2021 年 62 巻 5 号 p. 582-589
First-principles calculations were employed to evaluate the trapping energy of a H atom in a screw dislocation and an edge dislocation in Al. After obtaining the dislocation core structure in the absence of H, we calculated the trapping energy of H at several tens of possible trapping sites in the dislocation core and its vicinity. The maximum trapping energies were 0.08 and 0.15 eV/atom for the screw and edge dislocations without the zero-point vibrational energy correction, while they were 0.11 and 0.18 eV/atom for the screw and the edge dislocations with the correction. The calculation conditions employed in the present work correspond to the line density of approximately 1.0 H atom/nm along the dislocation line, which is sufficiently low to exclude H-H interaction.
Various lattice defects trap H atoms in Al. First-principles calculations have revealed the trapping energy of the H atom in the solid-solution state,1) vacancy,2) grain boundary,3) precipitate-matrix interface,4,5) and free surface.3,6) Woodward et al. performed first-principles calculations to obtain the detailed atomic structures of a screw and edge dislocation core in Al, and the resulting splitting widths of the partial dislocation cores were in good agreement with the experimental observations.7) Regarding the H-trapping energy of Al dislocations, there is a preliminary result (Screw: −0.01 eV/atom H, Edge: 0.17 eV/atom H) calculated from first-principles by Matsumoto et al.8) They have evaluated the H trapping energy for many kinds of defects in Al.9,10) As for dislocations, however, no detailed report on the calculation methods and their results has been published.
Some researchers have attributed one of the peaks observed in the thermal desorption spectra (TDS) of H from the Al alloys to the H trapping in the dislocation. Young and Scully estimated the H-trapping energy of Al dislocation to be 27.3 kJ/mol (0.28 eV/atom)11) and Izumi and Ito 31.1 kJ/mol (0.32 eV/atom).12,13) Ebihara et al.14) suggested, using numerical analysis on the TDS, that the H-trapping energy could be relatively low considering the annihilation of dislocations by an increase in temperature. The TDS of H from Al alloys has many unclear points. Although the H-trapping energy may differ between screw and edge dislocations, experiments cannot reveal the difference. Therefore, it is necessary to calculate the H-trapping energy due to the dislocations in Al using first-principles calculations.
It is still difficult to evaluate the number of H atoms a dislocation line can trap per unit length in the dislocation core and the elastic strain field around it. Conversely, it is now possible to calculate the H-trapping energy by a dislocation core in a low-density trapping state of approximately 1.0 atom/nm, which is sufficient to exclude H-H interactions. This study calculates such a H-trapping energy in a screw and edge dislocation in pure Al using first-principles calculations. Before this study, Shimizu et al. applied our preliminary calculated H-trapping energy of dislocations on their H-partitioning calculations in Al alloys.15–17) In the present paper, we describe the details of the calculation method, the calculated results, and their discussion.
To obtain the dislocation core structures of the screw and edge dislocations in an Al crystal using first-principles calculations, we constructed periodic boundary cells in which two dislocation cores form a quadrupole configuration. Figures 1(a) and (b) show an overview of the periodic boundary cells containing screw dislocations, and Figs. 1(c) and (d) edge dislocations. The dislocation core structures shown in the figures are the structures after the structural relaxation calculations described later. The VESTA18) software was used to visualize the atomic structures.
Periodic cells, including dislocations, for determining the core structures. (a) [1 1 0]/2 screw dislocations. (b) The DDM20) for the screw dislocations. The arrow size and length between two atoms show the displacement along the Burgers vector. (c) [1 1 0]/2 edge dislocations. (d) Lines show the atomic rows, which guide the eye.
First, we describe the process of constructing a periodic cell containing two screw dislocations. The starting point is a perfect face-centered cubic (fcc) crystal cell of Al having a lattice constant of 0.4039 nm, including four atoms. A supercell of the perfect crystal having an e1 axis of [5 −5 10], e2 axis of [6 −6 −6], and e3 axis of [1 1 0]/2 was formed. As shown in Fig. 1(b), the atomic displacement of the screw dislocation was given at two points: the lower-left and upper-right parts of the cell. In the lower-left and upper right parts of the cell, the Burgers vectors of the dislocation are [110]/2 and [−1 −1 0]/2. Between the two dislocation cores, the Burgers vectors have opposite signs to obtain a periodic boundary cell. We set the initial value of the split width of the partial dislocation cores to 0.742 nm. To reduce the elastic strain due to the dislocations before the structure relaxation, we inclined the e1 and e2 axes by 1/2 of the Burgers vector.
Next, we describe the process of constructing a cell containing two edge dislocations. A perfect crystal lattice supercell an e1 axis of [9 9 0], e2 axis of [−8 8 8], and e3 axis of [1 −1 2]/2 was formed based on the basic Al lattice. Next, we extended the e1 axis by 1/2 of the Burgers vector and inserted extra-half atomic planes so that the two dislocation cores had opposite signs in the lower-left and the upper-right parts. Even if we introduce a perfect dislocation core as the initial state, the dislocation core splits after the structural relaxation, as shown in Fig. 1(d).
We constructed periodic boundary cells of the quadrupole arrangement with two dislocations having different signs, as described above. Table 1 shows the structural parameters of the cells. Subsequently, we performed structural relaxations by first-principle calculations using the Vienna ab initio simulation package (VASP)19–22) to obtain the dislocation core structures. The cutoff energy (ENCUT) of the plane-wave basis was 325 eV, and the threshold of the structure relaxations was 0.01 eV/Ang., which is the force acting on the atoms. The total energies converged to less than 1.0 meV. The smearing parameter at the Fermi surface was 0.2 eV. The mesh of the reciprocal lattice space includes the origin (Γ point) and has a division number of 1 × 1 × 16 for the screw-dislocation cell and 1 × 1 × 10 for the edge-dislocation cell.
Figure 1(a) shows the screw dislocation cell after the structural relaxation. Figure 1(b) shows the detailed structure of the screw dislocation core using the arrows of the differential displacement map (DDM),23) in which the direction and length of the arrows show the displacement along the Burgers vector between two atoms. Figure 1(c) shows the edge-dislocation cell after the structural relaxation, while Fig. 1(d) shows the dislocation core structure in an easy-to-understand manner, by drawing lines on the atomic rows. We determined the splitting width to be approximately 0.8 nm for the partial screw dislocation cores and ∼1.0 nm for the partial edge-dislocation cores. Woodward et al.7) found that the splitting width of the partial dislocations in Al was in the range of 0.50–0.75 nm for the screw dislocation and 0.70–0.95 nm for the edge dislocation, although their definition of the width varies. Their results agree with our calculated splitting widths. Table 1 shows the cell parameters after relaxation and the number of atoms.
2.2 H-trapping energyThe cell used to obtain the dislocation core structure in the previous section has a short length in the e3-axis direction; thus, the distance between the trapped H atom and its image in the neighboring cell would be short because of the periodic boundary conditions. Here there is a concern that a repulsive interaction may exist between the H atoms. Therefore, we wish to perform calculations using a relatively long e3 axis. However, if we increase the length of the e3 axis, the number of atoms increases considerably. Therefore, we constructed the unit cell for the H-trapping energy by cutting out the atoms with one dislocation core into a polygonal shape and increasing the cell size in the e3-axis direction. By this method, we successfully reduced the number of atoms in the cell.
Figure 2(a) shows the cell for calculating the H-trapping energy for the screw dislocation core. We cut out the atoms with a pair of partial screw dislocation cores into a hexagonal shape and tripled the length of the e3 axis. As shown in Fig. 2(b), fixing the displacement of atoms near the surfaces to about three atomic layers reduces the influence of the surface. For the edge-dislocation core, as shown in Fig. 2(c), we cut out the atoms with a pair of partial edge-dislocation cores into a quadrangular shape and doubled the length of the e3 axis. As shown in Fig. 2(d), fixing the atom displacement near the surfaces reduces the surface influence. Table 1(b) summarizes the length of the e3 axis and the number of atoms in the cell for H-trapping calculations.
Polygonal cells surrounded by a vacuum region for calculating H-trapping energy. (a), (b) [1 1 0]/2 screw dislocation. (c), (d) [1 1 0]/2 edge dislocation. See text.
We calculated the H-trapping energies using the above cells by first-principles calculations as follows. By placing one H atom at a possible trapping site, we conducted structure relaxation of atomic positions. Here, we fixed the e1, e2, and e3 axes. In this case, the line densities of the trapping H atom are 1.2 and 1.0 atom/nm for the screw and edge dislocations, respectively. From our experience in H-trapping calculations on the grain boundary,3) we know that if the distance between two trapping H atoms is more than 0.4–0.5 nm, there will be no appreciable repulsive interaction. Therefore, the line density of approximately 1.0 atom/nm is sufficiently low for estimating the trapping energy of a single H atom.
We calculated the dissolution energy of the H atom in the Al perfect lattice of a 4 × 4 × 4 supercell (256 Al atom/cell), with a 3 × 3 × 3 reciprocal space mesh, including the origin (Γ point). The total energy is indicated by Etot(256Al) without H and Etot(256Al, H) with one H atom at the interstitial tetrahedral site. The total energy of the above cell, including one dislocation line, is Etot(Al disl.) without H and Etot(Al disl., H) with one H atom in or near the dislocation core. Using these energies, we obtain the trapping energy of H (Etrap) as follows.
| \begin{align*} E_{\text{trap}} &= E_{\text{tot}}(\text{Al disl., H}) - E_{\text{tot}}(\text{Al disl.}) \\ &\quad - \{E_{\text{tot}}(\text{256Al, H}) - E_{\text{tot}}(\text{256Al})\} \end{align*} |
When an H atom can be trapped, Etrap is negative. However, if we refer to the literature’s trapping energy, it is conventionally assumed to be positive (−Etrap). The larger the positive value, the stronger the trapping. We consider the zero-point vibrational energy (ZPE) correction only at the highest trapping energy site.
The possible trapping sites in a dislocation core and its vicinity were generated from the tetrahedral and octahedral sites in the starting Al lattice’s unit cell when we constructed the cell containing dislocation cores. Although we have many possible trapping sites, we calculate the H-trapping energies only at several tens of the sites in the dislocation core and its vicinity, due to the calculation time limitation.
Figure 3 shows the approximate magnitude of the trapping energy and the location of the trapping site. The blue cross symbol (×) in Fig. 3(a) indicates the site where the trapping energy was negative (trapping is impossible). Most of the sites had a similar configuration to the octahedral site in the fcc Al lattice. Although these are not H-trapping sites, it shows how far away from the dislocation core the trapping energy has been calculated. Figure 3(b) indicates the sites that trap H atoms with red circle symbols. The larger the radius of the circle, the larger the trapping energy. We obtained the maximum trapping energy of 0.08 eV/atom without the ZPE correction.
Distribution of the trapping sites and trapping H atom energies around the [1 1 0]/2 screw dislocation core in Al. (a) Blue cross symbols (×) indicate that the H-trapping energy is negative. (the H atom cannot be trapped.) (b) Red circle symbols indicate the position of the H-trapping site in the cell. The radius of the circle corresponds to the size of the trapping energy. The maximum trapping energy (*) is 0.08 eV/atom (0.11 eV/atom with ZPE correction). This figure does not show the movement of the dislocation core due to the H atom presence. It shows the DDM arrows of the screw dislocation core in the absence of trapping H atoms. See text.
We calculated the ZPE correction only for the highest energy-trapping site. Table 2 shows the results. In our calculations, the ZPE of the solid-solution state of an H atom at a tetrahedral site calculated in the 4 × 4 × 4 supercell of fcc Al was 0.158 eV/atom, which agrees with Wolverton’s result of 0.153 eV/atom.1) The ZPE of the highest energy-trapping site in the screw dislocation core was 0.124 eV/atom. These ZPEs resulted in the correction of 0.034 eV/atom. The highest trapping energy was estimated to be 0.11 eV/atom with the ZPE correction for the screw dislocation.
The H-trapping site position shown in Fig. 3 is displayed based on the trapping site relaxed coordinates obtained using first-principles calculations. Here, we do not show the movement of the dislocation core affected by H trapping. In other words, the structure of the partial dislocation cores was drawn by the DDM arrows when there is no H atom. In some cases, the H-atom presence shifts the dislocation core position by a distance of approximately 1–2 atoms. However, the details are not shown in the present paper because they are complicated.
We investigated the atomic configuration around the highest energy-trapping site in the screw dislocation core. For reference, Fig. 4(a) shows the H atom trapped at a tetrahedral site in the fcc Al perfect lattice and the surrounding Al atoms. The Al–H and Al–Al distances are 1.88 Å and 3.07 Å, respectively. These distances in the absence of an H atom are 1.75 Å and 2.86 Å, respectively, indicating that the surrounding Al atoms are pushed apart by the trapping of the H atom. Contrarily, Fig. 4(b) shows the highest energy-trapping site in the screw dislocation. The Al atoms surround the trapping H atom with the maximum Al–H distance of 2.09 Å. This maximum distance is approximately 11% longer than when an H atom is at the tetrahedral site in fcc Al. The long Al–H distance may appear to be one of the causes of the increase in the trapping energy. On the other hand, the volume of a tetrahedron of four Al atoms containing a H atom is smaller in the case of screw dislocation (9.67 Å3) than in the case of tetrahedral site in the Al lattice (10.25 Å3). In the case of screw dislocation, the size of trapping energy may not correlate with the size of free volume, because the trapping energy is very small.
H-trapping sites. (a) Solid-solution state of the H atom at a tetrahedral site in the fcc Al lattice. (b) Trapping state with the highest trapping energy (0.11 eV/atom with ZPE) in the screw dislocation core. Figure 3(b) shows this site. Numbers present the interatomic distances in angstrom (Å). The volume of tetrahedron containing a H atom is also shown in Å3.
As in the previous section, we calculated the H-trapping energy at several tens of possible trapping sites near the edge-dislocation core. Figure 5(a) shows the approximate magnitude of the trapping energy and the location of the sites. The blue cross symbol (×) in Fig. 5(a) indicates the sites that cannot trap the H atom because of zero or negative trapping energy. Most of the sites are similar to the octahedral sites in the fcc Al lattice. Figure 5(b) indicates the H-trapping sites having a positive trapping energy with red circle symbols. The larger the radius of the circle, the larger the trapping energy. The maximum trapping energy was 0.15 eV/atom without the ZPE correction. As with the screw dislocation core described in the previous section, the dislocation core movement due to the H-atom influence is not shown in this figure because it is complicated.
Distribution of the trapping sites and trapping H atom energies around the [1 1 0]/2 edge-dislocation core in Al. (a) Blue cross symbols (×) indicate that the H trapping energy is negative. (H atom cannot be trapped.) (b) Red circles indicate the H-trapping site. The radius of the red circle corresponds to the size of the trapping energy. The maximum trapping energy (*) is 0.15 eV/atom (0.18 eV/atom with ZPE correction). This figure does not show the movement of dislocation core due to the H atom presence. See text.
The ZPE at the site of the highest trapping energy in the edge-dislocation core was 0.131 eV/atom, and thus, the correction of the ZPE was 0.027 eV/atom. Finally, the highest trapping energy at the edge-dislocation core was determined to be 0.18 eV/atom with the ZPE correction. Table 2 summarizes these results.
Figure 6 shows the configuration of the Al atoms around the highest trapping energy site. At this site, the maximum Al–H distance is 2.16 Å, which is longer than the Al–H distance of 1.88 Å for the H atom in a solid-solution state at the tetrahedral site in fcc Al. In particular, the Al–H distance of 2.16 Å is longer than the maximum Al–H distance of 2.09 Å in the screw dislocation core. The longer A–H distance is accountable for the higher H-trapping energy in the edge-dislocation core compared to that in the screw-dislocation core, due to the larger free volume. The volume of Al tetrahedron containing a H atom is 11.26 Å3 for edge dislocation while 9.67 Å3 for screw dislocation at the strongest trapping sites.
Trapping state with the highest trapping energy (0.18 eV/atom with ZPE) in the edge-dislocation core. Figure 3(b) shows this site. Numbers present interatomic distances in angstrom (Å). The volume of tetrahedron containing a H atom is also shown in Å3.
The calculated results show that the maximum H-trapping energies without the ZPE correction were 0.08 and 0.15 eV/atom for the screw and edge-dislocation cores, respectively. The trapping energies with the ZPE correction were 0.11 and 0.18 eV/atom H for the screw and edge-dislocation cores, respectively, as shown in Table 2. At this time, the trapping density was approximately 1.0 atom/nm along the dislocation line.
The peak observed in the TDS measurement is attributed to the H trapping at the dislocation core, and its trapping energy was estimated to be 0.28 eV/atom by Young and Scully11) and 0.32 eV/atom by Izumi and Ito.12,13) Their experimentally estimated values are approximately 1.5–2.0 times larger than our calculated values (edge: 0.18 eV/atom). In their experimental estimation, the usual Choo-Lee plot24) was used to determine the trapping energy. As pointed out by Ebihara et al.,14) this method may overestimate the trapping energy because it does not consider the influence of the reduction in the number density of the dislocations during the temperature rise in the TDS measurement.
Our calculated trapping energies of dislocations smaller than the experimental values appear reasonable, considering the H-trapping energies of other defects. In the previous calculations, the H-trapping energy of a single vacancy in Al was ∼0.30 eV/atom,2) and that of the grain boundary was ∼0.25 eV/atom,3) which are larger than the calculated H-trapping energy of a dislocation (edge: 0.18 eV/atom). The size of the free volume in the defect decreases in the order of single vacancy > grain boundary > edge-dislocation core > screw-dislocation core. It appears that the size of the trapping energy correlates well with the size of the free volume in defects. The volume of Al tetrahedron containing a H atom is 11.26 Å3 for edge dislocation while 9.67 Å3 for screw dislocation as shown in Fig. 4 and 6. It should be noted here that the volume of Al tetrahedron in Al lattice (10.25 Å3) is a little larger than that in screw dislocation. It indicates that a small trapping energy may not correlate with a free volume.
Dislocations, particularly edge dislocations, have relatively large elastic strains away from the dislocation core; thus, there may be many interstitial sites that can weakly trap H atoms. If the number of trap sites is high, there is a possibility that an appreciable number of sites trap H atoms as a whole, even if the trap is weak. It was impossible to sufficiently calculate the trapping energy at many positions away from the dislocation core due to computational time limitations.
The peculiar behavior may occur when the dislocation core traps many H atoms with a high density, although its possibility of realization is not high because the concentration of H in Al lattice is very low in a realistic situation. A trial calculation conducted by Itakura, one of the authors of the present paper, suggested that a high-concentration H trapping could merge two partial edge-dislocation cores.25) Thus, there is a concern that the splitting width of two partial dislocation cores can change depending on the arrangement of trapping H atoms. To investigate this behavior, we should calculate the multiple H-trapping states near the dislocation core and the barrier energy of partial core movement.
Therefore, it is not clear how many H atoms a single dislocation can trap per unit length of the dislocation line. To clarify this, it is necessary to calculate the multiple H trapping state where the trapped H atoms are close. These calculations are very time consuming and remain a challenge to be tackled in the future.
Before the publication of the present paper, Shimizu et al. applied our preliminary calculated results (screw: 0.08 eV/atom, edge: 0.17 eV/atom) in their studies.15–17) The trapping energies cited in the literature are almost the same as those in the present paper, resulting in almost no difference in their H-partitioning calculations.
We determined the H-trapping energy near dislocation cores in Al using first-principles calculations. The maximum H-trapping energies were 0.08 and 0.15 eV/atom for the screw and edge-dislocation cores, respectively, without the ZPE correction. While with the ZPE correction, they were 0.11 and 0.18 eV/atom, respectively. The size of the ZPE correction was as small as 0.02–0.03 eV/atom. Note that we calculated the trapping energies within several tens of possible trapping sites and found the highest trapping energy sites in this limit. These trapping energies are estimated when the dislocation lines are trapped by H at a line density of approximately 1.0 atom/nm. The calculations for the lower and higher trapping densities would be conducted in future work.
This work was supported by Japan Science and Technology Agency (JST) under the Collaborative Research Based on Industrial Demand Heterogeneous Structure Control: Toward Innovative Development of Metallic Structural Materials, and JST CREST Grant Number JPMJCR1995 Japan. The authors thank Hiroyuki Toda, Kazuyuki Shimizu, Kyosuke Hirayama, and Kenji Matsuda for helpful discussion.
