Volume 106, Issue 12 p. 7704-7727
RESEARCH ARTICLE
Open Access

The MgO–TiO2–SiO2 system: Experiments and thermodynamic assessment

Mariia Ilatovskaia

Mariia Ilatovskaia

Institute of Materials Science, Technical University Bergakademie Freiberg, Freiberg, Germany

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André Treichel

André Treichel

Institute of Materials Science, Technical University Bergakademie Freiberg, Freiberg, Germany

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Olga Fabrichnaya

Corresponding Author

Olga Fabrichnaya

Institute of Materials Science, Technical University Bergakademie Freiberg, Freiberg, Germany

Correspondence

Olga Fabrichnaya, Technical University Bergakademie Freiberg, Institute of Materials Science, Freiberg, 09599, Germany.

Email: [email protected]

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First published: 03 August 2023

Abstract

Phase relations in the MgO–TiO2–SiO2 system have been investigated in air over a wide temperature range using the equilibration method. X-ray powder diffraction, scanning electron microscopy combined with wave length X-ray spectroscopy (SEM/EPMA), and differential thermal analysis (DTA) have been used for sample characterization. Based on the obtained experimental results, isothermal sections of the system at 1523, 1673, and 1773 K have been established. The solid-state invariant reaction MgTi2O5 + T-SiO2⇋P-MgSiO3 + TiO2 has been detected at 1625 ± 8 K by step-wise heat treatment. A partial liquidus projection has been suggested, and the temperatures and compositions of three eutectic invariant reactions have been experimentally measured by DTA and ex-situ analysis of the sample microstructures after melting using SEM/EPMA. Considering the newly obtained experimental data, thermodynamic parameters describing the system have been thermodynamically evaluated within the CALPHAD approach.

1 INTRODUCTION

The phases and the phase relations within the MgO–TiO2–SiO2 system are of interest for various application fields, including refractory materials, mineralogy, pyrometallurgy, and the development of ceramic filter materials. The aluminum industry nowadays demands strict metal quality standards, and filtration is the most feasible method for reducing the inclusions level entering a filter system before metal casting as the final inline treatment. Inclusions, primarily as Al2O3 or spinel MgAl2O4, arise from the oxidation of molten aluminum combined with dopants such as Mg or from furnace refractory. TiO2 coating deposited on corundum is supposed to filter actively spinel MgAl2O4 from Al-based molten alloy.1-3 For the Si,Mg-containing Al alloys, the reduction of TiO2 can then lead to the formation of MgTiO3, Al3Ti, or (Al,Si)3Ti.4 Therefore, to model the Al melt filtration process, in addition to the melt itself, one should consider the complex Al–Mg–Ti–Si–O system. Focusing on the oxide part of the complex system, the Al2O3–TiO2–MgO5, 6 and Al2O3–TiO2–SiO27, 8 subsystems were described in the authors preceding studies. As a part of the mentioned complex system and a linking bridge to combine the Al2O3-based systems, the MgO–TiO2–SiO2 subsystem is essential. Notably, the understanding of the phase relations in the MgO–TiO2–SiO2 system is of interest for the development of refractory materials technology, as well as for elucidating the geological processes that occur between the minerals that constitute this system (including both silicates and titanates).

Experimental information on phase relations in the MgO–TiO2–SiO2 system is available from the literature, however, this information is limited and mainly related to the melting behavior at 1 atm or to the effect of pressure on the melting behavior. Subsolidus phase relations have not been studied in detail so far. Nevertheless, those data have subsequently been used for thermodynamic assessments of the MgO–TiO2–SiO2 system. However, the thermodynamic models used for the phase descriptions are not consistent with those used in preceding studies of the present authors. Details are discussed in the next section. Therefore, the aim of the present work is an experimental study of phase relations in the MgO–TiO2–SiO2 system in a wide range of temperatures and compositions and a critical evaluation of available data on phase relations as well as thermodynamic assessment of thermodynamic parameters describing the system to derive a self-consistent thermodynamic database using the CALPHAD approach.

2 LITERATURE REVIEW

2.1 Binary systems

The phase relations in the MgO–TiO2, TiO2–SiO2, and MgO–SiO2 systems were repeatedly studied experimentally and their thermodynamic databases are available. The MgO–TiO2 system was recently assessed thermodynamically by Ilatovskaia et al.5 applying their own experimental data obtained in the range of 473−1710 K in air. This description, applying the ionic two-sublattice liquid model and considering the order-disorder phenomena in both Mg2TiO4 and MgTi2O5, is used in this study. The TiO2–SiO2 system was also recently assessed by Ilatovskaia and Fabrichnaya.8 The applied two-sublattice liquid model was reduced to the (SiO2,TiO2) substitutional model, which is consistent with the two-sublattice partially ionic model, and the corresponding interaction parameters were assessed.

The MgO–SiO2 system was first thermodynamically described by Hillert and Wang.9 The two-sublattice partially ionic model was applied for the liquid phase, while the solid phases were described as stoichiometric compounds. Huang et al.10 reoptimized the thermodynamic parameters describing the solid MgSiO3 phases (low-clinopyroxene, orthopyroxene, and protopyroxene), and thus a better fit to the available experimental data was achieved. No new data have been published so far. Since the thermodynamic models used to describe all three quasi-binary systems are consistent, the assessment for the MgO–SiO2 system by Huang et al.10 is accepted in this work without modifications.

2.2 Ternary MgO–TiO2–SiO2 system

Massazza and Sirchia11 constructed the liquidus projection of the system using the pyrometric cone method and microscopy investigation of the samples. Seven invariant reactions, including three ternary eutectics, L⇋MgTiO3+MgTi2O5+Mg2SiO4 at 1793 K, L⇋Mg2SiO4+MgSiO3+MgTi2O5 at 1713 K, and L⇋MgTi2O5+TiO2+MgSiO3 at 1663 K, and the monotectic LA+TiO2⇋LB+C-SiO2 at 1803 K, were indicated on the liquidus, as well as two maxima belong to the sections of Mg2SiO4–MgTi2O4 (1813 K) and MgSiO3–TiO2 (1693 K). This also revealed an information on phase fields on the solidus projection (triangulation). No ternary compound was found. MacGregor12 investigated the effect of pressure on the melting behavior along the MgSiO3–TiO2, Mg2SiO4–TiO2, and MgSiO3–MgTi2O5 sections at 1773−2173 K. The eutectic reaction in the MgSiO3–TiO2 section was suggested to be about 10 K lower than previously indicated.11 Two ternary eutectic reactions on the liquidus were verified at 1 atm: L⇋MgTi2O5+TiO2+MgSiO3 at 1663 K and L⇋Mg2SiO4+MgSiO3+MgTi2O5 at 1713 K. Moreover, the subsolidus reaction was found to occur at 1762 K and 15.2 kb, MgSiO3+MgTi2O5⇋Mg2SiO4+2TiO2. Using also the pyrometric cone method followed by microstructure investigation, Berezhnoi13 suggested the lowest crystallization temperature of 1688 K by the ternary eutectic reaction L⇋MgSiO3+TiO2+SiO2 rather than L⇋MgTi2O5+TiO2+MgSiO3 at 1663 K reported in Ref. [11] Panek et al.14 constructed the Mg2SiO4–Mg2TiO4 section indicating the peritectic decomposition L+MgO⇋Mg2SiO4+Mg2TiO4 at 1793 ± 10 K and 63.72MgO-24.94TiO2-11.34SiO2 (in mol.%). Later, Petrov et al.15 investigated the phase relations along the MgTiO3–Mg2SiO4, MgTi2O5–Mg2SiO4, MgTi2O5–MgSiO3, MgTiO3–MgSiO3, Mg2TiO4–MgSiO3, and Mg2TiO4–Mg2SiO4 sections at 1073−1773 K. No invariant reactions were observed within that temperature range. Sarver and Hummel16 pointed out the same phase relationships in the MgO–TiO2–SiO2 system at 1573 K as those presented on the solidus in Ref. [11]. An isothermal section at 1773 K with an emphasis on the liquid domain was recently reported.17 At 1883 K, the phase relations in the MgO–SiO2–TiOX were investigated under reducing conditions (p(O2) from 1.94∙10−9 to 2.75∙10−13 atm)18 and at high pressure of 10−24 GPa.19 Moreover, a wide area of the liquid immiscibility was investigated at 1883 K.20. Moreover, the cation substitutions and solubilities in MgTiO321, 22 and Mg2SiO423, 24 were observed.

The thermodynamic description of the MgO–TiO2–SiO2 system using the CALPHAD approach was first undertaken by Kaufman25 based on the experimental data of Massazza and Sirchia.11 The calculated isothermal sections in the temperature range of 800−2800 K indicated a joint extended miscibility gap in the liquid emanated from the MgO–SiO2 and TiO2–SiO2 systems. The liquidus projection of the MgO–TiO2–SiO2 system was also calculated by Kirschen and DeCapitani20 with an emphasis on the immiscibility in the liquid using the thermodynamic description derived on the basis of his own experimental data and those of Massazza and Sirchia.11 The liquid phase in Refs. [20, 25] was described by a substitutional model, which is incompatible with the two-sublattice partially ionic liquid model used in the present work. Also, the inversion in spinel and pseudobrookite was not modeled in Refs. [20, 25]. It should be noted that the calculations of the system at 1773 K using the FactSage software with corresponding oxide database were carried out by Chen et al.17 showing a rather good consistency between experimental and calculated data. However, a modified quasi-chemical model used to describe the liquid phase is not compatible with the model used in the present work and details of the thermodynamic description are not available. Therefore, a proper thermodynamic description of the MgO–TiO2–SiO2 system has not been suggested so far.

In summary, all published experimental data on phase relations in the MgO–TiO2–SiO2 system are mostly consistent to each other. Wherein, limited ranges of temperature and composition were only verified,12, 17 which in turn indicated good consistency with the original work11, while some data in their invariable and unverified form after Massazza and Sirchia11 were also widely used in further studies. Therefore, it turned out that the MgO–TiO2–SiO2 system has not been systematically investigated in a wide range of temperatures and compositions, and a new comprehensive study of phase equilibria in the MgO–TiO2–SiO2 system is necessary. Given this, the present study is devoted to an experimental study of phase relations in the MgO–TiO2–SiO2 system in air at 1500−2000 K. The obtained phase diagram data are the fundamental basis for the thermodynamic description of the system and, along with reliable published data, are taken into account in the thermodynamic evaluation.

3 MATERIALS AND METHODS

Samples within the MgO–TiO2–SiO2 system were prepared by coprecipitation followed by thermal decomposition of aqueous solutions, similarly to the procedure described by Fidancevska and Vassilev.26 The starting materials were magnesium nitrate Mg(NO3)2·6H2O (99.97%), titanium(IV) isopropoxide C12H28TiO4 (M = 284.23 g/mol, ρ = 0.955 g/cm3, 97%), and tetraethoxysilane C8H20SiO4 (M = 208.33 g/mol, ρ = 0.934 g/cm3, 98%), all produced by Alfa Aesar. The calculated volumes of the metal-organic precursors were first dissolved in absolute ethanol with magnetic stirring to obtain solutions 1A and 2A, respectively, while magnesium nitrate was dissolved in a small amount of distilled water to get solution 3A. The solution 1A consequently relates to the Ti-based precursor, 2A relates to Si, and 3A relates to Mg. Then, the solutions A1 and A2 were mixed with magnetic stirring at room temperature, and the solution A3 was added to the mixture (1A + 2A) with magnetic stirring and heating at 50−60°C in the presence of NH4OH to keep the pH above 9.0 during the process and to ensure complete precipitation of the corresponding hydroxides. The precipitate was visible as a cloudy white suspension. The precipitate was consistently evaporated at 80°C, dried at 90°C for 48 h, and two-step calcined at 500−800°C for 5 h to obtain the desired mixed oxide powder.

After, the obtained MgO–TiO2–SiO2 powders were ball milled, pressed into tablets at 300 MPa (tablet size: 8 mm in diameter and about 2−3 mm in height), and heat treated in a muffle furnace (Nabertherm) in air at chosen temperatures followed by furnace cooling. The temperature inside the muffle furnace was controlled (±3 K measuring accuracy) with the B-type thermocouple placed by the sample holder (PtRh crucible).

After heat treatment, the powdered samples were investigated at room temperature using an URD63 X-ray diffractometer (Seifert, FPM) with CuKα radiation (λ = 1.5418 Å). ICSD (Inorganic Crystal Structure Database, 2017, Karlsruhe, Germany) was used for the interpretation of the powder diffraction patterns. Qualitative and quantitative analyses of the XRD patterns were performed by Rietveld analysis using MAUD software.27, 28 To establish the phase distribution at the equilibrium state as well as invariant reactions, the microstructures of the samples after prolonged heat treatment and DTA observation were examined using JSM-7800 F (JEOL Ltd., Japan) equipped with an EDX detector. The phase compositions were accurately determined using JXA-8230 SuperProbe (JEOL Ltd., Japan) equipped with a wavelength-dispersive X-ray spectroscope (EPMA/WDX). The standards employed for EPMA were rutile for Ti Kα, periclase for Mg Kα, and quartz for Si Kα. The standard deviation of the EPMA measurements did not exceed 2−3 %.

Temperatures of the solid-state transformations and melting behavior were determined by TG-DTA SETSYS Evolution-1750 (SETARAM, France) using a B-type tricouple DTA rod (PtRh 6%/30% thermocouple). The ceramic specimens placed in open Pt crucibles were heated and cooled under a dynamic air atmosphere at rates of 10 and 30 K/min, respectively.

4 THERMODYNAMIC MODELING

The Thermo-Calc software29 and PARROT module were used to optimize the thermodynamic parameters applying for the description of the MgO–TiO2–SiO2 system. The thermodynamic descriptions of the binary subsystems of MgO–TiO2,5 MgO–SiO2,10 and TiO2–SiO28 are accepted without any alterations from the available thermodynamic descriptions. However, considering the ternary interactions in the studied MgO–TiO2–SiO2 system, the descriptions of the liquid phase and some solid phases were refined based on available experimental data (see details below). The list of the phases and the models used for their description are summarized in Table 1 and discussed in the following sections. The interaction parameters that are actually used in the present work including accepted and newly derived are included in Table 2.

TABLE 1. Phases in the MgO–TiO2–SiO2 system.
Phase Abbreviations Model
Liquid L (or Ionic) (Mg+2,Ti+2,Ti+3)P(O−2,Va,SiO4−4,SiO2,O,TiO2)Q
MgO Halite Hal (Mg+2)1(O−2)1
SiO2 Quartz Q-SiO2 or Qua SiO2
SiO2 Tridymite T-SiO2 or Tr SiO2
SiO2 Cristobalite C-SiO2 or Cr SiO2
TiO2 Rutile Rut ( T i + 4 ) 1 ( O 2 , V a 2 ) 2 ${( {{\mathrm{T}}{{\mathrm{i}}^{ + 4}}} )_1}{( {{{\mathrm{O}}^{ - 2}},{\mathrm{V}}{{\mathrm{a}}^{ - 2}}} )_2}$
Mg2TiO4 Spinel Sp ( M g + 2 , T i + 4 ) 1 ( M g + 2 , T i + 4 ) 2 ( O 2 ) 4 ${( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}},{\mathrm{T}}{{\mathrm{i}}^{ + 4}}} )_1}{( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}},{\mathrm{T}}{{\mathrm{i}}^{ + 4}}} )_2}{( {{{\mathrm{O}}^{ - 2}}} )_4}$
MgTiO3 Ilmenite Ilm ( M g + 2 ) 1 ( T i + 4 , S i + 4 ) 1 ( O 2 ) 3 ${( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1}{( {{\mathrm{T}}{{\mathrm{i}}^{ + 4}},{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_1}{( {{{\mathrm{O}}^{ - 2}}} )_3}$
MgTi2O5 Pseudobrookite Psbk ( M g + 2 , T i + 4 , S i + 4 ) 1 ( M g + 2 , T i + 4 , S i + 4 ) 2 ( O 2 ) 5 ${( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}},{\mathrm{T}}{{\mathrm{i}}^{ + 4}},{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_1}{( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}},{\mathrm{T}}{{\mathrm{i}}^{ + 4}},{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_2}{( {{{\mathrm{O}}^{ - 2}}} )_5}$
Mg2SiO4 Olivine Ol ( M g + 2 ) 1 ( M g + 2 ) 1 ( S i + 4 , T i + 4 ) 1 ( O 2 ) 4 ${( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1}{( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1}{( {{\mathrm{S}}{{\mathrm{i}}^{ + 4}},{\mathrm{T}}{{\mathrm{i}}^{ + 4}}} )_1}{( {{{\mathrm{O}}^{ - 2}}} )_4}$
MgSiO3 Orthopyroxene O-MgSiO3 or opx ( M g + 2 ) 1 ( S i + 4 ) 1 ( O 2 ) 3 ${( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1}{( {{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_1}{( {{{\mathrm{O}}^{ - 2}}} )_3}$
MgSiO3 Protopyroxene P-MgSiO3 or ppx ( M g + 2 ) 1 ( S i + 4 ) 1 ( O 2 ) 3 ${( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1}{( {{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_1}{( {{{\mathrm{O}}^{ - 2}}} )_3}$
MgSiO3 Low-clinopyroxene C-MgSiO3 or low-cpx ( M g + 2 ) 1 ( M g + 2 ) 1 ( S i + 4 ) 2 ( O 2 ) 6 ${( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1}{( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1}{( {{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_2}{( {{{\mathrm{O}}^{ - 2}}} )_6}$
TABLE 2. Thermodynamic description for the MgO–TiO2–SiO2 system.
Phase and Sublattice model Thermodynamic parameter References
Ionic Liquid ( M g + 2 , T i + 2 , T i + 3 ) P ( O 2 , SiO 4 4 , Va , O , Si O 2 , Ti O 2 ) Q ${({\mathrm{M}}{{\mathrm{g}}^{ + 2}},{\mathrm{T}}{{\mathrm{i}}^{ + 2}},{\mathrm{\;T}}{{\mathrm{i}}^{ + 3}})_P} {( {{{\mathrm{O}}^{ - 2}},{\mathrm{SiO}}_4^{ - 4},{\mathrm{Va}},{\mathrm{O}},{\mathrm{Si}}{{\mathrm{O}}_{2,{\mathrm{\;}}}}{\mathrm{Ti}}{{\mathrm{O}}_2}} )_Q}$ 0 G M g + 2 : O 2 I o n i c = + 2 · GMGOLIQ $\;{}^0G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{Ionic\;} = \; + 2\cdot{\mathrm{GMGOLIQ}}$ [30]
0 G T i + 2 : O 2 I o n i c = + 2 · GTI 1 O 1 + 141000 61.4 · T $\;{}^0G_{{\mathrm{T}}{{\mathrm{i}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{Ionic\;} = \; + 2\cdot{\mathrm{GTI}}1{\mathrm{O}}1 + 141000 - 61.4\cdot T$ [31]
0 G T i + 3 : O 2 I o n i c = + GTI 2 O 3 + 114147.4 45.6 · T $\;{}^0G_{{\mathrm{T}}{{\mathrm{i}}^{ + 3}}:{{\mathrm{O}}^{ - 2}}}^{Ionic\;} = \; + {\mathrm{GTI}}2{\mathrm{O}}3 + 114147.4 - 45.6\cdot T$ [31]
0 G M g + 2 : Va I o n i c = + GMGLIQ $\;{}^0G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{Va}}}^{Ionic\;} = \; + {\mathrm{GMGLIQ}}$ [32]
0 G T i + 2 : Va I o n i c = + GLIQTI $\;{}^0G_{{\mathrm{T}}{{\mathrm{i}}^{ + 2}}:{\mathrm{Va}}}^{Ionic\;} = \; + {\mathrm{GLIQTI}}$ [32]
0 G T i + 3 : Va I o n i c = + GLIQTI + 200000 $\;{}^0G_{{\mathrm{T}}{{\mathrm{i}}^{ + 3}}:{\mathrm{Va}}}^{Ionic\;} = \; + {\mathrm{GLIQTI}} + 200000$ [31]
0 G O I o n i c = 2648.9 + 31.44 · T + GHSEROO $\;{}^0G_{\mathrm{O}}^{Ionic\;} = \; - 2648.9 + 31.44\cdot T + {\mathrm{GHSEROO}}$ [33]
0 G Si O 2 I o n i c = + GSIO 2 LIQ $\;{}^0G_{{\mathrm{Si}}{{\mathrm{O}}_{2{\mathrm{\;}}}}}^{Ionic\;} = \; + {\mathrm{GSIO}}2{\mathrm{LIQ}}$ [10]
0 G Ti O 2 I o n i c = + GTIO 2 + 62222.4 28.2 · T $\;{}^0G_{{\mathrm{Ti}}{{\mathrm{O}}_2}}^{Ionic\;} = \; + {\mathrm{GTIO}}2 + 62222.4 - 28.2\cdot T$ [31]
0 G M g + 2 : SiO 4 4 I o n i c = + GMGSIO 4 + 4 · GMGOLIQ + 2 · GSIO 2 LIQ $\;{}^0G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{SiO}}_4^{ - 4}}^{Ionic\;} = \; + {\mathrm{GMGSIO}}4 + 4\cdot{\mathrm{GMGOLIQ}} + 2\cdot{\mathrm{GSIO}}2{\mathrm{LIQ}}$ [10]
0 L M g + 2 : O 2 , Va I o n i c = + 182000 + 26.8 · T $\;{}^0L_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:\;{{\mathrm{O}}^{ - 2}},\;{\mathrm{Va}}}^{Ionic\;} = \; + 182000 + 26.8\cdot T$ [5]
0 L M g + 2 : O 2 , Si O 2 I o n i c = + LMGSIO 0 $\;{}^0L_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:\;{{\mathrm{O}}^{ - 2}},{\mathrm{\;Si}}{{\mathrm{O}}_2}}^{Ionic\;} = \; + {\mathrm{LMGSIO}}0$ [10]
1 L M g + 2 : O 2 , Si O 2 I o n i c = + LMGSIO 1 $\;{}^1L_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:\;{{\mathrm{O}}^{ - 2}},{\mathrm{\;Si}}{{\mathrm{O}}_2}}^{Ionic\;} = \; + {\mathrm{LMGSIO}}1$ [10]
2 L M g + 2 : O 2 , Si O 2 I o n i c = + LMGSIO 2 $\;{}^2L_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:\;{{\mathrm{O}}^{ - 2}},{\mathrm{\;Si}}{{\mathrm{O}}_2}}^{Ionic\;} = \; + {\mathrm{LMGSIO}}2$ [10]
3 L M g + 2 : O 2 , Si O 2 I o n i c = + LMGSIO 3 $\;{}^3L_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:\;{{\mathrm{O}}^{ - 2}},{\mathrm{\;Si}}{{\mathrm{O}}_2}}^{Ionic\;} = \; + {\mathrm{LMGSIO}}3$ [10]
0 L M g + 2 : SiO 4 4 , Si O 2 I o n i c = + 2 · LMGSIO 0 $\;{}^0L_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{SiO}}_4^{ - 4},\;{\mathrm{Si}}{{\mathrm{O}}_{2{\mathrm{\;}}}}}^{Ionic\;} = \; + 2\cdot{\mathrm{LMGSIO}}0$ [10]
1 L M g + 2 : SiO 4 4 , Si O 2 I o n i c = + 2 · LMGSIO 1 $\;{}^1L_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{SiO}}_4^{ - 4},\;{\mathrm{Si}}{{\mathrm{O}}_{2{\mathrm{\;}}}}}^{Ionic\;} = \; + 2\cdot{\mathrm{LMGSIO}}1$ [10]
2 L M g + 2 : SiO 4 4 , Si O 2 I o n i c = + 2 · LMGSIO 2 $\;{}^2L_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{SiO}}_4^{ - 4},\;{\mathrm{Si}}{{\mathrm{O}}_{2{\mathrm{\;}}}}}^{Ionic\;} = \; + 2\cdot{\mathrm{LMGSIO}}2$ [10]
3 L M g + 2 : SiO 4 4 , Si O 2 I o n i c = + 2 · LMGSIO 3 $\;{}^3L_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{SiO}}_4^{ - 4},\;{\mathrm{Si}}{{\mathrm{O}}_{2{\mathrm{\;}}}}}^{Ionic\;} = \; + 2\cdot{\mathrm{LMGSIO}}3$ [10]
0 L M g + 2 : O 2 , Ti O 2 I o n i c = 181798.44 + 28.102167 · T $\;{}^0L_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:\;{{\mathrm{O}}^{ - 2}},{\mathrm{Ti}}{{\mathrm{O}}_2}}^{Ionic\;} = \; - 181798.44\; + 28.102167\cdot T$ [5]
1 L M g + 2 : O 2 , Ti O 2 I o n i c = 29264.098 $\;{}^1L_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}},{\mathrm{Ti}}{{\mathrm{O}}_2}}^{Ionic\;} = \; - 29264.098$ [5]
0 L T i + 2 , T i + 3 : O 2 I o n i c = 19431.27 $\;{}^0L_{{\mathrm{T}}{{\mathrm{i}}^{ + 2}},\;{\mathrm{T}}{{\mathrm{i}}^{ + 3}}:{{\mathrm{O}}^{ - 2}}}^{Ionic\;} = \;19431.27$ [31]
0 L T i + 2 : O 2 , Va I o n i c = + 168314.44 100 · T $\;{}^0L_{{\mathrm{T}}{{\mathrm{i}}^{ + 2}}:{{\mathrm{O}}^{ - 2}},\;{\mathrm{Va}}}^{Ionic\;} = \; + 168314.44 - 100\cdot T$ [31]
1 L T i + 2 : O 2 , Va I o n i c = + 190655.57 95 · T $\;{}^1L_{{\mathrm{T}}{{\mathrm{i}}^{ + 2}}:{{\mathrm{O}}^{ - 2}},\;{\mathrm{Va}}}^{Ionic\;} = \; + 190655.57 - 95\cdot T$ [31]
0 L T i + 3 : O 2 , Ti O 2 I o n i c = 36682.45 $\;{}^0L_{{\mathrm{T}}{{\mathrm{i}}^{ + 3}}:{{\mathrm{O}}^{ - 2}},{\mathrm{Ti}}{{\mathrm{O}}_2}\;}^{Ionic\;} = \; - 36682.45$ [31]
1 L T i + 3 : O 2 , Ti O 2 I o n i c = 2382.5262 $\;{}^1L_{{\mathrm{T}}{{\mathrm{i}}^{ + 3}}:{{\mathrm{O}}^{ - 2}},{\mathrm{Ti}}{{\mathrm{O}}_2}}^{Ionic\;} = \;2382.5262$ [31]
0 L Si O 2 , Ti O 2 I o n i c = + 54860.4 4.927958 · T $\;{}^0L_{{\mathrm{Si}}{{\mathrm{O}}_2},\;{\mathrm{Ti}}{{\mathrm{O}}_{2{\mathrm{\;}}}}}^{Ionic\;} = \; + 54860.4 - 4.927958\cdot T$ [8]
1 L Si O 2 , Ti O 2 I o n i c = 26345.39 + 12.837078 · T $\;{}^1L_{{\mathrm{Si}}{{\mathrm{O}}_2},\;{\mathrm{Ti}}{{\mathrm{O}}_{2{\mathrm{\;}}}}}^{Ionic\;} = \; - 26345.39 + 12.837078\cdot T$ [8]
0 L O , Si O 2 I o n i c = + 50000 $\;{}^0L_{{\mathrm{O}},{\mathrm{\;Si}}{{\mathrm{O}}_{2{\mathrm{\;}}}}}^{Ionic\;} = \; + 50000$ [8]
0 L M g + 2 : O 2 , Si O 2 , Ti O 2 I o n i c = 275835.96 $\;{}^0L_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:\;{{\mathrm{O}}^{ - 2}},{\mathrm{Si}}{{\mathrm{O}}_2},{\mathrm{Ti}}{{\mathrm{O}}_2}}^{Ionic\;} = \; - 275835.96$ This work
1 L Mg + 2 : SiO 4 4 , Ti O 2 Ionic = 6508.87 $\hspace*{0.28em}{}^{1}{L}_{{\mathrm{Mg}}^{+2}:\hspace*{0.28em}{\mathrm{SiO}}_{4}^{-4},\mathrm{Ti}{\mathrm{O}}_{2}}^{\textit{Ionic}\hspace*{0.28em}}=\hspace*{0.28em}-6508.87$ This work
2 L Mg + 2 , Ti + 3 : O 2 , Si O 2 Ionic = 104973.92 $\hspace*{0.28em}{}^{2}{L}_{{\mathrm{Mg}}^{+2},{\mathrm{Ti}}^{+3}:\hspace*{0.28em}{\mathrm{O}}^{-2},\mathrm{Si}{\mathrm{O}}_{2}}^{\textit{Ionic}\hspace*{0.28em}}=\hspace*{0.28em}-104973.92$ This work
Mg2TiO4 (sp) ( M g + 2 , T i + 4 ) 1 ( M g + 2 , T i + 4 ) 2 ( O 2 ) 4 ${( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}},{\mathrm{T}}{{\mathrm{i}}^{ + 4}}} )_1}{( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}},{\mathrm{T}}{{\mathrm{i}}^{ + 4}}} )_2}{( {{{\mathrm{O}}^{ - 2}}} )_4}$ 0 G T i + 4 : M g + 2 : O 2 s p = + SPINNORM $\;{}^0G_{{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{sp} = \; + {\mathrm{SPINNORM}}$ [5]
0 G M g + 2 : T i + 4 : O 2 s p = + 2 · SPININV GMGMG + 23.05244 · T $\;{}^0G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{sp} = \; + 2\cdot{\mathrm{SPININV}} - {\mathrm{GMGMG}} + 23.05244\cdot T$ [5]
0 G M g + 2 : M g + 2 : O 2 s p = + GMGMG $\;{}^0G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{sp} = \; + {\mathrm{GMGMG}}$ [5]
0 G T i + 4 : T i + 4 : O 2 s p = + 3 · SPINNORM + 2 · INVSP 2 · GMGMG + 23.05244 · T $\;{}^0G_{{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{sp} = \; + 3\cdot{\mathrm{SPINNORM}} + 2\cdot{\mathrm{INVSP}} - 2\cdot{\mathrm{GMGMG}} + 23.05244\cdot T$ [5]
MgTiO3 (ilm) ( M g + 2 ) 1 ( T i + 4 , S i + 4 ) 1 ( O 2 ) 3 ${( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1}{( {{\mathrm{T}}{{\mathrm{i}}^{ + 4}},{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_1}{( {{{\mathrm{O}}^{ - 2}}} )_3}$ 0 G M g + 2 : T i + 4 : O 2 i l m = 1617319.3142 + 744.617699 · T 119.39 · T · ln T 0.0045852 · T 2 + 1499000 · T 1 $\;{}^0G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{ilm} = \; - 1617319.3142 + 744.617699\cdot T - 119.39\cdot T \cdot \ln T - 0.0045852\cdot{T^2} + 1499000\cdot{T^{ - 1}}$ [5]
0 G M g + 2 : S i + 4 : O 2 i l m = + GMGOSOL + GSIO 2 S + 30000 $\;{}^0G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{ilm} = \; + {\mathrm{GMGOSOL}} + {\mathrm{GSIO}}2{\mathrm{S}} + 30000$ This work
MgTi2O5 (psbk) ( M g + 2 , T i + 4 , S i + 4 ) 1 ( M g + 2 , T i + 4 , S i + 4 ) 2 ( O 2 ) 5 ${( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}},{\mathrm{T}}{{\mathrm{i}}^{ + 4}},{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_1} {( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}},{\mathrm{T}}{{\mathrm{i}}^{ + 4}},{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_2}{( {{{\mathrm{O}}^{ - 2}}} )_5}$ 0 G M g + 2 : T i + 4 : O 2 p s b k = + PSBNORM $\;{}^0G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk} = \; + {\mathrm{PSBNORM}}$ [5]
0 G T i + 4 : M g + 2 : O 2 p s b k = + 2 · PSBINV GTI 3 O 5 + 23.0524439 · T $\;{}^0G_{{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{psbk} = \; + 2\cdot{\mathrm{PSBINV}} - {\mathrm{GTI}}3{\mathrm{O}}5{\mathrm{\;}} + 23.0524439\cdot T$ [5]
0 G M g + 2 : M g + 2 : O 2 p s b k = + 3 · PSBNORM + 2 · INVPSB 2 · GTI 3 O 5 + 23.05244 · T $\;{}^0G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{psbk} = \; + 3\cdot{\mathrm{PSBNORM}} + 2\cdot{\mathrm{INVPSB}} - 2\cdot{\mathrm{GTI}}3{\mathrm{O}}5 + 23.05244\cdot T$ [5]
0 G T i + 4 : T i + 4 : O 2 p s b k = + GTI 3 O 5 $\;{}^0G_{{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk} = \; + {\mathrm{GTI}}3{\mathrm{O}}5$ [5]
0 G T i + 4 : S i + 4 : O 2 p s b k = + 0.333 · GTI 3 O 5 + 0.667 · GSI 3 O 5 + 10195 $\;{}^0G_{{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk} = \; + 0.333 \cdot {\mathrm{GTI}}3{\mathrm{O}}5 + 0.667 \cdot {\mathrm{GSI}}3{\mathrm{O}}5 + 10195$ [8]
0 G S i + 4 : T i + 4 : O 2 p s b k = + 0.667 · GTI 3 O 5 + 0.333 · GSI 3 O 5 + 45975 $\;{}^0G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk} = \; + 0.667 \cdot {\mathrm{GTI}}3{\mathrm{O}}5 + 0.333 \cdot {\mathrm{GSI}}3{\mathrm{O}}5 + 45975$ [8]
0 G S i + 4 : S i + 4 : O 2 p s b k = + GSI 3 O 5 $\;{}^0G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk} = \; + {\mathrm{GSI}}3{\mathrm{O}}5$ [8]
0 G M g + 2 : S i + 4 : O 2 p s b k = + PSBNORMSI $\;{}^0G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk} = \; + {\mathrm{PSBNORMSI}}$ This work
0 G S i + 4 : M g + 2 : O 2 p s b k = + 2 · PSBINVSI GSI 3 O 5 + 23.0524439 · T $\;{}^0G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{psbk} = \; + 2\cdot{\mathrm{PSBINVSI}} - {\mathrm{GSI}}3{\mathrm{O}}5{\mathrm{\;}} + 23.0524439\cdot T$ This work
MgSiO3 (low-cpx) ( M g + 2 ) 1 ( M g + 2 ) 1 ( S i + 4 ) 2 ( O 2 ) 6 ${( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1}{( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1}{( {{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_2}{( {{{\mathrm{O}}^{ - 2}}} )_6}$ 0 G M g + 2 : M g + 2 : S i + 4 : O 2 l o w c p x = + 2 · GCLMGSIO $\;{}^0G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{low - cpx} = \; + 2\cdot{\mathrm{GCLMGSIO}}$ [10]
MgSiO3 (opx) ( M g + 2 ) 1 ( S i + 4 ) 1 ( O 2 ) 3 ${( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1}{( {{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_1}{( {{{\mathrm{O}}^{ - 2}}} )_3}$ 0 G M g + 2 : S i + 4 : O 2 o p x = + GCLMGSIO 2051.4 + 25.093 · T 3.3414 · T · ln T $\;{}^0G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{opx} = \; + {\mathrm{GCLMGSIO}} - 2051.4 + 25.093\cdot T - 3.3414\cdot T \cdot \ln T$ [10]
MgSiO3 (ppx) ( M g + 2 ) 1 ( S i + 4 ) 1 ( O 2 ) 3 ${( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1}{( {{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_1}{( {{{\mathrm{O}}^{ - 2}}} )_3}$ 0 G M g + 2 : S i + 4 : O 2 p p x = + GCLMGSIO + 11831 34.218 · T + 3.1114 · T · ln T + 0.0017629728 · T 2 $\;{}^0G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{ppx} = \; + {\mathrm{GCLMGSIO}} + 11831 - 34.218\cdot T + 3.1114\cdot T \cdot \ln T + 0.0017629728\cdot{T^2}$ [10]
Mg2SiO4 (ol) ( M g + 2 ) 1 ( M g + 2 ) 1 ( S i + 4 , T i + 4 ) 1 ( O 2 ) 4 ${( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1} {( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1}{( {{\mathrm{S}}{{\mathrm{i}}^{ + 4}},{\mathrm{T}}{{\mathrm{i}}^{ + 4}}} )_1}{( {{{\mathrm{O}}^{ - 2}}} )_4}$ 0 G M g + 2 : M g + 2 : S i + 4 : O 2 o l = + GOLIVMG $\;{}^0G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{ol} = \; + {\mathrm{GOLIVMG}}$ [10]
0 G M g + 2 : M g + 2 : T i + 4 : O 2 o l = + 2 · GMGOSOL + GTIO 2 + 60000 $\;{}^0G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{ol} = \; + 2 \cdot {\mathrm{GMGOSOL}} + {\mathrm{GTIO}}2 + 60000$ This work
MgO (hal) (MgO)1 0 G MgO h a l = + GMGOSOL $\;{}^0G_{{\mathrm{MgO}}}^{hal} = \; + {\mathrm{GMGOSOL}}$ [30]
TiO2 (rut) ( T i + 4 ) 1 ( O 2 , V a 2 ) 2 ${( {{\mathrm{T}}{{\mathrm{i}}^{ + 4}}} )_1}{( {{{\mathrm{O}}^{ - 2}},{\mathrm{V}}{{\mathrm{a}}^{ - 2}}} )_2}$ 0 G T i + 4 : O 2 r u t = + GTIO 2 $\;{}^0G_{{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:\;{{\mathrm{O}}^{ - 2}}}^{rut} = \; + {\mathrm{GTIO}}2$ [33]
0 G T i + 4 : V a 2 r u t = + GHSERTI + 40000 $\;{}^0G_{{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:\;{\mathrm{V}}{{\mathrm{a}}^{ - 2}}}^{rut} = \; + {\mathrm{GHSERTI}} + 40000$ [33]
0 L T i + 4 : O 2 , V a 2 r u t = 90233.9 22.7954 · T $\;{}^0L_{{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:\;{{\mathrm{O}}^{ - 2}},{\mathrm{V}}{{\mathrm{a}}^{ - 2}}}^{rut} = \; - 90233.9 - 22.7954\cdot T$ [33]
1 L T i + 4 : O 2 , V a 2 r u t = 89395.3 15.9034 · T $\;{}^1L_{{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:\;{{\mathrm{O}}^{ - 2}},{\mathrm{V}}{{\mathrm{a}}^{ - 2}}}^{rut} = \; - 89395.3 - 15.9034\cdot T$ [33]
C-SiO2 (cr) (SiO2)1 0 G Si O 2 c r = + GCRISTOB $\;{}^0G_{{\mathrm{Si}}{{\mathrm{O}}_{2{\mathrm{\;}}}}}^{cr} = \; + {\mathrm{GCRISTOB}}$ [10]
Q-SiO2 (qua) (SiO2)1 0 G Si O 2 q u a = + GSIO 2 S $\;{}^0G_{{\mathrm{Si}}{{\mathrm{O}}_{2{\mathrm{\;}}}}}^{qua} = \; + {\mathrm{GSIO}}2{\mathrm{S}}$ [10]
T-SiO2 (tr) (SiO2)1 0 G Si O 2 t r = + GTRIDYM $\;{}^0G_{{\mathrm{Si}}{{\mathrm{O}}_{2{\mathrm{\;}}}}}^{tr} = \; + {\mathrm{GTRIDYM}}$ [10]
Function Temperature range, K
GMGOLIQ (298.15−1700) 549098.33 + 275.724634 · T 47.4817 · T · ln T 0.00232681 · T 2 + 4.5043 · 10 8 · T 3 + 516900 · T 1 $ - 549098.33 + 275.724634\cdot T - 47.4817\cdot T\cdot\ln T - 0.00232681\cdot{T^2} + 4.5043 \cdot {10^{ - 8}}\cdot{T^3} + 516900\cdot{T^{ - 1}}$
(1700−2450) 585159.646 + 506.06825 · T 78.3772 · T · ln T + 0.0097344 · T 2 8.60338 · 10 7 · T 3 + 8591550 · T 1 $ - 585159.646 + 506.06825\cdot T - 78.3772\cdot T\cdot\ln T + 0.0097344\cdot{T^2} - 8.60338 \cdot {10^{ - 7}}\cdot{T^3} + 8591550\cdot{T^{ - 1}}$
(2450−3100) + 9110429.75 42013.7634 · T + 5298.548 · T · ln T 1.30122485 · T 2 + 5.8262601 · 10 5 · T 3 3.24037416 · 10 9 · T 1 $ + 9110429.75 - 42013.7634\cdot T + 5298.548\cdot T\cdot\ln T - 1.30122485\cdot{T^2}\; + 5.8262601 \cdot {10^{ - 5}}\cdot{T^3} - 3.24037416 \cdot {10^9}\cdot{T^{ - 1}}$
(3100−5100) 632664.468 + 589.239555 · T 84 · T · ln T $ - 632664.468 + 589.239555\cdot T - 84\cdot T\cdot\ln T$
GTI1O1 (298.15−2500) 551056.766 + 252.169378 · T 41.994808 · T · ln T + 327015.164 · T 1 0.00889792452 · T 2 + 1.0970448 · 10 8 · T 3 $ - 551056.766 + 252.169378\cdot T - 41.994808\cdot T\cdot\ln T + 327015.164\cdot{T^{ - 1}} - 0.00889792452\cdot{T^2} + 1.0970448 \cdot {10^{ - 8}}\cdot{T^3}$
GTI2O3 (298.15−470) 1545045.78 117799.056 · T 1 + 185.96227 · T 30.3934128 · T · ln T 0.099958898 · T 2 5.93279345 · 10 6 · T 3 $ - 1545045.78 - 117799.056\cdot{T^{ - 1}} + 185.96227\cdot T - 30.3934128\cdot T\cdot\ln T - 0.099958898\cdot{T^2} - 5.93279345 \cdot {10^{ - 6}}\cdot{T^3}$
(470−2115) 1586585.8 + 2395423.68 · T 1 + 937.087 · T 147.673862 · T · ln T 0.00173711312 · T 2 1.53383348 · 10 10 · T 3 $ - 1586585.8 + 2395423.68\cdot{T^{ - 1}} + 937.087\cdot T - 147.673862\cdot T\cdot\ln T - 0.00173711312\cdot{T^2} - 1.53383348 \cdot {10^{ - 10}}\cdot{T^3}$
GHSERMG (298.15−923) 8367.34 + 143.677875 · T 26.1849782 · T · ln T + 4.858 · 10 4 · T 2 1.393669 · 10 6 · T 3 + 78950 · T 1 $ - 8367.34 + 143.677875\cdot T - 26.1849782\cdot T\cdot\ln T + 4.858 \cdot {10^{ - 4}}\cdot{T^2} - 1.393669 \cdot {10^{ - 6}}\cdot{T^3} + 78950\cdot{T^{ - 1}}$
(923−3000) 14130.185 + 204.718543 · T 34.3088 · T · ln T + 1.038192 · 10 28 · T 9 $ - 14130.185 + 204.718543\cdot T - 34.3088\cdot T\cdot\ln T + 1.038192 \cdot {10^{28}}\cdot{T^{ - 9}}$
GMGLIQ (298.15−923) + 8202.243 8.83693 · T 8.0176 · 10 20 · T 7 + GHSERMG $ + 8202.243 - 8.83693\cdot T - 8.0176 \cdot {10^{ - 20}}\cdot{T^7} + {\mathrm{GHSERMG}}$
(923−3000) + 8690.316 9.392158 · T 1.038192 · 10 28 · T 9 + GHSERMG $ + 8690.316 - 9.392158\cdot T - 1.038192 \cdot {10^{28}}\cdot{T^{ - 9}} + {\mathrm{GHSERMG}}$
GHSERTI (298.15−900) 8059.921 + 133.615208 · T 23.9933 · T · ln T 0.004777975 · T 2 + 1.06716 · 10 7 · T 3 + 72636 · T 1 $ - 8059.921 + 133.615208\cdot T - 23.9933\cdot T\cdot\ln T - 0.004777975\cdot{T^2} + 1.06716 \cdot {10^{ - 7}}\cdot{T^3} + 72636\cdot{T^{ - 1}}$
(900−1155) 7811.815 + 132.988068 · T 23.9887 · T · ln T 0.0042033 · T 2 9.0876 · 10 8 · T 3 + 42680 · T 1 $ - 7811.815 + 132.988068\cdot T - 23.9887\cdot T\cdot\ln T - 0.0042033\cdot{T^2} - 9.0876 \cdot {10^{ - 8}}\cdot{T^3} + 42680\cdot{T^{ - 1}}$
(1155−1940) + 908.837 + 66.976538 · T 14.9466 · T · ln T 0.0081465 · T 2 + 2.02715 · 10 7 · T 3 1477660 · T 1 $ + 908.837 + 66.976538\cdot T - 14.9466\cdot T\cdot\ln T - 0.0081465\cdot{T^2} + 2.02715 \cdot {10^{ - 7}}\cdot{T^3} - 1477660\cdot{T^{ - 1}}$
(1940−6000) 124526.786 + 638.806871 · T 87.2182461 · T · ln T + 0.008204849 · T 2 3.04747 · 10 7 · T 3 + 36699805 · T 1 $ - 124526.786 + 638.806871\cdot T - 87.2182461\cdot T\cdot\ln T + 0.008204849\cdot{T^2} - 3.04747 \cdot {10^{ - 7}}\cdot{T^3} + 36699805\cdot{T^{ - 1}}$
GLIQTI (298.15−1300) + 12194.415 6.980938 · T + GHSERTI $ + 12194.415 - 6.980938\cdot T + {\mathrm{GHSERTI}}$
(1300−1940) + 368610.36 2620.99904 · T + 357.005867 · T · ln T 0.15526855 · T 2 + 1.2254402 · 10 5 · T 3 65556856 · T 1 + GHSERTI $ + 368610.36 - 2620.99904\cdot T + 357.005867\cdot T\cdot\ln T - 0.15526855\cdot{T^2} + 1.2254402 \cdot {10^{ - 5}}\cdot{T^3} - 65556856\cdot{T^{ - 1}} + {\mathrm{GHSERTI}}$
(1940−6000) + 104639.72 340.070171 · T + 40.9282461 · T · ln T 0.008204849 · T 2 + 3.04747 · 10 7 · T 3 36699805 · T 1 + GHSERTI $ + 104639.72 - 340.070171\cdot T + 40.9282461\cdot T\cdot\ln T - 0.008204849\cdot{T^2} + 3.04747 \cdot {10^{ - 7}}\cdot{T^3} - 36699805\cdot{T^{ - 1}} + {\mathrm{GHSERTI}}$
GSIO2LIQ (298.15−2980) 923689.98 + 316.24766 · T 52.17 · T · ln T 0.012002 · T 2 + 6.78 · 10 7 · T 3 + 665550 · T 1 $ - 923689.98 + 316.24766\cdot T - 52.17\cdot T\cdot\ln T - 0.012002\cdot{T^2} + 6.78 \cdot {10^{ - 7}}\cdot{T^3} + 665550\cdot{T^{ - 1}}$
(2980−4000) 957614.21 + 580.01419 · T 87.428 · T · ln T $ - 957614.21 + 580.01419\cdot T - 87.428\cdot T\cdot\ln T$
GHSERSI (298.15−1687) 8162.609 + 137.227259 · T 22.8317533 · T · ln T 0.001912904 · T 2 3.552 · 10 9 · T 3 + 176667 · T 1 $ - 8162.609 + 137.227259\cdot T - 22.8317533\cdot T\cdot\ln T - 0.001912904\cdot{T^2} - 3.552 \cdot {10^{ - 9}}\cdot{T^3} + 176667\cdot{T^{ - 1}}$
(1687−6000) 9457.642 + 167.271767 · T 27.196 · T · ln T 4.20369 · 10 30 · T 9 $ - 9457.642 + 167.271767\cdot T - 27.196\cdot T\cdot\ln T - 4.20369 \cdot {10^{30}}\cdot{T^{ - 9}}$
GHSEROO (298.15−1000) 3480.87 25.503038 · T 11.136 · T · ln T 0.005098888 · T 2 + 6.61846 · 10 7 · T 3 38365 · T 1 $ - 3480.87 - 25.503038\cdot T - 11.136\cdot T\cdot\ln T - 0.005098888\cdot{T^2} + 6.61846 \cdot {10^{ - 7}}\cdot{T^3} - 38365\cdot{T^{ - 1}}$
(1000−3300) 6568.763 + 12.65988 · T 16.8138 · T · ln T 5.95798 · 10 4 · T 2 + 6.781 · 10 9 · T 3 + 262905 · T 1 $ - 6568.763 + 12.65988\cdot T - 16.8138\cdot T\cdot\ln T - 5.95798 \cdot {10^{ - 4}}\cdot{T^2} + 6.781 \cdot {10^{ - 9}}\cdot{T^3} + 262905\cdot{T^{ - 1}}$
(3300−6000) 13986.728 + 31.259625 · T 18.9536 · T · ln T 4.25243 · 10 4 · T 2 + 1.0721 · 10 8 · T 3 + 4383200 · T 1 $ - 13986.728 + 31.259625\cdot T - 18.9536\cdot T\cdot\ln T - 4.25243 \cdot {10^{ - 4}}\cdot{T^2} + 1.0721 \cdot {10^{ - 8}}\cdot{T^3} + 4383200\cdot{T^{ - 1}}$
GTIO2 (298.15−4000) 976986.6 + 484.74037 · T 77.76175 · T · ln T 67156800 · T 2 + 1683920 · T 1 $ - 976986.6 + 484.74037\cdot T - 77.76175\cdot T\cdot\ln T - 67156800\cdot{T^{ - 2}} + 1683920\cdot{T^{ - 1}}$
GMGOSOL (298.15−1700) 619428.502 + 298.253571 · T 47.4817 · T · ln T 0.00232681 · T 2 + 4.5043 · 10 8 · T 3 + 516900 · T 1 $ - 619428.502 + 298.253571\cdot T - 47.4817\cdot T\cdot\ln T - 0.00232681\cdot{T^2} + 4.5043 \cdot {10^{ - 8}}\cdot{T^3} + 516900\cdot{T^{ - 1}}$
(1700−3100) 655489.818 + 528.597187 · T 78.3772 · T · ln T + 0.0097344 · T 2 8.60338 · 10 7 · T 3 + 8591550 · T 1 $ - 655489.818 + 528.597187\cdot T - 78.3772\cdot T\cdot\ln T + 0.0097344\cdot{T^2} - 8.60338 \cdot {10^{ - 7}}\cdot{T^3} + 8591550\cdot{T^{ - 1}}$
(3100−5000) 171490.159 1409.43369 · T + 163.674142 · T · ln T 0.044009535 · T 2 + 1.374896 · 10 6 · T 3 172665403 · T 1 $ - 171490.159 - 1409.43369\cdot T + 163.674142\cdot T\cdot\ln T - 0.044009535\cdot{T^2} + 1.374896 \cdot {10^{ - 6}}\cdot{T^3} - 172665403 \cdot {T^{ - 1}}$
(5000−5100) 722412.718 + 617.657452 · T 84 · T · ln T $ - 722412.718 + 617.657452\cdot T - 84\cdot T\cdot\ln T$
GMGSIO4 (298.15−6000) 288116 + 43.557 · T $ - 288116 + 43.557\cdot T$
LMGSIO0 (298.15−6000) + 38548 15.732 · T $ + 38548 - 15.732\cdot T$
LMGSIO1 (298.15−6000) −8402
LMGSIO2 (298.15−6000) + 158005 60.977 · T $ + 158005 - 60.977\cdot T$
LMGSIO3 (298.15−6000) −9405
SPINNORM (298.15−6000) 2211877.7407 + 1146.906951 · T 160.88 · T · ln T 0.009985 · T 2 + 1735500 · T 1 $ - 2211877.7407 + 1146.906951\cdot T - 160.88\cdot T\cdot\ln T - 0.009985\cdot{T^2} + 1735500\cdot{T^{ - 1}}$
SPININV (298.15−6000) + SPINNORM + INVSP $ + {\mathrm{SPINNORM}} + {\mathrm{INVSP}}$
INVSP (298.15−6000) 2824.083 28.335016 · T $ - 2824.083 - 28.335016\cdot T$
GCORUND (298.15−600) 1707351.3 + 448.021092 · T 67.4804 · T · ln T 0.06747 · T 2 + 1.4205433 · 10 5 · T 3 + 938780 · T 1 $ - 1707351.3 + 448.021092\cdot T - 67.4804\cdot T\cdot\ln T - 0.06747\cdot{T^2} + 1.4205433 \cdot {10^{ - 5}}\cdot{T^3} + 938780\cdot{T^{ - 1}}$
(600−1500) 1724886.06 + 754.856573 · T 116.258 · T · ln T 0.0072257 · T 2 + 2.78532 · 10 7 · T 3 + 2120700 · T 1 $ - 1724886.06 + 754.856573\cdot T - 116.258\cdot T\cdot\ln T - 0.0072257\cdot{T^2} + 2.78532 \cdot {10^{ - 7}}\cdot{T^3} + 2120700\cdot{T^{ - 1}}$
(1500−3000) 1772163.19 + 1053.4548 · T 156.058 · T · ln T + 0.00709105 · T 2 6.29402 · 10 7 · T 3 + 12366650 · T 1 $ - 1772163.19 + 1053.4548\cdot T - 156.058\cdot T\cdot\ln T + 0.00709105\cdot{T^2} - 6.29402 \cdot {10^{ - 7}}\cdot{T^3} + 12366650\cdot{T^{ - 1}}$
GALAL (298.15−6000) + 1.5 · GCORUND 0.5 · GHSEROO $ + 1.5\cdot{\mathrm{GCORUND}} - 0.5\cdot{\mathrm{GHSEROO}}$
NSPINEL (298.15−6000) + GMGOSOL + GCORUND 23204.5143 32.2303298 · T + 4.31045184 · T · ln T $ + {\mathrm{GMGOSOL}} + {\mathrm{GCORUND}} - 23204.5143 - 32.2303298\cdot T + 4.31045184\cdot T\cdot\ln T$
ISPINEL (298.15−6000) + NSPINEL + 18002.9059 0.398945666 · T $ + {\mathrm{NSPINEL}} + 18002.9059 - 0.398945666\cdot T$
GMGMG (298.15−6000) + NSPINEL + 2 · ISPINEL 2 · GALAL + 23.0525839 · T $ + {\mathrm{NSPINEL}} + 2\cdot{\mathrm{ISPINEL}} - 2\cdot{\mathrm{GALAL}} + 23.0525839\cdot T$
GSIO2S (298.15−540) 900936.64 360.892175 · T + 61.1323 · T · ln T 0.189203605 · T 2 + 4.9509742 · 10 5 · T 3 854401 · T 1 $ - 900936.64 - 360.892175\cdot T + 61.1323\cdot T\cdot\ln T - 0.189203605\cdot{T^2} + 4.9509742 \cdot {10^{ - 5}}\cdot{T^3} - 854401\cdot{T^{ - 1}}$
(540−770) 1091466.54 + 2882.67275 · T 452.1367 · T · ln T + 0.428883845 · T 2 9.0917706 · 10 5 · T 3 + 12476689 · T 1 $ - 1091466.54 + 2882.67275\cdot T - 452.1367\cdot T\cdot\ln T + 0.428883845\cdot{T^2} - 9.0917706 \cdot {10^{ - 5}}\cdot{T^3} + 12476689\cdot{T^{ - 1}}$
(770−848) 1563481.44 + 9178.58655 · T 1404.5352 · T · ln T + 1.28404426 · T 2 2.35047657 · 10 4 · T 3 + 56402304 · T 1 $ - 1563481.44 + 9178.58655\cdot T - 1404.5352\cdot T\cdot\ln T + 1.28404426\cdot{T^2} - 2.35047657 \cdot {10^{ - 4}}\cdot{T^3} + 56402304\cdot{T^{ - 1}}$
(848−1800) 928732.923 + 356.218325 · T 58.4292 · T · ln T 0.00515995 · T 2 2.47 · 10 10 · T 3 95113 · T 1 $ - 928732.923 + 356.218325\cdot T - 58.4292\cdot T\cdot\ln T - 0.00515995\cdot{T^2} - 2.47 \cdot {10^{ - 10}}\cdot{T^3} - 95113\cdot{T^{ - 1}}$
(1800−2960) 924076.574 + 281.229013 · T 47.451 · T · ln T 0.01200315 · T 2 + 6.78127 · 10 7 · T 3 + 665385 · T 1 $ - 924076.574 + 281.229013\cdot T - 47.451\cdot T\cdot\ln T - 0.01200315\cdot{T^2} + 6.78127 \cdot {10^{ - 7}}\cdot{T^3} + 665385\cdot{T^{ - 1}}$
(2960−4000) 957997.4 + 544.992084 · T 82.709 · T · ln T $ - 957997.4 + 544.992084\cdot T - 82.709\cdot T\cdot\ln T$
PSBNORM (298.15−6000) 2580078.3886 + 1179.4171263 · T 190.76 · T · ln T 0.00608 · T 2 + 2094000 · T 1 $ - 2580078.3886 + 1179.4171263\cdot T - 190.76\cdot T\cdot\ln T - 0.00608\cdot{T^2} + 2094000\cdot{T^{ - 1}}$
PSBINV (298.15−6000) + PSBNORM + INVPSB $ + {\mathrm{PSBNORM}} + {\mathrm{INVPSB}}$
INVPSB (298.15−6000) + 13497.083 4.1090551 · T $ + 13497.083 - 4.1090551\cdot T$
GTI3O5 (298.15−6000) + 3 · GTIO 2 GHSEROO $ + 3\cdot{\mathrm{GTIO}}2 - {\mathrm{GHSEROO}}$
GSI3O5 (298.15−6000) + 3 · GANDAL + 5475.42 + 29.104 · T 2 · GAL 3 O 5 + 23.05244 · T $ + 3 \cdot {\mathrm{GANDAL}} + 5475.42 + 29.104 \cdot T - 2 \cdot {\mathrm{GAL}}3{\mathrm{O}}5 + 23.05244 \cdot T$
GAL3O5 (298.15−6000) + 1.5 · GTIAL _ NO 0.5 · GTI 3 O 5 + 4.44159874 0.0050421 · T $ + 1.5 \cdot {\mathrm{GTIAL}}\_{\mathrm{NO}} - 0.5 \cdot {\mathrm{GTI}}3{\mathrm{O}}5 + 4.44159874 - 0.0050421 \cdot T$
GTIAL_NO (298.15−6000) 2676129.071049 + 1162.4925771 · T 182.6374 · T · ln T 0.011098 · T 2 + 2346281 · T 1 $ - 2676129.071049 + 1162.4925771 \cdot T - 182.6374 \cdot T \cdot \ln T - 0.011098 \cdot {T^2} + 2346281 \cdot {T^{ - 1}}$
GANDAL (298.15−6000) 2656008.38 + 1091.19544 · T 171.253095 · T · ln T 0.0136670814 · T 2 + 2523931.61 · T 1 $ - 2656008.38 + 1091.19544 \cdot T - 171.253095 \cdot T \cdot \ln T - 0.0136670814 \cdot {T^2} + 2523931.61 \cdot {T^{ - 1}}$
PSBNORMSI (298.15−6000) + GMGOSOL + 2 · GSIO 2 S + 34328 $ + {\mathrm{GMGOSOL}} + 2 \cdot {\mathrm{GSIO}}2{\mathrm{S}} + 34328$
PSBINVSI (298.15−6000) + PSBNORMSI + 10028 + 1.7 · T $ + {\mathrm{PSBNORMSI}} + 10028 + 1.7 \cdot T$
GCLMGSIO (298.15−6000) 1584678 + 1287726 · T 1 + 649.221 · T 104.3011 · T · ln T 0.009291338 · T 2 $ - 1584678 + 1287726\cdot{T^{ - 1}} + 649.221\cdot T - 104.3011\cdot T\cdot\ln T - 0.009291338\cdot{T^2}$
GOLIVMG (298.15−6000) 2232236 + 1904117 · T 1 + 959.629 · T 153.2804 · T · ln T 0.01257532 · T 2 $ - 2232236 + 1904117\cdot{T^{ - 1}} + 959.629\cdot T - 153.2804\cdot T\cdot\ln T - 0.01257532\cdot{T^2}$
GCRISTOB (298.15−373) 601467.73 8140.2255 · T + 1399.8908 · T · ln T 2.8579085 · T 2 + . 0010408145 · T 3 13144016 · T 1 $ - 601467.73 - 8140.2255\cdot T + 1399.8908\cdot T\cdot\ln T - 2.8579085\cdot{T^2} + .0010408145\cdot{T^3} - 13144016\cdot{T^{ - 1}}$
(373−453) 1498711.3 + 13075.913 · T 2178.3561 · T · ln T + 3.493609 · T 2 0.0010762132 · T 3 + 29100273 · T 1 $ - 1498711.3 + 13075.913\cdot T - 2178.3561\cdot T\cdot\ln T + 3.493609\cdot{T^2} - 0.0010762132\cdot{T^3} + 29100273\cdot{T^{ - 1}}$
(453−543) 3224538.7 + 47854.938 · T 7860.2125 · T · ln T + 11.817149 · T 2 0.0033651832 · T 3 + 1.2750272 · 10 8 · T 1 $ - 3224538.7 + 47854.938\cdot T - 7860.2125\cdot T\cdot\ln T + 11.817149\cdot{T^2} - 0.0033651832\cdot{T^3} + 1.2750272 \cdot {10^8}\cdot{T^{ - 1}}$
(543−3300) 943127.51 + 493.26056 · T 77.5875 · T · ln T + 0.003040245 · T 2 4.63118 · 10 7 · T 3 + 2227125 · T 1 $ - 943127.51 + 493.26056\cdot T - 77.5875\cdot T\cdot\ln T + 0.003040245\cdot{T^2} - 4.63118 \cdot {10^{ - 7}}\cdot{T^3} + 2227125\cdot{T^{ - 1}}$
(3300−4000) 973891.99 + 587.05606 · T 87.373 · T · ln T $ - 973891.99 + 587.05606\cdot T - 87.373\cdot T\cdot\ln T$
GTRIDYM (298.15−388) 918008.73 + 140.55925 · T 25.1574 · T · ln T 0.0148714 · T 2 2.2791833 · 10 5 · T 3 + 66331 · T 1 $ - 918008.73 + 140.55925\cdot T - 25.1574\cdot T\cdot\ln T - 0.0148714\cdot{T^2} - 2.2791833 \cdot {10^{ - 5}}\cdot{T^3} + 66331\cdot{T^{ - 1}}$
(388−433) 921013.31 + 224.53808 · T 37.8701 · T · ln T 0.02368535 · T 2 1.6835 · 10 7 · T 3 $ - 921013.31 + 224.53808\cdot T - 37.8701\cdot T\cdot\ln T - 0.02368535\cdot{T^2} - 1.6835 \cdot {10^{ - 7}}\cdot{T^3}$
(433−900) 919633.42 + 210.51651 · T 35.605 · T · ln T 0.03049985 · T 2 + 4.6255 · 10 6 · T 3 162026 · T 1 $ - 919633.42 + 210.51651\cdot T - 35.605\cdot T\cdot\ln T - 0.03049985\cdot{T^2} + 4.6255 \cdot {10^{ - 6}}\cdot{T^3} - 162026\cdot{T^{ - 1}}$
(900−1668) 979377.7 + 848.3098 · T 128.434 · T · ln T + 0.03387055 · T 2 3.786883 · 10 6 · T 3 + 7070800 · T 1 $ - 979377.7 + 848.3098\cdot T - 128.434\cdot T\cdot\ln T + 0.03387055\cdot{T^2} - 3.786883 \cdot {10^{ - 6}}\cdot{T^3} + 7070800\cdot{T^{ - 1}}$
(1668−3300) 943685.26 + 493.58035 · T 77.5875 · T · ln T + 0.003040245 · T 2 4.63118 · 10 7 · T 3 + 2227125 · T 1 $ - 943685.26 + 493.58035\cdot T - 77.5875\cdot T\cdot\ln T + 0.003040245\cdot{T^2} - 4.63118 \cdot {10^{ - 7}}\cdot{T^3} + 2227125\cdot{T^{ - 1}}$
(3300−4000) 974449.74 + 587.37585 · T 87.373 · T · ln T $ - 974449.74 + 587.37585\cdot T - 87.373\cdot T\cdot\ln T$
  • All values are given in SI units per mole of formula unit.
In general, solid solution phases are described within the compound energy formalism (CEF),34 in which various constituents of a phase are distributed between different crystallographic sites (sublattices). The molar Gibbs energy G m ${G_m}$ per formula unit for the three-sublattice model ( i 1 , i 2 , i 3 i n ) n 1 ( j 1 , j 2 , j 3 j n ) n 2 ( k 1 , k 2 , k 3 k n ) n 3 ${( {{i_1},{i_2},{i_3} \ldots {i_n}} )_{{n_1}}}{( {{j_1},{j_2},{j_3} \ldots {j_n}} )_{{n_2}}}{( {{k_1},{k_2},{k_3} \ldots {k_n}} )_{{n_3}}}$ is given as
G m = i n j n k n y i n 1 y j n 2 y k n 3 G i n j n k n T S C + G E , $$\begin{equation}{\mathrm{\;}}{G_m} = \mathop \sum \limits_{{i_n}} \mathop \sum \limits_{{j_n}} \mathop \sum \limits_{{k_n}} y_{{i_n}}^{\left( 1 \right)}y_{{j_n}}^{\left( 2 \right)}y_{{k_n}}^{\left( 3 \right)}{G_{{i_n}{j_n}{k_n}}}\; - T{S^C} + {G^E},\end{equation}$$ (1)
where y i n ( 1 ) $y_{{i_n}}^{( 1 )}$ , y j n ( 2 ) $y_{{j_n}}^{( 2 )}$ , and y k n ( 3 ) $y_{{k_n}}^{( 3 )}$ are the mole fractions of constituents in, jn, and kn on the first, second, and third sublattices, respectively; G i n j n k n ${G_{{i_n}{j_n}{k_n}}}$ is the Gibbs energy of an end member of the solution, in which the first sublattice is only occupied by constituent in, the second only by jn, and the third only by kn; T is the temperature in K; G E ${G^E}$ is the excess Gibbs energy; and S C ${S^C}$ is the configurational entropy which is defined as follows:
S C = R n 1 i n y i n 1 ln y i n 1 + n 2 j n y j n 2 ln y j n 2 + n 3 k n y k n 3 ln y k n 3 , $$\begin{eqnarray} &&\hspace*{-10pt}{\mathrm{\;}}{S^C} = \; - {\mathrm{R}}\nonumber\\ &&\hspace*{-10pt}\quad\left[ {{n_1}\mathop \sum \limits_{{i_n}} y_{{i_n}}^{\left( 1 \right)}{\mathrm{ln}}y_{{i_n}}^{\left( 1 \right)} + \left. {{n_2}\mathop \sum \limits_{{j_n}} y_{{j_n}}^{\left( 2 \right)}{\mathrm{ln}}y_{{j_n}}^{\left( 2 \right)} + {n_3}\mathop \sum \limits_{{k_n}} y_{{k_n}}^{\left( 3 \right)}{\mathrm{ln}}y_{{k_n}}^{\left( 3 \right)}} \right]} \right.,\nonumber\\ \end{eqnarray}$$ (2)
where n1, n2, and n3 are the stoichiometric numbers of each sublattice and R is the gas constant.
The excess Gibbs energy is expressed using the Redlich–Kister equation as
G E = 1 y i n 1 3 y k n 3 2 y j n 2 y j m 2 L i n : j n , j m : k n 2 + 2 y j n 2 3 y k n 3 1 y i n 1 y i m 1 L i n , i m : j n : k n 1 + 1 y i n 1 2 y j n 2 3 y k n 3 y k m 3 L i n : j n : k n , k m 3 , $$\begin{eqnarray} {G}^E &=& \mathop \sum \limits_{\left( 1 \right)} y_{{i}_n}^{\left( 1 \right)}\ \mathop \sum \limits_{\left( 3 \right)} y_{{k}_n}^{\left( 3 \right)}\mathop \sum \limits_{\left( 2 \right)} y_{{j}_n}^{\left( 2 \right)}y_{{j}_m}^{\left( 2 \right)}L_{{i}_n:{j}_n,{j}_m:{k}_n}^{\left( 2 \right)}\nonumber\\ &&+\, \mathop \sum \limits_{\left( 2 \right)} y_{{j}_n}^{\left( 2 \right)}\mathop \sum \limits_{\left( 3 \right)} y_{{k}_n}^{\left( 3 \right)}\mathop \sum \limits_{\left( 1 \right)} y_{{i}_n}^{\left( 1 \right)}y_{{i}_m}^{\left( 1 \right)}L_{{i}_n,{i}_m:{j}_n:{k}_n}^{\left( 1 \right)}\nonumber\\ &&+\, \mathop \sum \limits_{\left( 1 \right)} y_{{i}_n}^{\left( 1 \right)}\mathop \sum \limits_{\left( 2 \right)} y_{{j}_n}^{\left( 2 \right)}\mathop \sum \limits_{\left( 3 \right)} y_{{k}_n}^{\left( 3 \right)}y_{{k}_m}^{\left( 3 \right)}L_{{i}_n:{j}_n:{k}_n,{k}_m}^{\left( 3 \right)},\end{eqnarray}$$ (3)
where the parameter L i n , i m : j n : k n ( 1 ) $L_{{i_n},{i_m}:{j_n}:{k_n}}^{( 1 )}$ is related to interactions between constituents in and im on the sublattice (1), while the sublattice (2) is fully occupied by constituent jn, and the sublattice (3) by kn, and can be signified as
L i n , i m : j n : k n 1 = ν ν L i n , i m : j n : k n y i n 1 y i m 1 ν , $$\begin{equation}{\mathrm{\;}}L_{{i_n},{i_m}:{j_n}:{k_n}}^{\left( 1 \right)} = \mathop \sum \limits_\nu {}^\nu {L_{{i_n},{i_m}:{j_n}:{k_n}}}{\left( {y_{{i_n}}^{\left( 1 \right)} - y_{{i_m}}^{\left( 1 \right)}} \right)^\nu }\;,\end{equation}$$ (4)
where v is an integer number.

4.1 Liquid phase

The liquid phase is described by the two-sublattice model for ionic liquids35 with the formula (Mg+2,Ti+2,Ti+3)P(O−2,Va,SiO4−4,SiO2,O,TiO2)Q, where P and Q indicate the number of sites on each sublattice, which can vary with composition to maintain electroneutrality. The two-sublattice model for ionic liquids was also developed within the framework of CEF, Equations (1) to (4) are applied in their reduced form.

4.2 MgO–TiO2 phases

Mg2TiO4 has an inverse spinel structure (Strukturbericht H11). In Mg2TiO4, the degree of inversion is close to 1 at room temperature and decreases slightly with increasing temperature.36 Since there are no reports on any solubility of SiO2 in spinel Mg2TiO4, its description using the formula ( Mg χ + 2 , Ti 1 χ + 4 ) 1 T ( Mg 1 χ / 2 + 2 , Ti χ / 2 + 4 ) 2 O ( O 2 ) 4 $( {{\mathrm{Mg}}_{{\chi}}^{ + 2},{\mathrm{Ti}}_{1 - {{\chi}}}^{ + 4}} )_1^T( {{\mathrm{Mg}}_{1 - {{\chi}}/2}^{ + 2},{\mathrm{Ti}}_{{{\chi}}/2}^{ + 4}} )_2^O{( {{{\mathrm{O}}^{ - 2}}} )_4}$ , where χ is the inversion degree, is accepted in this work after Ilatovskaia et al.5

MgTiO3 has an ordered ilmenite structure in the space group R 3 ¯ $R\bar 3$ (Strukturbericht E22). While data on solubility of SiO2 in MgTiO3 are absent, according to Verhoogen et al.21 and Kuganathan et al.22, Si+4 ions can incorporate into the ilmenite structure at the Ti+4 positions. Therefore, the possible solubility of SiO2 can be modeled as follows: ( M g + 2 ) 1 ( T i + 4 , S i + 4 ) 1 ( O 2 ) 3 ${( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1}{( {{\mathrm{T}}{{\mathrm{i}}^{ + 4}},{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_1}{( {{{\mathrm{O}}^{ - 2}}} )_3}$ . This gives a new neutral end-member MgSiO3 with the ilmenite-type structure. The Gibbs energy of this end-member is adjusted as
G M g + 2 : S i + 4 : O 2 i l m = GMGOSOL + GSIO 2 S + a 1 , $$\begin{equation}{\mathrm{\;}}^\circ G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{ilm} = {\mathrm{\;GMGOSOL}} + {\mathrm{GSIO}}2{\mathrm{S}} + {a_1},\end{equation}$$ (5)
where GMGOSOL is the parameter related to the Gibbs energy of periclase MgO,30 GSIO2S is the parameter related to the Gibbs energy of quartz,37 and a1 is a large positive value to make ilmenite MgSiO3 unstable at this composition and atmosphere pressure. However, ilmenite MgSiO3 becomes stable at pressure above 18 GPa,38 which is not considered in this work.
MgTi2O5 has a pseudobrookite structure crystallizing in the space group C m c m $Cmcm$ (Strukturbericht E41). Assuming a partially disordered distribution of both types of cations between M1 and M2 octahedral positions (χ is about 0.4 at 1800 K), the model ( Mg 1 χ + 2 , Ti χ + 4 ) 1 M 1 ( Mg χ / 2 + 2 , Ti 1 χ / 2 + 4 ) 2 M 2 ( O 2 ) 5 $( {{\mathrm{Mg}}_{1 - {{\chi}}}^{ + 2},{\mathrm{Ti}}_{{\chi}}^{ + 4}} )_1^{M1}( {{\mathrm{Mg}}_{{{\chi}}/2}^{ + 2},{\mathrm{Ti}}_{1 - {{\chi}}/2}^{ + 4}} )_2^{M2}{( {{{\mathrm{O}}^{ - 2}}} )_5}$ was suggested.5 Although there are no reports on any solubility of SiO2 in pseudobrookite MgTi2O5, Si+4 must be incorporated into the model at the same positions as Ti+4 to make this description consistent with the description of pseudobrookite Al2TiO5(+Si) in the earlier work8 where Si+4 is present on both the M1 and M2 sublattices. The resulting formula is, thus, ( M g + 2 , T i + 4 , S i + 4 ) 1 ( M g + 2 , T i + 4 , S i + 4 ) 2 ( O 2 ) 5 ${( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}},{\mathrm{T}}{{\mathrm{i}}^{ + 4}},{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_1}{( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}},{\mathrm{T}}{{\mathrm{i}}^{ + 4}},{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_2}{( {{{\mathrm{O}}^{ - 2}}} )_5}$ . The parameters G M g + 2 : M g + 2 : O 2 p s b k $^\circ G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ , G M g + 2 : T i + 4 : O 2 p s b k $^\circ G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ , G T i + 4 : M g + 2 : O 2 p s b k $^\circ G_{{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ , G T i + 4 : T i + 4 : O 2 p s b k $^\circ G_{{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ , G T i + 4 : S i + 4 : O 2 p s b k $^\circ G_{{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ , G S i + 4 : T i + 4 : O 2 p s b k $^\circ G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ , and G S i + 4 : S i + 4 : O 2 p s b k $^\circ G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ have been defined in the previous assessments of the authors.5, 8 To define new unknown parameters, a simplified model can be considered: ( M g + 2 , S i + 4 ) 1 M 1 ( M g + 2 , S i + 4 ) 2 M 2 ( O 2 ) 5 $( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}},{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_1^{M1}( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}},{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_2^{M2}{( {{{\mathrm{O}}^{ - 2}}} )_5}$ , which could be represented by a reciprocal system giving a reciprocal reaction
G M g + 2 : S i + 4 : O 2 p s b k + G S i + 4 : M g + 2 : O 2 p s b k G M g + 2 : M g + 2 : O 2 p s b k G S i + 4 : S i + 4 : O 2 p s b k = Δ G 1 . $$\begin{eqnarray} &&^\circ G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk} + ^\circ G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{psbk}\nonumber\\ &&\quad - ^\circ G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{psbk} - \;^\circ G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk} = \Delta {G_1}\;.\end{eqnarray}$$ (6)
In this reciprocal system, the endpoints of the neutral line correspond to a normal hypothetical compound MgSi2O5 described by the Gibbs energy parameter G M g + 2 : S i + 4 : O 2 p s b k $^\circ G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ and an inverse hypothetical compound Si(Si0.5Mg0.5)O5 being the middle point of the line between end members S i + 4 Mg 2 + 2 O 5 2 ${\mathrm{S}}{{\mathrm{i}}^{ + 4}}{\mathrm{Mg}}_2^{ + 2}{\mathrm{O}}_5^{ - 2}$ and S i + 4 Si 2 + 4 O 5 2 ${\mathrm{S}}{{\mathrm{i}}^{ + 4}}{\mathrm{Si}}_2^{ + 4}{\mathrm{O}}_5^{ - 2}$ described by the Gibbs energy parameters G S i + 4 : M g + 2 : O 2 p s b k $^\circ G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ and G S i + 4 : S i + 4 : O 2 p s b k $^\circ G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ , respectively. The G M g + 2 : S i + 4 : O 2 p s b k $^\circ G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ term is expressed as
G M g + 2 : S i + 4 : O 2 p s b k = PSBNORMSI = GMGOSOL + 2 · GSIO 2 S + a 2 , $$\begin{eqnarray} &&{\mathrm{\;}}^\circ G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk}\nonumber\\ &&\quad =\, {\mathrm{\;PSBNORMSI\;}} = {\mathrm{\;GMGOSOL}} + 2 \cdot {\mathrm{GSIO}}2{\mathrm{S}} + {a_2},\nonumber\\ \end{eqnarray}$$ (7)
where a2 is the optimized parameter.
The G S i + 4 : M g + 2 : O 2 p s b k $^\circ G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ term could be determined by Equation (6). However, in this case, Δ G 1 $\Delta {G_1}$ in Equation (6) is not equal to zero since the G M g + 2 : S i + 4 : O 2 p s b k $^\circ G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ , G M g + 2 : M g + 2 : O 2 p s b k $^\circ G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ , and G S i + 4 : S i + 4 : O 2 p s b k $^\circ G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ parameters have certain values (i.e., last two parameters are accepted from Refs. [5, 8], respectively). Therefore, G S i + 4 : M g + 2 : O 2 p s b k $^\circ G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ can be determined by the electroneutrality reaction for inverse MgSi2O5 as
1 2 S i + 4 M 1 M g + 2 2 M 2 O 5 + 1 2 S i + 4 M 1 S i + 4 2 M 2 O 5 = S i + 4 M 1 Mg 0.5 + 2 Si 0.5 + 4 2 M 2 O 5 . $$\begin{eqnarray} &&\frac{1}{2}{\left( {{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} \right)^{M1}}\left( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} \right)_2^{M2}{{\mathrm{O}}_5} + \frac{1}{2}{\left( {{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} \right)^{M1}}\left( {{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} \right)_2^{M2}\;{{\mathrm{O}}_5}\nonumber\\ &&\quad =\, {\left( {{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} \right)^{M1}}\;\left( {{\mathrm{Mg}}_{0.5}^{ + 2}{\mathrm{Si}}_{0.5}^{ + 4}} \right)_2^{M2}{{\mathrm{O}}_5}.\end{eqnarray}$$ (8)
The Gibbs energy of inverse MgSi2O5 is then
PSBINVSI y M g + 2 M 2 = 0.5 , y S i + 4 M 2 = 0.5 , T = 1 2 G S i + 4 : M g + 2 : O 2 p s b k + G S i + 4 : S i + 4 : O 2 p s b k 4 R T ln 2 , $$\begin{eqnarray} &&{\mathrm{PSBINVSI\;}}\left( {\left( {\;y_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}}^{M2} = \;0.5,\;y_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}}^{M2} = \;0.5} \right),T} \right)\nonumber\\ &&\quad =\, \frac{1}{2}\;\left( {^\circ G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{psbk} + ^\circ G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk} - 4{\mathrm{R}}T{\mathrm{ln}}2} \right),\nonumber\\ \end{eqnarray}$$ (9)
with
PSBINVSI = PSBNORMSI + a 3 , $$\begin{equation}{\mathrm{PSBINVSI\;}} = {\mathrm{\;PSBNORMSI}} + {a_3},\end{equation}$$ (10)
where a3 is the optimized parameter.
As a result, the G S i + 4 : M g + 2 : O 2 p s b k $^\circ G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ term could be expressed from Equation (9) as follows:
G S i + 4 : M g + 2 : O 2 p s b k = 2 · PSBINVSI G S i + 4 : S i + 4 : O 2 p s b k + 4 R T ln 2 , $$\begin{eqnarray} &&{\mathrm{\;}}^\circ G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{{\mathrm{O}}^{ - 2}}}^{psbk}\nonumber\\ &&\quad = \;2 \cdot {\mathrm{PSBINVSI}} - ^\circ G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk} + 4{\mathrm{R}}T{\mathrm{ln}}2,\end{eqnarray}$$ (11)
G S i + 4 : S i + 4 : O 2 p s b k = GSI 3 O 5 = 3 · GANDAL 2 · GAL 3 O 5 + 5475.42 + 52.156 · T , $$\begin{eqnarray} &&{\mathrm{\;}}^\circ G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk} = \;{\mathrm{GSI}}3{\mathrm{O}}5{\mathrm{\;}} = \;3 \cdot {\mathrm{GANDAL}}\nonumber\\ &&\quad -\, 2 \cdot {\mathrm{GAL}}3{\mathrm{O}}5 + 5475.42 + 52.156 \cdot T,\end{eqnarray}$$ (12)
GAL 3 O 5 = 1.5 · GTIAL _ NO 0.5 · TI 3 O 5 + 4.4416 0.005 · T , $$\begin{eqnarray} &&{\mathrm{GAL}}3{\mathrm{O}}5{\mathrm{\;}} = {\mathrm{\;}}1.5 \cdot {\mathrm{GTIAL}}\_{\mathrm{NO}} - 0.5 \cdot {\mathrm{TI}}3{\mathrm{O}}5\nonumber\\ &&\quad +\, 4.4416 - 0.005 \cdot {\mathrm{T}},\end{eqnarray}$$ (13)
TI 3 O 5 = 3 · GTIO 2 GHSEROO , $$\begin{equation}{\mathrm{TI}}3{\mathrm{O}}5{\mathrm{\;}} = {\mathrm{\;}}3 \cdot {\mathrm{GTIO}}2 - {\mathrm{GHSEROO}},\end{equation}$$ (14)

where GSI3O5 (or G S i 3 O 5 $^\circ {G_{{\mathrm{S}}{{\mathrm{i}}_3}{{\mathrm{O}}_5}}}$ ) is the Gibbs energy of a fictive compound Si3O5 with the pseudobrookite structure accepted after Ilatovskaia et al.8 and GANDAL is the Gibbs energy of andalusite,37 GAL3O5 (or G A l 3 O 5 $^\circ {G_{{\mathrm{A}}{{\mathrm{l}}_3}{{\mathrm{O}}_5}}}$ ) is the Gibbs energy of a fictive compound Al3O5 with the pseudobrookite structure,6 and GTIO2 (or G Ti O 2 $^\circ {G_{{\mathrm{Ti}}{{\mathrm{O}}_2}}}$ ) is the Gibbs energy of rutile TiO2.33 It may seem strange that the G S i + 4 : S i + 4 : O 2 p s b k $^\circ G_{{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{\mathrm{S}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ term involves parameters evaluated in the assessments both of Al3O5 and Ti3O5 although it does not contain any Al or Ti. This arises because there could only be one G $^\circ G$ parameter acting as a reference in a phase, and the reference in the pseudobrookite phase in the complex Al2O3–MgO–TiO2–SiO2 system is agreed to be G T i + 4 : T i + 4 : O 2 p s b k $^\circ G_{{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{psbk}$ (or TI3O5). It would also be possible to adjust the values of Si3O5 and Al3O5 independently, and this would give different values for all G $^\circ G$ parameters that have a net charge.

4.3 MgO–SiO2 phases

Forsterite Mg2SiO4 of the olivine series crystallizes in the orthorhombic system (space group P b n m $Pbnm$ , Strukturbericht S12). The olivine structure consists of isolated (SiO4)−4 tetrahedrons that cross-link two symmetrically nonequivalent octahedral sites, M1 and M2, here occupied by Mg+2. Four sublattices were, thus, included in the model, ( M g + 2 ) 1 M 1 ( M g + 2 ) 1 M 2 ( S i + 4 ) 1 T ( O 2 ) 4 $( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1^{M1}( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1^{M2}( {{\mathrm{S}}{{\mathrm{i}}^{ + 4}}} )_1^T{( {{{\mathrm{O}}^{ - 2}}} )_4}$ . Considering that Ti+4 can directly substitute for Si+4 with increasing temperature under anhydrous conditions, reaching the maximum Ti solubility in Mg2SiO4 when it is buffered with the spinel Mg2TiO4,23, 24 the model for the olivine phase is therefore ( M g + 2 ) 1 M 1 ( M g + 2 ) 1 M 2 ( S i + 4 , T i + 4 ) 1 T ( O 2 ) 4 $( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1^{M1}( {{\mathrm{M}}{{\mathrm{g}}^{ + 2}}} )_1^{M2}( {{\mathrm{S}}{{\mathrm{i}}^{ + 4}},{\mathrm{T}}{{\mathrm{i}}^{ + 4}}} )_1^T{( {{{\mathrm{O}}^{ - 2}}} )_4}$ . The Gibbs energy of a new end-member is adjusted as
G M g + 2 : M g + 2 : T i + 4 : O 2 o l i v = 2 · GMGOSOL + GTIO 2 + a 4 , $$\begin{eqnarray} {\mathrm{\;}}^\circ G_{{\mathrm{M}}{{\mathrm{g}}^{ + 2}}:{\mathrm{M}}{{\mathrm{g}}^{ + 2}}: {\mathrm{T}}{{\mathrm{i}}^{ + 4}}:{{\mathrm{O}}^{ - 2}}}^{oliv} = {\mathrm{\;}}2 \cdot {\mathrm{GMGOSOL}} + {\mathrm{GTIO}}2 + {a_4},\nonumber\\ \end{eqnarray}$$ (15)
where a4 is a positive value in order to avoid making Mg2TiO4 stable on the MgO–SiO2 side.

Enstatite MgSiO3 (or orthopyroxene) is of the pyroxene series, and it is stable at low temperatures. The other forms are protoenstatite (or protopyroxene; both enstatite and protoenstatite crystallize in the orthorhombic system, P b c n $Pbcn$ ), which occurs at high temperatures, and clinoenstatite (or low clinopyroxene; in the monoclinic system, P 121 / c 1 $P121/c1$ ), which occurs in an unstable form at low temperatures. Since there are no reports on the solubility of TiO2 in MgSiO3, the descriptions of all three MgSiO3-based solid solutions are accepted after Huang et al.10

4.4 Rutile, halite, and SiO2-based phases

The rutile phase is tetragonal TiO2 crystallizing in the space group P 4 2 / m n m $P{4_2}/mnm$ (Strukturbericht C4). Periclase MgO has the NaCl-type structure in the space group F m 3 ¯ m $Fm\bar 3m$ (Strukturbericht B1) with the common name halite. Generally, the halite or rutile phase is described using a model within CEF with two sublattices, one for metal ions and one for oxygen ions. Since there is no information on any solubility in TiO2 and MgO, TiO2 rutile and MgO halite are described as stoichiometric phases.

SiO2 exhibits polymorphism; there are forms of α-quartz1 (or Q-SiO2; in trigonal P3221 space group) at low temperatures and tridymite (or T-SiO2; in orthorhombic C2221 space group) and cristobalite2 (or C-SiO2; in tetragonal P41212 space group) at high temperatures. Regardless the experimentally observed solubility in T-SiO2 or C-SiO2 which could be caused by a nonequilibrium state due to high viscosity of SiO2, all SiO2-based phases are also described as stoichiometric. The description of TiO2 rutile is taken from Hampl and Schmidt-Fetzer,33 the description of MgO halite is from Hallstedt,30 and the descriptions of Q-SiO2, C-SiO2, and T-SiO2 are all from Huang et al.10

5 RESULTS AND DISCUSSION

5.1 Experimental results

The series of sample compositions were chosen to determine all possible phase relations occurring in the ternary system within the temperature range from 1523 K to melting. The chemical compositions of the heat-treated samples were measured using EDX and were found to be consisted with the nominal ones within an admissible error of 3%. The results of XRD and EPMA/WDX investigations of the heat-treated samples are summarized in Table 3.

TABLE 3. List of the sample compositions for the MgO–TiO2–SiO2 system along with the SEM/EPMA and XRD results.
Nominal sample composition, mole fraction Volume fraction and lattice parameters, nm, by XRD Phase composition by SEM/EPMA, mole fraction
Sample MgO TiO2 SiO2 Annealing temperature, K/annealing time, hour Equilibrium phases Vol.% a b c beta MgO TiO2 SiO2
MTS-1 0.6253 0.2498 0.1250 1523/192 Mg2SiO4 37 0.4755 1.0231 0.5985
MgTiO3 33 0.5053 1.3905
Mg2TiO4 30 0.8441
1673/120 Mg2SiO4 38 0.4756 1.0228 0.5984
MgTiO3 32 0.5052 1.3900
Mg2TiO4 30 0.8438
1873/10 min Mg2SiO4 0.6617 0.0164 0.3219
MgTiO3 0.4797 0.5194 0.0009
Mg2TiO4 0.6502 0.3480 0.0018
MgO 0.9964 0.0033 0.0004
TiO2 0.0014 0.9976 0.0010
Eutectic 1 0.6453 0.1080 0.2468
Eutectic 2 0.5621 0.2918 0.1461
MTS-3a 0.5334 0.0666 0.4000 1523/192 Mg2SiO4 32 0.4760 1.0219 0.5989
P-MgSiO3 55 0.9258 0.8765 0.5321
MgTi2O5 13 0.9764 1.0033 0.3741
1673/120 Mg2SiO4 26 0.4759 1.0235 0.5984
C-MgSiO3 60 0.9622 0.8830 0.5181 108.33
MgTi2O5 14 0.9745 1.0007 0.3745
1893a Mg2SiO4 0.6545 0.0012 0.3440
C-MgSiO3 0.4875 0.0122 0.5003
MgTi2O5b
Liquid 0.3651 0.2662 0.3687
MTS-4a 0.5000 0.4000 0.1000 1523/192 Mg2SiO4 32 0.4755 1.0205 0.5985
MgTiO3 47 0.5054 1.3905
MgTi2O5 21 0.9729 1.0006 0.3742
1673/120 Mg2SiO4 32 0.4757 1.0207 0.5985
MgTiO3 45 0.5055 1.3903
MgTi2O5 23 0.9748 0.9999 0.3740
1873a Mg2SiO4 0.6546 0.0125 0.3329
MgTiO3 0.4930 0.5060 0.0011
MgTi2O5 0.3229 0.6762 0.0010
Liquid 0.4839 0.3831 0.1330
MTS-5 0.5000 0.2500 0.2500 1523/192 Mg2SiO4 59 0.4754 1.0207 0.5985
MgTi2O5 41 0.9746 1.0014 0.3731
1673/120 Mg2SiO4 62 0.4756 1.0200 0.5983
MgTi2O5 38 0.9738 1.0004 0.3741
1773/24 Mg2SiO4 0.6531 0.0045 0.3425
MgTi2O5 0.3306 0.6637 0.0057
Liquidb
1893a Mg2SiO4 0.6534 0.0054 0.3412
MgTi2O5 0.3252 0.6683 0.0055
Liquidb
MTS-7 0.4202 0.2098 0.3700 1523/192 Mg2SiO4 3 0.4759 1.0223 0.5960
P-MgSiO3 60 0.9244 0.8751 0.5315
MgTi2O5 37 0.9746 1.0013 0.3736
1673/120 Mg2SiO4 4 0.4773 1.0257 0.5964
C-MgSiO3 66 0.9613 0.8822 0.5179 108.35
MgTi2O5 30 0.9737 1.0003 0.3742
1773/120 Mg2SiO4 0.6743 0.0017 0.3239
MgTi2O5 0.3326 0.6637 0.0036
Liquid 0.4521 0.1929 0.3596
1923a Liquid 0.4480 0.2138 0.3383
MTS-8 0.3669 0.1832 0.4500 1523/192 P-MgSiO3 72 0.9243 0.8746 0.5314
C-SiO2 4 0.4997 0.6952
TiO2 25 0.4592 0.2960
1616/240 C-MgSiO3 52 0.9609 0.8818 0.5175 108.36
MgTi2O5 28 0.9728 0.9995 0.3739
C-SiO2 20 0.4985 0.6943
TiO2 <1 0.4594 0.2960
1633/120 C-MgSiO3 59 0.9611 0.8821 0.5176 108.34
MgTi2O5 25 0.9734 1.0003 0.3742
C-SiO2 16 0.4986 0.6949
1673/120 C-MgSiO3 59 0.9610 0.8820 0.5177 108.34
MgTi2O5 23 0.9734 1.0002 0.3742
C-SiO2 18 0.4986 0.6955
1773/120 MgTi2O5b
C-SiO2b
Liquid 0.3884 0.2058 0.4058
1923a C-SiO2 0.0026 0.0202 0.9772
Liquid 0.3827 0.1779 0.4395
MTS-9 0.1700 0.0900 0.7400 1598/120 P-MgSiO3 28 0.9242 0.8743 0.5315
C-SiO2 63 0.4998 0.6983
TiO2 9 0.4591 0.2958
1673/120 C-MgSiO3 24 0.9615 0.8823 0.5179 108.27
MgTi2O5 11 0.9742 1.0010 0.3743
C-SiO2 22 0.4991 0.6969
T-SiO2 43 0.5018 0.8209
1773/1 C-SiO2 0.0002 0.0175 0.9823
Liquid 0.3725 0.1746 0.4529
MTS-10 0.1200 0.2400 0.6400 1673/120 MgTi2O5 25 0.9736 1.0002 0.3742
C-SiO2 70 0.4988 0.6953
T-SiO2 5 0.5012 0.8193
1773/24 C-SiO2 0.0000 0.0256 0.9744
TiO2 0.0001 0.9968 0.0031
Liquid 0.3781 0.1833 0.4386
MTS-12 1673/120 Mg2SiO4 33 0.4759 1.0224 0.5983
Mg2TiO4 32 0.8437
MgO 35 0.4213
2003a Mg2SiO4 0.6617 0.0164 0.3219
Mg2TiO4 0.6502 0.3480 0.0018
MgO 0.9820 0.0070 0.0110
MTS-13 0.150 0.600 0.250 1523/192 P-MgSiO3 22 0.9243 0.8756 0.5318
C-SiO2 14 0.5001 0.7010
TiO2 64 0.4593 0.2960
1673/120 MgTi2O5 38 0.9735 1.0001 0.3741
TiO2 25 0.4594 0.2959
C-SiO2 37 0.4988 0.6954
1773/120 C-SiO2 0.0004 0.0251 0.9745
TiO2 0.0003 0.9983 0.0013
Liquid 0.3639 0.1792 0.4569
1923a C-SiO2 0.0066 0.0901 0.9033
TiO2 0.0002 0.9989 0.0009
Liquid 0.3823 0.1755 0.4422
MTS-14 0.220 0.690 0.090 1523/192 P-MgSiO3 14 0.9239 0.8764 0.5313
MgTi2O5 36 0.9741 1.0008 0.3739
TiO2 50 0.4593 0.2960
1673/120 MgTi2O5 62 0.9736 1.0001 0.3741
C-SiO2 16 0.4987 0.6950
TiO2 22 0.4594 0.2959
1773/24 MgTi2O5 0.3258 0.6711 0.0031
C-SiO2b
TiO2 0.0002 0.9985 0.0013
Liquid 1 0.3476 0.0697 0.5827
Liquid 2 0.1973 0.3813 0.4215
1923a Mg2SiO4 0.5869 0.0996 0.3135
MgTiO3 0.4848 0.4359 0.0793
TiO2 0.0001 0.9989 0.0010
Liquid 0.4665 0.2960 0.2375
MTS-15 0.280 0.420 0.300 1523/192 P-MgSiO3 38 0.9243 0.8761 0.5317
C-MgSiO3 17 0.9612 0.8816 0.5177 108.37
MgTi2O5 <1 0.9729 1.0006 0.3742
TiO2 44 0.4593 0.2960
1673/120 C-MgSiO3 16 0.9616 0.8823 0.5177
MgTi2O5 53 0.9735 1.0005 0.3742
C-SiO2 31 0.4988 0.6951
1773/2 MgTi2O5 0.3439 0.6468 0.0093
C-SiO2 0.0000 0.0189 0.9811
TiO2 0.0003 0.9969 0.0028
Liquid 0.4353 0.1088 0.4559
1823a TiO2 0.0004 0.9979 0.0017
Liquid 0.4331 0.1489 0.4180
  • a Heated in air without holding at the reached temperature.
  • b Areas are too small to be quantified by EPMA.
  • Notably, P-MgSiO3 and C-MgSiO3 are reported as high- and low-temperature phases, respectively; however, in the present study, P-MgSiO3 is observed at 1523 K, while C-MgSiO3 is observed above 1616 K. A similar issue was reported by Song et al.,39 who indicated P-MgSiO3 at 1653 K and C-MgSiO3 above 1673 K.

Since a fine microstructure (with a grain size of less than 2−5 μm) is specific for the ceramic samples heat treated at low temperatures, it is impossible to accurately measure the phase compositions using EDX (or WDX). Therefore, XRD was used to identify the phases present in the samples heat treated at both 1523 and 1673 K. However, the microstructures of the samples heat treated at 1673 K were also investigated by EPMA/WDX, but they were only sufficient for the semiquantitative WDX analysis, so the EPMA/WDX data are omitted in this case. Considering the lack of relevant literature data on the solubility and stoichiometry of the intermediate compounds in the binary MgO–SiO2 and MgO–TiO2 systems, the extension of the intermediate compounds into the ternary system was neglected. Additionally, no new phases within the ternary system were observed. Thus, based on the XRD results obtained (see Table 3), the isothermal sections at 1523 and 1673 K are presented in Figure 1A,B, respectively. Four three-phase equilibria (MgO + Mg2SiO4+ Mg2TiO4, Mg2SiO4+ Mg2TiO4+ MgTiO3, Mg2SiO4+ MgTiO3+ MgTi2O5, and Mg2SiO4+ P-MgSiO3+ MgTi2O4) in the MgO–MgSiO3–MgTi2O5 part of the system remain stable in the range of 1523−1673 K, while the tie lines change in the TiO2–MgTi2O5–MgSiO3–SiO2 part was observed. These results indicate the occurrence of a solid-state reaction of a transitional type, MgTi2O5 + C-SiO2⇋P-MgSiO3 + TiO2. Consequently, a stepwise heat treatment followed by XRD examination was carried out for sample MTS-8 to clarify the reaction temperature. As shown in Figure 2 (see also Table 3), the reaction is completed at 1633 K, revealing the presence of a three-phase area (P-MgSiO3 + MgTi2O5 + C-SiO2), while traces of TiO2 were still present in MTS-8 after heat treatment at 1616 K. Thus, the reaction temperature was assumed to be 1625 ± 8 K.

Details are in the caption following the image
Isothermal sections of the MgO–TiO2–SiO2 system determined experimentally (left) and calculated (right) at p(O2) = 0.21 bar and (A) 1523 K, (B) 1673 K, and (C) 1773 K. Small circles by numbers indicate the samples compositions; solid circles and open circles are for investigated and noninvestigated sample compositions at a certain temperature, respectively. Numbers in circles denote the three-phase fields. In Figure 1C (left), small open squares denote the EPMA data obtained in this work (see Table 3), while all in red—small open triangles (EDX data), big solid tie triangles (three-phase fields), and dotted tie lines (two-phase field tie lines)—are the data of Chen et al.17.
Details are in the caption following the image
XRD patterns for sample MTS-8 heat-treated at (A) 1523 K, (B) 1616 K, and (C) 1633 K.

At 1773 K, partial melting of the samples within the area of TiO2–MgTi2O5–Mg2SiO4–SiO2 was observed. It is worth noting that the experimental samples after heat treatment were not subjected to quenching, but were cooled down with the furnace. Therefore, the composition of the liquid phase could not be maintained with respect to 1773 K. Nonequilibrium solidification for the series of samples heat treated at 1773 K was subsequently observed by the microstructure investigation. The corresponding isothermal section, together with the EPMA/WDX data marked with open squares, is shown in Figure 1C. A three-phase equilibrium (Mg2SiO4 + MgTi2O5 + L) for samples MTS-5 and -7, a three-phase equilibrium (C-SiO2 + TiO2 + L) for samples MTS-10 and -13, and a two-phase equilibrium (C-SiO2 + L) for sample MTS-9 were established (Figure 3 and Table 3). Three other three-phase equilibria could thus tentatively exist: Mg2SiO4 + P-MgSiO3 + L, P-MgSiO3 + C-SiO2 + L, and MgTi2O5 + TiO2 + L. It was expected that the three-phase equilibrium (MgTi2O5 + TiO2 + L) could be observed for sample MTS-14. However, the equilibrium state at 1773 K could not be preserved due to slow cooling (furnace cooling) applied in this work. Thus, the crystallization of sample MTS-14 heat treated at 1773 K finished at a substantially lower temperature and the liquid composition shifted to a composition close to E3 on the liquidus (see below).

Details are in the caption following the image
Microstructures of the samples heat-treated at 1773 K: (A) MTS-7, (B) MTS-9, and (C) MTS-10.

This is also a case for sample MTS-8. A three-phase equilibrium (C-SiO2 + MgTi2O5 + L) for sample MTS-8 heat treated at 1773 K was established (see Table 3), although the three phases of C-SiO2, MgTi2O5, and liquid could be in equilibrium below the temperature of the invariant reaction U3 (1715 K; see below) until crystallization at E3 (1690 K; see below). Considering that faster diffusion occurs in the liquid state than in solid and the sample was not quenched but subjected to furnace cooling, it can be assumed that the liquid composition could correspond to the temperature below than 1773 K (even below 1715 K). Thus, there is uncertainty regarding the liquid composition of MTS-8 that was heat treated at 1773 K, since the temperature corresponding to the measured composition of the liquid is not exactly known. On the other hand, the liquid composition of MTS-8 heat treated at 1773 K followed by furnace cooling is within the liquid area suggested for this isothermal section. Such a statement on nonequilibrium crystallization is quite relevant for all samples investigated in this work at 1773 K.

Nevertheless, although the results obtained at 1773 K in this work are limited and not enough to construct the complete equilibrium isothermal section, they are only qualitatively compared with those presented by Chen et al.17 The authors17 investigated samples of the MgO–TiO2–SiO2 system equilibrated at 1773 K and then quenched (their data are shown in red in Figure 1C). Considering all the above features of the nonequilibrium solidification of the samples in this work, the comparison of the data with the data of Chen et al.17 shows reasonable agreement.

DTA followed by SEM/EPMA investigations were performed in this work to get information about invariant reactions involving the liquid phase in the MgO–TiO2–SiO2 system. The experimental results obtained in this work are compared with the available literature data in Table 4. It should be mentioned that the temperatures and compositions of liquids of invariant reactions are in reasonable agreement with those in Ref. [11]. However, considering also the solid-state reaction discovered in this study, the character of the reactions in the TiO2–SiO2-rich side is different in comparison to those in Ref. [11]: eutectic L⇋P-MgSiO3 + SiO2 + MgTi2O5 in this work against eutectic L⇋MgSiO3 + TiO2 + MgTi2O5 in Ref. [11] and transitional-type L + TiO2⇋SiO2 + MgTi2O5 in this work against eutectic L⇋TiO2 + SiO2 + MgSiO3 in Ref. [11]. The liquidus projection of the MgO–TiO2–SiO2 system is shown in Figure 4A. Notably, an area of the liquid immiscibility and related monotectic reaction L1⇋L2 + TiO2 + C-SiO2 (shown tentatively by black dashed line in Figure 4A) were not studied in this work and can be a subject of a further study, while all other invariant reactions occurred in the MgO–TiO2–SiO2 system are discussed below.

TABLE 4. Invariant reactions in the MgO–TiO2–SiO2 system.
Reaction Type, definition T, K Composition of the liquid phase, mole fraction References
MgO TiO2 SiO2
L + MgO⇋Mg2SiO4 + Mg2TiO4 Transitional, U1 1853 0.5899 0.2963 0.1138 11
1910 13
1793 ± 10 0.6372 0.2494 0.1134 14
1921 Experimenta
1862 0.6087 0.2100 0.1791 Calculateda
L + Mg2TiO4⇋Mg2SiO4 + MgTiO3 Transitional, U2 1813 0.5093 0.3705 0.1202 11
1835 Experimenta
1800 0.5548 0.2808 0.1622 Calculateda
L⇋Mg2TiO4 + Mg2SiO4 + MgTiO3 Eutectic, E1 1793 0.4938 0.3810 0.1252 11
1822 0.4839 0.3831 0.1330 Experimenta
1783 0.5223 0.3066 0.1689 Calculateda
L⇋Mg2SiO4 + P-MgSiO3 + MgTi2O5 Eutectic, E2 1713 0.4476 0.2427 0.3096 11
1713 0.4435 0.2367 0.2998 12
1704 0.4480 0.2138 0.3383 Experimenta
1717 0.4397 0.2246 0.3341 Calculateda
L⇋TiO2 + SiO2 + MgSiO3 Eutectic 1673 0.3846 0.2068 0.4086 11
1650 13
L + TiO2⇋T-SiO2 + MgTi2O5 Transitional, U3 1715 Experimenta
1728 0.3652 0.2816 0.3518 Calculateda
L⇋MgTi2O5 + TiO2 + MgSiO3 Eutectic 1663 0.4369 0.2345 0.3286 11
1663 0.4375 0.2299 0.3326 12
C-SiO2⇋T-SiO2 + (TiO2 + L) Degenerated, D1 1743 0.3091 0.3286 0.3623 11
1744 0.3477 3046 0.3462 Calculateda
C-SiO2⇋T-SiO2 + (P-MgSiO3 + L) Degenerated, D2 1743 0.3956 0.1377 0.4667 11
1744 0.4219 0.1231 0.4539 Calculateda
L⇋P-MgSiO3 + T-SiO2 + MgTi2O5 Eutectic, E3 1690 0.3827 0.1779 0.4395 Experimenta
1712 0.3394 0.2301 0.3746 Calculateda
L1⇋L2 + C-SiO2 + TiO2 Monotectic, M-M 1803 0.2503 0.3741 0.3756 11
0.0276 0.1145 0.8579
1809 0.2803 0.3797 0.3385 Calculateda
0.0065 0.0492 0.9464
L⇋Mg2SiO4 + MgTi2O5 emax1 1813 0.4806 0.3747 0.1448 11
1827 Experimenta
Calculateda
L⇋P-MgSiO3 + MgTi2O5 emax2 1698 0.4423 0.2386 0.3191 11
1688 12
1715 Experimenta
Calculateda
MgTi2O5 + T-SiO2⇋P-MgSiO3 + TiO2 Transitional 1625 ± 8 Experimenta
1626 Calculateda
  • a This work.
Details are in the caption following the image
Liquidus projection of the MgO–TiO2–SiO2 system determined experimentally (A) and calculated at p(O2) = 0.21 bar (B).

The melting temperature of the three-phase assemblage (MgO + Mg2SiO4 + Mg2TiO4), as determined by sample MTS-12, was found to be 1918 K on heating (hereinafter, referred to as onset point as a transformation temperature), as shown in Figure 5A. The microstructure of this sample after melting in DTA (Figure 6A) indicated the coexistence of three phases with no traces of eutectic: primary MgO with Mg2SiO4 and Mg2TiO4. This indicates the occurrence of an invariant reaction of the transition type U1, L + MgO ⇋ Mg2SiO4 + Mg2TiO4. Even though the composition of the liquid phase at point U1 on the liquidus cannot be determined from the microstructural analysis, the thermal event at 1918 K is assigned to this U1 reaction. Notably, Berezhnoi13 reported a pseudobinary eutectic between Mg2SiO