Grand canonical partition function of a serial metallic island system

We present a calculation of the grand canonical partition function of a serial metallic island system by the imaginary-time path integral formalism. To this purpose, all electronic excitations in the lead and island electrodes are described using Grassmann numbers. The Coulomb charging energy of the system is represented in terms of phase fields conjugate to the island charges. By the large channel approximation, the tunneling action phase dependence can also be determined explicitly. Therefore, we represent the partition function as a path integral over phase fields with a path probability given in an analytically known effective action functional. Using the result, we also propose a calculation of the average electron number of the serial island system in terms of the expectation value of winding numbers. Finally, as an example, we describe the Coulomb blockade effect in the two-island system by the average electron number and propose a method to construct the quantum stability diagram.

Фазовые превращения в однокомпонентных многофазных системах: фазово-полевой подход

The problems of constructing a multiphase model of the phase field for the processes of phase transitions of the first kind are considered. Based on the Gibbs energy of the complete system expressed in terms of antisymmetrized combinations of phase fields, it is shown that the equations of dissipative dynamics of a locally nonequilibrium system follow from the condition of its monotonic decrease, preserving the normalization of the sum of variables by one and the following properties of the previously known two-phase model.

Experimental Determination of Phase Equilibria in the Na2O-SiO2-WO3 System

The present study investigated phase equilibria in the Na2O-SiO2-WO3 system experimentally using high-temperature equilibration, quenching, and electron probe X-ray microanalysis (EPMA). New thermodynamic information on the Na2O-SiO2-WO3 system was derived based on the newly obtained experimental results and data from the literature. The primary phase fields of sodium metasilicate, sodium disilicate, and tridymite were determined along with the isotherms at 1073, 1173, and 1273 K. The solubilities of WO3 in SiO2, Na2Si2O5, and Na2SiO3, and the solubility of SiO2 in Na2WO4 were accurately measured using EPMA. Comparisons between the existing and newly constructed phase diagram were carried out and the differences are discussed. The phase equilibrium data will be beneficial to the future development of sustainable tungsten industries and thermodynamic modelling in WO3 related systems.

Phase Equilibria in the System CaO-SiO2-La2O3-Nb2O5 at 1400 °C

CaO-SiO2-La2O3-Nb2O5 system is of great significance for the pyrometallurgical utilization of Bayan Obo tailing resources. In the present work, the phase equilibrium of this quaternary system at 1400 °C was determined by a thermodynamic equilibrium experiment. On the basis of the recently determined CaO-La2O3-Nb2O5 phase diagram, some boundary surfaces of primary phase fields of CaO-SiO2-La2O3-Nb2O5 phase diagram were modified; then, the 1400 °C isothermal surface in the primary phase fields of SiO2, CaNb2O6, Ca2Nb2O7, and LaNbO4 was constructed, respectively. On this basis, CaO-SiO2-Nb2O5 pseudo-ternary phase diagrams with w(La2O3) = 5%, 10%, 15%, and 20% were determined, respectively. Considering the importance of equilibrium crystallization reaction type, we proposed a new rule named Tangent Line Rule to judge the univariant reaction type in the quaternary phase diagram. By applying Tangent Line Rule and Tangent Plane Rule previously proposed, some univariant and bivariant crystallization reaction types in the CaO-SiO2-La2O3- Nb2O5 phase diagram were determined, respectively. The current work can provide original data for the establishment of a thermodynamic database of Nb-bearing and REE-bearing slag system; the proposed Tangent Line Rule will promote the application of a spatial quaternary phase diagram.

Element selection for crystalline inorganic solid discovery guided by unsupervised machine learning of experimentally explored chemistry

The selection of the elements to combine delimits the possible outcomes of synthetic chemistry because it determines the range of compositions and structures, and thus properties, that can arise. For example, in the solid state, the elemental components of a phase field will determine the likelihood of finding a new crystalline material. Researchers make these choices based on their understanding of chemical structure and bonding. Extensive data are available on those element combinations that produce synthetically isolable materials, but it is difficult to assimilate the scale of this information to guide selection from the diversity of potential new chemistries. Here, we show that unsupervised machine learning captures the complex patterns of similarity between element combinations that afford reported crystalline inorganic materials. This model guides prioritisation of quaternary phase fields containing two anions for synthetic exploration to identify lithium solid electrolytes in a collaborative workflow that leads to the discovery of Li3.3SnS3.3Cl0.7. The interstitial site occupancy combination in this defect stuffed wurtzite enables a low-barrier ion transport pathway in hexagonal close-packing.

Extended Larché–Cahn framework for reactive Cahn–Hilliard multicomponent systems

At high temperature and pressure, solid diffusion and chemical reactions between rock minerals lead to phase transformations. Chemical transport during uphill diffusion causes phase separation, that is, spinodal decomposition. Thus, to describe the coarsening kinetics of the exsolution microstructure, we derive a thermodynamically consistent continuum theory for the multicomponent Cahn–Hilliard equations while accounting for multiple chemical reactions and neglecting deformations. Our approach considers multiple balances of microforces augmented by multiple component content balance equations within an extended Larché–Cahn framework. As for the Larché–Cahn framework, we incorporate into the theory the Larché–Cahn derivatives with respect to the phase fields and their gradients. We also explain the implications of the resulting constrained gradients of the phase fields in the form of the gradient energy coefficients. Moreover, we derive a configurational balance that includes all the associated configurational fields in agreement with the Larché–Cahn framework. We study phase separation in a three-component system whose microstructural evolution depends upon the reaction–diffusion interactions and to analyze the underlying configurational fields. This simulation portrays the interleaving between the reaction and diffusion processes and how the configurational tractions drive the motion of interfaces.

Phase equilibria in the CeO2–La2O3–Ln2O3 system at 1250 and 1500 °С

Materials based on cerium oxide, stabilized by oxides of rare earth elements, are promising for use in medicine, energy and mechanical engineering due to the uniqueness of their properties. State diagrams of CeO2–La2O3–Ln2O3 systems are the physicochemical basis for the creation of solid electrolytes for fuel cells, oxygen gas sensors, catalyst carriers, protective coatings on alloys, etc. Phase equilibria and structural transformations in CeO2–La2O3–Gd2O3 systems at temperatures 1250 and 1500 °С and in the binary system La2O3–Gd2O3 at temperatures 1100, 1500 and 1600 ° С in the whole range of concentrations were investigated using X-ray phase and microstructural analyzes. It was found that solid solutions based on cubic (F) modification with CeO2 fluorite type, monoclinic (B) and cubic (C) modifications of Gd2O3 and hexagonal (A) modification of La2O3 are formed in the ternary system CeO2–La2O3–Gd2O3. The boundaries of the phase fields and the periods of the crystal lattices of the formed phases are determined. It is established that in the CeO2–La2O3 –Gd2O3 system at 1250 and 1500 °С the phases of cubic symmetry are in equilibrium: on the basis of F–CeО2 with the spatial group Fm3m and C-phase on the basis of Gd2O3 with the spatial group Ia3. As the temperature decreases, there is a narrowing of all areas of homogeneity.

SYNTHESIS OF ALLOYS OF THE Sb2Se3-CuCr2Te4 SYSTEM AND PHYSICO-CHEMICAL PROPERTIES

The interaction in the quasi-ternary system Sb2Se3-CuTe-Cr2Te3 along the Sb2Se3-CuCr2Te4 section was studied by methods of physicochemical analysis: differential thermal (DTA), X-ray phase (XRD), microstructure (MSA), as well as by measuring the density and its microhardness and plotted. The phase diagram of the system is quasi-binary, eutectic type. The composition of the double eutectic formed in the system is 20 mol % CuCr2Te4 and a temperature of 490°C. As a result of the analysis of the microstructure, it was determined that there are single-phase fields based on the original components. It was found that at room temperature, solid solutions based on Sb2Se3 extend to 5 mol % CuCr2Te4, and on the basis of CuCr2Te4 solid solutions reach 13 mol % Sb2Se3.

Coupled diffusion and phase transition: Phase fields, constraints, and the Cahn–Hilliard equation

We develop a constrained theory for constituent migration in bodies with microstructure described by a scalar phase field. The distinguishing features of the theory stem from a systematic treatment and characterization of the reactions needed to maintain the internal constraint given by the coincidence of the mass fraction and the phase field. We also develop boundary conditions for situations in which the interface between the body and its environment is structureless and cannot support constituent transport. In addition to yielding a new derivation of the Cahn–Hilliard equation, the theory affords an interpretation of that equation as a limiting variant of an Allen–Cahn type diffusion system arising from the unconstrained theory obtained by considering the mass fraction and the phase field as independent quantities. We corroborate that interpretation with three-dimensional numerical simulations of a recently proposed benchmark problem.