The thermodynamics of iron and magnesium partitioning between olivine and liquid: criteria for assessing and predicting equilibrium in natural and experimental systems

Abstract

The equations for the chemical potentials of fayalite and forsterite components in olivine and liquid may be used to derive a thermodynamic expression for the exchange coefficient $K^{{\;\;\;{\text{Mg - Fe}}}}_{{D{\text{Ol - Liq}}}}$ (defined as the molar Mg/Fe²⁺ of the liquid divided by Mg/Fe of coexisting olivine). This expression, well known in the literature, shows that $K^{{\;\;\;{\text{Mg - Fe}}}}_{{D{\text{Ol - Liq}}}}$ is a function of temperature, pressure, and compositions of the olivine and liquid. Quantitative application of this equation requires knowledge of the free energies and volumes of liquid and crystalline fayalite and forsterite, and a description of the non-ideality of Fe–Mg mixing in olivine. Independent measurements of these parameters reported in the literature have been used to constrain the influence of temperature, pressure and olivine composition on iron–magnesium partitioning. The other requirement for calculation of $K^{{\;\;\;{\text{Mg - Fe}}}}_{{D{\text{Ol - Liq}}}}$ is knowledge of the ratio of activity coefficients of Fe²⁺ and Mg in the liquid phase (γFe²⁺/γMg), but no independent predictive model for this exists. To assess and predict the variation of γFe²⁺/γMg as a function of liquid composition, a database of almost 400 olivine–liquid pairs at 1 bar and 200 experiments at pressures to the upper stability limit of olivine has been considered. Measured values of $K^{{\;\;\;{\text{Mg - Fe}}}}_{{D{\text{Ol - Liq}}}}$ in this data base vary from 0.17 to 0.45, and the data were chosen to cover wide ranges of temperature and olivine composition, in addition to liquid composition. Within the framework of the thermodynamic equation, γFe²⁺/γMg is found to be a function of silica content and alkali content of the liquid. Water is also inferred to have a direct influence at high pressure. A predictive model for $K_{D} {\kern 1pt} ^{{{\text{Mg}} - {\text{Fe}}}}_{{{\text{Ol}} - {\text{Liq}}}}$ is proposed which recovers all input data at the 1σ level of uncertainty. It is used to assess equilibrium in published experimental studies of mantle melting and to illustrate the influence of individual parameters such as temperature and pressure. Application of the model to the field of melt inclusions is also discussed.

Appendices

Appendix 1

Thermodynamic properties of MgSi_0.5O₂ and FeSi_0.5O₂ liquid and solid

Crystalline fayalite melts incongruently at 1,490 K. The enthalpy of fusion at the melting temperature was first determined by Orr (1953) who proposed a value of 92 kJ/mol (of Fe₂SiO₄). Based upon new measurements Stebbins and Carmichael (1984) proposed a revised value of 89.3±1.1 kJ/mol for the hypothetical enthalpy of congruent fusion at 1,490 K taking into account the presence of ferric iron in the liquid. This latter value is preferred here, but values of 89.3 and 92 kJ/mol have both been considered (Fig. 2b). Concerning the heat capacities of fayalite liquid and solid, Stebbins and Carmichael (1984) provide a value for the liquid, while Orr (1953) provides data for the solid. However, for the present purposes values of the solid need to be extrapolated to temperatures much higher than those actually measured. As noted by Bottinga (1991), the equation fitted to the data of Orr (1953) given by Robie et al. (1979) incorporates the effect of a certain degree of premelting and does not extrapolate well, rapidly predicting the heat capacity of solid fayalite to be greater than that of liquid fayalite. Berman (1988) and to a lesser extent Bottinga (1991) propose equations for the heat capacity of crystalline fayalite which do not experience this problem. However, despite the uncertainty in the heat capacity difference between liquid and solid fayalite, it is found that equations of Robie et al. (1979), Berman (1988) and Bottinga (1991) all give indistinguishable results for ΔG ^fusion of Fe₂SiO₄ as a function of temperature because the ΔC _p term can be effectively neglected.

Crystalline forsterite melts congruently at a temperature generally taken to be 2,163 K (e.g. Robie et al. 1979). The equations for the heat capacity of crystalline forsterite of Robie et al. (1979) and Gillet et al. (1991) are in excellent agreement and the equation from the latter study has been adopted here. No direct measurement of the heat capacity of forsterite liquid exists but the temperature independent value of 252 J/mol predicted by the model of Richet and Bottinga (1985) is perfectly consistent with extrapolated C _p measurements in the system MgO–Al₂O₃–SiO₂ by Courtial and Richet (1993) and is used here. Only one direct calorimetric measurement has been made of the enthalpy of fusion at the melting temperature, reported to be 142±12 kJ/mol at 2,174 K (Richet et al. 1993). An alternative estimate based upon extrapolated heats of solution of glasses at 1,773 K, themselves extrapolated to the melting temperature predicts a value of 114±20 kJ/mol at 2,169 K (Navrotsky et al. 1989). Further constraints on the enthalpy of fusion are provided by the measurements of the enthalpy of vitrification of forsterite glass just above the glass transition temperature (60.1±2 kJ/mol at 1,040 K) determined by Tangeman et al. (2001). Combining this value with the enthalpy change of crystalline forsterite from 2,174 to 1,040 K (217 kJ/mol, using Gillet et al. 1991), and the enthalpy change for liquid forsterite over the same temperature range (289 kJ/mol, calculated using Richet and Bottinga 1985) it may be shown that the enthalpy of fusion of forsterite at 2,174 K is 132±2 kJ/mol. Based upon this analysis, this range of values, consistent with the direct measurements of Richet et al. (1993), has been considered in our calculations (see Fig. 2b).

Appendix 2

Adjusting silica content for the effect of alkalis and water

The case of alkalis

To predict γFe²⁺/γMg for alkali-bearing liquids, the simplest strategy is to adjust the observed silica content to bring it onto the correlation observed in the alkali-free systems (Fig. 6). To do this the offset between the observed silica content and that predicted at the derived value of γFe²⁺/γMg based upon the alkali-free correlation has been defined. This offset is called ΔSi, and qualitatively it may be expected to increase with increasing alkali concentration. The parameter [ΔSi/(XNa₂O+XK₂O)] has therefore been considered (called Ψ), such that the adjusted silica content (%SiO ^A₂ ) may be calculated using the equation:

$$ \% {\text{SiO}}_2^A = \% {\text{SiO}}_2 + \Psi \times \left( {\% {\text{Na}}_2 {\text{O}} + \% {\text{K}}_2 {\text{O}}} \right) $$

(11)

Application of Eq. 11 therefore requires a mathematical expression for Ψ, which in the simplest case would be a constant.

The data set used to constrain Ψ contains liquids at 1 bar and higher pressure, with total alkali concentration in the range 3–14 mol% and silica contents in the range 40–80 mol%. However, it may be noted that data at 1 bar only reach the highest values of alkali content at less than 60 mol% SiO₂, while the high pressure data base contains liquids rich in both alkalis and silica, although some considerable overlap of the two data sets exists.

When Ψ is calculated for these liquids, values are variable, but show clear systematic variations as a function of silica concentration and alkali content (Fig. 16). No difference is observed between Na-bearing liquids (e.g. Shi 1993) and K-bearing liquids (e.g. Takahashi 1978). Furthermore, values from 1 bar and high pressure experiments are perfectly consistent (Fig. 16a), thus all data have been considered simultaneously when defining the mathematical expression for Ψ. In detail, Fig. 16a shows that for liquids with 40–60 mol% SiO₂, subsets of the data at constant silica content have a variation of Ψ which is a linear function of total alkali content. The gradients and intercepts of these linear correlations have been determined then described as a function of silica content. For %SiO₂≤60 mol% the resulting expression for Ψ is:

$$ \Psi = \left( {0.46\left( {\frac{{100}} {{100 - \% {\text{SiO}}_2 }}} \right) - 0.93} \right)\left( {\% {\text{Na}}_2 {\text{O}} + \% {\text{K}}_2 {\text{O}}} \right) + \left( { - 5.33\left( {\frac{{100}} {{100 - \% {\text{SiO}}_2 }}} \right) + 9.69} \right) $$

(12)

At silica content >60 mol%, a non-linear dependence of Ψ on alkali content is inferred and an alternative expression for Ψ is proposed:

$$ \Psi = \left( {11 - 5.5\left( {\frac{{100}} {{100 - \% {\text{SiO}}_2 }}} \right)} \right) \times \exp ^{ - 0.13\left( {\% {\text{Na}}_2 {\text{O}} + \% {\text{K}}_2 {\text{O}}} \right)} $$

(13)

Examples of functions 12 and 13 are illustrated in Fig. 16b. It is of note that Eqs. 12 and 13 provide reasonable fits to the data outside of the suggested range (particularly 13) and that the distinction of liquids with or less than 60 mol% silica is arbitrary. Even so, greatest agreement is obtained when the two different expressions are used based upon the silica content of the liquid.

Fig. 16

ΔSi per alkali (Ψ, as defined in Appendix 2) as a function of alkali content. a All data distinguishing those at high pressure and 1 bar. b Examples of the proposed functional forms for calculating Ψ. Thin lines are Eq. 12 to be used for SiO₂ ≤60 mol%, thick lines are Eq. 13 to be used for SiO₂ >60 mol%, (shown dashed for 60 mol%).

The case of water content

As illustrated in Fig. 11, water would appear to have a direct effect on partitioning behaviour. This requires a second adjustment of silica content to bring the derived values of γFe²⁺/γMg onto the correlation observed in the alkali-free, anhydrous systems. In the case of hydrous liquids a new value of adjusted silica content (called %SiO ^#₂ ) is calculated from %SiO ^A₂ :

$$ \% {\text{SiO}}_2^\# = \% {\text{SiO}}_2^A + 0.8 \times {\text{wt}}\% {\text{H}}_2 {\text{O}} $$

(14)

For this purpose mole fractions used to determine %SiO ^A₂ (in Eqs. 11 and 12 or 13) are calculated on an anhydrous basis.