Phase stability for monolayer boron-carbon-nitride ͑BNC͒ ͑l-BNC͒ ternary system was examined by Monte Carlo simulations and the cluster expansion technique based on first-principles calculations. All the possible atomic arrangements exhibit positive formation energies, indicating phase separation into monolayer BN and graphene. The atomic arrangements in lowest formation energy have strong preferences for B-N and C-C atoms while disfavor with B-C, C-N, B-B, and N-N bonds along the first-nearest-neighbor coordination, which have a similar tendency for cubic BNC ternary alloys predicted in our previous study. Lattice vibration significantly enhances solubility limits for l-BNC: within the framework of harmonic approximation, complete miscibility achieves at around T = 3500 K, which is below melting lines between hexagonal BN and graphite.

Although metals strengthened by alloying have been used for millennia, models to quantify solid solution strengthening (SSS) were first proposed scarcely seventy years ago. Early models could predict the strengths of only simple alloys such as dilute binaries and not those of compositionally complex alloys because of the difficulty of calculating dislocation-solute interaction energies. Recently, models and theories of SSS have been proposed to tackle complex high-entropy alloys (HEAs). Here we show that the strength at 0 K of a prototypical HEA, CrMnFeCoNi, can be scaled and predicted using the root-mean-square atomic displacement, which can be deduced from X-ray diffraction and first-principles calculations as the isotropic atomic displacement parameter, that is, the average displacements of the constituent atoms from regular lattice positions. We show that our approach can be applied successfully to rationalize SSS in FeCoNi, MnFeCoNi, MnCoNi, MnFeNi, CrCoNi, CrFeCoNi, and CrMnCoNi, which are all medium-entropy subsets of the CrMnFeCoNi HEA.

The cluster expansion technique in conjunction with first-principles calculations has been applied in Monte Carlo simulations to derive the configurational thermodynamics of the bulk and ͑111͒ surface of Pt-Rh alloys. Lattice-dynamics calculations reveal that the vibrational contribution to Pt-Rh bulk phase stability is fairly negligible. Calculated short-range-order parameter, ground state, and ordering transition temperature T c of bulk Pt 50 Rh 50 are in satisfactory agreement with experimental values in the literature. Calculated composition profiles of the ͑111͒ surface at T = 1373 K show the enrichment of Pt at the top layer and Pt depleted at the second layer for the entire composition, which is in agreement with experimental observations. At low temperatures, a significant difference is found in the temperature dependence of the layer composition profile between Pt 25 Rh 75 and Pt 50 Rh 50. While Pt composition of the Pt 25 Rh 75 subsurface shows positive temperature dependence, that of Pt 50 Rh 50 has a minimum at T ϳ 300 K. The former can be qualitatively interpreted by taking account of the on-site energy only. The latter is due to the occurrence of sublayer-confined phase transition from ͑ ͱ 3 ϫ ͱ 3͒R30°order to disorder alloys.

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The phase diagram for the cubic BNC ternary system in a heterodiamond structure was examined by Monte Carlo simulations and the cluster expansion technique based on first-principles calculations. All the atomic arrangements exhibit positive formation energies, indicating phase separation into cubic BN ͑c-BN͒ and diamond. These arrangements show a strong preference for B-N and CC bonds and disfavor B-C, C-N, B-B, and N-N bonds along the first nearest neighbor coordination. This can be naturally attributed to the oversaturation and undersaturation of the number of electrons for respective first nearest neighbor bonds. Firstprinciples-based lattice-dynamics calculations reveal that the formation of a solid solution between c-BN and diamond decreases vibrational free energy, resulting in a significant enhancement of the solubility for both c-BN and diamond-rich phases. Complete miscibility is achieved over T = 4500 K, which is higher than the melting points of both diamond and c-BN.

Ground-state structures of six II-III spinel oxides are predicted by combining the cluster expansion method and first principles calculations. The ground states of MgGa 2 O 4 and MgIn 2 O 4 are found to be inverse spinels with a tetragonal lattice, whereas those of MgAl 2 O 4 , ZnAl 2 O 4 , ZnGa 2 O 4 , and ZnIn 2 O 4 are normal spinels with a cubic lattice. Order-disorder transition behaviors are examined using Monte Carlo simulations. The orderdisorder transition to exchange octahedral and tetrahedral cations takes place as commonly accepted. In inverse spinels, a new kind of transition to exchange II and III cations in octahedral sites can be recognized, which has not been reported experimentally. Their transition temperatures are evaluated.

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