Four-sublattice Model Research Articles

A thermodynamic description of the Al–Co–Ni system and site occupancy in Co + AlNi3 composite binder phase

Phase equilibria of the Al–Co–Ni system is extremely useful for understanding phase behavior of AlNi3-based L12 phases, regarding its attractive properties, such as high-temperature oxidation resistance and corrosion resistance, high hardness and strength. In the present work, all the experimental data about the phase equilibria and thermodynamic properties of the Al–Co–Ni system were critically evaluated, and a thermodynamic description for this system over the whole composition and temperature ranges was obtained. Most importantly, the ordered/disordered transition between fcc_A1 and L12 phases was described using a four-sublattice (4SL) model. Based on the crystallographic and experimental data, three stable ternary intermetallic phases were described using sublattice model. Two of them were treated as semi-stoichiometric compounds with homogeneity ranges for Co and Ni, and the other one was treated as a stoichiometric compound. The other phases were described by using substitutional solution model. The present work provides a more complete and accurate thermodynamic description compared with previous ones. A reasonable agreement was obtained between calculations and experimental data in the ternary system. The occupancy behavior of Co in AlNi3 is thermodynamically predicted, and the computed occupancy shows a much better fit to the experimental data than previous first-principles calculations. In addition, the composition range of the Co + AlNi3 two-phase region, concerning a new type of composite binder for cemented carbides, was found to shift to the Co-rich side with the decrease of the temperature.

Statistical theory of thermodynamic and dynamic properties of the RbHSO_{4} ferroelectrics

Within the modified four-sublattice model of RbHSO$_{4}$ with taking into account the piezoelectric coupling to the strains $\varepsilon_i$, $\varepsilon_4$, $\varepsilon_5$, and $\varepsilon_6$, the polarization components, static and dynamic dielectric permittivity of clamped and free crystal are calculated in the mean field approximation. At the proper choice of the values of the theory parameters, a satisfactory quantitative description of the available experimental data is obtained.

Ordering in ternary BCC alloys applied to the Al–Fe–Mn system

TWIP (TWinning Induced Plasticity) steels are attracting a lot of attention due to their combination of strength and ductility. In a previous work [1] (B. Lindahl, M. Selleby, Calphad 43 (2013) 86–93) a thermodynamic assessment of the Al–Fe–Mn system, which forms the basis of TWIP steels, was presented. The previous assessment treated the A2/B2 order-disorder transformation in the bcc phase using a two-sublattice model. In the present work a four-sublattice model has been used in order to also be able to describe the transition into the ordered D03 compound that occurs at lower temperatures. pair interaction energies for the Fe–Mn system are evaluated which prove crucial to the extrapolations into the Al–Fe–Mn system. Along with this various aspects of modeling chemical ordering using the Calphad approach are discussed. Equations for determining the ternary compound energies from binary pair interactions energies are presented and equations for determining the parameter values from the ordered parameters are derived.

Experimental investigation and thermodynamic modeling of the V–Ga system

Phase equilibria and thermodynamic properties of the V–Ga system were investigated through experiments and thermodynamic modeling. In the experimental part, eight alloys with different phase constituents were prepared by arc melting. The annealed alloys were analyzed by X-ray diffraction, scanning electron microscopy, electron probe microanalysis and differential scanning calorimetry. The homogeneity ranges of V3Ga and V6Ga5 at 873K and 1173K were determined. Six invariant reactions were confirmed and the reaction temperatures were measured. Based on the experimental phase diagram and thermodynamic data from the present work as well as those from literature, the V–Ga system was assessed by means of CALPHAD approach. The intermetallic compound V6Ga7 was described with a four-sublattice model (V)2(V)3(V, Ga)2(V, Ga)6, while V6Ga5 and V3Ga were described with a two-sublattice model according to their crystal structures and solubility ranges. A set of self-consistent thermodynamic parameters were obtained, and all calculations are in satisfactory agreement with the experimental observations.

Influence of External Pressures of Different Symmetry on the Phase Transitions and Thermodynamic of the RbHSO4 Ferroelectrics

We use the four-sublattice model of the RbHSO4 crystal modified by taking into account the piezoelectric coupling to the strains , . Within the mean field approximation we calculate the components of the polarization vector and static dielectric permittivity tensor of mechanically clamped and free crystals, as well as their piezoelectric characteristics. We study the influence of hydrostatic pressure and uniaxial stresses on the phase transition and physical characteristics of the crystal. A good quantitative description of the experimental data is obtained.

Inelastic neutron scattering studies ofYFeO3

Spin waves in the the rare earth orthorferrite YFeO$_3$ have been studied by inelastic neutron scattering and analyzed with a full four-sublattice model including contributions from both the weak ferromagnetic and hidden antiferromagnetic orders. Antiferromagnetic (AFM) exchange interactions of $J_1 = -4.23 \pm 0.08$ (nearest-neighbors only) or $J_1 = -4.77 \pm 0.08$ meV and $J_2 = -0.21 \pm 0.04$ meV lead to excellent fits for most branches at both low and high energies. An additional branch associated with the hidden antiferromagnetic order was observed. This work paves the way for studies of other materials in this class containing spin reorientation transitions and magnetic rare earth ions.

Thermodynamic modeling of the Cr–Ni–Ti system using a four-sublattice model for ordered/disordered bcc phases

A thermodynamic assessment of the ternary Cr–Ni–Ti system together with a refinement for the thermodynamic description of the binary Ni–Ti sub-system is performed following the CALPHAD method and using the compound energy formalism (CEF). The ordered/disordered transition between disordered Bcc_A2 and ordered Bcc_B2 phases is described using a four-sublattice (4SL) model. An optimal set of thermodynamic parameters for the Cr–Ni–Ti system is obtained by considering the experimental data from the literature. Comparisons between the calculated and measured phase diagrams indicate that almost all of the reliable experimental information is satisfactorily accounted for by the present modeling. The complete liquidus projection and reaction scheme of the Cr–Ni–Ti system are also presented.

Thermodynamic assessment of the Cu–Fe–Ni system

• The thermodynamic description of the Cu–Fe–Ni system has been updated. • The new experimental data have been used to refine thermodynamic model of the system. • The four-sublattice model has been adopted to predict the equilibria involving the ordered L1 2 phase. • A significant improvement in comparison with the previous assessments has been achieved. • The liquidus and solidus projections have been presented. The thermodynamic description of the Cu–Fe–Ni system has been updated considering the newly available experimental data, as well as compatibility of the present modeling with those used for the Cu and Fe systems. All of the experimental data available in the literature have been critically reviewed, and the inconsistent information has been excluded. The thermodynamic parameters have been evaluated in order to properly describe the thermodynamic properties of the liquid phase and miscibility gap in the solid state. A significant improvement in comparison with the previous thermodynamic descriptions has been achieved. Additionally, for the ordered L1 2 phase the four-sublattice model has been adopted to predict the ternary phase equilibria involving this phase. A set of thermodynamic parameters for the phases is given.

Spin configurations of the four-sublattice model on rectangular lattice

The spin configurations of the four-sublattice model with seven exchange interaction parameters (four of them are taken as direct-exchange interactions, the other three as super-exchange interactions) on a two-dimensional rectangular lattice have been investigated using a matrix method. For the four sublattices, using the two sets of the exchange parameters, we obtain collinear and non-collinear spin configurations for the given propagation vectors in the ground and the first excited states. When k=0, spin configuration is collinear ferromagnetic mode in the ground state and collinear antiferromagnetic mode in the first excited state. When k=[0.5, 0.5], spin configurations are non-collinear, that is, canted structures.

The Cu–Sn phase diagram part II: New thermodynamic assessment

A thermodynamic description for the Cu–Sn system was developed using the CALPHAD approach taking into account all available literature data as well as the experimental results presented in Part I of this work. The higher order transformation from the disordered A2 phase to the ordered D03 phase has been described using the (Cu,Sn)0.25(Cu,Sn)0.25(Cu,Sn)0.25(Cu,Sn)0.25 four-sublattice model for the first time. The calculated A2/D03 phase boundary is in good agreement with literature data. Additionally, the enthalpy of mixing of the liquid phase and the activity of Cu and Sn in the liquid alloys are well reproduced and enthalpies of formation of solid alloys and the activity of Sn in the (Cu) solid solution are calculated.

A thermodynamic assessment of the Cd–Mg system

On the basis of the experimental thermochemical and phase equilibrium data, the computational thermodynamic description of the Cd–Mg binary system is performed using CALPHAD technique. The solution phases, liquid and disordered Hcp_A3, are described as a substitutional model, and the excess Gibbs energies formulated with Redlich–Kister expression. The ordered phases, Cd3Mg, CdMg and CdMg3, are treated as the AB3 phase and modeled using four-sublattices. The order–disorder transformation between the AB3 and the Hcp_A3 phases is considered, so that the substitutional model for the Hcp_A3 disordered phase is extended into the four-sublattice model for the AB3 ordered phase. The thermochemical properties as well as the phase diagram of the Cd–Mg binary system are predicted. A set of self-consistent thermodynamic parameters has been obtained. The consistency between the calculated results and the reported data of both the phase equilibria and the thermochemical properties has been achieved.

Experimental Determination and Thermodynamic Assessment of Phase Equilibria in the Co–Mo System

Phase equilibria of the solid phases including the magnetic and martensitic transformation temperatures in the Co­Mo system were investigated using two-phase alloys, the diffusion couple technique, differential scanning calorimetry, and vibrating sample magnetometry. Furthermore, ab initio calculations of D019-Co3Mo and several fcc-base ordered structures, including metastable compounds, were carried out to estimate the formation energy. Based on these results, a thermodynamic assessment using the CALPHAD method was performed. A foursublattice model was used for the fcc-base phase to describe the order­disorder phase transformation. For the ® phase, both a three and a foursublattice model were applied. The set of thermodynamic values describing the Gibbs energy of the Co­Mo system reproduces the experimental phase diagram well. The four-sublattice model for the ® phase reproduces the site fractions as well as the phase boundaries better than the threesublattice model. The calculated metastable fcc-base phase diagram considering chemical and magnetic ordering is also reasonable. This is important for estimating the phase stability of the L12 phase in Co-base £/£A superalloys. [doi:10.2320/matertrans.M2012149]

Thermodynamic modeling of the LiCoO 2–CoO 2 pseudo-binary system

The thermodynamic modeling of the LiCoO 2–CoO 2 pseudo-binary system, a positive electrode material of Li-ion batteries, was performed using the CALPHAD technique. The O3-LiCoO 2 and the O1-CoO 2 phases were described using the four-sublattice model with the formula (Li,V a) 1/2(Li,V a) 1/2(Co) 1(O) 2, and the three-sublattice model with the formula (Li,V a) 1(Co) 1(O) 2. The H1_3 hybrid phase was treated as a non-stoichiometric compound. The thermodynamic quantities, such as the phase equilibria, formation enthalpies and cell voltage (vs. Li/Li +), were in agreement with data reported in the literature.

Ferroelastic phase transitions induced by the orbital and rotation ordering in perovskites

The phases formed in cubic crystals upon ferrodistortion and antiferrodistortion orientational ordering of anion polyhedra distorted as a result of the Jahn-Teller effect are established in terms of the statistical model in the mean-field approximation. For the four-sublattice three-minimum model, there are four types of ordered phases, and one of them is the intrinsic ferroelastic (ferrodistortion) phase. The conditions providing the appearance of ordering of a particular type are determined. The effect of orbital ordering on the rotational structural distortions is discussed. The conditions of the formation of ordered phases (D4h5, D4h18) that are most frequently encountered in perovskites are found within the eight-sublattice three-minimum model. It turns out that, under certain conditions, these phases can coexist in the same crystal and transform into each other. The variants of their mutual transformations are considered.

Thermodynamic assessments of the Ni–Pt and Al–Ni–Pt systems

The Ni–Pt system is assessed using the CALPHAD method. The four fcc-based phases, i.e. disordered solid solution phase, Ni 3Pt–L1 2, NiPt–L1 0 and NiPt 3–L1 2, are described by a four-sublattice model. The calculated thermodynamic properties and order/disorder phase transformations are in good agreement with the experimental data. In order to facilitate the assessment, first-principles pseudopotential calculations are also performed to calculate the enthalpy of formation at 0 K, and comparison with the assessed values is discussed. By combining the assessments of Al–Ni and Al–Pt, the Al–Ni–Pt ternary system is assessed within a narrow temperature range, focusing on the fcc-based phases and their phase equilibria with B2 phase.

An assessment of the entire Al–Fe system including D0 3 ordering

The Al–Fe system is important for many alloys and the new interest in iron aluminides makes it necessary to improve the modeling of the different ordered forms on the body-centered cubic lattice in this system. This has now been done using a four-sublattice model based on the compound energy formalism, which can describe disordered A2 and the B2, D 0 3 and B32 ordering. The chemical and ferromagnetic ordering transitions can be both first and second-order and they have a strong interaction. Almost all available experimental and theoretical data for all phases in the system have been fitted within estimated uncertainties.

The effect of transverse electric fields on dielectric, piezoelectric, elastic and thermal properties of the Rochelle salt NaKC_{4}H_{4}O_{6} · 4H_{2}O

Modied four-sublattice model for Rochelle salt by taking into account piezoelectric interactions with shear strain "4, "5 and "6 is proposed. Components of polarization vector and static dielectric permittivity tensor for both mechanically clamped and free crystals, their piezoelectric characteristics and elastic modules are derived in the mean eld approximation. A comprehensive study of transverse eld effect on phase transition temperatures, dielectric and elastic properties of Rochelle salt has been performed for the rst time.

Pseudospin model of above-barrier states in a three-component multiple-sublattice crystal

A model of a structurally disordered crystal with three equivalent states per site is studied. An effective pseudospin model Hamiltonian which takes account of above-barrier motion is constructed. An expression is obtained for the nonequilibrium free energy for one- and four-sublattice ordering. The state of metastable disorder for a one-sublattice structure and an equilibrium phase transition for a four-sublattice model are studied. The results are used to interpret Mössbauer studies of the Jahn–Teller crystal Cu(H2O)6⋅SiF6. A theoretical explanation is given for the statics-dynamics transition, observed in the disordered phase of this crystal as the structural phase transition point is approached.

Combined ab initio/CALPHAD modeling of fcc-based phase-equilibria in the Ir–Nb system

A preliminary thermodynamic assessment of the Ir–Nb system, one of the key binary systems of the Ir-based refractory superalloys, has been performed by combining ab initio calculations and the CALPHAD (CALculation of PHAse Diagrams) technique. The ground-state formation enthalpies have been calculated by the full-potential linearized augmented plane wave method. The free energies at finite temperatures have been estimated using the cluster variation method, where the effective cluster interaction energies have been extracted from the formation enthalpies by the cluster expansion method. The liquid and A1 phases are modeled as substitutional solutions. The L1 0 and L1 2 phases are described using the four-sublattice model with the formula (Ir,Nb) 1/4(Ir,Nb) 1/4(Ir,Nb) 1/4(Ir,Nb) 1/4, while other solid phases are not considered in the present assessment. The obtained parameter set reproduces well the characteristic features of the experimental phase diagram and thermodynamic quantities.

Phase-field simulation with the CALPHAD method for the microstructure evolution of multi-component Ni-base superalloys

In order to simulate multi-component diffusion in a Ni-base superalloy using the phase-field method, the Gibbs free energy of the γ and γ ′ phases in the quaternary Ni–Al–Re–Ta superalloy system was linked to the CALPHAD method using a four-sublattice model. The free energy curve and phase equilibrium compositions in this work were in good agreement with the results of the Thermo-calc calculation, indicating that the model was successfully linked to the CALPHAD method. A one-dimensional phase-field simulation showed a pileup of Re at the γ / γ ′ interface during the growth of precipitates as well as a spike of the Re concentration in a γ ′ phase, accompanied by precipitate coalescence.