Static Lattice Energy Calculations Research Articles

First-principles study of high-pressure stability, structure, and elasticity of FeS2 polymorphs

The pressure-dependent elastic properties of the Fe–S system are important to understand the dynamic properties of the Earth’s interior. We have therefore undertaken a first-principles study of the structural and elastic properties of FeS2 polymorphs under high pressure using a method based on plane-wave pseudopotential density function theory. The lattice constants, elastic constants, zero-pressure bulk modulus, and its pressure derivative of pyrite are in good agreement with the previous experiments and theoretical approaches; the lattice constants of marcasite are also consistent with the available experimental data. Calculations of the elastic constants of pyrite and marcasite have been determined from 0 to 200 GPa. Based on the relationship between the calculated elastic constants and the pressure, which can provide the stability of mineral, it would appear that pyrite is stable, whereas marcasite is unstable when the pressure rises above 130 GPa. Static lattice energy calculations predict the marcasite-to-pyrite phase transition to occur at 5.4 GPa at 0 K.

Rh2O3(II)-type structures inGa2O3andIn2O3under high pressure: Experiment and theory

High-pressure transitions of ${\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ and ${\mathrm{In}}_{2}{\mathrm{O}}_{3}$ were examined by using a laser-heated diamond-anvil cell combined with in situ x-ray diffractometry at pressures up to $108\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ and $20\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$, respectively. To predict the transition pressure to high-pressure phases, first principles static lattice energy calculations based on the density functional theory were also performed. ${\mathrm{Rh}}_{2}{\mathrm{O}}_{3}(\mathrm{II})$ phases were confirmed as post corundum phases for both ${\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ and ${\mathrm{In}}_{2}{\mathrm{O}}_{3}$ at about $37\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ and $7\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$, respectively. The transition pressures determined by high-pressure experiments are almost consistent with those by theoretical computations. In regard to the present sesquioxides, a link of transition from corundum to perovskite was not confirmed. The ${\mathrm{Rh}}_{2}{\mathrm{O}}_{3}(\mathrm{II})$ phases converted to corundum phases under decompression at room temperature. The bulk modulus of ${\mathrm{Rh}}_{2}{\mathrm{O}}_{3}(\mathrm{II})$ phase in ${\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ and ${\mathrm{In}}_{2}{\mathrm{O}}_{3}$ were determined as $271\ifmmode\pm\else\textpm\fi{}10\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ and $169\ifmmode\pm\else\textpm\fi{}4\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ with their pressure derivatives fixed at 4, respectively.

Subsolidus phase relations in Ca2Mo2O8–NaEuMo2O8-powellite solid solution predicted from static lattice energy calculations and Monte Carlo simulations

Thermodynamic mixing properties and subsolidus phase relations of Ca2Mo2O8-NaEuMo2O8 powellites were modelled in the temperature range of 423-1773 K with static lattice energy calculations based on empirically constrained interatomic potentials. Relaxed static lattice energies (SLE) of a large set of randomly varied structures in a 4 x 4 x 2 supercell of I4(1)/a powellite (a = 5.226 A, c = 11.433 A) containing 128 exchangeable (Ca, Na and Eu) atoms were calculated using the general utility lattice program (GULP). These energies were cluster expanded in the basis set of 69 pair-wise effective interactions and three configuration-independent parameters. Temperature-dependent enthalpies of mixing were calculated using the Monte Carlo method. Free energies of mixing were obtained by thermodynamic integration of the Monte Carlo results. The simulations suggest that the NaEuMo2O8 end-member is nearly fully ordered and has I4[combining macron] symmetry. The calculated subsolidus temperature-composition phase diagram is dominated by three miscibility gaps which are separated by narrow fields of stability of two ordered phases with the compositions of x = 4/9 and x = 2/3, where x is the mole fraction of the NaEuMo2O8 end-member.

Thermodynamics of mixing in diopside–jadeite, CaMgSi2O6–NaAlSi2O6, solid solution from static lattice energy calculations

Static lattice energy calculations (SLEC), based on empirical interatomic potentials, have been performed for a set of 800 different structures in a 2 × 2 × 4 supercell of C2/c diopside with compositions between diopside and jadeite, and with different states of order of the exchangeable Na/Ca and Mg/Al cations. Excess static energies of these structures have been cluster expanded in a basis set of 37 pair-interaction parameters. These parameters have been used to constrain Monte Carlo simulations of temperature-dependent properties in the range of 273–2,023 K and to calculate a temperature–composition phase diagram. The simulations predict the order–disorder transition in omphacite at 1,150 ± 20°C in good agreement with the experimental data of Carpenter (Mineral Petrol 78:433–440, 1981). The stronger ordering of Mg/Al within the M1 site than of Ca/Na in the M2 site is attributed to the shorter M1–M1 nearest-neighbor distance, and, consequently, the stronger ordering force. The comparison of the simulated relationship between the order parameters corresponding to M1 and M2 sites with the X-ray refinement data on natural omphacites (Boffa Ballaran et al. in Am Mineral 83:419–433, 1998) suggests that the cation ordering becomes kinetically ineffective at about 600°C.

Coupled Al/Si and O/N order/disorder in BaYb[Si4–x Al x O x N7–x ]sialon: neutron powder diffraction and Monte Carlo simulations

The fractions of aluminium, [Al]/[Al + Si], and oxygen, [O]/[O + N], in crystallographically distinct sites of BaYb[Si4–x Al x O x N7–x ] oxonitridoaluminosilicate (space group P63 mc, No. 186) were refined based on the results of neutron powder diffraction for a synthetic sample with the composition of x = 2.2(2) and simulated as functions of temperature for the compositions x = 2 and x = 2.3 using a combination of static lattice energy calculations (SLEC) and Monte Carlo simulations. The SLEC calcu lations have been performed on a set of 800 structures differing in the distribution of Al/Si and O/N within the 2 × 2 × 2 supercell containing 36 formula units of BaYb[Si4–x Al x O x N7–x ]. The SLEC were based on a transferable set of empirical interatomic potentials developed within the present study. The static lattice energies of these structures have been expanded in the basis set of pair-wise ordering energies and on-site chemical potentials. The ordering energies and the chemical potentials have been used to calculate the configuration energies of the oxonitridoaluminosilicates (so-called sialons) using a Monte Carlo algorithm. The simulations suggest that Al and O are distributed unevenly over two non-equivalent T(Si/Al) and three L(N/O) sites, respectively, and the distribution shows strong dependence both on the temperature and the composition. Both simulated samples exhibit order/disorder transitions in the temperature range 500–1000 K to phases with partial long-range order below these temperatures. Above the transition temperatures the Si/Al and N/O distributions are affected by short-range ordering. The predicted site occupancies are in a qualitative agreement with the neutron diffraction results.

Coupled Al/Si and O/N order/disorder in BaYb[Si4–xAlxOxN7–x]sialon

The fractions of aluminium, [Al]/[Al + Si], and oxygen, [O]/[O + N], in crystallographically distinct sites of BaYb[Si4–xAlxOxN7–x] oxonitridoaluminosilicate (space group P63mc, No. 186) were refined based on the results of neutron powder diffraction for a synthetic sample with the composition of x = 2.2(2) and simulated as functions of temperature for the compositions x = 2 and x = 2.3 using a combination of static lattice energy calculations (SLEC) and Monte Carlo simulations. The SLEC calcu lations have been performed on a set of 800 structures differing in the distribution of Al/Si and O/N within the 2 × 2 × 2 supercell containing 36 formula units of BaYb[Si4–xAlxOxN7–x]. The SLEC were based on a transferable set of empirical interatomic potentials developed within the present study. The static lattice energies of these structures have been expanded in the basis set of pair-wise ordering energies and on-site chemical potentials. The ordering energies and the chemical potentials have been used to calculate the configuration energies of the oxonitridoaluminosilicates (so-called sialons) using a Monte Carlo algorithm. The simulations suggest that Al and O are distributed unevenly over two non-equivalent T(Si/Al) and three L(N/O) sites, respectively, and the distribution shows strong dependence both on the temperature and the composition. Both simulated samples exhibit order/disorder transitions in the temperature range 500–1000 K to phases with partial long-range order below these temperatures. Above the transition temperatures the Si/Al and N/O distributions are affected by short-range ordering. The predicted site occupancies are in a qualitative agreement with the neutron diffraction results.

Thermodynamics of mixing in pyrope-grossular, Mg3Al2Si3O12-Ca3Al2Si3O12, solid solution from lattice dynamics calculations and Monte Carlo simulations

Static lattice energy calculations (SLEC), based on empirical pair potentials have been performed for a set of 125 different structures with compositions between pyrope and grossular, and with different states of order of the exchangeable Mg and Ca cations. Total energies of a subset of these configurations have been calculated with a density functional electronic structure method (ab initio). The excess energies derived from ab initio and SLEC results agree well with each other. Excess free energies of the 125 structures have been calculated at 300 and 1000 K and at 0 and 3 GPa and cluster expanded in a basis set of 8 pair-interaction parameters. These ordering parameters have been used to constrain Monte Carlo simulations of temperature-dependent properties in the ranges of 300-1500 K and 0-3 GPa. The free energies of mixing have been calculated using the method of thermodynamic integration. The calculations predict the development of a significant short-range and long-range ordering at the intermediate 50/50 composition. The long-range ordered phase with I4 1 22 symmetry becomes stable below 600 K. Two miscibility gaps driven by the stability of the intermediate phase develop at both sides of the 50/50 composition. Activity-composition relations in the range of 600-1500 K and 0-3 GPa are described with high-order Redlich-Kister polynomials.

New high-pressure phase relations in CaSnO3

High-pressure phase stabilities of the CaSnO 3 system were investigated using density functional theory static lattice energy calculations. We have identified six phase changes in this system: perovskite-post-perovskite, post-perovskite-B2+cotunnite in CaSiO 3 , B1-B2 in CaO, cassiterite-pyrite, pyrite-orthorhombic-I, and orthorhombic-I-cotunnite in SnO 2 . Our results demonstrate a new high-pressure phase sequence of perovskite → post-perovskite → B2 + cotunnite. The post-perovskite transformation occurs at ~12 GPa with ~2.5% volume reduction and then it decomposes into the oxide mixture of B2 CaO and cotunnite SnO 2 at ~70 GPa with 8.1% volume reduction. The results also predict a few percent bulk velocity variations across these phase changes.

Monte Carlo simulation of mixing in Ca3Fe2Ge3O12–Ca4Ge4O12 garnets and implications for the thermodynamic stability of pyrope–majorite solid solution

Static lattice energy calculations, based on empirical pair potentials, were performed for a large set of structures differing in the arrangement of octahedral cations within the garnet 2 × 2 × 2 supercell. The compositions of these structures varied between Ca3Fe2Ge3O12 and Ca4Ge4O12. The energies were cluster expanded using pair and quaternary terms. The derived ordering constants were used to constrain Monte Carlo simulations of temperature-dependent mixing properties in the ranges of 1,073–3,673 K and 0–10 GPa. The free energies of mixing were calculated using the method of thermodynamic integration. The calculations predict a wide miscibility gap between Fe-rich (cubic) and Fe-pure (tetragonal) garnets consistent with recent experimental observations of Iezzi et al. (Phys Chem Miner 32:197–207, 2005). It is shown that the miscibility gap arises due to a very strong cation ordering at the Fe-pure composition, driven by the charge difference between Ca2+ and Ge4+ cations. The structural and thermodynamic analogies between Ca–Ge and Mg–Si systems suggest that a similar miscibility gap should exist between pyrope and Mg–Si-majorite.

Thermodynamics of pyrope–majorite, Mg3Al2Si3O12–Mg4Si4O12, solid solution from atomistic model calculations

Static lattice energy calculations, based on empirical pair potentials have been performed for a large set of different structures with compositions between pyrope and majorite, and with different states of order of octahedral cations. The energies have been cluster expanded using pair and quaternary terms. The derived ordering constants have been used to constrain Monte–Carlo simulations of temperature-dependent properties in the ranges of 1073–3673 K and 0–20 GPa. The free energies of mixing have been calculated using the method of thermodynamic integration. At zero pressure the cubic/tetragonal transition is predicted for pure majorite at 3300 K. The transition temperature decreases with the increase of the pyrope mole fraction. A miscibility gap associated with the transition starts to develop at about 2000 K and x maj = 0.8, and widens with the decrease in temperature and the increase in pressure. Activity–composition relations in the range of 0–20 GPa and 1073–2673 K are described with the help of a high-order Redlich–Kister polynomial.

Static lattice energy calculations of mixing and ordering enthalpy in binary carbonate solid solutions

Enthalpies of mixing and ordering in the binaries of the quaternary carbonate (Ca,Mg,Fe,Mn)CO 3 solid solution have been modeled using static lattice energy minimization calculations. A set of self-consistent empirical potentials has been specially developed for this task. The calculations illustrate the importance of size mismatch in determining magnitudes of the enthalpies of mixing and ordering in the binaries. The enthalpy effects show positive correlation with the predicted temperatures of the order/disorder transitions in intermediate compounds with the dolomite structure. It is shown that the general behavior of the mixing enthalpy in high- and low- T limits can be constrained with a minimum set of sampled configurations.

Ab-initio calculations of the proton location in topaz-OH, Al 2 SiO 4 (OH) 2

The position of hydrogen in the structure of topaz-OH was determined by means of ab-initio quantum-mechanic calculations. Static lattice energy calculations predict the existence of four non-equivalent positions of protons, which are characterized by O4–H1... O1, O4–H2... O2, O4–H3... O3 and O4–H4... O4 hydrogen bonds. The distribution of the protons between positions of local equilibrium is controlled by the proton–proton avoidance rule and the strength of the hydrogen bonds. The most favourable configuration of hydrogen atoms is achieved for adjacent protons, which form O4–H3... O3 and O4–H4... O4 hydrogen bonds, respectively. The thermal excitation of atoms at a temperature of 55 K is large enough for the hydrogen atoms occasionally to change their positions to form O4–H1... O1 and O4–H2... O2 bonds. At ambient pressures and higher temperatures the protons are in a dynamic exchange between the allowed positions of local minima. As a consequence, for nearly room-temperature conditions, the dynamic change between different structural configurations leads to the violation of all possible symmetry elements and with that to space group #E5/E5#1. The flipping of the protons between different sites is achieved by simple rotation of the OH-dipole and does not produce any significant distortion of the framework of topaz, whose symmetry remains that of the space group Pbnm. Therefore, no reduction of symmetry has been observed in former X-ray studies on topaz-OH. Calculated IR absorption spectra of topaz-OH were found to be in good agreement with measured spectra. According to the calculations, the two favourable configurations of protons might correspond to the measured peak splitting within the OH-stretching range. An experimentally observed low-frequency band at 3520 cm−1 was assigned to the OH-stretching of the O4–H3... O3 bond, while the band at 3600 cm−1 was attributed to OH-stretching of the O4–H4... O4 hydrogen bond. The broad peak in FAR-IR frequency range at 100–150 cm−1 is attributed to the stretching of H3... O3 and H4... O4 contacts. The rate of proton exchange at 670 K among different sites was estimated by ab-inito molecular dynamic simulations. The calculations predict that flipping of adjacent protons between O4–H3... O3 and O4–H4... O4 bonds at 670 K occur at a rate of about 1.96 THz.

Simulation studies on the pyrope-grossular garnet solid solution

The local structural response of Ca/Mg substitution and the energetic effects associated with dodecahedral ordering in the pyrope-grossular garnet solid solution are derived from a combination of static lattice energy calculations and Monte Carlo simulations. We start with a thorough analysis of the goodness of the empirical potential models used for the modelling of aluminosilicate garnets. The degree of polyhedral distortion was found to be a sensitive indicator for the quality of the model and, by comparison with experimental data, was used to select the best of several available empirical potentials. The Ca/Mg substitution on the dodecahedral site in garnet was found to produce strong local distortions in the surrounding tetrahedral and octahedral polyhedra. This arises from the absence of rigid unit modes (RUMS) in the garnet structure, because local rotations of otherwise rigid SiO4 tetrahedra and AlO6 octahedra cannot occur in order to accommodate different-sized divalent cations in the dodecahedral sites. Strain effects, therefore, mainly govern the dodecahedral substitution, and the corresponding strain field around a dodecahedral site has a minimum radius of 5 A. Pyrope-grossular solid solution compositions were modelled using a supercell approach. For several garnet compositions many different configurations representing individual disordered arrangements were relaxed. The resulting energies were analyzed in terms of different-neighbour interactions to determine the parameters of a model Hamiltonian. The corresponding interaction energies were found to be virtually independent of composition. Surprisingly, the nearest-neighbour interaction between edge-sharing dodecahedra is of no particular significance in the garnets. Instead, the strongest interaction is only via the third-nearest neighbours, i.e. dodecahedra that are edge-shared to a common SiO4 tetrahedron. This cannot lead to dodecahedral long-range order in garnets, but can produce significant amounts of short-range order. Monte Carlo simulations were performed on several compositions to determine the macroscopic effects such as NMR-based cluster occupancy, ordering energy and configurational entropy of the short-range ordering process. As expected, the samples tend to random disorder at high temperatures, and at low temperatures it is compositions nearer Py50Gr50 that depart most strongly from random mixing. For example, a maximum reduction of 3.5 J mol−1 K−1 is predicted for Py75Gr25 and ∼10 J mol−1 K−1 for Py50Gr50. A comparison of NMR cluster occupancy with experimental 29Si MAS NMR resonance intensity is partly successful. However, the changes in NMR cluster occupancy are relatively low (∼5%) compared to changes in configurational entropy (∼30%), implying that it might be difficult to estimate exact entropy data from 29Si MAS NMR line intensities.

Van der Waals energy contour map describing the orthorhombic hydrocarbon chain packing of symmetry Pbnm

The packing of long aliphatic chains was classified by static lattice energy calculations. For the orthorhombic space group 62 (Pbnm), the van der Waals energy of subcells with systematically varied lattice constants aSC and bSC was minimized by changing the dihedral angle between the backbone planes of zigzag chain fragments (–CH2–CH2–). The calculated energies relate to infinite aliphatic chains because of the unlimited periodicity of the fragments in the C–C backbone direction. In order to facilitate the application of the energy criterion in the trial and error based indexing process, the resulting set of 383 energy data was plotted as a van der Waals energy contour map projected onto a plane of the reciprocal space, which is defined by the reciprocal spacings s110s. s020. The absolute and local energy minima correspond to different packing modes of orthorhombic subcells. The energy hyper-surface is divided by lines corresponding to special cell geometries into different areas. The lines are localized near saddle points of the energy surface. For the first time, the geometrical and energetic relations between the different packing modes of symmetry Pbnm are clearly demonstrated using van der Waals energy calculations. Based on these calculations, a new nomenclature for chain packing modes is suggested.

Phonon stabilized polytypism in PbI2: in situ Raman spectroscopy and transferable core-shell model calculations

Frequency shifts observed by in situ Raman spectroscopy of the reversible phase transition between the 2H and 12R polytype of PbI2 suggest that softening of phonons in the 12R polytype may stabilize this phase relative to the 2H phase at high temperatures. A core-shell model using short-range pair potentials and long-range electrostatic interactions for PbI2 was used for static lattice and lattice dynamics calculations. The potential parameters, which are fully transferable to other polytypes, were derived empirically for the 2H polytype and then transferred to the 12R polytype. The experimentally observed absence of dielectric anomalies is reproduced by static lattice energy calculations. The phonon contributions to the free energies of the polytypes were calculated for both polytypes, showing that the stabilization of the 12R relative to the 2H polytype is due to the phonon contribution to the free energy.