First-principles study on phase stability of MoSi${}_2$-NbSi${}_2$ pseudobinary alloys

The phase stability of MoSi${}_2$-NbSi${}_2$ pseudobinary alloys was examined by Monte Carlo simulation and the cluster expansion technique based on first-principles calculations. We found that formation energies of all possible atomic arrangements exhibited a positive sign, indicating that no stable intermediate phase exists between MoSi${}_2$ with C11${}_{\mathrm{b}}$ and NbSi${}_2$ with C40 structures. The C40 phase has significantly greater solubility as well as higher temperature dependence of solubility than C11${}_{\mathrm{b}}$, which agrees with previous experimental reports. Lattice vibration is found to significantly affect the solubility of both C11${}_{\mathrm{b}}$ and C40 phases, where its impact naturally increases at higher temperatures. From the analysis of Warren-Cowley short-range-order parameters, the C11${}_{\mathrm{b}}$ single phase can be interpreted as a nearly disordered state, while the C40 phase exhibits explicit deviation from the disordered state: C40 prefers Mo-Mo and Nb-Nb like-atom pairs for first-nearest-neighbor coordination, especially around equiatomic composition.

Figure 1

Selected multibody clusters for C11${}_{\mathrm{b}}$ (upper figure) and C40 (lower figure) structures. Large and small spheres are Mo (Nb) and Si sites for the C11${}_{\mathrm{b}}$ (C40) structure, and bold circles connected with bold lines denote clusters. Axes for conventional cells are described together.

Figure 2

Upper figure: Electronic contribution $E_{\mathrm{el}}$ to ECIs of multibody clusters for C11${}_{\mathrm{b}}$ and C40 structures. Lower figure: Vibrational contribution $F_{\mathrm{vib}}$ to ECIs as a function of temperature, $T$.

Figure 3

Simulated composition $x$ at $T=1400$ K on C11${}_{\mathrm{b}}$ and C40 as a function of the difference in chemical potential $\Delta \mu$. The curves for C11${}_{\mathrm{b}}$ and C40 are obtained by increasing and decreasing $\Delta \mu$ during the MC simulation, respectively.

Figure 4

Calculated phase diagram for MoSi${}_2$-NbSi${}_2$ pseudobinary alloys in the composition range of (Mo${}_{\left(1-x\right)}$Nb${}_x$)Si${}_2$ ($0\le x\le 1$). Broken and solid curves denote present calculation results without and with taking account of lattice vibrational effects. Chain curves denote theoretical prediction based on experimental thermodynamic data by Geng et al. (Ref. 22). Two-dot chain curves denote experimental study based on microstructural observation and DTA method by Wei et al. (Ref. 9). Filled squares represent experimental study of isothermal section at 800 ${}^{\circ }$C by Savitskiy et al. (Ref. 17).

Figure 5

Calculated Warren-Cowley SRO parameters for 1-NN (open circles) and 2-NN (filled circles) pairs as a function of composition $x$. Left: $T=1200$ K. Right: $T=1800$ K.