Fe²⁺–Mg partitioning between olivine and liquid at low oxygen fugacity: an experimental and thermodynamic framework

Abstract

A set of 1-atm gas-mixing experiments (n = 31) was run using Re wire loops at low-oxygen fugacity (ƒO₂, within one half-log unit of the Fe–FeO buffer) and temperatures of 1175–1400 °C to explore the compositional dependence of the Fe²⁺–Mg olivine–liquid exchange coefficient, ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$, under conditions where corrections for liquid Fe³⁺ are small. The bulk compositions used for these experiments include a picrite, a high-alumina basalt, and a suite of three MORB compositions with variable Fe/Mg. The ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ values from the n = 14 experiments run on pre-saturated Re wire loops were fit to a regular solution model that relates ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ to temperature, melt composition, and olivine composition. Fe²⁺–Mg exchange in the experiments is well-described using two parameters: the SiO₂ content of the liquid and the coexisting olivine composition. Combining our experiments with a literature compilation of low-pressure, low-ƒO₂ experiments (largely on non-terrestrial bulk compositions) produced a combined data set spanning a broad region of composition space (e.g., liquid TiO₂ and Na₂O + K₂O contents up to 18.4 wt% and 8.4 wt%, respectively). Fitting this expanded experimental database required two additional liquid compositional terms: Ti, and a Si-(Na + K) cross-term. Because the Fe³⁺ content of all of the experimental liquids is low, the compositional variation seen in ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ is effectively independent of the compositional effects on liquid Fe³⁺/Fe²⁺ ratios. Given olivine–liquid Mg or Fe²⁺ partition coefficient information, it is possible to eliminate the explicit dependence of ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ on olivine composition, enabling a simple iterative approach for calculating the composition of coexisting olivine given only the bulk composition (and Fe³⁺/Fe²⁺ ratio) of an olivine-saturated liquid.

Introduction

Experimental determinations of element partitioning between coexisting solid and liquid phases are essential for understanding crystallization and melting processes, such as modeling the chemical evolution of melts, their residues, and cumulate mushes. Olivine is the predominant mineral phase in the upper mantle (e.g., Stixrude and Lithgow-Bertelloni 2012; Warren 2016) and is the primary liquidus phase during low-pressure crystallization of primitive mafic melts (e.g., O’Hara 1968; Hess 1992; Grove and Brown 2018). Consequently, substantial effort has been devoted to quantifying chemical equilibrium between olivine and liquid (e.g., Bédard 2005), and in particular, the partitioning behavior of Fe²⁺ and Mg (e.g., Longhi et al. 1978; Gee and Sack 1988; Toplis 2005; Putirka 2016). The goal of these studies was to provide a basis for modeling the melting of olivine-bearing mantle assemblages and for reconstructing basaltic liquid lines of descent. In addition, determining whether olivine–liquid pairs have approached equilibrium in experimental studies and in nature has often been based on measuring Fe and Mg in both phases, calculating a Fe²⁺–Mg olivine–liquid exchange coefficient,

$${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}} \equiv \frac{{\left( {\frac{{{\text{X}}_{{{\text{FeO}}}} }}{{{\text{X}}_{{{\text{MgO}}}} }}} \right)^{{{\text{ol}}}} }}{{\left( {\frac{{{\text{X}}_{{{\text{FeO}}}} }}{{{\text{X}}_{{{\text{MgO}}}} }}} \right)^{{{\text{liq}}}} }},$$

(1)

and comparing it to an accepted ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ value. Equation (1) describes Fe²⁺–Mg partitioning between olivine and silicate melt, where X_MgO and X_FeO are oxide mole fractions in the liquid and the olivine (Roeder and Emslie 1970; note that the oxide mole fractions in Eq. (1) can be replaced by oxide weight percents, w_MgO and w_FeO). In Eq. (1), ${\mathrm{X}}_{\mathrm{FeO}}^{\mathrm{liq}}$ refers to iron present in the liquid as Fe²⁺ (we assume that all Fe in olivine is Fe²⁺); later references to FeO* or Fe* refer to all iron (Fe²⁺and Fe³⁺) present in the liquid or in a bulk composition as Fe²⁺. Hereafter, we only consider partitioning between olivine and liquid, and so the “ol/liq” specifier is dropped from “${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$” and the Fe²⁺–Mg olivine–liquid exchange coefficient is referred to as ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$.

Applying ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ to petrological problems is rooted in the study of Roeder and Emslie (1970), who found that for Hawaiʻian basalts equilibrated at atmospheric pressure and temperatures (T) of 1150–1300 °C, ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ is approximately constant with a value of 0.30. This value has been (and still is) widely used in the petrological literature (see Matzen et al. 2011 for re-evaluation of the original Roeder and Emslie 1970 data). However, it has long been recognized that 0.30 is only a convenient reference point and, as discussed below, that the exchange coefficient displays resolvable and systematic variations as a function of bulk composition. For example, experiments on high-Ti basaltic compositions showed that ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ is systematically lower than 0.30, decreasing to 0.22 in melts with up to 19 wt% TiO₂ (lunar basalts: Longhi et al. 1978; Grove and Beaty 1980; Delano 1980; Jones 1988; picrites at 1.2 and 2.8 GPa: Xirouchakis et al. 2001). Note that the experiments on lunar basalts were run under reducing conditions, often in Fe-metal capsules, thereby constraining Fe³⁺ in the melt to low values and effectively eliminating the need to correct electron microprobe FeO* glass values for their Fe³⁺ contents to calculate ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$. Although Roeder (1974) found that the FeO contents of liquids saturated with Fe-metal and olivine decreased with increasing alkali and alumina contents, he concluded that ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ remained ~ 0.30 ± 0.03 in melts with up to 4.39 wt% Na₂O, 3.51 wt% K₂O, and 18.24 wt% Al₂O₃. Later work on six component basalts at low ƒO₂ demonstrated that liquids with elevated Na₂O display systematically lower ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values (e.g., down to 0.23 in melts with 7.5 wt% Na₂O; Shi 1993), consistent with experiments on natural alkalic basalts run at higher ƒO₂ near the fayalite–magnetite–quartz buffer (FMQ) (e.g., Gee and Sack 1988).

In addition to the effects of TiO₂, alkalis, and alumina, variations in the concentrations of major elements in the liquid (e.g., SiO₂: Longhi et al. 1978; Gee and Sack 1988; Toplis 2005; Putirka 2016; FeO + MgO: Kushiro and Walter 1998; Filiberto and Dasgupta 2011) and in the composition of the olivine (Toplis 2005; Blundy et al. 2020) have also been shown to correlate with variations in ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$. As a result, several models have been developed to describe variations in ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ as functions of melt composition ± olivine composition ± T ± pressure (P) (e.g., Longhi et al. 1978; Ford et al. 1983; Gee and Sack 1988; Snyder and Carmichael 1992; Kushiro and Walter 1998; Kushiro and Mysen et al. 2002; Herzberg and O’Hara 2002; Toplis 2005; Filiberto and Dasgupta 2011; Putirka 2016; Blundy et al. 2020). Such models provide a more accurate description of olivine-melt equilibria than is possible by simply adopting a constant value for ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$, and such descriptions are useful for applications including reconstructing primary melts (e.g., Herzberg and O’Hara 2002; Herzberg and Asimow 2015; Brown Krein et al. 2021), and correcting for post-entrapment crystallization in olivine-hosted melt inclusions (e.g., Danyushevsky et al. 2002; Gaetani and Watson 2002).

For melts coexisting with olivine, ferrous oxide in the quenched liquid (i.e., FeO^liq) must be known to calculate ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ from electron microprobe analyses that generally report all Fe as FeO*. Fe³⁺/Fe²⁺ ratios vary significantly over geologically relevant ƒO₂ values (e.g., Carmichael 1991; Cottrell et al. 2022 and references therein), thus the analyzed FeO* of natural or experimental glasses often requires a correction by subtracting the amount of Fe³⁺ in the glass to obtain the Fe²⁺ content. Only a relatively small number of experimental olivine–liquid pairs have direct measurements of Fe³⁺/Fe²⁺ in the quenched glasses, and for these, ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ can be calculated directly (Mysen and Dubinsky 2004; Partzsch et al. 2004; Mysen and Shang 2005; Mysen 2006, 2007; Matzen et al. 2011; Blundy et al. 2020; Waters et al. 2020). Most 1-atm experiments on terrestrial bulk composition have been done in the vicinity of the FMQ, where liquid FeO* is not a good approximation for FeO^liq. Thus, for these and other relatively oxidized experiments, the Fe³⁺/Fe²⁺ ratio of the melt must be calculated using one of the many algorithms that relate Fe³⁺/Fe²⁺ to melt composition, ƒO₂, and T (e.g., Sack et al. 1980; Kilinc et al. 1983; Kress and Carmichael 1988, 1991; Borisov and Shapkin 1989; Nikolaev et al. 1996; Jayasuriya et al. 2004; Ghiorso and Kress 2004; Putirka 2016; Borisov et al. 2018; O’Neill et al. 2018). For a given liquid composition at FMQ, the different Fe³⁺/Fe²⁺ parameterizations are not in agreement, and this leads to significant differences in calculated ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values (see Matzen et al. 2011, Fig. 4); as shown in Supplementary Figure S1, these differences in calculated Fe³⁺/Fe²⁺ increase with increasing ƒO₂. Consequently, the need to correct for the presence of Fe³⁺ in most experimental glasses, particularly at ƒO₂ > FMQ, represents a source of uncertainty in reported ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values (e.g., Matzen et al. 2011). For example, in experiments run over a wide range of ƒO₂, the fraction of Fe²⁺ to Fe* in the melt can vary from > 95% (at very low ƒO₂, e.g., Bowen and Schairer 1935) to 20–30% (in air, e.g., Kress and Carmichael 1991; Jayasuriya et al. 2004; Mysen et al. 2004; Borisov et al. 2018), making it difficult to deconvolve the effects of melt and olivine composition on the ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ from our incomplete understanding of how melt composition and ƒO₂ affect the Fe³⁺/Fe²⁺ ratios of the liquids—a point emphasized by Blundy et al. (2020).

One way to circumvent the necessity of correcting for Fe³⁺ in quenched glasses is to focus on experiments run under sufficiently reducing conditions such that nearly all the Fe present in the liquid is Fe²⁺. For liquids, where X_FeO ≈ X_FeO*, ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ can be determined with minimal correction from electron microprobe analyses of the glasses. Here, we present 1-atm experiments designed to generate olivine–liquid pairs at ƒO₂ within ± 0.5 log units of the iron–wüstite buffer (i.e., IW ± 0.5; buffer equation from Huebner 1971) using terrestrial basaltic compositions, which are underrepresented among previous experiments run at similarly reducing conditions. At ƒO₂ levels ≤ IW + 0.5, the concentration of FeO in the melt approaches that of FeO*. Importantly, absolute differences in Fe³⁺/Fe²⁺ ratios predicted by the available models at IW are also small (for a single composition, Fe³⁺/Fe²⁺ models predict a maximum difference of 2–4% absolute for our experiments). Thus, for such experiments, ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{*}-\mathrm{Mg}}$ (referred to as K_D*, where ${\mathrm{X}}_{\mathrm{FeO}}^{\mathrm{liq}}$ in Eq. (1) is replaced by ${\mathrm{X}}_{\mathrm{FeO*}}^{\mathrm{liq}}$, see Eq. (2)) closely approaches ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$, and correlations between liquid composition and ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{*}-\mathrm{Mg}}$ can be used to isolate compositional effects on the partitioning of FeO and MgO between olivine and melt, independent of any Fe³⁺/Fe²⁺ correction. Combining our experimental data with low-ƒO₂ experiments from the literature, we present a thermodynamically based model to describe the compositional and temperature effects on ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ that spans most of the range of common terrestrial and extraterrestrial magmas.

Methods

Starting compositions

The bulk compositions used in this study are based on three basalt types [ocean island basalts (OIB); high-Al basalts (HAB); and mid-ocean ridge basalts (MORB)], and they were designed to have olivine as a liquidus phase and to span a range of Mg# (100[Mg/(Mg + Fe*)], molar). Bulk compositions are reported in Table 1. Mg# ranges from 40 to 80, and SiO₂, TiO₂, and CaO/Al₂O₃ values are 43.8–50.5 wt%, 0.55–1.41 wt%, and 0.65–0.81, respectively. SynHP1 is a synthetic, alkali-free picritic composition based on whole-rock compositions from Mauna Kea (see Matzen et al. 2011 and Table 1, this publication, for further details). As part of an earlier study, we had added Cr₂O₃ and Mn₂O₃ to synHP1 (the former to increase the stability field of spinel) to generate synHP1 + Cr + Mn (1.05 wt% Cr₂O₃ and 0.81 wt% MnO). All of the remaining bulk compositions have MnO contents of ~ 0.5 wt%; although this is higher than those typical of terrestrial basalts (0.1–0.2 wt%; e.g., Le Maitre 1976; Gale et al. 2013), we chose to elevate MnO concentrations to facilitate more precise determinations of Mn concentrations in olivine and melt without having to use a high microprobe beam current on the glasses. The HAB + Ol + Mn composition is a natural high-alumina basalt from Medicine Lake Highland (82–72c; Baker et al. 1991) to which 15% synthetic Mg₂SiO₄ powder was added to expand the olivine stability field to temperatures above 1300 °C (based on calculations using MELTS; Ghiorso and Sack 1995). The added forsterite resulted in a bulk composition with Mg# = 76 and the addition of Mn₂O₃ raised the MnO content from 0.16 to 0.49 wt%. In contrast to the synHP1 ± Cr + Mn compositions and the MORB compositions (discussed below) that are alkali-free, HAB + Ol + Mn has 1.88 wt% Na₂O and 0.08 wt% K₂O.

Table 1 Bulk compositions of the starting mixes

To create the synthetic MORB-like compositions, we used the database of Fo₉₀-corrected mid-ocean-ridge segment average compositions from Gale et al. (2014) and fit each of the major and minor oxides as a linear function of MgO. An initial target composition (see notes to Table 1) was calculated by projecting these fits to an MgO value of 14 wt% (the mean and median MgO contents of the Fo₉₀-corrected compositions from Gale et al. 2014 are 13.8 and 13.9, respectively). This base composition has an Mg# of 73, ~ 14.4 wt% MgO when normalized to a sum of 100 wt% on an alkali- and MnO-free basis, and a molar sum of MgO + FeO* of 0.491. The bulk compositions synMORB80 (Mg# = 80) and synMORB60 (Mg# = 60) were generated by varying MgO/FeO* at constant molar MgO + FeO*, i.e., by varying MgO/FeO* until the bulk composition had an Mg# of either 80 or 60. MELTS calculations (Ghiorso and Sack 1995) at an ƒO₂ of IW + 0.5 using a bulk composition with Mg# = 40 derived in the manner described above predicted pigeonite as the liquidus phase, followed by plagioclase, and then olivine at T ~ 1100 °C. To expand the stability field for olivine in the synMORB40 composition, its total molar MgO + FeO* content was increased by a factor of 1.2 relative to synMORB80 and synMORB60 (molar sum of MgO + FeO* = 0.589, on an alkali- and MnO-free basis). MELTS calculations on this FeO* and MgO enriched composition predicted olivine as the primary liquidus phase at T = 1215 °C. Note that like the synHP1 and synHP1 + Cr + Mn bulk compositions, the synMORB compositions are alkali-free.

High-purity oxides SiO₂, TiO₂, Al₂O₃, Fe₂O₃, Mn₂O₃, MgO, and Cr₂O₂, synthetic Mg₂SiO₄ powder, CaCO₃, and the preexisting synHP1 (synthetic) and HAB (natural basalt) powders were dried to remove adsorbed water prior to weighing to produce each of the bulk mixes (see caption to Table 1 for drying schedule). The powdered components were weighed, then ground and mixed under ethanol in an alumina mortar for at least 1 h. The homogenized powders were then pressed into ~ 1 cm diameter pellets under vacuum, with acetone as a binding agent.

Experimental design

All experiments were run at 1 atm in a vertical Deltech furnace using H₂–CO₂ gas mixtures to control ƒO₂. The type-S thermocouple used to set the experimental run temperatures was calibrated at the melting point of gold. The ƒO₂ of the furnace atmosphere was set using a yttria-stabilized zirconia oxygen sensor, which was calibrated by bracketing the ƒO₂ of the IW reaction (Fe + ½ O₂ = FeO) at 1350 °C—the bracketed reaction was found to lie within < 0.1 log units of the accepted value (Huebner 1971). Experiments were conducted at either IW + 0.5 or IW–0.5 (Table 2). Due to how the Re wire loop plus sample was introduced into the hot spot (see discussion below), the thermocouple and ƒO₂ sensor were not present in the furnace during the runs. The experimental ƒO₂ and T were thus set prior to each experiment, the furnace was run down to 1000 °C, and then the sample was introduced into the furnace which was run up to the target T based on the relationship between hotspot temperature and the set point temperature. For experiments at T ≥ 1225 °C, the ƒO₂ was set at the experimental run temperature; for the experiments at 1175 °C, the appropriate H₂/CO₂ ratio in the gas was set with the furnace hotspot held at 1250 °C to ensure that the gas mixture was in equilibrium (Beckett and Mendybaev 1997), then at 1175 °C the gas mixture was adjusted slightly to achieve the correct ƒO₂.

Table 2 Experimental run conditions and products

For each experiment, an ~ 80–100 mg chip was removed from the pellet of a given starting composition and attached to a Re wire loop using polyvinyl alcohol. The Re loop and sample were suspended from thin Pt quench wire attached to two 0.039″ diameter Pt wires within a two-hole ceramic rod. Given that Re is volatile in an oxidizing atmosphere at high temperatures (i.e., ƒO₂ > FMQ; Borisov and Jones 1999) and that the furnace tube was initially filled with air, the ceramic rod was positioned so that the loop and sample were at the top (and cool) part of the furnace tube when inserted into the furnace. With the controller set at 1000 °C, the H₂–CO₂ gas mixture was then introduced into the sealed furnace tube and allowed to equilibrate for ~ 30 min. The Re loop was then slowly lowered into the hot spot over the following 20 to 30 min. Once the silicate chip was positioned in the hotspot, the furnace was ramped up to the target run temperature at a rate of ~ 10 °C/min. The experiments were then held isothermally at temperatures of 1175–1400 °C for 6–48 h. All experiments were terminated by electrically fusing the thin Pt hanging wire and dropping the silicate and wire loop into deionized water. Temperature, run time, ƒO₂, and other notes on particular experiments are reported in Table 2. After quenching, the silicate bead was removed from the wire loop by gently tapping it with a small-diameter drill blank and hammer—this was sufficient to remove nearly all of the silicate material from the loop.

In the T-ƒO₂ region where Re is stable, it alloys with Fe to a much lesser extent than Pt (e.g., Grove 1982; Borisov and Jones 1999; Kessel et al. 2001). Nevertheless, at ƒO₂ ≲ IW + 1, Fe-loss to a Re sample container becomes non-trivial (Borisov and Jones, 1999). For this reason, the Re loops were preconditioned for each bulk composition and ƒO₂ by running one or two “saturation” experiments for each loop (denoted “S” in Table 2). The preconditioned Re loops were then reused for one or more subsequent experiments, and those experiments are referred to as “equilibration” experiments (“E” in Table 2), which were the only experiments used in modeling ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ (see Discussion). The loop number used for each experiment is listed in Table 2, along with the experimentally produced phases, and the % change in bulk FeO* as determined by mass balance (see Approach to equilibrium). The acronym used for our low-ƒO₂ experiments is “RKD”, which stands for Reduced K_D.

Analytical techniques

Chips of silicate material recovered from each experiment were mounted in epoxy, ground down with alumina papers, polished with < 2 µm and 0.25 µm diamond powder, and then ultrasonicated in ethanol. All experiments and secondary standards were carbon coated at the same time to ensure that they had the same carbon-coat thickness. Quantitative wavelength-dispersive (WDS) analyses of glasses, silicates, and oxides were collected with a JEOL JXA-8200 electron microprobe at Caltech operating at an accelerating voltage of 15 keV. Backgrounds on primary standards, secondary standards, and unknowns were determined using the mean-atomic-number procedure of Donovan and Tingle (1996) and raw X-ray counts were reduced with a modified ZAF procedure (Armstrong 1988). See Table 3 for a list of primary standards and count times for each analyzed element. Glasses were analyzed using a 10 nA beam current, and a 10 µm diameter beam. Glass analyses with oxide sums of 100 ± 1.5% were considered acceptable. Olivines were analyzed using a 40 nA beam current with a 1 µm diameter beam; olivine analyses were accepted if they had analytical totals of 100 ± 1.5%, a tetrahedral cation sum of 1.000 ± 0.015, and an octahedral cation sum of 2.000 ± 0.015, both calculated on a four-oxygen basis (Supplementary Figure S2). Spinels were analyzed using a 10 nA beam current and a focused beam (< 1 µm) due to their small size, typically < 10 µm across. The Fe³⁺/Fe²⁺ values of the spinels were calculated assuming ideal stoichiometry. Because all of the spinel analyses contained between 0.2 and 0.6 wt% SiO₂, likely due to Si contamination from beam interaction with the surrounding glass or olivine (e.g., Matzen et al. 2011; Davis and Cottrell 2018 and references therein), we corrected the spinel analyses following the procedure discussed in Matzen et al. (2011). A 10 nA and 1 µm beam was used to analyze plagioclase and pyroxene; analyses were accepted if the oxide sum was 100 ± 2%, and the cation sums for plagioclase and pyroxene were 5.00 ± 0.02 per 8 oxygens and 4.00 ± 0.05 per 6 oxygens, respectively. Analyses of metal (see section Fe–Pt metal blebs) used a focused 25 nA beam, along with pure metal standards and off-peak backgrounds (Supplementary Section S1.4).

Table 3 Experimental glass and olivine rim compositions in wt%

Euhedral olivine grains > 20 µm in longest dimension were selected for analysis and analyses of the rims were taken within 2–5 µm of the crystal edge. The cores in a subset of olivine grains from each experiment were also analyzed to check for compositional zoning. Due to stage drift and beam alignment issues, some rim analyses have Al₂O₃ contents higher than those expected in olivine phenocrysts (e.g., Sobolev et al. 2007). These elevated values are most likely due to X-ray counts from the adjacent glass. The alumina contents in olivine cores from each of the experiments showed limited variation, with typical 1σ values of ~ 0.01 wt%. For a given experiment, we used the maximum alumina content in the olivine core analyses plus 0.01 wt% Al₂O₃ to define the maximum acceptable alumina content in rim compositions from the same experiment. For example, in runs RKD-5 and RKD-14, the maximum alumina core values were 0.07 and 0.03 wt%, respectively, and thus olivine rim analyses with > 0.08 wt% and > 0.04 wt% Al₂O₃ were rejected. Thus, all olivine rim compositions used to generate the averages listed in Table 3 were subjected to three filters: oxide sum, stoichiometry, and Al₂O₃ content.

Secondary standards were used to assess the accuracy and precision of Fe/Mg ratios, and to monitor instrumental drift within an analytical session and between sessions. Three secondary standard glasses were used (BHVO-2g, BIR-1g, and GOR-128g) as well as two olivines from Caltech’s mineral collection (San Carlos, CIT–SC, and Guadalupe Island, CIT–GI). To monitor instrumental drift, the secondary standards were measured before and after measuring a particular phase in each equilibration experiment; e.g., the same glass chip of each secondary glass standard was measured 2–3 times before and after the 14–20 glass analyses of each equilibration experiment. A similar procedure (using the olivine secondary standards) was applied to analyzing the olivines in the equilibration experiments.

Results

Average compositions of spinel, plagioclase, pyroxene, and rare Fe–Pt blebs (interpreted to be a contaminant; see below) are reported in the Supplement (Section S1 and Supplementary Table S1). Although spinel, plagioclase, and pyroxene were used in the mass balance calculations (see Approach to equilibrium), they are not discussed further here.

Glass and olivine

For each of the saturation and equilibration experiments, the glass and olivine rim analyses that passed the various quality filters were averaged to obtain mean compositions (listed in Table 3); the number of analyses ranged from 14 to 48 for the glasses and 5 to 25 for the olivines. Standard deviations (Table 3) refer to the distribution of multiple microprobe analyses of each phase in each experiment (i.e., not to the standard error of the mean), and unless specified otherwise, references to standard deviations and the use of the ± symbol in the text refers to 1σ errors.

Figure 1 shows variations in the wt% concentrations of liquid SiO₂ and Al₂O₃ as a function of MgO and variations in glass Mg# as a function of run temperature. MgO contents range from ~ 6 to 17.5 wt% and SiO₂ and Al₂O₃ vary from 45 to 52 wt% (Fig. 1A) and from 11 to 18 wt% (Fig. 1B), respectively. The Mg#s of the glasses range from 31.5 to 78.1, and they are higher in experiments run on the same bulk composition at higher T (Fig. 1C). Variations in other oxide concentrations include: TiO₂ from 0.6 to 2.14 wt%, CaO from 8.6 to 13 wt%, and Na₂O + K₂O from 0.04 to 1.65 wt%. All of the experimental glasses are hypersthene normative (calculated using the CIPW norm, all Fe as FeO*). The fact that all experiments, including those on the alkali-free synHP1, synHP1 + Cr + Mn, and synMORB bulk compositions, contained some Na and K can be attributed to the gradual depletion of contaminant Na + K on the inner wall of the furnace tube (see Supplementary Section S2 and Matzen et al. 2011).

Fig. 1

Compositional trends in experimental glasses. A wt% SiO₂ vs. wt% MgO in the glasses, B wt% Al₂O₃ vs. wt% MgO, C Mg# (100[Mg/(Mg + Fe)], molar) in the experimental glasses vs. experimental T (°C). Larger symbols are equilibration experiments (labeled E in Table 2); smaller symbols are saturation experiments (labeled S in Table 2). Bulk composition color codes: synHP1 = dark green, (n = 1); synHP1 + Cr + Mn = light green; HAB + Ol + Mn = dark blue; synMORB80 = purple; synMORB60 = red; synMORB40 = light blue. Triangles – 1175 °C; Circles – 1225 °C; Squares – 1300 °C; Diamonds – 1350 °C; Star – 1400 °C. 1σ errors are smaller than the symbol sizes

Fo contents (Fo = 100[Mg/(Mg + Fe)], molar) of the experimentally produced olivines range from 57.7 to 92.3 (Fig. 2; the minimum value is from experiment RKD-24, synMORB40 at IW + 0.5 and 1175 °C; the maximum value is from experiment RKD-30, synMORB80 at IW–0.5 and 1300 °C). Olivine MnO contents range from 0.14 to 0.74 wt%, and have Cr₂O₃ and NiO contents up to 0.50 wt% and 0.73 wt%, respectively (Table 3). All the olivine crystals in experiments conducted at T ≥ 1225 °C have core and rim Fo contents that are indistinguishable within 1σ error (the analytical precision in Fo content ranges from ~ 0.1 for the more magnesian compositions to ~ 0.3 for synMORB40 olivines). However, in the IW–0.5 and IW + 0.5 experiments on the synMORB40 bulk composition run at T = 1175 °C, high-contrast backscattered electron (BSE) images revealed two olivine populations: olivines that were unzoned based on the BSE images, and olivines with cores that were darker than their rims in BSE (Fig. 3). Based on counting visibly zoned grains in the field of view of BSE images, the zoned olivines comprise less than 2% of the olivine grains exposed on the polished chips from each synMORB40 experiment. Microprobe traverses of the zoned olivines show that the cores are approximately 5–6 Fo units more magnesian than the rims; these grains are also zoned in MnO and CaO (this zoning is discussed in Supplementary Section S1.5 and shown in Supplementary Figure S3). However, it is important to stress that in each 1175 °C synMORB40 equilibration experiment, the rim Fo contents of zoned olivines are indistinguishable from the rim compositions of unzoned olivines and it is the rim compositions of unzoned olivines which were used to calculate equilibrium K_D,Fe*-Mg values (see Table 3 and Fig. 4). Microprobe traverses across olivine grains identified as unzoned in BSE show, with the exception of CaO, no core-to-rim gradients; CaO is elevated in analyses close to the surrounding glass, likely due to secondary fluorescence (Llovet and Salvat 2017). Since we have not corrected rim compositions for Ca fluorescence in any of the experiments, these data should not be used to evaluate olivine–liquid calcium partitioning. In summary, the Fe/Mg ratios of all analyzed olivine rims from a given experiment overlap within analytical uncertainty, and it is the averages of these analyses that are reported in Table 3. We also emphasize that the only equilibration experiments where any Fe/Mg zoning in olivines was observed, either in BSE images or by comparing core and rim analyses, were in the two synMORB40 experiments run at 1175 °C (RKD-24 and RKD-35; for additional details on these two experiments, see Supplementary Section S1.5). We infer that the remaining equilibration experiments approached bulk Fe/Mg equilibrium and were populated entirely by unzoned olivines (note that this includes the 1175 °C experiment on synMORB60).

Fig. 2

Average Fo% (100[Mg/(Mg + Fe)], molar) in olivine rims vs. run T (°C). Symbols as in Fig. 1. 1σ errors are smaller than the symbol sizes

Fig. 3

Backscattered electron (BSE) image of experimental run products from experiment RKD-22 (synMORB40) with contrast increased to enhance the grayscale zoning in some olivine phenocrysts (labeled “zoned ol”). Scale bar is 40 µm. Phase labels: ol – olivine, gl – glass, pl – plagioclase. Note that most of the olivines in the field of view are unzoned—this area was selected for imaging because it had several zoned olivines visible in the field of view, but they only constitute < 2% of the olivines exposed on the polished chip from the experiment (see Glass and olivine and Supplementary Section S1.5). In the zoned olivines, the cores are ~ Fo₆₇ and the rims are ~ Fo₆₀; rim compositions of the zoned and unzoned olivines overlap within 1σ

Fig. 4

Experimentally determined ${\mathrm{K}}_{\mathrm{D},\mathrm{Fe}^*-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ (K_D*) for RKD experiments run at ƒO₂ ≤ IW + 0.5 plotted vs. A Temperature, B FeO* wt% in glass, C Fo% = 100(Mg/[Mg + Fe], molar) in olivine. Symbols as in Fig. 1. The olivine compositions used to calculate ${\mathrm{K}}_{\mathrm{D},\mathrm{Fe}^*-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ all come from analyses of olivine rims. Error bars are 1σ and errors for the quantities plotted on the x-axis are smaller than the symbol sizes

Fe–Pt metal blebs

In 19 out of 31 experiments, trace amounts of an Fe–Pt ± Ni metal alloy were observed as isolated blebs in the polished charges (Supplementary Figure S4A and S4B), despite the fact that nominally there was no Pt in any experiments and none of the starting materials made from oxide mixes contained NiO (yet some experiments on these mixes contain Ni-bearing metallic blebs). The proportion of blebs in the experiments is difficult to estimate given their low abundances, but we estimate that they comprise << 0.1% of the experimental volume (see Supplementary Section S1.4). Normalized to 100 wt%, Pt and Fe comprise ≥ 98.4% of the average bleb compositions, and for a majority of the averages, this sum is > 99 wt%. Ni, when present, comprises between ~ 0.09 and 1.6 wt% (detection limit ~ 0.06 wt%) of the average alloy compositions. There are no qualitative correlations between the presence or absence of the Fe–Pt blebs and Fe loss or gain calculated via mass balance (see Approach to equilibrium and Supplementary Section S3). More details on the blebs and their compositions can be found in Supplementary Section S1.4 and Supplementary Table S1. We speculate that these blebs are produced by contamination from the Pt quench legs from which the Re loops hung, and we note that we have found no evidence that they have measurable effects on the liquid compositions or partitioning results.

Secondary glass and olivine standards

Analyses of the secondary glass standards (BIR-1g, BHVO-2g, and GOR-128g) constrain the precision and accuracy of the analyses of the experimental glasses. Importantly, for the analytical session in which the RKD glasses were analyzed, the average Fe*/Mg ratio of each secondary glass standard overlaps at 1σ with the global mean Fe*/Mg value for that standard from other sessions in which it was analyzed. The mean and 1σ values of the Fe*/Mg molar ratios measured in the three glasses from the RKD probe session are 0.612 ± 0.005 (BIR-1g, n = 69), 0.877 ± 0.009 (BHVO-2g, n = 73), and 0.219 ± 0.002 (GOR-128g, n = 69). The mean Fe*/Mg values for BIR-1g and BHVO-2g overlap within 1σ of the preferred values for these two glasses listed in the GeoRem database (http:// http://georem.mpch-mainz.gwdg.de; Jochum et al. 2006): 0.621 ± 0.006 (BIR-1g), 0.889 ± 0.008 (BHVO-2g). The Fe*/Mg ratio for GOR-128g overlaps within 2σ of the GeoRem preferred value for GOR-128g (0.212 ± 0.004). Based on data compiled in the GeoRem database, there appear to be relatively small but consistent differences between bulk rock FeO* and MgO values for BIR-1 and BHVO-2 (largely determined by XRF) and the preferred values for BIR-1g and BHVO-2g (determined by electron microprobe). Until this issue is resolved and given the fact that our probe Fe*/Mg values for these standard glasses overlap with the GeoRem preferred values, we have chosen to forego any post-analysis corrections to the glass compositions such as those described in Sack et al. (1980), Reynolds and Langmuir (1997), Stolper et al. (2004), and Gale et al. (2013).

Multiple analyses of nominally homogeneous secondary olivine standards CIT–SC and CIT–GI were used to assess the precision of the analyses of the experimentally produced olivines. In addition to measuring these standards during the same sessions in which the experimental olivines were analyzed, the same standard olivines were measured over several other sessions using different analytical protocols (e.g., counting times, spectrometer choices) and beam conditions. We found no drift at the 1σ level in the Fo contents of the secondary standards within individual sessions or between the sessions in which the RKD experiments were analyzed. Both secondary standards define a narrow range in the Fe/Mg contents calculated from repeat measurements in the analytical sessions where equilibration experiments were measured: CIT–SC, Fo = 89.89 ± 0.05 (n = 85) and CIT–GI, Fo = 89.95 ± 0.05 (n = 90). Note that published analyses of San Carlos olivines (Jarosewich et al. 1979; Houlier et al. 1990; Spandler and O’Neill 2010; Batanova et al. 2015; Tollan et al. 2018; Lambart et al. 2022) range from Fo = 88.4–91.5, with a mean of 90.5; repeat measurements of the same chip in the CIT–SC olivine mount are slightly more Fe-rich than the mean value above, but are well within the reported range.

Approach to equilibrium

Several lines of evidence suggest that our experimental results represent close approaches to Fe/Mg equilibrium: for example, (1) olivine and glass are homogeneous within the spatial resolution and analytical precision of the probe analyses (except for the zoned olivine subpopulation in the synMORB40 experiments); (2) time-series experiments converge to a constant value of the FeO*–MgO exchange coefficient; and (3) the overlap between the compositions of the rims of the rare normally zoned olivine grains in the synMORB40 runs and the cores and rims of homogeneous olivine grains in the same experiments suggests progressive conversion of initially more Mg-rich olivine to more Fe-rich olivine with time in the experiments. Further discussion of these three criteria for a close approach to equilibrium is given in Supplementary Section S2.

The proportions of phases in each experiment and estimates of the extent to which FeO*, Na₂O, and K₂O were gained or lost during each experiment was determined by mass-balance using a non-linear minimization routine based on Albarède and Provost (1977) (see Supplementary Section S3 for details and Supplementary Table S2 for results). All of the RKD experiments could be mass balanced, yielding acceptable fits at the 95% confidence level. The equilibration experiments run on pre-saturated Re loops lost ≤ 8.3% (relative) of their total FeO* and the average relative change in bulk FeO* was  – 3.5%. Only one experiment (a saturation run, RKD-22) was found to have gained iron (∆FeO* =  + 1.4% relative). Experiments using the same bulk composition run under identical conditions but with different degrees of FeO* loss have ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{*}-\mathrm{Mg}}$ values that overlap at the 1σ level (within 0.003, e.g., compare S and E experiments in Table 2). For the 19 experiments where Fe–Pt blebs were observed, ∆FeO* ranges from < 1% (essentially no change) to –13.4% (in a saturation run), with an average ∆FeO* of –2.9% and is uncorrelated with X_Fe in the blebs. The calculated FeO* loss or gain for each experiment is listed in Table 2. Experiments using the alkali-bearing mix HAB + Ol + Mn lost up to 52% of the original Na₂O present in the bulk mixture, and the two equilibration experiments (RKD-12 and RKD-20) lost approximately 36% and 40% of their initial Na₂O, respectively (Supplementary Figure S6 and Supplementary Table S2). The K_D* values for all of the HAB + Ol + Mn experiments (run at 1225 °C and 1300 °C and with Na₂O contents from 1.00 wt% to 1.56) overlap at the 1σ level.

Olivine–liquid partitioning

We proceed by defining the apparent Fe*–Mg exchange coefficient, K_D* for an olivine–liquid pair:

$${\text{K}}_{{{\text{D}},{\text{ FeO*}} - {\text{MgO}}}}^{{{\text{ol}}/{\text{liq}}}} = \left( {\frac{{{\text{w}}_{{{\text{MgO}}}} }}{{{\text{w}}_{{\text{FeO*}}} }}} \right)^{{{\text{liq}}}} { }\left( {\frac{{{\text{w}}_{{{\text{FeO}}}} }}{{{\text{w}}_{{{\text{MgO}}}} }}} \right)^{{{\text{ol}}}} \equiv {\text{K}}_{{\text{D}}}*$$

(2)

where w_MgO, w_FeO, and w_FeO* are the weight percent of MgO and FeO in each phase; as discussed above, all Fe is assumed to be FeO in the olivine and FeO* refers to all Fe as FeO in the liquid. K_D* is the shorthand used hereafter and corresponds to the quantity that can be routinely measured using the electron microprobe, where all Fe present in the glass is expressed as FeO. Because FeO^liq is always less than FeO*^liq (e.g., Bowen and Schairer 1935), ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ will always be greater than K_D*, and the magnitude of the difference will depend on the Fe³⁺/Fe²⁺ of the melt. Based on the available parameterizations of Fe³⁺/Fe²⁺ vs. ƒO₂, T, and liquid composition, the amount of Fe³⁺ in glasses from our experiments at IW + 0.5 is expected to be sufficiently low such that K_D* and ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ will generally differ by less than 0.010. The Fe³⁺/Fe²⁺ models of Nikolaev et al. (1996) and Jayasuriya et al. (2004) lead to slightly larger maximum differences in [${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ – K_D*] at IW + 0.5 of 0.022 and 0.019, respectively. Values of K_D* and their associated 1σ values (calculated by summing the FeO and MgO fractional errors for olivine and the FeO* and MgO fractional errors for liquid in quadrature) for the RKD experiments are listed in Table 2.

All bulk compositions except for synMORB40 were run at two or more temperatures that spanned a maximum interval of 100 °C (synMORB40 was only run at 1175 °C). The K_D* values for experiments using the same bulk composition but run at different temperatures all overlap within 1σ error (Fig. 4A). Thus, based on our experiments, at constant bulk composition and under reducing conditions, temperature changes of ≤ 100 °C do not produce changes in K_D* larger than our analytical uncertainty (typical 1σ ≈ 0.003, absolute). However, there are statistically significant differences in K_D* between experiments run using different bulk compositions. For example, the HAB + Ol + Mn experiments (dark blue symbols, Fig. 4A) have a uniformly lower mean K_D* value than the MORB or picritic bulk compositions run at the same T. The MORB suite of bulk compositions was designed to vary the molar Fe*/Mg ratio while keeping the relative molar proportions of the other oxides approximately constant, and the K_D* values for the MORB suite are positively correlated with liquid FeO* (Fig. 4B), consistent with previous studies with liquid FeO* + MgO < 30 wt% (e.g., Longhi et al. 1978; Jones 1988; Kushiro and Walter 1998; Filiberto and Dasgupta 2011). A linear fit weighted by the errors in K_D* to the synMORB experiments run at IW + 0.5 (r² = 0.90) indicates that K_D* increases by ~ 0.01 for every 6 wt% increase in liquid FeO*. Likewise, K_D* is systematically higher in the experiments with more Fe-rich olivines (Fig. 4C, note the x-axis shows decreasing Fo), reflecting the trend in FeO* in the glass and the observation that higher bulk Fe leads to higher K_D*.

The HAB + Ol + Mn, synHP1, and synHP1 + Cr + Mn experiments run at IW + 0.5 are displaced to lower K_D* values relative to the trend defined by the MORB suite. The HAB glasses from equilibration experiments have significantly more Na₂O (~ 1.35 wt%) than the other liquid compositions (< 0.5 wt% Na₂O), and the synHP1 ± Cr + Mn glasses contain higher TiO₂ contents (~ 2 wt%) than the other glasses (with the exception of RKD-21, synMORB60 run at 1175 °C with 2 wt% TiO₂). It is well known that increasing the concentrations of either of these two oxides results in a decrease in ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ (e.g., Longhi et al. 1978; Grove and Beaty 1980; Delano 1980; Gee and Sack 1988; Jones 1988; Shi 1993; Xirouchakis et al. 2001), and our K_D* results are consistent with these expected trends. Although our bulk compositions have up to ~ 0.8 wt% MnO (compared to typical basaltic values of ~ 0.1–0.2 wt%; e.g., Le Maitre 1976; Gale et al. 2013), our experimental results suggest that such elevated MnO contents have minimal effect on K_D*; e.g., RKD-25 (synHP1; 0.18 wt% MnO) and RKD-5 (synHP1 + Cr + Mn; 0.81 wt% MnO), run at the same T and ƒO₂, have indistinguishable K_D* values: 0.315 ± 0.004 and 0.315 ± 0.006.

Discussion

Using the RKD experiments, we develop a quantitative thermodynamic treatment of the compositional effects on ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ described in the previous section, and then expand this treatment by combining our data with a large set of literature experiments with olivine–liquid pairs run at comparably reduced conditions.

Parameterizing the compositional dependence of ${\mathrm{K}}_{{{\mathrm{D}},{\mathrm{Fe}}^{2 + } - {\mathrm{Mg}}}}^{{{\mathrm{ol}}/{\mathrm{liq}}}}$ using the RKD data

Here we briefly introduce the theoretical framework and equations that describe Fe²⁺–Mg equilibrium between olivine and melt. The exchange reaction can be written as:

$${\text{FeO}}^{{{\text{liq}}}} + {\text{MgSi}}_{{{1}/{2}}} {\text{O}}_{{2}}^{{{\text{ol}}}} \rightleftarrows {\text{FeSi}}_{{{1}/{2}}} {\text{O}}_{{2}}^{{{\text{ol}}}} + {\text{MgO}}^{{{\text{liq}}}} ,$$

(3)

which has an equilibrium constant, K₃:

$${\text{K}}_3 = \left( {\frac{{{\text{a}}_{{{\text{MgO}}}} }}{{{\text{a}}_{{{\text{FeO}}}} }}} \right)^{{{\text{liq}}}} { }\left( {\frac{{{\text{a}}_{{{\text{FeSi}}_{1/2} {\text{O}}_{2} }} }}{{{\text{a}}_{{{\text{MgSi}}_{1/2} {\text{O}}_{2} }} }}} \right)^{{{\text{ol}}}} ,$$

(4)

where ${\mathrm{a}}_{\mathrm{i}}^{\Phi }$ is the activity of component i in phase Φ. Combining Eq. (4) with Eq. (1) and using the definition ${\mathrm{a}}_{\mathrm{i}}^{\Phi }$ = ${\mathrm{X}}_{\mathrm{i}}^{\Phi } {\upgamma }_{\mathrm{i}}^{\Phi }$, where ${\mathrm{X}}_{\mathrm{i}}^{\Phi }$ and ${\upgamma }_{\mathrm{i}}^{\Phi }$ refer to the mole fraction and activity coefficient of component i in phase Φ, Eq. (4) can be re-written as:

$${\text{K}}_3 = \left( {\frac{{{\text{X}}_{{{\text{MgO}}}} }}{{{\text{X}}_{{{\text{FeO}}}} }}} \right)^{{{\text{liq}}}} { }\left( {\frac{{{\text{X}}_{{{\text{FeO}}}}}}{{{\text{X}}_{{{\text{MgO}}}}}}} \right)^{{{\text{ol}}}} \left[ {\left( {\frac{{\gamma_{{{\text{MgO}}}} }}{{\gamma_{{{\text{FeO}}}}}}} \right)^{{{\text{liq}}}} \left( {\frac{{\gamma_{{{\text{FeO}}}}}}{{\gamma_{{{\text{MgO}}}}}}} \right)^{{{\text{ol}}}} } \right]$$

(5)

In the special case where the activity coefficients all equal one, or if the product of the γ ratios of both phases equals one [i.e., the term in brackets in Eq. (5)], then ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ as defined in Eq. (1) is equal to the equilibrium constant, K₃. Because Fe* and Mg can routinely be measured precisely in both olivine and liquid (and, as emphasized above, for experiments at ~ IW the correction for Fe³⁺ in the liquid is small), the unknown parameters in Eq. (5) are the activity coefficients in each phase, which are functions of composition, T, and P. It has long been recognized that using exchange reactions to model mineral-melt equilibria is useful, because the temperature dependence of the ratio of partition coefficients (i.e., ${\mathrm{D}}_{\mathrm{i}}^{\mathrm{ol}/\mathrm{liq}}= \frac{{\mathrm{X}}_{\mathrm{i}}^{\mathrm{ol}}}{{\mathrm{X}}_{\mathrm{i}}^{\mathrm{liq}}}$, and $\frac{{\mathrm{D}}_{{\mathrm{Fe}}^{2+}}^{\mathrm{ol}/\mathrm{liq}}}{{\mathrm{D}}_{\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}}$ = ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$) is significantly smaller than that of each individual partition coefficient (e.g., Roeder and Emslie 1970; Longhi et al. 1978; Ford et al. 1983; Matzen et al. 2011). In the case of Fe–Mg olivine, this is a consequence mainly of the small differences in the enthalpies of fusion of fayalite and forsterite (Toplis 2005). Therefore, it is often assumed that the effect of temperature on the ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ can be ignored (e.g., Blundy et al. 2020), except through the indirect influence of T on activity-composition relations in the olivine and liquid (e.g., Ghiorso et al. 1983). The pressure dependence of ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ largely reflects the differences in the volumes of fusion of the olivine end members, and on this basis, the pressure effect on the ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ is predicted to be small (on the order of 0.008 absolute per 10 kbar, Ulmer 1989; Herzberg and O’Hara 1998; Toplis 2005). Based on this small effect, we have not included a pressure term in our expression for ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$, because the pressures in our combined data set discussed below (our experiments and those from the literature) only range from 1 to 260 bars (nevertheless, adding a pressure term would be straightforward, e.g., Eq. (9) in Toplis 2005).

To parameterize the compositional dependence of ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$, we have adopted the general approach of Toplis (2005), specifically, their Eq. (5):

$${\text{lnK}}_{{{\text{D}},{\text{Fe}}^{2 + } - {\text{Mg}}}}^{{{\text{ol}}/{\text{liq}}}} = \frac{{\Delta {\text{G}}^\circ }}{{{\text{RT}}}} + {\text{ln}}\left( {\frac{{\gamma_{\text{FeO}}}}{{\gamma_ {\text{MgO}}}}} \right)^{{{{{\text{liq}}}} }} + \frac{{{\text{W}}_{{{\text{Fe}} - {\text{Mg}}}}^{{{\text{ol}}}} }}{{{\text{RT}}}}\left( {{1 }{-}{\text{ 2 X}}_{{{\text{Fo}}}}^{{{\text{ol}}}} } \right)$$

(6)

where ${\mathrm{W}}_{\mathrm{Fe}-\mathrm{Mg}}^{\mathrm{ol}}$ is an interaction parameter describing Fe–Mg non-ideality in a symmetric binary olivine solid solution and $\Delta \mathrm{G}^\circ$ is the standard state Gibbs free energy change of reaction (3). Equation (6) explicitly takes into account non-ideality in both the olivine and the liquid. Toplis (2005) inferred that the ratio $(\frac{\mathrm{\gamma_{FeO}}}{\mathrm{\gamma_{MgO}}}$)^liq is a function of the SiO₂ and alkali contents of the liquid and constructed empirical functions that describe that relationship (see Appendix 2 in Toplis 2005). We have adopted a functional form for the activity coefficients in the liquid based on a truncated symmetric, strictly regular solution model (e.g., Ghiorso 1983) excluding, for now, any cross-terms in the Margules parameters (i.e., all of the W interaction terms are symmetric and each pair of binary interaction terms is independent of the others and of temperature):

$$ RT1n\gamma _{i}^{{{\text{liq}}}} = \sum\limits_{j} {W_{{i - j}}^{{{\text{liq}}}} } X_{j}^{{{\text{liq}}}} ;j \ne 1 $$

(7)

where i = MgO or FeO, ${\mathrm{W}}_{\mathrm{i}-\mathrm{j}}^{\mathrm{liq}}$ is the binary interaction parameter, ${\mathrm{X}}_{\mathrm{j}}^{\mathrm{liq}}$ is the single-cation mole fraction of component j in the liquid, and j ≠ i indicates that there are no terms for MgO–MgO or FeO–FeO in the summation. Because only the ratio $(\frac{\mathrm{\gamma_{FeO}}}{\mathrm{\gamma_{MgO}}}$)^liq is considered, Eq. (6) can be simplified by taking the difference of the expressions for i = FeO and MgO, and defining B_j ≡ [${\mathrm{W}}_{\mathrm{FeO}-\mathrm{j}}^{\mathrm{liq}}$– ${\mathrm{W}}_{\mathrm{MgO}-\mathrm{j}}^{\mathrm{liq}}$], giving:

$$ RT1n\left( {\frac{{\gamma _{{{\text{FeO}}}} }}{{\gamma _{{{\text{MgO}}}} }}} \right)^{{{\text{liq}}}} = \sum\limits_{j} {B_{j} X_{j}^{{{\text{liq}}}} } $$

(8)

Note that because the formulation is a symmetric solution, FeO–MgO and MgO–FeO terms would be equivalent and thus would cancel when expressed in terms of the B_FeO–MgO parameter. Equation (8) can be substituted into Eq. (6) to give the following thermodynamically based equation, where the B_j terms are adjustable parameters that describe the ratio of the compositional dependences of the MgO and FeO activity coefficients in the liquid as a result of interactions with other liquid components (e.g., SiO₂, alkalis, etc.), and where ${\mathrm{W}}_{\mathrm{Fe}-\mathrm{Mg}}^{\mathrm{ol}}$ is an adjustable parameter for non-ideality in the olivine. Following the discussion in Jayasuriya et al. (2004), we explored using either B_jX_j/RT or B_jX_j terms in Eq. (6). Preliminary fits to the RKD data and the literature data (discussed in the next section) had smaller average absolute deviations when we used B_jX_j terms. Thus, Eq. (6) becomes:

$$ 1nK_{{{\text{D,Fe}}^{{{\text{2 + }}}} {\text{ - Mg}}}}^{{{\text{ol/liq}}}} = \frac{{\Delta G^{0} }}{{{\text{RT}}}} + \sum\limits_{j} {B_{j} X_{j}^{{{\text{liq}}}} } + \frac{{W_{{{\text{Fe - Mg}}}}^{{{\text{ol}}}} }}{{{\text{RT}}}}\left( {1 - 2X_{{{\text{Fo}}}}^{{{\text{ol}}}} } \right) $$

(9)

The expression for ∆G° in Eq. (9) (∆G° = –6766 – 7.34 T(K), J/mol) is taken directly from Toplis (2005).

As discussed above, another consideration in fitting the compositional dependence of ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ is that routine microprobe measurements of Fe are reported as FeO*, but the exchange coefficient is based on the FeO contents of the liquids—which are generally unknown but can be calculated. Although the low ƒO₂ of the RKD experiments minimized Fe³⁺/Fe²⁺, Fe³⁺ will still be present even in silicate melts that reach metal saturation (e.g., Bowen and Schairer 1935). Using nine of the available expressions for Fe³⁺/Fe²⁺ in silicate liquids (Sack et al. 1980; Kilinc et al. 1983; Borisov and Shapkin 1989; Kress and Carmichael 1991; Nikolaev et al. 1996; Jayasuriya et al. 2004; Putirka 2016; Borisov et al. 2018; O’Neill 2018, with the correction from Berry and O’Neill 2022), the maximum Fe³⁺/Fe* (i.e., Fe³⁺/[Fe³⁺ + Fe²⁺], molar) values calculated in the IW + 0.5 experimental liquids range from 0.02 (O’Neill 2018) to 0.06 (Nikolaev et al. 1996); the median Fe³⁺/Fe* value for all the IW + 0.5 experiments using all of the models is 0.03. For the IW + 0.5 experiments, using the minimum estimate of Fe³⁺/Fe* = 0.02 in the liquid results in a correction, [${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ – K_D*], of ~ 0.006 absolute, which is comparable to our 2σ measurement error; the correction for a liquid with Fe³⁺/Fe* = 0.03 is ~ 0.010 absolute. For those experiments run at IW–0.5 the maximum estimated Fe³⁺/Fe* ranges from 0.01 (O’Neill 2018) to 0.04 (Nikolaev et al. 1996) with a median of 0.02, consistent with estimates of 1.5–2.6% Fe³⁺ in Fe-saturated systems (Bowen and Schairer 1935) and implies that corrections for these more reducing experiments would be within the analytical uncertainty. Importantly, for any of the individual parameterizations, the variation in calculated Fe³⁺/Fe* for experiments with different bulk (and thus, melt) compositions equilibrated at the same T and ƒO₂ is less than 0.01 absolute; a difference of 0.01 in Fe³⁺/Fe²⁺ translates to a [${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ – K_D*] value of ~ 0.003, which is ⪅ the 1σ analytical uncertainty on K_D*. Thus, based on the nine Fe³⁺/Fe²⁺ models that we have considered, we conclude that any variations in K_D* in our experiments run at the same T and ƒO₂ that are larger than the analytical uncertainties, as well as the overall range in K_D* of ~ 0.05 absolute observed in the RKD experiments (Fig. 4), must reflect the effects of liquid composition on ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ and not the effects of liquid composition and T on the Fe³⁺/Fe²⁺ ratio.

Equation (9) was fit to the 14 RKD equilibration experiments listed in Table 2. Although, as described in the previous paragraph, at ƒO₂ ≤ IW + 0.5 liquid Fe³⁺/Fe²⁺ contents are low, we have nevertheless chosen to make the small correction to K_D* and fit ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$. Based on the analysis of Blundy et al. (2020) and Cottrell et al. (2022), we chose the Borisov et al. (2018) model to calculate liquid Fe³⁺/Fe²⁺. A weighted linear least-squares routine was used to calculate the best-fit values of the unknown coefficients in Eq. (9). Errors on FeO^liq were taken from those on FeO*^liq and scaled with the calculated ratio (FeO/FeO*)^liq from Borisov et al., i.e., uncertainties associated with the Borisov et al. (2018) coefficients were not considered.

Different combinations of ${\mathrm{X}}_{\mathrm{j}}^{\mathrm{liq}}$ in the $\sum_{\mathrm{j}}{\mathrm{B}}_{\mathrm{j}} {\mathrm{X}}_{\mathrm{j}}^{\mathrm{liq}}$ summation (Eq. 9) were tested to find the minimum number of liquid compositional terms required to explain the variability in our experimental ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values, subject to the constraint that all coefficients satisfy a t test at the 95% confidence level. This exercise showed that, in addition to the ${\mathrm{W}}_{\mathrm{Fe}-\mathrm{Mg}}^{\mathrm{ol}}$ term, only one liquid compositional variable, ${\mathrm{X}}_{\mathrm{Si}}^{\mathrm{liq}}$, was required to fit the RKD experimental data. Si is a critical component of melt structure (e.g., Hess 1971; Mysen et al. 1982; Mysen 1990; Mills et al. 2012) and it has been an important component in several previous parameterizations of ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ (e.g., Longhi et al. 1978; Gee and Sack 1988; Toplis 2005; Putirka 2016). This exercise confirms the conclusion of Toplis (2005) that SiO₂ is correlated with $(\frac{\mathrm{\gamma_{FeO}}}{\mathrm{\gamma_{MgO}}}$)^liq (see their Fig. 5). Note that this fitting exercise does not preclude that other interaction terms (and cross-terms) in the liquid are non-zero (which they surely must be), only that including them does not improve the quality of the fit to our experimental ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values, which are based on liquids that have a small range of alkali and TiO₂ contents. Below, we return to the effects of other liquid components on ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$.

Fig. 5

Predicted ${\mathrm{K}}_{\mathrm{D},\mathrm{Fe}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ using Model 1 (function of T, ${\mathrm{X}}_{\mathrm{Si}}^{\mathrm{liq}}$, and olivine composition) vs. the experimentally measured values for A the RKD experiments (n = 14), symbols are the same as in Fig. 1, and B for a subset of the low-ƒO₂ literature (Lit.) experiments with liquids that contained ≤ 1.45 wt% Na₂O + K₂O and ≤ 2.14 wt% TiO₂ (n = 82) plus the RKD experiments (n = 14). Alkali and TiO₂ bounds are based on the maximum Na₂O + K₂O and TiO₂ contents in the RKD glasses. Gray circles – chondrites; gray triangles (right facing) – eucrites; blue diamonds – lunar; white diamonds – lunar high-Ti; red triangles (down-facing) – Martian; maroon stars – terrestrial; green triangles (left-facing) – ureilites; yellow triangles (down-facing) – RKD experiments shown in panel A. Fe³⁺/Fe²⁺ in the glasses was calculated using Borisov et al. (2018) at the T and ƒO₂ of each experiment; a.a.d. is the average absolute deviation in measured ${\mathrm{K}}_{\mathrm{D},\mathrm{Fe}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ – predicted ${\mathrm{K}}_{\mathrm{D},\mathrm{Fe}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$; in panel B, a.a.d. reflects only the literature data. The black diagonal line in each panel is a 1:1 line. Error bars are 1σ. Best-fit parameters for Model 1 are listed in Table 4

Figure 5A compares the measured and calculated ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values for the RKD experiments given the best-fit parameters (B_Si = 1.0259 ± 0.0213 and ${\mathrm{W}}_{\mathrm{Fe}-\mathrm{Mg}}^{\mathrm{ol}}$ = 3291 ± 182, where the uncertainties are based on applying weights to the linear regression using the 1σ errors on ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ calculated for each experiment). This result, referred to as Model 1, yields an average absolute deviation (a.a.d.) between the measured and predicted ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values of 0.0026, which is the same as the typical 1σ analytical precision in ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ (± 0.003). The best-fit value for ${\mathrm{W}}_{\mathrm{Fe}-\mathrm{Mg}}^{\mathrm{ol}}$ (3291 ± 182) overlaps previous literature estimates (3700 ± 800 J/mol, Wiser and Wood 1991; 2600 ± 500 J/mol, O’Neill et al. 2003; 3000 J/mol, Toplis 2005; and 3060 ± 980 J/mol, Tuff and O’Neill 2010).

It is worth reemphasizing that given the relatively low concentrations and restricted range in Na₂O + K₂O and TiO₂ in the glasses from our equilibration experiments (0.07–1.45 wt% and 0.60–2.14 wt%, respectively), these oxides are not needed to describe the compositional dependence of ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ in the RKD experiments. Including B_j terms for the alkalis and/or Ti does improve the fit, as would be expected by the addition of one or two additional parameters, but the added variables fail a t test for significance and thus the improved fit is indistinguishable at the 95% confidence level from the fit, where these B_j values are set to zero. Nevertheless, Model 1 (with B_Ti and B_Na+K = 0) overestimates the ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values for the two HAB + Ol + Mn equilibration experiments by 0.001 to 0.005 absolute (blue symbols in Fig. 5A). As discussed above, these two experiments have alkali contents in the liquid that are higher than the other RKD runs and thus the model misfit, although small, is consistent with the expected effect of alkalis on ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ (e.g., Toplis 2005). To model the effect of alkalis and Ti on ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$, in the following discussion we fit Eq. (9) to a data set that includes both our experiments and a compilation of low-pressure olivine-bearing experimental liquids run at ƒO₂ values ≤ IW + 0.5 that span a wide range of Na₂O + K₂O and TiO₂ contents.

Literature data: olivine–liquid experiments at ƒO₂ ≤ IW + 0.5

Previous experimental studies at low ƒO₂ have generally focused on lunar, martian, and asteroidal bulk compositions, because most igneous processes on those bodies occur under more reducing conditions compared to those on Earth (e.g., Wadhwa 2008; Putirka 2016). A literature search for nominally anhydrous experiments run at or near atmospheric pressure and at ƒO₂ ≤ IW + 0.5 returned 42 studies published between 1975 and 2021 containing 305 individual experiments with quantitative olivine and liquid data. Of these 305 experiments, 276 provided complete compositional information on all silicate and oxide phases, a prerequisite for mass balance, which is one of our tests for evaluating the consistency of the experiments and phase analyses. These 276 experiments can be categorized by provenance and bulk composition, i.e., high- and low-Ti lunar basalts, martian basalts, eucrites, etc.; the complete list and associated references are reported in Supplementary Section S4. The literature data set covers a much broader region of composition space than that sampled by the RKD experiments and includes liquid compositions with up to 18.4 wt% TiO₂, 8.4 wt% Na₂O, 1.94 wt% K₂O, and experimental olivines with Fo contents between 32.6 to 96.2 mol percent. Liquid compositions range from quartz normative (78% of the experiments) to nepheline normative (6%); 94% of the liquids have normative hypersthene and 22% contain normative olivine (CIPW norms calculated with all Fe as FeO). The differentiation index (D.I.; Thornton and Tuttle 1960) of these glasses range from 15 to 88 with a median value of 38, i.e., roughly basaltic in composition.

For each literature experiment, all phase compositions (and uncertainties where provided) were compiled into a database along with the experimental run conditions. The ƒO₂ of experiments run in pure Fe capsules was calculated using the activity model of Snyder and Carmichael (1992) and using a ∆G° for the reaction Fe + ½ O₂ = FeO calculated from the thermodynamic data of Robie et al. (1979), assuming that the activity of Fe in the metal phase was unity. Otherwise, the ƒO₂ values entered into the database are those reported in each study: oxygen sensor measurements; calculated from the buffer along which the experiment was run (e.g., IW); or, for graphite capsules, calculated from C–O equilibria (French and Eugster 1965). These ƒO₂ values were used to calculate liquid FeO and Fe₂O₃ values for each of the experimental glasses (using Borisov et al. 2018) and the FeO^liq concentrations were then used to calculate the ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values.

Each of the 276 literature experiments was subjected to the same quality control tests as were applied to the RKD experiments, albeit with slightly more lenient bounds for glass and olivine oxide sums, olivine stoichiometry, and mass balance (details are given at the end of the Supplementary Section S3). Of the 276 experiments that had sufficient compositional information to run the mass balance calculation, 200 (72%) had suitable fits at the 95% confidence level. Experiments that failed mass balance were flagged as potentially problematic and were not included in the fitting exercises. For those experiments that did not crystallize an Fe metal phase and passed mass balance, the mean calculated Fe-loss was  – 4.0% relative (the median value was –0.7% relative; 14 out of the 200 experiments had Fe-loss greater than  – 10%, with a maximum loss of  – 39.1%; and only two experiments gained more than + 5% Fe (maximum gain was + 12.8% relative)). Although such large amounts of Fe loss or gain could lead to olivine–liquid Fe²⁺–Mg disequilibrium, an Fe-loss rejection criterion was not applied to the literature data. Depending on the run temperature and duration and whether the loss occurred early or continuously during an experiment, it is possible that the olivine rim compositions re-equilibrated with the melt, thus making it difficult to choose a cutoff that is not arbitrary. Application of the data quality filters (see Supplementary Section S3) resulted in a preferred data set of 187 experimental olivine–liquid pairs that cover a temperature range of 1050 to 1503 °C and a compositional range that is comparable to that of the full set of 276 experiments.

Before using the preferred literature data set to determine which other liquid compositional terms in the summation in Eq. (9) are significant, we applied Model 1 (which was fit only to the RKD experiments) to the 82 literature experiments with TiO₂ and Na₂O + K₂O concentrations less than or equal to the maximum concentrations present in the RKD glasses used to calibrate Model 1 (≤ 2.14 wt% TiO₂ and ≤ 1.45 wt% Na₂O + K₂O). As discussed above, a large body of experimental data has shown that ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values correlate with liquid alkalis and TiO₂ contents (e.g., Longhi et al. 1978; Grove and Beaty 1980; Delano 1980; Gee and Sack 1988; Jones 1988; Shi 1993; Xirouchakis et al. 2001). Predicted (Model 1) vs. experimental ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values (Fe²⁺ calculated using Borisov et al. 2018) are shown in Fig. 5B. The average absolute deviation (a.a.d.) between the measured and predicted ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values for these low-Ti and low-alkali literature experiments is 0.0106, which is substantially higher than the 0.0026 from the RKD experiments (shown in Fig. 5B as yellow triangles). It is important to emphasize that the analytical precision of the literature data is typically worse than that associated with the RKD experiments—only 49 out of the 82 low-alkali and low-TiO₂ literature experiments report FeO* and MgO uncertainties in both phases, and for these 49 experiments the mean 1σ for ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ is 0.0105 (compared to 0.0026 for the RKD experiments). Therefore, for both the high-precision RKD data and those literature data with reported compositional errors and similar ranges in TiO₂ and Na₂O + K₂O, Model 1 recovers ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values to approximately the reported analytical precision. More specifically, for 42 out of these 49 experiments, the model ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values overlap their experimental values at 2σ.

Modeling the compositional dependence of ${\mathbf{K}}_{\mathbf{D},{\mathbf{F}\mathbf{e}}^{2+}-\mathbf{M}\mathbf{g}}^{\mathbf{o}\mathbf{l}/\mathbf{l}\mathbf{i}\mathbf{q}}$ using the RKD and literature data sets

In this section we describe a fit that combines both the RKD experiments and the preferred low-ƒO₂ literature data set (n = 201) that, as discussed above, covers a much wider compositional range than the RKD experiments. Based on exploring different combinations of compositional terms, we found that, in addition to terms for Ti and Na + K, an Si-(Na + K) cross-term¹ (i.e., X_Si multiplied by the sum [X_Na + X_K] in the liquid) was required to describe the non-ideality of FeO and MgO in liquids with high alkali and silica contents (SiO₂ ⪆ 55 wt% and Na₂O + K₂O ⪆ 4 wt%). Other liquid compositional terms were tested including a Si–Si interaction term, individual coefficients for Na and K, and cross-terms including Si-Ti and Ti-(Na + K), but only Si-(Na + K) led to a significant improvement in the fits to the measured exchange coefficients (p < 0.05 based on comparing the t-statistic for the model fits). The need for an Si-(Na + K) cross-term to fit the high-alkali and high-silica olivine–liquid exchange data is not surprising given the significant effect that alkalis have on the activity of SiO₂ in silicate melts (e.g., Kushiro 1975; Watson 1982; Ryerson 1985; Grove and Juster 1989; Hirschmann et al. 1998) and that both alkalis and silica have on the activity of FeO in melts (e.g., O’Neill 2022). When fitting ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ to the literature data set, it was found that the B_Na+K term could be omitted while retaining B_Si–(Na+K) without reducing the overall quality of the fit. Thus, the following set of liquid compositional terms were included in Eq. (9): B_Si, B_Ti, and B_Si–(Na+K).

Given that the literature data set is substantially larger than the RKD data set, we used an iterative approach for fitting the three liquid compositional terms and one olivine compositional term that provides sufficient weight to the much smaller set of RKD experiments:

• (1) Best-fit coefficients B_Si and ${\mathrm{W}}_{\mathrm{Fe}-\mathrm{Mg}}^{\mathrm{ol}}$ were first determined by fitting the high-precision RKD experimental data to Eq. (9) with B_Ti, and B_Si–(Na+K) set to zero.

• (2) With the B_Si and ${\mathrm{W}}_{\mathrm{Fe}-\mathrm{Mg}}^{\mathrm{ol}}$ values held constant, a second least-squares fit to Eq. (9) was done using the only literature data (n = 187) to constrain the values of B_Ti and B_Si–(Na+K).

• (3) Now fixing B_Ti and B_Si–(Na+K) to the values determined in step (2), the RKD data alone were used to refit the B_Si and ${\mathrm{W}}_{\mathrm{Fe}-\mathrm{Mg}}^{\mathrm{ol}}$ coefficients.

• (4) Finally the literature data were refit for B_Ti and B_Si–(Na+K), using the values for B_Si and ${\mathrm{W}}_{\mathrm{Fe}-\mathrm{Mg}}^{\mathrm{ol}}$ from step (3). Although this procedure could be repeated if necessary, the coefficients were found to converge after steps 1 through 4.

As was the case with the Model 1 fit, the iterative least-squares fits were weighted by the propagated measurement errors in FeO and MgO in the olivines and glasses (each glass FeO error is based on the FeO* error scaled by the calculated (FeO/FeO*)^liq ratio). Where this information was not available, the 1σ FeO and MgO values in each phase were calculated using the empirical FeO* and MgO fractional error equations described in the literature mass balance routine (Supplementary Section S3), with the 1σ FeO*^liq values scaled by (FeO/FeO*)^liq.

The best-fit values for the B_j coefficients, where j = Si, Ti, and Si–(Na + K), and the coefficient for the olivine term, ${\mathrm{W}}_{\mathrm{Fe}-\mathrm{Mg}}^{\mathrm{ol}}$, using the iterative procedure described above are referred to as Model 2 and are given in Eq. (10) and in Table 4:

$$ nK_{{{\text{D}},{\text{Fe}}^{{{\text{2 + }}}} {\text{ - M}}g}}^{{{\text{ol/liq}}}} = \frac{{\Delta G^{^\circ } }}{{RT}} + \sum\limits_{j} {B_{j} } X_{j}^{{{\text{liq}}}} + \frac{{W_{{{\text{Fe - Mg}}}}^{{ol}} }}{{RT}}\left( {1 - 2X_{{Fo}}^{{ol}} } \right){\text{ }} = [ - 6766 - 7.34T]/RT + 1.0445 \pm 0.0202X_{{Si}}^{{{\text{liq}}}} - 1.3125 \pm 0.1076X_{{Ti}}^{{{\text{liq}}}} - 3.0550 \pm 0.1851X_{{Si}}^{{{\text{liq}}}} (X_{{Na}}^{{{\text{liq}}}} + X_{K}^{{{\text{liq}}}} ) + 3040 \pm 158(1 - 2X_{{{\text{Fo}}}}^{{{\text{ol}}}} )/{\text{RT}} $$

(10)

Table 4 Values of fit parameters for ${\mathrm{ln K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$, $\mathrm{ln }{\mathrm{D}}_{\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$, and $\mathrm{ln}{\mathrm{ D}}_{\mathrm{Fe}^{2+}}^{\mathrm{ol}/\mathrm{liq}}$

where T is in Kelvin, ${\mathrm{X}}_{\mathrm{j}}^{\mathrm{liq}}$ is the single-cation mole fraction of element j in the liquid, ${\mathrm{X}}_{\mathrm{Fo}}^{\mathrm{ol}}$ is the mole fraction of forsterite in olivine, R is the gas constant in J/(mol K), and the values after the ± symbols are 1σ errors. Note that calculating all four parameters using only the literature data results in coefficients that are within error of those shown in Eq. (10), e.g., ${\mathrm{W}}_{\mathrm{Fe}-\mathrm{Mg}}^{\mathrm{ol}}$ calculated using only the literature data is 3044 ± 297 compared to 3040 ± 158 for Model 2. Both of these values for ${\mathrm{W}}_{\mathrm{Fe}-\mathrm{Mg}}^{\mathrm{ol}}$ overlap at 1σ with the value calculated using just the RKD data (3291 ± 182, Model 1) and all three are within the range of previously reported values (2600–3700 J/mol).

Applying Model 2 to the RKD data set results in an a.a.d. of 0.0018 (Fig. 6A), which is slightly lower than the a.a.d. of 0.0026 calculated using Model 1 (Fig. 5A). Model 2 improves the fit to the HAB + Ol + Mn experiments which have liquid Na₂O + K₂O ≈ 1.4 wt% (compare blue symbols in Figs. 5A and 6A), consistent with the previously observed effect of lower ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ for liquids with higher alkali contents (e.g., Ford et al. 1983; Gee and Sack 1988; Shi 1993; Toplis 2005; Putirka 2016). For the preferred literature data alone (n = 187), the a.a.d. is 0.0150 (Fig. 6B); adding those experiments that did not pass mass balance nor the oxide sum and/or stoichiometric constraints (n = 89) to the preferred data set (n now equal to 276), the a.a.d. is 0.0159. The similarity between the model residuals for the preferred and complete literature data set suggests that the errors in the 89 experiments that did not pass our mass balance and quality filters are approximately normally distributed in composition space. Finally, for the combined low-ƒO₂ data set (n = 201), the average ∆${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ value (measured – calculated) is  – 0.0016 indicating that the distribution is not skewed towards positive nor negative values.

Fig. 6

Predicted ${\mathrm{K}}_{\mathrm{D},\mathrm{Fe}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ values using Model 2 (function of T, ${\mathrm{X}}_{\mathrm{Si}}^{\mathrm{liq}}$, ${\mathrm{X}}_{\mathrm{Ti}}^{\mathrm{liq}}$, ${\mathrm{X}}_{\mathrm{Si}}^{\mathrm{liq}}({\mathrm{X}}_{\mathrm{Na}}^{\mathrm{liq}}+{\mathrm{X}}_{\mathrm{K}}^{\mathrm{liq}}$), and olivine composition) vs. experimentally measured values for A the RKD experiments (n = 14); symbols as in Fig. 1, and B for the combined low-ƒO₂ data set (n = 187 literature (Lit.) experiments plus the n = 14 RKD experiments); symbols are the same as in Fig. 5B, and are given at the bottom of the figure. For all the experimental glasses, Fe³⁺/Fe²⁺ was calculated using Borisov et al. (2018) at the T and ƒO₂ of each experiment; a.a.d. is the average absolute deviation in measured ${\mathrm{K}}_{\mathrm{D},\mathrm{Fe}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ – predicted ${\mathrm{K}}_{\mathrm{D},\mathrm{Fe}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$; in panel B, a.a.d. reflects only the literature data. The black diagonal line in each panel is a 1:1 line. Error bars are 1σ. Best-fit parameters for Model 2 are listed in Table 4

There is no statistically significant correlation between Fe-loss and the Model 2 residuals, suggesting that although a melt may initially lose Fe to the container, the olivine and liquid can then re-equilibrate if the run duration is sufficiently long and/or the temperature is sufficiently high; i.e., under such circumstances K_D* could closely approach the equilibrium value despite the experiment not being a perfectly closed system. A failure to closely approach equilibrium, while Fe is being actively lost from the experiment would lead to the interior of the olivine grain having a high Fe/Mg ratio relative to the equilibrium value given the Fe²⁺/Mg ratio in the melt; this in turn would lead to a K_D* value higher than the equilibrium value. Crystallization of a metal phase could also produce an elevated K_D*, even though the bulk FeO* content of the experimental charge is constant. The opposite effect would be observed if the sample were actively gaining Fe from an Fe capsule, while the olivine cores were shielded by slow diffusion from the rims in contact with the Fe-enriched melt. A lower than expected K_D* value would also result if early formed Mg-rich olivines did not completely re-equilibrate with melt whose Fe/Mg ratio was increasing due to continued crystallization—note that in this case (as with the case of metal crystallization), the experiment is closed with respect to Fe loss or gain. All of these effects are likely to be more important in lower temperature experiments, where Fe–Mg diffusion in olivine is slower than in higher temperature runs, and indeed the absolute values of the residuals between measured ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values for the combined RKD and literature data sets and those calculated using Model 2 are inversely correlated with temperature, albeit with considerable scatter. Although the Spearman rank order correlation coefficient is not strong (r_s =  – 0.213), the correlation is significant at greater than the 99% confidence level. Although other explanations are possible—for example, liquids from lower T experiments may share compositional characteristics that are not fully captured by Model 2—the inverse correlation is consistent with the inference that some of the lower temperature literature data did not closely approach bulk olivine–liquid Fe–Mg equilibrium.

Comparison to previous parameterizations of ${\mathbf{K}}_{\mathbf{D},{\mathbf{F}\mathbf{e}}^{2+}-\mathbf{M}\mathbf{g}}^{\mathbf{o}\mathbf{l}/\mathbf{l}\mathbf{i}\mathbf{q}}$

Figure 7A compares ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values from the composite database of the RKD experiments (n = 14) and low-ƒO₂ literature experiments (n = 187) with predicted values based on the output of Model 2. Also shown in Fig. 7 are comparisons of the same database of low-ƒO₂ ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values with predicted values from three other thermodynamically motivated parameterizations of the compositional dependence of ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$: Fig. 7B, Ford et al. (1983); Fig. 7C, Toplis (2005); and Fig. 7D, Blundy et al. (2020). In contrast to Model 2, each of these models was calibrated using available data from olivine–liquid experiments run over a wide range of ƒO₂ values extending to oxidizing conditions: (Ford et al. 1983: IW–1 to ~ FMQ + 1.5; Toplis 2005: ~ IW to ~ FMQ + 1.5; Blundy et al. 2020: IW + 1.2 to air²). Of the 42 studies used to calibrate Model 2, nine overlap with those used in Ford et al. (1983), three overlap with Toplis (2005), and there is no overlap with the data set used in Blundy et al. (2020) (overlapping studies are listed in Supplementary Section S4).

Fig. 7

Predicted vs. experimentally measured ${\mathrm{K}}_{\mathrm{D},\mathrm{Fe}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ values. The data set consists of the RKD experiments (n = 14) and the preferred low-ƒO₂ experiments (n = 187). A Model 2 (this study), B Ford et al. (1983), C Toplis (2005), D Blundy et al. (2020). The expression used to calculate Fe³⁺/Fe²⁺ for each model is: Borisov et al. (2018), (A); Sack et al. 1980, B; Kilinc et al. (1983) plus P₂O₅ term from Toplis et al. (1994), C. Symbols are the same as in Figs. 5B and 6A; a.a.d. refers to the average absolute deviation between the measured value and the value calculated using each model. The diagonal line in each panel is a 1:1 line. Error bars are 1σ

The Ford et al. (1983) model contains terms for 1/T and P/T, and compositional terms for Mg, Fe²⁺, Ca, Mn, Cr, Ni, Si, Al, Fe³⁺, Na + K, Ti, and P³ in the liquid and utilizes the Sack et al. (1980) model to calculate Fe³⁺/Fe²⁺ for each experimental liquid used to calibrate their model. The Toplis (2005) model is the basis of our Eq. (9) and is a function of ${\mathrm{X}}_{\mathrm{Fo}}^{\mathrm{ol}}$ and SiO₂ and Na₂O + K₂O in the liquid; Fe³⁺/Fe²⁺ was calculated using Kilinc et al. (1983), modified by the P₂O₅ term from Toplis et al. (1994). A feature of the Toplis model is that the silica and alkali-based equation used to describe variations in $(\frac{\mathrm{\gamma_{FeO}}}{\mathrm{\gamma_{MgO}}})$^liq changes from linear to non-linear at 60 mol% silica. The Blundy et al. (2020) model (their Eq. (7)) parameterizes ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ only in terms of olivine composition using 52 experiments, where Fe³⁺/Fe²⁺ was measured in the coexisting glasses and thus is independent of any Fe³⁺/Fe²⁺ algorithm. In Fig. 7, ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values were calculated after first calculating FeO^liq using the Fe³⁺/Fe²⁺ scheme associated with the Ford et al. and Toplis models, respectively.

The average of the measured minus model ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values (${\overline{\mathrm{\Delta K}}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$) and the a.a.d. values for the four models are: Model 2, ${\overline{\mathrm{\Delta K}}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ = – 0.0016, a.a.d. = 0.0141; Ford et al. (1983), ${\overline{\mathrm{\Delta K}}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}=\hspace{0.17em}$– 0.0044, a.a.d. = 0.0160; Toplis (2005), ${\overline{\mathrm{\Delta K}}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}=\hspace{0.17em}$– 0.0059, a.a.d. = 0.0200; and Blundy et al. (2020), ${\overline{\mathrm{\Delta K}}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ = – 0.0045, a.a.d. = 0.0236. Histograms of the distributions of $\Delta{\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ for each model are shown in Supplementary Figure S7. Although the mean values of these distributions are all essentially the same, the model of Blundy et al. (2020) does not permit ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values much less than ~ 0.30, yet such values are a significant fraction (43 out of 201, or 21%) of our low-ƒO₂ data set, especially in experiments that generated high-alkali and high-TiO₂ liquids. The low concentrations of Fe₂O₃ in all of the experimental liquids in our database makes it difficult to attribute these low ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values to a compositional dependence of Fe³⁺/Fe²⁺ that is not captured by the Borisov et al. (2018) model which is used in Model 2. As emphasized by Toplis (2005), the ratio of the FeO and MgO activity coefficients in the melt cannot be treated as a constant except over limited ranges in liquid composition. Thus, liquid compositional terms (such as the B_j coefficients in Eq. (9)) are necessary to describe the effects of variations in liquid composition—especially variations in silica, alkalis, and titania—on ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$. We emphasize, however, that it is unclear which, if any, of the functional forms used to describe the liquid solution behaviors is preferable; resolving this issue will likely involve applying independently constrained activity–composition relationships for FeO^liq and MgO^liq as guides to the proper functional form for the ratio of activity coefficients in Eq. (9).

Model 2 and the Ford et al. (1983) model recover the low-ƒO₂ experimental ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values with similar degrees of accuracy, but we note that Model 2 uses less than half the number of coefficients of the Ford et al. model. Model 2 and the model of Toplis (2005) differ only in how they parameterize $(\frac{\mathrm{\gamma_{FeO}}}{\mathrm{\gamma_{MgO}}})$^liq, and their overall fits to the data are similar, although the points in Fig. 7C (the Toplis model) have a larger spread in the predicted values and are systematically displaced above the 1:1 line relative the points in Fig. 7A (Model 2). The difference in the parameterization of the activity coefficient ratio does lead, however, to some potentially important differences in the performance of the models. One difference is that the Toplis (2005) model has a change at 60 mol% silica in the functional form used to model $(\frac{\mathrm{\gamma_{FeO}}}{\mathrm{\gamma_{MgO}}})$^liq and this can lead to a discontinuity in the calculated ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ value: for example, if the SiO₂ content of the liquid in run CHS 63 (Collinet and Grove 2020) is increased by ~ 0.25 wt%, the calculated ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ value increases from 0.31 to 0.35.

Both the Ford et al. and the Toplis models were calibrated using 1-atm experimental data sets that covered a much wider range of ƒO₂ than that used to determine the coefficients given in Eq. (10). Thus, for both of these earlier studies, the Fe³⁺/Fe²⁺ algorithms used to calculate FeO^liq exert more leverage on the model coefficients than is the case for this study, where the difference in FeO*^liq and calculated FeO^liq is small. Since in the comparison above, we are applying the Ford et al. and Toplis models to data that is at the low-ƒO₂ end of their calibration data sets, it is worth considering how well Eq. (10) is able to calculate olivine–liquid ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values for experiments at higher ƒO₂ (e.g., at FMQ). We used Model 2 to predict ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values for a database of 454 experimental olivine–liquid pairs run at FMQ ± 0.25 (Matzen et al. 2011); Borisov et al. (2018) was used to calculate Fe³⁺/Fe²⁺ in the experimental liquids. The average of the ∆${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values is + 0.0114, and the a.a.d. is 0.0187. Recall that the corresponding values for the combined low-ƒO₂ calibration data set are ${\overline{\mathrm{\Delta K}}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ = – 0.0016 and a.a.d. = 0.0141. Thus, applying Eq. (10) to a data set of experiments run at ƒO₂ values three to six log units more oxidizing than those used in the calibration and that have liquids with substantially higher total alkalis (14.3 vs. 8.4 wt%) leads to a small increase in the a.a.d. for the measured vs. modeled ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values and a small shift towards a positive value for the average of the distribution. For comparison, the Ford et al., Toplis, and Blundy et al. average $\Delta {\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ and a.a.d. values for the FMQ ± 0.25 data are: ${\overline{\mathrm{\Delta K}}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ =  – 0.0007, a.a.d. = 0.0168; ${\overline{\mathrm{\Delta K}}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ =  + 0.0029, a.a.d. = 0.0149; and ${\overline{\mathrm{\Delta K}}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ =  – 0.0103, a.a.d = 0.0237. We conclude that Eq. (10) can be used to model olivine–liquid Fe²⁺–Mg exchange under conditions much more oxidizing than IW. Supplementary Section S5 discusses how the different Fe³⁺/Fe²⁺ models affect experimental ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values at ≤ IW + 0.5 and FMQ ± 0.25 (also see Supplementary Figures S8 and S9). In the next section, we show how to use Eq. (10) to solve for ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ and Fo given only a liquid composition and ƒO₂ (or Fe³⁺/Fe²⁺ ratio in the liquid).

Practical application of the model for ${\mathbf{K}}_{\mathbf{D},{\mathbf{F}\mathbf{e}}^{2+}-\mathbf{M}\mathbf{g}}$

One of the practical advantages of a constant ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ (e.g., as assumed by Roeder and Emslie 1970) is that because it is independent of temperature and phase composition, one can simply and directly calculate the equilibrium olivine composition coexisting with a melt (assuming the cation fractions of Mg and Fe in olivine sum to 2/3). Given the olivine and coexisting liquid compositions, one can then calculate the partition coefficient for Mg (i.e., ${\mathrm{D}}_{\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$), and from the simple linear relationship observed between ${\mathrm{lnD}}_{\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ and 1/T (or alternatively, ln ${\mathrm{D}}_{\mathrm{Fe}2+}^{\mathrm{ol}/\mathrm{liq}}$ and 1/T), calculate T (Roeder and Emslie 1970). Application of Eq. (9) is less straightforward, because ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ is no longer a constant, but rather is a function of liquid composition (i.e., the ${\mathrm{X}}_{\mathrm{j}}^{\mathrm{liq}}$ s), olivine composition, and T. A simple solution is possible if the liquidus temperature for a given liquid can be predicted based on the melt composition alone or if the T is known independently. If so, Eq. (9) directly relates ${\mathrm{X}}_{\mathrm{Fo}}^{\mathrm{ol}}$ and ln(${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$), and it is straightforward to iterate on ${\mathrm{X}}_{\mathrm{Fo}}^{\mathrm{ol}}$ to calculate ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$. This is the approach used by Herzberg and Asimow (2015) in implementing an olivine–liquid calculation in their PRIMELT program—for each melt composition, T was calculated using Beattie (1993) and the ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ expression in Toplis (2005) was solved iteratively.

Alternatively, Eq. (9) can be rewritten so as to eliminate the composition of olivine from the equation such that the ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ only depends on the liquid composition and temperature. This is done by replacing ${\mathrm{X}}_{\mathrm{Fo}}^{\mathrm{ol}}$ in Eq. (9) with a temperature and composition dependent expression for either the Mg or Fe²⁺ partition coefficient for olivine–liquid (i.e., ${\mathrm{D}}_{\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ or ${\mathrm{D}}_{\mathrm{Fe}^{2+}}^{\mathrm{ol}/\mathrm{liq}}$) and using the stoichiometric constraint that the cation fractions of Mg and Fe in olivine (referred to here as $\upeta$) equal 2/3, i.e., ${\mathrm{X}}_{\mathrm{Fo}}^{\mathrm{ol}}$ = $\frac{{(\mathrm{D}}_{\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}} {\mathrm{X}}_{\mathrm{Mg}}^{\mathrm{liq}})}{\upeta }$. Note that while we use a value of η = 2/3, which is valid for olivine along the forsterite–fayalite join, $\upeta$ could be adjusted to lower values to account for other divalent cations such as Mn, Ca, and Ni, in the olivine (e.g., Takahashi and Irvine 1981; Beattie 1993).

Putirka et al. (2007) reviewed six olivine–liquid ${\mathrm{D}}_{\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ geothermometers and concluded that the expression in Beattie (1993) was the most accurate. The coefficients given in Beattie (1993) were determined using olivine-saturated experimental liquids run over a wide range of P, T, and ƒO₂ (liquid Fe³⁺/Fe²⁺ ratios were calculated using Kilinc et al. 1983). For internal consistency, we have refit Eq. (10) in Beattie (1993) using our low-ƒO₂ (n = 201) data set with liquid Fe³⁺/Fe²⁺ ratios calculated using Borisov et al. (2018). The equations for ${\mathrm{D}}_{\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ and ${\mathrm{D}}_{\mathrm{Fe}^{2+}}^{\mathrm{ol}/\mathrm{liq}}$ are:

$$\text{lnD}_{{\text{Mg}}}^{{\text{ol/liq}}} = 0.5({\text{A}}_{\text{Mg}}/\text{T} + (\text{P} - 1)\text{F}_{\text{Mg}}/\text{T} - \text{C}_{\text{Mg}} - 2\text{ln(1.5NM)} - 2\text{ln(3}{\text{X}}_{{{\text{Si}}}}^{{{\text{liq}}}} ) + \text{NF})$$

(11)

$$\text{lnD}_{{\text{Fe}^{2{+}}}}^{{\text{ol/liq}}} = 0.5({\text{A}}_{\text{Fe}^{2+}}/\text{T} + (\text{P} - 1)\text{F}_{\text{Fe}^{2+}}/\text{T} - \text{C}_{\text{Fe}^{2+}} - 2\text{ln(1.5NM)} - 2\text{ln(3}{\text{X}}_{{{\text{Si}}}}^{{{\text{liq}}}} ) + \text{NF})$$

(12)

where ${\mathrm{D}}_{\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ and ${\mathrm{D}}_{\mathrm{Fe}^{2+}}^{\mathrm{ol}/\mathrm{liq}}$ are defined as ($\frac{{\mathrm{X}}_{\mathrm{Mg}}^{\mathrm{ol}}}{{\mathrm{X}}_{\mathrm{Mg}}^{\mathrm{liq}}})$ and ($\frac{{\mathrm{X}}_{\mathrm{Fe}2+}^{\mathrm{ol}}}{{\mathrm{X}}_{\mathrm{Fe}2+}^{\mathrm{liq}}})$, respectively (as above, the ${\mathrm{X}}_{\mathrm{i}}^{\Phi }$ s are calculated on a single-cation, mole fraction basis); T is in degrees K; P is in bars; and the equations for NM and NF are given in the notes to Table 4. The best-fit values and ± 1σ uncertainties of the coefficients are A_Mg = 13,428 ± 284; F_Mg = 0.049; C_Mg = 6.224 ± 0.191; ${\mathrm{A}}_{\mathrm{Fe}^{2+}}=12{,}592 \pm 335$; ${\mathrm{F}}_{\mathrm{Fe}^{2+}}=0.048$; and ${\mathrm{C}}_{\mathrm{Fe}^{2+}}=7.916 \pm 0.225$. Note that the pressure terms (F_Mg and F_Fe2+) were taken directly from Beattie (1993) and were not fitted parameters. Supplementary Figure S10 compares the measured vs. predicted values for the n = 201 low-ƒO₂ experiments using Eqs. (11) and (12); both the Mg and Fe²⁺ data are well-described by the equations; however, Eq. (12) systematically overestimates the high-alkali and high-silica chondrite melting experiments (gray circles in Supplementary Figure S10), whereas Eq. (11) does a better job in predicting the partitioning behavior for these experiments. Finally, although written with ${\mathrm{D}}_{\mathrm{i}}^{\mathrm{ol}/\mathrm{liq}}$ as the dependent variable, both equations were solved for T using a non-linear fitting routine. For each equation, the sum of squared residuals (for T), average ∆T values, and a.a.d. for ∆T and ${\mathrm{D}}_{\mathrm{i}}^{\mathrm{ol}/\mathrm{liq}}$ are given in Table 4. As noted by earlier authors (e.g., Roeder and Emslie 1970), the fact that the ratio of A_Fe2+ and A_Mg in Eqs. (12) and (11) is close to unity (0.95 for the two equations above) explains the relative insensitivity of ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ to changes in T.

Substituting either Eq. (11) or (12) into Eq. (10) yields an expression for ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ with only one unknown, T (note that T also appears in the ${\mathrm{D}}_{\mathrm{i}}^{\mathrm{ol}/\mathrm{liq}}$ expressions):

$$\text{lnK}_{{{\text{D,Fe}}^{{2 + }} - {\text{Mg}}}}^{{\text{ol/liq}}} = \frac{{\Delta {\text{G}}^\circ }}{{{\text{RT}}}} + \Sigma_{{\text{j}}} {\text{B}}_{{\text{j}}} {\text{X}}_{{\text{j}}}^{{{\text{liq}}}} + \frac{{{\text{W}}_{{{\text{Fe}} - {\text{Mg}}}}^{{{\text{ol}}}} }}{{{\text{RT}}}}\left( {1 - 2\frac{{{\text{D}}_{{{\text{Mg}}}}^{{{\text{ol}}/{\text{liq}}}} {\text{ X}}_{{{\text{Mg}}}}^{{{\text{liq}}}} }}{{\upeta }}} \right)$$

(13)

$$\text{lnK}_{{{\text{D,Fe}}^{{2+}} - {\text{Mg}}}}^{{\text{ol/liq}}} = \frac{{\Delta {\text{G}}^\circ }}{{{\text{RT}}}} + \Sigma_{{\text{j}}} {\text{B}}_{{\text{j}}} {\text{X}}_{{\text{j}}}^{{{\text{liq}}}} + \frac{{{\text{W}}_{{{\text{Fe}} - {\text{Mg}}}}^{{{\text{ol}}}} }}{{{\text{RT}}}}\left( {1 - 2\left[ {1 - \frac{{{\text{D}}_{{{\text{Fe}^{2+}}}}^{{{\text{ol}}/{\text{liq}}}} {\text{ X}}_{{{\text{Fe}^{2+}}}}^{{{\text{liq}}}} }}{{\upeta }}} \right]} \right)$$

(14)

Either of these two equations can be solved iteratively along with either Eq. (11) or (12) to yield a unique value of T and ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ at the olivine liquidus, from which ${\mathrm{X}}_{\mathrm{Fo}}^{\mathrm{ol}}$ can be determined using either the calculated ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ and the liquid Mg/Fe²⁺, i.e., ${\mathrm{X}}_{\mathrm{Fo}}^{\mathrm{ol}}=1/(1+ \frac{{\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}}{{\mathrm{X}}_{\mathrm{Mg}}^{\mathrm{liq}}/ {\mathrm{X}}_{\mathrm{Fe}2+}^{\mathrm{liq}}} )$; or the expression ${\mathrm{X}}_{\mathrm{Fo}}^{\mathrm{ol}}= \frac{{(\mathrm{D}}_{\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}} {\mathrm{X}}_{\mathrm{Mg}}^{\mathrm{liq}})}{\upeta }$; or ${\mathrm{X}}_{\mathrm{Fo}}^{\mathrm{ol}}= 1- \frac{{(\mathrm{D}}_{\mathrm{Fe}2+}^{\mathrm{ol}/\mathrm{liq}} {\mathrm{X}}_{\mathrm{Fe}2+}^{\mathrm{liq}})}{\upeta }$. Supplementary Section S6 gives additional details of this calculation and a worked example using a representative MORB composition from Gale et al. (2013).

The sets of equations based on ${\mathrm{D}}_{\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ and ${\mathrm{D}}_{\mathrm{Fe}^{2+}}^{\mathrm{ol}/\mathrm{liq}}$ perform nearly identically when used to iteratively calculate the T, ${\mathrm{X}}_{\mathrm{Fo}}^{\mathrm{ol}}$, and ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ for the olivine–liquid pairs in our experimental database (histograms are plotted in Supplementary Figure S11). For the RKD data (n = 14) both expressions yield a.a.d. values for temperature (experimental – calculated) of ~ 20 °C; for the complete low-ƒO₂ data set (n = 201) the temperature a.a.d. values for the Mg expression and the Fe²⁺ expression are 18 and 23 °C, respectively. For the combined data set, average absolute deviations for calculated ${\mathrm{X}}_{\mathrm{Fo}}^{\mathrm{ol}}$ and ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values using the ${\mathrm{D}}_{\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ and ${\mathrm{D}}_{\mathrm{Fe}^{2+}}^{\mathrm{ol}/\mathrm{liq}}$ expressions are very similar: ${\mathrm{X}}_{\mathrm{Fo}}^{\mathrm{ol}}$, 0.0082 vs. 0.0085 and ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$, 0.0150 vs. 0.0149, respectively. For reference, the average of the 1σ values for the experimental ${\mathrm{X}}_{\mathrm{Fo}}^{\mathrm{ol}}$ and ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values based on the 149 experiments that report errors for Fe and Mg in both phases are 0.0104 and 0.0199, respectively. Thus, the average absolute deviations between the model values and the experimental values are on the same order as the average uncertainties associated with the experimental values. As we have stated above, at ƒO₂ values around IW, calculated FeO^liq is insensitive to the choice of Fe³⁺/Fe²⁺ algorithm. However, this is not the case with increasing ƒO₂ (see Supplementary Figure S1 and Supplementary Section S5). We expect that ${\mathrm{D}}_{\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ will be less sensitive to any systematic errors in Fe³⁺/Fe²⁺ than ${\mathrm{D}}_{\mathrm{Fe}^{2+}}^{\mathrm{ol}/\mathrm{liq}}$, and for this reason we suggest that Eq. (13) be used in preference to Eq. (14).

Note that the ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ calculation is rather insensitive to the cation sum of Mg and Fe²⁺ in the olivine. Lowering $\upeta$ from 2/3 to 0.6600, the median cation sum of (Fe + Mg)^ol in the combined low-ƒO₂ data set, only increases the calculated ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}$ values by, on average, 0.00055 (range = 0.00043–0.00063). Olivine phenocrysts tend to have higher Fe plus Mg cation sums than the olivines in our experimental data set: based on 12,015 high-precision olivine analyses from MORBs and OIBs in Sobolev et al. 2007, (Fe + Mg)^ol ranges from 0.6581 to 0.6653 (median = 0.6619) and is essentially uncorrelated with ${\mathrm{X}}_{\mathrm{Fo}}^{\mathrm{ol}}$. Thus, not explicitly including olivine Ni, Mn, and Ca contents in an Fe²⁺–Mg olivine fractionation or addition calculation will not have a substantial effect on the FeO and MgO contents of the calculated liquids.

Conclusions

• A set of 1-atm Re wire loop experiments (referred to as “RKD” for Reduced K_D,) containing olivine + liquid were run at IW–0.5 or IW + 0.5 on bulk compositions spanning a range of common terrestrial rock types related to Hawaiʻian picrite, high-alumina basalt, and MORB. The Fe²⁺–Mg olivine–liquid exchange coefficient, ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$, varies systematically with liquid and olivine composition in these experiments and the results have been used to contribute to quantifying the compositional dependence of this important parameter. A key feature of the experiments is that the Fe³⁺/Fe* ratios of the melts were low (Fe³⁺/Fe* $\lesssim$ 0.04, Borisov et al. 2018), allowing us to isolate the effects of bulk composition independent of any significant correction for the presence of Fe³⁺ in the quenched glass. This is important, because with increasing ƒO₂, the magnitude of this correction increases and varies (at constant liquid composition, T, and ƒO₂) depending on which Fe³⁺/Fe²⁺ algorithm is used.

• Fourteen low-ƒO₂ experiments run on pre-saturated Re loops were fit to a version of the model proposed by Toplis (2005), modified using a strictly regular solution model after Ghiorso et al. (1983) to treat the non-ideality of the liquid phase (Eq. (9)). The resulting expression relates ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ to T, olivine composition, and melt composition using two fitted parameters: one for the effect of melt SiO₂ on the ratio of the activity coefficients of FeO and MgO in the liquid (B_Si) and one for Fe–Mg mixing in the coexisting olivine (${\mathrm{W}}_{\mathrm{Fe}-\mathrm{Mg}}^{\mathrm{ol}}$); and a term for ∆G° that was taken from Toplis (2005). This model (Model 1) produces an average absolute deviation in ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ for the RKD experiments of 0.0026, which is on order of the analytical precision (1σ ≈ 0.003).

• We combined our RKD data with a literature data set of olivine–liquid pairs from experiments also run at ƒO₂ ≤ IW + 0.5 (n = 187) to expand the range of liquid compositions. Applying our modeling approach to this combined data set required two terms in addition to those in Model 1—B_Ti and a cross-term, B_Si-(Na+K)—to adequately fit the ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ values from the literature experiments. This model (Model 2; Eq. (10)) yields an average absolute deviation of 0.0141 in ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ between the measured and modeled values for the RKD and literature data (n = 201). A key feature of this model is that it captures the significant decreases in ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ associated with increasing TiO₂ content (up to 18.35 wt%) and total alkalis (up to 8.43 wt%).

• Model 2 gives an expression from which ${\mathrm{K}}_{\mathrm{D},{\mathrm{Fe}}^{2+}-\mathrm{Mg}}^{\mathrm{ol}/\mathrm{liq}}$ can be calculated as a function of T, liquid composition [Si, Ti, and Si-(Na + K)], and coexisting olivine composition. It is more convenient for most applications to calculate an olivine composition given only the liquid composition, T, and ƒO₂ (from which Fe³⁺/Fe²⁺ in the liquid can be calculated using Borisov et al. 2018). We recast Model 2 as a function of only T and liquid composition assuming the formulation of Beattie (1993) for the T-dependence of olivine–liquid MgO (or FeO) partition coefficients. This eliminates the composition of the coexisting olivine from Eq. (10), thereby making it straightforward to solve for the liquidus temperature for a given melt and the composition of olivine coexisting with the melt at this temperature.