DI44A-01 INVITED
Static and Dynamic Equations of State: Key Challenges and Solutions
Together with the crystal structure, the equation of state is the most fundamental parameter obtained from high- pressure investigations. Each year a large number of papers are published whose main objective is accurate determination of the equation of state of various materials. However, challenges in equation of state measurement abound, and it is fair to say that many of these papers fail to achieve their goal, or at least fail to demonstrate that they have achieved their goal. In this paper, I will review the current status of equation of state measurement on geomaterials, focusing especially on the challenges associated with measurements to Mbar pressures. A significant dilemma in equation of state measurements is the trade-off between measurement accuracy and pressure range. The crux of the problem is the need to maximize pressure range while minimizing the contribution from deviatoric stress. Recent developments in radial x-ray diffraction have greatly advanced our understanding of differential stresses, and their application to equations of state will be discussed. The usefulness of direct elasticity measurements such as Brillouin scattering for accurate equations of state will be demonstrated with several recent examples. Shock compression methods are evolving rapidly and continue to play an important role in equation of state measurement. New dynamic quasi-isentropic compression measurements to multi-megabar pressures are promising but further highlight the importance of characterizing shear stresses at ultrahigh pressures. New Hugoniot data for metals such as gold can be used to sort out uncertainties in pressure scales at Mbar conditions. Finally, it has been noted that some materials such calcium fluoride and oxide garnets adopt virtually incompressible high-pressure phases above 1 Mbar under shock loading (Nellis and Petach, 2007). New static high-pressure measurements can shed light on the equation of state and structure of these materials under such conditions.
DI44A-02
Thermodynamic Properties of the Magnesium-Olivine-Pyroxene System Derived From a Lattice Vibrational Technique
We are currently constructing a thermodynamic database providing phase diagrams, thermophysical and thermochemical properties for materials with a geophysical relevance, applicable in the pressure and temperature regime of the Earth's mantle. The computational technique is based on Kieffer's (1979) approach to model the vibrational density of states of a substance, a key property to derive the Helmholtz energy. The developed thermodynamic framework, which allows the calculation of Vp and Vs sound wave velocities, uses model-input properties related to Raman and infrared spectroscopic data. It puts tighter constraints on thermodynamic properties compared to methods based on polynomial parameterizations of thermal expansivity, heat capacity and isothermal bulk modulus. Jacobs & de Jong (2005, 2007) showed that this framework discriminates, based on internal consistency, between the quality of disparate sets of experimental thermochemical, thermophysical and phase diagram data. The present work focuses on the application of vibrational modeling to the magnesium-olivine-pyroxene system, a system relevant to Earth's mantle. We show how our approach is used to point to inconsistencies in experimental datasets. Pressure calibration problems affecting the derivation of phase diagrams are discussed. The results, presented here, were used in a numerical model of convection in the Earth's mantle to reveal, effects of phase transitions on the degree of layering, mineral distribution and sound wave velocities in the transition zone, around 660 km depth in the Earth. References Kieffer S.W. (1979), Rev. Geophys. Space Physics, 17, 35-59. Jacobs M.H.G. and B.H.W.S. de Jong (2005), Phys. Chem. Minerals, 32, 614-626. Jacobs M.H.G., and de Jong B.H.W.S. (2007), Geochim. Cosmochim. Acta, 71, 3630-3655.
DI44A-03
Crystal Chemical Controls on Equation of State
Minerals are known to compress through a number of mechanisms, ranging from polyhedral distortion to electronic transitions. Two mechanisms which can produce significant volume decreases are angle-bending and bond compression. The crystal chemical effects of these two mechanisms have been studied and documented for years. With more recent advances in theory and software enabling the accurate determination of bonding topologies, M-O bonding to bridging oxygens has been shown to modify compressibility by changing angle-bending force constants. Minerals that compress mainly through angle-bending tend be soft. Good examples are quartz and cristobalite, minerals composed solely of corner-sharing silicate tetrahedra with bulk moduli of 37 and 12 GPa, respectively. Rock salt structured oxides must compress strictly by bond compression, and are much stiffer – lime and periclase have bulk moduli of 111 and 156 GPa, respectively. Feldspars have bulk moduli intermediate to the above examples. Based solely on the presence of Al-O-Si angles, theoretically softer than Si-O-Si angles, feldspars should be softer than quartz or cristobalite, but the T-O-T angles are stiffened by bonds to interstitial cations. The number and nature of these bonds affects compressibility sufficiently to create exceptions to Bridgman's law, which correlates bulk modulus with ambient unit cell volume in isostructural materials. In this paper, we present new high-pressure refinements of the crystal structures of jadeite, aegirine, and NaGa- clinopyroxene. Bulk moduli of these pyroxenes and all other end-member clinopyroxenes we could find in the literature (19 total) are plotted vs. unit cell volumes to test Bridgman's law. The data fall along two trends, each of which is separately consistent with Bridgman's law. Pyroxenes in one trend are dramatically stiffer than those in the other trend, with bulk moduli that differ by approximately 40 GPa. The only difference between the topologies of the structures in the two trends is in the bonding around M2. Structures in the less compressible trend have M2-O3 bonds that oppose Si-O-Si angle-bending in the tetrahedral chains. This angle-bending is an important compression mechanism in pyroxenes. McCarthy et al. (in press) term these bonds "antipathetic". Pyroxenes in the more compressible trend lack these bonds. There are other M2-O3 bonds that visual inspection suggests might tend to encourage angle-bending, but do not appear to have an effect. McCarthy et al. term these bonds "apathetic," and suggest the term "sympathetic" for M-O bonds that actually soften angles. Other examples from the literature will be presented including one from the feldspars that may be a truly sympathetic bond. McCarthy, A.C., Downs, R.T., and Thompson, R.M. (in press) Compressibility trends of the clinopyroxenes, and in- situ high-pressure single-crystal X-ray diffraction study of jadeite. American Mineralogist.
DI44A-04 INVITED
Some Theoretical Issues on the Equation of State of Silicate Melts
Properties of silicate melts, particularly their densities under deep Earth conditions (i.e., the equation of state), are critical to the understanding of evolution of terrestrial planets. However, the inspection of the existing experimental data reveals fundamental differences in the mechanisms of compression between melts and solids that poses important questions as to the interpretation and applications of experimental data on compression of melts to Earth science problems. Common to all complex liquids, the X-ray studies of radial distribution function (RDF) show the presence of short-range-order (SRO) but absence of long-range-order (LRO). Furthermore, the classic analysis by Bottinga-Weill (1970) showed that a silicate melt can be modeled as a mixture of oxide components, where the (partial) molar volumes of component oxides are close to those of solid counterparts. However, the observed RDF indicates that the bond-length of "oxide component" does not shrink as much as expected from the volume reduction under compression. Also the observed bulk moduli for individual "oxide" components in melts are much less than those of solid counterparts, indicating that much of the compression occurs through the geometrical rearrangement of oxide units. In addition, RDF and NMR observations show that in many silicate melts, coordination numbers of oxygen surrounding cations increase (continuously) with pressure. These two types of structural changes at different scales, namely the geometrical rearrangement of oxide units and the coordination changes of individual oxide units themselves, contribute significantly to the compression of silicate melts. We present a simple model to incorporate these structural details in the equation of state of silicate melts. We find that the coordination change affects the internal energy (and the vibrational entropy) whereas the geometrical arrangement of oxide units contributes to the configurational entropy. Consequently, the former affects the effective compressibility of silicate melts, while the latter affect the thermal component of equation of state such as the Grüneisen parameter and the temperature dependence of bulk modulus. We will discuss the implications of the present model for the validity of ideal mixing model.
DI44A-05 INVITED
Anelastic behaviour of FexO at high pressure
Elastic properties of solids are among the most important for solid Earth geophysics, engineering, and solid- state physics. Elastic moduli can be determined from pressure-volume relations (static or shock-wave compression), from acoustic waves velocity measurements (ultrasonic interferometry), or from dispersion curves of acoustic phonon branches (neutron, x-ray or light inelastic scattering). In the case of an ideal elastic solid the elastic moduli determined using different techniques should coincide within the experimental error (with the conversion factor between isothermal and adiabatic moduli). However, if any anelastic relaxation exists and the equilibrium strain for a given stress is achieved only after certain finite time interval, the effective elastic moduli measured by different methods would systematically vary, depending on sampling frequency. Anelasticity in a solid appears due to defects or other crystal imperfections when the energy minimum is achieved not only by varying the atomic geometry but also by changes in the materials' mesostructure (e.g. structure with a characteristic length larger than the crystal unit cell size). We have chosen non-stoichiometric FexO as a candidate for a strongly anelastic material. Although FexO has been intensively studied for several decades, many of its properties including elastic behaviour remain controversial and not well understood. A combined single-crystal inelastic x-ray scattering (IXS) and x-ray diffraction (XRD) study of synthetic wüstite Fe0.95O at elevated pressure revealed increasing the difference in the bulk modulus determined from static (XRD) and dynamic (IXS) measurements upon compression. We explain and quantitatively describe this observation by anelastic relaxation in wüstite. The reanalysis of previous results put some more evidence for relative good coincidence for bulk moduli and systematic difference for K' between static and dynamic measurements.
DI44A-06
High-Accuracy Pressure Scale Computed With Quantum Monte Carlo
There has recently been considerable interest in improving pressure calibration standards for ultrahigh pressure studies in the diamond anvil cell. Developing such a standard requires reliable equation of state (EOS) data, however there are at present no primary pressure standards for the megabar range. Recently, cubic boron nitride (cBN) has been suggested as an accurate pressure gauge [1]. Unlike ruby, its structure is highly constrained by symmetry and stable beyond 100 GPa, and it has a well-separated Raman spectrum with sufficient pressure dependence to allow accurate pressure calibration. Calculation of the cBN EOS with density functional theory (DFT) gives reasonable agreement with experiment, but the results from different approximate functionals disagree. Quantum Monte Carlo is a first principles simulation method which circumvents approximate functionals and is widely regarded to provide the most accurate predictions of the properties of solids available. We utilize state-of-the-art QMC methods to obtain the static EOS for cBN. To this, we apply corrections for phonon contributions computed within DFT. The resulting theoretical EOS agrees well with experiment at pressures below 50 GPa, with a deviation from the zero-pressure lattice constant of only +0.25%. Above 50 GPa, our EOS begins to differ from recent measurements by Datchi et al. [2], with a maximum deviation of approximately -10 GPa at 160 GPa, the highest compression achieved experimentally. Possible reasons for this discrepancy, both theoretical and experimental, will be discussed. [1] Phys. Rev. B 72, 100104R (2005) [2] Phys. Rev. B 75, 214104 (2007)
DI44A-07
Thermal Equations of State in the Fe-FeS, Fe-FeO, and Ammonia-Water Systems
With the maturing of laser heating systems for the diamond anvil coupled to 3rd generation synchrotron radiation sources, studies at high pressures are moving from end member to binary and even more complex systems. Interesting discoveries are being made, for instance the addition of Si to pure Fe stabilizes the bcc phase to exceptionally high pressures [Lin, et al., 2002, Science]. More recently Ni has been found to also stabilize the bcc structure of iron at pressures in excess of 200 GPa [Dubrovinsky, et al., 2007, Science]. Thus minor elements can significantly change the topology of phase diagrams. The observations of Lin et. al. sparked the interests of theorticians who modeled the effects of impurities [Vocadlo, et al., 2003, Nature]. Various compositions in the binary system Fe-FeS have been studied to 160 GPa and in the Fe-FeO system to 100 GPa. In addition to thermal equation of state data we have also collected melting data and subsolidus phase relations. This data provide information on how these candidate minerals would behave at the elevated temperatures and pressures, and thus have implications for the current state of the interior of terrestrial planets and their thermal evolution. On a lighter note (in terms of atomic weight, and icy planets), the addition of ammonia to water affects the phase diagram of pure water in unexpected ways. For example at high ammonia concentrations, (>25 wt. %) Ice IV, a metastable phase in the pure water phase diagram is the first phase to crystallize. On further compression Ice IV transforms directly to Ice VII with a staggering 13 % density increase. The equation of state of Ice VI has also been studied, and has an impressively low bulk modulus of only 6 GPa. Pressure, temperature, volume and composition data in conjunction with the associated phase equilibria provide strong constraints for quantum mechanical calculations.
DI44A-08
The effect of iron concentration on the compressibility of (Fe,Mg)(Al,Si)O3 perovskite.
Recent studies have indicated that the substitution of Al into (Mg,Fe)(Al,Si)O3 perovskite is coupled to ferric Fe and that a significant portion of the Fe in the lower mantle perovskite is, therefore, likely to be in the ferric state. The substitution of trivalent cations into silicate perovskite can occur either by coupled substitution onto both Mg and Si sites or by substitution onto the Si site charge balanced by the creation of an oxygen vacancy. The results of previous studies are inconclusive as to the effect of these substitutions on elastic properties. To resolve these issues we have investigated the effect of bulk iron concentration on equation of state properties of Al bearing magnesium silicate perovskite using single crystal x-ray diffraction in a diamond anvil cell. Three single crystals with the compositions, (Mg2+0.93,Fe3+0.034,Fe2+0.035,Al3+0.012)(Si4+0.89,Al3 +0.096)O2.96 (Mg2+0.872,Fe3+0.088,Fe2+0.042,Al3+0.007)(Si4+0.892,Al 3+0.108)O3 (Mg2+0.782,Fe3+0.192,Fe2+0.048,Al3+0.031)(Si4+0.863,Al {3+}0.137)O2.95 were synthesized using a multianvil apparatus at 25 GPa and 2000°C. Ferric iron concentrations were determined using Mössbauer and Electron energy loss spectroscopic techniques. The single crystals were compressed to 9 GPa at ambient temperature. The compression data were fitted to a third order Birch Murnaghan equation of state. The bulk modulii of the three crystals decreases with increasing Fe content from 243(3) GPa to 240(2) GPa to 234(2) GPa. In addition, increasing the Fe content raises the K'.