Cluster Expansion Hamiltonians Research Articles

Phase stability of V–Ta alloy using cluster expansion and Monte Carlo techniques

The ground states and the solid state part of the phase diagram of V–Ta alloys are examined by combining the Density Functional Theory based total energy calculations with the cluster expansion and the Monte Carlo techniques. In agreement with experiments, our cluster expansion of V–Ta alloy, based on the bcc lattice, does not find any compound ground state. Cluster expansion of V–Ta, based on C15 Laves lattice, on the other hand, clearly establishes that V2Ta is the ground state composition. Phase equilibrium computed by Monte Carlo simulations, using these cluster expansion Hamiltonians, produces phase boundaries in good agreement with CALPHAD and experimental literature.

First-principles calculation of phase equilibrium of V-Nb, V-Ta, and Nb-Ta alloys

In this paper, we report the calculated phase diagrams of V-Nb, V-Ta, and Nb-Ta alloys computed by combining the total energies of 40–50 configurations for each system (obtained using density functional theory) with the cluster expansion and Monte Carlo techniques. For V-Nb alloys, the phase diagram computed with conventional cluster expansion shows a miscibility gap with consolute temperature T_c=1250 K. Including the constituent strain to the cluster expansion Hamiltonian does not alter the consolute temperature significantly, although it appears to influence the solubility of V- and Nb-rich alloys. The phonon contribution to the free energy lowers T_c to 950 K (about 25%). Our calculations thus predicts an appreciable miscibility gap for V-Nb alloys. For bcc V-Ta alloy, this calculation predicts a miscibility gap with T_c=1100 K. For this alloy, both the constituent strain and phonon contributions are found to be significant. The constituent strain increases the miscibility gap while the phonon entropy counteracts the effect of the constituent strain. In V-Ta alloys, an ordering transition occurs at 1583 K from bcc solid solution phase to the V_(2)Ta Laves phase due to the dominant chemical interaction associated with the relatively large electronegativity difference. Since the current cluster expansion ignores the V_(2)Ta phase, the associated chemical interaction appears to manifest in making the solid solution phase remain stable down to 1100 K. For the size-matched Nb-Ta alloys, our calculation predicts complete miscibility in agreement with experiment.

No miscibility gap in Pt–Rh binary alloys: A first-principles study

Recently, diffraction experiments and bond-order calculations gave given cause for distrust in the widely accepted phase diagram of binary Pt–Rh, suggesting it forms a solid solution below 1100K. This is in contrast to the phase diagram published by Raub in 1959. In order to clarify this situation, we use a first-principles approach to conduct an exhaustive ground-state search and set up a cluster-expansion Hamiltonian. This allows for grand-canonical Monte-Carlo simulations in order to investigate the phase behavior. We show that the miscibility gap does not in fact exist. We also present simulated diffraction patterns extracted from the cluster-expansion Hamiltonian, which are in good agreement with experimental data.

Accurate coverage-dependence incorporated into first-principles kinetic models: Catalytic NO oxidation on Pt (1 1 1)

The coverage of surface adsorbates influences both the number and types of sites available for catalytic reactions at a heterogeneous surface, but accounting for adsorbate–adsorbate interactions and understanding their implications on observed rates remain challenges for simulation. Here, we demonstrate the use of a density functional theory (DFT)-parameterized cluster expansion (CE) to incorporate accurate adsorbate–adsorbate interactions into a surface kinetic model. The distributions of adsorbates and reaction sites at a metal surface as a function of reaction conditions are obtained through Grand Canonical Monte Carlo simulations on the CE Hamiltonian. Reaction rates at those sites are obtained from the CE through a DFT-parameterized Brønsted–Evans–Polyani (BEP) relationship. The approach provides ready access both to steady-state rates and rate derivatives and further provides insight into the microscopic factors that influence observed rate behavior. We demonstrate the approach for steady-state O2 dissociation at an O-covered Pt (111) surface—a model for catalytic NO oxidation at this surface—and recover apparent activation energies and rate orders consistent with experiment.

Atomic Arrangements in AuPt/Pt(100) and AuPd/Pd(100) Surface Alloys: A Monte Carlo Study Using First Principles-Based Cluster Expansions

We have constructed model Hamiltonians for AuPt/Pt(100) and AuPd/Pd(100) surface alloys based on the cluster expansion method and density functional theory. The cluster expansion Hamiltonians were used in Monte Carlo simulations to study the equilibrium arrangements of surface atoms in these two systems for a range of compositions and temperatures. We report on and explain results from these simulations in terms of the differing interatomic interactions present in each alloy. In AuPt surface alloys, homonuclear Pt–Pt interactions are favored over heteronuclear Au–Pt interactions, whereas in AuPd the opposite is true. Accordingly, our simulations show that Pt prefers to agglomerate, whereas Pd prefers to form smaller contiguous ensembles, such as monomers and dimers. Our simulations also reveal that the AuPd surface alloy can adopt c(2 × 2) ordering at low temperatures and 50% Pd coverage and exhibits a tendency for Pd monomers to occupy sites at the second nearest-neighbor distance from one another. Finally, we compare experimental data available in the literature to our results and find them in good qualitative agreement.

Transmission electron microscopy and first-principles calculations of hydrogen ordering inβ-YH2+x

In ${\text{YH}}_{2+x}$ hydrides stoichiometric deviations are associated with various interesting phenomena such as magnetic transitions and metal-insulator transitions; ordering of hydrogen on the octahedral sites (O sites) was suggested to be responsible. Long-range order was discovered in hydrogenated Y films by electron diffraction and high-resolution transmission electron microscopy. Ordering on interstitial O sites in the fcc-based ${\text{Y}}_{\text{fcc}}{\text{H}}_{2+x}$ solid solution $(0<x<1)$ was investigated with computational tools from the Alloy Theoretic Automated Toolkit. The resulting set of 94 structure energies was fit to a cluster-expansion Hamiltonian that was used to perform a ground-state analysis for the mostly metastable fcc-based solid solution and for the more stable ${\text{Y}}_{\text{fcc}}{\text{H}}_{2+x}+{\text{Y}}_{\text{hcp}}{\text{H}}_{2+x}$ two-phase mixture. The calculated structures were tested by the observed diffraction conditions and two possible ordered structures were suggested, a ground-state triclinic $x=0.0625$ and an I-centered cubic $x=0.375$.

Cluster expansion models for Fe–Cr alloys, the prototype materials for a fusion power plant

We compare two approaches to modelling the phase stability of iron and Fe–Cr binary alloys: Cluster expansion and magnetic cluster expansion. The first, based on a cluster expansion Hamiltonian, describes the effects of configurational disorder in an alloy on its thermodynamic properties. Cluster expansion can be used for studying alloys by both equilibrium and kinetic Monte Carlo methods. The second, recently proposed, “magnetic” cluster expansion (MCE) method extends cluster expansion treatment to magnetic degrees of freedom by including magnetic moments of individual atoms as variables. MCE has a unique capability for modelling the properties of a magnetic alloy in a broad range of compositions ranging from pure ferromagnetic Fe to antiferromagnetic Cr. We describe applications of both methods to modelling various properties of candidate fusion materials.

First-principles phase diagram calculations for the HfC–TiC, ZrC–TiC, and HfC–ZrC solid solutions

We report first-principles phase diagram calculations for the binary systems HfC--TiC, TiC--ZrC, and HfC--ZrC. Formation energies for superstructures of various bulk compositions were computed with a plane-wave pseudopotential method. They in turn were used as a basis for fitting cluster expansion Hamiltonians, both with and without approximations for excess vibrational free energies. Significant miscibility gaps are predicted for the systems TiC--ZrC and HfC--TiC, with consolute temperatures in excess of 2000 K. The HfC--ZrC system is predicted to be completely miscibile down to 185 K. Reductions in consolute temperature due to excess vibrational free energy are estimated to be $\ensuremath{\sim}7%$, $\ensuremath{\sim}20%$, and $\ensuremath{\sim}0%$, for HfC--TiC, TiC--ZrC, and HfC--ZrC, respectively. Predicted miscibility gaps are symmetric for HfC--ZrC, almost symmetric for HfC--TiC and asymmetric for TiC--ZrC.

First principles phase diagram calculations for the system NaNbO3–KNbO3: Can spinodal decomposition generate relaxor ferroelectricity?

First principles based phase diagram calculations were performed for the system (1−X) NaNbO3–(X) KNbO3. Plane wave pseudopotential calculations of formation energies were used as a basis for fitting a cluster expansion Hamiltonian, and a phase diagram was calculated. The predicted phase diagram has an unusual asymmetric miscibility gap, which suggests that it might be possible to synthesize a lead-free compositionally modulated sample with bulk composition X≈0.23; such a sample is likely to exhibit relaxor characteristics.

First principles phase diagram calculations for the wurtzite-structure systems AlN–GaN, GaN–InN, and AlN–InN

First principles phase diagram calculations were performed for the wurtzite-structure quasibinary systems AlN–GaN, GaN–InN, and AlN–InN. Cluster expansion Hamiltonians that excluded, and included, excess vibrational contributions to the free energy, Fvib, were evaluated. Miscibility gaps are predicted for all three quasibinaries, with consolute points, (XC,TC), for AlN–GaN, GaN–InN, and AlN–InN equal to (0.50, 305K), (0.50, 1850K), and (0.50, 2830K) without Fvib, and (0.40, 247K), (0.50, 1620K), and (0.50, 2600K) with Fvib, respectively. In spite of the very different ionic radii of Al, Ga, and In, the GaN–InN and AlN–GaN diagrams are predicted to be approximately symmetric.

First-principles phase diagram calculations for the system NaCl–KCl: The role of excess vibrational entropy

First principles phase diagram calculations were performed for the system NaCl–KCl. Plane-wave pseudopotential calculations of formation energies were used as a basis for fitting cluster expansion Hamiltonians, both with and without an approximation for the excess vibrational entropy ( S VIB). Including S VIB dramatically improves the agreement between calculated and experimental phase diagrams: experimentally, the consolute point is { X C = 0.348, T C = 765 K} Exp; without S VIB, it is { X C = 0.46, T C ≈ 1630 K} Calc; with S VIB, it is { X C = 0.43, T C ≈ 930 K} Calc.

Unexpected Ground State Structures in Relaxor Ferroelectrics

Cluster expansion Hamiltonians derived from pseudopotential total energies were used to predict ground-state (GS) cation configurations for some A ( B 1/2 3+ B ' 5+ 1/2 ) O 3 , A ( B 1/3 2+ B ' 5+ 2/3 ) O 3 , and [ A 1/2 1+ A ' 3+ 1/2 ] BO 3 perovskites. Predicted GS structures for Pb ( Mg 1/3 Nb 2/3 ) O 3 (PMN), Pb ( Mg 1/3 Ta 2/3 ) O 3 , [ Na 1/2 Bi 1/2 ] TiO 3 , and [ K 1/2 Bi 1/2 ] TiO 3 differ from those predicted by a purely ionic model. Monte Carlo simulation of cation order-disorder phenomena in PMN predicts the transition sequence: [001] NCC ' M 1 : 1 M Disordered .

Cation ordering in some ABO3 perovskites

Nonempirical phase diagram calculations were performed for one A(B′,B′′)O3, and two [A', A'']BO3 perovskite-based systems: Pb(Sc1/2Ta1/2)O3-PbTiO3, PbTiO3-CaTiO3, and PbTiO3-SrTiO3. In addition, total energy calculations were performed for some other A(B', B'')O3 and [A', A'']BO3 systems. Cluster expansion Hamiltonians were derived from Potential Induced Breathing (PIB) total energy calculations for large supercells (up to 80 atoms), and then used as input for cluster variation method (CVM) phase diagram calculations. These calculations give general insight into the thermodynamics of solid solutions that are based on coupled substitution and into the limits of the computational approach. In addition, spherical self consistent atomic deformation (SSCAD) total energy calculations were performed for several Pb(Mg1/3Nb2/3)O3, and Ba(Zn1/3Ta2/3)O3 supercell configurations. Also presented, are the results of a neutron powder difraction study of cation ordering in a [Pb0.5469 Ca0.4531]TiO3 sample.

First-principles study of multiple order-disorder transitions in Cd2AgAu Heusler alloys.

The bcc-based Heusler alloys exhibit a series of order-disorder phase transitions as a function of temperature. The high-temperature phase is a disordered bcc solid solution, and the low-temperature phase is the Heusler structure. An intermediate ordered phase is also typically observed in real systems. A prototype cluster variation method (CVM) analysis is presented that shows that the relative stabilities of the Heusler and intermediate phases can vary continuously, depending on a fine balance between ordering tendencies in the constituent binary systems. Given these basic conclusions, a first-principles analysis of order-disorder transitions in ${\mathrm{Cd}}_{2}$AgAu was performed. A cluster expansion Hamiltonian was constructed based on a series of linearized muffin-tin orbital calculations in the atomic sphere approximation. CVM calculations were then performed in the ternary bcc tetrahedron approximation. In addition to the transition temperatures, long-range order parameters, and sublattice occupations for ${\mathrm{Cd}}_{2}$AgAu, an isoplethal section of the ternary phase diagram was also calculated. In general, agreement with experiment is excellent, given the first-principles nature of the calculation. This study clearly demonstrates the ability of first-principles statistical-mechanical calculations to treat complex ordering phenomena in ternary systems. \textcopyright{} 1996 The American Physical Society.