Geodynamic models commonly assume equations of state as a function of pressure and temperature. This form is legitimate for homogenous materials, but it is impossible to formulate a general equation ...of state for a polyphase aggregate, e.g., a rock, as a function of pressure and temperature because these variables cannot distinguish all possible states of the aggregate. In consequence, the governing equations of a geodynamic model based on a pressure‐temperature equation of state are singular at the conditions of low‐order phase transformations. An equation of state as a function of specific entropy, specific volume, and chemical composition eliminates this difficulty and, additionally, leads to a robust formulation of the energy and mass conservation equations. In this formulation, energy and mass conservation furnish evolution equations for entropy and volume and the equation of state serves as an update rule for temperature and pressure. Although this formulation is straightforward, the computation of phase equilibria as a function of entropy and volume is challenging because the equations of state for individual phases are usually expressed as a function of temperature and pressure. This challenge can be met by an algorithm in which continuous equations of state are approximated by a series of discrete states: a representation that reduces the phase equilibrium problem to a linear optimization problem that is independent of the functional form used for the equations of state of individual phases. Because the efficiency of the optimization decays as an exponential function of the dimension of the function to be optimized, direct solution of the linearized optimization problem is impractical. Successive linear programming alleviates this difficulty. A pragmatic alternative to optimization as an explicit function of entropy and volume is to calculate phase relations over the range of pressure‐temperature conditions of interest. Numerical interpolation can then be used to generate tables for any thermodynamic property as a function of any choice of independent variables. Regardless of the independent variables of the governing equations, a consistent definition of pressure, and the coupling of equilibrium kinetics to deformation, is only possible if the continuity equation accounts for dilational strain.
An algorithm for the construction of phase diagram sections is formulated that is well suited for geodynamic problems in which it is necessary to assess the influence of phase transitions on rock ...properties or the evolution and migration of fluids. The basis of the algorithm is the representation of the continuous compositional variations of solution phases by series of discrete compositions. As a consequence of this approximation the classical non-linear free energy minimization problem is trivially solved by linear programming. Phase relations are then mapped as a function of the variables of interest using bisection to locate phase boundaries. Treatment of isentropic and isothermal phase relations involving felsic and mafic silicate melts by this method is illustrated. To demonstrate the tractability of more complex problems involving mass transfer, a model for infiltration driven-decarbonation in subduction zones is evaluated. As concluded from earlier closed system models, the open-system model indicates that carbonates are likely to persist in the subducted oceanic crust beyond sub-arc depths even if the upper section of the oceanic mantle is extensively hydrated. However, in contrast to more simplistic models of slab devolatilization, the open-system model suggests slab fluid production is heterogeneous and ephemeral. Computed seismic velocity profiles, together with thermodynamic constraints, imply that for typical geothermal conditions serpentinization of the subducted mantle is unlikely to extend to >
25 km depth and that the average water-content of the serpentinized mantle is <
2 wt.%.
Phase equilibrium models are used routinely to predict geophysically relevant mantle properties. A limitation of this approach is that nonlinearity of the phase equilibrium problem precludes direct ...assessment of the resultant uncertainties. To overcome this obstacle, we stochastically assess uncertainties along self‐consistent mantle adiabats for pyrolitic and basaltic bulk compositions to 2000 km depth. The dominant components of the uncertainty are the identity, composition and elastic properties of the minerals. For P wave speed and density, the latter components vary little, whereas the first is confined to the upper mantle. Consequently, P wave speeds, densities, and adiabatic temperatures and pressures predicted by phase equilibrium models are more uncertain in the upper mantle than in the lower mantle. In contrast, uncertainties in S wave speeds are dominated by the uncertainty in shear moduli and are approximately constant throughout the model depth range.
Key Points
Uncertainties in mantle geophysical properties to 2000 km depth are assessed stochastically
Uncertainty in P wave speed and density is greatest in the upper mantle
Uncertainty in S wave speed is roughly constant with depth
The observation that primitive arc magmas are more oxidized than mid-ocean-ridge basalts has led to the paradigm that slab-derived fluids carry SO
and CO
that metasomatize and oxidize the sub-arc ...mantle wedge. We combine petrography and thermodynamic modelling to quantify the oxygen fugacity (fO
) and speciation of the fluids generated by serpentinite dehydration during subduction. Silicate-magnetite assemblages maintain fO
conditions similar to the quartz-fayalite-magnetite (QFM) buffer at fore-arc conditions. Sulphides are stable under such conditions and aqueous fluids contain minor S. At sub-arc depth, dehydration occurs under more reducing conditions producing aqueous fluids carrying H
S. This finding brings into question current models in which serpentinite-derived fluids are the cause of oxidized arc magmatism and has major implications for the global volatile cycle, as well as for redox processes controlling subduction zone geodynamics.
The compilation of thermodynamic models for geophysical applications is such a tedious and complex process that it is generally impractical for researchers to refit parameters in existing models in ...light of new constraints. To mitigate this difficulty, we develop a Bayesian algorithm that permits the modification of a thermodynamic model to account for additional observational constraints. This algorithm can be applied to any thermodynamic dataset and can utilize a wide variety of experimental constraints. To demonstrate the applicability of the algorithm it is used to revise the Stixrude and Lithgow‐Bertelloni (2011, https://doi.org/10.1111/j.1365‐246x.2010.04890.x), whole‐mantle terrestrial thermodynamic model, using phase equilibrium constraints provided by Bertka and Fei (1997, https://doi.org/10.1029/96jb03270), for the more iron‐rich compositions that are thought to be relevant to the Martian mantle. The revised thermodynamic model provides a more reliable prediction of phase equilibria in the Martian mantle. Seismic properties are calculated in an internally self‐consistent manner along hot and cold areotherms to constrain the upper and lower bounds of these properties for different bulk silicate Mars compositional models.
Plain Language Summary
Thermodynamic models capable of predicting geophysical properties, such as seismic wave velocities and density are useful in determining the structure of planetary interiors. We have developed an algorithm capable of refining parameters in thermodynamic models in light of new experimentally determined constraints. This algorithm is applied to the widely used Stixrude and Lithgow‐Bertelloni (2011, https://doi.org/10.1111/j.1365‐246x.2010.04890.x), thermodynamic model with data for a Mars‐like composition and pressure‐temperature conditions. Our results show that small changes to the thermodynamic dataset can result in a marked improvement in the agreement between the predicted and experimental results, providing estimates for geophysical properties that are consistent with experimental data. This will enable us to more reliably constrain the composition of the Martian mantle from the seismic data provided by NASA's InSight mission.
Key Points
Bayesian algorithm capable of optimizing existing thermodynamic models to account for new observational constraints
Experimentally determined phase equilibria are reproduced with minor refinements to existing thermodynamic model
Revised bounds for seismic velocity and density profiles in the Martian mantle
Swift monitoring of NGC 4151 with an ∼6 hr sampling over a total of 69 days in early 2016 is used to construct light curves covering five bands in the X-rays (0.3-50 keV) and six in the ultraviolet ...(UV)/optical (1900-5500 Å). The three hardest X-ray bands (>2.5 keV) are all strongly correlated with no measurable interband lag, while the two softer bands show lower variability and weaker correlations. The UV/optical bands are significantly correlated with the X-rays, lagging ∼3-4 days behind the hard X-rays. The variability within the UV/optical bands is also strongly correlated, with the UV appearing to lead the optical by ∼0.5-1 days. This combination of 3 day lags between the X-rays and UV and 1 day lags within the UV/optical appears to rule out the "lamp-post" reprocessing model in which a hot, X-ray emitting corona directly illuminates the accretion disk, which then reprocesses the energy in the UV/optical. Instead, these results appear consistent with the Gardner & Done picture in which two separate reprocessings occur: first, emission from the corona illuminates an extreme-UV-emitting toroidal component that shields the disk from the corona; this then heats the extreme-UV component, which illuminates the disk and drives its variability.
Alpine deformation in the Grimsel granodiorite (Aar massif, Central Alps) at greenschist facies conditions (6.5 ± 1 kbar for 450°C ± 25°C) is characterized by the development of a network of ...centimetre to decametre localized shear zones that surround lenses of undeformed granodiorite. Localization of deformation is assumed to be the result of a first stage of extreme localization on brittle precursors (nucleation stage) followed by a transition to ductile deformation and lateral propagation into the weakly deformed granodiorite (widening stage). A paradox of this model is that the development of the ductile shear zone is accompanied by the crystallization of large amounts of phyllosilicates (white mica and chlorite) that maintains a weak rheology in the localized shear zone relative to the host rock so that deformation is localized and prevents shear zone widening. We suggest that chemical processes, and more particularly, the metamorphic reactions and metasomatism occurring during re‐equilibration of the metastable magmatic assemblage induced shear zone widening at these P–T–X conditions. These processes (reactions and mass transfer) were driven by the chemical potential gradients that developed between the thermodynamically metastable magmatic assemblage at the edge of the shear zone and the stable white mica and chlorite rich ultramylonite formed during the first stage of shear zone due to localized fluid infiltration metasomatism. P–T and chemical potential projections and sections show that the process of equilibration of the wall rocks (μ–μ path) occurs via the reactions: kf + cz + ab + bio + MgO + H2O = mu + q + CaO + Na2O and cz + ab + bio + MgO + H2O = chl + mu + q + CaO + Na2O. Computed phase diagram and mass balance calculations predict that these reactions induce relative losses of CaO and Na2O of ∼100% and ∼40% respectively, coupled with hydration and a gain of ∼140% for MgO. Intermediate rocks within the strain gradient (ultramylonite, mylonite and orthogneiss) reflect various degrees of re‐equilibration and metasomatism. The softening reaction involved may have reduced the strength at the edge of the shear zone and therefore promoted shear zone widening. Chemical potential phase diagram sections also indicate that the re‐equilibration process has a strong influence on equilibrium mineral compositions. For instance, the decrease in Si‐content of phengite from 3.29 to 3.14 p.f.u, when white mica is in equilibrium with the chlorite‐bearing assemblage, may be misinterpreted as the result of decompression during shear zone development while it is due only to syn‐deformation metasomatism at the peak metamorphic condition. The results of this study suggest that it is critical to consider chemical processes in the formation of shear zones particularly when deformation affects metastable assemblages and mass transfer are involved.
Very early stages of crack evolution from corrosion pit. Internal view (i.e., looking at the surface skin of the pit and crack as viewed from inside the bulk of the specimen), via X-ray tomographic ...rendering, of the pit-to-crack transition within a steel specimen stressed at 90%
σ
0.2 and exposed to simulated condensate at 90
°C for 668
h (the surface of the sites rendered on the left are shown in the SEM image on the right for comparison). Two pits are imaged showing crack initiation on the pit wall near the pit mouth. Finite element analysis of U-shaped pit. Maximum principal stress and maximum principal strain of a 500
μm deep U-shaped pit in a 6.4
mm diameter cylindrical specimen at 90%
σ
0.2 viewed in cross section. The maximum stress, as expected, is located at the base of the pit while the maximum strain occurs along the pit wall close to the pit mouth.
Display omitted
► X-ray tomography is a powerful technique for pit-to-SCC transition characterisation. ► The preferred site for crack initiation is on the pit wall close to the pit mouth. ► FEA indicate that localised plastic strain provides explanation for SCC nucleation. ► Kondo’s criteria at the point of transition are not supported for SCC in this system. ► There is a need to reassess quantitative modelling of SCC growth from pits.
SCC in turbine disc steels exposed to simulated steam-condensate tends to nucleate preferentially from corrosion pit precursors. The evolution of these cracks is not straightforward and not well understood. In this work, unique three-dimensional X-ray microtomographic images have confirmed that cracks develop predominantly at the shoulder of the pit, near the pit/surface interface, for specimens stressed to 50–90%
σ
0.2. In support of this observation, FEA of model pits indicate that strain is a maximum on the pit wall just below the pit mouth. Implications of these observations for the pit-to-crack transition and predictive-modelling of crack nucleation and growth are discussed.
Recent crystallographic data indicate that in biotite Ti orders preferentially onto the M2 octahedral site rather than onto the M1 site as assumed in previous solution models for ...K2O–FeO–MgO–Al2O3–SiO2–H2O–TiO2–O2 (KFMASHTO) biotite. In view of these data, we reformulate and reparameterize former biotite solution models. Our reparameterization takes into account Fe–Mg order–disorder and ferric iron contents of natural biotite as well as both natural and experimental observations on biotite Ti‐content over a wide range of physicochemical conditions. In comparison with previous biotite models, the new model reproduces the Ti‐content and stability field of biotite as constrained by experiments with significantly better accuracy. The predictive power of the model is tested by comparison with petrologically well‐characterized natural samples of SiO2‐saturated and SiO2‐undersaturated rocks that were not used in the parameterization. In all these tests, the reformulated model performs well.
PDF
