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Publication Figures

Publication Date:
2019-08-19

First Author:
Daniel R. Mason

Title:
Relaxation volumes of irradiation-induced defects in tungsten

Paper Identifier:
CP/19/273

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Figure Reference Title Description Number of Figure Data Items Identifier Download Figure Details
fig 1 a Example convergence study for relaxation volume of the 19 interstitial cluster computed using three methods. Example convergence study for relaxation volume of the 19 interstitial cluster computed using three methods. All three methods converge to very similar values, with a system size error inversely proportional to system size n. Best t lines are indicated to guide the eye. Calculations were performed with the MNB potential. Other potentials and other defects show qualitatively similar results. 0 CF/19/274 Download
fig2b fig2b Below: Geometry used for free-surface calculations. An unsupported free sphere of atoms was constructed and relaxed, then a single defect was generated in the center, and the atoms relaxed again. The volume before and after the defect was placed was computed using qhull.50 One half of the sphere is shown,37 together with high-energy atoms of the defect. Atoms are colored by excess potential energy from 0 eV (blue) to 2 eV (red). Atoms with energy under 0.1 eV are not shown. In this image, a 55 interstitial loop is embedded in a sphere of 180 000 atoms and relaxed with the MNB potential. 0 CF/19/275 Download
fig 1 A typical configuration of defects produced by a collision cascade event in tungsten, A typical configuration of defects produced by a collision cascade event in tungsten, initiated by a 150 keV primary knock-on atom (PKA) and simulated using the method described in Sec. II D. Vacancies (white spheres), and interstitials (red), were identified using a Wigner-Seitz defect analysis.37 The von Mises stress in a [211] plane intersecting the cascade is also shown. Note that close to the defects, the stresses can be as high as 100 GPa, comparable to the shear modulus (? ¼ 160 GPa). In the study below, we analyze the complex stress fields of individual defects and clusters of defects formed in cascades, similar to those shown in this figure. 0 CF/19/276 Download
Relaxation volumes of randomly generated interstitial defect clusters for different numbers of interstitial atoms in the defect, N Relaxation volumes of randomly generated interstitial defect clusters for different numbers of interstitial atoms in the defect, N. DFT values for energies and relaxation volumes from Ref. 13 are shown with filled circles. Crosses: the values computed with MNB potential; open circles are the values computed using the DND potential. Shaded ellipses are drawn to guide the eye to the regions covered by data generated using the relevant potentials. Note that the DND potential tends to predict a higher formation energy and lower relaxation volume of a defect cluster than the MNB potential. 0 CF/19/277 Download
Relaxation volumes of randomly generated vacancy defect clusters for different numbers of vacant sites in the defect, N. Relaxation volumes of randomly generated vacancy defect clusters for different numbers of vacant sites in the defect, N. DFT values for energies are from Ref. 46, the relaxation volumes were computed in this study and are shown with filled circles. Crosses: the values computed with MNB potential; open circles with DND potential. Shaded ellipses are drawn to guide the eye to the regions covered by the potentials. Note that the DND potential tends to predict a lower formation energy and smaller magnitude relaxation volume than the MNB potential. The CEA potential predicts the smallest relaxation volume. The MNB potential data have a high degree of overlap with the DFT relaxation volume data. 0 CF/19/278 Download
fig 5 Formation energy and relaxation volume of low-energy interstitial defect clusters. Formation energy and relaxation volume of low-energy interstitial defect clusters. All clusters and loops are of circular shape. DFT values for formation energies are extrapolated with lines fitted to Ef ¼ a0 ffiffiffi N p ln N þ a1 ffiffiffi N p þ a2 (see Table IV). DFT values for relaxation volumes are extrapolated with lines fitted to ?rel=?0 j j¼ N þ b0 ffiffiffi N p ln N þ b1 ffiffiffi N p þ b2 (see Table V). Note that the energies for ideal interstitial defects computed with the potentials are very similar, but the relaxation volumes differ considerably with the DND potential typically predicting smaller values and CEA4 larger. 0 CF/19/279 Download
fig 6 Formation energy and relaxation volumes of low-energy vacancy defect clusters. Formation energy and relaxation volumes of low-energy vacancy defect clusters. The loops were generated with circular shapes, and the voids as spheres. DFT computed energies of the low-energy vacancy clusters from Ref. 46, with an extrapolated line fitted to Ef ¼ a0N2=3 þ a1. DFT computed relaxation volumes for the same structures in Ref. 46 fitted to ?rel=?0 ¼ b0N2=3 þ b1. 0 CF/19/280 Download
fig 8 Formation energy and relaxation volumes of defect clusters generated by MD cascade simulations Formation energy and relaxation volumes of defect clusters generated by MD cascade simulations. Filled symbols: computed with MNB potential; open circles with DND potential. The formation energy of the cascades computed with the two potentials is very similar, but MNB tends to produce a larger relaxation volume. Note that the cascade configurations were generated with the DND potential, then relaxed with both MNB and DND. Note the wide range of number of Frenkel pairs generated in a single cascade, a characteristic of the stochastic process of loop generation,6 has recently been confirmed experimentally.43 0 CF/19/281 Download
fig10 The first derivative of the pairwise part of selected empirical potentials, in the effective gauge47,83 where ?eq ¼ 1, F[0] ¼ F[1] ¼ 0. first derivative of the pairwise part of selected empirical potentials, in the effective gauge47,83 where ?eq ¼ 1, F[0] ¼ F[1] ¼ 0. The MNB potential is descended from the smooth Ackland-Thetford form,47,48 whereas the DND potential is a piecewise cubic-spline which did not consider the first derivative during fitting. Vertical lines are drawn at first and second nearest neighbor positions. Note that the DND potential is not unstable at short separation, as might be inferred from this plot, as it is stabilized by its many-body part. In MD simulations, the Ziegler- Biersack-Littmark (ZBL) correction is also generally applied at short range.76,84 The Ackland-Thetford pairwise potential is very similar to the MNB. 0 CF/19/282 Download
fig 7 The relaxation volume anisotropy ?, as defined by the ratio of the smallest to the largest partial relaxation volume [see Eq. (10)], for idealized circular prismatic loops. The relaxation volume anisotropy ?, as defined by the ratio of the smallest to the largest partial relaxation volume [see Eq. (10)], for idealized circular prismatic loops. 0 CF/20/308 Download
fig 9 A comparison of the relaxation volume computed using a full relaxation of the cascade (x-axis) to the relaxation volume A comparison of the relaxation volume computed using a full relaxation of the cascade (x-axis) to the relaxation volume predicted using the tabulated fits to the data (Tables III, V, and VII). The diagonal line indicates a 1:1 match— i.e., a perfect reproduction of the relaxation volume. 0 CF/20/307 Download