Machine-learning interatomic potential for radiation damage and defects in tungsten

We introduce a machine-learning interatomic potential for tungsten using the Gaussian approximation potential framework. We specifically focus on properties relevant for simulations of radiation-induced collision cascades and the damage they produce, including a realistic repulsive potential for the short-range many-body cascade dynamics and a good description of the liquid phase. Furthermore, the potential accurately reproduces surface properties and the energetics of vacancy and self-interstitial clusters, which have been longstanding deficiencies of existing potentials. The potential enables molecular dynamics simulations of radiation damage in tungsten with unprecedented accuracy.

Figure 1

Top: short-range repulsion of a W–W dimer given by our DFT, the all-electron DFT-DMol data [38], the screened Coulomb potential $V_{\mathrm{pair}}$, and the trained GAP. The inset highlights the divergence of DFT compared to DFT-DMol at around 1 $\AA$, which is used when sampling structures for the training database. Bottom: The near-equilibrium part of the dimer curve.

Figure 2

Energy-volume curves of the various crystal structures included in the training database. The data points are DFT data and the solid lines are the GAP predictions.

Figure 3

Volume-conserving deformation paths of bulk W computed with GAP and DFT.

Figure 4

Phonon dispersion of bcc W as given by the GAP and compared with DFT [54, 70] and experimental data [71].

Figure 5

Linear thermal expansion (top) and the expansion coefficient (bottom) of bcc W predicted by the GAP and compared with experimental results [72]. GAP data is obtained from both MD simulations and by using the quasiharmonic approximation (QHA).

Figure 6

Elastic constants and bulk modulus of bcc W as functions of temperature. Experimental data are from Ref. [74]. GAP data for the bulk modulus is obtained from both MD simulations and by using the quasiharmonic approximation (QHA). DFT data are shown at 0 K for the elastic constants and as obtained by the QHA for the bulk modulus. The solid lines connecting the GAP points are polynomial fits to guide the eye.

Figure 7

Surface energies predicted by the GAP and compared with DFT. Only the first four surfaces were included in the training database. The experimental values are from Ref. [59].

Figure 8

Reconstruction of the $\left(100\right)$ surface. Top left: initial unrelaxed surface, top right: reconstructed surface. Only the top two surface layers are shown and atoms are coloured according to height, so that atoms in the top layer are green and atoms in the second layer purple. Red arrows show the direction of the displacement vectors with respect to the unrelaxed surface. The bottom figure shows a side-view of the zigzag surface layer.

Figure 9

Total energy difference (left) and force components (right) for stepwise movement of one atom along various crystal directions in bcc W. None of the structures along these specific paths were included in the training database. Results using a previous GAP [54], not trained to any repulsive data, are shown for a few crystal directions for comparison.

Figure 10

Threshold displacement energies obtained with the GAP at 0 K. The colours are linearly interpolated between the (light gray) data points. The average value of the uniformly sampled points is $106.2\pm 5.6\mathrm{eV}$ and the minimum values 45.5 eV for the $\langle 100\rangle$ direction, 51.5 eV for $\langle 111\rangle$ and 78 eV for $\langle 110\rangle$, compared with the experimental values $42\pm 1\mathrm{eV}$ for $\langle 100\rangle$ and $44\pm 1\mathrm{eV}$ for $\langle 111\rangle$ [80].

Figure 11

Formation energies of self-interstitial atoms in bcc W. The formation energy of the ground-state $\langle 11\xi \rangle$ is 10.34 eV in the GAP compared to 10.25 eV in DFT [63].

Figure 12

Top: Energy difference for a $\langle 110\rangle$ dumbbell SIA rotating to the $\langle 111\rangle$ direction, passing through the global minimum $\langle 11\xi \rangle$. Bottom: migration path between adjacent $\langle 11\xi \rangle$ configurations along the $\langle 111\rangle$ direction. DFT data are from Ref. [64].

Figure 13

Formation energies of self-interstitial clusters in W compared between GAP and DFT data from Ref. [82]. Note that only sizes 1 and 2 were fit, so all the other data points serve as tests of the potential.

Figure 14

Binding energies of a divacancy in bcc W at different nearest-neighbor separations compared with DFT from Refs. [22, 85] and our own. Only the 1NN and 2NN configurations were included in the training database. The EAM results are obtained using the potential from Ref. [13] but is representative of the trend reproduced by most other EAM potentials as well.

Figure 15

Formation energies of vacancy clusters in W obtained with the GAP.

Figure 16

The energies and forces given by the GAP at short interatomic distances with and without the external pair potential. The zoomed-in insets show the numerical instability of GAP at extremely short interatomic distances.