Tetrahedron approximation of the cluster variation method for b.c.c. alloys

Ackermann, H.; Inden, Gerhard; Kikuchi, Ryoichi

doi:10.1016/0001-6160(89)90259-9

“…Shinoda et al [131] analysed the stability of the B2 phase, connecting it to the free-energy minimum curves inside the ternary system, by means of the CVM in the irregular tetrahedron (IT) approximation [132]. The parameters for the calculations were obtained from heat of sublimation for the pure elements and alloy formation energies for the B2 phase [131].…”

Section: Cvm Calculationsmentioning

confidence: 99%

Assessment of the Fe–Ni–Al system

Eleno

¹

,

Frisk²,

Schneider

³

2006

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“…Similarly, there is a two-phase region that exists between the D0 3 and B2 phases in the compound unit model that is absent in the pairwise model. Other pairwise models simulating a system with D0 3 and B2 compounds produce a second order phase transition between the D0 3 and B2 phases [7,8], which would also violate the third law of thermodynamics, as one or both phases would have to have a non-zero solubility at 0 K. Thus, the presence of this two-phase region also seems to be an improvement made by the inclusion of compound unit energies.…”

Section: Phase Diagramsmentioning

confidence: 99%

“…Lattice models provide a convenient framework for studying the evolution and stability of alloy microstructures [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. These models typically focus on the mesoscale behavior of the system, describing the state of the alloy on a fixed lattice, with an interatomic potential that expresses the relative preference for different arrangements of species on the lattice.…”

Section: Introductionmentioning

confidence: 99%

“…The pairwise model does not produce this result for negative enthalpy of mixing alloys, instead predicting a disordered alloy state (solid solution) to be thermodynamically stable. Moreover, this limitation is inherent to the pairwise potential, and thus exists regardless of whether the equilibrium state is determined via the cluster variation method [7][8][9] or a Monte Carlo simulation [1,6].…”

Section: Introductionmentioning

confidence: 99%

“…For example, the nearest neighbor model exhibits an equiatomic compound in the 2D square lattice because at this stoichiometry the only way to attain all solute-solvent bonds is to form an ordered phase. While the above discussion mainly focuses on the nearest neighbor model to simplify the explanation, this behavior is expected to be present at absolute zero for any pairwise model with negative interaction parameters [1,[7][8][9]. The inability to capture accurate ordering tendencies limits the usefulness of such a description for designing alloys, where it is often desirable to study the precipitation of an ordered compound at off-stoichiometric compositions, and in systems where ordering has additional energetic implications beyond merely the geometric ones.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A compound unit method for incorporating ordered compounds into lattice models of alloys

Kalidindi

¹

,

Schuh

²

2016

Computational Materials Science

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a b s t r a c tLattice models can be a basic tool for alloy design, due to their ability to capture the most important thermodynamic and kinetic phenomena of a wide-range of alloys at a low computational cost. However, in order to correctly treat ordered precipitates at off-stoichiometric compositions requires multi-body potentials, and these can be challenging to calibrate to known alloy behaviors. Here we introduce a simple means of capturing the multi-body terms needed to treat ordered compounds in a lattice model based on defining ''compound units". This approach is particularly designed for, and easily calibrated in, cases where the structure and formation energy of equilibrium compounds are already known. This is accomplished by defining a compound unit that derives its energy from the formation energy of the compound as an a priori input. The method is illustrated for a binary alloy with D0 3 and B2 stable compounds.

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Atomic Ordering

Inden

¹

2013

Materials Science and Technology

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The sections in this article are Introduction Definition of Atomic Configurations Configurational Variables Point Variables Point Correlation Functions and Point Probabilities Pair Variables, Correlation Functions, and Probabilities Generalized Cluster Variables, Correlation Functions, and Probabilities Short‐Range Order ( sro ) Configurations–Long‐Range Order ( lro ) Configurations The Existence Domain and Configuration Polyhedron F . C . C . Structure, First‐Neighbor Interactions F . C . C . Structure, First and Second Neighbor Interactions B . C . C . Structure, First and Second Neighbor Interactions Ground States Pair Interactions Ground State Energies F . C . C . Structure, First‐Neighbor Interactions F . C . C . Structure, First and Second Neighbor Interactions B . C . C . Structure, First and Second Neighbor Interactions Energy Minimum at Constant Composition Canonical Energy of lro States Relevant Literature Effective Cluster Interactions ( ECIs ) Phase Equilibria at Finite Temperatures Cluster Variation Method Calculation of Phase Diagrams with the CVM Phase Diagram Calculation with the Monte Carlo Method Examples of Prototype Diagrams F . C . C . Structure, First Neighbor Interactions F . C . C . Structure, First and Second Neighbor Interactions B . C . C . Structure, First and Second Neighbor Interactions Hexagonal Lattice, Anisotropic Nearest‐Neighbor Interactions The Cluster Site Approximation ( CSA ) Application to Real Systems The Au  Ni System The Thermodynamic Factor of Ordered Phases The B . C . C . Fe  Al System The F . C . C . Ni – Al System Ternary Systems B . C . C . Fe – Ti – Al B . C . C . Ferromagnetic Fe – Co – Al H . C . P . Cd – Mg Concluding Remarks Appendix Acknowledgements

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Tetrahedron approximation of the cluster variation method for b.c.c. alloys

Cited by 78 publications

References 11 publications

Assessment of the Fe–Ni–Al system

Assessment of the Fe–Ni–Al system

A compound unit method for incorporating ordered compounds into lattice models of alloys

Atomic Ordering

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