Liquid gallium-lead mixture spinodal, binodal, and excess thermodynamic properties

A statistical mechanical based theory is developed for incorporating the effect of the long-range forces in Ga-Pb alloy. In particular, the simplified random-phase approximation is employed in conjunction with the Grosdidier et al. [Phys. Rev. B 72, 024207 (2005)] model for GaGa and PbPb interactions while a suitable nonadditive pair potential is introduced between unlike atoms. We present analytical expressions for the equation of state and for the concentration fluctuations $S_{CC}\left(0\right)$, and then the role of the nonadditivity parameter is established by consulting the empirical critical parameters of the spinodal curve. It becomes possible to deduce the behavior of different thermodynamic functions such as Gibbs energy of mixing, excess Gibbs energy, isobaric heat capacity, and $S_{CC}\left(0\right)$ in the vicinity of the liquid-liquid critical point and under extremely high pressure. The impact of temperature and pressure on segregation, and compound-forming tendencies is also investigated. Moreover, the immiscibility gap of the alloy is calculated and compared with the empirical results. The results suggest that: (i) the nonadditivity of the potential tails plays a dominant role in the determination of the spinodal curve, (ii) $S_{CC}\left(0\right)$ is a very sensitive function in triggering the chemical short-range order, and (iii) the segregation or phase separation in Ga-Pb alloy is an outcome of the temperature dependence of the energy mismatch parameter. In conclusion, the present equation of state is sufficiently accurate that it provides a binodal curve, an excess Gibbs energy of mixing and isobaric heat capacity in quantitatively good agreement with the empirical results.

Figure 1

Temperature dependence of size ratio ${\sigma }_r={\sigma }_{22}/{\sigma }_{11}$, energy ratio ${\epsilon }_r={\epsilon }_{22}/{\epsilon }_{11}$, and range ratio ${\lambda }_r={\lambda }_{22}/{\lambda }_{11}$ of the SS-potential parameters.

Figure 2

The temperature dependence of the ordering potential, depicted from Eqs. (19, 18, 16).

Figure 3

Impact of the pressure and the nonadditivity parameter on the stability function, $S_{CC}{\left(0\right)}^{-1}$, at critical gallium concentration of $C_{1c}^{\text{Ga}}=0.557$ and critical temperature of $T_c=879.15\text{K}$ (Ref. 35).

Figure 4

Comparison of the calculated spinodal curves, the present work (full line) with theoretical calculations (Ref. 32) (dashed line), and the empirical binodal curve (Ref. 35) (circles).

Figure 5

Effect of temperature on the concentration fluctuation in the long-wavelength limit $\left[S_{CC}^{\ast }\left(0\right)=S_{CC}\left(0\right)/S_{CC}^{id}\left(0\right)\right]$ for Ga-Pb mixture as a function of gallium concentration at fixed pressure $P_c=27.6644\text{kbars}$.

Figure 6

The impact of pressure on $S_{CC}^{\ast }\left(0\right)$ for ${\text{Ga}}_{0.557}{\text{-Pb}}_{0.443}$ alloy at several isotherms.

Figure 7

Concentration fluctuations $S_{CC}^{\ast }\left(0\right)$ versus temperature for ${\text{Ga}}_{0.557}{\text{-Pb}}_{0.443}$ alloy at (a) low pressures, $P\le P_c\left(=27.6644\text{kbars}\right)$, and (b) high pressures, $P\ge P_c$.

Figure 8

Gibbs free energy of mixing, $G_{\text{mix}}$, as a function of Ga concentration for different isotherms.

Figure 9

Comparison of the calculated binodal curves using the present EOS (full line) with the empirical binodal curve (Ref. 35) (solid squares).

Figure 10

Comparison between the calculated excess Gibbs free energy of mixing [Eq. (38)] (solid line) and the experimental data (dashed line) (Ref. 6) at four isotherms.

Figure 11

Calculated molar heat capacity at constant pressure [Eq. (39)] (solid line) compared to the empirical values (Ref. 5) (solid circles) at different isotherms.