- Email: [email protected]
Binding of multiple H atoms to solute atoms in bcc Fe using first principles
Available online at www.sciencedirect.com
Acta Materialia 59 (2011) 5812–5820 www.elsevier.com/locate/actamat
Binding of multiple H atoms to solute atoms in bcc Fe using first principles W. Counts a,⇑, C. Wolverton a, R. Gibala b a b
Department of Materials Science and Engineering, Northwestern University, Evanston, IL, USA Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI, USA Received 16 March 2011; received in revised form 27 May 2011; accepted 29 May 2011 Available online 27 June 2011
Abstract We previously performed a series of density functional theory calculations to investigate the interaction between single H atoms and point defects in body-centered cubic (bcc) Fe (Counts W, Wolverton C, Gibala R. Acta Mater 2010;58:4730). Here, we extend that work to a systematic study of binding between multiple H atoms and solute atoms in bcc Fe. We investigate the binding of multiple H atoms to one another, to interstitial C and to substitutional solutes. Our study shows the following: (i) H–H interactions are weak. The maximum attractive H–H binding energy is around 0.03 eV, which agrees with experimental values. (ii) The maximum attractive incremental binding energy of a second H atom to a C–H defect pair is 0.07 eV. (iii) We investigate the ability of 3d transition metal solutes to bind up to five H atoms. The binding energy of the second H to a 3d transition metal solute is attractive with a value 0.03 eV greater than binding of energy of the first, independent of solute. The binding energies of the third to fifth H atoms vary but are generally positive. Based on a stability analysis of the H binding energies, we find that the largest H–solute defect complex for V, Cr, Co, Ni and Zn contains two H atoms, while for Sc, Ti, Mn, and Cu the largest defect complex contains four H atoms. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Density functional; Hydrogen embrittlement; Ferritic steels; Iron
1. Introduction Hydrogen embrittlement (HE) of body-centered cubic (bcc) iron (a-Fe) is a well-known but still not well under stood phenomenon. A number of different HE mechanisms have been proposed, including decohesion [1,2], hydrogenenhanced localized plasticity [3,4], and the hydrogen-enhan ced strain-induced vacancy [5,6] models. While these mechanisms are all distinct, each of them requires that H accumulates within the material to form a high H concentration region that enables the respective HE mechanism. Such HE-susceptible regions form as a result of H diffusion, making each of these different HE mechanisms dependent on H diffusion.
⇑ Corresponding author.
E-mail address: [email protected] (W. Counts).
H readily diffuses through a-Fe [7], but lattice defects impede the motion of H [8]. However, experiments are less clear concerning the strength of the H–defect interactions because it is not possible to directly measure the H–defect binding energy. In the literature, H–defect binding energies have been probed using a variety of different experimental techniques: magnetic relaxation [9], H permeation [10], thermal detrapping [11,12] and internal friction [13] among others. The magnitude of the H–defect interaction in each of the aforementioned experiments is then determined with the aid of a model that includes a binding energy term. Using this general approach, experimental H–point defect interactions like H–H self-interactions [9], as well as H–C [9], H–Ti [10] and H–h binding [11,12] in bcc Fe have been quantified (h = vacancy). Experiments are often also unable to extract details about the geometry of the H–point defect cluster. In one case, Meyers et al. [11] used ionchanneling to obtain additional information about the
1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.05.058
W. Counts et al. / Acta Materialia 59 (2011) 5812–5820
position of the D atom. They showed that D atoms occu˚ from the octahedral site pied a position displaced 0.4 A (o-site). However, this technique did not explicitly identify the lattice defect that had bound D. Theoretical and computational investigations into H– point defect interactions are an important complement to experimental studies because they can provide information to validate and interpret experimental results and can also provide details that are difficult to access experimentally. Computational approaches, if sufficiently accurate, are valuable because they can directly quantify the binding energy of H with a particular point defect in a given configuration. One example where insight from theory was valuable is the D–h defect pair. Initially, experiments and theory both revealed strong D–h binding in bcc Fe. Results from an effective medium theory (EMT) based study by Besenbacher et al. [12] revealed that the D–h binding energy for one to six D atoms was favorable. Guided by these theoretical results, the authors of Ref. [12] were able to explain the emergence of a second plateau region in their detrapping data at higher D concentrations. They argued that when the concentration of D is larger than that of vacancies, vacancies bind multiple D atoms. Thus, the second detrapping plateau observed in the experiment was due to multiple D atoms binding to the vacancy. Both EMT and a diffusion model were then used to qualitatively determine that the incremental vacancy binding energy for the first two D atoms is 0.63 eV,1 and 0.43 eV1 for the subsequent (3–6) D atoms. Density functional theory (DFT) has been successfully used to model h–point defect interactions in a variety of systems. There is good agreement between available experimental data and DFT concerning h–solute interactions in Al [14,15], Mg [16], bcc Fe [17,18] and facecentered cubic (fcc) Fe [19]. DFT has also shown that vacancies are effective H traps in various metal systems. DFT has shown that it is energetically favorable for a vacancy to bind up to between 10 [20] and 12 [21] H atoms in fcc Al, up to 9 H atom in hcp Mg [20], and up to 6 H atoms in bcc Fe [22]. In the case of bcc Fe, the DFT-based binding energies agree well both qualitatively and quantitatively with the experimental results of Besenbacher et al. [12]. It should be noted that binding energies alone are not enough to determine how many H atoms a defect will bind because the defect configuration depends on both the binding energies and defect concentrations. For example, the H–h binding energy in Al is 0.35 eV,1 and the binding energy of a second H to the vacancy is also positive, around 0.30 eV.1 In cases when the concentration of H (cH) is much larger than the concentration of vacancies (ch), there is a large probability that H atoms will find vacancies along their diffusion path. These H atoms will
1 A positive binding energy refers to an attractive interaction, and negative value refers to a repulsive interaction between defects.
5813
bind to the vacancy and form H–h defects because the H–h binding energy is positive. The excess H atoms will find these H–h defects and form defect clusters containing a vacancy and two H atoms defects (2H–h) because the binding energy of a second H to the vacancy is also positive. In cases when ch cH, it is far more likely that a H atom will encounter a vacancy rather than a H–h defect. Thus it is more likely in this case to form H–h defects over 2H–h defect clusters even though the binding energy of a second H is positive. A number of studies have used a thermodynamic formalism to account for this defect concentration effect. All of these thermodynamic evaluations use DFT binding energies as input. As mentioned earlier, DFT calculated binding energies indicate that it is energetically favorable for a vacancy to bind 10–12 H atom in Al and 9 H atoms in Mg. Based on an analytical thermodynamic analysis, Ismer et al. [20] found that vacancies trap multiple H atoms in fcc Al only when the H2 gas pressure is of the order of 10 GPa and in hexagonal close-packed (hcp) Mg when the H2 gas pressure is nearly 1 GPa. Gunaydin et al. [23] used Born–Oppenheimer molecular dynamics to probe the effect of defect concentrations on H–h in Al. In spite of the fact that the H–h binding energies reveal that it is energetically favorable for a vacancy to bind up to 12 H atoms, they observed that only H–h defect complexes with 1, 2 or 6 H atoms formed depending on cH. While a number of the aforementioned DFT studies have focused on h–point defect (including h–H) interactions, there are fewer theoretical studies focused on H–solute interactions. Monasterio et al. [24] explored the interplay of binding processes when H, C and vacancies are present in bcc Fe. Using DFT, they calculated the H–C binding energy to be 0.02 eV, and the binding energy of a second H to a pre-existing H–C defect to be 0.05 eV.1 In a previous study, we investigated binding of a single H to a number of different solutes [25]. We found that the maximum H–solute binding energies ranged from 0.00 to 0.25 eV.1 The fact that these binding energies are positive indicates that H–solute defects are stable. An open question concerning H–solute interactions is the following: can solute atoms, like vacancies, bind multiple H atoms? Solutes that bind multiple H atoms can have a greater effect on the H diffusion rate than solutes that can bind only a single H atom and thus may also play larger role in HE. Furthermore, the vacancy formation energy in bcc Fe is quite large, of the order of 2 eV [25,26], making the equilibrium vacancy concentration low. In bcc Febased steels, the concentration of alloying elements is normally much larger than that of vacancies. Therefore, it is important to fully understand character of H–solute interactions. In this paper, we use first-principles DFT calculations to quantify the binding energy of multiple H atoms to various solute atoms in bcc Fe. We investigate the binding of multiple H atoms to one another, to interstitial C and to substitutional solutes.
5814
W. Counts et al. / Acta Materialia 59 (2011) 5812–5820
2. Methodology Our first-principles calculations are based on DFT [27,28] as implemented in the Vienna Ab-initio Simulation Package (VASP) [29,30]. All calculations were performed using the projector augmented wave (PAW) [31] potentials, and the generalized gradient approximation (GGA), with the exchange–correlation functional of Perdew–Burke–Ernzerhof (PBE) [32]. All calculations employed spin-polarization to account for the ferromagnetic state of bcc Fe. Unless otherwise stated, the DFT calculations were relaxed with respect to supercell shape and volume as well as all atomic positions in order to find the minimum energy. Two different bcc supercells were used: 3 3 3 (54 atoms) and 4 4 4 (128 atoms) conventional bcc cells. The 3 3 3 supercell was used for the majority of the calculations in this work, and cases where the 4 4 4 supercell was used are explicitly noted. The Monkhorst– Pack scheme was used to sample k-points in the Brillouin zone. A 6 6 6 k-point mesh was used with the 3 3 3 supercell and a 4 4 4 k-point mesh was used with the 4 4 4 supercell. All calculations utilized Fermi smearing of the electronic occupancy with a width of 0.05 eV and a plane wave cut-off energy of 450 eV. In addition, the large supercells made the evaluation of the projector operators in real space more computationally efficient. The accuracy of the computational parameters was tested on bulk bcc Fe. These parameters yielded a predicted lat˚ and a local magnetic moment tice parameter (a0) of 2.82 A on each atom of 2.2 lB; these values are in good agreement with experiments [33]. In an effort to minimize and quantify the error in the DFT calculations, a 3 3 3 supercell was used to test the following: k-point convergence (6 6 6 vs. 8 8 8 k-point mesh); energy cut-off convergence (330–450 eV); and the effect of evaluating the projector operators in real space vs. reciprocal space. We found that the change in the formation energy of a single H impurity atom due to these parameters was 0.01 eV. We also tested the convergence between the 3 3 3 and 4 4 4 supercells. We found that both the formation and binding energies of H–H defects as well as H–transition metal defects in 3 3 3 and 4 4 4 supercells were converged to within 0.015 eV. It should be noted that the k-point meshes of the two supercells are different (11,664 k points atom for the 3 3 3 supercell, and 8192 k points atom for the 4 4 4 supercell). Thus the 0.015 eV uncertainty refers to the 4 4 4 supercell results. The substitutional solute point defects (Sc, Ti, V, Cr, Mn, Co, Ni, Cu, Zn) were created by substituting a transition metal atom for a single Fe atom in the supercell. Sc, Ti, V, Cr and Mn all couple antiferromagnetically with Fe. Co, Ni, Cu and Zn all couple ferromagnetically with Fe. For
each of the solutes except Mn, the aforementioned magnetic state was the only stable electronic solution. For systems containing Mn, we observed stable electronic solutions for Mn atoms both ferromagnetically aligned and antiferromagnetically aligned with the surrounding Fe atoms [25,34]. In all cases, antiferromagnetically aligned Mn had a lower energy. H interstitial atoms were placed in tetrahedral sites (t-sites), in agreement with indirect experimental evidence [35] and DFT calculations [25,36]. C interstitial atoms were placed in octahedral sites (o-sites), in agreement with both experiments [37] and DFT calculations [25,38]. For defect clusters containing three or more point defects, it is possible to define both an incremental and a total binding energy. The total binding energy is defined as the difference between the energy of the defect cluster and the sum of the individual defect energies, which combined to form the defect cluster. The incremental binding energy quantifies the energy change due to a single point defect interacting with a pre-existing defect complex to form a larger defect cluster. In order to avoid confusion, only incremental binding energies are presented in Section 3 of this paper. Note that when a defect cluster contains only two point defects, the incremental and total binding energies are the same. In this paper, we investigate incremental binding energies associated with three different H defects. The first type is the H–H defect and its binding energy (EH;H b ) is defined via: ¼ 2 EðFen ; HÞ EðFen ; H–HÞ EðFen Þ; EH;H b
ð1Þ
where EðFen ; HÞ is the energy of a computational cell containing n Fe atoms and one interstitial H atom, EðFen ; H–HÞ is the energy of a computational cell containing n Fe atoms and two interstitial H atoms constituting a defect pair, and EðFen Þ is the energy of a computational cell containing n Fe atoms. The second defect type is the H–C defect. The binding energy of a single H to a single C to form a H–C defect pair is defined as: EH;C b EH;C ¼ EðFen ;CÞ þ EðFen ;HÞ EðFen ;H–CÞ EðFen Þ; b
ð2Þ
where EðFen ; CÞ is the energy of a computational cell containing n Fe atoms and 1 C atom, and EðFen ; H–CÞ is the energy of a computational cell contain n Fe atoms and a H–C defect pair. The binding energy of a single H to a to form a 2H–C defect pre-existing H–C defect pair E2H;C b complex is defined as: E2H;C ¼ EðFen ;H–CÞ þ EðFen ;HÞ EðFen ;2H–CÞ EðFen Þ; b ð3Þ where EðFen ; 2H–CÞ is the energy of a computational cell containing n Fe atoms and a 2H–C defect complex. The final defect type involves H and a substitutional solute atom (Sol). The binding energy of a single H to a single Sol to form a H–Sol defect pair is defined as: EH;Sol b EH;Sol ¼ EðFen1 ;SolÞ þ EðFen ;HÞ EðFen1 ;H–SolÞ EðFen Þ; b ð4Þ
W. Counts et al. / Acta Materialia 59 (2011) 5812–5820
5815
where EðFen1 ; SolÞ is the energy of a computational cell containing n 1 Fe atoms and 1 Sol atom, and EðFen1 ; H–SolÞ is the energy of a computational cell contain n 1 Fe atoms and a H–Sol defect pair. The binding energy ofthe ith H to a pre-existing (i 1)H–Sol defect pair to form a iH–Sol defect complex is defined as: EiH;Sol b EiH;Sol ¼ EðFen1 ; Hi1 –SolÞ þ EðFen ; HÞ b EðFen1 ; H i –SolÞ EðFen Þ;
ð5Þ
where EðFen1 ; Hi1 –SolÞ is the energy of a computational cell containing n 1 Fe atoms and a Hi1–Sol defect complex, and EðFen1 ; Hi –SolÞ is the energy of computational cell containing n 1 Fe atoms and a Hi–Sol defect complex. It is important to note that a positive binding energy indicates a favorable, energy-lowering attraction between defects. This convention is used throughout this paper. 3. Results and discussion 3.1. H–H interactions We begin with the interaction between H atoms in bcc Fe. We have calculated H–H binding at several different H–H distances. Because H prefers a tetrahedral site (t-site) bcc Fe, the smallest possible separation distance between two H atoms is in neighboring tetrahedral interstitial sites, ˚ . In this case, one H which are only separated by 1.0 A occupies the 1st nearest neighbor (1NN) t-site relative to the other. We will call this H–H defect a “1NN defect pair” and use this NN notation to identify other H–H defects. The 1NN defect pair is not stable. Both H atoms spontaneously relax to different t-sites and form a 3NN defect pair configuration. We have also calculated the H–H interactions at a variety of distances longer than the 1NN geometry. The largest achievable H–H separation distance in a ˚ . At this separation distance, 4 4 4 supercell is 9.8 A H;H Eb = 0.01 eV, which we take as the uncertainty in the calculated H–H binding energies. The calculated EH;H are shown in Fig. 1 as a function of b H–H separation distances. The EH;H results also show that b a H–H defect pair is not energetically favorable until the ˚ , which corH–H separation distance is greater than 2.0 A responds to a 4NN defect pair. The maximum calculated EH;H is around 0.03 eV for the 4NN, 8NN-a and 17NN b defect pair. Generally speaking, the calculated H–H interactions are not very strong. Our DFT calculated maximum EH;H agrees well with Au and Birnbaum’s [9] experimenb tally measured value of 0.04 eV. The results in Fig. 1 also show that EH;H vs. nearest neighb bor is not a single-valued function. Specifically, there are two different EH;H for the 8NN and 10NN defect pairs. b The different values correspond to different H–H defect geometries that have the same initial H–H separation distance. The two different geometries for both the 8NN and 10NN defect pairs are shown in Fig. 2. In the –a defects, each of the H atoms has four separate nearest-neighbor Fe
Fig. 1. DFT calculated H–H binding energies in bcc Fe. The 2NN–7NN were computed using a 3 3 3 supercell, while the others were calculated using a 4 4 4 supercell. The calculated H–H binding energy ˚ is 0.01 eV. For clarity, this value is not at a separation distance of 9.8 A plotted. The “a” and “b” labels represent two different H–H defect geometries (at 8NN and 10NN). In defect free bcc Fe, the t-sites labeled “a” and “b” have the same separation distances, however in the relaxed configuration the H–H separation distances are slightly different.
atoms, while in the –b defects, the two H atoms share a Fe atom. For both the 8NN and 10NN defects, the –b geomethan the –a geometries. For a given tries have a lower EH;H b H–H separation distance, the H–H defect geometry that contains two separate H–Fe tetrahedra is energetically favorable over the defect geometry where the H atoms share a nearest-neighbor Fe atom. 3.2. H–C interactions We next turn to the binding of H atoms to an interstitial C. In a previous study, we used DFT calculations to explore the binding between H and C as a function of distance between the two atoms [25]. We found that an H–C defect pair is stable and energetically favorable when the ˚ (1.2a0), and that two atoms are separated by at least 3.5 A the H–C binding energy decreased to almost zero when
Fig. 2. 8NN and 10NN H–H defect geometries. In defect-free bcc Fe, the t-sites labeled “a” and “b” have the same separation distances; however, the EH;H associated with each geometry is different. The EbH;H for 8NNb a > 8NN-b, and EH;H for 10NN-a > 10NN-b (i.e. 8NN-a and 10NN-a are b the respective lowest energy configurations).
5816
W. Counts et al. / Acta Materialia 59 (2011) 5812–5820
˚ (1.3a0). We build on these they were separated by 3.8 A results to construct plausible geometries for a C atom binding more than one H atom. Specifically, we construct three different defect clusters containing 2 H and 1 C atom (2H– C) where all the H–C separation distances are between 3.5 ˚ . These 2H–C defects are shown in Fig. 3. For and 3.8 A each of the three 2H–C defects, the lowest-energy 1H–C defect geometry found by Ref. [25] (the H atom labeled “A” in Fig. 4) was used as a basis to which a 2nd H atom (the H atoms labeled “X”, “Y” and “Z”) was added. The binding energies of the 2nd H atom for each of the H positions are shown in Fig. 3, along with corresponding 1H–C binding energy of each H position for comparison. For each of the 2H–C defects, the binding energy of the 2nd H atom E2H;C is positive, indicating that the second b H is favorably bound to an existing H–C pair. The binding is similar to the individual H–C of the 2nd H atom E2H;C b HðÞ;C binding energy of the second H atom (Eb where = X, Y or Z). The fact that these two binding energies are similar shows that there is no large synergistic effect between H atoms surrounding the C atom. Rather, the presence of an H atom near a C atom has little effect on the binding of a 2nd H atom. The H–H separation distances in the geometries of ˚ , respectively. At these sepFig. 3a and b are 4.9 and 6.3 A aration distances, the H–H interactions are not strong: between 0.00 and 0.02 eV. Thus it is not surprising that the presence of an H atom near a C atom has little effect on the binding of a 2nd H atom. The H–H separation dis˚ . However, this H– tance in Fig. 3c is smaller, around 2.9 A H configuration has a EH;H = 0.01 (see Fig. 2b). Here b again, the H–H interactions without C present are small. HðZÞ;C Therefore it follows that Eb is similar to E2H;C . b Our theoretical results agree qualitatively with experimental results that binding between H and C is favorable, but not strong. Au and Birnbaum [9] measured a H–C binding energy of 0.03 eV, whereas our maximum calculated 1H
and 2H binding energies are around 0.07 eV. A DFT study = 0.02 eV and by Monasterio et al. [24] calculated EH;C b E2H;C = 0.05 eV. Their negative E2H;C value is qualitab b tively different to our positive E2H;C value found for all b 2H–C geometries. However, Monasterio et al. [24] do not provide details about the respective H–C defect geometries, making a complete comparison with their results difficult. 3.3. H–Sol interactions In this section, we investigate the binding between multiple H atoms and substitutional solute atoms in bcc Fe. We have previously found [25] that the maximum, positive H– solute binding energy for Sc, Ti, V, Ni, Cu and Zn solutes occurred when the H atom occupied the 2NN t-site relative to the solute atom. Based on these results, H–Sol defects containing multiple H atoms were created by placing H atoms in other 2NN t-sites surrounding the solute atom. We also found [25] that the H–solute binding energy was slightly negative for Cr, Mn and Co when H occupied the 2NN t-site, and nearly zero when H occupied the 3NN t-site. Despite the negative binding energy, H–solute defects with Cr, Mn and Co and multiple H atoms were investigated to gain insights into trends across all the 3d transition metals. It is possible to generate a large number of different H– Sol defect cluster geometries with multiple H atoms even if the H atoms are restricted to the 2NN t-sites surrounding the solute atom. Hence, in order to better understand H– H interactions in the presence of a solute atom, a number of these triple defects containing 2H (2H–Sol) were constructed and calculated from DFT. In all cases (except where noted), the H–Sol distance was at the 2NN position, but we differentiate the various geometries in terms of the H–H separation distances. Two “far” configurations (Fig. 4a and b), with H–H separation distances of 4.3 ˚ , and three “close” configurations (Fig. 4c–f), and 5.1 A
Fig. 3. The geometry and binding energy of the 2H–C defect complexes. The letters correspond to different H positions around the C atom. The H atom labeled “A” forms the lowest-energy 1H–C defect pair to which the second H atom was added. The binding energies for each individual H atom to the C HðÞ;C atom are noted as Eb , where * indicates the H location. The incremental binding energy of the second H atom (either X, Y or Z) is Eb2H;C . Note that the H atoms “A” and “X” are in symmetrically identical positions. All 2H–C was based on a 4 4 4 supercell.
W. Counts et al. / Acta Materialia 59 (2011) 5812–5820
5817
Fig. 4. The 2H–Sol defect geometries investigated in this work. “H–Sol-H” refers to two H atoms surrounding the solute in 2NN t-sites. “H–H–Sol” refers ˚ . Black lines to one H atom in a 2NN t-site and the other not bound to the solute atom. The numerical values refer to the H–H separation distance in A between Fe atoms outline a cubic unit cell. Gray lines are Fe–H bonds. The H–H–Sol: 2.8 results are based on a 4 4 4 supercell.
˚ were investiwith H–H separation distances of 2.0–2.8 A gated to determine whether H would prefer a larger or smaller spacing in the presence of a solute atom. In addition, one 2H–Sol configuration where the H atoms did not surround the solute atom (Fig. 4f) was tested to determine whether H–H interactions were preferred over H–Sol interactions. The binding energy of the 2nd H E2H;Sol to a H–Sol b defect pair in the six configurations shown in Fig. 4 was investigated for four solutes that have varying E1H;Sol : Sc, b Ti, Mn and Ni. The E2H;Sol for these four solutes is shown b in Fig. 5. Considering only the H–Sol–H results in Fig. 5, one sees that the E2H;Sol for a given solute do not vary b greatly. The H–Sol–H energy spreads are between 0.03 eV for Ti, Mn and Ni and 0.05 eV for Sc. In cases where E1H;Sol is stronger, configurations with larger H–H b separations have larger binding energies. For Ti and Sc, E2H;Sol is largest for the H–Sol–H: 4.3 and H–Sol–H: 5.1 b configurations. In cases where E1H;Sol is weaker, there is b not much difference in the binding energies between configurations with larger and smaller H–H separation distances. For Mn and Ni, E2H;Sol for the H–Sol–H: 4.3 and H–Sol–H: b 2.0 configurations is similar. The binding energies for the H–H–Sol: 2.8 configuration is relatively constant (0.04–0.05 eV) for each of the solutes. The E1H;Sol for each of the solutes differs, ranging from b 0.2 eV for Sc to about 0.02 eV for Mn (see Fig. 7). If the solute atom affected H binding in the H–H–Sol: 2.8 configuration, then E2H;Sol for this configuration would be differb ent for each solute. The binding energies for H–H–Sol: 2.8
Fig. 5. The DFT calculated E2H;Sol for four different solutes. The Fe solute b results are the H–H interactions without a substitutional solute for the various configurations. The H–H–Sol: 2.8 results are based on a 4 4 4 supercell.
configurations are similar, suggesting that a solute atom does not affect the binding of a 2nd H atom when it is not bound to the solute atom. A comparison of the H– Sol–H and H–H–Sol results in Fig. 5 reveals that H atoms do not always prefer binding to a solute atom over another H atom. For Mn and Ni, H–H–Sol binding energy is larger than all the H–Sol–H binding energies, suggesting that a 2nd H atom would prefer binding to the H atom over the solute atom. For Sc and Ti, the opposite is true and a 2nd H atom would prefer binding to the solute atom. One would expect H–H–Sol binding to be preferred for
5818
W. Counts et al. / Acta Materialia 59 (2011) 5812–5820
solutes that have small (<0.04 eV) E1H;Sol . The 3d transition b metals that fit such a criterion are V, Cr, Mn, Co and Ni [25]. The 2H results were then used as a guide to construct a series of related H–Sol defects with 1–5 H atoms as shown in Fig. 6. The series starts with one H occupying a 2NN tsite, which is the lowest-energy 1 H configuration for most 3d transition metals. The 2H–Sol configuration is the H– Sol–H: 4.3 configuration from Fig. 5. This configuration was chosen because it is either the most stable configuration for solutes that strongly bind H or its energy is very close to the most stable for solutes that weakly bind H. For the same reason an attempt was made to keep the H–H separation distances large within this series of related H–Sol defects. The subsequent 3H–5H defects were constructed by adding a 2NN H to the previous H–Sol defect. The series of configurations in Fig. 6 represent a pathway for a solute atom to bind up to 5 H atoms. A solute will bind multiple H atoms only when the H concentration is sufficiently high. Recently, Sanchez et al. [39] showed that it is energetically favorable for Fe to undergo a large body-centered-tetragonal (bct) distortion when the concentration of H is very high (33 at.%). Under these conditions, H prefers the o-site over the t-site. For lower H concentrations, these authors find (as we do) that the t-site is preferred. Our calculations of multiple H atoms binding to a defect are based on low overall H concentrations, and we only considered t-site occupation of H. Even so, it is possible that a local increase in the H concentration could result in a local distortion analogous to the bct distortion found by Sanchez et al. However, we did not observe any evidence for this type of significant bct distortion in any of our defect cells, even though we relax cell vectors for all defect supercells. The H binding energies for each of the configurations in Fig. 6 is shown in Fig. 7. In all cases, we see that a single solute has a favorable binding for more than one H. In fact, the binding energies of multiple H atoms is often greater than the binding of the initial H. For example, a comparison of E1H;Sol and E2H;Sol for each of the solutes b b 2H;Sol reveals that Eb is around 0.03 eV greater than E1H;Sol . b The H binding behavior of the third and fourth H atoms can be broken into two different categories. For the solutes Sc and Mn, and for pure Fe, ExH;Sol increases for x = 3–4, b meaning it is energetically favorable for these solutes to add both the third and fourth H atoms. For the solutes Ti, V, Cr, Co, Ni, Cu and Zn, ExH;Sol decreases for x = 3 b followed by an increase for x = 4 to a level near E2H;Sol . b
Fig. 7. DFT calculated EbxH;Sol , where x = 1–5, for the 3d transition metal solutes.
The 5H configuration was investigated for most of the solutes (apart from Cr, Co and Zn) and in each case it was found that the binding energy of the fifth H atom decreased . to values below E1H;Sol b The dependence of the binding energy on the number of H atoms is qualititatively different for solute atoms compared to vacancies. At a vacancy, E1H; and E2H; are simb b ilar and then binding energy drops significantly (between 0.1 and 0.2 eV) as the vacancy binds 3–6 H atoms [22], while at a substitutional solute the H binding energy potentially increases as more H is bound. The reason the H binding energy does not drop in the presence of a solute is related to the H–H separation distances. In the case of a solute atom, the shortest H–H separation distance is ˚ when 2–4 H atoms are present. The EH;H data around 2.8 A b in Fig. 1 shows that two H atoms occupying t-sites sepa˚ do not have a negative binding energy and rated by 2.8 A thus H–H interactions will not cause a reduction in ExH;Sol . b There is little experimental data with which to compare the DFT results in Fig. 7. Pressouyre and Bernstein [10] measured the H–Ti binding energy and reported a value of 0.14–0.19 eV. The DFT calculated E1H;Ti is 0.08 eV and b E4H;Ti is 0.13 eV. The agreement between experiment and b theory improves when the possibility that Ti binds more than one H atom is considered. As discussed earlier, the issue of how many H atoms will bind to a particular solute depends both on the calculated binding energies as well as the defect concentrations. Without performing a full thermodynamic analysis, we cannot determine the H concentrations needed to form various types of H–solute defects. However, we can use the conclusions of Gunaydin et al. [23] to describe which of the energetically stable H–solute defects are also stable
Fig. 6. H–Sol defect geometries containing 1–5 H atoms. Black lines between Fe atoms outline a cubic unit cell. Gray lines are Fe–H bonds.
W. Counts et al. / Acta Materialia 59 (2011) 5812–5820 Table 1 Energetically stable and decomposition stable H–solute complexes. Solute
Stable H–solute complexes
Sc Ti V Cr Mn Co Ni Cu Zn
1H–Sc; 2H–Sc; 4H–Sc 1H–Ti; 2H–Ti; 4H–Ti 1H–V; 2H–V 2H–Cr 2H–Mn, 4H–Mn 2H–Co 1H–Ni; 2H–Ni 1H–Cu; 2H–Cu; 4H–Cu 1H–Zn; 2H–Zn
with respect to decomposition into other H–solute defects. Based on their Born–Oppenheimer molecular dynamics investigation of H–h binding in Al, Gunaydin et al. [23] noted defect complexes that did not lie on the convex (if a favorable Eb is defined as negative) or concave (if a favorable Eb is defined as positive) hull would either absorb or release H. Thus defect complexes can be energetically stable, but not stable with respect to decomposition into other H–solute defect complexes. Applying the concept of a concave hull to the data in Fig. 7 reveals that only a few H–solute defect complexes are both energetically stable and stable with respect to decomposition, as shown in Table 1. For all the 3d transition metals, the 2H–Sol defect clusters fulfill both the energy and decomposition stability requirements, while all the 3H–Sol and 5H–Sol defect complexes fail one or both stability requirements. For Cr, Mn and Co, the 1H–Sol defect is not energetically stable, but other defect complexes do fulfill both stability requirements. 4. Conclusions We used DFT calculations to explore H–H interactions as well as multiple H binding to interstitial and substitutional solute atoms in bcc Fe. H–H interactions are not energetically favorable when the H–H separation distance ˚ , after which the EH;H is generally positive, is less than 2.0 A b but small. The maximum EH;H of 0.03 eV is associated with b three different H–H defect pairs whose separation distances ˚ and agrees well with the experirange from 2.0 and 4.2 A mentally measured value. The lowest energy H–C defect has a EC;H of 0.07 eV and b ˚. the H–C separation distance for this defect pair is 3.5 A We then calculated the binding energy of a second H atom in three different locations to the lowest energy H–C defect. The incremental binding energy of the second H atom E2H;C ranges from 0.01 to 0.07 eV. In each case, the bindb ing energy of the second H atom is similar to that of the first, suggesting that there is no synergistic effect between H atoms surrounding the C atom. Thus, the presence of an H atom near a C atom has little effect on the binding of a second H atom. Finally, we investigated the ability of 3d transition metal solutes to bind up to five H atoms. A better understanding
5819
of H–H interactions in the presence of a solute atom was gleaned by investigating a number of defect complexes with two H atoms placed in various tetrahedral sites around a solute atom. Generally, we found that larger H–H separation distances are favored over shorter ones. With this insight, a series of related H–solute defects containing 1– 5 H atoms was constructed and the H binding energies were calculated. The binding energy of the second H to a 3d transition metal solute is 0.03 eV greater than binding of energy of the first, independent of solute. The binding energies of the third to fifth H atoms vary but are generally positive. The overall stability of a H–solute defect complex depends on two separate criteria. First the binding energy must be positive, and second the defect complex must be stable with respect to decomposition to other H–solute defect complexes. The decomposition stability was determined by analyzing the concave hull formed by the incremental binding energies. Defect complexes that did not lie on the concave hull will either absorb or release H to form a defect complex that does lie on the concave hull. Based on this stability analysis, we find that the largest H–solute defect complex for V, Cr, Co, Ni and Zn contains two H atoms, while for Sc, Ti, Mn and Cu, the largest defect complex contains four H atoms. Acknowledgements The authors would like to acknowledge funding from General Motors Corporation. The authors also acknowledge funding from the Department of Energy under grant DEFG36-08GO1813. The authors would like to acknowledge helpful discussions with Scott Jorgensen. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
Troiano AR. Trans ASM 1960;52:54. Oriani RA, Josephic PH. Acta Metall 1974;22:1065. Beachem CD. Metall Trans 1972;3:437. Birnbaum HK, Sofronis P. Mater Sci Eng A 1994;176:191. Nagumo M, Nakamura M, Takai K. Metall Trans A 2001;32:339. Takai K, Shoda H, Suzuki H, Nagumo M. Acta Mater 2008;56:5158. Oriani RA. Acta Metall 1970;18:147. Darken LS, Smith RP. Corrosion 1949;5:1. Au JJ, Birnbaum HK. Acta Metall 1978;26:1105. Pressouyre GM, Bernstein IM. Metall Trans A 1978;9:1571. Meyers SM, Picraux ST, Stoltz RE. J Appl Phys 1979;50:5710. Besenbacher F, Myers SM, Nordlander P, Nørskov JK. J Appl Phys 1987;61:1788. Gibala R. Trans AIME 1967;239:1574. Wolverton C. Acta Mater 2007;55:5867. Wolverton C, Ozolins V, Asta M. Phys Rev B 2004;69:144109. Shin D, Wolverton C. Acta Mater 2010;58:531. Ohnuma T, Soneda N, Iwasawa M. Acta Mater 2009;57:5947. Domain C, Becquart CS, Foct J. Phys Rev B 2004;69:144112. Slane JA, Wolverton C, Gibala R. Metall Trans A 2004;35:2239. Ismer L, Park MS, Janotti A, Van de Walle CG. Phys Rev B 2009;80:184110. Lu G, Kaxiras E. Phys Rev Lett 2005;94:155501. Tateyama Y, Ohno T. Phys Rev B 2003;67:174105.
5820
W. Counts et al. / Acta Materialia 59 (2011) 5812–5820
[23] Gunaydin H, Barabash SV, Houk KN, Ozolins V. Phys Rev Lett 2008;101:075901. [24] Monasterio PR, Lau TT, Yip S, Van Vliet KJ. Phys Rev Lett 2009;103:85501. [25] Counts W, Wolverton C, Gibala R. Acta Mater 2010;58:4730. [26] De Schepper L, Segers D, Dorikens-Vanpraet L, Dorikens M, Knuyt G, Stals LM. Phys Rev B 1983;27:5257. [27] Hohenberg P, Kohn W. Phys Rev 1964;136:B864. [28] Kohn W, Sham LJ. Phys Rev 1965;140:A1133. [29] Kresse G, Hafner J. Phys Rev B 1993;47:558. [30] Kresse G, Fu¨rthmuller J. Phys Rev B 1996;54:11169.
[31] Blo¨chl PE. Phys Rev B 1994;50:17953. [32] Perdew JP, Burke K, Ernzerhof M. Phys Rev Lett 1996;77:3865. [33] Kittel C. Introduction to solid state physics. 7th ed. New York: Wiley; 1996. [34] Vincent E, Becquart CS, Domain C. J Nucl Mater 2006;351:88. [35] Hirth JP. Metall Trans A 1980;11:861. [36] Jiang DE, Carter EA. Phys Rev B 2004;70:064102. [37] Jack DH, Jack KH. Mater Sci Eng 1973;11:1. [38] Jiang DE, Carter EA. Phys Rev B 2003;67:214103. [39] Sanchez J, Fullea J, Andrade C, de Andres PL. Phys Rev B 2010;78:014113.
Acta Materialia 59 (2011) 5812–5820 www.elsevier.com/locate/actamat
Binding of multiple H atoms to solute atoms in bcc Fe using first principles W. Counts a,⇑, C. Wolverton a, R. Gibala b a b
Department of Materials Science and Engineering, Northwestern University, Evanston, IL, USA Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI, USA Received 16 March 2011; received in revised form 27 May 2011; accepted 29 May 2011 Available online 27 June 2011
Abstract We previously performed a series of density functional theory calculations to investigate the interaction between single H atoms and point defects in body-centered cubic (bcc) Fe (Counts W, Wolverton C, Gibala R. Acta Mater 2010;58:4730). Here, we extend that work to a systematic study of binding between multiple H atoms and solute atoms in bcc Fe. We investigate the binding of multiple H atoms to one another, to interstitial C and to substitutional solutes. Our study shows the following: (i) H–H interactions are weak. The maximum attractive H–H binding energy is around 0.03 eV, which agrees with experimental values. (ii) The maximum attractive incremental binding energy of a second H atom to a C–H defect pair is 0.07 eV. (iii) We investigate the ability of 3d transition metal solutes to bind up to five H atoms. The binding energy of the second H to a 3d transition metal solute is attractive with a value 0.03 eV greater than binding of energy of the first, independent of solute. The binding energies of the third to fifth H atoms vary but are generally positive. Based on a stability analysis of the H binding energies, we find that the largest H–solute defect complex for V, Cr, Co, Ni and Zn contains two H atoms, while for Sc, Ti, Mn, and Cu the largest defect complex contains four H atoms. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Density functional; Hydrogen embrittlement; Ferritic steels; Iron
1. Introduction Hydrogen embrittlement (HE) of body-centered cubic (bcc) iron (a-Fe) is a well-known but still not well under stood phenomenon. A number of different HE mechanisms have been proposed, including decohesion [1,2], hydrogenenhanced localized plasticity [3,4], and the hydrogen-enhan ced strain-induced vacancy [5,6] models. While these mechanisms are all distinct, each of them requires that H accumulates within the material to form a high H concentration region that enables the respective HE mechanism. Such HE-susceptible regions form as a result of H diffusion, making each of these different HE mechanisms dependent on H diffusion.
⇑ Corresponding author.
E-mail address: [email protected] (W. Counts).
H readily diffuses through a-Fe [7], but lattice defects impede the motion of H [8]. However, experiments are less clear concerning the strength of the H–defect interactions because it is not possible to directly measure the H–defect binding energy. In the literature, H–defect binding energies have been probed using a variety of different experimental techniques: magnetic relaxation [9], H permeation [10], thermal detrapping [11,12] and internal friction [13] among others. The magnitude of the H–defect interaction in each of the aforementioned experiments is then determined with the aid of a model that includes a binding energy term. Using this general approach, experimental H–point defect interactions like H–H self-interactions [9], as well as H–C [9], H–Ti [10] and H–h binding [11,12] in bcc Fe have been quantified (h = vacancy). Experiments are often also unable to extract details about the geometry of the H–point defect cluster. In one case, Meyers et al. [11] used ionchanneling to obtain additional information about the
1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.05.058
W. Counts et al. / Acta Materialia 59 (2011) 5812–5820
position of the D atom. They showed that D atoms occu˚ from the octahedral site pied a position displaced 0.4 A (o-site). However, this technique did not explicitly identify the lattice defect that had bound D. Theoretical and computational investigations into H– point defect interactions are an important complement to experimental studies because they can provide information to validate and interpret experimental results and can also provide details that are difficult to access experimentally. Computational approaches, if sufficiently accurate, are valuable because they can directly quantify the binding energy of H with a particular point defect in a given configuration. One example where insight from theory was valuable is the D–h defect pair. Initially, experiments and theory both revealed strong D–h binding in bcc Fe. Results from an effective medium theory (EMT) based study by Besenbacher et al. [12] revealed that the D–h binding energy for one to six D atoms was favorable. Guided by these theoretical results, the authors of Ref. [12] were able to explain the emergence of a second plateau region in their detrapping data at higher D concentrations. They argued that when the concentration of D is larger than that of vacancies, vacancies bind multiple D atoms. Thus, the second detrapping plateau observed in the experiment was due to multiple D atoms binding to the vacancy. Both EMT and a diffusion model were then used to qualitatively determine that the incremental vacancy binding energy for the first two D atoms is 0.63 eV,1 and 0.43 eV1 for the subsequent (3–6) D atoms. Density functional theory (DFT) has been successfully used to model h–point defect interactions in a variety of systems. There is good agreement between available experimental data and DFT concerning h–solute interactions in Al [14,15], Mg [16], bcc Fe [17,18] and facecentered cubic (fcc) Fe [19]. DFT has also shown that vacancies are effective H traps in various metal systems. DFT has shown that it is energetically favorable for a vacancy to bind up to between 10 [20] and 12 [21] H atoms in fcc Al, up to 9 H atom in hcp Mg [20], and up to 6 H atoms in bcc Fe [22]. In the case of bcc Fe, the DFT-based binding energies agree well both qualitatively and quantitatively with the experimental results of Besenbacher et al. [12]. It should be noted that binding energies alone are not enough to determine how many H atoms a defect will bind because the defect configuration depends on both the binding energies and defect concentrations. For example, the H–h binding energy in Al is 0.35 eV,1 and the binding energy of a second H to the vacancy is also positive, around 0.30 eV.1 In cases when the concentration of H (cH) is much larger than the concentration of vacancies (ch), there is a large probability that H atoms will find vacancies along their diffusion path. These H atoms will
1 A positive binding energy refers to an attractive interaction, and negative value refers to a repulsive interaction between defects.
5813
bind to the vacancy and form H–h defects because the H–h binding energy is positive. The excess H atoms will find these H–h defects and form defect clusters containing a vacancy and two H atoms defects (2H–h) because the binding energy of a second H to the vacancy is also positive. In cases when ch cH, it is far more likely that a H atom will encounter a vacancy rather than a H–h defect. Thus it is more likely in this case to form H–h defects over 2H–h defect clusters even though the binding energy of a second H is positive. A number of studies have used a thermodynamic formalism to account for this defect concentration effect. All of these thermodynamic evaluations use DFT binding energies as input. As mentioned earlier, DFT calculated binding energies indicate that it is energetically favorable for a vacancy to bind 10–12 H atom in Al and 9 H atoms in Mg. Based on an analytical thermodynamic analysis, Ismer et al. [20] found that vacancies trap multiple H atoms in fcc Al only when the H2 gas pressure is of the order of 10 GPa and in hexagonal close-packed (hcp) Mg when the H2 gas pressure is nearly 1 GPa. Gunaydin et al. [23] used Born–Oppenheimer molecular dynamics to probe the effect of defect concentrations on H–h in Al. In spite of the fact that the H–h binding energies reveal that it is energetically favorable for a vacancy to bind up to 12 H atoms, they observed that only H–h defect complexes with 1, 2 or 6 H atoms formed depending on cH. While a number of the aforementioned DFT studies have focused on h–point defect (including h–H) interactions, there are fewer theoretical studies focused on H–solute interactions. Monasterio et al. [24] explored the interplay of binding processes when H, C and vacancies are present in bcc Fe. Using DFT, they calculated the H–C binding energy to be 0.02 eV, and the binding energy of a second H to a pre-existing H–C defect to be 0.05 eV.1 In a previous study, we investigated binding of a single H to a number of different solutes [25]. We found that the maximum H–solute binding energies ranged from 0.00 to 0.25 eV.1 The fact that these binding energies are positive indicates that H–solute defects are stable. An open question concerning H–solute interactions is the following: can solute atoms, like vacancies, bind multiple H atoms? Solutes that bind multiple H atoms can have a greater effect on the H diffusion rate than solutes that can bind only a single H atom and thus may also play larger role in HE. Furthermore, the vacancy formation energy in bcc Fe is quite large, of the order of 2 eV [25,26], making the equilibrium vacancy concentration low. In bcc Febased steels, the concentration of alloying elements is normally much larger than that of vacancies. Therefore, it is important to fully understand character of H–solute interactions. In this paper, we use first-principles DFT calculations to quantify the binding energy of multiple H atoms to various solute atoms in bcc Fe. We investigate the binding of multiple H atoms to one another, to interstitial C and to substitutional solutes.
5814
W. Counts et al. / Acta Materialia 59 (2011) 5812–5820
2. Methodology Our first-principles calculations are based on DFT [27,28] as implemented in the Vienna Ab-initio Simulation Package (VASP) [29,30]. All calculations were performed using the projector augmented wave (PAW) [31] potentials, and the generalized gradient approximation (GGA), with the exchange–correlation functional of Perdew–Burke–Ernzerhof (PBE) [32]. All calculations employed spin-polarization to account for the ferromagnetic state of bcc Fe. Unless otherwise stated, the DFT calculations were relaxed with respect to supercell shape and volume as well as all atomic positions in order to find the minimum energy. Two different bcc supercells were used: 3 3 3 (54 atoms) and 4 4 4 (128 atoms) conventional bcc cells. The 3 3 3 supercell was used for the majority of the calculations in this work, and cases where the 4 4 4 supercell was used are explicitly noted. The Monkhorst– Pack scheme was used to sample k-points in the Brillouin zone. A 6 6 6 k-point mesh was used with the 3 3 3 supercell and a 4 4 4 k-point mesh was used with the 4 4 4 supercell. All calculations utilized Fermi smearing of the electronic occupancy with a width of 0.05 eV and a plane wave cut-off energy of 450 eV. In addition, the large supercells made the evaluation of the projector operators in real space more computationally efficient. The accuracy of the computational parameters was tested on bulk bcc Fe. These parameters yielded a predicted lat˚ and a local magnetic moment tice parameter (a0) of 2.82 A on each atom of 2.2 lB; these values are in good agreement with experiments [33]. In an effort to minimize and quantify the error in the DFT calculations, a 3 3 3 supercell was used to test the following: k-point convergence (6 6 6 vs. 8 8 8 k-point mesh); energy cut-off convergence (330–450 eV); and the effect of evaluating the projector operators in real space vs. reciprocal space. We found that the change in the formation energy of a single H impurity atom due to these parameters was 0.01 eV. We also tested the convergence between the 3 3 3 and 4 4 4 supercells. We found that both the formation and binding energies of H–H defects as well as H–transition metal defects in 3 3 3 and 4 4 4 supercells were converged to within 0.015 eV. It should be noted that the k-point meshes of the two supercells are different (11,664 k points atom for the 3 3 3 supercell, and 8192 k points atom for the 4 4 4 supercell). Thus the 0.015 eV uncertainty refers to the 4 4 4 supercell results. The substitutional solute point defects (Sc, Ti, V, Cr, Mn, Co, Ni, Cu, Zn) were created by substituting a transition metal atom for a single Fe atom in the supercell. Sc, Ti, V, Cr and Mn all couple antiferromagnetically with Fe. Co, Ni, Cu and Zn all couple ferromagnetically with Fe. For
each of the solutes except Mn, the aforementioned magnetic state was the only stable electronic solution. For systems containing Mn, we observed stable electronic solutions for Mn atoms both ferromagnetically aligned and antiferromagnetically aligned with the surrounding Fe atoms [25,34]. In all cases, antiferromagnetically aligned Mn had a lower energy. H interstitial atoms were placed in tetrahedral sites (t-sites), in agreement with indirect experimental evidence [35] and DFT calculations [25,36]. C interstitial atoms were placed in octahedral sites (o-sites), in agreement with both experiments [37] and DFT calculations [25,38]. For defect clusters containing three or more point defects, it is possible to define both an incremental and a total binding energy. The total binding energy is defined as the difference between the energy of the defect cluster and the sum of the individual defect energies, which combined to form the defect cluster. The incremental binding energy quantifies the energy change due to a single point defect interacting with a pre-existing defect complex to form a larger defect cluster. In order to avoid confusion, only incremental binding energies are presented in Section 3 of this paper. Note that when a defect cluster contains only two point defects, the incremental and total binding energies are the same. In this paper, we investigate incremental binding energies associated with three different H defects. The first type is the H–H defect and its binding energy (EH;H b ) is defined via: ¼ 2 EðFen ; HÞ EðFen ; H–HÞ EðFen Þ; EH;H b
ð1Þ
where EðFen ; HÞ is the energy of a computational cell containing n Fe atoms and one interstitial H atom, EðFen ; H–HÞ is the energy of a computational cell containing n Fe atoms and two interstitial H atoms constituting a defect pair, and EðFen Þ is the energy of a computational cell containing n Fe atoms. The second defect type is the H–C defect. The binding energy of a single H to a single C to form a H–C defect pair is defined as: EH;C b EH;C ¼ EðFen ;CÞ þ EðFen ;HÞ EðFen ;H–CÞ EðFen Þ; b
ð2Þ
where EðFen ; CÞ is the energy of a computational cell containing n Fe atoms and 1 C atom, and EðFen ; H–CÞ is the energy of a computational cell contain n Fe atoms and a H–C defect pair. The binding energy of a single H to a to form a 2H–C defect pre-existing H–C defect pair E2H;C b complex is defined as: E2H;C ¼ EðFen ;H–CÞ þ EðFen ;HÞ EðFen ;2H–CÞ EðFen Þ; b ð3Þ where EðFen ; 2H–CÞ is the energy of a computational cell containing n Fe atoms and a 2H–C defect complex. The final defect type involves H and a substitutional solute atom (Sol). The binding energy of a single H to a single Sol to form a H–Sol defect pair is defined as: EH;Sol b EH;Sol ¼ EðFen1 ;SolÞ þ EðFen ;HÞ EðFen1 ;H–SolÞ EðFen Þ; b ð4Þ
W. Counts et al. / Acta Materialia 59 (2011) 5812–5820
5815
where EðFen1 ; SolÞ is the energy of a computational cell containing n 1 Fe atoms and 1 Sol atom, and EðFen1 ; H–SolÞ is the energy of a computational cell contain n 1 Fe atoms and a H–Sol defect pair. The binding energy ofthe ith H to a pre-existing (i 1)H–Sol defect pair to form a iH–Sol defect complex is defined as: EiH;Sol b EiH;Sol ¼ EðFen1 ; Hi1 –SolÞ þ EðFen ; HÞ b EðFen1 ; H i –SolÞ EðFen Þ;
ð5Þ
where EðFen1 ; Hi1 –SolÞ is the energy of a computational cell containing n 1 Fe atoms and a Hi1–Sol defect complex, and EðFen1 ; Hi –SolÞ is the energy of computational cell containing n 1 Fe atoms and a Hi–Sol defect complex. It is important to note that a positive binding energy indicates a favorable, energy-lowering attraction between defects. This convention is used throughout this paper. 3. Results and discussion 3.1. H–H interactions We begin with the interaction between H atoms in bcc Fe. We have calculated H–H binding at several different H–H distances. Because H prefers a tetrahedral site (t-site) bcc Fe, the smallest possible separation distance between two H atoms is in neighboring tetrahedral interstitial sites, ˚ . In this case, one H which are only separated by 1.0 A occupies the 1st nearest neighbor (1NN) t-site relative to the other. We will call this H–H defect a “1NN defect pair” and use this NN notation to identify other H–H defects. The 1NN defect pair is not stable. Both H atoms spontaneously relax to different t-sites and form a 3NN defect pair configuration. We have also calculated the H–H interactions at a variety of distances longer than the 1NN geometry. The largest achievable H–H separation distance in a ˚ . At this separation distance, 4 4 4 supercell is 9.8 A H;H Eb = 0.01 eV, which we take as the uncertainty in the calculated H–H binding energies. The calculated EH;H are shown in Fig. 1 as a function of b H–H separation distances. The EH;H results also show that b a H–H defect pair is not energetically favorable until the ˚ , which corH–H separation distance is greater than 2.0 A responds to a 4NN defect pair. The maximum calculated EH;H is around 0.03 eV for the 4NN, 8NN-a and 17NN b defect pair. Generally speaking, the calculated H–H interactions are not very strong. Our DFT calculated maximum EH;H agrees well with Au and Birnbaum’s [9] experimenb tally measured value of 0.04 eV. The results in Fig. 1 also show that EH;H vs. nearest neighb bor is not a single-valued function. Specifically, there are two different EH;H for the 8NN and 10NN defect pairs. b The different values correspond to different H–H defect geometries that have the same initial H–H separation distance. The two different geometries for both the 8NN and 10NN defect pairs are shown in Fig. 2. In the –a defects, each of the H atoms has four separate nearest-neighbor Fe
Fig. 1. DFT calculated H–H binding energies in bcc Fe. The 2NN–7NN were computed using a 3 3 3 supercell, while the others were calculated using a 4 4 4 supercell. The calculated H–H binding energy ˚ is 0.01 eV. For clarity, this value is not at a separation distance of 9.8 A plotted. The “a” and “b” labels represent two different H–H defect geometries (at 8NN and 10NN). In defect free bcc Fe, the t-sites labeled “a” and “b” have the same separation distances, however in the relaxed configuration the H–H separation distances are slightly different.
atoms, while in the –b defects, the two H atoms share a Fe atom. For both the 8NN and 10NN defects, the –b geomethan the –a geometries. For a given tries have a lower EH;H b H–H separation distance, the H–H defect geometry that contains two separate H–Fe tetrahedra is energetically favorable over the defect geometry where the H atoms share a nearest-neighbor Fe atom. 3.2. H–C interactions We next turn to the binding of H atoms to an interstitial C. In a previous study, we used DFT calculations to explore the binding between H and C as a function of distance between the two atoms [25]. We found that an H–C defect pair is stable and energetically favorable when the ˚ (1.2a0), and that two atoms are separated by at least 3.5 A the H–C binding energy decreased to almost zero when
Fig. 2. 8NN and 10NN H–H defect geometries. In defect-free bcc Fe, the t-sites labeled “a” and “b” have the same separation distances; however, the EH;H associated with each geometry is different. The EbH;H for 8NNb a > 8NN-b, and EH;H for 10NN-a > 10NN-b (i.e. 8NN-a and 10NN-a are b the respective lowest energy configurations).
5816
W. Counts et al. / Acta Materialia 59 (2011) 5812–5820
˚ (1.3a0). We build on these they were separated by 3.8 A results to construct plausible geometries for a C atom binding more than one H atom. Specifically, we construct three different defect clusters containing 2 H and 1 C atom (2H– C) where all the H–C separation distances are between 3.5 ˚ . These 2H–C defects are shown in Fig. 3. For and 3.8 A each of the three 2H–C defects, the lowest-energy 1H–C defect geometry found by Ref. [25] (the H atom labeled “A” in Fig. 4) was used as a basis to which a 2nd H atom (the H atoms labeled “X”, “Y” and “Z”) was added. The binding energies of the 2nd H atom for each of the H positions are shown in Fig. 3, along with corresponding 1H–C binding energy of each H position for comparison. For each of the 2H–C defects, the binding energy of the 2nd H atom E2H;C is positive, indicating that the second b H is favorably bound to an existing H–C pair. The binding is similar to the individual H–C of the 2nd H atom E2H;C b HðÞ;C binding energy of the second H atom (Eb where = X, Y or Z). The fact that these two binding energies are similar shows that there is no large synergistic effect between H atoms surrounding the C atom. Rather, the presence of an H atom near a C atom has little effect on the binding of a 2nd H atom. The H–H separation distances in the geometries of ˚ , respectively. At these sepFig. 3a and b are 4.9 and 6.3 A aration distances, the H–H interactions are not strong: between 0.00 and 0.02 eV. Thus it is not surprising that the presence of an H atom near a C atom has little effect on the binding of a 2nd H atom. The H–H separation dis˚ . However, this H– tance in Fig. 3c is smaller, around 2.9 A H configuration has a EH;H = 0.01 (see Fig. 2b). Here b again, the H–H interactions without C present are small. HðZÞ;C Therefore it follows that Eb is similar to E2H;C . b Our theoretical results agree qualitatively with experimental results that binding between H and C is favorable, but not strong. Au and Birnbaum [9] measured a H–C binding energy of 0.03 eV, whereas our maximum calculated 1H
and 2H binding energies are around 0.07 eV. A DFT study = 0.02 eV and by Monasterio et al. [24] calculated EH;C b E2H;C = 0.05 eV. Their negative E2H;C value is qualitab b tively different to our positive E2H;C value found for all b 2H–C geometries. However, Monasterio et al. [24] do not provide details about the respective H–C defect geometries, making a complete comparison with their results difficult. 3.3. H–Sol interactions In this section, we investigate the binding between multiple H atoms and substitutional solute atoms in bcc Fe. We have previously found [25] that the maximum, positive H– solute binding energy for Sc, Ti, V, Ni, Cu and Zn solutes occurred when the H atom occupied the 2NN t-site relative to the solute atom. Based on these results, H–Sol defects containing multiple H atoms were created by placing H atoms in other 2NN t-sites surrounding the solute atom. We also found [25] that the H–solute binding energy was slightly negative for Cr, Mn and Co when H occupied the 2NN t-site, and nearly zero when H occupied the 3NN t-site. Despite the negative binding energy, H–solute defects with Cr, Mn and Co and multiple H atoms were investigated to gain insights into trends across all the 3d transition metals. It is possible to generate a large number of different H– Sol defect cluster geometries with multiple H atoms even if the H atoms are restricted to the 2NN t-sites surrounding the solute atom. Hence, in order to better understand H– H interactions in the presence of a solute atom, a number of these triple defects containing 2H (2H–Sol) were constructed and calculated from DFT. In all cases (except where noted), the H–Sol distance was at the 2NN position, but we differentiate the various geometries in terms of the H–H separation distances. Two “far” configurations (Fig. 4a and b), with H–H separation distances of 4.3 ˚ , and three “close” configurations (Fig. 4c–f), and 5.1 A
Fig. 3. The geometry and binding energy of the 2H–C defect complexes. The letters correspond to different H positions around the C atom. The H atom labeled “A” forms the lowest-energy 1H–C defect pair to which the second H atom was added. The binding energies for each individual H atom to the C HðÞ;C atom are noted as Eb , where * indicates the H location. The incremental binding energy of the second H atom (either X, Y or Z) is Eb2H;C . Note that the H atoms “A” and “X” are in symmetrically identical positions. All 2H–C was based on a 4 4 4 supercell.
W. Counts et al. / Acta Materialia 59 (2011) 5812–5820
5817
Fig. 4. The 2H–Sol defect geometries investigated in this work. “H–Sol-H” refers to two H atoms surrounding the solute in 2NN t-sites. “H–H–Sol” refers ˚ . Black lines to one H atom in a 2NN t-site and the other not bound to the solute atom. The numerical values refer to the H–H separation distance in A between Fe atoms outline a cubic unit cell. Gray lines are Fe–H bonds. The H–H–Sol: 2.8 results are based on a 4 4 4 supercell.
˚ were investiwith H–H separation distances of 2.0–2.8 A gated to determine whether H would prefer a larger or smaller spacing in the presence of a solute atom. In addition, one 2H–Sol configuration where the H atoms did not surround the solute atom (Fig. 4f) was tested to determine whether H–H interactions were preferred over H–Sol interactions. The binding energy of the 2nd H E2H;Sol to a H–Sol b defect pair in the six configurations shown in Fig. 4 was investigated for four solutes that have varying E1H;Sol : Sc, b Ti, Mn and Ni. The E2H;Sol for these four solutes is shown b in Fig. 5. Considering only the H–Sol–H results in Fig. 5, one sees that the E2H;Sol for a given solute do not vary b greatly. The H–Sol–H energy spreads are between 0.03 eV for Ti, Mn and Ni and 0.05 eV for Sc. In cases where E1H;Sol is stronger, configurations with larger H–H b separations have larger binding energies. For Ti and Sc, E2H;Sol is largest for the H–Sol–H: 4.3 and H–Sol–H: 5.1 b configurations. In cases where E1H;Sol is weaker, there is b not much difference in the binding energies between configurations with larger and smaller H–H separation distances. For Mn and Ni, E2H;Sol for the H–Sol–H: 4.3 and H–Sol–H: b 2.0 configurations is similar. The binding energies for the H–H–Sol: 2.8 configuration is relatively constant (0.04–0.05 eV) for each of the solutes. The E1H;Sol for each of the solutes differs, ranging from b 0.2 eV for Sc to about 0.02 eV for Mn (see Fig. 7). If the solute atom affected H binding in the H–H–Sol: 2.8 configuration, then E2H;Sol for this configuration would be differb ent for each solute. The binding energies for H–H–Sol: 2.8
Fig. 5. The DFT calculated E2H;Sol for four different solutes. The Fe solute b results are the H–H interactions without a substitutional solute for the various configurations. The H–H–Sol: 2.8 results are based on a 4 4 4 supercell.
configurations are similar, suggesting that a solute atom does not affect the binding of a 2nd H atom when it is not bound to the solute atom. A comparison of the H– Sol–H and H–H–Sol results in Fig. 5 reveals that H atoms do not always prefer binding to a solute atom over another H atom. For Mn and Ni, H–H–Sol binding energy is larger than all the H–Sol–H binding energies, suggesting that a 2nd H atom would prefer binding to the H atom over the solute atom. For Sc and Ti, the opposite is true and a 2nd H atom would prefer binding to the solute atom. One would expect H–H–Sol binding to be preferred for
5818
W. Counts et al. / Acta Materialia 59 (2011) 5812–5820
solutes that have small (<0.04 eV) E1H;Sol . The 3d transition b metals that fit such a criterion are V, Cr, Mn, Co and Ni [25]. The 2H results were then used as a guide to construct a series of related H–Sol defects with 1–5 H atoms as shown in Fig. 6. The series starts with one H occupying a 2NN tsite, which is the lowest-energy 1 H configuration for most 3d transition metals. The 2H–Sol configuration is the H– Sol–H: 4.3 configuration from Fig. 5. This configuration was chosen because it is either the most stable configuration for solutes that strongly bind H or its energy is very close to the most stable for solutes that weakly bind H. For the same reason an attempt was made to keep the H–H separation distances large within this series of related H–Sol defects. The subsequent 3H–5H defects were constructed by adding a 2NN H to the previous H–Sol defect. The series of configurations in Fig. 6 represent a pathway for a solute atom to bind up to 5 H atoms. A solute will bind multiple H atoms only when the H concentration is sufficiently high. Recently, Sanchez et al. [39] showed that it is energetically favorable for Fe to undergo a large body-centered-tetragonal (bct) distortion when the concentration of H is very high (33 at.%). Under these conditions, H prefers the o-site over the t-site. For lower H concentrations, these authors find (as we do) that the t-site is preferred. Our calculations of multiple H atoms binding to a defect are based on low overall H concentrations, and we only considered t-site occupation of H. Even so, it is possible that a local increase in the H concentration could result in a local distortion analogous to the bct distortion found by Sanchez et al. However, we did not observe any evidence for this type of significant bct distortion in any of our defect cells, even though we relax cell vectors for all defect supercells. The H binding energies for each of the configurations in Fig. 6 is shown in Fig. 7. In all cases, we see that a single solute has a favorable binding for more than one H. In fact, the binding energies of multiple H atoms is often greater than the binding of the initial H. For example, a comparison of E1H;Sol and E2H;Sol for each of the solutes b b 2H;Sol reveals that Eb is around 0.03 eV greater than E1H;Sol . b The H binding behavior of the third and fourth H atoms can be broken into two different categories. For the solutes Sc and Mn, and for pure Fe, ExH;Sol increases for x = 3–4, b meaning it is energetically favorable for these solutes to add both the third and fourth H atoms. For the solutes Ti, V, Cr, Co, Ni, Cu and Zn, ExH;Sol decreases for x = 3 b followed by an increase for x = 4 to a level near E2H;Sol . b
Fig. 7. DFT calculated EbxH;Sol , where x = 1–5, for the 3d transition metal solutes.
The 5H configuration was investigated for most of the solutes (apart from Cr, Co and Zn) and in each case it was found that the binding energy of the fifth H atom decreased . to values below E1H;Sol b The dependence of the binding energy on the number of H atoms is qualititatively different for solute atoms compared to vacancies. At a vacancy, E1H; and E2H; are simb b ilar and then binding energy drops significantly (between 0.1 and 0.2 eV) as the vacancy binds 3–6 H atoms [22], while at a substitutional solute the H binding energy potentially increases as more H is bound. The reason the H binding energy does not drop in the presence of a solute is related to the H–H separation distances. In the case of a solute atom, the shortest H–H separation distance is ˚ when 2–4 H atoms are present. The EH;H data around 2.8 A b in Fig. 1 shows that two H atoms occupying t-sites sepa˚ do not have a negative binding energy and rated by 2.8 A thus H–H interactions will not cause a reduction in ExH;Sol . b There is little experimental data with which to compare the DFT results in Fig. 7. Pressouyre and Bernstein [10] measured the H–Ti binding energy and reported a value of 0.14–0.19 eV. The DFT calculated E1H;Ti is 0.08 eV and b E4H;Ti is 0.13 eV. The agreement between experiment and b theory improves when the possibility that Ti binds more than one H atom is considered. As discussed earlier, the issue of how many H atoms will bind to a particular solute depends both on the calculated binding energies as well as the defect concentrations. Without performing a full thermodynamic analysis, we cannot determine the H concentrations needed to form various types of H–solute defects. However, we can use the conclusions of Gunaydin et al. [23] to describe which of the energetically stable H–solute defects are also stable
Fig. 6. H–Sol defect geometries containing 1–5 H atoms. Black lines between Fe atoms outline a cubic unit cell. Gray lines are Fe–H bonds.
W. Counts et al. / Acta Materialia 59 (2011) 5812–5820 Table 1 Energetically stable and decomposition stable H–solute complexes. Solute
Stable H–solute complexes
Sc Ti V Cr Mn Co Ni Cu Zn
1H–Sc; 2H–Sc; 4H–Sc 1H–Ti; 2H–Ti; 4H–Ti 1H–V; 2H–V 2H–Cr 2H–Mn, 4H–Mn 2H–Co 1H–Ni; 2H–Ni 1H–Cu; 2H–Cu; 4H–Cu 1H–Zn; 2H–Zn
with respect to decomposition into other H–solute defects. Based on their Born–Oppenheimer molecular dynamics investigation of H–h binding in Al, Gunaydin et al. [23] noted defect complexes that did not lie on the convex (if a favorable Eb is defined as negative) or concave (if a favorable Eb is defined as positive) hull would either absorb or release H. Thus defect complexes can be energetically stable, but not stable with respect to decomposition into other H–solute defect complexes. Applying the concept of a concave hull to the data in Fig. 7 reveals that only a few H–solute defect complexes are both energetically stable and stable with respect to decomposition, as shown in Table 1. For all the 3d transition metals, the 2H–Sol defect clusters fulfill both the energy and decomposition stability requirements, while all the 3H–Sol and 5H–Sol defect complexes fail one or both stability requirements. For Cr, Mn and Co, the 1H–Sol defect is not energetically stable, but other defect complexes do fulfill both stability requirements. 4. Conclusions We used DFT calculations to explore H–H interactions as well as multiple H binding to interstitial and substitutional solute atoms in bcc Fe. H–H interactions are not energetically favorable when the H–H separation distance ˚ , after which the EH;H is generally positive, is less than 2.0 A b but small. The maximum EH;H of 0.03 eV is associated with b three different H–H defect pairs whose separation distances ˚ and agrees well with the experirange from 2.0 and 4.2 A mentally measured value. The lowest energy H–C defect has a EC;H of 0.07 eV and b ˚. the H–C separation distance for this defect pair is 3.5 A We then calculated the binding energy of a second H atom in three different locations to the lowest energy H–C defect. The incremental binding energy of the second H atom E2H;C ranges from 0.01 to 0.07 eV. In each case, the bindb ing energy of the second H atom is similar to that of the first, suggesting that there is no synergistic effect between H atoms surrounding the C atom. Thus, the presence of an H atom near a C atom has little effect on the binding of a second H atom. Finally, we investigated the ability of 3d transition metal solutes to bind up to five H atoms. A better understanding
5819
of H–H interactions in the presence of a solute atom was gleaned by investigating a number of defect complexes with two H atoms placed in various tetrahedral sites around a solute atom. Generally, we found that larger H–H separation distances are favored over shorter ones. With this insight, a series of related H–solute defects containing 1– 5 H atoms was constructed and the H binding energies were calculated. The binding energy of the second H to a 3d transition metal solute is 0.03 eV greater than binding of energy of the first, independent of solute. The binding energies of the third to fifth H atoms vary but are generally positive. The overall stability of a H–solute defect complex depends on two separate criteria. First the binding energy must be positive, and second the defect complex must be stable with respect to decomposition to other H–solute defect complexes. The decomposition stability was determined by analyzing the concave hull formed by the incremental binding energies. Defect complexes that did not lie on the concave hull will either absorb or release H to form a defect complex that does lie on the concave hull. Based on this stability analysis, we find that the largest H–solute defect complex for V, Cr, Co, Ni and Zn contains two H atoms, while for Sc, Ti, Mn and Cu, the largest defect complex contains four H atoms. Acknowledgements The authors would like to acknowledge funding from General Motors Corporation. The authors also acknowledge funding from the Department of Energy under grant DEFG36-08GO1813. The authors would like to acknowledge helpful discussions with Scott Jorgensen. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
Troiano AR. Trans ASM 1960;52:54. Oriani RA, Josephic PH. Acta Metall 1974;22:1065. Beachem CD. Metall Trans 1972;3:437. Birnbaum HK, Sofronis P. Mater Sci Eng A 1994;176:191. Nagumo M, Nakamura M, Takai K. Metall Trans A 2001;32:339. Takai K, Shoda H, Suzuki H, Nagumo M. Acta Mater 2008;56:5158. Oriani RA. Acta Metall 1970;18:147. Darken LS, Smith RP. Corrosion 1949;5:1. Au JJ, Birnbaum HK. Acta Metall 1978;26:1105. Pressouyre GM, Bernstein IM. Metall Trans A 1978;9:1571. Meyers SM, Picraux ST, Stoltz RE. J Appl Phys 1979;50:5710. Besenbacher F, Myers SM, Nordlander P, Nørskov JK. J Appl Phys 1987;61:1788. Gibala R. Trans AIME 1967;239:1574. Wolverton C. Acta Mater 2007;55:5867. Wolverton C, Ozolins V, Asta M. Phys Rev B 2004;69:144109. Shin D, Wolverton C. Acta Mater 2010;58:531. Ohnuma T, Soneda N, Iwasawa M. Acta Mater 2009;57:5947. Domain C, Becquart CS, Foct J. Phys Rev B 2004;69:144112. Slane JA, Wolverton C, Gibala R. Metall Trans A 2004;35:2239. Ismer L, Park MS, Janotti A, Van de Walle CG. Phys Rev B 2009;80:184110. Lu G, Kaxiras E. Phys Rev Lett 2005;94:155501. Tateyama Y, Ohno T. Phys Rev B 2003;67:174105.
5820
W. Counts et al. / Acta Materialia 59 (2011) 5812–5820
[23] Gunaydin H, Barabash SV, Houk KN, Ozolins V. Phys Rev Lett 2008;101:075901. [24] Monasterio PR, Lau TT, Yip S, Van Vliet KJ. Phys Rev Lett 2009;103:85501. [25] Counts W, Wolverton C, Gibala R. Acta Mater 2010;58:4730. [26] De Schepper L, Segers D, Dorikens-Vanpraet L, Dorikens M, Knuyt G, Stals LM. Phys Rev B 1983;27:5257. [27] Hohenberg P, Kohn W. Phys Rev 1964;136:B864. [28] Kohn W, Sham LJ. Phys Rev 1965;140:A1133. [29] Kresse G, Hafner J. Phys Rev B 1993;47:558. [30] Kresse G, Fu¨rthmuller J. Phys Rev B 1996;54:11169.
[31] Blo¨chl PE. Phys Rev B 1994;50:17953. [32] Perdew JP, Burke K, Ernzerhof M. Phys Rev Lett 1996;77:3865. [33] Kittel C. Introduction to solid state physics. 7th ed. New York: Wiley; 1996. [34] Vincent E, Becquart CS, Domain C. J Nucl Mater 2006;351:88. [35] Hirth JP. Metall Trans A 1980;11:861. [36] Jiang DE, Carter EA. Phys Rev B 2004;70:064102. [37] Jack DH, Jack KH. Mater Sci Eng 1973;11:1. [38] Jiang DE, Carter EA. Phys Rev B 2003;67:214103. [39] Sanchez J, Fullea J, Andrade C, de Andres PL. Phys Rev B 2010;78:014113.