Continuous Displacement Cluster Variation Method Research Articles

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

Conversion of Magnetic Freedoms into Atomic Configurational Freedoms within the Cluster Variation Method

The continuous displacement cluster variation method (CDCVM) has introduced local atomic displacements into the theoretical framework of the cluster variation method (CVM) by viewing an atom displaced from a Bravais lattice point as a particular atomic species located at the lattice point. This idea of conversion from a freedom of local displacements into configurational freedom is extended in this paper to magnetic freedoms. Various magnitudes of local magnetic moments are considered, as well as two spin directions, on up-spins and down-spins. The approach is applied to pure Ni and its Curie temperature is explored with the entropy formula of the tetrahedron approximation in the CVM, using the first-nearest-neighbor pair interaction energies extracted from the total energies of various spin configurations, which are estimated from electronic-structure calculations.

Modelling of a displacive transformation in two-dimensional system within four-body approximation of Continuous Displacement Cluster Variation Method

Following our previous study on distorted to non-distorted displacive phase transformation in the two-dimensional square lattice based on the continuous displacement cluster variation method (CDCVM) within the Bragg–Williams (BW) approximation, we performed a higher order approximation, four-body approximation, in the entropy term and compared the results obtained by the two approximations. The transformation temperature decreases with the higher order approximation, which shares the common feature with conventional CVM studies on replacive transformations. The present study predicts the first-order transformation, which is markedly different from the previous study based on the BW approximation. Furthermore, by employing the four-body approximation, we are able to reproduce the o-type distribution of displaced atoms around a Bravais lattice point by changing the atomic interaction energy, which was by no means possible by the BW approximation.

Cluster Variation Method as a Theoretical Tool for the Study of Phase Transformation

Cluster variation method (CVM) has been widely employed to calculate alloy phase diagrams. The atomistic feature of the CVM is consistent with first-principles electronic structure calculations, and the combination of CVM with electronic structure calculation enables one to formulate free energy from the first-principles. CVM free energy conveys affluent information of a given system, and the second-order derivative traces the stability locus against configurational fluctuation. The kinetic extension of the CVM is the path probability method (PPM) which is utilized to calculate transformation and relaxation kinetics associated with the temperature change. Hence, the CVM and PPM are coherent methods to perform a synthetic study from initial non-equilibrium to final equilibrium states. By utilizing CVM free energy as a homogeneous free energy density term, one can calculate the time evolution of ordered domains within the phase field method. Finally, continuous displacement cluster variation method (CDCVM) is discussed as the recent development of CVM. CDCVM is capable of introducing the local lattice displacement into the free energy. Moreover, it is shown that CDCVM can be extended to study collective atomic displacements leading to displacive phase transformation.

Modelling of a displacive transformation in two-dimensional system within a single-site approximation of continuous displacement cluster variation method

It is attempted to model a displacive phase transformation by Continuous Displacement Cluster Variation Method. The main focus of the present study is placed on distorted to non-distorted phase transformation in the two-dimensional square lattice. The entropy is formulated within the single-site approximation (point approximation), while the pair-wise atomic interaction energies are combined to stabilize a distorted phase at low temperatures. The distorted to non-distorted phase transformation in the present study is of the second order, and the calculated distribution of atoms suggests that the transformation is of the displacive type in the classification.

SHORT RANGE ORDERING AND LOCAL DISPLACEMENT OF ALLOYS STUDIED BY CVM

Cluster Variation Method (CVM) is a powerful statistical mechanics means to investigate phase equilibria of an alloy. The advantageous feature of the CVM stems from the fact that wide range of atomic correlations which play an important role at the phase transition is efficiently incorporated into the free energy formula. Hence, configurational fluctuation can be systematically studied through the calculations of correlation functions in the real space and short range order diffuse intensity spectrum in the k-space. However, one of the deficiencies of the conventional CVM is the fact that local lattice distortion (local atomic displacement) is not correctly dealt with. In order to improve such shortcomings, Continuous Displacement Cluster Variation Method (CDCVM) has been developed. In the CDCVM, local lattice distortion is mapped onto the configurational freedom of a multi-component alloy on a rigid (uniformly deformable) lattice. With CDCVM, the applicability of CVM is enlarged and the calculations of diffuse intensity spectrum originating from local lattice distortion can be performed.

First-principles cluster variation calculations of tetragonal-cubic transition in ZrO2

Abstract It is attempted to extend the basic idea of continuous displacement cluster variation method (CDCVM) to the study of a displacive phase transition . As a preliminary study, we focus on cubic to tetragonal transition in ZrO 2 in which oxygen atoms on the cubic lattice are displaced alternatively in the opposite direction (upward and downward) along the tetragonal axis. Within the CDCVM, displaced atoms are regarded as different atomic species, and two distinguished atoms, A -oxygen (upward shifting) and B -oxygen (downward shifting), are introduced in the description of the free energy. FLAPW electronic structure total energy calculations are performed to extract effective interaction energies among displaced oxygen atoms, and by combing them with CDCVM, the transition temperature is calculated from the first-principles.

Theoretical Investigation of Alloy Phase Equilibria by Continuous Displacement Cluster Variation Method

Continuous Displacement Cluster Variation Method is employed to study binary phase equilibria on the two dimensional square lattice with Lennard-Jones type pair potentials. It is confirmed that the transition temperature decreases significantly as compared with the one obtained by conventional Cluster Variation Method. This is ascribed to the distribution of atomic pairs in a wide range of atomic distance, which enables the system to attain the lower free energy. The spatial distribution of atomic species around a Bravais lattice point is visualized. Although the average position of an atom is centred at the Bravais lattice point, the maximum pair probability is not necessarily attained for the pairs located at the neighboring Bravais lattice points. In addition to the real space information, k-space information are calculated in the present study. Among them, the diffuse intensity spectra due to short range ordering and atomic displacement are discussed.

CVM-based first-principles calculations for Fe-based alloys

ABSTRACTCluster Variation Method (CVM) has been recognized as one of the most reliable theoretical tools to incorporate wide range of atomic correlations into a free energy formula. By combining CVM with electronic structure total energy calculations, one can perform first-principles calculations of alloy phase equilibria. The author attempted such CVM-based first-principles calculations for various alloy systems including noble metal alloys, transition-noble alloys, III-V semiconductor alloys and Fe-based alloy systems. Furthermore, CVM can be extended to two kinds of kinetics calculations. One is Path Probability Method (PPM) which is the natural extension of the CVM to time domain and is quite powerful to investigate atomistic kinetic phenomena. The other one is Phase Field Method (PFM) with the CVM free energy as a homogeneous free energy density term in the PFM. The author’s group applied the latter procedure to study time evolution process of ordered domains associated with disorder-L10 transition in Fe-Pd and Fe-Pt systems. CVM has, therefore, a potential applicability for the systematic studies covering atomistic to microstructural scales. It has been, however, pointed out that the conventional CVM is not able to include local lattice relaxation effects and that the resulting order-disorder transition temperatures are overestimated. In order to circumvent such inconveniences, Continuous Displacement Cluster Variation Method (CDCVM) has been developed. Since first-principles CDCVM calculations are still beyond the scope at the present stage, preliminary results on the two dimensional square lattice and an fcc lattice with primitive Lennard-Jones type potentials are demonstrated in the last section.

Application of Continuous Displacement Cluster Variation Method to Study Phase Equilibria

Cluster Variation Method (CVM) has been widely recognized as one of the most reliable theoretical tools to study phase equilibria in metallic alloy systems. The conventional CVM, however, does not allow atomic local displacements and, therefore, calculated results often encounter various inconveniences such as the overestimation of transition temperatures. Continuous Displacement Cluster Variation Method (CDCVM) was proposed to circumvent such deficiencies of the conventional CVM. Preliminary studies on an order-disorder phase diagram based on CDCVM indicate that the transition temperature is shifted downward reproducing experimental tendencies. In the present study, lattice thermal vibration effects are also incorporated through Morse potential. It is concluded that the local lattice distortion effects are quite effective to reduce the transition temperature.

First-principles calculation of microstructural processes in alloys

By combining Cluster Variation Method with FLAPW electronic structure total energy calculations and Phase Field Method, time evolution of Anti Phase Boundary associated with L1 0 ordering process in Fe– Pd was calculated from the first-principles. The theoretical framework of these calculations is reviewed, and it is pointed out that the introduction of the local lattice relaxation effects is indispensable to achieve higher accuracy. Preliminary calculations based on Continuous Displacement Cluster Variation Method are attempted on two-dimensional square lattice to examine the significance of the local lattice relaxation effects.

Theoretical investigation of phase equilibria by the continuous displacement cluster variation method

Abstract By employing the continuous displacement cluster variation method, order – disorder transition behavior of a two-dimensional square lattice is investigated. It is demonstrated that the distribution of displaced atoms around a Bravais lattice point continuously changes with temperature. The additional freedom of atomic displacement contributes to the entropy and thermal lattice expansion is confirmed. Finally, it is shown that the order – disorder transition temperature decreases as compared with the one by the conventional cluster variation method in which no local atomic displacements are allowed.

Thermal Expansion Calculated by Continuous Displacement Cluster Variation Method

Thermal expansion of a single component system in a two dimensional square lattice is calculated by employing Continuous Displacement Cluster Variation Method. It is demonstrated that the entropy effect arising from the freedom of local atomic displacements drives the thermal expansion. The coefficient of thermal expansion curve is compared with the one calculated based on Debye-Gruneisen model within the quasi-harmonic approximation. The two curves demonstrate opposite behavior.

Continuous Displacement Cluster Variation Method in Fourier Space

The diffuse scattering spectra originated from both short range order and local atomic displacements for a binary alloy in a two dimensional square lattice are calculated within the realm of Continuous Displacement Cluster Variation Method (CDCVM). The key ingredient of the present scheme is to regard a displaced atom as a distinguished species and to reduce the displacement spectrum to the Short Range Order Diffuse Intensity (SROI) for a multicomponent alloy. The peak of the diffuse intensity appears at the wave vector (1/2, 1/2). By comparing with the SROI calculated by the conventional Cluster Variation Method (CVM), it is realized that the majority of the intensity originates from the contribution of short range order configurational fluctuation. The topology of the intensity distribution of the present result is similar to that derived by the conventional CVM, however the magnitude of the peak intensity differs. This is due to the fact that the directionalily of the atomic displacement is averaged out and is renormalized in the present calculation.

Continuous Displacement Cluster Variation Method and Diffuse Scattering Intensity Calculation

Continuous Displacement Cluster Variation Method within the pair approximation is combined with Lennard-Jones type potential and we investigate local atomic displacements for a binary ordering system in the two dimensional square lattice at 1:1 stoichiometry. The order-disorder transition temperature is reduced by the introduction of the local atomic displacement, while the second order nature of the transition is conserved. The distribution of unlike pair is localized in the vicinity of the rigid lattice points, which is explained as the deepest pair potential characteristic for the ordering system. The obtained information of atomic displacements in the real space is Fourier transformed to calculate diffuse scattering intensity spectrum in k-space. The maximum intensity contribution from the linear term is attained at (kx, k * y ) with k * x = k * y < 0.5 in k-space, while the maxima of the short range order diffuse intensity appears at (0.5, 0.5). This is interpreted based on the concept of displacement wave.