Modeling vaporized silicates overview
To quantitatively study rock vaporization processes, we must have a complete set of accurate thermodynamic models for both condensed and vapor phases sampling broad ranges in composition and temperature. It is critical that these solution models faithfully represent geologically realistic mixed compositions, including minor components which play key roles in vaporization due to their high volatilities, reflecting strong preferences for the vapor phase. Fortunately, this burden is eased by the typically low-pressure conditions of vaporization, where thermodynamic models can neglect the complications introduced by compression (with the notable exception of supercritical vapors in high-energy giant impacts). The MELTS model (Ghiorso and Sack 1995), for instance, provides an accurate description of igneous geologic systems in this low-pressure regime, and is thus well-suited to the task. Calculating the composition, speciation, and abundance of the gas phase likewise requires an extensive and detailed thermodynamic database of vapor species, conveniently provided by standard thermodynamic tables, e.g. NIST-JANAF or similar tables (Lamoreaux and Hildenbrand 1984; Lamoreaux, Hildenbrand, and Brewer 1987). As most vaporization processes occur at low-pressure conditions, we can likewise rely on empirical thermal models of pure 1 bar gasses that use the ideal gas approximation to extend to arbitrarily low partial pressures and mixed compositions. This database of phase models is finally used with an appropriate equilibrium-seeking algorithm—depending on the vapor abundance regime—to predict the equilibrium phase assemblage by minimizing the total Gibbs energy of the system.
Vapor-abundance regimes
Finding an equilibrium assemblage requires a comprehensive search of composition/abundance space for the global minimum Gibbs energy, where the optimal strategy depends on the relative total abundance of vapor in the system. In the two endmember regimes, vapor-limited and vapor-dominated, the bulk composition of the system is housed nearly exclusively in the condensed phases or the mixed vapor phase, respectively, with negligible mass residing in other low-abundance phases. This simplifies computation, since equilibrium is established independently by either the condensed phases or mixed vapor alone. These dominant phases thus determine the equilibrium conditions, setting the chemical potentials of the system that control the properties of coexisting minor phases. Far more complex is the intermediate vapor-abundant case, in which both vapor and condensed phases coexist in significant quantities. Finding equilibrium in this case requires simultaneous energy minimization of all available condensed phases and vapor species.
The vapor-limited regime is the most common across a range of geoscience contexts, as it occurs at much lower temperatures and higher pressures, thus spanning a larger range of environmental conditions. A system is vapor-limited when the amount of vapor (by mass) is vastly outweighed by condensed phases. This ensures that the equilibrium state of the system is largely set by the condensed phases alone, allowing us to model these systems in two steps, first considering only condensed phases and then adding vapor species as a second-order correction. With this approach, we can use existing thermodynamic models (like the equilibration algorithm embedded in MELTS) to establish condensed-phase equilibrium, after which we can then determine the coexisting abundances of vapor species. The task is even further simplified at higher temperatures where the condensed phase is fully molten. In this case, condensed-phase equilibrium is trivial, since the composition and abundance of liquid are just equal to the bulk composition and total mass of the system. The conditions of equilibrium are therefore controlled by the condensed phase assemblage, dictating the chemical potentials of the system (\(\mu_j\)). Vapor species abundances are finally determined directly from these chemical potentials, providing a complete picture of all the stable phases in the system.
Modeling equilibrium vapor compositions
In the vapor-limited regime, thermodynamics provides a straightforward framework for predicting vapor compositions, where the abundance of each vapor species is independently adjusted to equalize its chemical potential with the bulk system. Using mass-balance constraints, we write the balanced chemical reaction that forms the \(i^{th}\) gaseous species from the condensed phase assemblage by exchange of basic system components (e.g. the oxides with additional oxygen consumed or produced as needed): \[ \phi_{iv} = \sum_{j} \nu_{ij} c_j + \nu_{iO_2} c_{O_2} \] where \(\phi_{iv}\) is the \(i^{th}\) vapor species, \(c_j\) & \(c_{O_2}\) are the vector of basic system components (like oxides) plus oxygen, and \(\nu_{ij}\) & \(\nu_{iO_2}\) give the stoichiometry of the vapor species in question, expressing its composition in terms of system components. With the law of mass action, the corresponding equilibrium condition is written: \[ \begin{aligned} \mu_{iv} &= \sum_{j} \nu_{ij} \mu_j + \nu_{iO_2} \mu_{O_2}\\ \mu^0_{iv} + RT\log P_{iv} &= \sum_{j} \nu_{ij} \mu_j + \nu_{iO_2} \cdot ( \mu_{O_2}^0 + RT\log f_{O_2}) \end{aligned} \] where \(P_{iv}\) is the equilibrium partial pressure of vapor species \(i\), \(f_{O_2}\) is the oxygen fugacity, and \(\mu_{iv}\), \(\mu_j\), & \(\mu_{O_2}\) are the chemical potentials of vapor species \(i\), component \(j\), & molecular oxygen, respectively. This expression of equilibrium imposes equality of chemical potentials for gas species \(i\) in terms of system components that are freely exchanged with coexisting liquid or solid phases. The above formulation uses chemical potentials directly, but is equivalent to the activity-focused formulation of Sossi and Fegley Jr (2018). Rearranging, we obtain the governing equation for the abundances of each vapor species in terms of its partial pressure: \[ \log P_{iv} = \frac{1}{RT} \left[ \sum_{j} \nu_{ij} \mu_j + \nu_{i O_2}\mu_{O_2}^0 - \mu_{iv}^0 \right] + \nu_{iO_2} \log f_{O_2} \] The equilibrium abundance of each vapor species is thus determined by environmental conditions (temperature and oxygen fugacity), the condensed phase assemblage (that dictates the chemical potentials of the system), and the 1 bar thermal properties of each vapor species (\(\mu_{iv}^0\) & \(\mu^0_{O_2}\)).
Leveraging ideal behavior of vapor species
Calculation of species abundances using the governing expression above, rests upon the simplified behavior of ideal gases in the low density limit, where species in a mixed vapor do not interact with one another. The ideal gas approximation imposes a simple link between pressure and temperature effects through the ideal gas law, enabling experimental constraints at a 1 bar reference pressure to trivially extend to arbitrarily low-density pressure conditions. For this application, the primary benefit of the ideal gas approximation is the absence of compositional mixing terms, ensuring that the fugacity of any gaseous species is simply equal to its partial pressure, \(f_{i}^v=P_i^v\), enabling the abundance of each vapor species to be determined independently of all others.
To evaluate the chemical potential of each vapor species, the ideal gas approximation must be coupled with an analytic expression for energy as a function of temperature, like the Shomate equation used by the thermochemical tables of JANAF and Lamoreaux and Hildenbrand (1984). The Shomate equation empirically captures energy variations for a wide selection of materials over a broad temperature range, and multiple piecewise models are sometimes combined to retain desired accuracy over 1000+ degree intervals. The generalized polynomial form of the Shomate equation describes molar enthalpy in kJ/mol: \[ \Delta \bar{H}^0 = \bar{H}^0 - \bar{H}^0_{298.15} = At + \frac{B}{2} t^2 + \frac{C}{3} t^3 + \frac{D}{4} t^4 - E t^{-1} + F \] molar entropy in J/mol/K: \[ \bar{S}^0 = A\ln{t} + Bt + \frac{C}{2} t^2 + \frac{D}{3} t^3 - \frac{E}{2} t^{-2} + G \] and molar heat capacity in J/mol/K: \[ C^0_P = A + Bt + C t^2 + D t^3 + E t^{-2} \] where \(t\) is temperature given in kilo-Kelvin (\(t = T/1000\)). These are combined to evaluate the molar Gibbs energy (or chemical potential): \[ \bar{G}^0 \equiv \mu^0 = \Delta \bar{H}^0 - T \bar{S}^0 \] These numerical coefficients (\(A,B,C,D,E,F,G\)) are provided for a huge set of phases in the JANAF thermochemical database, as well as the tables of Lamoreaux and Hildenbrand (1984) and Lamoreaux, Hildenbrand, and Brewer (1987). Using these expressions, we can rapidly evaluate the chemical potential reference state for every vapor species in the system, allowing us to determine the equilibrium partial pressure of each one.