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Computational Petrology and Pyroxene Thermodynamics

S. Sommacal, M. Sambridge and H.StC. O'Neill

A new way to represent the Gibbs free energy (G) for any multi-phase system has been formulated for silicate/oxide systems. The chemical composition of each phase has been unambiguously expressed by the number of cations per formula unit (‘constituents', symbol ‘N'), which are subject to constraints by both stoichiometry and charge balance. Related to the constituents are the site occupancies for each element i in phase f . For each constituent we can write an expression of the type

where 

represents the site occupancies of element i in site j, the summation to be extended to all sites n in phase f . As a result for every phase f in the system the molar Gibbs free energy (Gf ) assumes the form of

with respectively i = 1, …, c number of element and j = 1,.., n number of sites present in phase f . The total G of a system will be then given by

 

where nf = number of moles of phase f,Gf = free energy per mole of phase f , and p = number of phases f in the system.

 For each phase, the molar free energy (Gf ) is given by the sum of contributions from 1) the Gibbs free energy of the end-members, 2) ideal mixing on sites, and 3) excess mixing terms. The principle underlying the formulation of the term due to the Gibbs free energy of the end-members has been elucidated. As an example, the expression of this term for a general (Na-Ca-Mg-Fe2+-Al-Cr-Fe3+-Si-Ti) pyroxene system (32 end-members) has been derived.

The Gibbs Free Energy Minimum Principle states that the equilibrium value of any unconstrained parameter in a system in contact with a temperature and pressure reservoir minimizes the Gibbs free energy (G) at constant temperature and pressure. It follows that at any given temperature and pressure a closed multi-phase system is at its equilibrium condition when the chemical composition of the phases present in the system and the number of moles of each are such that the Gibbs free energy of the system reaches its minimum value.

In order to compute phase equilibria in pressure-temperature-compositional space a computer program (Gib) has been written to find the minimum in the Gibbs free energy. In the program the unknowns sought are the constituents of each phase together with the numbers of moles of that phase. The system's Gibbs free energy is minimized under mass balance, stoichiometry, charge balance and positivity constraints. The minimization is carried out by making use of a ‘Feasible iterate sequential quadratic programming' method (FFSQP) which is specifically designed for general constrained optimization problems of this kind. 

Initial application of the program is to a system of coexisting pyroxenes (orthopyroxene, and low Ca- and high Ca clinopyroxene). Future work will focus on extending the calculations to the subsolidus upper-mantle systems consisting of olivine, plagioclase, spinel and garnet in addition to pyroxenes. This will provide the basis for a more comprehensive thermodynamic modeling tool for investigating phase/melting relationships in the Earth's mantle at high pressure and temperature.