A simple and generalised P–T–V EoS for continuous phase transitions, implemented in EosFit and applied to quartz

Abstract

Continuous phase transitions in minerals, such as the α–β transition in quartz, can give rise to very large non-linear variations in their volume and density with temperature and pressure. The extension of the Landau model in a fully self-consistent form to characterize the effects of pressure on phase transitions is challenging because of non-linear elasticity and associated finite strains, and the expected variation of coupling terms with pressure. Further difficulties arise because of the need to integrate the resulting elastic terms over pressure to achieve a description of the P–T–V equation of state. We present a fully self-consistent simplified description of the equation of state of minerals with continuous phase transitions based on a purely phenomenological adaptation of Landau theory. The resulting P–T–V EoS includes the description of the elastic softening occurring in both phases with the minimum number of parameters. By coupling the volume and elastic behaviour of the mineral, this approach allows the EoS parameters to be determined by using both volume and elastic data, and avoids the need to use data at simultaneous P and T. The transition model has been incorporated in to the EosFit7c program, which allows the parameters to be determined by simultaneous fitting of both volume and elastic data, and all types of equation of state calculations to be performed. Quartz is used as an example, and the parameters to describe the full P–T–V EoS of both α- and β-quartz are determined.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

References

  1. Anderson OL (1995) Equations of state of solids for geophysics and ceramic science. Oxford University Press, Oxford

    Google Scholar 

  2. Angel RJ (2000) Equations of state. In: Hazen RM, Downs RT (eds) High-pressure and high-temperature crystal chemistry, vol 41. Reviews in Mineralogy and Geochemistry. MSA, Chantilly, VA, USA, pp 35–60

  3. Angel RJ, Bismayer U (1999) Renormalization of the phase transition in lead phosphate, Pb3(PO4)2, by high pressure: lattice parameters and spontaneous strain. Acta Crystallogr B-Struct Sci 55:896–901

    Article  Google Scholar 

  4. Angel RJ, Allan DR, Miletich R, Finger LW (1997) The use of quartz as an internal pressure standard in high-pressure crystallography. J Appl Crystallogr 30:461–466

    Article  Google Scholar 

  5. Angel RJ, Jackson JM, Reichmann HJ, Speziale S (2009) Elasticity measurements on minerals: a review. Eur J Mineral 21:525–550

    Article  Google Scholar 

  6. Angel RJ, Gonzalez-Platas J, Alvaro M (2014a) EosFit7c and a Fortran module (library) for equation of state calculations. Z Kristallogr 229:405–419

    Google Scholar 

  7. Angel RJ, Mazzucchelli ML, Alvaro M, Nimis P, Nestola F (2014b) Geobarometry from host-inclusion systems: the role of elastic relaxation. Am Mineral 99:2146–2149

    Article  Google Scholar 

  8. Ashley KT, Caddick MJ, Steele-MacInnis MJ, Bodnar RJ, Dragovic B (2014) Geothermobarometric history of subduction recorded by quartz inclusions in garnet. Geochem Geophys Geosyst 15:350–360 doi:10.1002/2013GC005106

    Article  Google Scholar 

  9. Ashley K, Steele-MacInnis M, Bodnar RJ, Darling RS (2016) Quartz-in-garnet inclusion barometry under fire: reducing uncertainty from model estimates. Geology 44:699–702. doi:10.1130/G38211.1

    Article  Google Scholar 

  10. Axe JD, Shirane G (1970) Study of the α–β quartz phase transformation by inelastic neutron scattering. Phys Rev B 1:342–348

    Article  Google Scholar 

  11. Berman RG (1988) Internally-consistent thermodynamic data for minerals in the system Na2O–K2O–CaO–MgO–FeO–Fe2O3–Al2O3–SiO2–TiO2–H2O–CO2. J Petrol 29:445–522

    Article  Google Scholar 

  12. Boffa-Ballaran T, Angel RJ, Carpenter MA (2000) High-pressure transformation behaviour of the cummingtonite-grunerite solid solution. Eur J Mineral 12:1195–1213

    Article  Google Scholar 

  13. Camara F, Carpenter M, Domeneghetti MC, Tazzoli V (2003) Coupling between non-convergent ordering and transition temperature in the C2/c–P21/c phase transition in pigeonite. Am Mineral 88:1115–1128

    Article  Google Scholar 

  14. Carpenter MA (2000) Strain and elasticity at structural phase transitions in minerals. In: Transformation processes in minerals, vol 39. Reviews in mineralogy and geochemistry. pp 35–64

  15. Carpenter MA, Salje EKH (1998) Elastic anomalies in minerals due to structural phase transitions. Eur J Mineral 10:693–812

    Article  Google Scholar 

  16. Carpenter MA, Salje EKH, Graeme-Barber A (1998a) Spontaneous strain as a determinant of thermodynamic properties for phase transitions in minerals. Eur J Mineral 10:621–691

    Article  Google Scholar 

  17. Carpenter MA, Salje EKH, Graeme-Barber A, Wruck B, Dove MT, Knight KS (1998b) Calibration of excess thermodynamic properties and elastic constant variations associated with the alpha–beta phase transition in quartz. Am Mineral 83:2–22

    Article  Google Scholar 

  18. Carpenter MA, Hemley RJ, Mao HK (2000) High-pressure elasticity of stishovite and the P4(2)/mnm–Pnnm phase transition. J Geophys Res Solid Earth 105:10807–10816

    Article  Google Scholar 

  19. Demuth T, Jeanvoine Y, Hafner J, Angyan J (1999) Polymorphism in silica studied in the local density and generalized-gradient approximations. J Phys Condens Matter 11:3833–3874

    Article  Google Scholar 

  20. Dolino G (1990) The α-inc-β transitions of quartz: a century of research on displacive phase transitions. Phase Transit 21:59–72

    Article  Google Scholar 

  21. Dorogokupets PI (1995) Equation of state for lambda transition in quartz. J Geophys Res 100:8489–8499

    Article  Google Scholar 

  22. Ehrenfest P (1933) Phase changes in the ordinary and extended sense classified according to the corresponding singularities of the thermodynamic potential. Proc Acad Sci Amst 36:153–157

    Google Scholar 

  23. Heine V, Welche PRL, Dove MT (1999) Geometrical origin and theory of negative thermal expansion in framework structures. J Am Ceram Soc 82:1793–1802

    Article  Google Scholar 

  24. Hellfrich G, Connolly JAD (2009) Physical contradictions and remedies using simple polythermal equations of state. Am Mineral 94:1616–1619

    Article  Google Scholar 

  25. Holland TJB, Powell R (1998) An internally consistent thermodynamic data set for phases of petrological interest. J Metamorph Geol 16:309–343

    Article  Google Scholar 

  26. Holland TJB, Powell R (2011) An improved and extended internally consistent thermodynamic dataset for phases of petrological interest, involving a new equation of state for solids. J Metamorph Geol 29:333–383. doi:10.1111/j.1525-1314.2010.00923.x

    Article  Google Scholar 

  27. Hosieni KR, Howald RA, Scanlon MW (1985) Thermodynamics of the lambda transition and the equation of state of quartz. Am Mineral 70:782–793

    Google Scholar 

  28. Kimizuka H, Kaburaki H, Kogure Y (2003) Molecular-dynamics study of the high-temperature elasticity of quartz above the α–β phase transition. Phys Rev B 67:024105

    Article  Google Scholar 

  29. Kouketsu Y, Nishiyama T, Ikeda T, Enami M (2014) Evaluation of residual pressure in an inclusion–host system using negative frequency shift of quartz Raman spectra. Am Mineral 99:433–442

    Article  Google Scholar 

  30. Kroll H, Kirfel A, Heinemann R, Barbier B (2012) Volume thermal expansion and related thermophysical parameters in the Mg, Fe olivine solid-solution series. Eur J Mineral 24:935–956

    Article  Google Scholar 

  31. Lakshtanov DL, Sinogeilin SV, Bass JD (2007) High-temperature phase transitions and elasticity of silica polymorphs. Phys Chem Miner 34:11–22. doi:10.1007/s00269-006-0113-y

    Article  Google Scholar 

  32. Landau LD, Lifshitz EM (1969) Statistical Physics. Pergamon Press, Oxford

    Google Scholar 

  33. McConnell JDC, McCammon CA, Angel RJ, Seifert F (2000) The nature of the incommensurate structure in akermanite, Ca2MgSi2O7, and the character of its transformation from the normal structure. Z Kristallogr 215:669–677

    Google Scholar 

  34. Milani S, Angel RJ, Scandolo L, Mazzucchelli ML, Boffa-Ballaran T, Klemme S, Domeneghetti MC, Miletich R, Scheidl KS, Derzsi M, Tokar K, Prencipe M, Alvaro M, Nestola F (2017) Thermo-elastic behaviour of grossular garnets at high pressures and temperatures. Am Mineral 102:851–859

    Article  Google Scholar 

  35. Mirwald PW, Massonne H-J (1980) The low-high quartz and quartz-coesite transition to 40 kbar between 600 and 1600 °C and some reconnaissance data on the effect of NaAlO2 component on the low quartz-coesite transition. J Geophys Res B 85:6983–6990

    Article  Google Scholar 

  36. Müser M, Binder K (2001) Molecular dynamics study of the α–β transition in quartz: elastic properties, finite size effects, and hysteresis in the local structure. Phys Chem Miner 28:746–755

    Article  Google Scholar 

  37. Orear J (1982) Least squares when both variables have uncertainties. Am J Phys 50:912–916

    Article  Google Scholar 

  38. Peng Z, Chien S-Y, Redfern SAT (2012) Dynamic mechanical relaxation and loss in the incommensurate phase of quartz. J Phys Condens Matter 24:255403. doi:10.1088/0953-8984/24/25/255403

    Article  Google Scholar 

  39. Perez-Mato JM, Orobengoa D, Aroyo MI (2010) Mode crystallography of distorted structures. Acta Crystallogr A 66:558–590

    Article  Google Scholar 

  40. Pippard AB (1956) Thermodynamic relations applicable near a lambda-transition. Phil Mag 1:473–476

    Article  Google Scholar 

  41. Raz U, Girsperger S, Thompson AB (2002) Thermal expansion, compressibility and volumetric changes of quartz obtained by single crystal dilatometry to 700 °C and 3.5 kilobar (0.35 GPa). Schweiz Mineral Petrogr Mitt 82:561–574. doi:10.5169/seals-62381

    Google Scholar 

  42. Rodriguez-Carvajal J, Gonzalez-Platas J (2003) Crystallographic Fortran 90 Modules Library (CrysFML): a simple toolbox for crystallographic computing programs. IUCr Comput Commission Newslett 1:50–58

    Google Scholar 

  43. Rosenfeld JL, Chase AB (1961) Pressure and temperature of crystallization from elastic effects around solid inclusion minerals? Am J Sci 259:519–541

    Article  Google Scholar 

  44. Salje EKH (1985) Thermodynamics of sodium feldspar I: order parameter treatment and strain induced coupling effects. Phys Chem Minerals 12:93–98

    Article  Google Scholar 

  45. Salje E, Wruck B, Thomas H (1991) Order-parameter saturation and low-temperature extension of Landau theory. Zeitschrift für Physik B. Condens Matter 82:399–404

    Article  Google Scholar 

  46. Scheidl K, Kurnosov A, Trots DM, Boffa-Ballaran T, Angel RJ, Miletich R (2016) Extending the single-crystal quartz pressure gauge to hydrostatic pressures of 19 GPa. J Appl Crystallogr 49:2129–2137. doi:10.1107/S1600576716015351

    Article  Google Scholar 

  47. Schranz W, Havlik D (1999) Acoustic dispersion near structural phase transitions. Phase Transit 68:557–566

    Article  Google Scholar 

  48. Schranz W, Tröster A, Koppensteiner J, Miletich R (2007) Finite strain Landau theory of high pressure phase transformations. J Phys Condens Matter 19:275202

    Article  Google Scholar 

  49. Shen AH, Bassett WA, Chou I-M (1993) The α–β quartz transition at high temperatures and pressures in a diamond-anvil cell by laser interferometry. Am Mineral 78:694–698

    Google Scholar 

  50. Slonczewski JC, Thomas H (1970) Interaction of elastic strain with the structural phase transition of strontium titanite. Phys Rev B 1:3599–3608

    Article  Google Scholar 

  51. Sochalski-Kolbus LM, Angel RJ, Nestola F (2010) The effect of Al/Si disorder on the bulk moduli of plagioclase feldspars. Mineral Mag 74:943–950. doi:10.1180/minmag.2010.074.6.943

    Article  Google Scholar 

  52. Tröster A, Schranz W, Miletich R (2002) How to couple Landau theory to an equation of state. Phys Rev Lett 88:055503-055501-055504

    Article  Google Scholar 

  53. Tröster A, Schranz, W., Karsai F, Blaha, P. (2014) Fully consistent finite-strain Landau theory for high-pressure phase transitions. Physi Rev X 4:031010

    Google Scholar 

  54. Wang Z, Liu Y, Song W, Bi Y, Xie H (2011) A broadband spectroscopy method for ultrasonic wave velocity measurement under high pressure. Rev Sci Instrum 82:014501. doi:10.1063/1.3518953

    Article  Google Scholar 

  55. Wang J, Mao Z FJ, Duffy T (2015) Elasticity of single-crystal quartz to 10 GPa. Phys Chem Miner 42:203–212. doi:10.1007/s00269-014-0711-z

    Article  Google Scholar 

  56. Welche PRL, Heine V, Dove MT (1998) Negative thermal expansion in beta-quartz. Phys Chem Miner 26:63–77

    Article  Google Scholar 

Download references

Acknowledgements

Software development for this project was supported by ERC starting grant 307322 to Fabrizio Nestola, and by the MIUR-SIR Grant “MILE DEEp” (RBSI140351) to Matteo Alvaro. We thank Andrea D’Alpaos and Mario Putti (University of Padova) for advice on least-squares minimisation, Tim Holland (Cambridge), Tom Duffy (Princeton) and Kyle Ashley (Texas) for discussions on various aspects of equations of state and phase transitions, Aaron Wolf (Ann Arbor) for a thorough and thought-provoking review, and Javier Gonzalez-Platas (La Laguna) for continuing collaboration on the development of the cfml library.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ross J. Angel.

Additional information

Communicated by Timothy L. Grove.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Angel, R.J., Alvaro, M., Miletich, R. et al. A simple and generalised P–T–V EoS for continuous phase transitions, implemented in EosFit and applied to quartz. Contrib Mineral Petrol 172, 29 (2017). https://doi.org/10.1007/s00410-017-1349-x

Download citation

Keywords

  • Quartz
  • Continuous phase transition
  • Equation of state
  • Elasticity
  • EosFit