Elsevier

Calphad

Volume 26, Issue 1, March 2002, Pages 33-54
Calphad

Cluster variation method in the computational materials science

https://doi.org/10.1016/S0364-5916(02)00023-8Get rights and content

Abstract

Cluster Variation Method (CVM) has been very successful in the computations of alloy phase diagrams as well as in many problems of the materials science related to the phase transitions. Originally, CVM was developed in the framework of the so-called rigid lattice approximation, but it has recently been extended to include continuous atomic displacements due to thermal lattice vibration and local atomic distortion due to size mismatch of the constituent atoms. In the present study, we focus our attention on the latter continuous displacement treatment of CVM. The continuous displacement (CD) formulation of the CVM is applied to study the phase stability of the binary alloys. The basic idea is to treat an atom which is displaced by r from its reference lattice point as a species designated by r. The effects of continuous atomic displacement on the thermodynamic quantities and phase transitions of binary alloys are investigated in detail. We also discuss the extension of the CD treatment of CVM to the calculations of solid-liquid and gas liquid phases transitions.

References (30)

  • I.Z. Fisher et al.

    Sov. Phys., Dokl.

    (1960)
  • R. Kikuchi

    Prog. Theor. Phys.

    (1966)
  • K. Terakura et al.

    Mat. Sci. Forum.

    (1989)
  • C. Bichara et al.

    Physica

    (1992)
    C. Bichara et al.

    Physica

    (1992)
    R. Kikuchi et al.

    Physica

    (1992)
  • K. Masuda-Jindo et al.

    Surf. Sci.

    (1998)
    V.L. Vinograd et al.

    Phys. Chem. Minel.

    (1998)
  • R. Kikuchi et al.

    Comp. Mat. Sci.

    (1999)
    G. Treglia et al.

    Comp. Mat. Sci.

    (1999)
  • R. Kikuchi et al.

    Modell. Simul. Mater. Sci. Eng.

    (2000)
    T. Mohri

    Modell. Simul. Mater. Sci. Eng.

    (2000)
    Y Rosenfeld

    Phys. Rev. Lett.

    (2000)
    M. Shimono et al.

    Phys. Rev.

    (2000)
  • H.A. Bethe

    Proc. Roy. Soc.

    (1935)
    J.G. Kirkwood

    J. Chem. Phys.

    (1935)
  • Y. Takagi

    Proc. Phys.-Math. Soc. Jap.

    (1941)
  • Y.-Y. Li

    J. Chem. Phys.

    (1949)
  • R. Kikuchi

    Phys. Rev.

    (1951)
  • J.A. Barker

    Proc. Roy. Soc.

    (1953)
  • J. Hijmans et al.

    Physica

    (1955)
  • J. Hijmans et al.

    Physica

    (1956)
    J. Hijmans et al.

    Physica

    (1956)
  • Cited by (23)

    • Evaluation of the genetic algorithm performance for the optimization of the grand potential in the cluster variation method

      2018, Calphad: Computer Coupling of Phase Diagrams and Thermochemistry
      Citation Excerpt :

      The cluster variation method is a very efficient tool used not only in the computation procedure of alloy phase diagrams but also in many applications of materials science in connection to phase transitions. In fact, such method can be used with a satisfactory accuracy to calculate phase equilibrium configurations in solid solutions characterized by the nature of the involved phases, which varies only in the permutation of atoms and clusters at the level of the lattices sites [4–7]. The cluster variation method has been proposed initially by Kikuchi in a series of pioneering works as an approximate approach to model order-disorder phenomena by providing analytical formulations for the configuration entropy, internal and free energies of the system as a function of the cluster probability variables [8–11].

    • Quantitative modeling and experimental verification of carbide precipitation in a martensitic Fe-0.16 wt%C-4.0 wt%Cr alloy

      2016, Calphad: Computer Coupling of Phase Diagrams and Thermochemistry
      Citation Excerpt :

      Unfortunately, it is impossible to measure the interfacial energy directly and thus there are large uncertainties to the reported values. The available data is mainly derived from indirect methods such as inverse modeling of experimental data [35], first-principles calculations [36], the cluster variation method (CVM) [37], the cluster/site approximation (CSA) [38] and the embedded atom method (EAM) [39]. However, all of the above calculation methods are not easy to apply to multicomponent commercial alloys.

    • Ab initio calculation of the BCC Fe-Al-Mo (Iron-Aluminum-Molybdenum) phase diagram: Implications for the nature of the τ<inf>2</inf> phase

      2009, Calphad: Computer Coupling of Phase Diagrams and Thermochemistry
      Citation Excerpt :

      As already mentioned, the present model uses a rigid lattice approximation and, therefore, important contributions to the free energy are neglected. The most important is, probably, the entropic effect of the vibrational degrees of freedom, which has been found to cause a homogeneous depression of the temperature scale of the calculated phase diagram of the order of 40% to 60% compared to the corresponding rigid lattice case [24,54]. The temperature scale reported in the present work, therefore, must be analyzed with consideration to this effect of the missing vibrational entropy.

    • Thermodynamic self-consistency issues related to the Cluster Variation Method: The case of the BCC Cr-Fe (Chromium-Iron) system

      2008, Calphad: Computer Coupling of Phase Diagrams and Thermochemistry
      Citation Excerpt :

      In fact, two indirect evidences point to this relevant aspect of the vibrational degrees of freedom in the alloy thermodynamics. The first evidence is based on the results reported by Kikuchi and Masuda-Jindo, using the CD-CVM in the pair approximation [20,21]. These authors showed that the relaxation of the rigid lattice constraint results in the reduction of about 40%–50% in the temperature scale of prototype phase diagrams (both phase separating and alloying systems).

    • A generalized defect correlation model for B2 compounds

      2008, Solid State Sciences
      Citation Excerpt :

      Moreover, these approaches assume random distribution of point defects and are therefore not able to describe phenomena which are related to point defect cluster formation in highly ordered crystals. Within the framework of the Cluster-Variation Method (CVM) [17,18], where entropy and free energy of a binary alloy system are written in terms of cluster probability variables, cooperative phenomena leading to SRO can be described properly. However, to our knowledge, CVM has not been applied to point defect cluster formation in highly ordered B2 structure alloy phases up to now.

    View all citing articles on Scopus
    View full text