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The effects of chemical and mechanical interactions on the thermodynamic pressure for mineral solid solutions

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Abstract

We use a coupled thermodynamically consistent framework to model reactive chemo-mechanical responses of solid solutions. Specifically, we focus on chemically active solid solutions that are subject to mechanical effects due to heterogeneous stress distributions. The stress generation process is driven solely by volume changes associated with the chemical processes. We use this model to describe the underlying physics during standard geological processes. Furthermore, simulation results of a three-species solid solution provide insights into the phenomena and verify the interleaving between mechanical and chemical responses in the solid. In particular, we show the evolution of the thermodynamic pressure as the system goes to a steady state.

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References

  1. Almgren, F., Wang, L.: Mathematical existence of crystal growth with Gibbs–Thomson curvature effects. J. Geom. Anal. 10(1), 1–100 (2000)

    MathSciNet  MATH  Google Scholar 

  2. Bennethum, L.S., Weinstein, T.: Three pressures in porous media. Transp. Porous Media 54(1), 1–34 (2004)

    MathSciNet  Google Scholar 

  3. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28(2), 258–267 (1958)

    MATH  ADS  Google Scholar 

  4. Carpenter, M.A.: A “conditional spinodal’’ within the peristerite miscibility gap of plagioclase feldspars. Am. Mineral. 66(5–6), 553–560 (1981)

    Google Scholar 

  5. Clavijo, S., Sarmiento, A., Espath, L., Dalcin, L., Cortes, A.M., Calo, V.M.: Reactive n-species Cahn–Hilliard system: a thermodynamically-consistent model for reversible chemical reactions. J. Comput. Appl. Math. 350, 143–154 (2019)

    MathSciNet  MATH  Google Scholar 

  6. Clavijo, S.P., Espath, L., Calo, V.M.: Extended Larché–Cahn framework for reactive Cahn–Hilliard multicomponent systems. Contin. Mech. Thermodyn. 33(6), 2391–2410 (2021)

    MathSciNet  ADS  Google Scholar 

  7. Clavijo, S.P., Espath, L., Sarmiento, A., Calo, V.M.: A continuum theory for mineral solid solutions undergoing chemo-mechanical processes. Contin. Mech. Thermodyn. 34(1), 17–38 (2022)

    MathSciNet  ADS  Google Scholar 

  8. Dal, H., Miehe, C.: Computational electro-chemo-mechanics of lithium-ion battery electrodes at finite strains. Comput. Mech. 55(2), 303–325 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Dalcin, L., Collier, N., Vignal, P., Côrtes, A., Calo, V.M.: PetIGA: a framework for high-performance isogeometric analysis. Comput. Methods Appl. Mech. Eng. 308, 151–181 (2016)

    MathSciNet  MATH  ADS  Google Scholar 

  10. Droubay, T.C., Pearce, C.I., Ilton, E.S., Engelhard, M.H., Jiang, W., Heald, S.M., Arenholz, E., Shutthanandan, V., Rosso, K.M.: Epitaxial Fe\(_{3-x}\)Ti\(_x\)O\(_4\) films from magnetite to ulvöspinel by pulsed laser deposition. Phys. Rev. B 84(12), 125443 (2011)

    ADS  Google Scholar 

  11. Eberl, D.D., Środoń, J., Kralik, M., Taylor, B.E., Peterman, Z.E.: Ostwald ripening of clays and metamorphic minerals. Science 248(4954), 474–477 (1990)

    ADS  Google Scholar 

  12. Elliott, C.M., Garcke, H.: Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix. Physica D 109(3–4), 242–256 (1997)

    MathSciNet  MATH  ADS  Google Scholar 

  13. Gibbs, J.W.: On the equilibrium of heterogeneous substances. Am. J. Sci. 3(96), 441–458 (1878)

    MATH  ADS  Google Scholar 

  14. Gurtin, M.E., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  15. Hobbs, B.E., Ord, A.: Does non-hydrostatic stress influence the equilibrium of metamorphic reactions? Earth Sci. Rev. 163, 190–233 (2016)

    ADS  Google Scholar 

  16. Hobbs, B.E., Ord, A.: Coupling of fluid flow to permeability development in mid-to upper crustal environments: a tale of three pressures. Geol. Soc. Lond. Spec. Publ. 453(1), 81–120 (2018)

    ADS  Google Scholar 

  17. Howell, D., Wood, I., Dobson, D., Jones, A., Nasdala, L., Harris, J.: Quantifying strain birefringence halos around inclusions in diamond. Contrib. Miner. Petrol. 160(5), 705–717 (2010)

    ADS  Google Scholar 

  18. Johnson, C.A.: Generalization of the Gibbs–Thomson equation. Surf. Sci. 3(5), 429–444 (1965)

    ADS  Google Scholar 

  19. Keller, L.: Mineral growth in metamorphic rocks: relationships between chemical patterns, mineral microstructure and reaction kinetics. In: AGU Fall Meeting Abstracts, vol. 2008, p. MR14A-04 (2008)

  20. Larche, F., Cahn, J.: Thermochemical equilibrium of multiphase solids under stress. Acta Metall. 26(10), 1579–1589 (1978)

    Google Scholar 

  21. Larché, F., Cahn, J.W.: A linear theory of thermochemical equilibrium of solids under stress. Acta Metall. 21(8), 1051–1063 (1973)

    Google Scholar 

  22. Larché, F., Cahn, J.W.: A nonlinear theory of thermochemical equilibrium of solids under stress. Acta Metall. 26(1), 53–60 (1978)

    Google Scholar 

  23. Larche, F., Cahn, J.W.: The interactions of composition and stress in crystalline solids. J. Res. Natl. Bur. Stand. 89(6), 467 (1984)

    Google Scholar 

  24. Lindsley, D.H.: Some experiments pertaining to the magnetite–ulvöspinel miscibility gap. Am. Miner. 66(7–8), 759–762 (1981)

    Google Scholar 

  25. Miehe, C., Dal, H., Schänzel, L.-M., Raina, A.: A phase-field model for chemo-mechanical induced fracture in lithium-ion battery electrode particles. Int. J. Numer. Methods Eng. 106(9), 683–711 (2016)

    MathSciNet  MATH  Google Scholar 

  26. Milke, R., Dohmen, R., Becker, H.-W., Wirth, R.: Growth kinetics of enstatite reaction rims studied on nano-scale, part I: methodology, microscopic observations and the role of water. Contrib. Miner. Petrol. 154(5), 519–533 (2007)

    ADS  Google Scholar 

  27. Milke, R., Abart, R., Kunze, K., Koch-Müller, M., Schmid, D., Ulmer, P.: Matrix rheology effects on reaction rim growth I: evidence from orthopyroxene rim growth experiments. J. Metamorph. Geol. 27(1), 71–82 (2009)

    ADS  Google Scholar 

  28. Miyazaki, K.: Ostwald ripening of garnet in high \(P/T\) metamorphic rocks. Contrib. Miner. Petrol. 108(1), 118–128 (1991)

    ADS  Google Scholar 

  29. Miyazaki, K.: A numerical simulation of textural evolution due to Ostwald ripening in metamorphic rocks: a case for small amount of volume of dispersed crystals. Geochim. Cosmochim. Acta 60(2), 277–290 (1996)

    ADS  Google Scholar 

  30. Moulas, E., Podladchikov, Y.Y., Aranovich, L.Y., Kostopoulos, D.: The problem of depth in geology: when pressure does not translate into depth. Petrology 21(6), 527–538 (2013)

    Google Scholar 

  31. Nemchin, A., Giannini, L., Bodorkos, S., Oliver, N.: Ostwald ripening as a possible mechanism for zircon overgrowth formation during anatexis: theoretical constraints, a numerical model, and its application to pelitic migmatites of the Tickalara Metamorphics, northwestern Australia. Geochim. Cosmochim. Acta 65(16), 2771–2788 (2001)

    ADS  Google Scholar 

  32. Noll, W., Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. In: Coleman, B.D., Noll, W. (eds.) The Foundations of Mechanics and Thermodynamics: Selected Papers, pp. 145–156. Springer, Berlin (1974)

    MATH  Google Scholar 

  33. Perez, M.: Gibbs–Thomson effects in phase transformations. Scr. Mater. 52(8), 709–712 (2005)

    ADS  Google Scholar 

  34. Powell, R., Evans, K.A., Green, E.C., White, R.W.: On equilibrium in non-hydrostatic metamorphic systems. J. Metamorph. Geol. 36(4), 419–438 (2018)

    ADS  Google Scholar 

  35. Sarmiento, A., Espath, L., Vignal, P., Dalcin, L., Parsani, M., Calo, V.M.: An energy-stable generalized-\(\alpha \) method for the swift-Hohenberg equation. J. Comput. Appl. Math. 344, 836–851 (2018)

    MathSciNet  MATH  Google Scholar 

  36. Sekerka, R.F., Cahn, J.W.: Solid-liquid equilibrium for non-hydrostatic stress. Acta Mater. 52(6), 1663–1668 (2004)

    ADS  Google Scholar 

  37. Tajčmanová, L., Podladchikov, Y., Powell, R., Moulas, E., Vrijmoed, J., Connolly, J.: Grain-scale pressure variations and chemical equilibrium in high-grade metamorphic rocks. J. Metamorph. Geol. 32(2), 195–207 (2014)

    ADS  Google Scholar 

  38. Tajčmanová, L., Vrijmoed, J., Moulas, E.: Grain-scale pressure variations in metamorphic rocks: implications for the interpretation of petrographic observations. Lithos 216, 338–351 (2015)

    ADS  Google Scholar 

  39. Truesdell, C.: Historical introit the origins of rational thermodynamics. In: Truesdell, C. (ed.) Rational Thermodynamics, pp. 1–48. Springer, New York (1984)

    MATH  Google Scholar 

  40. Tsagrakis, I., Aifantis, E.C.: Thermodynamic coupling between gradient elasticity and a Cahn–Hilliard type of diffusion: size-dependent spinodal gaps. Contin. Mech. Thermodyn. 29(6), 1181–1194 (2017)

    MathSciNet  MATH  ADS  Google Scholar 

  41. Vrijmoed, J.C., Podladchikov, Y.Y.: Thermodynamic equilibrium at heterogeneous pressure. Contrib. Miner. Petrol. 170(1), 1–27 (2015)

    ADS  Google Scholar 

  42. Wheeler, J.: Dramatic effects of stress on metamorphic reactions. Geology 42(8), 647–650 (2014)

    ADS  Google Scholar 

  43. Zhong, X., Vrijmoed, J., Moulas, E., Tajčmanová, L.: A coupled model for intragranular deformation and chemical diffusion. Earth Planet. Sci. Lett. 474, 387–396 (2017)

    ADS  Google Scholar 

Download references

Acknowledgements

This publication was also made possible in part by the CSIRO Professorial Chair in Computational Geoscience at Curtin University and the Deep Earth Imaging Enterprise Future Science Platforms of the Commonwealth Scientific Industrial Research Organisation, CSIRO, of Australia. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 777778 (MATHROCKS). The Institute for Geoscience Research (TIGeR) and the Curtin Institute for Computation kindly provide continuing support at Curtin University.

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Authors and Affiliations

  1. Ali I. Al-Naimi Petroleum Engineering Research Center, King Abdullah University of Science and Technology (KAUST), 23955-6900, Thuwal, Saudi Arabia

    Santiago P. Clavijo

  2. School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK

    Luis Espath

  3. School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Kent Street, Bentley, Perth, WA, 6102, Australia

    Victor M. Calo

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  1. Santiago P. Clavijo
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  3. Victor M. Calo
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Correspondence to Santiago P. Clavijo.

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Communicated by Andreas Öchsner.

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Clavijo, S.P., Espath, L. & Calo, V.M. The effects of chemical and mechanical interactions on the thermodynamic pressure for mineral solid solutions. Continuum Mech. Thermodyn. 35, 1821–1840 (2023). https://doi.org/10.1007/s00161-023-01200-4

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