Relationship between activity and three phase boundary in the ternary phase diagram

Theoretical formulae have been derived for determining the activity along a three-phase boundary in a three component system. The activities can be calculated from ternary phase diagrams from the partial thermodynamic properties and the use of a single integration constant. The latter is calculated from the binary phase diagrams. The formulae are rigorous and no assumption is made.     

Formalism of new ternary model expressed in terms of binary regular-solution type parameters

In the previous paper, we have presented a new method for calculating ternary thermodynamic properties based on three binaries. This new model can overcome the inherent defects for both current symmetrical and asymmetrical models and can combine two kinds of current models into one. More important, it can eventually realize the computerization for calculating thermodynamic properties and phase diagrams for multicomponent systems. In this paper we will give this new model a new formalism based on binary regular-solution type parameters. When this model is applied, all we need to do is substitute these parameters into formulae presented here and evaluate them by using a simple calculator. Since a large number of real systems can be approximately fit through a regular-solution type model, therefore, this new formalism is significant.     

Thermodynamic modeling of the Mg–Al–Sb system

Thermodynamic modeling of the Mg–Al–Sb system is carried out for the first time in this work. Among the constituent binaries in this system, only the Al–Sb and Mg–Sb are re-optimized. Liquid phases are described by the Redlich–Kister polynomial model, whereas the high temperature modification of Mg₃Sb₂ compound in the Mg–Sb system is described by the sublattice model. The constructed database is used to calculate and predict thermodynamic properties, binary phase diagrams of Al–Sb and Mg–Sb, and liquidus projections of the ternary Mg–Al–Sb. The calculated phase diagrams and the thermodynamic properties such as enthalpy, entropy, Gibbs free energy of mixing, and activities are found to be in good agreement with the experimental data from the literature. The established Mg–Al–Sb database predicted a closed ternary liquid miscibility gap, six ternary eutectics, two ternary peritectics, four saddle points and a critical point.     

The phase equilibria in the Mg–Ni–Ca system

The three binary systems Mg–Ni, Ca–Ni and Mg–Ca have been re-optimized. A self-consistent thermodynamic database of the Mg–Ni–Ca system is constructed by combining the optimized parameters of these three constituent binaries. Lattice stability values are not added to the pure elements Mg-hcp, Ni-fcc, Ca-fcc and Ca-bcc to construct this database. The Redlich–Kister polynomial model is used to describe the liquid and the terminal solid solution phases, and the sublattice model is used to describe the non-stoichiometric phase, in this system. The constructed database is used to calculate the three binary and the ternary systems. The calculated binary phase diagrams along with their thermodynamic properties such as Gibbs energy, enthalpy, entropy and activities are found to be in good agreement with experimental data from the literature. This is the first attempt to construct the ternary phase diagram of the Mg–Ni–Ca system. The established database for this system predicted three ternary eutectic, five ternary quasi-peritectic, two ternary peritectic and two saddle points.     

The theorem of corresponding relation between neighboring phase regions and their boundaries in phase diagrams (I)

Several fundamental concepts in phase diagrams have been discussed. The theorem of corresponding relation between the number of different phases in the neighboring phase regions Φ and the dimension R₁ of their phase boundary in the phase diagrams is derived from the phase rule. The five corollaries of this theorem for isobaric phase diagrams are derived also. By applying this theorem and its corollaries, the following results can be obtained. Both the boundary rule and contact rule of phase regions in phase diagrams may be deduced. The ten empirical rules for constructing the complex ternary isobaric phase diagrams from phase diagram units may be put on a theoretical basis. The relation between neighboring phase regions and their boundary in phase diagrams of one-, two-, three- and four-components of all types and in the different horizontal and vertical sections of ternary phase diagrams may be explained without introducing any supplementary concept.     

A Calphad-compatible method to calculate liquid/liquid interfacial energies in immiscible metallic systems

For the calculation of liquid/liquid interfacial energies in monotectic metallic alloys only simplified models are known, which suppose the validity of the regular solution model for the liquid solution. In the present paper a Calphad-compatible method is developed to calculate the liquid/liquid interfacial energies in binary, ternary and multicomponent liquid monotectic alloys. The method can be considered as the extension of the Butler equation to the liquid/liquid interfaces. The first order interfacial phase transition has been predicted at a certain critical temperature, which is about 18% of the bulk critical temperature. Below this critical temperature the compositions of the two sides of the liquid/liquid interface are different, while above this critical temperature the compositions at the two sides of the interface are identical. The latter case is valid for the majority of liquid metallic systems above their monotectic temperatures. The method has been validated against the experimentally measured liquid/liquid interfacial energies in the Ga–Pb and Al–Bi systems. The method is found to be very sensitive to the correctness of the Calphad-assessment of the partial excess Gibbs energies of the components in the liquid solution. The excess Gibbs energy function of the liquid Al–Bi solution was reassessed using our recent exponential temperature dependence of the interaction energies.     

The equilibrium phase diagram of the pentaerythritol-pentaglycerine-2-amino-2methyl-1,3, propanediol (PE-PG-AMPL) system

Polyalcohols and amines have been considered as potential thermal energy storage materials, owing to their energetic solid-solid phase transitions. In this paper, we present equilibrium phase diagram of Pentaerythritol (PE)–Pentaglycerine (PG)– 2-amino-2methyl-1,3, propanediol (AMPL) ternary system that has been thermodynamically assessed using the CALPHAD method. A special class of, so called, “Plastic Crystals” have tetrahederally configured molecules with O–H⋯O or N–H⋯O layered or chained bonds in the low temperature phase, and store latent heat of transformation during solid-solid phase change in orientationally disordered crystals at higher temperatures. For example, in polyalcohols, there are O–H⋯O bond rotations around the C–C bonds that are responsible for storing large amounts of solid state phase change energy. Several binary equilibrium phase diagrams of polyalcohols, amines and combination thereof have been calculated and experimentally validated. We only know of three ternary phase diagram of these plastic crystals reported, to the best of knowledge. In the thermodynamic calculations of thePE-PG-AMPL system, we used the binary phase diagram experimental data for the optimization and calculation of excess energies. The binary systems have been optimized using regular and sub-regular solution models. The binaries as well as the ternary system have been calculated from room temperature to the liquid phase. The solution phases are modeled as substitutional solutions, in which the excess Gibbs energies are expressed by the Redlich–Kister–Muggianu polynomial. There is a very good agreement between previously reported experimental data and the calculated phase diagrams.     

A new treatment for calculating activities from phase diagrams involving two liquid or solid coexisting phases

New formulae for calculating activities and activity coefficients from phase diagram at a fixed temperature are presented for a binary system involving a two liquid or solid coexisting phases. In the new formulae. Richardson's assumption that for many systems is approximately equal to 3000, has been adopted. In terms of the new formulae, activities of components in Zn-Pb and Zn-Bi binary systems are calculated and the results show that the formulae are efficient.     

Phase diagrams and elastic properties of the Fe-Cr-Al alloys: A first-principles based study

The phase diagrams and elastic properties of the Fe-Cr-Al alloys in full-temperature and all-compositional ranges are calculated. By combining first-principles calculations and cluster variation method, binary and ternary phase diagrams are obtained. A new ternary ordered phase B32 which is different from ternary extension of binary phases appears in the ternary section around temperature of 600 K. The binary FeAl phases show an extremely high solubility for Cr, while the binary CrAl phase solid solution has a low solubility for Fe. By combining first-principles calculations and cluster expansion method, the bulk modulus, shear modulus and Poisson's ratio are calculated. The shear modulus and Poisson's ratio show a strong ordering dependency, while the ordering dependency in bulk modulus is weak. Disordered Fe-Cr alloys with a little Al solvent shows ductile property, the Al-rich corner has brittle property.     

Critical evaluation of ternary phase diagram data: Important considerations in the scrutiny of the correctness,coherence, and interpretation

Thermodynamic optimization of ternary phase diagrams via Calphad approach is a complex procedure. The success and quality of such a Calphad optimization depend on the reliability of the experimental data and on the scrutiny of the critical evaluation of the experimental datasets. With this in mind, we provide a set of recommendations that might facilitate the critical evaluation procedure and improve the quality of the thermodynamic datasets for the calculations of the phase diagrams. The particulars regarding the consistency between binary and ternary phase diagrams as well as the internal agreement between the different ternary datasets, are discussed.     

Thermodynamic modeling of the Al–Bi,Al–Sb,Mg–Al–Bi and Mg–Al–Sb systems

All available thermodynamic and phase diagram data of the binary Al–Bi and Al–Sb systems and ternary Mg–Al–Bi and Mg–Al–Sb systems were critically evaluated, and all reliable data were used simultaneously to obtain the best set of the model parameters for each ternary system. The Modified Quasichemical Model used for the liquid solution shows a high predictive capacity for the ternary systems. The ternary liquid miscibility gaps in the Mg–Al–Bi and Mg–Al–Sb systems resulting from the ordering behaviour of the liquid solutions can be well reproduced with one additional ternary parameter. Using the optimized model parameters, the experimentally unexplored portions of the Mg–Al–Bi and Mg–Al–Sb ternary phase diagrams were more reasonably predicted. All calculations were performed using the FactSage thermochemical software package.     

Multivectors: Nine components represented on a ternary diagram

Methods of graphical representation of data need to keep pace with increasingly complex data manipulation. Conventional ternary diagrams allow the presentation of three components as a point in a plane. Multivector diagrams allow plotting of up to nine components in one ternary diagram by representing compositions not as points but as vectors sharing a “triple-point”. The triple point represents the composition of three sets of three components (triads). The vector arms represent the composition within triads. Multivector diagrams can be used to illustrate individual samples or multiple samples. The techniques for computing, plotting, reading and interpreting multiple vectors follow the same rules as a conventional ternary diagram such that any of the nine components or groups of components can be compared directly or with the aid of simple projections. Although the mathematical and statistical implications of this graphical representation are not presented, it is demonstrated that the technique is useful for visual assessment of compositions and key compositional ratios. Multivector diagrams can be applied to any system where a graphical representation of up to nine components may be useful. Multivectors are applied to examples from whole rock geochemistry, whole rock modal mineralogy, mineral alteration, chemical weathering and groundwater chemistry.     

Assessment of thermodynamic properties in pure polymers

The thermodynamic properties including enthalpy, entropy and Gibbs free energy of 15 types of pure polymers were derived from experimental data of heat capacity, as a function of temperature. The calculated results are presented in both polynomial expressions and graphs. This database of pure polymers will provide basic parameters in assessments of phase diagrams in binary, ternary and higher-order polymer systems.     

Thermodynamic description of the Cu-Ge-Pb system: Experiment and modeling

Decades of scientific work dedicated to the investigation of phase diagrams gave significant benefit to industry and science. After all those years of phase diagram investigation still there is missing information about phase diagram of some ternary systems. One of those systems is Cu-Ge-Pb. It is known importance of Cu-based alloys and Ge-based alloys in electro industry. Since such combination is not tested before this work will provide information about phase diagram of ternary Cu-Ge-Pb system. In this work ternary Cu-Ge-Pb system has been tested experimentally and analytically by using Calphad model. Two isothermal sections at 600 and 400 °C and three vertical sections are experimentally tested and results were compared with calculated corresponding phase diagrams. None of the ternary compound and large solubility of third element in binary compound is not confirmed. Liquidus projection, invariant reaction and scheme of invariant reaction are presented. Scheil and Lever simulation of solid phases for Cu₈₀Ge₁₀Pb₁₀ alloy were calculated.     

Thermodynamic assessment of the Bi-alkali metal (Li,Na, K,Rb) systems using the modified quasichemical model for the liquid phase

Bi-alkali metal (Li, Na, K, Rb) binary systems have been systematically assessed based on the available phase diagrams and thermodynamic data. The modified quasichemical model, which takes short-range ordering into account, is used to describe the liquid phase. All intermetallic phases are treated as stoichiometric compounds. A set of self-consistent model parameters is obtained and the experimental data are reproduced well within experimental error limits. The enthalpy of mixing, entropy of mixing, and activity of element are calculated, showing the liquid phase exhibits maximum short-range ordering at 75 at% X (X=Li, Na, K, Rb). Some systematic variations and regularities are presented, indicating the enthalpy and entropy of mixing for the liquid phase at the maximum short-range ordering along with the enthalpy of formation and melting temperature of intermediate compound BiX₃ change with the atomic radius of alkali metals regularly.     

Thermodynamic properties of nickel-sulfur melts

A correlation is presented for the thermodynamic properties of nickel-sulfur melts with sulfur contents less than 45 atom percent in good agreement with available experimental data. The correlation is mathematically equivalent to assuming the presence of distinct chemical species in the liquid, with physical interactions between the species being described by the three-suffix Margules equations with zero ternary interactions along with the regular solution assumption. The model furnishes directly information on the saturation with solid nickel, and is apparently able to describe the properties of unstable phases over a surprisingly large range of temperature (908–1726 K). This best correlation was found with species that are present in the solid phase (Ni₃S₂,NiS), along with monoatomic nickel.