Cluster Expansion of Alloy Theory: A Review of Historical Development and Modern Innovations

Abstract

The parameterization of a physical or empirical model from a set of highly accurate but expensive calculations or measurements to generate less precise but cheaper predictions is common in many disciplines. In computational materials science and informatics-enabled design of materials, the cluster expansion (CE) method provides a direct approximation of the free energy of a lattice, or any other thermodynamic variable, in terms of a discrete cluster function, making it one of the most widely used approaches for phase diagram calculations, including order–disorder phase transitions. In this article, we review the theoretical developments that culminated in the formulation of the CE method, numerous statistical techniques currently used to fit and optimize the parameters of the CE model, the convergence of the CE method with modern machine learning and data science techniques, and recent developments that push the field beyond the conventional CE, including for structural alloy design.

Abbreviations and Variables

Abbreviations

APB:

Antiphase boundaries

ASE:

Atomic Simulation Environment (see Ref. 28)

ATAT:

Alloy Theoretic Automated Toolkit (see Refs. 27,107,108)

CASM:

Cluster Approach to Statistical Mechanics (see Refs. 109,110)

CLEASE:

CLuster Expansion in Atomic Simulation Environment (see Ref. 113)

CE:

Cluster expansion

CVM:

Cluster variation method

DAPB:

Diffuse antiphase boundary

DFT:

Density functional theory

ECI:

Effective cluster interaction

HEA:

High-entropy alloy

ICET:

Integrated Cluster Expansion Toolkit (see Ref. 112)

KECI:

Kinetic effective cluster interaction

KRA:

Kinetically resolved activation

PINN:

Physics-informed neural network

P$^{4}$:

Piecewise Polynomial Potential Partitioning (see Ref. 111)

QE:

Quantum Espresso

SQS:

Special quasirandom structure

SVM:

Support vector machine

VASP:

Vienna Ab Initio Simulation Package

Variables

C :

A cluster configuration

$F_{\mathrm{cvm}}$ :

Free energy in the cluster variation method (CVM) (defined in Eq. 3)

$F(\rho )$ :

Free energy functional in terms of the probability distribution function $\rho$

$F^*_{\sigma }$ :

Constrained vibrational free energy (see Eq. 21)

H :

Ising model Hamiltonian

$J_{\alpha }$ :

Effective cluster interaction (ECI) coefficient for cluster $\alpha$

$J({\mathbf {k}})$ :

Fourier transform of J (see Eq. 18)

$\mathbf {J_{\alpha }}$ :

A vector of ECIs collected for different configurations

$K_{\alpha }$ :

Kinetic effective cluster interaction (see Eq. 22)

${L}_{\text {aug}}$ :

An augmented lattice that includes local distortions of the ideal lattice in addition to the ideal (high-symmetry) lattice (see Eq. 20)

M :

Number of components (typically distinct chemical species) in a crystal lattice

N :

Number of sites (lattice points) in a crystal lattice

$N_{\alpha }$ :

Number of subclusters in cluster $\alpha$ (see Fig. 2)

$P(\boldsymbol{\sigma })$ :

A property of the lattice expanded as a function of the collection of spin variables $\boldsymbol{\sigma }$ (defined in Eq. 1)

$\mathbf {P}$ :

A vector of the scalar property of a lattice collected for different configurations

$Q(\mathbf {v}|X,\mathbf {y})$ :

Bayesian probability density for an ECI over its possible values $\mathbf {v}$ trained with dataset $X,\mathbf {y}$

$S(\rho )$ :

Entropy functional in terms of the probability distribution function $\rho$

T :

Temperature

$U(\rho )$ :

Internal energy functional in terms of the probability distribution function $\rho$

$V_{\alpha }$ :

Contribution of cluster $\alpha$ to the internal energy in the cluster variation method (CVM)

X :

A matrix of dependent variables (“input values”) used in a training or test dataset (each row corresponds to an output value in $\mathbf {y}$, see Eq. 14)

Z :

Partition function

$a_{\alpha }$ :

Weight of the cluster $\alpha$ towards the total configurational entropy in the cluster variation method (CVM)

f :

A basic figure

$i,i', i'',\dots$ :

An individual lattice site

k :

Boltzmann constant

m :

A possible value for $\sigma _i$; it can take the value of any integer between $-M/2$ and M/2 except for 0 if M is even, and any integer between $-(M-1)/2$ and $(M-1)$ $\div 2$ including 0 if M is odd

$m_{\alpha }$ :

Multiplicity of cluster $\alpha$, i.e., the number of symmetrically equivalent clusters $\alpha$

$n, n',n'', \dots$ :

Individual indices used in a Chebyshev polynomial $\Theta _n$

$n_{\alpha }$ :

The number of lattice sites in cluster $\alpha$

$n_{\beta }$ :

The number of lattice sites in cluster $\beta$

s :

The set of indices $\{n, n',n'',\dots \}$ for the order of the Chebyshev polynomial $\Theta _n$ used in building a cluster function $\phi _{\alpha s}$ (see Eq. 4)

$x_\mathrm{A}, x_\mathrm{B}$ :

Normalized concentration of atom of type A (B) in an alloy

$\mathbf {y}$ :

A vector of dependent variables (“output values”) used in a training or test dataset (see also X)

${\bar{\Pi }}_f (x, \eta )$ :

Generalized correlation function for basic figure f in terms of concentration x and long-range order parameter $\eta$, (see, e.g., Eq. 16)

$\boldsymbol{\Pi }$ :

Matrix of cluster correlation functions

$\Theta _n(\sigma _i)$ :

Discrete Chebyshev polynomial of order n as a function of the spin variable $\sigma _i$

$\alpha , \beta , \gamma$ :

Clusters of lattice sites, relationships between them are explained for each model in the text (see Figs. 1 and 2)

$\alpha '$ :

A set of clusters that are symmetrically equivalent to cluster $\alpha$, i.e., clusters that belong to the same orbit as cluster $\alpha$

$\zeta _{\sigma }$ :

Vicinity of lattice distortions in an augmented lattice ${L}_{\text {aug}}$ (see Eq. 20)

$\eta$ :

Long-range order parameter

$\rho _{\alpha }$ :

Cluster (probability) distribution function in the cluster variation method (CVM)

$\rho _\mathrm{ex}$ :

Equilibrium probability distribution function

$\sigma _i$ :

The spin or occupation variable of site i, associated with the type of atom at site i in a cluster, e.g., in a binary alloy, +1 if site i is occupied by atom A and −1 if occupied by atom B

$\boldsymbol{\sigma }$ :

The vector of spin variables for all the N sites in the crystal

${\hat{\sigma }}({\mathbf {k}},\sigma )$ :

Fourier transform of $\sigma$ (see Eq. 18)

$\varsigma$ :

The neighbor order parameter in the Bethe model; $\varsigma = 1$ for perfect neighbor order and $\varsigma =0$ for perfect neighbor disorder

${\phi }_{\alpha s}$ :

The cluster function corresponding to cluster $\alpha$ and polynomial order s (defined in Eq. 4)

${\bar{\phi }}_{\alpha }(\boldsymbol{\sigma })$ :

The multisite or cluster correlation function for the collection of spin variables $\boldsymbol{\sigma }$ corresponding to cluster $\alpha$ (defined in Eq. 2)