Statistical mechanics of coupled solid solutions in the dilute limit

A statistical mechanical analysis of the limiting laws for coupled solid solutions shows that the random model, in which the configurational entropy is calculated as if atoms mix randomly on each crystallographic site, is correct as a first approximation. In coupled solid solutions, since atoms of different valence substitute on the same sites, significant short-range order which reduces the entropy can be expected. A first-order correction is rigorously obtained for the entropy in dilute binary short-range ordered coupled solid solutions: $$\bar S^{{\text{XS}}} {\text{/R = }}Q\left( {{\text{e}}^{--H_{\text{A}} /{\text{R}}T} \left( {\frac{{H_{\text{A}} }}{{{\text{R}}T}} + 1} \right) - 1} \right)N_2^a N_4^b ,$$ where Q is the number of positions an associated cation pair can assume per formula unit, H _A is the association energy per formula unit, and N ₂ ^a and N ₄ ^b are the site occupancy fractions for atoms 2 and 4 that are dilute on sites a and b. S ^XS is the configurational entropy minus the random model entropy. Aluminous pyroxenes on the joints diopside-jadeite and diopside-CaTs are examined as examples. A generalization for dilute multiple component solutions, including possible long-range ordering variations is given by: $$\frac{{\bar S^{{\text{XS}}} }}{{\text{R}}}{\text{ = }}\sum\limits_i {\sum\limits_j {\sum\limits_k {Q_i } } \left( {{\text{e}}^{--H_{\text{A}}^{j{\text{ }}k{\text{, }}i} /{\text{R}}T} \left( {\frac{{H_{\text{A}}^{j{\text{ }}k{\text{, }}i} }}{{{\text{R}}T}} + 1} \right) - 1} \right)N_j^l N_k^m ,} $$ where i labels each crystallographically distinct pair, j and k label atomic species, l and m label crystallographic sites, and the N's are site occupancy fractions for the solute atoms. A total association model is examined as well as the partial association and random models. Real solution behavior must lie between the total association model and the random model. Molecular models in which the ideal activity is proportional to a mole fraction, which in itself is not always unambiguously defined, do not lie in this range and furthermore have no physical justification.     

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.     

The reaction forsterite+cordierite=aluminous orthopyroxene+spinel in the system MgO-Al2O3-SiO2

Reversal experiments at 1,150–1,300°C on the reaction forsterite+cordierite=aluminous orthopyroxene+spinel in the system MgO-Al₂O₃-SiO₂ show the equilibrium to have a negativedT/dP. The slope andT-P location of this equilibrium have been modelled using available heat capacity data and various structural models which explore the configurational entropy contributions to the totalΔS. The experimental data are consistent with the aluminous orthopyroxene model of Ganguly and Ghose (1979) where limited Al disorder occurs between theM1 andM2 sites, Al-Si mixing occurs on the tetrahedralB site with the ‘aluminum avoidance’ principle maintained, and Mg-Al disorder occurs in spinel with an interchange enthalpy of 9–12 kcal mol^?1. Additionally, Al-Si disordering which occurs in the indialite structure of cordierite is inconsistent with the experimental data and all pyroxene and spinel mixing models; consequently, Si and Al in anhydrous cordierites to 1,300°C in the system MgO-Al₂O₃-SiO₂ must be largely ordered.     

A computational study of Al/Si ordering in cordierite

The ordering of Al and Si in Mg cordierite Mg₂Al₄Si₅O₁₈ is considered using computer simulation. First the enthalpy of interaction J _ij between sites is derived by computer modelling 101 different Al/Si configurations and analysing their energies. They are compared with similar results for three other minerals and with ab initio calculations to assess the whole approach. Secondly the ordering process is studied using Monte Carlo simulation applied to the J _ij . The ordering phase transition temperature T _c is found as 1800°C in reasonable agreement with the experimental estimate of 1450° C. These are much lower than the estimate T _c(ABW)≈7600°C obtained from Bragg-Williams theory. Strong short-range order sets in below T _c(ABW), and the reasons for much lower temperature T _c of long-range ordering are discussed. Strong short-range also sets in very rapidly in a simulated anneal, in agreement with experiment. Thirdly an attempt is made to compare our calculated enthalpies directly with the results of NMR and calorimetry experiments, not completely successfully. A free energy ΔG≈4.6 eV for the activation barrier for ordering is suggested.     

Short-range order and frustration in omphacite: Comparison of three CVM approximations

We compare three cluster variation method (CVM) models for order-disorder in omphacite (Al_0.5Mg_0.5)[Na_0.5Ca_0.5]Si₂O₆: the generalized point approximation (GPA), generalized pair approximation (GPaA) and an approximation that is based on two eight-body clusters (2X8A). The same Hamiltonian (set of pairwise interactions) is used for all three approximations. Pair probabilities predicted by the GPA obey the geometric constraints of the crysal structure (the frustration constraint), but the exclusion of short-range order (SRO) leads to overestimates of the configurational internal energy (〈E〉), and the critical temperature for cation order-disorder (T _c ). The GPaA violates the frustration constraint, and it therefore fails to predict a P2 ₁/n → C2/c order-disorder transition and yields values for 〈E〉 and SRO that are inconsistent with the omphacite crystal structure. The 2X8A predicts SRO that is consistent with the frustration constraint, and it yields improved (lower) estimates of 〈E〉 and T _c relative to the GPA: 〈E〉_2X8A<〈E〉_GPA, and T _c (2X8A)≈0.741 T _c (GPA).     

Thermodynamics of open networks: Ordering and entropy in NaAlSiO4 glass,liquid, and polymorphs

The thermodynamic properties of carnegieite and NaAlSiO₄ glass and liquid have been investigated through C _p determinations from 10 to 1800 K and solution-calorimetry measurements. The relative entropies S ₂₉₈-S₀ of carnegieite and NaAlSiO₄ glass are 118.7 and 124.8 J/mol K, respectively. The low-high carnegieite transition has been observed at 966 K with an enthalpy of transition of 8.1±0.3 kJ/mol, and the enthalpy of fusion of carnegieite at the congruent melting point of 1799 K is 21.7±3 kJ/mol. These results are consistent with the reported temperature of the nepheline-carnegieite transition and available thermodynamic data for nepheline. The entropy of quenched NaAlSiO₄ glass at 0 K is 9.7±2 J/mol K and indicates considerable ordering among AlO₄ and SiO₄ tetrahedra. In the liquid state, progressive, temperature-induced Si, Al disordering could account for the high configurational heat capacity. Finally, the differences between the entropies and heat capacities of nepheline and carnegieite do not seem to conform to current polyhedral modeling of these properties     

Viscosity and configurational entropy of silicate melts

With the configurational entropy theory of relaxation processes of Adam and Gibbs (1965), one predicts that the viscosity depends on temperature according to log $\text{η = A}{}_{\mathrm{e}}\text{+}\text{B}{}_{\mathrm{e}}\text{TS}{}_{\mathrm{<mtext>conf</mtext>}}$, where S_conf is the configurational entropy of the liquid. Thermochemical calculations of S_conf performed for some mineral compositions show the importance of non-configurational contributions to the entropy differences between amorphous and crystalline phases. Except for the case of SiO₂, the available thermodynamic data indicate that the above equation for viscosity accounts quantitatively for the experimentally determined temperature dependence of the viscosity of silicate melts. The Adam and Gibbs theory also provides a simple rationale for the non linear variation of the logarithmic viscosity with composition in mixed alkali silicate liquids at low temperatures, the minimum of viscosity resulting from the contribution of the entropy of mixing to S_conf.     

The configurational entropies of Ca2MgSi2O7-Ca2SiAl2O7 melilites and related minerals

Equations for the configurational entropy and homogeneous equilibria in Ca₂MgSi₂O₇-Ca₂SiAl₂O₇ melilites are derived for a site constraint that does not permit Mg on the smaller T₂ tetrahedral sites. This constraint leads to one ordering parameter and one composition parameter. The maximum configurational entropy for perfectly ordered crystalline solutions is 3.795 cal K^–1 mol^–1, corresponding to the structural formula Ca₂(Mg_0.33Al_0.67) (Si_0.67Al_0.33)₂O₇ and not the equimolar composition Ca₂(Mg_0.5Al_0.5)(Si_0.75Al_0.25)₂O₇. Similarly, the configuration Ca₂(Si_0.33Al_0.67)(Si_0.33Al_0.67)₂O₇ has the maximum entropy for the gehlenite end-member composition. The tabulated entropy of end-member gehlenite at 298.15 K must be corrected by at least 2Rln2, which corresponds to a substantial correction to its Gibbs energy at high temperature. The same corrections are applicable to other minerals having two crystallographically distinct sites in a 21 ratio and where the same configurational entropy equation applies: MgFe₂O₄ (magnesioferrite), NiFe₂O₄ (trevorite), TiFe₂O₄ (ulvospinel), TiMg₂O₄, TiZn₂O₄, CuFe₂O₄, and TiFe₂O₅ (pseudobrookite) inasmuch as these substances prefer the same inverse ordered state as gehlenite at low temperatures. The effects of the ideal enthalpy and entropy on temperatures of homogeneous equilibria in gehlenite and MgTi₂O₅ are evaluated. Geological-thermometer phase diagrams of the long-range ordering parameter plotted against temperature have a sigmoidal shape for these minerals owing to the fact that this type of disordering is non-convergent.     

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?Å. 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.     

Investigation of superstructures in mullite by high resolution electron microscopy and electron diffraction

Superstructures in synthetic mullite, Al_4+2x Si_2?2x O_10?x , prepared using the zone melting technique, are studied by high resolution electron microscopy (HREM), electron diffraction, and in situ energy dispersive X-ray spectroscopy. For x=0.40 composition HREM images indicate only short-range order. Near x=0.5 composition mullite has an antiphase domain structure with antiphase boundaries (APB's) oriented parallel to (100). For higher Al-content the APB's run in non-rational orientations which are slightly inclined against the {601}-planes. We propose models for the superstructures in which oxygen vacancies are arranged in channels parallel to the crystal b-axis. The models are supported by a one-to-one match between observed and computer-simulated images. An Al-rich limit of x=2/3 for mullite is deduced on the basis of stereochemical considerations. Different ordering schemes for the tetrahedral cations are believed to result in the orientation change of the APB's near x=0.5 composition.     

X-Ray radial distribution function analysis of acid volcanic glasses from the Eastern Rhodopes,Bulgaria

Five natural acid volcanic glasses (perlites) from the Eastern Rhodope mountains, Bulgaria, have been studied by X-ray diffraction. The quantity of the microlites varies from 1–3.5 weight percent. It is higher in the glasses from the rhyolite-perlite transition zone. Total pair correlation functions have been calculated for three of the glasses with less than 2 weight percent microlites. All total pair correlation functions are quite similar and have six well defined peaks up to 8 Å. Beyond 8 Å they are practically featureless. The general form of the curves and peak positions suggests that the short-range order in all the three glasses is compatible with a 6-membered tetrahedral ring polymerization scheme with some contribution of fourmembered rings. The T-01 (T=Si, Al) distance shows linear correlation with the weight percent ratio Al₂O₃/SiO₂. The averaged first nearest neighbour distances T-01, O-01 and T-T1 are 1.615±0.005 Å, 2.66±0.02 Å and 3.16±0.02 Å, respectively. The mean T-O-T bond angle is 157±4°. Energy minimization and topology considerations of the possible distribution of different tetrahedral rings are discussed.     

Computer simulations of the Al-Si disorder in synthetic cordierites: Configuraltional entropy constrained by 29Si NMR data

The published ²⁹Si NMR data on synthetic Mg-cordierites have been used to estimate the changes in configurational Al-Si entropy of the samples due to metastable disorder. The results show that with the increase of the time of annealing in the range of 2 min-2000 h at 1185 °C the entropy of disorder in cordierite decreases from 17.1 to 6.4 J/mol K, while at 1400 °C in the range of 2 min–88 h the entropy changes from 15.4 to 8.8 J/mol K. The decrease in entropy is followed by the appearance and increase of long-range ordering which is reflected in changes of T₁ and T₂ site occupancies, decrease in the number of Si-O-Si and Al-O-Al groupings around O1 oxygens and in a decrease in the number of Al-O-Si-O-Al contacts among hexagonal 6T₂-rings. The derived values of entropy effects together with published calorimetric data suggest that the enthalpy of metastable disordering strongly depends on the temperature of annealing.     

Magnetic structure and cation distribution in (Fe,Mn)2SiO4 (olivine) by neutron diffraction

The cation distribution in the synthetic samples of olivine-type structure with composition (Fe _x Mn_1?x )₂SiO₄ was determined at room temperature and confirms previous Mössbauer results. At low temperature an antiferromagnetic ordering is observed. The magnetic structures can be described in the crystallographic cell (i.e. k=0). They are interpreted on the basis of the irreducible representations (modes) of the symmetry groups which are compatible with Pnma. The dominant modes observed for all compounds, including Fe₂SiO₄ and Mn₂SiO₄, only differ in their direction. The main direction of magnetization is dominated by the Fe²⁺ single-ion anisotropy. At 4.2K, for x=0.29, it is parallel to the c-axis, whereas for x=0.76 the direction is parallel to the b-axis. The anisotropy of the M1-sites dominates in the first case, whereas M2-anisotropy dominates in the second case. The influence of temperature is demonstrated for x=0.50 where c is the main direction at 4.2K, when it is b at 38K.     

Thermodynamic modelling of non-convergent ordering in orthopyroxenes: a comparison of classical and Landau approaches

The excess Gibbs free energy due to non-convergent ordering is described by a Landau expansion in which configurational and non-configurational entropy contributions are separated:

Theoretical studies of charge relaxation effects on the statics and dynamics of oxides

Inclusion of spherical charge relaxation in response to the long-range electrostatic potential (potential induced breathing, or PIB) gives improved results for the static and dynamic properties of oxides. PIB is a Gordon-Kim type model, in which the crystal charge density is estimated by overlapping ionic charge densities. No experimental data are used, except for the values of universal constants, and in this sense the results are from first principles. In contrast to earlier models which include some form of charge relaxation, we explicitly include the breathing effects on the self-energy and pair potentials in the model Hamiltonian. PIB is a many-body effect that couples the long-and short-range forces in a way that is not present in any other first principles or empirical models. It leads to the observed violations of the Cauchy relations for the elastic constants whereas central force rigid ion models cannot violate the Cauchy relations. PIB also reduces the predicted LO-TO splitting because the breathing effect introduces dynamical effective charges that are lower in magnitude than the ionic charges. Some results are shown and discussed for MgO (periclase), BeO (bromellite), Al₂O₃ (corundum), TiO₂ (rutile) and SiO₂ (quartz and stishovite).     

Microsommite: crystal chemistry, phase transitions, Ising model and Monte Carlo simulations

Microsommite, ideal formula [Na₄K₂(SO₄)] [Ca₂Cl₂][Si₆Al₆O₂₄], is a rare feldspathoid that occurs in volcanic products of Vesuvius. It belongs to the cancrinite–davyne group of minerals, presenting an ABAB… stacking sequence of layers that contain six-membered rings of tetrahedra, with Si and Al cations regularly alternating in the tetrahedral sites. The structure was refined in space group P6₃ to R=0.053 by means of single-crystal X-ray diffraction data. The cell parameters are a=22.161?Å=√3a _dav, c=5.358?Å=c _dav; Z=3. The superstructure arises due to the long-range ordering of extra-framework ions within the channels of the structure. This ordering progressively decreases with rising temperature until it is completely lost and microsommite transforms into davyne. The order–disorder transformation has been monitored in several crystals by means of X-ray superstructure reflections and the critical parameters T _c?≈?750?°C and β?≈?0.12 were obtained. The kinetics of the ordering process were followed at different temperatures and the activation energy was determined to be about 125?kJ?mol^?1. The continuous order–disorder phase transition in microsommite has been discussed on the basis of a two-dimensional Ising model in a triangular lattice with nn (nearest neighbours) and nnn (next-nearest neighbours) interactions. Such a model was simulated using a Monte Carlo technique. The theoretical model well matches the experimental data; two phase transitions were indicated by the simulated runs: at low temperature only one of the three sublattices begins to disorder, whereas the second transition involves all three sublattices.     

Empirical models relating viscosity and tracer diffusion in magmatic silicate melts

The Adam-Gibbs equations describing relaxation in silicate melts are applied to diffusion of trace components of multicomponent liquids. The Adam-Gibbs theory is used as a starting point to derive an explicit relation between viscosity and diffusion including non-Arrhenian temperature dependence. The general form of the equation is D_iη = A_iexp{Δ(s_cE_i)/TS_c}, where D is diffusivity, η is melt viscosity, T is absolute temperature, Δ(s_cE_i) is the difference between the products of activation energies and local configurational entropies for viscous and diffusive relaxation, A_i is a constant that depends on the characteristics of the diffusing solute particles, and S_c is configurational entropy of the melt. The general equation will be impractical for most predictive purposes due to the paucity of configurational entropy data for silicate melts. Under most magmatic conditions the proposed non-Arrhenian behaviour can be neglected, allowing the general equation to be simplified to a generalized form of the Eyring equation to describe diffusion of solutes that interact weakly with the melt structure: D_iη/T = Q_iexp{ΔE_i/RT}, where Q_i and ΔE_i depend on the characteristics of the solute and the melt structure. If the diffusing solute interacts strongly with the melt structure or is a network-forming cation itself, then ΔE_i = 0, and the relation between viscosity and diffusion has the functional form of the classic Eyring and Stokes-Einstein equations; D_iη/T = Q_i. If the diffusing solute can make diffusive jumps without requiring cooperative rearrangement of the melt structure, the diffusivity is entirely decoupled from melt viscosity and should be Arrhenian, i.e., D_i = Q_iexp{B_i/T}. A dataset of 594 published diffusivities in melts ranging from the system CAS through diopside, basalt, andesite, anhydrous rhyolite, hydrous rhyolite, and peralkaline rhyolite to albite, orthoclase, and jadeite is compared with the model equations. Alkali diffusion is completely decoupled from melt viscosity but is related to melt structure. Network-modifying cations with field strength Z_i²/r between 1 and 10 interact weakly with the melt network and can be modelled with the extended form of the Eyring equation. Diffusivities of cations with high field strength have activation energies essentially equal to that of viscous flow and can be modelled with a simple reciprocal Eyring-type dependence on viscosity. The values of Q_i, ΔE_i and B_i for each cation are different and can be related to the cation charge and radius as well as the composition of the melt through the parameters Z_i²/r, M/O, and Al/(Na + K + H). I present empirical fit parameters to the model equations that permit prediction of cation diffusivities given only charge and radius of the cation and temperature, composition and viscosity of the melt, for the entire range of temperatures accessible to magmas near to or above their liquidus, for magmas ranging in composition from basalt through andesite to hydrous or anhydrous rhyolite. Pressure effects are implicitly accounted for by corrections to melt viscosity. Ninety percent of diffusivities predicted by the models are within 0.6 log units of the measured values.     

Computational study of tetrahedral Al–Si ordering in muscovite

 The nature of Al–Si ordering across the tetrahedral sites in muscovite, K₂Al₄(Si₆Al₂O₂₀)(OH)₄, was investigated using various computational techniques. Values of the atomic exchange interaction parameters J _l were obtained. From these parameters, a two-dimensional Al–Si ordering scheme was deduced. The transition temperature T _c for this two-dimensional ordering is 1900 K. There are several possible ordering schemes in three dimensions, based on different stacking sequences of ordered sheets of tetrahedral sites. Monte Carlo simulations of both two-dimensional and three-dimensional ordering were performed, but in the three-dimensional simulation only the two-dimensional ordering is seen, implying that three-dimensional ordering is too slow to be attained during the timescale of the simulation. The effect of the three-dimensional interactions is to raise the two-dimensional ordering temperature to 2140 K. From the three-dimensional Monte Carlo simulation, the frequency of occurrence of 4Si0Al, 3Si1Al, 2Si2Al and 1Si3Al clusters was determined, which match those inferred by ²⁹Si MAS–NMR measurements reasonably well. In fact, the match suggests that the cation ordering seen in experiments corresponds to a configuration with considerable short-range order but no long-range order, similar to a state that is at a temperature just above an ordering phase transition. Received: 28 August 2000 / Accepted: 12 March 2001     

Hyperfine interactions and spin transfer between Cr3+ and Al3+ in a synthetic single crystal of Mg2SiO4. A theoretical approach to the interpretation of EPR data

The occurrence of Cr-Al pairs in Mg₂SiO₄ has been detected by EPR spectroscopy. In the case where Cr³⁺ replaces Mg at the M2 position three different neighboring Si sites may be substituted by Al³⁺, which should yield different superhyperfine interactions. A new spectrum is presented which shows the presence of two of these possible pair configurations. An assignment of the spectral features to a specific Cr-Al pair with Cr at M2 from the experimental data alone was not possible, therefore, MSX _α cluster calculations have been performed from which the differences in the superhyperfine interaction for the various pair configurations could be obtained. Best agreement with the data of the Cr³⁺(M2)-Al pair exhibiting the most intense group of lines in the EPR spectrum was obtained for the situation where Al³⁺ is at the Si position with the shortest distance to M2. The second observed Cr³⁺(M2)-Al pair, which is significantly weaker in intensity, could not yet be assigned.