Thermodynamics of ordered and disordered phases in the binary Mo-Ru system

We have performed ab initio calculations of the mixing enthalpy for the Mo-Ru alloy system. Both completely random alloys on the fcc, bcc, and hcp lattices as well as ordered and partially ordered structures based on the hcp lattice and a $\sigma$ phase have been examined. Further, we have performed a ground-state search for the Ru-rich region using ab initio derived effective interactions, and find a series of structures below the tie line of the simple compounds. Using the structures from this ground-state search, we are able to make an estimation of the contribution to the total energy due to ordering effects in this system. We find unusually large deviations between calculated and experimental values of the mixing enthalpy for Ru-rich hcp alloys. Our calculations indicate, in agreement with experiment, that there are ordering trends in the system. However, even under assumption of maximal order theoretical results differ substantially from the experiment. Possible reasons for the disagreement are discussed.

Figure 1

Mixing enthalpy for random ${\mathrm{Ru}}_x{\mathrm{Mo}}_{1-x}$ alloys. Results calculated in this work by means of EMTO-CPA method for bcc (filled circles), fcc (filled squares), and hcp (filled diamonds) alloys are shown together with experimental data for Ru-rich hcp alloys from Ref. 41 (open diamonds). Theoretical results for hcp Mo-Ru alloys simulated by supercells with atoms distributed over sites of ideal underlying crystal lattice (filled triangles) and with fully relaxed atomic positions (crosses) obtained by PAW method in Ref. 8 are also shown for comparison.

Figure 2

Effective pair interactions of the Ising Hamiltonian determined for the hcp Ru-Mo alloys by the screened generalized perturbation method. The number stands for the shell to which the interactions belong. For the hcp lattice, we have two inequivalent first nearest-neighbor shells, denoted 1 and $1^{\prime }$ in the figure, respectively.

Figure 3

(Color online) Results of a ground-state search for in Ru-rich hcp region. Shown are results for ordered and partially ordered $D{O}_{19}$ (open squares) and $B19$ (open diamonds) alloys, as well as for the lowest energy structures based on 8 (filled diamonds), 16 (filled squares), and 24 (filled circles) atom supercells at each concentration. Crosses represent the energies of many different configurations possible in the large supercells (see text for more explanation).

Figure 4

16 atom long period structure found as the lowest-energy state for ${\mathrm{Mo}}_{0.50}{\mathrm{Ru}}_{0.50}$ hcp alloy. White atoms are Ru and black atoms denote Mo.

Figure 5

24 atom long period structure found as the lowest-energy state for ${\mathrm{Mo}}_{0.375}{\mathrm{Ru}}_{0.625}$. White atoms are Ru and black atoms denote Mo.

Figure 6

Calculated mixing enthalpies of completely random hcp ${\mathrm{Ru}}_x{\mathrm{Mo}}_{1-x}$ alloy (filled diamonds, solid line), partially ordered $D{O}_{19}$ (filled circles), and $B19$ (dashed line) alloys, as well as for the lowest-energy structures based on 16 (dot-dashed line) and 24 (dotted line) atom supercells at each concentration obtained as a result of the ground-state search described in Sec. 4. Calculated mixing enthalpy for the most stable $\sigma$ phase is shown by the star. Experimental results from Ref. 41 (open diamonds) are also presented. The inset magnifies the Ru-rich part of the graph.