A Néri

J Guignard

M Monnereau

M J Toplis

G Quitté

Metal segregation in planetesimals: constraints from experimentally determined interfacial energies

Keywords: Metal-silicate differentiation, interfacial energies, high temperature experiments, planetesimals, primitive achondrites

High temperature experiments have been performed to constrain interfacial energies in a three-phase system (metal -forsterite -silicate melt) representative of partially differentiated planetesimals accreted early in the solar system history, with the aim of providing new insights into the factors affecting the interconnection threshold of metal-rich phases. Experiments were run under controlled oxygen fugacity (∆ Ni-NiO = -3) at 1440°C, typically for 24h. Quantification of the true dihedral angles requires a resolution of at least 30 nm per pixel in order to reveal small-angle wedges of silicate melt at crystal interfaces.

At this level of resolution, dihedral angle distributions of silicate melt and olivine appear asymmetric, an observation interpreted in terms of anisotropy of olivine crystals. Based upon the theoretical relation between dihedral angles and interfacial energies in a three-phase system, the relative magnitudes of interfacial energies have been determined to be: γ Melt-Ol <

Introduction

Metal-silicate differentiation in early-accreted bodies of the solar system involves processes that are still poorly understood. Metal connectivity is a key factor because efficient phase separation requires an interconnected network of the metal-rich phase. In other words, the interconnection threshold (i.e. the volumetric phase fraction at which the phase forms a continuous network) has to be reached. This threshold is often assumed to have a very low value (3 to 6 vol%, e.g. Šrámek et al., 2012). However, the meteoritic record shows evidence of incomplete metal segregation. For example, primitive achondrites (e.g. Acapulcoites and Lodranites) are partially differentiated natural samples that exhibit signs of silicate partial melting, up to 25 vol% (e.g. [START_REF] Bild | The Lodran meteorite and its relationship to the ureilites[END_REF][START_REF] Mccoy | A petrologic and isotopic study of lodranites: Evidence for early formation as partial melt residues from heterogeneous precursors[END_REF][START_REF] Mccoy | A petrologic, chemical, and isotopic study of Monument Draw and comparison with other acapulcoites: Evidence for formation by incipient partial melting[END_REF][START_REF] Palme | The Acapulco meteorite: Chemistry, mineralogy and irradiation effects[END_REF][START_REF] Papike | The Lodran primitive achondrite: petrogenetic insights from electron and ion microprobe analysis of olivine and orthopyroxene[END_REF][START_REF] Zipfel | Chemical composition and origin of the Acapulco meteorite[END_REF] and have FeNi alloy contents up to ~20 vol% and troilite contents of ~3 vol% [START_REF] Bild | The Lodran meteorite and its relationship to the ureilites[END_REF][START_REF] Palme | The Acapulco meteorite: Chemistry, mineralogy and irradiation effects[END_REF].

Under these conditions, the metal content is larger than commonly assumed interconnection thresholds and these bodies should be fully differentiated, which they are clearly not. To rationalize this observation, either metal-rich phases did not readily form a connected network and/or the kinetics of metal segregation were slow compared to cooling timescales. While this latter possibility cannot be excluded for the case of primitive achondritesbecause the parent bodies were planetesimals of modest radius [START_REF] Neumann | Modeling the evolution of the parent body of acapulcoites and lodranites: A case study for partially differentiated asteroids[END_REF], the gravitational driving force for segregation was low -the question of relevant values of segregation threshold must be treated first.

In general terms, the interconnection threshold is a function of interfacial energies that drive the geometry of local phase boundaries to minimize the energy of the system [START_REF] Bulau | Mechanical and thermodynamic constraints on fluid distribution in partial melts[END_REF][START_REF] Jurewicz | The distribution of partial melt in a granitic system: The application of liquid phase sintering theory[END_REF][START_REF] Smith | Some elementary principles of polycrystalline microstructure[END_REF]. Thus, the system will evolve to shorten high-energy interfaces at the expense of lower energy geometries. Physical expression of these effects corresponds to the set of dihedral angles that form at phase triple junctions (i.e. where three phases meet). In a two-phase system (e.g. melt-mineral), the dihedral angle θ, corresponding to the angle of the melt between two solid grains, is described by equation [START_REF] Šrámek | Thermal evolution and differentiation of planetesimals and planetary embryos[END_REF] [START_REF] Smith | Some elementary principles of polycrystalline microstructure[END_REF].

cos =

(1)

where γ SS is the solid-solid interfacial energy and γ SL the solid-liquid interfacial energy (in J.m -2 ).

Measuring dihedral angles is thus a way to quantify the ratios of interfacial energies and to constrain variations in interconnection threshold. Two methods are commonly used to measure dihedral angles: the first one is the sessile drop technique using in-situ X-Ray imaging a droplet on a solid substrate (e.g. [START_REF] Humenik | Metal-Ceramic Interactions: III, Surface Tension and Wettability of Metal-Ceramic Systems[END_REF][START_REF] Jung | A study on the interfacial tension between solid iron and CaO-SiO2-MO system[END_REF]. The second one consists in a more statistical approach with repeated measurements on 2D scanning electron images. In this work we use the second method was preferred as it allows more accurate imaging of the triple junctions. In the case of a two-phase system, such as a metallic melt in an olivine matrix, the relation between interconnection threshold and dihedral angle is well established (e.g. [START_REF] Laporte | The Grain-Scale Distribution of Silicate, Carbonate and Metallosulfide Partial Melts: a Review of Theory and Experiments[END_REF]. For instance, an irondominated metallic liquid typically has a high dihedral angle of 110-130° in an olivine matrix [START_REF] Gaetani | Wetting of mantle olivine by sulfide melt: implications for Re/Os ratios in mantle peridotite and late-stage core formation[END_REF][START_REF] Shannon | High pressure constraints on percolative core formation[END_REF], corresponding to an interconnection threshold of roughly 10 to 15 vol% [START_REF] Laporte | The Grain-Scale Distribution of Silicate, Carbonate and Metallosulfide Partial Melts: a Review of Theory and Experiments[END_REF]. Adding light elements (C, O, S) to this metallic melt decreases its dihedral angle, and therefore its interconnection threshold [START_REF] Ballhaus | Mobility of core melts during Earth's accretion[END_REF][START_REF] Gaetani | Wetting of mantle olivine by sulfide melt: implications for Re/Os ratios in mantle peridotite and late-stage core formation[END_REF][START_REF] Holzheid | Textural equilibria of iron sulfide liquids in partly molten silicate aggregates and their relevance to core formation scenarios[END_REF][START_REF] Minarik | Textural Entrapment of Core-Forming Melts[END_REF]. For compositions in the vicinity of the Fe-FeS eutectic, dihedral angles span a range from 60° up to 90°, yielding a broad range of interconnection thresholds from 0 to 7 vol%, as estimated from [START_REF] Laporte | The Grain-Scale Distribution of Silicate, Carbonate and Metallosulfide Partial Melts: a Review of Theory and Experiments[END_REF]. However, as mentioned above, segregation is not efficient in such systems, core formation being too slow to match with the meteoritic record [START_REF] Bagdassarov | Centrifuge assisted percolation of Fe-S melts in partially molten peridotite: Time constraints for planetary core formation[END_REF]. Adding a silicate melt makes the experimental system more realistic and tends to promote more efficient segregation [START_REF] Bagdassarov | Centrifuge assisted percolation of Fe-S melts in partially molten peridotite: Time constraints for planetary core formation[END_REF]. However, equilibrium geometries of such three-phase systems (crystalline silicate + silicate melt + pure metal or sulfide) have received little attention. To our knowledge, data are only available for a Fe-FeS eutectic composition [START_REF] Holzheid | Textural equilibria of iron sulfide liquids in partly molten silicate aggregates and their relevance to core formation scenarios[END_REF] in a partially molten system of San Carlos olivine and basalt.

During the evolution of a planetesimal, composition of the metallo-sulfide phase ranges from the eutectic composition, representing the first metal-rich liquids produced, to pure metal (FeNi). The present study focuses on the measurement of dihedral angles and equilibrium geometries in that latter end-member. Constraints are obtained using high temperature experiments in a one-atmosphere furnace, a pressure relevant given the lowpressure environment in early-accreted planetesimals. Equilibrium textures in the experimentally produced samples containing partially molten silicate and pure metal are then used to construct a global metal-silicate differentiation model that can help to rationalize the observed mineralogy of primitive achondrites.

Experimental setup

Experiments were performed in a high temperature, 1 bar, controlled-atmosphere vertical furnace, using a three-phase system. The experiments allow to determine the equilibrium geometries between forsterite, silicate melt and nickel metal with accurate quantification of the dihedral angles and associated interfacial energies.

Synthesis of starting material

Samples are mixtures of crystalline forsterite (Fo), silicate melt (Liq) and nickel metal (Ni).

The starting material was prepared in two steps.

First, a silicate glass was synthesized, with a composition in equilibrium with pure forsterite at 1430°C as determined from the Anorthite-Diopside-Forsterite ternary phase diagram in the CMAS (CaO -MgO -Al 2 O 3 -SiO 2 ) system (Morse, 1980), a good analogue of iron-free silicate melts in meteorites. Thus, the proportion of silicate melt (of fixed composition) can be varied at a single temperature. Oxide powders (SiO 2 99.8% -Alfa Aesar, Al 2 O 3 99.999% extra pure -Acros) and carbonates (CaCO 3 99+% -Acros, 4(MgCO 3 ).Mg(OH) 2.5 (H 2 O) -Acros) were mixed in the appropriate proportions to produce the desired composition of silicate melt (Table 1). This mixture was finely ground, placed in a platinum crucible and held for two hours at 1100°C to allow complete dehydration and decarbonation, then heated to 1500°C for two hours for melting. The material was quenched, ground and put back in the furnace at 1500°C for two additional hours to ensure chemical homogeneity. After quenching, glass composition was checked using Electron Micro Probe Analysis (EMPA, Table 1). Finally, the glass was finely crushed before mixing with the other components.

Although metal is dominated by iron in natural samples, nickel has been chosen in these experiments in light of its lower melting point and its stability at higher oxygen fugacity (ƒO 2 ). This choice allowed performing experiments (at conditions of temperature and ƒO 2 that are easily obtained and controlled) in which the metallic subsystem does not interact significantly with the silicates, i.e. the diffusion of Ni in forsterite and silicate melt is low (see section 2.2).

In the second stage of sample synthesis, the glass was mixed with commercial powders of forsterite (Neyco CERAC, Inc., 325 mesh, 99% pure; grain size 0-40 µm) and nickel (Goodfellow, 99.8% pure, grain size 1-70 µm), with fixed metal content but varying forsterite / glass ratios in order to investigate the effect of the relative abundance of these two phases (Fo : Liq : Ni expressed in vol%): 20:75:5, 47.5:47.5:5, 62.5:32.5:5, 75:20:5. The obtained mixtures were then sintered to reduce porosity as the gas trapped between grains may act as a fourth phase with its own interfacial energy. Spark Plasma Sintering (SPS) was used at the PNF 2 platform (University Paul Sabatier, Toulouse, France), in an argon vacuum, at 100 MPa and 1200°C for 5 minutes, as described by [START_REF] Guignard | Dense fine-grained aggregates prepared by spark plasma sintering (SPS), an original technique in experimental petrology[END_REF]. The 20:75:5 mixture could not be sintered by SPS due to devitrification around 900°C that caused the graphite mold to break due to the large change in volume of the sample. In this case, the wire loop technique was employed using a platinum wire (⌀ = 0.2 mm, 99.9% pure -Johnson Matthey) on which ~30 mg of sample was placed and put for 30 seconds at 1450°C to form a droplet held on the loop by surface tension.

Annealing experiments

Experiments were performed in the vertical furnace at a fixed temperature of 1440°C, ~15°C below the melting point of nickel. Temperature was measured (± 1°C) using an Stype thermocouple (PtRh 10% /Pt). The oxygen fugacity was fixed by mixing appropriate proportions of CO/CO 2 yielding a log(ƒO 2 ) of -8.46 atm. (i.e. ∆ Ni-NiO = -3). Such reducing conditions allow nickel grains to remain in their metallic state and limit diffusion and dissolution into silicates. Annealing experiments were run for 24 hours for all samples and an additional experiment of 10 days was performed on the 75:20:5 mixture to check attainment of textural equilibrium.

Analytical techniques

After experiments, samples were mounted in epoxy resin and polished down to 0.3 micron using Al 2 O 3 grit for 2 hours. FEG-SEM (Field Emission Gun Scanning Electron Microscope) and EPMA (Electron Probe Micro Analysis) were then performed.

Textures and dihedral angles were analyzed with the image processing software Fiji [START_REF] Eliceiri | NIH Image to ImageJ : 25 years of image analysis[END_REF][START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF] from digital images acquired by backscattered electron detection, secondary electron detection, or stacking both kinds of images, on a JEOL JSM-7800F Prime FEG-SEM at the Raimond Castaing center in Toulouse (France).

Images were acquired using a 10 keV accelerating voltage and a 10 nA beam current. There are two levels of organization of triple junctions depending on the scale of observation. At macroscale, the minimization of interfacial energies has a weak effect on the geometry that is principally controlled by crystallographic orientation. On the other hand, at microscale, at the apex of the triple junctions, geometry is dictated by the minimization of interfacial energies between the three phases. To quantify a potential shift from a macroscopic to a microscopic behavior, various magnifications and thus resolutions were investigated: x250

(376 nm per pixel), x1000 (94 nm per pixel), x3000 (30 nm per pixel) and x10,000 (9.4 nm per pixel).

Quantitative analyses of the chemical composition of the phases were performed using a

Cameca SXFive EPMA at the Raimond Castaing Center (Université Paul Sabatier, Toulouse, France). The operating conditions were: accelerating voltage 15 keV; beam current 20 nA;

analysed surface of ~2x2 µm 2 . Periclase, corundum, wollastonite, platinum and NiO were used as standards for Mg, Al, Ca and Si, Pt, and Ni respectively.

3 Dihedral angle measurement

Chemical and textural equilibrium

Chemical equilibrium was checked using EPMA on the two silicate phases (Table 1). The composition of the glass is found to not perfectly match the equilibrium composition expected at 1450°C. This difference can be explained by crystallization of forsterite (~4 wt%) during sample quench. NiO was detected in forsterite and silicate glass, at levels of 2 wt% and 0.6 wt%, respectively (Table 1). No concentration profile was observed in either of these phases, which is in agreement with diffusion and partitioning coefficients previously obtained at similar experimental conditions [START_REF] Borisov | Experimental study of the effect of SiO2 on Ni solubility in silicate melts[END_REF][START_REF] Dingwell | The solubility and oxidation state of nickel in silicate melt at low oxygen fugacities: Results using a mechanically assisted equilibration technique[END_REF][START_REF] Matzen | The effect of liquid composition on the partitioning of Ni between olivine and silicate melt[END_REF][START_REF] Petry | Experimental determination of Ni diffusion coefficients in olivine and their dependence on temperature, composition, oxygen fugacity, and crystallographic orientation[END_REF], indicating chemical equilibrium.

Textural equilibrium was checked using SEM images (Figure 1). In the 20:75:5 and 47.5:47.5:5 samples, nickel grains are elongated in comparison with the other samples. A remarkable feature of the 20:75:5 sample is that nickel is clustered at the center of the sample and forms an interconnected network. Looking at textures at the grain scale, similar features and olivine triple junctions at 120° are observed in all samples. There are no differences in the textures or in the dihedral angle distributions (see 3.3) between 24 hourand 10 day-experiments. Hence, at triple junction scale, textural equilibrium is reached.

True dihedral angle measurement

Dihedral angles of silicate melt and forsterite were measured using the "angle tool" of the Fiji distribution of ImageJ, while the dihedral angle of nickel was inferred from the two others, considering that the sum of the three dihedral angles is 360°. In order to obtain sufficient statistics, measurements were repeated on at least 100 different triple junctions.

These measured distributions are called apparent dihedral angle distributions, as they are representative of a 2D slice in a 3D volume. For a single population of dihedral angles, these apparent distributions can be fitted with a Gaussian function centered on the median value [START_REF] Jurewicz | Distribution of apparent angles on random sections with emphasis on dihedral angle measurements[END_REF], that corresponds to the true dihedral angle, i.e. the dihedral angle in the 3D volume.

From macroscopic to microscopic behavior of triple junctions

Figures 2a, 2b and 2c display the true dihedral angle for each phase in each sample as a function of pixel size. With increasing detail (i.e. decreasing pixel size), the dihedral angle of the melt decreases from ~70° to ~25°, and then reaches a steady value within uncertainty, as shown by the measurements at a resolution of 9.4 nm per pixel. The dihedral angles of forsterite and Ni exhibit the opposite trend upon increasing resolution: they increase from 125° to 160° for forsterite and from 160° to 175° for Ni, until a constant value is reached.

These observations indicate that micro-films and/or micro-channels of silicate melt are present between forsterite and/or Ni grains, as previously suggested by [START_REF] Cmíral | A close look at dihedral angles and melt geometry in olivine-basalt aggregates: a TEM study[END_REF].

For low melt fractions, distribution of the melt can be anisotropic [START_REF] Faul | Constraints on the Melt Distribution in Anisotropic Polycrystalline Aggregates Undergoing Grain Growth[END_REF]: some grain boundaries are still dry whereas the others are wetted by silicate melt, and the width of channels between grains may vary from 10 to 100 nm, making resolution a key parameter to image melt distribution (Garapić et al., 2013). In the present study, close to triple junctions, these structures represent hooks between forsterite and nickel grains that are thinner and thinner toward the apex of the triple junction (Figure 1g and1h). Typically, the width of these structures at half-length of the hook is about 100 nm. To be accurately measured, we considered that at least 9 pixels are needed, i.e., a resolution threshold of 30 nm per pixel corresponding to a x3000 magnification in our case. At lower resolution, the continuous evolution of dihedral angle as a function of magnification indicates a continuous evolution from the microscopic to the macroscopic energy balance.

It is also of note that at low resolution (376 and 94 nm per pixel), dihedral angles appear to be a function of phase proportions. For instance, the 20:75:5 and the 47.5:47.5:5 samples (i.e., where Ni grains are elongated and seem to form a network) show dihedral angles of roughly 60° for the silicate melt and 170° for the nickel, whereas the 62.5:32.5:5 and the 75:20:5 samples (i.e. samples displaying isolated beads of nickel) exhibit angles that are 20° higher for silicate melt and 15° lower for nickel (Figure 2a and2c). At higher resolution this difference is no longer apparent, highlighting the same microscopic behaviors but different macroscopic ones.

Once all the small-scale details are unraveled, all samples display the same dihedral angle distribution and the same values of true dihedral angles, regardless of the proportions of each phase. In light of this result, the different datasets for a same magnification were merged to provide better statistics on the dihedral angle distribution. In the remainder of the text, the median values of these new merged distributions are considered as the true dihedral angles (Figure 2, Table 2).

Macroscale geometry: apparent isotropy

Figures 2d and 2e

show the merged dihedral angle distributions at 376 and 94 nm per pixel and the associated Gaussian fits for each phase. These distributions are fitted with an unimodal Gaussian curve, as suggested by [START_REF] Jurewicz | Distribution of apparent angles on random sections with emphasis on dihedral angle measurements[END_REF]. The quality of the fit is better for the 376 nm per pixel distributions than for that at 94 nm per pixel, which is due to a slight asymmetry of the latter and an intermediate behavior between macro-and micro-scales. The good agreement between data and unimodal Gaussian curves suggests that there is a single population of dihedral angles, so at this macroscale, olivine grains appear isotropic, as commonly assumed in most previous studies of dihedral angles in this type of system.

Microscale geometry: anisotropy of forsterite grains

At the microscale, i.e. close to the apex of the triple junctions (Figure 2f, 30 and 9.4 nm per pixel), distributions for forsterite and silicate melt become clearly asymmetric, with the emergence of a narrow peak and a shoulder towards lower values for forsterite, and higher values for silicate melt. In this case, data cannot be fitted by a unimodal Gaussian function.

Similar features appear when angle distributions are mixed [START_REF] Jurewicz | Distribution of apparent angles on random sections with emphasis on dihedral angle measurements[END_REF], indicating a possible difference of interfacial energies along crystal faces. In this respect, it has been shown that wetting behavior of olivine depends on crystal orientation (e.g. [START_REF] Jung | Olivine crystallographic control and anisotropic melt distribution in ultramafic partial melts[END_REF][START_REF] Wanamaker | The effect of melt composition on the wetting angle between silicate melts and olivine[END_REF]. Hence, the asymmetry of the distributions may be linked to different populations of dihedral angles reflecting the anisotropic behavior of forsterite. On the other hand, dihedral angle distribution of nickel remains very symmetric, which can be explained by its cubic crystalline system for which all crystal planes are equivalent. Finally, due to its low viscosity, silicate melt accommodates the anisotropy of forsterite and has thus different dihedral angle populations due to different olivine orientations. Three principal populations of interfacial energies exist in forsterite crystals [START_REF] Jung | Olivine crystallographic control and anisotropic melt distribution in ultramafic partial melts[END_REF], corresponding to Miller indices of ( 010), ( 110) and a family of minor crystal planes (( 021), ( 120), ( 001), ( 101) and ( 100)). Hence, three Gaussian functions were fitted on both silicate melt and forsterite dihedral angle distributions (Figures 2f,2g and 2h).

The agreement between measured and calculated distributions is characterized by a correlation greater than 0.95 (inset in Figures 2g and2h). and 3 respectively (Table 3). Calculating the area under each Gaussian provides the proportion of each population of interfacial energies, corresponding roughly to ~33 % for each distribution. These values are in good agreement with the proportions of the crystallographic planes given by [START_REF] Jung | Olivine crystallographic control and anisotropic melt distribution in ultramafic partial melts[END_REF].

4 Interfacial energies and implications for metal segregation in planetesimals 4.1 Three-phase system: theoretical approach

In a three-phase system, the equilibrium geometry of a triple junction can be represented as in Figure 3. When textural equilibrium is reached, the three different dihedral angles θ 1 , θ 2 and θ 3 vary as a function of the interfacial energies γ 12 , γ 13 and γ 23 according to equation ( 2), assuming crystalline isotropy [START_REF] Laporte | The Grain-Scale Distribution of Silicate, Carbonate and Metallosulfide Partial Melts: a Review of Theory and Experiments[END_REF][START_REF] Smith | Some elementary principles of polycrystalline microstructure[END_REF].

( ) = ( ) = ( ) (2)
In detail, given that olivine crystals are anisotropic (see above), this equation can be used for each pair of olivine -silicate melt dihedral angle population. Rearranging this equation yields a set of three equations:

( ) ( ) = ( ) ( ) = ( ) ( ) = (3)
Based on this equation, the ratio of two interfacial energies can be inferred by calculating the ratios of sinuses of dihedral angles. From a theoretical point of view, it is thus possible to illustrate how variations in phase geometry translate into variations in the ratio of interfacial energies (Figure S1). Given that an individual dihedral angle cannot be smaller than 0° and no larger than 180°, and that the sum of the three dihedral angles is constrained to be 360°, the variation of interfacial energy ratio can be represented using a ternary diagram. Due to the aforementioned conditions, only the inner triangle of the ternary diagram is of interest (Figure 4a). From equation (3), equality between two interfacial energies can be achieved when two dihedral angles are equal. Plotting these lines in the ternary diagram defines six different energetic regimes. In each of these six domains, the relative order of interfacial energies differs (Figure 4b). The central point of this diagram represents the case where each phase has a dihedral angle of 120° (i.e. all interfacial energies are equal).

While a given ratio of interfacial energies always leads to the same geometry at the scale of an individual triple-junction, the larger-scale distribution of material will be affected by phase proportions, and limited by the least abundant phase. Given that olivine is always the major phase, the range of potential geometries is illustrated for the cases of nickel (Figure 4c) or silicate melt (Figure 4d) as the minor phase. For instance, we consider the case where two phases have dihedral angles of 180° and the third one has a dihedral angle of 0°. As displayed on Figure 4c and4d, in this situation the third phase entirely wets the two others, preventing contact between them. If the third phase is present in small proportion, it will form a thin film along the interface between the two others. Alternatively, if one of the two other phases is limited in proportion, this phase will form an isolated sphere into the third phase (Figure 4c and4d).

Figures 4c and 4d also represent a first order estimation of the interconnection threshold -dihedral angle dependence. When a phase has a dihedral angle close to zero, it has a thin lens shape, spreading along the interface of the two other phases. In this case, its interconnection threshold is very low. Increasing the value of the angle leads to an increasingly spherical shape and higher interconnection threshold.

Interfacial energy ratios calculation

Interfacial energies can theoretically be calculated from the knowledge of the dihedral angles, if at least one interfacial energy is known independently. In this respect, the interfacial energy between silicate melt and olivine measured by [START_REF] Cooper | Interfacial Energies in the Olivine-Basalt System[END_REF] seems a suitable reference point (0.5 ± 0.2 J.m -2 ). However, this value assumes isotropy of crystalline olivine, a situation that does not reflect reality as discussed above. To our knowledge, no study reports different interfacial energies for different orientations of olivine crystals, so that it is preferable to stick to calculating ratios as follows:

= ( ) ( ) = ( ) ( ) = ( ) ( ) (4)
Results of these calculations are shown in Table 3 for the different crystal plane families (i.e. subsets) and for the complete dataset (referred to as "merged"). When the raw data are treated, the sum of the three dihedral angles is greater than 360° for each subset. Given that this result is impossible, angles have been recalculated by fixing the nickel dihedral angle to 176.1° (corresponding to the measured value), then normalizing the two remaining values such that the sum is 360°.

The three-phase system considered composed of silicate melt, forsterite and nickel plots in the γ Melt-Fo < γ Melt-Ni < γ Fo-Ni domain as shown in Figure 5 (using data from Table 2, column "Merged", and from Table 3). At low resolution, nickel grains seem to be almost equally in contact with the forsterite and the silicate melt, whereas at high resolution, they are more rounded and barely in contact with forsterite crystals. This behavior is illustrated through the variation of the interfacial energy ratios as a function of magnification. At low resolution (376 and 94 nm per pixel) both γ Melt-Fo /γ Melt-Ni and γ Melt-Fo /γ Fo-Ni ratios are almost equal, while at higher resolution (30 and 9.4 nm per pixel) these two energy ratios are lower and Ni has a higher interfacial energy than the rest of the system.

The energy ratios involving olivine are quite different depending on crystal orientation.

From subset 1 to subset 3, the γ Melt-Fo /γ Melt-Ni and γ Melt-Fo /γ Fo-Ni ratios decrease from 0.3 to 0.1, and the γ Melt-Ni /γ Fo-Ni ratio increases toward values close to 1, illustrating an evolution in the geometry toward the γ Melt-Ni = γ Fo-Ni isovalue line (Figure 5). The changes of equilibrium geometries for the different subsets can be illustrated using Figure 4. For example, in case of low metal fraction, the metal grain becomes less embedded in forsterite from subset 3 to subset 1, making it easier to form an interconnected network in the latter case. This leads to an interconnection threshold that varies locally, increasing from subset 1 to subset 3. As interfacial energies are the lowest for subset 1, this geometry will most likely dominate the overall distribution. However, the energetic effect is countered by samples composed of an aggregate of randomly oriented crystals leading to metal grains in contact with various olivine crystal planes, including crystalline planes of high energy. To reach a state of lower energy, a dynamic process such as deformation would be required to re-orientate olivine grains. Despite these differences, it is of note that all subsets of olivine orientation fall in the same energetic regime where γ Melt-Fo < γ Melt-Ni < γ Fo-Ni .

In order to infer the mobility of sulfur-rich and pure metallic melts in an olivine matrix, the experimental results of [START_REF] Ballhaus | Mobility of core melts during Earth's accretion[END_REF] can be used. Calculating the interfacial energy ratios from their measured dihedral angles gives γ Ol-Ol /γ Ol-FeS = 1.463 to 1.658 and γ Ol- Ol /γ Ol-Fe = 1.075 to 1.104. Taking these ratios at face-value (note that calculated values are based on angles measured by [START_REF] Ballhaus | Mobility of core melts during Earth's accretion[END_REF] at a resolution that was not specified) suggests that γ Ol-Ol ≈ γ Ol-Fe > γ Ol-FeS . In such a two-phase system, the olivinemetal-rich phase interfacial energy is lower in the case of sulfide than of pure metal. Thus sulfide melts appear to wet olivine grains more than pure metallic ones do, and sulfur-rich melts thus have a lower interconnection threshold (~3 to 7 vol% from [START_REF] Laporte | The Grain-Scale Distribution of Silicate, Carbonate and Metallosulfide Partial Melts: a Review of Theory and Experiments[END_REF] than pure metallic melts in an olivine matrix (~12 vol% from [START_REF] Laporte | The Grain-Scale Distribution of Silicate, Carbonate and Metallosulfide Partial Melts: a Review of Theory and Experiments[END_REF], although the high dihedral angles determined in this study suggest a much higher interconnection threshold on the order of 25 vol%).

In natural systems (i.e. planetesimals in the early solar system), silicate and metal-rich phases fractions have close melting temperatures (~1000°C). Furthermore, the first melt compositions produced upon heating of the metal-rich fraction are sulfur-rich, close to the Fe-FeS eutectic composition. To our best knowledge, a single study reports for measurements of dihedral angles in a three-phase system hosting sulfide melts [START_REF] Holzheid | Textural equilibria of iron sulfide liquids in partly molten silicate aggregates and their relevance to core formation scenarios[END_REF]. As dihedral angle measurement resolution is not specified, these values have to be used with caution. In their study, the γ Melt-Ol /γ Melt-Sulfide ratio is greater than 1, whereas the γ Melt-Ol /γ Melt-Ni is lower than 1 in the case of a pure metal implying that γ Melt-Sulfide <γ Melt-Ni . This switch in the value of the ratio γ Melt-Fo /γ Melt-Sulfide compared to γ Melt-Fo /γ Melt-Ni indicates a change in the order of interfacial energies, i.e. of energetic regime, relative to the case of pure metal, which then becomes γ Melt-Sulfide < γ Melt-Ol < γ Ol-Sulfide (Figure 5). The sulfur-rich melt thus appears to have a low interfacial energy with silicate melt and a higher one with olivine so that the contact between sulfur-rich melt and solid silicates tends to be limited. In this case, the sulfide melts produced at low temperature near the eutectic will tend to spread along the surface of solid silicate grains in order to increase the contact surface with the silicate melt, thus having a tendency to interconnect in the presence of a silicate melt (Figure 5).

Metal-silicate differentiation in planetesimals

Comparing dihedral angle measurements carried out in this study for a pure metal with previous ones for a sulfur-rich phase shows that increasing the sulfur content of the metallic liquid increases its wettability, reducing its dihedral angle and its percolation threshold.

Once this threshold exceeded, the sulfur-rich metallic liquid is able to percolate, leaving a metallic residue whose threshold increases due to decreased sulfur content. Therefore, in the absence of a large degree of silicate melting, the complete extraction of metal in a small body remains unlikely.

To explore the consequences of a dependence of the interconnection threshold on sulfur content, the evolution of a body heated by decay of 26 Al has been modeled. Upon heating, silicate and iron-sulfur begin to melt at similar temperature (see below). The behavior of each subsystem thus has to be modeled as their relative proportions depend on each other.

Silicate melt is likely to form an interconnected network even at very low melt fraction (<1 vol% estimated from [START_REF] Laporte | The Grain-Scale Distribution of Silicate, Carbonate and Metallosulfide Partial Melts: a Review of Theory and Experiments[END_REF][START_REF] Wray | The geometry of two-phase aggregates in which the shape of the second phase is determined by its dihedral angle[END_REF], thus it is assumed to be continuously extracted. As the fraction of silicate decreases, that of sulfide melt increases until the interconnection threshold is reached. This behavior depends at the same time on the relation between the sulfur content of the sulfide melt and its interconnection threshold and on the melting curve of silicates.

Modeling the metal-sulfide subsystem

The metallic subsystem is assumed to have an initial composition of H-type ordinary chondrites as they are a possible precursor material of primitive achondrites [START_REF] Palme | The Acapulco meteorite: Chemistry, mineralogy and irradiation effects[END_REF]. Such a composition corresponds to volumetric phase proportions of 3 vol% troilite and 7 vol% metal [START_REF] Guignard | Textural properties of iron-rich phases in H ordinary chondrites and quantitative links to the degree of thermal metamorphism[END_REF], or 7 wt% S in the metallic subsystem [START_REF] Wasson | Compositions of Chondrites[END_REF]. As temperature rises, removal of silicate melt increases the volume fraction of metal, but the sulfur content of the sulfide melt decreases due to equilibrium melting of the metallic subsystem. The overall effect is an increase of the interconnection threshold of the molten fraction, from a lower limit at the Fe-FeS eutectic (3 vol%, [START_REF] Roberts | Fe-Ni-S melt permeability in olivine: Implications for planetary core formation[END_REF][START_REF] Yoshino | Core formation in planetesimals triggered by permeable flow[END_REF] to an upper limit for pure metal (25 vol% estimated from the large value of nickel dihedral angle measured in the present study, see 3.2.3). Between these two endmembers, the sulfur content -interconnection threshold dependance is poorly constrained. For this reason, several different functions have been considered (Figure 6a). Threshold function (a) increases linearly with sulfur content from the Fe-S eutectic composition to pure iron, while other functions investigate non-linear trends.

Altogether, they fall into two categories: a first one with with a sharp increase of the threshold upon decreasing sulfur content in the vicinity of the eutectic (threshold functions a and b), and a second one with a smooth increase of the threshold upon decreasing sulfur content in the vicinity of the eutectic (threshold functions c, d and e). The liquidus curve of the Fe-S phase diagram is taken from Buono and Walker (2011) at 1 bar (Figure 7, black curve) and allows plotting the different percolation threshold functions as a function of temperature (Figure 6b, full lines). The percolation thresholds increase with temperature until all the metallic subsystem is entirely molten (~1400°C at 7 wt% of sulfur, dashed black thin vertical line Figure 6b), then the threshold functions reach constant values. In the following, it is considered that once the interconnection threshold reached, the network forming sulfide melt is extracted.

Another important issue is the efficiency of draining once the percolation threshold has been reached. [START_REF] Ghanbarzadeh | Percolative core formation in ć planetesimals enabled by hysteresis in metal connectivity[END_REF] demonstrated that drainage is affected by an hysteresis effect, where not only the molten fraction exceeding the percolation threshold flows out, but almost all the fluid is lost, except for a small fraction trapped in network irregularities. This amount corresponds to the trapping threshold, which is zero for a dihedral angle less than 60° as in the case of silicate melts, but up to 2 vol% for a dihedral angle of 120° [START_REF] Ghanbarzadeh | Percolative core formation in ć planetesimals enabled by hysteresis in metal connectivity[END_REF]. In the present calculation, the trapping threshold is set to 1 vol%.

Modeling the silicate subsystem

The melting curve of the silicate fraction has been calculated using a fractionate melting simulation on the Rhyolite-Melts thermodynamic software, with continuous extraction of the melt [START_REF] Asimow | Algorithmic modifications extending MELTS to calculate subsolidus phase relations[END_REF][START_REF] Ghiorso | An H2O-CO2 mixed fluid saturation model compatible with rhyolite-MELTS[END_REF][START_REF] Ghiorso | Chemical mass transfer in magmatic processes IV. A revised and internally consistent thermodynamic model for the interpolation and extrapolation of liquid-solid equilibria in magmatic systems at elevated temperatures and pressures[END_REF][START_REF] Gualda | Rhyolite-MELTS: a Modified Calibration of MELTS Optimized for Silica-rich, Fluid-bearing Magmatic Systems[END_REF]. An initial composition of H-type chondrites [START_REF] Wasson | Compositions of Chondrites[END_REF] is also assumed and temperature ranges from 950°C to 2000°C, at 1 bar, along the Iron-Wüstite (IW) buffer. The FeO content of the silicates is calculated from the FeO/(FeO+MgO) ratio [START_REF] Wasson | Compositions of Chondrites[END_REF], Table 1). The melting curve is displayed in Figure 6c and 6d as a black curve. The different kinks in the curve correspond to the complete melting of the different silicate phases. This simulation shows a silicate eutectic around 1000°C.

Metal segregation

Upon removal of the silicate melt produced, the quantity of metal increases (whole metal, i.e. solid iron and sulfide on Figure 6c and sulfide melt on Figure 6d). Once the quantity of sulfide melt reaches its interconnection threshold, all but 1 vol% is assumed to be extracted as explained above. This removal causes the loss of sulfur, changing the composition of the metallic subsystem (Figure 7, color curves for the different threshold functions) if it occurred while there was still some solid iron. In this case, the interconnection threshold is shifted towards higher values; this tendency is represented in Figure 6b (dashed lines) and S2.

The different threshold functions described here fall in two categories. In the first case, the interconnection threshold of the sulfide cannot be reached until a large fraction of silicate is melted. For example, threshold function (a) leads to the result that the high interconnection threshold of pure metal is preserved over a wide range of compositions (Figure 6a). In this pessimistic case, the conditions to extract the sulfur-bearing component would require 67% melting of the silicate fraction, corresponding to an extreme temperature of ~1850°C (Figure 6b, 6c and 6d). In the case of threshold function (b) (linear increase with sulfur content), the threshold is constantly higher than the volume fraction of sulfide melt over a wide range of temperatures (Figure 6b, 6c and 6d), such that the threshold is not reached before extraction of 55% of the silicates, i.e. a temperature of 1600°C. The same sort of behavior is also true for threshold function (c), that is a non-linear function of Scontent, with lower interconnection threshold at high S-content. In this case, the onset of drainage occurs at 13 vol% of molten sulfide and a degree of silicate melting of 30 wt%, i.e.

1450°C (Figure 6b, 6c and 6d)., For the three cases described above (threshold functions (a), (b) and (c)), sulfide melt extraction requires the removal of large volume of silicate melt and temperatures above 1420°C, meaning that the metallic subsystem is fully molten and its extraction takes place in a single event, leaving only a small residual trapped fraction of 1 vol%. Such a scenario appears unlikely given that the loss of silicate melt also takes away thre principal heat source, 26 Al that is transferred to the melt by plagioclase melting near 1200°C. In the case of fractional melting assumed here, the 26 Al heat sources will thus be totally extracted for a melting degree no higher than 15%. However, models on the percolation process in Vesta [START_REF] Mizzon | The magmatic crust of Vesta[END_REF] show that silicate melt migration is not instantaneous. Even though the heat source is transferred towards the surface of the body, the degree of mantle melting can be as high as 50% due to time-scale of melt extraction.

For the second family of cases, threshold functions (d) and (e) allow interconnection and extraction of the sulfide melt at lower temperatures, i.e. at lower silicate melting degrees and below the liquidus of the metallic subsystem. In details, function (d) is at first glance similar to (c), but it yields a very different extraction behavior (Figure 6b, 6c and 6d). In this case, a first episode of drainage occurs at 3 vol% of liquid sulfide, after extraction of 19% of silicate melt (1250°C). This concerns 2% of the total volume, corresponding to 20% of the metallic phase. This loss of iron-sulfide liquid reduces the sulfur content of the residual metallic subsystem by two-thirds of its initial value, thus raising the interconnection threshold to 18% (Figure 6b and7). Both factors inhibit a possible second extraction of sulfide melt that would occur at a silicate melting degree of 65% and a temperature of 1800°C (Figure 6b, 6c and 6d), conditions that cannot be reached easily for the aforementioned reasons. As a consequence, only the first episode of loss of molten sulfide remains plausible. In this case, it will leave behind a material with a metal volume fraction close to the initial state, i.e. 10 vol%, but depleted in sulfur (~2.5 wt%, Figure 7), and a silicate highly depleted in its feldspar and clinopyroxene components. Function (e) assumes that the interconnection threshold of the eutectic is preserved for a wide range of sulfur contents and allows three succesive extractions at 1220°C, 1380°C and 1600°C corresponding respectively to 17%, 24% and 50% of silicate partial melting (Figure 6b, 6c and 6d). These stages result in a metal volume fraction of 10%, 8% and 1% (trapped fraction) respectively, the final episode of drainage taking place after complete fusion of the metal. During these stages the sulfur content decreases to 2.3 wt% and then to 1 wt% (Figure 7). Interestingly, function (e) belongs to both threshold function categories, since it allows partial metal extraction at moderate silicate partial melting (17%), and then complete drainage at a higher but perhaps still possible degree of silicate melting (50%).

Comparison with experimental data and primitive achondrites

The present model indicates that the S-content dependence of the interconnection threshold plays a critical role on the drainage of sulfur-bearing liquids. In the absence of direct experimental determination concerning the effect of S-content, we turn to experiments in similar systems and data from natural samples. For example, experiments conducted by [START_REF] Rose | Wetting Properties of Fe-Ni-Co-Cu-O-S Melts against Olivine: Implications for Sulfide Melt Mobility[END_REF] addressed the effect of oxygen on the wettability of metallic melts. They observed a sharp decrease in the dihedral angle from 0 to 5 mol% of oxygen, followed by a smooth decreasing linear trend at higher oxygen content, a behavior similar to that described by functions (c), (d) or (e). [START_REF] Holzheid | Textural equilibria of iron sulfide liquids in partly molten silicate aggregates and their relevance to core formation scenarios[END_REF] built a data compilation addressing the effect of sulfur, but it lacks data on pure metal. Adding the data from the present study, a sharp decrease of the interconnection threshold is observed at low sulfur content, followed by a smooth decreasing linear behavior, again similar to curves (c), (d) and (e). In light of these considerations, it is concluded that this sort of dependence is the most plausible.

In turn, this sheds new light on the processes experienced by primitive achondrites and may potentially explain their composition. These meteorites are widely accepted to represent partial melting residues for both metallic and silicate subsystems. Sulfur contents of the Fe-FeS system is plotted in Figure 7 for different primitive achondrites. Note that the composition can be highly heterogeneous in the same meteorite, from one section to another: for instance Acapulco displays S-contents ranging from ~6 wt% up to ~15 wt% [START_REF] Guignard | Textural properties of iron-rich phases in H ordinary chondrites and quantitative links to the degree of thermal metamorphism[END_REF][START_REF] Palme | The Acapulco meteorite: Chemistry, mineralogy and irradiation effects[END_REF][START_REF] Zipfel | Chemical composition and origin of the Acapulco meteorite[END_REF]. Primitive achondrites can nonetheless be assigned to two categories following their S-content: they are either depleted or enriched relative to the H-chondrite primitive material (Figure 7). Depletion can be due to the extraction of a sulfide melt, whose composition is different from that of the bulk metallic subsystem. In a similar way, S-enriched meteorites can be viewed as regions that have locally concentrated sulfide melts draining down from upper parts of the parent body. Indeed, it should be appreciated that differentiation at moderate melt fraction operates through liquid migration inside a porous solid matrix. This migration is affected by compaction waves that form alternating horizons enriched and depleted in liquid (Šrámek et al., 2012) that may freeze if the heat source (i.e. 26 Al content) is insufficient to allow complete separation. S-enriched and S-depleted meteorites can thus sample different horizons of the same planetesimal. This feature may be simply understood in the case of a S-dependent threshold corresponding to function (d) or (e). Recent studies on Hf-W systematics [START_REF] Kruijer | Protracted core formation and rapid accretion of protoplanets[END_REF] also support the idea that different sulfide melts extraction events may occur during differentiation. More precisely, data on IVB iron meteorites suggest two segregation steps: a first one with an early extraction of a S-rich liquid, and a second one, later, with an iron-rich melt. However, the present study outlines that the extraction of a silicate melt is a necessary condition to allow the metallic melt fraction to exceed its interconnection threshold.

Finally, the formation of a small core with a heterogeneous sulfur content is a natural outcome of the present model depending on the temperature of extraction, as melts with various compositions are extracted. This model thus provides a plausible explanation for the diversity of sulfur content in iron meteorites, even though one should keep in mind that these meteorites most likely formed by larger degrees of melting and are expected to have lost some sulfide due to the sulfur volatility and the mechanical weakness of troilite.

Conclusion

Laboratory experiments were performed to study the local geometry of a three-phase system composed of forsterite, silicate melt and nickel grains relevant to partially differentiated meteorites. The dependence of equilibrium geometries (dihedral angles) on interfacial energies was also considered from a theoretical point of view. Finally, the evolution of the metal fraction in a planetesimal was modeled assuming an interconnection threshold that is a variable function of S-content. The main conclusions can be summarized as follows:

• From an experimental and analytical point of view, a resolution of 30 nm per pixel is necessary for accurate measurements of dihedral angles, in particular the complex shape of forsterite grains that precludes simple prediction of the different interconnection thresholds of the system. A theoretical approach shows that different energetic regimes exist and that the accurate measurement of dihedral angles is needed to predict the geometry.

• Fe-FeS eutectic melt (first metallic liquid to form) and pure metal (last metallic liquid to form) belong to two distinct interfacial energy regimes. The comparison between the equilibrium geometries in a system containing Fe-FeS eutectic melt

shows that the sulfide -silicate melt interfacial energy is energetically favorable when the metallic melt is sulfur-rich, meaning that such liquids will spread along the faces of silicate crystals, decreasing the interconnection threshold of the metallic melt. On the other hand, in the case of pure metal, grains remain largely spherical, increasing the interconnection threshold.

• The variation of sulfur content of primitive achondrites from one sample to the other is best explained when the interconnection threshold significantly decreases at low sulfur content and remains close to the value of the Fe-FeS eutectic for a wide range of compositions at higher S contents. Different sulfide melt extractions are then possible at temperatures of 1200-1400°C, requiring ~25 vol% of silicate melt. These conditions can be reached even if the mobility of the heat source is taken into account. Successive extraction events provide a good explanation for the variability of sulfur content observed in primitive achondrites, which are either representative of a region that experienced sulfide extraction or from a region that accumulated sulfide melt from the outer parts of the body. is the minor phase, (d) silicate melt is the minor phase. If one angle is equal to 180°, the phase either forms a sphere or a film. When the dihedral angle is equal to 0°, the phase is highly wetting, its contact is punctual or inexistent, and this phase prevents the two others from being in contact with each other. (merged data at resolutions of 30 and 9.4 nm per pixel). Grey symbols correspond to the data from [START_REF] Holzheid | Textural equilibria of iron sulfide liquids in partly molten silicate aggregates and their relevance to core formation scenarios[END_REF]. Error bars are smaller than the symbols. Sketches show the geometry of the three phases according to their dihedral angles. Color code is the same as in Figure 3, except for sulfide, in red. Data with pure metal fall in the γ12 < γ13 < γ23 (γMelt-Ol < γMelt-Ni < γOl-Ni) domain, even when crystalline anisotropy is taken into account (shaded pentagons), which is clearly different from a system with sulfide: γ13 < γ12 < γ23

(γMelt-Sulfide < γMelt-Ol < γOl-Sulfide). [START_REF] Guignard | Textural properties of iron-rich phases in H ordinary chondrites and quantitative links to the degree of thermal metamorphism[END_REF][START_REF] Zipfel | Chemical composition and origin of the Acapulco meteorite[END_REF] and TIL99002 [START_REF] Patzer | Evolution and classification of acapulcoites and lodranites from a chemical point of view[END_REF]) and Sdepleted ones in yellow (Acapulco [START_REF] Palme | The Acapulco meteorite: Chemistry, mineralogy and irradiation effects[END_REF], Lodran [START_REF] Bild | The Lodran meteorite and its relationship to the ureilites[END_REF][START_REF] Guignard | Textural properties of iron-rich phases in H ordinary chondrites and quantitative links to the degree of thermal metamorphism[END_REF] and NWA 725 [START_REF] Patzer | Evolution and classification of acapulcoites and lodranites from a chemical point of view[END_REF]). The discrepancy in the sulfur content of the different primitive achondrites can be accounted for given that the evolution of the interconnection threshold is close to threshold functions (d) or (e), allowing the fractionation of sulfur within the planetesimal.

Figure 1 :

 1 Figure 1: SEM images representative of the different samples. White: nickel; dark grey:forsterite and light grey: silicate melt. (a.) 47.5:47.5:5 sample at x1000 (94 nm per pixel), (b.) 32.5:62.5:5 sample at x1000 (94 nm per pixel), (c.) 20:75:5 at x1000 (94 nm per pixel), (d.) 75:20:5 sample at x1000 (94 nm per pixel), (e.) 20:75:5 sample at x3000 (30 nm per pixel), (f.) 75:20:5 sample at x3000 (30 nm per pixel), (g.) 20:75:5 sample at x10 000 (9.4 nm per pixel), (h.) 75:20:5 sample at x10 000 (9.4 nm per pixel). Images on (g.) and (h.) highlight the hook shape of the silicate melt toward the apex. This kind of structure is typically of 100 nm at half-height of the hook, requiring a resolution threshold of 30 nm per pixel to have a proper imaging.

Figure 2 :

 2 Figure 2: Measured median dihedral angle as a function of pixel size (in nm) for (a) silicate melt, (b) forsterite, (c) nickel and merged dihedral angles distributions at different pixel sizes: (d) 376 nm per pixel, (e) 94 nm per pixel and (f) 30 and 9.4 nm per pixel. Circle color symbols on (a), (b), (c) correspond to different samples, black circle represents the median value of all datasets merged at a given magnification and gray lines are guides for the eyes. These three diagrams show that the dihedral angle reaches steady value at a pixel size of 30 nm per pixel, corresponding to the resolution at which the fine microstructure is entirely imaged. Scatter between different samples at lower resolution is due to measurement artifacts, as this small-scale geometry is not entirely imaged. Step curves on (d), (e) and (f) correspond to the dihedral angle distribution, curly curves correspond to Gaussian fits, single Gaussian function for (d) and (e) and triple Gaussian function for (f); black dotted lines represent the median value of the distributions. The agreement between data and triple Gaussian fit is shown in (g) for the silicate melt distribution and in (h) for the olivine distribution. Orange, green and blue dotted vertical lines correspond to median values of each subset of dihedral angle. Increasing the resolution makes the distribution more asymmetric, meaning that different populations of dihedral angle exist, representative of the forsterite crystalline anisotropy at microscale.

Figure 3 :

 3 Figure 3: Schematic view of a triple junction between the silicate melt, a forsterite grain and a nickel grain. Dihedral angles θ 1 , θ 2 and θ 3 are related to interfacial energies γ 12 , γ 13 and γ 23 when textural equilibrium is achieved, following equation 2.

Figure 4 :

 4 Figure 4: (a) Ternary diagram showing the inner triangle of possible space given the conditions of θ > 0° and θ < 180°. (b) Reduced ternary diagram showing the different energetic regimes. Full lines separate these regimes and represent equality between two interfacial energies. Sketches explaining the geometry in remarkable points of the ternary diagram are represented in (c) and (d). Two end-member cases are considered: (c) metal

Figure 5 :

 5 Figure 5: Ternary diagram of the dihedral angles with the different energetic regimes. Purple symbols represent data from this study. Pentagon symbols stand for the different populations of dihedral angles taking into account forsterite crystalline anisotropy

Figure 6 :

 6 Figure 6: Evolution of the metal volume fraction of a silicate-metal mixture as a function of temperature. Silicate melt is assumed to be continuously extracted so that the metal fraction increases with the melting degree of silicate. The metallic melt is completely drained when its percolation threshold is reached, except a trapped fraction set to 1 vol%. The percolation threshold is a function of the sulfur content. (a) Different interconnection threshold evolutions as a function of sulfur content of the sulfide melt. (b) Percolation threshold functions as a function of temperature. Full lines are for simulations that experienced sulfide melt extraction once all the iron is molten, while dashed lines represent the case where the extraction occurred when a solid iron residue was present. (c) Evolution of the metal volume fraction (solid iron and sulfide melt) as the silicate is extracted through fractionated melting. (d) Evolution of the sulfide melt volume fraction as a function of temperature. In (c) and (d) the black line represents the volume of silicate melt extracted as a function of temperature, the shaded area indicating the residual silicate. The different kinks in the silicate melting curves represent the disappearance of mineral species, arrows pointing them out. Color lines draw the volume fraction of the metallic subsystem in (c) and of the sulfide melt in (d) for the different percolation threshold functions depicted in (a) and (b).

Figure 7 :

 7 Figure 7: Evolution of the sulfur content of metal plotted on the phase diagram of the Fe-FeS system. Color dots represent the composition of the extracted melt, following the color code for the different threshold functions of Figure 6a. Color bars represent the composition of different primitive achondrites, sulfur enriched ones in pink (Acapulco

  The newly obtained median values for each population are 15.2°, 35.5° and 52.9° for the silicate melt and 173.1°, 162.2° and 137.5° for the olivine, each pair of dihedral angles corresponding to a new subset called 1, 2

Table 1 : Composition in wt% of the starting materials and composition of the different phases after experiments. Error bars, expressed as 2σ standard deviation, were estimated based on repeated measurements of at least 30 points for each phase.

 1

	Phase composition (wt%)	MgO	SiO2	CaO	Al2O3	NiO	Sum
	Commercial Fo	57.55 ± 0.78 43.03 ± 0.55 0.55 ± 0.16	0.00	0.00	101.17
	Silicate melt aimed composition	22.92	45.49	14.55	17.04	0.00	100.00
	Starting silicate melt	23.83 ± 0.32 44.72 ± 0.84 14.70 ± 0.31 16.75 ± 0.49 0.00	100.00
	Fo in Fo:Melt:Ni (20:75:5)	54.68 ± 0.58 42.50 ± 0.52 0.50 ± 0.05	0.20 ± 0.07	2.12 ± 0.73	100.00
	Fo in Fo:Melt:Ni (47.5:47.5:5)	54.78 ± 0.45 42.47 ± 0.53 0.50 ± 0.06	0.20 ± 0.08	2.04 ± 0.054 100.00
	Silicate melt in Fo:Melt:Ni (20:75:5)	21.32 ± 0.57 44.58 ± 0.22 15.23 ± 0.44 18.26 ± 0.33 0.60 ± 0.26	100.00
	Silicate melt in Fo:Melt:Ni (47.5:47.5:5) 21.43 ± 0.29 43.94 ± 0.19 15.50 ± 0.13 18.53 ± 0.15 0.60 ± 0.11	100.00

Table 2 : Median dihedral angles (in °) at the triple junctions for different magnifications and all samples. The merged sections correspond to all dataset put together. In each column the value of the dihedral angle is reported with its 2SE

 2

( = √"#$

where SD

is the standard deviation and N the number of measurements) uncertainty.

	Composition		×250	×1,000	×3,000	×10,000
		Fo	126.7 ±11.0	149.6 ±5.1	161.1 ±5.0	164.7 ±6.3
	20:75:5	melt	62.6 ±1.1	37.4 ±5.5	24.1 ±6.1	21.2 ±5.2
		Ni	170.9 ±0.6	175.3 ±2.3	177.3 ±4.0	179.2 ±4.2
		Fo	126.2 ±1.5	145.9 ±6.1	166.8 ±7.2	164.4 ±4.3
	47.5:47.5:5	melt	63.4 ±1.7	38.4 ±6.3	19.3 ±6.8	18.3 ±7.4
		Ni	167.7 ±0.8	174.2 ±2.6	176.6 ±3.0	180.0 ±5.1
		Fo	126.9 ±1.5	150.2 ±5.1	156.2 ±5.9	157.6 ±6.2
	62.5:32.5:5	melt	80.0 ±1.6	46.9 ±5.2	27.8 ±5.6	24.5 ±5.5
		Ni	155.6 ±0.8	165.8 ±1.8	175.0 ±1.7	175.6 ±2.3
		Fo	122.4 ±1.1	145.5 ±4.5	162.6 ±5.5	163.6 ±9.4
	75:20:5	melt	84.3 ±1.2	50.8 ±4.6	26.5 ±5.0	22.6 ±6.6
		Ni	155.7 ±0.5	164.5 ±1.5	174.1 ±2.2	176.2 ±4.8
		Fo	125.6 ±1.2	148.4 ±2.6	161.1 ±3.0	161.7 ±3.8
	Merged	melt	70.5 ±1.4	42.2 ±2.7	24.7 ±2.9	21.4 ±3.1
		Ni	162.3 ±0.7	168.8 ±1.1	175.3 ±1.3	176.9 ±1.8

Table 3 : Dihedral angles of the different phases when crystalline anisotropy is considered and corresponding interfacial energies. * correspond to values calculated from dihedral angle data. Uncertainties are 2SE calculated using the derivative method.

 3

	Populations	Phase	Dihedral angle (degree)	Interfacial energy ratios
		Fo	125.6 ±1.2	γMelt-Fo/γMelt-Ni	0.374 ±0.016
	x250	Melt	70.5 ±1.4	γMelt-Fo/γFo-Ni	0.323 ±0.013
		Ni	162.3 ±0.7	γMelt-Ni /γFo-Ni	0.862 ±0.015
		Fo	148.4 ±2.6	γMelt-Fo/γMelt-Ni	0.372 ±0.044
	x1000	Melt	42.2 ±2.7	γMelt-Fo/γFo-Ni	0.290 ±0.031
		Ni	168.8 ±1.1	γMelt-Ni /γFo-Ni	0.781 ±0.070
		Fo	173.1 ±0.7	γMelt-Fo/γMelt-Ni	0.358 ±0.104
	Subset 1	Melt	15.2 ±1.1	γMelt-Fo/γFo-Ni	0.265 ±0.077
		Ni	176.1 ±1.1	γMelt-Ni /γFo-Ni	0.741 ±0.071

Acknowledgements

Authors wish to thank two anonymous reviewers for their valuable comments. Authors also thank Geoffroy Chevallier (PNF 2 , Université Paul Sabatier, Toulouse, France) for his help for samples sintering, Philippe de Parseval and Stéphane le Blond du Plouy (Centre Raimond Castaing, Toulouse, France) for the EPMA analyses and SEM imaging, respectively. This work is part of the PALLAS project funded by the ANR (grant ANR-14-CE33-006-01). Authors also thank the Université Paul Sabatier for its financial support (contribution to the PhD grant to A. Néri).

Temperature (°C)