Here, the results of our investigations are presented and it is discussed how they can be interpreted in the context of grain boundary hydrogen embrittlement. On the phenomenological side, our main interest is in the competition between dislocation and grain boundaries in the aggregation of hydrogen. On the methodological side, the main goal is a quantitative scale transfer from the quantum mechanical description of hydrogen at dislocations and grain boundaries to an elastoplastic deformation model on the macroscale that reflects hydrogen aggregation. Apparently, one of the core challenges in such approaches is the reduction of complexity of the model while still catching the essential effects. We reduce the complexity step by step, which allows us to consider in each stage of simplification the loss of accuracy we inevitably tolerate in the description.

#### 4.2. Modelling Hydrogen Aggregation Considering Dislocation and Grain Boundary Effects

We distinguish two contributions to the hydrogen aggregation at grain boundaries here: on the one side, the stress concentration at the grain boundaries when the system is subjected to mechanical load and, on the other hand, the binding energies and binding length scales to the grain boundaries when the system is free from external stresses. For the latter, we introduce a voxel averaging scheme that allows us to include results from the ab initio calculations that are reported in [

15] (see

Figure 6).

Here,

${L}_{\mathrm{voxel}}$ is the length of the voxel and

${l}_{\mathrm{GB}}$ the range of attractive sites associated with the grain boundary. Then, the volume fraction of the grain boundary is defined as

The average hydrogen concentration in this voxel can then be expressed as

Via the ab initio results from [

15], the concentration becomes dependent on the distance to the grain boundary as

${E}_{\mathrm{GB}}={E}_{\mathrm{GB}}\left(r\right)$ when

r is the distance from the grain boundary, thus

However, the grain boundary fraction we introduce via this scheme is then fixed for a given voxel length, which leads to undesirable effects. When the voxel length is set to

${L}_{voxel}=1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m, we obtain a hydrogen concentration profile as shown in

Figure 7, corresponding to an estimated grain boundary fraction of ∼10

${}^{-3}$. This scenario shows basically no remaining hydrogen segregation at the dislocations in the grains. To reduce the grain boundary fraction to values that show less dominant hydrogen aggregation at the grain boundaries, a voxel length of about 100 micrometres has to be chosen (see the simulation result obtained for that case in

Figure 8). However, in that case, the numerical parametrisation implies a solution of the elastoplastic equations on an undesirably large spatial scale.

At this point, we have to recognise that defining a representative grain boundary binding energy, which can be used to obtain a composite model, i.e., analytical description, is a rather ineffective approach in comparison to the composite model for the dislocations. The spatial distribution of dislocations can be assumed to suffice the requirements to be fulfilled for a representative volume element definition, but the distribution of grain boundaries is subject to complex geometric constraints. Furthermore, the characteristic length scale of the grain boundary distribution is typically several orders of magnitude larger than the characteristic length scale of the dislocation distribution. This comparably small density requires larger representative volumes for the composite model, which would introduce further inaccuracies to the model.

To reflect the simulation results, we relate them to classical McLean [

32] segregation profiles based on the same ab initio data sets. This includes estimates for the hydrogen segregation from dislocations to grain boundaries and the effect of hydrogen enrichment at the grain boundaries on segregation to further increased concentration levels.

We estimate the influence of a locally hydrogen enriched grain boundary region on the segregation behaviour of additional hydrogen to those remaining attractive sites at the grain boundary. For this sake, we assume that the work of volume expansion due to the hydrogen is the dominant contribution to the solution energies. Furthermore, we assume that the change in the work of volume expansion under hydrogen solution with increasing hydrogen content is dominated by the change of the bulk modulus of the region at the grain boundary.

Therefore, we consider the following two limiting cases. First, when no other hydrogen atoms are present at the GB, the resulting segregation profile in a simple McLean picture is just based on the energies reported in [

15]. In the second limiting case, we assume that half of all locally available, attractive sites, i.e., with a formation enthalpy

$\le 0.25$ eV, are populated. For the

${\mathsf{\Sigma}}_{3}\left[1\overline{1}0\right]\left(112\right)$ bcc grain boundary, this corresponds to a minimum of three hydrogen atoms per volume

$V\le {10}^{-28}$ m

${}^{3}$, which is in the order of 10% at hydrogen at the grain boundary. To estimate the difference in the segregation energy, assume

$\mathsf{\Delta}{E}_{el}=\nu \mathsf{\Delta}{B}_{0}$ is assumed, with bulk modulus contrast

$\mathsf{\Delta}{B}_{0}$ and partial molar volume of hydrogen

$\nu $. The exact elastic grain boundary data we need is not available, but the change of the bulk modulus due to the hydrogen aggregation at the grain boundary is estimated based on the bulk results reported in [

33]. The resulting change

$\mathsf{\Delta}{B}_{0}$ is approximated as 15 GPa. For the partial molar volume of hydrogen, we refer to the comprising studies summarized in [

34], which suggest a constant value of 1.7 × 10

${}^{-6}$ m

${}^{3}$/mol of atomic hydrogen (half of a H

${}_{2}$ molecule) over wide temperature and pressure ranges. The shift for the segregation energy then amounts to

$\mathsf{\Delta}{E}_{el}\approx 15\times 1.7\times {10}^{3}/{N}_{A}$ eV/J, i.e.,

$\mathsf{\Delta}{E}_{el}\approx 255$ meV.

This value certainly represents a grain boundary that is very densely populated by hydrogen. As the most attractive sites are restricted to a distance of 1–2 Angstroms away from the grain boundary, we assume that the elastic shift of the segregation energies only affects those sites that lie in this area. For the

${\mathsf{\Sigma}}_{3}$ grain boundary investigated in [

15], this corresponds to the four sites within approx. 2 Angstroms distance.

The resulting segregation profiles are shown in

Figure 9, and the effect of the local elastic softening on the segregation profile is partially compensated when also dislocations as hydrogen traps are taken into account. While the detailed data presented in

Figure 4 shows a maximal attraction of 0.34 eV in the area close to the dislocation core, we use here the averaged dislocation binding energy of 0.14 eV, corrected by an entropic contribution about

${k}_{B}Tln({V}_{dislocation}/{V}_{voxel})$. This is consistent as it reproduces an occupation of the dislocation-associated sites, which is reasonable for hydrogen enriched regions, and we make a similar assumption for the estimate of the occupation shift due to the hydrogen induced elastic softening. For an interpretation in the context of macroscopic metallurgical processes, a more detailed description of the ambient hydrogen chemical potential is required. Apart from the surface properties of the samples, especially surface roughness and surface porosity, the humidity of the atmosphere is also essential. When the samples are subjected to large thermal gradients due to heat treatment, the distribution of hydrogen at grain boundaries and dislocations close to the surface will change depending on the distance to the surface.

This resulting effect of hydrogen binding to dislocations and hydrogen enrichment due to reduced mechanical resistance to hydrogen aggregation is a high local hydrogen density at the grain boundary. Though these results are based on estimates for the bulk modulus contrast and an effective dislocation binding energy, which result from a site occupation median, they exhibit the weakness of ferritic grain boundaries to hydrogen accumulation. For increasingly high levels of hydrogen aggregation to the grain boundary, as they are required for hydrogen embrittlement, a kinetic transport of hydrogen is required in addition to thermodynamically driven transport. As recently pointed out in [

35], hydrogen shielded slip transfer to grain boundaries might offer not only a source for grain boundary stress concentration, but also this non-thermodynamical hydrogen transport process.

At this point, it is worth discussing the effect of the presence of hydrogen atoms along the grain boundaries on the plastic behavior on the polycrystalline metals. Recent simulations [

35] show that hydrogen atoms have multiple effects of the dislocation–GB interactions. First of all, the segregated hydrogen atoms can develop stress field around the grain boundaries. These stress fields which stem from the misfit volumetric strain of the H atoms can attract/repel the dislocations. Thus, the average slip along the GBs can change [

35]. This change promotes the accumulation of slip in local regions along the boundary that can lead to the formation of nano-cracks and voids.

Moreover, the presence of H atoms can not only significantly increase the critical shear stress needed for resolving the lattice dislocation in the grain boundaries, but it can also change the nature of the GB–dislocation interaction. The presence of H atoms can block the dissociation of the lattice dislocations into GB-dislocation. Thus, the slip either remains along the grain boundary or is transited to the adjacent grain at significantly higher stresses. This leads to the formation of more populated pile-ups and eventually leads to intergranular fracture of the grain boundary surfaces. The dislocations that are present in the pile ups can attract hydrogen atoms and deliver it to the grain boundary. As shown in previous studies in nickel, these extra hydrogen atoms can reduce the fracture energy significantly [

36] and make brittle fracture favourable. This intergranular fracture cannot be achieved by considering only the equilibrium segregation hydrogen atoms along the boundary [

37]. However, clarification of the kinetic aspects of this process needs further investigation.