Constraints on the Equations of State of stiff anisotropic minerals: rutile, and the implications for rutile elastic barometry

Abstract

We present an assessment of the thermo-elastic behaviour of rutile based on X-ray diffraction data and direct elastic measurements available in the literature. The data confirms that the quasi-harmonic approximation is not valid for rutile because rutile exhibits substantial anisotropic thermal pressure, meaning that the unit-cell parameters change significantly along isochors. Simultaneous fitting of both the diffraction and elasticity data yields parameters of KTR0= 205.14(15) GPa, KSR0= 207.30(14) GPa, $K_{TR0}^{\prime} $= 6.9(4) in a 3rd-order Birch-Murnaghan Equation of State for compression, αV0= 2.526(16) × 10–5 K–1, Einstein temperature θE = 328(12) K, Anderson-Grüneisen parameter δT = 7.6(6), with a fixed thermal Grüneisen parameter γ = 1.4 to describe the thermal expansion and variation of bulk modulus with temperature at room pressure. This Equation of State fits all of the available data up to 7.3 GPa at room temperature, and up to 1100 K at room pressure within its uncertainties. We also present a series of formulations and a simple protocol to obtain thermodynamically consistent Equations of State for the volume and the unit-cell parameters for stiff materials, such as rutile. In combination with published data for garnets, the Equation of State for rutile indicates that rutile inclusions trapped inside garnets in metamorphic rocks should exhibit negative residual pressures when measured at room conditions.

Original Calibration of a Garnet Geobarometer in Metapelite

“…There are other precise, minor element geothermometers applied to estimate metamorphic temperature including the Ti-in-quartz geothermometer [8,9], Zr-in-rutile geothermometer [10][11][12][13][14], and the Ti-in-zircon thermometer [12,15], etc. In recent years, based on the physical properties of the minerals, Raman-spectra geobarometers have also been developed [16][17][18][19][20][21][22][23][24][25][26][27].…”

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.…”

“…Gonzalez et al 2019;. However, calculations based on the equations of state of garnet (Milani et al 2015) and rutile indicate that rutile inclusions trapped inside garnets at typical metamorphic conditions should exhibit negative residual pressures when measured at room conditions provided they did not undergo some non-elastic process after entrapment (Zaffiro et al 2019). Ab initio calculations did not reveal any anomalous behavior of the structure in the studied range of strains which could result in a deviation of the behavior described by the EoS, nor any non-linearity in the Grüneisen relationship between strains and Raman peak shifts (Eqn.…”

“…However, the vibrational frequencies do not depend on the pressure but on the strains (Key 1967) and in our work were found to be linear with volume. The pressure dependence of the volume of rutile is described by the non-linear Burch-Murnaghan equation of state of the third order (Zaffiro et al 2019). Thus, fitting the Δω (cm -1 ) against pressure with a linear equation is not correct.…”

Limits to the Validity of Thermal-Pressure Equations of State

“…This leads to changes of their phonon-mode frequencies along the isochors, and thus invalidates a fundamental assumption behind thermal-pressure EoS including the MGD and the Holland-Powell EoS [16] widely used for petrological and planetary thermodynamics. Fits of a thermal-pressure EoS to volume and elastic data of such materials are often poor, and at the same time can lead to extreme values of EoS parameters [27]. Second, at low pressures below the isochor of the reference volume V 0 (and thus at high temperatures), the static pressure P ref may become sufficiently negative to make the compressional part of the EoS invalid.…”