The particle-in-cell model for ab initio thermodynamics: implications for the elastic anisotropy of the Earth’s inner core

Gannarelli, C. M. S.; Alfè, Dario; Gillan, M. J.

doi:10.1016/j.pepi.2003.09.001

Cited by 38 publications

(31 citation statements)

References 43 publications

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“…DFT results for the perfect lattice free energy for c/a = 1.6 were reported earlier by Alfè et al (2001), and were more recently reported by Gannarelli et al (2003) for q in the range 1.48-1.72. The present calculations are closely related to the previous ones, but we have introduced some refinements.…”

Section: Perfect Lattice Free Energysupporting

confidence: 78%

The axial ratio of hcp iron at the conditions of the Earth’s inner core

Gannarelli

Alfè

Gillan

2005

Physics of the Earth and Planetary Interiors

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We present ab initio calculations of the high-temperature axial c/a ratio of hexagonalclose-packed (hcp) iron at Earth's core pressures, in order to help interpret the observed seismic anisotropy of the inner core. The calculations are based on density functional theory, which is known to predict the properties of high-pressure iron with good accuracy. The temperature dependence of c/a is determined by minimising the Helmholtz free energy at fixed volume and temperature, with thermal contributions due to lattice vibrations calculated using harmonic theory. Anharmonic corrections to the harmonic predictions are estimated from calculations of the thermal average stress obtained from ab initio molecular dynamics simulations of hcp iron at the conditions of the inner core. We find a very gradual increase of axial ratio with temperature. This increase is much smaller than found in earlier calculations, but is in reasonable agreement with recent high-pressure, high-temperature diffraction measurements. This result casts doubt on an earlier interpretation of the seismic anisotropy of the inner core.

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Section: Perfect Lattice Free Energysupporting

confidence: 78%

The axial ratio of hcp iron at the conditions of the Earth’s inner core

Gannarelli

Alfè

Gillan

2005

Physics of the Earth and Planetary Interiors

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“…Low temperature, high pressure calculations (Stixrude and Cohen, 1995;Laio et al, 2000;Steinle-Neumann et al, 2001;Gannarelli et al, 2003Gannarelli et al, , 2005Vočadlo et al, 2009;Sha and Cohen, 2010;Chen et al, 2011) suggest that Pwave propagation is faster along the c-axis than along the a-axis, in agreement with low pressure/low temperature hcp analogs. Experimental determination of the elastic constants suggest a fast direction lying at an intermediate angle between the a and c-axes (Mao et al, 1998;Merkel et al, 2005), but the technique that was used includes serious artifacts related to stress heterogeneities induced by plastic deformation (Antonangeli et al, 2006;Merkel et al, 2009).…”

Section: Stratified Flow Bsupporting

confidence: 60%

“…Mao et al, 1998;Fiquet et al, 2001;Merkel et al, 2005;Antonangeli et al, 2006;Badro et al, 2007;Mao et al, 2008) while they remain a challenge for the most advanced ab initio calculations (e.g. Stixrude and Cohen, 1995;Laio et al, 2000;Steinle-Neumann et al, 2001;Gannarelli et al, 2003Gannarelli et al, , 2005Vočadlo et al, 2009;Sha and Cohen, 2010;Chen et al, 2011). From those experiments and calculations, the anisotropy of hcp iron remains unclear.…”

Section: Stratified Flow Bmentioning

confidence: 99%

Texturing in Earth’s inner core due to preferential growth in its equatorial belt

Deguen¹,

Cardin²,

Merkel³

et al. 2011

Physics of the Earth and Planetary Interiors

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We propose an extension of the model by Yoshida et al. (1996), where deformation in the inner core is forced by preferential growth in the equatorial belt, by taking into account the presence of a stable compositional stratification. Stratification inhibits vertical motion, imposes a flow parallel to isodensity surfaces, and concentrates most deformation in a shallow shear layer of thickness ∼ B −1/5 , where B is the dimensionless buoyancy number.The localization of the flow results in large strain rates and enables the development of a strong alignment of iron crystals in the upper inner core. We couple our dynamical model with a numerical model of texture development and compute the time evolution of the lattice preferred orientation of different samples in the inner core. With sufficient stratification, texturing is significant in the uppermost inner core. In contrast, the deeper inner core * Corresponding authorEmail address: philippe.cardin@ujf-grenoble.fr (Philippe Cardin ) Preprint submitted to Physics of the Earth and Planetary Interiors June 10, 2011is unaffected by any flow and may preserve a fossil texture. We investigate the effect of an initial texture resulting from solidification texturing at the ICB. In the present inner core, the deformation rate in the shallow shear layer is large and can significantly alter the solidification texturing, but the solidification texture acquired early in the inner core history can be preserved in the deeper part. Using elastic constants from ab initio calculations, we predict different maps of anisotropy in the modern inner core. A model with both solidification texturing and subsequent deformation in a stratified inner core produces a global anisotropy in reasonable agreement with seismological observations.

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“…Two earlier calculations used the particle-in-cell (PIC) model to obtain the lattice vibrational contributions, and predicted a rapid increase in the c/a axial ratio to above 1.7 at the core conditions. 28,33 However, later theoretical work by Alfe and coauthors using ab initio molecular dynamics simulations [34][35][36][37] and experimental measurements up to 2000K 12 both gave much smaller temperature dependences of the c/a ratio. We found that the results from the first-principles linear response calculations and the PIC model usually agree well except when the lattice approaches instability, and both theoretical techniques predicted a slight increase in the axial ratio with temperature, in contradiction to the earlier PIC computations.…”

Section: Introductionmentioning

confidence: 99%

First-principles thermal equation of state and thermoelasticity of hcp Fe at high pressures

Sha

Cohen

2010

Phys. Rev. B

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We investigate the equation of state and elastic properties of hcp iron at high pressures and high temperatures using first principles linear response linear-muffin-tin-orbital method in the generalized-gradient approximation. We calculate the Helmholtz free energy as a function of volume, temperature, and volume-conserving strains, including the electronic excitation contributions from band structures and lattice vibrational contributions from quasi-harmonic lattice dynamics. We perform detailed investigations IntroductionIron is one of the most abundant elements in the Earth, and is fundamental to our world. The study of iron at high pressures and high temperatures is of great geophysical interest, since both the Earth's liquid outer core and solid inner core are composed mostly of this element. Although the crystal structure of iron at the extremely high temperature (4000 to 8000 K) and high pressure (330 to 360 GPa) conditions found in the inner core is still under intensive debate, 1-10 the hexagonal-close-packed phase (ε-Fe) is commonly believed to have a wide stability field extending from deep mantle to core conditions, and serves as a starting point for understanding the nature of the inner core. 11 Significant experimental and theoretical efforts have been recently devoted to investigate various properties of hcp iron at high pressures and high temperatures. New high-pressure diamond-anvil-cell techniques have been developed or significantly improved, which makes it possible to reach higher pressures and provide more valuable information on material properties in these extreme states. First-principles based theoretical techniques have been improved in reliability and accuracy, and have been widely used to predicate the high pressure-temperature behavior and provide fundamental understandings to the experiment.Despite intensive investigations, numerous fundamental problems remain unresolved, and many of the current results are mutually inconsistent. 11 The melting line at very high pressures has been one of the most difficult and controversial problems. 12-19Other major problems include possible subsolidus phase transitions 2,4,5,11,20 and the magnetic structure of the dense hexagonal iron. In section II we detail the theoretical methods to perform the first-principles calculations and obtain the thermal properties and elastic moduli. We present the results and related discussions about the thermal equation of state in section III, and about the thermoelasticity in section IV. We conclude with a brief summary in Section V. II. Theoretical methodsThe Helmholtz free energy F for many metals has three major contributionswith V as the volume, T as the temperature, and δ as the strain. E static is the zerotemperature energy of a static lattice, F el is the thermal free energy arising from electronic excitations, and F ph is the lattice vibrational energy contribution. We obtain both E static and F el from first-principles calculations directly, assuming that the eigenvalues for given lattice and nuclear pos...

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The particle-in-cell model for ab initio thermodynamics: implications for the elastic anisotropy of the Earth’s inner core

Cited by 38 publications

References 43 publications

The axial ratio of hcp iron at the conditions of the Earth’s inner core

The axial ratio of hcp iron at the conditions of the Earth’s inner core

Texturing in Earth’s inner core due to preferential growth in its equatorial belt

First-principles thermal equation of state and thermoelasticity of hcp Fe at high pressures

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