Abstract
Gibbs energy minimization is the means by which the stable state of a system can be computed as a function of pressure, temperature and chemical composition from thermodynamic data. In this context, state implies knowledge of the identity, amount, and composition of the various phases of matter in heterogeneous systems. For seismic phenomena, which occur on time-scales that are short compared to the timescales of intra-phase equilibration, the Gibbs energy functions of the individual phases are equations of state that can be used to recover seismic wave speeds. Thermodynamic properties relevant to modelling of slower geodynamic processes are recovered by numeric differentiation of the Gibbs energy function of the system obtained by minimization. Gibbs energy minimization algorithms are categorized by whether they solve the non-linear optimization problem directly or solve a linearized formulation. The former express the objective function, the total Gibbs energy of the system, indirectly in terms of the partial molar Gibbs energies of phase species rather than directly in terms of the Gibbs energies of the possible phases. The indirect formulation of the objective function has the consequence that although these algorithms are capable of attaining high precision they have no generic means of treating phase separation and expertise is required to avoid local minima. In contrast, the solution of the fully linearized problem is completely robust, but offers limited resolution. Algorithms that iteratively refine linearized solutions offer a compromise between robustness and precision that is well suited to the demands of geophysical modeling.
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References
Asimow, P.D. and Ghiorso, M.S., Algorithmic modifications extending MELTS to calculate subsolidus relations, Am. Mineral., vol. 83, nos. 9–10, pp. 1127–1132.
Bina, C.R., Free energy minimization by simulated annealing with applications to lithopsheric slabs and mantle plumes, Pure Appl. Geophys., 1998, vol. 151, pp. 605–618.
Bina, C.R. and Helffrich, G., Calculation of elastic properties from thermodynamic equation of state principles, Ann. Rev. Earth Planet. Sci., 1992, vol. 20, pp. 527–552.
Bucher, K. and Frey, M., Petrogenesis of Metamorphic Rocks, Berlin: Springer, 1994.
Callen, H.B., Thermodynamics: an Introduction to the Physical Theories of Equilibrium Thermostatics and Irreversible Thermodynamics, New York: Wiley, 1960.
Connolly, J.A.D., Computation of phase equilibria by linear programming: a tool for geodynamic modeling and its application to subduction zone decarbonation, Earth Planet. Sci. Lett., 2005, vol. 236, nos. 1–2, pp. 524–541.
Connolly, J.A.D., The geodynamic equation of state: what and how, Geochem. Geophys. Geosyst., 2009, vol. 10, no. (Q10014). doi 10.1029/2009gc002540
Connolly, J.A.D. and Kerrick, D.M., An algorithm and computer-program for calculating composition phase-diagrams, Calphad-Computer Coupling of Phase Diagrams and Thermochemistry, 1987, vol. 11, no. 1, pp. 1–55.
Connolly, J.A.D. and Khan, A., Uncertainty of mantle geophysical properties computed from phase equilibrium models, Geophys. Rev. Lett., 2016, vol. 43, no. 10, pp. 5026–5034. doi 10.1002/2016gl068239
Dahlen, F.A., Metamorphism of nonhydrostatically stressed rocks, Am. J. Sci., 1992, vol. 292, no. 3, pp. 184–198.
DeCapitani, C. and Brown, T.H., The computation of chemical equilibria in complex systems containing nonideal solutions, Geochim. Cosmochim. Acta, 1987, vol. 5, pp. 12639–2652.
Fuhrman, M.L. and Lindsley, D.H., Ternary-feldspar modeling and thermometry, Am. Mineral., 1988, vol. 73, nos. 3–4, pp. 201–215.
Gerya, T.V., Maresch, W.V., Willner, A.P., et al., Inherent gravitational instability of thickened continental crust with regionally developed low- to medium-pressure granulite facies metamorphism, Earth Planet. Sci. Lett., 2001, vol. 190, nos. 3–4, pp. 221–235.
Ghiorso, M.S., Carmichael, I.S.E., Rivers, M.L., and Sack, R.O., The Gibbs free-energy of mixing of natural silicate liquids–an expanded regular solution approximation for the calculation of magmatic intensive variables, Contrib. Mineral. Petrol., 1983, vol. 84, nos. 2–3, p. 107–145. doi 10.1007/bf00371280
Hacker, B.R. and Abers, G.A., Subduction factory 3: an Excel worksheet and macro for calculating the densities, seismic wave speeds, and H2O contents of minerals and rocks at pressure and temperature, Geochem. Geophys. Geosystem, 2004, vol. 5, no. 1. doi 10.1029/2003GC000614
Hillert, M., Phase Equilibria, Phase Diagrams and Phase Transformations: their Thermodynamic Basis, Cambridge: Cambridge University Press, 2008.
Karpov, I.K., Chudnenko, K.V., and Kulik, D.A., Modeling chemical mass transfer in geochemical processes: thermodynamic relations, conditions of equilibria, and numerical algorithms, Am. J. Sci., 1997, vol. 297, no. 8, pp. 767–806.
Khan, A., Connolly, J.A.D., and Olsen, N., Constraining the composition and thermal state of the mantle beneath europe from inversion of long-period electromagnetic sounding data, J. Geophys. Res. Sol. Earth, 2006, vol. 111, B10102. doi 10.1029/2006JB004270
Powell, R., Equilibrium Thermodynamics in Petrology: an Introduction, London: Harper and Row, 1978.
Prausnitz, J.M., Molecular Thermodynamics of Fluid-Phase Equilibria, New York: Englewood Cliffs, Prentice-Hall, 1969.
Ricard Y., Mattern E., and Matas J., Synthetic tomographic images of slabs from mineral physics, VanDerHilst R.D., Bass J.D., Matas J., Trampert J. (eds). Earth’s Deep Mantle: Structure, Composition, and Evolution. 2005. V. 160. P. 283–300.
Saxena, S.K. and Eriksson, G., Theoretical computation of mineral assemblages in pyrolite and lherzolite, J. Petrol., 1983, vol. 24, pp. 538–555.
Sobolev, S.V. and Babeyko, A.Y., Modeling of mineralogical composition, density, and elastic-wave velocities in anyhdrous magmatic rocks, Surv. Geophys., 1994, vol. 15, pp. 515–544.
Stixrude, L. and Lithgow-Bertelloni, C., Thermodynamics of mantle minerals: I. Physical properties, Geophys. J. Int., 2005, vol. 162, no. 2, pp. 610–632.
Stixrude, L. and Lithgow-Bertelloni, C., Influence of phase transformations on lateral heterogeneity and dynamics in earth’s mantle, Earth Planet. Sci. Lett., 2007, vol. 263, nos. 1–2, pp. 45–55.
Stixrude, L. and Lithgow-Bertelloni, C., Thermodynamics of mantle minerals: II. Phase equilibria, Geophys. J. Int., 2011, vol. 184, no. 3, pp. 1180–1213. doi 10.1111/j.1365-246X.2010.04890.x
Tirone, M., Ganguly, J., and Morgan, J.P., Modeling petrological geodynamics in the earth’s mantle, Geochem. Geophys. Geosyst., 2009, vol. 10. Q04012 doi 10.1029/2008GC002168
White, W.B., Johnson, S.M., and Dantzig, G.B., Chemical equilibrium in complex mixtures, J. Chem. Phys., 1958, vol. 28, no. 5, pp. 751–755.
Wood, B.J. and Holloway, J.R., A thermodynamic model for subsolidus equilibria in the system CaO-Mgo-Al2O3-SiO2, Geochim. Cosmochim. Acta, 1984, vol. 66, pp. 159–176.
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Published in Russian in Petrologiya, 2017, Vol. 25, No. 5, pp. 533–542.
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Connolly, J.A.D. A primer in gibbs energy minimization for geophysicists. Petrology 25, 526–534 (2017). https://doi.org/10.1134/S0869591117050034
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DOI: https://doi.org/10.1134/S0869591117050034