Quantifying multipoint ordering in alloys

A central problem in multicomponent lattice systems is to systematically quantify multipoint ordering. Ordering in such systems is often described in terms of pairs, even though this is not sufficient when three-point and higher-order interactions are included in the Hamiltonian. Current models and parameters for multipoint ordering are often only applicable for very specific cases or require approximating a subset of correlated occupational variables on a lattice as being uncorrelated. In this paper, cluster order parameters are introduced to systematically quantify arbitrary multipoint ordering motifs in substitutional systems through direct calculations of normalized cluster probabilities. These parameters can describe multipoint chemical ordering in crystal systems with multiple sublattices, multiple components, and systems with reduced symmetry. These are defined in this paper and applied to quantify four-point chemical ordering motifs in platinum/palladium alloy nanoparticles that are of practical interest to the synthesis of catalytic nanocages. Impacts of chemical ordering on nanocage stability are discussed. It is demonstrated that approximating four-point probabilities from superpositions of lower-order pair probabilities is not sufficient in cases where three- and four-body terms are included in the energy expression. Conclusions about the formation mechanisms of nanocages may change significantly when using common pair approximations.

Figure 1

(a) Chemical ordering is often described in terms of pairs (dark red motifs) and how much the chemistry of a given neighbor shell deviates from its nominal stoichiometry on average in the crystal. (b) Chemical ordering across multiple shells and/or sublattices can be difficult to quantify and often requires approximation.

Figure 2

A spectral expansion of the energy is enabled by defining a complete basis over the entire domain of alloy configurations. The alloy configuration is given by a collection of spin variables that specify the occupation of the lattice sites. The basis functions are defined as all possible products of the spin variables (or appropriate site basis function to maintain orthogonality conditions). For a cluster expansion in this basis, there are terms corresponding to not only single sites and pairs (of different ranges) similar to an Ising model but also all high-order terms such as triplets and beyond.

Figure 3

The palladium (111) face-centered-cubic nanoparticles coated with a few atomic layers of a Pt-Pd alloy (a), prepared in Ref. [6], rely on specific multipoint chemical ordering motifs, palladium channels that span the surface alloy region (b), to generate high-surface-area catalyst cages for the oxygen reduction reaction (c). A sufficient amount of Pd is needed to observe the “channel” order motif [shown in (b)] to allow for the etching of the palladium core but not so much that it diminishes the catalyst durability.

Figure 4

Examples of surface cells used to fit the cluster expansion coefficients (grouped as multiples of the primitive surface cell). All symmetrically distinct configurations in the $1\times$ and $2\times$ cells are used for fitting along with larger randomized surface cells.

Figure 5

Effective cluster interactions and expansion coefficients for the surface alloy system in contact with the implicit solvent. The blue bars mark the magnitude and sign of the coefficient for pair, triplet, and quadruplet clusters, order 2, 3, and 4, respectively.

Figure 6

The pair parameters for all of the nearest-neighbor (NN) shells contained in the four-point channel are provided along with the combined single-site correlations at the surface and bottom of the alloy.

Figure 7

The exact four-point order parameter corresponding to the channel-shaped cluster from direct calculation during Monte Carlo simulations (blue circles) is compared with that calculated from Clapp's implementation of the Kirkwood superposition (brown diamonds). A visualization of low-order probabilities used to approximate the four-point channel probability is shown for each case.

Figure 8

The relative error in the measured cluster probabilities using the cluster order parameter formalism compared with a constituent pair superposition approximation.