We use ternary systems as an example to discuss the need for a clear identification of invariant phase transformations. It is becoming more and more challenging to distinguish clearly and unambiguously among four-phase reactions, for example, in cases of polymorphism of reagents, decomposition or formation of compounds, and liquid–liquid immiscibility, as opposed to the currently used schematic and incomplete classification into eutectic, peritectic, all other reactions. A nomenclature of phase transformations (exemplified for isobaric phase diagrams of ternary systems) is proposed to help to differentiate such transformations by the type of phase reactions, better understand the processes occurring in the system, and more clearly present the results of experimental investigations of the system. The results can be used to design phase diagrams of ternary systems that can be calculated thermodynamically, studied experimentally, and prepared for computer modeling.

Variants of development of digital twins of the isobaric phase diagram of the ternary system LiCl–UCl3–KCl in the form of computer 3D models are considered. Three variants of the phase diagram model are proposed, depending on the number of binary compounds in the UCl3–KCl system. The first variant accounts for the formation of one binary compound K2UCl5, wherein the ternary system is triangulated by the quasi-binary section LiCl–K2UCl5 into two eutectic subsystems. The second variant considers the formation of three compounds: K2UCl5, K3UCl6, and KU2Cl7. The system is triangulated by the LiCl–K2UCl5 section into two subsystems, in each of which the eutectic invariant transformation is preceded by the decomposition of binary compounds. In a third, hypothetical variant, based on the formation of two compounds, K2UCl5 and K3UCl6, the occurrence of three invariant transformations is assumed. The models are constructed across the entire temperature range and account for both the high-temperature phase regions with a melt and the subsolidus regions. The presented digital twins are not only a tool for further study of the LiCl–UCl3–KCl system but also provide predictive opportunities for experimental investigation, depending on the chosen variant of the UCl3–KCl system structure.

Система Ag-Sb-Sn представляет интерес для микроэлектроники и является перспективным материалом, используемым при высокотемпературной пайке. Анализ фазовых равновесий в системе осложнен отсутствием согласованного мнения о ее строении: различные версии обусловлены неоднозначным описанием образующихся соединений и условий их существования. Так, соединение Sb3Sn4 либо считается устойчивым вплоть до комнатной температуры, либо существует в ограниченном диапазоне температур 323-242oC. Целью работы был анализ фазовых равновесий в системе Ag-Sb-Sn в зависимости от способа образования и температурных границ существования бинарного соединения Sb3Sn4. Для этого были построены две версии трехмерной (3D) компьютерной модели фазовой диаграммы Ag-Sb-Sn. Использовалась технология сборки фазовой диаграммы из поверхностей и/или фазовых областей. Изо- и политермические разрезы рассчитывались по обоим вариантам 3D-модели. Полученные версии обеспечивают полное геометрическое описание фазовой диаграммы, в том числе в твердофазных областях. Кроме того, модель может быть скорректирована при получении новой экспериментальной информации и даже радикальном изменении понимания свойств соединений, образующихся в системе. Обе версии 3D-модели расширяют прогностические функции фазовой диаграммы, и в дальнейшем помогут понять структуру фазовых диаграмм других тройных систем, сформированных на основе бинарной системы Sb-Sn.

Three-dimensional (3D) computer model of isobaric phase diagram of the Ag-Au-Sb system, constructed by assembling from surfaces and phase regions, is presented. Calculations of material balances permit to study the history of crystallization of a particular alloy, formation of a microstructure with detailed analysis and qualitative and quantitative assessment of its constituent elements. The latter opens up opportunities for creating digital passports of materials received in the system.

Three-dimensional (3D) computer models of Ag–Cu–Ni and Ag–Cu–Pb isobaric phase diagrams, designed based on 23 and 31 base points, respectively, assembled from 14 and 32 surfaces, 9 and 15 phase fields, respectively, and intended to digitize information on these diagrams, are used to verify the adequacy of interpretation of published isothermal and polythermal sections, both calculated and experimentally studied ones. The geometric features of the phase diagram regions that relate to liquid–liquid miscibility gaps and solid solution decomposition are refined in the 3D models. Mistakes arising from an incorrect imaging of the decay of copper–nickel solid solutions and from discrepancies in the values of the Ag–Cu–Pb ternary eutectic temperature are shown on polythermal sections.

A four-dimensional (4D, in concentration–temperature coordinates) computer model of the isobaric phase diagram of uranium, sodium, magnesium, and plutonium chlorides, as well as four three-dimensional (3D) computer models of the phase diagrams of the ternary systems forming it, has been constructed. The technology of assembling a 4D model of 46 hypersurfaces and 17 phase regions was used in the design. The obtained 4D model of the UCl 3 –NaCl–MgCl 2 –PuCl 3 phase diagram makes it possible to visualize a four-dimensional object as a whole (with all its hypersurfaces and phase regions) by any arbitrarily given 2D and 3D sections, as well as it is able to reproduce published (experimental or thermodynamically calculated) 2D sections. The scope of application of the results of the work is the development of materials for fuel components of fourth-generation molten salt reactors and pyrochemical recycling of spent fuel rods. For the first time, a comprehensive, complete description of phase diagrams composed of uranium, plutonium, sodium, and magnesium chlorides has been obtained.

The reliability of modeling of the ZrO2-SiO2-Al2O3 phase diagram up to the temperature 3000 K (melting of ZrO2) was illustrated using simulation by the three-dimensional (3D) computer model in comparison with the results of calculation by the NUCLEA database. For this purpose, the 3D computer model with three polymorphic modifications of ZrO2 and four modifications of SiO2 has been designed. The satisfactory agreement between both approaches as well as experimental studies of the ZrO2-SiO2-Al2O3 phase diagram was demonstrated while considering the features of isothermal sections and isopleths in the system under study.

A computer 3D-model was presented for the isobaric phase diagram of the ZrO 2 –SiO 2 –Al 2 O 3 system with the formation of the ZrSiO 4 and Al 6 Si 2 O 13 compounds. Its geometric structure was derived through the sequential construction of a scheme of phase reactions, including all polymorphic transitions in the subsolidus and the rearrangement of the interaction of binary compounds, as well as zirconium and aluminum oxides; its transformation into a scheme of uni- and invariant states in the tabular and graphical (3D) forms; and the construction of a prototype and its transformation into a spatial model of the phase diagram of the real ZrO 2 –SiO 2 –Al 2 O 3 system. Features of the iso- and polythermal sections of the phase diagram of the considered system, which were calculated using the thermodynamic NUCLEA database, were discussed in comparison with the sections of the obtained 3D-model.

The three-dimensional (3D) computer model of the isobaric phase diagram of the ZrO2–SiO2–Al2O3 system with formation of the ZrSiO4 and Al6Si2O13 compounds is presented. The development of its geometric structure was carried out through the sequential construction of the phase reaction scheme, including all polymorphic transitions in the sub-solidus and the rearrangement of the interaction of binary compounds as well as zirconium and aluminum oxides, its transformation into the scheme of uni- and invariant states in the tabular and graphical (3D) forms, the construction of the prototype, and its transformation into a spatial model of the phase diagram of the real ZrO2–SiO2–Al2O3 system. Features of the isothermal sections and isopleths of the phase diagram of the considered system calculated using the thermodynamic NUCLEA database are discussed in the comparison with the 3D model sections.

A spatial (3D) computer model of an isobaric phase diagram of the LiCl–PrCl3–KCl system was designed. The model was assembled of 66 surfaces and 27 phase regions, of which 31 surfaces and 14 phase regions degenerate to verticals or vertical planes because of the limited solubilities of the initial chlorides and their compounds. Published experimental isothermal sections and thermoanalytical curves for 33 salt melts were used to improve the quality of the 3D model. The results can be used to improve the molten-salt refining technology of nuclear fuel waste.

A four-dimensional (4D, in concentration–temperature coordinates) computer model of the isobaric phase diagram of uranium, sodium, magnesium, and plutonium chlorides, as well as four three-dimensional (3D) computer models of the phase diagrams of the ternary systems forming it, has been constructed. The technology of assembling a 4D model of 46 hypersurfaces and 17 phase regions was used in the design. The obtained 4D model of the UCl3–NaCl–MgCl2–PuCl3 phase diagram makes it possible to visualize a four-dimensional object as a whole (with all its hypersurfaces and phase regions) by any arbitrarily given 2D and 3D sections, as well as it is able to reproduce published (experimental or thermodynamically calculated) 2D sections. The scope of application of the results of the work is the development of materials for fuel components of fourth-generation molten salt reactors and pyrochemical recycling of spent fuel rods. For the first time, a comprehensive, complete description of phase diagrams composed of uranium, plutonium, sodium, and magnesium chlorides has been obtained.

A spatial (3D) computer model of an isobaric phase diagram of the LiCl–PrCl3–KCl system was designed. The model was assembled of 66 surfaces and 27 phase regions, of which 31 surfaces and 14 phase regions degenerate to verticals or vertical planes because of the limited solubilities of the initial chlorides and their compounds. Published experimental isothermal sections and thermoanalytical curves for 33 salt melts were used to improve the quality of the 3D model. The results can be used to improve the molten-salt refining technology of nuclear fuel waste.

Three-dimensional (3D) computer model of the Al-Sn-Zn isobaric phase diagram is presented. It is shown that the T-x-y diagram consists of 64 surfaces and 25 phase regions. Features of Al-Zn binary system phase diagram related to decomposition of aluminum solid solution and its influence on geometric structure of liquidus and solidus surfaces in the ternary system, formed by this binary system and tin, are considered. Critical analysis of the published data was carried out and errors in visualization of those fragments of the Al-Zn T-x diagram, in which phase regions are formed with the participation of new aluminum solid solutions, appeared after the decomposition of the initial solid solution, are discussed. To design this 3D model, literary data were used, and the assessment of its quality is based on isopleths and isothermal sections from the same primary sources.

Because of the many contradictions, connected with the literature data about the binary systems, which form the ternary LiF–KF–RbF system, three possible versions of the T–x–y diagram geometric structure of this system were proposed and their three-dimensional computer models had been designed.

The purpose of the paper is to establish the position of the tie-lines in the Sb-Sm-Se system at 450°C and 620°C, to determine the optical band gap of the phases.The Sb component is in equilibrium with the γ-Sm 2 Se 3-X -Sm 3 Se 4 (ST Th 3 P 4 ) solid solution region with α-Sm 2 Se 3 . A continuous solid solution forms between the SmSb and SmSe (ST NaCl) phases, with which the Sm 3 Se 4 and Sm 4 Sb 3 phases are in equilibrium. The SmSb-Sm 3 Se 4 , Sm 3 Se 4 -SmSb 2 , SmSe-Sm 3 Sb 2 phases are also in equilibrium.In the Sb-Sm 2 Se 3 -Se system at 450°С, the tie-line passes between the Sb 2 Se 3 -Sm 2 Se 3 , Sb 2 Se 3 -SmSe 1.9 phases.In the Sb-Se system based on Sb 2 Se 3 , a solid solution of the subtraction type Sb 2-X ◻ X Se 3 (X = 0–0.04) is formed. In the Sb-Sm-Se system, there is a solid solution of the substitution type along the cuts from Sb 2 Se 3 to the Sm 2 Se 3 (7 mol. % Sm 2 Se 3 ), SmSe 1.9 (4 mol. % SmSe 1.9 ) phases. The extreme compositions of solid solutions have a peritectic point.Due to the change in the position of the tie-lines in the Sb-Sm 2 Se 3 -Se system at 620°C, additional phases appear in the equilibrium samples from the Sb 2 Se 3 -Sm 2 Se 3 section (annealing at 450°C) when heated above 620°C: Sb, SmSe 1.9 .The optical band gap of the phases is: Sb 2-x Sm x Se 3 solid solution 1.17–1.19 eV, α-Sm 2 Se 3 1.62 eV, SmSe 1.9 1.68 eV. Optical properties of incommensurate SmSe 1.9 crystal that were investigated for the first time for this class of crystals indicate complex electronic structure that can be characterized as a multi-bandgap one. Sm ions in SmSe 1.9 are proved to be predominantly in 3+ oxidation state.Previously, the formation of ternary compounds in the system was reported in the literature. Carefully conducted research allows us to assert their absence.

The aim of this article is to predict possible invariant reactions in a quaternary system formed by oxides of titanium, aluminum, silicon, and zirconium. As a result, based on the data on invariant transformations in faceting ternary systems, a scheme of phase reactions with the participation of a liquid is derived, and then the contours of the liquidus hypersurfaces are described for basic invariant points belonging to binary and ternary systems, as well as the six obtained points of the quaternary system: two are peritectic, two are eutectic, and two are intermediate, corresponding to phase reactions of two variants of the peritectic type.

In view of the contradictory opinions on the formation of zircon ZrSiO4 in the ZrO2–SiO2 system (by a peritectic or peritectoid reaction), two variants of prototypes of a three-dimensional (3D) computer model of the T–x–y diagram of the ZrO2–SiO2–Al2O3 system were constructed and used to consider in detail the possible invariant transformations in the subsolidus with the participation of the polymorphs of ZrO2 and SiO2. Because the later publication tended toward the formation of zircon by the peritectoid reaction, a 3D model of the corresponding T–x–y diagram of the real ZrO2–SiO2–Al2O3 system, which is constituted by 177 surfaces and 67 phase regions, was designed and can be used in practice.

The research analyses the controversies surrounding the technique for the formation of a CaO-Al2O3 binary system and the nature of melting of compounds in it, i.e. whether the 12:7 compound is technically possible and whether the 1:1 and 1:2 compounds are congruently or incongruently melting compounds. It also discusses whether in the CaO-MgO-Al2O3 ternary system the following compounds can be formed: a 3:1:1 compound alone or, in addition to it, two more compounds of 1:2:8 and 2:2:14. A 3D model of the T-x-y diagram was created for the most common version, with six binary and three ternary compounds. Its high-temperature portion (above 1300°C) consisted of 234 surfaces and 85 phase regions. Ternary compounds were formed as a result of three peritectic reactions. Besides them, six quasi-peritectic and three eutecticinvariant reactions occurred in the system with the participation of the melt. The principle of construction for a threedimensional model involved a gradual transition from a phase reaction scheme (which is transformed into a scheme of uni- and invariant states) presented in a tabulated and then in a graphical form (a template of ruled surfaces and isothermal planes corresponding to invariant reactions) to a T-x-y diagram prototype (graphic images of all liquidus, solidus, and solvus surfaces). The design was concluded with the transformation of the prototype into a 3D model of the real system after the input of the base points coordinates (concentrations and temperatures) and the adjustment of curvatures of lines andsurfaces. The finished model provides a wide range of possibilities for the visualisation of the phase diagram, including the construction of any arbitrarily assigned isothermal sections and isopleths. The 3D model was designed with the help of the author’s software PD Designer (Phase Diagram Designer). To assess the quality of the 3D model, two versions of an isothermal section at 1840 °C were compared: model section and a fragment of an experimental section near Al2O3.

Experimental study of phase equilibria in quaternary Fe-Ni-Co-Cu system has been carried out using differential thermal analysis, scanning electron microscopy and energy dispersive x-ray spectrometry. Experimental data obtained for several synthesized alloys in the Fe-Ni-Co-Cu system have been used for plotting the liquidus and solidus hypersurfaces projections also for design and verification of the T-x-y-z phase diagram. The 4D geometrical description of the Fe-Ni-Co-Cu system, plotted with applying fundamental geometrical principles of phase diagrams design, facilitates visualization of the liquidus and solidus hypersurfaces in 2D and 3D formats, and the entire four-dimensional T-x-y-z phase diagram using any 2D or 3D sections.