Redox-controlled mechanisms of C and H isotope fractionation between silicate melt and COH fluid in the Earth’s interior

The budget, recycling, and evolution of COH volatiles are central to our understanding of many geochemical and geophysical properties of the interior of the Earth. The behavior of COH volatiles, therefore, is a very important factor in the processes that governed the formation and evolution of the solid Earth and perhaps other terrestrial planets (Ardia et al. 2013; Armstrong et al. 2015).

The speciation and activity/concentration of volatiles are particularly relevant to melting and crystallization. For carbon-bearing species, for example, reduced and oxidized C behaves quite differently (Saxena and Fei 1988; Ulmer and Luth 1991). These factors can govern melt composition (see also Kushiro 1974, 1998; Eggler 1975; Eggler and Baker 1982; Gaetani and Grove 1998; Foley et al. 2009). For example, the clearly different effects of CO₂ and CH₄ on melting and crystallization of model magma systems are evident in the example in Fig. 1. The response of the liquidus boundary between forsterite and enstatite in the NaAlSiO₄–Mg₂SiO₄–SiO₂–COH system to changing volatile species at high pressure illustrates such effects (Taylor and Green 1987). This boundary defines the activity of SiO₂ in the system and moves from SiO₂ deficient to SiO₂-enriched when carbon is reduced from CO₂ to CH₄ (Fig. 1). In fact, the effect of CH₄ is not that different from the effect of H₂O. This difference is also the underlying explanation for redox melting in the mantle (Song et al. 2009).

The stable isotope behavior is critical to monitor budgets and recycling of volatiles (Van Soest et al. 1998; Dixon et al. 2002; Kingsley et al. 2002; Javoy 2004). Volatile components in the COH system, when dissolved in magmatic liquids, can affect stable isotope fractionation between the melt and coexisting fluids and crystalline materials (Dobson et al. 1989; Mattey et al. 1990; Deines 2002). Moreover, the redox state of the volatile components such as hydrogen, carbon, and perhaps sulfur can affect their influence on stable isotope fractionation in magmatic systems (Poulson 1996; Deines 2002).

In order to employ stable isotope behavior such as those of hydrogen and carbon to deduce processes of formation and evolution of the Earth’s interior, experimental data are necessary. With such tools, we can address how, in particular, COH volatiles and their isotopes fractionate between magmatic liquids, fluids, and crystallizing materials as a function of redox conditions temperature and pressure. Data from samples analyzed after quenching to ambient conditions have been reported (Dobson et al. 1988; Mattey et al. 1990; Mattey 1991). However, extrapolation of such data to conditions during equilibration of magmatic liquids with fluids and crystals at high temperature and pressure is quite challenging because both fluids and melts commonly alter their structure, and sometimes their composition, during quenching to ambient temperature and pressure conditions (Kuroda et al. 1982; Baker and Stolper 1994; Zhang and Frantz 2000; Mysen and Yamashita 2010). That nature of those alterations can vary depending on the temperature–pressure quenching path. It is, therefore, better to determine the isotope fractionation while the samples are at the high temperature and pressure under controlled redox conditions relevant to magmatic processes in the Earth’s interior. In this report, I will present and discuss recent experimental data relevant to these questions and illustrate with a few examples how the composition and redox state of COH volatiles affect carbon and hydrogen stable isotope fractionation between melts and fluids under conditions corresponding to those of the deep crust and upper mantle.

A description and discussion of how carbon and hydrogen isotopes fractionate between fluids and melts at high temperature and pressure and with variable redox conditions is the focus of this presentation. Most of the experimental work on this subject have been carried out by using externally heated hydrothermal anvil cell method (Bassett et al. 1996) where the samples can be probed by vibrational spectroscopy while at the desired conditions. It should be noted that in these diamond cell experiments, temperature is an independent variable. Pressure is generated by the fluid in sample chambers of approximately constant volume and measured with probes such as the one-phonon shift of carbon-13 diamond embedded in the sample (Schiferl et al. 1997). It is for this reason that pressure is included as an upper horizontal axis in some of the figures in this presentation.

Vibrational spectroscopies, such as Raman and infrared not only can be used to probe structure but may also be employed to determine isotope ratios. Most of the experimental data discussed in this report were obtained by such methods. This technique can be used because the frequency ratio of vibrations of two oscillators that differ only in their mass (e.g., two isotopes) v₁/v₂, is proportional to the square root of their mass ratio, √m₁/m₂. Provided that the cross section for a given vibration is not dependent on the type of isotope, 1 and 2, the ratio of integrated areas of the vibrational spectra, A₁/A₂, equals their abundance ratio, X₁/X₂. Data in support of this conclusion exist for the oxygen isotopes, ¹⁶O and ¹⁸O, (McKay et al. 2013) and hydrogen isotopes (Dalou et al. 2015).

With those caveats in mind, isotope ratios can be obtained by both transmission infrared absorption and by Raman spectroscopy. Moreover, the samples can be probed with those methods while at the desired conditions. Raman spectroscopy has an advantage over infrared spectroscopy in that Raman spectroscopic intensities do not depend on sample density, whereas infrared absorption intensity does. Densities of coexisting phases at high temperature and pressure often are not available. Laser beams used in microRaman measurements typically are 1–2 μm across and with confocal optics penetrate less than 30 μm into the sample (depending on optical transparency). Measurements with transmission microinfrared absorption have less spatial resolution with a minimum aperture of approximately 30 × 30 μm. In addition, in infrared transmission spectroscopy, one has to ensure that only the sample of interest is in the optical path.

In addition to isotope ratios, it is also possible to ascertain the speciation and abundance ratio of the volatiles and the silicate components in silicate–COH systems from Raman and infrared spectroscopic data. Sometimes, it has also been possible to determine isotope fractionation between individual volatile species such as CO₃ and HCO₃ groups in silicate melts and coexisting, silicate-saturated melts, while these phases were at the temperature, pressure, and redox conditions of interest (Mysen 2015a, 2016, 2017). Further details of the experimental and analytical methods used for this purpose can be found in those papers and Dalou et al. (2015).

Although solubility of volatile components in silicate melts is not the focus of this report, solubility information is necessary in order to characterize the isotope behavior. Solubility of COH components can also be determined spectroscopically, using either infrared or Raman spectroscopy (Scholze 1960; Behrens et al. 2004, 2006). Traditionally, analyses have been conducted on quenched materials, but recent technical advances now allow infrared and Raman spectroscopy to be used to determine the concentration of at least some volatiles while the samples are at the desired temperatures and pressures (Behrens and Yamashita 2008; Le Losq et al. 2015).

Fractionation of stable isotopes between coexisting melts and fluids can be linked to solubility and solution mechanisms of volatiles in melts and of silicate in coexisting COH fluids. The principles that govern solubility in coexisting melts and fluids have been reviewed extensively elsewhere (Paillat et al. 1992; Newton and Manning 2008) and will not be discussed further here.

COH volatiles in melts

Determination of the solubility of H₂O and CO₂ in silicate–COH melts requires sufficiently oxidizing conditions so that carbon-bearing species more reduced than that of carbon in CO₂ are unimportant. Those conditions correspond to oxygen fugacities at or above that somewhere between that defined by the NNO (nickel-nickel oxide) and QFM (quartz-magnetite-fayalite) buffer (Mysen et al. 2011). In other words, CO₂–H₂O mixtures will be the principal species in environments such as those of subduction zone melting, for example (Wood et al. 1990; Arculus 1994).

The solubilities of neither H₂O nor CO₂ are simple linear functions of the CO₂/H₂O abundance ratio of the system (Fig. 2a; see also Behrens et al. 2009). This non-linear solubility behavior is at least in part because of non-deal mixing behavior of CO₂–H₂O fluids (Eggler et al. 1979; Aranovich and Newton 1999). In addition, silicate melt–CO₂–H₂O mixtures also are non-ideal (Papale et al. 2006) which adds to the non-linear behavior of CO₂ and H₂O solubility in Fig. 2a.

Reduced carbon species may be encountered in the deep Earth where redox conditions approach those described by the IW (iron-wüstite) buffer and below (Frost and McCammon 2008; Rohrbach et al. 2011). Even lower oxygen fugacity appears to have existed during the early core-forming stages of the Earth (O'Neill 1991; Righter and Drake 1999). Under such conditions, the dominant C-bearing species in the COH system likely is methane, CH₄. Molecular H₂ can also play a significant role and contribute more than 10% of the volatile species under such very reducing conditions (Kadik et al. 2014). Its solubility in silicate at high temperature and pressure is greater than that which would be expected if hydrogen simply existed as molecular H₂. There is some evidence to suggest that H₂ interacts chemically with the silicate melt to form OH-groups (Luth and Boettcher 1986; Luth et al. 1987).

More experimental solubility data exist for carbon in melts as a function of redox conditions (Morizet et al. 2010; Mysen et al. 2011; Ardia et al. 2013). Some of the data are summarized for various melt and magma compositions in Fig. 2b. It is evident that as carbon undergoes reduction, its solubility in silicate melt decreases. However, there is considerable variation in available solubility data so that, for example, even the solubility of CH₄ in basalt composition melt at several GPa pressure and upper mantle temperatures has been reported between about a hundred ppm C (Fig. 2b; see also Ardia et al. 2013; Armstrong et al. 2015) to several thousand ppm (Kadik et al. 2004, 2014). The reasons for this wide solubility range are not clear, but may be related to experimental difficulties controlling the speciation of COH volatiles under extremely reducing conditions.

In a situation where there is no interaction between the molecules, isotope fractionation factors can be calculated (Bigeleisen and Mayer 1947; Bottinga 1968). However, in most magmatic environments, there are indeed interactions between various silicate and COH complexes (Chacko et al. 2001). Changes in structural complexes in fluids and brines are known to affect isotope behavior (Horita 1988; O'Neil and Truesdell 1993; Horita et al. 1995; O'Neil et al. 2004). Isotope substitution also can affect materials properties (Horita et al. 2010). It should not be surprising, therefore, that significant isotope fractionation can be observed between melts or fluids with variable structural complexes. For example, hydrogen and carbon isotope ratios in melts in equilibrium with fluid of fixed composition vary as a function of changes on solution mechanisms of COH species in the melt (Fig. 4). The data in Fig. 4 were obtained by analyzing glasses along the join Na₂O–SiO₂ after quenching from equilibration with a COH fluid at 1400 °C and 2 GPa and with the hydrogen fugacity buffered with the iron-wüstite-H₂O buffer. Isotope variations are expressed relative to composition, NS5 (Na₂O · 5SiO₂). Subsequent spectroscopic examination of the fluid, which remained of constant composition in all experiments, was a mixture of CH₄, H₂, and H₂O. It can be seen quite clearly that the D/H and ¹³C/¹²C ratios respond to changes in water and methane speciation in the melt, respectively. Similar effects have been reported for D/H fractionation in melts formed in the silicate–H₂O–D₂O systems determined with proton and deuteron MAS NMR (Wang et al. 2015). Analogous experimental data were reported by Dobson et al. (1989) for D/H fractionation between fluid and rhyolite and feldspar melts and Mattey et al. (1990) and Mattey (1991) for carbon isotope fractionation between fluid, basalt, and sodamelilite melt. In light of the more recent data such as summarized in Fig. 4, those earlier data likely reflected variations in speciation of the volatiles in the synthetic and natural melts.

In studies such as those described in the previous paragraph, mass spectrometric measurements were made on glasses and fluids subsequent to quenching from high temperature and pressure. However, fluids quenched from pressures in the GPa pressure range and high temperature such as was the case with those experiments often comprise several molecular silicate components. These silicate-rich fluids are prone to precipitation of solid materials during quenching. There are also the problems that derive from exsolution of fluid from melts as these are cooled during quenching thus changing the content and possibly proportion of volatile components.

In order to circumvent quenching problems, more recent experiments have combined high-temperature/high-pressure experiments conducted in diamond anvil cells with vibrational spectroscopy as a means to obtain isotope ratios of elements whose isotopic mass differences are relatively large (e.g., D and H, ¹³ and ¹²C; see (Foustoukos and Mysen 2012; Mysen 2013; Dalou et al. 2015).

With this method, it is taken advantage of the fact that vibrational frequencies are discreet functions of oscillator mass:

where v₁ and v₂ are vibrational frequencies of a specific mode for isotopes 1 and 2. The m₁ and m₂ are the atomic masses of the ¹²C and ¹³C oscillator. In the case of an elemental phase such as graphite or diamond, m₁ and m₂ could be ¹²C and ¹³C.

We emphasize that the frequency difference between vibrations with two different isotopes does not change with abundance, but the vibrational intensity, expressed as integrated areas, A₁ and A₂, does. From vibrational spectra, the isotope ratio, X₁/X₂, of a sample then becomes

Such vibrational spectroscopic methods have been used to determine hydrogen and carbon isotope ratios in coexisting silicate melts and fluids in silicate–COH systems contained in hydrothermal diamond anvil cells at pressures and temperatures up to those of the Earth’s upper mantle. In some of these experiments, redox conditions were controlled with metal/oxide buffers (Re/ReO₂, Ti/TiO₂).

The method has the significant advantage that the analyses are conducted in situ while the samples are the conditions of interest, the method is non-destructive and with spatial resolution on the micrometer scale (see above). There is, therefore, no concern for sample changes during quenching because quenching is not involved. It can also probe quite small samples without interference from adjacent phases. Moreover, in this experimental environment, pressure/temperature conditions can be changed quickly (on the time scale of minutes) without changing the sample. A potential limitation is the sensitivity of the method to isotope abundances relevant to natural conditions. In an effort to assess whether or not this could be a problem, Dalou et al. (2015) conducted experiments with several series of fluid/melt D/H fractionation with different bulk D/H ratios in silicate–H₂O–D₂O systems, but did not find any concentration-dependent variations (Fig. 5).

Hydrogen isotopes

In silicate–COH melt and fluid systems, the principal hydrogen-bearing species are H₂O, OH, H_2, CH₄, CH₃, and HCO₃ where redox conditions and chemical composition of the fluids govern which species will dominate. The molecular species, H₂O and H₂, when present likely occupy 3-dimensional cavities in glass and melts much like that reported for noble gases, for example (Carroll and Stolper 1993; Zhang et al. 2010; Guillot and Sator 2012). Hydrogen isotopes in those molecular species likely will not show significant fractionation effects. As to bicarbonate, HCO₃, no clear isotopic effects in either infrared or Raman spectra (Mysen 2015a) which may suggest that the C–O bond is not affected detectably whether H⁺ or D⁺ forms the bicarbonate complex.

For the exchange equilibrium,

$$ \mathrm{H}\left(\mathrm{fluid}\right)+\mathrm{D}\left(\mathrm{melt}\right)=\mathrm{D}\left(\mathrm{fluid}\right)+\mathrm{H}\left(\mathrm{melt}\right), $$

(7)

under both oxidizing and reducing conditions, the dominant contribution is via OD/OH fractionation. The appearance of the vibrational modes assigned to O–D and O–H stretch vibrations is shown in the insert in Fig. 6, where the exchange equilibrium coefficient, \( {K}_{D/H}^{fluid/ melt} \), is defined as;

$$ {K}_{D/H}^{fluid/ melt}=\left[\frac{\left(D/H\right)(fluid)}{\left(D/H\right)(melt)}\right], $$

(8)

where D/H denotes abundance ratio. From experiments conducted under oxidizing conditions, the D/H ratio simply is the ratio of integrated intensity such as indicated in the insert in Fig. 6, whereas for reducing conditions, information from deuterated methane and methyl complexes also is incorporated.

The D/H evolution with temperature in fluids and melts under oxidizing and reducing conditions differ significantly (Table 1). In general, under oxidizing conditions the ∆H is greater, whether in melts or coexisting fluids. The D/H abundance ratio in fluid always is greater than that in melt. This behavior leads to a D/H exchange coefficient that not only differs at given temperature and pressure, but the rate of change with temperature is different under oxidizing and reducing conditions (Fig. 6; see also Table 1). Notably, experimentally determined D/H fractionation data for rhyolite–H₂O and feldspar composition–H₂O (Dobson et al. 1989) are quite similar to those of silicate–COH under reducing conditions (Fig. 6). Furthermore, the temperature-dependent D/H fractionation in silicate–H₂O–D₂O, using the same silicate composition and working in similar temperature and pressure ranges, result in ∆H = 6.5 ± 0.7 kJ/mol (Mysen 2013). These groups have somewhat similar structural behavior (see discussion above—Eq. (2)) so a closer similarity between silicate–COH under reducing conditions and silicate–H₂O than between silicate–COH under oxidizing conditions and silicate–H₂O may not be so surprising.

	Oxidizing	Reducing
∆H(fluid)*	9.7 ± 2.0 kJ/mol	− 4.9 ± 1.1 kJ/mol
∆H(melt)*	34 ± 4 kJ/mol	− 1.2 ± 0.2 kJ/mol
∆H(Eq. 8)**	24.8 ± 6 kJ/mol	− 4.5 ± 0.5 kJ/mol

In all cases, ideal mixing is assumed

^*Enthalpy change from temperature-dependent D/H abundance ratio, ln(X_OD/X_OH) vs. 1/T (K⁻¹), in fluid and melt

^** Enthalpy change from temperature-dependent exchange equilibrium coefficient between fluid and melt, ln[(X_OD/X_OH)^fluid/[(X_OD/X_OH)^melt] vs. 1/T (K⁻¹)

The variations of D/H ratios in fluids and melts and, therefore, in the D/H exchange coefficient are because of changing silicate and COH speciation with changing redox conditions (Kadik et al. 2004; Mysen et al. 2011; Ardia et al. 2013). Additionally, speciation change as a function of temperature and pressure takes place partly because silicate solute concentration in the fluid increases and partly because of changing C-bearing species abundance in the melt (Manning 2004; Newton and Manning 2008; Mysen et al. 2009; Mibe and Bassett 2008; Mysen et al. 2013). Network-modifying cations, which include H⁺ and D⁺ (which form bonding with nonbridging oxygen), will show preference for specific silicate species depending on the ionization potential of the metal cation as has been demonstrated with ¹H and ²H MAS NMR spectroscopy (Wang et al. 2015). These effects govern H⁺– and D⁺–oxygen bonding, with the temperature (and pressure)-dependent D/H ratio summarized in Fig. 6 as the result.

D/H behavior in COH-fluid-saturated magmatic systems

Variations in COH speciation in silicate–COH systems at high temperature and pressure affect melt polymerization. Silicate speciation and melt polymerization affect all physical and chemical properties of magmatic liquids, including D/H fractionation among species in the phases (Wang et al. 2015) which, in turn, will affect melt/mineral/fluid D/H fractionation behavior. This feature is evident in Fig. 6. Both redox conditions and abundance of COH volatiles are important.

Experimental data such as summarized in Fig. 6 (see also Table 1) can be employed to illustrate how temperature and redox conditions affect D/H fractionation factors, α_D/H^fluid/melt, and how the fractionation factors, in turn, influence D/H evolution of melts and fluids in magmatic silicate–COH systems in the upper mantle and the deep crust. In doing this, I emphasize, however, that the data to be used are from model compositions in the Na₂O–Al₂O₃–SiO₂–COH system and should, therefore, not be applied quantitatively to natural magmatic processes. The behavior does, however, illustrate principles that should be kept in mind when applying hydrogen isotope data to deduce petrogenetic processes in the Earth’s interior.

The exchange equilibrium coefficient, \( {K}_{\mathrm{D}/\mathrm{H}}^{\mathrm{fluid}/\mathrm{melt}} \), in Eq. (8) equals the D/H fraction factor, of \( {\alpha}_{\mathrm{D}/\mathrm{H}}^{\mathrm{fluid}/\mathrm{melt}} \), provided that the \( {K}_{\mathrm{D}/\mathrm{H}}^{\mathrm{fluid}/\mathrm{melt}} \) is independent of bulk composition. According to the results in Fig. 5, this seems a reasonable assumption. Then, we can write its temperature-dependence;

$$ \ln\ {K}_{\mathrm{D}/\mathrm{H}}^{\mathrm{fluid}/\mathrm{melt}}=\ln\ {\upalpha}_{\mathrm{D}/\mathrm{H}}^{\mathrm{fluid}/\mathrm{melt}}=a/T+b, $$

(9)

where T is temperature (kelvin) and a and b are the regression coefficients in a plot such in Fig. 6. The fractionation factor is a strong non-linear function of temperature and is more sensitive to temperature under oxidizing conditions than under reducing conditions (Fig. 7). That fact that these curves do indeed pass through 1 as the result of changing silicate content in the fluid and COH fluid content in the melt with increasing temperature.

As an example of how the D/H relationships may affect the D/H evolution of magmatic liquids in (COH)-bearing mantle environments, let us start with the assumption that the COH fluid retains a fixed D/H = ratio, δD, of − 100‰. This value was chosen because it is reasonable near the δD value range of the Earth’s upper mantle (Bell and Ihinger 2000). The δD values of the melt in equilibrium with such a fluid is

$$ {\updelta \mathrm{D}}^{\mathrm{melt}}=\frac{{\updelta \mathrm{D}}^{\mathrm{fluid}}}{a_{\mathrm{D}/\mathrm{H}}^{\mathrm{fluid}/\mathrm{melt}}}+1000\cdotp \frac{1-{a}_{\mathrm{D}/\mathrm{H}}^{\mathrm{fluid}/\mathrm{melt}}}{a_{\mathrm{D}/\mathrm{H}}^{\mathrm{fluid}/\mathrm{melt}}}. $$

(10)

In Eq. (10), α_D/H^fluid/melt is defined in Eq. (9).

As can be seen from the calculated results (Fig. 8), this leads to the suggestion that a magmatic liquid in equilibrium with COH fluid under oxidizing conditions can be as much as 100% heavier than a melt equilibrated with COH under reducing conditions. The δD^melt difference is not significantly dependent on temperature of equilibration.

For carbon isotopes, ¹³C and ¹²C, the relationship between vibrational frequency and oscillator mass in Eq. (5) would suggest a frequency difference near 4% if the masses simply were those of the carbon isotopes. However, this is not the case. For oxidized carbon, the frequency shift of C–O stretch vibrations, which for natural carbon (nearly pure ¹²C) occur near 1070 cm⁻¹ in the Raman spectra (Frantz 1998) is only about 1.5%. Under reducing conditions, that of C–H stretch vibrations is less than 1% (see inserts in Fig. 9). This happens because oxygen and hydrogen vibrations, respectively, contribute to the oscillator mass.

The variations with temperature of the ¹³C/¹²C ratios in the CO₃ and HCO₃ species also differ in fluids and melts (Table 2). In fluid, the ¹³C/¹²C ratio in HCO₃ species decreases with temperature (and pressure) whereas for the CO₃ groups, the opposite temperature trend is observed (Table 2). For (C–O–H)-saturated melt coexisting with fluid, on the other hand, the ¹³C/¹²C ratio of HCO₃ complexes increases and that of CO₃ complexes decreases with increasing temperature and pressure (Table 2). The absolute ∆H values for carbonate complex variations in melts is greater than for fluids, but do, of course, approach one another as the temperature (825 °C) and pressure (1303 MPa) conditions of supercritical phase stability field are reached (Mysen 2017).

The different temperature trajectories of the ¹³C/¹²C in fluids and melts probably reflect the much higher silicate and aluminosilicate content of melts compared with fluid. This difference is responsible for the presence of essentially only Q⁰ species in fluids compared with significant abundance of more polymerized aluminosilicate species in the melt (Mysen 2017). Given the interaction between the carbonate groups and oxygen in the Q-species in the melts and the fact that nonbridging oxygen in Q-species of different degree of polymerization likely are energetically diggerent (Kohn and Schofield 1994; Mysen 2007), it follows that the energetics of oxygen bonds associated with CO₃ and HCO₃ groups is also different. These difference, in turn, means that energetics of the C–O bonds in CO₃ and HCO₃ groups differ. As a result, the ¹³C/¹²C abundance evolution with temperature (and pressure) of these groups in silicate melt will differ from that in fluid. These differences in silicate speciation in fluid and melt change, however, with temperature and pressure. In particular, the silicate content of the fluid likely increases, which will cause changes in Q-speciation in the fluid (Mysen et al. 2013). As this occurs, the influence of fluid composition on its ¹³C/¹²C ratio will also become increasingly important, a conclusion which is similar to that made for temperature- and pressure-dependent changes D/H fractionation in silicate–H₂O and silicate–C–O–H systems (Wang et al. 2015; Dalou et al. 2015; Le Losq et al. 2016).

The bulk ¹³C/¹²C, ∑ ¹³C/∑¹²C, comprises the contributions from both the CO₃²⁻ and HCO₃⁻ groups, ¹³CO₃²⁻, ¹²CO₃²⁻, ¹³HCO₃²⁻, and ¹²HCO₃²⁻ together with the molecular fraction of the species, X_CO3 and X_HCO3, where X_HCO3 = 1 − X_CO3, so that

and with a similar expression for fluid. The K_D^melt and K_D^fluid differ (Fig. 9a; Table 2) because the values and temperature-dependence of ¹³C/¹²C of fluid and melt differ.

Therefore, for the carbon isotope exchange equilibrium between melt and fluid under oxidizing conditions we have the exchange equilibrium

for which,

where K_D^melt and K_D^fluid are defined in Eq. (11) for melt and fluid, respectively.

The temperature-dependence of this K_D^melt/fluid (Fig. 10b) leads to ∆H = 9.5 ± 0.8 kJ/mol. Under the assumption that the carbon isotope ratio is not dependent on total carbon concentration, the K_D^fluid/melt values equal the fractionation factors. The carbon isotope fractionation factor under reducing conditions is about three times as sensitive to temperature than under oxidizing conditions. This means the carbon isotope behavior also differs from the D/H fractionation behavior where oxidizing conditions are those under which the D/H fractionation factor is the most sensitive to temperature.

Carbon isotope behavior in COH-fluid-saturated magmatic systems

Before addressing an example of how redox conditions during melting and crystallization may affect the carbon isotope behavior, the reader is reminded that the data to be used for this illustration were obtained in simple three component systems and with only one C/O/H ratio of the fluid component. We must also remember that the melts examined in this study were (C–O–H)-saturated and the coexisting (C–O–H) fluids were silicate-saturated. The speciation behavior is not, therefore, the same as in silicate-free carbonate–H₂O system such as in the in situ studies of Facq et al. (2014) and Foustoukos and Mysen (2015).

The example below, therefore, is intended to illustrate principles, but not necessarily quantitative behavior. Calculated δ¹³C-evolutions in a melt in equilibrium with a (C–O–H)-bearing source with δ¹³C of fixed at − 7 and − 30 ‰ and in a temperature corresponding to those of magmatic processes in the deep crust and upper mantle are shown in Fig. 11. The − 7 and − 30 ‰ δ¹³C values correspond to the range in δ¹³C of mantle-derived peridotite and eclogite nodules in basalt and kimberlite (see Deines 2002, for review).

For a melt, its δ¹³C value can differ from that of the source region by more than 100‰ and this difference increases with increasing temperature (Fig. 11). This also means that exsolution of C–O–H fluid from a melt in the upper mantle and deep crust, the δ¹³C of an evolving CO₂-rich fluid can be equally enriched in carbon-13 relative to the magma. This isotope behavior is, however, quite different under reducing and oxidizing conditions regardless of the δ¹³C value of the source rock (Fig. 11). Under reducing conditions, (COH)-saturated melts are isotopically heavier than silicate-saturated fluids. Under oxidizing conditions, the δ¹³C of the magmatic liquid at lower temperature is also heavier than fluid, but the temperature effect shifts this value to become isotopically lighter at higher temperature. This different behavior under reducing and oxidizing conditions reflects the different structural roles and effects on melt and fluid structure depending on the oxidation state of carbon (Mysen 2015a) (see also Eqs. (1) and (2)). It follows that changing redox conditions during melting and crystallization of (C–O–H)-bearing deep crust and upper mantle can have profound effect on the carbon isotopic signature of the melt.

The behavior of COHN fluids, their isotopes, and their interaction with magmatic liquids are at the core of understanding formation and evolution of the Earth.

The solubility of individual COHN components in silicate melts can differ by several orders of magnitude and ranges from several hundred ppm to several wt% at upper mantle pressure and temperature. Silicate solubility in fluid can reach several molecular. Different solubility of oxidized and reduced C-bearing species in melts reflects different solution equilibria. The oxidized species are H₂O, OH⁻, CO₃²⁻, HCO₃⁻, CO₂, and N₂, whereas reduced species are H₂O, OH⁻, H₂, CH₄, CH₃, NH₃, and NH₂. The C-solubility decreases and N-solubility increases with f_O2 below the QFM oxygen buffer.

The structural changes with variable redox conditions of carbon and silicate in magmatic melts and fluids result in D/H, ¹³C/¹²C, and ¹⁵N/¹⁴N fractionation between melt, fluid, and crystalline materials that depend on redox conditions and can differ significantly from 1 even at magmatic temperatures. The D/H fractionation between aqueous fluid and magma yields ∆H between − 5 and 25 kJ/mol depending on redox conditions. The ∆H values for ¹³C/¹²C fractionation factors are near − 3.2 and 1 kJ/mol under oxidizing and reducing conditions, respectively. These differences are because energetics of O–D, O–H, O–¹³C, and O–¹²C bonding environments are governed by different solution mechanisms in melts and fluids.

It is suggested that (COH)-saturated partial melts in the upper mantle can have δD values of 100‰, or more, lighter than coexisting silicate-saturated fluid. This effect is greater under oxidizing than under reducing conditions. Analogous relationships exist for ¹³C/¹²C. For example, at magmatic temperatures in the Earth’s upper mantle, ¹³C/¹²C of melt in equilibrium with COH-bearing mantle in the − 7 to − 30‰ range increases with temperature from about 40 to > 100‰ and 80–120 ‰ under oxidizing and reducing conditions, respectively.