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.
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References
Anderson OL (1995) Equations of state of solids for geophysics and ceramic science. Oxford University Press, Oxford
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
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
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
Angel RJ, Jackson JM, Reichmann HJ, Speziale S (2009) Elasticity measurements on minerals: a review. Eur J Mineral 21:525–550
Angel RJ, Gonzalez-Platas J, Alvaro M (2014a) EosFit7c and a Fortran module (library) for equation of state calculations. Z Kristallogr 229:405–419
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
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
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
Axe JD, Shirane G (1970) Study of the α–β quartz phase transformation by inelastic neutron scattering. Phys Rev B 1:342–348
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
Boffa-Ballaran T, Angel RJ, Carpenter MA (2000) High-pressure transformation behaviour of the cummingtonite-grunerite solid solution. Eur J Mineral 12:1195–1213
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
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
Carpenter MA, Salje EKH (1998) Elastic anomalies in minerals due to structural phase transitions. Eur J Mineral 10:693–812
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
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
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
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
Dolino G (1990) The α-inc-β transitions of quartz: a century of research on displacive phase transitions. Phase Transit 21:59–72
Dorogokupets PI (1995) Equation of state for lambda transition in quartz. J Geophys Res 100:8489–8499
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
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
Hellfrich G, Connolly JAD (2009) Physical contradictions and remedies using simple polythermal equations of state. Am Mineral 94:1616–1619
Holland TJB, Powell R (1998) An internally consistent thermodynamic data set for phases of petrological interest. J Metamorph Geol 16:309–343
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
Hosieni KR, Howald RA, Scanlon MW (1985) Thermodynamics of the lambda transition and the equation of state of quartz. Am Mineral 70:782–793
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
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
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
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
Landau LD, Lifshitz EM (1969) Statistical Physics. Pergamon Press, Oxford
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
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
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
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
Orear J (1982) Least squares when both variables have uncertainties. Am J Phys 50:912–916
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
Perez-Mato JM, Orobengoa D, Aroyo MI (2010) Mode crystallography of distorted structures. Acta Crystallogr A 66:558–590
Pippard AB (1956) Thermodynamic relations applicable near a lambda-transition. Phil Mag 1:473–476
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
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
Rosenfeld JL, Chase AB (1961) Pressure and temperature of crystallization from elastic effects around solid inclusion minerals? Am J Sci 259:519–541
Salje EKH (1985) Thermodynamics of sodium feldspar I: order parameter treatment and strain induced coupling effects. Phys Chem Minerals 12:93–98
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
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
Schranz W, Havlik D (1999) Acoustic dispersion near structural phase transitions. Phase Transit 68:557–566
Schranz W, Tröster A, Koppensteiner J, Miletich R (2007) Finite strain Landau theory of high pressure phase transformations. J Phys Condens Matter 19:275202
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
Slonczewski JC, Thomas H (1970) Interaction of elastic strain with the structural phase transition of strontium titanite. Phys Rev B 1:3599–3608
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
Tröster A, Schranz W, Miletich R (2002) How to couple Landau theory to an equation of state. Phys Rev Lett 88:055503-055501-055504
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
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
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
Welche PRL, Heine V, Dove MT (1998) Negative thermal expansion in beta-quartz. Phys Chem Miner 26:63–77
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.
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Communicated by Timothy L. Grove.
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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
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DOI: https://doi.org/10.1007/s00410-017-1349-x