Effects of aluminum on hydrogen solubility and diffusion in deformed Fe-Mn alloys

We discuss hydrogen diffusion and solubility in aluminum alloyed Fe-Mn alloys. The systems of interest are subjected to tetragonal and isotropic deformations. Based on ab initio modelling, we calculate solution energies, then employ Oriani's theory which reflects the influence of Al alloying via trap site diffusion. This local equilibrium model is complemented by qualitative considerations of Einstein diffusion. Therefore, we apply the climbing image nudged elastic band method to compute the minimum energy paths and energy barriers for hydrogen diffusion. Both for diffusivity and solubility of hydrogen, we find that the influence of the substitutional Al atom has both local chemical and nonlocal volumetric contributions.


I. INTRODUCTION
High strength structural steels are a material class key to further developments in automotive light-weight construction [1][2][3]. At Mn weight fractions between 15-25 percent, high manganese steels offer a combination of high strength and desirable plasticity characteristics. The stable austenitic structure offers high cost efficiency [4][5][6][7][8], and investigations on Fe-Mn systems revealed the relation of high work hardening rates and ductility to the austenitic microstructure [9,10]. In addition, high manganese steels exhibit a lower susceptibility to hydrogen embrittlement [11] than ferritic and martensitic steels [12,13].
The systematic investigation of hydrogen embrittlement in high manganese steels is challenged by the microstructural, interfacial and chemical complexity of these materials. The scientific focus has been on intergranular failure modes and general heterophase interface failure [14][15][16][17][18]. Hydrogen induced degradation [19] is a major hindrance for the implementation of promising steel designs for automotive applications [20]. While the austinitic phase itself exhibits low diffusivity and high solubility of hydrogen, the transformation induced plasticity (TRIP) moderated martensitic transformation yields strong gradients of solubility and diffusivity due to the presence of ferritic phases.
An improved resistance to hydrogen induced delayed fracture can be realised adding temper softeners like Cr, Mo and V in martensitic steels. This highly desirable effect originates from the suppressed fracture by carbide precipitation, shifting the intergranular fracture to intragranular fracture [21]. In austenitic steels, the addition of Ni and Si is an established method to increase the ductility of hydrogen-loaded steels [22][23][24][25], and the beneficial effect of Mn on hydrogen solubility has been investigated for instance in [26,27]. The successful alloying of high manganese steels with Al, which reduces the weakness against hydrogen embrittlement, suggests to study the effect of Al on the different aspects of hydrogen in Fe-Mn systems. As it is an integral part of production processes, we also study the influence of mechanical loading, in practice isotropic and tetragonal distortions, on solubility and mobility of hydrogen.
The article is organized as follows: In the second section we describe the methodical basis of our approach, including several aspects of the systematic approximations we make. The third section contains the results of our investigations, where we distinguish between stress-free, isotropically distorted and tetragonally distorted systems and discuss their influence on hydrogen solubility and diffusivity. Finally we conclude with a brief discussion of the presented results in the context of different material configurations and future perspectives.

II. METHODS
The methodical basis of our investigation is electronic structure density functional theory (DFT). It allows to predict the quantities of interest for exactly defined crystal geometries and chemical compositions. The systems which we investigate are introduced in Figs. 1 and 2. They consist of 2×2×2 supercells of an austentitic Fe 24 Mn 8 matrix and the resulting system when we substitute one Fe atom by an Al atom, Fe 23 Mn 8 Al. Hydrogen is introduced on interstitial sites. We obtained our ab initio results from DFT calculations employing the Vienna ab initio simulation package (VASP) [28]. For the PAW pseudo-potentials [29] we use the GGA-PBE exchange and correlation functional [30]. The convergence of DFT related parameters has been carefully tested. We find that an 8×8×8 k-point Monkhorst-Pack [31] mesh for the 2×2×2 supercell and a plane wave energy cutoff of 600 eV are sufficient to ensure a convergence of the calculated formation energies to better than 10 meV. An electronic smearing of 0.1 eV was chosen within the Methfessel-Paxton scheme [32]. Further, subject-specific numerical details will be discussed at the appropriate places of this publication.

Hydrogen solubility
The goal of this section is to determine the concentration of hydrogen as solid solution in the fcc Fe-Mn-Al system. Two aspects are of primary interest here: First, the local arrangement of the hydrogen atoms in the interstitial sites in different environments of iron, manganese and aluminum, to find a site preference for the hydrogen. Second, the dependence of the solubility on mechanical deformation conditions, in particular for pure volume changes due to isotropic straining, and for volume-conserving tetragonal distortions.
The central quantity required to determine the equilibrium hydrogen concentrations is the solution energy, which is defined as Here, E M H [σ] is the energy of the metal-hydrogen system, where σ denotes the occupied interstitial site of the hydrogen, as illustrated in Fig. 2. E M is the energy of the purely metallic alloy system, and E H2 /2 is the reference potential for half an isolated H 2 molecule. We note that for vanishing external stress the solution energy and enthalpy coincide. This energy difference E σ will in the following be computed using ab initio techniques. It serves as an input for the calculation of the equilibrium hydrogen concentration (or occupation probability for the different interstices), which is in the dilute limit of non-interacting hydrogen atoms given by Sievert's law Not shown here are the tetrahedral sites. We note that the oc2 configuration has the same next neighbour configuration as our reference octahedral site in the Fe24Mn8 system, 4 Fe and 2 Mn atoms, but differs by the substitution of one Fe by Al. The oc1 case corresponds to the substitution of one Fe atom at the reference site with an Al atom, changing the next neighbourhood configuration as 4 Fe 2 Mn → 3 Fe 2 Mn 1 Al. [33], where p H2 is the partial pressure of the surrounding H 2 gas. We have to distinguish this pressure from a mechanical pressure P acting on the solid solution phase, which can be independently controlled by mechanical constraints. In the following, we will focus on the mechanical part and not further discuss the gas pressure p H2 . The value of the (suppressed) proportionality constant in the above expression (2) depends on the choice of a reference pressure and the precise definition of the hydrogen concentration.

Stress free systems
We calculate the energy contributions E M H and E M for the supercells shown in Figs. 1 and 2, using full ionic   Table I.
For the Fe-Mn reference system we only consider the energetically most favorable configuration shown in Fig. 1, using a Fe 24 Mn 8 supercell, where the hydrogen atom in the octahedral position is surrounded by 4 iron and 2 manganese atoms. Such a site exists 24 times in the supercell. The other type of octahedral sites, where the H atom is surrounded by 6 Fe atoms, exists 8 times, but has a higher solution energy when the system is deformed.
This energy is compared to different atomic arrangements for a Fe 23 Mn 8 Al system, where one iron atom is replaced by aluminum. For this configuration, there are four chemically inequivalent environments for the H atom, as depicted in Fig. 2. For the configuration denoted by 'oc1' the nearest neighbors of the the H atom are 3 Fe, 2 Mn and one Al atom, which exists four times in the supercell. The configuration 'oc2' has the neighborhood 4 Fe, 2 Mn, 0 Al, with a multiplicity of 20. 'oc3' has the environment 6 Fe, 0 Mn, 0 Al, and it exists six times. Finally, 'oc4' has 5 Fe, 0 Mn, 1 Al neighbors and exists twice. Apart from the chemical environment, also the local geometric arrangement influences the solution energies. In the case of the 'oc2' configuration, which is of specific interest due to its energetic preferrability, we find the most attractive local arrangement in 9 of the 20 sites. The resulting 11 sites are less attractive and split into three subgroups of local geometric arrangements. The resulting solution energies for the most attractive sites for each chemical environment are listed in Table I.
By comparison to the reference system Fe 24 Mn 8 , we see that there are two types of interstices with identical next neighbour configuration there and in Fe 23 Mn 8 Al. The oc2 site is energetically more favourable by about 60 meV than the reference site with 4 Fe and 2 Mn atoms as nearest neighbor. This is attributed to the nonlocal effect of the substitutional Al atom. The oc3 site is about 70 meV less attractive than the reference site with 6 Fe atoms as nearest neighbor, which has nearly the same energy as the reference site with 4 Fe and 2 Mn atoms in the immediate neighborhood. This discrepancy shows that the solution energy is lowest in a manganse rich environment, which is in agreement with the predictions in [26,27]. It also exhibits that in the Fe 23 Mn 8 Al system, the volumetric effect of the Mn atoms is limited to the direct vicinity of the Mn atoms.
The general observation is therefore, that Al atoms as next neighbour make octahedral sites energetically less favorable. We can therefore conclude that for low temperatures and hydrogen concentrations the sites close to aluminum atoms will be not be populated. In the room temperature regime with k B T ≈ 25 meV the occupation probability ratio of the energetically most attractive oc2 configuration of the Fe-Mn-Al system in comparison to the Fe-Mn reference system is therefore according to Eq. (2) given by where the factor r oc2/ref takes into account the multiplicity of the amount of reference sites and relevant 'oc2' sites.
Since for chemical trends only differences of solution energies are important, not the absolute values, the contribution of quantum mechanical zero point vibrations is neglected. In fact, they turn out to be only about 0.02 eV as upper limit e.g. for Fe 1−x Mn x at x = 0, 1 as reported in [26]. Similar to the arguments in [26] we can therefore ignore this contribution.
In the calculations we have ignored magnetic effects. The antiferromagnetic double layer ordering has been shown to be the energetically most stable collinear configuration [34] and a good approximation to the true ground state in γ Fe. However, we recall that two important effects of magnetism, namely the repulsion of the hydrogen atoms by the magnetic moment and the increased interstitial volume due to the larger lattice constant with increasing magnetic ordering -from nonmagnetic, via antiferromagnetic to ferromagnetic -are large by magnitude, but they mainly compensate each other. We refer to Refs. [26,35] for details about the nonmagnetic approximation.

Isotropic deformation
In contrast to the calculations above, where we used a fixed pressure P = 0, we now fix the volume of the supercell. This means, that we evaluate the energies E M H [σ] and E M both at the same lattice constant. Specifically, for the equilibrium lattice constant of the system with hydrogen, the purely metallic system is therefore under tension and has a higher energy due to this elastic effects, compared to the results in Table I, where both energies were calculated at their individual equilibrium volumes. Consequently, the solution energies are lower for the present fixed volume boundary conditions. For isotropic deformations we strain the supercell equally in all three principal directions, which can be described by the diagonal transformation tensor The resulting solution energies are shown in Fig. 3 for the different configurations in Figs. 1 and 2. The first observation is that to good accuracy all energies follow a linear behavior in the interval of investigated lattice constants a lat with almost the same slope, hence only having a parallel offset. This is in agreement with the generic picture that hydrogen mainly leads to an isotropic expansion of the stress free system by an amount ∆V , which differs only weakly for different metals [36]. Hence the coupling to an isotropic strain leads to an energy term proportional to P ∆V . With the pressure P being proportional to the deviation of the lattice constant from its equilibrium value, we therefore expect a linear dependence of the solution energy on the lattice constant. We can further support this picture by a separation of the solution energy in a volumetric (elastic) and a chemical contribution, as indicated by the vertical and horizontal arrows in Fig. 3 respectively. For the volumetric contribution we consider the hydrogen solution energy difference of the chemically identical system, but evaluated once at its equilibrium lattice constant and once at the lattice constant of the Al free reference system. Hence, this energy difference contains the release of mechanical work due to the adjustment to the appropri-ate lattice constant. In contrast, the chemical contribution is calculated from the solution energy difference of two chemically different systems, namely the Fe-Mn-Al and the Fe-Mn systems, but this time evaluated at the same lattice constant. Hence, this energy contribution expresses the energy difference for insertion of a single hydrogen atom into the same rigid atomic structure, but with different chemical environment due to the exchange of Fe by Al.
As can be seen from Fig. 3 exemplarily for the configurations oc1 and oc2 is that volumetric contribution is basically identical, in agreement with the above argument, but the chemical contributions differ significantly. This difference is in line with the stress free results in Table I. Therefore, also under isotropic strain we find that hydrogen prefers a direct environment which does not contain aluminum.

Tetragonal deformation
Again, the starting point for the consideration is the system with a Fe 24 Mn 8 supercell with cubic symmetry. By the substitution of one of the Fe atoms by Al, the supercell experiences in equilibrium a shape relaxation with a slight tetragonal distortion. On a larger scale the Al atoms will be randomly distributed, and therefore the effects from the different specific arrangement will average out. As a result, the supercells will mainly preserve their cubic shape, which we take as basis for additional external tetragonal distortions.
In contrast to the isotropic deformations in the previous section we consider here a case where the distortion is volume-conserving, to clearly demonstrate the different behavior. For that, the supercell undergoes a transformation as described by the tensor which is volume conserving up to third order in the tetragonal strain δ. The supercell is stretched in two principal directions and compressed in the other. The aforementioned dependence of the small intrinsic tetragonal distortion is equivalent to the application of different directions of the compression. The resulting energy curves as function of the tetragonal strain δ are shown in Fig. 4 for the Fe 23 Mn 8 AlH supercell in configuration oc1. It shows the characteristic quadratic dependence of the elastic energy as function of strain when deformed from the cubic reference state. The dashed curve is the arithmetic means of the cases with the three different orientations of the principal axes and reflects the averaged elastic response for a larger system with random distributions. Such an averaging will therefore also be employed  for the following calculations, while we still distinguish between the different configurations shown in Fig. 2. Similar to the isotropic case we computed the hydrogen solution energy as function of the tetragonal distortion, and the results are shown in Fig. 5. Obviously, the behavior is very different than for the isotropic deformation in Fig. 3. For that we note that the considered strain interval ∆δ = 0.06 in the present plot is much larger than in the previous one, where he had ∆δ ≈ 0.017. Still, the energetic variation is much smaller here, despite the larger strain interval. Together with the shape of the curves we can therefore conclude that strain effects to the solution energy are a higher order correction here. This is in agreement with the argument above that the interstitial hydrogen mainly leads to an isotropic expansion of the system, and therefore it only weakly interacts with a volume conserving external strain.
As a result, the chemical influence of the Al atom is clearly dominant compared to the contributions from the tetragonal deformation. The hydrogen sites in the vicinity of the Al atom, oc1 and oc4, exhibit solution energies which are not distinguishable within the accuracy limits of our approach. The oc3 site is comparably attractive as the reference site, while the oc2 site exhibits the shift of about -60 meV relative to the reference system basically over the full range of tetragonal deformations.

Hydrogen diffusion
In this section we intend to obtain information on the hydrogen diffusion coefficient for different strain states and to understand the influence of aluminum. For that, we consider diffusion of hydrogen in the Fe-Mn(-Al) matrix via trap site and Einstein diffusion. To discuss the influence of aluminum induced trapping we apply Oriani's theory [37], which is applicable when the trap sites and the regular lattice sites for the diffusing species are in local equilibrium. The term local equilibrium means that the kinetics of the equilibration of the regular sites and trap sites are described by sufficiently separated timescales. Then, important parameters are the solution energies and the trap densities. This allows to draw conclusions on the relative change of the diffusion coefficients depending on composition and strain.
For Einstein diffusion, we obtain estimates for the changes of the diffusion coefficients based on the diffusion barriers of the involved transition paths. We employ climbing image nudged elastic band calculations (CINEB) [38,39], which search the saddle point of the reaction path, i.e. the barrier between two connected sites.
In the spirit of the above discussion we use initial octahedral and final tetrahedral states of the reaction path with cubic cell symmetry and averaged lattice constants, and therefore can refrain from using solid state nudged elastic band calculations [40]. The energy difference due to the averaged lattice constant -e.g. a lat = 3.502Å for Mn 32 H with hydrogen in an octahedral position and a lat = 3.506Å for occupation of a tetrahedral site, leading to an averaged valueā lat = 3.504Å -is in the range of 5 meV. We used Mn 32 H to test the validity of this approximation against available data on transition points and found good agreement [26]. We also have checked the convergence of the CINEB calculations with respect to the number of images which initially discretise the transition path.

Stress free systems
The oc2 sites are traps with a strain dependent binding energy. In absence of any deformation they have a depth of 60 meV, relative to the sites in the reference system Fe 24 Mn 8 . There are 20 possible hydrogen oc2 sites which are preferential and we assume that at most half of them are accessible due to the indirect hydrogen repulsion when placed in adjacent sites. This leads to the estimate of the trap density as N t = 1.34 × 10 28 m −3 , and the N l is the density of regular sites, with the ratio N t /N l ≈ 0.28. The apparent diffusivity relative to the trap-free diffusivity (aluminum free) diffusivity is according to Oriani's theory which is plotted in Fig. 6 as function of temperature. We see that the presence of aluminum suppresses the hydrogen diffusivity in the room temperature range by about one order of magnitude.

Isotropic deformation
Before discussing the influence of an isotropic deformation on trap site diffusion, we mention that differences in the bulk moduli between the hydrogen free and hydrogen loaded systems are typically small, see Table II. Thus, equal strains correspond in good approximation to equal pressures. Fig. 6 shows additionally to the stress free case the ratio of the trap site diffusion coefficient according to the Oriani model for isotropic compression (δ = −0.01) and tension (δ = 0.01). This leads to a change of the diffusion coefficient relative to the case without aluminum by up to one order of magnitude at room temperature, based on the consideration of the oc2 traps. In general, a tensile strain increases the diffusion coefficient, as the larger interatomic spacing allows for easier jumps of the interstitial hydrogen via a reduction of the energy barriers.
For Einstein diffusion we focus on the configurations oc1, oc2 and the reference case of the supercell without Al, as these are most illustrative to discuss the influence of the Al alloying with respect to deformation effects.
For isotropically strained systems with only one Al atom in the 32 atom supercell, the diffusion coefficient behaves as and both changes of the exponent and the preexponential factor for variations of composition and strain state have to be considered. Here, n tb is the number of transition bonds to the next sites, a is the distance to these sites, and E Diff is an effective diffusion barrier which reflects the dominant transitions contributing to the diffusion process.
In Fig. 7 we show the barriers, i.e. the energy difference E Diff between the transition and starting point, as function of lattice constant. The starting point is a hydrogen atom in position oc1, which then crosses a tetrahedral site close to the Al atom, with one Fe, two Mn and one Al atom as next neighbours. This initial site is relevant when local accumulations of Al yield a low density of available more attractive sites. The barrier of this transition close to the Al atom is increased by about 126 meV, which means that this transition will play a negligible role when other transitions are accessible. Under expansion, the system yields a substantial decrease of the barrier height.
The density of high-barrier transitions in the system with an Al atomic concentration of 1/32 is still low, and therefore one cannot assume that they efficiently reduce the number of transition paths from the considered site in the spirit of shutting down percolation paths. Therefore, this leads only to a weak decrease of the pre-exponential factor n tb . However, when the number of high-barrier We recall that at zero pressure, the shift in barrier height due to substituting an Fe atom in the reference system by an Al atom is about 126 meV. transition paths is increased, i.e. when the atomic concentration of Al is increased to 2/32 or 3/32, the probability that only the high barrier transitions along sites of type oc1 are accessible from a considered site increases. Under these circumstances the energy barriers related to jump processes starting at oc1 type positions become dominant, which leads to a change of the relevant exponential factor. It leads to a reduction of the diffusion coefficient relative to the aluminum free system by a factor which can reach ∼ 10 −3 at room temperature.

Tetragonal strain
We can proceed in a similar way for systems under tetragonal strain. We distinguish between the three different possible strains relative to the elastically stiff axis, as they all appear in a random distribution of Al in an Fe-Mn-Al alloy. As expected, the effect of the deformation on the trap site diffusion is weak, see Fig. 8. At room temperature, the ratio of the apparent diffusion coefficients is changed by less than order of magnitude.
The picture is different when we consider the effect of the deformation on Einstein diffusion processes. As shown in Fig. 9, variations in the relative orientation of the preferred elongation axis can result in a shift of the diffusion barriers by two orders of magnitude at low temperatures. The chemical shift at zero pressure is larger, yielding a possible decrease of the diffusion coefficient in the vicinity of Al atoms up to three orders of magnitude. , the diffusion barrier for the alloyed system from the octahedral site oc1 in direct vicinity of an Al atom to a tetrahedral site with 1 neighbouring Fe, 2 neighbouring Mn and 1 neighbouring Al atoms (three upper curves oc1), and the diffusion barriers for the alloyed system from the octahedral site oc2 distant to an Al atom to a tetrahedral site with 2 neighbouring Fe and 2 neighbouring Mn atoms (three lower curves oc2). Note that the shift in the diffusion barrier for the sites in the vicinity of the Al atom is about 126 meV. We recognise that the orientation of the tetragonal strain relative to the stiff axis of the system has appreciable influence especially for sites in the neighbourhood of the Al atom.
Again, the chemical contribution to the solution energy is dominant in the non-stressed state.

IV. SUMMARY & CONCLUSIONS
In the present paper we have discussed the influence of Al alloying and deformations on hydrogen absorption and diffusion in high manganese steels, represented by Fe 24 Mn 8 and Fe 23 Mn 8 Al systems. In the considered configurations, the substitution with Al leads to two counteracting effects on the hydrogen dissolved on interstitial sites in the matrix. On the one hand, at low Al concentrations of about 3at%, the number of trap sites, which increase the hydrogen solubility and decrease the diffusivity, is large. On the other hand, at larger Al concentrations an effective blocking of paths for the hydrogen diffusion could lead to a drastic reduction of the diffusivity, caused by the high barrier of the transition path between octahedral and tetrahedral sites in the direct vicinity of aluminum atoms. Here we note that the high Mn contents in TWIP steels allow for high Al additions without destabilizing the austenite.
In general, the influence of volumetric changes during isotropic deformations has larger influence on the diffusivity of hydrogen compared to volume preserving tetragonal strains. While the volumetric deformation typically results in a linear dependence of the solution energy on the volume, the influence of tetragonal strains yield effects of second order. This is in agreement with the notion that hydrogen leads to an isotropic expansion of the host alloy.
Spatially inhomogeneous Al distributions lead to an orientation dependent influence of a tetragonal distortion on the diffusion of hydrogen. High local aluminum concentrations have the potential to reduce the diffusion coefficient locally by up to three orders of magnitude in the room temperature regime. For a careful investigation of this effect, large scale Monte Carlo simulations with an emphasis on percolation effects may shed further light on this issue.