Theoretical calculation of the melting curve of Cu-Zr binary alloys

Helmholtz free energies of the dominant binary crystalline solids found in the Cu-Zr system at high temperatures close to the melting curve are calculated. Our theoretical approach combines fundamental measure density functional theory (applied to the hard-sphere reference system) and a perturbative approach to include the attractive interactions. The studied crystalline solids are Cu(fcc), ${\mathrm{Cu}}_{51}{\mathrm{Zr}}_{14}$($\beta$), CuZr(B2), ${\mathrm{CuZr}}_2$(C11b), Zr(hcp), and Zr(bcc). The calculated Helmholtz free energies of crystalline solids are in good agreement with results from molecular-dynamics (MD) simulations. Using the same perturbation approach, the liquid phase free energies are calculated as a function of composition and temperature, from which the melting curve of the entire composition range of this system can be obtained. Phase diagrams are determined in this way for two leading embedded atom method potentials, and the results are compared with experimental data. Theoretical melting temperatures are compared both with experimental values and with values obtained directly from MD simulations at several compositions.

Figure 1

Effective pair potentials for Cu-Cu, Cu-Zr, and Zr-Zr interactions calculated using the MKOSYP potential [25]. The dashed lines represent the Cu-Cu interactions in fcc Cu (black) and Zr-Zr interactions in bcc Zr (blue). Solid lines represent the pair interactions in the CuZr(B2) structure.

Figure 2

The Helmholtz free energy $F(T,\rho )$ plotted as a function of atomic number density ($\rho =N/V$) for three binary crystals in the Cu-Zr system: (a) ${\mathrm{Cu}}_{51}{\mathrm{Zr}}_{14}$, (b) CuZr, and (c) ${\mathrm{CuZr}}_2$. The minimum of each curve corresponds to the relaxed structure at that temperature. Connected blue circles represent results for the MKOSYP potential, and connected red squares represent the results for the CMS potential.

Figure 3

Comparison between the theoretically calculated Helmholtz free energy, $F$ (eV/atom), and the MD simulation. The filled symbols connected by solid lines are from our perturbation calculations using the MKOSYP potential. The open symbols connected by dashed lines refer to MD simulation results using the same EAM potential [28].

Figure 4

Contributions to the Helmholtz free energy from the HS reference system (dashed lines), the perturbative correction (solid lines), and the one-body term (dotted dashed line) determined by Eq. (15) as a function of temperature.

Figure 5

Gibbs free energy at zero pressure for crystalline CuZr (B2) (dashed line) and the left-hand side of Eq. (16) for the liquid mixture (solid lines). The crossing points of the above two lines give the coexistence temperature at the corresponding $x_{\text{Zr}}$.

Figure 6

Melting curve (liquidus) predicted from the two EAM potentials (dashed line) compared with the experiment (solid lines) [27]. The filled diamonds represent the melting temperature obtained from the MD simulations.