Evaporation and the Isotopic Composition of Type a and B Refractory Inclusions

Calculations are described which predict the evolution of chemical compositions of non-ideal CaO-MgO-Al2O3SiO2 (CMAS) liquids undergoing open system evaporation into pure H2 gas. This evaporation model is used to explain the degree of mass fractionation of Mg and Si isotopic compositions observed in refractory inclusions[1, 2]. Equilibrium Vapor Pressure: Assume an infinite reservoir of CMAS liquid, at fixed P(H2) and temperature (T). For every gaseous species (see [3] for a complete list), a balanced chemical reaction can be written involving only that species, monatomic H and O, and a liquid oxide component, for example: 0.5*Al2O3 (l) + H(g) = AlOH(g) + 0.5*O(g). (eq. 1) Determining the equilibrium constants, K, for these reactions from the Gibbs energies of the various species, and using Berman’s model [4] for the activities, a, of CMAS liquid oxides, a mass-action law can be written for each gaseous species, e.g. from eq. 1: PAlOH = K*(aAl2O3) *PH*PO . (eq. 2) Expressions in PH and PO can be written for P(O2), P(H2O), etc. The dissociation constant for H2 fixes PH at a fixed P(H2). A mass-balance expression can be written expressing a "total pressure" of each element, for example, PAl tot = PAl + PAlOH + PAlO + ..., (eq. 3) resulting in 5 equations in 6 unknowns: PO and PCa , PMg , PAl , PSi , PO . Finally, it is apparent that evaporated metal atoms cannot leave their oxygen behind in the liquid, so the sixth equation necessary to solve the system is PO tot = PCa tot + PMg tot + 1.5*PAl tot + 2*PSi . (eq. 4) Substitution of equations like eq.2 into eq.3, and of eq.3 into eq.4 results in a single, non-linear equation in PO, which can be solved quickly by iteration, allowing calculation of the partial pressures of all gas species in equilibrium with the fixed liquid composition. One may, alternatively, fix the total pressure of the system, P, rather than P(H2). An initial estimate of PH is chosen, and PO determined as above. Then PH alone is adjusted so that the sum of all partial pressures is equal to P. A new PO is computed with this new PH, then a new PH is computed, and so forth to a stable solution. Evaporation Model: The process of open system evaporation can be approximated by calculation of the vapor pressure over the liquid at each of a series of small, equilibrium evaporation steps. Once the vapor composition over a particular liquid composition is determined, sufficient vapor of that composition is removed from the system such that one percent of the initial Mg is lost from the liquid. That is, the liquid composition is adjusted to reflect the loss of vapor having the computed composition. The new liquid composition becomes the initial composition for the next evaporation step. The "evaporated" increment is here assumed to contain CMAS oxides in proportion to their vapor pressures over the liquid. In a full kinetic treatment of evaporation rates from a CMAS droplet into a CMAS-free gas, the Hertz-Knudsen equation states, Ji = αiPi mikT) (eq. 5) in which k is the Boltzmann constant, T is in Kelvins and J, α, P and m are the flux from the droplet, the evaporation coefficient, the equilibrium vapor pressure and mass, respectively, of each evaporating species i. At all conditions investigated, the vapor pressures of Ca and Al species are ~10 times those of Mg and Si, and Mg(g) and SiO(g) contain >99% of the Mg and Si in the vapor. The evolution of the SiO2/MgO ratio in the droplet is, therefore, directly related to the relative loss rates of SiO(g) and Mg(g), JSiO/JMg = (mMg/mSiO) (αSiO/αMg)(PSiO/PMg). (eq. 6) Little is known about the relative α values of different molecules evaporating from a liquid, but α for the congruent evaporation of forsterite varies by a factor of 5 with T, gas composition and crystallographic orientation [5]. Because of this uncertainty and the fact that measured activities [6] sometimes differ by several tens of percent from those predicted by the model [4] for CMAS liquids, we have taken the relative loss rates of SiO2 and MgO simply as the calculated ratio PSiO/PMg. The justification for doing so is the excellent agreement between model results and experimental observations seen in Fig. 1.