Elsevier

Computational Materials Science

Volume 83, 15 February 2014, Pages 207-211
Computational Materials Science

Adaptive cluster expansions and redox-dependent atomic ordering

https://doi.org/10.1016/j.commatsci.2013.10.013Get rights and content

Highlights

  • Adaptive cluster expansions (ACE) improve prediction of atomic ordering.

  • ACE allows unprecedented modeling control of the local redox environment.

  • Compact expansions permit physical interpretations of 2- and 3-body interactions.

  • The approach is tested for oxide-fuel cell materials YSZ and LSCR.

Abstract

An adaptive cluster expansion (ACE) methodology is presented which enables exploration of atomic ordering interactions in solids as a function of the redox environment. A previously developed cluster expansion methodology is augmented via inclusion of explicit effective charge dependence within the topological cluster basis. This augmentation produces an enhanced fit precision across a wide composition range and the ability to directly control the model’s redox state during Monte Carlo system equilibrations. The approach is validated in applications to yttria-stabilized zirconia (YSZ) and the perovskite (La0.8, Sr0.2)(Cr0.8, Ru0.2)O2.9 (LSCR), where significant variability in atomic ordering is seen across redox space. A locally adaptive lattice Monte Carlo sampling, utilizing the ACE methodology, is developed and validated in applications to determine the 0 K ground state configurations of YSZ and LSCR supercells with varying redox conditions. These equilibrations have direct relevance to solid-oxide fuel cell applications, whose components are subject to widely varying redox environments. The superior convergence of ACE results in a smaller number of numerically significant expansion terms, not only speeding the analysis but also permitting a physical interpretation of their meaning.

Introduction

The starting point for many material models is to express the system internal or cohesive energy as a simple function (fit) of a cumulative set of local topological interactions, such as structurally distinct pairs, triplets, and quartets. The Cluster Expansion (CE) methodology is a highly successful example, in which structural features such as interatomic distances and angles are implicitly contained in the expansion coefficients. CE methods project libraries of first-principles and/or experimentally derived material properties (e.g. energy, magnetic moment, dielectric constants) onto an orthogonal set of topological clusters of sites, typically consisting mostly of short-range pair and triplet interactions [1]. What results from this projection is a semi-analytical equation (Eq. (1)) for the desired property using concentration and configuration space of a single crystallographic topological structure as component variables. While the summation in Eq. (1) is formally infinite and complete, the CE is necessarily truncated to achieve a desired precision [2].E(σ)=αmαJαφα(σ)

Here mα is the multiplicity of cluster α, Jα are the variational coefficients of cluster α, and φα(σ) are the lattice averages of the cluster functions φα(σ) defining distinct site groupings. For systems with only two possible lattice occupations, the calculation of the cluster function values is directly analogous to lattice spin models such as the Heisenberg, Ising, and Potts models. A graphic representation of clusters on a binary lattice is given in Fig. 1.

In the standard CE approach, the expansion coefficients are independent of component concentrations; considerable success has been obtained in representation of ordering and alloy effects in binary systems [2], [3]. In a typical application CE and Monte Carlo (MC) sampling are combined to survey a wide concentration field, which would be inaccessible to purely first-principles calculations. Successful applications of the CE/MC methodology have been extended to ternary and quaternary systems [4], [5], [6], [7]. However, application of expansions like Eq. (1) across a wide range of redox environments, phase spaces, and concentration spaces can lead to a lengthy series (for a specified level of precision, see below) with increasingly complex cluster topologies which are not susceptible to physical interpretation. For example, it would be naïve to assume that the interaction energy of two oxygen atoms within the yttria-stabilized zirconia (YSZ) electrolyte system is static across Y concentration space, within both highly reductive and oxidative environments; thus terms with as many as six or seven sites are required to capture the variation of local stoichiometry [5]. Fitting the multidimensional parameter space with more system-appropriate, flexible functions leads to more compact expansions with improved precision, as will be demonstrated.

A novel materials modeling scheme should be validated by application to nontrivial problems: in the present case, to materials for which redox variability plays a key role. Solid Oxide Fuel Cell (SOFC) materials are chosen here precisely due to their particular critical responses to variable redox environments. SOFCs are under continual development as a technology to efficiently generate clean electricity from domestic fuel sources. Solid Oxide Electrolyser Cells (SOECs), essentially SOFCs operated in reverse, are also being evaluated as a means to produce fuel from excess electricity produced by renewable sources [8]. SOFC and SOEC performance is governed by the component materials’ stability and conductivity (both electronic and ionic) at high temperatures and in various redox environments. In a previous work [5], an energy cluster expansion was developed for yttria stabilized zirconia (YSZ) to help clarify the relationship between dopants and defect structures. The atomic ordering properties of lanthanum perovskites (La1−x, Srx)(Cr1−y, Fey)O3−δ (LSCF) and (La1−x, Srx)(Cr1−y, Ruy)O3−δ (LSCR) were also previously explored [6] using energy CEs to elucidate connections between atomic ordering and composition. These SOFC/SOEC materials thus provide a significant and severe test of the extended modeling formalism which permits direct manipulation of a system’s redox environment.

Section snippets

Fitness criterion

Testing the degree of convergence and precision of an expansion requires definition of a fitness criterion. The fitness criterion used in this work is equal to the CE crossvalidation score (CV) [2] (Eq. (2)).CV=1Ni=1N(Ei-E(σ))21/2where Ei is the input database (typically ab initio) energy and i runs over the entire dataset. The associated predicted CE energy E(σ) is found by a least-squares error minimization over N-1 structures (by deleting the ith element) of the training set. The CV score

Adaptive cluster expansions for YSZ and LSCR

Cluster selections for two YSZ ACEs, with first through third order charge expansion parameters (x = 1, 2, 3 in Eq. (3)), were optimized using the genetic algorithm (GA) from a training library of 702 structures. Building upon training libraries from previous works [5], [6], [11], all LSCR training structures were calculated using the PW91 version of the generalized gradient approximation (GGA), and all YSZ training structures were calculated using the simplest Local Density Approximation (LDA).

Conclusions

A novel adaptive cluster expansion methodology for predicting solid materials atomic ordering, which employs a structure’s effective charge as an expansion parameter, has been developed and tested. Redox-dependent atomic ordering tendencies were calculated for YSZ and LSCR prototypical fuel cell materials. Comparison between standard Cluster Expansion (CE) and Adaptive Cluster Expansion (ACE) results for these two systems permits a meaningful evaluation of benefits gained from the more flexible

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