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

  • Symposium: PTM 2015: Solid-Solid Phase Transformations in Inorganic Materials
  • Published:
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Abstract

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.

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Abbreviations

a :

Lattice constant

E :

Internal energy

\( \Delta E \) :

Heats of formation

\( E_{\hbox{max} } \) :

Largest cluster in the expansion of internal energy

\( e_{ij} (e_{ij}^{0} ) \) :

Pair interaction energy between species i and j

F, ∆F :

Free energy

F 0 :

Homogeneous free energy

F T :

Fourier transform

\( F_{ll'} ,\,F^{\prime\prime} \) :

Second-order derivative of free energy F

\( F_{\text{chem}} \) :

Chemical free energy of the entire system

\( f\left( {{\bf{r}}_{i} } \right) \) :

Point distribution function, probability of finding an atomic species i at \( {\bf{r}}_{i} g_{ij} \left( {{\bf{r}}_{i} ,{\bf{r^{\prime}}}_{j} } \right) \) pair distribution function, probability of finding a pair of species i and j at \( {\bf{r}}_{i} \) and \( {\bf{r^{\prime}}}_{j} \), respectively

H :

Hamiltonian

\( I_{\text{SRO}}^{{}} \left( {\bf{k}} \right) \) :

Short range order diffuse scattering intensity

\( L_{ij} \) :

Relaxation constant

\( k_{B} \) :

Boltzmann constant

M :

Mobility

N :

Total number of atoms (lattice points)

\( {\tilde{\bf{R}}} \) :

Rotation operator

S :

Entropy

S max :

Largest cluster involved in the entropy formula

T :

Temperature

T 0 :

Instability temperature (spinodal-ordering temperature)

T s :

Spinodal-ordering temperature

T t :

Transition temperature

\( t,\,\Delta t \) :

Time

t′:

Normalized time with respect to relaxation constant

\( u_{ijklmn} \) :

Octahedron cluster probability of finding atomic species specified by subscripts

\( v_{j} \) :

Effective cluster interaction energy for a cluster designated by j

V(r):

Volume (atomic distance)

\( \tilde{V}_{{\left\{ J \right\}}} \left( {l,l^{\prime}} \right) \) :

V-matrix for atomic configuration specified by J. l’ is a sub-cluster involved in the cluster l

\( w_{ijkl} \) :

Tetrahedron cluster probability of finding atomic species specified by subscripts

\( w_{{_{ijkl} }}^{(s)} \) :

Square cluster probability of finding atomic species specified by subscripts

\( w_{{_{ijkl} }}^{(r)} \) :

Rectangle cluster probability of finding atomic species specified by subscripts

\( W_{ijkl,mnop}^{\alpha \alpha \beta \beta } \) :

Tetrahedron path variable indicating time transition from atomic species ijkl at time t to mnop at time \( t + \Delta t \) on the sublattice points \( \alpha \alpha \beta \beta \)

\( x_{i} \) :

Point cluster probability/concentration of species i

\( X_{{\left\{ J \right\}}} \) :

Cluster probability of configuration specified by J

\( X_{i,j} \) :

Point path variable indicating time transition from atomic species i at time t to j at time \( t + \Delta t \)

\( y_{ij} \) :

Pair cluster probability for atomic pair i–j/concentration of pair cluster i–j

\( Y_{ij,kl}^{\alpha \alpha } \) :

Pair path variable indicating time transition from atomic species ij at time t to kl at time \( t + \Delta t \)on the sublattice points αα

\( \delta X_{l}^{*} \left( {\bf{k}} \right) \) :

Fourier transformation of correlation function of \( \delta \xi_{l} \)

z:

Coordination number

\( z_{ijk} \) :

Triangle cluster probability for triangle arrangement of atomic species i–j–k

Z :

Partition function

\( \left| {\delta Z_{l} \left( {\bf{k}} \right)} \right| \) :

Normal mode amplitude

α, β :

Sublattice point \( \beta_{\text{B}} = {1 \mathord{\left/ {\vphantom {1 {k_{\text{B}} }}} \right. \kern-0pt} {k_{\text{B}} }} \cdot T \)

γ, δ :

Cluster

\( \xi_{i} \) :

Correlation function for a cluster i

\( \delta \xi_{l} \) :

Deviation of correlation function from equilibrium value

η :

Long range order parameter

η i :

ith field variable

θ :

Spin flop probability per unit time

κ i :

Gradient energy coefficient for field variable η i

λ :

Lagrange multiplier

\( \varLambda_{l} \left( {\bf{k}} \right) \) :

Eigen value of the determinant of the second-order derivative matrix of the free energy

μ i :

Chemical potential for species i

\( \rho ,\,\rho_{\text{eq}} \) :

The density of the state

\( eq \) :

Stands for equilibrium state

σ :

Spin variable

\( \left\{ {\varvec{\upsigma}} \right\} \) :

Atomic configuration

\( \phi_{ij} \left( {{\bf{r}}_{i} ,{\bf{r}}'_{j} } \right) \) :

Pair potential between species i and j

\( \varPhi \equiv {{\beta_{\text{B}} F} \mathord{\left/ {\vphantom {{\beta_{\text{B}} F} N}} \right. \kern-0pt} N} \) :

Thermodynamic potential per lattice point

\( \varphi_{\text{S}} \) :

Cluster probability within PFM

\( \chi_{ijklmnop} \) :

Cubic cluster probability

\( \varPsi \equiv {F \mathord{\left/ {\vphantom {F N}} \right. \kern-0pt} N} \) :

Thermodynamic potential

ω :

z/2

\( \varXi \) :

Path variable

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Acknowledgment

This work is partly supported by JSPS Grant-in-Aid for Scientific Research (B) 26289227. The author acknowledges Professors Y. Chen at Tohoku University and M. Ohno for their stimulating discussions and also Ms Kadowaki for the help of careful preparation of the manuscript. The author also sincerely acknowledges Professor Militzer for inviting him to PTM conference, which motivates to write this manuscript. Finally, the author is grateful to Center for Computational Materials Science at IMR, Tohoku University, for the machine time of SR16000.

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  1. Center for Computational Materials Science, Institute for Materials Research, Tohoku University, Katahira 2-1-1, Aoba-ku, Sendai, 980-8577, Japan

    Tetsuo Mohri

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Manuscript submitted January 14, 2016.

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Mohri, T. Cluster Variation Method as a Theoretical Tool for the Study of Phase Transformation. Metall Mater Trans A 48, 2753–2770 (2017). https://doi.org/10.1007/s11661-017-3989-x

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