Cluster Variation Method Research Articles

Software Tools for Integrating Special Quasirandom Structures and the Cluster Variation Method into the CALPHAD Formalism

Software Tools for Integrating Special Quasirandom Structures and the Cluster Variation Method into the CALPHAD Formalism

A game-theoretic approach to managing the composition and structure of a bearing-only measurement system in conditions of a priori uncertainty

The problem of managing the composition and structure of a bearing-only measurements system (BOMS) in a game-theoretic formulation is considered. An approach of cooperative search for the placement of BOMS points and a method for estimating the work-time indicator of the system are proposed. The search for the placement of BOMS points uses the toolkit of multi-agent potential games. The criteria for selecting the placement of points and the type of potential function are determined. The management of the composition and structure of the BOMS is based on the results of the cluster-variation method. A structural and functional description of the simulation model is presented. The above results of simulation modeling confirm the practical effectiveness of the proposed approaches.

A cluster-based computational thermodynamics framework with intrinsic chemical short-range order: Part I. Configurational contribution

Exploiting Chemical Short-Range Order (CSRO) is a promising avenue for manipulating the properties of alloys. However, existing modeling frameworks are not sufficient to predict CSRO in multicomponent alloys (>3 components) in an efficient and reliable manner. In this work, we developed a hybrid computational thermodynamics framework by combining unique advantages from Cluster Variation Method (CVM) and CALculation of PHAse Diagram (CALPHAD) method. The key is to decompose the cumbersome cluster variables in CVM into fewer site variables of the basic cluster using the Fowler-Yang-Li (FYL) transform, which considerably reduces the number of variables that must be minimized for multicomponent systems. CSRO is incorporated into CALPHAD with a novel cluster-based solution model called FYL-CVM. This new framework brings more physics into CALPHAD while maintaining its practicality and achieves a good balance between accuracy and computational cost. It leverages statistical mechanics to yield a more physical description of configurational entropy and opens the door to cluster-based CALPHAD database development. The application of the FYL-CVM model in a prototype fcc AB alloy demonstrates its capability to calculate the phase diagram and thermodynamic properties with remarkable accuracy comparable to CVM. The hybrid CVM-CALPHAD framework represents a new methodology for thermodynamic modeling that enables atomic-scale order to be exploited for materials design.

Nonequivalent Antiferromagnetically Coupled Sublattices Induce Two-Step Spin-Crossover Transitions: Equilibrium and Nonequilibrium Aspects

As a continuation to the previously published work (Yalçın et al. (2022)), we investigate the equilibrium and nonequilibrium properties of the spin-crossover systems, with a specific focus on the nonequivalent sublattice, and compare these properties with those of the equivalent sublattices. We used the lowest approximation of the cluster variation method (LACVM) to derive the static equations for the order parameters of the two sublattices and determine high-spin fraction in relation to temperature and external magnetic field in a spin-crossover system. At a low temperature, the transition from stable high-spin (HS) state where nHS=1 occurs in the plateau region, where nHS=0.5 for nonequivalent sublattices. The order parameters for non-equivalent sublattices exhibit different states at the transition temperature. Also, we study the nonequilibrium properties of the order parameters and high-spin fraction using the path probability method (PPM). With the current model, we obtain and analyze the relaxation curves for the order parameters Sa, Sb, and high-spin fraction. These curves demonstrate the existence of bistability at low temperatures. At the end of this study, we present the flow diagram that shows the order parameters for different temperature values. The diagram exhibits states that are stable, metastable, and unstable.

Continuous Displacement Cluster Variation Method for the Study of Local Lattice Distortion in an Alloy

Continuous Displacement Cluster Variation Method for the Study of Local Lattice Distortion in an Alloy

Magnetocaloric effect in the Jx−Jy Blume–Capel model

The magnetocaloric properties of the Jx−Jy Blume–Capel model with Jx/Jy=−0.5 are studied using the two point approximation of the Cluster Variation Method. The phase diagrams exhibit different phases depending on the anisotropy parameter, magnetic field, temperature and exchange interaction ratio. It is shown how the nature of the phase transition changes as parameters are varied. For an anisotropy parameter value of D=1.45, a tricritical point and a reentrant behavior are identified. It was found inverse and direct magnetocaloric effect. The inverse magnetocaloric effect of the model is enhanced at low fields and temperatures around jumps between plateaus of magnetization.

Interaction models and configurational entropies of binary MoTa and the MoNbTaW high entropy alloy

We introduce a simplified method to model the interatomic interactions of high entropy alloys based on a lookup table of cluster energies. These interactions are employed in replica exchange Monte Carlo simulations with histogram analysis to obtain thermodynamic properties across a broad temperature range. Kikuchi's cluster variation method entropy formalism is applied to directly calculate entropy from statistics on short- and long-range chemical order, and we discuss the convergence of the entropy as clusters of differing size are included. A high temperature series expansion aids in our understanding of the convergence. Computer codes implementing these methods, and supporting data, are freely available on the internet.

Synthetic first-principles studies from phase equilibria to microstructural formation in the Fe-Pt L10 phase

Electronic-structure calculations, the cluster-variation method of statistical mechanics, and the phase-field method were combined in attempted first-principles calculations of phase equilibria and microstructural evolution associated with the disorder-$\mathrm{L}{1}_{0}$ transition of the Fe-Pt system. The calculated disorder-$\mathrm{L}{1}_{0}$ transition temperature was within $\ensuremath{\sim}10$ K difference from the experimental value, and the locus of spinodal ordering temperature is placed in the phase diagram. The calculated microstructure demonstrates preferential growth of the ordered domain along the 100> direction and, in the later period, an anisotropic morphology of an antiphase domain structure develops. We offered an interpretation from the atomistic point of view for this morphology. We therefore achieved consistent first-principles multiscale calculations of phase equilibria and microstructural evolution, bridging microscopic to mesoscopic scales without any adjusting parameters.

Cluster variation method for investigation of multi-principal-element metallic alloys

We present atomic-scale investigations of the thermodynamic properties of quinary bcc-based multi-principal-element alloys (MPEAs) within {Al,Co,Cr,Fe,Mn,Mo,Ni}, by means of short-range pair energetics and the cluster variation method (CVM) in the irregular tetrahedron approximation. We focus on an essential couple of key-properties detrimental for potential applications of MPEAs, namely the trends to (i) long-range ordering (LRO) and (ii) phase separation (PS). While the CVM has been more commonly employed in a grand canonical frame controlled by chemical potentials (δμ), we propose an original δμ-driven CVM scheme which allows an easier exploration of wide composition spaces typical of MPEAs. This modified scheme yields a slice-by-slice analysis of this space, by means of pseudo-ternary isothermal sections, which demonstrates the ability of the CVM to lead, with moderate computational effort, to full phase diagrams of MPEAs. In particular, our CVM calculations provide elements to interpret recent unexpected experimental trends of AlCrFeMnMo. Comparison with earlier studies of the same MPEAs using the less accurate point-mean-field (PMF) approximation indicates that CVM and PMF have global agreement for LRO at higher temperatures, whereas CVM should definitely be preferred as soon as PS begins to occur. Both approaches, employed more systematically in future MPEA studies in conjunction with ab initio electron theory and dedicated experiments, may offer convenient tools to check the merits of, and even improve, the various sets of effective pair interactions nowadays increasingly available for atomic-scale simulations of MPEAs, thus helping to identify composition domains relevant to avoid trends (i) and (ii) and design more reliable MPEAs.

Dynamic behavior of the antiferromagnetically coupled bilayer Ising model

Using the path probability and lowest approximation of cluster variation method, we study the dynamic and equilibrium properties of a bilayer magnetic system, consisting of two ferromagnetic monolayers antiferromagnetically coupled for different spins (σ = 1/2 and S = 1). Firstly, numerical results of the monolayer and total magnetizations are presented under the effect of the diverse physical parameters, and the phase diagrams of bilayer system are discussed. Then, since it is well established that the path probability method is an effective method for the existence of metastable states, the time evolution of monolayer- and total magnetizations is investigated.

Frustrated mixed-spin chains: Three-site interactions and flat-band magnons

We investigate the ground-state phase diagram of frustrated mixed-spin (1, 1/2) chains with three-site four-spin interactions employing different approaches, such as cluster mean field theory, cluster variational method, and spin-wave theory. The interplay of next-nearest-neighbor and three-site interactions leads to a plethora of magnetic and nonmagnetic phases in the ground-state phase diagram. We show that aside from the ferromagnetic and Neel orders, the ground state possesses a magnetic phase in which magnon excitations are dispersionless. The emergence of flat-band magnons is a consequence of the inhomogeneity of our frustrated mixed-spin chains and does not occur in similar uniform spin chains. We also demonstrate that the presence of three-site interaction gives rise to various nonmagnetic phases, such as antiferroquadrupole order and quantum spin-liquid phase.

Stable, metastable and unstable states of the mixed spin (1, 3/2) Ising model in the absence and presence of constant magnetic fields

Stable, metastable and unstable states of the mixed spin (1, 3/2) Ising system in the absence and presence of constant external magnetic fields (CEMFs) were investigated both by the cluster variation method (CVM) and the path probability method (PPM). First, we examine the thermal behavior of the magnetizations by using the CVM and search the metastable and unstable branches of magnetizations in addition to the stable branches. We present the metastable phase diagrams besides to equilibrium phase diagrams in the absence of the CEMFs. We also constructed only the metastable phase diagram under the presence of CEMFs. Since the metastability is a stable state of a dynamical system, we used the PPM on the model and found the dynamic equations. Dynamic equations were solved by means of the flow diagrams (FDs). The FDs clearly display the stable, metastable and unstable states in which also obtained within the CVM. The FDs display the role of the metastable, unstable states and initial conditions as well as how the system trapped into the metastable state. Some properties of the metastable states were observed from the FDs. These studies were made both for the ferrimagnetic and antiferrimagnetic cases.PACS: 05.50. + q, 05.70.Fh, 05.70.Ln, 64.60.Cn, 64.60. My, 75.10.Hk

Static and dynamic properties of two-sublattice spin-crossover systems

We investigate the static and dynamic properties of spin-crossover systems in the presence of an external magnetic field using two-sublattice Ising-like Hamiltonian. At low temperatures both sublattices are in the LS state, and in the HS state at high temperatures. The new feature of spin-crossover is depicted for our model when we take the non-equivalent sublattices JA≠JB, the two-step spin-crossover phase transition is obtained with the presence of the hysteresis loop. In order to study the properties of two-step spin-crossover, we use the lowest approximation of the cluster variation method for static properties. The dynamic properties of the system with bilinear interactions are studied by the path probability method. We present high-spin state fraction nHS vs temperature and magnetic field variations for constant values of the degeneracy ratio between high-spin and low-spin states. In the present model, we obtained the relaxation curves of the order parameters SA and SB using different temperature values. Finally, the flow diagram of the order parameters is depicted for different values of rate constant k1 and temperature, which displays the stable, metastable, and unstable properties. Static and dynamic self-consistent equations of the system were performed by using Newton-Raphson and Runge-Kutta methods.

First-principles calculations and thermodynamic assessment of the Nb–V system using CE-CVM

A thermodynamic assessment of the Nb–V system has been carried out based on first-principles calculations and all the available experimental thermodynamic and phase equilibria data. Mixed-parametric ab initio calculations were performed to calculate the enthalpy of mixing data with Special quasirandom structures (SQS) and cluster expansion (CE) method. The Gibbs energy of the bcc solid phase was described using tetrahedron approximation of the Cluster Expansion and Cluster Variation Methods (CE-CVM). A sub-regular solution model was used for describing Gibbs energy of the liquid phase. An optimal set of parameters has been obtained for describing the Gibbs energies of the liquid and solid phases in the Nb–V binary system. The calculated phase diagram and thermodynamic properties agree with experimental data. The relevance of CE-CVM has been demonstrated in the present case by calculating the solid phase's short-range order (sro) parameters as a function of composition and temperature.

Random Crystal and Magnetic Field Effects on Hysteresis Loops and Phase Transition of Blume-Capel Model in Cluster Variation Method

Random Crystal and Magnetic Field Effects on Hysteresis Loops and Phase Transition of Blume-Capel Model in Cluster Variation Method

Cluster Variation Method as a Powerful Tool for Theoretical and Computational Materials Science

Cluster Variation Method as a Powerful Tool for Theoretical and Computational Materials Science

Comparison between Monte Carlo and Cluster Variation method calculations in the BCC Fe–Al system including tetrahedron interactions

Abstract The BCC Fe –Al system has been modeled with the Monte Carlo (MC) method in the grand-canonical ensemble calculation using the Ising model. The present calculation was performed in isothermal conditions with a computer crystal with 483 atoms at 1000 K. The interaction parameters for this calculation were taken from a previous Cluster Variation Method (CVM), which included tetrahedron interactions. The comparison of the integral thermodynamic properties (internal energy and equilibrium alloy composition) obtained in both calculations shows good agreement. Small deviations are observed only in the neighborhood of the second-order B2/A2 critical composition.

Experimental study and Cluster Variation modelling of the A2/B2 equilibria at the titanium-rich side of the Ti–Fe system

AbstractThe experimental data concerning the A2 (β-Ti)/B2 (TiFe) equilibrium have been reviewed and four new tie-lines atT= 1343 K, 1173 K, 1073 K and 973 K were determined. These new experimental data confirm the assessed phase boundary A2/B2 at the A2 side, except for temperatures near the eutectic. No data were available of the solubility range of the intermetallic B2 phase. The present data of the phase boundary at the B2 side give a lower solubility than previously assumed. The set of experimental data and additional new information on the Fe–Rh–Ti phase diagram lead to new pair and tetrahedron interaction energies by Cluster Variation Method calculations in the tetrahedron cluster approximation.

Thermodynamics of Binary bcc and fcc Phases for Exclusive Second-Neighbour Pair Interactions Using Cluster Variation Method: Analytical Solutions

Thermodynamics of Binary bcc and fcc Phases for Exclusive Second-Neighbour Pair Interactions Using Cluster Variation Method: Analytical Solutions

Bimodal-random field Blume–Capel model in the cluster variation method

The outcome of the bimodal random magnetic field acting up or down along the z-axis on the spins of spin-1 with equal probability is investigated by using the lowest approximation of the cluster-variation method in the Blume–Capel model. The equal probability, p=12, causes the model to give phase transitions of the form either second- or first-order. The phase diagrams of the model are computed on the (D,kBT/J) and (h,kBT/J) planes by taking into account the thermal variations of the order-parameters. The obtained phase diagrams reveal different phase regions with ground state values of 1 and 1/2 in addition to tricritical and critical end points. Simple and double hysteresis loops are also observed in the model.