Thermodynamic optimization of Si-Zr-N system using Calphad approach coupled with ab initio methods

Gibbs energy model parameters for Si-Zr-N system were obtained using Calphad approach coupled with ab initio calculations. The enthalpies of formation of $\alpha$ and $\beta$Si₃N₄ in Si-N system and Zr₅Si₃N end-member in Si-Zr-N system were calculated using density functional theory (DFT). The finite temperature thermodynamic properties were calculated using quasiharmonic approximation (QHA). The computed heat capacities were fitted to appropriate expressions valid down to 0 K. The ab initio thermochemical data obtained in the present work and the experimental thermochemical and constitutional data from literature were used for the thermodynamic optimization of Si-N and Si-Zr-N systems. The calculated phase equilibria and thermodynamic properties are in good agreement with the input data.     

Re-modeling of Laves phases in the Cr–Nb and Cr–Ta systems using first-principles results

Total energies of Laves phases Cr₂X, CrX₂, CrCr₂ and XX₂ (X=Nb,Ta) in all three structural forms C14, C15 and C36 have been calculated ab initio by pseudopotential VASP code with a complete relaxation of structural parameters. The calculated values were used in a two-sublattice model for re-modeling of Gibbs energies of Laves phases and subsequently for calculation of phase diagrams of Cr–Nb and Cr–Ta systems by CALPHAD method. It turns out that application of ab initio calculated values of total energy of hypothetical “end-members” in a two-sublattice model substantially simplifies the modeling and lowers the number of necessary parameters. Comparison of phase diagrams obtained by a model using first-principles results with previous empirical approach as well as relative stability of Cr₂X polytypes is presented.     

Ab initio simulations on the pure Cr lattice stability at 0K: Verification with the Fe-Cr and Ni-Cr binary systems

Significant discrepancies have been observed and discussed on the lattice stability of Cr between the predictions from the ab initio calculations and the CALPHAD approach. In the current work, we carefully examined the possible structures for pure Cr and reviewed the history back from how Kaufman originally determined the Gibbs energy of FCC-Cr in the 1970s. The reliability of Cr lattice stability derived by the CALPHAD and ab initio approaches was systematically discussed. It is concluded that the Cr lattice stability based on the CALPHAD approach has large uncertainty. Meanwhile, we cannot claim that the ab initio H_FCC-Cr is error-free as FCC-Cr is an unstable phase under ambient conditions. The present work shows that the ab initio H_FCC-Cr can be a viable scientific approach. As both approaches have their limitations, the present work propose to integrate the ab initio results into the CALPHAD platform for the development of the next generation CALPHAD database. The Fe-Cr and Ni-Cr binary systems were chosen as two case studies demonstrating the capability to adopt the ab initio Cr lattice stability directly into the current CALPHAD database framework.     

Electron density modeling of large systems using the transferable atom equivalent method

The transferable atom equivalent (TAB) modeling method is a resource-efficient alternative to routine HF/SCF ab initio calculations. Electron density representations created by TAE reconstruction are designed to allow numerous molecular properties to be quickly assessed with results similar to those obtained at the HF/6-31 + G¹ level of theory. While Hartree-Fock calculations using this basis set do not provide state-of-the-art results in many areas, they do give a good representation of molecular geometries and of most ground-state electronic properties. In contrast to traditional ab initio methods, the CPU and disk resources required for TAE reconstruction are comparable to those utilized by molecular mechanics geometry optimizations. TAE modeling involves retrieval of a set of appropriate atomic electron density representations from the TAE library, followed by a self-consistent assembly process in which each atom adjusts slightly to its new environment. Atomic electronic properties are affected by this adjustment, and can be queried separately to give atom-centered results or combined to yield molecular properties. An automated version of our RECON assembly program is under development which can be used to rapidly scan large databases for specific combinations of molecular electron density properties such as the spatial distributions of electrostatic potential fields or other electron density properties. The TAE library currently contains 64 atom types which were derived by 18-dimensional cluster analysis of the electron density properties of over 6000 integrated atoms. Good agreement with ab initio results is observed when the molecules under investigation are small enough to permit direct comparison with HF/6-31 + G¹ Gaussian92 calculations. TAE reconstructions of very large molecular systems cannot be calibrated by direct comparison with ab initio results, so validation of the method is done using piece wise comparison with smaller model compounds. A brief analysis of the binding of the immunosuppressant drug FK506 with FKBP is presented as an example of large-molecule/small-molecule interaction modeling using the TAE method.     

Gibbs energy modeling of Fe–Ta system by Calphad method assisted by experiments and ab initio calculations

In this work we report the Gibbs energy model parameters of equilibrium phases of Fe–Ta system obtained by the Calphad approach. In order to assist the Gibbs energy modeling new constitutional and enthalpy increment data were obtained by experiments. Further, the energy of formation of intermediate phases and the energy of mixing of bcc solid solution were calculated by ab initio method. These results were combined with selected experimental data from the literature in order to optimize the model parameters of the Gibbs energy functions. Calculated phase diagram and the thermochemical properties are in good agreement with the experimental results.     

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 L1₀ and L1₂ 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.     

Structural stability of intermetallic phases in the Ga-Ti system

The total energies of intermetallic compounds in the Ga-Ti system are calculated employing electronic density functional theory (DFT) using pseudopotentials constructed by the projector augmented waves (PAW) method in the generalized gradient (GGA) approximation for the exchange and correlation energy. The calculations are performed for the experimentally observed compounds at their ideal stoichiometry as well as for structures which are stable in systems of early transition metals or rare earth elements with p elements of columns IIIB, IVB, and VB. The calculated formation enthalpy of the hexagonal B8₂-GaTi₂ compound is in excellent agreement with the value obtained by direct reaction calorimetry. The composition dependence of the enthalpies of formation is slightly asymmetric, the values of the enthalpies of formation being slightly more negative in the Ga-rich side. For the stable intermetallic compounds, the calculated zero-temperature lattice parameters agree well with those obtained experimentally at ambient temperature. Furthermore, for stable phases with unit-cell-internal degree(s) of freedom, the results of ab initio calculations show good agreement when compared with data obtained by structural analysis of X-ray diffraction.     

Fitting of interatomic potentials without forces: A parallel particle swarm optimization algorithm

We present a methodology for fitting interatomic potentials to ab initio data, using the particle swarm optimization (PSO) algorithm, needing only a set of positions and energies as input. The prediction error of energies associated with the fitted parameters can be close to 1 meV/atom or lower, for reference energies having a standard deviation of about 0.5 eV/atom. We tested our method by fitting a Sutton–Chen potential for copper from ab initio data, which is able to recover structural and dynamical properties, and obtain a better agreement of the predicted melting point versus the experimental value, as compared to the prediction of the standard Sutton–Chen parameters.     

Introducing k-point parallelism into VASP

For many years ab initio electronic structure calculations based upon density functional theory have been one of the main application areas in high performance computing (HPC). Typically, the Kohn–Sham equations are solved by minimisation of the total energy functional, using a plane wave basis set for valence electrons and pseudopotentials to obviate the representation of core states. One of the best known and widely used software for performing this type of calculation is the Vienna Ab initio Simulation Package, VASP, which currently offers a parallelisation strategy based on the distribution of bands and plane wave coefficients over the machine processors. We report here an improved parallelisation strategy that also distributes the k-point sampling workload over different processors, allowing much better scalability for massively parallel computers. As a result, some difficult problems requiring large k-point sampling become tractable in current computing facilities. We showcase three important applications: dielectric function of epitaxially strained indium oxide, solution energies of tetravalent dopants in metallic VO₂, and hydrogen on graphene.     

Structural stability of intermetallic phases in the Sn–Ti system

The total energies of intermetallic compounds in the Sn–Ti system are calculated employing electronic density-functional theory (DFT) using pseudopotentials constructed by the projector augmented waves (PAW) method in the generalized gradient (GGA) approximation for the exchange and correlation energy. The calculations are performed for the experimentally observed compounds at their ideal stoichiometry as well as for structures which are stable in systems of early transition metals or rare earth elements with p-elements of columns IIIB, IVB, and VB. The calculated formation enthalpy of the hexagonal Sn₅Ti₆ compound is slightly less exothermic than the value obtained by direct reaction calorimetry. For the stable intermetallic compounds, the calculated zero-temperature lattice parameters agree well with those obtained experimentally at ambient temperature. More, for stable phases with unit cell-internal degree(s) of freedom, the results of ab initio calculations show good agreement when compared with data obtained by structural analysis of X-ray diffraction. The composition dependence of the enthalpies of formation is slightly asymmetric. The electronic densities of state of the D8₈- Sn₃Ti₅ compound have been computed; the curve shows the hybridization of Sn 5p states with Ti 3d states. The stability of the intermetallic compounds in the Ti–Sn system is due to this hybridization.