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SUMMARY Phase transitions play an important role for the style of mantle convection. While observations and theory agree that a substantial fraction of subducted slabs and rising plumes can move through the whole mantle at present day conditions, this behaviour may have been different throughout Earth’s history. Higher temperatures, such as in the early Earth, cause different phase transitions to be dominant, and also reduce mantle viscosity, favouring a more layered style of convection induced by phase transitions. A period of layered mantle convection in Earth’s past would have significant implications for the secular evolution of the mantle temperature and the mixing of mantle heterogeneities. The transition from layered to whole mantle convection could lead to a period of mantle avalanches associated with a dramatic increase in magmatic activity. Consequently, it is important to accurately model the influence of phase transitions on mantle convection. However, existing numerical methods generally preclude modelling phase transitions that are only present in a particular range of pressures, temperatures or compositions, and they impose an artificial lower limit on the thickness of phase transitions. To overcome these limitations, we have developed a new numerical method that solves the energy equation for entropy instead of temperature. This technique allows for robust coupling between thermodynamic and geodynamic models and makes it possible to model realistically sharp phase transitions with a wide range of properties and dynamic effects on mantle processes. We demonstrate the utility of our method by applying it in regional and global convection models, investigating the effect of individual phase transitions in the Earth’s mantle with regard to their potential for layering flow. We find that the thickness of the phase transition has a bigger influence on the style of convection than previously thought: with all other parameters being the same, a thin phase transition can induce fully layered convection where a broad phase transition would lead to whole-mantle convection. Our application of the method to convection in the early Earth illustrates that endothermic phase transitions may have induced layering for higher mantle temperatures in the Earth’s past.

SUMMARY We derive exact expressions for the thermal expansivity, heat capacity and bulk modulus for assemblages with arbitrarily large numbers of components and phases, including the influence of phase transformations and chemical exchange. We illustrate results in simple two-component, two-phase systems, including Mg–Fe olivine-wadsleyite and Ca–Mg clinopyroxene-orthopyroxene and for a multicompontent model of mantle composition in the form of pyrolite. For the latter we show results for the thermal expansivity and heat capacity over the entire mantle pressure–temperature regime to 40 GPa, or a depth of 1000 km. From the thermal expansivity, we derive a new expression for the phase buoyancy parameter that is valid for arbitrarily large numbers of phases and components and which is defined at every point in pressure–temperature space. Results reveal regions of the mantle where the magnitude of the phase buoyancy parameter is larger in magnitude than for those phase transitions that are most commonly included in mantle convection simulations. These regions include the wadsleyite to garnet and ferropericlase transition, which is encountered along hot isentropes (e.g. 2000 K potential temperature) in the transition zone, and the ferropericlase and stishovite to bridgmanite transition, which is encountered along cold isentropes (e.g. 1000 K potential temperature) in the shallow lower mantle. We also show the bulk modulus along a typical mantle isentrope and relate it to the Bullen inhomogeneity parameter. All results are computed with our code HeFESTo, updates and improvements to which we discuss, including the implementation of the exact expressions for the thermal expansivity, heat capacity and bulk modulus, generalization to allow for pressure dependence of non-ideal solution parameters and an improved numerical scheme for minimizing the Gibbs free energy. Finally, we present the results of a new global inversion of parameters updated to incorporate more recent results from experiment and first principles theory, as well as a new phase (nal phase), and new species: Na-majorite and the NaAlO2 end-member of ferropericlase.

We present first principles predictions of the absolute entropy of a silicate liquid (CaSiO3) over a wide pressure-temperature range encompassing the Earth’s mantle (2000–6000 K, 0–270 GPa). The results are derived from molecular dynamics simulations based on density functional theory and the two-phase thermodynamic (2PT) method, which divides the vibrational density of states into solid-like and gas-like parts, and which we describe in detail. The heat capacity derived from the absolute entropy agrees well with experimental measurements at low pressure. We find that the vibrational contribution accounts for more than 75% of the total entropy over the range of our study. We find that the two-body approximation to the excess entropy (relative to the ideal gas), which is computed with knowledge only of the radial distribution function, is excellent over most of the range considered, except at the lowest pressures and temperatures. We also compute the entropy of CaSiO3 perovskite and use the entropy of liquid and solid phases to determine their absolute free energies and the melting curve. The melting curve agrees well with experiment and independent theoretical determinations based on Clausius-Clapeyron integration and the ZW method, showing a melting temperature of 5600 K at the Earth’s core-mantle boundary. We find a nearly universal scaling of the self-diffusion coefficients with the excess entropy D ∗ = 0.9 exp 1.2 S ex , where D* is the self-diffusion coefficient suitably non-dimensionalized using macroscopic thermodynamic properties, and Sex is the excess entropy in units of the Boltzmann constant per atom. We use our results to estimate the temperature distribution in an isentropic, molten silicate Earth, finding 4.4 kJ/kg/K to be the lowest entropy that is completely molten, and producing a temperature at the core-mantle boundary of 6000 K.

Under equilibrium conditions, the mineral olivine (α) transforms to wadsleyite (β) near 410 km depth1,2. The phase transition is a global boundary known as the “410-km discontinuity”, where seismic wave speed and density increase by 3–10 percent3-6. We present new observations of triplicated P-waves from dense seismic arrays in northeastern China that constrain the structure of the Pacific slab near the 410-km discontinuity beneath the northern Sea of Japan. Our analysis of P-wave travel times and waveforms at periods as short as T = 2 s indicates the presence of an ultra-low-velocity layer within the cold slab, with a P-wave velocity that is at least 20% lower than in the ambient mantle and an apparent thickness of ~ 16 km along the wave path. This ultra-low-velocity layer could contain unstable material (e.g., poirierite) with reduced grain size where diffusionless transformations are favored7-12

The Earth’s magnetic field has operated for at least 3.4 billion years, yet how the ancient field was produced is still unknown. The core in the early Earth was surrounded by a molten silicate layer, a basal magma ocean that may have survived for more than one billion years. Here we use density functional theory-based molecular dynamics simulations to predict the electrical conductivity of silicate liquid at the conditions of the basal magma ocean: 100–140 GPa, and 4000–6000 K. We find that the electrical conductivity exceeds 10,000 S/m, more than 100 times that measured in silicate liquids at low pressure and temperature. The magnetic Reynolds number computed from our results exceeds the threshold for dynamo activity and the magnetic field strength is similar to that observed in the Archean paleomagnetic record. We therefore conclude that the Archean field was produced by the basal magma ocean., Cooling of the iron core in the early Earth may have been too slow to allow for the generation of a magnetic field. Based on quantum mechanical and geodynamical modelling approaches, the authors find that the electrical conductivity of silicate liquid at high pressure and temperature conditions could have been sufficient to generate a silicate dynamo and a magnetic field in the early Earth.

The discovery of more than 4500 extrasolar planets has created a need for modeling their interior structure and dynamics. Given the prominence of iron in planetary interiors, we require accurate and precise physical properties at extreme pressure and temperature. A first-order property of iron is its melting point, which is still debated for the conditions of Earth’s interior. We used high-energy lasers at the National Ignition Facility and in situ x-ray diffraction to determine the melting point of iron up to 1000 gigapascals, three times the pressure of Earth’s inner core. We used this melting curve to determine the length of dynamo action during core solidification to the hexagonal close-packed (hcp) structure. We find that terrestrial exoplanets with four to six times Earth’s mass have the longest dynamos, which provide important shielding against cosmic radiation.

Ab initio molecular dynamics simulations predict that CaSiO3 perovskite melts at 5600 K at 136 GPa, and 6400 K at 300 GPa, significantly higher than MgSiO3 perovskite. The entropy of melting (1.8 kB per atom) is much larger than that of many silicates at ambient pressure and of simple liquids and varies little with pressure. The volume of melting decreases rapidly with increasing pressure, to 3 % at 136 GPa, producing a melting slope that diminishes rapidly with pressure. We determine the melting temperature via the ZW method, combining the Z method, for which we clarify the theoretical basis, with a waiting time analysis. The ZW method results are internally confirmed by integrating the Clausius‐Clapeyron equation, which also yields our results for the entropy and volume of melting. We find the eutectic composition on the MgSiO3‐CaSiO3 join to be x Ca = 0.26 at 136 GPa and that metasilicate melt is denser than coexisting silicates., Key Points Calcium silicate perovskite melts at 5600 K at the core mantle boundaryMetasilicate liquids are denser than coexisting crystalline silicates at the core‐mantle boundaryThe Z method allows precise ab initio melting temperature determination

Finding the Emperor's head Volcanic island and seamount chains form from deep-seated plumes of hot material upwelling through the mantle. The most famous of these is the Hawaiian-Emperor seamount chain. However, a large volcanic structure associated with a plume head that should precede the chain has long been missing. Wei et al. finally identified the likely location of this structure in the mantle under eastern Russia. The structure was likely subducted 20 million to 30 million years ago, and the location helps constrain several geodynamic processes. Science , this issue p. 983

Significance During accretion, giant impacts may partially vaporize planets, with profound consequences for energy and mass transfer between the growing planet and its satellites. We find that giant impacts produce much more supercritical fluid than previously thought, with a much lower liquid–vapor critical temperature. The vapor is depleted in Mg with respect to coexisting liquid and has very different chemical bonding, being dominated by covalent molecules (O 2 and SiO) up to the critical point. These molecules also appear in significant fractions within the near-critical liquid phase, making the liquid structure fundamentally different from that near the melting point.

Ferropericlase (Mg, Fe)O is an abundant mineral of Earth's lower mantle and the liquid phase of the material was an important component of the early magma ocean. Using quantum-mechanical, finite-temperature density-functional theory calculations, we compute the electronic component of the electrical and thermal conductivity of (Mg0.75, Fe0.25)O crystal and liquid over a wide range of planetary conditions: 0–200 GPa, 2000–4000 K for the crystal, and 0–300 GPa, 4000–10,000 K for the liquid. We find that the crystal and liquid are semi-metallic over the entire range studied: the crystal has an electrical conductivity exceeding 103 S/m, whereas that of the liquid exceeds 104 S/m. Our results on the crystal are in reasonable agreement with experimental measurements of the electrical conductivity of ferropericlase once we account for the dependence of conductivity on iron content. We find that a harzburgite-dominated mantle with ferropericlase in combination with Al-free bridgmanite agrees well with electromagnetic sounding observations, while a pyrolitic mantle with a ferric-iron rich bridgmanite composition yields a lower mantle that is too conductive. The electronic component of thermal conductivity of ferropericlase with XFe=0.19 is negligible (

Ab intio molecular dynamics simulations show that the electrical conductivity of liquid SiO2 is semimetallic at the conditions of the deep molten mantle of early Earth and super-Earths, raising the possibility of silicate dynamos in these bodies. Whereas the electrical conductivity increases uniformly with increasing temperature, it depends nonmonotonically on compression. At very high pressure, the electrical conductivity decreases on compression, opposite to the behavior of many materials. We show that this behavior is caused by a novel compression mechanism: the development of broken charge ordering, and its influence on the electronic band gap.

Several theoretical studies indicate that a substantial fraction of the measured seismic anisotropy could be interpreted as extrinsic anisotropy associated with compositional layering in rocks, reducing the significance of strain‐induced intrinsic anisotropy. Here we quantify the potential contribution of grain‐scale and rock‐scale compositional anisotropy to the observations by (i) combining effective medium theories with realistic estimates of mineral isotropic elastic properties and (ii) measuring velocities of synthetic seismic waves propagating through modeled strain‐induced microstructures. It is shown that for typical mantle and oceanic crust subsolidus compositions, rock‐scale compositional layering does not generate any substantial extrinsic anisotropy (, Key Points Rock‐scale layering in the Earth's mantle produces negligible extrinsic anisotropyRock‐scale layering cannot be detected with seismic anisotropy but mainly with seismic wave scatteringGrain‐scale layering can generate substantial extrinsic anisotropy around the transition zone

Recent geochemical studies document a wide range of 182W anomalies in mantle-derived rocks, the origin of which is unknown. Here we explore the consequences of basal magma ocean crystallization while 182Hf was extant. We determine the partition coefficient of Hf between bridgmanite and silicate melt DHf throughout the pressure-temperature regime of Earth's lower mantle from first principles molecular dynamics coupled with thermodynamic integration. The calculations are based on Hf4+ entering the Mg-site, which we show is energetically preferable to the Si-site. We find that DHf increases by more than an order of magnitude from 25 GPa to 140 GPa, making Hf much more compatible with increasing pressure. The larger DHf at greater depths produces a strong fractionation of 182Hf between crystallized bridgmanite and silicate melt. We use a simple geochemical model to show that heterogeneous tungsten anomalies are a natural outcome of the formation and crystallization of a basal magma ocean, with the crystallized products enriched in 182W, while the remaining liquid is depleted by a similar amount. Our results show that the observed large variability of 182W/184W in terrestrial samples may be due to the solidification of a basal magma ocean.

Seismic tomography models reveal two large low shear velocity provinces (LLSVPs) that identify large-scale variations in temperature and composition in the deep mantle. Other characteristics include elevated density, elevated bulk sound speed, and sharp boundaries. We show that properties of LLSVPs can be explained by the presence of small quantities (0.3-3%) of suspended, dense Fe-Ni-S liquid. Trapping of metallic liquid is demonstrated to be likely during the crystallization of a dense basal magma ocean, and retention of such melts is consistent with currently available experimental constraints. Calculated seismic velocities and densities of lower mantle material containing low-abundance metallic liquids match the observed LLSVP properties. Small quantities of metallic liquids trapped at depth provide a natural explanation for primitive noble gas signatures in plume-related magmas. Our model hence provides a mechanism for generating large-scale chemical heterogeneities in Earth's early history and makes clear predictions for future tests of our hypothesis.

The regime governing the growth of Mercury's core is unknown, but the dynamics of core growth are vital to understanding the origin and properties of the planet's weak magnetic field. Here, we use advanced first-principles methods, which include a magnetic entropy contribution, to investigate the magnetic and thermo-elastic properties of liquid Fe-S-Si and of pure liquid iron at the conditions of Mercury's core. Our results support a ‘top-down’ evolution of the core, whereby solid iron-rich material crystallises at shallow depths and sinks. This process would likely result in a compositionally driven dynamo within a stably stratified uppermost liquid layer, providing an explanation for the observed properties of the weak magnetic field of Mercury.

Author(s): Garber, JM; Maurya, S; Hernandez, JA; Duncan, MS; Zeng, L; Zhang, HL; Faul, U; McCammon, C; Montagner, JP; Moresi, L; Romanowicz, BA; Rudnick, RL; Stixrude, L | Abstract: Some seismic models derived from tomographic studies indicate elevated shear-wave velocities (≥4.7nkm/s) around 120–150nkm depth in cratonic lithospheric mantle. These velocities are higher than those of cratonic peridotites, even assuming a cold cratonic geotherm (i.e., 35nmW/m2 surface heat flux) and accounting for compositional heterogeneity in cratonic peridotite xenoliths and the effects of anelasticity. We reviewed various geophysical and petrologic constraints on the nature of cratonic roots (seismic velocities, lithology/mineralogy, electrical conductivity, and gravity) and explored a range of permissible rock and mineral assemblages that can explain the high seismic velocities. These constraints suggest that diamond and eclogite are the most likely high-Vs candidates to explain the observed velocities, but matching the high shear-wave velocities requires either a large proportion of eclogite (g50nvol.%) or the presence of up to 3nvol.% diamond, with the exact values depending on peridotite and eclogite compositions and the geotherm. Both of these estimates are higher than predicted by observations made on natural samples from kimberlites. However, a combination of ≤20nvol.% eclogite and ~2nvol.% diamond may account for high shear-wave velocities, in proportions consistent with multiple geophysical observables, data from natural samples, and within mass balance constraints for global carbon. Our results further show that cratonic thermal structure need not be significantly cooler than determined from xenolith thermobarometry.

Ab initio molecular dynamics simulations carried out for tetragonal and orthorhombic distortions of iron closely follow the results of static-lattice electronic-structure calculations in revealing that the body-centered cubic (bcc) phase of Fe is mechanically unstable at pressures of 0.3–1.5 TPa and temperatures up to 7000 K. Crystal-structural instabilities originate in the static lattice for the bcc configuration, and are consistent with recent results from both static and dynamic high-pressure experiments. Both theory and experiment thus show that the close-packed (hexagonal, hcp and face-centered cubic, fcc) crystal structures of iron are those relevant to the cores of Earth-like planets.

We find that Earth’s earliest magnetic field may have been produced in a deep magma ocean, and that silicate dynamos may exist in super-Earth exoplanets as well. Ab initio molecular dynamics simulations show that silicate liquids are semimetallic at the extreme pressure and temperature conditions characteristic of planetary interiors. The electrical conductivity shows a remarkable nonmonotonic dependence on pressure that reveals connections to the underlying atomic structure, and highlights broken charge ordering as a novel compression mechanism.

We combine spin-polarized density functional theory with first principle molecular dynamics (FPMD) to study the spin crossover in liquid Fe2SiO4, up to 300 GPa and 6000 K. In contrast to the much sharper transition seen in crystals, we find that the high- to low-spin transition occurs over a very broad pressure interval (>200 GPa) due to structural disorder in the liquid. We find excellent agreement with the experimental Hugoniot. We combine our results with previous FPMD calculations to derive the partial molar volumes of the oxide components MgO, FeO, and SiO2. We find that eutectic melts in the MgO-FeO-SiO2 system are denser than coexisting solids in the bottom 600 km of Earth's mantle.

Article history: Elastic properties of MgSiO3-perovskite are of fundamental importance for our understanding of the thermal and chemical state of the lower mantle. However, the elastic moduli and especially their derivatives with respect to temperature and pressure at lower mantle conditions are still uncertain. In this study, we have carried out extensive first principles molecular dynamics simulations to determine the equation of state, elastic constants, moduli and velocities of MgSiO3-perovskite over a wide temperature and pressure regime (from static conditions to 3500 K and from 36 GPa to 140 GPa). Systematic errors arising from approximations to the exchange-correlation functional in density functional theory have been essentially eliminated with a generalized re-scaling method. Molecular dynamics trajectories were carefully converged with respect to duration and system size and analyzed to distinguish the effects of anharmonicity. Based on the new elastic properties derived in this study, the pyrolite mineralogical model is found to predict wave velocities in close agreement with those of seismic observations in most of lower mantle regime.

[1] Seismic refraction studies of deep-crustal and upper mantle structure beneath some oceanic hotspot provinces reveal the presence of ultramafic bodies with P-wave velocities of Vp ~ 7.4–8.0 km/s lying at or above the Moho, e.g., Hawaii, the Marquesas, and La Reunion. However, at other hotspot provinces such as the Galapagos, Nazca Ridge, and Louisville the lower crust is intruded by large volumes of gabbroic (mafic) rocks (Vp ~ 6.8–7.5 km/s). Ultramafic primary melts formed beneath mature oceanic lithosphere at pressures of ~2–3 GPa (60–90 km depth), and ponded at the Moho due to their relatively high density, can explain the observed ultramafic deep-crustal bodies. By contrast, plume melts formed at depths of ~15–30 km beneath thin lithosphere crystallize assemblages that are more gabbroic. The velocity and density gradient is particularly strong in the pressure range 0.6–1.5 GPa due to the replacement of plagioclase by olivine as melts become more MgO-rich with increasing pressure (and degree) of melting. This anomalous density gradient suggests a possible filtering effect whereby plume melts equilibrated at relatively shallow depths beneath very young and thin oceanic lithosphere may be expected to be of nearly gabbroic (mafic) composition (~6–10% MgO), whereas ultramafic melts (MgO ~ 12–20%) formed beneath older, thicker oceanic lithosphere must pond and undergo extensive olivine and clinopyroxene fractionation before evolving residual magmas of basaltic composition sufficiently buoyant to be erupted at the surface. A survey of well-studied hotspot provinces of highly-varying lithospheric age at the time of emplacement shows that deep-crustal and upper mantle seismic refraction data are consistent with this hypothesis. These results highlight the importance of large-volume intrusive processes in the evolution of hotspot magmas, with intrusive volumes being significantly larger than those of the erupted lavas in most cases. Pyrolite melting can account, to first order, for the total crustal column of magmatic products, whereas alternative models such as selective melting of pyroxenite blobs probably cannot.

[1] The viscosity of silicate liquids at high temperature is crucial to our understanding of chemical and thermal evolution of the Earth since its early stages. First-principles molecular dynamics simulations of seven liquids across the MgO-SiO2 binary show that the viscosity varies by several orders of magnitudes with temperature and composition. Our results follow a compensation law: on heating, the viscosity of all compositions approaches a uniform value at 5000 K, above which pure silica becomes the least viscous liquid. Viscosity depends strongly on composition (fourth power), implying a strong nonlinear dependence of the configurational entropy on composition. Using the simulation results, we derive and evaluate different types (Arrhenius and non-Arrhenius) of models for accurate description of the viscosity-temperature-composition relationship. Our results span the thermal regime expected in a magma ocean, and indicate that melt migration is important for understanding the generation and preservation of melts from frictional heating at very fast slip in impact processes.

Knowledge of the multi-component thermodynamics and phase equilibria of silicate melts in Earth's deep interior are key to understanding the thermal and chemical evolution of the planet, yet the melting phase diagram of the lower mantle remains poorly constrained, with large uncertainties in both eutectic composition and temperature. We use results from first-principles molecular dynamics of nine compositions along the MgO–SiO 2 binary to investigate the compositional dependence of liquid state thermodynamics, applying our results to describe incongruent melting for the system at deep lower mantle pressures. Our phase diagram is bi-eutectic throughout the lower mantle, with no liquid immiscibility. Accounting for solid–liquid partitioning of Fe, we find partial melts of basaltic and peridotitic lithologies to be gravitationally stable at the core–mantle boundary, while liquidus density contrasts predict that perovskite will sink and periclase will float in a crystallizing pyrolytic magma ocean.

We use first-principles free-energy calculations to predict a pressure-induced spin crossover in the liquid planetary material (Mg,Fe)O, whereby the magnetic moments of Fe ions vanish gradually over a range of hundreds of GPa. Because electronic entropy strongly favors the nonmagnetic low-spin state of Fe, the crossover has a negative effective Clapeyron slope, in stark contrast to the crystalline counterpart of this transition-metal oxide. Diffusivity of liquid (Mg,Fe)O is similar to that of MgO, displaying a weak dependence on element and spin state. Fe-O and Mg-O coordination increases from approximately 4 to 7 as pressure goes from 0 to 200 GPa. We find partitioning of Fe to induce a density inversion between the crystal and melt, implying separation of a basal magma ocean from a surficial one in the early Earth. The spin crossover induces an anomaly into the density contrast, and the oppositely signed Clapeyron slopes for the crossover in the liquid and crystalline phases imply that the solid-liquid transition induces a spin transition in (Mg,Fe)O.

Chemical heterogeneity, produced by the near-surface rock cycle and dominated volumetrically by subducted oceanic crust and its depleted residue, is continuously subducted into the mantle. This lithologic-scale chemical heterogeneity may survive in the mantle for as long as the age of Earth because chemical diffusion is inefficient. Estimates of rates of subduction and mantle processing over geologic history indicate that most or all of the mantle may be composed of lithologically heterogeneous material. Mineralogical models of the mantle show that chemical heterogeneity over many decades in length scale may be detectable by geophysical probes via its influence on seismic-wave propagation. Grain-scale heterogeneity influences the aggregate absolute seismic velocity and its lateral variation with temperature. The elastic-wave velocity contrast associated with lithologic-scale heterogeneity may be sufficient to produce observable scattering of short-period seismic waves.

We use first principles molecular dynamics simulations based on density functional theory in the local density approximation to investigate CaMgSi 2 O 6 liquid over the entire mantle pressure regime. We find that the liquid structure becomes much more densely packed with increasing pressure, with the mean Si–O coordination number increasing nearly linearly with volume from fourfold near ambient pressure to sixfold at the base of the mantle. Fivefold Si–O coordination environments are most abundant at intermediate compression. The properties of Mg and Si coordination environments are nearly identical to those in MgSiO 3 liquid, whereas Ca is more highly coordinated with larger mean Ca–O bond length as compared with Mg. The density increases smoothly with increasing pressure over the entire range studied. The Gruneisen parameter increases by a factor of three on twofold compression. The density contrast between diopside composition liquid and the isochemical crystalline assemblage is less than 2% at the core mantle boundary, less than that in the case of MgSiO 3 . Thermodynamic properties are described in terms of a liquid-state fundamental thermodynamic relation.

P>We complete the development and description of a thermodynamic method for the computation of phase equilibria and physical properties of multiphase mantle assemblages. Our previous paper focused on the computation of physical properties. In this paper, our focus shifts to the phase equilibria. We further develop our theory to specify the ideal and excess contributions to solution properties and derive properties of multiphase assemblages. We discuss our global inversion strategy for determining the values of the free parameters in our theory and compare inverted parameter values with expectations based on scaling arguments. Comparisons between our method and experimental phase equilibria data encompass the pressure-temperature regime of Earth's mantle. Finally, we present applications of our method that illustrate how it may be used to explore the origins of mantle structure and mantle dynamics. Continuing rapid advances in experimental and theoretical petrology and mineral physics have motivated an expansion of the scope of our model via the addition of several new phases, and of the soda component: an appendix lists all parameters in our model and references to the experimental and theoretical studies that constrain them. Our algorithm for global minimization of the Gibbs free energy is embodied in a code called HeFESTo, and is detailed in a second appendix.

The full elastic tensors of the sheet silicates muscovite, illite-smectite, kaolinite, dickite, and nacrite have been derived with first-principles calculations based on density functional theory. For muscovite, there is excellent agreement between calculated properties and experimental results. The influence of cation disorder was investigated and found to be minimal. On the other hand, stacking disorder is found to be of some relevance for kaolin minerals. The corresponding single-crystal seismic wave velocities were also derived for each phase. These revealed that kaolin minerals exhibit a distinct type of seismic anisotropy, which we relate to hydrogen bonding. The elastic properties of a shale aggregate was predicted by averaging the calculated properties of the contributing mineral phases over their orientation distributions. Calculated elastic properties display higher stiffness and lower p-wave anisotropy. The difference is likely due to the presence of oriented flattened pores in natural samples that are not taken into account in the averaging.

Central to the goals of mineral physics is an elucidation of how material behavior governs planetary processes. Planetary accretion, differentiation into crust, mantle, and core, ongoing processing by magmatism, dynamics and thermal evolution, and the generation of magnetic fields, are all processes controlled by the physical properties and phase equilibria of planetary materials. Knowledge of these processes has advanced in large measure by progress in the study of material behavior at extreme conditions of pressure and temperature. Thinking of a planet as an experimental sample emphasizes the relationship between planetary processes and material behavior via properties that control its response to natural perturbations. We may explore the response of the planet to a sudden increase in energy, provided, for example, by a giant impact of the type that is thought to have formed Earth’s moon (Canup 2004). Responses include an increase in temperature, controlled by the heat capacity, a decrease in density controlled by the thermal expansivity, and phase transformations controlled by the free energy. Phase transformations also contribute to changes in temperature and density via the heats and volumes of transformation. A giant impact also perturbs the stress state, to which the planet responds via compression (controlled by the bulk modulus), adiabatic heating (Gruneisen parameter) and phase transformations (free energy). A planet is an unusual experimental sample because its pressure is self-generated via gravitational self-compression, and its temperature via adiabatic compression and radioactive decay. Indeed one of the major challenges in understanding planetary-scale processes is that the characteristic pressure (1 Mbar) and temperature (several thousand K) are so large. Consider an Earth-like planet in which the mantle makes up 2/3 of the mass and the core makes up the remainder. The pressure P M at the base of the mantle depends approximately linearly on planetary mass M: P M …

The thermal conductivity of the lower mantle plays a major role in shaping the structure and dynamics of the region. However, the thermal conductivity of lower mantle minerals are, at present, not well constrained, because of difficulties in making measurements at such high pressures. Here we describe the most common theoretical methods available to calculate the thermal conductivity of materials, which provide an invaluable alternative to experimental techniques. In each case the general scheme is given and particular considerations highlighted. The advantages and disadvantages and applicability of the methods are then discussed. We conclude with a short review of theoretical studies of the lattice thermal conductivity of periclase.

We have calculated the high-pressure properties of fayalite, including band structure, magnetism, equation-of-state and elastic properties using density functional theory within the generalized gradient approximation (GGA) and using the GGA+U method. We show for the first time that the addition of an on-site Hubbard repulsion term to the GGA leads to improved agreement between calculated and experimental values of structural and elastic properties, as well as electronic band gap, particularly at high pressure. High-pressure elastic instability, originating in the vanishing of the elastic constant c(44) and previously predicted on the basis of extrapolated experimental data, is found with the GGA+U method. Experimental measurements of the elastic constants agree better with the GGA+U method, than with the GGA, which calls for a reassessment of the structural and elastic properties of iron-bearing minerals calculated using standard density functional theory, which have hitherto been used to interpret the structure and dynamics of the mantle. The improvement can be related to the better description of magnetic structure: without the Hubbard U the magnetic moments on the iron ions decrease with pressure, whereas when it is included they remain almost constant, at least, up to the highest pressure studied. This also leads to better predictions of equation-of-state. Our calculated partial density-of-states indicate that the lowest energy excitation across the electronic band gap lies within the d-orbital manifold, and we argue that this is not observed at ambient pressure because the signal in the optical spectrum is too weak, accounting for apparent discrepancies with reported experimental values. Upon compression the nature of the gap changes due to the broadening of the 3d manifold, and this likely renders the fundamental gap observable in optical spectroscopy at high pressure, accounting for the better agreement between theory and experiment above 20 GPa. We associate the changes in the character of the bands with a change in the crystal structure. (C) 2009 Elsevier B.V. All rights reserved.

Chemical diffusion is a fundamental process in the evolution of planets. Equilibration within and among phases in response to changes in physical conditions requires the comprising chemical species to be spatially rearranged, over distances comparable to the grain size. Quantitative description of such processes demands diffusivities of these chemical species to be accurately known, while detailed insight into the mechanisms of diffusion at the atomic scale elucidate their dependence on pressure, temperature and composition. By applying our understanding of chemical bonding in condensed systems to numerically simulate diffusivity over a range of pressures and temperatures of planetary interest, we can obtain direct constraints on diffusivities at these extreme conditions, and self consistently assess the models used to extrapolate experimental data. Such computations further serve as a proving ground for testing the robustness of the various levels of theory applied in the characterization of bonding and dynamics of a material. The mathematical description of diffusion was first developed in the context of thermal transport by Fourier (1822). Its applicability to chemical transport was recognized by Fick (1855), who cast Fourier’s law of thermal conduction in terms of chemical transport and applied it to experiments on the diffusion of salt in a column of water. In an anisotropic system of n components, Fick’s description is most generally given by \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\frac{{\partial}\mathit{c}\_{\mathit{A}}}{{\partial}\mathit{t}}\ =\ {{\sum}\_{\mathit{i},\mathit{j}}^{3}}{{\sum}\_{\mathit{B}}^{\mathit{n}\ {-}\ 1}}\ {-}\ \mathit{D}\_{\mathit{AB}}^{\mathit{ij}}\frac{{\partial}^{2}\mathit{c}\_{\mathit{B}}}{{\partial}\mathit{x}\_{\mathit{i}}{\partial}\mathit{x}_{\mathit{j}}}\] \end{document}(1) where c A is the concentration of species A , and D AB ij is the relevant component of the diffusivity matrix (e.g., Crank 1975; Lasaga 1998; Zhang 2010). However, this description of diffusion is entirely in the continuum limit, and does not give any insight into the mechanisms by which chemical transport occurs at the atomic level. The description of chemical diffusivity was cast in an atomistic context by studies of Brownian motion (Einstein 1905; Smoluchowski …

We have investigated the antigorite m = 17 structure [Mg48Si34O85(OH)62] by density functional theory (DFT), with the aim to probe the method on such a large and low symmetry ( Pm ) system. We found a satisfactory match with the experiments using both LDA and GGA approximations to exchange-correlation, although the former performs slightly better here. Predicted cell constants are within 0.3% for LDA, and 0.5% for GGA, of the experimental values. Average atomic displacements after relaxation are within ~0.06 A. All the fine structural details of antigorite are reproduced: apical Si-O bonds shorter than basal Si-O bonds; external Mg-O distances shorter than internal Mg-O distances; pronounced tetrahedral ditrigonal distortion. Where palpably biased bond distances were present in the experimental data because of disorder, theoretical methods promptly recover the structure into more reliable bond geometry. These findings let one envisage the employment of DFT calculations as a promising tool for refining or validating complex structures, whenever the experiments suffer of limitations due to the poor quality of the material being investigated. As an additional benefit, we also compare the total energies of two competing models for the m = 17 antigorite structure, the one refined by Capitani and Mellini (2004) and that proposed by Dodony et al. (2002). We found the former more stable, both as published (Δ E = 1.1 kJ/mol·atom−1) and after full cell relaxation at constant volume (Δ E = 0.5 kJ/mol·atom−1).