A thermodynamic analysis of the Fe-Nb-B ternary system has been carried out by estimating the unknown thermodynamic properties of various borides and solid solutions with using a first-principles method. The calculations showed that, at least in the ground state, the solid solution, in which the B atoms substitute for the Fe atoms, was more probable than that where the B atoms dissolved in the octahedral interstitial sites. Furthermore, the calculated values for the binary borides in the Fe-B and Nb-B systems were in reasonable agreement with those found by experiment. The thermodynamic properties such as formation enthalpies for the FeNbB and Nb 3 B 2 phases were also determined for ternary borides. The thermodynamic functions determined using these theoretical values, as well as the available experimental information on the phase boundaries, successfully revealed the phase equilibria in the Fe-Nb-B ternary system over the entire composition and temperature ranges.KEY WORDS: phase diagram; thermodynamic analysis; CALPHAD; first-principles calculations; borides.
835© 2008 ISIJ using first-principles calculations, and the results were introduced into our thermodynamic analysis. The calculation procedure will be explained in this section.The free energy of the various lattice configurations is often depicted using the cluster expansion method (CEM). 7) This method is summarized as follows. The total energy of a selected ordered structure having a bcc or fcc lattice is computed. Then, the total energy is used to extract the chemical interaction energy between the Fe and B atoms in various crystalline environments. This established method of extracting interaction parameters from the total energy has already been applied to a large number of alloy systems, as discussed in the literature, 8) and we followed this procedure in our study.The energy of formation, DE f form , of a stoichiometric ordered compound, f, was extracted from the total energy by subtracting the concentration-weighted average of the energy of the pure elements in the f state, as:..... (1) where c i is the concentration of element i. The term E f total denotes the total energy of the f superstructure, while the terms E total boc-Fe and E total rhombohedral-B refer to the total energy of bcc Fe and rhombohedral-type B, respectively. Furthermore, the free energy rather than energy of formation is more appropriate for applying thermodynamic properties to phase diagram calculations. The free energy of formation, DF f form , of a structure, f, is described by the following equation:........ (2) where DE f vib (T) is the lattice vibration energy and DS f vib (T) is the vibration entropy. To obtain the free energy at a finite temperature, the obtained free energy of formation is described using the effective cluster interaction, v j , for cluster j, and the cluster correlation function, x j f , as defined in Ref. The internal energy of a given phase, DF 0 , can then be calculated from the extracted effective cluster interaction energy using Eq. (4), as de...
