MafiCH: a general model for H₂O–CO₂ solubility in mafic magmas

Abstract

The solubility of CO₂ in mafic magmas is strongly dependent on magma composition, which ultimately affects magma storage conditions and eruptive behavior. Recent experimental work showed that previously published volatile solubility models for mafic magmas are not well calibrated at mid-crustal pressures (400–600 MPa). Using a simple thermodynamic model, here we construct a general CO₂ solubility model for mafic magmas by establishing the compositional dependence of two key thermodynamic parameters. The model is calibrated using experimental data from 10 magma compositions that span a range of pressures as well as silica (44–53 wt.%) and total alkali (2–9 wt.%) contents. We also survey the experimental literature for relevant H₂O solubility data to determine how to model H₂O solubility for these magmas. We combine these separate CO₂ and H₂O solubility models into a single general model for mixed-fluid (H₂O–CO₂) solubility in mafic magmas called MafiCH. We test the MafiCH model using experiments from three compositions that fall both within and beyond the calibrated range, and find that the model accurately constrains the CO₂ solubility of depolymerized magmas. Sensitivity tests identify that Na, Ca, and Al have the largest effect on CO₂ solubility while Si and Mg do not play a strong role in CO₂ solubility in mafic, depolymerized melts. Overall, saturation pressures calculated using the new model presented here are typically lower than those predicted by previous models. The model provides a new framework to interpret volcanic data from mafic magma compositions for which no experimental data is available.

Introduction

The solubility of volatiles in magmas plays a critical role in the storage conditions and eventual eruption of magma. At magma storage depths, carbon is the primary volatile element in the exsolved phase because it has a low solubility compared to H₂O and its solubility is very strongly pressure dependent (e.g., Holloway and Blank 1994). Mafic magmas dissolve carbon as carbonate (e.g., Fine and Stolper 1986; Blank and Brooker 1994), and experimental work has shown that mafic magmas with variable alkali contents exhibit a very wide range of CO₂ solubilities (e.g., Dixon 1997; Lesne et al. 2011b; Shishkina et al. 2014; Vetere et al. 2014; Allison et al. 2019). While the compositional dependence of CO₂ solubility in mafic magmas is notable, H₂O solubility does not vary significantly by composition in mafic magmas (e.g., Moore et al. 1998; Lesne et al. 2011a; Iacono-Marziano et al. 2012). Of course, most natural magmas contain a mixed (H₂O–CO₂) volatile phase, so while we focus primarily on CO₂ solubility in this work due to its significant compositional variability, we also assess H₂O solubility in order to model behavior in natural systems.

Knowledge of volatile solubility is required to constrain characteristics of the volcanic plumbing system using pre-eruptive volatile contents (e.g., Lowenstern 1995). It is not typically feasible or reasonable, however, to conduct solubility experiments on an unstudied composition solely to interpret volcanic data. Instead, well-calibrated volatile solubility models can be used to determine these properties in unstudied magma compositions. Numerous models exist (e.g., Newman and Lowenstern 2002; Papale et al. 2006; Iacono-Marziano et al. 2012; Duan 2014; Shishkina et al. 2014; Ghiorso and Gualda 2015), but it is not always clear which model is most appropriate for the volcanic system of interest (e.g., Wieser et al. 2022). The choice between similar volatile solubility models may ultimately be based on familiarity and ease of use. For example, the VolatileCalc model (Newman and Lowenstern 2002), which uses the basalt model from Dixon (1997), is currently commonly used to calculate fluid-saturated isobars and melt inclusion saturation pressures for basaltic compositions. In the two decades since that model was published, however, additional experiments have provided updated solubility relationships (e.g., experiments on alkali-rich mafic magmas from Botcharnikov et al. 2005, Behrens et al. 2009, Pichavant et al. 2009, Lesne et al. 2011a, b, Iacono-Marziano et al. 2012, Iacovino et al. 2013, Pichavant et al. 2014, Shishkina et al. 2014, Vetere et al. 2014, Iacovino et al. 2016, Allison et al. 2019, and Schanofski et al. 2019).

The applicability of volatile solubility models may be limited as a result of the pressure–temperature–compositional (P–T–X) range of experiments used to calibrate the models, which may partly explain why volatile solubility models can yield drastically different results (e.g., saturation pressures) for a single composition (refer to Wieser et al. 2022 for detailed discussion regarding volatile solubility model differences). For instance, Allison et al. (2019) presented new H₂O–CO₂ experiments in six alkali-rich mafic magmas at pressures from 400 to 600 MPa and found that previously published models did not completely describe the CO₂ results. Allison et al. (2019) instead calibrated individual thermodynamic models for CO₂ solubility that showed strong agreement with the experimental data for each of the six compositions. In Fig. 1, we show the 400 MPa fluid-saturated isobar (at 1200 °C) calculated using two of the Allison et al. (2019) individual models compared with results using six earlier volatile solubility models (colored curves; models described in Table 1). In the Etna composition (Fig. 1a), for a completely anhydrous magma (at H₂O = 0 wt%), the models predict a sizeable range of dissolved CO₂ contents from ~ 2500 to ~ 4500 ppm at 400 MPa and 1200 °C. The models show even more disagreement for the Vesuvius composition (Fig. 1b), with calculated dissolved CO₂ contents from ~ 2500 to ~ 9000 ppm at anhydrous conditions. These results illustrate that the selection of volatile solubility model can strongly impact the interpretation of a volcanic system.

Fig. 1

Comparison of previously published solubility models for mafic magmas. 400 MPa isobars calculated using different models for the (a) Etna and (b) Vesuvius compositions at 1200 °C. Isobars from Allison et al. (2019) are calculated using the individual thermodynamic models. For models that allow for fO₂ input, Fe speciation was calculated using Kilinc et al. (1983) at NNO + 1 (see supplementary material Table S2)

Table 1 Brief summary of previously published solubility models for mafic magmas that can be applied to alkali-rich compositions

In this study, we develop a method to calculate mixed-fluid (H₂O–CO₂) solubility calibrated for a wide range of mafic magma compositions at pressures up to 700 MPa. The model, called MafiCH, can be thought of as a combination of models. For CO₂ solubility, we use a thermodynamic approach adapted from Holloway and Blank (1994) and incorporate the full multicomponent composition into the model to achieve robust solubility relationships. We use experiments from Allison et al. (2019) and additional sources to calibrate the model. H₂O solubility is calculated using an empirical method adapted from Lesne et al. (2011a) based on an evaluation of H₂O solubility models using experimental data from the literature. To test the accuracy and applicability of the MafiCH solubility model, we compare model calculations with experimental data from Fanara et al. (2015) and Schanofski et al. (2019) that were not used in the model calibration. Two of the tested compositions (trachybasalt and leucitite) are largely within the calibrated range of the model, while a third more silicic magma (phonolite) was tested to explore the compositional limits of the model. We next evaluate how the choice of solubility model can impact volcanic interpretation by comparing saturation pressures calculated by different models. Finally, we assess what might control CO₂ solubility in alkali-rich mafic magmas based on insights from this new general model.

Previously published models for H₂O–CO₂ solubility in mafic magmas

Before building a new model, we first examine some of the approaches that have been used to model mixed-fluid (H₂O–CO₂) solubility in mafic compositions. We focus on six solubility models that calculate changes in solubility as a result of compositional variability over a wide range of conditions. The details of the six models examined here, including how they were accessed for this study, are summarized in Table 1.

Broadly speaking, these models vary as a result of how they are constructed and calibrated. First, solubility models are calibrated using experimental data, so the P–T–X range for which they are valid depends on the experiments incorporated into each model. Some models are constructed as a result of new experimental data, and thus have a very specific compositional and pressure range. Some other models are comprehensive, in that they constrain volatile solubility for all magmas, from basalt to rhyolite, across a temperature and pressure range that encompasses conditions relevant for most natural systems. The treatment of carbon in comprehensive models is variable, as some models account for the fact that mafic end members dissolve carbon solely as carbonate while felsic magmas store carbon as carbon dioxide. Second, models vary in their mathematical construction, which can range from full multi-component models based in thermodynamics to purely empirical models with single-parameter inputs. One additional manner in which models can vary is in how they are available to users. Options may include applications that are specific to an operating system (i.e., .app for Mac or .exe for Windows), other downloadable file formats such as spreadsheets, or web-based calculators. However, some published models may not include any sort of calculator at all, or the calculator may be missing a desired function (e.g., a method to calculate saturation pressures). Fortunately, the recent publication of open-source tool VESIcal (Iacovino et al. 2021) for Python3 has addressed many of these issues of accessibility for a number of the previously published volatile solubility models.

VolatileCalc (Newman and Lowenstern 2002) is a very commonly used tool for determining H₂O–CO₂ solubility relationships, and it employs the oldest model that we discuss in this work. VolatileCalc is available as an excel macro spreadsheet that incorporates separate solubility models for rhyolitic and basaltic magmas. The model for basalt in VolatileCalc comes from the work of Dixon (1997), in which a simple thermodynamic model (Fine and Stolper 1986; Stolper and Holloway 1988; Dixon et al. 1995) was calibrated using previously published experiments from four mafic compositions. In this approach, major element cation fractions from these compositions were combined into a single parameter called Π, where cations capable of bonding with carbonate (Ca⁺², Na⁺¹, K⁺¹, Mg⁺², Fe⁺²) contribute to higher Π values and polymerizing cations (Al⁺³ and Si⁺⁴) lower the value of Π. Π was then linearly correlated with CO₂ solubility at a single pressure (100 MPa) to generate a relationship between Π and thermodynamic equilibrium. While applying this model to natural samples, Dixon (1997) found that Π varied linearly with SiO₂ content in glasses from an alkalic ocean island suite, and so SiO₂ content was used as a proxy for Π. This simplified SiO₂-based algorithm of Dixon (1997) was the model incorporated into VolatileCalc (Newman and Lowenstern 2002). While very straightforward to use, this model is restricted to calculation of solubility for magmas with < 49 wt% SiO₂ and is not recommended for use at pressures > 500 MPa.

A more recent model by Iacono-Marziano et al. (2012) provided updated solubility relationships specific to alkali-rich mafic magmas. Iacono-Marziano et al. (2012) presented new experimental data on a trachybasalt composition from Etna volcano, and combined the new data with experiments from the literature to calibrate a volatile solubility model. They generated a multi-parameter fit to thermodynamic expressions to produce a semi-empirical model. The Iacono-Marziano et al. (2012) models for H₂O and CO₂ are largely controlled by a calculated parameter, NBO/O, which had been suggested in previous studies to strongly control CO₂ solubility (e.g., Brooker et al. 2001). NBO refers to non-bridging oxygens, or oxygens associated with network-modifying cations (i.e., oxygens bonded with K⁺¹, Na⁺¹, Ca⁺², Mg⁺², and Fe⁺²) rather than polymerizing tetrahedra (i.e., oxygens bonded with Si⁺⁴, Ti⁺⁴, and Al⁺³). The CO₂ model from Iacono-Marziano et al. (2012) also requires inputs of FeO + MgO, H₂O, and total alkali content.

Even more recently, Shishkina et al. (2014) conducted experiments across eight different mafic compositions to assess volatile solubility. Like Iacono-Marziano et al. (2012), Shishkina et al. (2014) combined their new experiments with literature data to calibrate a model. The Shishkina et al. (2014) study published purely empirical fits for H₂O and CO₂ solubility at 1200–1250 °C. The Shishkina et al. (2014) equation for CO₂ is an exponential relationship with an updated Π parameterization, termed Π*, for its compositional factor, while the equation for H₂O is a polynomial fit that depends on the Na + K cation fraction composition of the magma. The Shishkina et al. (2014) study did not provide a recommendation for how to combine the equations for mixed fluid compositions (i.e., how to calculate H₂O and CO₂ partial pressures). The details of how we perform calculations for mixed-fluids using the Shishkina et al. (2014) model in this study (e.g., isobars in Fig. 1) are described in a footnote to Table 1.

The final three volatile solubility models examined here are comprehensive thermodynamic models. The Papale model (Papale 1997; Papale and Polacci 1999; Papale et al. 2006) was one of the first comprehensive H₂O–CO₂ solubility models, while Duan (2014) and MagmaSat (Ghiorso and Gualda 2015) were developed more recently. Each of these models surveyed the experimental literature and used these data to calibrate binary interaction parameters for their thermodynamic models that describe how the volatile species interact with different compositional components in the melt. The main ways these three models vary from one another are the specific experiments that are used to calibrate each model, the equations of state used for the fluids (i.e., determination of fluid fugacity), and how different compositional elements are used in the interaction parameters. For example, both Papale et al. (2006) and MagmaSat (Ghiorso and Gualda 2015) include an influence of H₂O on CO₂ solubility, though Papale et al. (2006) shows higher CO₂ solubility at low H₂O for basalts while MagmaSat (Ghiorso and Gualda 2015) shows the opposite effect. Both Duan (2014) and MagmaSat (Ghiorso and Gualda 2015) use the same equation of state (Duan and Zhang 2006), but differ in other ways, such as the treatment of carbon speciation (CO₃ and CO₂), which is accounted for in MagmaSat (Ghiorso and Gualda 2015) but not Duan (2014).

There are at least three additional models that can be used to constrain volatile solubility in mafic magmas, though we do not examine them in detail for this study for the reasons discussed here. The Lesne et al. (2011a, b) study measured solubility in three alkali-rich compositions, but the CO₂ model is calibrated only at low pressures (< 200 MPa). The Lesne et al. (2011b) experiments were later incorporated into the Allison et al. (2019) individual thermodynamic fits for CO₂ solubility. SolEx (Witham et al. 2012) is a program that uses the Π formulation from Dixon (1997), as well as solubility relationships for other volatile elements (S, Cl). However, it does not include a function to calculate saturation pressure, which is a particularly meaningful way to compare models used in this work. Finally, Eguchi and Dasgupta (2018) is a model that constrains only the solubility of CO₂ at relatively anhydrous compositions, and so it is not relevant for many of the mixed-fluid magmatic systems we discuss here.

The MafiCH model

We construct a new general model, called MafiCH, for calculating H₂O–CO₂ solubility as a function of mafic magma composition. The strength of the new CO₂ model we present in MafiCH comes from its thermodynamic basis, use of the full multicomponent magma composition, and calibration across a wide P-X range within the realm of mafic magmas that dissolve carbon as CO₃. The CO₂ model is calibrated with thermodynamic data from 10 compositionally variable mafic magmas for which a wide pressure range of experimental data are available, including the six compositions from Allison et al. (2019). The full major element composition of these magmas was used to derive a relationship describing the thermodynamic parameters. Our primary focus is on CO₂ solubility, but the solubility of H₂O is critical for the interpretation of natural magmas that typically contain mixed H₂O–CO₂ fluids. Thus, we also explore H₂O solubility in mafic magmas based on the experimental literature and previously published models. Note that an excel spreadsheet is provided in the supplementary material to perform solubility calculations using the model. This spreadsheet does not use macros, so it should be compatible with a range of spreadsheet programs. We also include two Python3 scripts for functions that cannot be calculated in excel without the use of macros: degassing paths and a series of saturation pressures (i.e., for a large dataset). Full details for using these scripts are included in the supplementary material. We test the new MafiCH model against recent experimental data for magma compositions both within and beyond the calibrated compositional range.

CO₂ solubility in mafic magmas

The thermodynamic model for CO₂ solubility in mafic magmas

We use the simple thermodynamic model that was described by Fine and Stolper (1986) and Stolper and Holloway (1988) to constrain the concentration of dissolved carbon in equilibrium with a pure CO₂ fluid. We selected this particular model because numerous studies of CO₂ solubility in mafic magmas (e.g., Holloway and Blank 1994; Thibault and Holloway 1994; Dixon et al. 1995; Lesne et al. 2011b) including Allison et al. (2019) found that the CO₂ solubility data from experiments showed strong agreement with this thermodynamic treatment. Complete details of the model are included in Holloway and Blank (1994) and Allison et al. (2019) and briefly explained here. Note that because the vast majority of solubility experiments for a single composition are performed at a constant temperature (typically 1200 or 1250 °C for mafic magmas), we do not possess the appropriate data to explore the temperature dependence of this model and thus present the model formulation at a constant temperature.

In this model, because CO₂ is stored in mafic magma as CO₃, CO₂ solubility is defined by the equilibrium of the reaction:

$${\text{CO}}_{2} \left( {{\text{vapor}}} \right) + {\text{O}}^{2-} \left( {{\text{melt}}} \right) = {\text{CO}}_{3}^{2-} \left( {{\text{melt}}} \right).$$

(1)

The equilibrium constant (K) of this reaction at pressure P and constant temperature T is given by:

$$K\left( {P,T} \right) = K_{0} \left( {P_{0} ,T_{0} } \right) \times \exp \left[ { - \Delta V_{r}^{0,m} \times \left( {P - P_{0} } \right) \times \left( {R \times T} \right)^{ - 1} } \right].$$

(2)

Note that because this model assumes equilibrium with a pure CO₂ fluid, the pressure term in Eq. 2 refers to the partial pressure of CO₂ in the system when dealing with a mixed-fluid (H₂O–CO₂) system. In Eq. 2, K₀ (P₀, T₀) is the equilibrium constant at a reference pressure and temperature. $\Delta V_{r}^{0,m}$ is the partial molar volume change of the condensed (melt) components of the reaction in Eq. 1 and is considered to be independent of pressure and temperature. Thermodynamic parameters K₀ (P₀, T₀) and $\Delta V_{r}^{0,m}$ are determined empirically for different magma compositions.

To calculate CO₂ contents from the equilibrium constant K (P, T) determined in Eq. 2, first calculate K_f:

$$K_{f} = K\left( {P,T} \right) \times f_{{{\text{CO}}_{2} }} \left( {P,T} \right),$$

(3)

then $X_{{{\text{CO}}_{3} }}^{m} :$

$$X_{{{\text{CO}}_{3} }}^{m} = K_{f} \times \left( {1 - K_{f} } \right)^{ - 1} ,$$

(4)

and finally wt% CO₂:

$${\text{wt}}\% \;{\text{CO}}_{2} = \left( {44.01 \times X_{{{\text{CO}}_{3} }}^{m} } \right) \times \left\{ {\left( {44.01 \times X_{{{\text{CO}}_{3} }}^{m} } \right) + \left[ {\left( {1 - X_{{{\text{CO}}_{3} }}^{m} } \right) \times FW_{{{\text{one}}}} } \right]} \right\}$$

(5)

In Eq. 5, 44.01 is the formula weight of CO₂ (in g mol⁻¹) and FW_one is the formula weight of the magma on a one-oxygen basis; for alkali basalt the value of FW_one is 36.594 g mol⁻¹ (Holloway and Blank 1994). In this work we treat FW_one as a constant value since it does not vary significantly for mafic magmas; values of FW_one for the 10 compositions we use to calibrate our general model are within ± 1.4 g mol⁻¹ of the alkali basalt value.

The thermodynamic parameters $\Delta V_{r}^{0,m}$ and K₀ (P₀, T₀) can be determined for different compositions from a linear regression of experimental data (i.e., Fig. 10 in Allison et al. 2019). On a plot with $\ln \left[ {f_{{{\text{CO}}_{2} }} \left( {P,T} \right) \times \left( {X_{{{\text{CO}}_{2} }}^{m} } \right)^{ - 1} } \right]$ versus $\left[ {\left( {P - P_{0} } \right)\times\left( {R \times T} \right)^{ - 1} } \right]$, where $X_{{{\text{CO}}_{2} }}^{m}$ is the mole fraction of dissolved CO₂, the slope of a linear regression through experiment data yields $\Delta V_{r}^{0,m}$ and the y-intercept corresponds to $- \ln K_{0}$. As explained above, P refers to the partial pressure due to CO₂ for a mixed-fluid experiment. Note that the values of $\ln \left[ {f_{{{\text{CO}}_{2} }} \left( {P,T} \right) \times \left( {X_{{{\text{CO}}_{2} }}^{m} } \right)^{ - 1} } \right]$ will differ based on the units of pressure used for the fugacity term. This difference will be a constant value for every experiment data point: the natural logarithm of the conversion between pressure units (i.e., values using MPa = ln (0.1) + values using bars). However, the value of $\left[ {\left( {P - P_{0} } \right)\times\left( {R \times T} \right)^{ - 1} } \right]$ does not vary by pressure units since the value of the gas constant R will also change to account for different units. Consequently, the y-intercept and therefore K₀ will change depending on what pressure units are used, but $\Delta V_{r}^{0,m}$ depends only on the volume unit used for R.

The previous studies that have used this model published $\ln K_{0}$ values determined from pressure in bars, so we also use units of bars for calculations to maintain legacy consistency and then convert to SI units of MPa. In this work the reference conditions are 1000 bars (i.e., 100 MPa) and 1200 °C. The presence of the gas constant R necessitates that temperature be in units of Kelvin for calculations. Thus, the value of R used throughout this work is 83.144621 cm³ bar K⁻¹ mol⁻¹, and the units of $\Delta V_{r}^{0,m}$ are cm³ mol⁻¹.

Figure 2 shows the effect of the thermodynamic parameters on CO₂ solubility. More negative values of $\ln K_{0}$ correlate with lower CO₂ solubilities for the same value of $\Delta V_{r}^{0,m}$. $\Delta V_{r}^{0,m}$ changes the curvature of the solubility curve such that larger values of $\Delta V_{r}^{0,m}$ result in lower CO₂ solubilities at pressures greater than P₀ for the same value of K₀. Changes in K₀ and $\Delta V_{r}^{0,m}$ have greater impact on CO₂ solubility as CO₂ fugacity increases. For the reasons described above, values of $\ln K_{0}$ shown in Fig. 2 were calculated using pressure in bars, and the solubility curve results were then converted to MPa.

Fig. 2

Impact of the thermodynamic parameters on CO₂ solubility. Properties of the CO₂ fluid (plotted on the x-axis) required for calculation of the curves are determined using the modified Redlich–Kwong equation of state (Holloway 1977, 1981, 1987) with the Saxena and Fei (1987) high-pressure correction as detailed in the appendix to Holloway and Blank (1994). a At constant value of $\Delta V_{r}^{0,m}$ decreasing $\ln K_{0}$ values correlate with lower CO₂ solubilities. b At constant value of K₀ increasing $\Delta V_{r}^{0,m}$ correlates with lower slope and lower CO₂ solubilities. Note that values of $\ln K_{0}$ shown here were calculated using pressure in bars to maintain legacy consistency, and the solubility curve results were converted to MPa for the plot

This thermodynamic model requires information about the properties of the fluid (i.e., an equation of state) to perform calculations, even for a pure CO₂ system. We use the modified Redlich–Kwong equation of state (Holloway 1977, 1981, 1987) with the Saxena and Fei (1987) high-pressure correction as detailed in the appendix of Holloway and Blank (1994). Because the thermodynamic model is calibrated by correlating CO₂ solubility with CO₂ fugacity (see Fig. 2), any volatile solubility calculations made using the model presented in this study must use this same equation of state (modified Redlich–Kwong with high-pressure correction). This equation of state is internally consistent with this simple thermodynamic model, and it is specifically calibrated in the P–T region of interest for crustal magmas. One of the more recent equations of state for H₂O and CO₂ (Duan and Zhang 2006) was designed to be applicable over a very wide range of crustal conditions, and it yields very similar values of pure CO₂ and H₂O fugacity to our selected equation of state (see supplementary material Table S1). For CO₂, these two equations of state begin to diverge slightly only at the highest pressures considered by this study (> 600 MPa), with Duan and Zhang (2006) predicting slightly higher CO₂ fugacities.

Development of a general thermodynamic model for CO₂ solubility in mafic magmas

To produce a general model for CO₂ solubility in mafic magmas, we develop a compositional fit for parameters $\Delta V_{r}^{0,m}$ and K₀ (P₀, T₀) (as $\ln K_{0}$) in the thermodynamic model (see Eq. 2). To ensure a wide compositional range for this model, we calibrate this general model using the six compositions from Allison et al. (2019), which also incorporated experimental data from Iacono-Marziano et al. (2012) and Lesne et al. (2011b), as well as four additional compositions from the experimental literature (see Table 2 and supplementary material Table S2). We selected these 10 compositions not only for their compositional diversity, but also because the available experiments were conducted over a wide pressure range (including experiments ≥ 500 MPa) and had well-constrained fluid compositions, particularly at low $X_{{{\text{H}}_{2} {\text{O}}}}^{f}$.

Table 2 Thermodynamic parameters, $\Delta V_{r}^{0,m}$ and $\ln K_{0}$for each composition with associated cation fractions used to construct the general model for CO₂

From the literature, we incorporate experiments from a basanite (Holloway and Blank 1994), leucitite (Thibault and Holloway 1994), basalt (N72 from Shishkina et al. 2010), and phonotephrite (AH3 from Vetere et al. 2014). The basanite and leucitite experiments were both previously used by Dixon (1997) to define Π, and contain lower SiO₂ than the compositions studied by Allison et al. (2019). We include the basalt (N72) experiments from Shishkina et al. (2010) to extend the compositional range of the model to lower total alkali contents. The phonotephrite composition (AH3) from Vetere et al. (2014) contains high total alkali contents (~ 9 wt%), with different proportions of Na₂O and K₂O compared to the high-alkali compositions from Allison et al. (2019), in addition to higher SiO₂.

The compositional variability of the 10 magmas used to calibrate the general model for CO₂ is shown in Fig. 3. The magmas encompass a wide range of compositions, from 44 to 53 wt% SiO₂ and 2 to 9 wt% total alkalis (Na₂O + K₂O; Fig. 3a). The compositions not only show a wide range of silica and total alkali contents, but also different relative proportions of each alkali component. Importantly, the compositions also vary significantly in the other major elements (Fig. 3b). In Fig. 3b, the relative cation fractions of each composition are plotted compared to the Etna composition. No two compositions follow the same pattern in major element composition. For example, while the Sunset Crater composition shares similar Si⁺⁴, Ti⁺⁴, Al⁺³, and Fe⁺²* contents to the Etna composition, the two magmas vary in Mg⁺² and Ca⁺². The Stromboli and SFVF compositions follow similar trends in Ti⁺⁴, Al⁺³, and Fe⁺²*, but SFVF has higher Na⁺¹ and lower Mg⁺² and Ca⁺² content than Stromboli. This strong variability in the calibration compositions serves to generate a robust model that can predict CO₂ solubility in numerous mafic magmas.

Fig. 3

Compositions included in the general model for CO₂. a Total alkali and silica compositions of experiments. b Cation fractions of each composition as referenced to the Etna composition

The thermodynamic parameters $\Delta V_{r}^{0,m}$and K₀ (P₀, T₀) have already been determined for eight of these compositions in the original publications of those experiments. Values of $\Delta V_{r}^{0,m}$ and K₀ (P₀, T₀) were calibrated in Allison et al. (2019) for all six compositions in that study, in Holloway and Blank (1994) for the basanite, and in Thibault and Holloway (1994) for the leucitite. Here we calibrate values of $\Delta V_{r}^{0,m}$ and K₀ (P₀, T₀) for the N72 basalt (Shishkina et al. 2010) and AH3 phonotephrite (Vetere et al. 2014) using the linear regression method described in the previous sub-section for inclusion in this model.

We used the following criteria to select experiments from the N72 basalt (Shishkina et al. 2010) and AH3 phonotephrite (Vetere et al. 2014) to generate the regressions for the thermodynamic parameters $\Delta V_{r}^{0,m}$ and K₀ (P₀, T₀). First, all pure H₂O experiments from both compositions and four experiments in Shishkina et al. (2010) without successful fluid composition determinations were excluded by necessity. Second, we excluded experiments with $X_{{{\text{H}}_{2} {\text{O}}}}^{f}$ values below 0.1. These studies determined fluid composition by the mass loss method, and for $X_{{{\text{H}}_{2} {\text{O}}}}^{f}$ values below 0.1, there is only a very small mass of H₂O in the fluid (generally < 0.1 mg), so any uncertainty in this measurement can yield large errors in fluid compositions (see supplementary material). Third, we also excluded some experiments with very low values of $\left[ {\left( {P - P_{0} } \right)\times\left( {R \times T} \right)^{ - 1} } \right]$, because the parameter determination method does not perform well at very low partial pressures of CO₂ (Holloway and Blank 1994). In particular, experiments with low $\left[ {\left( {P - P_{0} } \right)\times\left( {R \times T} \right)^{ - 1} } \right]$ values and CO₂ abundances below ~ 175 ppm were targeted for exclusion. The experiments used to calibrate the thermodynamic parameters for AH3 (Vetere et al. 2014) and N72 (Shishkina et al. 2010) are listed in the supplementary material (Table S3).

To produce compositional relationships for the thermodynamic parameters ($\Delta V_{r}^{0,m}$ and $\ln K_{0}$), we use a multiple linear regression of cation fractions (Table 2) from each composition. Note that because oxygen fugacity is not always measured in experimental or natural samples, we calculate cation fractions with all iron as FeO (Fe⁺²). Also, because the reporting of phosphorus and manganese is inconsistent in the literature, these elements are not included in the normalization of cation fractions. A series of multiple linear regressions yielded the following best-fit compositional relationships:

$$\Delta V_{r}^{0,m} = \beta + \sum {\left( {\alpha_{i} \times D_{i} } \right)}$$

(6)

$$\ln K_{0} = \psi + \sum {\left( {\theta_{i} \times D_{i} } \right)}$$

(7)

where D_i indicates the cation fraction of element i and α_i, β_i, θ, and ψ are coefficients determined from the multiple linear regressions. Values for the coefficients are listed in Tables 3–4. Values of $\ln K_{0}$ using Eq. 7 require pressure in bars for calculations; if units of MPa are desired, subtract ln (0.1) from the result of Eq. 7. The values of $\Delta V_{r}^{0,m}$ and $\ln K_{0}$ produced by Eqs. 6 and 7 are within ~ 3% and ~ 0.5%, respectively, of the values determined from the experimental data for each of the 10 calibration compositions (Table 2). This analysis suggests that the general model is robust and very accurately predicts CO₂ solubility in these 10 compositions.

Table 3 Coefficients for $\Delta V_{r}^{0,m}$ equation

Table 4 Coefficients for $\ln K_{0}$ equation

Using the general thermodynamic model for CO₂ solubility in mafic magmas

The general model for CO₂ is well-calibrated for magmas that contain 44 wt% to 53 wt% SiO₂ and 2 wt% to 9 wt% total alkalis (Na₂O + K₂O) at pressures ranging from ~ 50 MPa to 700 MPa. Here we summarize the general procedure for application of the model to calculate the CO₂ solubility for any mafic magma composition. Calculators are available in the supplementary material (an excel spreadsheet and two Python3 scripts).

We first describe how to use the CO₂ model for a basic system: mafic magma in equilibrium with a pure CO₂ fluid. To begin, calculate the cation fractions for the composition of interest and use Eqs. 6–7 to compute the thermodynamic parameters $\Delta V_{r}^{0,m}$ and K₀. Next, calculate K (P, T) from Eq. 2; recall that P must be in units of bars, unless the calculated value of K₀ is converted to alternate pressure units. P₀ is 1000 bars and T is 1200 °C (convert T to Kelvin for calculations since gas constant R is present). Finally, use Eqs. 3–5 to transform K (P, T) to wt% CO₂. Note that ${f}_{{\mathrm{CO}}_{2}}$(P, T) in Eq. 3 must be in the same pressure units as P and calculated using the modified Redlich–Kwong equation of state (Holloway 1977, 1981, 1987) with the Saxena and Fei (1987) high-pressure correction as detailed in the appendix of Holloway and Blank (1994). Use of a different equation of state would require recalibration of the thermodynamic parameters. In Eq. 5, the value of FW_one should be 36.594 g mol⁻¹, since this constant value was used to determine the thermodynamic parameters for this model.

For mixed-fluid (H₂O–CO₂) compositions, the pressure and fugacity terms in Eqs. 2 and 3, respectively, must be determined based on the proportion of CO₂ in the fluid. To determine the fugacity of CO₂ in a mixed-fluid magma, calculate the pure CO₂ fugacity at the total pressure of the system and then multiply the pure fugacity by the fluid composition $\left( {1 - X_{{{\text{H}}_{2} {\text{O}}}}^{f} } \right)$. The partial pressure of CO₂ can then be determined from this value of CO₂ fugacity using the equation of state (modified Redlich–Kwong with high pressure correction).

H₂O solubility in mafic magmas

Previous studies have suggested that H₂O is relatively insensitive to melt composition (Moore et al. 1998; Lesne et al. 2011a; Iacono-Marziano et al. 2012). Partly for this reason, the Allison et al. (2019) experiments were not designed to assess H₂O solubility, and those experiments did not show separate trends in H₂O solubility according to composition. To explore the effect of composition on H₂O solubility, we compiled experimental data for 22 mafic magma compositions (anhydrous SiO₂ < 56 wt%) with a wide variety of major element compositions (Fig. 4a and supplementary material Table S4) from 14 studies (Cocheo 1994; Dixon et al. 1995; Moore et al. 1995, 1998; Ohlhorst et al. 2001; Berndt et al. 2002; Botcharnikov et al. 2005; Di Matteo et al. 2006; Shishkina et al. 2010, 2014; Lesne et al. 2011a; Vetere et al. 2011, 2014; Iacono-Marziano et al. 2012). The experimental data from these studies is included in the supplementary material (Table S5).

Fig. 4

H₂O solubility experiments compared with models for mafic magmas. a Total alkali and silica compositions of experiments. b Experiments for compositions in the grey region of (a) compared with the Lesne et al. (2011a) power-law equation for the Etna composition. c Same as (b) except the experiments shown are for the compositions beyond the grey region of (a). d, e Evaluation of the accuracy of different models for H₂O solubility, shown as differences between the experimental data and predictions from models. Results from the Lesne et al. (2011a) power-law equation for Etna are compared with the Iacono-Marziano et al. (2012) model in (d) and the Shishkina et al. (2014) model in (e)

We first examine this data compilation for possible compositional controls on H₂O solubility. We compare the H₂O contents of the experiments with H₂O fugacity, calculated using the modified Redlich–Kwong equation of state (Holloway 1977, 1981, 1987) with the Saxena and Fei (1987) high-pressure correction as detailed in the appendix of Holloway and Blank (1994). For mixed-fluid experiments, we calculate the pure H₂O fugacity at the total pressure of the experiment and then multiply it by the experiment fluid composition ($X_{{{\text{H}}_{2} {\text{O}}}}^{f}$) to determine H₂O fugacity. For clarity the data are separated into two groups: roughly half of the experiments, those with compositions falling within the grey region of Fig. 4a, are shown in Fig. 4b, while the remaining experiments are shown in an identical plot in Fig. 4c. In Fig. 4b, c, we also plot the power-law fit for the Etna composition from Lesne et al. (2011a) for comparison:

$$f_{{{\text{H}}_{2} {\text{O}}}}^{{1200\;^{ \circ } {\text{C}}}} = 104.98 \times {\text{wt}}\% \;{\text{H}}_{2} {\text{O}}^{1.83}$$

(8)

in which fugacity is in units of bars.

Overall, the H₂O solubility data (Fig. 4b, c) do not show any clear relationship to composition, and most of the experiments closely follow the Lesne et al. (2011a) power-law fit for Etna (Eq. 8). Note that the experimental data (Fig. 4b, c) is quite limited above ~ 5 wt% H₂O, which likely reflects the difficulty in quenching hydrous glasses. This data compilation lends further support to the conclusion of previous studies (e.g., Moore et al. 1998; Lesne et al. 2011a; Iacono-Marziano et al. 2012) that H₂O solubility in mafic magmas is not strongly affected by melt composition. If there is an effect, it cannot be resolved over the broad compositional range of the current experimental literature, but it merits future careful investigation.

There exist several H₂O solubility models calibrated specifically for alkali-rich mafic magmas (e.g., Iacono-Marziano et al. 2012; Shishkina et al. 2014), and we next evaluate how these models perform for this compilation of data. We plot the difference between the experimental data and the model-calculated H₂O solubility in Fig. 4d, e. The average difference between measured H₂O and calculations of H₂O content using the Lesne et al. (2011a) power-law fit for Etna for all experiments in this compilation is 0.35 wt% (median: 0.25 wt%), with a maximum difference of 1.68 wt%. Using the Iacono-Marziano et al. (2012) model, however, the average difference is 0.70 wt% (median: 0.41 wt%), with two notable outliers at 7.43 and 7.30 wt% (Fig. 4d). We note that these two outliers are experiments published in 2014 (after the Iacono-Marziano et al. 2012 model), and the maximum difference between this model and measured H₂O for all other experiments in this compilation is 3.30 wt%. For the Shishkina et al. (2014) model (Fig. 4e), the average and median differences between the experiments and this model are identical to those for the Lesne et al. (2011a) Etna fit, with a slightly higher maximum difference of 1.85 wt%. However, due to the polynomial formulation of the empirical Shishkina et al. (2014) model, it yields non-zero values for H₂O solubility at 0 MPa that depend solely on Na + K cation fraction. For the compositions in Fig. 4, the Shishkina et al. (2014) model calculates concentrations of H₂O at 0 MPa ranging from 0.97 to 1.08 wt%. Previous experimental work suggests that H₂O solubility in mafic magmas (i.e., tholeiite) near 0 MPa is much lower than 1 wt% (~ 0.1 wt% at 1 atm; Baker and Grove 1985).

It is apparent that models relating H₂O solubility in mafic magmas to compositional components generally show more disagreement with the experimental data than calculations using a single empirical fit (Fig. 4d, e). Our data compilation shows an uneven distribution of experiments by pressure (more experiments at lower pressures) as well as differences in the number of experiments available for each composition. For these reasons, we do not calibrate a new empirical equation from the data compilation and instead use the Lesne et al. (2011a) power-law fit for Etna (Eq. 8) to calculate H₂O solubility in mafic magmas. The calculators available in the supplementary material use Eq. 8 for the H₂O model (in conjunction with the thermodynamic CO₂ model) for all mixed-fluid determinations.

To use Eq. 8 for H₂O solubility in conjunction with the general thermodynamic model for CO₂ solubility to determine volatile solubilities for mixed-fluid (H₂O–CO₂) magmas, we determine partial pressures of H₂O and CO₂ using the equation of state and fluid composition of the system. To determine the fugacity of H₂O in a mixed-fluid magma, we calculate the pure H₂O fugacity at the total pressure of the system and then multiply the pure fugacity by the fluid composition $\left( {X_{{{\text{H}}_{2} {\text{O}}}}^{f} } \right)$. The partial pressure of H₂O can then be determined from this value of H₂O fugacity using the equation of state (modified Redlich–Kwong with high pressure correction). For example, a magma at 500 MPa and 1200 °C with a pure fluid composition (i.e., 100% H₂O or CO₂) would have a pure H₂O fugacity of 607 MPa $\left( {X_{{{\text{H}}_{2} {\text{O}}}}^{f} = 1} \right)$ or a pure CO₂ fugacity of 1610 MPa $\left( {X_{{{\text{H}}_{2} {\text{O}}}}^{f} = 0} \right)$. At these same P–T conditions, a mixed-fluid experiment with fluid composition of 0.2 $X_{{{\text{H}}_{2} {\text{O}}}}^{f}$ will have H₂O fugacity of 121 MPa, which corresponds to a partial pressure of 122 MPa for H₂O. The CO₂ fugacity for this fluid composition is 1288 MPa, corresponding to a partial pressure of 451 MPa for CO₂.

Testing the MafiCH model for H₂O–CO₂ solubility in mafic magmas

A robust general model should be capable of predicting solubility relationships for a variety of mafic compositions for which volatile solubility has not been experimentally determined. To test the validity of the MafiCH model for unstudied compositions, we compare the model calculations with recent experimental data at 1200–1250 °C between 50 and 500 MPa from Fanara et al. (2015) and Schanofski et al. (2019). The magma compositions are a “trachybasalt” with 49 wt% SiO₂ and 4.4 wt% Na₂O + K₂O from Fanara et al. (2015) and from Schanofski et al. (2019), a leucitite with 44.5 wt% SiO₂ and 12 wt% Na₂O + K₂O and phonolite with 56 wt% SiO₂ and 15.4 wt% Na₂O + K₂O. Note that Fanara et al. (2015) refer to this composition as “trachybasalt” because it was synthesized to reproduce a natural trachybasalt from the Campanian Ignimbrite, but the starting glass composition is alkali basalt. The “trachybasalt” magma from Fanara et al. (2015) is within the calibrated compositional range of the MafiCH model (Fig. 5). The leucitite magma from Schanofski et al. (2019) has higher K content than the calibration compositions but is otherwise within the calibrated range. The phonolite composition from Schanofski et al. (2019) is beyond the calibrated compositional range of the CO₂ model (Fig. 5) for all elements except Ti and Na. We don’t specifically intend for MafiCH to be used to constrain volatile solubility in polymerized melts such as a phonolite, but we test this composition to examine the performance of the model when extrapolated beyond its calibrated range. Results from the comparison between the MafiCH model and experiments from these three compositions are shown in Fig. 6.

Fig. 5

Compositions from Fanara et al. (2015) and Schanofski et al. (2019) compared with the compositional range of experiments used to calibrate H₂O and CO₂ models. Cation fractions are plotted relative to the Etna composition from Allison et al. (2019)

Fig. 6

Comparison between the model presented here (curves) and experiments (symbols) from Fanara et al. (2015) and Schanofski et al. (2019). Colors of symbols correspond to the total pressure of the experiment. Fugacities of the experiments were calculated using the equations of state for H₂O and CO₂ used in Allison et al. (2019). Panels in the left column show CO₂ data and panels in the right column show H₂O data. Top row shows the trachybasalt composition from Fanara et al. (2015); middle row shows the leucitite composition from Schanofski et al. (2019); bottom row shows the phonolite composition from Schanofski et al. (2019). Error bars correspond to the values given in Fanara et al. (2015) and Schanofski et al. (2019). No error bars are shown for fugacity as these studies did not provide error estimates for fluid compositions, aside from noting specific experiments of the Phonolite composition from Schanofski et al. (2019) with a large error in the determination of $X_{{{\text{H}}_{2} {\text{O}}}}^{f}$ (identified here by symbols containing an x). Phonolite experiments with $X_{{{\text{H}}_{2} {\text{O}}}}^{f}$ < 0.4 are shown with outlined symbols to indicate experiments with the most polymerized melt structures

Of note, both Fanara et al. (2015) and Schanofski et al. (2019) determine fluid compositions using the mass-loss method, though neither study provided error estimates for the $X_{{{\text{H}}_{2} {\text{O}}}}^{f}$ values. In the mass-loss method, the experiment capsule is first weighed, and then frozen in liquid nitrogen prior to and during puncture. The capsule is next allowed to warm to room temperature to release the CO₂ fluid and re-weighed. Finally, the capsule is stored in a drying oven to release the H₂O fluid and weighed again. There are a few sources of uncertainty involved in mass-loss fluid determinations that were mentioned in these studies, including balance error, incomplete fluid separation, and mixed-fluid bubbles bursting during oven heating. These sources of uncertainty are discussed in further detail in the supplementary material and Figs. S1, S2. Overall, larger uncertainties in $X_{{{\text{H}}_{2} {\text{O}}}}^{f}$ are expected for experiments that include small masses (< 1 mg) of one or both fluids. Additionally, uncertainty in $X_{{{\text{H}}_{2} {\text{O}}}}^{f}$ propagates to greater uncertainty in fugacity at higher pressure. So, while we do not show error bars for fugacities in Fig. 6, we note that based on the calculations shown in the supplementary material, uncertainty in fluid composition can produce fugacity errors of hundreds of MPa, particularly at high fugacities.

The MafiCH model shows strong agreement with the experimental data from the “trachybasalt” composition from Fanara et al. (2015) (Fig. 6a, b). The vast majority of these experiments define trends that align closely with both the CO₂ and H₂O models. The experiment with the highest CO₂ fugacity appears to notably deviate from the CO₂ model, but this discrepancy could perhaps be partly explained by errors in the fluid measurement $\left( {X_{{{\text{H}}_{2} {\text{O}}}}^{f} } \right)$. This particular experiment contained ~ 10 mg of CO₂ in the fluid and only 1 mg of H₂O.

The MafiCH model shows excellent agreement with the experimental data from the leucitite composition (Fig. 6c, d). The CO₂ model in particular is remarkably well suited to this experimental data. All but two experiments are within 500 ppm CO₂ of the model calculations, and most overlap with the model when analytical error is accounted for. The H₂O data is generally offset to slightly higher H₂O (average: 0.34 wt%) compared with the model, but many experiments, especially those with < 5 wt% H₂O, do agree with the model within error.

The phonolite experiments illustrate the performance of the CO₂ model in MafiCH beyond its calibrated range as well as its limitations. Roughly one third of the experiments for the more evolved phonolite composition show strong agreement with the CO₂ model, but the others diverge to lower CO₂ contents than predicted by the model (Fig. 6e). The experiments that follow the modeled solubilities all exhibit fluid compositions of $X_{{{\text{H}}_{2} {\text{O}}}}^{f}$ > 0.4. At drier compositions, however, the model and experiments deviate significantly (up to 1100 ppm difference), with the experiments showing lower CO₂ solubility than what is calculated by the model. Schanofski et al. (2019) discuss this difference in CO₂ solubility at dry vs. wet conditions, and attribute it to the effect of H₂O on the polymerization of the melt. At high H₂O contents and $X_{{{\text{H}}_{2} {\text{O}}}}^{f}$, H₂O breaks up tetrahedra in the melt structure, and thus more CO₂ can be incorporated into the melt structure (e.g., Mysen 1976). These results suggest that the CO₂ model works well for relatively depolymerized melts (i.e., mafic magmas and hydrous intermediate magmas with CO₂ stored as CO₃), but does not accurately constrain solubility for polymerized melts beyond its calibrated range.

For H₂O, the phonolite experiments generally follow the MafiCH model, with some variability (Fig. 6f). Note that some of the phonolite experiments were determined by Schanofski et al. (2019) to have large errors in their fluid compositions as a result of mixed-fluid bubbles bursting during oven drying, and these experiments are marked with an x in Fig. 6e, f. Excluding the experiments with large fluid composition errors, the average difference between the H₂O model and phonolite experiments is 0.34 wt%, identical to that of the leucitite experiments. In the phonolite experiments, however, the deviation from the model differs based on the amount of H₂O in the experiments. At low H₂O contents (< 3 wt%), the model predicts slightly higher H₂O than the experiments, while at higher H₂O (> 5 wt%) the model predicts slightly lower H₂O. The experiments and H₂O model show strong agreement between 3 and 6 wt%.

Overall, these tests suggest that the MafiCH model is accurate within its calibrated range, and in some cases (i.e., depolymerized melts) may apply beyond this compositional range. We recommend comparing the cation fractions of the composition of interest to the calibrated ranges shown in Fig. 5 to determine whether the MafiCH model will be useful for a specific volcanic system.

Implications

Volcanic plumbing systems: saturation pressures from previously published solubility models

Melt inclusion volatile contents are commonly used to interpret the depth of melt inclusion formation and crystal residence and therefore magma storage and structure of the volcanic plumbing system (e.g., Lowenstern 1995). Volatile solubility models are used to calculate the pressure at which the melt inclusion volatiles are saturated in the magma to provide minimum depth estimates. Different solubility models will yield different melt inclusion entrapment pressures (e.g., Fig. 1), which will affect interpretation of the volcanic system.

To compare how different solubility models affect the interpretation of volcanic data, we determine saturation pressures at 1200 °C for hypothetical melt inclusion volatile compositions. We chose conditions for which the new MafiCH model is particularly well calibrated to serve as a reference. In particular, the hypothetical melt inclusion volatile composition is relatively CO₂-rich to take advantage of the 400–600 MPa experiments from Allison et al. (2019) that form the core data set for model calibration. In the supplementary material, we show the results of this same test for two additional melt inclusion volatile compositions (Figs. S3, S4). We focus on only one volatile composition here, as the overall trends in calculated saturation pressures are largely the same for these three different melt inclusion volatile compositions.

We show saturation pressures calculated for a melt inclusion with 2.5 wt% H₂O and 4000 ppm CO₂ in Fig. 7. The saturation pressures are plotted for each of the 10 calibration compositions for the MafiCH CO₂ model according to the wt% total alkali content of the magma for ease of visualization. The major element compositions of these magmas are included in the supplementary material (Table S2). For models that require Fe speciation, we convert from total Fe assuming an oxidation state of NNO + 1 using Kilinc et al. (1983).

Fig. 7

Saturation pressures calculated from six previously published volatile solubility models compared with the new MafiCH model for a theoretical melt inclusion composition at 1200 °C with 2.5 wt% H₂O and 4000 ppm CO₂. Saturation pressures are shown in panels in the left column and the differences in saturation pressure from the MafiCH model are shown in panels in the right column. The MafiCH model (squares) is compared with two previously published models in each row: (top) VolatileCalc (Newman and Lowenstern 2002) and MagmaSat (Ghiorso and Gualda 2015); (middle) Duan (2014) and Papale et al. (2006); (bottom) Shishkina et al. (2014) and Iacono-Marziano et al. (2012). Results are plotted according to total alkali content of the magma compositions

First, we discuss the results from the oldest and newest models being tested in comparison to the new MafiCH model (Fig. 7a, b): VolatileCalc (Newman and Lowenstern 2002) and MagmaSat (Ghiorso and Gualda 2015). VolatileCalc is unable to calculate saturation pressures for three compositions (basalt N72, basaltic andesite SFVF, and phonotephrite AH3) as they have > 49 wt% SiO₂. For the SiO₂ input for VolatileCalc calculations, we use the SiO₂ value that results from normalizing to 100% anhydrous after conversion of total Fe to FeO and Fe₂O₃. There is a significant misfit between VolatileCalc and the new MafiCH model for the Vesuvius (~ 280 MPa) and Stromboli (~ 190 MPa) compositions, and lower, but still notable misfit for the leucitite (~ 115 MPa) and Etna (~ 90 MPa) compositions. The results from VolatileCalc are largely similar to the new MafiCH model for the remaining three compositions (Sunset Crater, Erebus, and the basanite; < 30 MPa difference). For the most volatile-rich melt inclusion composition (4.5 wt% H₂O and 4000 ppm CO₂; see supplementary material Fig. S4), increasing divergence between VolatileCalc and the new MafiCH model occurs, and noticeable misfits appear for all compositions. The VolatileCalc model is not intended for use at pressures above 500 MPa, and these results illustrate that it is not well calibrated at these high pressures as it calculates saturation pressures that are much higher than what are indicated by the latest experimental data. MagmaSat agrees very closely with the new MafiCH model for these three hypothetical melt inclusion compositions for the magmas with > 5.5 wt% alkalis. There is some disagreement, however, between MagmaSat and MafiCH for the compositions with lower total alkali contents, particularly the basalt (N72) and basaltic andesite (SFVF) magmas.

Next, we examine the two other comprehensive models (Fig. 7c, d): Papale et al. (2006) and Duan (2014). The Papale et al. (2006) model calculates very similar saturation pressures for magmas across this entire compositional range, with only ~ 132 MPa difference between the lowest and highest saturation pressures (Fig. 7c). The range of saturation pressures predicted by MafiCH is nearly 3× wider (371 MPa difference between the lowest and highest pressures) than the Papale et al. (2006) calculations. The Papale et al. (2006) model shows strong agreement with the new MafiCH model for the Sunset Crater alkali basalt composition and moderate agreement (~ 50–60 MPa difference) for SFVF, Etna, Stromboli, and Erebus. For the remaining compositions, the Papale et al. (2006) model shows poor agreement with the MafiCH model (> 100 MPa difference), with particular misfits for the AH3 phonotephrite (~ 285 MPa difference) and leucitite (~ 205 MPa difference). The Duan (2014) model generally calculates higher saturation pressures for each composition compared with the new MafiCH model, though it calculates lower saturation pressures for the Vesuvius composition. The Duan (2014) model agrees with the new MafiCH model for two very different compositions: the Stromboli alkali basalt and Erebus phonotephrite. The Duan (2014) model also shows moderate agreement with the new MafiCH model for the Vesuvius composition (~ 55 MPa difference), but poor agreement (> 100 MPa difference) for the remaining seven compositions. Another interesting note is that both the Duan (2014) and Papale et al. (2006) models calculate nearly identical saturation pressures for the Erebus and AH3 compositions, which have similarly high total alkali contents but significant compositional differences in other elements. Experimental data and the new MafiCH model show drastically different CO₂ solubilities for the Erebus and AH3 magmas.

Finally, we compare pressures calculated using the two models that are specific to mafic magmas (Fig. 7e, f): Iacono-Marziano et al. (2012) and Shishkina et al. (2014). The Shishkina et al. (2014) model consistently calculates higher pressures than the new MafiCH model, while the Iacono-Marziano et al. (2012) model consistently calculates lower pressures. There are noticeable misfits between the Shishkina et al. (2014) model and the new MafiCH model, specifically for the basaltic andesite (SFVF; ~ 260 MPa difference), basalt (N72; ~ 130 MPa difference), as well as the compositions with the highest alkali contents, AH3 (~ 190 MPa difference) and Erebus (~ 160 MPa difference). The Iacono-Marziano et al. (2012) model shows strong divergence from the new MafiCH model for the Erebus (~ 180 MPa difference) composition and moderate disagreement for the basanite (~ 95 MPa difference) and Sunset Crater (~ 75 MPa difference) compositions. For the remaining compositions, the pressures from Iacono-Marziano et al. (2012) agree with the MafiCH model within < 50 MPa.

Overall, VolatileCalc (Newman and Lowenstern 2002), Duan (2014), and Shishkina (2014) tend to calculate higher saturation pressures than the new model, while the Iacono-Marziano et al. (2012) model consistently yields lower saturation pressures. The Papale et al. (2006) model returns very similar saturation pressures for these 10 mafic compositions of variable alkali contents, suggesting that it is not well suited to distinguish solubility differences in this compositional range. Saturation pressures in Fig. 7 calculated using the Iacono-Marziano et al. (2012) and MagmaSat (Ghiorso and Gualda 2015) models have the smallest average difference (~ 50 MPa) from the MafiCH model for these 10 compositions. However, we note that we chose to focus on relatively hydrous volatile compositions for this test to avoid bias in saturation pressures as a result of variable isobar shapes at low H₂O contents (see Fig. 1). The Iacono-Marziano et al. (2012) and MagmaSat (Ghiorso and Gualda 2015) models in particular show a strong decrease in CO₂ solubility at low H₂O contents, and would therefore show larger deviations from MafiCH at relatively anhydrous compositions. The behavior of CO₂ solubility at low H₂O is poorly understood at present, so the different trends shown by models at H₂O-poor fluid compositions are relatively unconstrained.

The degree of consensus between models also varies based on the overall magma composition. Pressures calculated for the Sunset Crater alkali basalt show the best agreement for all models; in Fig. 7, the average difference between each model and the MafiCH model is just ~ 55 MPa for Sunset Crater. The models show significant disagreement (> 100 MPa average difference from MafiCH in Fig. 7) for the highest silica composition (SFVF basaltic andesite), as well as the compositions at the extremes of total alkali contents in this range (< 3 wt% and > 7 wt%). These significant disagreements may indicate that these compositional ranges are at or beyond the calibrated limits of the previous models. Some of these differences are likely due to the fact that some models attribute most of the compositional dependence of CO₂ solubility in mafic magmas to total alkali contents rather than the full multicomponent magma composition. The Erebus magma, for example, is one of the compositions that show a wide range of saturation pressures predicted by the different models. Erebus has high total alkali contents, corresponding to high CO₂ solubility in some models, but it also has the lowest Ca cation fraction and highest Al cation fraction in our calibration dataset, resulting in lower CO₂ solubility in other models.

One final interesting observation from Fig. 7 is the relative solubility differences between the magmas. In the compositions assessed in Fig. 7, saturation pressures generally decrease (i.e., volatile solubility increases) with increasing total alkali content, with the exception of magmas with higher silica contents (i.e., the SFVF composition) and very high total alkali contents (> 6 wt%). With the exception of VolatileCalc (Newman and Lowenstern 2002) and the Papale et al. (2006) model, the newer models generally replicate these same overall trends of volatile solubility with total alkali content. This may suggest that models have now identified the major elemental impacts on volatile solubility in mafic magmas, even though the models do vary in their overall calibration. If that is the case, future work to refine the models should focus on compositions for which the recent models do not converge (i.e., basaltic andesites and magmas with high total alkali contents).

Relative influence of compositional components on CO₂ solubility (what affects CO₂ solubility?)

Previous models specific to mafic magmas have correlated solubility relationships with compositional parameters that range from single elements to multicomponent calculations. For example, VolatileCalc (Newman and Lowenstern 2002) uses the SiO₂ content (wt%) as a proxy for the compositional parameter Π from Dixon (1997). Iacono-Marziano et al. (2012) and Shishkina et al. (2014) use more complex compositional parameters (NBO/O and Π*, respectively) to correlate with CO₂ solubility. In Fig. 8 we compare the calculated CO₂ solubility from MafiCH at 500 MPa and 1200 °C for the 10 calibration compositions compared with these compositional parameters.

Fig. 8

Comparison of CO₂ solubility at 500 MPa and 1200 °C calculated by the MafiCH model with compositional parameters from other works. CO₂ solubilities are calculated for the 10 calibration compositions in the CO₂ model of MafiCH. The parameters are: a SiO₂ content in wt% from VolatileCalc (Newman and Lowenstern 2002), b NBO/O from Iacono-Marziano et al. (2012), c Π* from Shishkina et al. (2014), and d agpaitic index (Ca⁺² + Na⁺¹ + K¹)/Al⁺³ as described in Mangan et al. (2021)

Figure 8a shows essentially no correlation between CO₂ solubility and SiO₂ content (wt%). This is unsurprising, because the SiO₂ parameterization incorporated into VolatileCalc (Newman and Lowenstern 2002) was based on suite of samples from a single submarine location and not intended for broad application. The NBO/O parameter used by the Iacono-Marziano et al. (2012) model also shows very little correlation with CO₂ solubility as determined by MafiCH (Fig. 8b). In fact, three of these compositions have nearly identical values of NBO/O but very different CO₂ solubilities (~ 6000–13,000 ppm CO₂ at 500 MPa and 1200 °C). The Π* parameter used by Shishkina et al. (2014) does appear to capture much of the variability in CO₂ solubility by these magmas, although the relationship between Π* and CO₂ solubility is non-linear (Fig. 8c). Recent work by Mangan et al. (2021) suggested that CO₂ solubility may be related to the agpaitic index of a magma, defined as (Ca⁺² + Na⁺¹ + K⁺¹)/Al⁺³. The MafiCH model supports this conclusion, showing a strong positive correlation between CO₂ solubility and agpaitic index (Fig. 8d). The agpaitic index correlation helps to explain why two compositions with similar total alkali contents (i.e., Erebus and AH3) can have very different CO₂ solubilities.

Because the CO₂ model in MafiCH does provide accurate solubility relationships for a variety of mafic compositions (e.g., Fig. 6a and c), we make an assumption that it can be used to test how small changes in composition impact CO₂ solubility, especially when interpolating between calibrated compositions. While this is a significant assumption, this exercise may provide useful insights about the mechanism of CO₂ dissolution in mafic magmas and changes in volatile solubility during magma evolution or for closely associated magmas, as well as identify priorities for future experimental investigations.

To investigate the influence of compositional components on CO₂ solubility, we determine the expected change in solubility when the abundance of one cation component is varied (Figs. 9, 10). We first calculate the CO₂ solubility at 500 MPa and 1200 °C for a specific composition, represented by the asterisk. We then change just one cation component of this composition in 0.005 increments and re-normalize all cations at each step. Then, using the MafiCH model, we recalculate $\ln K_{0}$ and $\Delta V_{r}^{0,m}$ and determine the new CO₂ solubility as a result of this change in composition. Note that symbols in grey in Figs. 9, 10 indicate that the calculated cation fraction is beyond the calibrated range of the model.

Fig. 9

Predicted influence of Na⁺¹, K⁺¹, Ca⁺², and Si⁺⁴ on CO₂ solubility according to the MafiCH model. CO₂ solubility (dissolved CO₂ content) calculated by the model at 500 MPa and 1200 °C for the a Sunset Crater, b Erebus, c Stromboli, and d Etna compositions is shown by the asterisk symbol. Additional symbols show how calculated solubility (at 500 MPa and 1200 °C) changes as one component is varied (single cation fraction in 5% intervals). At each step the modified composition is re-normalized to determine new values of $\ln K_{0}$ and $\Delta V_{r}^{0,m}$. Symbols in grey indicate that the calculated cation fraction is beyond the calibrated range of the model. Diagonal lines are drawn to aid in comparisons between panels

Fig. 10

Predicted influence of Ti⁺⁴, Al⁺³, Fe*, and Mg⁺² on CO₂ solubility according to the MafiCH model. CO₂ solubility (dissolved CO₂ content) calculated by the model at 500 MPa and 1200 °C for the Stromboli composition is shown by the asterisk symbol. Additional symbols show how calculated solubility (at 500 MPa and 1200 °C) changes as one component is varied (single cation fraction in 5% intervals). At each step the modified composition is re-normalized to determine new values of $\ln K_{0}$ and $\Delta V_{r}^{0,m}$. Symbols in grey indicate that the calculated cation fraction is beyond the calibrated range of the model. Diagonal lines are included for visual reference and easy comparison to Fig. 9

Figure 9 depicts the influence of sodium, potassium, calcium, and silicon content on CO₂ solubility according to the MafiCH model. We show the results of these tests for the Sunset Crater (Fig. 9a), Erebus (Fig. 9b), and Stromboli (Fig. 9c) magmas because they exhibit nearly identical CO₂ solubilities despite their variable chemical compositions. We also examine the Etna composition (Fig. 9d) as it is fairly similar in composition to the Sunset Crater magma yet has a higher CO₂ solubility. The remaining compositions used to calibrate the CO₂ model display similar overall trends to those in Fig. 9. According to this test, Si⁺⁴, which is the sole compositional input (as SiO₂) in the VolatileCalc model (Newman and Lowenstern 2002), has essentially no impact on CO₂ solubility in this compositional range. This is an interesting, but perhaps expected result as SiO₂ content was only used as a proxy for the compositional parameter Π from Dixon (1997) for a specific set of samples. This SiO₂ proxy was not intended to be broadly applied to a wide range of mafic magmas.

The effect of potassium on CO₂ solubility according to the MafiCH model is variable and non-linear. Its effect on CO₂ solubility is low in the Sunset Crater composition (Fig. 9a), slightly stronger for Stromboli (Fig. 9c) and Etna (Fig. 9d), and it has a very strong effect in the Erebus magma, nearly equal to the effect to calcium and sodium (Fig. 9b). However, the impact of potassium on CO₂ solubility appears to be much lower at low K⁺¹ contents. For example, in the Stromboli composition (0.023 initial K⁺¹; Fig. 9c), the addition of potassium has a positive effect on CO₂ solubility, but there is essentially no change in solubility if potassium is subtracted. In the Erebus magma (0.033 initial K⁺¹; Fig. 9b), however, CO₂ solubility does show a notable decrease as potassium content is decreased.

Calcium and sodium exhibit the strongest positive effects on CO₂ solubility based on the results shown in Fig. 9, but the effects vary across these four compositions. The magnitude of the solubility increase due to increased calcium or sodium is smaller for Sunset Crater (Fig. 9a) and Erebus (Fig. 9b) compared to Stromboli (Fig. 9c) and Etna (Fig. 9d). Furthermore, in the Sunset Crater and Erebus compositions, addition of calcium has a slightly stronger effect on CO₂ solubility than the addition of sodium, whereas for the Stromboli composition (Fig. 9c), addition of sodium has the larger effect. In the Etna composition (Fig. 9d), addition of sodium has a nearly equal impact on CO₂ solubility as the addition of calcium.

We also examine the influence of titanium, aluminum, iron, and magnesium content on CO₂ solubility according to the MafiCH model in Fig. 10. Figure 10 was constructed following the same method that was used for Fig. 9. For these elements, the overall trends are the same for all 10 magma compositions used to calibrate the CO₂ model, and so we focus only on the results for the Stromboli composition. Aluminum has a very strong negative effect on solubility, nearly as strong as the positive effect of calcium and sodium. This is likely because in order for Al⁺³ to form a tetrahedron with oxygen, it requires an additional cation, like K⁺¹ or Na⁺¹ to charge balance the tetrahedron. So not only does aluminum serve to polymerize the melt by forming a tetrahedron, but it also prevents one of these alkali cations from bonding with carbonate, leading to a very strong effect on CO₂ solubility. The remaining elements in Fig. 10 appear to have very little effect on CO₂ solubility in these compositions. Overall, this analysis (Fig. 9, 10) illustrates the complexity of CO₂ solubility. The influence of each individual component appears to depend upon the abundance of other components. Also, this result demonstrates that it is necessary to use the full multi-component melt composition to build well-calibrated solubility models.

Conclusions

We developed a general thermodynamic model for CO₂ solubility in mafic magmas across a wide range of total alkali contents. We combine this new thermodynamic model with an empirical model for H₂O solubility from Lesne et al. (2011a) to yield a general model for H₂O–CO₂ solubility in mafic magmas called MafiCH.

• 1. The CO₂ model was calibrated using recent experiments on six compositions from Allison et al. (2019) as well as four additional compositions from the literature. Using multiple linear regression analysis, we have derived two empirical compositional relationships (Eqs. 6–7) that describe the thermodynamic parameters $\Delta V_{r}^{0,m}$ (partial molar volume change) and $\ln K_{0}$ (equilibrium constant at reference conditions).

• 2. The MafiCH model is calibrated for compositions with 44 to 53 wt% SiO₂ and 2 to 9 wt% total alkalis (Na₂O + K₂O), at pressures between ~ 50 and 700 MPa. A comparison with experimental data from magmas both within and beyond this compositional space suggests that MafiCH is very accurate within its calibrated range (i.e., mafic magmas with cation fractions within the ranges shown in Fig. 5). In some cases, MafiCH can be used beyond the calibrated compositional range, as long as the melt is a relatively depolymerized composition with CO₂ stored as CO₃.

• 3. One of the primary benefits of the MafiCH model is that it accurately describes CO₂ solubility behavior in mafic magmas at a range of crustal pressures. The CO₂ model is particularly robust because it is based in thermodynamics, calibrated across a wide P–X range of mafic magmas, and employs the full multicomponent magma composition to determine CO₂ solubility relationships.

• 4. The new MafiCH model yields melt inclusion saturation pressures that are typically lower than the pressures predicted by previously published volatile solubility models relevant to mafic magmas. The previously published models show the most divergence from MafiCH in less constrained compositional regions such as in magmas with higher SiO₂ contents (i.e., basaltic andesites), as well as magmas with relatively high (> 7 wt%) and low (< 3 wt%) total alkali contents. Many of the more recent models show similar trends in volatile solubility by magma composition, suggesting that models are beginning to agree on the elemental controls on volatile solubility.

• 5. Previously published volatile solubility models have developed compositional indices such as SiO₂, Π (or Π*) and NBO/O to describe solubility within restricted compositional regions. CO₂ solubility determined using the MafiCH model does not directly correlate with any of these indices. However, the agpaitic index, (Ca⁺² + Na⁺¹ + K¹)/Al⁺³, does generally scale with CO₂ solubility calculated by the MafiCH model across a wide compositional range of mafic magmas. Additional analysis suggests that the influence of individual compositional components will be required to further improve volatile solubility models.

• 6. Based on calculations using the MafiCH model, Na⁺¹ and Ca⁺² appear to have the strongest positive effect on CO₂ solubility and Al⁺³ has a nearly equal negative effect. K⁺¹ also has a strong positive effect. Other elements (e.g., Si⁺⁴, Mg⁺²) individually have little effect on CO₂ solubility in this compositional range.

• 7. The absolute solubility influence of each element varies depending on the abundance of other elements. This assessment suggests that the full multi-component magma composition is necessary to constrain CO₂ solubility for mafic magmas. It also suggests that additional experiments are required to constrain compositional regions that extend beyond the calibrated range of the MafiCH model.