Lattice Vibrations Change the Solid Solubility of an Alloy at High Temperatures

We develop a method to accurately and efficiently determine the vibrational free energy as a function of temperature and volume for substitutional alloys from first principles. Taking ${\mathrm{Ti}}_{1-x}{\mathrm{Al}}_x\mathrm{N}$ alloy as a model system, we calculate the isostructural phase diagram by finding the global minimum of the free energy corresponding to the true equilibrium state of the system. We demonstrate that the vibrational contribution including anharmonicity and temperature dependence of the mixing enthalpy have a decisive impact on the calculated phase diagram of a ${\mathrm{Ti}}_{1-x}{\mathrm{Al}}_x\mathrm{N}$ alloy, lowering the maximum temperature for the miscibility gap from 6560 to 2860 K. Our local chemical composition measurements on thermally aged ${\mathrm{Ti}}_{0.5}{\mathrm{Al}}_{0.5}\mathrm{N}$ alloys agree with the calculated phase diagram.

Figure 1

Comparison of harmonic and anharmonic vibrational Gibbs free energies as a function of temperature, at zero pressure. Free energies were calculated in the harmonic approximation (HA) with symmetry-imposed force constants (SIFC) via the TDEP (SIFC HA) and small displacement (SMD HA) methods. Including thermal expansion as the only anharmonic effect results in the quasiharmonic approximation curve (SIFC QHA). The free energy including intrinsic anharmonic effects via the terms $\mathrm{\Delta }{U}_0$ and $g(\omega ,T)$ from Eq. (3) (SIFC TDEP) is compared with a single thermodynamic integration data point (TI) as a benchmark. The differences between these methods are summarized in the Supplemental Material [27].

Figure 2

Calculated phase diagram for $B1\text{−}{\mathrm{Ti}}_{1-x}{\mathrm{Al}}_x\mathrm{N}$. Comparing the dotted and solid lines shows the effect of including the anharmonic vibrational entropy for binodal using the SIFC TDEP method. The dash-dotted lines correspond to the spinodal metastable region including the anharmonic contribution and in the solid solution region to the spinodal line without the vibrational contribution. The binodal was also computed in the quasiharmonic approximation using the TDEP method (SIFC QHA) to demonstrate the effect of including thermal expansion but omitting temperature-induced intrinsic anharmonicity on the phase diagram.

Figure 3

Two-dimensional contour plots extracted from atom probe tomography data for different annealing times (a) 2 h, (b) 4 h, (c) 12 h and (d) 100 h.

Figure 4

The limits of the concentration of metal sublattices as a function of aging time.