## Abstract

The behaviour of hydrogen at structural defects such as grain boundaries plays a critical role in the phenomenon of hydrogen embrittlement. However, characterization of the energetics and diffusion of hydrogen in the vicinity of such extended defects using conventional *ab initio* techniques is challenging due to the relatively large system sizes required when dealing with realistic grain boundary geometries. In order to be able to access the required system sizes, as well as high-throughput testing of a large number of configurations, while remaining within a quantum-mechanical framework, an environmental tight-binding model for the iron–hydrogen system has been developed. The resulting model is applied to study the behaviour of hydrogen at a class of low-energy {110}-terminated twist grain boundaries in *α*-Fe. We find that, for particular Σ values within the coincidence site lattice description, the atomic geometry at the interface plane provides extremely favourable trap sites for H, which also possess high escape barriers for diffusion. By contrast, via simulated tensile testing, weakly trapped hydrogen at the interface plane of the bulk-like Σ3 boundary acts as a ‘glue’ for the boundary, increasing both the energetic barrier and the elongation to rupture.

This article is part of the themed issue ‘The challenges of hydrogen and metals’.

## 1. Introduction

The embrittlement of metallic systems by hydrogen has long been a research topic of interest. In conventional steels, problems related to hydrogen embrittlement (HE) were significantly ameliorated by technological improvements in processing, such as improved cleanliness during industrial processing, thus limiting excess oxide and nitride formation within the material. With the development of advanced high-strength steels, with ultimate tensile strengths exceeding 1 GPa, the problem of HE is again a topic of major industrial and academic interest, as seen by a number of European-based consortia [1–3] which aim to study and understand various aspects of the phenomenon.

A key issue in understanding HE is in describing the behaviour of hydrogen at structural defects. Since the solubility of H in bulk *α*-Fe is extremely low (less than 3 at. ppm at room temperature [4]), it is typically assumed that H will segregate to available sinks in the material, such as grain and phase boundaries, dislocations and nanovoids. A common feature between all of the structural defects is a change in available coordination and/or volume for H in comparison with the tetrahedral and octahedral sites available in the bulk. The interplay between coordination number and, for example, available Voronoi volume for H must, therefore, play an important role in understanding HE.

The theoretical modelling of hydrogen behaviour at structural defects is made challenging by the system sizes required to accurately treat, for example, grain boundaries, particularly in order to avoid significant chemical interactions between the solute hydrogen atoms. The system size, as well as the lowered symmetry of the simulation box, leads to considerable challenges for *ab initio* simulations. In many cases, particularly with phase boundaries, it is difficult to *a priori* determine the possible locations of interstitial hydrogen atoms, thus requiring additional calculational effort. Furthermore, the effects of incoherency, and the explicit treatment of misfit dislocations, is beyond the capacity of standard density functional theory (DFT) calculations, apart from very specific conditions.

As a result, the *ab initio* literature [5–8] of hydrogen behaviour at grain boundaries has been limited to the study of highly symmetrical grain boundaries, where the periodic cell can be limited to a few dozen atoms. In *α*-Fe, particular emphasis has been placed on the study of the twin grain boundary, which provides a bulk-like environment, but which also provides a weak trapping site for hydrogen. The Σ5[001](310) and Σ5[001](210) symmetric tilt grain boundaries (STGBs) have been taken as models for ‘open’ grain boundaries, and do indeed present potential trap sites for hydrogen where psuedo-voids are created. In [9], the mobility of H in the vicinity of the Σ5[001](310) was studied extensively using kinetic Monte Carlo (KMC) simulations, where segregation to such voids and eventual saturation of the grain boundaries with H has a significant effect on the mobility. However, a particular problem with the Σ5 STGBs is their extremely high energy with respect to other grain boundaries; the likelihood of these boundaries being formed is rather low, and it is difficult to assess how transferable the hydrogen behaviour at such model boundaries is to realistic materials, where grain boundaries typically have both tilt and twist character, with a wide variety of misorientation angles.

On the other hand, a wide family of empirical and semi-empirical atomistic approaches such as the well-known embedded-atom method [10] or the more recent Gaussian-process-derived [11,12] atomistic potentials are more than capable of reaching simulation sizes suitable for the study of even low-angle grain boundaries. There has indeed been a large degree of work towards investigating the interactions of H in Fe; however, the neglect of explicit chemistry and, perhaps even more significantly, explicit treatment of magnetism in such approaches places limitations on the transferability and applicability of such methods to study systems with more than one component.

The tight-binding (TB) approximation [13], based on the linear combination of atomic orbital framework for the electronic structure, offers an alternative route to solving the Schrödinger equation efficiently. In its traditional two-centred orthogonal (Slater–Koster) form, the TB method offers a good description of chemical and structural trends across the periodic table [14,15]. Moreover, extensions to include charge-transfer [16] and magnetic effects [17–19] are readily achievable. However, the simplicity and tractability offered by the two-centred approximation is tempered by the experience that the orthogonal approach has limitations when simulating systems whose coordination significantly differs from bulk, such as for surfaces and boundaries. As a result, significant efforts have been made in recent years [20,21] to extend the TB approach in order to include the effects of screening and/or environmental contributions. A key result of this work is the development of ‘environmentally dependent’ TB models which are applicable to complex defect structures.

In terms of determining possible structural defects, it is immediately clear that, when considering even pure *α*-Fe, there are an infinite number of possible structures, the relative relevance of which is *a priori* unclear. In this work, we have concentrated on a class of {110}-terminated twist grain boundaries in Fe, the grain boundary energies of which are both low and relatively independent of the angle of rotation. The low energies of these boundaries in comparison with other boundaries of similar degrees of fit (Σ within the coincidence site lattice theory) indicate that these boundaries are of interest. By varying the angle of rotation of the twist boundaries, a wide variety of atomistic environments at the grain boundary plane are generated, which have a significant effect on the binding and diffusion properties of H located in the vicinity of the grain boundary plane.

The overall outline of the paper is, therefore, as follows. In §2, the environmental TB approach for the Fe–H binary system is outlined, along with specific details of the calculations performed. In §3, the {110}-terminated twist grain boundaries are discussed, and the choice of these particular boundaries is motivated. The TB model is then applied to the behaviour of H in the vicinity of the various grain boundaries (§4). The characterization of possible H binding sites and the effect of the twist angle on the trapping energies themselves are discussed. In addition, the diffusion behaviour of H to and from the grain boundary plane, as well as between trap sites within the plane of the grain boundaries, is evaluated. In §5, the mechanical properties of the Σ3{110} twist boundary in the absence and presence of H are studied via simulated rupture tests. The influence of H on the local bonding properties is examined and some implications for H-induced damage of this particular boundary are outlined. An outlook on further potential uses of the model concludes the paper.

## 2. Environmental tight-binding approach

The environmental TB method can be seen as an extension to the conventional TB approximation, via the inclusion of three-centred local environmental contributions. The approach was developed first to study the electronic structure of small clusters of Ni and Co [21], metals for which the interplay between the free-electron-like *sp*-valent electrons and the tightly bound 3*d* electrons is critical for understanding the stability of different crystal phases. A key finding of the paper [21] was that, when a non-orthogonal description is used, it is necessary to consider both the corrections to the effective onsite levels of the atoms via the potentials of the other surrounding atoms and the so-called ‘three-centred’ contributions to the intersite (or ‘hopping’) matrix elements of the Hamiltonian. Omission of either term, or using simply a two-centred non-orthogonal description, leads to a model with no qualitative improvement over the orthogonal description. Here, we extend the analysis of that work to the study of the Fe–H system. The central ideas of the methodology will be sketched here, with a more detailed description of the approach appearing elsewhere [22].

The starting point of the analysis is the cohesive energy within an *ab initio* description, defined as
2.1where *E*_{band} is the total *band* energy of the system, *E*_{Hxc} is the second-order contribution arising from charge-transfer contributions and magnetism, *E*_{dc} is the usual double-counting correction arising from the overlap of the free atomic densities and *E*_{free-atom} is the total energy of the free atoms separated at infinity.

The principal object to evaluate in TB is the Hamiltonian matrix **H**_{TB}, and in particular the matrix elements of **H**_{TB} within an atom-centred one-electron basis {|*ϕ*_{Iμ}〉}, where capital letters denote an atom and Greek letters denote an orbital on that particular atom. Within an *ab initio* framework, these matrix elements can be formally written as
2.2where is the one-electron kinetic energy operator (in atomic units) and *V* _{eff}[*ρ*] is the effective one-electron Kohn–Sham potential, which is a functional of the one-electron density *ρ*. Since in general the basis {|*ϕ*_{Iμ}〉} will be non-orthogonal, we also have the TB overlap matrix **S**_{TB}, defined in terms of its matrix elements
2.3If one has a particular set of {|*ϕ*_{Iμ}〉}, then one can evaluate equations (2.2) and (2.3) and solve the generalized eigenvalue problem,
2.4where ** Ψ** is the matrix of eigenvectors and

**is a vector of corresponding eigenvalues. From this, one could easily evaluate the band energy**

*ϵ**E*

_{band}and one-electron density

*ρ*(

**r**), and thus solve the Kohn–Sham equations within this particular basis—a DFT calculation within an atomic-orbital basis.

However, in order to make the problem tractable, we wish to reduce the nonlinear complexity of equation (2.2). To make a connection with the conventional TB approach, we wish to reduce *H*_{IμJν} and the total energy functional itself into one-, two- and three-centred contributions. Without loss of generality, one can partition the one-electron effective potential *V* _{eff} into atom-centred contributions,
2.5where {**R**_{I}} are the set of atomic positions. However, the nonlinear part of the effective potential, namely that arising from the exchange and correlation contributions, means that the total potential of a multi-atom system cannot be described as the sum of the *individual* potentials of each atom,
2.6Here, however, we use a multi-centre expansion [23] of the exchange-correlation potential *V* _{xc}[*ρ*], such that
2.7and then truncate the expansion to include up to three-body terms, such that the expansion equation (2.7) contains only one-, two- and three-centred terms. This truncation up to three-centred terms is made consistently for the Hamiltonian and the total energy functional, which means that, given a particular choice of basis, all parameters of the total energy functional can be evaluated using clusters of no more than three atoms. This approximation is found to give errors of less than 5 meV per atom in the band energy for ferromagnetic *α*-Fe (at the equilibrium lattice constant) with respect to the untruncated exchange-correlation potential. The exchange and correlation itself is treated with the PBE [24] implementation of the generalized gradient approximation. As indicated in equation (2.1), we use the second-order expansion of the spin-DFT [25], which is consistent with a Stoner-like description of the magnetism.

In order to perform the calculations in this paper, a couple of additional details are required. The Hamiltonian matrix elements are evaluated within an analytical basis set, namely a set of Slater-type orbitals,
2.8where *n*_{μ},*l*_{μ},*m*_{μ} are the quantum numbers associated with basis function *μ*, *r*=|**r**−**R**_{I}|, *Y* _{lm} are the usual real-space spherical harmonics, *N*_{μ} is a normalization constant and *f*(*r*) is a cut-off function, defined as
which leads to smoothly decaying functions of finite range. The exponential parameters *α*_{Iμ} are obtained from calculations of *α*-Fe at the theoretical lattice parameter *a*∼2.84 Å and the H_{2} molecule (*d*_{H-H}=0.73 Å). For the present calculations, the 4*s*, 4*p* and 3*d* states of Fe are chosen as the valence state, with 1*s* being the only valence state included for H.

While all Hamiltonian matrix elements could be calculated ‘on the fly’, it is more efficient to evaluate the matrix elements for all dimers and trimers whose bond lengths lie within the cut-off range of the basis functions, and then to interpolate the results. In contrast with McEniry *et al.* [21], we eschew fitting functions such as exponentials, as the fitting errors led to numerical instabilities in some simulations. Since our basis functions themselves are well-defined functions, the Hamiltonian matrix elements can be obtained to within numerical precision; therefore, we use a fine (approx. 0.05 Å) grid of interatomic distances, and use cubic splines to interpolate between the tabulated results for the two- and three-centred contributions.

All calculations are performed with periodic supercells, with full structural relaxation, including relaxation of the cell shape, unless otherwise stated. To achieve a desired convergence of less than 0.1 meV per atom, necessary for the description of the hydrogen diffusion barriers in Fe, Monkhorst–Pack *k*-point grids, with density of approximately 8000 kpts ⋅ atoms, are used. Owing to the fact that hydrogen is extremely light, the zero-point vibrations of the H atom are taken into account in the evaluation of the trapping energies; however, the variation of the zero-point vibrational energy along diffusion paths, within the nudged-elastic band scheme, is neglected. In some cases, the results are compared with *ab initio* values; these are obtained with the plane-wave VASP package [26], using an equivalent *k*-point mesh, and an energy cut-off consistent with a convergence of less than 0.1 meV per atom.

## 3. {110}-Terminated twist grain boundaries

Within an atomistic modelling framework, it is not feasible to screen all possible classes of grain boundaries, from near-bulk Σ3-type boundaries to low-angle grain boundaries characterized by periodic arrays of edge dislocations, while encompassing the whole range of twist and tilt boundaries. It is the view of the authors that a more systematic approach can be achieved by considering a particular class of boundaries, which can be well characterized by a particular order parameter, and then to examine properties of the boundaries as a function of that parameter.

Holm *et al.* [27] and Ratanaphan *et al.* [28] have evaluated the grain boundary energies of a set of FCC and BCC metals using the embedded-atom method, for a broad range of atomistically accessible boundaries. In the case of the BCC metals Mo and Fe, it was found that, as a function of Σ, the {110} twist grain boundaries were typically among the lowest in energy, and that the energies of such boundaries were virtually independent of the angle of misorientation. Because of this low energy, as well as the low energy of the corresponding [110] surface in *α*-Fe, one may anticipate that such boundaries are experimentally relevant, and that the role of H in the vicinity of such boundaries is hence of interest.

The atomic arrangements of the {110} twist boundaries are illustrated in figure 1. The initial bi-crystal is aligned along the [110] direction, with the two parts of the crystal rotated by a particular angle *ϕ* with respect to one another. For arbitrary values of *ϕ*, the resultant structure is aperiodic; however, for particular values of *ϕ*, coincidence site lattice structures can be obtained, from a Σ3 structure for a rotation angle of 70.53^{°} to a Σ17 structure for *ϕ*∼87^{°}. Unsurprisingly, in order to reach higher values of Σ, the symmetry of the system is lowered, which corresponds, within a periodic supercell, to larger effective unit cells.

In this work, we consider the four grain boundaries with the lowest Σ values, namely the Σ3, Σ9, Σ11 and Σ17, the structures of which are shown in figure 2. In order to reduce the impact of interactions between two parallel interfaces within our periodic supercell, the distance between adjoining interfaces is chosen to be at least 16 Å. We find through numerical testing that this distance is sufficient to converge the grain boundary energies to approximately 10 mJ m^{−2}. As the Σ value is increased, the size of the simulation cell will also increase, going from 48 atoms for the Σ3 (in the absence of H) to 544 atoms for the Σ17. Although we perform benchmark *ab initio* simulations with these system sizes for pure Fe, the addition of H increases the complexity of the *ab initio* simulation in two ways. Firstly, the lowering of symmetry adds to the computational expense by reducing the number of available symmetry operations. Secondly, in the absence of H, the electronic structure within each grain of the system is rather bulk-like; the insertion of H perturbs the electronic structure, leading to longer-ranged effects on the magnetic structure of each ‘grain’, which in turn impacts the electronic convergence of the system.

## 4. Energetics and diffusion for twist grain boundaries

The first test of the Fe–H TB model is to evaluate the H-free grain boundary energies *γ*_{GB} for the four considered grain boundaries; these energies are obtained using the usual formula
4.1where *E*_{cell} is the total energy of the TB simulation cell, *N*_{cell} is the total number of atoms in the cell, is the energy per atom of pure *α*-Fe at the simulation cell volume and *A*_{GB} is the area of the considered grain boundary. The factor of 2 arises from the fact that the simulation cell has two equivalent grain boundaries. In order to obtain the lowest-energy structure, rigid translations of the grains with respect to one another parallel to the grain boundary plane were performed, but were not found to lower the energy of the system.

The summary of the results for *γ*_{GB} is given in table 1. The bulk-like Σ3 grain boundary has the lowest energy of the four boundaries: the value of *γ*^{Σ3}_{GB}∼0.32 J m^{−2} is in excellent agreement with both the *ab initio* results obtained and the embedded-atom method modelling of Ratanaphan *et al.* [28]. The Σ9, Σ11 and Σ17 boundaries have higher grain boundary energies (0.58−0.61 J m^{−2}). The values are in good agreement with *ab initio* approaches; although the energetic ordering of the structures is different, the absolute errors are of the order of 2 meV per atom^{}, which is well within the expected accuracy of the TB model.

The next stage is to obtain the energetics of H in the vicinity of the various grain boundaries. As it is not straightforward to determine possible binding sites at the boundary plane, a brute-force approach of evaluating the energetics at all plausible interstitial sites was performed. The result of one such high-throughput screening effort is illustrated in figure 3, where the solution energies of H for over 400 configurations at the grain boundary plane of the Σ3 boundary were calculated in order to assess the potential energy surface. Even for the simplest boundary considered, the topology of the energy surface at the boundary is far from trivial. For the larger cells of the Σ11 and Σ17 boundaries, approximately 900 initial structures were considered per boundary, with full relaxation performed for all situations where the hydrogen solution energy was less than 1 eV. Such vast configurational explorations for these boundaries are practically unfeasible with *ab initio* approaches.

The most interesting feature that we find is that, in contrast with our expectations of observing a large number of possible trap sites, we find just two energetically favourable trapping sites per grain boundary, which are found to be symmetrically equivalent. All other interstitial sites in the vicinity of the boundary plane relax to sites that are either higher in energy than or energetically equivalent to the tetrahedral and octahedral sites found in the bulk. The energetics of the trapping sites are summarized in table 2. The trapping sites at the Σ3 boundary are fourfold coordinated tetrahedral-like sites (figure 4), where the trapping energy of −0.26 eV with respect to hydrogen in bulk *α*-Fe arises from an elongation of the tetrahedron perpendicular to the boundary plane, and a subsequent increase of around 10% in the available Voronoi volume. For the other three grain boundaries, the H trapping energies are much higher in magnitude; however, the local structures of the binding sites are ostensibly the same, with a fivefold coordination (figure 4). The observed trapping energies are large (*E*_{trap}≈−1 eV per atom) in comparison with the typically calculated energies at grain boundaries or dislocations. The very high trapping energies appear to arise from the combination of the enhanced available volume, and the asymmetry of the coordination shell, which lowers the repulsive contributions to the total energy from *E*_{dc} in equation (2.1) while maintaining a large chemical contribution. By analysing the charge densities (see figures in the electronic supplementary material), we found that the binding of H to Σ3 is rather isotropic, while the binding in the other grain boundaries is much more complex, with relatively strong directional effects. We attempted to insert a second H atom into the deep traps, but this was found to be energetically unfavourable. Since the area of the repeat cells of the grain boundaries is much larger for Σ>3, the *density* of these ‘point’ trapping sites is rather low, and since there are no additional trapping sites at these boundaries, these boundaries therefore can only act as sinks for a limited amount of H. The effect of this low density of trapping sites can be seen graphically in figure 5, where the grain boundary energy is plotted as a function of chemical potential *μ*_{H}, assuming that all possible sites at the boundary are occupied. In the case of Σ9 and Σ17, although the binding is strong, the effect on the grain boundary energy, even for relatively high *μ*_{H}, is rather weak. In the cases of Σ3 and Σ11, the effect is rather extreme, due to the lower area of the repeat unit cell, and hence the higher density of traps. Indeed, in the case of the Σ3 boundary, the effective grain boundary energy *γ*_{GB} can become negative for high values of *μ*_{H}.

In order to further investigate the impact of such boundaries on the behaviour of H, we performed a series of nudged-elastic-band simulations to calculate the diffusion barriers for H around the trap sites (table 2). For the Σ3 boundary, the diffusion barrier from the bulk into the trap sites is found to be approximately 0.11 eV, which is comparable in magnitude to the diffusion barrier between tetrahedral sites within the bulk itself (approx. 0.09 eV). For the Σ9,Σ11 and Σ17 boundaries, the diffusion barrier into the trap sites is of the order of approximately 0.25 eV, which indicates that, at room temperature, it should be reasonably easy to occupy these deep trap sites. The *depopulation* of these trap sites, on the other hand, is extremely difficult, as the escape barriers in each case are of the order of 1 eV. Moreover, since there are no additional low-energy sites in the vicinity of the deep traps in the boundary planes, it is difficult to use the grain boundary plane itself as an aid to diffusion. The calculated barriers between the isolated deep traps in the Σ9–Σ17 boundaries are all of the order of 1 eV.

Hence, the {110} twist grain boundaries, other than the Σ3, have very specific features with regards to hydrogen behaviour. The picture of the boundaries is one of isolated deep traps for H, which are relatively easy to occupy under normal conditions, but for which the hydrogen atoms have little possibility of rapidly escaping. Indeed, for hydrogen atoms which are not trapped by the deep traps, or in situations where the traps are already filled, the diffusion barrier across the boundary plane, calculated to be of the order of 0.1–0.2 eV, is rather low, meaning that the boundary plane is only a very weak barrier to hydrogen motion. For the Σ3 boundary, the trapping sites are weaker, but considerably denser, so that the ability of the Σ3 boundary to attract hydrogen may be more significant, even if the lifetime of hydrogen there is rather short.

## 5. Tensile testing of Σ3

In the previous section, we have studied the trapping and diffusion behaviour of H at the various twist boundaries; in this section, we wish to understand the effect that H present at these boundaries has on the mechanical properties of the interface. In order to do this, we perform simulated tensile tests, in a similar spirit to those of Tahir *et al.* [7]. As the Σ3 boundary has the smallest repeat cell, and a high density of trap sites per unit area, we choose to study the effect of H on the tensile behaviour of this boundary. The relaxed interface is strained in a stepwise manner, and at each stage of the elongation the internal coordinates of the system are relaxed. At some applied strain, the interface will break and split into two unconnected grains. The simulations were performed firstly without H, and then repeated for various H concentrations; here, we report only the results for the fully saturated boundary, namely where the two symmetry-equivalent trapping sites on each of the two grain boundaries within the periodic cell are occupied. This corresponds to conditions of H chemical potential of the order of −0.2 eV and above.

The effective stress–strain curves are illustrated in figure 6. There are a number of key features that are apparent here. We find that the presence of hydrogen at the grain boundary slightly increases Young’s modulus (by approx. 3%), therefore stiffening the grain boundary. More notably, we find that the presence of H leads to an increase of around 25% in the elongation to fracture. By looking at the obtained geometries at a strain of 25% (figure 7), we see that the H atoms at the boundary hinder the elongation of the bonds at the boundary plane in comparison with the bonds within the grain interior, and therefore act as a ‘glue’ to keep the interface together.

This has a number of important implications in terms of the embrittlement problem. Firstly, for this particular grain boundary, the theory of hydrogen-enhanced decohesion does not apply; the presence of hydrogen enhances the cohesive properties of the grain boundary. Moreover, the conventional thermodynamical Rice–Wang model [29] would indicate that, since the adsorption energy of H to the [110] free surface of Fe (*E*_{H}=−0.4 eV) is much larger than that to the grain boundary, thus lowering the work of separation, hydrogen should have a detrimental effect on the stability of this phase boundary. Our simulations show that this is not the case. On the other hand, Rice & Wang [29] pointed out that changes both in the work of separation and in the maximum value of the stress as a function of strain, in the presence of hydrogen, were the key effects to take into account. In our case, *σ*_{max} is increased by the presence of H. Therefore, an interesting direction for future work is to examine the relative importance of changes in the work of separation and changes to the stress–strain curve; in other words, a comparison between the importance of thermodynamic and mechanical effects.

It is of course clear that the present rupture tests can only capture part of the physics of hydrogen-induced failure. The current simulations are performed at zero temperature, and hence the role of thermally activated processes such as dislocation nucleation are absent from the theory. Indeed, the ‘glue’ effect which helps to enhance the cohesion of the interface at low temperature, may hinder the motion of dislocations at higher temperatures, which may lead to a loss of ductility. In order to understand the problem of HE in such materials more generally, one should indeed consider both the ‘bond-breaking’ or decohesion-driven failure mechanism as well as plasticity effects, where the interaction of H with the dislocation emitted from the grain boundaries under strain may be the dominant effect in some conditions. The extension of the simulations to include temperature via real-time molecular dynamics is work in progress.

## 6. Conclusion and outlook

In this work, a systematic and transferable environmentally dependent TB modelling has been outlined, and applied to the Fe–H system. Through extensive high-throughput calculations with the TB model, we have examined the behaviour of H in the vicinity of a set of energetically relevant {110}-terminated twist grain boundaries. In the case of the lowest-energy Σ3, we find a relatively high density of energetically weak traps, which allow for reasonably fast diffusion of H in the vicinity. In the other three cases examined, we find the surprising result that we find a low density of very deep traps, which are very likely to be occupied, but for which hydrogen escape is extremely difficult.

In the case of the Σ3 boundary, we have examined the effect that hydrogen has on the tensile properties of the grain boundary itself. We find that, for this particular boundary, the presence of H at the interface plane acts as a glue to keep the boundary together, and that the elongation to fracture is increased in the presence of H. Therefore, the mechanism of hydrogen-enhanced decohesion appears to be unlikely for this grain boundary.

In terms of the modelling methodology, the success of the current model has opened up a variety of avenues for future research. Ongoing work includes the examination of H behaviour at low-angle tilt grain boundaries, where we expect more complex binding and diffusion behaviour due to the lowered symmetry. A further topic of interest is the effect of co-segregration of carbon and hydrogen to grain boundaries and dislocations, and the influence that co-segregation has on the energetics and mechanics of such structural defects. The systematic and flexible nature of the present modelling methodology also allows multi-component systems to be included in a straightforward manner, with relatively low effort required.

We have demonstrated the applicability of the present TB model to high-throughput simulations of geometrically complex systems, which are effectively beyond the domain of *ab initio* methods. Since the study of extended defects is crucial in the understanding of hydrogen embrittlement, the extension of atomistic simulations illustrated here, to allow fully quantum-mechanical simulations to be performed on large simulation cells, is of vital importance in attempting to understand specific features of the H interactions in determining failure mechanisms in Fe-based materials.

## Authors' contributions

E.J.M.E. developed the tight-binding approach and performed the simulations and analysis. The manuscript was written by E.J.M.E. and T.H., with comments by J.N. All authors read and approved the final manuscript.

## Competing interests

We declare that we have no competing interests.

## Funding

The authors acknowledge financial support from thyssenkrupp Steel Europe AG.

## Acknowledgements

The authors acknowledge engaging discussions with Mike Finnis, Gerard Leyson, Jutta Rogal, Thomas Schablitzki and Ralf Drautz. In addition, the authors thank the referee for the suggestions of further analysis.

## Footnotes

One contribution of 24 to a discussion meeting issue ‘The challenges of hydrogen and metals’.

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.3784598.

- Accepted December 16, 2016.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.