Computer simulations of the structural and physical properties of the olivine and spinel polymorphs of Mg2SiO4

The aim of the work presented is to develop a computer simulation technique which will predict the structure and physical properties of forsterite and ringwoodite, the major mantle-forming polymorphs of Mg₂SiO₄. The technique is based upon energy minimization, in which all structural parameters are varied until the configuration with the lowest energy is achieved. The lattice energy and physical properties (e.g. elasticity and dielectric constants) are calculated from interatomic potentials, which generally include electrostatic and short-range terms. We investigate several types of traditional potential models, and present a new type of model which includes partial ionic charges and a Morse potential to describe the effect of covalency on the Si-O bond. This new form of potential model is highly successful, and not only reproduces the zero-pressure structural, elastic and dielectric properties of forsterite and ringwoodite, but also accurately describes their pressure dependence.     

Charge distribution from least-energy structure in Ca - Mg orthosilicates

Calcium-olivine, γ-Ca₂SiO₄, larnite, β-Ca₂SiO₄, merwinite, Ca₃Mg(SiO₄)₂, and monticellite, CaMgSiO₄, are considered. According to a rigid oxyanion scheme, eulerian orientation angles of the SiO₄ tetrahedra and translation coordinates of Ca and Si atoms are specified as structural variables τk. All derivatives of the static energy (Born model) contain atomic charges and repulsive parameters as unknowns; the minimum energy conditions ?E _L/?τ_k=0 yield 34 equations which are solved by a least-squares method. The set of energy parameters fitting structural properties of all four phases together is: z _Ca=1.50, z _o=?1.10 e, r _Ca=1.05, ρ=0.25 Å; the Mg charge was fixed at 1.38 e, from a previous study on forsterite. An average shift of 0.04 Å is observed between experimental and least-energy calculated atomic positions. Results are compared with those of Mg₂SiO₄, where the fit was based both on thermoelastic and on structural properties. If no charge values were fixed “a priori”, just ratios between charges could be determined by fitting them to structural data only.     

A transferable interatomic potential for crystalline phases in the system MgO—SiO2

A central interatomic potential model is presented for compounds in the binary system MgO-SiO₂. The potential, of a simple form which consists of a Coulombic term, a Born repulsive term, and a Van der Walls term for oxygen-oxygen interactions, is designed to predict the properties of magnesium silicates containing Si in octahedral and tetrahedral coordination. This is achieved by fitting simultaneously to forsterite and MgSiO₃ ilmenite crystal structure data, and fixing the partial ionic charges using elastic data for forsterite. The potential is found to transfer successfully to γ-Mg₂SiO₄ and MgSiO₃ perovskite. The potential results in local structural errors around the bridging oxygen ions in clinoenstatite and β-Mg₂SiO₄. The predicted structure for MgSiO₃ garnet is similar to the experimentally measured structure of the MnSiO₃ analogue. Calculated elastic constants average to K=2.41 Mbar and μ=1.44 Mbar for the bulk and shear moduli of MgSiO₃ perovskite, and K=1.87 Mbar and μ=1.10 Mbar for the bulk and shear moduli of MgSiO₃ garnet.     

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.     

The lattice energy of forsterite. Charge distribution and formation enthalpy of the SiO 4 4− ion

In the lattice energy expression of forsterite, based on a Born-Mayer (electrostatic+repulsive+dispersive) potential, the oxygen charge z _o, the hardness parameter ρ and the repulsive radii r _Mg and r _Si appear as unknown parameters. These were determined by calculating the first and second partial derivatives of the energy with respect to the cell edges, and equalizing them to quantities related to the crystal elastic constants; the overdetermined system of equations was solved numerically, minimizing the root-mean-square deviation. To test the results obtained, the SiO ₄ ^4? ion was assumed to move in the unit-cell, and the least-energy configuration was sought and compared with the experimental one. By combining the two methods, the optimum set of parameters was: z _o=?1.34, ρ=0.27 Å, r _Mg=0.72 Å, r _Si=0.64 Å. The values ?8565.12 and ?8927.28 kJ mol^?1 were obtained, respectively, for the lattice energy E _Land for its ionic component E _L ⁰ ,which accounts for interactions between Mg²⁺ and SiO ₄ ^4? ions only. The charge distribution calculated on the SiO ₄ ^4? ion was discussed and compared with other results. Using appropriate thermochemical cycles, the formation enthalpy and the binding energy of SiO ₄ ^4? were estimated to be: ΔH _f(SiO ₄ ^4? )=2117.6 and E(SiO ₄ ^4? )=708.6 kJ mol^?1, respectively.     

Ab initio calculated geometries and charge distributions for H4SiO4 and H6Si2O7 compared with experimental values for silicates and siloxanes

Ab initio STO-3G molecular orbital theory has been used to calculate energy-optimized Si-O bond lengths and angles for molecular orthosilicic and pyrosilicic acids. The resulting bond length for orthosilicic acid and the nonbridging bonds for pyrosilicic acid compare well with Si-OH bonds observed for a number of hydrated silicate minerals. Minimum energy Si-O bond lengths to the bridging oxygen of the pyrosilicic molecule show a close correspondence with bridging bond length data observed for the silica polymorphs and for gas phase and molecular crystal siloxanes when plotted against the SiOSi angle. In addition, the calculations show that the mean Si-O bond length of a silicate tetrahedron increases slightly as the SiOSi angle narrows. The close correspondence between the Si-O bond length and angle variations calculated for pyrosilicic acid and those observed for the silica polymorphs and siloxanes substantiates the suggestion that local bonding forces in solids are not very different from those in molecules and clusters consisting of the same atoms with the same coordination numbers. An extended basis calculation for H₄SiO₄ implies that there are about 0.6 electrons in the 3d-orbitals on Si. An analysis of bond overlap populations obtained from STO-3G* calculations for H₆Si₂O₇ indicates that Si-O bond length and SiOSi angle correlations may be ascribed to changes in the hybridization state of the bridging oxygen and (d – p) π-bonding involving all five of the 3d AO's of Si and the lone-pair AO's of the oxygen. Theoretical density difference maps calculated for H₆Si₂O₇ show a build-up of charge density between Si and O, with the peak-height charge densities of the nonbridging bonds exceeding those of the bridging bonds by about 0.05 e Å^?3. In addition, atomic charges (+1.3 and ?0.65) calculated for Si and O in a SiO₂ moiety of the low quartz structure conform reasonably well with the electroneutrality postulate and with experimental charges obtained from monopole and radial refinements of diffraction data recorded for low quartz and coesite.     

Raman scattering and lattice vibrations of Ni2SiO4 spinel at elevated temperature

Raman spectra of Ni₂SiO₄ spinel (O _h ⁷ Z=8) have been measured in the temperature range from 20 to 600 °C and the Raman active vibrations (A _1g +E _g +3F _2g ) have been assigned. A calculation of the optically active lattice vibrations of this spinel has been made, assuming a potential function which combines general valence and short range force constants. The values of the force constants at 20 and 500 °C have been calculated from the vibrational frequencies of the observed Raman spectra and infrared (IR) spectral data. The Ni spinel at 20 °C has a prominently small Si-O bond stretching force constant of K(SiO)=2.356 ～ 2.680 md/Å and a large Ni-O bond stretching constant of K(NiO)=0.843 ～ 1.062 md/Å and these force constants at 500 °C decrease to K(SiO)=2.327 ～ 2.494 md/Å and K(NiO)=0.861 ～ 0.990 md/Å. The Si-O bond is noticeably weakened at high temperatures, despite the small thermal expantion from 1.657 Å (20 °C) to 1.660 Å (500 °C). These changes of the interatomic force constants of the spinel at high temperatures are in accord with the thermal structure changes observed by X-ray diffraction study. The weakened Si-O bond is consistent with the fact that Si atoms in the spinel lattice can diffuse at significant rates at elevated temperature.     

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.     

Synthetic Mn3+-kyanite and viridine, (Al2-xMn x 3+ ) SiO5, in the system Al2O3-MnO-MnO2-SiO2

Experiments on the join Al₂SiO₅-“Mn₂SiO₅” of the system Al₂O₃-SiO₂-MnO-MnO₂ in the pressure/temperature range 10–20 kb/900–1050° C with gem quality andalusite, Mn₂O₃, and high purity SiO₂ as starting materials and using /_O2-buffer techniques to preserve the Mn³⁺ oxidation state had following results: At 20 kb/1000°C orange-yellow kyanite mixed crystals are formed. The kyanite solid solubility is limited at about (Al_1.88Mn _0.12 ³⁺ )SiO₅ and, thus, equals approximately that on the join Al₂SiO₅-“Fe₂SiO₅” (Langer and Frentrup, 1973) indicating that there is no Jahn-Teller stabilisation of Mn³⁺ in the kyanite matrix. 5 mole % substitution causes the kyanite lattice constants a _o, b _o, c _o, and V _o to increase by 0.015, 0.009, 0.014 Å, and 1.6 ų, resp., while α, β, γ, remain unchanged. Between 10 and 18 kb/900°C, Mn³⁺-substituted, strongly pleochroitic (emeraldgreen-yellow) andalusite_ss (viridine) was obtained. At 15 kb/900°C, the viridine compositional range is about (Al_1.86Mn _0.14 ³⁺ )SiO₅-(Al_1.56Mn _0,44 ³⁺ )SiO₅. Thus, Al→Mn³⁺ substitutional degrees are appreciably higher in andalusite than in kyanite, proving a strong Jahn-Teller effect of Mn³⁺ in the andalusite structure, which stabilises this structure type at the expense of kyanite and sillimanite and, thus, enlarges its PT-stability range extremely. 17 mole % substitution cause the andalusite constants a _o, b _o, c _o, and V _o to increase by 0.118, 0.029, 0.047 Å and 9.4 ų, resp. At “Mn₂SiO₅”-contents smaller than about 7 mole %, viridine coexists with Mn-poor kyanite. At “Mn₂SiO₅”-concentrations higher than the maximum kyanite or viridine miscibility, braunite (tetragonal, ideal formula Mn²⁺Mn³⁺[O₈/Si0₄]), pyrolusite and SiO₂ were found to coexist with the Mn³⁺-saturated ky _ss or and _ss, respectively. In both cases, braunites were Al-substituted (about 1 Al for 1 Mn³⁺). Pure synthetic braunites had the lattice constants a _o 9.425, c _o, 18.700 Å, V _o 1661.1 ų (ideal compn.) and a _o 9.374, c _o 18.593 ų, V _o 1633.6 ų (1 Al for 1 Mn³⁺). Stable coexistence of the Mn²⁺-bearing phase braunite with the Mn⁴⁺-bearing phase pyrolusite was proved by runs in the limiting system MnO-MnO₂-SiO₂.     

Ab initio valence force field calculations for quartz

We have derived valence force constants for the tetrahedral SiO₄ unit and the inter-tetrahedral SiOSi linkage from previous ab initio molecular orbital calculations on H₄SiO₄ and H₆Si₂O₇ using a split-valence polarized Gaussian basis set (6-31G^*), and used these to calculate the infrared and Raman active vibrational modes of α-quartz. The calculation gives frequencies approximately 15% greater than experiment, as expected from harmonic force constants obtained at this level of Hartree-Fock theory, but the calculation gives the correct distribution of modes within each frequency range. Calculated 28–30 Si and 16–18 O isotope shifts and pressure shifts to 6 GPa are also in reasonable agreement with experiment. We have also used our ab initio force field to calculate the vibrational spectrum for β-quartz. The results suggest either that inclusion of a torsional force constant is important for determining the stability of this high temperature polymorph, or that the β-quartz has a disordered structure with lower symmetry (P6₂) domains, as suggested by earlier diffraction studies.     

The electric field gradient at the position of copper in Cu2O and electronic charge density analysis by means of K-factors

The multipole moments of the charge density in Cu₂O are related to the electric field gradient (EFG) at the Cu position through crystallographic K-factors. The experimental EFG imposes certain restrictions on models for the charge distribution, which are discussed in detail. Nominal charges of +1 for Cu and ?2 for O on the basis of the ideal ionic model can be excluded. The sign of the field gradient is predicted to be negative.     

Effects of pH on isotherm modeling and cation competition for Cd(II) and Cu(II) biosorption on Myriophyllum spicatum from aqueous solutions

The biosorption characteristics of Cd(II) and Cu(II) ions from aqueous solutions obtained using submerged aquatic plant (Myriophyllum spicatum) biomass were investigated in terms of equilibrium, kinetics, thermodynamics, and cation competition. Langmuir and Freundlich models were applied to describe the biosorption isotherm of metal ions by M. spicatum biomass and isotherm constants considering the most important parameter, pH. The variation of sorption isotherm constants showed pH dependence. The Langmuir and Freundlich models fitted the equilibrium data well. The maximum biosorption capacity (q _m) of M. spicatum biomass was determined to be 29.07 mg/g for the Cd(II) ion at pH 5.0 and 12.12 mg/g for the Cu(II) ion at pH 6.0. Chi square analysis showed that the Freundlich model fitted the equilibrium data better than the Langmuir isotherm. Competition of Cd(II) and Cu(II) in a binary solution showed that the Langmuir monolayer capacity of Cd(II) decreased from 29.07 mg/g with only Cd(II) in solution to 12.02 mg/g in the presence of Cu(II). Kinetics results showed that the biosorption processes of both metal ions followed the pseudo-second-order kinetics well. The calculated thermodynamic parameters (?G ⁰, ?H ⁰, and ?S ⁰) showed that biosorption of Cd(II) and Cu(II) ions onto M. spicatum biomass was feasible, spontaneous, and endothermic in nature. Fourier transform infrared spectroscopy spectrum analysis revealed that Cd(II) and Cu(II) sorption was mainly ascribed to carboxyl, hydroxyl, amine, and C–N groups in M. spicatum.     

Thermodynamic calculation of peridotite phase relations in the system MgO-Al2O3-SiO2-Cr2O3, with some geological applications

The system MgO-Al₂O₃-SiO₂(MAS) comprises 88–90% of the bulk composition of an average peridotite. The MAS ternary is thus a suitable starting point for exploring peridotite phase relations in multicomponent natural systems. The basic MAS phase relations may be treated in terms of the reactions (see list of symbols etc).

1. py (in Gt)=en (in Opx)+mats (in Opx),
2. en (in Opx)+sp (in Sp)=mats (in Opx)+fo (in Ol), and
3. py (in Gt)+fo (in Ol)=en (in Opx)+sp (in Sp).

Extensive reversed phase equilibria data on these three reactions by Danckwerth and Newton (1978), Perkins et al. (1981), and Gasparik and Newton (1984) employing identical experimental methods in the same laboratory have been used by us to deduce the following internally consistent thermodynamic data applying the technique of linear programming:ΔH ₂₉₈₍₁₎ ⁰ = 2536 J, ΔS ₂₉₈₍₁₎ ⁰ =? 6.064 J/K;ΔH ₂₉₈₍₂₎ ⁰ = 29435 J, ΔS ₂₉₈₍₂₎ ⁰ = 8.323 J/K; andΔH ₂₉₈₍₃₎ ⁰ =?26899 J, ΔS ₂₉₈₍₃₎ ⁰ =?14.388 J/K.These data are also found to be consistent with results of calorimetry. Figure 2 shows the calculated phase relations based on our thermodynamic data; they are consistent with the phase equilibria experiments. Successful extension of the MAS phase relations to multicomponent peridotites pivots on the extent to which the effects of the “non-ternary” (i.e. other than MAS) components can be quantitatively handled. Particularly hazardous in this context is Cr₂O₃, although it barely makes up 0.2 to 0.5 wt% of such rocks. This is because Cr⁺³ fractionates extremely strongly into Sp. This study focuses on the peridotite phase relations in the MgO-Al₂O₃-SiO₂-Cr₂O₃ (MASCr) quaternary. Thermodynamic calculations of the MASCr phase relations have been accomplished by using ΔH ₂₉₈ ⁰ and ΔS ₂₉₈ ⁰ values for the reactions (1) through (3) indicated above, in conjunction with data on thermodynamic mixing properties of

1. binary Sp (sp-pc) crystalline solution (Oka et al. 1984),
2. ternary Opx (en-mats-mcts) crystalline solution (this study), and
3. binary Gt (py-kn) crystalline solution (this study).

The results are shown in P-T projections (Figs. 3a and b) and isobaric-isothermal sections of MASCr in a projection through the component fo onto the SiO₂-Al₂O₃-Cr₂O₃ ternary (Figs. 4a and b). The most important results of this work may be summarized as follows:

1. With increasing incorporation of Cr⁺³ into Sp and Gt, the X _mats isopleths of the reactions (1) and (2) are shifted to higher temperatures (Fig. 3a); simultaneously, the spinel-peridotite to garnet-peridotite phase transition is moved to higher pressures (Fig. 3b).
2. At identical P and T, the X _mats values of Opx coexisting in equilibrium with Ol and Sp is strongly dependent upon the X _pc value in the latter phase (Figs. 4a and b). Accurate correction for the composition of Sp is, therefore, a necessary precondition for geothermometry of the spinelperidotites.
3. The discrepant temperatures reported by Sachtleben und Seck (1981, Fig. 5) from the spinel-peridotites of the Eifel area (systematically too high temperatures as a function of X _pc in Sp) are demonstrated to be the result of ignoring the nonideality in the chromian spinels.

     

Optimization of CNDO for molecular orbital calculation on silicates

The use of approximate molecular orbital (MO) calculations [particularly complete neglect of differential overlap (CNDO)] as a tool in understanding chemical bonding in silicates is investigated. This requires first a detailed analysis of the parametrization employed by the CNDO theory when third row atoms are involved. The accuracy of the CNDO calculations is tested by calculations on the equilibrium bond lengths, orbital energies, and bond stretching force constants of simple third row molecules, for which we have experimental data and/or ab initio results. The effects of an optimization of the parameters in the theory on the calculated properties are then analyzed. The theory is subsequently applied to a sequence of silicate prototypes: silicic acid, H₄SiO₄, disiloxane, (SiH₃) - O - (SiH₃), and disilicic acid, (SiO₃H₃) - O - (SiO₃H₃). With proper tuning of the parameters, the CNDO method can be useful in further elucidating the details of the bonding in silicates.     

Phonon spectra and rigid-ion model calculations on andalusite

Polarised Raman and infrared spectra of (ir) andalusite (Al₂SiO₅) single crystals have been measured and interpreted on the basis of a rigid-ion model calculation. The Al-O bond strength is found to be about 70% ionic in character whereas the mainly covalently bound SiO₄ tetrahedra show ca. 40% ionicity. The interatomic short range forces are strongest between silicon and oxygen and rather weak around the fivefold coordinated aluminium. Thermal soft modes appear above 200°C and are correlated with a weakening of the Al-O bonds.     

Dielectric constants of BeO,MgO, and CaO using the two-terminal method

Using fused SiO₂, CaF₂, and SrF₂ samples with accurately known dielectric constants, we have evaluated the accuracy and precision of two-terminal dielectric constant measurements on small single crystals using empirically determined edge corrections. Values of κ′ at 1 MHz of 3.836±0.05 for silica, 6.814±0.07 for CaF₂ and 6.463±0.09 for SrF₂ indicate an accuracy and precision of 1.0–1.5% for samples having areas of 0.05–1.0 cm². Dielectric constants of BeO, MgO, and CaO measured by this technique are: BeO, κ′_a=6.87 and κ′_c=7.74; MgO, κ′= 9.90; and CaO, κ′=11.95 where κ′_a and κ′_c are the dielectric constants parallel to the a and c axes, respectively. Dielectric loss measurements on CaO in vacuum between 5–400 K at 10–10⁵ Hz indicate significant dispersion at temperatures higher than 300 K, but the effect of the losses on the dielectric constant is less than 1% at 1 MHz and 300 K.     

Variational stabilization of the ionic charge densities in the electron-gas theory of crystals: Applications to MgO and CaO

The electron-gas theory of crystals is extended to include the effects of many-body forces that arise from both electrostatic and overlap interactions. These effects are incorporated through a self-consistent spherical relaxation of the ionic charge distributions such that the crystal binding energy is minimized. This variational model is used to compute the elastic constants and equations of state of MgO and CaO, and we compare its results to those derived from earlier electron-gas models. In the variational model, the anion charge distributions are markedly more sensitive to the local crystal environment than they are in the PIB or other electron-gas models. We find that for these oxides the variational model gives the best overall agreement with experiment for lattice constants, equations of state, dissociation energies, and elastic moduli.     

Computer modelling of Al/Si ordering in sillimanite

Computer simulation is used to investigate the effect of Al/Si disordering over the tetrahedral sites on the lattice energy and the lattice constants of the mineral sillimanite Al₂SiO₅. A methodology for an atomistic assessment of the energy of the reaction 2(Si-O-Al)→(Si-O-Si)+(Al-O-Al) and its various contributions is established. This ordering energy is 0.97 eV for nearest neighbour sites in the ab-plane and 0.56 eV for those separated in the c-direction. The large difference is due to a greater constraint on the atomic relaxation in the ab-plane and shows the structural dependence of the ordering energy. Its magnitude appears to be determined by a complicated balance between Coulomb and short-range repulsive energy involving strain over many bonds, both in the ordered and disordered structures. There is also a significant interaction between second neighbour sites whereas the contribution of more distant neighbours is negligible. The lattice energies of most of the 154 configurations studied show a linear behaviour as a function of short-range order, specified by the number of Al-Al pairs. The ordering temperature T_c, estimated on the basis of a statistical mechanical model of disordering, and the calculated ordering energies are in semi-quantitative agreement with experimental values.     

Raman spectroscopic study of clinopyroxenes in the system CaScAlSiO6 - CaAl2SiO6

Seven clinopyroxenes in the system CaScAlSiO₆- CaAl₂SiO₆ synthesized at 1 atm and under high pressure have been studied by Raman spectroscopy. The T-O-T stretching band of CaScAlSiO₆ pyroxene can be deconvoluted into three bands corresponding to Al-O-Al, Al-O-Si, and Si-O-Si stretching vibrations, although that of CaAl₂SiO₆ can be deconvoluted into the two bands (Al-O-Al+Al-O-Si) and Si-O-Si. The Al-O-Si Raman shifts of CaScAlSiO₆ and CaAl₂SiO₆ pyroxenes are found to fall on the linear plot of the relationship between T-T distance and Raman shifts in ABSi₂O₆-type pyroxenes, suggesting that the Al-O-Si chains are relatively long. Variation of areal fractions of the Raman bands demonstrates that the partial disordering of Al/Si depends on the ionic radius and electronegativity of the octahedral ion.