Alloy Metallurgy

Al-Zn-Cu-Mg Regression

Summary

Performance on a multi-property Al-Cu-Mg-Zn alloy metallurgy regression suite benchmarked against DFT (PBE) reference data from the OQMD and the evalpot legacy dataset. The benchmark covers bulk formation energies, lattice geometry, elastic constants, solute-solute binding curves, surface energies, stacking fault energies, generalised stacking fault (GSF) curves, and solute-stacking fault interaction energies, targeting precipitate phases (Θ and Θ’’) relevant to Al-alloy design.

Metrics

Bulk properties

Calculated for 8 OQMD structures (Al, Al-Cu, Mg, Mg-Zn, Cu, Zn, and two Al-Cu-Mg precipitate cells). Each structure is relaxed with the LBFGS optimiser with FrechetCellFilter applied, until the maximum force component is below 1×10⁻⁶ eV/Å or 1000 steps are reached. Elemental reference energies are derived from the relaxed pure-element structures and used to compute formation energies.

  1. Formation Energy MAE (meV/atom)

    Mean absolute error of the DFT formation energy per atom.

  2. Volume per Atom MAE (ų/atom)

    Mean absolute error of the relaxed volume per atom.

  3. Lattice Constant MAE (Å)

    Mean absolute error of the relaxed lattice constants.

  4. Beta Angle MAE (°)

    Mean absolute error of the relaxed monoclinic beta lattice angle.

Surface and fault energies

Calculated for FCC Al (OQMD 8100) and Cu (OQMD 635950) and HCP Mg (OQMD 9226) and Zn (OQMD 122929). Each reference structure is first relaxed fully (atoms and cell) with fmax=1×10⁻⁶ eV/Å.

  1. Surface Energy MAE (J/m²)

    Mean absolute error of surface energies γ = (Eslab − N × Ebulk) / (2A). FCC surfaces: (111), (100), (110) for Al and Cu. HCP surfaces: (0001), (10m10), (11m20), (1m101), (10m12), (11m21), (11m22) for Mg and Zn.

  2. Stacking Fault Energy MAE (mJ/m²)

    Mean absolute error of the stable (2/3 Burgers vector) and unstable (5/6 Burgers vector) stacking fault energies for FCC Al and Cu. Atoms are allowed to relax in the direction perpendicular to the fault plane (fmax=5×10⁻³ eV/Å, up to 100 steps).

  3. GSF Energy MAE (mJ/m²)

    Mean absolute error of the generalised stacking fault energy curves for the Θ precipitate {111} slip system (12 displacement points, base cell AIIDA_339739) and the Θ’’ precipitate {0m11} slip system (8 displacement points, base cell AIIDA_481617). Atoms are allowed to relax perpendicular to the fault plane; for Θ’’ the cell z-dimension is also relaxed (fmax=5×10⁻³ eV/Å, up to 200 steps per point).

  4. Solute-Stacking Fault MAE (mJ/m²)

    Mean absolute error of the solute-stacking fault interaction energies for Cu, Mg, Zn, and Si solutes in an Al-FCC matrix (4×4 in-plane, 4 z-planes) and Al solutes in a Cu-FCC matrix (3×4 in-plane, 4 z-planes).

Elastic constants

Calculated for all 8 bulk structures after full cell relaxation. Elastic constants are derived from a stress-strain approach using strain magnitude ±0.5%.

  1. Bulk Modulus MAE (GPa)

    Mean absolute error of the Voigt-Reuss-Hill bulk modulus.

  2. Shear Modulus MAE (GPa)

    Mean absolute error of the Voigt-Reuss-Hill shear modulus.

  3. Elastic Constant MAE (GPa)

    Element-wise mean absolute error across all 21 independent Cij components of the 6×6 Voigt elastic tensor.

Solute-solute binding

Binding energy curves as a function of solute-solute separation distance. Pure-element FCC supercells are constructed and fully relaxed, then pairs of solutes are introduced and the binding energy is computed for all symmetry- inequivalent neighbour shells up to max_index=8.

  1. Solute-Solute Binding MAE (eV)

    Mean absolute error of the solute-solute binding energies. Pairs computed: Zn-Zn, Zn-Cu, Zn-Mg, Zn-Vac, Cu-Cu, Cu-Mg, Cu-Vac, Mg-Mg, Mg-Vac, Vac-Vac in Al-FCC (4×4×4 supercell); Al-Al, Al-Vac, Vac-Vac in Cu-FCC (3×3×3 supercell).

Benchmark speed

Slow: all four tests are marked slow, as the analysis combines their outputs into a single table. Their individual costs are:

  • Bulk properties (test_alzncumg_regression): Fast — 8 bulk cell relaxations; likely less than a minute on GPU.

  • Surface and fault energies (test_alzncumg_fault_surfaces): Medium — surface slab and stacking fault calculations across ~30 surface/fault/GSF combinations; likely minutes on GPU.

  • Elastic constants (test_alzncumg_elasticity): Medium — stress-strain elastic tensor for 8 structures; likely minutes to tens of minutes on GPU.

  • Solute-solute binding (test_alzncumg_solute_solute): Slow — many FCC supercell relaxations per solute pair; likely hours on GPU.

Data availability

Input structures:

  • OQMD structures: Al (8100), Al-Cu alloy (635950), Mg (9226), Mg-Zn (122929), Cu (695020), Zn (10434) — from the Open Quantum Materials Database (https://oqmd.org/) under the Creative Commons Attribution 4.0 International licence.

  • Non-OQMD precipitate cells: Θ’’ (NOTINOQMD_00001) and Θ (NOTINOQMD_00002) — from the legacy evalpot dataset.

  • Special GSF base cells: AIIDA_339739 (Θ {111}) and AIIDA_481617 (Θ’’ {0m11}) — VASP POSCAR cells from AiiDA DFT workflows included in the S3 data bundle.

Reference data:

  • DFT (PBE) values from the evalpot legacy dataset, stored in AlZnCuMg_regression/references/DFT.json.

  • OQMD:

    • Saal, J.E., Kirklin, S., Aykol, M., Meredig, B., and Wolverton, C., “Materials Design and Discovery with High-Throughput Density Functional Theory: The Open Quantum Materials Database (OQMD)”, JOM 65, 1501–1509 (2013). https://doi.org/10.1007/s11837-013-0755-4

    • Kirklin, S., Saal, J.E., Meredig, B., Thompson, A., Doak, J.W., Aykol, M., Rühl, S., and Wolverton, C., “The Open Quantum Materials Database (OQMD): assessing the accuracy of DFT formation energies”, npj Computational Materials 1, 15010 (2015). https://doi.org/10.1038/npjcompumats.2015.10