A computer simulation study of OH defects in Mg2SiO4 and Mg2GeO4 spinels

Classical atomistic simulation techniques have been used to investigate the energies of hydrogen defects in Mg₂SiO₄ and Mg₂GeO₄ spinels. Ringwoodite (γ-Mg₂SiO₄) is considered to be the most abundant mineral in the lower part of the transition zone and can incorporate large amounts of water in the form of hydroxyls, whereas the germanate spinel (γ-Mg₂GeO₄) corresponds to a low-pressure structural analogue for ringwoodite. The calculated defect energies indicate that the most favourable mechanisms for hydrogen incorporation are coupled either with the reduction of ferric iron or with the creation of tetrahedral vacancies. Hydrogen will go preferentially into tetrahedral vacancies, eventually leading to the formation of the hydrogarnet defect, before associating with other negatively charged point defects. The presence of isolated hydroxyls is not expected. The same trend is observed for germanate, and thus γ-Mg₂GeO₄ could be used as a low-pressure analogue for ringwoodite in studies of water-related defects and their effect on physical properties.     

Pseudopotential periodic Hartree-Fock study of Mg2SiO4 polymorphs: Olivine,modified spinel and spinel

Pseudopotential periodic Hartree-Fock calculations have been performed on the three polymorphs of Mg₂SiO₄ with a polarized split valence basis set. The energy differences between polymorphs at their experimental geometries are correctly predicted. The olivine to modified spinel and olivine to spinel phase transition pressures have been estimated and agree within a few GPa with their experimental values. The bonding in Mg₂SiO₄ is discussed from the point of view of the, band structures, projected density of states, electron density and electron localization function (ELF) curves. It is concluded that both Mg-O and Si-O bonds are highly ionic.     

Comparison of the raman microprobe spectra of (Mg,Fe)2SiO4 and Mg2GeO4 with olivine and spinel structures

Raman microprobe (RMP) spectra were produced for each of the olivine and spinel structured phases of Mg₂GeO₄ and (Mg, Fe)₂SiO₄. The assembled data show that bands due to the tetrahedra in silicate and germanate olivines shift in a way that indicates a dominant mass effect. This correspondence is difficult to make in spinels due to differences in structural type. Differences in Fe/Mg content of olivine shift the tetrahedral vibration bands only slightly, but their linear shifts could be used to indicate the composition of the phase.     

The rheology of olivine and spinel magnesium germanate (Mg2GeO4): TEM study of the defect microstructures

Synthetic polycrystals of α-Mg₂GeO₄ (with the olivine structure) and γ-Mg₂GeO₄ (with the spinel structure) deformed at high temperature and pressure in their respective stability fields were investigated by analytical transmission electron microscopy. Specimens with a mean grain size of 20–30 µm deform by dislocation glide and/or climb. The predominance of glide versus climb depends on stress and grain orientation. The defect microstructures of both polymorphs are very similar to those observed in their respective silicate analogues, α- and γ-(Mg,Fe)₂SiO₄, and, in the case of the spinel phase, very similar to those observed in magnesium aluminate spinels. These observations suggest that Mg₂GeO₄ is a good rheological analogue for the Earth’s upper mantle. A spinel specimen deformed under the same conditions of temperature and strain rate as an olivine specimen was approximately three times stronger than olivine. In specimens of both phases deformed at or above 1400 K, a thin amorphous film composed of Mg, Ge, and O was detected along some grain boundaries. Grains ≤10 µm diameter surrounded by a film of amorphous phase (>10 nm thick) exhibited low dislocation densities, and deformation appeared to have occurred by grain boundary sliding.     

Importance of considerations of mixing properties in establishing an internally consistent thermodynamic database: thermochemistry of minerals in the system Mg2SiO4-Fe2SiO4-SiO2

A thermodynamic solution model is developed for minerals whose compositions lie in the two binary systems Mg₂SiO₄-Fe₂SiO₄ and Mg₂Si₂O₆-Fe₂Si₂O₆. The formulation makes explicit provision for nonconvergent ordering of Fe²⁺ and Mg²⁺ between M1 and M2 sites in orthopyroxenes and non-zero Gibbs energies of reciprocal ordering reactions in both olivine and orthopyroxene. The calibration is consistent with (1) constraints provided by available experimental and natural data on the Fe-Mg exchange reaction between olivine and orthopyroxene ± quartz, (2) site occupancy data on orthopyroxenes including both crystallographic refinements and Mössbauer spectroscopy, (3) enthalpy of solution data on olivines and orthopyroxenes and enthalpy of disordering data on orthopyroxene, (4) available data on the temperature and ordering dependence of the excess volume of orthopyroxene solid solutions, and (5) direct activity-composition determinations of orthopyroxene and olivine solid solutions at elevated temperatures. Our analysis suggests that the entropies of the exchange [Mg(M2)Fe(M1)Fe(M2)Mg(M1)] and reciprocal ordering reactions [Mg(M2)Mg(M1)+ Fe(M2)Fe(M1)Fe(M2)Mg(M1)+Mg(M2)Fe(M1)] cannot differ significantly (± 1 cal/K) from zero over the temperature range of calibration (400°–1300° C). Consideration of the mixing properties of olivine-orthopyroxene solid solutions places tight constraints on the standard state thermodynamic quantities describing Fe-Mg exchange reactions involving olivine, orthopyroxene, pyralspite garnets, aluminate spinels, ferrite spinels and biotite. These constraints are entirely consistent with the standard state properties for the phases-quartz,-quartz, orthoenstatite, clinoenstatite, protoenstatite, fayalite, ferrosilite and forsterite which were deduced by Berman (1988) from an independent analysis of phase equilibria and calorimetric data. In conjunction with these standard state properties, the solution model presented in this paper provides a means of evaluating an internally consistent set of Gibbs energies of mineral solid solutions in the system Mg₂SiO₄-Fe₂SiO₄-SiO₂ over the temperature range 0–1300° C and pressure interval 0.001–50 kbars. As a consequence of our analysis, we find that the excess Gibbs energies associated with mixing of Fe and Mg in (Fe, Mg)₂SiO₄ olivines, (Fe, Mg)₃Al₂Si₃O₁₂ garnets, (Fe, Mg)Al₂O₄ and (Fe, Mg)Fe₂O₄ spinels, and K(Mg, Fe)₃AlSi₃O₁₀(OH)₂ biotites may be satisfactory described, on a macroscopic basis, with symmetric regular solution type parameters having values of 4.86±0.12 (olivine), 3.85±0.09 (garnet), 1.96±0.13 (spinel), and 3.21±0.29 kcals/gfw (biotite). Applications of the proposed solution model demonstrate the sensitivity of petrologic modeling to activity-composition relations of olivine-orthopyroxene solutions. We explore the consequences of estimating the activity of silica in melts forming in the mantle and we develop a graphical geothermometer/geobarometer for metamorphic assemblages of olivine+orthopyroxene+quartz. Quantitative evaluation of these results suggests that accurate and realistic estimates of silica activity in melts derived from mantle source regions,P-T paths of metamorphism and other intensive variables of petrologic interest await further refinements involving the addition of trace elements (Al³⁺ and Fe³⁺) to the thermodynamic formulation for orthopyroxenes.     

Calorimetric study of olivine solid solutions in the system Mg2SiO4-Fe2SiO4

Solution enthalpies of synthetic olivine solid solutions in the system Mg₂SiO₄-Fe₂SiO₄ have been measured in molten 2PbO·B₂O₃ at 979 K. The enthalpy data show that olivine solid solutions have a positive enthalpy of mixing and the deviation from ideality is approximated as symmetric with respect to composition, in contrast to the previous study. Applying the symmetric regular solution model to the present enthalpy data, the interaction parameter of ethalpy (W_H) is estimated to be 5.3±1.7 kJ/mol (one cation site basis). Using this W_h and the published data on excess free energy of mixing, the nonideal parameter of entropy (W_s) of olivine solid solutions is estimated as 0.6±1.5 J/mol·K.     

The lattice dynamics and thermodynamics of the Mg2SiO4 polymorphs

We use an approach based upon the Born model of solids, in which potential functions represent the interactions between atoms in a structure, to calculate the phonon dispersion of forsterite and the lattice dynamical behaviour of the beta-phase and spinel polymorphs of Mg₂SiO₄. The potential used (THB1) was derived largely empirically using data from simple binary oxides, and has previously been successfully used to model the infrared and Raman behaviour of forsterite. It includes ‘bond bending’ terms, that model the directionality of the Si-O bond, in addition to the pair-wise additive Coulombic and short range terms. The phonon dispersion relationships of the Mg₂SiO₄ polymorphs predicted by THB1 were used to calculate the heat capacities, entropies, thermal expansion coefficients and Gruneisen parameters of these phases. The predicted heat capacities and entropies are in outstandingly good agreement with those determined experimentally. The predicted thermodynamic data of these phases were used to construct a phase diagram for this system, which has Clausius-Clapeyron slopes in very close agreement with those found by experiment, but which has predicted transformation pressures that show less close agreement with those inferred from experiment. The overall success, however, that we have in predicting the lattice dynamical and thermodynamic properties of the Mg₂SiO₄ polymorphs shows that our potential THB1 represents a significant step towards finding the elusive quantitative link between the microscopic or atomistic behaviour of minerals and their macroscopic properties.     

Single crystal hydrostatic compression of (Mg,Mn,Fe,Co)2SiO4 olivines

Four synthetic endmember olivines (Mg,Mn, Fe,Co)₂SiO₄ with space group Pbnm were loaded together in one diamond cell mount. Their unit-cell parameters were determined by single crystal X-ray diffraction to 10 GPa. The linear compressibilities β_a, β_b, β_c were 1.53, 2.90, 2.32; 1.45, 3.48, 1.98; 1.35, 3.29, 1.76; and 1.25, 2.82, 2.01×10⁻³ GPa⁻¹ for Mg₂SiO₄, Mn₂SiO₄, Fe₂SiO₄ and Co₂SiO₄, respectively. The b axis is the most compressible direction in all crystals studied. Bulk modulus K_T0 and its first pressure derivative were simultaneously determined for Mg₂SiO₄, Fe₂SiO₄ and Co₂SiO₄ crystals respectively by fitting volume data to a third order Birch-Murnaghan equation of state. They are 127(4) and 4.2(8), 136(3) and 4.1(7), and 144(2) and 4.1(5). The K_T0 and could not simultaneously be determined unambiguously for Mn₂SiO₄. Direct comparisons of unit-cell volumes at high pressure among pairs of olivines reveal anomalous compression behavior of the Mg₂SiO₄ crystal regarding the bulk modulus-volume relationship. This behavior, however, could not be observed in the transition metal olivines (Mn,Fe,Co)₂SiO₄. The distinct electronic configurations of Mg²⁺ and the transition metal cations Mn²⁺, Fe²⁺, and Co²⁺ result in the different compression behaviors of Mg₂SiO₄ and (Mn,Fe,Co)₂SiO₄. Received: 14 April 1997 / Revised, accepted: 29 July 1997     

Molecular dynamics simulations of pressure and temperature effects on MgSiO3 and Mg2SiO4 melts and glasses

Ab-initio interionic potentials for Mg²⁺, Si⁴⁺, and O^2– have been used in molecular dynamics (MD) simulations to investigate diffusivity changes, pressure-induced structural transitions, and temperature effects on polymerization in MgSiO₃ and Mg₂SiO₄ melts and glasses. The potential gives reasonable agreement with the 0.1 MPa radial distribution function of MgSiO₃ glass. Maxima in the diffusion coefficients of Si⁴⁺ and O^2– occur as pressure is increased on the MgSiO₃ melt. The controlling structural mechanism for this behavior is the Q¹ species of SiO₄ tetrahedra. Mg²⁺ diffusion coefficients decrease monotonically with pressure in both melt compositions. Increasing Mg²⁺ coordination number and population of 3- and 4-membered SiO₄ rings with pressure combine to hinder translation of the Mg²⁺ ions. The dominant changes in structure with pressure are a decrease in the intertetrahedral (Si-O--Si) angle up to approximately 4 g/cm³ and coordination changes of the ions above this density. Temperature effects on viscosity in these simulated melts are indirectly studied by analyzing polymerization changes with temperature. Polymerization and coordination numbers increase with decreasing temperature and a small quench rate effect is observed. Fair agreement is found between the MD simulations and experimental equation of state for Mg₂SiO₄, but the equation of state predictions for MgSiO₃ melts are much less accurate. The zero pressure volume, V ₀, is significantly higher and K ₀ is lower in the simulations than empirical values. The inadequacies reflect error in using the ionic approximation for polymerized systems and a need to collect more data for a variety of molecular configurations in the development of ab-initio potentials.     

High pressure phase transformations in the joins Mg2SiO4-Ca2SiO4 and MgO-CaSiO3

High pressure phase transformations for all the mineral phases along the joins Mg₂SiO₄-Ca₂-SiO₄ and MgO-CaSiO₃ in the system MgO-CaO-SiO₂ were investigated in the pressure range between 100 and 300 kbar at about 1,000 °C, by means of the technique involving a diamond-anvil press coupled with laser heating. In addition to the four end-members, there are three stable intermediate mineral components in these two joins. Phase behaviour of all the end-member components at high pressure have been reported earlier and are reviewed here. Results of this study reveal that the three intermediate components are all unstable relative to the end-members at pressures greater than 200 kbar. Ultimately, monticellite (CaMgSiO₄) decomposes into CaSiO₃ (perovskite-type)+MgO; merwinite (Ca₃MgSi₂O₈) decomposes into Ca₂SiO₄(K₂NiF₄-type)+CaSiO₃ (perovskite-type)+MgO; and akermanite (Ca₂MgSi₂O₇) decomposes into CaSiO₃ (perovskite-type)+MgO. Note that the decomposition reactions of all phases studied here result in the formation of MgO. Intermediate Ca-Mg silicates transform to pure Ca-silicates plus MgO, while pure Mg₂SiO₄ transforms to MgSiO₃+MgO.     

A MEG study of the olivine and spinel forms of Mg2SiO4

In this paper we present a theoretical investigation of the structures and relative stability of the olivine and spinel phases of Mg₂SiO₄. We use both a purely ionic model, based on the Modified Electron Gas (MEG) model of intermolecular forces, and a bond polarization model, developed for low pressure silica phases, to investigate the role of covalency in these compounds. The standard MEG ionic model gives adequate structural results for the two phases but incorrectly predicts the spinel phase to be more stable at zero pressure. This is mainly because the ionic modeling of Mg₂SiO₄ only accounts for 95 percent of the lattice energy. The remainder can be attributed to covalency and many-body effects. An extension of the MEG ionic model using “many-body” pair potentials corrects the phase stability error, but predicts structures which are in poorer agreement with experiment than the standard ionic approach. In addition, calculations using these many-body pair potentials can only account for 10 percent of the missing lattice energy. This model predicts an olivine-spinel phase transition of 8 GPa, below the experimental value of 20 GPa. Therefore, in order to understand more fully the stability of these structures we must consider polarization. A two-shell bond polarization model enhances the stability of both structures, with the olivine structure being stabilized more. This model predicts a phase transition at about 80 GPa, well above the observed value. Also, the olivine and spinel structures calculated with this approach are in poorer agreement with experiment than the ionic model. Therefore, based on our investigations, to properly model covalency in Mg₂SiO₄, a treatment more sophisticated than the two-shell model is needed.     

Thermodynamic properties of the forsterite-fayalite (Mg2SiO4-Fe2SiO4) solid solution. determination of heat of formation

A high temperature calorimetric solution method involving the use of a sodium and lithium metaborate (NaBO₂+LiBO₂) molten bath at 1,180 K, has been developed in order to measure the enthalpy of formation of the forsterite-fayalite (Mg₂SiO₄-Fe₂SiO₄) system. The samples used for these measurements have been obtained by synthesis. They have been carefully controlled by chemical and X-ray analysis and Mössbauer spectroscopy. The results obtained for forsterite and fayalite agree closely with the data commonly found in the literature. Owing to the uncertainty of the experimental results, it is not possible to determine conclusively whether an enthalpy of mixing exists in this system.     

Molecular dynamics simulations of interdiffusion in MgSiO3-Mg2SiO4 melts

Molecular dynamics simulations based on ab initio interatomic potentials have been performed to study the kinetics and mechanisms of interdiffusion in MgSiO₃ and Mg₂SiO₄ melts. Diffusion coefficients in the presence of a concentration gradient are lower than self-diffusion coefficients due to the requirements of local charge-balance within the system. Extrapolations of Mg²⁺ diffusion coefficients obtained at high temperatures (5000 to 3000 K) into geologically relevant temperatures (1500 to 2500 K) are reasonably accurate compared to experimental Mg²⁺ diffusion coefficients in a compositionally similar system. A significant system-size effect is also observed on the rates of diffusion obtained from the molecular dynamics simulations. Diffusion mechanisms involve movement of individual Mg²⁺ and O^2- ions and anionic [SiO_n]^(4–2n) complexes for both self-diffusion and interdiffusion.     

Electric field gradient calculation in forsterite,Mg2SiO4

The electric field gradient (EFG) in Mg₂SiO₄ is calculated on the basis of the extended point ion model, including the local term of the overlap contribution. The agreement with experimental data deduced from the quadrupole coupling constants and principal axes at the Mg sites is quite good. The results of the present calculation exhibit a small overlap contribution to the EFG at M1 and a clearly bigger one at M2, whereas the lattice contribution to the EFG at M1 and M2 is reversed. The distinct overlap effects are assumed to be due to the particular Mg₂SiO₄ crystal structure and the different point symmetry at M1 and M2. The oxygen polarizability and charge used to calculate the EFG tensor were found to be smaller than the theoretical polarizability and formal charge, respectively. The sign of the Mg quadrupole coupling constants at M1 and M2, which has not been determined experimentally, results from the EFG calculation as positive.     

A study of the polymorphic transformations in Co2SiO4

The olivine-beta phase transformation in Co₂SiO₄ has been studied in a large-volume high-pressure apparatus (USSA-2000). The experimental conditions straddle the univariant phase boundaries due to the presence of a substantial temperature gradient (150–200° C over the 3-mm length of the sample). At conditions close to equilibrium, the transformation mechanism is one of limited nucleation with rapid growth of large crystals of the beta phase. Transmission electron microscopy (TEM) analysis shows that the cation stacking faults in Co₂SiO₄-spinel are identical to those seen in numerous spinels. The 010 faults in beta-Co₂SiO₄ are found to be identical to those seen in beta-(Mg,Fe)₂SiO₄ found in shocked veins in meteorites.     

The effect of Fe on the crystal structure of wadsleyite β-(Mg1 − x Fe x )2SiO4, 0.00≤x≤0.40

Single crystals of five wadsleyite compositions, β-(Mg,Fe)₂SiO₄ with Fe/(Fe+Mg)=0.00, 0.08, 0.16, 0.25 and 0.40, have been synthesized at high temperature and pressure in a uniaxial, split-sphere apparatus. Crystal structures of these samples, determined by x-ray diffraction techniques, reveal that iron is significantly ordered: Fe is depleted in the M2 octahedron, while it is enriched in M1 and M3. The most iron-rich synthetic sample, which falls well outside previous estimates of wadsleyite stability, raises questions regarding published Mg₂SiO₄-Fe₂SiO₄ phase diagrams at transition zone conditions.     

Lattice dynamics and inelastic neutron scattering from forsterite,Mg2SiO4: Phonon dispersion relation,density of states and specific heat

Magnesium-rich olivine (Mg_0.9Fe_0.1)₂SiO₄ is considered to be a major constituent of the Earth's upper mantle. Because of its major geophysical importance, the temperature and pressure dependence of its crystal structure, elastic and dielectric constants, long-wavelength phonon modes and specific heat have been measured using a variety of experimental techniques. Theoretical study of lattice dynamics provides a means of analyzing and understanding a host of such experimental data in a unified manner. A detailed study of the lattice dynamics of forsterite, Mg₂SiO₄, has been made using a crystal potential function consisting of Coulombic and short-range terms. Quasiharmonic lattice dynamical calculations based on a rigid molecular-ion model have provided theoretical estimates of elastic constants, long-wavelength modes, phonon dispersion relation for external modes along the three high symmetry directions in the Brillouin zone, total and partial density of states and inelastic neutron scattering cross-sections. The neutron cross-sections were used as guides for the coherent inelastic neutron scattering experiment on a large single crystal using a triple axis spectrometer in the constant Q mode. The observed and predicted phonon dispersion relation show excellent agreement. The inelastically scattered neutron spectra from a powder sample have been analyzed on the basis of a phonon density of states calculated from a rigid-ion model, which includes both external and internal modes. The experimental data from a powder sample show good agreement with the calculated spectra, which include a multiphonon contribution in the incoherent approximation. The computed phonon densities of states are used to calculate the specific heat as a function of temperature using both the rigid molecular-ion and rigid ion models. These results are in very good agreement with the calorimetric measurement of the specific heat. The interatomic potential developed here can be used with some confidence to study physical properties of forsterite as a function of pressure and temperature.     

Cation diffusion in olivine—II. Ni-Mg,Mn-Mg,Mg and Ca

Diffusivities of bivalent cations. Mg, Ni. Mn and Ca, in olivine were determined experimentally. The diffusivities of Ni and Mn in forsterite were determined by couple annealing between Ni₂SiO₄ and Mg₂SiO₄, and Mn₂SiO₄ and Mg₂SiO₄, respectively. The diffusivities of Mg and Ca in forsterite were determined by the use of ²⁶Mg and Ca tracers, respectively. Combined with other published results, the diffusion coefficients for bivalent cation diffusion in pure forsterite along the c crystallographic axis range from 2.45 × 10^?11 to 1.4 × 10^?13 cm² sec^?1 at 1200°C, in the order of Fe > Mn > Co > Ni > Mg > Ca. The results suggest that the diffusivity is governed by at least two factors, i.e. the size of the diffusing ionic species and the change of defect density in the crystal structure which is induced by substitution of diffusing ion for Mg ion.     

Crystal structures and cation distributions in simple spinels from powder XRD structural refinements: MgCr2O4, ZnCr2O4, Fe3O4 and the temperature dependence of the cation distribution in ZnAl2O4

The crystal structure and cation distributions in the spinels MgCr₂O₄, ZnCr₂O₄, Fe₃O₄ and a suite of ZnAl₂O₄ samples annealed at 900 to 1400° C and then rapidly quenched, have been determined by powder X-ray diffraction, using several different X-ray procedures and both conventional structure-factor refinement and whole-pattern (or Rietveld) refinement methods. The chromite spinels are expected from crystal chemical considerations to have an almost completely normal cation distribution (inversion parameter, x, equal to zero). In agreement with this expectation, three samples of MgCr₂O₄ annealed at 900, 1100 and 1300° C, and ZnCr₂O₄ were all found to have x=0 within two estimated standard deviations (esd), suggesting that the accuracy with which cation distributions in spinels may be determined by powder XRD is close to the estimated precision. Slightly better results are obtained assuming neutral-atom scattering curves rather than half-ionized or fully ionized, but the differences are small (within the esd). The results from the Rietveld refinements are similarly in good agreement with those using the conventional structure factor refinement approach (agreement within the combined esd's), although in detail the Rietveld procedure sometimes produces small systematic differences in refined parameters. The suite of ZnAl₂O₄ spinels show a smooth increase in x from 0.01 at 900° C to 0.05 at 1300° C, and this behaviour is well described by the simple thermodynamic model for disordering in spinels with α_Zn-Al=89 kJ/mol, assuming β=?20 kJ/mol. The oxygen positional parameters for Fe₃O₄ are similar to those from published single crystal studies, indicating that the powder method also yields accurate interatomic distances in spinels.