Cluster Variation Method Research Articles

Impact of chain fragmentation on charge transfer scenario and two-plateaus-like behavior of [formula omitted] in Y Ba 2Cu 3O 6+ x

A model of charge transfer mechanism from CuO chains to CuO 2 planes has been proposed to account for doping of the planes, assuming that only chains containing more than three oxygen atoms can contribute to hole transfer. Only chains with l ≥ 4 are assumed to have transferred a certain fraction, approximately 40%, of the holes created by oxygen added to the chain beyond the first three oxygen atoms. Using the so obtained x dependence of doping, p ( x ) , at constant (room) temperature and utilizing empirical parabolic phase relation T c ( p ) ( T c ( p ) = T c , max [ 1 − 82.6 ( p ( x ) − 0.16 ) 2 ] ), the T c versus x dependence is found to have two clearly distinguished plateaus at 60 and 90 K, remarkably fitting to experimental T c ( x ) . The effect of statistics of CuO chain fragmentation has been included by applying cluster variation method to two dimensional asymmetric next-nearest-neighbor Ising model that is employed to describe oxygen-chain ordering in basal planes. The obtained results indicate that plateaus, coinciding with T c ′ ( x ) = 0 , emerge either when p ′ ( x ) = 0 ( p ( x ) ≈ const ), in the region of OII phase formation (the 60 K plateau), or when p = 0.16 , representing the optimal doping at x ≈ 0.91 in OI phase (the 90 K plateau).

Theoretical Investigation of Phase Equilibria for Metal-Hydrogen Alloy

Cluster variation method (CVM) was applied to calculate phase equilibria of metal-hydrogen systems. Two subjects are introduced in the present report. One is the summary of previous studies on the Pd-H system, and it is demonstrated that a single CVM free energy formula can systematically derive information of phase equilibria, intrinsic stability, and short range order diffuse intensities. The second subject is the theoretical calculations of superabundant vacancy (SAV) formation. Within the square approximation of the CVM, it is shown that abundant vacancies are introduced with the absorption of hydrogen when the interaction between vacancy and hydrogen is considered.

Role of relativistic effects in the electronic structure and chemical bond of lead hexacyanoferrate

The electronic structure of a large fragment of the Pb2Fe(CN)6 crystal lattice with the trigonal structure is studied in the framework of a fully relativistic discrete variational cluster method. On the basis of comparing the results obtained with the data of previous non-relativistic calculations we have considered the relativistic effects on the electronic energy spectrum, the charge density distribution, and Pb-N, N-C, and C-Fe chemical bonding.

Phase stabilities of Pd-based alloys for membranes for hydrogen gas separation: A statistical thermodynamics approach

A statistical thermodynamics approach based on the cluster variation method (CVM) is put forward for designing new Pd-based membrane materials. The tetrahedron approximation of CVM coupled with effective pair potentials is used to describe the phase boundaries between α- and β-phases in Pd-alloys–hydrogen systems. The model takes into account the volume changes during α–β transitions, and the possible order–disorder transitions. In principle, the method allows to predict the phase boundaries of new, not yet synthesized, ternary alloys using effective pair potentials determined based on experimental data of known binary Pd-alloys.

Modeling the microstructural evolution of Ni-base superalloys by phase field method combined with CALPHAD and CVM

A quantitative simulation with phase field method (PFM) in a real alloy system was performed with a new strategy of modeling the microstructure evolution of Ni-base superalloys, in which, the calculation of phase diagrams (CALPHAD) method and cluster variation method (CVM) are combined with the PFM. In this strategy, the four sub-lattice model is used to evaluate the chemical free energy density while CVM to calculate some parameters, such as the lattice misfit and gradient coefficients in phase field equations. With this strategy, the microstructure evolution of a Ni–Al binary alloy at the temperature of 1000K has been simulated; the elastic energy due to the lattice misfit and elastic inhomogeneity between γ and γ′ phases has also been taken into account. Moreover, the directional coarsening phenomenon of Ni–Al alloy has been reoccurred when the various external stress conditions are applied, which shows good agreement with Pineau’s directional coarsening map.

Thermodynamic properties and phase diagram by the statistical moment and cluster variation methods: Application to pure metals and Ta-W alloys

The thermodynamic quantities of metals and alloys are studied using the moment method in the quantum statistical mechanics, going beyond the quasi-harmonic (QH) approximations. Including the power moments of the atomic displacements up to the fourth order, the free energies and the related thermodynamic quantities are derived explicitly in closed analytic forms. The configurational entropy term is taken into account by coupling the moment expansion scheme with the cluster variation method (CVM). The energetics of the binary (TaW) alloys are treated within the framework of the first-principles tight-binding linear muffin-tin orbital (TB-LMTO) method coupled to the coherent potential approximation (CPA) and generalized perturbation method (GPM). The equilibrium phase diagrams are calculated for the refractory Ta-W bcc alloys.

Cluster variation investigation of phase equilibria in Fe–Co system using simulated annealing approach

The cluster variation method (CVM) coupled with the constrained simulated annealing (CSA) algorithm for global free energy minimization is proposed as an efficient approach for the study of alloy phase diagram. The chemical order–disorder transitions in Fe–Co alloy system between 300 and 1100K are investigated using the proposed CSA–CVM approach using the irregular tetrahedron approximation in a bcc lattice. The calculations have been carried out for a hypothetical paramagnetic Fe–Co alloy, i.e. with chemical interactions only, as well as for Fe–Co alloy with both magnetic and chemical interactions acting simultaneously, i.e., considering the quaternary Fe↑Fe↓Co↑Co↓ system. The calculated A2/B2 boundary is found to match well with that determined experimentally. The incorporation of magnetic interaction is also found to have a large influence on the order–disorder transitions. The main advantages of the CSA–CVM approach is that it works efficiently with no initial value dependence on optimization parameters, and in contrast to some of the previously used minimization techniques, it is always found to converge to the global minimum. Another new aspect of the present investigation is that the phase diagram is obtained here by plotting long range order versus composition for the most ordered phase (B2 in this study) and not the commonly used grand potential versus chemical potential curves for various phases, thereby saving a lot of labor. Precise predictions for the variation of heat capacity with temperature have also been made.

Interatomic potentials consistent with thermodynamics: The Fe–Cu system

A methodology is developed to fit semi-empirical interatomic potentials aimed at obtaining a consistent thermodynamic behavior. The procedure is based on the cluster variation method theory which is seamlessly integrated to the other more standard equations of the fitting technique. A new interatomic potential for the Fe–Cu system is thus built within the framework of the embedded atom method, to be used in studies of the microstructure evolution of Fe–Cu alloys under irradiation. The potential is shown to reproduce very reasonably the Cu solubility curve in the Fe matrix as well as to lead to better description of the point defect kinetics with respect to previous interaction models. Limitations of the fitting technique and possible ways of improvement are discussed.

Thermodynamic, alloying and defect properties of plutonium: Density-functional calculations

We show that thermodynamic properties of Pu metal as well as those of Pu (1− x) M x (M = Al, Ga, In) compounds are poorly described by traditional DFT but rather well modeled when including magnetic interactions. The cluster variation method is used to emphasize that the presence of Ga atoms in the δ-Pu phase leads to an important chemical short-range order which stabilizes this phase as a function of temperature and composition. Our results indicate also that Ga atoms introduce local strain fields in δ phase which relax with increasing distance from the Ga atoms. Finally we show that local relaxations are also essential to predict correct vacancy formation and migration energies in this phase.

Deviation of congruent composition in Fe-Pd system

The first-principles calculation of the disordered L1 0 phase boundary for the Fe-Pd system is attempted by combining the FLAPW electronic structure total energy calculations and the cluster variation method via the cluster expansion method. The lattice vibration effects are taken into account based on the Debye-Gruneisen model within quasi-harmonic approximation. The transition temperature is reproduced with very high accuracy. However, the experimental congruent composition of disordered L1 0 phase that significantly deviates from 1:1 stoichiometry is not reproduced. Further calculations are attempted based on the phenomenological Lennard-Jones type pair potential, which is capable of introducing both tetragonality of the L1 0 ordered phase and additional configurational freedom because of the magnetic spins. The preliminary calculations indicate that the tetragonality enhances the stability of the L1 0 ordered phase and the magnetic contributions also change the transition temperature. Despite these findings, the shift of the congruent composition still remains as a subject that needs further research. The electronic origin of the shifting of the congruent composition is briefly discussed.

First-principles calculation of phase equilibria and phase separation of the Fe-Ni alloy system

Theoretical investigation of the phase equilibria of the Fe-Ni alloy has been performed by combining the FLAPW total energy calculations and the Cluster Variation Method through the Cluster Expansion Method. The calculations have proved the stabilization of the L1 2 phase at 1:3 stoichiometry, which is in agreement with the experimental result, and predicted the existence of L1 0 as a stable phase below 550 K; this L1 0 phase has been missing in the conventional phase diagram. The calculations are extended to the Fe-rich region that is characterized by a wide range phase separation and has drawn considerable attention because of the intriguing Invar property associated with a Fe concentration of 65%. To reveal the origin of the phase separation, a P-V curve in an entire concentration range is derived by the second derivative of free energy functional of the disordered phase with respect to the volume. The calculation confirmed that the phase separation is caused by the breakdown of the mechanical-stability criterion. The newly calculated phase separation line combined with the L1 0 and L1 2 order-disordered phase boundaries provides phase equilibria in the wider concentration range of the system. Furthermore, a coefficient of thermal expansion (CTE) is attempted by incorporating the thermal vibration effect through harmonic approximation of the Debye-Gruneisen model. The Invar behavior has been reproduced, and the origin of this anomalous volume change has been discussed.

The Relationship between Plefka's Expansion and the Cluster Variation Method

We expand some approximate free energies in the cluster variation method for random Ising models with respect to exchange interactions, and compare them with the Plefka's expansion. It can be clarified that some approximate free energies in the cluster variation method include all terms relating to specific clusters (or diagrams) in Plefka's expansion. Revealing the relationship between Plefka's expansion and the cluster variation method allows us to understand how the cluster variation method treats correlations among nodes.

Unified approach for cluster variation method calculations of phase diagrams in fcc substitutional alloys with interstitial species

Some alloys show interstitial-induced phase transitions and order-disorder transitions due to the mutual interactions between the interstitial I species and the substitutional S host lattice. An innovative approach, based on the cluster variation method CVM, that takes this coupling into account is proposed here for the calculation of thermodynamic data and phase boundaries. In the case of fcc substitutional alloys with interstitial species a simple cube is chosen as the basic cluster. The cube is defined such that it explicitly accounts for the mutual interaction between the S and the I sublattices comprising the system. Expressions for the configurational entropy in the cube approximation and the internal energy are derived. Phase diagrams for several hypothetical binary host alloys with interstitials are calculated. The results obtained using the proposed simplecube approximation demonstrate the effect of mutual interactions on the phase boundaries.

First Principles Calculations for Alloy Design of Moderate Temperature Age-Hardenable Al Alloys

For aerospace structural applications of age-hardenable aluminum at temperatures above 100°C, a primary alloy-design criterion is creep resistance which depends on the strengthening effect and thermal stability of the second phases.. First principle calculations can be used to study fundamental properties of these phases and, therefore, help to identify the desired ones and their precipitate structures. In order to produce the desired phases, which are usually thermodynamically metastable, and to suppress the undesired phases, computational analysis (combining first principle calculations, cluster variation methods and CALPHAD) can assist in identifying beneficial trace additions and deleterious impurities that must be eliminated. This paper, using Al-Cu-Mg as an example, illustrates this approach, which if successful, should shorten the normal alloy development period.

Ab initiocalculation of the phase stability in Au-Pd and Ag-Pt alloys

The phase stability of superstructures based on the fcc lattice in the Au-Pd and Ag-Pt alloy systems are examined from the fully relativistic electronic density functional theory. The electron-ion interaction is described by the projector augmented-wave PAW method and the exchange-correlation effects are treated in the generalized gradient approximation GGA. The cluster expansion method is used to obtain effective cluster interactions on the fcc lattice and is used also to guide a systematic ground state search for both alloy systems. The ground state analysis reveals a multitude of ground states in Au-Pd, especially at the Au-rich side. Possibly long-period super-structures occur near the Au70Pd30 composition. The ground state analysis indicates a uniquely stable AgPt compound with the L11 structure CuPt prototype and it also suggests a marginally stable ordered compound for Ag3Pt. However, our ab initio study rules out the existence of the remarkably stable Ag3Pt phase with L12 structure, reported first more than half a century ago and since then included in many assessments. We also find no indication for a stable ordered state at the AgPt3 composition. The cluster variation method CVM with a large maximal cluster is used to compute the enthalpy of mixing of the disordered solid solutions and the solid portion of the Au-Pd and Ag-Pt phase diagrams. These results are critically compared with experimental data and phase diagram assessments. It is shown that cluster expansions cannot account for the high-temperature miscibility gap in the Ag-Pt system when the effective cluster interactions do not reach beyond the second nearest neighbor. Only when third nearest neighbors are included in the cluster expansion is it possible to obtain a phase diagram that agrees qualitatively with the assessed Ag-Pt phase diagram.

Using first-principles results to calculate finite-temperature thermodynamic properties of the Nb–Ni μ phase in the Bragg–Williams approximation

Results of first-principles (FP) total energy calculations for 32 different configurations of the μ phase in the binary system Nb–Ni are used in the compound energy formalism (CEF) to model finite-temperature thermodynamic properties. A comparison with Cluster Expansion Hamiltonian–Cluster Variation Method (CEH-CVM) calculations indicates that the CEF describes temperature-dependent site occupancies as well as the CEH-CVM within the temperature range of interest for applications. This suggests that the Bragg–Williams–Gorsky approximation (BWGA) used in the CEF is sufficient to describe site occupancies and thermodynamics of the μ phase. A phase diagram is calculated using the μ phase description derived in the present work together with a previous Calphad description for the other phases of this system. The FP-CEF approach significantly improves the description of the thermodynamic properties as a function of composition compared to the Calphad procedure generally used up to now.

The Cluster Variation Method for Efficient Linkage Analysis on Extended Pedigrees

BackgroundComputing exact multipoint LOD scores for extended pedigrees rapidly becomes infeasible as the number of markers and untyped individuals increase. When markers are excluded from the computation, significant power may be lost. Therefore accurate approximate methods which take into account all markers are desirable.MethodsWe present a novel method for efficient estimation of LOD scores on extended pedigrees. Our approach is based on the Cluster Variation Method, which deterministically estimates likelihoods by performing exact computations on tractable subsets of variables (clusters) of a Bayesian network. First a distribution over inheritances on the marker loci is approximated with the Cluster Variation Method. Then this distribution is used to estimate the LOD score for each location of the trait locus.ResultsFirst we demonstrate that significant power may be lost if markers are ignored in the multi-point analysis. On a set of pedigrees where exact computation is possible we compare the estimates of the LOD scores obtained with our method to the exact LOD scores. Secondly, we compare our method to a state of the art MCMC sampler. When both methods are given equal computation time, our method is more efficient. Finally, we show that CVM scales to large problem instances.ConclusionWe conclude that the Cluster Variation Method is as accurate as MCMC and generally is more efficient. Our method is a promising alternative to approaches based on MCMC sampling.

First-principles calculations of phase equilibria and transformation dynamics of Fe-based alloys

Theoretical procedures of first-principles calculations of phase stability and phase equilibria are summarized. The present scheme is shown to be able to reproduce the transition temperatures with surprisingly high accuracy for Fe−Pd and Fe−Pt systems. The main emphasis of the present report is placed on the extension of the first-principles calculation to transition dynamics calculations. This is performed by combining the cluster variation method with the phase-field method via a coarse graining operation. The time evolution process of antiphase boundaries associated with L10 ordering for Fe−Pd system is demonstrated.

Ab initio calculation of the bcc Fe–Al phase diagram including magnetic interactions

Abstract The metastable phase diagram of the body-centered cubic-based ordering equilibria in the Fe–Al system has been calculated by the cluster expansion method, through the combination of the full potential-linear augmented plane wave and cluster variation methods. The results are discussed with reference to the effect of including the spin polarizations of Fe in the thermodynamic model.

Multicritical phase diagrams of the antiferromagnetic spin-3/2 Blume–Capel model

The antiferromagnetic spin-3/2 Blume–Capel model in an external magnetic field is investigated, and the phase diagrams are obtained in detail by using the cluster variation method. The model exhibits distinct critical regions, including the first-order, second-order and special points: two double critical points, a critical end point, a tricritical point and a zero-temperature critical point. The new phase diagram topology is also found that was not obtained previously. Comparison of the results with those of other studies on this, and closely related systems, is made.