Analysis of the subsolidus structure in the Al2O3 – FeO – TiO2 system

The study of the subsolidus structure of multicomponent systems for the synthesis of composite materials with specified phase composition and properties is urgent. Insufficient knowledge of the Al2O3 – FeO – TiO2 system arouses research interest in the structure of the system, as well as in the processes that occur in the system in different temperature ranges. A thermodynamic analysis of the Al2O3 – FeO – TiO2 system was carried out and it was found that the partition of the system into elementary triangles changes in five temperature ranges: I – up to a temperature of 1413 K, II – in the temperature range 1413 – 1537 K, III – 1537 – 1630 K, IV – 1630 – 2076 K and V – above the temperature of 2076 K. The main geometrical-topological characteristics of the subsolidus structure of the system and its phases were analyzed: the areas of elementary triangles, the degree of their asymmetry, the area of regions in which phases exist and the probability of the existence of phases. It was found that the FeAl2O4 – Fe2TiO4 – FeO elementary triangle with a relatively large area and a fairly small degree of asymmetry remained unchanged up to a temperature of 2076 K and the FeAl2O4 phase had the highest probability of existence above a temperature of 1413 K; all this indicates the reliability of predicting the phase composition of synthesized materials in this area and does not require special technological conditions for the accuracy of dosing and the time for homogenization of precursors. In the temperature range 1537 – 1630 К, the Al2TiO5 – FeAl2O4 – TiO2 elementary triangle has the largest area, but rearrangement of the connections occurs above a temperature of 1630 K. In this range, researchers may be interested in the FeTi2O5 – Al2TiO5 – FeTiO3 elementary triangle, which has the smallest area and the greatest degree of asymmetry. Of course, it is possible to perform additional calculations to determine whether the compositions belong to the joint area of two elementary triangles Al2TiO5 – FeAl2O4 – TiO2 and FeTi2O5 – Al2TiO5 – FeTiO3, special technological methods of mass preparation and synthesis must be strictly observed in working in this area. For corundum refractories and corundum-based materials with increased heat resistance, it is advisable to calculate whether the compositions belong to the joint region Al2O3 – Al2TiO5 – FeAl2O4 (in the temperature range 1537 – 1630 K) and Al2TiO5 – FeTiO3 – Al2O3 or FeTiO3 – Al2O3 – FeAl2O4 (above a temperature of 1630 K). The calculated data obtained above a temperature of 2076 K, as a consequence of non-proving the existence of the Al4TiO8 compound, are of recommendatory nature and require further theoretical and practical studies. Based on the results obtained, recommendations are given on the range of compositions that are optimal for obtaining new materials with the required phase composition and desired properties. This will contribute to the development of the latest resource-saving technologies for the manufacture of composite materials.

GEOMETRICAL–TOPOLOGICAL CHARACTERISTICS OF THE SUBSOLIDUS STRUCTURE IN THE MgO – Al2O3 – TiO2 SYSTEM

Among the materials that attract attention from the point of view of creating refractory products with increased heat resistance, one can single out materials based on compositions of the MgO – Al2O3 – TiO2 system. As a result of the thermodynamic analysis of the MgO – Al2O3 – TiO2 system, it was found that the partition of the system into elementary triangles will change in three temperature ranges: I – up to 1537 K, II – in the temperature range 1537 – 2076 K and above 2076 K. It has been established that up to a temperature of 2076 K there is a concentration range of spinel phases: magnesium aluminate spinel – quandylite. Above 1537 K, there is a concentration range: tialite – karroite, which meets the requirements for materials with high heat resistance. The elementary triangle TiO2 – Al2TiO5 – MgTi2O5 can be used to obtain heat–resistant materials based on Al2TiO5 stabilized by MgTi2O5. To obtain heat–resistant periclase–spinel materials, an elementary triangle Mg2TiO4 – MgAl2O4 – MgO is recommended, in which only compounds with a cubic crystal lattice are present. Thus, the division of the MgO – Al2O3 – TiO2 system into elementary triangles and the analysis of the geometrical–topological characteristics of the phases of the system made it possible to select in the system under study the regions of compositions that have optimal properties for obtaining materials with the specified optimal properties.

A NOTE ON THE CONSTRUCTION OF THE EQUILATERAL TRIANGLE WITH SCALENE ELEMENTARY TRIANGLES IN PLATO'S TIMAEUS: PL. TI. 54A-B

In the Timaeus Plato says that, among the infinite number of right-angled scalene elementary triangles, the best (τὸ κάλλιστον) is that ἐξ οὗ τὸ ἰσόπλευρον ἐκ τρίτου συνέστηκε. Apart from few exceptions to be mentioned shortly, the translations of the Timaeus, which I am aware of spanning the period from the second half of the nineteenth century up to recent times, have usually rendered this passage as meaning that such an elementary triangle is that which, when two are combined, the equilateral triangle forms as a third figure. For instance, Bury and Zeyl respectively translate: out of which, when two are conjoined, the equilateral triangle is constructed as a third. and from [a pair of] which the equilateral triangle is constructed as a third figure. I shall refer to this sort of translation as the Prevailing Translation (hereafter PT).