Mixing Properties Of Solid Solutions Research Articles

Microscopic strain in a grossular-pyrope solution anti-correlates with excess volume through local Mg-Ca cation arrangement, more strongly at high Ca/Mg ratio

Unit-cell volume and microstrain of Py 40 Gr 60 garnets vary with synthesis temperature and annealing time, showing a strong negative correlation, as is also seen in another garnet solid solution, Py 20 Gr 80 . This anti-correlation is explained by local Ca-Mg cation arrangement in which Ca-Ca and Mg-Mg third-nearest-neighbor (Same 3NN = S3NN) pairs form at rates other than those expected from random Ca-Mg distribution. S3NN pairs cause microstrain (Bosenick et al. 2000) but allow more efficient packing than random Ca-Mg pairings that contribute to excess volume, hence smaller cell volumes correlate with more microstrain. Both longer annealing time and higher heating temperature cause more S3NN formation, larger microscopic strain, and smaller unit-cell volume. The anti-correlation of microstrain and excess volume is weaker in our previous study of pyrope-rich solutions (i.e., Py 80 Gr 20 , Du et al. 2016) because excess volume varies little from Ca-Ca S3NN pairings in pyrope-rich solutions, whereas Mg-Mg S3NN pairings in grossular-rich solutions studied here are effective at reducing excess volume. Heating to 600 °C under room pressure or cold hydrostatic compression to 10 GPa does not reset microstrain. Margules’ formulations for microstrain and volume as a function of Ca/Mg ratio captures these features, especially the two-peaked distribution of microstrain with composition discovered by Du et al. (2016). The similar two-peaked distributions of microstrain and excess energies derived from ab initio calculation with short-range ordering of Mg and Ca cations (Vinograd and Sluiter 2006) indicate that the macroscopic thermodynamic mixing properties of solid solutions are directly related to arrangement of cations with large size misfit. The observed changes of microstrain with annealing temperature suggest that mixing properties measured from our pyrope-grossular garnet solid solutions synthesized at same temperature can serve as better experimental constrains for computational work.

Thermodynamics of mixing in an isostructural solid solution: Simulation methodologies and application to the rutile-cassiterite system

The accuracies of two different approaches to model thermodynamic mixing properties of solid solutions are explored using the rutile-cassiterite solid solution as an example. Both methods employ an expansion of the configurational enthalpy in terms of pairwise interactions energies. In the first method the partition function is directly computed from the excess energies of all Ti/Sn configurations within a 2 × 2 × 4 supercell. In the second method the free energy of mixing is calculated by a thermodynamic integration of the thermally averaged enthalpies computed with the Monte Carlo method using an 8 × 12 × 16 supercell. The phase relations derived from Monte Carlo simulations agree well with the available experimental data, under the condition that the free energy is corrected for the effect of the excess vibrational entropy. The direct calculation of the partition function provides reasonable phase relations only when the configurational entropy is corrected to be consistent with the ideal mixing in the high-temperature limit. Advantages and drawbacks of the both approaches are discussed. The findings are generally applicable to models of isostructural solid solutions.

Stability of sapphirine-bearing mineral assemblages in the system FeO-MgO-Al2O3-SiO2 and metamorphic P-T parameters of aluminous granulites

Consistent thermodynamic data on the properties of pure mineral end members and the mixing properties of solid solutions in the system FeO-MgO-Al2 O 3 -SiO 2 were employed to simulate phase rela� tions of sapphirine, garnet, spinel, orthopyroxene, cordierite, quartz, Al silicates, and corundum. Composi� tional variations of the solid solutions with temperature notably modify the topology of the P-T diagrams, which differ from the petrogenetic grids widely used in the literature. It is worth noting that the evaluation of PT metamorphic conditions based on reaction relations in sapphirinebearing assemblages cannot be so far considered reliable enough. The lower stability limit of the sapphirine + quartz assemblage in the system in question is possibly located at much lower P-T parameters: at least 835°C and ~6 kbar. The sapphirine + kya� nite assemblage can be stable at temperatures below 860°C and a pressure of ~11 kbar, and the stability field of the sapphirine + olivine assemblage is narrow and constrained to temperatures no higher than ~800°C.

Scaling of thermodynamic mixing properties in garnet solid solutions

The volumes and enthalpies of mixing, DeltaV(Mix) and DeltaH(Mix), of binary solid-solution aluminosilicate garnets have been studied by computer simulation. The use of "average atoms" to simulate solid solution was found to give results that are considerably different from those obtained by calculating and averaging over many configurations of cations at a given composition. Although we expect mineral properties calculated from model calculations to be correct only on a qualitative rather than a quantitative scale, fair agreement with experiment was obtained where carefully tested potential parameters were used. The results show that mixing behaviour in these materials is controlled by local strain and relaxation effects resulting from the atomic size mismatch of the mixing divalent cations. In particular, DeltaV(Mix) and DeltaH(Mix) shown to scale quadratically with the volume difference between the end members, and to vary essentially symmetrically with composition, with a moderate dependence on the degree and nature of cation order. We conclude that computer modelling should be useful in providing detailed qualitative information about the mixing properties of solid solutions, which can help to better constrain and interpret experimental results.

Comparison of Semi-Empirical and Ab Initio Calculations of the Mixing Properties of MO–M′O Solid Solutions

Theoretical predictions of the mixing properties of solid solutions can be made, in principle, via three different approaches: through the use of phenomenological, semi-classical atomistic, and ab initio techniques. Comparison of the results, obtained by application of all of the above-mentioned ways, is carried out for an example of the oxide solid solutions with the NaCl-type structure: CaO–MgO, CaO–MnO, MnO–NiO, and MgO–MnO. From this it is inferred that the predicted mixing energies for the whole range of compositions are generally in reasonable agreement with each other and with available experimental measurements for the systems under consideration. However, closer inspection shows marked systematic differences between the values of mixing energy and the structure relaxation parameters predicted with the use of different methods. In particular, this is due to the use of rather small and ordered unit cells for the atomistic and ab initio calculations. This allows one to make approximate estimates of disordering energies and propose some ways to improve the theoretical simulation of the solid solutions' mixing properties.

Tie-lines and mixing properties of solid solutions in the system CaO-SrO-PbO-O at 1100 K

Phase relations in the system CaO-SrO-PbO-O2 at 1100 K have been determined by equilibrating samples with different compositions in air, oxygen, or evacuated ampoules for 7 days and characterizing quenched specimens by optical and scanning electron microscopy, energy-dispersive analysis of x-rays (EDX), and x-ray diffraction (XRD). There is a solid-state miscibility gap in the pseudo-binary system CaO-SrO, and continuous solid solubility between Ca2PbO4 and Sr2PbO4 at 1100 K. Substitution of Ca for Sr occurs only to a limited extent (∼2 mol.%) in SrPbO3. The calcium-rich solid solutions (Ca1−y Sr y )2PbO4 characterized by y≤0.255 are in equilibrium with PbO in air; compositions with y≥0.255 coexist with (Ca1−z Sr z )PbO3. There is a three-phase region involving the two monoxide solid solutions (Ca1−x Sr x )O on either side of the miscibility gap with x=0.24 and 0.71 and (Ca1−y Sr y )2PbO4 with y=0.96. Accurately determined are the locations of tie-lines between the solid solutions. Attainment of equilibrium was checked by the conventional tie-line rotation technique. The excess Gibbs energy of mixing of the solid solution with orthorhombic structure is obtained by an analysis of tie-line data; for the mixing of one mole of Ca and Sr represented by (Ca1−y Sr y )Pb0.5O2, ΔG E=y(1−y) [15,840−2950 y] J/mol. The thermodynamic properties suggest the onset of immiscibility in this solid solution below 884 (±5) K. The miscibility gap is asymmetric with a critical composition at y=0.43 (±0.02). Inside the triangle (Ca1−y Sr y )Pb0.5O2−(Ca1−z Sr z )PbO3−PbO, a small liquid-phase region is present close to the PbO corner, surrounded by three two-phase fields. Each corner of the approximately triangular liquid-phase region is associated with a three-phase field.

Computing Lippmann Diagrams from Direct Calculation of Mixing Properties of Solid Solutions: Application to the Barite-Celestite System

The Lippmann diagram for the system(Ba, Sr)SO4-H2O was computed at 25 °Cby determining the solid-phase activity coefficientsfrom first principles calculations. Directcalculations of the mixing properties of thebarite-celestite series indicate this solid solutionbehaves as non-ideal and non-regular. At 25 °C,the enthalpy of mixing shows a minimum around 50 mole% SrSO4 due to an ordering tendency. Thefree energy of mixing shows two minima that delimit awide and symmetric miscibility gap (from 2.1 to 97.9 mole% SrSO4) at this temperature. The excessfree energy of mixing requires a Guggenheim expansionseries of 5 terms to be described, where the termswith odd exponents are null as a consequence of thesymmetric distribution of the mixing properties withcomposition. The Lippmann diagram shows a peritecticpoint that corresponds to the composition of an aqueoussolution which is simultaneously at equilibrium withthe two extremes of the miscibility gap. The largedifference between the solubility products of theendmembers involves a strong preferential partitioningof the less soluble endmember towards the solid phase,which explains the extremely Ba-poor composition ofthe aqueous solution (aqueous activity fraction forBa2+ = 0.000446 ) at the peritectic point.

Ｂｒｉｔｔ‐Ｌｕｅｃｋｅ法による固溶体の熱力学的性質を表すパラメータの計算

This review describes the method of Britt and Luecke (1973), which is applicable to the calculations of parame-ters relevant to the mixing properties of solid solutions. Based on the experimental results on ion-exchange reactions of (Fe, Mn, Mg)TiO3-(Fe, Mn, Mg)Cl2(aq) at 600°C and 98MPa (Kubo et al., 1992), this paper computes the Wilson parame-ters for the (Fe, Mn, Mg)TiO3 solid solution and the Gibbs energy changes of the individual binary exchange reactions.

Geothermobarometry of high-grade metapelites: simultaneously operating reactions

S U M M A R Y : Experimental data on equilibria involving rock-forming minerals in highgrade metapelites (garnet, orthopyroxene, cordierite and plagioclase) have been used to derive an internally consistent system of mineralogical thermometers and barometers based on component reactions and excess mixing properties of solid solutions. Application of these data to a granulite-facies metapelite sample reveals that compositional inhomogeneity of minerals on a thin-section scale is likely to be a result of several reactions opening simultaneously at the retrograde stage of metamorphic evolution. The widely accepted practice of estimating pressure and temperature from separate reactions may thus lead to errors and misinterpretation of P T paths.