General purpose potential for glassy and crystalline phases of Cu-Zr alloys based on the ACE formalism

A general purpose machine-learning interatomic potential (MLIP) for the Cu-Zr system is presented based on the atomic cluster expansion formalism [R. Drautz, Phys. Rev. B 99, 014104 (2019)]. By using an extensive set of Cu-Zr training data generated withdensity functional theory, this potential describes a wide range of properties of crystalline as well as amorphous phases within the whole compositional range. Therefore, the machine learning interatomic potential (MLIP) can reproduce the experimental phase diagram and amorphous structure with considerably improved accuracy. A massively different short-range order compared to classica interatomic potentials is found in glassy Cu-Zr samples, shedding light on the role of the full icosahedral motif in the material. Tensile tests of B2-CuZr inclusions in an ${\mathrm{Cu}}_{50}{\mathrm{Zr}}_{50}$ amorphous matrix reveal the occurrence of martensitic phase transformations in this crystal-glass nanocomposite.

Figure 1

Energy-volume relation of the training and test data sets (a), energy error with respect to the distance from the convex hull (b), force scatter plot (c), and cumulative energy RMSE as function of the distance from the convex hull (d). In (b) and (d), the impact of using training data that covers a wide range of energies and of the applied energy based weighting scheme can be observed.

Figure 2

Cu GSF energy along $\langle 112\rangle$ on a {111} plane and $\gamma$ surface calculated with DFT and ACE.

Figure 3

Phase diagrams based on the work of He et al. [10] (a) and calculated with our potential (b), along with $0\mathrm{K}$ formation energies calculated with DFT and ACE (c), and Gibbs energies of ${\mathrm{Cu}}_5\mathrm{Zr}$ and CuZr phases compared to the stable two phase mixtures at corresponding concentrations as calculated with ACE. For (b) and (d), formation energies at $0\mathrm{K}$ were shifted to the DFT values.

Figure 4

Finite-temperature phonon dispersion including anharmonic interactions calculated with the self-consistent phonon method (a). For comparison, the phonon dispersion as calculated with the harmonic approximation is shown as a black line. Also shown is the energy along the transformation path from B2 to B19' used by Hatcher et al. [88] for NiTi (b). Contrary to their findings for NiTi, our potential predicts a small barrier for CuZr.

Figure 5

Errors in formation energy of our ACE and several classical IPs compared to DFT data for structures included in training (white background) and extrapolating to Laves phases (gray shaded area). All formation energies are calculated with respect to ${\mathrm{Cu}}_{\mathrm{FCC}}$ and ${\mathrm{Zr}}_{\mathrm{HCP}}$.

Figure 6

Total structure factor of MGss at $300\mathrm{K}$ simulated with the ACE MLIP (a) and Mendelev EAM potential (b) in comparison to XRD data from Ref. [98]. Experimental compositions were ${\mathrm{Cu}}_{65}{\mathrm{Zr}}_{35}$ and ${\mathrm{Cu}}_{35}{\mathrm{Zr}}_{65}$ instead of ${\mathrm{Cu}}_{64}{\mathrm{Zr}}_{36}$ and ${\mathrm{Cu}}_{36}{\mathrm{Zr}}_{64}$, respectively. S(q) are shifted by $+2$ and $+1$ for ${\mathrm{Cu}}_{64}{\mathrm{Zr}}_{36}$ and ${\mathrm{Cu}}_{50}{\mathrm{Zr}}_{50}$ to make them more visible. The data obtained with the Mendelev potential do not show an accurate splitting of the second peak. Additionally, the height of the first peak of ${\mathrm{Cu}}_{64}{\mathrm{Zr}}_{36}$ is much more accurate for the ACE potential.

Figure 7

Population of Voronoi polyhedra in ACE MGs samples.

Figure 8

Formation energy of MGs samples produced with ACE and Mendelev potentials as calculated with DFT, ACE, and Mendelev potentials. ${\mathrm{Cu}}_{\mathrm{FCC}}$ and ${\mathrm{Zr}}_{\mathrm{HCP}}$ were used as reference states.

Figure 9

Atomic strain in tensile tests of Cu-Zr glasses with different compositions and at different temperatures. Snapshots were taken at $12\%$ strain. At low temperatures, the strain is localized in shear bands. At high temperatures, multiple shear transition zones are activated, leading to a more homogeneous deformation mechanism.

Figure 10

Shear band formation and structure evolution of a ${\mathrm{Cu}}_{50}{\mathrm{Zr}}_{50}$ matrix with a crystalline CuZr-B2 inclusion. The strain localizes at the interface until a shear band is formed. A martensitic phase transition from the B2 to B19' structure of roughly the same width as the shear band can be observed in the crystalline phase.