Commentary on “Constraints on the Equations of State of stiff anisotropic minerals: rutile, and the implications for rutile elastic barometry” [Miner. Mag. 83 (2019) pp. 339–347]

Abstract

The conclusion of Zaffiro et al. (2019; Constraints on the Equations of State of stiff anisotropic minerals: rutile, and the implications for rutile elastic barometry. Mineralogical Magazine, 83, 339–347) that the Mie–Grüneisen–Debye (MGD) Equation of State (EoS) cannot fit the available data for rutile is shown to be incorrect, even though rutile exhibits significant anisotropic thermal pressure which invalidates the quasi-harmonic approximation used as the basis for the MGD EoS. The refined parameters for the MGD EoS of rutile are: KTR0= 205.05(25) GPa, $K_{TR0}^{\prime} $ = 7.2(5), θD = 399(20) K, γ0= 1.40(2) and q = 1.5(7). This EoS predicts volumes, bulk moduli and volume thermal expansion coefficients for rutile at metamorphic conditions that are statistically indistinguishable from those predicted by the ‘isothermal’ type of EoS reported previously.

A Grüneisen tensor for rutile and its application to host-inclusion systems

“…The zero-pressure bulk modulus determined by fitting the volumes from the hydrostatic simulations against static pressures with a 3 rd order Birch-Murnaghan equation of state (BM3-EoS), is 233.78(24) GPa with the ′ = 4.54(5). The calculated bulk modulus is stiffer than that determined by Zaffiro et al (2019) and Angel et al (2020) from the available data in the literature (K0T = 205.14(15) GPa, ′=6.9(4)), obtained by using the same EoS, partially because our simulations are performed at the static limit and therefore do not account either for the zero-point pressure nor for thermal pressure that would soften the bulk modulus on passing from 0K to 300K (Prencipe et al 2011). In addition, the rutile structure has a soft mode whose effect on the bulk modulus will not be accounted for in our static DFT simulations.…”

Thermal equation of state of Fe3O4 magnetite up to 16 GPa and 1100 K

“…The entire P-V-T dataset was fitted by a BM2 EoS combined with a Mie-Grüneisen-Debye (MGD) model for the thermal pressure, using the EoSfit7 GUI software (Angel et al 2014;Gonzalez-Platas et al 2016) to obtain V0 and KT0 and the high-temperature parameters, namely the Debye temperature 𝜃D, the Grüneisen parameter 𝛾 and its logarithmic volume derivative q. We took advantage of the q-compromise Mie-Grüneisen-Debye approach implemented in EoSfit7 GUI (Angel et al 2020) in order to reduce the number of fitting parameters and the otherwise large correlation of q with 𝛾 in the relatively limited pressure and temperature range targeted in this study. Still, refining K0, 𝜃D, and 𝛾 was yielding larger correlations and thus uncertainties, especially for 𝜃D and γ (𝜃D = 1397(865) K with V0 = 44.604(7) cm 3 /mol, K0 = 182(1) GPa and 𝛾 = 2.3( 8)).…”

Elasticity of mixtures and implications for piezobarometry of mixed-phase inclusions

“…Further small increases in amphibole content (e.g. 9 vol % and 10 vol %) continue to reduce the bulk modulus of the inclusion towards (Angel et al, 2022b), rutile (Angel et al, 2020) and cummingtonite (Holland and Powell, 2011).…”

“…(b) Calculated pressures of rutile plus amphibole inclusions trapped in pyrope garnet at 3 GPa and 800 • C. The grey area indicates the compositions for which P foot is either not defined or exceeds ±10 GPa. Calculations with the EoSs of pyrope(Angel et al, 2022b), rutile(Angel et al, 2020) and cummingtonite(Holland and Powell, 2011).…”

“…The intersection of the entrapment isomekes with 25 • C (vertical dashed line) correspond to the P foot values shown in Fig.11; note that the entrapment isomekes for inclusion with 9 vol %-12 vol % amphibole do not arrive to 25 • C and therefore do not have a P foot . Calculated with the EoSs of pyrope(Angel et al, 2022b), rutile(Angel et al, 2020) and cummingtonite(Holland and Powell, 2011).…”