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For the first time, the formation enthalpies at 0 K of every ordered configuration of a ternary σ-phase, i.e. 5³ = 243 configurations, have been calculated using the electronic density functional theory. The Cr–Mo–Re system has been chosen for the present investigation since the two binary Cr–Re and Mo–Re σ-phase are known to show opposite Re site preference: high coordination number sites in Cr–Re and low coordination number sites in Mo–Re.

The ternary Cr–Mo–Re σ-phase diagram has been computed at 0 K. It presents at this temperature several distinct single-phase regions separated by a large number of miscibility gaps. Finite temperature properties have been calculated using only the configurational entropy, in the Bragg–Williams approximation. Only above 800 K, the Gibbs energy surface becomes convex. The occupancies of the inequivalent sites have been computed as a function of composition at several temperatures. Re site preference is shown to change progressively in the ternary field when passing from Cr–Re to Mo–Re binary borders.