On the enumeration of phase diagrams

The topological isomorphism of polyhedra with trivalent vertices and solid-liquid diagrams of three-component systems allows the problem of constructing the complete set of the topological types of diagrams with a given set of characteristics to be reduced to the problem of the generation of marked cubic graphs, which are the Schlegel projections of polyhedra. The problem of the enumeration of possible topological types of melting diagrams containing M binary and N ternary stoichiometric congruently melting compounds is considered. Relations between the topological characteristics of such diagrams are given. The total number of topologically different types of melting diagrams with one binary and one ternary congruently melting compounds was found to be 64.     

Topology of subsolidus sections of phase diagrams of quaternary reciprocal systems without solid solutions

A problem of constructing subsolidus isobaric–isothermal sections of phase diagrams of quaternary reciprocal systems with constant-composition compounds and limited solid solutions was considered. Relations between the topological characteristics of such sections were derived. Their topological classification was proposed. An algorithm for constructing sections using the theory of finite graphs was developed. A problem of enumerating sections for diagrams with a given set of classification characteristics was solved.     

Topology of melting diagrams of ternary reciprocal systems without continuous solid solutions

The topological properties were considered for melting diagrams of ternary reciprocal systems with constant-composition phases or with isomorphic diagrams with limited solid solutions. An algorithm for constructing the diagrams was described, and an example was given for a melting diagram with one binary peritectic compound and two ternary stoichiometric compounds, all melting without decomposition. A problem of enumeration of all the phase diagrams with such a set of compounds was solved.     

Enumeration of melting diagrams of three-component systems with stoichiometric compounds

The topologic isomorphism of polyhedra having trivalent vertexes and phase diagrams of three-component systems allows the reduction of the problem of constructing the complete set of topologic types of diagrams having given characteristics to the problem of generating labeled cubic graphs. Each of them is the Schlegel projection of some polyhedron on one of its faces. This work is concerned with the problem of enumeration possible topologic types of phase diagrams containing M binary and N ternary stoichiometric compounds. Relations between the topologic characteristics of these diagrams are presented. A great variety of types of diagrams of the class in question is demonstrated.     

Melting diagrams of quaternary systems with a single compound

Topological analysis was made of the structure of the melting diagrams of quaternary systems with a single compound. Graphs of phase diagrams were constructed, which show the relative positions of invariant points and monovariant lines on the liquidus hypersurface. It was shown that there exist two types of diagrams with a binary compound and six types of diagrams with a ternary compound. Quaternary compound C can form by congruent crystallization reaction L = C (I) or by the interaction of melt with components: L + K₁ = C (II), L + K₁ + K₂ = C (III), or L + K₁ + K₂ + K₃ = C (IV). There are 23 types of diagrams with reaction (I) and 23 types of diagrams with reaction (II), which described by identical graphs; 25 types of diagrams with reaction (III); and 6 types of diagrams with reaction (IV). Fragments of the obtained diagrams correspond to various topological situations, which may occur in more-component systems.     

Partition of the composition tetrahedron during construction of subsolidus isobaric-isothermal sections of the phase diagrams of quaternary systems

A problem of constructing subsolidus isobaric-isothermal sections of phase diagrams with constant-composition compounds and limited solid solutions was considered. Relations between the topological characteristics of such sections were derived. Their topological classification was proposed. An algorithm for constructing sections using the theory of finite graphs was developed. A problem of enumerating sections for diagrams with a given set of classification characteristics was solved.     

Phase diagrams in materials science of topological insulators based on metal chalcogenides

The literature data on topological insulators (TIs) based on metal chalcogenides, which constitute a new unique class of functional materials, are systematized here in the context of physicochemical analysis and crystal chemistry. An accent is on the phase diagrams of relevant systems and the crystal structures of the main TI types. We show that, for search and design of new phases having TI properties, it will be expedient to revise earlier constructed phase diagrams for some systems, especially B^V–Se(Te) and A^IV–B^V–Te (A^IV = Ge, Sn, Pb; B^V = Sb, Bi) systems, using new approaches to studies of phase equilibria. There is also a need for systematic studies of complex systems involving binary and ternary TIs.     

Solid-solute phase equilibria in aqueous solution. IV. Calculation of Phase Diagrams

The thermodynamic principles of conventional (T-x, P-T) phase diagrams and solubility (log ΣK-x) diagrams depicting solid-solute phase equilibria in aqueous solution are derived from a unifying point of view. It is shown that thermodynamic quantities necessary for the construction of conventional phase diagrams can be obtained from solubility measurements. The unary system calcite-aragonite and the binary system aragonite-strontianite, where solubility data are available over the whole compositional range, have been selected as examples. In the latter case, the constraint of constant composition of the solid phase leading to a metastable equilibrium with the respective solute species is an essential point in the thermodynamic derivation and was observed experimentally as well.     

Structural Features of Solid Phases and Topology of Phase Diagrams

The paper considers model phase diagrams of binary and ternary systems involving transformations (transitions) of phase equilibria. The relationship between the type of structural solidstate transformation and the type of phase diagram is shown. Topological series of phase diagrams of systems with continuous and limited types of solid solutions are considered, including phase diagrams with polymorphic transitions between intermediate solid phases of variable composition.     

Enumeration of melting diagrams of ternary reciprocal systems without continuous solid solutions

The topological properties were considered for melting diagrams of ternary reciprocal systems with constant-composition phases or limited solid solutions. Relations between the topological parameters of the diagrams were found. Algorithms for solving problems of generation of liquidus surface schemes and their enumeration and classification were presented.     

恒温截面法构制三维相图判误用数点分析法

以相图拓扑学规律为基础，提出了利用系列恒温截面分析相平衡空间结构以及绘制立体相图时的判误数点分析法，以实例说明数点法在相图分析中的应用，并提出了三维相图的构造的实用判误方法。     

Theoretical consideration of bulk free-radical copolymerization with allowance for the preferential sorption of monomers into globular nanoreactors

The thermodynamics of a globular solution of binary copolymer in two-component solvent is theoretically considered. Classification of phase diagrams of such a solution is performed by their topological appearance. Based on the thermodynamic theory, binary bulk free-radical copolymerization under low conversions is examined in systems where macroradicals being in globular conformational state represent isolated nanoreactors. The theoretical results achieved permit an explanation of all qualitative peculiarities observed experimentally under copolymerization in such anomalous systems.     

Topological transformation of the phase diagram for the ternary system cesium nitrate-water-acetonitrile

In the ternary system cesium nitrate-water-acetonitrile, in which liquid-liquid phase separation with an upper critical solution point (UCSP) exists in the liquid binary subsystem, was studied in the range from ?5 to 120°C using visual polythermal analysis. Liquid-liquid phase separation in the ternary system is observed above 96.0°C and below 2.8°C. Acetonitrile distribution coefficients between the aqueous and organic phases of monotectic equilibrium were calculated for various temperatures. Phase isotherms of the system confirm the general scheme of the topological transformation of phase diagrams in salt-binary solvent ternary systems with salting out.     

Phase equilibria and critical phenomena in the rubidium nitrate-water-acetonitrile system

Phase equilibria and critical phenomena were studied from -5 to 120°C in the rubidium nitrate-water-acetonitrile system, in which the liquid binary subsystem is characterized by liquid-liquid phase separation with an upper critical solution point (UCSP), using a visual polythermal method. We found that rubidium nitrate has a salting out effect on water-acetonitrile solutions and causes them to demix at any temperature in the specified range. Acetonitrile distribution coefficients between aqueous and organic monotectic phases were calculated for various temperatures. The minimum value was observed for 20.0°C. Six isothermal phase diagrams of the system were plotted to verify a fragment of the global scheme of the topological transformation of phase diagrams for salt-binary solvent ternary systems with salting out. The salting out effects of potassium, rubidium, and cesium nitrates on water-acetonitrile mixtures were comparatively analyzed.     

The global phase behavior of the two-component systems with intracomponent association: Flory approach

Within the Flory approach we study the phase diagrams of two-component fluids, the molecules of each component A(f(A)), B(f(B)) bearing f(A) (f(B)) functional groups capable of forming thermoreversible A-A and B-B bonds. We develop a general procedure to classify these diagrams depending on the values of four governing parameters -- entropies and normalized energies of A-A and B-B bonds, and give full topological classification of phase diagrams with f(A,B)> or =3. We show that these phase diagrams can have immiscibility loops and up to four critical points.     

Phase equilibria and critical phenomena in the cesium nitrate–water–pyridine ternary system

The solubilities of components, phase equilibria, and critical phenomena in the cesium nitrate–water–pyridine ternary system are studied in the 5–100°C temperature range by the visual–polythermal method. Cesium nitrate is found to exhibit a salting-out effect at temperatures above 79.9°C causing phase separation in homogeneous water–pyridine solutions. The temperature of formation of the critical monotectic tie line (79.9°C) and the compositions of solutions corresponding to the liquid–liquid critical points at three temperatures are determined. The pyridine distribution coefficients between the aqueous and organic phases of the monotectic state at 85.0, 90.0, and 100.0°C are calculated. Their values demonstrate that salting-out of pyridine from aqueous solutions by cesium nitrate increases at higher temperatures. The plotted isotherms of phase diagrams confirm the fragment of the scheme of topological transformation of the phase diagrams of salt–binary solvent ternary systems with salting-in and salting-out phenomena.     

Topological transformation of a phase diagram for the sodium nitrate-water-isopropanol ternary system

Phase equilibria and critical phenomena in the sodium nitrate-water-isopropanol ternary system, where a boundary binary liquid system shows no immiscibility over the entire temperature range of its liquid state, were studied in the range from 5 to 90°C using a visual polythermal method. The formation temperature of a monotectic critical tie-line was determined to be 6.1°C, and the solution compositions corresponding to critical solution points at various temperatures were determined. Isopropanol partition coefficients between the aqueous and organic phases of monotectic equilibrium were calculated for seven temperatures. The isopropanol salting out from aqueous solutions by sodium nitrate was shown to be enhanced by rising temperature. Isothermal phase diagrams of the title system were constructed to verify a fragment of the general scheme of topological transformations of phase diagrams for salt-binary solvent ternary systems with salting out.     

Phase equilibrium in polymer systems

The topological analysis of phase equilibria in polymer systems, which was developed by S.P. Papkov; the classification of the types of phase equilibrium; the principle of mutual independence of the two types of equilibrium; the concept of the generality of the phase equilibrium in polymer-solvent, polymer-plasticizer, polymer-oligomer, and polymer-polymer systems; and the processes of water sorption by polymers are considered. It is shown that the problems that Papkov dealt with remain topical and the concepts that he elaborated can underlie new studies in the field of phase equilibria in polymer systems.     

A simple statistical mechanical theory for the mutual solubility of fluid mixtures of water and organic solvents under high pressure

A simple statistical mechanical theory is presented to explain phase diagrams of fluid mixtures with both a lower critical solution temperature and an upper critical solution temperature under pressure. By postulating a temperature dependence for the interaction free energy parameter of the constituent molecules and a pressure dependence for the excess volume, phase diagrams with both lower critical solution temperature, and upper critical solution temperature and their pressure dependence can be reproduced by quadratic surfaces in temperature-concentration-pressure space. The topological aspects of the observed phase diagrams in this space have been related to our theoretical model, and the thermodynamical meaning of the topologies has been interpreted based on our model. Experimental data for the mutual solubility of water and 2-butanol under pressure and that of water and 3-methylpyridine with added salts have been analyzed quantitatively and theoretical parameters are determined.