | Evaporation metamorphism in the early solar nebula-Evaporation experiments on the melt FeO-MgO-SiO 2 -CaO-Al 2 O 3 and chemical fractionations of primitive materials[END_REF][START_REF] Floss | Elemental and isotopic fractionations produced through evaporation of the Allende CV chondrilte: Implications for the origin of HAL-type hibonite inclusions[END_REF][START_REF] Davis | Isotope mass fractionation during evaporation of Mg 2 SiO 4[END_REF]; Table S8). On the other hand, the peak temperatures inferred by experimental analogs of chondrules range from < 1800 K to 2200 K [START_REF] Connolly | The formation of chondrules: Petrologic tests of the shock wave model[END_REF][START_REF] Hewins | Temperature conditions for chondrule formation[END_REF][START_REF] Connolly | Chondrules: The canonical and non-canonical view[END_REF]. In the cases of high peak temperatures of 1800-2200 K under vacuum (or a canonic partial pressure of Mg), chondrule melts would inevitably be subjected to Mg loss by evaporation.

It is also noticeable that some chondrules have only been partially molten, as evidenced by the presence in type I chondrules of 16 O-rich olivines that are considered to be "relict" survivors from the chondrule precursors [START_REF] Weinbruch | Oxygen isotopic composition of individual olivine grains from the Allende meteorite[END_REF][START_REF] Jones | Oxygen isotope heterogeneity in chondrules from the Mokoia CV3 carbonaceous chondrite[END_REF]Pack et al., 2004;[START_REF] Chaussidon | Oxygen isotopic constraints on the origin of magnesian chondrules and on the gaseous reservoirs in the early Solar System[END_REF][START_REF] Rudraswami | Oxygen isotope systematics of chondrules in the Allende CV3 chondrite: high precision ion microprobe studies[END_REF][START_REF] Tenner | Oxygen isotope systematics of chondrule phenocrysts from the CO3.0 chondrite Yamato 81020: evidence for two distinct oxygen isotope reservoirs[END_REF]Marrocchi et al., 2019a), and by the chemical disequilibrium between olivines/orthopyroxenes and mesostases (e.g., [START_REF] Libourel | Role of gas-melt interaction during chondrule formation[END_REF][START_REF] Libourel | Oxygen isotopic constraints on the origin of Mg-rich olivines from chondritic meteorites[END_REF]. This partially molten state for some chondrules can be corroborated by the existence of two internal 26 Al-26 Mg isochrons in the Leoville chondrule showing that all the phases in this chondrule were not re-equilibrated for Mg isotopes at the time of last melting event (Fig. 7I).

Therefore, partial melting of a Mg-rich precursor phase such as olivine must also be taken into account in the Mg budget of chondrule melts, together with potential Mg vaporization.

In a scenario combining melt evaporation and relict olivine dissolution, the Mg isotopic compositions of the chondrule melt are expected to be controlled by the thermal history of the chondrule. In detail, heating events would likely result in a competition between (i) Mg loss , which is 0.984-0.987 for evaporation in vacuum [START_REF] Davis | Isotope mass fractionation during evaporation of Mg 2 SiO 4[END_REF][START_REF] Mendybaev | Experimental evaporation of Mg-and Si-rich melts: Implications for the origin and evolution of FUN CAIs[END_REF] but can be close to unity at high Mg partial pressures in the ambient gases (e.g., [START_REF] Galy | The formation of chondrules at high gas pressures in the solar nebula[END_REF][START_REF] Yu | Experimental study of evaporation and isotopic mass fractionation of potassium in silicate melts[END_REF]. For a chosen  gasmelt 25/24 value (e.g., 0.999 in Fig. 9A), peak temperatures and cooling rates can affect the Mg fluxes by evaporation and olivine dissolution, thus impacting the δ 25 Mg values in chondrules. The higher peak temperatures or lower cooling rates result in more vaporization of Mg, and produce larger δ 25 Mg differences between chondrule melt and olivines (Fig. 9A).

It is important to note that magnesium evaporation associated with olivine dissolution can which are supposed to be depleted in radiogenic 26 Mg for their very low 27 Al/ 24 Mg ratios. This evolution of δ 26 Mg* in chondrule is sensitive to the peak temperatures and the cooling rates of the heating events. For instance, starting from a peak temperature of 2123 K can lead to decreases in bulk δ 26 Mg* values of chondrules by 0.015‰ at a cooling rate of 500 K/h and by 0.008‰ at 1500K/h (Fig. 9B). The decreases in bulk δ 26 Mg* values of chondrules would be significantly smaller when starting from a lower peak temperature of 2023 K, i.e., 0.005‰ at 500 K/h and by 0.002‰ at 1500K/h (Fig. 9B). Once the initial Mg in chondrule melt has been almost replaced by Mg from olivines, there would be (δ 26 Mg*) bulk ≈ (δ 26 Mg*) melt ≈ (δ 26 Mg*) olivine_0 (Fig. 9B). If 26 Al is not extinct yet at the time of chondrule quenching after the last heating event, the chondrule bulk δ 26 Mg* value would increase till 26 Al extinction (Fig. 9C), and show at present-day an internal 26 Al-26 Mg isochron, which dates the last melting event for chondrules.

Implications for the formation of chondrules

The presence in chondrule melts of δ 25 Mg values 0.68-0.94‰ higher than in olivines for three of the studied chondrules implies, most likely, that these chondrules have experienced some degrees of Mg evaporation, meaning that the peak temperatures of these chondrules have surpassed 1800 K over the last melting events. The lack of resolvable δ 25 Mg difference between olivines and mesostases within other chondrules (B-Ch1, Leo-4337-t2-ps2B-c2 and Leo-4337-t2-ps2B-c7) can be ascribed either to a completely molten state of the chondrule precursors or to a low peak temperature (< 1800 K) during a possible re-heating event. The uncertainty of the effects from Mg partial pressure on the Mg evaporation rate and  gasmelt 25/24 prevents us from defining the exact Mg fluxes by evaporation and those by olivine dissolution from the δ 25 Mg records. However, more information can be gained by knowing the δ 26 Mg* data of bulk chondrules at a precision of 0.001-0.003‰, which is approachable thanks to improvements in solution MC-ICP-MS method [START_REF] Larsen | Evidence for magnesium isotope heterogeneity in the solar protoplanetary disk[END_REF][START_REF] Schiller | Early accretion of protoplanets inferred from a reduced inner solar system 26 Al inventory[END_REF][START_REF] Olsen | Magnesium and 54 Cr isotope compositions of carbonaceous chondrite chondrules-Insights into early disk processes[END_REF][START_REF] Van Kooten | Isotopic evidence for primordial molecular cloud material in metal-rich carbonaceous chondrites[END_REF][START_REF] Luu | Bulk chondrite variability in mass independent magnesium isotope compositions -Implications for initial solar system 26 Al/ 27 Al and the timing of terrestrial accretion[END_REF][START_REF] Gregory | Primordial formation of major silicates in a protoplanetary disc with homogeneous 26 Al/ 27[END_REF].

There have been high-precision δ 26 Mg* data in literature showing that chondrules with super-solar 27 Al/ 24 Mg ratios (> 0.101) can exhibit negative δ 26 Mg* values down to -0.018 ± 0.002‰ [START_REF] Olsen | Magnesium and 54 Cr isotope compositions of carbonaceous chondrite chondrules-Insights into early disk processes[END_REF][START_REF] Van Kooten | Isotopic evidence for primordial molecular cloud material in metal-rich carbonaceous chondrites[END_REF], which the authors have attributed to a reduced initial 26 Al/ 27 Al ratios (1.5-2.2 × 10 -5 ) in the chondrule forming regions relative to the canonical 26 Al/ 27 Al ratios of ≈ 5 × 10 -5 . The distribution of initial 26 Al abundance in the Solar System is so far a highly controversial topic with the co-existing arguments for either heterogeneity at ± 80% scale (e.g., [START_REF] Schiller | Early accretion of protoplanets inferred from a reduced inner solar system 26 Al inventory[END_REF][START_REF] Olsen | Magnesium and 54 Cr isotope compositions of carbonaceous chondrite chondrules-Insights into early disk processes[END_REF][START_REF] Van Kooten | Isotopic evidence for primordial molecular cloud material in metal-rich carbonaceous chondrites[END_REF][START_REF] Bollard | Early formation of planetary building blocks inferred from Pb isotopic ages of chondrules[END_REF][START_REF] Bollard | Combined U-corrected Pb-Pb dating and 26 Al-26 Mg systematics of individual chondrules-Evidence for a reduced initial abundance of 26 Al amongst inner Solar System chondrules[END_REF] or homogeneity with variations smaller than ± 20% (e.g., [START_REF] Jacobsen | 26 Al-26 Mg and 207 Pb-206 Pb systematics of Allende CAIs: Canonical solar initial 26 Al/ 27 Al ratio reinstated[END_REF][START_REF] Villeneuve | Homogeneous distribution of 26 Al in the solar system from the Mg isotopic composition of chondrules[END_REF][START_REF] Budde | Hf-W chronology of CR chondrites: Implications for the timescales of chondrule formation and the distribution of 26 Al in the solar nebula[END_REF][START_REF] Luu | Bulk chondrite variability in mass independent magnesium isotope compositions -Implications for initial solar system 26 Al/ 27 Al and the timing of terrestrial accretion[END_REF][START_REF] Gregory | Primordial formation of major silicates in a protoplanetary disc with homogeneous 26 Al/ 27[END_REF]. We cannot solve this debate here, but we stress that a combination of Mg evaporation and olivine dissolution during chondrule melting can lead to a preferential loss of Mg that is rich in radiogenic 26 Mg. This mechanism is capable of producing negative bulk δ 26 Mg* values in some chondrules with super-solar 27 Al/ 24 Mg ratios (Fig. 9C). All these observations highlight the necessity of scrutinizing the phase relation inside chondrules from a metal stable isotope perspective (e.g., Mg, Si and Fe), which can help trace the thermal histories of chondrules during their formation.

It is noteworthy that if 26 Al was homogeneously distributed in the solar nebula, forming chondrules with negative bulk δ 26 Mg* values of -0.018‰ from the precursors with initial δ 26 Mg* value of -0.020‰ would imply a strong vaporization of Mg, thus requiring the peak temperatures at chondrule melting to be significantly higher than 1800 K (up to 2123 K) and the cooling rates not to be too high (≤ 500 K/h) (Fig. 9B,C). Applying the Mg fluxes of evaporation inferred from the δ 26 Mg* records and the δ 25 Mg difference of +0.68 to +0.94‰ between chondrule melts and olivines, the  gasmelt 25/24 value can be anchored to ≈ 0.999. This value is much closer to unity compared to the values of 0.984-0.987 expected for evaporation in vacuum [START_REF] Davis | Isotope mass fractionation during evaporation of Mg 2 SiO 4[END_REF][START_REF] Mendybaev | Experimental evaporation of Mg-and Si-rich melts: Implications for the origin and evolution of FUN CAIs[END_REF], and rather points to a high Mg partial pressure in the ambient gases during chondrule formation [START_REF] Galy | The formation of chondrules at high gas pressures in the solar nebula[END_REF][START_REF] Yu | Experimental study of evaporation and isotopic mass fractionation of potassium in silicate melts[END_REF].

Such thermal and gas conditions are the most consistent with the conditions required for the shock wave model [START_REF] Desch | The importance of experiments: Constraints on chondrule formation models[END_REF][START_REF] Fedkin | Mineralogical and isotopic constraints on chondrule formation from shock wave thermal histories[END_REF]. In addition, the presence of Mg evaporation supports an "open" system for chondrules during their formation. This calls for revisiting the records of moderately volatile elements in chondrules. Earlier observations for the limited K isotopic variations [START_REF] Humayun | Potassium isotope cosmochemistry: Genetic implications of volatile element depletion[END_REF][START_REF] Alexander | The lack of potassium-isotopic fractionation in Bishunpur chondrules[END_REF] and absence of Na depletion [START_REF] Alexander | The formation conditions of chondrules and chondrites[END_REF] suggest a "closed" system for chondrules during their last melting events. However, latest studies unravel that (i) chondrules have been depleted in Zn by two orders of magnitude and as well isotopically fractionated in Zn compared to the matrix [START_REF] Pringle | The origin of volatile element depletion in early solar system material: Clues from Zn isotopes in chondrules[END_REF]van Kooten and Moynier, 2019), (ii) chondrules have higher Mn/Na ratios (up to 2) than bulk chondrites (around 0.3), likely pointing to depletion in Na [START_REF] Mahan | Volatile element evolution of chondrules through time[END_REF], and (iii) chondrules are significantly depleted in volatile elements relative to bulk chondrites (van Kooten et al., 2019).

Conclusions

(1) For in situ Mg isotopic measurements by LA-MC-ICP-MS, 48 Ca 2+ is the major doubly charged species interfering with the ions of Mg isotopes, and the productions of 48 Ti 2+ and 52 Cr 2+ are negligible. Monitoring together the intensities of 44 Ca 2+ and 24 Mg + under a sub cup configuration permits a sufficient correction of the 48 Ca 2+ interferences on 24 Mg + . The matrixinduced Mg isotopic fractionations are 1.0-1.5‰/amu for standards including olivines and komatiitic/basaltic/andesitic glasses, which are significantly smaller than those by SIMS (8‰/amu; [START_REF] Chaussidon | In situ analysis of non-traditional isotopes by SIMS and LA-MC-ICP-MS: Key aspects and the example of Mg isotopes in olivines and silicate glasses[END_REF]. After the correction of 48 Ca 2+ interferences and matrix effects, the LA-MC-ICP-MS technique in this study is able to provide accurate Mg isotopic measurements with 2SD uncertainties of ± 0.04-0.06‰ for olivines and of ± 0.10-0.60‰ for mesostasis and melilite phases on the δ 26 Mg* values, and of ± 0.25-0.30‰ for olivines and of ± 0.30-0.55‰ for mesostasis phases on the δ 25 Mg values.

(2) The internal 26 Al-26 Mg isochrons provide initial 26 Al/ 27 Al ratios of 5.46 (± 0.38) × 10 -5 to 6.14 (± 0.92) × 10 -5 for the studied refractory inclusions and 0.16 (± 0.08) × 10 -5 to 1.87 (± 0.92) × 10 -5 for the chondrules, which are identical with uncertainty to literature values. The presence of Mg-rich olivines with low δ 25 Mg values in some type I chondrules implies that chondrule precursors may have formed via condensation of rapid-cooling gases, as similarly recorded by AOAs. Moreover, the higher δ 25 Mg values in mesostases than in olivines for three of the studied chondrules likely support Mg evaporation during the heating events.

(

   1          1  mtr            10 3
Substituting equations A5 and A6 into equation A7 provides:

(A8)  26 Mg CaTiCr  48 Ca 2  48 Ti 2 24 Mg   1  mtr  1        52 Cr 2 26 Mg           10 3
Since  mtr = 0.517 has been chosen in this study, we have:

(A9) 1  mtr  1  0.934
The 48 Ca 2+ / 24 Mg + ratios of samples can be written: The vaporization of Mg in vacuum [START_REF] Hashimoto | Evaporation metamorphism in the early solar nebula-Evaporation experiments on the melt FeO-MgO-SiO 2 -CaO-Al 2 O 3 and chemical fractionations of primitive materials[END_REF][START_REF] Floss | Elemental and isotopic fractionations produced through evaporation of the Allende CV chondrilte: Implications for the origin of HAL-type hibonite inclusions[END_REF][START_REF] Davis | Isotope mass fractionation during evaporation of Mg 2 SiO 4[END_REF] can be fitted by an empirical fourth-power function of temperature (Table S8):

(A12) r evap  0.000959(0.000044)  T 1.702(0.093)     4 with the evaporation rate of Mg that is described as the fraction of total Mg per minute r evap and the temperature in K. The Mg evaporation rate starts to be significant above 1800 K, T and rises rapidly with temperature. In a reservoir with high dust-enrichments, vaporization can increase partial pressures of Mg in the gases, and therefore reduce the vaporization rate of Mg or lead to some re-condensation. For instance, based on cathodoluminescence mapping of Mgrich olivines in a set of type I chondrules, [START_REF] Libourel | Chondrules as direct thermochemical sensors of solar protoplanetary disk gas[END_REF] inferred that the olivines had likely grown in a melt experiencing Mg and Si exchanges with the surrounding gas.

However, experiments investigating Mg evaporation from silicate melts at non-vacuum gaseous environments are currently not available. Hence, we assume that equation A12 also applies to the case of chondrule melting, but we mitigate the uncertainty from this assumption by considering various gas-melt Mg isotopic fractionation factors during evaporation.

As described in [START_REF] Soulié | Olivine dissolution in molten silicates: An experimental study with application to chondrule formation[END_REF], the olivine dissolution rate ( in μm/min after a  dissol spherical approximation) in a melt follows:

(A13)  dissol  A 0  e  E a /RT
where A 0 and E a are Arrhenius parameters, representing pre-exponential factor and activation energy, respectively. Note that A 0 and E a would vary with different melt compositions (such as anorthite, spinel and tridymite; [START_REF] Soulié | Olivine dissolution in molten silicates: An experimental study with application to chondrule formation[END_REF].

For a given thermal history, (A14)

T  f (t)
where t represents the time in minute since the beginning of the selected time window, the rates of Mg evaporation and olivine dissolution can be calculated from equations A12 and A13.

To compare the effects from Mg evaporation and olivine dissolution, the Mg fluxes from the two processes can be both described as the fraction of the total Mg in the initial system, e.g., the Mg budget of the system will be smaller than "1" after some Mg evaporation. To start the modeling, we assume for the chondrule precursors: δ 25 Mg non-olivine or olivine_0 = 0‰, ( 27 Al/ 24 Mg) bulk_0 = 0.101, ( 27 Al/ 24 Mg) olivine_0 = 0.001, ( 27 Al/ 24 Mg) non-olivine or melt_0 = 2.011, F Mg- non-olivine_0 = 0.05 and F Mg-olivine_0 = 0.95, where F represents the fraction of total Mg in the precursors (olivine and non-olivine). The choice of F Mg-olivine_0 = for the precursors is based on the fact that the Mg budget of a type I PO chondrule is dominated by olivine. The simplest scenario is to assume that the non-olivine phases will be preferentially melted upon heating, i.e. F Mg-melt_0 ≈ F Mg-non-olivine_0 . The Mg flux per minute to the melt due to olivine dissolution (

) over an incremental time, Δt, interval between t i and t j (t j = t i + Δt) can be quantified F dissol as: instead of equations A13, the Mg fluxes of olivine dissolution between t i and t j can be approximated to be the same as those of vaporization:

(A15) F dissolt i t j  F Mgolivinet i  F Mgolivinet i 3  R   dissolt i  t F Mgolivinet i 3  R         3          
(A17) F dissolt i t j  F evapt i t j
The Mg fractions and 27 Al/ 24 Mg ratio in olivine, melt and bulk system at t j should follow mass conservation (assuming that Al is not evaporated from the chondrule melt):

(A18) F Mgolivinet j  F Mgolivinet i  F dissolt i t j (A19) F Mgmeltt j  F Mgmeltt i  F evapt i t j  F dissolt i t j (A20) 27 Al 24 Mg       meltt j  27 Al 24 Mg       meltt i  F Mgmeltt i F Mgmeltt j         (A21) 27 Al 24 Mg       bulkt j  27 Al 24 Mg       bulk _ 0  1 F Mgolivinet j  F Mgmeltt j
Once Mg evaporation occurs, the 27 Al/ 24 Mg ratio in bulk chondrule would increase, whereas the 27 Al/ 24 Mg ratios of chondrule melt should be stable due to the cancel-out of the Mg fluxes from evaporation and olivine dissolution. For the δ 25 Mg values, can be approximated into a two-step process in a small time-interval. The remaining Mg in chondrule melt at t j after evaporation would tend to be more enriched in the heavy isotopes following:

(A22)  25 Mg meltevapt i   25 Mg meltt i  1000   gasmelt 25/24  1    F evapt i t j F Mgmeltt i
The Mg gain from olivine dissolution brings the δ 25 Mg values of chondrule melt at t j to:

(A23)  25 Mg meltt j   25 Mg meltevapt i  F Mgmeltt i  F evapt i t j F Mgmeltt j   25 Mg olivine _ 0  F dissolt i t j F Mgmeltt j
The δ 25 Mg values of bulk chondrule at t j would then be:

(A24)  25 Mg bulkt j   25 Mg olivine _ 0  F Mgolivinet j   25 Mg meltt j  F Mgmeltt j F Mgolivinet j  F Mgmeltt j
We model below, for the sake of the exercise, a case where the precursors (olivine + matrix phases) were condensed from a canonic gas at 0.717 Myr after CAIs (i.e., one half-life for 26 Al)

and the chondrule heating took place 2 Myr after CAIs, in agreement with general 26 Al-26 Mg chronologies of chondrules reconstructed assuming a homogeneous 26 Al/ 27 Al as a function of time (i.e., last melting events at 1-3 Myr after CAIs and precursors which can be nearly as old as CAIs, e.g., [START_REF] Kita | A short duration of chondrule formation in the solar nebula: Evidence from 26 Al in Semarkona ferromagnesian chondrules[END_REF][START_REF] Villeneuve | Homogeneous distribution of 26 Al in the solar system from the Mg isotopic composition of chondrules[END_REF][START_REF] Luu | Short time interval for condensation of high-temperature silicates in the solar accretion disk[END_REF][START_REF] Gregory | Primordial formation of major silicates in a protoplanetary disc with homogeneous 26 Al/ 27[END_REF].

We consider thereafter for the precursors: ( 27 Al/ 24 Mg) bulk_0 = 0.101, ( 27 Al/ 24 Mg) olivine_0 = 0.001, ( 27 Al/ 24 Mg) non-olivine or melt_0 = 2.011, F Mg-non-olivine_0 = 0.05 and F Mg-olivine_0 = 0.95. Assuming initial 26 Al/ 27 Al ratio and δ 26 Mg* value of 5.23 × 10 -5 and -0.040‰, respectively, for the solar nebula [START_REF] Jacobsen | 26 Al-26 Mg and 207 Pb-206 Pb systematics of Allende CAIs: Canonical solar initial 26 Al/ 27 Al ratio reinstated[END_REF][START_REF] Villeneuve | Homogeneous distribution of 26 Al in the solar system from the Mg isotopic composition of chondrules[END_REF][START_REF] Villeneuve | Magnesium isotopes constraints on the origin of Mg-rich olivines from the Allende chondrite: Nebular versus planetary?[END_REF][START_REF] Luu | Short time interval for condensation of high-temperature silicates in the solar accretion disk[END_REF][START_REF] Chen | Chronology of formation of early solar system solids from bulk Mg isotope analyses of CV3 chondrules[END_REF][START_REF] Gregory | Primordial formation of major silicates in a protoplanetary disc with homogeneous 26 Al/ 27[END_REF]

(A25)  26 Mg *   meltt j   26 Mg *   meltt i  F Mgmeltt i  F evapt i t j     26 Mg *   olivine _ 0  F dissolt i t j F Mgmeltt j (A26)  26 Mg *   bulkt j   26 Mg *   olivine _ 0  F Mgolivinet j   26 Mg *   meltt j  F Mgmeltt j F Mgolivinet j  F Mgmeltt j
For any other chondrule precursors and T-t paths, the time-integrated 27 Al/ 24 Mg ratios as well as the δ 25 Mg and δ 26 Mg* values of chondrule melt or bulk chondrule can be obtained via iteration of equations A12, A14 and A16 to A26. Note that under the approximation of , the Mg evaporation histories of chondrules can be reconstructed from F dissolt i t j  F evapt i t j their Mg isotopic records.

Figures and captions

Fig S3 to S6), with the circle for the Allende session with a linear regression of y = (0.097 ± 0.005) x + (0.000017 ± 0.000019), the diamond for the Leoville session with y = (0.097 ± 0.005) x + (0.000017 ± 0.000019), and the triangle and square for the other two sessions including only mineral and glass standards with y = (0.091 ± 0.004) x + (0.000073 ± 0.000058) and y = (0.100 ± 0.001) x + (0.000042 ± 0.000014)]. The CMAS glasses have been marked with the lighter colors to be distinguished from the olivines and komatiitic/basaltic/andesitic glasses.

Fig. 2. (A)

Plot of measured 44 Ca 2+ / 24 Mg + ratios versus true 44 Ca/ 24 Mg ratios and (B) plot of measured 27 Al + / 24 Mg + ratios versus true Al/ 24 Mg ratios for mineral and glass standards.

Symbols are the same as those in Fig. 1. The sessions using spot sizes of 40 μm to 65 μm (i.e., the circle, diamond and square) are providing a single well-defined linear regression for each session. For the session with a large spot size of 110 μm (i.e., the triangle), the CMAS glasses (i.e., the light green triangle) form a calibration line different from that defined by olivines and komatiitic/basaltic/andesitic glasses (i.e., the green triangle). It is noteworthy that the 27 Al/ 24 Mg and 44 Ca/ 24 Mg ratios for individual ablation spots on chondrules can be derived using the correlations in (A) and (B), which together with the Mg ion intensities can help discriminate well the ablated phases. The regression data are calculated with the Model 1 in Isoplot 4.15 [START_REF] Ludwig | Isoplot version 4.15: a geochronological toolkit for microsoft Excel[END_REF]. Probably due to late disturbance, three spots have been excluded from the 26 Al-26 Mg isochron of the Allende melilite Mel-1 in (A). For chondrule Leo-4337-t2-ps2B-c7, there are possibly two internal 26 Al-26 Mg isochrons, and those spots plotting close to the bulk CAI 26 Al-26 Mg isochron [START_REF] Jacobsen | 26 Al-26 Mg and 207 Pb-206 Pb systematics of Allende CAIs: Canonical solar initial 26 Al/ 27 Al ratio reinstated[END_REF][START_REF] Larsen | Evidence for magnesium isotope heterogeneity in the solar protoplanetary disk[END_REF] have been highlighted in orange symbols. for bulk chondrules [START_REF] Galy | The formation of chondrules at high gas pressures in the solar nebula[END_REF][START_REF] Bizzarro | Mg isotope evidence for contemporaneous formation of chondrules and refractory inclusions[END_REF][START_REF] Bouvier | Magnesium isotopic fractionation in chondrules from the Murchison and Murray CM2 carbonaceous chondrites[END_REF][START_REF] Luu | Short time interval for condensation of high-temperature silicates in the solar accretion disk[END_REF][START_REF] Olsen | Magnesium and 54 Cr isotope compositions of carbonaceous chondrite chondrules-Insights into early disk processes[END_REF][START_REF] Van Kooten | Isotopic evidence for primordial molecular cloud material in metal-rich carbonaceous chondrites[END_REF][START_REF] Chen | Chronology of formation of early solar system solids from bulk Mg isotope analyses of CV3 chondrules[END_REF] and AOAs [START_REF] Larsen | Evidence for magnesium isotope heterogeneity in the solar protoplanetary disk[END_REF] (Table S7) are shown for comparison.

  ) covering the compositional range of meteoritic samples, and having various CaO, TiO 2 and Cr 2 O 3 contents: synthetic olivines (devoid of CaO, TiO 2 and Cr 2 O 3 ), komatiitic, basaltic and andesitic glasses (with various amounts of CaO, TiO 2 and Cr 2 O 3 ), and CaO-MgO-Al 2 O 3 -SiO 2 (CMAS) glasses having only

  Al/ 27 Al ratios and thus biased 26 Al-26 Mg isochrons. We also figure out that the matrix-induced Mg isotopic fractionations on CMAS glasses are quite sensitive to the choice of laser fluence for ablation. Compared to the systematics defined from olivines, komatiitic to andesitic glasses, diopside and crystalline MORB, the measurements of CMAS glasses yield lower [δ 25 Mg measured -δ 25 Mg true ] values at a low laser fluence of 3.37 or 4.23 J/cm 2 , and higher [δ 25 Mg measured -δ 25 Mg true ] values at a high laser fluence

3. 5 .

 5 Error estimate for the measured Mg isotopic resultsFor the Mg isotopic measurements from LA-MC-ICP-MS measurements, three sources of analytical uncertainty have to be considered: (i) internal uncertainty of the measurements, (ii) uncertainty from 48 Ca 2+ interference correction and (iii) uncertainty from matrix effect correction. Each laser ablation spot on a sample forms an individual analysis. The internal uncertainty can be estimated using duplicate analyses of mineral and glass standards. For the Allende and Leoville sessions, the internal uncertainties are related to the number of Mg ions detected and can be assessed from the 2SD of the delta values of the raw Mg isotopic ratios of olivine and glass standards. The uncertainty from 48 Ca 2+ correction is controlled by the measured 44 Ca 2+ / 24 Mg + ratios and the uncertainty on the estimated 48 Ca 2+ / 44 Ca 2+ ratios. The additional uncertainty from matrix effect correction can be estimated from fitting the [δ 25 Mg measured -δ 25 Mg true ] values and MINBS values in each session following equation 7. Note that an additional uncertainty on the δ 26 Mg* values due to matrix effect correction can arise from the choice of β mtr , which can range between 0.511 for kinetic fractionation law and 0.521 for equilibrium fractionation law(Young et al., 2002b;[START_REF] Davis | Isotopic mass fractionation laws for magnesium and their effects on 26 Al-26 Mg systematics in solar system materials[END_REF]. However, this effect should be negligible at an analytical precision of ±0.04‰ on δ 26 Mg*. The final uncertainties of the δ 25 Mg, δ 26 Mg and δ 26 Mg* values can be obtained as the sums of the uncertainties from these different sources.After error propagation, the 2SD uncertainties on δ 26 Mg* are ±0.04-0.06‰ for olivines and ± 0.10-0.60‰ for mesostasis and melilite phases. For δ 25 Mg, the 2SD uncertainties are ±0.25-0.30‰ for olivines whereas ±0.30-0.55‰ for mesostasis and melilite phases. Note that the larger uncertainty on mesostasis and melilite phases than on olivines is due to their lower MgO contents and less Mg ion counts at a fixed laser ablation setting. It is noteworthy that if aiming to measure precisely the δ 26 Mg* values of olivine phases in natural samples, San Carlos olivines or other olivines should be used as the bracketing standards instead of basaltic glasses to reduce as much as possible the uncertainty from sample-to-standard normalization.3.6. Measurements of δ 26 Mg* and δ 25 Mg values for standards after correction

  . Nonetheless, this late aqueous alteration is unlikely to affect much the Mg isotopic compositions from chondrule formation. (i) Refractory inclusions and chondrules from the same sample sections show no significant disturbance in their 26 Al-26 Mg isochrons, suggesting that the 27 Al/ 24 Mg variations in the studied chondrules are likely inherited from the melting events prior to26 Al extinction, implying that the mobility of Mg by aqueous alteration has been weak.(ii) Experiments and studies of different types of terrestrial rocks weathered to various degrees have shown the existence of a small Mg isotopic fractionation between fluid and mineral (during either the dissolution of primary minerals or the formation of secondary minerals) of  26 Mg fluid-mineral < ≈ -1‰ (e. g.[START_REF] Ryu | Experimental investigation of Mg isotope fractionation during mineral dissolution and clay formation[END_REF]. Even though 10% in mass of Mg in the mesostases has been removed during secondary aqueous alteration, this would result in an upwards shift on the δ 25 Mg value by only ≈ 0.10‰.

  from chondrule melts by evaporation and (ii) Mg gain into chondrule melts by olivine dissolution (see the model in Appendix). The model results show that vaporization of Mg should lead to higher δ 25 Mg values in both the melt and the bulk chondrule. For a set T-t paths, the scale of the changes in δ 25 Mg value is controlled by  gasmelt 25/24

  also impact the bulk δ 26 Mg* values of chondrules. Before the extinction of 26 Al, various degrees of radiogenic 26 Mg enrichments can develop within the different phases of chondrule precursors before the last melting event took place. Under a partial melting scenario that has Mg-rich olivines as the relict components, preferential melting of the high-27 Al/ 24 Mg phases within the chondrule precursors during heating events would produce melts carrying Mg enriched in radiogenic 26 Mg (in other words, the positive and high δ 26 Mg* values), and Mg vaporization from such melts would deplete the radiogenic 26 Mg from bulk chondrules, along with the increase in 27 Al/ 24 Mg ratio. The replenishments of Mg from dissolution of olivines together with Mg losses by evaporation would force the δ 26 Mg* values in the melt and the bulk of chondrule migrating towards the δ 26 Mg* values of the olivines of precursors (Fig.9B),

  It is however noteworthy that the temperature-dependent function for olivine dissolution in equation A13 is valid only when the melt composition did not change much with olivine dissolution. In other words, the melt reservoir has to be large and merely affected by input from the dissolving olivine. Nonetheless, most type I chondrules have high olivine/melt ratios, in which olivine dissolution would have a strong impact on the composition of chondrule melts, and therefore also on the olivine dissolution rate. As shown by the experiments using starting materials with chondrule-like olivine/melt ratios in[START_REF] Villeneuve | Relationships between type I and type II chondrules: Implications on chondrule formation processes[END_REF], olivine dissolution rate can be reduced by 2-3 orders of magnitude when the SiO 2 and MgO contents start to be high in the melts due to olivine dissolution. Substituting R = 1000 μm and the olivine dissolution parameters of an 'anorthite'-like melt composition from[START_REF] Soulié | Olivine dissolution in molten silicates: An experimental study with application to chondrule formation[END_REF] into equation A15 shows that the Mg fluxes of olivine dissolution in a melt that is strongly undersaturated in olivine can surpass the Mg fluxes of evaporation by 2 orders of magnitude. As such, a more realistic situation for the Mg fluxes in chondrule melt should be: (i) olivine dissolution occurs but ceases quickly through saturating the melt in olivine and (ii) olivine dissolution proceeds with olivine under-saturation in the melt created by evaporation. As such,

Fig. 3 .

 3 Fig. 3. Plot of [δ 25 Mg measured -δ 25 Mg true ] values caused by matrix effects versus MINBS values

Fig. 5 .

 5 Fig. 5. (A) Plot of δ 25 Mg bracketing standard versus δ 26 Mg bracketing standard and (B) plot of measured

Fig. 6 .

 6 Fig. 6. Comparing the δ 25 Mg values by LA-MC-ICP-MS and those by solution MC-ICP-MS

Fig. 7 .

 7 Fig. 7. Internal 26 Al-26 Mg isochrons of the studied Allende and Leoville refractory inclusions

Fig. 8 .

 8 Fig. 8. Plot comparing δ 25 Mg variations in the studied chondrules and AOA. The literature data

Fig. 9 .

 9 Fig. 9. Modeling of the changes of (A) melt-olivine δ 25 Mg difference and (B) bulk δ 26 Mg*

  26 Mg   52 Cr 2 24 Mg   48 Ca 2  48 Ti 2As the 44 Ca 2+ signals have been monitored, the 48 Ca 2+ / 24 Mg + ratios of samples can be written:The apparent non-mass dependent Mg isotopic effects from doubly charged ion interferences of Ca, Ti and Cr (noted hereafter Δ 26 Mg Ca+Ti+Cr ) should follow a linear relation against the measured 44 Ca 2+ / 24 Mg

	 26 Mg CaTiCr  0.934    	44 Ca 2 24 Mg  	48 Ca 2 44 Ca 2  0.934 	48 Ti 2 24 Mg  	52 Cr 2 26 Mg 	    10 3	(6)
		25 Mg / 24 Mg		meas		25 Mg  24 Mg   48 Ca 2  48 Ti 2
		26 Mg / 24 Mg				(4)
	48 Ca 2 24 Mg  	44 Ca 2 24 Mg  	48 Ca 2 44 Ca 2	(5)

meas

 + ratio (see equations A1 to A11 in Appendix for demonstration):

  ) In a partial melting for chondrule formation, peak temperature and cooling rate can impact on the Mg isotopic inventories of chondrules, via influencing the Mg fluxes by vaporization and by olivine dissolution. To produce the existing Mg isotopic observations under an assumption of homogeneous 26 Al/ 27 Al as a function of time in the solar system,

	(noted as Δ 26 Mg Ca+Ti+Cr ) should be:
	 26 Mg CaTiCr 	26 Mg / 24 Mg 26 Mg  / 24 Mg   meas   	 1		25 Mg / 24 Mg 25 Mg  / 24 Mg   meas	(A7)
	chondrules must have been heated up to high peak temperatures (up to 2123 K) and stayed
	above 1800 K for several tens of minutes during cooling to vaporize Mg, likely pointing to a
	heat source from shock wave for chondrule formation.

  , the chondrule precursor would have (δ 26 Mg*) bulk = (δ 26 Mg*) olivine = (δ 26 Mg*) non-olivine = -0.020‰ at 0.717 Myrs after CAIs, evolving to (δ 26 Mg*) bulk ≈ -0.003‰, (δ 26 Mg*) olivine ≈ -0.020‰ and (δ 26 Mg*) non-olivine or melt ≈ +0.326‰ at the time of heating (2 Myr after CAIs). The δ 26 Mg* values of the melt and the bulk of chondrule at t j after melt evaporation and olivine dissolution are given by:

  . 1. Plot of Δ 26 Mg Ca+Ti+Cr /1000 versus measured 44 Ca 2+ / 24 Mg + ratios for mineral and glass standards. The raw Mg isotopic ratios have included the isobaric interferences from doubly charged ions including 48 Ca 2+ , 48 Ti 2+ and 52 Cr 2+ . The four types of symbols (i.e., circle, diamond, square, triangle) represent the data from four sessions (Tables

Table 1

 1 Chemical and Mg isotopic composition of the standards, with CMAS glasses from this study and others from[START_REF] Chaussidon | In situ analysis of non-traditional isotopes by SIMS and LA-MC-ICP-MS: Key aspects and the example of Mg isotopes in olivines and silicate glasses[END_REF].

			Type						δ 25
			Al 2 O 3	MgO	SiO 2	44 Ca/ 24	48 Ti/ 24	52 Cr/ 26	Mg
	Sample		wt%	wt%	wt%	Mg a	Mg a	Mg a	(‰)
	San	Carlos	olivi						-
	olivine		ne n.a.	48.88 40.91 0.0000 0.0000 0.0000	0.16
	Ol synth Watson	olivi						-
	#01		ne n.a.	49.33 41.02 0.0000 0.0000 0.0000	0.62
	Ol synth Watson	olivi						-
	#02		ne n.a.	49.33 41.02 0.0000 0.0000 0.0000	0.62
	Ol synth Watson	olivi						-
	#04		ne n.a.	45.02 40.00 0.0000 0.0000 0.0000	0.62
	Ol synth Watson	olivi						-
	#05		ne n.a.	40.22 38.88 0.0000 0.0000 0.0000	0.62
	Ol synth Watson	olivi						-
	#06		ne n.a.	32.83 37.14 0.0000 0.0000 0.0000	0.62
			glass						-
	StHs6/80-G	17.80 1.97	63.70 0.0509 0.1681 0.0051	0.18
			glass						-
	BCR-2G		13.80 3.59	54.80 0.0382 0.2955 0.0030	0.16
			glass						-
	TB-1G		16.70 3.70	54.30 0.0344 0.1133 0.0000	0.16
			glass						-
	T1-G		17.10 3.75	58.60 0.0359 0.0948 0.0033	0.15
			glass						-
	ML3B-G		13.60 6.59	51.40 0.0302 0.1522 0.0158	0.17
			glass						-
	BHVO-2G	13.60 7.13	49.30 0.0303 0.1807 0.0232	0.12
			glass						-
	KL2-G		13.30 7.34	50.30 0.0282 0.1642 0.0236	0.14
	MORB		glass						-
	(CLDR01RV)	15.32 8.70	49.94 0.0267 n.a.	n.a.	0.13
			glass						-
	BIR-1G		15.60 9.37	47.30 0.0271 0.0464 0.0254	0.15
			glass						-
	GOR132-G	11.00 22.40 45.50 0.0072 0.0064 0.0666	0.09
			glass						-
	GOR128-G	9.91	26.00 46.10 0.0046 0.0052 0.0515	0.15
			glass						-
	CMAS-synth-I	33.55 3.66	24.06 0.2008 0.0000 0.0000	1.99

Acknowledgements

We thank Pascale Louvat, Claire Rollion-Bard and Jessica Dallas for helping with the MC-ICP-MS and laser ablation system at IPGP. David Madre is appreciated for his help with the sample preparation for SIMS measurements at CRPG (Nancy). The Associate Editor Dimitri A. Papanastassiou and three anonymous reviewers are thanked for their constructive comments, which have improved significantly the manuscript. This work was supported by ANR grant CRADLE (ANR-15-CE31-0004-01), and by the UnivEarthS Labex program (ANR-10-LABX-0023 and ANR-11-IDEX-0005-2).

Appendix

Formalism of non-mass dependent Mg isotopic effects from 48 Ca 2+ , 48 Ti 2+ and 52 Cr 2+ After transformation, equations 3 can represent a multiplication factor on the true Mg isotope ion ratio: (A1) 25

Supplementary data tables

Table S1 Chemical and Mg isotopic compositions of the used standards.

Table S2 Analytical parameters of laser system and mass spectrometry.

Table S3 Chemical and Mg isotopic compositions of olivine/glass standards and Allende

chondrules measured with laser fluence of 3.37 J/cm 2 and spot size of 40 μm.

Declaration of interests

 The authors declare that they have known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: