Ab initio electronic structure calculations may provide valuable input to the CALPHAD method, especially when complex phases are present in the phase diagram. Our approach to modelling thermodynamic properties of alloys combines the CALPHAD treatment, which requires the knowledge of the dependence of Gibbs energy of particular structures on the temperature, pressure and composition, and ab initio electronic structure calculations, which yield correct total energies of individual phases. As an example, we model thermodynamic properties of the Co-Mo and Fe-Mo systems containing the sigma phase. It is impossible to measure the Gibbs energy of the sigma phase of pure constituents, therefore we take it from the ab initio electronic structure calculations based on the density functional theory (DFT) and employing the full crystal potential. We utilize the full-potential linear augmented plane wave (FLAPW) method incorporated in the WIEN97 code [1] together with the general gradient approximation (GGA) [2] for the exchange-correlation energy term. This treatment gives us the possibility to evaluate reliable total energy differences between the structures exhibiting different crystal symmetry, i.e. between the standard element reference (SER) states and sigma phase of all included elements. These energy differences are calculated at the equilibrium volumes, and, therefore, we avoid the uncertainty connected with the use of experimental volume of binary sigma phases for total energy calculation of hypothetical (i.e. unstable) sigma phase structure of pure constituents.
The calculated total energy differences, called also lattice stability differences, represent one of the most important parts of the Gibbs energy of the sigma phase in the two-sublattice model [3]. This model also includes the entropy of transformation of SER to the sigma phase and the part of Gibbs energy representing non-ideal mixing, which both has to be adjusted to phase equilibrium data. Using this model we can calculate the phase diagram of the chosen systems, which can be compared with the phase diagram calculated by means of the three-sublattice model [3, 4] and with the experimental data. We can also predict the behaviour of the Gibbs energy and the enthalpy of the sigma phase in the whole composition range. The enthalpy of formation of sigma phase found on the basis of the phase equilibria calculations at higher temperature can be compared with the energy of formation calculated by means of the LMTO-ASA [5, 6] and FLAPW methods. Both calculations provide positive values of enthalpy of formation with respect to the SER state of pure components.
This research was supported by the Grant Agency of the Czech Republic (Projects Nos. 106/03/P002 and 106/02/0877). The use of computer facilities of the MetaCenter of Masaryk University, Brno, is acknowledged; phase equilibria were calculated by means of Thermocalc code.
1 P. Blaha, K. Schwarz, J. Luitz, WIEN97, Vienna University of Technology 1997 (improved and updated Unix version of the original copyrighted WIEN code, which was published by P. Blaha, K. Schwarz, P. Sorantin and S.B. Trickey in Comput. Phys. Commun. 59, 399 (1990)).
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