Predicting activation energies for vacancy-mediated diffusion in alloys using a transition-state cluster expansion

Kinetic Monte Carlo models parameterized by first principles calculations are widely used to simulate atomic diffusion. However, accurately predicting the activation energies for diffusion in substitutional alloys remains challenging due to the wide variety of local environments that may exist around the diffusing atom. We address this challenge using a cluster expansion model that explicitly includes a sublattice of sites representing transition states and assess its accuracy in comparison with other models, such as the broken bond model and a model related to Marcus theory, by modeling vacancy-mediated diffusion in Pt-Ni nanoparticles. We find that the prediction error of the cluster expansion is similar to that of other models for small training sets, but with larger training sets the cluster expansion has a significantly lower prediction error than the other models with comparable execution speed. Of the simpler models, the model related to Marcus theory yields predictions of nanoparticle evolution that are most similar to those of the cluster expansion, and a weighted average of the two approaches has the lowest prediction error for activation energies across all training set sizes.

Figure 1

(a) A schematic of transition-state cluster expansion. Large gray and brown spheres represent Pt and Ni, respectively, on lattice sites (local minima on the potential energy surface). Small green dots represent saddle points for hops between two local minima. For illustration purposes a two-dimensional lattice is shown, but the model in this paper is built on a three-dimensional fcc lattice. (b) A set of three training structures representing a diffusive hop, with the diffusing species at the transition-state site circled in green. From left to right: initial, transition, and final state.

Figure 2

Leave-one-out cross validation (LOOCV) of (a) the formation energies and (b) activation energies from the transition-state cluster expansion. The dashed lines in panel (b) indicate $\pm 0.2$ eV deviation from perfect agreement between DFT-calculated and cluster-expansion-predicted activation energies.

Figure 3

Leave-one-out cross validation (LOOCV) of the activation energy from the four methods against the known DFT activation energy in the training set. In panel (a) the black and red data points are the simple broken bond and the KRA broken bond model, respectively. The dashed lines indicate $\pm 0.2$ eV deviation from perfect agreement.

Figure 4

Test set root-mean-square (RMS) error as a function of training set size, expressed as a percentage of the remaining data set (excluding the test set), which are 234 structures. The test set contains 60 structures. The standard deviations of the test set errors are colored as shaded regions.

Figure 5

(a)–(e) Snapshots of Pt-Ni nanoparticles after the KMC simulations at 1000 K using the (a) cluster expansion, (b) parabolic potential model, (c) constant activation energy model, (d) simple broken bond model, and (e) KRA broken bond model. Gray spheres represent Pt and green spheres represent Ni. (f), (g) The first layer Ni composition and overall Ni composition from the KMC simulations as a function of KMC time.

Figure 6

Cluster-expansion-predicted activation energies ($E_a$), averaged over 20,623 hops in randomly generated nanoparticles, as a function of number of nearest-neighbor atoms $n_{\mathrm{Ni}}$ and $n_{\mathrm{Pt}}$ around the diffusing species. (a) The activation energies for Pt. (b) The activation energies for Ni. (c) The differences between the activation energies in panels (a) and (b).