Effect of mixed partial occupation of metal sites on the phase stability of γ-Cr23−xFexC6 (x = 0–3) carbides

The effect of mixed partial occupation of metal sites on the phase stability of the γ-Cr23−xFexC6 (x = 0–3) carbides is explored as function of composition and temperature. Ab initio calculations combined with statistical thermodynamics approaches reveal that the site occupation of the carbides may be incorrectly predicted when only the commonly used approach of full sublattice occupation is considered. We found that the γ-M23C6 structure can be understood as a familiar sodium chloride structure with positively charged rhombic dodecahedron (M(4a) M12(48h)) and negatively charged cubo-octahedron (M8(32f) C6(24e)) super-ion clusters, together with interstitial metal atoms at the 8c sites. The stability of the partially occupied phase can be easily rationalized on the basis of a super-ion analysis of the carbide phase. This new understanding of γ-M23C6 carbides may facilitate further development of high-chromium heat-resistant steels.

. Description of the energetically most stable structures !-Cr 23−x Fe x C 6 (x = 0-3). Wyckoff sites, atomic occupancy, and fractional coordinates are specified. Each structure is denoted in the form MMMM where sequentially reference is made to the 4a, 8c, 32f, 48h sites and M = −(Fe) when a site is occupied by Cr (partially by Fe). Subscript for Fe indicates fraction Fe atoms with the remaining fraction Cr atoms.  Figure S1. (a) Per atom volume (Å 3 ) and (b) per atom magnetic moment (µ B / atom) of !-Cr 23−x Fe x C 6 (x = 0-3) as a function of x Fe = x/23 and partial site occupation. Adopted notation for each structure is as indicated in Table S1. In panel (a) the configurations with the same Fe content are sorted in boxes in downward descending order.

1.! Details of the cluster expansion method
For our cluster expansion, we first defined an initial pool of clusters. This cluster pool consisted of all clusters in which no two sites are farther apart than 0.31 a 0 , where a 0 is the lattice parameter of the M 23 C 6 carbide. This yields as largest possible clusters five 4-body clusters. Of course, all the subclusters are considered as well, such as four single-point clusters, seven pair clusters, and eight 3-body clusters, yielding a total of 24 clusters. When the actual cluster expansion was carried out it was found that the predictive error, or leave-one-out cross validation score, was lowest when two of the 4-body terms were excluded, thus yielding a cluster expansion with 22 rather than 24 effective cluster interactions. The 24e carbon occupied sites were not included in the

2.! Thermodynamic parameters derived using phonon calculations and
Debye model.
The configurations studied, together with the thermodynamic parameters are listed in Table S2. For structures with low symmetry shown in blue italic type, the Debye temperature was estimated from a quadratic function considering the bulk modulus, cell volume, composition, and their cross terms. !
The quadratic function was obtained by fitting to the Debye temperatures of 23 structures for which a full ab initio phonon calculation was feasible, where the harmonic heat capacity was used to extract the Debye temperature. ! S8! Table S2. Properties needed for the Debye model for configurations with both full and partial occupations: Debye temperature θ D (K), equilibrium lattice parameter a 0 (Å), per atom volume V 0 (Å 3 ), pressure derivative of the volume dV/dP, bulk modulus B 0 (GPa), and pressure derivative of the bulk modulus at zero pressure B' 0 . Notation for each configuration is as indicated in Table S1. For structures shown in blue italic type, the Debye temperature was estimated from a 10-parameter quadratic function of B 0 , V 0 , composition, and their cross terms. Structures with B' 0 = 4.0 are estimated with a second-order Birch-Murnaghan equation of state. Such a second-order fit always yields B' 0 = 4.0.

Structure
Formula