Application of CALPHAD to high pressures

The current methods of performing CALPHAD (CALculation of PHAse Diagrams) calculations of high-pressure phase equilibria often lead to spurious predictions of negative thermal expansion or negative heat capacity at high pressures. It is shown that the origin of the problem lies in an incompatibility between the temperature dependence of the widely used SGTE (Scientific Group Thermodata Europe) database and that of typical equations of state of the Mie–Grüneisen type. This inconsistency is also linked to the general problem of describing mechanical instability in CALPHAD. In the present work, a new free energy formulation is developed specifically for inclusion of pressure effects in CALPHAD methodology. It is based on an interpolation between SGTE data at low pressures and the quasiharmonic lattice model at high pressures. The new formulation is constrained to physically credible predictions of the thermophysical properties, while preserving the simplicity of the CALPHAD method. Examples are given of calculations of thermophysical properties and high-pressure phase equilibria in Al, Si, MgO, Fe and the Al–Si binary alloy system.     

Cu–Ni nanoalloy phase diagram – Prediction and experiment

The Cu–Ni nanoalloy phase diagram respecting the nanoparticle size as an extra variable was calculated by the CALPHAD method. The samples of the Cu–Ni nanoalloys were prepared by the solvothermal synthesis from metal precursors. The samples were characterized by means of dynamic light scattering (DLS), infrared spectroscopy (IR), inductively coupled plasma optical emission spectroscopy (ICP/OES), transmission electron microscopy (TEM, HRTEM), and differential scanning calorimetry (DSC). The nanoparticle size, chemical composition, and Cu–Ni nanoparticles melting temperature depression were obtained. The experimental temperatures of melting of nanoparticles were in good agreement with the theoretical CALPHAD predictions considering surface energy.     

Thermodynamic assessments of binary phase diagrams in organic and polymeric systems

The Sanchez–Lacombe (SL) model and the Flory–Huggins model were used for the calculation of binary phase diagrams in organic and polymer systems, respectively. The thermodynamic parameters of the liquid and gas phases in acetone–carbon disulfide (CS₂), butane–heptane, cyclohexane–aniline systems, and liquid phases in polystyrene–polybutadiene and polystyrene–bisphenol A poly-carbonate systems were optimized, based on the experimental data. The calculated results with various pressures are in good agreement with the experimental data. It is hoped that this method will be widely applied in the prediction of binary phase diagrams in organic and polymer systems.     

Prediction of liquid–liquid phase diagrams of aqueous salt+PEG systems using a thermodynamic model

In this study a thermodynamic model for the phase behavior of aqueous salt+polymer solutions is developed. The model is based on the solution theory of Hill, which included scaling laws for the polymer molecular mass dependence and Pitzer–Debye–Hückel theory. This model was tested for systems composed of two different molecular mass of polyethylene glycols (PEG) and five different inorganic salts. All the model parameters were determined from independent measurements. The agreement between the experimental and predicted phase diagrams by this model is good.     

Experimental investigation and thermodynamic assessment of phase equilibria in the Cr-Re-Ru ternary system

The phase equilibria of the Cr-Re-Ru ternary system at 1100 °C and 1200 °C have been experimentally investigated using electron probe microanalyzer and X-ray diffraction. The σ₁-Cr₂Re₃ and σ₂-Cr₂Ru phases with the same D8_b crystal structure do not form a continuous intermetallic compound phase at 1100 °C and 1200 °C in the Cr-Re-Ru ternary system confirmed by experimental results. According to the experimental results in this work, the Cr-Re-Ru ternary system has been assessed by means of the CALPHAD (CALculation of PHAse Diagrams) method. The calculated isothermal section at 1400 °C shows that the two independent phases σ₁-Cr₂Re₃ and σ₂-Cr₂Ru are replaced by the continuous intermetallic compound σ. The current established thermodynamic database of the Cr-Re-Ru ternary system may provide the essential information, and support for the thermodynamic assessment of multicomponent system and the development of Ni-based superalloys.     

Algorithms useful for calculating multi-component equilibria,phase diagrams and other kinds of diagrams

Phase diagrams are important in materials science. They consist of lines separating regions with different sets of stable phases and different kind of axes can be used. Many available phase diagrams are drawn directly from experimental information. However, they are closely related to the thermodynamic properties of the phases and can be calculated from thermodynamic models of the phases using the Calphad technique. This paper presents a set of algorithms to calculate equilibria and several kinds of diagrams using a very flexible set of conditions and axes. The algorithms can be applied to multi-component systems using model parameters in thermodynamic databases.     

GIBBS: isothermal-isobaric thermodynamics of solids from energy curves using a quasi-harmonic Debye model

Given the energy of a solid (E) as a function of the molecular volume (V), the gibbs program uses a quasi-harmonic Debye model to generate the Debye temperature Θ(V), obtains the non-equilibrium Gibbs function G(V;p,T), and minimizes G to derive the thermal equation of state (EOS) V(p,T) and the chemical potential G(p,T) of the corresponding phase. Other macroscopic properties are also derived as a function of p and T from standard thermodynamic relations. The program focuses in obtaining as much thermodynamical information as possible from a minimum set of (E,V) data, making it suitable to analyse the output of costly electronic structure calculations, adding thermal effects at a low computational cost. Any of three analytical EOS widely used in the literature can be fitted to the p−V(p,T) data, giving an alternative set of isothermal bulk moduli and their pressure derivatives that can be fed to the Debye model machinery.

Program summary

Title of the program:gibbsCatalogue number: ADSYProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADSYProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: Persons requesting the program must sign the standard CPC non-profit use licenseComputers on which the program has been tested: Intel Pentium, Alpha, Sun Sparc/Ultra/BladeOperating system under which the program has been tested: Unix, GNU/LinuxProgramming language used: Fortran 77Memory required to execute with typical data: 700 KBNo. of bits in a word: 32No. of processors used: 1No. of bytes in distributed program, including test data, etc.: 277 497No. of lines in distributed program, including test data, etc.: 7390Distribution format: tar gzip fileKeywords: Quasi-harmonic Debye model, equation of stateNature of physical problem: Derivation of the static and thermal equation of state, chemical potential, and thermodynamic properties of a crystal from energy-volume data only.Method of solution: A quasi-harmonic Debye model is used to obtain the vibrational Helmholtz free energy as a function of temperature at the molecular volumes of input. The non-equilibrium Gibbs energy is then minimized at any temperature T and pressure p to obtain the EOS and the chemical potential. Several standard EOS parameters can be derived by fitting analytical forms to the pressure-volume data. Finally, some thermodynamic properties are computed for each (p,T).Restrictions on the complexity of the problem: Thermal effects are assumed to be well represented by a quasi-harmonic Debye model, in which the temperature dependence of the internal parameters is embedded into the temperature dependence of the volume.Typical running time: less than 1 s (Pentium III, 800 MHz) for 25 (E,V) pairs, 10 pressure and 10 temperature values.     

Experimental investigation and thermodynamic calculation of the Cu–In–Sb phase diagram

The phase diagram of the Cu–In–Sb ternary system is of importance in predicting the interface reaction between In-based solder materials and the Cu substrate.     

Experimental investigation and thermodynamic prediction of the Ni-Pb-Sb phase diagram

The phase diagram of the ternary Ni-Pb-Sb system was investigated experimentally by differential scanning calorimetry (DSC) and scanning electron microscopy (SEM) with energy dispersive spectrometry (EDS) methods, and predicted using the calculation of phase diagrams (CALPHAD) method. The phase transition temperatures of alloys along three predicted vertical sections of the Ni-Pb-Sb ternary system, with molar ratio Ni:Sb=1:3, Ni:Pb=1, and x(Sb)=0.6, were measured by DSC. The predicted isothermal section at 700 °C was compared with the results of the SEM-EDS analysis in this work.     

Experimental study and thermodynamic assessment of the Cu–Ni–Ti system

The Cu–Ni–Ti ternary system has been systematically investigated combining experimental measurements with thermodynamic modeling. With selected equilibrated alloys, the equilibrium phase relations in the Cu–Ni–Ti system at 850 °C were obtained by means of SEM/EDS (Scanning Electron Microscopy/Energy Dispersive Spectrum), EPMA (Electron Probe Micro-Analysis) and XRD (X-ray Diffractometry). Phase transformation temperatures were measured by DSC (Differential Scanning Calorimetry) analysis in order to construct various vertical sections in the Cu–Ni–Ti system. The liquidus projection of the ternary system was determined by the identifying primary crystallization phases in the as-cast alloys and from the liquidus temperatures obtained from the DSC analyses. Based on the available data of the binary systems Cu–Ni, Cu–Ti, Ni–Ti and the ternary system Cu–Ni–Ti from the literature and the present work, thermodynamic modeling of the Cu–Ni–Ti ternary system was performed using the CALculation of PHAse Diagram (CALPHAD) approach. A new set of self-consistent thermodynamic parameters for the Cu–Ni–Ti ternary system was obtained with an overall good agreement between experimental and calculated results.     

Thermodynamic optimizations of the Nd-Sn and Sn-Tb systems

The thermodynamic optimizations of the Nd-Sn and Sn-Tb binary systems were carried out by means of the Calculation of Phase Diagram (CALPHAD) method on the basis of the available experimental data including the thermodynamic properties and phase equilibria. The Gibbs free energies of the liquid, bcc, bct, dhcp and hcp phases were described by the substitutional solution model with the Redlich-Kister equation, while all of the intermetallic compounds (Nd₅Sn₃, Nd₅Sn₄, Nd₁₁Sn₁₀, NdSn, Nd₃Sn₅, NdSn₂, Nd₃Sn₇, Nd₂Sn₅, NdSn_3, Sn₃Tb, βSn₇Tb₃, αSn₇Tb₃, Sn₂Tb, Sn₅Tb₄, SnTb₄, Sn₁₀Tb₁₁, Sn₄Tb₅ and Sn₃Tb₅) were described by the sublattice model. A set of self-consistent thermodynamic parameters of each phase in the Nd-Sn and Sn-Tb binary systems has been obtained, and the calculated results are in good agreement with the available experimental data.     

Use of chemical potential of a compound in potential phase diagrams

Potential phase diagrams, where the chemical potential of a compound is used as an axis, do not obey the usual rules for potential phase diagrams. Examples are presented in order to help the interpretation of such diagrams. The thermodynamic background of the special properties is examined.     

A thermodynamic reassessment of the Si-Sr system

The Si-Sr binary system has been thermodynamically reassessed in the present work based on the critically reviewed experimental data available in the literature, especially newly published experimental phase diagram data in the Si-rich side. The liquid phase has been modeled with both the substitutional solution model and the associate model, and two sets of self-consistent thermodynamic parameters that describe the system are thus obtained. The shortcomings of the previous assessment are removed, and a better agreement with experimental data is achieved compared with the previous assessment.     

Thermodynamic assessment of Cu–Eu and Cu–Yb system

The Cu–Eu and Cu–Yb binary systems have been assessed with CALPHAD method. Liquid, BCC and FCC phases are treated as substitutional solution phases, of which the excess Gibbs energies are modeled by Redlich–Kister polynomial functions. The binary intermetallic compounds are treated as stoichiometric phases. Thermodynamic parameters of various phases have been optimized and the calculated results are in agreement with experimental data.     

Thermodynamic reassessment of the Cu–V system supported by key experiments

The temperature of the degenerated invariant reaction in the Cu–V system was accurately determined by means of Differential Scanning Calorimetry (DSC) measurements. On the basis of the experimental data from the present work and those critically assessed from the literature, an optimal thermodynamic data set for the Cu–V system was obtained. Significant improvements have been made, compared with the previous assessments.     

Modelling and calculation of the phase diagrams of the LiF–NaF–RbF–LaF3 system

Four ternary phase diagrams of the quaternary system LiF–NaF–RbF–LaF₃ were calculated from the data of LiF–NaF, LiF–RbF, LiF–LaF₃, NaF–RbF, NaF–LaF₃ and RbF–LaF₃ binary phase diagrams using the Kohler symmetric and Kohler–Toop asymmetric approximation. Excess Gibbs parameters of all six mentioned binaries were optimized using the experimental results taken from the literature. For the LiF–RbF system our own data were used. In all cases very good agreement between the experimental data and our optimized values was achieved. Excess Gibbs functions for the liquid phases were obtained using the modified quasi-chemical method based on quadruplet interactions and the excess Gibbs function for the solid solution was calculated by a sublattice model. The quaternary eutectic was determined and a set of pseudo-ternary systems with fixed ratio of LaF₃ was calculated in order to find the optimal composition for a molten salt fuel.     

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.