Thermodynamic optimization of the Ni–Sn binary system

Through the CALPHAD method and based on experimental data of thermodynamic properties and phase boundaries, the phase diagram of the Ni–Sn binary system has been reassessed. The liquid and fcc_A1 (terminal rich nickel solid solution) phases were described by using a simple substitutional model, the excess Gibbs energy being formulated with a Redlich–Kister expression. The other intermediate phases (Ni₃Sn_HT, Ni₃Sn₂_HT, Ni₃Sn₂_LT, Ni₃Sn₄), were described with a several sublattice model with different formula; the Gibbs energy of the reference compounds was assumed to be linear, and the binary interaction terms on the sub-lattices to be constant. Ni₃Sn_LT was treated as a stoichiometric compound. The solubility of Ni in the terminal phase bct_A5(Sn) was neglected because it is very small. Finally a set of self-consistent thermodynamic parameters for all condensed phases in the Ni–Sn binary system was obtained, which can reproduce most of the experimental data.     

The experimental study of the Bi–Sn, Bi–Zn and Bi–Sn–Zn systems

The binary Bi–Sn was studied by means of SEM (Scanning Electron Microscopy)/EDS (Energy-Dispersive solid state Spectrometry), DTA (Differential Thermal Analysis)/DSC (Differential Scanning Calorimetry) and RT-XRD (Room Temperature X-Ray Diffraction) in order to clarify discrepancies concerning the Bi reported solubility in (Sn). It was found that (Sn) dissolves approximately 10 wt% of Bi at the eutectic temperature.

The experimental effort for the Bi–Zn system was limited to the investigation of the discrepancies concerning the solubility limit of Zn in (Bi) and the solubility of Bi in (Zn). Results indicate that the solubility of both elements in the respective solid solution is approximately 0.3 wt% at 200 C.

Three different features were studied within the Bi–Sn–Zn system. Although there are enough data to establish the liquid miscibility gap occurring in the phase diagram of binary Bi–Zn, no data could be found for the ternary. Samples belonging to the isopleths with w(Bi) 10% and w(Sn) 5%, 13% and 19% were measured by DTA/DSC. The aim was to characterize the miscibility gap in the liquid phase. Samples belonging to the isopleths with w(Sn) 40%, 58%, 77/81% and w(Zn) 12% were also measured by DTA/DSC to complement the study of Bi–Sn–Zn. Solubilities in the solid terminal solutions were determined by SEM/EDS. Samples were also analyzed by RT-XRD and HT-XRD (High Temperature X-Ray Diffraction) confirming the DTA/DSC results for solid state phase equilibria.     

Thermodynamic assessment of Fe–Ti–S ternary phase diagram

A thermodynamic analysis of the Fe-Ti-S ternary system was performed by incorporating first-principles calculations into the calculation of phase diagrams (CALPHAD) method. To evaluate the Gibbs energy, the Debye-Grüneisen model was applied for some sulfides of the Ti-S binary system. In addition, the cluster expansion and cluster variation methods were used for the solid solution phases in the Ti-S binary and (Fe,Ti)S phases. The calculated Ti-S binary phase diagram showed good agreement with the experimental results. The very low solubility of the Ti solid solution in the Ti-S system, as reported by Murray, agreed well with our calculated results. A binodal phase decomposition of the liquid phase was expected in the S-rich region. The Gibbs energy curve of (Fe,Ti)S between FeS and TiS was found to be convex downward. This is characteristic of an isomorphic solid solution, attributed to the attractive interaction between Fe and Ti in (Fe,Ti)S. The vertical phase diagram between FeS and TiS, obtained using the thermodynamic database, was in good agreement with the experimental results of Mitsui et al. The solubility products of (Fe,Ti)S have been experimentally estimated previously. The calculated solubility product agreed with the experimental value of TiS.     

Thermodynamic assessment of the CaO–P2O5–SiO2–ZnO system with special emphasis on the addition of ZnO to the Ca2SiO4–Ca3P2O8 phase

The CaO–P₂O₅–SiO₂–ZnO system including all binary and ternary sub-systems has been thermodynamically assessed using all available experimental data. Particular attention was given to the phase C2S–C3P which forms a complete solid solution with end-members α-Ca₂SiO₄ and α′-Ca₃P₂O₈. In addition, the present modelling of the phase C2S–C3P allows the inclusion of experimentally determined solubility values of zinc oxide in both end-members of the phase C2S–C3P. The mutual solubility between different crystallographic modifications of calcium and zinc phosphates is also described in this work using available experimental data. 24 phospates as stoichiometric phases have also been included in the database.     

A cluster expansion for Cu-Au alloys based on experimental data

The calculation of effective cluster interactions (ECIs) from experimental data rather than from calculated total energies is described. The procedure is illustrated for Cu-Au alloys with the enthalpy of mixing for random alloys being calculated from experimental data for the ordered alloys.

The cluster expansion formalism from which the ECIs are obtained could direct the conventional thermodynamic evaluation of phase diagrams to a physically more fundamentel viewpoint and, consequently, become an important new component to be considered in the CALPHAD-like assessments.     

Thermodynamic description of solution phases of systems Fe-Cr-Si and Fe-Ni-Si with low silicon contents and with application to stainless steels

A thermodynamic optimization has been made for the solution phases of the ternary systems Fe-Cr-Si and Fe-Ni-Si in the iron-rich corner. Ternary substitutional-solution interaction parameters were optimized for the liquid, fcc and bcc phases, and the lower-order data were taken from the earlier assessed sub-systems. The calculated results were validated with experiments of phase equilibria and component activities in liquid. The new data were then added to the existing database of multicomponent steels and calculations were carried out for certain silicon containing stainless steels. As a result, clearly improved agreement was obtained between calculations and experiments concerning the liquidus temperatures and the solid/liquid partitioning of silicon in these steels.     

Thermodynamics of solid solutions with ordering tendencies

This paper describes a new procedure for the calculation of phase diagrams for quasibinary systems that show ordering phenomena near special stoichiometric compositions. The diagrams are calculated from excess Gibbs energy functions that are modulated in the solid state at the specified stoichiometric compositions by distribution functions of the Gauss type. These ‘Gaussian’ terms consider the contribution of the composition and temperature depending ordering energy to the excess Gibbs energy of the solid solution.

As the solid state is described in this way by a single continuous Gibbs function, the thermodynamic factor for interdiffusion in such quasibinary systems can also be derived as a single continuous function yielding negative values inside the miscibility gaps of the phase diagram.     

Ferrite — M2C coherent phase equilibrium in AF1410 Steel

A Model for coherent ferrite/M₂C carbide equilibrium is applied to AF1410 steel. The contribution fron coherent elastic energy to the Gibbs free energy has been expressed in a format compatible with the Redlich-Kister-Muggianu extrapolation formula for multicomponent phases. An equilibrium calculation at the standard tempering temperature of 510C predicts a substantial deviation from stoichiometry and a measurable solubility of iron in the M₂C phase, both of which have been verified experimentally.     

An understanding of high entropy alloys from phase diagram calculations

The concept of High Entropy Alloy (HEA) is understood from the point of view of phase diagram calculation. The role of entropy of mixing on the phase stability is discussed for both ideal and non-ideal solid solution phases. The relative stability of a solid solution phase and line compounds is illustrated using hypothetical systems. Calculated binary and multicomponent phase diagrams are used to explain the phenomena observed experimentally for HEAs. The potential of using the CALPHAD (CALculation of PHAse Diagrams) approach in aiding the design of alloys with multiple key components is also discussed.     

Experimental study and thermodynamic modelling of the ZrO2–LaO1.5 system

Phase relations in the ZrO₂–LaO_1.5 system were studied experimentally in the temperature range from 1673 to 1973 K. X-ray diffraction and scanning electron microscopy were employed to obtain the structural information and the compositions of the tetragonal and pyrochlore (La₂Zr₂O₇) phases. The solubility of LaO_1.5 in the tetragonal phase was determined to be very small. The homogeneity range of the pyrochlore phase is estimated to be less than 2 mol% at 1973 K, and less than 1 mol% at 1673 K according to the present work. Based on the experimental results obtained in this work, as well as the available phase diagram and thermodynamic data in literature, a self-consistent thermodynamic assessment was carried out by using the ionic sublattice solution model.     

Thermodynamic Study of the Chemical Vapour Deposition in the Si---B---C---H---Cl System

This paper presents the results of a thermodynamic study concerning the codeposition of phases of the silicon, boron and carbon system by a classical C.V.D. technique from an initial gaseous mixture composed of methyltrichlorosilane (MTS), boron trichloride (BCl₃) and hydrogen. The thermodynamic approach developed here is specific in the sense that for the first time the non-stoichiometry of boron carbide is described from B₄C up to B₁₀C. The Gibbs' energy of the solid solution phase is assumed to follow the sublattice model.

A deposition diagram is determined as a function of the initial gas composition together with the equilibrium yields of the different gaseous and solid species. Finally, an interesting deposition domain could be chosen for future experiments of deposition on C/C, C/SiC and SiC/SiC composites.     

Predicting miscibility gaps in reciprocal liquids

There is a strong correlation between the appearance of liquid miscibility gaps in reciprocal phases and the standard Gibbs energy of the reciprocal reaction, Δ°G. The associate solution model does not predict such a correlation but the two-sublattice model does.

Physically, the critical temperature of a reciprocal miscibility gap is related to Δ°G and is also affected by the presence of short range order. The latter effect can be estimated from Δ°G using the conformal solution theory which was developed for reciprocal systems with univalent ions. In that approximation the short range order effect is described with a reciprocal parameter. The estimate of that parameter for systems with different valencies is now discussed.

The calculated phase diagram may be reasonable for low values of the reciprocal parameter, relative to Δ°G, but is not satisfactory for large values because two asymmetric miscibility gaps will appear when the critical temperature has been suppressed to of the value predicted without a reciprocal term. An alternative form of the reciprocal term is now proposed, which does not produce the splitting into two miscibility gaps.     

Predictions on the solubility and equiscale line of water content for the quaternary system (Li+Na+Cl+SO4+H2O) at 298.15 K

Pitzer ion-interaction theory is widely used to calculate the solubilities of salt–water systems. In the paper, the solubilities and equiscale lines of water contents of the reciprocal quaternary system (Li+Na+Cl+SO₄+H₂O) at 298.15 K were calculated using Pitzer theory and Harvie, Møller, Weare solubility modeling approach (HMW model). In this predictive calculation, the quasi-Newton and dichotomy methods were used to solve the system of simultaneous equations of the equilibrium expressions among solid and liquid phases, and to obtain the ionic molalities, Jänecke indices of phase diagram and the Jänecke indices of the equiscale lines of water contents in the reciprocal quaternary system. Based on the Jänecke indices, the dry-salt phase diagram and the equiscale lines of water-phase diagram were plotted. The predictive phase diagram via the theoretical calculation agrees well with the experimental phase diagram except for the phase region of double salt 2 (Li₂SO₄·Na₂SO₄, Db2). Distributions of equiscale lines of water-phase diagram on the dry-salt diagram of the reciprocal quaternary system reflect that phase areas of double salts of Db1 and Db2 bear maximal water contents. The predictive phase diagram and equiscale lines of water-phase diagram are often necessary in the technological design of chemical engineering for the utilization of inorganic salts in salt lake brines.     

A sequential method for system identification in hierarchical structure

Implicit in all of hierarchical systems theory is the idea that it is generally easier to deal with several low order systems than with one system of high order. The basic idea of hierarchical systems theory is to decompose a large dimensional system into smaller dimensioned sub-systems in such a way that the overall system objectives can be met.

This paper is concerned with the application of hierarchical system theory to the identification problem. Specifically, the equations associated with a given identification problem are recast such that they may be decomposed into infimal subproblems of system identification which can be coordinated using hierarchical systems theory.

The maximum a posteriori approach to system identification is taken. This leads to a two point boundary value problem solution which determines optimum state and parameter estimates and estimates of any unknown prior statistics. Invariant imbedding is used to resolve this two point boundary value problem such that a recursive or sequential solution to the identification problem is obtained. Several examples indicate the use of the identification algorithms.     

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.     

Experimental investigation and thermodynamic description of the Cu-Cr-Zr system

The Cu-Cr-Zr ternary system was investigated via thermodynamic modeling coupled with key experiments. The isothermal section of the Cu-Cr-Zr system at 1123 K was determined by means of optical microscopy, X-ray diffraction and electron probe microanalysis, and the phase transformation temperatures were measured by differential scanning calorimetry. The Cu-Cr sub-binary system was re-assessed with a substitutional solution model for the solution phases to ensure its compatibility in multi-component system. A set of self-consistent thermodynamic parameters for the Cu-Cr-Zr systems was obtained using the CALPHAD (CALculation of PHAse Diagram) approach, and the calculated phase diagram is in a satisfactory agreement with the present experimental results and literature information. An increase of the thermodynamic stability for the CuZr and Cr₂Zr phases due to the ternary solubility is verified by calculation.