Simulating Segregation in a Ternary Cu-Pd-Au Alloy with Density Functional Theory, Machine Learning, and Monte Carlo Simulations
Y. Yang, Z. Guo, A. J. Gellman and J. R. Kitchin, “Simulating Segregation in a Ternary Cu-Pd-Au Alloy with Density Functional Theory, Machine Learning, and Monte Carlo Simulations”, The Journal of Physical Chemistry C 126(4), 1800-1808 (2022). https://doi.org/10.1021/acs.jpcc.1c09647
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Question
Pd-based alloys are used in hydrogen separation membranes and in catalysis (for example vinyl acetate synthesis on AuPd), and their performance depends on surface composition, which differs from the bulk because of segregation. Most simulations of segregation treat binary alloys, and the few ternary studies relied on empirical potentials or very small DFT slabs. The paper asks whether a neural network potential trained on DFT can make Monte Carlo simulations of a ternary Cu-Pd-Au fcc(111) surface feasible across the whole composition triangle, and how the predictions compare with high-throughput measurements on composition spread alloy films (Yang et al. 2022).
Approach
DFT energies came from VASP with PBE, PAW potentials, a 400 eV cutoff and about 5 k-points per reciprocal angstrom (convergence error about 2 meV/atom). The authors generated 5278 fcc(111) slabs: 5100 random slabs of several small cell sizes at six lattice constants between pure Cu (3.637 Å) and pure Au (4.174 Å), split 9:1 into training and validation, plus 178 larger 12 × 12 × 5 slabs held out as a generalization test. A Behler-Parrinello network with one subnetwork per element (two hidden layers of ten neurons, 221 parameters each) used G2 symmetry functions at three η values and a 6 Å cutoff, trained with AMP and ASE. They analyzed the fingerprint space directly, showing that for a given η the fingerprints lie on triangular planes set by lattice constant and by layer (surface, subsurface, bulk), and that the training data nearly fill those planes. Monte Carlo atom swaps with Metropolis acceptance were then run at 600 K on 10 × 10 × 15 slabs with Vegard’s-law lattice constants, at 24 bulk compositions, using 15,000 accepted swaps and averaging the last 6000. Results were compared with LEIS and EDX data on Cu-Au-Pd composition spread films from Gellman’s group (Yang et al. 2022).
Key findings
Training and validation mean absolute errors were both about 2 meV/atom with no sign of overfitting, and the larger test slabs had a comparable MAE with a small systematic bias. In the simulations Au was enriched at the surface at every composition, Cu was depleted everywhere (most strongly near the CuAu edge), and Pd was depleted except near the PdCu binary, where it segregated to the surface. Expressing pairwise segregation as exchange free energies, the Au-Pd values were close to parity with experiment, Au over Cu was overestimated, and Pd versus Cu had the opposite sign from experiment, which shows Cu enrichment over Pd (Yang et al. 2022).
The authors then tested explanations for the discrepancies. The CuPd B2 phase (0.4 < xCu < 0.7) reduces but does not invert Cu segregation in experiment, so it cannot explain the sign error. Surface relaxation was checked on 285 swaps in a 2 × 2 × 5 slab: unrelaxed network and relaxed DFT swap energies differed by under 0.2 eV (mostly under 0.1 eV) and always agreed in sign for interlayer swaps, so relaxation largely cancels, as found for AuPd. Vibrational free energy at 1000 K changed dilute Pd-in-Cu and Cu-in-Pd segregation energies only slightly (for example -0.014 to -0.063 eV and 0.146 to 0.168 eV) without changing sign. The overestimated Au segregation in CuAu was linked to PBE: computed formation energies of Cu3Au, CuAu and CuAu3 are much less negative than experiment (for example -56 versus -93 meV/atom for CuAu), and PBE underestimates surface energies, especially for Au (0.87 versus 1.51 J/m2). For CuPd, the authors point to surface orientation, because the experimental films are polycrystalline. On fcc(110), unlike fcc(111), relaxation reverses the sign of dilute segregation energies, and Monte Carlo on CuPd(110) with a second network trained on 1000 slabs to predict relaxation energy reproduced the experimental Cu segregation trend, while the unrelaxed simulation did not (Yang et al. 2022).
Significance
This extends the neural network plus Monte Carlo approach used for AuPd to a three-component alloy over its full composition space, and uses fingerprint-space analysis to argue that small training cells suffice for larger simulation cells. Equally useful is the systematic accounting of why simulation and experiment disagree: relaxation and vibrations matter little on fcc(111), functional errors explain the CuAu overestimate, and the single-facet slab model is the likely culprit for CuPd (Yang et al. 2022).